ASTRONOMY A POPULAR HANDBOOK HAROJ D JACOBY &RARY i K1VERSJTY OF V CALIFORNIA ASTRONOMY THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS ATLANTA SAN FRANCPSCO MACMILLAN & CO., LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, LTD. TORONTO Photo by Barnard, Oct. 19, 1911. Exposure, 46 min. (see p. 14). PLATE 1. Comet c 1911, discovered by Brooks. ASTRONOMY A POPULAR HANDBOOK BY HAROLD JACOB Y RUTHERFURD PROFESSOR OF ASTRONOMY IN COLUMBIA UNIVERSITY WITH THIRTY-TWO PLATES AND MANY FIGURES IN THE TEXT got* THE MACMILLAN COMPANY 1915 All rights reserved COPYRIGHT, 1913, BY THE MACMILLAN COMPANY. Set up and electrotyped. Published September, 1913 Reprinted August, 1915. Norfoooto J. 8. Cashing Co. Berwick & Smith Co. Norwood, Mass., U.S.A. 915 ASTRONOMY LIBRARY To know That which before us lies in daily life Is the prime wisdom." Paradise Lost, VIII, 192. 502 PREFACE THE present volume has been prepared with a double purpose, and upon a plan somewhat unusual. First, an effort has been made to meet the wishes of the ordinary reader who may desire to inform himself as to the present state of astronomic science, or to secure a simple explanation of the many phenomena constantly exhibiting themselves in the universe about him ; and the further purpose has been to produce a satisfactory textbook for use in high schools and colleges. Thus, for the general reader, it has been thought necessary to eliminate all formal mathematics ; for the student, on the other hand, the occasional use of elementary algebra and geometry are essential. To satisfy these two apparently contradictory conditions, the book has been written in two parts ; the first free from mathematics, the second a series of extended elementary mathematical notes and explanations to which appropriate references are made in the first part of the book. Thus the general reader may confine his attention to the non-mathematical part ; the student should master the whole volume. Attention is directed especially to Chapter I, in which is presented a brief summary of the entire science. It is hoped that this will serve to strengthen in most readers a desire for further and more detailed information. To the student this chapter should furnish as much knowledge as he must have in his possession before beginning a direct vii PREFACE study of the sky with a telescope. In the author's extended experience as a college teacher of elementary astronomy, he has found it most desirable to give life to the subject by requiring frequent evening visits of students at the obser- vatory. These should begin almost immediately upon commencing the study of the science ; and the first chapter is therefore intended to give the -students something to work upon, even in their earliest observatory visits. At Columbia and Barnard colleges, these visits are required on frequent dates, regularly assigned throughout the year, without regard to the state of the weather. When clear, the tele- scope is used ; when this is impossible, oral and informal discussion takes place upon the work done in the classroom. Attention is also given to the daylight study of solar shadows, all students being required to construct a prac- tical sundial, as explained in Chapter V. The author has, of course, drawn freely upon many other books, especially in the preparation of numerous diagrams, and in arranging the various parts of the subject in order. But most of the diagrams are new, and all have been sim- plified as much as possible. In a few cases, illustrations were copied from very old astronomic textbooks: references are then always given, in the hope that some readers, at least, will be led to examine these fine venerable classics of the science. Almost all the inserted plates are photographic repro- ductions of actual photographs. For these the author is under deep obligations to Professor E. E. Barnard, of the Yerkes Observatory, and to the astronomers of the Lick Observatory. COLUMBIA UNIVERSITY, May, 1913. Vlll TABLE OF CONTENTS )H AFTER PAGE I. THE UNIVERSE . .... . . . f ' . 1 Introductory. General view of the science. Its practical use : navigation, coast and boundary surveying, timekeeping for mankind. Value as a culture study. II. THE HEAVENS . , . . . , . . . . 22 What we can see by examining the sky without a tele- scope. The celestial sphere with its points, lines, and cir- cles. Diurnal phenomena; day and night; rising and setting of the stars. Aspect of the heavens from New York, from the equator, and from the polar regions. III. HOW TO KNOW THE STARS . . . .... 45 The planets and the principal fixed stars and constella- tions. Maps, globes, and planispheres. IV. TIME . . . . . . . . " . . .. . 65 Star-time or sidereal time. Solar time and standard time. Differences of time between different places on the earth. The international date line. V. THE SUNDIAL . . . . . . ... . 78 How to make one, and how to use it. VI. MOTHER EARTH .. . . ... ... . . 86 Notions of the ancients. Proof of curvature and rotation. The Foucault experiment. Measurement of the earth's size and shape ; by the ancients, and by the moderns. Geodesy. The earth's mass: weighing the earth. Experiments of Maskelyne and Cavendish. The terrestrial interior. Varia- tion of latitudes. The atmosphere : twilight ; refraction. VII. THE EARTH IN RELATION TO THE SUN . . . . . 116 Orbit around the sun : how it might be determined. The seasons. Astronomic explanation of the geologic ice age. The length of the year determined by the ancients. Trop- ical and sidereal years. Precession of the equinoxes. Nu- tation. Age of the great pyramid. Equation of time. Aberration of light ; its discovery by Bradley, ix TABLE OF CONTENTS CHAPTER PAGK VIII. THE CALENDAR 138 History of the calendar. How to find the day of the week for any date, past or future. Perpetual calendars : how to make and use them. How to find the date of Easter Sunday in any year. IX. NAVIGATION .151 How ships find their way across the ocean. Method used before the days of chronometers. X. MOONSHINE 160 Source of the moon's light. The lunar months, sidereal and synodic. Phases of the moon. Phases of the earth. Air and water absent on the moon. Occultations. Meas- urement of the distance of the moon from the earth. Axial rotation. Librations. Determination of lunar diameter, volume and weight. Lunar day. Harvest rnoon. Sun and moon in the almanac. Moon's true orbit. Measure- ment of the height of lunar mountains. XI. THE PLANETS 183 Kepler and Newton. Central forces. Ptolemaic theory and Copernican theory. Planetary periods. Bode's law. Modern orbit work. Elements of orbits. Measuring and weighing planets. Satellites. Mechanical stability and perturbations in the solar system. Conjunctions and oppo- sitions : visibility of planets. XII. THE PLANETS ONE BY ONE .217 Each planet's characteristics considered separately. Hab- itability of Mars. XIII. THE TIDES 251 Explanation of tidal phenomena due to lunar attraction modified by solar attraction. Effect of the tides on the moon itself. XIV. THE SOLAR PARALLAX . 260 Distance from the earth to the sun. Modern investiga- tions : minor planet method; Eros; transit of Venus; Halley's method ; indirect methods. TABLE OF CONTENTS CHAPTER PAGE XV. ASTRONOMIC INSTRUMENTS . . . . . / . . 272 The telescope : magnifying power ; cross-threads and mi- crometers. The meridian circle and chronograph. The equatorial. Photographic telescopes. The spectroscope. XVI. SUNSHINE. . . . / .' . . . . . 286 Constitution of the sun. Sunspots. Measuring and weighing the sun. Theories as to durability of the sun. Photosphere, chromosphere and corona. Axial rotation of the sun. XVII. ECLIPSES. . . .. . : . . . ^ . .297 Explanation of their cause. Eclipse limits. Umbra and penumbra. Annular eclipses. Prediction of eclipses by means of the Saros. Transits of Mercury and Venus. XVIII. COMETS 307 The coma, nucleus and tail. Size and mass. Danger of collision with the earth. Light pressure theory. Comet hunting. Naming comets. Their orbits. Families and relationships of comets. XIX. METEORS AND AEROLITES . . . . . . . 315 Shooting stars. Showers. Radiant. Cause of light. Fragments of comets. Height above the earth's surface. Aerolites : their chemical composition. XX. STARSHINE 322 Magnitude and brilliancy of the stars. Variable and temporary stars. Stellar eclipses. Star distances : meas- urement of parallax ; the light-year. Motions of the fixed stars : proper motion and radial velocity. Stellar chemis- try. The sun's own motion in space : the apex. Shall we reach Vega? Statistical studies of the universe. Stel- lar distribution. Kinetic theory of stars. Binary stars. Clusters and nebulae. The galaxy. XXL THE UNIVERSE ONCE MORE 356 Origin of the universe. Laplace's nebular hypothesis. Chamberlin's planetesimal hypothesis. APPENDIX .363 Elementary mathematical explanations. xi LIST OF PLATES PLATE 1. Comet c 1911, discovered by Brooks .... Frontispiece TO FACE PAGE 2. Spiral Nebula . . . ;. ..... . . , . . 4 3. The Moon in the First Quarter Phase . . . . . . 17 4. The Samrat Yantra, " Prince of Dials," at Jaipur .... 84 5. Precessional Motion of the Pole 133 6. Full Moon and Crescent Moon 162 7. Lunar Enlargement . 182 8. Mars and the Crescent Venus 225 9. Discovery of Planetoids ....... 234 10. Saturn 242 11. Saturn 244 12. The Lick Observatory, Mt. Hamilton, Cal. . . 272 13. The Lick Telescope ..... . . . .279 14. The Crossley Reflector . . . . . . . .281 15. Lick Spectroscope . 283 16. Various Spectra 284 17. The Sun 288 18. Great Sunspot .290 19. The Prominences . . . . . . . . . . 293 20. Total Solar Eclipse, with Corona . 295 21. The Morehouse Comet, Nov. 18, 1908 . . . . . . 307 22. Halley's Comet . . . . . . . . .' . 311 23. Meteor Trail . . . . 318 24. The Constellation Serpentarius . . . . ... .323 25. Nova Persei . . .'.-.' . . . ... 328 26. Spectra 337 27. A Star Cluster in Hercules and the Double Star Krueger 60 . .349 28. The Pleiades . . . - . 351 29. The North America Nebula . . . . . :. ..354 30. Nebulae . .... . . . . . . ... 359 31. The Trifid Nebula . . . . . . . . . .360 32. Title-page of Newton's Principia . . . . ... . 399 xiii ASTRONOMY CHAPTER I THE UNIVERSE IF a company of men and women should chance to be gathered together on some clear, quiet evening under the dome of the sky, and if there should happen to come into that company one known to possess an acquaintance with the facts and the theories of astronomic science, if these things should occur, inevitably there would descend upon that astronomer a shower of questions. These he would answer in simple language, after a kindly fashion for many centuries the habit of his guild; and, as he passed on, he would once more marvel, as he had done many times before, at the changelessness of man's desires. For these ques- tions of the multitude, welcome ever to the star-man, to-day still resemble those that were laid of old as problems before his predecessors at the side of the pyramids. Why, for instance, does the moon appear at times as a full round disk, at others as a tiny crescent? Why do certain bright stars called planets seem to wander about among the multitude of their fellows ? Why is summer hot and winter cold ? How do navigators find their way across the trackless ocean by observing the heavenly bodies ? Let us begin by attempting to set forth as best we may the answer to some of these many eternal questions from the skies. For the astronomer is not always present ; and even B 1 ASTRONOMY if he were, it is often better to gather our information in silence, by means of the process called reading. The very name Astronomy tells us what our science is. Derived from two Greek words, Astronomy means "the law of the stars. 77 Where does the law of the stars hold sway? Throughout all space. What is space ? Space is the place where astronomy has its being. When does astronomy enforce its laws ? Throughout all time. What is time ? Time is the period during which astronomy has its being. Astronomy needs no logical definitions of space and time. They belong to it ; they are part of it. Somewhere, then, in the endless void of space our universe is suspended ; the visible universe. Is that visible universe but one of many ? Are there invisible universes without number scattered through the vastness of space like conti- nents in an endless ocean? The human mind loses itself in speculations such as this : nor do such speculations here concern us ; for astronomers consider only the ascertainable phenomena of the universe that unfolds itself to our senses. There is in existence a vast quantity of matter and a vast quantity of active energy, or force. It is not necessary at this point to define these terms ; but we should remember that according to accepted theory the total of matter and the total of energy in the universe do not change. Matter is never destroyed ; and the accepted law of the conservation of energy tells us that the quantity of energy in the universe is likewise constant and unvarying in amount. None ever disappears out of existence. But both matter and energy may and do undergo changes in form and appearance. Thus, water may appear as steam or as ice ; and the energy of a moving body may be transformed into heat, light, or electricity. 2 THE UNIVERSE When we examine this visible universe of ours at night with the unaided eye, we "see several different kinds of ob- jects : nebulae, or small luminous clouds ; star clusters, like the famous group called the Pleiades ; individual stars ; the moon; and, occasionally, comets or meteors. In the day we see the sun; sometimes the moon; and very rarely indeed a particularly bright star or comet. We shall give here a brief outline of existing knowledge concerning these various celestial objects, leaving a detailed description of their peculiarities to later chapters. They are all com- posed of matter ; all, if in motion, move in accordance with the laws of mechanical science which govern the operation of energy ; and all, if they change, undergo only changes such as accord with the laws of physics and chemistry. First, then, the nebulae. We shall begin with these because they probably represent the form in which matter shows itself to us in its most primitive stage of development. Only one or two can be seen with the unaided eye; and these only on very clear nights when the moon is invis- ible. In the telescope they appear as patches of luminous cloud, often more or less irregular in form. They were once thought to be simply conglomerations of small stars, so close to each other that the optical powers of existing telescopes were unable to separate them into constituent units. This view gained in probability for a long time, because, as the power of telescopes increased with the in- crease of skill among opticians, astronomers were con- stantly resolving new nebulae, as they used to call it ; sepa- rating them into simple close clusters of faint stars. But the invention of an instrument called the spec- troscope, in the middle of the nineteenth century, put us in possession of a means, previously non-existent, for dis- 3 ASTRONOMY tinguishing with certainty between the light of incandescent gases and that derived from incandescent or luminous matter in the liquid or solid stage. With the spectroscope astrono- mers have been able to ascertain that there are many nebulae in a truly gaseous condition; that probably most of these objects are gaseous bodies ; that they could not be resolved into stars, even if terrestrial man possessed to-day tele- scopes more powerful than he is likely ever to have at his command. According to many modern theorists, we may take the nebulae to be matter not yet fashioned into stars. This means, of course, that certain forces are at work in the nebulae; forces of irresistible power, slow in action, as all cosmic changes must be slow when measured by the life of human generations ; but sure in action, too, with that in- finite sureness which belongs in celestial spaces. These forces doubtless produce motions of vast import within the body of the nebula ; heat is doubtless engendered ; conden- sations occur at certain points ; nuclei are formed ; prob- ably, finally, one or more stars take the place of these nuclei ; and so, perhaps, is the original nebular material transformed into stars such as men see clustered upon the sky of night. Certainly the force of gravitation must be active. Since the time of Newton, in the seventeenth century, it has been known that there is a force of gravitation ; that under the influence of that force every particle of matter in the uni- verse attracts or pulls every other particle of matter ; that the combined effect is always motion of some sort, each particle pursuing in space some determinate path or orbit under the influence of gravitational attraction exerted by all the particles. The most recent observations of nebulae have brought out 4 Photo by Keeler, May 10, 1899. Exposure, 4 hours. PLATE 2. Spiral Nebula. THE UNIVERSE the fact that they are extremely numerous ; probably many hundreds of thousands exist, although only about ten thou- sand have been catalogued. This fact is of importance; for if we are to regard the stars as a product of development or evolution from the nebulae, we should expect these gaseous bodies to exist in numbers comparable with the number of the stars themselves. But of even greater interest is another recent observation of the nebulae. The predominant type seems to be spiral in form; a species of central hub, carrying two attached curved spires, like a whirling wheel with two very flexible spokes but no rim. There can be little doubt that these nebulae are subject to internal motions, probably rapid in themselves, but appearing infinitely slow to us because of the almost inconceivably vast distance by which we are sundered from them. According to the foregoing theory, which admits the exist- ence of irregular as well as spiral nebulae, we should expect to find the stars in groups, a certain number assembled comparatively close together near certain former nebulous regions within the sidereal universe. And this is precisely what we do find. Usually the number of stars thus belong- ing together is small ; very frequently but a single star can be detected with the telescope. But many of the constit- uent stars of a group may be too faint to show themselves on account of their distance ; often they are all probably too faint, except the large one that may have resulted from the former hub or center of the parent nebula, if it was one of true spiral form. Furthermore, gravitational attractions and orbital motions may have commenced among the stars of every group even before they had become separate bodies. While they were still numerous, frequent collisions must 5 ASTRONOMY have brought about the coalescence of two or more into a single larger unit. In short, we have here an outline of a fairly consistent explanation to carry us forward from the nebular stage to that of stellar development, a theory that leads us to expect star-groups ranging all the way from a single visible luminous object to a detached assemblage closely packed in a globular cluster. And the stars are simply suns ; our sun is a star. There can be no doubt that the stars are not the tiny twinkling points of light they seem to be. Their apparent lack of size or volume is simply a result of the distance by which they are separated from us ; their twinkling comes from undulations or other irregularities in the ocean of terrestrial atmosphere, or a air," through which we are compelled to view them. We know the stars to be self-luminous, masses of glowing incandescent solids, liquids, or gases. We know the stars to be composed of chemical elements practically identical with those found on the earth. We know the stars to be subject to the law of gravitation ; that every particle of matter in each one of them is endowed with that mys- terious quality postulated by Newton, the power of pulling all other matter in the universe. From all these known facts, and reasoning by analogy, we are led to believe the stars to be suns, more or less like our own sun, though by no means necessarily in the same stage of cosmic develop- ment. All are doubtless cooling gradually and steadily by the constant radiation of heat into space ; some have probably reached temperature conditions similar to those existing in our sun ; and there may very probably be some that are attended by planets like our earth. The stars are classified according to their so-called magni- tudes, by which astronomers mean simply their lucidity or 6 THE UNIVERSE brightness, not their actual dimensions ; the first-magnitude stars are the brightest, the fifth-magnitude stars the faintest usually visible to the unaided eye under ordinary conditions. There are in all about sixteen hundred stars of the first five magnitudes ; and only about one half of these can be seen at any one time, when the sky is perfectly cloudless, because the other half are always concealed from view below the horizon. The stars are also divided into a series of so-called constellations ; very irregular, even grotesque imaginary figures of men, animals, and other objects, placed in the sky by the astronomers of old, and retained there in a somewhat simplified form by the moderns, principally on account of an unwillingness to destroy the ancient landmarks of this venerable and venerated science. The stars so far described are called fixed stars, which means that they do not change their relative positions in space ; that any two of them now close together have been thus in proximity from the beginning, and will remain so to the end. But modern researches have brought out the fact that these apparently fixed stars are not really fixed absolutely. They have motions in space ; these motions seem to us extremely slow and minute simply because the stellar distances are so vast. For at a sufficiently great distance, even large and rapid motions will necessarily appear reduced and retarded. And it is, in fact, quite inconsistent with what we know of the laws governing gravitational attraction to suppose any par- ticle of matter in the universe to be really fixed in position absolutely. Everything must move; must be following some duly appointed path, ever contrasting the intricate com- plexity of nature with the wondrous simplicity of nature's order and nature's law. Even our sun, regarded as a star, cannot be fixed in space, but must be -moving majestically 7 ASTRONOMY through the void, drawing with it our attendant earth, and ourselves upon it. And if the stars are incandescent suns, we must expect to find, and we do find, that some among them undergo inter- nal changes that make their visible brightness vary. In certain cases slowly, in others more rapidly, their luminosity waxes and wanes with a more or less periodical regularity. Now and again, rarely and at long intervals, some special catastrophe takes place ; some convulsion of nature, whereby a new star is made to blaze forth into view where previously had been only darkness. Possibly we witness in such cases the result of a sudden collision in space between two ancient suns previously cooled through the ages, and long since bereft of luminosity and of life. The stars that change their brilliancy are called variable stars; those that blaze forth suddenly are "new stars, " or novce. As we have already stated, the sky contains stellar sys- tems other than those involving but a single visible object. Of these probably the most interesting are the double stars, composed of two individuals, often of different colors. These double stars appear but single to the unaided eye; only when the powers of a telescope of some size are brought into play, is it possible to resolve them into their component parts. In the field of view of such an instrument the stars all appear as brilliant points of light, occasionally glittering and sparkling, but the glitter and sparkle are imperfections caused by terrestrial atmospheric effects, and by the impos- sibility of constructing telescope lenses whose surfaces are ground to the right theoretic shape with absolute exactness. In other words, the stars appear in the telescope much as they do to the eye : only when the star is a double, the telescope often shows it as such, while the eye is unable to 8 THE UNIVERSE see between the two components. And it is a very impres- sive sight, when we turn a telescope upon one of these double stars, to see the two tiny points of light projected on the deep, fathomless background of the night sky, and to realize that the speck of darkness between them is a bit of abysmal space. Sometimes the close proximity of the components of a double star is fortuitous merely. The two objects may simply appear close together through happening to lie in almost exactly the same direction from us. But one of them may in reality be behind the other, and at a distance from us immeasurably greater than the first. In this respect astronomic observation differs from the viewing of ordinary objects on the earth. If, for instance, we should happen to notice two men, both standing at points almost exactly north of us, but one ten times as far away as the other, we would at once detect a difference of distance from the fact that the distant man would appear much smaller than the near one. But in the case of the stars, which we see as points of light merely, we could gather no such information. Even if one of the stars should be brighter than the other, this extra brilliancy might be due to a higher intrinsic light- giving power, and in no sense a result of greater proximity. When two stars thus appear close together, though in reality separated by a great distance, they probably have nothing in common, and are of lesser interest. But in certain cases the two stars will appear close together through really being near each other in space. Then they must belong to a single system; have probably originated in a single nebula ; true twin suns, bound one to the other and the other to the one ; held by the invisible, intangible, but indestructible power of gravitational attraction. 9 ASTRONOMY In addition to these fixed stars, whose motions were un- known to the ancients ; whose motions are so slow that generations of men must come and go before they can reveal themselves to the unaided eye, in addition to these fixed stars, the night sky contains five other bright stars called of old the planets, from the Greek word TrXavrJTrjs, the wanderer. They have been named Mercury, Venus, Mars, Jupiter, and Saturn. The most conspicuous thing about them, when viewed without a telescope, is their peculiar and rapid motion among the fixed stars. They can be seen to make an entire circuit of the heavens, traveling apparently among the fixed stars, in brief periods of time ranging from about a year to about thirty years. Of course we now know the cause. These planets are not properly stars at all ; they are like the earth, attendants of our sun, revolving around the sun in perfectly definite paths or orbits, and in perfectly definite periods of time. Compared with the fixed stars, they are all extremely near the sun. And being all thus com- paratively near the sun, they are of course also all compara- tively near each other ; and our earth being one of the num- ber, they are all comparatively near the earth, too. But we have just seen that the extreme apparent slowness of stellar motion is really only a result of the extraordinarily great distance by which we are separated from the stars ; as this immensity of distance does not exist in the case of the planets, of course their apparent motions must and do appear to us comparatively rapid. Their apparent motions are also complex in a high degree. Two of them, Mercury and Venus, move around the sun in orbits smaller than that of the earth, and therefore entirely within the earth's orbit ; the other three, Mars, Jupiter, and Saturn, are exterior to the earth. Mercury has the 10 THE UNIVERSE smallest orbit of all. It is always actually quite close to the sun, and therefore always appears near the sun when seen projected on the sky. Of course, it cannot be seen when the sun is visible on account of the overwhelming luminosity of the sun itself. Therefore we can observe Mercury occa- sionally only, just after sunset, near the point of the horizon where the sun has disappeared ; or just before sunrise, near the point of the horizon where the sun is about to make its appearance. It is thus always seen in the evening or morn- ing twilight, and was called of old the evening star or the morning star. The same is true of the planet Venus, which attains, however, a much greater apparent distance from the sun. The exterior planets, Mars, Jupiter, and Saturn, may be seen at certain times throughout a wide range of space on the sky, and at any hour of the night, all of which phe- nomena will be explained in detail in a later chapter. Still other planets exist ; but they are mostly too faint for the unaided eye; they have been discovered telescopically in modern times. All, together with our sun itself, are proba- bly the result of gradual changes in a parent nebula. The planets are unlike the stars in still another important particular. We have seen that the stars are self-luminous, incandescent; the planets are quite different, and give out no light of their own. They shine only by reflected light which they receive from the sun. The light goes from the sun to the planet ; illumines it ; and then we see the planet by solar light, just as we see objects in a room by reflected solar light, which we call daylight. This produces a rather curious telescopic planetary phenomenon called phase, a phenomenon which is most conspicuous also in the case of our moon. The planets are globular in shape, and 11 ASTRONOMY therefore only one hemisphere can be illumined by the sun at any one time. But the planet does not usually happen to turn its illuminated hemisphere directly towards the earth. Therefore we usually see only a part of the bright hemisphere, and this often looks more or less like what is called a half-moon. In other words, we always see a hemi- sphere of the globular planet, but it is not the same hemi- sphere which is turned toward the sun, and which is therefore bright. If the hemisphere we see and the bright hemi- sphere are mutually exclusive, we see a dark or " new-moon" phase. If the bright hemisphere and the one we see over- lap, we see a crescent, half-moon, or other phase, as the case may be. Among the planets, Mercury, Venus, and Mars show the most conspicuous phase phenomena. Sir John Herschel has given a good illustration of dimen- sions in our solar and planetary system. Represent the sun by a globe two feet in diameter. Then Mercury will be a grain of mustard seed on a circle 164 feet in diameter with the sun near its center ; Venus, a pea, 284 feet distant ; the earth, also a pea, 430 feet away; Mars, a pin's head, 654 feet ; Jupiter and Saturn, oranges, distant respectively half a mile and four-fifths of a mile. The nearest fixed star, on the same scale, would be distant about 8000 miles, not feet. This illustration brings out clearly the compara- tively minute dimensions of the solar system in relation to the vastness of stellar distances. In actual appearance the planets differ greatly in the tele- scope; and they differ especially from the fixed stars. For even our most powerful optical apparatus will not suffice to magnify the latter so as to make them appear otherwise than as minute points of light. Many of them doubtless possess globular dimensions greatly exceeding any- 12 THE UNIVERSE thing we find in the solar system ; but the vast distances cause these dimensions to shrink into mere nothingness, even in our largest telescopes. But in the case of the planets these great distances do not exist, and therefore the telescope shows their spherical size in the plainest possible way. But the planets differ greatly one from the other. Jupiter shows a bright, nearly round disk, crossed by a few dark straight lines or bands. It is accompanied, in small telescopes, with four satellites or moons, which can be seen to revolve around the planet. At times they pass behind the planet and disappear; and again, one or other of them is so placed that the planet inter- poses between it and the sun. Then, too, it disappears; for the satellites also shine by reflected solar light ; and, of course, they receive none when Jupiter is placed between them and the sun. Finally, at certain other times a satellite may pass between Jupiter and the sun ; and then it can be seen to cast a small round shadow dot on the bright surface of the planet. Such phenomena are called eclipses. Saturn is the most beautiful of the planets, viewed with a telescope of moderate size. It has a number of moons or satellites, mostly too small to be seen in a glass of low power ; but its most conspicuous feature is the famous ring of Saturn. This is a flat disk surrounding the planet, and, in the words of Huygens, who was the first to explain it correctly, nowhere " sticking to" the planet. The ring, like the other bodies of our system, shines by reflected solar light ; and it is always distorted in appearance, as seen from the earth, into a flat- tened oval or ellipse, like a cart-wheel seen nearly edgewise. At certain times we actually do see it exactly edgewise, and then it appears, of course, like a thin, straight, bright line against the dark sky background. And when the ring 13 ASTRONOMY appears opened up to a considerable extent, we can see this dark background of the sky by looking through the openings between the ring and the ball of the planet. Mars and Venus show us plain bright disks of moderate size, exhibiting in small telescopes little or no detail of any kind in the way of markings or bands. Their most conspic- uous feature is the phase, which is much more marked than it is in the case of Jupiter and Saturn, whose phase phenomena are practically altogether unnoticeable. This follows, of course, from the fact that the quantity of visible phase is due to proximity ; and Mars and Venus, being the planets nearest to our earth, must, of course, show more phase than the distant planets Jupiter and Saturn. Mercury, as we know, is seen only in the twilight, showing in the telescope a small disk with marked phases. Comets are occasional visitors to the solar system. They come presumably from outer space in the course of their orbital motions under the influence of gravitational and perhaps other forces ; remain for a time in the vicinity of the solar system ; are consequently visible to us ; and finally retire again into the depths of space whence they came. When bright enough to be observed without the tele- scope, they commonly exhibit to our view a brilliant come- tary "head," containing a central condensation or nucleus surrounded by a mass of tenuous luminous haze, and to it often attached a long visible streamer or tail, in olden times dreaded by all as a possible harbinger of wars and pestilence. All these cometary phenomena are well seen in the photograph reproduced as a frontispiece in the present volume. The tail in this case has more than one streamer ; and its length, as photographed, is about 11, or nearly one-eighth the distance from the horizon to the zenith. 14 THE UNIVERSE The tail actually seen by astronomers was at one time twice as long. The little curved lines on the photograph are star-images. We should of course expect these to be round dots in the picture ; but in photographs of this kind they are almost always drawn out into little curves, for a very simple reason. The telescope is aimed accurately at the comet when the exposure of the photographic plate is commenced, and it is kept thus pointed at the comet during the whole duration of the exposure. This of course makes a "moved picture" of the stars, as photographers would call it. For the comet will " wander " among the stars, like a planet, in consequence of its orbital motion in space ; and if the telescope's movement upon its stand is adjusted cor- rectly to allow for the comet's motion, the photographic images of the stars must suffer. The earth, considered as an astronomic body, is but one of the smaller planets; yet in one respect it is the most important of all, since it is the one upon which we live. Astronomers have been able to ascertain many facts about the earth, which we shall for the present summarize with the utmost brevity, postponing all detailed description to a later chapter. We know, first, that the earth rotates once daily on an axis ; that this rotation carries us around, too ; that in consequence of it the sun, stars, and other heavenly bodies seem to rise in the east, climb upward in the sky, and finally sink down again and set in the west. We also know that our earth, like the other planets, travels around the sun in an orbit ; that it requires a whole year to complete a circuit of that orbit ; that in consequence of the daily axial rotation and the yearly orbital revolution, we experience the phenomena of night and day, summer and winter, phenomena to be explained fully 15 ASTRONOMY later ; finally, we know from actual measures made upon the surface of our planet that the earth is a slightly flattened globe about 8000 miles in diameter. The moon is the only satellite of our earth, and by far the most conspicuous object in the night sky; the most beauti- ful and interesting of all the heavenly bodies when observed through small telescopes ; and important especially as being our nearest neighbor in the whole wide domain of cosmic space. Again summarizing existing knowledge as briefly as possible, the moon is now thought by astronomers to have once formed a part of the earth ; to have been set free in some very distant age in the past by the action in some way of gravitational and possibly other forces. It revolves around the earth in an orbit somewhat similar to the earth's own annual orbit around the sun ; completes such an orbital revolution in about twenty-seven and one-quarter days ; and, in consequence thereof, appears to make a complete circuit among the far more distant fixed stars and planets in the same period, traveling around from a position of apparent proximity to any given fixed star back to the same star again in the twenty-seven and one-quarter day period. It is not self-luminous or incandescent, but shines by reflected sunlight like the planets ; in consequence of its nearness to the earth, it exhibits the most pronounced phase phenomena, varying all the way from the full-moon, down through the half-moon stage, to actual invisibility at the time of new-moon. It is about 240,000 miles distant from the earth ; is about 2000 miles in diameter ; and the gravi- tational attraction of its mass upon the waters of terrestrial oceans gives rise to the ebb and flow of the tides. The physical or actual appearance of the moon is not unlike that of the earth. The surface, as seen in the tele- 16 Photo at Lick Observatory. PLATE 3. The Moon in the First Quarter Phase. THE UNIVERSE scope, is very much broken ; there are several mountain ranges, and, especially prominent, a great number of large craters, apparently of volcanic origin, and usually having a mountain peak in the center. Extremely conspicuous features of the lunar surface, as seen in small telescopes, are the very black shadows which are cast on the surface when sunlight falls obliquely on the mountains and craters. There is no air or other atmosphere, and no water ; nor have we any reliable evidence that any perceptible changes have taken place in the volcanic surface features since accu- rate records of telescopic observation were begun by men. To complete this preliminary brief outline survey of our subject, it remains to add a few words about the sun, the central body of our solar system. The sun is our source of light and heat ; without it life, as we know it, would be im- possible on our earth. It is about ninety-three million miles distant from us, and nearly a million miles in diameter; within its vast bulk might be placed the earth and moon, together with the entire lunar orbit in which, as we have said, the moon revolves around the earth in twenty-seven and one- quarter days. The sun turns on an axis in a period of about twenty-five terrestrial days ; its surface is usually marked by the well-known sun spots, visible in small telescopes plainly, and first seen by Galileo, when he turned upon the sun probably the first telescope ever made. These spots are now known to have periods of special frequency. Every eleven years they occur in greater numbers than usual; and this period of eleven years is in some mysterious way connected with the known frequency periods of auroras and magnetic storms on our earth. The bulk and mass of the sun are so great that its gravitational attraction far exceeds that of all the planets combined. It thus becomes c 17 ASTRONOMY the gravitational ruler of the whole solar system ; around it all the planets may be said to revolve in their duly appointed paths or orbits. It is hoped that the foregoing brief summary of astronomic science may help to awaken a desire in the reader to possess more detailed knowledge ; and this we shall endeavor to give in later chapters ; perhaps we may be permitted to conclude the present one by calling attention to the value of astronomy for practical purposes as well as for mental dis- cipline and study. It is often said that astronomy is a somewhat detached subject; of interest certainly, but hav- ing little or no close and intimate relation to the everyday affairs of human life. But in reality the converse is the truth. Probably no other of the more abstruse sciences enters so directly and so frequently into our daily affairs as does astronomy. There are at least three services per- formed by astronomy that are essential, and without which civilization, as we know it, would be impossible. These things are : first, the regulation of time ; second, the exe- cution of boundary surveys and the making of maps and charts ; third, navigation. Few persons stop to think when they enter a jeweler's shop to correct their watches by comparison with the jeweler's " regulator," or when they communicate by telephone with a central telephone station to ask for the correct time, that both the jeweler and the telephone operator must themselves have some source of correct time by which to regulate their regulators. This source of correct time is the astronomical observatory. The standard observatory clock is itself but a fallible piece of machinery fabricated by fallible human hands, and it can be kept right only by con- stant comparisons, made on every clear night, with the 18 THE UNIVERSE unvarying time standards provided by nature, the stars themselves in their courses. For instance, time observations of the stars are made regularly and nightly in the United States Naval Observatory at Washington, the chief official astronomic station of the United States government. With these observations the standard clocks in the clock room of the observatory are corrected and timed ; and from these standard clocks electric signals are sent out daily in accord- ance with a pre-arranged schedule so that time-balls can be made to indicate the exact instant of noon to the people, and jewelers and others may correct their regulators. Thus is every citizen in touch with the astronomic observatory almost daily and of necessity, although he does not generally realize the fact until it is brought specially to his attention. And the matter of mapping and charting is equally depend- ent upon astronomy. Ordinary small surveys of farms or towns may be made by ordinary surveyor's instruments without constantly having recourse to astronomers. But of what value would be a map of an entire continent unless the customary latitude and longitude lines were inscribed upon it ? And these essential lines cannot be so inscribed without astronomic observations. Such observations must necessarily be made specially for the purposes of each survey, and the consequent calculations always depend, too, upon cer- tain prior astronomic data contained in published astronomic " tables" or printed books, themselves in turn based on aver- age or mean results obtained in the great observatories of the world during the last couple of centuries by steady con- tinuous systematic study and observation of the stars. Even more important than continental maps for the progress of civilization are the coast charts published by the various governments of the maritime nations. These also 19 ASTRONOMY require very precise latitude and longitude lines ; and here, as before, recourse must be had to astronomic observations and accumulated astronomic results. Finally, navigation itself, upon the open sea, could not proceed successfully without astronomy. Those of our readers who have crossed the ocean in a magnificent modern steamer may have seen at times the captain or navigating officer "take the sun," as it is called, with a sextant. Pos- sibly they have thought that after making such an obser- vation the navigator could read on the face of the sextant the exact position of the ship at the moment, its latitude and its longitude on the earth, as ordinarily understood in geography. But such is by no means the fact. Before they can be made to yield this essential information, sextant observations must be subjected to a somewhat laborious process of numerical calculation, or " reduction," as it is called. This is an astronomic process ; and in carry- ing it to completion the navigator again requires certain printed tables of a purely astronomic character. These are contained in a book called the " nautical almanac," which is published annually in various languages by the several civilized governments of the world. And again, as before, for the preparation of such nautical almanacs, these governments must maintain, and do maintain, astronomic computing bureaus, manned by astronomers, and employing in their calculations once more the published results obtained by astronomers of the past in the various great fixed obser- vatories. The details of all these astronomic activities must, of course, be postponed to later chapters ; but it is hoped that enough has been said here to remove from the reader's mind the possible notion that astronomy is of little or no practical utility in the ordinary affairs of men. 20 THE UNIVERSE But far beyond and above all this, the study of astronomy possesses a value peculiarly its own, as a means of mental training. On account of venerable age and consequent approximate perfection of knowledge, this science is char- acterized especially above all others by the peculiar intri- cacy of the elementary problems it presents, and by the unusual exactness of which their solutions admit. Further- more, notwithstanding the importance of its direct practical applications, which have been mentioned, the study of astron- omy is peculiarly free from any materialistic tendency, from any connection, in short, with utilitarian motives. It is not a vocational study, giving knowledge which can be sold for money by the young college graduate upon his entry into practical affairs. But it is preeminently a study which will give a clearer outlook upon the universe in which we pass our lives, preeminently one that will make that universe seem a pleasanter place in which to live. So that if a certain portion of our time is to be devoted to studies that are not strictly vocational, astronomy will surely be found a profitable and desirable subject. And surely also there is much to be gained in our choice of studies from the selection of such as are likely to arouse a real interest in the student ; to arouse that desire for knowledge which, once awakened, will make the task of the teacher an easy one. Here again astronomy holds a most favorable place. That which has its being within the confines of a single drop of water is as wonderful as are the motions within a planetary or sidereal system. But the animalcules within that drop of water, though their number be myriad, can never stir our deepest interest, for they are without that strong appeal to the imag- ination, without those vast distances and mighty forces, the materials of astronomic study alone. 21 CHAPTER II THE HEAVENS PROBABLY the best method of approaching the study of astronomy is to begin with those observations and problems that do not require the use of any instruments whatever. These problems are surely the earliest problems, since men of old must have begun to discuss the mysterious events they could see about them in the universe long before they had invented even the rudest instruments of measure- ment. Astronomy is a study of the sky ; and the first thing to be noticed in a study of the sky is the sky itself. To us it ap- pears at night like a great, round, blue, hollow dome within which we are standing. To its interior surface seem to be attached the apparently numberless bright twinkling points of light we call stars. In the day it carries only the sun, and perhaps, too, the moon rather faintly visible ; and in the intermediate periods which we call twilight, and which occur at dawn and at dusk, we can see perhaps two or three dim stars, called morning and evening stars. We know that these morning and evening stars are certain of the planets, which, as we have already seen, are members of the solar system like our earth, circling around the sun, each in its proper path or orbit. But there is no real dome of the sky above and around us ; it is simply an optical illusion, a creation of our own imagina- tion. Nevertheless, it is most convenient to imagine it to 22 THE HEAVENS be real, because we can thus fix our first astronomical ideas to something tangible ; and by a consideration of this round dome as if it actually existed, we shall be able to clarify and to solve many interesting problems. Granting, then, that there is such a dome above us, we have no reason to imagine it other than perfectly round. Let us regard it as a great hollow ball or sphere; astronomers have given it the name Celestial Sphere. The next question is whether this celestial sphere is the same sphere everywhere. Is the celestial sphere surround- ing New York identical with that surrounding the city of Capetown, South Africa ? The answer is : yes. The sphere is the same sphere everywhere. Theoretically, the center of the sphere is at the center of the earth ; and since the diameter of the earth is about eight thousand miles, an observer on the earth's surface will be distant about four thousand miles from the true center of the sphere. But such a distance as four thousand miles is literally a .mere nothing compared with the infinitely vast distance of the celestial sphere. The whole planet earth shrinks into a mere dot in comparison. It makes absolutely no difference whether you are on the earth's surface, or could be transferred to its center, you would s.ee identically the same imaginary celestial sphere. The stars and other heavenly bodies, wherever they may be situated around us in space, seem to be projected upon that distant celestial sphere, and attached to its interior surface. Even if you could make a sudden jump of about ninety- three million miles from the earth to the sun, you would still see the same identical sphere, much too far away to be affected by such a little change in the observer's position. Not only the earth, but its entire orbit, including the sun, shrink into a dot. Astronomy is 23 ASTRONOMY truly a science of vast distances. But there is this essential difference between the distance of the celestial sphere and all other distances in the science. The far-ness (if we may use such a word) of this imaginary sky sphere is in- finitely greater than any other actually known and meas- ured by men. The accompanying Fig. 1 is intended to illustrate this notion of the celestial sphere. The large circle is supposed to represent the sphere; only, of course, its size cannot be made big enough ; the reader must imagine it ex- tended to infinity. The dot E at the cen- ter of the big circle is the earth; the reader and the author are sup- posed to be standing on the surface of that dot. The tiny circle represents the earth's FIG. 1. The Celestial Sphere. annual path around the sun, the sun itself being the larger dot at the center of the tiny circle. The crosses represent stars scattered through sidereal space at all sorts of distances from the earth. The lines with arrows passing through the crosses indicate the points on the interior surface of the celestial sphere where the stars will appear to be projected, and where they will seem to be attached to the interior or supposedly visible surface of the sphere. The longest arrow indicates the point on the sphere where the sun will appear projected, that arrow 24 THE HEAVENS being, of course, merely a straight line passing from the earth to the sun and thence continued outward to the sphere. For the sun will also appear to us as if attached to the interior surface of the sphere, like the stars, at the point indicated by its arrow. This elementary notion, that the various celestial bodies will appear to be located on the sphere at the points shown by their arrows is an important idea, and one that is not at all difficult to grasp. We must not forget that the arrows are all supposed to be infinitely long ; even the solar arrow is infinite, although the sun dot and the earth dot are very near each other, cosmically speaking. Having thus fixed our ideas as to the celestial sphere, we must next study it in its relation to the various objects that appear projected upon it ; and the first important thing to consider more in detail is the position of the sun on the sphere. We have already seen that the earth travels around the sun once in a year. The path or orbit in which the earth thus travels is an oval or ellipse; but for the purpose of a first approximation such as we shall here con- sider, we can take this path of our earth to be a circle, with the sun at its center. Now this circular orbit, like every circle, must lie entirely in a single plane or flat surface. The accompanying Fig. 2 shows this circular approximate orbit of the earth E moving around the sun at the center S FlG - 2 - The Earth's in the direction shown by the arrow. The single plane or flat surface in which the entire orbital path lies is here of course the flat plane of the paper on which this page is printed. It is evident that the earth, being always in its orbit, 25 ASTRONOMY must likewise always be situated in the plane of the paper. And the sun, being at the center of the circular orbit, must also be in the same plane. From these . considerations fol- lows the important preliminary principle that earth and sun are both constantly in a single plane. To this important fundamental plane has been given the name Plane of the Ecliptic. The plane of the ecliptic is defined, then, as the plane in which are situated at all times the sun, the earth, and the earth's orbit around the sun. Now let us extend our ideas so as to include the celestial sphere in our consideration of the earth's orbit. Imagine the orbital plane, but not the orbit, extended or stretched outward, indefinitely, farther and farther, approaching gradually an infinite bigness, until at last it meets the imaginary celestial sphere. Evidently, it will cut out a circle on the celestial sphere, just as though one were to slice a round orange with a flat cut. The line in which the rind of the orange would be severed by such a cut would then be a circular line ; and so also must the line cut out on the celestial sphere by the ecliptic plane be a circle. The fact that the sphere is an immense globe and the orange a small ball here makes no difference. The principle is the same. It is possible to draw a little more information from the analogy of the orange. Wherever we slice the orange, we obtain a circle ; but if it was sliced through the center, the orange would be cut in two equal halves, and then the circle would be the largest circle that could possibly be drawn around the rind of the orange. Applying this to the case of the celestial sphere cut by the ecliptic plane, we see at once that here also the sphere is cut in two equal halves. For the earth, as we have seen, is at the center of the celes- 26 THE HEAVENS tial sphere ; and therefore the ecliptic plane, which passes through the earth, is also a cut or slice through the center of the celestial sphere. Consequently, the circle cut out on the celestial sphere by the ecliptic plane produced to infinity is a circle as large as can possibly be drawn on the celestial sphere, and it divides that sphere in two equal halves. Such a circle drawn on a sphere, dividing it into halves, is called a Great Circle of the sphere. The particular great circle of the celestial sphere, cut out by the plane of the ecliptic produced to infinity, is called simply the Ecliptic. The ecliptic, then, is defined as a great circle of the celestial sphere cut out by the plane of the earth's orbit around the sun, produced to infinity. It would be a convenience if some one could go up to the sky and mark out the ecliptic circle upon it with a big paint-brush. While this is im- possible, it is perfectly easy to mark it upon a celestial globe ; and the reader is advised to examine such a globe, when he will surely find the ecliptic plainly drawn upon it. The important peculiarity of the ecliptic circle is this : the sun must always at all times appear to lie in that circle. And the reason is quite simple, as shown again in Fig. 3. Here we have once more drawn a large circle to represent the infinite celestial sphere ; and the dot which should represent the combined sun, earth, and earth's orbit around the sun is shown at the center, magnified into a circle. The observant reader will notice, upon comparing Figs. 1 and 3, that in the former figure the earth occupies the center of the sphere, whereas in Fig. 3 the sun is at the center. But the figures are interchangeable, as we already know, because of our having assigned infinite size to the celestial sphere. In Fig. 3 the smaller circle represents the earth's orbit around the sun, E' and E" being two positions of the earth 27 ASTRONOMY in its orbit. The corresponding apparent positions of the sun, as projected on the celestial sphere, are shown at S f and S". For, as we already know, if an imaginary line be drawn from the earth to the sun, we must necessarily see FIG. 3. The Ecliptic Circle. the sun from the earth along the direction of that imaginary line; and if the line be extended outward until it pierces the celestial sphere, the sun will appear to us projected on the sphere at the point where the sphere is pierced by the line. Now this sight line from the earth to the sun will neces- sarily lie entirely in the plane of the earth's orbit, for in that plane both the earth and the sun are at all times situated. Consequently, the sight line, when extended to pierce the celestial sphere, must necessarily always pierce that sphere somewhere on the circle cut out on the sphere by the plane 28 THE HEAVENS of the earth's orbit produced outward to infinity. But this circle is the ecliptic ; and thus we have a proof that the sun must always appear on the sky projected upon the ecliptic circle. And it is certainly a most remarkable thing that it should thus be possible to draw an imaginary circle on the sky such that at all hours of the day, on every day of the year, and of every year, when we look at the sun, it will appear to be situated at some point of that circle. Yet it all follows quite simply from the above elementary con- siderations concerning our earth's orbital motion around the sun. And it is furthermore already equally evident that as the earth progresses around its orbit, as shown by the curved arrow, the sun will appear to progress around the ecliptic circle with a rate of motion corresponding to the earth's own motion in its orbit. Figure 3 also gives a good opportunity to explain the meaning of the terms "angle" and " angular distance," which we shall have frequent occasion to use. An angle is defined as the difference in direction between two lines. Thus, if we consider the lines SS' and SS" in Fig. 3, the angle between them is indicated by the combination of letters S'SS". Every angle is thus indicated by a combination of three letters ; the middle letter of the three always indicating the point of the angle, or its so-called " vertex." The cor- responding angular distance on the celestial sphere between S' and S" is the arc S'S" ; and such angular distances must of course be measured in degrees. In Fig. 3 the angular distance S'S" is about 120, or one-third of an entire circum- ference of 360. These facts about the ecliptic constitute one of the most important discoveries of the very earliest astronomers. The hazy records of extreme antiquity indicate that the Chinese 29 ASTRONOMY knew the ecliptic and had measured its position on the sky as early as 1100 B.C. The early Greek astronomers of Alex- andria certainly knew of it ; for instance, we have fairly reliable records showing that Eratosthenes (276-196 B.C.) measured its position quite accurately. The next important phenomenon to which our attention must be directed results from still another motion of our earth ; namely, its axial rotation. As we all know, the earth turns on its axis once daily ; a motion which is quite distinct from its orbital revolution around the sun. Both motions take place simultaneously, the earth traveling around the sun in its orbit while it is at the same time spinning on its axis, much as a couple of waltzing dancers move from end to end of the room while at the same time spinning rapidly around each other. This terrestrial rotation has an immediate effect upon the celestial sphere and all the heavenly bodies which appear projected upon it. For the astronomer, being fastened to the earth, turns around with it, perforce. And as the earth turns, with the astronomer attached, it is constantly pre- senting him to a new part of the celestial sphere. Just so a dancing couple face every point of the compass in suc- cession, in consequence of their spinning motion, and quite independent of the fact that they are also moving about in the room at the same time. This turning of the astronomer successively toward dif- ferent parts of the celestial sphere makes that sphere appear to him as though it were turning around the earth instead of the earth turning within it, precisely as a railway passenger sees fields and trees apparently flying past his train, although he knows these objects are really fixed in position, and him- self in rapid motion. 30 THE HEAVENS The axial rotation of the earth takes place from west to east; and the consequent seeming rotation of the celestial sphere is from east to west. Objects projected on the sphere partake of this seeming motion ; the sun, the moon, the morning and evening stars, and all the other stars. This is the cause of day and night ; of the rising and setting of all heavenly bodies, including the sun. As the earth rotates from west to east, they all seem to revolve in the opposite direction daily, rising from beneath the eastern horizon, slowly climbing the sky, and again sinking down to set in the west. These facts are quite generally known with respect to the sun and moon, but comparatively few are aware that the stars also rise and set. It is reported that Sir George Airy, a recent astronomer royal of England, used to say that not more than one person in a thousand knows that the stars, like the sun, rise and set. Most people think the stars are always the same, simply a uniform countless assemblage of thickly clustered luminous points. Having thus explained the earth's rotation, we must next consider its rotation axis. Our planet earth, in its rotation, turns about an imaginary line or axis passing through its center and meeting the earth's surface at the north and south poles of the earth. Now imagine for a moment this rotation axis extended outward in both directions, farther and farther, until at last the two ends pierce the celestial sphere itself. They would, of course, mark out on the sphere two points corresponding exactly to the two terrestrial poles. These two points are called the north and south Poles of the heavens, or the celestial poles. The long line joining them is the axis of the celestial sphere, and a very short bit near the middle of the line is the terrestrial rotation axis. Figure 4 again shows the celestial sphere, this time with the earth 31 ASTRONOMY at the center, magnified from its proper size of a mere dot, so as to exhibit the earth's rotation axis and its prolongation, the axis of the celes- tial sphere. N and S are the north and south poles of the earth; NS is the terrestrial rotation axis ; and its pro- longation to the ce- lestial sphere marks out N' and S', the north and south poles of the celestial sphere. Now since the ap- parent rotation of the celestial sphere is merely a result of the earth's turning, and since the latter takes place around the axis, so also the great sphere's seeming turning must take place about this same axis. In other words, all the stars must seem to re- volve nightly around the two poles of the heavens. Stars very near the poles on the sky will seem to turn in little circles; those farther from the pole will seem to turn in larger and larger concentric circles. Figure 5 shows a few of these circles in which the stars appear to revolve nightly, and indicates that those near the two poles of the heavens are small. As we go farther from the poles the circles become larger, until at last we come to stars halfway between the two celestial poles, where the largest of all the circles occurs. The circles are of course all parallel ; and they are concentric in the sense 32 FIG. 4. The Celestial Poles. THE HEAVENS FIG. 5. Diurnal Circles. that their real centers all lie on a single straight line, the axis of the celestial sphere. The stars, as they appear to revolve in the circles, of course complete a revolution every twenty-four hours, since the axial rotation of the earth within the sphere is the true cause of the whole phenomenon ; and this axial rotation occupies exactly one day of twenty-four hours. And because of this daily period, the circles are called Diurnal Circles of the celestial sphere. Diurnal circles are defined, then, as par- allel circles on the celestial sphere in which the stars com- plete their daily apparent rotation around the celestial poles. We have just seen that the largest of all the diurnal circles is the one halfway between the two celestial poles; and it is a particularly important one. It of course divides the entire celestial sphere in two halves, which are called the northern and southern celestial hemispheres, and this largest diurnal circle is itself called the Celestial Equator. It corresponds exactly to the equator on the earth, which similarly divides our planet into northern and southern hemispheres. In fact, it is clear that as the terrestrial and celestial poles correspond exactly, so also the terrestrial and celestial equators must correspond exactly. And it is therefore also possible to define the celestial equator in a D 33 ASTRONOMY manner quite analogous to the definition of the other impor- tant great circle of the celestial sphere, the ecliptic. For if, as in the case of the ecliptic plane, we imagine the plane of the earth's equator stretched out and extended until it finally reaches the celestial sphere, it will cut out a great circle on the sphere, and this great circle is the celestial equator. So we might define the celestial equator as a great circle on the celestial sphere cut out by the plane of the earth's equator produced to infinity, and this definition is equivalent to the former one, which describes the celestial equator simply as the largest of all the diurnal circles. Having thus defined the celestial poles and equator, it is easy to carry analogy a little farther, and inquire what corresponds on the sky to latitude and longitude on the earth. The reader will recall from geography that when we desire to define the position of a place on the earth we do so by giving its latitude and longitude. Terrestrial latitude is defined as the angular distance of a place north or south of the earth's equator, and terrestrial longitude is its angular distance east or west from some so-called " prime meridian," such as that of Greenwich, England. Exactly analogous methods are used for defining a star's place on the sky, or the location of the point where it ap- pears to us projected on the celestial sphere. Unfortunately, the terms celestial latitude and longitude have not been used for this purpose. Instead of these terms, astronomers use the words " declination " and "right-ascension" ; which bear the same signification with respect to the celestial equator that terrestrial latitude and longitude bear to the equator on our earth. It is of interest to consider here the initial point from which astronomers reckon right-ascensions ; for there is no prime 34 THE HEAVENS meridian on the sky like that of Greenwich on the earth. Instead, astronomers use an initial point on the celestial equator, and from it the right-ascensions of all celestial objects are counted. This point is called the Vernal Equi- nox, and its location will be understood easily from the fol- lowing considerations. We have so far defined two great circles of the celestial sphere, each dividing the sphere in two halves. They are the ecliptic circle and the celestial equator. Now these two circles, as shown in Fig. 6, must inter- sect at two opposite points on the sphere ; for any pair of great circles on any sphere must evidently do this. These two op- posite points are called equinoctial points; one is the Vernal Equinox, the other the Autumnal FIG. 6. Two Great Circles intersecting at Opposite have occasion farther Points of the Sky. on to explain the im- (After Cassini ' a A * tronomie > * 78 - Pari3 - 1740 -> portance of these two points a little more in detail ; for our present purpose we need merely remember that the vernal equinox point is by universal convention selected as the initial point for measuring all right-ascensions. 1 At the risk of seeming somewhat tiresome, we must still add to these rather prolix preliminary explanations a very 1, Appendix. 35 ASTRONOMY few more necessary definitions. For there is still another important great circle on the celestial sphere, again dividing it into a pair of halves, but a different pair from the two hemispheres north and south of the celestial equator. This important great circle is the Horizon. In astronomy the horizon is precisely the same thing as the horizon in ordinary life. It is defined accurately as a great circle on the celestial sphere cut out by an infinitely extended level plane touching the earth at the point where the observer stands. 1 Of course, in the interest of exactness, we should note in passing that the same horizon circle would be cut out on the sky by a plane parallel to the first, but passing through the earth's center beneath the observer's feet. This is, of course, again a result of the fact that the earth's radius of four thousand miles, by which distance these two planes are separated, is a perfectly negligible quantity in comparison with the infinite distance of the celestial sphere. Having defined the horizon, it is easy to add two other definitions, both of which refer to astronomical terms having also the same signification precisely that they bear in ordi- nary English. These are the Zenith, which is simply the point of the celestial sphere directly overhead, and therefore exactly 90 distant from every part of the horizon ; and Altitude, or angular elevation above the horizon. Altitude is defined accurately thus : the altitude of a celestial body is its angular distance (p. 29) above the horizon. The altitude of the zenith is thus evidently 90. As we now know the meaning of the two points on the celestial sphere called the celestial north pole and the zenith, it is possible to define next the Celestial Meridian. This is a great circle drawn on the sphere from the celestial north pole 1 This plane, in mathematical language, is a plane tangent to the earth. 36 THE HEAVENS to the zenith, and thence extended completely around the sphere until it returns again to the pole. Very simple con- siderations show that the celestial meridian must pass through the north and south points of the horizon. 1 The accompanying Fig. 7, representing a celestial globe, may make the foregoing description clearer. The circle HVO is generally made of wood, and represents the celes- tial horizon. HPZAO is usually made of brass, and represents the celestial meridian, passing through the celestial pole P, the zenith Z, the north point of the horizon H, and the south point of the horizon O. The circle ASQ is the celes- tial equator, every- where 90 distant from the pole P. The cir- cle BC is a diurnal circle. ZV is a flexible strip of brass marked with degrees and pivoted at Z. It can be turned to any part of the horizon, and, by means of the degree divisions marked upon it, we can measure the altitude or angular elevation of any star above the horizon. Some of the constellation figures (p. 7) are also drawn on the globe. Having now defined the principal circles and points upon the celestial sphere, let us next investigate the position of the 1 For additional definitions and explanations, see Note 2, Appendix. 37 FIG. 7. The Celestial Globe. (From Lalande's Astronomic, 3 ed., Tome 1, p. 74. 1792.) Paris, ASTRONOMY north pole of the sphere with respect to our horizon. We shall first imagine an observer standing at the north pole of the earth. It is evident from the ac- companying Fig. 8 that such an ob- server would see the celestial pole directly H overhead, in the zenith. For P being the observer's posi- tion at the north pole of the earth, PS will be the earth's rota- tion axis, passing through the two poles of the earth. And if this axis is lengthened out to an infinite size, it will meet the celestial sphere at P', the north pole of the sphere, which will clearly be directly overhead. PR is a level plane touching the earth where the observer stands ; consequently H is a point of the horizon, in accordance with our definition (p. 36). OH' is a plane passing through the earth's center parallel to the level plane PR ; and the points R and R' will coincide on the celestial sphere because the distance PO is absolutely neg- ligible in comparison with the infinite distance of the sphere. These considerations show, that to an observer at the pole of the earth the celestial pole will be at the zenith, and its alti- tude, or angular elevation above the horizon, will be 90. To an observer standing at the terrestrial equator the position of the pole will be quite different, as shown in Fig. 9. 38 FIG. 8. Observer at the North Pole ; Celestial Pole in the Zenith. THE HEAVENS If we place the observer on the earth at E, and call E a point of the equator, the terrestrial rotation axis will be at PS, because any point on the terres- trial equator must be 90 distant from the terrestrial poles. This puts the celes- tial north pole at P', which coincides with K, a point on the horizon of an ob- server at E. It fol- lows from this that if we go to the equa- tor of our earth, we will there see the ce- lestial pole in our horizon, distant 90 from our zenith at Z, directly overhead. Having thus ascertained that to an observer at the pole of the earth the celestial pole appears overhead, and to one at the equator in the horizon, it is not difficult to realize that an observer traveling from the pole to the equator will see his celestial pole gradually seem to move down from his zenith to his horizon. For if the celestial pole occupies two extreme positions in the zenith and horizon when the observer is in two extreme terrestrial positions at the pole and equator, it is clear that as the observer occupies successive intermediate terrestrial positions, the celestial pole will seem to occupy successive positions also, intermediate between the zenith and horizon. This is the reason why travelers going south, and noting the pole 39 FIG. 9. Observer at the Equator ; Celestial Pole in the Horizon. ASTRONOMY star night after night, see that star gradually sinking lower in the sky; and if they continue southward quite to the equator, they see the pole star actually disappearing at the horizon. For the pole star is so placed in space as to be projected on the sky very near the imaginary celestial pole ; and consequently the visible pole star partakes of the changes which we have just explained. 1 In fact, the altitude, or angular elevation of the celestial pole above the horizon, is everywhere equal to the observer's terrestrial latitude, or angular distance from the terrestrial equator. This very important theorem enables us at once to study the very different appearance of the celestial sphere and its diurnal circles as seen from different places on the earth. At the equator, where the pole is in the horizon, the celestial sphere looks like Fig. 10, called the Right Sphere. Here the diurnal circles (p. 33) are all perpendicular to the horizon, and they are all bisected or halved by the horizon. Consequently, as the celestial bodies perform their daily apparent rotation with the sphere, in consequence of the corresponding daily axial rotation of the earth inside, their diurnal circles being all halved by the horizon, all the celestial bodies will be above the horizon just as long as they are below it. They will be "up" twelve hours, and "down" (or "set") twelve hours. 1 Note 3, Appendix. 40 FIG. 10. The Right Sphere. (After Long's Astronomy, Vol. 1, p. 91. Cambridge, 1742.) THE HEAVENS Now we have seen (p. 29) that as the earth travels around the sun in its annual orbit, the sun seems to travel around the ecliptic in a corresponding manner. But, wherever it may be projected on the ecliptic, it must always be on a diurnal circle ; in the light of what we have just learned about the right sphere, this diurnal circle must be halved by the horizon ; therefore, to an observer at the equator, the sun will be above the horizon twelve hours every day in the year. We therefore see that at the equator day and night are always equal throughout the whole year. Quite a different state of things holds at the pole, where we see what is called the Parallel Sphere, as indicated in Fig. 11. Here the ce- lestial pole is at the zenith, and the diur- nal circles are all par- allel to the horizon. If a celestial body is above the horizon at all, its entire diurnal circle is above the horizon ; it will re- main "up" twenty- four hours during each axial rotation of the earth. The largest diurnal circle, the equator, here coincides with the horizon ; to an observer in the northern hemisphere, stars between the celestial equator and the north pole never set ; those between the equator and the south pole never rise. How would these facts affect the sun, which is always seen in the ecliptic, as we know ? We also know that the ecliptic is halved or bisected by the celestial equator (p. 35). 41 FIG. 11. The Parallel Sphere. (After Long's Astronomy.) ASTRONOMY Therefore, during half the year the sun will be between the equator and the north pole. During that half-year its successive diurnal circles on the parallel sphere will be entirely above the horizon, and the sun will not set. This explains the important and well-known fact that at the north pole the sun remains above the horizon six months, and day, as well as night, is six months long. To observers situated on the earth in places like New York, intermediate between the pole and the equator, the sky appears in the form called the Ob- lique Sphere, shown in Fig. 12. Here the diurnal circles are neither perpendicular to the horizon, nor parallel to it. Being parallel to each other, they all make the same angle with the horizon, an angle which is different in different terrestrial latitudes. And the diurnal circles are not halved by the horizon, either. Each such circle is divided by the horizon in two unequal parts. If the circle is between the celestial equator and the north celestial pole, as B, Fig. 12, the part above the horizon is the longer. If the circle is between the equator and the south pole, as ^-the part below the horizon is the longer. Thus it follows that stars projected on the sky between the equator and the north celestial pole are above the horizon each day longer than they are below it, and vice versa. Only stars on the celestial equator itself have a 42 FIG. 12. The Oblique Sphere. (After Long's Astronomy.) THE HEAVENS halved diurnal circle, and are above and below the horizon equal twelve-hour periods. Some of the diurnal circles quite near the north pole, as A, do not reach the horizon at all. Stars projected on these diurnal circles will there- fore never set; and stars with corresponding diurnal circles near the south pole will never rise. 1 Observers in the south- ern hemisphere of the earth, of course, have these conditions reversed. The sun, always projected on the ecliptic, may have its diurnal circle divided either way. We have seen (Fig. 6, p. 35) that the ecliptic is bisected or halved by the equator. Consequently, when the sun is seen in one half of the ecliptic, it is between the equator and the pole, and therefore above the horizon longer than below it ; and when it is seen in the other half of the ecliptic, it is below the horizon longer than above it. In the one case, the days are longer than the nights ; in the other, the nights are longer than the days. As the sun is seen in one half of the ecliptic during about half of each year, it follows that during half of each year our days are longer than our nights in the temperate regions of the earth, where the oblique sphere prevails. Only when the sun is exactly on the equator, at one of the two points where it is intersected by the ecliptic, does the sun have a halved diurnal circle, giving us equal periods of light and darkness, equal days and nights. We have already seen that these two points of intersection of the equator and ecliptic are called equinox points. We now know the origin of the name ; when the sun is seen projected at either of these points of the ecliptic, we have equal days and nights. It may facilitate the comprehension of these facts if the reader will again examine Fig. 7, p. 37, the Celestial Globe. 1 Note 4, Appendix. 43 ASTRONOMY From the above elementary considerations follows at once a preliminary understanding of the phenomenon called the Seasons. For it is clear that the half-year during which our days are longer than our nights will be a summer or hot half-year, since we obtain our heat from the sun ; and the half-year with the long nights will be a cold or winter half-year. Near the terrestrial equator, where the right sphere gives constantly equal days and nights, there must be, and is, a complete uniformity of seasons. Near the pole, with its parallel sphere, there is a long six months' summer day, and a corresponding winter night. But the polar summer is itself cold, because even in summer the sun never rises to a great altitude above the polar horizon. 44 CHAPTER III HOW TO KNOW THE STARS ANY ONE beginning the study of astronomy quite naturally desires* to proceed as quickly as possible from the reading of books about the stars to an examination of the stars themselves in the sky. And in a first preliminary survey it is of interest to learn the names of the principal stars and constellations as they have been handed down to us from olden times. It is not at all difficult to acquire this knowl- edge, now that we have become acquainted (Chapter II) with the celestial sphere, and the more important lines and circles which astronomers imagine to be drawn upon that sphere. There are four objects that often puzzle beginners, when they attempt to compare star maps with the night sky for the purpose of identifying the more important bright stars. These are the same four things that so puzzled the ancients, the four bright planets, Venus, Mars, Jupiter, and Saturn. Mercury, the only other bright planet, is rarely seen ; but one or more of the above four is almost sure to be conspicu- ous in the sky, to a certain extent impairing the correctness of our star maps. For these star maps do not show the planets ; and for a very simple reason. We know (p. 10) that the planets seem to wander among the fixed stars ; they appear, now here, now there, at very widely varying points of the sky. On the other hand, the fixed stars (p. 7) retain relative posi- 45 ASTRONOMY tions practically unchanging; if, for instance, any three are located in a straight line, they will continue to lie on that straight line for centuries, so far as observations with the unaided eye can ascertain. It would be possible to make a star map showing both stars and planets as they appear on any given date. But such a map would not be correct six months later; while it would still show the fixed stars in their proper relative positions, the planets would be wrongly placed, on account of their wanderings. For this reason, astronomers omit the planets altogether from their star charts ; and beginners are puzzled. For the beginner, upon looking at the sky, always ob- serves the planets first of all, because they appear as bright as, or brighter than, the most brilliant fixed stars on account of their proximity to the earth. For instance, the beginner may see three lucid stars forming a small triangle, with the brightest star of the three at the apex of the triangle. He at once looks at the star map, to identify this triangle. Finding none in the proper place, he always concludes that he has misunderstood the printed directions ; packs up his books and lantern, and returns indoors, discouraged. The beginner in astronomy is always modest as to his abilities, and blames himself if the universe fails to fit the printed directions. Nor does any real astronomer ever lose this modest characteristic of the beginner; for he who has studied this science most deeply is ever most of all convinced that he is still a beginner. Of course the absence, of the triangle from the star map was simply due to the extremely brilliant object at the apex being a planet, and therefore properly absent from the map. The triangle on the sky appears on the map as a simple straight line with but two stars upon it. 46 HOW TO KNOW THE STARS Therefore the beginner should first of all learn to know the planets, so that he can eliminate them in comparing his star map with the sky. And it is fortunately easy to become familiar with the planets, perhaps even easier than to learn the stars. We have merely to take advantage of the planets' superior brilliancy in order to identify them. The best way is to make observations in the dusk, after sunset, before the stars begin to become visible. If there is any bright planet above the horizon at that time, it will be the first to show itself in the twilight ; it will be the evening star. But this priority of appearance in the evening is not nec- essarily a sure test for distinguishing the planets; for if no planet is above the horizon at the moment of sunset, the first object seen in the dusk of the twilight sky will, of course, be the brightest of the fixed stars then above the horizon. Therefore it is important to have another criterion for identifying the planets. It is a fact that the planets always appear projected on the sky rather near the ecliptic circle (p. 27). Therefore, if we could locate the position of the ecliptic circle on the celestial sphere, we should have addi- tional evidence as to whether the evening star appearing first is really a planet. If a planet, it must be near the ecliptic circle. The following method will enable the beginner to locate the ecliptic circle approximately on the sky. One point of the circle is, of course, determined by the position of the sun, which, as we know (p. 27), is always seen projected on that circle. Consequently, as we are making these ob- servations in the evening twilight, it follows that one point of the ecliptic is near that point of the horizon where the sun has just set. If we can now locate on the sky one other point of the 47 ASTRONOMY ecliptic, we can determine roughly the location of the entire ecliptic circle ; for two points are sufficient to locate any great circle on the sky. This can be done best by mak- ing use of the celestial meridian, which we recall as a great circle drawn on the sky from the zenith directly overhead down to the south point of the horizon (p. 36). The point of the meridian crossed by the ecliptic can be ascertained from the following little table, which gives the roughly approximate altitude, or angular elevation above the hori- zon, of that point on the meridian which is crossed by the ecliptic. To use this table, it is merely necessary to face the south point of the horizon, and imagine the meridian drawn ver- tically upward on the sky from that point to the zenith overhead. Next we must imagine the entire distance on the meridian from horizon to zenith divided into ninety equal degrees or spaces. Then the table gives us for various terrestrial latitudes, and for various dates, the number of degrees between the horizon and the point of the meridian at which the ecliptic crosses it. To facilitate the practical use of the table we have placed in it, next to each number of degrees, a simple fraction which will perhaps be more con- venient in making actual observations. Thus, where the table gives 46, we find also the fraction ^, meaning that the ecliptic crosses the meridian approximately halfway up from the south point of the horizon to the zenith. The fraction J belongs with 46, because 46 is approximately half of 90, the total angular distance from horizon to zenith. For example, if we should observe at New York (approxi- mate latitude 40) at sunset on January 1, we would imagine a great circle drawn around the sky from the sunset point of the horizon to a point on the meridian halfway between 48 HOW TO KNOW THE STARS TABLE FOR FINDING THE ECLIPTIC AT SUNSET Angular Altitude of its Intersection with the Meridian above the South Point of the Horizon LATITUDE 30 LATITUDE 40 LATITUDE 50 January 1 . . 58 f u n 46 \ u n 32 \l February 1 ; '. 73 U 62 | 49 \l March 1 . . . 82 | 71 U 61 IS April 1 ... -83 tj 73 |J 63 IS May 1 ... 78 f J 66 |{ 54 |J June 1 . . . 75 f? 52 |? 39 H Julyl . .;. 52 |? 40 *f 26 *f August 1 . . 42 |f 31 I? 20 if September 1 38 |f 28 If 18 if October 1 . . 37 |? 27 f s M 17 If November 1 39 U 28 |" 18 *? December 1 . . 45 4? 34 | 22 | the zenith and tftte south point of the horizon. This imag- ined line would be part of the ecliptic; extending it beyond the meridian, and around the sky to the eastern horizon, would give us the remaining visible portion of the ecliptic. And any object suspected of being a bright planet would necessarily be found very near this ecliptic circle. If the moon should chance to be visible at sunset, it would give us an additional point near the ecliptic; for the moon likewise always appears in the immediate vicinity of that circle. Actual observations of this kind will of course be extended in the twilight for about an hour after sunset. As the posi- tion of the ecliptic changes somewhat during that hour, we have added two letters to each number in the table. One of the letters is either a u or a d, and shows whether the ecliptic point on the meridian is moving up or down at the moment of sunset. The other letter is either an n or E 49 ASTRONOMY an s, and indicates whether the ecliptic point on the horizon is moving north or south at the moment of sunset. Thus, an hour or more after sunset on^ January 1 at New York (lati- tude 40), we should draw the ecliptic a little to the north of the observed sunset point in the horizon, and a little above the 46 point on the meridian. If we find a brilliant object in the dusk in this way on the ecliptic, we may still further test its planetary character by the absence of twinkling, for planets do not twinkle as much as stars. If the suspected object shines quietly, serenely, almost without scintillation, we may be tolerably sure it is a planet. Still another important aid is at the service of the begin- ner in his planetary search, the ordinary almanac. This will tell him what " evening stars" or planets are visible on the date when he makes his observations ; and it is certainly a great help to know in advance whether any planets are to be in sight. The almanac will also inform him as to the names of the planets he may expect to see. But even without an almanac it is generally easy to dis- tinguish between the different planets. Mercury, when visible, always appears very near the horizon, close to the point where the sun has set. The best date to look for it may be found by adding successive periods of one hundred and sixteen days to the initial date, Nov. 2, 1913. The planet can usually be seen for a few days before and after the dates obtained in this way, if the horizon is unusually free from cloud or mist. Conditions are .most favorable when the computed dates occur in the early part of the year, from January to May. And in these months espe- cially it is important to begin looking for Mercury at least a week before the predicted dates. 50 HOW TO KNOW THE STARS Venus should be sought after sunset on the ecliptic ; its angular distance from the sun is never more than 47 (about one-quarter of a great semicircle of the sky) ; and it may be much less. In looking for it, about an hour after sunset, we must remember that in an hour the sun will have moved a considerable distance below the horizon ; therefore, even if Venus is 47 distant from the sun, we must expect its distance from the sunset point of the horizon to be con- siderably less. An initial date when Venus attains its greatest distance from the sun is Feb. 12, 1913. Subse- quent occurrences of the same phenomenon may be expected at intervals of 584 days thereafter (1.60 years). These dates are, of course, highly favorable for observing the planet. Both Mercury and Venus are extremely bright. Mars, Jupiter, and Saturn also always appear near the ecliptic, but they may attain very great angular distances from the sun. They are, in fact, directly opposite the sun in the sky at certain dates, which are the most favorable dates for finding these planets. The dates are : Mars, Jan. 5, 1914, and thereafter at intervals of 780 days Jupiter, July 5, 1913, and thereafter at intervals of 399 days Saturn, Dec. 7, 1913, and thereafter at intervals of 378 days When thus opposite the sun, the planets are easily found. It is merely necessary to imagine a straight line drawn from the sun to the observer, and thence continued outward to the celestial sphere at a point opposite the sun. And if we imagine the line drawn an hour after sunset, we must not draw it from the sunset point of the horizon, but from the sun itself, making an approximate allowance for the sun's having moved some distance below the horizon during the interval of an hour since sunset. On these critical dates 51 ASTRONOMY Mars, Jupiter, and Saturn are on the meridian, due south, at midnight. For some time after the critical dates, these three planets, always remaining near the ecliptic, diminish their angular distances from the sun at the approximate monthly average rate of 25 for Mars, 30 for Jupiter, and 32 for Saturn. In estimating such angular distances it is well to remember that the angular diameter of the full moon is about one-half a degree. Furthermore, all the above numbers vary some- what in different years. The interval of 780 days between successive critical dates for Mars is especially variable : it is usually only about 750 days when the predicted date occurs in the early months of the year. A final test as to the planets may be obtained if the ob- server has a small telescope or good field glass at his disposal. In such an instrument the planets show their round disks quite plainly, while the fixed stars appear in the field of view as mere points of light without any visible extension into disks. In a three-inch telescope Jupiter shows moons, usually four, and Saturn usually exhibits the ring. Most observers detect in Mars a sort of reddish or ruddy color. Coming now to the identification of the fixed stars, we shall employ a method resembling somewhat our procedure in the case of the planets. It is not our purpose to include in the present volume detailed charts showing all stars visible to the unaided eye, but rather to confine our atten- tion to the stars of especial brilliance, and the more con- spicuous constellations with which every one should have an acquaintance. The first things to find in the sky are the pole star and the constellation Ursa Major (Great Bear or " Dipper"). These objects are near the north celestial pole, and very 52 HOW TO KNOW THE STARS far from the ecliptic ; consequently, the planets never appear among them to confuse the visible configurations of stars. The pole star, close to the north celestial pole, is always elevated above our horizon by an angular altitude very nearly equal to the observer's latitude (p. 40). To find it, we must therefore face the north, and imagine the celestial meridian drawn on the sky vertically upward from the north point of the horizon to the zenith. The pole star will then be found almost exactly on the meridian, and elevated above the horizon by an angle equal to the observer's terrestrial latitude. In New York, for instance, it will be elevated 41, or about f of the total angular dis- tance from horizon to zenith. The pole star is not very bril- liant ; being of the second magnitude, it will be inferior to sev- eral of the brightest stars visible in vari- ous parts of the sky. To verify this identification of the pole star we make use of Ursa Major. This constellation con- tains seven stars, not of the first magnitude, arranged as shown in Fig. 13. This figure exhibits the constellation as it appears in the sky at 9 P.M. about April 21 in each year. The reader will notice that the two end stars of the seven are in the me- ridian directly above the pole star, and that they point almost 53 FIG. 13. The Pole Star and Ursa Major as seen at 9 P.M. on April 21. ASTRONOMY exactly toward the pole star. For this reason these two stars are called "The Pointers." If these seven stars appear on the sky occupying the position shown in Fig. 13 with respect to the pole star at 9 P.M. about April 21, there is no doubt that the pole star has been identified correctly. In using Fig. 13, the reader should bear in mind that the constellation Ursa Major will appear much larger on the sky than it does in the figure. The scale of the figure has been so chosen that the distance of Ursa Major from the pole star is proportioned correctly to the elevation of the pole star above the horizon ;. and this choice of scale makes the constellation appear rather small. The other con- stellation figures, 14, 15, 17, 18, 19, 20, 21, 22, are all drawn to the same scale, to avoid confusion ; and the reader must expect all these constellations to be larger on the sky than they appear in the figures. In consequence of the seeming rotation of the celestial sphere about the pole (p. 32), the pointers will further occupy the positions shown in Fig. 14, at 9 P.M. on the several dates indicated in the figure. On intermediate dates the pointers will of course occupy intermediate positions ; and with the help of these figures the reader should have no difficulty in finding the pole star and making certain of its identification by means of the pointers. There is one other interesting constellation near the celestial pole: Cassiopeia, the "Lady in the Chair." It is found easily, also, by the, aid of the pointers. Imagine a straight line drawn from the pointers to the pole star, and continued beyond the pole star an angular distance equal to the distance between the pointers and the pole star. The end of the line will then be in Cassiopeia, and the appear- 54 HOW TO KNOW THE STARS 55 ASTRONOMY ance of that constellation is shown in Fig. 15. It looks like the letter W. The arrow shown in the figure indicates the direction of the pole star from Cassiopeia, and is approxi- mately a continuation of the line by means of which Cassio- peia was found. In comparing Fig. 15 with the sky, it is therefore necessary to turn the book around until the arrow is nearly parallel to the direction of the pointers from the pole star. This would make the arrow vertical upwards, as shown in Fig. 15, at 9 P.M. on May 18, and verti- FIG. 15. Cassiopeia ca j downwards at 9 P.M. on November 18. It would be horizontal to the right on February 18, at 9 P.M. ; and horizontal to the left on August 18, 9 P.M. On intermediate dates the arrow would of course occupy positions intermediate between these vertical and horizontal ones ; always, of course, at the hour of 9 P.M. Having thus indicated a method of finding the two impor- tant polar constellations, we shall next show how to identify the brightest fixed stars of the first magnitude visible in the United States and Europe. They are fifteen in number; in the following list we have arranged them in the order of luminosity, the brightest of all being placed first. To find these stars, we shall use a method similar to that employed for locating the ecliptic circle on the sky. Let the observer face the south at 9 P.M., and imagine the merid- ian drawn on the sky vertically upward from the south point of the horizon to the zenith, directly overhead. Let him once more imagine the meridian divided into ninety degrees or spaces, beginning at the south point of the hori- zon, and ending at the zenith. The following table will then tell him the dates on which the various stars in question 56 HOW TO KNOW THE STARS FIRST-MAGNITUDE STARS NAME CONSTELLATION COLOR Sirius Canis Major (Big Dog) Blue-white Vega Lyra (Harp) Blue-white Arcturus Bootes (Bear-keeper) Orange Capella Auriga (Charioteer) Yellow Rigel Orion (Hunter) White Procyon Canis Minor (Little Dog) White Betelgeuse Orion (Hunter) Red Altair Aquila (Eagle) Yellow Aldebaran Taurus (Bull) Red Antares Scorpius (Scorpion) Red Pollux Gemini (Twins) Orange Spica Virgo (Virgin) White Fomalhaut Piscis Australis (Southern Fish) Orange Regulus Leo (Lion) White Deneb Cygnus (Swan) White appear on the meridian, and their altitude or angular eleva- tion above the south point of the horizon when they are thus situated on the meridian, always at the hour of 9 P.M. The date of reaching the meridian at 9 P.M. is the same for all terrestrial latitudes ; but the altitudes vary in different latitudes, and are therefore given in the table for latitudes 30, 40, and 50. If the observer's latitude is intermediate between 30 and 40, or between 40 and 50, he can of course use altitudes intermediate between those given in the table. Sometimes the tabular altitudes are a little greater than 90. This indicates that the stars in question cross the meridian north of the zenith. To see them, an observer facing south would need to bend his head back so as to see a little beyond his zenith. A better way is to turn around and face the north, when the stars in question will be seen very near the zenith. 57 ASTRONOMY The identification of the bright stars will, of course, in- clude an identification of the important constellations in which they are situated, as indicated in the preceding table. TABLE TO BE USED IN FINDING FIRST-MAGNITUDE STARS ON THE MERIDIAN AT 9 P.M. STAR DATE ON MERID- IAN, 9 P.M. ALTITUDE ABOVE SOUTH POINT OF HORIZON Lat. 30 Lat. 40 Lat. 50 Sirius . . . Feb. 15 43 33 23 Vega . . . Aug. 15 99 89 79 Arcturus . * . June 10 80 70 60 Capella . . Jan. 23 106 96 86 Rigel . . . Jan. 23 52 42 32 Procyon . . Mar. 1 65 55 45 Betelgeuse Feb. 2 67 57 47 Altair . ... Sept. 3 69 59 49 Aldebaran . Jan. 13 76 66 56 Antares . . July 13 34 24 14 Pollux . . , Mar. 2 88 78 68 Spica . . . May 28 49 39 29 Fomalhaut Oct. 20 30 20 10 Regulus . . Apr. 8 72 62 52 Deneb . . - . Sept. 16 105 95 85 The above table is correct at 8 P.M. instead of 9 P.M. on dates two weeks later than those given in the table. It is correct at 10 P.M. on dates two weeks earlier than the tabular dates. To facilitate finding the bright stars on dates other than those on which they reach the meridian at 9 P.M., we now give another table containing the dates when these stars rise and set at 9 P.M. as seen from the three terrestrial lati- tudes 30, 40, and 50. In addition to the dates of rising and setting, the table contains the direction (as N.E., S.W., 58 HOW TO KNOW THE STARS NNW NNE WNW WSW etc.), to which the observer must turn in order to see his star rise or set. In making these observations it is impor- tant to remember that the immediate vicinity of the ho- rizon is usually ob- structed by trees, houses, etc., and that even when these obstructions are absent, the ho- rizon itself is sel- dom entirely free from clouds or mist. Therefore the ob- server should not expect a rising star to be visible for some time (possibly as much as an hour) after 9 P.M. on the tabular date of rising; and he may expect it to disappear from view some time before 9 P.M. on the tabular date of setting. The directions N.W., S.E., etc., to which the observer must turn, are roughly approximate only ; but accurate enough to facilitate finding the stars. The accompanying Fig. 16 shows the order in which these directions follow each other around the horizon. The table on the next page is correct at 8 P.M. instead of 9 P.M. on dates two weeks later than those given in the table. It is correct at 10 P.M. on dates two weeks earlier than the tabular dates. To aid still further in the identification of the finest con- 59 ssw FIG. 16. The " Points of the Compass." ASTRONOMY M Mars completes a revolu- tion around the sun. In the accompanying Fig. 52, therefore, Mars must occupy the same position FIG. 52. Kepler's Mars Observations. M on each of the above dates, while the earth will occupy the successive positions E, F, G, H. These terrestrial positions will be equidistant points on a circle with its center at B, if we suppose that the earth moves uniformly in a circular orbit. Under this supposition, these points must be equidistant, since they are separated by a series of equal time-intervals, each equal to the Martian period. And it is important to notice that the successive returns of Mars to the same point M are independent of any assumption as to the form or 1 Frisch, Kepler, Vol. 3, p. 275 ; Small, Kepler, p. 202. 194 THE PLANETS position of the Martian orbit. Whatever and wherever this orbit may be, Mars must return to the same point after each complete orbital revolution has been terminated. For the date 1590, Mar. 5, when the earth was at E, Kepler had Tycho's observation of the position of Mars as pro- jected on the ecliptic circle, or rather the position of that point on the ecliptic circle which was nearest to Mars. This gave the direction of the sight-line EM from the earth to Mars. The directions of the lines from the center B to the earth, and from the center to Mars, were furnished by the tables of planetary motion in Tycho's possession. Thus the directions of the three sides of the triangle EBM were known, and from these the three angles of the triangle were obtained by subtraction. But when the three angles of a triangle are known, it, is possible to calculate the relative lengths of the triangle's sides. 1 By successive applications of this process, Kepler computed that 2 BE = .66774 X BM. BF = .67467 X BM. BG = .67794 X BM. BH = .67478 X BM. These numbers should all be equal if the point around which the earth describes equal angles in equal times were at B, the center of the circle in which the earth is supposed to move. So Kepler's numbers show that the point about which the earth's angular motion is uniform is not at the center B of the earth's orbit, supposed circular ; but that it is at some point C, outside the center of the orbit. Kepler was 1 In the language of trigonometry, the sides are proportional to the sines of the opposite angles. 2 Frisch, Kepler, Vol. 3, p. 275. 195 ASTRONOMY able to compute the position of this point C ; and the corre- sponding changing distance between the earth and the sun. These results were of course obtained long before he perfected his three laws ; they are regarded justly as marking one of the most difficult and important advances ever made in human knowledge. There is still another remarkable peculiarity about the planetary distances from the sun; like the foregoing, of historic interest only. When we compare these distances, we find an accidental relation between them. Let us number the planets consecutively, from the sun outward, calling Mercury, 1 ; Venus, 2 ; Earth, 3 ; Mars, 4 ; the Planetoids, 5 ; Jupiter, 6 ; Saturn, 7 ; Uranus, 8 ; Neptune, 9. Let us then multiply the number 2 by itself four times, say for Mars, which is planet number 4. This gives 16. Then ta'ke three-quarters of this number, giving 12. Increase this result by 4, giving 16. Divide this by 10, giving 1.6. The result is an approximate value for the distance of Mars from the sun, counting the earth's distance from the sun as 1. This curious arbitrary rule is known as Bode's law ; astronomers have been acquainted with it for more than a century ; but we know of no physical reason why it should have a real existence. The following little table contains a comparison of the known planetary distances with their values calculated as above. In the case of the planetoids an average value is given. The table shows that the law is quite accurate until we reach Neptune ; then the error increases suddenly ; and we must conclude that the whole thing is one of those rare and jremarkable Coincidences that nature sometimes provides, apparently to mislead scientific investigators. 196 THE PLANETS PLANET No. KNOWN DISTANCE "BODE" DISTANCE ERROR " BODE " Mercury 1 0.4 0.5 0.1 Venus 2 0.7 0.7 0.0 Earth 3 1.0 1.0 0.0 Mars 4 1.5 1.6 0.1 Planetoids .... Jupiter 5 6 2.6 5.2 2.8 5.2 0.2 0.0 Saturn ...... 7 9.5 10.0 0.5 Uranus 8 19.2 196 04 Neptune 9 30.0 38.8 8.8 Having thus described certain famous historic methods of studying the planetary distances, etc., we shall next give a somewhat more detailed description of the planetary orbits, and the exact nature of the observations by means of which we study them in modern times. When we deter- mine the position of a planet by observation, we really deter- mine only the direction in which we see it projected on the celestial sphere. We point a telescope at the planet, and, by moving the telescope, bring the center of the plane- tary disk very accurately into the middle point of the field of view, which, for this purpose, may be supposed to be fitted with a very fine pair of cross threads to mark the center. Then, if the telescope mounting be provided with suitable " graduated" 1 circles, we can read the angles measured by those circles, and thus ascertain the direction of the planet in space, referred to certain points and lines, such as the celestial poles and equator. In other words, we measure the planet's right-ascension and declination (p. 34), as it is seen projected on the celestial sphere. We can also note the exact time when this observation 1 Brass circles divided into degrees, minutes, and seconds of arc. 197 ASTRONOMY was made, thus fixing the moment when the planet's direc- tion from the earth was measured. There are other methods of making these observations in addition to direct measure- ment with graduated circles attached to the telescope ; but all are alike in this : they furnish us with the direction in space of the sight-line joining the earth with the planet, and the instant of time when that line had the direction in question. Direct observation gives no information whatever as to the planet's distance from the earth. It tells us nothing about the length of the line joining earth and planet ; only its direction in space. If several observations of this kind have been made at different times, separated perhaps by a number of days, or even months, the earth will itself have moved consider- ably in its own orbit in the interval between the observa- tions. The planet will also have moved in its own orbit. Consequently, both ends of the line will have moved in different orbits and with different velocities; so that the changes in direction of the line will have been of an extremely complicated nature. But the changes in space of one end of the line are well known to us, the earth end. For we know the orbit of the earth around the sun, and can calculate the terrestrial position in space accurately for each moment of time when an observation was made. Knowing thus, from calculation, the position of one end of the line, and, from observation the direction of the line, the line itself becomes fully known, all but its length. Thus, in Fig. 53, if at a certain time t\ the earth was at a known point of its orbit EI, and the planet was seen in the observed direction PI, we know the line EiPi, all but its length. If, at a second observation, made at the time t 2 , the earth was at E z , and the planet 198 THE PLANETS was seen in the direction P Z) we again know the line E^Pi, excepting its length. And the same is true of a third line E 3 P Z . But it is to be remarked that these three lines will not lie in a single plane, unless the terrestrial and plane- tary orbits around the sun should happen to lie in the same plane, which is not accurately the case in our solar system. The problem now is to determine the planetary orbit from observations of this kind. But we know certain additional things about this planetary orbit. We know that it is an ellipse or oval ; that the sun is in the focus; and we know the position of the sun with respect to the earth from our knowledge of the terres- trial orbit, since the sun is also in the focus of that orbit. Both orbits have the same point for a focus, and the sun is in that point. Furthermore, we know the planet's orbital motion must be such as to satisfy Kepler's laws of FlG " 53 va ^ a n n s et Obser - planetary motion (p. 187), and so we know the planet must have moved in such a way as to cause its radius vector to sweep over areas proportional to the known time-intervals between the observations. It is a fact that an unknown planetary orbit can be thus determined from three observations such as have been de- scribed. Our geometric problem may therefore be stated thus : Given three observed straight lines in space ; it is re- quired to find an ellipse, cutting these three lines in three points, such that the radii vectores to the sun or focus from these three points will satisfy Kepler's laws. It would carry us too far afield in mathematical astronomy 199 ASTRONOMY to deduce here the methods by which this problem can be solved; but several interesting things about it can be enumerated. In the first place, the problem always has two solutions : there are always two ellipses in space that satisfy the problem. One of these is the planetary orbit; the other is the earth's own orbit. For the latter is also an ellipse ; it cuts the three lines because they are sight lines from the earth to the planet ; and the earth's motions in its own elliptic orbit, of course, satisfy Kepler's law. But suppose the problem to have been solved for the planet, also ; let us see what we need to know about the orbit in order to say that we know the orbit completely. Six different things must become known, and six only; these are called the six " elements" of a planetary orbit. First we must know the length of the largest diameter of the ellipse, and the degree of flattening, the eccentricity, as it is called. These two elements being known, we know the size and shape of the orbit. We could draw it to scale. Next we must know two more things, to define where the orbit is located in space. These two elements fix or deter- mine the position in space of the plane in which the orbit lies. To fix this plane, we must know the angle it makes with the plane of the earth's orbit, the ecliptic plane ; and we must know the position in the ecliptic plane of the line along which the planetary orbit plane cuts the ecliptic plane. This, as we have seen, is called the "line of nodes" ; and the angle between the two planes is called the " inclina- tion" of the planet's orbit. Having thus fixed the size and shape of the orbit in its plane, and the position of the plane itself in space, we must still know two more elements. We must know where the planet was in its orbit at some definite time ; and we must 200 THE PLANETS know the position of the orbit in its own plane. As we have already seen (p. 120) the planet is said to be in peri- helion when it is so placed in its orbit as to be at its nearest possible approach to the sun. The perihelion point is that point of the orbit which is nearest the sun. Therefore we use for one of the orbital elements the exact time of peri- helion passage. This element fixes the position in the orbit occupied by the planet at a definite moment of time. Finally, to locate the orbit in its own plane, we must know the direction in that plane of the " major axis," or longest diameter of the oval orbit. The six elements of a planetary orbit are therefore the following : 1. Longest diameter of oval 1 These fix the size and shape of 2. Eccentricity, or degree of flattening J the orbit. 3. Inclination of orbit plane 1 These fix the position in space of the orbit 4. Position of lines of nodes ] plane. 5. Time of perihelion passage. 6. Direction of orbital major axis in its own plane. A seventh orbital element is usually added : the Period, or time required for a complete orbital revolution of the planet. But this element is not really an independent one ; for the planetary periods and the diameters of the orbits are connected by Kepler's harmonic law (p. 188), by means of which either may be calculated from the other. The elements of an orbit once computed from three com- plete observations of the planet's apparent position, as pro- jected on the celestial sphere, and seen from the earth, the problem can be inverted, and the subsequent apparent pro- jected positions of the planet calculated from the elements. Thus is it possible to predict exactly where each planet may be seen in the sky. If a series of such calculated predictions 201 ASTRONOMY are tabulated for every day in the year, the tabulation is called a planetary Ephemeris. The United States govern- ment publishes such a tabulation annually, under the title "The American Ephemeris and Nautical Almanac." In it the planetary positions are printed for each day in the form of right-ascensions and declinations; and by means of these printed numbers it is easy to find the planets in the sky. The measurement of a planet's axial rotation period, cor- responding to the terrestrial sidereal day, is not a very easy matter. The best method of doing it is to observe with the telescope any spot or mark that may be distinctly visible on the planet's surface. As the planet turns on its axis, this spot will alternately appear and disappear; for it will of course be invisible when the planet's rotation carries it around to the side which is turned away from our earth. If we note the exact time elapsing between two successive returns of the spot to the apparent center of the planet's disk, this interval will be the planet's rotation period, or day. Such an observation must, of course, be corrected for any effects produced by variations in the relative positions of the planet and the earth, due to their respective orbital motions. And the result can also be much improved by allowing a considerable number of rotations to elapse between the two observations. If this can be done, the effect of any error in noting the exact time when the spot arrives at the center of the disk will be greatly diminished. But none of the planets, with the exception of Mars, have spots sufficiently perfect to admit of precise observation. Our knowledge as to the duration of the planetary days is therefore still very defective. If the paths of the spots, as they move across the visible 202 THE PLANETS planetary disk, can be mapped with sufficient accuracy, we can further ascertain from them the location of the plane- tary rotation poles, the inclination of the planetary equator to the plane of the orbit, and other related matters. Unfor- tunately, information of this kind is still very meager, on account of the lack of suitable spots on the surfaces of most planets. We shall next consider the measurement of a planet's size, its diameter, surface area, and volume. We have seen that ordinary astronomical instruments enable us to measure FIG. 54. Planetary Diameter. only the directions in which we observe the heavenly bodies projected on the celestial sphere. Thus, for instance, we can determine whether a star lies in the direction of the celestial equator, or whether its direction makes an angle of 10 with the direction of the celestial equator. If the former, the declination (p. 34) of the star would be ; if the latter, 10. Now if we thus observe the difference in direction of the two sides of a planetary disk (pp. 13, 52), we have at once the " angular diameter," or the angle subtended by the planet to an observer on the earth. Figure 54 explains this matter. E is the observer on the earth, P the disk of the planet. The two arrows show the directions in which the observer sees the two sides of the planetary disk projected 203 ASTRONOMY on the celestial sphere. The small angle at E is the differ- ence of these two directions, and it is the angular diameter of the planet, which is measured by observation. A knowledge of this angular diameter tells us nothing about the actual diameter of the planet in miles, unless we know also the distance D between the earth and the planet. For it is obvious that it would require twice as big a planet to subtend the angular diameter observed at E, if the planet were removed to double the distance D. But the distance Z), at the moment of observation, can always be calculated, if we know the dimensions and other particulars of the orbits pursued by the earth and the planet around the sun. And with the distance D available, it is easy to calculate the planet's diameter in miles from the observed angular diam- eter. 1 Having thus found the planet's diameter in miles, it is frequently convenient to represent it in terms of the earth's diameter as a unit. We can then find the surface area of the planet, as compared with that of the earth, by simply squaring the planet's diameter expressed in terms of the earth's diameter as unity. And the same number cubed will give us the planet's volume, as compared with the earth's. For it is a well-known mathematical principle that the areas of spherical bodies are proportional to the squares of their diameters ; and their volumes are proportional to the cubes of the diameters. A somewhat more difficult problem is the determination of a planet's mass. If there happens to be a satellite re- volving around the planet, the problem is comparatively 1 This involves merely a trigonometric solution of the long, narrow triangle shown in Fig. 54, using the angle at E, which has been measured, and the two including sides, which are both equal to D in length. 204 THE PLANETS easy. We can then determine by observation the period of the satellite's revolution in its orbit around the planet ; and its distance from the planet in miles can also be observed by precisely the same process just used to ascertain the planet's own diameter in miles. From these data the planet's mass can be computed. 1 With regard to the planet's satellites in general, there is not much more to be said. Their distances from the planets are determined, as we have just seen, by means of angular measures. Their periods of revolution around the planets are best found by noting the time elapsing between successive " elongations," or occasions when the satellite's orbital motion around its planet carries it to its greatest apparent angular distance from the planet. Most satellite orbits are almost exact circles : our own moon has an exceptionally flattened or elliptic one. And the planes of the satellite orbits are mostly very near the planes of the planets' equators ; indeed, the equatorial bulg- ing of the planet itself should suffice to pull the orbit plane of a close satellite into the planetary equatorial plane, from gravitational causes alone. That the planets have an in- creased diameter at the equator, and a corresponding polar flattening, has been verified by direct measurements in the case of our earth (p. 97). For the other planets its exist- ence is proved by comparing separate determinations of polar and equatorial angular diameters, if the position of the poles has become known. When the satellites are un- usually far from their planets, as in the case of our moon, their orbits lie nearly in the planes of the planets' own orbits around the sun. Before leaving this subject of orbits in the solar system, 1 Note 26, Appendix. 205 ASTRONOMY we shall discuss briefly the permanence or " stability" of those orbits. Will they endure forever? Will the solar system change materially in the course of time ? The planets move primarily under the influence of solar attraction as if they were themselves mere particles devoid of more than an infinitesimal mass. They are, in fact, all extremely small in comparison with the great sun. Never- theless, they do possess mass in a certain degree ; and con- sequently there is an interaction between them, which shows itself in slight perturbative effects upon the planetary orbits. In other words, if, by any method, we determine the elements of a planetary orbit in any given year, we shall not find these elements remaining unchanged forever. After the lapse of sufficient centuries, the planetary inter- actions and perturbations effect changes in the orbital ele- ments of the solar system. These changes are of two kinds : 1. The Periodic perturbations. 2. The Secular perturbations. The periodic perturbations increase and diminish in com- paratively brief intervals of time, comparable in length to the orbital periods of the planets themselves. But the secular changes, produced, as it were, in each orbit by all the other orbits acting upon it, are extremely slow in period, requiring many thousands of years to complete a cycle. The periodic perturbations never displace the position in which we see a planet projected on the celestial sphere more than about one or two minutes of arc, except in the case of Jupiter and Saturn, which are at times displaced from their proper or unperturbed orbital positions as much as half a degree, more or less. The most interesting facts about the secular perturbations, 206 THE PLANETS known from the researches of Laplace and Lagrange, are as follows : 1. The major diameters and periods of the orbits do not change. 2. The inclinations and eccentricities vary in an oscilla- tory manner. 3. The nodal points and perihelion points move around the ecliptic and orbital planes, respectively. 4. All changes of whatever kind are probably oscillatory ; so that the solar system is stable and permanent. After the lapse of sufficient ages, it will always return again to its original condition, no matter what changes it may have undergone. Of this, however, there exists a slight doubt, due to a possible imperfection discovered recently in La- place's mathematical demonstrations. 5. There is in the solar system an " in variable plane," not subject to change, and containing the center of gravity of all the bodies composing the system. Throughout the foregoing explanations, the word " period" has been used to indicate the interval of time required by a planet to complete an orbital revolution around the sun. But there exists more than one kind of planetary period. When we were discussing the planet earth, the sidereal year (p. 128) was defined as the time required by the earth to complete one orbital revolution around the sun. Thus, if we imagine an observer situated on the sun, the sidereal year will be the time elapsing between two successive ap- parent returns of the earth to the same fixed star, if both star and earth are supposed to be seen from the sun, and projected on the celestial sphere. In the same way, the sidereal period of any planet is the time required for a com- plete orbital revolution, from any fixed star back to the 207 ASTRONOMY same star, and seen from the sun. So far as the sidereal period is concerned, then, the earth is in precisely the same condition as all the other planets. We also found (p. 128) that the earth has a tropical year, used especially in calendar making. Of course no other planet has a tropical year, so far as dwellers on the earth are concerned. But the other planets all have another important kind of year, which the earth does not have. It is called the Synodic year and corresponds to the synodic period (p. 161) in the case of the moon. To define it, sup- pose, in Fig. 55, we have the orbits of the earth and Jupiter. For both planets the sidereal year is the time required to complete revolutions from any two points such as E and J back again to the same points. But for Jupiter, which has a synodic year, this synodic year is defined as beginning when a straight FIG. 55. Synodic Year. , ,t , *i. line drawn from the earth to the sun at S passes through Jupiter at J. And, similarly, the synodic year ends when the revolutions of both bodies make it again possible to draw a straight line from the earth to Jupiter through the sun. We have here supposed the orbits of both earth and Jupiter to lie in a single plane. This may be done as a first approximation for all the planets, since none of their orbits lie in planes very greatly inclined to the ecliptic plane, in which the terrestrial orbit is situated. Both the sun and Jupiter are seen from the earth pro- jected on the background of the celestial sphere; conse- 208 THE PLANETS quently, when they are in this straight-line position, they should appear to us at the same point on the sky. Owing to the existing small angle between the orbit planes, it will happen only rarely that they will appear to occupy the same point quite exactly. So the synodic year is considered to commence when they are as nearly as possible in a straight- line position, and therefore in the closest possible apparent proximity, as seen by us projected on the sky. At such a time, we say that Jupiter is in Conjunction with the sun. In general, the term " conjunction " is thus used whenever two celestial bodies are in very close prox- imity, as seen from the earth, projected on the celestial sphere. A very simple mathematical relation exists between the synodic and sidereal periods of any planet. It is based on the fact that the synodic period depends on a line passing through the earth as well as the planet, and must therefore be affected by the terrestrial as well as the planetary rate of orbital motion ; while the sidereal period depends on the planetary motion alone. 1 The foregoing reasoning applies strictly to those planets only whose distances from the sun are greater than that of the earth from the sun. These are called Superior planets to distinguish them from Mercury and Venus, which are accordingly called Inferior planets, because their orbits lie within that of the earth. These inferior planets, of course, have sidereal and synodic periods defined in the same way as the corresponding periods of the superior planets. The accompanying Fig. 56 represents the case of an inferior planet such as Venus. The sidereal period of Venus, like that of Jupiter, is the 1 Note 27, Appendix, p 209 ASTRONOMY time required by Venus to complete an orbital revolution around the sun, from any fixed star back to the same star again, supposed seen from the sun. But when we draw our straight line passing through the sun, the earth, and Venus, Fig. 56 shows that such a line can be drawn when Venus is in the position V, or in the position V. In either case, Venus and the sun will be seen from the earth close together, as projected on the celestial sphere ; and will therefore be in conjunction. When Venus is thus in conjunction through being situated between the sun and the earth, we call the phenomenon Inferior Conjunction ; and when the sun is between Venus and the earth, we call it Superior Conjunction. Of course a superior planet, like Jupiter, whose orbit is entirely outside that of the earth, can never be placed between the FlG - f$; m ferior earth and the sun, and can therefore never have an inferior conjunction. Superior planets have superior conjunctions only; inferior planets have both inferior and superior conjunctions. The synodic period of Venus is, then, the time in days elapsing between two successive inferior conjunctions, or two successive superior conjunctions. But the mathe- matical relation connecting the synodic and sidereal periods is slightly different from that which holds in the case of a superior planet. 1 The following little table contains approximate plane- tary periods ; and exhibits the interesting fact that both kinds of periods increase from Mercury to Mars, inclu- sive. Also, for this part of the table, the synodic periods are always the greater periods. But for all the other 1 Note 28, Appendix. 210 THE PLANETS SIDEREAL PERIOD SYNODIC PERIOD 1 Mercury . ... . . 88 days 116 days Venus . - 225 " 584 " Earth -.-.- 365 " Mars . 687 " '' 780 " Jupiter . '". 12 years 399 " Saturn . 30 " 378 " Uranus * 84 " 370 " Neptune 165 " 368 " planets the synodic periods are far smaller than the si- dereal periods; and they are all nearly equal in dura- tion. 2 It is plain that when any planet is in conjunction with the sun, we shall be unable to see it. Sun and planet being then projected on the sky at nearly the same point, the bright solar light will, of course, overcome the faint planet, and make it invisible. In other words, the planet, appearing near the sun, will be above the horizon in daytime. To make the planet visible, it must be far from the sun, as seen projected on the sky ; i.e. there must have been consider- able synodic motion since the time of conjunction. Visi- bility from the earth depends on synodic motion, not actual motion in the orbit. It is customary to use the term " elongation" to designate a planet's angular distance from the sun, as we see it pro- jected on the sky. At the time of conjunction, the planet's elongation is very small ; it may even be zero. We have seen in Figs. 55 and 56 the state of affairs when a conjunc- tion with the sun occurs in the case of a superior and inferior 1 These periods have been used on pp. 50 and 51. 2 Note 29, Appendix. 211 ASTRONOMY planet. As the synodic motion advances after conjunction, the planets increase their elongation from the sun. Figures 57 and 58 show the maximum elongations the two kinds of planets can attain. For the superior planets, like /, repre- senting Jupiter (Fig. 57), the elongation may reach 180. For the inferior planets, like V, repre- senting Venus (Fig. 58), there is a certain definite maximum angle of elongation SEV, which occurs when there is a right angle at V ; i.e. when there is a right angle Fio. 57. Superior Planet. Greatest between the direct i ons o f earth and sun, as seen from the planet. When the elongation is 180 in the case of a superior planet (Fig. 57), the sun is directly opposite the planet, as seen from the earth, projected on the sky. Thus, in the figure, the sun would be seen from the earth E projected toward the upper part of the page, and Jupiter directly opposite, projected toward the lower part of the page. The planet is then said to be in Opposition. The greatest possible elon- gations (Fig. 58) for the inferior planets Mercury and Venus, which can never be in opposition, are 47 1 for Venus, and 28 for Mercury. These numbers may be verified by means of a simple mathematical calculation. 2 Let us still remember that for the purpose of a first ap- proximation we may consider all the planetary orbits to lie in a single plane, the plane of the ecliptic. It follows that 1 Cf. p. 51. 2 Note 30, Appendix. 212 FIG. 58. Inferior Planet. Greatest Elongation. THE PLANETS we must always see the planets projected on the sky near the great circle cut out by that plane, the ecliptic circle, in which we also see the sun projected. Now since Mercury is thus always near the ecliptic circle, and always within 28 of the sun, it must appear to us to oscillate back and forth near the ecliptic circle, appearing now on one side of the sun, now on the other. This is also true of Venus, the other inferior planet, though here the arc of oscillation is much greater, as we have seen. When Mercury is at either extreme of its oscillation, it is in greatest elongation. When it is an eastern elongation, Mercury being east of the sun, the planet is visible for a short time after sunset. When it is a western elongation, the planet is west of the sun, and is visible a short time before sunrise. But owing to the apparent proximity of the sun, Mercury is always projected against the rather bright background of the sky near the point where the sun rises or sets at the horizon. Thus Mercury is not very easy to see. Venus, with its much greater possible elongation angle, is a very easy object to the unaided eye. In general, we thus find that the visibility of an inferior planet depends on the production of these maxima of elonga- tion by the synodic motion (cf. p. 50). In the case of a superior planet the state of affairs is very different. Visibility still depends on synodic motion ; as before, the planets cannot be seen near the time of conjunc- tion. But as their synodic motion advances, these planets do not approach a moderate maximum elongation, and appear to oscillate back and forth across the sun. For, as we have already seen, the superior planets have their oppo- sitions when their elongation from the sun is 180 ; then they are directly opposite the sun ; and are therefore observ- 213 ASTRONOMY able on the visible part of the celestial meridian at mid- night, when the sun is on the lower and invisible part (cf. p. 51). But, nevertheless, the superior planets do have certain oscillations in their apparent motions among the stars, as seen from the earth. These oscillations cause them to per- form at times so-called " retrograde" motions, traveling apparently among the stars from east to west instead of west to east, which is their usual direction of apparent motion. Sometimes, too, they have temporary " station- ary points/' appearing immobile for a short time, like fixed stars. To understand this state of affairs, let us consider for a moment the orbits of the earth and a superior planet like Mars. The accompanying Fig. 59 shows these orbits, not drawn to scale, but again supposed to be in a single plane, and circular. Be- FIG. 59. Retrograde ginning at the time of opposition, Mars, Motion of Mars. . , . . .. -mmn earth, and sun are shown on the line MES. At the end of one month, Mars will be at M' and the earth at E'. After three months, Mars will be at M" and the earth at E". At these three dates, therefore, terrestrial observers will see Mars projected on the sky along the three successive directions EM, E'M', and E"M". Both planets have been constantly and uniformly moving in the direction of the curved arrows, yet from E J we see Mars along E'M', apparently retrograded back of the direction EM, or contrary to the direction of orbital motion for both planets. At E"M" , Mars has again begun to move forward in its ap- parent motion among the stars, and that forward motion will evidently become more rapid a little later. It is also 214 THE PLANETS clear that about the time the apparent motion changes from retrograde to direct, Mars will for a short time appear quite stationary among the stars. And it is further evident from the figure that the middle of the arc of apparent retro- gression must occur about the time of opposition, when the planet is nearest the earth. There is but one more peculiarity of importance in con- nection with this apparent motion of the planets as seen from the earth, and projected on the sky. It arises from the fact that the orbital planes do not coincide accurately with the ecliptic plane, and therefore the planets do not always appear to us on the sky projected accurately on the ecliptic circle. They have certain small apparent motions toward the ecliptic circle, and again away from it. It follows that a planet's arc of retrograde motion does not simply return along the same line over which it traveled in its direct motion, as would be the case if all planetary motions were accurately in the ecliptic plane. The actual retrograde apparent motions usually involve peculiar curves, both for the superior and inferior planets. We shall close this chapter with another reference to the Keplerian method of determining the planetary periods. The matter could not be explained fully until the synodic period had been made clear. By the aid of that period, astronomers of old possessed still another simple way of ascertaining the sidereal period by observation. They could observe the date when the planet was in opposition to the sun, when it comes to the meridian at midnight. Then the interval between two successive oppositions is the synodic period (p. 208) ; and from the synodic period they could calculate the sidereal period, which is the true period of orbital revolution, by means of a simple mathematical 215 ASTRONOMY equation. 1 This method cannot be used for the inferior planets, as they do not have oppositions. The accuracy of this measurement of period could be increased greatly by comparing two oppositions between which the planet had made many revolutions around the sun. Thus, by a comparison of two oppositions separated by five hundred synodic periods, the error of observation affecting the exact times of opposition would, of course, be divided by 500. This was actually possible in the case of the principal planets, by utilizing existing ancient records of opposition observations. Furthermore, it was necessary to compare distant oppo- sitions, to eliminate the effects of orbital flattening in the case of both the planet and the earth. For it is clear that successive synodic periods will not be accurately equal : they would be so, if the orbits were truly circular ; but from the average of a large number of successive revolutions this source of error is practically removed. 1 Note 27, Appendix. 216 CHAPTER XII THE PLANETS ONE BY ONE IT will now be of interest to consider separately the many details in which the planets differ amongst themselves ; and we shall begin with Mercury, the one nearest the sun. As we know, it always appears projected on the sky in the vicinity of the sun (p. 50) ; sometimes on one side of it, sometimes on the other. The ancients did not perceive that this planet, seen alternately on opposite sides of the sun, was a single body. They had two names for it, Apollo and Mercury. The seasons on Mercury must present a rather curious problem. We have no means of ascertaining with any degree of certainty the angle between this planet's rotation axis and the plane of its orbit (p. 203). On the earth this angle is 66 J ; and it is owing to the existence of such an angle that we have the regular terrestrial seasons (p. 120). Therefore we know very little about the seasons of Mercury, so far as they may be analogous to terrestrial seasons. But we know that the distance of this planet from the sun has so large a variation between perihelion and aphelion that a very variable quantity of solar heat must reach it at different times. There must exist a variability from this cause, great enough to make very appreciable temperature changes. The interaction of this with possible seasons of the terrestrial kind may give rise to hot summers and cold summers, etc., in different years. 217 ASTRONOMY In the telescope, Mercury exhibits phases like our moon, and due to the same cause. It has little or no atmosphere in all probability ; and most astronomers can see but the faintest surface markings. Lowell, however, has published drawings of Mercury showing many geometric lines and angles ; and he thinks they change their apparent positions on the planet very slowly. This would indicate that there is no rapid axial rotation like the earth's. If the planet turned quickly on its axis, the rotation would soon carry some of the markings out of sight around the edge of the planet (p. 202). Possibly, therefore, Mercury, like the moon, rotates on its axis but once, while making an orbital circuit around the sun in 88 days. If this be so, there must be a very hot hemisphere, always facing the sun, and a very cold opposite hemisphere. But this would be modified somewhat by the very large librations (p. 171), which would result from Mercury having an unusually flattened orbit around the sun. The surface of Mercury is not very brilliant. It has been calculated that it reflects only 13 per cent of the solar light falling upon it. This p^'^tflffp nf liflht-reflection is called the planet's Albedo ; and Mercury has the lowest albedo in the solar system. The planet Venus, the other inferior planet, is also seen alternately on opposite sides of the sun, appearing as morning and evening star. But it attains a much greater angular distance from the sun than does Mercury, and is also more brilliant. It is at times- the brightest of all the planets, and can even be seen by the unaided eye in full day- light near the occasions of its greatest elongation from the sun. The telescopic phases of Venus range all the way from a 218 FIG. 60. Ptolemy's Theory of Venus. THE PLANETS ONE BY ONE complete circle down to a narrow crescent. 1 According to old Ptolemy's theory (p. 189), we should never see the phase of Venus larger than the half- moon shape. For Ptolemy supposed Venus moving on a circle whose center was always near a line joining the earth and the sun. It is clear, from Fig. 60, that the angle at Venus be- tween the earth and the sun could never be as small as a right angle; and so Venus could never show a phase bigger than the half-moon, according to the accepted Aristotelian theory of phase phenomena (p. 163). This matter is most in- teresting ; the moment Galileo turned the first astronomic telescope upon Venus, about the year 1610, and saw a phase larger than the half -moon, he had at once a strong proof that something was wrong with this particular detail of the sacrosanct Ptolemaic theory. To FIG. 61. Greatest Luminosity of Venus. , remove this difficulty, however, it would only have been necessary for Ptolemy to lengthen the radius of the Venus epicycle. 1 Well shown in Plate 8, p. 225, a reproduced photograph. 219 ASTRONOMY Venus gives a good example to demonstrate that a planet does not attain its greatest luminosity when nearest the earth, nor when exhibiting the largest possible phase. Fig- ure 61 illustrates this problem. The points S, V, E repre- sent positions of the sun, Venus, and the earth at inferior conjunction (p. 210). Venus has then no perceptibly disk ; we see its dark side. As time goes on, the phase of Venus grows; and the light reflected toward us increases in pro- portion to the increasing area of the visible disk. But at the same time the distance from Venus to the earth is increasing; and the intensity of the planet's light, as received by the earth, of course diminishes rapidly with the increase of our distance from Venus. Thus, in the figure, at the moment when Venus has reached the point V, the earth is at E'. The area or phase of the visible disk has grown from zero at V to the segment shown unshaded at V ', but the distance between the two planets has increased from VE to V'E'. Thus we see that, beginning with inferior conjunction, the disk area grows much more rapidly than the distance ; consequently, Venus grows more brilliant to our eyes. But, later on, this is reversed ; so there must be a certain point where Venus suddenly begins to substitute a decrease of visible brilliancy for the previous increase. This is the moment of maximum luminosity, as seen from the earth ; it is a nice problem, requiring the infinitesimal calculus for its solution, to determine this moment exactly. It will suffice here to say that it occurs about 36 days from inferior conjunction, when Venus has a phase like the crescent moon. Venus is believed to possess an atmosphere, for it has a very high albedo, or light-reflecting power, which indicates a reflecting surface containing clouds. Moreover, Venus is 220 THE PLANETS ONE BY ONE occasionally seen to pass between the earth and the sun, -a phenomenon called a Transit of Venus. When these transits are about to occur, and just as the planet is be- ginning to encroach upon the solar disk, as seen from the earth, a ring of light becomes visible around the part of Venus not yet projected upon the sun. This cannot be ex- plained otherwise than as a refraction or reflection of solar light by the planetary atmosphere. Certain ill-defined shadings have been seen at times on the planet's surface : Lowell goes so far as to give a map of Venus showing very clear geometrical structures of straight lines. These are of interest because of their bearing on Lowell's observations and theories as to the Mars " canals." From observations of the markings he concludes that Venus (as he also found in the case of Mercury) has a very slow axial rotation ; that it probably turns on its axis in 225 days, which is also its sidereal period. If this be correct, Venus must always turn the same face toward the sun. The planet Mars, which we shall next consider, differs greatly from Mercury and Venus. Its orbit is exterior to that of the earth and varies quite considerably from an exact circular form, so that the planet's distance from the sun, and its distance from the earth, undergo very wide variations, corresponding to the planet's motion in its orbit. Furthermore, unlike Mercury and Venus, Mars has certain very well-defined and constantly visible surface markings. These have enabled astronomers to ascertain with precision the length of the Martian day, or period of axial rotation. It is found to be about 24f of our terrestrial hours, or nearly the same as the day of our earth. The diameter of Mars is about half that of the earth ; and the inclination of its axis to the plane of its orbit around the sun is 65. Since the 221 ASTRONOMY corresponding angle of inclination in the case of the earth is 66f , it is clear that the Martian seasons will resemble closely those experienced by ourselves. Thus there are many points of resemblance between the two planets, Mars and the earth ; and therefore is Mars the best hunting ground for those who seek a planet with intelligent inhabitants. In the telescope, Mars shows no crescent phases like the moon or the inferior planets, because its orbit is outside that of the earth; and so the angle at Mars between the earth and the sun can never be as big as a right angle. Its atmosphere should be less dense than that of the earth ; for the absence of clouds is indicated by our seeing con- stantly permanent markings on the planet's own surface, and by the observable fact that Mars has an unusually low albedo, or light-reflecting power. Moreover, owing to its small size and small mass, the attractive force of gravity on Mars is less than half that existing on the earth. Conse- quently, it is not improbable that Mars has been deprived of its atmosphere in the same way that the moon is believed to have lost its own air (p. 167). Mars has two satellites or moons ; and they are in some respects the most peculiar bodies in the solar system. Their special oddity arises from their close proximity to the planet, and the consequent shortness of their periods of orbital revolution about it. Deimos, the outer satellite, has an orbital period of 30 h 18 m . Phobos, the inner one, revolves in its orbit in 7 h 39 m . These brief intervals are the " lunar sidereal periods " (p. 161) for Mars. Now the planet itself takes about 24f hours to complete an axial rotation. There- fore the orbital motion of Phobos, as seen from Mars, makes it move among the stars from west to east much faster than the apparent diurnal motion of the Martian celestial sphere 222 THE PLANETS ONE BY ONE makes the satellite seem to move from east to west. In other words, Phobos rises in the west and sets in the east ! Deimos, however, with its period of 30 h 18 m , travels diur- nally from east to west. We can investigate easily its apparent diurnal motion. Its period being 30. 3 h , in one 360 hour it moves among the stars In the same time the 30.3 360 diurnal rotation of the Martian celestial sphere is ^j-=- The apparent motion of Deimos is therefore : 360 360 24.7 30.3 in an hour, east to west. It will therefore make an apparent rotation of 360 around the sky in a number of hours found by dividing 360, the circumference of an entire diurnal circle, by the above hourly motion. The result of this division is 128 hours; and this is the " lunar day" (p. 176) of Deimos. And so we have the unusual condition that the lunar day is far longer than the lunar "month," or sidereal period. Approaching now the question of Martian "inhabitants," and their canals, we must first inquire as to the existence of water vapor in the atmosphere of the planet. Is there any? This is a matter of prime importance in connection with the famous supposed canals. If there exists on the planet a network of geometric markings, their explanation as waterways must stand or fall by the water vapor in the planetary air. A flow of water in canals can be imagined only if we suppose also evaporation of that water into an atmosphere, and subsequent precipitation of it as snow or rain. If this precipitation occurs for some reason principally near the planet's poles, while the evaporation takes place all along the canals, we might imagine the latter to have 223 ASTRONOMY been constructed artificially to carry the water away from the poles, so as to fructify and irrigate the entire planetary surface. When these markings were first seen by Schiaparelli the weight of observational evidence favored the presence of this water vapor. Such observational evidence is all obtained by means of an instrument called the spectroscope (p. 282). If the solar light reflected from Mars passes through a plane- tary atmosphere containing water vapor, its " spectrum," as seen in the spectroscope, will show certain bands called water vapor bands. Unfortunately, we cannot observe the Martian light until it has passed through the terrestrial atmosphere, which always contains some water vapor. The difficulty is to determine whether any observed vapor bands are due to the Martian atmosphere, or to that of the earth. There is but one way to distinguish between the two : we must compare the Martian spectrum with that of our moon. The lunar spectrum will show no water vapor effects except such as are due to the earth's air, for the moon itself has no atmosphere. Consequently, a lunar observation gives us only terrestrial water vapor bands ; a Mars observation gives us the terrestrial plus the Martian effects. Any observable difference between the two is due to Martian vapor alone. It is clear that this method of observation cannot be suc- cessful if there is very much water vapor in our own air. If there is, the slight difference between the moon and Mars will be masked completely. As existing observations have been found discordant, Campbell made an expedition to the summit of Mt. Whitney (15,000 ft.) in 1909. He took the necessary instruments with him and photographed the spectra of Mars and our moon at the exact time when Mars was most favorably situated in proximity to the earth. 224 PLATE 8. Photos by Barnard. Mars and the Crescent Venus. THE PLANETS ONE BY ONE At the great elevation of Mt. Whitney there was but little terrestrial atmosphere above the observer. The lunar spectrum, which exhibits the effects caused by our own air, showed so little water vapor that Campbell concludes it would never have been detected at all by a person previously ignorant of its existence. And the Martian spectrum was equally destitute of water vapor bands, even the faintest. On account of this elimination of the earth's air, these observations must be considered by far the most reliable in our possession. They seem to settle the water vapor question in the negative ; with the Martian vapor the Martian water goes; and, without water, the canals are impossible as artificial waterways. The markings on Mars of which we are certain consist of various permanent patches, lines, and areas of different shades; and there are also two bright spots at the poles known certainly since the time of Herschel. The accompany- ing Plate 8 contains a series of photographs of Mars, showing these markings and white spots very plainly. There is also distinct evidence of axial rotation, the exposures having been made in sets of three, with an interval of l h 22 m between the first and last set. In that interval the markings have moved perceptibly across the disk (p. 202). The lower part of the plate shows a photograph of Venus, in the crescent phase (p. 219). The polar spots seem to increase in the Martian winter season, and to diminish in the summer. If so, they may be ice-caps ; and it is this notion that gives color to the canal theory. For the melting ice-caps, in this theory, are used as the source of water to be pumped through the canals. Later, the water evaporated out of the canals is supposed to be returned to the poles by atmos- pheric movements, and there again precipitated as snow. Q 225 ASTRONOMY But if the planet, in comparison with the earth, is as cold as it should be according to its distance from the sun, there is quite a possibility that the caps are not ice at all, but perhaps some other substance, such as solidified carbon dioxide. It is not difficult to calculate the theoretic temperature on the surface of Mars, and it is found to be 33 on the Fahrenheit scale. 1 Since this low theoretic temperature would negative the existence of water in an uncongealed state, the advocates of canals are compelled to assume a heavy blanket of atmosphere, charged with much water vapor, to keep Mars warm, as it were, and cause the actual surface temperature to be far above its theoretic value. Such an atmosphere might act in this way ; but here again we find the absolute necessity of assuming the presence of water vapor in spite of observational evidence to the con- trary; and we are asked to imagine Mars to be an arid desert requiring irrigation, and yet above this arid desert a wet, foggy atmosphere, highly charged with water vapor. In view of the great public interest in this Mars matter, we shall venture to quote briefly from an article published by the writer a few years ago. First, as to the question : Do the geometric markings really exist ? The evidence here is almost all positive. Most astronomers who have observed Mars under favorable conditions and with powerful telescopes have seen markings, but the number of lines reported varies from several hundreds down to two or three. Finally, a very few prominent markings have been photographed. Let us first consider the visual evidence. Let us examine the witnesses, for that is what these astronomers really are, eye-witnesses of the lines on Mars. Some years ago, Lowell observed on the planets Venus and Mercury certain 1 Note 31, Appendix. 226 THE PLANETS ONE BY ONE systems of geometric markings. As it is impossible to suppose that all planets possess intelligent engineers, it is essential to the Martian theory to show that these Venus markings are quite unlike those now seen on Mars. Ac- cordingly, in his book entitled Mars and its Canals, pub- lished in 1906, Lowell refers to these older Venus observa- tions in the following words : ' i The Venusian lines are hazy, ill-defined, and non-uniform." But in the original article in which he described what he saw on Venus, 1 we find the following : "They (the markings on Venus) are not shadings more or less definite, but perfectly distinct markings. I have seen them when their contours had the look of a steel engraving." The only way in which these two statements concerning Venus can be brought into accord is to suppose that in the interval of nine years the observer has, for some reason, changed his opinion. Additional important testimony is furnished by Mr. A. E. Douglass, who was chief assistant at the Lowell observatory in Flagstaff for seven years, from 1894 to 1901, and since then held the position of astronomer there for a considerable period. In May, 1907, he published an article in the Popular Science Monthly, entitled "Illusions of Vision, and the Canals of Mars." This title alone shows that he had changed his views ; and his actual words in the 1907 article are : "The ray illusion (sic) is to me a very satisfactory explana- tion of many faint canals . . . the only objective reality is the spot from which they start." Again, speaking of what he calls the "halo illusion," he says : 1 Monthly Notices of the Royal Astronomical Society, London, 1897. 227 ASTRONOMY "The double canals of Schiaparelli in 1881-2 and of Perrotin and Thallon in 1886 . . are . . due to this cause/' And again : "Thus, in conclusion, we see that there are fundamental defects in the human eye producing faint canal illusions." Having thus outlined briefly the apparent contradiction in Lowell's testimony and the reversal of Douglass', it will be of interest to explain how it may be possible for observers to be in error to such an extent. For this purpose, we must mention some of the possible causes that may impair the correctness of an observation. At least five im- perfections come into play : imperfections of the earth's atmosphere, the telescope, the eye, the optic nerve, and the imagination. The process of seeing a thing is not at all simple. Light waves coming from the object under examination, after passing through the atmosphere and telescope, fall upon the outer surface of the eye. They are concentrated, or focused, by the lens in the eye, and pro- duce an effect, which we do not quite understand, upon the retina at the back of the eye. This, in some un- explained way, results in an impression being received by the brain through the optic nerve. Then, the brain in its turn does an unexplained something with that im- pression ; what we think we see is equal to that which came through the eye and optic nerve, plus what the brain does with it later. The mind cannot distinguish between an impression caused by the eye and optic nerve and one produced by action of the brain itself. Now it is important to remember that imperfections of the atmosphere, such as clouds, and all imperfections of the telescope, generally tend to diminish or destroy the 228 THE PLANETS ONE BY ONE possibility of vision ; but those of the eye and imagination, if they act, are just as likely to increase the number of details we think we see. Especially when the object is faint and indistinct, trembling, as it were, on the very limit of visibility, then especially can a very slight activity of the imagination either prevent our seeing it, or bring it seemingly into view. And this extreme faintness ad- mittedly exists in the case of almost all the Martian mark- ings. This theory explains why highly experienced observers see so much more than beginners. They think they are training the eye, so as to increase its powers, while in reality they may only be training that slight imperfection of the imagination which tends to increase details thought to be visible. The theory also furnishes an explanation of the fact that so considerable a number of observers think they have seen the faint canals. Nothing more strongly increases the powers of imaginary seeing of seeing the unseen than the knowledge that others have already made the ob- servation. We are very prone to "see" what we are told by others is visible : we think we see what we desire and hope to see ; do what we will, we cannot prevent this. Coming now to a consideration of the photographic ob- servations, we must mention one or two matters that are not well known to the general public. In the first place the size of a Mars picture made by direct exposure of a photographic plate at the focus of the Lowell telescopes is not larger than the head of an ordinary pin. From so small a picture we could not even hope to discover any details. Therefore we must enlarge it as much as possible ; and there are two ways of doing this. The first is to place an enlarging lens in the telescope itself. Two disadvantages limit this 229 ASTRONOMY method. First, it complicates the optical system of the telescope, with consequent loss of distinctness in the image ; and secondly, it makes the image on the plate less brilliant. The cause of this loss of brilliancy is simple. The total quantity of light received from the planet is constant; if, therefore, we spread it over an enlarged surface, each part of that surface will receive less light. For instance, with an enlargement of five diameters, the surface of the image is 25 times as large. The resulting diminution of light makes necessary a longer exposure of the photograph, and a con- sequent increased difficulty in making the clock mechanism attached to the telescope follow with exactness the motion of Mars in the sky. Experiment has shown the greatest photo- graphic enlargements that can be made in this way with the Lowell telescopes ; and the negatives of Mars, including the most recent ones made in the Andes, never exceed three- sixteenths of an inch in diameter. The other method of increasing the size of photographs is to use an ordinary enlarging camera, after the telescopic negative has been finished. There is here no difficulty in securing sufficient light, as in the case of enlargements made in the telescope itself. For we can use artificial illumination of the original negative, and make this illumination as strong as may be necessary. But there is another serious difficulty. Every photographic negative is developed by placing the plate in a chemical bath, after it has been ex- posed to light. This results in the precipitation of silver particles upon the plate, wherever its sensitive surface has been exposed to light. The picture is thus built up of sep- arate particles of silver. These particles are so small that the eye cannot distinguish the separate ones ; they run together, as it were, to form the picture. But the case 230 THE PLANETS ONE BY ONE is very different if we magnify the negative. We then see the separate grains of silver, scattered here and there about the surface, and the picture itself is lost altogether. The same difficulty occurs if we attempt to examine any photo- graph with an ordinary microscope of considerable power. The separate silver grains at once appear, and the picture effect is lost. All this photographic experimentation, therefore, has not yet resulted in good pictures more than three-sixteenths of an inch in diameter and produced by purely photo- graphic processes, though somewhat larger negatives may possibly be made in the future. 1 All larger published pictures have been reproduced from hand drawings, and are therefore simply visual observations. The alleged photographic verifications have been made by the same observers who have studied Mars in the visual telescope; again the eye, optic nerve, and brain were brought into play, and exactly the same causes as before impair the reliability of these visual observations of photographs. We conclude that neither by visual nor by photographic evidence has the existence of an artificial network of markings been proven, or even rendered highly probable. Therefore the time has not yet come when we shall have to inquire whether geometric lines indicate the presence of intelligent inhabitants ; that time will arrive if the lines, themselves are ever shown to possess a real or even a highly probable existence. We shall next consider the Planetoids, or Minor Planets (pp. 183, 196). A large number of these tiny bodies travel in 1 Those shown in Plate 8, p. 225, made with the 40-inch Yerkes' tele- scope, are about twice as large as the Lowell photographs ; and they show no signs of geometric canal networks. 231 ASTRONOMY orbits situated between Mars and Jupiter ; up to the present time several hundreds have been discovered and their orbits and motions computed. Much interest attaches to the history of the first one ever observed, Ceres. It had long been noted that the space between Mars and Jupiter was an exceptionally large empty space in the solar system ; and it seemed strange that no planet should exist there. The matter appeared still more peculiar after Bode's empirical law was published (p. 196) in the latter part of the eighteenth century ; for this law indicated that there should be a planet between Mars and Jupiter. And in 1781 this indication received stronger confirmation, when the older Herschel found Uranus, one of the modern exterior planets, in or near the position predicted by the law. An astronomical society was accord- ingly organized to make a systematic search for the expected unknown planet. But not until the first day of the nine- teenth century did the long-sought object reveal itself; and to an independent observer in the Italian city of Palermo. There Piazzi was making an accurate catalogue of the fixed stars. Every night he made telescopic observations from which he could compute stellar right-ascensions and dec- linations (p. 34) ; and he planned to enter in a catalogue these two coordinates for every star in the sky, bright enough to come within the range of his telescopic power. But he did not confine himself to single observations. Each night's work was checked by careful repetition on several other nights. Sometimes he found an error, which usually consisted in the discovery of a star that had escaped his notice at some previous observation. But on this historic occasion, he found that a star was absent, although he had observed it on another night. And, strange to say, there 232 THE PLANETS ONE BY ONE was also an additional star close by, one that had apparently remained unobserved on the previous night. The conclusion was irresistible that the new star was the same one he had observed before, and that it must have moved among the other stars in the interval. This motion among the stars (p. 10) is the distinguishing characteristic of planets. A third observation made the matter sure : the second star was again absent, and a third new star once more appeared in a place previously vacant. The apparent motion between the second and third observations was proportional, both in magnitude and direction, to that between the first and second observations. So it must surely be a wandering star, a true planet. Discovery and fame were his. But Piazzi was able to observe the new planet during a few weeks only, on account of illness. When news reached the astronomers of northern Europe, Ceres had already passed so near conjunction (p. 209) with the sun that further observations were impossible. There was well-grounded fear that the planet would not be found again ; for astrono- mers at that time had no good method of determining a planet's orbit from observations extending through such a short time. The older planets had been observed through many complete revolutions, and there was never any danger of their being lost, because they are bright enough to be visible easily by the unaided eye. But there was a young astronomer at Gottingen, Gauss by name, who succeeded in solving this difficult problem ; and from his published orbit and ephemeris it was easy to find the planet again as soon as the apparent motion of the sun in the ecliptic had brought the planet to a position where it could again be sought in darkness. A year later, in 1802, the second minor planet Pallas was 233 ASTRONOMY found. In 1804 Juno was added ; and in 1807, Vesta. It was not until 1845 that another appeared ; and three more in 1847. From that time on discovery proceeded but slowly, because the method in use was still the tiresome and arduous process employed by Piazzi. But in 1891, Wolf attacked the problem photographically, the photographic method having just commenced to be widely applied to astronomic purposes. His procedure was perfectly simple. A photographic plate was exposed in the telescope for several hours ; and care was taken to make sure that clock- work attached to the telescope moved it accurately during those hours, so as to keep pace exactly with the diurnal rotation of the celestial sphere. The photograph, when developed, would of course show a round dot corresponding to each fixed star within the field of view of the telescope. But if there was a wandering planet in range, it would move slightly with respect to the stars during the period of photographic exposure ; and conse- quently its image in the picture would be drawn out into a short line, instead of appearing as a round dot like the stellar images. Thus the presence of a line would infallibly betray the existence of a planet. As many as seven plane- toids have been thus found on a single plate ; so the method is enormously effective. To it we owe an immense increase in our minor planet knowledge during the past twenty years. Plate 9 is a photographed field of stars, with two planetoid lines, or " trails. " They will be found near the middle of the picture, as indicated by the marginal arrows. The trails are not quite parallel, showing that the orbit planes of these two planetoids are inclined at slightly different angles to the ecliptic plane. The difference in thickness of the trails indicates a difference of luminosity in the two planetoids. 234 I Photo by Barnard. PLATE 9. Discovery of Planetoids. THE PLANETS ONE BY ONE The orbits of the small planets present some interesting peculiarities. There are several open spaces where practi- cally no orbits appear. Curiously enough, these open spaces occur at points where the minor planet periods of orbital revolution in accordance with Kepler's harmonic law (p. 188) bear a simple relation to the period of Jupiter. It was long ago explained by Lagrange that if two planets have periods connected by a simple proportion, such as f, 3, |, etc., then persistent perturbations (p. 206) must result, which will gradually change the orbits until the simple relation is destroyed. It is in accord with this principle that Jupiter has forced the minor planet orbits out of these critical positions in space, and made them congregate at intermediate positions. As to the size of the planetoids, it has been computed that the mass of the entire group can be but a small fraction of the earth's mass. The individual planetoids are probably not more than one ten-thousandth as massive as the earth, and their diameter will not average more than twenty miles. As to the evolution of these minor planets, there is not much doubt. If we accept the hypothesis of Laplace, usually known as the Nebular Hypothesis, the planets were formed by the concentration of matter thrown off from the sun in early ages, while it was still in a gaseous or nebulous con- dition. This matter is supposed to have been detached from the central mass in the form of a ring : we have only to imagine the minor planets an exceptional case, in which the ring, after breaking up, was prevented from concentrating into a single body by the perturbative action of the big planet Jupiter. Any other hypothesis as to the early history of the planets must, of course, also explain the planetoids as an exceptional case in cosmic evolution. 235 ASTRONOMY Among the minor planets is one very remarkable one, discovered by Witt in 1898 and by him named Eros. Its orbit comes well within that of Mars, and it approaches the earth at times nearer than any other planetary body except our own moon. It can pass within about 13 \ million miles of the earth ; and this makes it an especially valuable planet to observe for the purpose of ascertaining by certain indirect methods the distance from the earth to the sun. It is altogether probable that observations of Eros will give us ultimately the most accurate value of the sun's distance yet attained. There will be a very favorable opportunity to attempt the necessary observations in 1931 (cf. p. 263). Proceeding outward from the sun, we now reach the planet Jupiter, the largest in the solar system, and the most brilliant object in the sky at night, with the possible occa- sional exception of Venus. In the telescope Jupiter is a magnificent object, second only to Saturn in interest. It surpasses Saturn in size, but it lacks the splendid, calm mysterious ring. Markings of a more or less permanent character exist ; they look like cloud-belts running along the planet's equator. And clouds they doubtless are ; for Jupiter must have retained, and must possess, a deep layer of atmos- phere, on account of his very high gravitational attraction (see pp. 167, 222) ; and since there is also a high albedo, or reflecting power, we should expect the outer surface to be made of clouds, which have this power in a high degree. Jupiter's rotation period, or day, can of course be deter- mined from the markings. It is 9 h 55 m ; but there is some uncertainty in this period, because the cloud-markings probably have a drift of their own on the planet's surface, and thus do not determine the rotation with precision. The axis is only 3 out of perpendicularity with the orbit 236 THE PLANETS ONE BY ONE plane ; so that there should be no considerable seasonal differences of temperature. But the average Jovian tempera- ture must be very high, for the constant visibility of clouds indicates a hot surface temperature. If this be correct, Jupiter must be in a condition slightly resembling that of the sun. It must furnish its own heat, for it is too far from the sun to receive much thermal assistance from that body. Jupiter has eight moons, of which four can be seen with a small telescope. They are very interesting historically, for their discovery in 1610 by Galileo gave its death-blow to the old Ptolemaic theory of the universe. The follow- ing is a brief account of this great discovery, partly quoted from an article by the writer, first printed in the New York Evening Post. What must have been Galileo's feelings when he first found with his "new" telescope the satellites of Jupiter? They were seen on the night of Jan. 7, 1610. He had al- ready viewed the planet through his earlier and less powerful glass, and was aware that it possessed a round disk like the moon, only smaller. Now he saw also three objects that he took to be little stars near the planet. But on the following night, the three small stars had changed their positions, and were now all situated to the west of Jupiter, whereas on the previous night two had been on the eastern side. He could not explain this phenomenon, but he recognized that there was something peculiar at work. Long afterwards, in one of his later works, translated into quaint old English by Salusbury, he declared that "one sole experiment sufficeth to batter to the ground a thousand probable Arguments." The 9th was cloudy, but on the 10th he again saw his little stars, their, number now reduced to two. He guessed 237 ASTRONOMY that the third was behind the planet's disk. The positions of the two visible ones were altogether different from either of the previous observations. On the llth he became sure that what he saw was really a series of satellites accom- panying Jupiter on his journey through space, and at the same time revolving around him. In the 12th, at 3 A.M., he actually saw one of the small objects emerge from behind the planet ; on the 13th he finally saw four satellites. Two hundred and eighty-two years were destined to pass away before any human eye should see a fifth. It was Barnard, in 1892, who followed Galileo. To understand the effect of this discovery upon Galileo requires a person who has himself watched the stars, not as a dilettante, seeking recreation or amusement, but with that deep reverence that comes only to him who feels nay, knows that in the moment of observation just passed he, too, has added his mite to the great fund of human knowl- edge. Galileo knew, on that llth of January, 1610, that the memory of him would never fade ; that the very music of the spheres would thenceforward be attuned to a truer note, if any would but hearken to the Jovian harmony. For he recognized that the visible revolution of these moons around Jupiter, while that planet was itself visibly traveling through space, must end the old Ptolemaic theory of the universe. Here was a great planet, the center of a system of satellites, and yet not the center of the universe. Surely, then, the earth, too, might be a mere planet like Jupiter, and not the supposed motionless center of all things. The most interesting phenomena about the Jovian satel- lites are their frequent eclipses and transits (p. 13). Any satellite may be eclipsed to us, either through passing behind the ball of the planet, or by moving into such a 238 THE PLANETS ONE BY ONE position that the planet interposes between the satellite and the sun. In the latter case, the satellite receives no sunlight to reflect in our direction, and so becomes invisible. At other times a satellite will " transit" between us and Jupiter. Then it generally becomes invisible, too, unless it happens to be projected against a dark part of the Jovian surface, such as one of the cloud-belts. Finally, a satellite may pass between Jupiter and the sun, when its shadow is thrown on the planet's surface, and is plainly visible as a round black dot slowly crossing the bright planetary disk. None of the transits or eclipses occur suddenly : the satellites all have disks of sensible magnitude, and thus encroach upon the planet's edge very gradually. Observations of the exact time of these satellite eclipses are useful as an easy approximate method of determining terrestrial longitudes. If we note the instant of local time when an eclipse occurs, and compare it with the calculated Greenwich time, as given in the astronomical almanac, we ascertain at once the time difference of the observer's position on the earth, measured from Greenwich. And this, multiplied by 15, gives his longitude at once in degrees (p. 74). This method requires no instruments beyond an ordinary small telescope; but it is not very precise on ac- count of the impossibility of observing the exact instant when the eclipse happens. Somewhat higher precision, with equal simplicity in method, may be secured from ob- servations of star occultations (p. 166).. It was from observations of Jupiter's satellite eclipses that Roemer, in 1675, first ascertained that light is not propagated through space instantaneously, but requires an appreciable time for its transmission. He used a long series of satellite eclipses, and found they did not succeed each other at equal 239 ASTRONOMY intervals. During half the year they came too soon, by gradually increasing amounts; during the other half-year they came too late, by similar quantities of time. Roemer soon found that they came too soon when the earth was approaching Jupiter ; too late when it was increasing its dis- tance from Jupiter. He concluded correctly that they came too soon because the earth's approach to Jupiter diminished the distance the light traveled before reaching the earth, after the eclipse actually occurred. Consequently, the light arrived at the earth sooner, and observers saw the event sooner, too. It is clear from Fig. 62 that the extreme difference between ac- celerated and retarded eclipses must be the quantity of time required by light to cross a diame- ter EiE z of the earth's or- bit. This is found, by ob- servation of the eclipses, to be 998 seconds. Itfol- an eclipse will be observed 998 seconds sooner than it would have been observed if the earth had remained at E z instead of traveling around its orbit. Bradley kn^w of Roemer 's observations when he explained aberration (p. 136), nearly a century later, as a result depending on the velocity of light and the earth's velocity in its orbit. Since modern laboratory experiments have made known that light moves at the rate of 186,000 miles per second, we 240 FIG. 62. Roemer's Discovery. lows that when the earth is at THE PLANETS ONE BY ONE have only to multiply this number by 998 to obtain the diam- eter of the earth's orbit in miles. This gives 186,000,000 miles, approximately ; and this number is correct. The planet Saturn is the last of the large planets observed by the ancients. The most interesting thing about it is its magnificent ring system, a series of three disk-like rings, situated nearly in the plane of the planet's equator. The history of their discovery is worth noting. We hear of them first from Galileo, who saw a couple of " handles" or ansce attached to the planet in 1610. He was unable to explain them ; and, when he looked for them again on a later date, was unable to see them at all. The story is that he gave them up as inexplicable. Nearly half a century later, in 1656, Huygens published a book De Saturni Luna Observatio Nova, in which he announced the discovery of a satellite, and also gave a correct explana- tion of the mysterious ansce. But Huygens was not quite certain that his explanation was right. He was most anxious to secure for himself the priority of discovery, and yet he was unwilling to make a premature and possibly in- correct announcement. So he resorted to the ingenious device of a "logogriph," or puzzle. It appears in the De Saturni Luna as follows : 1 aaaaaaa ccccc d eeeee g h iiiiiii 1111 mm nnnnnnnnn oooo pp q rr s ttttt uuuuu It was not until 1659, three years later, in a book entitled Systema Saturnium, that Huygens rearranged the above letters in their proper order, reading : 2 1 It may be found in 's Gravesande's ^dition of Huygens, Lugduni Ba- tavorum, 1751, p. 526. 2 Same edition, p. 566. B 241 ASTRONOMY " Annulo cingitur, tenui piano, nusquam cohaerente, ad eclipticam inclinato.' ' At the same time/ he re-published a series of drawings exhibiting several incorrect interpretations of the ring phenomena, as observed by various older astronomers. These are reproduced in Plate 10 : Fig. 1 is by Galileo, ob- served 1610 ; Fig. 2, by Scheiner, 1614 ; Figs. 3, 8, 9, 13, by Ricciolus, 1640-1650; Figs. 4, 5, 6, 7 by Hevelius; Fig. 10, by "Eustachius de Divinis," 1646-1648 ; Fig. 11, by Fontana, 1646 ; Fig. 12, by Gassendi and Blancanus. Under these reproductions from Huygens we have placed a fine drawing made by Barnard with the Yerkes 40-inch telescope, Dec. 12, 1907. It will be observed that by the publication of the logo- griph of 1656, Huygens secured for himself the credit of what he had done. If any other astronomer had pub- lished the true explanation after 1656, Huygens could have proved his claim to priority by re-arranging the letters of his puzzle. On the other hand, if further researches showed that his explanation was wrong, he would never have made known the true meaning of his logogriph, and would thus have escaped the ignominy due to publishing an erroneous explanation. So the method of announcement was com- parable in ingenuity with the Huygenian explanation itself. The ring phases admit of easy explanation. The rotation axes of all revolving bodies maintain constant directions in space, unless disturbed by attractions such as cause the earth's axis to produce the precession of the equinoxes (p. 129). Therefore the plane of the rotating rings must like- wise always maintain an unvarying direction in space. Now if this plane of the rings is imagined extended outward, 1 Same edition, p. 634. 242 PLATE 10. Saturn. THE PLANETS ONE BY ONE until it cuts the celestial sphere, it will trace out a great circle there. This circle necessarily meets the ecliptic circle in two opposite points (cf. Fig. 6, p. 35), which are called nodes; and it so happens that the angle between the great ring circle and the ecliptic is 28 on the celestial sphere. Saturn revolves in its orbit around the sun in a period of about 30 years. Therefore, it must pass one of the nodes every fifteen years, approximately. When Saturn is thus FIG. 63. Phases of Saturn's Ring. projected at one of the nodes, the sun, in its apparent motion along the ecliptic, may happen to appear in the other node at the same time. These positions are illustrated in Fig. 63 ; but in using this figure, it must not be forgotten that Saturn's orbit around the sun is very nearly in the ecliptic plane, in which the earth's orbit is also located. Let the sun, then, be at $, and the earth at E. Thus the sun ap- pears projected on the celestial sphere at the node '. Saturn, located at H in its orbit, is projected at the same time in the other node at H' '. It is evident that the earth must then lie on the line HS', the intersection of the two planes, 243 ASTRONOMY so that it is temporarily in the plane of the ring as well as in the ecliptic plane. When the earth is thus in the ring plane, we must see the ring edgewise ; and it is so thin that it then becomes quite invisible, except to the most powerful modern telescopes. It disappears, as Galileo found. Furthermore, near these times of disappearance, the earth may be for a short time on either side of the ring plane. And unless Saturn is quite accurately at the node, the sun will also be a little on one side of the ring plane or on the other. But the ring is illuminated on one side only, the side toward the sun. Consequently, if the sun happens to be on one side of the ring plane while the earth is on the other, we observe the dark side of the ring- system. It should then also be invisible ; but powerful telescopes will still show it, appearing like a fine line of light. This is well seen in Barnard's drawing (Plate 10, p. 242), together with certain condensations, or thick places in the ring. This drawing was made with the ring in the edgewise phase : in Plate 11, we have added a fine photo- graph, also by Barnard, showing the open phase. This negative was made Nov. 19, 1911, with the 60-inch reflecting telescope at the Mt. Wilson observatory in California. As we have found, these times of ring-disappearance occur about once every fifteen years. In years near the periods of disappearance, the ring is seen nearly edgewise : it then looks like a very narrow oval or ellipse ; and it opens out to the widest extent about seven or eight years on either side of the date of disappearance. But the ring can never open into a circle, for the earth can never be elevated more than 28 above the plane of the ring, since 28 is the angle between the ring plane and the ecliptic plane, in which the earth is always situated. And the earth would need to be 244 THE PLANETS ONE BY ONE elevated 90 above the ring plane to enable us to see the ring as a circle. As to the constitution of the rings, we have very certain and most interesting knowledge. They are neither solid, liquid, nor gaseous, but consist of a dense swarm of tiny satellites, moving in orbits closely interwoven, and all lying in, or near, the plane of Saturn's equator. They are so numerous, and their orbits so closely packed and in- tertwined, that we cannot see between them, and so they look like a solid disk. They are not very unlike the group of planetoids, which are known to encircle the sun be- tween the orbits of Mars and Jupiter (p. 231). In 1857, Clerk-Maxwell proved mathematically that it is impossible for a system of solid or liquid rings to exist permanently. They would be in unstable equilibrium, and must infallibly break into a series of satellites. And this mathematical demonstration was abundantly verified observationally in 1896, by Keeler. He observed the ring on both sides of the planet with the spectroscope. With this instrument, to be described later, it is possible to measure the linear velocity with which the edges of the ring approach the earth, or re- cede from it, as the ring performs its axial rotation around the polar axis of Saturn. Now it can be shown mathemati- cally that if the ring is really a mass of satellites, its outside edge should rotate more slowly than its inside edge. 1 On the other hand, if the ring is solid, of course the outside must move faster than the inside. Keeler found by actual measurement that the outside of the ring was moving 10 miles per second ; the inside, 12J miles ; and he thus verified observationally the correctness of Clerk-Maxwell's mathematical conclusion. This observation of Keeler's is destined to rank as a 1 Note 32, Appendix. 245 ASTRONOMY classic observation. We are given to regard astronomy as an ancient science, long since perfected, and incapable of further progress of importance. But this analysis of the ring constitution by methods purely mathematical; this theoretic prediction of invisible relative velocities of rotation ; and, finally, the complete observational verification by a method essentially novel, all this constitutes a chain of scientific research worthy of standing at the side of the master work of the seventeenth century. Saturn has ten moons, in addition to the swarm composing the ring system. The largest (discovered by Huygens) is visible in small telescopes. Five were found before 1700; Herschel found two in 1789 ; Bond one in 1848 ; the other two were discovered photographically at Harvard College observatory within recent years. The next planet, Uranus, was discovered by Sir William Herschel in 1781. The histqry of Herschel, and of this dis- covery, is not without interest. He was the son of a Ger- man musician, was born in 1738, and came to England in 1757 to seek his fortune. He settled at Bath, where he supported himself successfully as a music teacher. Although he worked very hard at his music, he found time to study also his favorite sciences of mathematics and astronomy. Having no instrument, he decided to make one ; but it was not until 1774 that he succeeded in constructing a tolerable reflecting telescope. He wrote in 1783: "I determined to accept nothing on faith, but to see with my own eyes what others had seen before me." Four times he made a new telescope, each of greater size than the last ; and with each he made a re-survey of the entire visible heavens. On his tomb is graven the epitaph : " Coelorum perrupit claustra." 246 THE PLANETS ONE BY ONE It was in the second of his celestial reviews, made with an instrument only seven feet long, that, as he says, "in examining the small stars in Gemini, I perceived one that appeared visibly larger than the rest. I suspected it to be a comet." Within a short time, Lexell was able to show that the motions of the new object could not be explained by any cometary orbit, and that it must be a new planet. It was perhaps the most startling discovery ever made in astronomy : Herschel named it the Georgium Sidus, in com- pliment to the English king, who promptly honored him with an appointment at court, and made him rich with a pension of 200. He removed to Slough, near Windsor, where he built " Observatory House," and made it memor- able as the scene of endless important astronomical dis- coveries. Long afterwards, Arago characterized it as "le lieu du monde ou il a etefait le plus de decouverles." Uranus has four satellites, two discovered by the same Herschel in 1787, and two by Lassell in 1851. They have one important peculiarity : they revolve in their orbits around the planet from east to west instead of west to east, the usual direction of orbital motion in the solar system. They are thus an exceptional case, and constitute in a way an unexplained difficulty in the Laplacian nebular hypothe- sis (p. 235), which would seem to require all satellites to revolve in the same direction. The outermost known planet is Neptune, remarkable principally on account of the interest attaching to its dis- covery. Shortly after Uranus had been found, astronomers searched their old records, and ascertained that good ob- servations of it existed as early as 1690. But it had always passed for a star, its disk not being big enough to betray its 247 ASTRONOMY planetary character on sight, even in the telescope. But no orbit could be found which would bring these early observa- tions into accord with the numerous ones which began to be accumulated immediately after discovery. And the planet soon refused to live up to its modern observations, also. More than one astronomer suggested that there must be an unknown planet exterior to Uranus, and perturbing its mo- tion, so as to throw it alternately in advance of its proper orbital position and behind it. In 1845, a young Englishman, Adams, who had graduated from Cambridge University only two years before, succeeded in constructing an orbit for the hypothetical exterior planet, basing his calculations simply on the observed discrepancies in the orbital motion of Uranus. He wrote to the astronomer royal at Greenwich, asking him to look for the new object with his big telescope in a certain definite position on the sky.' We now know that this position given by Adams was correct within 2, so that a little careful " sweeping" with the tele- scope would undoubtedly have revealed the planet. But the astronomer royal made an unfortunate mistake ; the story is that he delayed attending to Adams' letter. But another astronomer, Leverrier, was also working at the problem in France ; by August, 1846, he, too, had worked out the new orbit. On the 23d of September a letter from him arrived in Berlin and was delivered to Galle at the observatory there. Galle had a new and very complete star-chart of the proper region of the sky ; and it was for this reason that Leverrier had written to him, rather than to any other astronomer. As soon as it became dark, Galle went into the observatory dome, and began to compare his chart with the sky. He very soon found a strange body within less than 1 of the exact spot indicated in Leverrier's 248 THE PLANETS ONE BY ONE letter. It was an exciting moment; the new planet had been seen at last. One curious fact is that both Adams and Leverrier made use of Bode's law (p. 196) to obtain an approximate value for the supposed planet's distance from the sun. This law has no foundation in theory ; but it had proved to be fairly exact for all planets then known, including Uranus. But it fails for Neptune ; and accordingly both the computers were very largely in error as to this important element of their new planet's orbit. There is ground for supposing that their success was due in some degree to accidental favoring circumstances. But the result was unquestionably a great triumph for mathematical science and for Newton's law of gravitation : at this distance of time it is proper to divide the honor of the discovery equally between Adams and Leverrier. It is certainly great enough for two men, but in the middle of the last century, an ascerbitous controversy raged about the assignment of priority in this matter. Neptune is so far away from the earth that but few de- tails have been discovered concerning it. There is but one known satellite. And beyond Neptune, no further planets have been found, though the existence of such "ultra-Nep- tunian" bodies has often been suspected. But none has ever been revealed, even to the most careful photographic surveys so far made in the heavens. But there is one other material substance in the solar system that requires mention here. The mysterious Zodiacal Light is observable as a faintly luminous band traceable along the ecliptic circle outward from the sun for a consid- erable distance both east and west. There is also at times a faint glow called by the German name " Gegenschein " discernible in the part of the ecliptic opposite the sun. The 249 ASTRONOMY whole thing may perhaps be best explained as a ring of excessively minute planets, revolving around the sun in an orbit larger than that of the earth. Those near the sun would, of course, be the brightest ; and the Gegenschein would be the combined effect of an infinitude of these particles acting like tiny full-moons in their position of opposition (p. 163) to the sun. 250 CHAPTER XIII THE TIDES KEPLER was probably the first man to notice that the tides of ocean are due to some form of attraction exerted by the moon. He looked upon the moon as a personal ally of the earth, and in his quaint Latin remarks that "a mutual affection between allied bodies tends towards their union. " It is possible that Kepler may have had some hazy idea of gravitation as a species of personal characteristic of celestial bodies. Let us begin by summarizing the facts easily observable by any one who examines the behavior of the ocean along its shores. In the first place, it will be found that the water level changes considerably. During six hours, approxi- mately, the waters rise ; and again, for about six hours, they fall. In each day there are ordinarily two high tides and two low tides. Furthermore, in addition to merely rising and falling, the water also flows along the coast, in one direc- tion during a period equal to the time of rising tide, more or less, and in the opposite direction during a period corre- sponding in duration to the falling tide. Thus strong tidal currents exist ; and navigators frequently take advantage of them to increase the speed of ships, especially in the case of sailing vessels engaged in the difficult business of beating (as it is called) against an adverse wind. We shall consider the earth for a moment as a globe uniformly covered with a shallow ocean. The most im- 251 ASTRONOMY portant cause of tides on such an imaginary globe would be the gravitational attraction of the moon. We know, from Newton's law, that such gravitational attraction dimin- ishes rapidly with an increase in the distance separating the attracted particle from the moon. Consequently, the moon attracts the water on the earth's surface more strongly than it does the more distant solid earth beneath it. We should there- fore expect the moon to heap The Tides. U P Watei> at that P mt On the earth which is nearest to the moon. But there is also water on the earth on the side opposite the moon ; and this water is attracted less than the solid earth ; it is attracted least of any terrestrial material, because it is farthest of all from the moon. In other words (Fig. 64), the moon should pull the water away from the earth at M, tending to heap it up ; and at 0, it should pull the earth away from the water, again tending to heap it up. But the tidal forces exerted by the moon do not act in the above very simple way ; in fact, the actual heaping up of the water due to the above cause would be quite insignificant. A far greater effect is produced at points not directly under the moon. Here the tidal force is not vertical, because the moon is not directly overhead ; it may be regarded as divided between the vertical and horizontal directions. And the horizontal fraction is then usually the important one. It tends to move particles of water horizontally along the earth's surface, and to move them toward the place where the moon is overhead. But owing to the earth's axial rotation, the moon rises in the east and sets in the west. While it is east of the meridian in any given place, it is pulling the 252 THE TIDES water particles eastward with the horizontal fraction of the force. After crossing the meridian, it of course pulls them westward ; and therefore the result should be an oscilla- tion of the water particles backward and forward, occupy- ing approximately half a day to go and come. And of course this means half a lunar day (p. 176) ; not twelve ordinary solar hours. Since the moon, at any given moment, is east of certain places on the earth, and west of other places, it follows that different parts of the ocean will be oscillating different ways at the same time. This must produce a tidal wave, with high tide at the place where the crest of the wave is situated. The crest would follow the moon around, as the earth ro- tates, but it would not be under the moon. It can be shown that it would, in fact, ordinarily be 90 distant from the point under the moon. And there would be a second crest opposite the first, according to reasoning similar to that of Fig. 64 ; and therefore two daily high tides, following the moon around. Having thus outlined the explanation of the semi-diurnal tide, of which the period is half the lunar day, it is now possible to explain that the two tides each day are of un- equal size. In general, one rises higher than the other. Although we have seen that the high tidal crest is not under the moon, we can still reason as if it were, in order to explain the above inequality. As the earth rotates on an axis perpen- FlG - 65 - Dmmai inequality of Tides. dicular to the plane of the terrestrial equator, every point on the earth's surface must rotate in a circle parallel to the equator. Now, in Fig. 65, supposing 253 ASTRONOMY the tidal crests to be under the moon and opposite it, and the earth rotating around the axis NS, a person at P will not have as high a high tide as he will have twelve hours later, when the earth's rotation has carried him around to P'. The difference is shown by the dotted lines at P and P', and is called the diurnal inequality of the tides. It is due, of course, to the moon's not being situated in the plane of the equator. If the moon were in the equator plane, the tidal crests would be placed symmetrically with respect to the equator, and the two high tides would be practically equal. In fact, it must happen on two days each month that the moon really is in the equator plane. For the moon's ap- parent motion on the sky, due to its orbital motion around the earth, appears to take place in a great circle of the celes- tial sphere (p. 160), which must, of course, cut the celestial equator at two points. And it is a fact in accord with actual tidal observation that the diurnal inequality disap- pears twice each month. Then the two tides are equal. To complete this part of the subject, two more details must be mentioned. First, we recall that the lunar orbit around our earth is an oval or ellipse, with the earth at one focus, at some distance from the center. The moon will therefore be especially near the earth at certain times. When it is at the nearest point of its orbit, the perigee (p. 169), its tide-raising force will be greater than at any other time. In fact, this force is about | greater at perigee than at apogee. Perigee and apogee, of course, both occur each month ; consequently, the high tides are by no means equally high during the entire month. The other matter requiring mention is the effect of the sun on the tides. It operates in a manner precisely similar to the moon, but its greater distance diminishes the tidal 254 THE TIDES force, so that the solar tide is only about T \ of the lunar tide. The sun is far larger than the moon, but its greater gravi- tational attraction due to mass or bulk is more than counter- balanced by its greater distance. But it is clear that when the sun and moon are so placed that they act together, we shall get especially high tides ; and when they act against each other, the tides will be especially feeble. Of course these two bodies pull together when sun, earth, and moon are situated more or less in a single straight line ; and this occurs at the dates of full and new moon (p. 163). We then have the great tides called Spring Tides ; and when the moon is in the first and third quarters we have the little tides called Neap Tides. Spring and neap tides have rela- tive heights in the ratio of 11 + 5 to 11 5, or 8 to 3, be- cause the solar pull is T \ of the lunar. The above brief outline of tidal theory is greatly modi- fied when applied to the actual earth, upon which the oceans are deep bowls, large, but still limited in size; and the gulfs, sounds, etc., small, shallow limited cups. The laws of wave motion in areas limited as to size and depth come into play : according to these laws, the rate of progress of a wave, or its time of oscillation, depends on the depth of water. For instance, in a basin like the north Atlantic Ocean, a wave would move 500 miles per hour if it were set in motion, and then left to itself. It would pass from Europe to America in about six hours. Thus its period for going and returning would be nearly equal to the tidal period of half a lunar day. It can be demonstrated that when these two periods are thus about equal, the tides will be large. The water will practically oscillate about a neutral line in the middle of the ocean, giving high tide at the European coast when it is low tide at the American. 255 ASTRONOMY But this explanation is complicated still further by the configuration of the coasts, whereby the Atlantic does not act as a single basin with a single neutral line, but as several basins overlapping, more or less. But the general result is nearly as stated. When we come to the peculiar tides belonging to limited areas of the coast, such a basin as Long Island Sound, for instance, still a different explanation is required. Here the tidal wave is not really due directly to the moon ; it is a special local oscillation, set up by contact or impact from the lunar tide in the ocean outside the sound. In such cases, conditions become quite complicated, and often lead to tides much higher than the ocean tides. For instance, in Long Island Sound the rise and fall is about seven feet ; high tide occurs at nearly the same time throughout the sound ; and the wave motion produces a rapid current, or motion of the water particles along the sound. Another interesting tidal modification is found in the funnel-shaped Bay of Fundy, where the tides rise and fall as much as forty feet, or even more. Tidal phenomena produce results of importance other than recurrent changes in the oceans. Tidal evolution is a term used to describe effects produced on the earth as a whole by tidal action continued throughout vast ages of time. It is clear that the tidal motions of great masses of water must consume a vast quantity of mechanical energy, especially where the great tidal currents occur along the ocean coasts. In such cases there must also be much friction between the land and water. Friction will generate heat, and consume more energy. Now all this energy must be derived from some source : the law of the conservation of energy (p. 2) tells us that there 256 THE TIDES can be no manifestation of new energy, such as we have just mentioned, without an equal and corresponding dim- inution of the manifestation of energy somewhere else. The place where we should expect to find this diminution is in the earth's rotation. In other words, we should expect tidal friction, etc., to act as a sort of brake on the earth's axial rotation, and to bring about a consequent minute lengthening of the terrestrial day, after the lapse of sufficient ages of time. But the most delicate astronomic observations have failed to detect any such lengthening of the day. It must therefore be extremely small, certainly not more than TOFO o" f a second in a century. But there is another interesting consequence of these con- siderations : how does terrestrial tidal friction affect the moon's motions ? In Fig. 66, the moon is shown in the celestial equator, and the two great tidal protuber- ances at H and H f . Ac- cording to theory, as we have seen (p. 253), these protuberances or tidal crests should be at A and B, 90 from the point . FIG. 66. Tidal Effect on the Moon. under the moon. But so much of the tidal effect as acts like friction, to retard the terrestrial rotation, must also make the two protuber- ances lag behind their proper positions at A and B. This brings H nearer the moon than H' ; increases the lunar attraction at H, as compared with H' ; and therefore ac- celerates the moon's motion in its orbit, as shown by the arrow. Now it can be demonstrated from the principles of s 257 ASTRONOMY mechanics, that increasing the velocity of a body moving in an orbit will increase also the size of the orbit and the period of revolution of the body in the orbit. Therefore, tidal friction must make the moon recede from the earth, and must also lengthen the lunar sidereal period. In the hands of G. H. Darwin, these simple principles have led to an extremely plausible theory as to the formation of our moon. According to Darwin, the moon once formed part of the earth ; the entire mass was in a semi-liquid or plastic con- dition ; and was in quite rapid rotation about an axis. There was a tremendous flattening of the earth at the poles, due to plasticity. It can be shown mathematically that such a rotatory flattened plastic body may assume any one of several shapes. One of these possible figures is pear-shaped. The fact that we have a moon is thought by Darwin to prove that the pear-shaped figure actually was the one that happened to prevail. The rotating pear-shaped figure should then pass over into an hour-glass ; from that to a dumb-bell, with unequal weights at the ends. Finally comes a separa- tion; a true planet with a moon, both revolving rapidly about their common center of gravity, and very near each other. Now come gigantic tides ; tides compared with which our present ocean tides are absolutely insignificant. For the plastic earth was subject to great bodily tides, not merely little oscillations of a thin shell of ocean. Frictional forces then produced no mere slight perturbative action they were dominating forces. The moon was driven farther and farther from the earth ; and the lunar sidereal period was lengthened, until both bodies reached the condition now existing. If this theory is correct, it enables us to predict for future 258 THE TIDES ages the final condition of our moon, when the last stage of equilibrium shall finally prevail. When that occurs, the lunar sidereal period and the terrestrial day will be equal, the earth rotating on its axis in 55 of our present days, and the moon making an orbital revolution around the earth in precisely the same period. The moon should then be always opposite the same point of the earth's surface; and both bodies should revolve as though both were attached rigidly to the ends of an unbending bar. 259 CHAPTER XIV THE SOLAR PARALLAX WE have had occasion to mention several times the im- portance of a correct knowledge of the distance separating the earth from the sun. In our discussions of planet &ry motions we have considered this distance to be known ; in fact, we have assumed all the elements (p. 200) of the earth's orbit to be within our knowledge. Until the latter part of the eighteenth century, astronomers had only a very rough knowledge of the sun's distance, or of its Parallax. This last term may be defined easily ; it is exactly analogous to the corresponding term already defined (p. 169) in the case of the moon. By the solar parallax we simply mean half the angular diame- ter (p. 118) of the Fzo.67. Solar Parallax. 6arth > ^Upposed to be seen from the sun. Thus, in Fig. 67, imagine an observer situated on the sun at S. Draw a straight line from S to the center of the earth at C and another to the surface of the earth at 0. Then the angle at S between these two lines is half the angu- lar diameter of the earth as seen from the sun, and is there- fore the solar parallax. A simple equation exists, 1 by means 1 If, in Fig. 67, we let D represent the sun's distance; TT, the parallax angle ; and r the earth's radius ; we have at once, from the triangle SCO : tan TT ^ -y or D = ^^' (Cf. Note 20, Appendix.) 260 THE SOLAR PARALLAX of which we can calculate the solar parallax from the solar distance, or the distance from the parallax. If either be known, we can at once find the other. So the term " solar parallax" is really a substitute for " solar distance." The two terms are interchangeable, in a way; but they are not synonymous. One is an angle, the other a linear dis- tance. The present accepted value of the solar parallax is 8."80. We shall now consider various ways of measuring it. The reader will remember the method already mentioned (p. 168) for ascertaining the moon's distance by simultaneous observations of that body from two observatories, one in a high northern latitude on the earth, and the other in a high southern latitude. Of course this same method might be applied to the sun, but there is an objection that renders it almost useless. This objection arises from the small size of the parallax angle, which is really the quantity to be measured. The lunar parallax is about 1, the solar only 8. "8 ; consequently, any small error of observation, such as one-tenth of a second of arc, will have a considerable effect in the case of the solar parallax, while it would be quite in- appreciable in the case of the moon. This difficulty can be obviated in some degree by measur- ing the solar distance in an indirect manner. The distance is really only used to ascertain the scale, or size, of the planetary orbits. For with the aid of Kepler's harmonic law (p. 188), we can find the relative distances of the various planets from the sun, after we have observed their periods. Then, knowing the distances, we can make a map of all the orbits, here once more supposed to be concentric circles. And, again with our knowledge of the periods, we can locate the planets themselves in their orbits on the map, for any 261 ASTRONOMY date, if we have also observed the dates of conjunctions, etc. (p. 209). Such a map will be correct in every respect, except that the scale remains unknown. The heavenly bodies will be shown in their proper relative places on the date in question, but we do not know the number of miles corresponding to an inch on the map ; in other words, the scale of the map. To ascertain this, it will be sufficient to measure observationally the distance from the earth to any other planet on the date for which the map was drawn. This distance once known from the observation in miles, every other distance on our map of the solar system also becomes known in miles. This work will be most accurate, if we select for measure- ment a planet which comes comparatively near the earth, and make our observations and our map at the time of closest approach. For, after all, distance from the earth can be measured only by using the earth's own diameter in the way sur- veyors use what they call a "base-line." Thus, in Fig. 68, for a planet at P, we can at best only measure the angles POM and PMO, so as to ascertain the planet's distance by constructing the triangle POM from the known base OM (the earth's diameter) and the two measured angles. 1 But the base OM is always necessarily wofully short, compared with the planet's distance; the slightest errors in the measured angles produce very large errors in the distance. Therefore 1 We can, of course, substitute a solution of the triangle by trigonometry for the geometrical construction. 262 THE SOLAR PARALLAX we must do this work when we can observe a planet that is as near to us as planets are ever found. The planet Mars has been used with advantage. . A time is selected when Mars is in opposition (p. 212) so that it comes to the meridian at midnight, and can therefore be observed almost all night. And an opposition is chosen, too, when the earth has one of its closest approaches to Mars. This combina- tion of conditions gives the most favorable state of affairs for the desired measurements. F 69 Favor Figure 69 shows the positions of the sun, earth, able Opposition and Mars at the time of opposition. The orbits are not circles, and therefore the distances of the two planets from the sun are variable. If the opposition is one at which the earth happens to be at its greatest distance from the sun (aphelion), and Mars at its closest possible approach to the sun (perihelion, p. 120), the distance between the earth and Mars is as small as it can ever be ; and the con- ditions are especially favorable for its precise measurement. Having thus secured a favorable opposition, there are two different ways in which the Martian distance can be observed. We may employ a modification of the method already described for the moon (p. 168), and observe Mars from two terrestrial observatories situated as far apart as possible. In that case our base-line is the line joining the two ob- servatories. Or we can use the "diurnal" method. In this method, the planet is observed from the same place on the earth at two different times on the same night : first, shortly after sunset ; and second, shortly before sunrise. In the interval, the rotation of the earth on its axis will carry the observer to a different position in space ; and the line joining his two positions becomes the base-line. 263 ASTRONOMY Thus, in Fig. 70, let us disregard the slow orbital motions of Mars and the earth, since these will amount to but little in the few hours elapsing between the two observations ; and consider the earth's diurnal rotation only. Let the earth's center be at (7, with the rotation axis passing through FIG. 70. Diurnal Method. C perpendicular to the printed page; and the observer at O w . Then the observer will be carried in about ten hours from O w to O e by the diurnal rotation, and the length of the line O w O e can be calculated easily from the known dimen- sions of the earth, and the time elapsing between the two observations. This line O w O e becomes the base-line. From the point O w we see the planet projected on the celestial sphere in the direction MW ; from O e we see it in the direc- tion ME. The difference of the two measured directions is the angle O w MO e ; from this, together with the known base- line O w O e , we can calculate the distance MC from Mars to the earth. In making the observations, both for the diurnal method and for the method with two observatories, the most accurate way to observe is to use as an auxiliary some small star appearing near Mars on the sky. In Fig.- 71, which represents a part of the sky, such a star is shown. In either method of observation, owing to the ob- server's change of position from one end of the base-line to the other, the observations show the planet 264 Observa- tion of Mars. THE SOLAR PARALLAX projected at two slightly different positions on the sky, as MI and M%. The star itself is always seen in the same posi- tion, because the stars are all practically infinitely distant in comparison with any base-line available on the earth. The angular distance on the sky between the star and Mars (or its equivalent, the difference in direction of the star and Mars as seen from the earth) can be measured in the same way that the angular diameter of a planet is measured (p. 203), with an instrument called a Micrometer, to be described later. Thus we obtain from the two observations the angular distances of MI and M 2 from the star. Their difference is the arc M iM z on the sky ; and this is the angle O e MO w of Fig. 70, or the angle subtended by the base-line O e O w at the distance of Mars from the earth. Comparatively simple calculations will then transform this angle into a knowledge of the Martian distance, the Martian parallax, and thence the solar parallax. The diurnal and the two-observatory methods each have advantages and disadvantages. In the diurnal method, the observations can all be made by one man in one place with one instrument. This eliminates those errors that arise from per- sonality of the observer, and differences between different in- struments, etc. On the other hand, the two observations are necessarily separated by several hours in time, while they can be made quite simultaneous in the two-observatory method. In our present discussion, we have neglected totally the slight orbital motions of Mars and the earth. This is without effect if the two observations are simultaneous ; but they never are so in the diurnal method. The two planetary motions must be taken into account by calculation ; and thus any slight existing errors in our supposed knowledge of 265 ASTRONOMY the two planetary orbits produce slight indirect inaccuracies in the resulting parallax determination. But in the two-observatory method it is by no means easy to secure perfectly simultaneous observations, either. The two stations are very far apart on the earth. The weather is quite likely to be cloudy at one station when it is clear at the other, thus preventing simultaneous results. Indeed, vagaries of weather sometimes seem especially designed to hinder astronomers in their work, particularly when simultaneous observations are required. But in the diurnal method, the astronomer carries his weather with him, as it were, while he is rotated by the earth from one end of his base-line to the other. If he begins with a good clear night, he is quite likely to secure the necessary corresponding second observation. The best measurements of Mars by the diurnal method were made by Gill in 1877. He organized a special astro- nomical expedition to the island of Ascension in the south Atlantic ; and this was an especially favorable spot for his purpose. It was desirable to be near the equator, so that the diurnal base-line might be a long one. For at the pole, of course, the diurnal circle shrinks into a mere point. Gill obtained the value 8/78 for the sun's parallax from the Ascension expedition, but it appeared that various causes interfered to render the result less exact than was desired. Chief among these causes was the difficulty of measuring the angular distance between the planet and a neighboring star in the nianner we have described. This difficulty arises from the fact that Mars appears in the telescope as a disk, while the stars, of course, show only tiny points of light : and there seems to be some kind of personal error introduced by the effort to measure from a 266 THE SOLAR PARALLAX disk to a point. For this reason, Gill decided to repeat the work, using certain of the planetoids (p. 231) whose orbits are located between Mars and Jupiter. These little planetoids, of course, appear in the telescope like star points, and the above cause of personal error does not arise. Having been appointed director of the great observatory maintained by the British admiralty at the Cape of Good Hope, Gill attacked the problem on a large scale, using three different planetoids, Iris, Sappho, and Victoria, all of which have orbits suitably situated for the purpose. He caused to be constructed a special instrument for measuring the angular distances between the planetoids and neighboring small stars. This instrument is called a Heliometer. Four were made, and mounted respectively at the Cape of Good Hope, New Haven, Leipzig, and Gottingen. With all these special instruments simultaneous observations were made in such a way that both the diurnal and the two-ob- servatory methods could be used in the subsequent calcula- tions. The final result of the whole campaign was to fix the solar parallax at 8. "80 : this value is now regarded as the best, and has been adopted by all authorities to determine the scale of the solar system, and to perform calculations of every kind relating to planetary motions. Since this work of Gill's was completed, a certain newly discovered planetoid, Eros (p. 236), was found to have an orbit so placed that it can at times approach the earth nearer than any other object in the heavens except our own moon. Consequently, its distance from the earth must admit of very accurate determination. One of the close approaches of Eros, or favorable oppositions (cf. p. 263), occurred in 1900 ; and extensive observations were then made, 267 ASTRONOMY this time by the newly perfected method of photographic observation. Results have been published only very re- cently, and they confirm Gill's value, 8. "80. This method of minor planet observation is so superior that all other methods are of historic interest only; but, historically, the famous determinations from transit of Venus observations (p. 221) are well worth a careful study. If we consider the motion of an inferior planet like Venus, and assume the orbits of both Venus and the earth to lie in a single plane, then, at each inferior conjunction, Venus will pass between the earth and the sun. This is evident from Fig. 72, which once more shows Venus at inferior conjunction. In point of fact, the orbits do not lie exactly in a plane, but, nevertheless, a passage between us and the sun does sometimes occur. It FIG. 72. Venus at inferior Conjunc- will happen whenever the inferior conjunc- tion takes place at about the time when Venus is at one of the nodes (p. 200) ; or, in other words, when the conjunction happens while Venus is on the line of intersection of the two orbit planes of Venus and the earth. When on this line of intersection, Venus is for the moment in both planes ; and if there is also an inferior conjunction at the same time, there must be a transit. Venus, during transit, is seen as a small round black dot projected on the bright disk of the sun. This dot appears to enter the solar disk on the western edge, transits the sun in a line approximately straight, and finally passes away from the sun again at the eastern edge. It then disappears ; for, of course, we cannot see Venus against the sky back- ground, when near the sun, since the illuminated side of the planet is then turned toward the sun. We see the dark side ; 268 THE SOLAR PARALLAX it is "new Venus," if we may borrow a term from the analogy of the moon. Halley, a famous astronomer royal of England, showed how to determine the solar parallax from transit of Venus observations. 1 His method is shown in Fig. 73. FIG. 73. Transit of Venus, Halley 's Method. (After HerschePs Outlines of Astronomy, London, 1851, p. 289.) Two observers on the earth, located at the points A and B, widely separated in latitude, are provided with good clocks, and observe the exact quantity of time required by Venus to complete a transit across the sun's disk. But these two observers will see Venus crossing the sun along two different lines or " chords" SP and sp. The lengths of these two chords can be calculated in seconds of arc ; and thence the solar parallax can be determined. 2 Halley thought it would be possible to observe the dura- tions belonging to the two chords within two seconds of time. In actual observable transits of Venus this would give the solar parallax correct within one-hundredth of a second of arc. But no such precision of observation has ever been possible, principally because Venus has an atmosphere (p. 220) which introduces errors in the observed durations of the chords. Even in the modern transits of 1874 and 1882, these errors were as great as 10 seconds of time. 1 See Phil. Trans. Roy. Soc. Lond., Vol. XXIX, p. 1716, or Button's Abridgment, Vol. VI, p. 243. 2 Note 33, Appendix. ASTRONOMY There is an interesting astronomical story connected with the first observed transit of Venus. It seems that in the year 1639 there was a young curate in England ; a man living in miserable circumstances, but nevertheless inspired with an extraordinary zeal in the study of astronomy. This man, Horrocks by name, and at the time but 22 years of age, united in his own person two of the most poorly paid pro- fessions, that of an unbeneficent clergyman of the estab- lished church, and an astronomer. A diligent student of Kepler's writings, he had been able to correct an error of the latter, and to predict that a transit of Venus would occur on Nov. 4, 1639. He was unable to fix the exact hour, but he had a little telescope, and prepared him- self to watch the sun through the entire day. Now comes the peculiarly human part of the story. He found that the eventful date would occur on a Sunday. It seemed of the last importance to secure the observation, which was at that time an unprecedented one ; the circum- stances therefore found him undecided between his duty at church and his keen desire to secure fame as an astronomer. His sense of duty prevailed : he decided to give to the tele- scope only the intervals between services. And he had his reward, after afternoon service. Hurrying to his poor home, he was in time to see the black round planetary dot on the sun just before sunset, which happens at a very early hour in the northern latitude of England, and in November. To-day, a tablet may be seen in Westminster Abbey, bearing a Latin inscription commemorating this famous observation. 1 1 A very good account of it is to be found in Cassini's Elements d' Astronomie, published in 1740. At that time the Horrocks observa- tion was still unique, and Cassini founds many calculations upon it. The tablet in Westminster bears a quotation from Horrocks' own work, Venus 270 THE SOLAR PARALLAX Having thus outlined several methods of determining the sun's parallax by observations of planets, it remains to mention certain indirect methods of arriving at its value. For instance, the sun's distance can be computed from the theory of the aberration of light (p. 136). 1 Another quite independent way is called the perturbation method. Briefly stated, it consists in measuring the slight perturba- tions (p. 403) produced in the regular elliptic motions of the planets. These perturbations are caused by gravita- tional attractions between the bodies concerned, and the mathematical equations expressing them involve the distance from earth to sun as a factor. If this distance is known, it is possible to compute the perturbations ; or, the perturba- tions being measured observationally, the solar distance may be computed. in Sole Visa, published by Hevelius in 1662 at Dantzig. The quotation reads, "Ad majora avocatus quae ob haec parerga negligi non decuit." 1 Note 34, Appendix. 271 CHAPTER XV ASTRONOMIC INSTRUMENTS IT is not easy to understand the details of instruments from printed descriptions and illustrations. A short verbal explanation, by an astronomer in an observatory, with the instrument under discussion before him, is the very best way to gain an insight into the methods and machinery of obser- vation. For those who have no opportunity to visit an observatory, we give here a brief account of the most impor- tant kinds of astronomic apparatus, prefacing it with Plate 12, a photograph of the famous Lick observatory buildings on the summit of Mt. Hamilton, in California. To begin with the telescope itself. In the popular imag- ination, it is a big tube more or less filled with lenses from end to end. But this notion is quite wrong. Theoretically, the telescope has two lenses only, one at each end of the tube. The large lens, which is turned toward the sky, is called the Object Glass : upon it falls the light coming from the celestial object under observation. This light is concen- trated or " focused" by the object glass, and forms an image of the celestial body near the small end of the tube where the observer places his eye. Between this focal image and the object glass, the tube is empty. The other telescope lens is placed at the small end of the tube, between the observer's eye and the focal image, but very near the latter. This lens is simply a magnifying glass, or microscope, and is intended to enlarge the focal image, so 272 ASTRONOMIC INSTRUMENTS that the observer will see more detail than would be possible with the eye alone. This eye-end lens of the telescope is called the Eye-piece. In modern instruments, both telescope lenses are of the kind called " Compound" lenses. Each is made up of two or three separate lenses, placed close together, or even in actual contact. By this compounding of the lenses it has been found possible to eliminate partially certain optical imperfections from which all lenses suffer. But each compound lens really acts like a simple lens, except that it does its work better. Galileo's telescope of 1610, which found the moons of Jupiter and the spots on the sun, had single lenses only. Telescopes intended for terrestrial use have an extra lens in the eye-piece, called an " erecting" lens. For the simple astronomic telescope reverses the image of the object we look at ; we see it with the top and bottom interchanged, and the right and left sides likewise inverted. This is of no conse- quence in astronomy, since there is no up or down in space, and a round planet may be observed just as well one way as the other. But for terrestrial purposes, we must have ob- jects represented to the eye in their true positions. This extra erecting lens diminishes slightly the efficiency of the telescope, because it introduces two additional glass surfaces, the two sides of the erecting lens itself. And as human hands cannot grind lenses with absolute accuracy to their correct theoretic shape, it follows that the erecting lens causes slight errors that do not exist in the astronomical eye-piece. Plate 3, p. 17, shows the moon as seen in an astronomical or inverting telescope (cf. p. 166). The question is often asked: "What is the magnifying power of a given telescope?" Or the same question occurs in another form: "How near does this telescope bring the T 273 ASTRONOMY moon?" These two questions are really one. The moon's distance is 240,000 miles (p. 169) ; a telescope magnifying 1000 times would therefore bring it within a distance of 240 miles ; or give us as good a view, approximately, as we would get with the unaided eye if the moon were only 240 miles away. From what has been said before, it is perfectly clear that, within certain limits, the magnifying power of a telescope is just as great as we care to make it. The magnifying power comes from the power of the eye-piece regarded as a mi- croscope : it is evident that we can use a microscope of high power, or one of low power, on the same telescope, at dif- ferent times; and thus we can vary the magnification af- forded by the instrument as a whole. But there is a definite practical difficulty that limits the available power of the eye-piece. Suppose we are observing a planet, such as Mars, for in- stance. The quantity of light received from Mars may be regarded as constant, and therefore a constant quantity of light from Mars reaches the focal image. This light is there spread uniformly over the surface of that image. Now if we double the magnifying power of the eye-piece, we shall see twice as large an image. The same quantity of light from Mars is therefore spread over a larger surface, and so the image is dimmer than before. Increase of magnifying power in the eye-piece enlarges the image and brings out more detail; but it makes that detail fainter (cf. p. 230). If we continue to increase the magnification, there must come a time when we shall increase the detail, but will be unable to see it on account of faintness. For these reasons, astronomic telescopes are provided with a " battery" of eye-pieces of different powers. It is customary for the 274 ASTRONOMIC INSTRUMENTS astronomer to try gradually increasing powers until he finds, by experiment, the one that gives the best result. It will not be the same one each night, because a little higher power than usual may be employed when the terrestrial atmosphere is especially clear. As soon as the above limit of power is reached, no further increase is possible, unless we enlarge the object glass; or, in other words, make the whole telescope bigger. With a larger object glass we can gather more light from Mars, because the " light-gathering " power of an object glass must increase with an increased area of the glass itself. And if we gather more light, we can have a larger focal image without making it too dim for practical use. Ex- perience has shown that under the most extremely favorable terrestrial atmospheric conditions, it is possible to use a magnification of about 100 for each inch in the diameter of the object glass. In the case of the great Yerkes telescope, the diameter of that glass is 40 inches; a power of 4000 should therefore be conceivably possible; and the moon should be brought within the equivalent of -!$$$-, or 60 miles. But this theoretic result is never quite attained in practice, because all imperfections of the atmosphere are magnified by increased optical power. We see the moon as we would see it with the unaided eye at a distance of 60 miles ; but through an atmosphere more like water than air. Among the most important accessories of a telescope, when it is to be used for accurate measurement, is a pair of " cross- threads" at the focus. These threads are usually made of spider web ; for they must be extremely delicate, so that magnification by the eye-piece will not prevent accuracy of observation. The field of view of a telescope provided with cross-threads would look like Fig. 74. When the focal image 275 ASTRONOMY of a star is brought to the exact intersection of the cross- threads, by moving the telescope, the instrument is aimed or " pointed" accurately at the star; and if the telescope is attached to brass circles divided into degrees and minutes, it is possible to measure the exact direction in which we see the star projected on FIG. 74. Cross- . threads in a the celestial sphere. This kind of measure- ment is fundamental in the astronomy of precision (cf. p. 197). Sometimes the single pair of fixed cross-threads is replaced by a pair of parallel threads aa f and W shown in Fig. 75, together with a cross-thread cc' '. In such an arrangement the two parallel threads aa f and W are made movable, while cc' is fixed. The two parallel threads can be moved nearer together or farther apart by means of suitable screws out- side the tube of the telescope ; and a method is also provided for measuring accurately their distance asunder. This ar- rangement is called a micrometer (p. 265), and with it short angular distances on the sky can be measured with very high precision. Examples of such measurements are the obser- vation of Mars to ascertain the solar parallax (p. 265), the angular diameter of the sun (p. 118) or of the planets (p. 203), etc. The methods of mounting telescopes for as- . . . ,. T- FIG. 75. The tronomical purposes are most interesting. It is Micrometer. of course essential that the tube be movable : it must be possible to turn it about pivots or "axes," in order that it may be pointed toward different parts of the sky. The most simple form of mounting is indicated in Fig. 76, which shows an instrument called a Meridian Circle. OE is the telescope, being the object-glass and E the eye-piece. At / is the focus, containing the cross-threads. AX is a 276 ASTRONOMIC INSTRUMENTS rotation axis, firmly attached to the telescope. There is no motion of the instrument, except rotation around this one axis ; but a complete rotation about that axis is possible. The line AX is made to point due east and west when in proper adjustment; and it is made perfectly level. So the telescope must point north or south accurately, since it is placed at right angles to the axis AX. It follows that if the " sight-line " EfO be con- tinued outward indefi- nitely beyond 0, until it reaches the celestial sphere, it will meet that sphere at some point of the celestial meridian (p. 36). And if the telescope is turned through a complete rotation around the FIG. 76. Meridian Circle. axis AX, the sight-line EfO may be imagined to trace out the celestial meridian on the sky. It results from these considerations that the meridian circle can observe stars on the celestial meridian only; and, conversely, if a star be observed on the cross-threads of the telescope, the observer knows that it is at that moment projected on the celestial meridian. If the exact time of the observation be noted, too, it is possible to calculate the right- ascension (p. 34) of the star ; and thus are the right-ascen- 277 ASTRONOMY sions of stars and planets determined observationally by astronomers. This meridian instrument is provided also with two brass circles c and c', divided into degrees and minutes of arc. By the aid of these circles it is possible to measure the altitude (p. 36) of the star, or its angular elevation above the horizon at the moment when it is observed to be on the meridian. From this measurement of altitude it is possible to calculate the star's declination (p. 34) . Thus the meridian circle makes known both the right-ascension and declination of the star or planet, and these give us its exact location on the sky at the moment of observation. It will be noticed that a precise record of the time of such observations is most essential. For this purpose astronomers employ " standard" pendulum clocks of the most extreme accuracy, usually kept in vaults and air-tight cases where the temperature and barometric pressure are not allowed to vary, so as to produce inaccuracy in the running of the clocks. For the actual record of the time of observation, the clock is connected with an electric " chronograph." This instrument maintains an automatic record of the running of the clock upon a sheet of paper attached to a revolving brass drum; and upon this same sheet the observer can record electrically the instant of time when he makes his observation ; and he can make this record without his error ever exceeding one-fifth of a second of time. By taking the mean of several observations, the average error can even be reduced below this small amount. The above process, as outlined, indicates the method of observing stars of unknown location on the sky in order to make known their right-ascensions and declinations. But the same meridian telescope, clock, and chronograph can 278 PLATE 13. The Lick Telescope. ASTRONOMIC INSTRUMENTS be used for the observation of known stars ; and will then furnish a check upon the time indicated by the standard clock. It is by this latter process that the astronomic standard clocks are kept correct in order that the observatories may be able to distribute correct time telegraphically for the control of the " regulator" clocks that are to be found in most jewelers' shops, where people "set" their watches (cf. p. 18). Having thus described briefly the astronomer's instrument of precision, the meridian circle, we must next consider the "equatorial" mounting, the arrangement with which almost all ordinary telescopic observations are made. This is the form of mounting usually fitted with a micrometer for the measurement of small angular diameters, distances, etc. A photograph of a large telescope, mounted equatorially, is reproduced in Plate 13. This instrument is set up at the Lick observatory; the diameter of the object-glass is 36 inches ; and the whole observatory floor is built like an ele- vator, so that it can be moved up and down, to accommodate the observer when the tube is directed to various parts of the sky. The supporting pillar of the telescope mounting passes down through a hole in the floor, so that the in- strument itself is not disturbed, when the floor is raised or lowered. James Lick, donor of the telescope, is buried under it. The first essential of such a mounting is some form of "universal joint," so that the tube may be aimed at any part of the sky. A single axis, such as that of the meridian circle, is not sufficient. Accordingly, in the equatorial, shown again in Fig. 77, we find two axes, A and A', perpendicular to each other. The telescope can be rotated around the axis A' ; and this axis itself, with the telescope attached, can be rotated around the axis A. A combination of the two rota- 279 ASTRONOMY tions furnishes a universal motion, giving access to any part of the sky. By means of these rotations, the tube can be moved from the position of Fig. 77 to that of Plate 13. The axis A is called the polar axis ; and the instrument is so constructed and adjusted that this axis points directly toward the celestial pole (p. 32). Since the stars perform their apparent diurnal rotations (p. 33) around that pole, it fol- lows that they will seem to perform them around the polar axis of the equatorial. This sim- plifies the use of the in- strument; for a star once brought into the field of view of the tele- scope, we can keep it there by moving the instru- ment around the polar axis only. For the tele- scope must be kept mov- ing to prevent the diur- nal rotation of the stars from carrying the object under observation out of the field of view. The necessary rotation around a single axis can be accomplished automatically by means of clock-work, shown inside the vertical supporting pillar in Plate 13. The as- tronomer is thus left free to pursue his observations without any further attention to the telescope. If there were no inclined polar axis, but, in its place, a pair of axes, one verti- 280 FIG. 77. The Equatorial. PLATE 14. The Crossley Reflector. ASTRONOMIC INSTRUMENTS cal and the other horizontal, this simple clock-work plan would be impossible. The equatorial carries two circles, c and c', divided into degrees, and attached to the axes A and A'. With the circle c' it is possible to measure the declination (p. 34) of an object in the field of view ; and with the circle c its hour-angle (cf . p. 68) can be measured. And if we wish to find a known object which is invisible to the unaided eye, we have merely to turn the telescope around the two axes, until the two circles indicate the object's known declination and hour- angle, when it will at once appear in the field of view. The foregoing description of the telescope applies to the " refractor" with its object-glass and eye-piece. But there is another form of instrument, the " reflector," in which the object-glass lens is replaced by a curved mirror. This forms a focal image similar to that given by a lens ; and the image is again examined with an eye-piece lens as before. Plate 14 shows the equatorially mounted Crossley reflector of the Lick observatory. The polar axis is shaped like a double cone ; and the reflector is at the bottom of the big tube. The focal image is formed near the top of the tube, and is there examined with an eye-piece shown in the plate. It was with this instrument that Keeler made his famous photographs of spiral nebulae, one of which is shown in Plate 2 (p. 4). Others will be described in a later chapter. Telescopes intended for astronomic photography are always mounted equatorially, and their clock-work mech- anism is made especially precise. For when it is desired to make long photographic exposures, the telescope, which takes the place of a camera, must of course be kept in motion, so as to neutralize the diurnal rotation of the stars. If this is not done quite exactly, we obtain only a worthless "moved / 281 ASTRONOMY negative. " To be certain of this essential element in astro- photography, such telescopes are usually made with two tubes, like a pair of opera-glasses. The one tube is a photo- graphic telescope, the other a visual one, provided with cross-threads. With this arrangement, the astronomer can watch an object with the visual telescope, while it is being photographed in the other. If the clock does not move quite perfectly, the error will at once show itself ; for the object will move away from the cross-threads in the visual telescope. The slightest tendency to such motion must be prevented ; and for this purpose the clocks of such instruments are provided with certain adjust- ing screws, or other devices, with which the astronomer can correct any possible errors of the clock while it is actually running, and while the photograph is being exposed. The last important astronomic instrument to be described here is called the Spectro- scope, of which Fig. 78 shows the essential parts. The reader will recall that if we look at a source of light through a glass prism, it will be spread out into a band of colors, violet, indigo, blue, green, yellow, orange, red. The actual spectroscope consists of a prism P (or a succession of several prisms) and two brass tubes. One, the "collimator" C, admits light to the prism. It has a narrow slit s, at one end, so that the light may enter as a thin line, parallel to the edge of the prism ; at the other end there is a lens, 0, to render the rays of light parallel. The 282 FIG. 78. The Spectroscope. ASTRONOMIC INSTRUMENTS other brass tube is merely a little view telescope O'E, with which to examine the " spectrum" and to magnify it. Fre- quently it is better to substitute for the prism a glass plate on which a great number of very close parallel lines have been ruled with a dividing engine. Such a glass spreads light into a spectrum similar to the one obtained with a prism. Plate 15 shows a spectroscope attached to the eye- end of the big Lick telescope. Three prisms are used ; . and the eye-piece of the view telescope is replaced by a plate- holder, so that stellar spectra may be photographed. If we send colored light into the spectroscope, the result is as follows : If the light is red, it goes to the red part of the spectrum, where it belongs, and we there see a bright red line, which is merely an image of the spectroscope slit. But if we send in white light, which is really .a mixture of all kinds of colored light, the prism analyzes it : the yellow part goes to the yellow part of the spectrum, etc. We then see the colored " continuous spectrum," made up of an enormous number of slit-images side by side (see Plate 16, 1). But if the light comes from an incandescent gas, instead of a solid or liquid, we see no continuous spectrum, but a series of bright-colored lines, or images of the slit, variously located throughout the spectrum; and the combination of lines is different for every different gas. We can actually deter- mine the name of the incandescent gas from the positions of the lines. This is well shown in Plate 16, 3, 4, and 5, for the incandescent vapors of sodium, hydrogen, and potassium. But the most singular thing of all relates to the " absorp- tion" of light by gases; and this is one essential thing in astronomic spectroscopy. If a beam of white light is passed through a layer of gas or vapor before entering the spectro- scope, this vapor will sift out, and absorb, precisely those 283 ASTRONOMY light-rays or colors which the gas or vapor would itself emit if it were incandescent, and which would then appear as a bright-line spectrum in a spectroscope. From the foregoing principles we shall find it possible to study the chemical constitution, or the nature of the vapor existing in the sun and stars. And there is also another prin- ciple, due to Doppler, by means of which we can obtain in- formation of still another kind. It is a fact that the spectral lines may at times be shifted out of their proper positions as ordinarily seen in the spectrum. We are taught in the science of physics that red light has comparatively long light-waves ; violet light, the shortest waves. Now if a source of light, such as a star, happens to be increasing its distance from us at the time of observation, we shall receive fewer light-waves per second than would be the case if the distance were stationary. But if we receive fewer light- waves per second, they will seem to be longer waves, and therefore more like the waves from red light. The effect is to shift the observable lines toward the red end of the spec- trum. This shift can be measured with a micrometer, and from such measurements it is actually possible to deter- mine the velocity with which a star is approaching us or receding from us in space (cf. p. 245). The planet Venus has often been observed to test this principle ; and the photographed spectrum of Venus in Plate 16, 6, shows plainly the shift of the spectral lines. The middle spectrum belongs to Venus ; the two outer ones are artificial spectra produced in the observatory for compari- son. Microscopic measurement of this " spectrogram," as it is called, enables us to compute that Venus was increasing its distance from the earth at the rate of eight miles per second when the spectrogram was made. From our knowl- 284 PLATE 16. Various Spectra. ASTRONOMIC INSTRUMENTS edge of the orbital motions of Venus and the earth, it is possible to check this result by an entirely independent method of calculation ; and thus no doubt remains as to the correct- ness of the Doppler principle. In addition to the spectroscope just described, astronomers also employ a ' ' slitless ' ' instrument. This is made by mount- ing one or more prisms outside the object-glass of a telescope, prisms large enough to cover the entire object-glass. With such an instrument it is possible to photograph on a single plate the spectra of all stars in the telescopic field of view, while the slit spectroscope will give only one star-spectrum at a time. It is therefore clear that the slitless instrument is best for statistical researches intended, for instance, to classify all spectra ; but the slit instrument is more accurate, and is also the only form permitting the measurement of line shift in accordance with Doppler's principle. 285 CHAPTER XVI SUNSHINE IN Chapter XIV we have considered at length the ancient problem of determining the distance of the sun from our earth : let us next attempt to describe the sun itself. Here we meet something distinctly modern in the venerable science; for almost all knowledge we have of the sun is knowledge obtained during the last hundred years. The ancients knew little or nothing about it. Our subject falls readily into two parts : first, information obtained by the use of the spectroscope (p*. 282) ; and secondly, investigations other than spectroscopic. Let us begin with the chemistry of the sun. We have already had an explanation (p. 283) of the manner in which gases absorb certain light-rays while passing through them, each gas ab- sorbing its own particular combination of such rays. This principle, applied to the sun, gives the following result. The body of the sun sends out white light which the spec- troscope would naturally split up into a continuous spec- trum (p. 283). But before this white light can pass through the outer gaseous layers of the sun, the absorption phenom- ena take place ; and so the regular solar spectrum appears in the spectroscope as continuous, but crossed by a vast number of black lines. These lines are simply the dark places where absorption has occurred. Thus, for instance, if there is iron vapor in the outer solar atmosphere, we shall see dark lines in the solar spectrum at exactly the points 286 SUNSHINE where we should see bright lines (p. 283) if we vaporized some iron in the laboratory, and examined the spectrum of its light. All these dark lines are merely bits of nothingness occupying the places where there should be images of the spectroscope slit, if absorption were absent. Such are the famous Fraunhofer dark lines, named from their discoverer. Plate 16, 2, shows a number of the principal ones, together with the letters by which spectroscopists designate them. The double ZMine, for instance, arises from absorption due to sodium vapor, as may be seen from a comparison with the sodium vapor spectrum, 3. So much being premised, we can now explain the method of using the spectroscope to ascertain the sun's chemical com- position. It is merely necessary to attach a spectroscope to an ordinary telescope in such a way that the slit will be in the telescopic focal plane (p. 272). Arrangements must then be made, by means of a tiny reflecting prism attached to the slit, to throw into the view telescope the spectrum of any desired substance vaporized and heated to incandes- cence in the observatory, near the telescope. We then see the solar spectrum and the artificial spectrum side by side. Now this artificial spectrum is a bright-line spectrum (p. 283) : and if opposite each of its bright lines we find a dark Fraunhofer line in the solar spectrum, we have conclusive proof that the substance vaporized in the observatory is actually present as a gas in the outer atmosphere of the sun. Many terrestrial chemical elements have been thus found in the sun : the general conclusion is that earth and sun have a similar constitution, which was to be expected, if we accept any hypothesis postulating a common origin for the sun and earth. (Cf. nebular hypothesis, p. 235.) Next we must consider a very interesting phenomenon 287 ASTRONOMY called the Reversing Layer. We have seen that the Fraun- hofer dark lines are due to absorption in the outer gases of the sun. But these gases are themselves so hot as to be incan- descent. It is only because the inner sun is so much hotter and so much more luminous that we ordinarily see only the light from the inner sun and not from the outer gases. The latter are dark by comparison only. There is just one occasion when it is possible to observe the light from the outer gases separately and directly. This occurs during a total solar eclipse, when the moon hap- pens to pass accurately in line between the earth and the sun. On such an occasion, when the lunar globe, advancing in its orbit around the earth, has almost covered the sun, just before it is covered absolutely, there must be a moment when a tiny sickle of the outermost layer of the sun is alone visible. At that exciting moment, and at that moment only, can we look upon the outermost incandescent gases. But their light suffers no further absorption ; and so, like all incandescent gases, should give a spectrum consisting of bright lines only. And this is precisely what occurs. If we observe the advancing eclipse, just for an instant before totality, the continuous solar spectrum with its myriad black Fraunhofer lines is suddenly replaced by a bright- line spectrum. Each bright line corresponds accurately to one of the vanished dark lines, since the dark lines were caused by absorption due to the very gases that are now furnishing the bright-line spectrum. The critical instant over, the sun is covered totally, and the bright lines in turn disappear, too. This phenomenon is appropriately termed the Flash Spectrum. The surface we see when we turn a telescope upon the sun is called the Photosphere. It is not uniformly brilliant, 288 PLATE 17. The Sun. Photo by Fox. SUNSHINE but shows certain brighter " nodules," and also especially bright points called "faculse," 1 which appear mostly at the edges of the sun and near the sunspots. They are perhaps suspended in the solar atmosphere at a high level, and owe their extra brightness to our observing them through a somewhat thinner layer of solar atmosphere. Probably everything we see when we examine the sun is " atmosphere." Young aptly compares the state of affairs to a gas-burner, in which the heated particles of the mantle are far more lumi- nous than the flame of gas which heats them. Let us next enumerate some of the principal facts known about the sunspots (p. 17). We are on sure ground when we speak of their size ; for we can, as usual, measure their angular diameters, and then compute their linear dimensions from our knowledge of the sun's distance. At times they are 50,000 miles in diameter; and exceptionally large ones can even be seen without a telescope. But our knowledge is less certain when we attempt to explain their cause. They are to be regarded probably as solar atmospheric disturbances or storms. In that case we should expect them to shift their positions on the sun's surface, much as storm-centers move across our earth. And we find by observation that all spots have a common drift; and those near the solar equator also drift toward it, while those far from the equator drift toward the solar poles. This might be analogous to our phenomena of the trade-winds, especially as spots never occur near the solar poles, or exactly at the equator. So the real cause of the spots must be regarded as un- known. They may be eruptions from the interior ; they may be gases rushing downward into hollows. But we cannot help thinking they are vast storms of some kind; 1 Plate 17. U 289 ASTRONOMY storms of which the materials are incandescent gases, moving with great velocities, and at enormously high temperatures. The duration of individual spots is not great, never more than 18 months ; and the central, apparently blackest part of the spots, called the " umbra," is not really dark, but appears so only though contrast with the much more lumi- nous surrounding solar material. They have also a periodic- ity, discovered in 1843 by Schwabe ; and this is perhaps the most interesting of the many unexplained observations of the sun. Schwabe found, by constant watching of the solar surface, that every eleven years there is a period of extra great spot frequency. This discovery owes its importance to the known fact that there exists also an eleven-year period in the frequency of terrestrial magnetic storms : and especially great sunspots are always accompanied with very strong magnetic disturbances and auroral displays on earth. This establishes the existence of some intimate magnetic relation between earth and sun; but it has not yet been possible to reach a satisfactory explanation of it. Nor have astronomers been able to make certain that any other terrestrial meteorological phenomena exhibit a real connection with the spots, though many efforts have been made to do so on account of the assistance such investigations might give in the matter of weather prediction. The accompanying Plate 18 is a photograph of a partic- ularly large sunspot which occurred July 17, 1905. It had an unusual and very brilliant " bridge" across the umbra. The next important question requiring consideration re- lates to the size of the sun. We have already determined its distance to be about 93 million miles. Observations very similar in principle to methods already explained for the moon and planets enable us to measure the sun's apparent 290 SUNSHINE angular diameter (cf . p. 203) ; and this we find to be 32' 4", on the average. In Fig. 79 we then have, as usual, a long, narrow triangle, of which the base is the sun's linear diameter AB. This we can calculate readily, because we know the sun's angular diameter, or the angle 32' 4" at the vertex FIG. 79. Sun's Diameter. of the triangle, situated on the earth. The result comes out nearly nine hundred thousand miles for the sun's linear diameter. Comparing this with the known terrestrial diameter (p. 97), we find that the sun's diameter is approximately 110 times that of the earth. To ascertain the sun's mass is a little more difficult than to find its diameter ; but it can be estimated by simple mathe- matical methods, 1 which show that it is about 330,000 times the earth's mass. We can also calculate the force of gravity that must exist on the solar surface as compared with the gravitational at- traction existing on our earth. For the gravity force on the surface of a sphere is, by Newton's law, proportional to the mass of the sphere, divided by the square of its radius. If we then consider all solar quantities expressed in terms of the corresponding terrestrial quantities as units, we have : B . -- , solar mass 330000 Solarforceof gravity = (golar radiug) 2 = ~ - 28, approximately. This means that an ordinary one-pound weight would 1 Note 35, Appendix. 291 ASTRONOMY weigh 28 pounds, if transported to the sun's surface, and there weighed with an ordinary terrestrial spring-balance. The solar volume or bulk compares with that of the earth in the proportion of the cubes of their radii ; that is, as 1 to (HO) 3 . This makes the solar volume 1,300,000 times the earth's. And since density or specific gravity is proportional to mass divided by volume, it follows that the solar density, as compared with the earth's, is : solar mass 330000 solar volume ~ 1300000 " ' 25 > or only about } the earth's density. The latter, com- pared with water, is about 5.5 ; so the solar density is only about 1^ times that of water. This means that a cubic foot of average solar material, transported to the earth's surface, would there weigh only about 1J times as much as a cubic foot of water. The quantity of light and heat received by us from the sun is certainly enormous ; and yet it cannot be more than a small fraction of the total quantity actually radiated into space. A most interesting question arises in connection with this matter : How does the sun maintain through the ages so gigantic an output of heat energy? What is the source of the sun's heat ? Helmholtz has proposed a plausible possible cause, the shrinkage of the sun's vast bulk under the influence of its own gravitational attraction. He computed that an annual shrinkage of only 300 feet in the solar diameter would be transformable into enough heat energy to keep radiation active as it now is. And it would require 8000 years for this diminution of size to reduce the sun's observable angular diameter by a single second of arc ; nor could any smaller diminution be discov- ered by observation with actual astronomic instruments. 292 SUNSHINE The fact that we have not observed a reduction of the sun's size is therefore no argument against the Helmholtz theory. Up to this point we have supposed the sun to consist of a highly heated interior of more or less unknown constitu- tion, surrounded by an atmosphere of incandescent gases which produce the Fraunhofer lines by absorption, and the bright-line flash spectrum during an eclipse. But we know much more than this. Beyond the photosphere and the reversing layer of gases is the Chromosphere, or color sphere, composed principally of great flaming masses of red hydrogen vapor. Sometimes great red jets burst out- ward to immense elevations from the solar surface. These are the Prominences (Plate 19) ; and while these various solar layers have different names, it must not be supposed that they are distinct. They intermingle, doubtless, at their boundaries, and melt into each other without sudden interruptions. The hydrogen prominences were first seen during a total solar eclipse, when the photosphere was completely covered by the moon. But just after the eclipse of 1868, Janssen and Lockyer for the first time succeeded in observing them with- out an eclipse. We cannot see them by merely looking at the sun with a telescope, and covering the central part of the solar disk at the telescopic focus, because the terrestrial atmosphere is strongly illuminated by the sun itself, arid so the prominences become invisible by contrast. But if we bring the slit of a spectroscope tangent to the sun's disk at the focus of a telescope, and open the slit wide, the prominences become visible. For we then see two spectra superposed, one upon the other. The first is an ordinary continuous solar spectrum derived from the diffused sunlight in the terrestrial atmos- 293 ASTRONOMY phere, the second a bright-line spectrum from the incandes- cent hydrogen of the prominences. Now, if we employ in the spectroscope a number of prisms, instead of a single one, both these spectra will be spread out to a great length. The continuous spectrum will be thereby rendered dimmer, but the bright-line spectrum will have its lines separated further, without rendering them less brilliant. If we con- tinue thus increasing the " dispersion" of the spectroscopic prisms, we shall finally diminish the luminosity of the at- mospheric continuous spectrum until it disappears practi- cally, and we see only the bright-line spectrum of the promi- nence. Now, as we know, these bright lines are ordinarily merely images of the slit. But if the slit has been opened wide enough to be wider than the angular diameter of the promi- nences, the bright lines become images of the prominences, instead of images of the slit. We have therefore merely to point the view telescope at a place in the spectrum where there is ordinarily a bright hydrogen line, and we shall see the prominence, if there happens to be one on the sun's edge at the point where we have placed the widely open slit tangent to the sun's image at the telescopic focus. In 1891 Hale invented an instrument called a spectrohelio- graph, with which the prominences may be photographed without an eclipse. Plate 19 was made with such an instru- ment. It utilizes the light of calcium gas, which, like hydro- gen, is plentiful in the prominences ; and can be made to give a fine bright line in the middle of the usual dark Fraun- hofer calcium line due to the photosphere and reversing layers. The spectrum is allowed to fall on a screen having a second narrow slit corresponding accurately to the bright calcium line from the prominence. Through this slit 294 SUNSHINE the calcium light which originated in the prominence passes to a photographic plate, so that the plate receives prominence light only. Now, by mechanical means, the original slit of the spectroscope is moved across the solar image at the telescopic focus ; and the second slit in the screen is moved in unison. The result is to build up a picture of the sun on the photographic plate with light from the outer solar layer only, and thus to secure a photograph of the prominences. Still another extraordinary solar phenomenon has been discovered during total eclipses. This is the Corona, which bursts into view when the sun is completely concealed by the moon, and appears as a faint luminous ring, of more or less irregular shape, around the sun. We know that it belongs to the sun, because its spectrum is that of an in- candescent gas, not a continuous solar spectrum, such as it would be if we had here to do merely with solar light reflected in some way from the moon ; and also because the coronal form always appears the same at the same moment, even when seen from observatories widely separated on the earth. Beyond this little is really known for certain as to the corona. Plate 20 is a photograph of the corona during a total eclipse, the sun's disk being entirely covered by the inter- posed moon. The form of the streamers indicates that the phenomenon may be electric or magnetic. A large promi- nence appears near the lower part of the plate, jutting out from the obscured solar disk. The height of this prominence is estimated by comparing it with the photographed diameter of the sun : this would make the prominence about 40,000 miles high. The question of solar axial rotation can be examined best by studying the apparent motions of the spots. (Cf. the 295 ASTRONOMY case of the planets, p. 202.) It is found that they always seem to cross the solar disk from east to west ; and when one of them makes a complete rotation, disappearing at the western, and later re-appearing at the eastern edge of the sun, the whole revolution takes about 27J days. But this is not the true period of solar axial rotation : the above observed period, and the true one, are related by an equation analogous to the equation connecting the synodic and sidereal periods of the planets (p. 209), since we must correct the observed period for the effect of the earth's orbital motion around the sun during the time occupied by the latter in turning on its axis. 1 This correction makes the true axial rotation period of the sun about 25| days. These sunspot motions not only tell us the period of the sun's axial rotation; they also enable us to ascertain the direction in space of the rotation axis (cf. p. 203). The apparent paths of the spots are generally curved ; but on June 3 and December 5 they appear quite straight. Referring to Fig. 80, we see that this straightness determines on these dates the axial FIG. so. Sun's or polar points A and B on the sun's edge, and also the angle by which the solar rotation axis is inclined to the plane of the ecliptic. The angle is about 83. 1 The planetary equation here takes the form : 11 1 true period of rotation length of year observed period of rotation 296 CHAPTER XVII ECLIPSES ECLIPSES are occasional phenomena; they are usually defined as temporary obscurations of the sun or moon, either wholly or in part. We have seen that these two bodies are visible from two very different causes : the sun is self-luminous, it is visib le because it sends ' ^ . . 1 1 i lourii \_J us its own light ; the y J Moon moon is merely ren- dered visible when il- luminated by the sun. /^"^N Lunar Eclipse Therefore a solar ( &un ) eclipse can occur only FlG 81 Eclipses . if the moon inter- poses between us and the sun, thereby preventing our see- ing it ; but a lunar eclipse happens when the earth passes between the moon and sun, so that solar light cannot reach the moon, and render it visible. Thus there is not neces- sarily any actual obstruction in the way of our seeing the eclipsed moon ; it is invisible merely because it is dark for the time being. Figure 81 makes all this plain. Hipparchus was the first to explain eclipses, and the method of making a fairly good approximate prediction of their occurrence. Figure 81 shows that if the orbits of the earth and moon were both situated in a single plane surface (here represented by the plane of the paper) there must result one eclipse of the sun and one of the moon during 297 ASTRONOMY each revolution of the moon in its orbit around the earth. This simple state of affairs is modified and com- plicated by the fact that the lunar orbital plane is actually inclined about 5 to the ecliptic plane, in which the earth's orbit is situated (p. 160). Figure 82 is supposed to represent S Ecliptic ,. * FIG. 82. Moon's Circle. a Portion of the SUr- face of the celestial sphere. NS is part of the ecliptic circle, in which the sun is always seen ; and NM is part of the great circle cut out on the sphere by the plane of the moon's orbit, in which great circle the moon is seen, for the same reason that the sun is always seen in the ecliptic (p. 160). N is the point of inter- section of these two great circles on the celestial sphere, the angle between them being 5 only. The point N is called the "node" of the lunar orbit; and there is another similar node on the opposite side of the sky, because any pair of great circles must necessarily intersect at two opposite points on the sphere. We have already seen that the moon moves around the earth, and therefore appears to travel around the sky among the stars, at the rate of about 13 per day, so that it overtakes and passes the sun once in each synodic period or lunar "month" (p. 161). When it thus passes the sun, the two bodies are said to be in conjunction (cf. p. 209). If this conjunction happens to occur exactly at the nodal point N, then sun, moon, and earth will lie in a single straight line, and a "central" eclipse will occur. Furthermore, it will be an eclipse of the sun; for if the sun and moon appear projected at a single point of the sky, they must both lie on the same side of the earth (Fig. 81, solar eclipse). 298 ECLIPSES But an eclipse can also happen when the sun and moon are in opposition, or 180 apart, as seen projected on the sky. If such an opposition takes place when the moon is exactly in the node N, and the sun in the other, or opposite, node, we once more have sun, earth, and moon in a single straight line, and a central eclipse takes place. Only, in this case, the earth is between the sun and moon (Fig. 81, lunar eclipse), and the central eclipse is a lunar eclipse. The problems connected with eclipse prediction would pre- sent but little of interest beyond the above, if the node N always remained at the same point of the ecliptic. But this node is constantly moving along the ecliptic, a phenomenon somewhat analogous to precession of the equinoxes (p. 126) in the case of the earth. Only, the lunar node, unlike the equinoxes of the terrestrial orbit, moves quite rapidly, making a circuit of the entire ecliptic once in about 19 years. It is this phenomenon that complicates the eclipse problem and makes it interesting. Up to this point we have supposed eclipses to occur at the exact nodes only ; and this would be the case if the sun and moon appeared to us on the sky as mere mathematical points, like the fixed stars. But both sun and moon have quite large disks, as we see them projected on the sky. Each disk has a diameter of about half a degree of arc. Consequently (Fig. 83), when FIG. 83. Contact a conjunction occurs, these two disks may just touch if their centers are half a degree apart, provided we suppose the terrestrial observer located at the earth's center. But an observer on the surface of the earth, as at the point in Fig. 84, will see the lunar and solar disks in contact when 299 ASTRONOMY the moon is at MI ; while to an observer at the earth's center c, there would be no contact until the moon had advanced to M 2 . It is clear from Fig. 83 that the angle ScM 2 is |. The angle ScMi (from the center of the sun to the center of the moon, as seen from the center of the earth) is 1 J, approximately, 1 Earth thus enlarging greatly the FIG. 84. Observer^the Surface of the poss ibility of an eclipse being actually visible from some point or other on the earth's surface. Now referring again to Fig. 82, we see that if the conjunc- tion occurs when the centers of the sun and moon are at the points S and M, just far enough from the node A^ to make these two points 1J apart, the two disks will just touch, and we shall barely escape the occurrence of an eclipse, visible from some point of the earth's surface. If M and S happen to be a little nearer the node, the two disks will overlap, and there must be at least a partial solar eclipse. Knowing the angle at N to be 5, there is no difficulty in calculating how great must be the angular distances NM and NS, to make the distance SM just 1|. This distance NM is thus found to be about 17 ; so that when conjunction occurs within about 17 of the node, there is an eclipse. But this number 17 may vary all the way from 15 to 19 in different years, largely on account of small periodic changes of the angle N, between the two orbital planes. These changes are of course due to orbital perturbations (p. 206). The number 17 is called the "solar eclipse limit." As we have seen, these eclipses at conjunction are solar eclipses; but corresponding eclipse limits exist also in the 1 Note 36, Appendix. 300 ECLIPSES case of oppositions of the sun and moon, when lunar eclipses occur. Referring again to Fig. 81, we see that a solar eclipse is possible, only if the disks of the sun and moon, as projected on the sky, actually overlap. But Fig. 85 makes plain that a lunar eclipse will happen if the moon enters or touches the shadow cast into space by the earth. But the apparent angular diameter of this shadow, as seen from the earth, FIG. 85. Lunar Eclipse. and at the distance of the moon from the earth, is larger than the moon's angular diameter. The result is that a lunar eclipse takes place if the center of the moon and the center of the shadow are separated by an angular distance of less than about 1, as seen from the earth. We can, of course, again calculate how far the sun must be from the node, at the time of opposition, to make this angular distance less than 1. We thus find the lunar eclipse limit of about 11, with a variation between 10 and 12, in round numbers. Since conjunction always happens at new-moon, and op- position at full-moon, it follows from the foregoing simple considerations that there will be a solar eclipse at the time of new-moon, if the sun is within about 17 of the node; and there will be a lunar eclipse at the time of full-moon if it is within about 11 of the node. Two other simple conclu- sions follow at once : (1) Since the sun appears to move about 1 daily in the ecliptic, there will be a solar eclipse if the date of new-moon falls within 17 days of the date when the sun appears in the node. And there will be a lunar eclipse if the date of full-moon falls within 11 days of the same date, all in round numbers. (2) The solar eclipse limit being 301 ASTRONOMY the larger, solar eclipses must be more frequent, on the whole, than lunar eclipses. Hipparchus was the first to explain correctly the seeming paradox that lunar eclipses are seen much more often than solar eclipses, although the latter occur more frequently. The reason is perfectly simple. When the earth interposes between the sun and moon, and the moon thus enters the earth's shadow, it becomes dark at once, because it gives no light of its own. Consequently, any one on the earth who should be able to see the moon will fail to see it on account of the eclipse. But at any given instant, the moon should be visible from half the earth's surface ; therefore, if there is a lunar eclipse, at least half the earth's inhabitants will see it. But in the case of a solar eclipse, the sun is not made dark. The sun's light is actually cut off by the interposed moon ; and it is never at any one time thus cut off from observers living in more than a small part of the earth's surface. In Fig. 86, an observer on the earth at A will see the moon projected in his zenith, just as it would be seen by an observer at the earth's center C. But FIG. 86. Solar Eclipse. an observer at B will see the moon in the direction BM, instead of AM. This will pro- ject it in the sky for the two observers at points whose angular distance apart is equal to the angle BMC. Now this angle may be as great as 1, approximately ; so that the disks of the moon and sun might easily overlap for an observer at A, but not for an observer at B (cf. Fig. 84, p. 300). In other words, solar eclipses are by no means 302 ECLIPSES visible throughout an entire hemisphere of the earth, like lunar eclipses. In fact, the distances of the sun and moon from the earth are such that any total solar eclipse can be seen from a very narrow strip of the earth's surface only : not more than 70 miles wide at the most, and ex- tending through a distance of much less than a hemisphere. Plate 20 (p. 295) is a photograph of a total solar eclipse. Having thus outlined the general nature of eclipses and their causes, we shall next describe certain special phenomena which are of sufficient interest to be mentioned. The earth's shadow, into which the moon enters when eclipsed, is not uniformly dark throughout. It is, in fact, made up of two parts, the ' ' umbra/ ' or shadow proper, and the ' ' penumbra, ' ' or partial shadow. Figure 87 shows how the penumbra is formed. There are two penumbral regions, as it were, and one umbral region. The latter is a central cone which re- FIG. 87. The Penumbra. ceives light from no part of the sun. The two penumbral regions receive light from part of the sun, while the rest of space behind the earth receives light from the entire solar surface. It is evident that the darkness of the penumbra will increase gradually from its outer edges to the boundary lines, where it gives the black umbra. The moon, when about to be eclipsed, will therefore enter the penumbra 303 ASTRONOMY first, and be partially darkened ; and the darkening will increase gradually until it becomes practically complete, as the moon enters the umbra. The same gradual phenomena will be repeated in the inverse order towards the end of the eclipse. In the case of solar eclipses (Fig. 88), if we consider the interposed moon as casting a shadow on the earth, the eclipse will be total where the true shadow cone cuts the earth, and partial where the penumbral regions meet the terrestrial surface. Owing to the ellipticity of the moon's orbit, and consequent variation in the distance between the earth and moon, it sometimes happens that the true shadow cone of Fig. 88 does not quite reach the FIG. 88. Solar Eclipse. earth * In SUch a Case, Ul that part of the terrestrial surface for which the eclipse is central, the sun will appear as a luminous "annulus," or ring, with the central part dark. Such eclipses are called Annular eclipses, and occur, of course, for the sun only. A total solar eclipse can never last longer than eight minutes at any one place on the earth, but totality in the case of the moon may last a couple of hours. There exists a peculiar periodicity in the recurrence of eclipses called the Saros. It was discovered by the Chal- dseans, who found, by actual observation, and comparison with ancient records in their possession, that after the lapse of a period of 6585 days after an eclipse, the phenomenon will be repeated ; and eclipse occurrences can thus be pre- dicted easily. The explanation is as follows : The reader will remember that in the case of the earth's 304 ECLIPSES motion around the sun we found the tropical year to be shorter than the sidereal year by about twenty minutes, on account of precession of the equinoxes (p. 126). In a similar way, on account of the motion of the moon's nodes, the time required by the moon to travel in its orbit from one node back to the same node again is shorter than the lunar sidereal period (p. 161). This nodal period is called the Draconitic period ; it is three hours shorter than the sidereal period, and two days seven hours shorter than the synodic period. So we have : 1 Sidereal period = 27 d 8 h Draconitic period = 27 5 approximately Synodic period =29 12 An inspection of these figures shows that 223 synodic periods equal 242 draconitic periods, very nearly ; and either includes 6585 days. But successive full-moons, or successive new-moons, follow each other at intervals of one synodic month, because the synodic month is the in- terval between two successive overtakings of the sun by the moon, in their respective apparent motions around the celestial sphere. And successive passages through either node succeed each other at intervals of one draconitic period. Therefore, any period of days like the Saros, containing exactly a definite number of synodic periods and also a definite number of draconitic periods, after the lapse of such a period of days, both the lunar phase and the node passage must both repeat. Therefore, if there was an eclipse at the first new or full moon of the Saros period, there must also be an eclipse at the first new or full moon of the succeeding Saros period. 1 Note 37, Appendix. x 305 ASTRONOMY In the light of the above explanation of eclipses, it may be possible to make somewhat clearer the allied phenome- non called a transit of Venus (p. 268). Such transits also occur in the case of Mercury, but they are then of lesser in- terest. The planetary nodes do not move around the ecliptic rapidly, like the lunar nodes; they remain almost stationary at a definite point. The sun, in its apparent motion around the ecliptic, reaches the nodal points of Venus on June 5 and December 7 ; so that transits of that planet happen (if at all) within a day or two of these dates. To ascertain the interval between successive transits, we note that : 5 synodic periods of Venus = 8 years, nearly ; 152 synodic periods of Venus = 243 years, very nearly. Since conjunctions of Venus occur at intervals of one sy- nodic period, any given transit may be followed by another at the same node eight years later. But there could not be a third transit sixteen years later; the eight-year period is not exact enough for that. We should then have to wait for the 243-year period to become effective. But at the other node, a transit, or an eight-year pair of transits, may happen after half the 243-year cycle has elapsed. 306 CHAPTER XVIII COMETS THE comets, stellce cometce, or stars with hair, must next receive our attention. These bodies usually move in very elongated orbits, with the sun at one focus. They often come as mere occasional visitors to the solar system, are seen during a short period only, while they are traversing that part of their orbit which is near the sun and therefore also near the terrestrial orbit. Occasionally comets have been as luminous as the brightest planet (Venus) ; have sometimes been seen in daylight; and very often have a long appendage streaming out from the head, the comet's "tail." It is easy to enumerate the chief known facts concerning the comet's physical appearance. The head usually con- sists of a "coma," or hazy nebula, containing a "nucleus," or central condensation. Attached to it is the tail, or, as it was sometimes formerly called, the "beard." The coma is the part that generally becomes visible first, as the body begins to approach the solar system. The nucleus forms later, or at least becomes visible later. The tail, strange to say, is always directed away from the sun; so that when the comet is receding from the sun, after passing the perihelion point (p. 120) of its orbit, it pushes its tail out ahead of it. But many comets were not discovered until after perihelion. It was then that the astronomers of old used the name "beard" for the tail. Comets are big; their volume is sometimes incredibly 307 ASTRONOMY large. The heads run up to a million miles in diameter, and the tails may be ten million miles long, or even much longer. But they have little mass, as is evidenced by the total absence of gravitational perturbations (p. 206) in the motions of the earth and Venus, even when a big comet passes very near these planets. Owing to this vast ness of bulk and extremely small mass, comets have, of course, a very low density, especially in their tails. Moreover, stars have at times been seen through the comets, even through their heads. In view of this extreme lack of mass, it may seem strange that modern science is compelled to admit the possibility, at least, of danger resulting from collision between the earth and a comet. If the cometary particles are infinitesi- mally small, no injury would follow ; but if the particles are rocks weighing tons, they might cause considerable local damage at the point of collision on the earth. But, on the whole, the most plausible theory is to suppose these bodies to be composed of tiny particles traveling together in swarms, and separated by distances many times greater than the diameter of the particles. And the particles may be surrounded by atmospheres of incan- descent gases ; for we know that comets are partly self- luminous, although they send us also a certain amount of reflected sunlight, like the planets. This is made certain by observations of their spectra, which generally show the existence of hydrocarbon gas in a luminous state, as well as a dim continuous spectrum containing Fraunhofer (p. 287) lines, the sure indication of solar light. We have a good theory to account for the repulsive forces that must come from the sun, so as to make the cometary tail always point away from that body. The researches of 308 COMETS Clerk-Maxwell (the same who proved mathematically the satellite construction of Saturn's ring, p. 245) have brought out the fact that light-rays exert a slight physical pressure upon any object they reach. This pressure is theoretically proportional to the area illuminated. Now if we imagine a small spherical cometary particle, and suppose its radius to diminish, the light pressure will diminish in proportion to the square of the radius, because the area of the circular cross-section illuminated diminishes in that proportion. But the volume and mass of the particle will diminish as the cube of the radius. Therefore the mass diminishes more rapidly than the area. But the solar gravi- tational attraction is proportional to the mass ; consequently, the solar attraction diminishes more rapidly than the light pressure, as the particle grows smaller. We have therefore merely to imagine the particle small enough, and the light pressure will balance the attraction. Still smaller particles will actually be repelled. If we now suppose the tail composed of particles smaller than those in the head, everything is explained : the head attracted, the tail repelled, by the sun. Probably the comet has no tail until it approaches the sun, when the light pressure sifts out the small particles ; repells them in an increasing degree as the comet comes near perihelion ; and thus makes the tail "grow." Perhaps the tail particles never rejoin the head, but are left scattered throughout the length of the cometary orbit. If we finally suppose both gravitational attraction and light repulsion to be exerted on the several particles, both by the sun and the comet's head, we have a combination of forces sufficiently flexible to account for the most complicated observed forms of cometary tails. (See Plate 21, p. 307.) 309 ASTRONOMY The number of comets is very great ; while only some four hundred were observed before 1610, when Galileo first used the telescope, at least four hundred more have been found in the succeeding three centuries. Of course this greater abundance of discovery in modern times has been brought about by the possibility of recording comets too faint to be seen by the unaided eye. Only thirteen naked- eye comets belong to the nineteenth century. The method used in discovering comets is interesting. This work is carried on by specialists ; except as a result of unusual chance, one can expect to find new comets only after a number of years' severe study of the heavens. The usual process is to "sweep" the sky with a telescope of moderate size and low magnifying power. Any hazy object may be a comet ; for when they are distant and dim, comets always look more or less like small nebulae. The only sure test to distinguish them is to watch for an hour or two, and ascertain whether there is motion relative to the fixed stars. If there is motion, a comet has been found. But this motion test occupies much time ; and at this point the experience of years is of value. For the comet hunter learns at last to know all the tiny configurations of stars seen in the telescope ; he knows the telescopic constellations at sight, as well as most astronomers know Orion and the Great Bear. It is said that Olbers, for instance, could tell the approximate right-ascension and declination of the point on the sky toward which his telescope was directed, by simply looking through the eye-piece, and noting the diagram of stars appearing in the field of view. And the comet hunter must know all the little nebulse too, as well as their positions on the sky relative to the sur- rounding small stars. When he sweeps a faint nebulous ob- 310 PLATE 22. Halley's Comet. Photo by Curtis. COMETS ject into view, he does not ordinarily need to delay his work by testing for motion; he recognizes the object at once, if it is one of the known small nebulae. Comets are usually named after their discoverer, though a few bear the name of some person who has explained their motions or peculiarities by a special investigation. Thus the famous comet of Halley, a photograph of which is shown in the accompanying Plate 22, was the first comet for which future returns to the solar system were predicted by means of orbital calculations. These were made by Halley (cf. p. 269) ; and his name was accordingly assigned to this comet. Small telescopic comets are at first designated by the year of discovery and a letter ; as, 1899 a, etc. Later, when orbits have been computed, they take the number of the year in which their closest approach to the sun, or perihelion, occurs; and a number in addition to show the order of cometary perihelion passages during that year. Thus Donati's great comet of 1858 was 1858 /, or the sixth comet discovered in 1858; and later it became 1858 VI, or the one whose perihelion passage was the sixth perihelion passage in 1858. The period of visibility does not usually last more than a. few months, although its average duration has been length- ened considerably in recent years, because modern giant tele- scopes can observe the comets long after their orbital motion has carried them quite beyond the range of ordinary glasses. Concerning these orbital motions, the mists .of antiquity certainly enshroud some very singular notions. There was a time when comets were regarded as material thrown out from the earth, possibly through volcanoes. It was not until 1577 that Tycho Brahe for the first time proved from actual measurements that the great comet of that year was 311 ASTRONOMY surely farther from the earth than is the moon. Kepler thought comets are alive. Hooke, in 1675, a century after Tycho Brahe, suggested that comet orbits might be para- bolic : a very few years later, Newton showed that they are " conic sections," and Halley calculated actual orbits for all the comets observed up to that time. There are three kinds of conic sections, the ellipse, para- bola, and hyperbola ; and it is easy to draw a figure illustrat- ing these three curves as comet orbits. Parabolic orbits are four times as frequent as elliptic orbits : hyperbolas are FIG. 89. Forms of Comet Orbits. very rare, and it is not absolutely certain that any such orbits really exist. Figure 89 exhibits the three orbital forms, together with the comparatively tiny circular terrestrial orbit, and the sun-dot at its center. We see especially how nearly alike all three kinds of comet orbits are, while 312 COMETS the comet is near the earth's path; and, after all, it is only then that we can see a comet, and observe its position, as projected on the sky. To construct a comet orbit from observations is often as difficult as trying to draw a circle of large radius through three points very close together. Thus, in Fig. 90, it is easy to draw the circle A accurately, through the three points PI, P 2 , P 3 . But if we were asked to draw a circle through P/, P 2 ', P 8 ', we might not be able to decide between the circle B and the circle C. Now the ellipse is a closed curve ; the others are open curves : therefore only elliptic orbits will produce so-called "periodic" comets, with future returns to the solar system. The parabolic comets visit us once, and never return. But since parabolic orbits closely resemble extremely elongated ellipses, it is not always possible to make certain whether any given object is periodic or not. But it is, after all, really immaterial whether a given comet orbit is truly parabolic, or elliptic, with a period of several hundred thousand years. The exact details of any comet orbit are defined by means of elements exactly analogous to the elements of a planet's orbit (p. 200). So much being premised about these orbits, we can now consider one of the most interesting things known about comets, their " families." For there are in existence most curious kinships between various groups of comets. Coming back to Kepler's amusing notion that they are alive, we must expect to find close relationships among them, and 313 ASTRONOMY also some that are merely distant cousins, as it were. The most remarkable " blood-relations " are the great comets of 1668, 1843, 1882, and 1887. They must be brother-comets, for they all pursue practically the same orbit, though travel- ing in different parts of it. They approached the solar system from the direction of the bright star Sirius, and left again in nearly the same direction, in a parabolic orbit. On the other hand, there are the comet-families belonging to the great planets, especially Jupiter. Here all the comets of a family have the peculiarity that the points of their orbits farthest from the sun, the aphelion points, all lie near the orbit of Jupiter. In other words, they recede from the sun just far enough to reach Jupiter's orbit. If Jupiter happens to be in the neighborhood when they get out there, he must exert a powerful gravitational attraction upon them. It is supposed that on their first arrival, per- haps in parabolic orbits, this attraction pulled them around into ellipses, having their node and aphelion point near the place where this disturbing pull took place. This is the well- known capture theory of comets, due to Laplace. So we see that the comets did not originally belong to our solar system; they come to us from outer space, possibly from among the fixed stars, possibly from some nearer region. If they come from interstellar spaces, we should, on the whole, expect to find a preponderance of orbits having their aphelion points lying in the direction of that point on the sky toward which the solar system's own motion in space is tending. For the solar system, as a whole, is drifting through cosmic space, as will be explained in a later chapter. But we have only slight indications of such a clustering of aphelion points : our whole theory as to comet origins is till hazy, very hazy. 314 CHAPTER XIX METEORS AND AEROLITES THE consideration of comets leads us directly to the closely related sub j ect of meteors or ' ' shooting stars . ' ' These look like stars falling from the sky ; actually, they are small particles of matter traveling in space, and passing through the earth's atmosphere. They give a bright light, and usually leave a long visible trail behind them. Sometimes they do not appear merely as isolated bodies; but regular showers occur, with the bright intermittent trails almost covering the sky, or a portion of it, for a con- siderable time. When this happens, it has been found that all the meteor trails are directed away from some single point on the sky. This point is called the Radiant ; and it has the peculiarity that the trails are always short in its vicinity. Figure 91 exhib- its this state of affairs. The point R on the celestial sphere is the radiant. FIG. 91. Radiant of Meteor Shower. All the trails are directed away from it, as shown by the arrows ; and the longer trails originate at points farther from the radiant than do the short trails. Figure 92 shows that the whole appearance is due simply to perspective. The meteors move in parallel lines to meet the earth. Suppose the observer to be on the surface of the earth, at 0, and two meteors moving along parallel lines, 315 ASTRONOMY such as AiBi and A^B^. To the observer at the meteors will seem to move along the lines A\C\ and A 2 (7 2 . And all the meteors will seem to move along lines that will appear to radiate out from a single point R, where AiCi and A 2 C 2 intersect, if produced backwards from Ai and A 2 . And this point will appear in the sky near the meteors that seem to have short trails. Each meteor shower can be distinguished s^ce of the Earth ~ f rO m all others ; not by FIG. 92. Radiant of Meteor Shower. , . ,, . , a difference in the ap- pearance of its constituent meteors, but by the position on the sky of its radiant. Thus the shower called the Leonids, the greatest of all the showers, has its radiant in the con- stellation Leo (Fig. 20, p. 62). These meteors occur always about November 12, and have been found especially numerous at intervals of 33 years. The cause of this fixity in the dates and recurrences of individual showers is quite simple. Each shower travels in a definite orbit around the sun, just like a periodic comet (p. 313). This orbit somewhere intersects the orbit of the earth; or, at least, passes very near it. But the earth must reach that point of intersection on the same date each year. Therefore the shower must occur on that particular date, if it occurs at all. And it will occur if the meteors hap- pen to be at the intersection point of the two orbits on the date when the earth also reaches that point. In the case of the November Leonids this happens only once in 33 years ; but the Perseids, or August meteors, are ready for us every year. We conclude, of course, that the Perseids are spread 316 METEORS AND AEROLITES C'irc/e out all along their orbit, so that we meet some of them when- ever we strike the orbit. But the Leonids must be concen- trated in a certain region in their orbit : this region comes around to the point of intersection in the proper way only once in 33 years. A very interesting fact about the meteors is that we or- dinarily observe more of them per hour just before sunrise than we do just after sunset. The reason is shown in Fig. 93. EI is the position of the earth in its orbit at about six in the morning, just be- fore sunrise. The sun is seen projected on the ecliptic at /Si, which is therefore near the east point of the horizon. The earth's orbital mo- tion is for the moment directed towards the point P, 90 west of S, where the sun appears on the ecliptic. The earth's orbital motion takes place in the direction of the curved arrow, so that at six in the morning we are on the front of the earth in respect to its orbital motion; we advance to meet the meteors. But on the opposite side of the earth it is at the same instant 6 o'clock in the evening. There they see only such meteors as overtake the earth, while on the front side we see them all. After the lapse of twelve hours, the earth has made a half-turn on its rotation axis ; conditions are reversed ; and we then have our clocks at six in the evening. 317 FIG. 93. 90 West off he Sun Meteors at Sunrise and Sunset. ASTRONOMY We are then, in our turn, on the back of the earth with respect to the direction of its orbital motion. Meteors are never seen until they enter the atmosphere of our earth, but their heat and light are not due to atmos- pheric friction in the ordinary sense. Sometimes it is said, erroneously, that they are "set on fire" in the same way as a friction match is lighted by being rubbed on a rough surface. Their light is really caused by the compression of air in advance of the moving meteor. It is harder for the meteor to move against the compressed air ; this retards its motion, and the motion energy is transformed into heat energy (p. 2). Doubtless both the meteor and the air are heated* When a moving body is retarded by the resistance of an atmosphere, the heat engendered is proportional to the square of the velocity of motion. At the usual meteoric ve- locity, the temperature produced is probably equivalent to several thousand degrees Fahrenheit; and this, of course, will melt almost any substance. The rapid motion through the air then tears off particles of heated incandescent matter from the melted meteoric surface ; and these particles are left behind to form the tail or trail of the meteor. It is not known just why it sometimes remains visible for many minutes. Plate 23 is a photographic reproduction of a meteor trail, showing two remarkable variations of brilliancy. The negative also contains a couple of interesting nebulae of irregular form. It is altogether probable that the meteors, and especially the meteoric showers, are nothing else but fragments of dis- integrated comets. As soon as periodicity in the recurrence of showers was recognized, and it thus became plain that the meteors travel in definite orbits, it was but a short step to compare those orbits with known cometary paths. And 318 PLATE 23. Meteor Trail. Photo by Barnard. METEORS AND AEROLITES soon after the great Leonid shower of 1866, Schiaparelli showed that the Perseids, or August meteors, are in the same orbit as a comet discovered by Tuttle in 1862. And it was not long before the Leonids were similarly identified with the comet of 1866, discovered by Temple. At least eight different meteor showers are now known to be connected with comets. The conclusion is possible that the comet is itself but a condensed place in the meteoric procession; the meteors themselves the disintegrated part of the material involved in the whole transaction. Certain it is that at least one comet (Biela's) has actually been seen to break up. It was discovered in 1826, re-appeared in 1846 according to prediction, and was seen to break in two during this period of visibility. In 1852 it was seen again, the two parts being now widely separated ; and it has never been visible since. In its orbit, however, moves one of the big meteor swarms. It is very important to determine as accurately as possible the height of meteors above the earth's surface. For this is about the only direct method we have to ascer- tain observationally the extent of the terrestrial atmosphere. Since the meteor becomes visible only when it penetrates , ' , , . FIG. 94. Height of Meteors. the earth s envelope of air, we shall know something about the height of the air if we can measure the height of the meteor. The only way to do this is to select a couple of stations on the earth and make simultaneous observations of the same meteor. Figure 94 shows how this is done. 319 ASTRONOMY If observers on the earth's surface, at Oi and 2 , see the meteor M projected on the celestial sphere near the stars Si and $2, it is clear that we can calculate the height MH of the meteor from the known distance 0i0 2 between the observers, and the angles $iOi0 2 and S 2 2 0i, which are, of course, known, if we know the positions of the stars Si and S% on the sky. But there is great difficulty in securing these observations in such a manner as to be certain that they apply to the same meteor. It is necessary to make the attempt on some night when numerous meteors are to be expected ; and it is then by no means easy to be certain that the ob- servations of M from the two stations are really simultaneous, and apply to but a single meteoric object. Such as they are, observations of this kind indicate an extreme height of 75 miles for our atmosphere. Having thus explained the principal facts about the meteors, we come next to the Aerolites. These are stones, or pieces of iron mixed with other materials, which fall upon the surface of the earth from time to time. There is little doubt of their being meteors that actually strike the earth, probably on account of unusually large size. For a very big meteor would not be entirely consumed, as it were, in its passage through the air, and might be attracted down to the surface of the earth by the gravitational pull of the planet. Those that are completely consumed perhaps fall on the earth finally as dust : certain it is that dust, probably meteoric, has been found on the surface of ancient Arctic ice. This is the so-called "cosmic dust." Many specimens of aerolites are preserved in museums. A number have actually been seen to fall, so there is no doubt whatever as to their origin being outside the earth itself. When seen 320 METEORS AND AEROLITES at night they exhibit a bright round head, with a luminous trail. Occasionally there is an audible explosion. The outer surf aces of the aerolitic specimens in our museums seem to have been melted, showing the effect of the high temperatures produced through their impeded motion in the air. Chemically, they contain only elements or sub- stances known on the earth. 321 CHAPTER XX STARSHINE IN the preceding chapters we have completed a somewhat detailed description of the solar system, and are now ready to proceed outward into space, to study the distant uni- verse of stars. Astronomers believe that an analogy exists, more or less close, between our sun and the stars (p. 6). Together with its system of planets, the sun may be regarded as a small isolated group suspended in space, and separated from other similar groups by distances almost incomparably greater than any existing within the solar system itself. We may be quite sure that the stars are all excessively dis- tant : we are troubled by no doubts in this respect, as were the ancients. For we now have the law of gravitation; from it we know that if there were a celestial body of any kind in space, as massive as the sun, and not more than ten thousand times as far away as the distance separating the earth from the sun, that body would surely reveal its existence through observable perturbations (p. 206) pro- duced in the motions within our solar system. And this it would do, even if it were a dead sun, no longer luminous, and quite invisible. To produce gravitational attraction, and consequent perturbative effects, merely the presence of matter would be necessary, not visible matter. Furthermore, actual measurements, to be described later, have shown that the nearest fixed star so far observed is more than 200,000 times as far from the sun as is the earth. 322 PLATE 24. The Constellation Serpentarius. (From Hevelius' Prodromus Astronomiae, Gedani, 1690.) STARSHINE But this last argument is not conclusive, because we can measure only visible stars, and the nearest one might con- ceivably be non-luminous. This objection, of course, does not apply to the gravitational argument. We have seen (p. 6) that the stars are classified according to their magnitudes, and that this term " magnitude" does not here have its usual meaning. It has no relation to size or bigness, but simply indicates the degree of luminosity or brightness of a star. We shall now consider this matter somewhat more in detail. Old Hipparchus (pp. 127, 189, 297) was the first to divide the stars into magnitude classes ; he simply selected arbitrarily the twenty brightest stars he could see, and called them first-magnitude stars. He then designated as sixth-magnitude all objects that belonged at his absolute lower limit of vision, that he could just see, though with difficulty. Stars of intermediate luminosity he placed in intermediate classes,, also somewhat arbitrarily. This gave a rather rough classification ; but it is still in use (with some improvements) down to the present day. Adopting Hipparchus' magnitude scale, we find the num- ber of stars of the various magnitudes situated between the north pole of the heavens and the circle of declination 35 south of the celestial equator to be as follows : 1st mag. 14, 3d mag. 152, 5th mag. 854, 2d mag. 48, 4th mag. 313, 6th mag. 2010, Total, 3391. The above rough system of classification was replaced by a more exact one about 1850. Sir John Herschel had remarked from his photometric observations that first- magnitude stars average just about 100 times the luminosity of sixth-magnitude stars. So it was suggested that we take the exact " fifth root " of 100 as the ratio between the lumi- 323 ASTRONOMY nosities of any two successive star-magnitudes. And this ratio is called the "light-ratio." In this way, we make the fifth-magnitude stars VlOO times as bright as the sixth-magnitude stars; the fourth VlOO times as bright as the fifth; etc. Consequently, the first-magnitudes would be VlOO X VlOO X VlOO X VlOO X VlOO, or 100, times as bright as the sixth-magnitudes, in exact accord with Herschel's observation. Now the fifth root of 100 is about 2|, so that stars of any magnitude are approximately two-and-one-half times as bright as those of the next fainter magnitude. To fix a definite zero for this scale, it has been decided to select certain stars as standards. Thus, Aldebaran is a standard first-magnitude : other stars can be compared with it ; their light- ratio measured by observation; and the magnitude difference then ascertained. 1 This process, of course, some- times assigns zero-magnitude, or even a negative magnitude, to an exceptionally bright star, like Sirius. On the above scale, Sirius actually comes out from photometric observa- tions as minus 1.4. The sun's stellar magnitude is about 26, the enormous luminosity being in this case due to proximity, not to intrinsic light-giving power. Observations of the relative luminosities of stars are made with an instrument called an astronomic photometer. The ordinary telescope may be so used in a very simple way. To estimate a star's brightness, we have only to place dia- phragms pierced with holes of various sizes outside the object-glass (p. 272) until we find one that will just allow us to glimpse the star. It is obvious that this method is possible, for if we use successively a series of diaphragms with apertures diminishing gradually, we shall in effect be 1 Note 38, Appendix. 324 STARSHINE making the telescope smaller and smaller, and there must come a time when it has been made so small that it will just fail to show the star under observation. From the size of the aperture in this last diaphragm, it is possible to cal- culate the luminosity of the star. 1 There are also other, and perhaps better, forms of pho- tometers, in which the star under examination is compared with an artificial star produced by a light in the observatory placed behind a screen having a very small hole. Varying the artificial star until it appears of the same luminosity as the real one enables the observer to measure accurately the brightness of the latter. Magnitudes may also be measured photographically. The little dots produced on a sensitive plate by prolonged exposure in the telescope vary in a sort of proportion to star-magnitudes : the bright stars produce larger dots. Therefore a microscopic measure- ment of the dot diameters on an astronomic negative enables us to estimate star-magnitudes. But all astronomic photometric measures are subject to considerable error on account of " light-absorption " in the terrestrial atmosphere. Some of the stellar light is lost in passing through our air. This effect is, of course, smallest for stars near the zenith; for there light passes straight through the atmospheric layer, and at right angles to it. The path through the air is thus the shortest possible. But for stars near the horizon the light enters the atmosphere at a rather small angle, and its path is much longer, before it reaches the observer's eye. Consequently, stars are bright- est when they are near the zenith. How much is the total light received from the stars ? This question has been widely studied, but only the very 1 Note 39, Appendix. 325 ASTRONOMY roughest results have been obtained. Possibly the whole sidereal heavens give about as much light as 2000 stars like Vega. This is approximately $ of full moonlight ; and it includes the considerable quantity of starlight coming from objects below the sixth magnitude, and therefore invisible to the unaided eye. The heat received from the stars is almost evanescent ; only the very slightest indications of it have been rendered perceptible by the most delicate thermometric apparatus so far invented. It is also possible to obtain a very rough comparison between the total light actually emitted by the stars and by the sun. It is found, for instance, that Vega emits about 49 times the light sent out by the sun. 1 Other stars give similar results : we see, therefore, that our sun is really a rather faint star, but not an infinitesimally small one, in comparison with Vega. There exists still another very remarkable phenomenon in connection with stellar luminosity, its variability. For it must not be supposed that the brightness of all stars is strictly constant : many increase or diminish their light from time to time; and these are called " variable stars." There are several distinct kinds. Some vary their light steadily, ever increasing or diminishing it ; others show no regularity whatever in their rise and fall; and a few, the " temporary stars," or novce } appear suddenly in the heavens, and last but a short time. Finally, there are stars that wax and wane with more or less accurate periodic regularity ; and wonderful variables, in which changes are caused by some form of eclipse phenomenon. Of these last the best known is the star called Algol (the Demon). 1 Note 40, Appendix. 326 STARSHINE The number of stars whose light alters steadily in one direction is very small : demonstrated permanent variations of brilliancy since the time of Hipparchus are extremely in- frequent. But we have records that Eratosthenes saw /3 Librae brighter than Antares, though the contrary is surely the fact at present. Similarly, in 1603, Bayer recorded Castor as brighter than Pollux, in the constellation Gemini ; but now Pollux gives us more light than Castor. The most prominent irregular variable is 77, in the con- stellation Argo Navis, in the southern celestial hemisphere. Sir John Herschel saw it as bright as Sirius in 1843, on the occasion of a visit to the Cape of Good Hope. It has been of the seventh magnitude since 1865, and is still in process of change. Temporary stars have blazed up about eighteen times since men began to write their records of the skies. The most famous is Tycho Brahe's star of 1572, which was brighter than any other star, and lasted only sixteen months all together. This is the star that first interested Tycho in astronomy : it is reported that he refused for a long time to describe his observations, because he thought it beneath the dignity of a Danish nobleman to write a book. A peculiarly interesting recent object was nova Aurigce (the new star in the constellation Auriga) , which appeared in 1891. It has been variously explained as the result of heat engendered by a collision between two dark stars, or as an explosion occurring in a single dark body. Another important new star was found in the constellation Perseus in 1901. Within a few days after discovery its brightness grew to the first magnitude, but it faded again with almost equal celerity. It developed a surrounding nebulosity after a time ; and is all together one of the most 327 ASTRONOMY wonderful astronomic objects ever observed. The accom- panying Plate 25 shows the manner in which this nebulosity grew in size. If this growth is the result of motion outward from the central star, the velocity must have been incredibly rapid. What cosmic process or catastrophe there occurred before our eyes, we can neither describe fully nor attempt to explain. The periodic variable stars are of three kinds. First, those like the star Mira (the Wonder) in the constellation Cetus, which is ordinarily invisible without a telescope, but increases rather suddenly every eleven months to the third magnitude, when it is of course a naked-eye star. The second variety of these, variables includes stars like fi, in the constellation Lyra, with changes completed in a very short period ; and with the alteration of light apparently com- pounded of two different variations superposed. And the third class of these stars are like Algol, in the constellation Perseus, which, at regular intervals, undergoes a partial eclipse. A possible explanation of regular variations might be of- fered from the analogy of sunspots. Certain stars may have very large permanent spot zones that are carried around by axial rotation ; and when these are turned toward the earth we may properly expect for a time a considerable diminution of light. The eclipse theory for the Algol stars is quite old ; but it was not proved correct in a convincing way until 1889, when Vogel successfully observed the spectrum according to the Doppler principle (p. 284). He showed that the visible star Algol has a velocity of recession from the earth before its period of minimum luminosity, and an equal velocity of approach after such minimum. The explanation now pos- 328 STARSHINE tulates (Fig. 95) a dark but massive companion to the visible star, and both revolving in nearly circular orbits about their common center of gravity. The orbit plane is sup- posed to be directed towards the solar system, so that we see the orbits edgewise. The dark body is supposed to be smaller in size than the luminous or visible one. In the course of their orbital revolutions, there will come a time when the smaller dark body will be at Z), and the lumi- nous body at L. An observer on the earth, in the direction shown by the straight arrow, will see the luminous body partly covered or eclipsed by the smaller dark body. As soon as the eclipse is over, the luminous body will regain its full brilliancy, as in the positions D r and L' '. From a combination of the velocities of approach and recession, observed spectroscopically, with the known light-variations of Algol, Vogel was able to prove that the distance between the two bodies is 3f million miles, and their diameters respec- tively 0.8 and 1.1 million miles. It will thus be seen that they are so near each other as to be almost in rolling con- tact ; but there exists no cosmic law preventing the occur- rence of orbital motion of this kind. The combined mass of the two bodies Vogel found to be about f that of the sun; his calculations were made by a process analogous to the method of determining a planet's mass from the observed orbital period of its satellite (p. 204). 329 FIG. 95. Explanation of Algol Stars. ASTRONOMY The more complicated light-variations of certain other stars may also be explained on the eclipse theory, the orbital planes being supposed inclined to the direction of the earth, and neither body dark. If both are luminous, there will still be a diminution of light during the eclipses. For one luminous body will then appear superposed on the other, and the total light will be the same as the larger body would emit alone. The next thing we have to consider is the important question of stellar distances from the solar system. After all, this is in a way the most interesting matter we have to dis- cuss in connection with the stars, for the question at issue really relates to the actual dimensions of the magnificent sidereal universe. In the first place, let us once more define " stellar par- FIG. 96. Stellar Parallax. Solar parallax (p. 260) has been explained to be half the earth's angular diam- eter, as seen from the sun. In a similar way, stellar par- allax is the angle SS'E, in Fig. 96, between two lines, one drawn from the star S' to the sun S, and the other from the star S', tangent to the earth's annual orbit around the sun at E. In other words, when the earth reaches such a posi- tion in its orbit around the sun that there is a right angle at the earth E, between the sun S and the star S' } then the angle SS f E is the star's parallax. We are compelled to use the radius of the earth's orbit in defining stellar parallax, whereas we used the radius of the 330 STARSHINE earth itself in the case of solar parallax. For the earth's own radius is far too small to subtend an angle at all appreciable, if the vertex is situated at the vast distance of the stars. Even when we use the orbital radius, the very nearest star, so far as we now know, has a parallax angle of only three- quarters of a second of arc. It is clear, from Fig. 96, that the star's parallax angle really equals the difference in direction of the star as seen from the earth, and as it would be seen by a supposed ob- server on the sun. As the earth goes around its orbit, this parallactic displacement, or change of the star's direction due to the observer's being on the earth instead of the sun - this parallactic displacement must change its direction. At intervals of six months, during which the earth traverses one-half of its orbit, the displacement reaches equal amounts, but opposite directions. In the interval, there is a constant change in the direction of the displacement ; so that, if a star is projected on the sky at a point perpendicular to the plane of the earth's orbit, it will appear to describe in a year a little circle on the sky, which is a miniature replica of the earth's own orbit around the sun. 1 A star in the plane of the earth's orbit will simply appear to swing back and forth through the year in a short, straight line, instead of describing a circle. Stars in intermediate positions will have apparent "parallactic orbits," which will be small ovals, intermediate in form between the circle and the straight line. . The measurement of a star's parallax is therefore nothing more than a measurement of the form and angular diameter of it's little apparent parallactic orbit on the sky. This may be measured by an " absolute" method, or a " differential " 1 A star so situated is, of course, at the "pole" of the ecliptic circle on the sky. 331 ASTRONOMY one. The absolute method requires a determination of the star's right-ascension and declination at various dates during a year. These being located on a celestial chart give the parallactic orbit at once. But the method is practically of little value, because we possess no instruments capable of measuring these declinations, etc., within the small fraction of a second of arc which is here necessary. The differential method is better, since it enables us to use a micrometer (p. 276) as illustrated in Fig. 97. The parallactic orbit of a star is here shown as an ellipse or oval (p. 331). Si and 82 are apparent positions of the star in its parallactic orbit on two different dates. Si and $ 2 ' are two small stars in the vicinity. The observer measures, on various dates through the year, the small angular distances, from the parallax star to the small stars Si and $ 2 '. These FIG. 97. Differen- latter, on account of their minuteness, may reasonably be supposed situated at practically an infinite distance from us, and therefore to have no appre- ciable parallaxes of their own, and no parallactic orbits. So the measures determine a series of points on the parallactic orbit ; and, from the size of the orbit, the parallax of the star S. It is clear that this method should bring out a value of the parallax possessing a high degree of precision : and microscopic measures of an astronomic photograph may here replace actual visual micrometric work at the telescope with advantage. If the small stars are not really at an infinite distance, the differential method fur- nishes what may be called " relative parallaxes"; or, in a sense, the parallax excess of the star under examination over the small stars supposed infinitely distant. The first 332 STARSHINE successful stellar parallax measurement was made by Bessel in 1838. He used the differential method, and applied it to a star of moderate magnitude in the constellation Cygnus (cf. p. 192). When we come to translate stellar parallax measures into terms of linear distances, We arrive at numbers so large as to be unmanageable. For this reason astronomers have invented a new and large linear unit to be used in sidereal measurements. It is called the "light-year," and is defined as the linear distance light would travel in a year. As the velocity of light is about 183,000 miles per second, the light- year, in miles, amounts to 183,000 multiplied by the number of seconds in a year. It is therefore an enormous unit of distance ; but it is none too large for use in describing stellar distances in space. Its length is about 60,000 times the dis- tance from the earth to the sun, and corresponds to a paral- lax of 3J seconds of arc. Closely connected with the above subject of stellar dis- tances is the question of the stars' motions. We have already seen (p. 7) that the objects called fixed stars are not really fixed in space. They are all actually drifting across the sky ; it is only because of their vast distance that their motions seem to us small and slow. In reality, their velocities are of the same order of magnitude as those existing among the planets of our solar system : but to the rough instruments of the ancients these motions remained unrevealed; and so the stars received their designation of "fixed," to distinguish them from the wandering planets. And within the last half-century it has become possible to do even more than merely measure this stellar drift across the face of the sky ; the drift which shows itself as a change of the star's right-ascensions and declinations. As we 333 ASTRONOMY have already seen (p. 284), we can now also evaluate, with the spectroscope, stellar velocities of motion in the "line of sight"; or, in other words, the star's linear velocities in a direction perpendicular to the celestial sphere. Now changes in a star's right-ascension and declination do not necessarily prove the existence of motion. For the pre- cession of the equinoxes (p. 126) moves the point from which we count right-ascensions, and it also shifts the celestial equator from which we count declinations. Since right- ascension is angular distance from the vernal equinox, measured on the equator, and declination angular dis- tance from the equator, it is clear that precessional changes of equinox and equator will change both these quantities. But all precessional effects can be calculated easily : and even after these effects have been eliminated from our observations by a suitable process of calculation, we still find small " residual" changes of right-ascension and decli- nation. These may be ascribed wholly or in part to real motions of the stars. It is this residual motion that is called a star's " proper motion." This term is now a century old in astronomy : it is applied only to motion across the celestial sphere; not to motion in the line of sight, revealed spec- troscopically. The latter has been separately denominated " radial velocity." Proper motion is measured in seconds of arc per annum ; radial velocity in linear miles per second. Less than two hundred stars have proper motions as large as one second of arc per annum. The largest known mo- tion of the kind belongs to a little star numbered V 243 in a great catalogue of stars made at Cordova in South America. It drifts nearly nine seconds annually. The next largest and, until 1898, the largest known belongs to a star numbered 1830 in a catalogue observed by Groombridge in England 334 STARSHINE about the year 1790. This star is often called the " run- away" ; it travels seven seconds in a year. The relation of proper motion and radial velocity is indi- cated in Fig. 98. If a star at Si moves to $ 2 in a unit of time (a year, let us say), and if the solar system is at E, we can draw SiSi perpendicular to SiE, to indicate the star's appar- ent motion across the celestial sphere as seen from E. In the same unit of time the star will have receded from the earth by a distance SiS 2 '', and this will represent its annual radial ve- locity. The true motion of the star, SiS 2 , may be regarded as really made up of two parts, -the radial motion and the proper motion. Now that we are able to measure these radial velocities, it is at last possible to ascertain from ob- servation both parts of the dis- tance SiSz, which is the star's true motion in a unit of time. Before we had the spectro- scope and Doppler's principle, we never knew, or could know more than the one part SiSi. These considerations empha- size the importance of the spectroscope in sidereal research : it has not only created a celestial chemistry ; it has also given us new and essential knowledge in the oldest department of astronomic science, the theory of sidereal motions. Not infrequently it happens that a certain number of stars are found to have proper motions, and probably real motions in space, nearly identical, both in amount and direc- tion. For instance, Proctor pointed out some years ago 335 FIG. 98. Proper Motion and Radial Velocity. ASTRONOMY that five of the bright stars in the Dipper (p. 52) probably possess such a community of motion. There can be little doubt that they are proceeding through space in parallel lines, and that they belong together. The same is true of a number of the brighter stars in the Pleiades group. They are not merely an apparent group, but a real cluster. A very surprising thing, discovered in 1909, is that Sirius is also probably a member of the Dipper group of stars. If this be so, the group in question is not a distant congeries passing us at a distance, perhaps, of billions of miles; but it is passing so close that we are actually now within the drifting group of stars itself. This follows from the fact that Sirius is in quite a different part of the sky from the Dipper region. We have seen quite enough to make clear the high value to science of these very modern measures of radial velocity : unfortunately, it has not been found possible as yet to apply the method to faint stars, because their light is not sufficient to give a measurable spectral image. This class of work was first attempted by Huggins in 1867 ; and he began by measur- ing the radial velocity of Sirius, the brightest of the stars, with a visual telescope and spectroscope. It was not until Vogel began to use photography in 1888 that any consider- able extension of the process became possible. Fainter stars then first became observable, for the exposure of a photographic plate can be lengthened within reasonable limits, so as to give even a small quantity of light time to impress a spectral image on the plate. Good examples of radial velocities are found in the star a Carinse, receding from the solar system 17 miles per second ; and o Ceti, receding 54 miles per second. As the earth's own orbital velocity around the sun is about 18| 336 STARSHINE miles per second, we see that the stellar motions are only of the same order of velocity as those existing within the con- fines of the solar system. Spectrum photographs of the above stars are shown in the lower part of Plate 26. In each case the stellar spectrum is placed between two artificial spectra, produced in the observatory for comparison. It will be seen that the stellar spectral lines of both stars are displaced in the same direction with respect to the artificial spectra, because both stars are receding; but the displacement is much greater in the case of o Ceti, on account of its greater velocity of recession. The upper photograph of Plate 26 is a comparison of the lunar spectrum with that of Saturn and its ring (cf. p. 245). It shows, as it should, that the outer part of the ring is moving slower than the inner part. This matter of stellar spectroscopy was first taken up by Huggins to study the chemical composition of the stars, though it led him also to the measurement of radial veloci- ties, as we have seen. At about the same time, Secchi undertook similar investigations, and to him we owe a sort of classification of stellar spectra, as follows : 1. White and blue stars, with strong evidence of hydrogen. Examples are Sirius and Vega. These stars are believed to be in an early stage of cosmic development. 2. Solar stars, showing in their spectra many dark lines due to absorption, as in the solar spectrum. Capella is one of those stars ; they are supposed to be somewhat older than those showing the hydrogen lines. 3. Red stars, with spectra showing dark, broadened lines. 4. Faint red stars, probably very old ; the spectra having a few bright lines. All the spectroscopic observations indicate a stellar z 337 ASTRONOMY chemistry similar to that of the solar system. The entire sidereal universe seems to contain but one set of chemical elements; and these are very widely distributed. So we see once more that our sun is a star; and if the other stars are traveling through space, our sun should also be in motion, carrying its planets with it, much as the earth moves in its orbit around the sun and carries the moon with it. Analogy would lead us to expect such a solar motion. Of course, the simplest way to study this question is to examine the radial velocities of a large number of stars. Sup- pose we find that those near a certain point on the sky are approaching us; those near the opposite point receding; and those halfway between, neither approaching nor reced- ing. Then we may conclude that this is merely a result of the solar system's own motion ; and that we are approaching the stars projected near the point of the sky where they seem to be approaching us. Towards this point on the sky, then, the solar motion is for the moment directed. 1 Campbell made such a research a few years ago, using spectroscopic results derived from 280 stars. The point on the sky indicated by his work is not very far from the first-magnitude star Vega. It is called the "apex of the sun's way." Naturally, Campbell used only bright stars whose spectra could be observed ; and of course brightness indicates nearness, other things being equal. It may there- fore be a fact that stars having a common drift with the sun predominate in Campbell's series of observations ; and if so, this might partly invalidate his result. But however this may be, he finds the region near Vega to contain the apex, and 13 miles per second as the " cosmic linear velocity" of the solar system. 1 Note 41, Appendix. 338 STARSHINE Another fact that may cause a slight error in such an in- vestigation as the foregoing is the necessity of making some assumption as to the average real motions of the stars whose radial velocities are observed. For we do not find all stars near the apex approaching us ; only a preponderance of motions of approach. So astronomers assume that in the average of so large a number of stars (in this case, 280) there will be as many motions in any one direction as in any other. Therefore, a preponderance of motions of approach near the apex must be due to solar motion, not to motion of the stars themselves. It is singular that as far back as 1783 Sir William Herschel obtained almost the same result from a discussion of the proper motions (p. 334) of various stars across the sky. His method is well illus- trated in Fig. 99. If there are two lamp- posts, LI and Z/2, on opposite sides of a . , , -,. FIG. 99. Determination of the Apex (Herschel). street, the angular dis- tance between them will seem much larger when they are viewed from HI than from Hi. A person walking from HI to H 2 will see this increase of the angular distance. Apply- ing this principle to the sky, Herschel concluded that near the apex the constellations must be opening out, as we ap- proach, and at the opposite point of the sky they must be closing in. In other words, near the apex stellar proper motions directed away from that point must predominate ; and near the op- posite point, called the " anti-apex,' 7 proper motions directed towards the critical spot must be most in evidence. Herschel had at his disposal the measured proper motions of only 339 ASTRONOMY thirteen stars all together. Yet with the insight of rare genius, he so sifted this meager evidence that he was able to find the right-ascension and declination of the apex with some approximation to correctness. Later investigators have of course repeated this work with much more elaborate modern material at command. They find a result in very fair accord with the spectroscopic one. But they also find this important peculiarity. If the proper motions are divided into groups, and the calculations made separately with stars of large and small proper motions, the apex comes out farther south on the sky for the stars of large proper motions. Now it is evident that any investigation of this kind must assume that if there were no solar motion, the stellar proper motions would be quite casual, and free from any tendency to congregate in direction towards or from any apical point. But such a tendency in the stars themselves is indicated by the peculiar result just mentioned. It is clear that the large proper motion stars must have a common path of their own. But largeness of proper motion should indicate nearness to us, other things being equal ; for at a sufficient distance, even large proper motions would shrink into ap- parent nothingness. Therefore it is within the bounds of possibility that our sun is a member of a drifting stream of stars, to which, in general, the large proper motion stars belong also. In the light of the above discussion of cosmic motions of the sun and stars, as well as stellar distances, it is possible to con- sider an interesting special problem which may be solved approximately with modern observational data. We have seen that the solar system is moving toward Vega at the rate of 13 miles per second (p. 338). Observations of Vega's 340 STARSHINE own radial velocity indicate that it is itself receding from us at the rate of 3 miles per second ; so we are overtaking it at the rate of 10 miles per second. But the number of seconds in a year is, approximately, 365 X 24 X 60 X 60 ; and the actual annual approach of the two stars, therefore, 365 X 24 X 60 X 60 X 10 miles. The parallax of Vega has been measured; it is O."ll. From the parallax, the distance between the solar system and Vega may be computed, 1 and it comes out, approximately : 93000000 X 200000 ^rr To ascertain the time in years required by the solar system to reach the position occupied by Vega, we must divide the distance of Vega by the rate of approach per year. We thus obtain, for the number of years required to over- take Vega in space : 93000000 X 200000 _1_ _, 0.11 < 365 X 24 X 60 X 60 X 10 or, approximately, 560,000 years ; and after the lapse of that period of time, the solar system should reach Vega. But in that interval Vega will have moved, too, for it has a proper motion across the sky of 0."5 per year, which is about four times its parallax angle. Figure 100 will explain this state of affairs. At a certain moment, the sun and earth are at S and E, with Vega at Vi. Then, by definition, the angle EViS is Vega's parallax. At the end of a year, the earth will be back at E, very nearly, but Vega will be seen at Vz, because its proper motion will have carried it across the sky, as seen from the solar system, through the angle Fi$F 2 . Since this angle is four times the parallax 1 Note 42, Appendix. 341 ASTRONOMY angle EViS, it must follow, approximately, that FiF 2 is four times ES. Therefore, ViV 2 is 4 X 93,000,000 miles, or 372,000,000 miles. So we see that when the sun reaches the point where Vega should be in 560,000 years, Vega will have moved at FIG. 100. Vega's Parallax and Proper Motion. least 372,000,000 X 560,000 miles, and this is about as near as we shall ever approach Vega. What will be Vega's parallax at that time? We can answer this question by comparing the present distance of Vega, which we have found to be : 93000000 X 200000 -air miles ' with its distance in 560,000 years as just obtained. The two numbers are not far from equal, and therefore Vega's future parallax will not be far from its present parallax. So there is no danger of a cosmic collision with Vega, so far as we may judge from the above rough calculation. Our present discussion would not be complete without a brief account of certain quite recent researches, made principally by Kapteyn. His idea is that we need extensive statistical knowledge of stellar distribution, more than direct measures of a few parallaxes. He therefore undertakes to compute the average parallax of the stars of any given magni- 342 STARSHINE tude, separating the stars thus by magnitudes for the obvious reason that the fainter ones may be expected to average greater distances than the brighter ones, and therefore smaller parallaxes. Figure 101 shows his method of attack upon this problem. The arrow SS' is intended to represent the annual proper motion of a star S across the face of the sky, the arrow, of course, indicating both the direction and quantity of such motion. The position of the apex on the sky is shown at A. The line S'Si is drawn perpendicular to SA, and the smaller arrow SSi then shows how much the proper motion SS' carried the star away from the apex. In fact, the proper motion SS' may be regarded as com- pounded of two motions : . FIG. 101. Kapteyn's Researches. &oi , which affects the an- gular distance from the star to the apex; and Si'S', which does not affect that distance, being at right angles to SA. Now Kapteyn " resolves" (as it is called) all known proper motions into two such " components," one directed away from the apex, the other at right angles to the first. But we have already seen (p. 339) that the effect of the solar system's own motion in space is to open out the constella- tions near the apex ; therefore SSi, the star's proper motion component away from the apex, must include the effect of the solar system's motion ; but Si'S', the other component, is free from such effect. In the general average of a large number of stars of any given magnitude, the two components should be equal, if the sun were at rest. For the effect of the solar motion would then disappear ; and we may assume 343 ASTRONOMY that the star's own motions are as likely to be in one direc- tion as another, so that the average components would balance, as it were. It follows that any difference of the two components, derived from observed averages, must be an effect of the sun's motion alone. Now this average difference is expressed in seconds of arc, being an observed angular proper motion across the sky. Arid we know the linear velocity of the solar system toward the apex to be 13 miles per second (p. 338), or 13 X 365 X 24 X 60 X 60 miles annually. Figure 102 shows the further FIG. 102. Kapteyn's Researches. procedure. The arrow SiS 2 represents the solar system's motion toward the apex, in a year. Kapteyn's " average star" is shown at S s . The little angle SiSA is the average star's proper motion component away from the apex, or the above-mentioned observed difference of the two com- ponents. Knowing, now, this little angle, and the linear velocity SiS z , we can calculate SiS z , or the average star's distance from the solar system Si. This beautiful method of investigation has enabled Kapteyn to obtain a table of approximate stellar distances. He gives the following results, for two types of stars sepa- rately (cf . p. 337) : I, bluish white stars, like Sirius ; II, solar stars, like Capella. The distances are expressed in light- years (p. 333), three of which correspond approximately to a parallax of one second of arc. The table shows that the 344 STARSHINE solar stars are decidedly nearer to us than are the Sirian stars. TABLE OF KAPTEYN'S DISTANCES (In light-years) STAB'S MAGNITUDE TYPE I II 1 101 43 2 130 56 3 166 71 4 213 91 5 273 117 10 948 405 15 3270 1404 Kapteyn has also obtained some further interesting results as to stellar distribution. He uses a "unit of sidereal space/' and for this space unit he imagines a cosmic sphere of "unit radius," which he defines as a sphere such that a star on its surface would have a parallax of one second of arc to an observer at its center. The length of this radius would be about three light-years. Then, since parallax angles are inversely proportional to distances, it follows that stars whose parallaxes are greater than O."20 are all within a sphere whose radius is 5 units. About 20 stars with such parallaxes are known; and we may assume that these are probably all that exist. Now the volumes of spheres are proportional to the cubes of their radii ; so that the above-mentioned sphere with radius 5 must have a volume of 125. And as there are 20 stars in it, there must exist about one star to each six units of space; and this is the approximate "star-density" in the cosmic vicinity of the solar system. This conclusion may 345 ASTRONOMY not be very accurate; but it constitutes a most important addition to our knowledge of sidereal astronomy, since even a rough approximation is better than a total absence of information. And Kapteyn has been able to proceed a step farther. Having found the average distance of stars of any given magnitude, and knowing the average proper motions and radial velocities as well, he has been able to compute the actual average linear velocities of the stars in space; and he finds them to be somewhat less than twice the cosmic velocity of the solar system. 1 Therefore the stars' average speed is about 26 miles per second; or, in a year, about seven times the distance between the earth and sun. But the sidereal unit, or radius of the unit sphere, is about 200,000 times the distance from earth to sun ; so the stars move a sidereal unit, on the average, in 27,000 years. Now we have found that, on the average, about one star exists in each six units of space. From this it may be com- puted, according to the theory of probabilities, that the average distance between the stars is about 3.5 linear sidereal units. Therefore the stars move through a distance equal to the 3,5 units that separate them in about 3.5 X 27,000, or 100,000 years, on the average. It follows from all these considerations that stars will approach each other infrequently, even within astronomical proximities, enormous as these are. The above conclusions all relate to averages ; and we know 1 Knowing the star's parallax, or distance, and the angular annual proper motion across the sky, we compute the linear velocity component parallel to the celestial sphere in the same way that we obtain a planet's linear diameter from its measured angular diameter. Then, knowing the radial velocity, and the linear velocity at right angles to it, as computed from the proper motion, we can finally calculate the actual velocity. 346 STARSHINE one or two stars that probably have far greater velocities than the average. For instance, the star 1830 Groombridge (p. 334) has a velocity of perhaps 140 miles per second. We must conclude that this particular star is passing through our sidereal universe, and will leave it altogether in a few million years. But in general, from what has been said, it would appear that the stars in our universe are much like the molecules of a gas, as indicated in the kinetic theory of gases. The difference between a glass vessel full of gas and a universe full of stars is merely one of scale. In either case, each star or molecule moves in a more or less straight line of random direction, until or unless a couple of them happen to collide. In both cases, such collisions are extremely fre- quent : only, in the gas, the word " frequent" signifies a very minute fraction of a second ; in stellar space, the same word may mean centuries. Perhaps this kinetic theory of stars would undergo some material modification if we admit that all observed stellar motions are not necessarily random ones, and that there may be star-groups of common motion, and star-streams of vast extent. But however this may be, these magnificent researches are all inspiring in a high degree : it is extraordi- nary that such can still be made in our own day in the oldest and most completely perfected domain of human knowledge, the science of astronomy. Not infrequent in the sky are the " binary" stars; 1 twin suns, they have been called (cf. p. 9). Each component star of such a pair moves around the center of gravity of both in an oval orbit, just as the earth and moon (p. 174) move around their center of gravity. The orbits are studied with a special micrometer (p. 276), with which astronomers can measure 1 The lower part of Plate 27 is a photograph of a binary. 347 ASTRONOMY the angular distance between the two components in seconds of arc, and also the angle between the line joining them on the sky and a line drawn from the principal one to the celes- tial pole. Thus, in Fig. 103, SiP is an arc of a great circle imagined on the sky, joining the principal component star Si and the pole. We then measure the small angular distance SiSz t be- tween the two components of the double star, ' and also the angle PSiS z . The latter is called the " position angle." If we continue these FIG. IDS. Binary measures, at intervals, for a number of years, and then draw the resulting orbital curve, we find it to be an oval or ellipse, often very much flattened. But this is only an apparent orbit ; for what we see is the real orbit, projected on the celestial sphere. It is only when the plane of the binary star's orbit happens to be perpen- dicular to our sight-line that the apparent orbit coincides with the real one. But it is always possible to calculate the location of the true orbital plane ; and, in fact, the elements of the real orbit, by applying Kepler's laws of motion (p. 187) to the apparent orbit. Yet, even after this has been done, we have only a "relative" orbit, representing the motion of one component star of the binary pair with respect to the other. For a very few binaries, the actual orbit of both compo- nents has been separately determined, by means of micro- metric comparisons with neighboring small stars. And when we are so fortunate as to know also the parallax of the binary, we can calculate the linear dimensions of the orbits in miles. Without the parallax, we can know only the angular dimensions of these orbits in seconds of arc. The relation of the two dimensions is like the relation of the 348 Photo by Barnard. Photo Mt. Wilson Observatory. PLATE 27. A Star Cluster in Hercules and the Double Star Krueger 60. STARSHINE angular diameter of the sun to its real linear diameter (p. 118). In the few cases for which such researches have been carried out with success, the linear size of the orbits appears to be comparable with the orbital radii we find in the solar system. So we conclude that the binary systems are not extremely large, speaking cosmically. Certain binary stars have been recognized as such by spectroscopic instead of micrometric observations. We have already described Vogel's discoveries with regard to Algol (p. 328), where the binary character of the star was betrayed by an observed periodic change in the direction of its radial velocity. But Pickering also found that certain spectra photographed with a slitless spectroscope (p. 285) showed a periodical doubling of the lines. Ordinarily single, they became double at uniform intervals of time. Pickering explained this correctly as indicating a binary system, in which, unlike Algol, the components are both luminous stars. When one component is approaching us, and the other reced- ing, in consequence of their orbital motions, the spectral lines are displaced in opposite directions, according to the Doppler principle, and we get two separate spectra and two sets of lines. When both components are moving at right angles to the sight-line, as they will do in another part of their orbit, the two spectra are superposed, and we get one set of lines only. It is remarkable that we can thus separate a pair of stars with certainty, although they appear so near each other on the sky that the most powerful telescope shows but a single object at all times. This was a great triumph for the spectroscope ; the observation, together with the correct explanation of it, will surely have a place in the classic annals of astronomy. It is interesting to note that we can calculate also the 349 ASTRONOMY masses of these binary stars for which both orbits and paral- laxes have been observed. As we have just seen, we then know the linear dimensions of the orbit in miles, and so we can apply the method used (p. 204) for obtaining the mass of a planet having an observable satellite. 1 The masses of these binaries, in the few cases where they have become known, are found to be of the same order of magnitude as the sun's mass. Before leaving the subject of binary stars, it may be of interest to touch on one possible theory as to their origin. It is not now believed by astronomers that the Laplacian theory of celestial development (p. 235) is the only pos- sible or even probable one. For the Laplacian idea leads to a single central sun, with many planets of far smaller size than the sun. But it is possible that an original whirling nebula may have undergone changes more or less approximating the formation of two nuclei. These, revolving, gave rise, first, to an egg-shaped, later, a dumb-bell shaped, revolving body. The latter, finally separating, should produce twin suns, at first revolving with their surfaces almost in contact. Such a condition might even explain some of the peculiar light-changes of certain variable stars. Later, there might arise perturbative action, similar to the tidal effects produced between the moon and terrestrial oceans. These would drive the two bodies farther apart (cf . p. 258), and possibly lead to a visible binary star. Nor is there any objection to our imagining some of these distant suns to be attended by planets. Only, if such planets are no larger than Jupiter, we could not possibly hope to see them with the most powerful of our telescopes. It is an easy transition from the binary systems to those 1 Note 43, Appendix. 350 PLATE 28. The Pleiades. Photo by Barnard. STARSHINE still more wonderful, in which three or more incandescent suns revolve about their common center of gravity in plain view of the telescope. For instance, the constellation Lyra contains an important double star, the " double double/ ' in which each component is itself a binary, forming all to- gether a quadruple star. And in addition to these " multiple stars," we find also various " clusters." Some contain comparatively few stars, spread over quite a considerable bit of the sky. The Pleiades group is a famous cluster of this kind. Two views of it are shown in the accompanying Plate 28 : one con- tains the stars only ; the other, made with a large telescope, indicates that most of these stars are still surrounded with nebulosities. Here and there we can see a nebulous lane running from one star to the next ; nor have these peculiar formations ever been thoroughly explained. There are also other clusters, like the close-packed globular one in the constellation Hercules (Plate 27, p. 349), consisting of many thousand stars separated from each other on the sky by very small angular distances only. We have two sure facts to indicate that the clusters are single objects, and not mere fortuitous groupings of stars, unconnected with each other, situated at all sorts of distances from the solar system, and appearing close together because they happen to be projected near a single point of the sky. First, in the Pleiades group, it is known that over forty stars have practically identical proper motions in the same direc- tion on the sky, pointing to a community of real motion in space. And secondly, many close clusters have been found to contain a most unusually large percentage of variable stars, again indicating a community of origin for the whole cluster. 351 ASTRONOMY Unfortunately, it has not yet been possible to measure the parallax of any cluster. We can but make guesses at their distance from the solar system, rough estimates based on the size of their proper motions. These are so small that we must assume the clusters to be very distant, probably not less than 400 light-years. If removed to that distance, our own sun would give us no more light than a star of the eleventh magnitude. It follows that the clustered stars are perhaps comparable in size with the sun, for they, too, average the eleventh magnitude, more or less. With the above estimate of distance, we can also estimate the linear size of the clusters from their angular diameter, in the usual way ; and we find them to be about two light- years in diameter. If such a cluster contained 10,000 stars, the average distance from one to another would be about 25,000 times the distance from the earth to the sun. At such distances gravitation would not be strong enough to bring all the constituent stars under the influence of a central force : it would not even produce velocities of interstellar orbital motion such as could become perceptible to our micrometric instruments during the comparatively short period since precise observations were commenced by ter- restrial man (cf. p. 322). Closely related to the clusters are the nebulae (p. 3). Indeed, certain clusters, like the Pleiades, are so completely interwoven with clouds of nebulous matter, and with nebulous lines connecting the several stars, that one is al- most inclined to regard them as nebulae partly condensed into stellar nuclei. But the true nebulae are undoubtedly gaseous : spectroscopic evidence on this point is conclusive (pp. 4, 283). " Planetary nebulae" are a class of nearly circular light- clouds possessing almost planetary central disks. Their 352 STARSHINE spectra contain certain lines belonging to nebulae only, and ascribed to incandescence of a hypothetical substance "nebulium." At their centers, nebulae of this class have at times nuclei that look like stars. One is tempted to imagine them in the last stage of nebular development, and on the verge of becoming starry. There exist also a few ring-formed nebulae. But the type-form of nebula is the spiral nebula (cf. Plate 2, p. 4). Keeler thought there are 120,000 objects of this form within the photographic range of his big reflect- ing telescope at the Lick observatory. His observations opened the eyes of astronomers to the probability that the Laplacian plan of cosmic evolution may not be the one generally active; that a single sun like ours is less likely to occur in space than some more complicated system of suns, resulting perhaps from a great apparently whirling complex by us seen as a spiral nebula. The Andromeda nebula is by far the largest of the spirals : for modern long-exposure photographs have proved its spiral character, though it was always supposed to be elliptical or ring-formed until celestial photography came into general use. We cannot measure its distance from us by any of our parallax methods : but it is possible to fix for its parallax a limit of O."01. The parallax cannot be much larger than this, or traces of it would reveal themselves to our instru- ments. But assuming this parallax, and the known angular diameter of the nebula (one and a half degrees of arc), it must have a linear diameter 540,000 times as great as the distance between the earth and the sun. 1 It is furthermore of interest to compare this nebula's possible attractive force with that exerted by the sun on 1 Note 44, Appendix. 2 A 353 ASTRONOMY the earth. It may be computed that if the nebular density is only lorowooir ^ ^ ne sun ' s > ^ ne ne bula will attract the earth as much as the sun does. 1 So the fact that we find no perturbative attraction whatever in the solar system resulting from the nebulae, proves that, though enormous in size, they are of an almost inconceivable tenuity ; in fact, almost without any density whatever. To the foregoing very recent researches must be added an observation of most ancient date, but one that all the modern theories have failed to explain quite fully. This is the Milky Way, or " galaxy," which shows itself as a band of small stars star /* dust - - encircling the sky almost like the celestial equa- FIG. 104. The Galaxy. tor, ecliptic, etc. It has many starless rifts and lanes, and several "holes" ; not- ably the " coal-sack," situated near the south pole of the heavens. It contains numerous star-clusters, but few nebulae. One of the most interesting is shown in Plate 29, the ' ' North America" nebula, so named because of its shape. The galaxy, resembling a great circle of the celestial sphere, must of course have two nodes, or points of intersection with the ecliptic circle. These are near the solstices, and the galactic circle makes an angle of about 60 with the eclip- tic. According to Sir William Herschel, the stars of the galaxy are spread out in a thin disk, in which our sun is also situated. In Fig. 104, the solar system is shown at S, and the galactic disk is within the rectangle. Outside the rectangle, the stars are fewer and farther asunder. It is evident that if we look in the direction A or B from S, we shall look through the 1 Note 45, Appendix. 354 PLATE 29. The North America Nebula. Photo by Barnard. STARSHINE thick part of the galaxy, and see an enormous number of stars projected on the celestial sphere ; but if we look toward (7, we shall see projected on the sky only a thin part of the disk, and the sparser stars, outside it. And the disk, of course, when produced outward to the celestial sphere, will cut out the galactic great circle. Actual counts of stars have been made by Herschel and others, to ascertain the number per square degree of sky surface at various angular distances from the galaxy. The numbers are found to vary in the following proportions : ANGULAR DISTANCE RELATIVE NUMBER OF STARS FROM GALAXY PER SQUARE DEGREE 90 4 60 7 30 18 122 The unexplained difficulty with Herschel's explanation of the galaxy still remains. The extreme minuteness of the galactic stars indicates immense distance, as does also their lack of observable proper motions across the sky. But if so enormously distant, how can the galaxy constitute a single disk-shaped cluster or universe ? But perhaps we are here inventing a difficulty, because of our inability even to imagine the scale of the sidereal universe. However this may be, we may close this part of our subject with a definite statement that there is no evidence in the possession of astronomers to indicate a " central sun "around which all the stars are circulating in their orbits. As already stated, we now believe their motions more nearly resemble the gyrations of the molecules in a gas under the kinetic theory. Gravitation probably takes hold in an appreciable degree, only when a couple of stellar molecules happen to pass near each other, speaking cosmically. 355 CHAPTER XXI THE UNIVERSE ONCE MORE THE reader may recall that we commenced our long ex- planation of astronomic science in the present volume with a chapter entitled "The Universe." Now that we at last approach the end, let us once again return to the beginning, and reexamine the evolutionary processes of the cosmic universe, in the light of the astronomic knowledge we have been able to gain. Cosmogony is a name given to the various theories of the universe and its life-history : there is no subject more entic- ing to the mind of man ; none in which he is more prone to be misled into fields of mere speculation, quite outside the domain of strictest logic, based on irrefragible observa- tional premises. We have already mentioned (p. 235) the Laplacian nebular hypothesis, with its rotating nebulous sun, forming planets by the successive separation of rings : it will now be proper to inquire a little more closely into the admissibility of La- place's idea. It will be well to begin by summarizing the known facts that are favorable to Laplace : 1 1. The planetary orbits all lie nearly in the same plane; and the direction of orbital motion is the same for all planets, and for the sun's axial rotation. 2. The orbits are all nearly circular. 1 Laplace, "Exposition du Systeme du Monde," p. 470, in Oeuvres de Laplace, Paris, MDCCCXLVI. 356 THE UNIVERSE ONCE MORE 3. With the exception of Uranus and Neptune, the equa- torial planes of the planets, and even the orbital planes of their satellites, all coincide approximately with the fundamental plane of all the orbits; and the direction of the satellites' orbital motion is in general also the direction of the planets' axial rotation. 4. We have accepted the Helmholtz theory (p. 294), that the sun's source of heat must be sought in the contrac- tion of its bulk : in that case the sun must have once been incomparably larger than at present, just as the nebular hypothesis requires. Laplace, of course, was not in posses- sion of Helmholtz' calculations when his own theory was published. Now, as a matter of logic, a correct theory must explain every observed fact within its range. A single contrary observation may destroy logically an entire theory, no matter how many other observations seem to confirm it. Let us then next enumerate some of the objections that have been advanced against the nebular hypothesis. 1. Phobos, the inner satellite of Mars (p. 222), and the inner edge of Saturn's ring (p. 245), revolve in their orbits faster than the axial rotation periods of Mars and Saturn respectively. This will not do : the contracting planet should rotate faster than any satellite revolves around it, just as the inner planets have shorter sidereal periods than the outer ones. 2. The kinetic theory of gases would lead us to expect (pp. 167, 222) that a gas like hydrogen would all be lost into space from each planet in the form of molecules, soon after the ring was thrown off from the sun, on account of the very small gravitational pull of the ring. Yet we still find hy- drogen in plenty on the earth. 357 ASTRONOMY 3. The throwing off of the rings is in itself an hypothesis difficult of acceptance, on account of the presumably extreme rarity of the outer parts of the solar nebula, and consequent lack of cohesion. And why was the process of expelling rings intermittent instead of continuous ? 4. The mechanical movements in the system present diffi- culties. For instance, the total quantity of rotary momen- tum now belonging to Jupiter is about -f^ of the total belong- ing to the entire solar system included within the orbit of Saturn. Yet Jupiter's mass is only about one-thousandth of the total mass remaining within Saturn's orbit, including the sun. Are we then to suppose that the present sun, at a single moment, parted with so small a fraction of its mass, which yet carried away almost all its rotary power ? Chamberlin and Moulton have endeavored recently to substitute a new and different theory for that of Laplace. They call it the " planetesimal hypothesis" ; and they think the recent evolution of stellar systems, and of our solar system, began with the accidental close approach (not collision) of two stars. If we imagine such an approach to have taken place, we must suppose the two bodies revolving for a time in orbits around their common center of gravity. If the approach was not very close, these orbits would be open curves, like the orbits of most comets when they approach the sun (p. 312). The two bodies would separate after a certain time, and would never again pass near each other. But while they were together, the gravitational effect of each on the other would be tremendous. Doubtless each would eject masses of highly heated gaseous matter, much as the great red hydrogen prominences (p. 293) are ejected from our sun. Upon these ejected gases, and upon the other outer gaseous envelopes of the two stars, gigantic tidal forces 358 Spiral Nebula, seen edgewise. Photo by Keeler. Owl Nebula. PLATE 30. Nebulae. Photo by Hale. THE UNIVERSE ONCE MORE would be exerted. Consequent gigantic tidal effects would be produced in each star by the other. Like the tidal crests caused by the moon in terrestrial oceans (p. 252), the ejected masses would travel outwards from each star, directly toward the other star, and directly away from it. Afterwards, these masses would move in strange orbits under the combined gravitational attractions of the two stars : we can in a way trace out these orbits by com- putational methods, which have been tried by Moulton. Chamberlin calls these masses " planetesimals " ; and it is assumed that they make their appearance in great num- bers and at short intervals. Figure 105 shows the probable orbits they would pursue. The dotted lines indicate various orbits ; the full lines show the . . , , . . FIG. 105. Planetesimal Hypothesis. points reached at a given in- stant of time by the several planetesimals pursuing the dotted orbits. When we look at the result of such a performance, what shall we expect to see? If we assume the instant of time when we make our observation to be that instant when the several planetesimals have just reached the full lines, we shall not see them traveling along their dotted orbits, but we shall see them all lying on the full lines. In other words, we shall see a spiral nebula. Now whatever strength there may be in this hypothesis, there can be no doubt that Keeler's observations of nebulae (p. 353) establish the fact that the spiral is the normal type of nebula. For comparison with Plate 2, p. 4, we give in Plate 30 a photograph of a nebula which is doubtless another spiral, but seen edgewise. The lower part of the plate 359 ASTRONOMY shows the "owl" nebula in Ursa Major, which looks more like a Laplacian nebulous sphere. Plate 31, p. 360, the "trifid" nebula, is a good example of quite irregular shape. So the new theory gives a notion as to the possibility of spirals resulting from a close approach of two stars, which may have been previously moving about in space aimlessly, after the fashion of the molecules belonging to a rarefied gas (p. 347). Thus the theory goes back to a very early stage of cosmic development, and shows how stars, even dark ones, can be transformed into nebulae. But how can the spiral nebula, in turn, develop into a sun and planets such as we have in our solar system? Of course, the sun is simply the remaining material from one of the original stars after the other had left it. If the approach was near enough to lead to a permanent orbital proximity, we might perhaps expect a binary system. And the sun provided, it is easy to imagine the origin of the planets. They are unusually large occasional planetesimals, increased gradually by the accretion of other smaller ones, swept up by them as they moved along in their dotted orbits. When there was no large dominating planetesimal for a long time, a group of minor planets might result. The satellites must be regarded as little planetesimals that were shot out near the larger ones that became planets. Chamberlin and Moulton have traced out in detail the bearing of their theory upon the various objections that have been enumerated against the Laplacian idea, and with considerable success. The great advantage of their hypoth- esis is that it gives us an origin antedating the nebular stage ; that it makes a cycle of cosmic life and death ; and especially a cycle in accord with the actual visible appearance of existing nebulae. This the Laplacian theory does not do. 360 PLATE 31. The Trifid Nebula. Photo by Hale. THE UNIVERSE ONCE MORE The future of the solar system seems fairly clear under either hypothesis. The present state of affairs is one of apparently stable equilibrium ; and should continue, unless an accident arrives from outside the system. But even without such accident, the solar system cannot be eternal, because the gradual shrinkage of the sun cannot continue forever. When the time comes for the sun to lose its heat- radiating power, the solar system must become cold and dead. If, after countless ages, it shall ever thereafter be revivified, the cause will be a fresh approach to some other star, dark or brilliant, whose vast disturbing attraction will once more break up the solar matter into a mist : and if a great part of the energy remaining in the system shall be transformed into heat, then that mist will once again be a fire-mist, which may once more pass through all the stages of cosmic life and death. 361 APPENDIX THE following notes contain explanations omitted in the text, and requiring occasionally a knowledge of elementary algebra, geometry, and trigonometry as far as the solution of plane right triangles. Note i. Declination and Right-ascension (p. 35). Declination corresponds exactly to latitude on the earth; the declination of a star is its angular distance on the celestial sphere measured north or south from the celestial equator. This angular declination, like all angles, must, of course, be measured on some circle ; for measuring it we must imagine a circle drawn upon the sky from the star to the equator, and perpendicular to the equator. Such a circle, drawn for the purpose of measuring declination, is called an Hour-circle. The point where the hour-circle meets the celestial equator may be called the foot of the hour-circle. The right-ascension of a star is then defined as the angular distance, measured on the celestial equator, from the vernal equinox point on the equator to the foot of the hour-circle drawn from the star to the equator. Note 2. Hour-angle, etc. (p. 37). We may now define also the term " hour-angle," which is closely related to the hour-circle used in measuring right-ascension. The hour-angle of a star is defined as the angular distance, measured like right-ascension on the celestial equator, from the meridian to the foot of the hour-circle drawn from the star to the equator. Thus hour-angle and right-ascension are both arcs measured on the equator ; both arcs have one end in common, the foot of the hour-circle; but the other ends are different, being respectively the meridian and the vernal equinox. All the astronomical terms so far defined are exhibited in Fig. 106. It represents half the celestial sphere, the half which is 363 ASTRONOMY above the horizon, and therefore visible to us. The large circle NESW represents the horizon; and the celestial hemisphere is shown projected down upon the plane of this horizon. The zenith, or point directly overhead, of course projects down into the center of the horizon circle. The great meridian circle appears as the line NPZS, since it must pass through the zenith Z as well as the north and south points of the horizon shown at N and S. The celestial north pole, which is, by definition, in the ce- lestial meridian, will project down to some point P. The celestial equator, everywhere 90 dis- tant from the pole P, will project into FIG. 106. The Celestial Sphere. , J , the circle WME. Any star selected at random may be supposed to be projected down at the point S'. Then S'D, an arc drawn on the sphere through the star and perpendicular to the equator, is by definition an hour-circle. It is evident that all hour-circles must pass through the pole P. The arc DM on the celestial equator, included be- tween the meridian at M and the foot of the hour-circle at D, is the hour-angle of the star. The arc S'A is the star's altitude, or angular elevation above the horizon. Finally, if we draw a short piece of the projected ecliptic circle, we may take V to be one of its points of intersection with the celestial equator WME, the other point of intersection being of course below the horizon. And if we let V be that one of the two points of intersection which we have called the vernal equinox, then the right-ascension of the star S' is the arc VD, measured on the equator, and included 364 APPENDIX between the vernal equinox V and the foot of the hour-circle at D. The arc DS' is the declination. The above astronomical terms may be divided into two classes ; viz. : those that retain a constant position among the stars on the celestial sphere, and those that are as constantly shifting their positions among the stars on account of the daily seeming rotation of the whole sphere. Thus, for instance, the zenith, which is the point directly overhead, does not partake of the seeming turning of the sphere. The following little table shows the two classes of terms : CHANGING POSITIONS AMONG THE STARS. Do NOT ROTATE WITH SPHERE Zenith Horizon Altitude Hour-angle Meridian UNCHANGING POSITIONS AMONG THE STARS. ROTATE WITH THE SPHERE Celestial Poles, Celestial Equator Ecliptic Circle All hour-circles Right-ascension Declination The stars, sun, etc. Vernal Equinox Note 3. Position of Celestial Pole (p. 40). The relative positions of the celestial pole and the horizon may be made clear by means of a simple diagram. Figure 107 shows a portion of the earth, with its center at C, and the observer on the surface at 0. The outer concentric circle HPZE is the celestial meridian, on the celestial sphere. The zenith Z will be directly over the observer at 0, on the FIG. 107. Position of Celestial Pole. prolongation of the observer's terrestrial radius CO. The celestial pole must be at some point of the celestial meridian, by definition. Let this point be P. The celestial equator will meet the meridian at E, 90 distant from P. The terrestrial pole will be at P f , and E f will be a point of the terrestrial equator. The angle E'CO will be the terrestrial latitude of the point 0, since it is the 365 ASTRONOMY angular distance of the point from the terrestrial equator at E f . The angle PCH is the altitude or angular elevation of the celestial pole above the horizon at H. For H, as we know, is the north point of the horizon for an observer at 0. But %ZCE = %-PCH, since PC is perpendicular to CE, and HC perpendicular to ZC. Hence we have a demonstration that the altitude of the celestial pole is everywhere equal to the terres- trial latitude of the observer. Thus, as stated in the text, this alti- tude will be 90 to an observer at the pole of the earth, where the latitude is 90 ; and it will be to an observer at the equator, where the terrestrial latitude is likewise 0. Note 4. Stars that never Set (p. 43). It is evident that these stars are the ones whose diurnal circles have an angular distance from the celestial pole less than PH, (Fig. 107) ; i.e. less than the observer's terrestrial latitude. These stars will have a declination greater than (90 latitude). Note 5. Sidereal Time (p. 67). The sidereal time, or hour-angle of the vernal equinox, is the arc VM in Fig. 106. To make the definition of sidereal time perfectly general, as- tronomers count all hour-angles westward from the meridian, and allow them to increase continuously to 24 h . Thus, an hour- angle l h east from the meridian, corresponding to 23 h sidereal time, would be called a west hour-angle of 23 b . Note 6. Right-ascension of the Meridian (p. 67). Again recurring to Fig. 106, it is clear from our definitions that the right-ascension of a star on the meridian is the arc VM ; and we have seen in Note 5 that this identical arc is also the sidereal time. Therefore the sidereal time and the right-ascension of the meridian at any instant are the same. The general relation of hour-angle, right-ascension, and sidereal time may also be deduced from Fig. 106. We have from our definitions : VD = right-ascension of star S r , DM = hour-angle of star $', VM = sidereal time. 366 APPENDIX And since, from Fig. 106 : VM = VD + DM, it follows that in general : Sidereal time = right-ascension + hour-angle. Hour-angle = sidereal time right-ascension. The last equation enables us to ascertain the hour-angle of a star at any instant, if we know its right-ascension, and have a correct sidereal clock at hand. Note 7. Hour-angle of Visible Sun (p. 68). In Fig. 106, if we let S r be the visible sun at any instant, its hour-angle is the arc DM, measured in hours, minutes, and seconds. This same arc is also the apparent solar time at that instant. . " Note 8. Terrestrial and Celestial Meridians (p. 73). If we imagine a line drawn from the center of the earth to the observer, and thence continued outward to the celestial sphere, it will pierce the sphere at the ob- server's zenith. The terrestrial meridian, by definition, passes through the north pole of the earth and the observer. The celestial meridian, also by definition, passes through the celestial north pole and the observer's zenith. Therefore the celestial merid- ian is a projection of the terrestrial meridian outward on the celestial sphere. Figure 108 is like Fig. 106, with the addition of a second celestial meridian. The figure represents the celestial sphere projected down upon the horizon of New York, 367 FIG. 108. Time Differences. ASTRONOMY of which the zenith appears as before at Z. The projection of the zenith of Greenwich at the same instant is at Z' . Therefore PZ'M' will be the projection of part of the celestial meridian of Greenwich. The sun and vernal equinox are projected at S f and V, as before. Then DM is the sun's hour-angle at New York; DM', its hour-angle at the same instant at Greenwich. MM', which measures the angle between the two celestial merid- ians, is also the difference of the two hour-angles, or the solar time difference between New York and Greenwich. And this is the same as the longitude difference, measured by the two cor- responding terrestrial meridians on the earth inside the celestial sphere. At the same moment, VM and VM f are the hour-angles of the vernal equinox at New York and Greenwich ; and MM' is also the sidereal time difference. Consequently, the sidereal and solar time differences are equal and identical ; they are both measured by the same arc MM'. Note 9. Angle of Gnomon (p. 79). It is evident that the "factor" in the table is simply the tangent of the latitude. In Fig. 24, be = ac tan bac, and if the tabular factor is tan latitude, the construction of the figure will make the angle bac equal to the latitude, as required for the gnomon. Note 10. Mathematical Principles of the Sundial (pp. 80, 84). To demonstrate the correctness of the construction given in the text for drawing a sundial, it is necessary to have recourse to the well-known formulas of spherical trigonometry relating to the solution of right-angled spherical triangles. The accompanying Fig. 109 represents the conditions of the problem. The large circle ZPNQS is the celestial meridian. The circle NIVS is the horizon, on the plane of which the dial is to be drawn. The center of the dial is at ; and QP is the axis of the celestial sphere. As the edge of the gnomon is parallel to the axis QP, we may regard it as lying in that axis, because the sun will appear to rotate around the edge of the gnomon (p. 84). So we may consider the edge of 368 APPENDIX the gnomon to start at 0, and to extend a short distance in the direction OP. Now suppose OIV, situated in the horizon plane NIVS, to be the direction in which the shadow falls at four o'clock. Then, remembering that solar time is simply the hour-angle of the sun, we recall that "four o'clock" means that the sun's hour-angle is four / ./ \? hours, or 60. We may suppose the sun to appear at the point S' at four o'clock. Then, from the definition of hour-angle (p. 363), the sun is then distant 60 from the meridian; or the angle ZPS' is 60. The op- posite angle NPIV, be- ing equal to ZPS', is thus also 60. Now let us consider the spherical triangle formed on the celestial sphere by the three points P, N, and IV. In it we know the side PN, for it is the elevation of the celestial pole above the horizon, and therefore equal to the latitude of the place where the dial is to be used (p. 365). As we have just seen, we also know the angle NPIV, which is 60. And we know the angle PNIV to be a right angle, because the celestial meridian is perpendicular to the horizon. According to the principles of spherical trigonometry, if we know one side and one acute angle of a right-angled spherical triangle, we can calculate all the other parts of the triangle. In the present problem, we need only calculate the side NIV. For this measures the angle NOIV, which is the dial angle for the four-o'clock line, or the angle which the four-o'clock line makes with the north-and-south line ON. In the same way, we can calculate the dial angles for the one- o'clock, three-o'clock lines, etc. The twelve-o'clock line, or noon- 2s 369 FIG. 109. The Sundial. ASTRONOMY line, is of course ON ; for at noon the sun is on the meridian, and the shadow of a gnomon pointing at the celestial pole will then fall due north. We might construct the dial by simply laying off the proper computed angles for the various hours from the dial center 0. The trigonometric formula for calculating the side NIV, or the dial angle NO IV, is : tan NIV = tan NPIV sin PN. And if we let : u = dial angle for any hour, t = corresponding hour-angle of the sun, I = latitude of the place, then the general formula is : tan u = tan t sin I. The dial angles calculated by this formula for the latitude of New York are as follows : XII. I. II. III. IV. V. VI. 0' 9 56' 20 40' 33 10' 48 32' 67 42' 90 0' It now remains to show that the construction given in the text (Fig. 25) is in accord with the above general formula. In this figure we have really drawn two half-dials, so as to allow for the thickness of the gnomon. To prove the construction of Fig. 25 correct, we have now to show, for instance, that : tan cal = tan 15 sin I. The factors given in the table on p. 80 are the sines of the latitudes I. Therefore, since we made Me (Fig. 25) equal to ca multiplied by the factor in the table, it follows that : Me = ca sin I. (1) We made the angle cMI (Fig. 25) equal to one-sixth of a right 370 APPENDIX angle, or 15. Therefore, from the right-angled plane triangle Mel, we have : tan 15, Me or: Ic = Me tan 15. Substituting the value of Me from equation (1) gives : Ic = ca tan 15 sin I, or: = tan 15 sin I. (2) ca Now from the right-angled plane triangle c/a, we have : = tan cal. (3) ca Substituting from equation (2) in equation (3), we have tan cal = tan 15 sin I ; and the correctness of the construction in Fig. 25 is proved, since the above equation accords with the general form : tan u = tan t sin I. Note ii. Theory of Foucault Experiment (p. 91). We have explained the conditions of the problem if the experi- ment were performed at the north pole of the earth. There the point of suspension of the pendulum's wire would of course be situated in the prolongation of the earth's axis, and would conse- quently not move as a result of the earth's axial rotation, which is the only motion of the earth here requiring consideration. In any other latitude, the point of suspension would go around as the earth rotates : it is therefore necessary to explain further the statement that the direction in space of the pendulum's plane of oscillation tends to remain constant. The fact is that when the point of suspension moves, the plane of oscillation moves also ; but it tends to occupy a position constantly parallel to itself. Any one can satisfy himself that this is correct by fastening a small metal ball to a string and letting it oscillate, the end of the string being held in the experimenter's hand. It will be found that the experimenter may walk across the room, carrying the end 371 ASTRONOMY of the string ; yet the plane of oscillation will remain constantly parallel to itself. So much being premised, we may now proceed to calculate the rate at which the marks under the pendulum should rotate. Let us suppose we start the pendulum swinging in a north-and-south direction, and therefore directly under the celestial meridian, and in the plane of the meridian. In Fig. 110, let NABCS be the meridian directly over the pendulum when we start it swinging, and suppose it swings between two points in the room corresponding to the points A and B of this meridian. In a second of time the earth's rotation will have brought a new celestial meridian over the swinging pendu- lum, and the old one will have gone to the position NA'B'C'S. But the pendulum will still swing parallel to the plane of the first meridian, and the rotation shown by the experiment will be equal to the angle between the two meridians. Let us draw Fig. Ill to show this angle. This figure is like a map in a geography book. If the original meridian was AB, and the meridian at the end of one second A'B f , the rotation shown by the pendulum will be the angle between these two lines. If we draw A'M perpendicular to BB', the rotation angle will be the angle MA'B 1 '. Let us call this angle a. It is well known that on a map of the earth's equatorial regions the terrestrial meridians are practically parallel : there is no " convergence of meridians" there ; and there would be no Fou- cault effect. Near the pole the angle between the meridians is a maximum : there the Foucault effect is also greatest. In this way we translate our astronomical problem into terms of geometry ; it is now merely a question of simple geometry to as- 372 s FIG. 110. The Foucault Experiment. BM B FIG. 111. The Fou- cault Experiment. APPENDIX certain the angle of convergence between two neighboring meridians on the earth in any latitude, such as that of New York, for instance ; and this angle will be the Foucault pendulum rate of rotation. We see at once from Fig. Ill that, in any latitude, we have from the triangle A'B'M: and since for very small angles like a the tangent and the arc are equal, we may write : Referring again to Fig. 110, which we may now take to represent the earth instead of the celestial sphere, we observe that the latitude arcs A A', BB', and CC' are all arcs of circles whose radii are PA', P'B', and OC'. The last radius OC' is the earth's radius, because we shall consider CC' to be an arc of the equator. Now suppose the point B' to correspond to the terrestrial latitude V . Then V is the angle B'OC', for the latitude is the angular distance of B f from the equator. But the line P'B' in the plane of the circle NA'B'S is proportional to the cosine of the angle B'OC'. Simi- larly, the radii of all arcs like A A', BB', etc., are simply proportional to the cosines of the latitudes corresponding to the points A', B' } etc. But the arcs themselves must be proportional to their radii. So it follows that the linear lengths of the arc A A', BB', are also proportional to the cosines of the corresponding latitudes. We have called I' the latitude corresponding to the point B'. Let us call I the latitude corresponding to A'. Now we have found that the arcs A A' and BB' are proportional in length to the cosines of the latitudes I and V . Therefore the difference between A A' and BB' must be proportional to the difference of the same cosines, which we may express by the following equation, in which K is simply a constant denoting proportionality : BB' - AA' = K (cos V - cos Z). But, from Fig. Ill : MB' = BB' - AA'. 373 ASTRONOMY Consequently, from the preceding equation : MB' = K (cos V - cos Z). (2) Now, in Fig. 110, draw the line A'Q perpendicular to P'B', com- pleting a little right-angled triangle A'B'Q. (We may regard the short arc A'B' as here equivalent to a straight line.) Then we have : QB' = cos I 1 - cos Z, and : QB' = A'B' sin QA'B'. But: QA'B' = B'OC' = V \ therefore: QB' = A'B'sml'. But: A'B' - (Z- V). Consequently : QB' = cos V - cos I = (Z - Z') sin Z'. It then follows from equation (2) that : MB' = K(l - I') sin V. (3) We also have, obviously, from Fig. Ill : MA' =1-1'. (4) Now substituting from equations (3) and (4) in equation (1), we have finally : = K sin I'. (5) This simple equation (5) establishes the important principle that the rate of rotation of the Foucault pendulum in one second must everywhere be proportional to the sine of the latitude of the place where the experiment is performed. It is further obviously indifferent whether the original impulse was given to the pendulum in the direction of the meridian ; for whatever angle the original impulse made with the original merid- ian, at the end of one second of time that angle will have changed by the same quantity with respect to the meridian. It is quite easy to find the value of the constant K in equation (5). For at the north pole, sin I' = 1, since V = 90. Therefore, at the pole, equation (5) becomes : a = K. But we already know that at the pole the pendulum must make one complete revolution of 360 in 24 hours. So it must 374 APPENDIX there revolve at the rate of 15 per hour, or 15' per minute. With this value of K we therefore have in any latitude V : Rate of revolution = (15' per minute) sin V. In New York, for instance, V = 40 48', sin V = 0.65. Rate of rotation = 9/75 per minute. The above demonstration of the Foucault pendulum theory is not rigorous, but it is sufficiently accurate for ordinary purposes, provided the duration of the experiment is not much greater than one hour. Note 12. The Torsion Constant (p. 108). The problem of ascertaining the torsion constant T from the time of oscillation of the torsion balance is quite analogous to the corresponding problem of determining the length of an ordinary pendulum from its time of vibration. It is shown in books on physics that if we let : t = vibration time of an ordinary simple pendulum, I = length of the pendulum, g = the force of gravity on the earth, * = the ratio 3.1416, then : An analogous formula exists in the case of the torsion balance, except that instead of g, the force of gravity, the formula involves T, the torsion constant. We now let I represent the entire length ab of the torsion balance arm (Figs. 32 and 33), and m the mass of either small ball a or b. Then the torsion balance formula is : and those readers who are acquainted with the science of mechanics will not balance. will note that 2m-\ is the "moment of inertia" of the entire 375 ASTRONOMY Solving this equation for T gives : and this equation will make known the value of T for any torsion balance after we have observed its vibration time t, measured the length of the arm ab, and ascertained m by weighing the small balls in an ordinary balance. Note 13. The Cavendish Experiment (p. 110). Returning now to Fig. 33, let us use the following notation : M = mass in grams of either big lead ball, m = mass in grams of either small ball, d = measured distance in centimeters from the position of rest b' to B f , the center of the big lead ball, g = the acceleration due to the "force of gravity," as used in physics, or 981 centimeters, I = length of torsion balance arm, or the distance ab, in centi- meters, / = the force with which both big balls turn the balance. Now, according to Newton's law, the attractive force between the balls B' and V is (p. 103) : G^j, (1) in which formula G is a so-called " gravitational constant," in- troduced to indicate that the attraction is proportional to' not equal to it. The distance from B f to a' is : consequently, the attractive force existing between B' and a' is : Both forces (1) and (2) tend to turn the torsion balance. They act against each other, however, tending to rotate the balance in opposite directions. And the force (1) is larger than (2) ; so that it will determine the final direction of rotation. 376 APPENDIX Furthermore, the entire force (1) tends to turn the balance, while only a small part or " component" of (2) has such an effect. We can easily find this component, which acts from B' upon a' so as to turn the balance. According to the so-called " parallelogram of forces" this component is : ^ ^sinB'o'6', or : Q Mm d d or, finally : Mmd .* (3) The effective force tending to rotate the balance, and resulting from the big ball B' ', will be the difference of (1) and (3). It will be J/, since, in our notation, / is the force with which both big balls tend to rotate the balance. By subtracting (3) from (1) we thus obtain the equation : %f = GMm(~ 2 - -\ (4) For brevity, let us put : Then we have : if = GMmD, (6) and, solving for M, we obtain : The force / must be determined from observations of the torsion balance, when under the influence of the big lead balls. Trans- ferring these big balls from the position A', B' to the position A", B" usually rotates the balance through a very small angle only. It is therefore necessary to measure this angle by very delicate means. For this purpose a small light mirror is attached to the center of the arm ab of the balance, and rotation is measured by allowing a strong beam of light to fall on this mirror, and to be 377 ASTRONOMY thence reflected upon a scale at some distance from the apparatus. The rotation of the balance is thus magnified, and can be measured without difficulty. To introduce these measures into our formulas, let : = the total change in centimeters of the light on the scale brought about by changing the big balls from A', B' to A", B". Q = the distance of the scale from the mirror. To employ the units usual in measures of this kind, we must reduce the motion a to what it would have been on a scale at unit distance from the mirror. This would be ^- We must also allow for the well-known fact that a moving reflected beam changes its direction twice as fast as the mirror turns. This reduces the motion on the scale at unit distance to Finally, we must again divide by 2 to obtain the effect corresponding to the half motion W, instead of the whole motion b"b', since we are calcu- lating the disturbance of the balance from a position of rest, and have measured its motion between two positions of extreme disturbance. This gives the observed motion on the scale, to be used in the further calculations, as : Tn (8) 4Q Now it is a principle underlying the torsion or twisting of rods or fibers, a principle verified easily by experiment, that the force required to twist the rod or fiber through any angle is proportional to that angle. For instance, if a certain force would turn the torsion balance through an angle of 10, it would require just twice as much force to turn it through 20. It follows from this principle, and from the definition of the torsional constant T, that the force required to turn the balance through the angle (8) where readers familiar with Mechanics will note that T is really the ''turning moment" for unit angle applied at unit distance from the center. 378 APPENDIX This expression (9) is not yet equal to the force /, because / is applied at the ends of the balance arms where the small balls are. The length of this balance arm being -, we see that (9) must be 2 equal to / - ; and so we may write the equation : T-fL . (10) From this we have : /(observed) = - (11) We have already obtained in Note 12 (p. 375) an expression for T as follows : -.2 72 m __ " f i to /-J n\ o /2 With the help of equations (11) and (12) we can compute the observed force / from our observations of and Q, and the known dimensions, etc., of the parts of the balance. Next we can establish easily an expression for the attractive force existing between either little ball and the earth. For this purpose, let R = the radius of the earth, in centimeters, E = mass of the earth, in grams. Then we have : Attractive force between small ball and earth = G-^' (13) Equation (13) follows directly from Newton's law, if we recall that the earth attracts bodies exterior to it precisely as if the entire mass of the earth were concentrated at its center. Thus the radius of the earth becomes the distance between the earth and the small ball, and its square appears in the denominator of equa- tion (13). Furthermore, according to the teaching of Physics, the attractive force existing between the small ball and the earth is also measured by the weight of the small ball, since weight is merely the result of such attractive force of the earth. And in physics^ the weight 379 ASTRONOMY of any object is shown to be equal to its mass multiplied by the force of gravity, g. So we have : Attractive force between small ball and earth = mg. (14) Equating the right-hand members of equations (13) and (14) gives : n Em /1 K v mg = G; (15) or : P2 E =^f (16) Cr If we now divide equation (16) by equation (7), we obtain : (17) We now obtain the value of / from equation (11) by the help of equation (12). This gives: , _ = Substituting from equation (18) in equation (17) gives, finally : (19) Equation (19) enables us to calculate the mass of the earth, E, in terms of the mass of either big lead ball, M. It will be noted that the only quantities used in equation (19) and actually observed in the Cavendish experiment are and Q. Most of the other quanti- ties are ascertained by measurements and weighings before the tor- sion balance is put together. The time of vibration, t, is found in seconds by observing the combined duration of a considerable num- ber of oscillations, made with the big lead balls entirely removed. In the actual apparatus mounted for use in the astronomical lecture-room at Columbia University, New York, the following numerical data exist : t = 281.5 seconds, d = 5.3 centimeters, / = 3.6 centimeters, g = 981 centimeters j TT = 3.1416 centimeters, M = 2750 grams, 380 APPENDIX and the radius of the earth is : R = 6.371 X 10 8 centimeters. With these numbers we obtain from equation (19) : E = 0.30 X 10 27 - 2 grams. In an actual experiment the writer found : Q = 189 centimeters, = 10.86 centimeters. Therefore : Q = 17.4, and : E = 5.22 X 10 27 grams. The present accepted value of the earth's mass is : 6 X 10 27 grams ; so that the result of the above lecture-room experiment is fairly satisfactory. Note 14. Linear Distances from Angles Alone (p. 119). The simple Figure 112 shows the correctness of the principle stated in the text. Suppose, for instance, that the three angles of a triangle are given, and it is required to draw the triangle. It is not possible to do so ; because, with the given angles, we do not know whether we should make it of the , i 7-> ., m FIG. 112. Distance from size A, or the size B, or any other size. lo Angles know the triangle fully, we must know the length of at least one side. Angles alone enable us to draw a figure which is geometrically similar to the required figure, but they do not enable us to draw the figure itself to scale. Note 15. Calendar Rule (p. 144). To demonstrate this rule, we begin by assuming that our era commenced with a year numbered 0, so that 1913 was the 1914th year of the era. Of course there was not really an initial year 0, but we can imagine the calendar extended to that time. Then the principle on which our rule is based consists in calculating the number of days from January 1 of the year to the date under in- 381 ASTRONOMY vestigation, and ascertaining how many weeks elapsed in the interval. It happens that January 1 of the year was a Sunday. Let us next compute the number of days between January 1 of the year and March 1 of any year, such as 1913. We select March 1, because it is desirable for the moment to use a date that follows the possible extra day inserted as February 29 in leap-years. Let us indicate the year number, such as 1913, by the letter y, and the century number, such as 19 in the year 1913, by the letter c. The total number of days from January 1 in the year 0, to March 1 of the same year, is 59, for the year was not a leap-year. Consequently, if there were no leap-years, we should have : No. of days from Jan. 1, year 0, to Mar. 1, year y = 365 y + 59. As each leap-year adds one day, we must increase this by the number of leap-years from the year to the year y } and including the year y, if it be a leap-year. To find this number, let us divide c and y by 4, and call the remainders after the division n and r 3 . Then it is clear that under the Gregorian rule the number of leap- years will be : i (y - r,) - c + \ (c - n). Furthermore, this number will be a whole number, because we can prove easily that y r 3 and c r\ are both divisible exactly by 4, without remainder. The proof of this is as follows : If we divide any number what- ever, N, by some other number Z>, and find from the division a quotient Q and a remainder R ; then, if we divide N R by D, we shall again find the same quotient Q, but the remainder will now be 0. Thus, if we divide 1913 by 4, we find the quotient Q is 478 and the remainder R is 1. If we now subtract this remainder 1 from the original number 1913, we have for N R, 1912. This being divided by the same divisor 4, gives the old quotient Q as 478, but the remainder is now 0. This shows that our expression for the number of leap-years is a whole number, as it should be. We then have, by addition of the number of leap-years : Total no. of days from Jan. 1, year 0, to Mar. 1, year y - 365 y + 59 + J (y - r g ) - c + i (c - n). 382 APPENDIX Now, if March 1 in the year y is a Sunday, like the first day of the era, the above number must be divisible exactly by 7. But if March 1 in the year y is Monday, one day later than Sunday, we can make the above number divisible by 7 if we subtract 1 from it; and 1 is the week-day number for Monday, minus 1. Similarly, for Wednesday, for which the week-day number is 4, we would subtract 3. In general, let us indicate by w the week-day number of March 1, whatever it may be in the year y, and subtract w 1 from the above total number of days. This gives : 365 y + 59 + I (y - r,) - c + J (c - n) - (w - 1), and this number is now always divisible exactly by 7. Our real problem is to determine (w 1) from the fact that the number just obtained is thus divisible exactly by 7. In doing this we may evidently increase or diminish our number by any exact multiple of 7 without impairing its divisibility by 7 or affecting the value of w. We shall introduce two new remainders r 2 and r 4 , by dividing the century number c, and the year number y, by 7, just as we have already divided them both by 4. This having been done, we may correct our total number of days as follows, noting, of course, that each number added or sub- tracted is divisible exactly by 7. We shall add : subtract 364 y + 56, add 7 TI + 7 r 3 , subtract 3 (y r 4 ) + (c r 2 ), and so our total number becomes : 3 + r 2 + 5 r 3 + 5 n + 3 r 4 - (w - 1). This number is now made up of remainders only. It will be a comparatively small number, as no remainder is larger than 6 ; and it is still divisible exactly by 7. It is therefore clear that (w l) is simply the remainder that will occur in the division by 7 of the number : 3 + 5 n + r 2 + 5 7-3 + 3 r 4 , and thus (IP 1) is determined for March 1 in the year y. 383 ASTRONOMY But we need to find (w 1) for any day in the year y, not merely for March 1. To accomplish this for any other day in March, say the 3d, for instance, we have merely to add 2 to the above number, before dividing by 7, because March 3 comes two days later than March 1. In general, if we indicate by d the date in March for which the week-day is required, we must add (d 1) to the above number. This gives, for March d th : 3 + 5 ri + r 2 + 5 r 3 + 3 r 4 + (d - 1), or : 2 + 5 ri + r 2 + 5 r 3 -f- 3 r 4 + d ; and this number being divided by 7 will give the (w 1) of March d for a remainder. The same expression will hold for April, if we add 31, because there are 31 days in March. Adding 31, and deducting 28, an exact multiple of 7, gives for April : 5 + 5 ri + r 2 + 5 r 3 + 3 r 4 + d. A similar expression holds for each month, a difference occurring only in the number at the beginning of the expression. If we in- dicate that month-number by m, we may write for any month : in + 5 7*1 + r 2 + 5 r 3 + 3 r 4 + d. The values of m for the various months may then be written in a little table (see Rule, p. 144). In forming this table it is necessary to remember that there will be a slight difference between the ra's for leap-years and ordinary years. We have started with the formula for March 1, in order to avoid this difference as much as possible. After that date in the year there is no difference. But in January and February the leap-year m's are smaller by 1 than those for ordinary years, on account of the interpolated February 29. The entire rule may be arranged in the accompanying tabular form. That part of the formula which does not vary in a whole century, namely, 5 r\ + r^ we have designated by K. In the Julian calendar K is evidently always 0, because there is no century ex- ception in the leap-year rule of that calendar. For the sake of symmetry, we have here indicated the final remainder (w 1) by r 8 . 384 APPENDIX CALCULATION OF WEEK-DAY, GREGORIAN OR JULIAN CALENDAR FORMULA TABLE OF m (d=Day of the Month) Divide by and call the remainder Ord'y Year Leap Year WEEK-DAY Nos. Century No. 4 Ti Jan. 6 5 1 Sunday Century No. 7 r* Feb. 2 1 2 Monday Year No. 4 r s March 2 2 3 Tuesday Year No. 7 n April 5 5 4 Wednesday 5r, + 3r-f-in 7 1 May 5 Thursday + m + d\ . r$ June 3 3 6 Friday July 5 5 7 Saturday Aug. 1 1 Sept. 4 4 Oct. 6 6 Nov. 2 2 Dec. 4 4 where K = 5 n + r 2 , Gregorian ; # = 0, Julian. (Gregorian K = 20, from 1900 to 1999.) Week-day No. = r 5 + 1. Note 16. Gauss' Rule for Easter (p. 148). To demonstrate the rule, we shall consider the Julian calendar first, and then modify our results to accord with the present Gregorian calendar. The lunar month of chronology, or the period of the moon's orbital revolution around the earth, is approximately 29 \ days long. In making the ecclesiastical calendar it was therefore decided to have lunar months of 29 and 30 days occur alternately as a general rule. But for a reason to be explained in a moment, an extra lunar month of 30 days is inserted at the end of every third year for six successive periods of three years each, or eighteen years in all. Then, one year later, at the end of the nineteenth year, an additional extra lunar month of 29 days is further in- serted in the calendar. 2c 385 ASTRONOMY The lunar calendar for nineteen years then stands as follows : 3 years (36 months) alternating 29 and 30 days, total . . 1062 days Extra months of 30 days 30 days The above repeated five times more (1092 x 5) 5460 days The 19th year of 12 months alternating 29 and 30 days . . 354 days The final extra month of 29 days 29 days Total 6935 days The above calculation takes no account of leap-years, which occur every fourth year in the Julian calendar. To get these leap-years into the lunar calendar, too, the ecclesiastical chronol- ogists adopted the simple plan of putting an extra day into the lunar month of February, whenever it is put into the civil month of February. In 19 years this will happen five times when any one of the first three years of the 19 is a leap-year ; but if the fourth year of the 19 is a leap-year, it will happen four times only. Thus, on the average : 19 years will have 6935 + 5, or 6940 days three times, and 19 years will have 6935 + 4, or 6939 days once. The mean of these figures is 6939f days ; and this is the average number of days in 19 lunar years, according to accepted chronologic rules. Now the length of a Julian tropical or calendar year is 365J days. Consequently, 19 Julian years will contain 365 \ X 19, or 6939f , days, agreeing exactly with the lunar figures just found. This agreement is evidently not accidental, but is the result of the above conventional and arbitrary rules for the extra lunar months. One important thing follows from this agreement : if we write the calendar dates of full-moon for a period of 19 years, these calen- dar dates will then be repeated in the next and in every subsequent period of 19 years. Now it so happens that the year 0, or the year next preceding the year 1 of our era, was the first year of a 19-year cycle. Consequently, the year 1 was the second of the 19-year cycle, the year 2 the third, and the year 19 the first of the next cycle. It is clear that, in general, if we divide the year number y by 19, and call the remainder r 6 , then r 6 + 1 will be the position of the year y in a 19-year cycle. 386 APPENDIX The next step is to find for any year the date of the Easter full- moon, which, according to the Nicene council's decree, is the first to fall on March 21 or thereafter. Let us call the date of this full-moon March 21 -f P, and suppose dates in March to be car- ried over into April, so that April 1 will be called March 32. Now it so happens that in the year preceding the year 1, the Easter full-moon, Julian calendar, fell on March 36, so that P was then 15. As there are 354 (12 X 29J) days' in a lunar twelve-month, it is clear that in the year 1 Easter full-moon must have occurred 11 (which is 365-354) days earlier. And in each succeeding year of the 19-year period, Easter full-moon must have occurred either 11 days earlier than in the preceding year, or 19 (which is 30-11) days later. Of course the" occasions when it occurs 19 days later are accounted for by the extra 30-day months inserted every three years. The following table exhibits the above state of affairs : YEAR T6 P 15 1 1 4 11 days earlier than year 2 2 23 19 days later than year 1 3 3 12 11 days earlier than year 2 4 4 1 11 days earlier than year 3 5 5 20 19 days later than year 4, etc. etc. It is clear that we shall have, in general, if we let v and x be two unknown whole numbers : P = 15 + 19 z - 11 v, or: P = 15 + 19 (x -h v) - 30 v. It is further clear that in this equation : x + v = r 6 , because, to get P in the table above, we have always added 19 r 6 and then subtracted the largest possible value of 30 v. So we may write : P = 19 r 6 + 15 - 30 v. 387 ASTRONOMY From this it appears that P is simply the remainder occurring in the division of 19 r& + 15 by 30. If we call this remainder r 7 we can therefore find the date of Easter full-moon in the Julian calendar thus : Divide by and call the remainder Year No., y, 19 r 6 + 15 19 30 ; And the date of Easter full-moon, Julian calendar, is March 21 + r 7 . The above method of calculation not only applies to the first period of 19 years, but is entirely general ; because, as we have seen, subsequent 19-year periods simply repeat the same dates of full- moon. We must now pass to the Gregorian calendar. It is evident that the two calendars are in accord at the beginning of the era, and do not diverge until the year 100, when the Gregorian calendar omits a Julian leap-year. This will of course change P by one day, and this same difference of one day will continue from the year 100 to the year 199. From 200 to 299 there will be a difference of two days, etc. It is clear that we can allow for this cause of difference between the two calendars by varying the number 15 that occurs in the quantity 19 r 6 + 15. Let us call this variable number M. Then, in both calendars, M is 15 from the beginning of the era to the year 99. In the Gregorian calendar M increases by 1 each century thereafter, except that for every fourth century this increase is omitted because of the Gregorian leap-year exception. Using our former notation, in which c is the century number, we have for the Gregorian calendar : M = 15 + c - i (c - n). But this value of M , thus corrected for the Gregorian leap-year, is not yet quite right. A further last correction is still necessary on account of a slight inaccuracy in the lunar period of 19 years. A lunar month is not exactly 29J days long; its true length is 29.530586 days. So the 235 lunar months of a 19-year period 388 APPENDIX really amount to 235 X 29.530588 days ; or 6939 days, 16 hours, 31 minutes, and not 6939f days, as already obtained. The error of 1* 29 W amounts to a day in 308 years. But the framers of the ecclesiastical calendar assumed this error to reach one day in 312 J years, or 8 days in 2500 years. So they directed that a correction be made, such that M be diminished by 1 seven times successively at the ends of 300-year periods, and an eighth time at the end of a 400-year period, or 8 times in 2500 years. The last period of 2500 years terminated at the end of the year 1799, and the correction was then 5 ; new corrections are therefore required in 2100, 2400, 2700, 3000, 3300, 3600, 3900, all at in- tervals of 300 years. But the next following correction does not come until 4300 instead of 4200, on account of the eighth period being one of 400 years. This condition will be satisfied for all time if we divide 8 c + 13 by 25, call the remainder r 8 , and subtract from M the correction : 8 c + 13 - r 8 . 25 This may be verified readily by drawing up a table of this cor- rection, when it will be found to have the value 5 for y = 1799, and to increase thereafter forever in the proper way. We have, then, finally, for the Gregorian calendar: M = 15 + c - i (c - n) - A- (8 c + 13 - r g ) ; and wnen M is greater than 30, we may subtract from it, if we choose, the largest possible exact multiple of 30. And in the Gregorian calendar the date of Easter full-moon is now March 21 + r 7 , where r 7 is the remainder resulting from the division by 30 of the number 19 r 6 + M. Having thus found a method of calculating the Gregorian date, March 21 + r 7 , of the Easter full-moon, we must now find the date of the Sunday next following, which will be Easter Sunday. We need therefore only calculate the week-day of the date March 21 + r 7 , to know the date of Easter. Referring to our former civil calendar formulas, we shall find the remainder r 5 for the Easter full-moon date, March 21 + r 7 , which remainder we shall call TQ for this special case, if we divide by 7 the quantity : 5 r 3 + 3 r 4 + K + 2 + r 7 . 389 ASTRONOMY Now if r 9 comes out 0, the Easter full-moon comes on Sunday, and Easter is 7 days later, according to the Nicene decree. If r 9 is 1, the full-moon day is Monday, and Easter is 6 days later. In general, we obtain the date of Easter Sunday by adding to March 21 + r^ the number : 7 -r 9 . Collecting all our formulas, we can now find the date of Easter Sunday as follows ; and thus the rule given on p. 148 is demon- strated. Divide by and call remainder Century No., c 4 r\ K = 5 ri -f- r 2 , Gregorian calendar. Century No., c 7 r-i K = 0, Julian calendar. Year No., y 4 r* M = 15 + c - 1 (c - n) -& (8 c + 13 - r 8 ), Greg, calen- Year No., y 7 r* dar. M = 15, Julian calendar. 8c + 13 25 r$ Easter Sunday is then March Year No., y 19 r& 28 + r 7 - r 9 , 19 r 6 + M 30 TI or April r 7 r 9 3. 5 r 3 + 3 n + K \ + O 1 7 7-9 The following are values of K and 2 + r-i } M for the Gregorian calendar : 1800-1899, K = 14, M = 23, 1900-1999, K = 20, M = 24. As an example, let us calculate the date of Easter Sunday for 1913. We have : r 3 = 1, r 4 = 2, K = 20, M = 24, r 6 = 13, r 7 = 1, r 9 = 6; Easter Sunday is on March 28 + 1 - 6 = March 23. We must now explain the two exceptions that occur in the Gregorian calendar only (p. 149). The first of these happens when r-i = 29. The formulas ' have been deduced on the supposition that the March and April full-moons occur at an interval of 30 days. But that interval may be 29 days only. The framers of the calendar have assumed, rather arbitrarily, that if there is a full-moon on March 19, or earlier in March, the April full-moon 390 APPENDIX will occur 30 days later. But if the March full-moon is on the 20th, or later, the April full-moon will happen 29 days later. Thus the ecclesiastical April full-moon will happen on the same day, no matter whether the March full-moon comes on the 19th or 20th. As this cannot occur in reality, the framers of the calendar have directed that when the March full-moon happens on the 20th, which occurs whenever r 7 = 29, then r 7 shall be diminished arbi- trarily by 1. That is, we must use 28 instead of 29, or move the April moon back one day. But the diminution of r 7 by 1 will ordinarily also diminish r 9 by 1. Consequently, r 7 rg will remain unchanged, and so will the date of Easter Sunday, which depends on r 7 r 9 . Only when r 7 . = 29 and r 9 = will the change of r 7 from 29 to 28 have any effect. For when r 9 = 0, a diminution by 1 will change it into 6, and r 7 r 9 will be diminished by 7, making Easter exactly one week earlier. But when r 7 = 29 and r 9 = 0, the rule always makes Easter come on April 26. Therefore the exception is as stated : whenever Easter comes by the rule on April 26, use April 19 instead. There will be an example of this in 1981. Unfortunately, the above exceptional case introduces another complication. The change of r 7 from 29 to 28 does not make it impossible for the value r 7 = 28 to occur again under the general rule, and during the same 19-year period. This might make two full-moons occur on the same date twice in a single 19-year period, which is, in fact, impossible. To avoid this, the framers of the calendar have ruled, again arbitrarily, that there shall be a second exception. Under this exception, 28 is changed to 27, whenever, in the same 19-year period, the first exception occurs. We must therefore investigate when the first exception can occur. In determining r 7 we performed a division by 30. Let us indicate the quotient of this division by v. Then we have, if r 7 is 29, according to the first exception : 19 r 6 + M = 30 v + 29. Now multiply this equation by 11, and add 11 to each member. This gives : 209 r 6 + 11 M -f 11 = 330*; + 319 + 11 = 330 v + 330. 391 ASTRONOMY The right hand member is now divisible exactly by 30 ; there- fore the left-hand member is also so divisible. But the division of 209 r 6 by 30 will leave a remainder of 29 r&. To make this dis- appear, the remainder in the division of 1 1 M + 1 1 by 30 must be r 6 . But r 6 is always less than 19 by its definition. Therefore the first exception will occur whenever, in the division of 11 M + 11 by 30, the remainder is less than 19. But again, as in the case of the first exception, the change of TI from 28 to 27 will make no difference in the date of Easter, unless r 9 = 0. When r 7 = 28 and r 9 = 0, Easter, according to the rule, comes on April 25. The change of r 7 moves this date to April 18. Therefore the second exception reads : whenever, in the division of 11 M + 11 by 30, the remainder is less than 19, and if r-i = 28 and r 9 = 0, Easter Sunday is on April 18, instead of April 25, as given by the rule. An example of this will occur in 1954. To complete this subject it is necessary to remark that r 7 can never be 29, 28, and 27 within a single period of 19 years. There- fore no further exception is necessary on account of the possibility that the above two exceptions might, by acting together, produce two cases of r 7 = 27 in a single 19-year period. Note 17. The Sextant (p. 154). To prove the fundamental principles of the sextant, that the angle between the mirrors is half the altitude of the sun, imagine the plane of the paper to be the plane of the circle of the sextant. Then, in Fig. 113, the plane of the circle is supposed to be held vertically, in such a way that it will pass through the sun at S. The navigator sees the horizon with the upper part of the telescope through the unsilvered part of the mirror m ; and he sees the sun along the line TmMS after reflection from both mirrors. The angle MTm is the altitude of the sun above the horizon; the angle at P is the angle between the mirrors. It is necessary to prove that : Angle MTm = 2 angle P. The lines MP' and mP' are drawn perpendicular to the mirrors M and m. Then, according to the optical principles governing 392 APPENDIX the Horizon \ FIG. 113. Theory of Sextant. the reflection of light from plane mirrors, the two angles a are equal and so are the two angles b. Furthermore, the angle SMm, or 2 a, is an exterior angle to the triangle mMT. Consequently : Angle 2 a = angle 26 + angle M Tm, or : Angle MTm = 2 (angle a angle b). Similarly, from the triangle mMP' : Angle P' = angle a angle b. But arigle P' = angle P, because their sides are perpendicular, each to each. Therefore, Angle P = angle a angle 6. And it follows that : Angle MTm = 2 angle P. Q. E. D. 393 ASTRONOMY Note 18. Longitude Determination (p. 158). A reference to Fig. 106 (p. 364) will show how the longitude may be computed from the sun's altitude, measured with the sextant. In the figure, if S f is the sun, the arc AS' is the measured altitude. If this be subtracted from 90, we have the arc ZS', or angular zenith distance of the sun. The arc S'D is the sun's declination, and may be ascertained for the date of the observation from the nautical almanac. Subtracting this declination from 90 makes known the arc PS', or the angular polar distance of the sun. The ship's latitude is also supposed to be known; without it, the longitude cannot be computed. But the ship's latitude always is known, because the navigator will have determined it at noon, and can easily allow for any slight change in the ship's latitude since the last noon observation, since he knows the com- pass course he is steering, and the approximate speed of the ship. But the latitude is the arc PN in the figure, or the altitude of the celestial pole above the horizon. This latitude being subtracted from 90, gives the arc ZP, or the angular distance from the ce- lestial pole to the zenith. Thus these three subtractions from 90 make known the three sides of the spherical triangle ZPS'. It is a principle of trigonometry that any spherical triangle can be solved completely, and all its parts made known, if we know its three sides. Thus we find the spherical angle S'PZ, of which the vertex is at the pole, and which is measured by the arc DM on the celestial equator. But DM is by definition the hour-angle of the sun S' ; and the sun's hour-angle is the local apparent solar time. This need merely be corrected by applying the equation of time (p. 134) to obtain the local mean solar time of the ship, ready for comparison with the Greenwich time taken from the face of the chronometer by an assistant at the instant when the sun was observed for longitude by the navigator. Note 19. Moon's Distance (p. 169). Figure 114 shows how the moon's distance is determined. We shall assume, as a sufficiently close first approximation, and to make the problem easy to understand, that the two observatories are situated on the same meridian of terrestrial longitude, but very widely separated in latitude. One should be in the northern 394 APPENDIX hemisphere; the other in the southern. The observatories of Greenwich, England, and the Cape of Good Hope, for instance, satisfy these conditions quite closely. In Fig. 114, then, and 0' are the two observatories, the circle representing the earth. The arc 00' is known, for it is simply the latitude difference of the two observatories. The angle OCO' is equal to the arc 00' ; and [the lines CO and CO' are each known radii of the earth. Therefore, by simple trigo- nometry, we can solve the tri- angle OCO', and gain a knowl- edge of the distance 00', which is to be our base-line, and of the two angles COO' and CO'O. We next measure at both observatories simultaneously, with suitable astronomic in- struments, the exact lunar alti- tude, or angular elevation of the moon above the horizon, at the instant when the diurnal rotation has brought the moon to the celestial meridian. These simultaneous observations will be possible, because the moon will reach the meridian of both places at the same instant, since we have imagined our two observatories lying on the same meridian of terrestrial longitude, and therefore having the same celestial meridian over them in the sky. Having measured the moon's altitude above the horizon, we can at once find its angular distance from the zenith. For the latter point is always 90 distant from the horizon ; so that we obtain the angular zenith distance of the moon by simply subtracting its measured altitude from 90. Those two angular zenith distances, thus known from the measured altitudes, are the angles MOZ and MO'Z' in Fig. 114. Next we subtract these angles from 180, giving us the angles MOC 395 FIG. 114. Moon's Distance. ASTRONOMY and MO'C. From these we again subtract the angles COO' and CO'O, found above, thus obtaining values of MOO' and MO'O. These now make possible a trigonometric solution of the triangle MOO', of which we now know the base 00' and the two adjoining angles. Thus we get OM and O'M in miles. After that we can solve the two triangles COM and CO'M, since we know the length of the two sides CO and OM, as well as the included angle COM ; and in the other triangle we know CO' and O'M as well as the in- cluded angle CO'M. A solution of either triangle gives us CM, the distance from the center of the earth to the moon. It is scarcely necessary to add that the ideal condition here assumed as to location of observatories does not exist in fact. But a slight divergence from this condition in no way impairs the prin- ciple of the method; it merely adds a certain additional com- plexity to the trigonometrical calculations. Note 20. Lunar Parallax (p. 169). Figure 42 shows that the moon's parallax and distance are con- nected by a very simple trigonometric formula : f sin parallax = -^- = fLM distance of moon This formula shows that we can calculate the parallax if we know the distance, or the distance if we know the parallax. The two are closely related; astronomers frequently speak of measuring the parallax of the moon or other heavenly body, when they merely mean a measurement of its distance. Note 21. The Moon's Mass (p. 175). Figure 115 is intended to make this matter clear. S is the sun ; the circle is the annual terrestrial orbit. When the center of gravity is at Ci, the earth at EI, and the moon at MI, the sun will appear from the earth projected in the direction Si. This is exactly the same as would be the case if there were no moon, for then the earth would itself be at C\. But when the center of gravity is at C 2 , the earth will be at E 2 ; and the sun will be seen projected in the direction $ 2 ', instead of $ 2 , which is its direction as seen from C 2 , and which would be its direction from the earth if there were no moon. 396 APPENDIX Thus the sun will be seen a certain angular distance in advance of its proper position ; and a half-month later it will be similarly retarded. The total range is 12", so that the angle S 2 SS 2 is 6". Therefore, in the triangle C 2 SE 2 , we know the angle C 2 SE 2 to be 6"; and we know the two sides C 2 S and E 2 S, the dis- tance from the earth to the sun, which can be meas- ured. Solving the triangle, we find the side C 2 E 2 to be about 2880 miles. We then form the propor- tion : E 2 C 2 : M 2 C 2 = moon's mass : Earth's mass ; from which we can compute the lunar mass, since the other quantities in the propor- tion are now all known. Note 22. Concavity of Moon's True Orbit with Respect to the Sun (p. 181). We can test this question by means of Fig. 116. It is evident from a mere glance at Fig. 46 (p. 181) that there is no doubt as to the concavity of the moon's orbit toward the sun at the time of full-moon, shown at Ms. Difficulty arises only in the case of the new-moon phase, shown at MI and M 5 . Therefore, in Fig. 116, we shall examine especially the new-moon phase. Let Ei t MI, and S be positions of the earth, moon, and sun at the time of new-moon. Let EiE 2 be a portion of the earth's orbit around the sun ; and let the small circles represent the lunar orbit around the earth. While the earth moves from E\ to E 2 , we may suppose the moon to move around the earth from C to M 2 . In other words, if the moon did 397 FIG. 115. Mass of the Moon. ASTRONOMY not revolve around the earth, it would be at C when the earth reached E 2 . Designate the angle EiSE 2 by the letter 0, and let r e and r m represent radii of the earth's orbit around the sun and the moon's orbit around the earth. Finally, let EiT be a tangent to the earth's orbit at El] draw E 2 P perpen- dicular to #1$; E 2 B parallel to EiS; and M 2 A parallel to E^T. If we let be a small angle, MiM 2 will be a small part of the moon's path near new-moon : it will evidently be con- cave towards the sun if M 2 is farther from the tangent EiT than is M\. While the moon was moving from MI to M 2 the entire lunar orbit fell away from the tan- gent the distance E\P ; but the moon rose toward the tangent a distance nearly equal to AB. Therefore the moon recedes from the tangent a total dis- tance of EiP AB. Now we have, evidently : FIG. 116. Moon's True Orbit. But: Also : EiP = r e - r e cos 6, AB = r m r m cos M 2 E 2 A. M 2 E 2 A = M 2 E 2 C + CE 2 A. (1) (2) M 2 E 2 C = 13 6, because the moon's angular motion in its orbit is about 13 times as fast as the earth's (p. 161) ; and : CE 2 A = 6, because AB is parallel to E& It follows that: M ^ A = 14 0. and, from equation (2) : AB = r m - r w cosl40. (3) PHILOSOPHISE N A T U R A L I S PR INC I PI A MATHEMATICA Autore J S N E ffTO N, Inn. Coll. Cantab. Sac. Mathefcos Profeflbre LncaftanO) Be Socictatis Regalis Sodali. IMPRIMATUR- S. P E P Y S, Reg. Sac. P R JE : S E y*iii 5. 1686. L N D I N /, fuflii Soeietatif Regime ac Typis Jofefhi Streater. Profiat apud plures Bibliopolas. Anno MDCLXXXVIL PLATE 32. Title-page of Newton's Principia. APPENDIX A simple calculation, using the value = 1 ( r 2 = 93,000,000, gives : EiP = 16,000 miles, AB = 7130 miles. r m = 240,000, It follows that the moon recedes from the tangent about 8870 miles in one day, while the earth is moving about 1 in its orbit around the sun. This proves that the moon's true orbit is con- cave towards the sun, even at the time of new-moon. Note 23. Law of Areas (p. 186). Figure 117 shows how the point P 3 ' is found. Draw parallel and equal to P 2 P 2 '. Then the actual motion of the planet in the second second will take place along the diagonal P 2 P 3 ' of the parallelogram P 2 P 3 P 3 'P 2 '. This theorem of the " parallelogram of forces" is demonstrated in works on elementary physics : l perhaps the easiest way 'to understand it is to notice that P 3 ' is point to which P 2 must go, if it actually completes separately the two motions P 2 P 2 ', and P 2 'P 3 ' equal and parallel to P 2 P 3 . FIG. 117. Law of Areas. Note 24. Law of Areas (p. 186). 2 We have still to prove the triangles SPiP 2 and *SP 2 P 3 ' equal in area. Referring again to Fig. 117, we see that the triangles $P 2 P 3 ' and $P 2 P 3 are equal, since they have the same base P 2 , and their altitudes are equal because their vertices P 3 and P/ lie on the line P 3 P 3 ', which is parallel to P 2 *S. And we have already found the triangle $P 2 P 3 equal to SPiP 2 . Therefore the triangle SP 2 P 3 ' is also equal to PiP 2 . 1 Figure 117 may be found in the first edition of Newton's immortal Prin- cipia, of which the title-page is reproduced as Plate 32. The president of the Royal Society, whose name appears on the title-page as having author- ized the printing, is the famous diarist. On p. 13 of the Principia ap- pears Corol. I : " Corpus viribus conjunctis diagonalem parallelogrammi eodem tempore describere, quo latera separatis." 2 On p. 37 of the same work of Newton appears Prop. I, Theorem I : "Areas quas corpora in gyros acta radiis ad immobile centrum virium ductis describunt . . . esse temporibus proportionates." 399 ASTRONOMY Note 25. Harmonic Law (p. 188). It would carry us too far afield in mathematical astronomy to give here the demonstrations by which Kepler's three laws may be derived from Newton's single law ; but there is little difficulty in considering by elementary methods the special case of a circular planetary orbit. The circle is, in fact, a close approximation to the actual planetary orbits in the solar system : none of these orbits are very much flattened from the circular form. We must first investigate the nature of the solar attractive force. In the case of a circular orbit this force is necessarily constant under Newton's law, because the planet is always at the same distance from the sun. Now consider the accompanying Fig. 118. Let PP' be a very short arc of a circle, whose center is at S. Draw the diameter PD and the chord P'D ; and let fall the perpendicular P'C upon PD. Draw the chord PP', the tangent PP"; and let fall the perpendicular P'P" upon PP" from P f . Then, from the similar right-angled triangles PP'C and PP'D, we have : FIG. 118. Solar Attraction. or: PP' : PC = PD : PP', PP 72 PC = PD Now let our circle be a planetary orbit, with the planet at P, the sun at S ; and suppose that in one second of time the planet would move along the orbit to P' . We may consider this very short arc PP' coincident with its chord PP'. From the principle of the parallelogram of forces (p. 399), the actual motion PP' may be regarded as the resultant of two motions : PP", which would be the planet's actual motion from P in a second if the solar attraction were to cease suddenly ; and 400 APPENDIX PC, which would be the planet's actual motion in a second if attraction toward the sun operated alone. Now PP' is the planet's velocity in its orbit per second, which we shall call V ; and PD is twice the radius of the orbit, which latter we shall call r. Let us also designate the distance PC by the letter x, and consider all distances to be measured in miles. Then, from the geometry of the figure, as we have just seen : But as we have said, PC or x is the distance the planet would move or "fall" toward the sun in a second, if the solar attraction acted alone, without any additional orbital or tangential impulse derived, perhaps, from the original catastrophe by which the planet was brought into existence. The question now is : How great must be the solar attractive force to cause a planet to fall from a position of rest at P through the distance PC or x in a second ? This raises the question of how forces are measured. What is a suitable unit of force? Now the solar attraction is not applied suddenly as a single impulse ; it is applied continuously. Conse- quently, the planet would fall the short distance x toward the sun with a uniformly increasing velocity, faster and faster, but begin- ning with zero velocity at P. Its average velocity would be attained halfway between P and C. But the actual distance it would move in a second is of course the same as if it traveled constantly with its average velocity. And as it would fall a distance x miles in a second, its average velocity must be x miles per second. Therefore it would be moving with the velocity x miles per second when halfway between P and C ; and upon reaching C its velocity would have increased to 2 x miles per second. But in astronomy, as in mechanics, our units are so chosen that force is always measured by the quantity of velocity accumulated in a second, multiplied by the quantity of mass in the moving body. The velocity thus accumulated in a second is called ''acceleration" ; and as the velocity of the falling planet increased from zero to 2 x miles per second, the acceleration produced by the solar attractive 2D 401 ASTRONOMY force must be represented by the number 2x. Calling this ac- celeration /, we thus have : /=2z; and this, combined with equation (1), gives : 172 / - V -- (2) Now the whole circumference of the circular orbit is 2 irr ; and the planetary orbital velocity V is of course equal to the circum- ference divided by the period of orbital revolution. It follows that if we call this period t, expressed in seconds of time, we shall have: and, therefore, from equation (2) : PI 111 ijjj:. f=^r- \ (3) If we now apply equation (3) to two separate planets, indicating by subscript numbers quantities belonging to the first and second of the two planets, we shall have : or : But we know from Newton's law that the attractive forces exerted by the sun on two different planets, if of equal mass, will be inversely proportional to the squares of the distances separating those planets from the sun. This may be written thus : /i:A-tf:ri*; or: ' s-j <5> Equating the right-hand members of equations (4) and (5) gives : *i 2 : tf = 402 APPENDIX This is the third (or harmonic) law of Kepler, which is therefore thus demonstrated as a consequence of Newton's law in the case of circular orbits. A similar proof is possible, by the aid of the higher mathematics, without this assumption as to the form of the orbit; but a small correction is always required, because we have taken the planets to have equal masses. Still retaining our circular orbit formulas as a sufficient first approximation, we are now in a position to understand Newton's famous test as to whether the force of gravitation observable on the earth also extends outward as far as the moon. We shall present this test here in a somewhat modernized form, based on the formulas just obtained. Resuming our equation (3), we have the acceleration exerted by the earth upon the moon : in which r is now the distance from the earth to the moon, and t the moon's sidereal period (p. 161). This equation is correct, if the Newtonian law of gravitation extends to the moon, and not other- wise. Newton's test now consists in comparing the value of / calculated by means of equation (6) with its value easily obtained by another method. It was known from laboratory experiments, even in the time of Newton, that the earth attracts an object situated on its surface with a force which is called the " force of gravity," and which produces an acceleration designated by the letter g in physics. It is also known that the earth's attraction upon any object exterior to it acts as if the entire mass of the earth were concen- trated at its center. 1 Now the distance from the earth's center to an object on its surface is equal to the earth's radius, and may be designated by R ; while the distance from the earth's center to the moon is r. It follows that if the earth's attraction varies inversely as the square of its distance from the object attracted, as postulated by Newton, we may write the following simple proportion involving 1 This was demonstrated by Newton. 403 ASTRONOMY /, due to the earth's attraction upon the moon, and g. due to the earth's attraction upon surface objects : f:g= 7* : R?' from which we have at once : The values of / in equations (6) and (7) must be equal, if both g and / result from the same identical law of Newtonian gravitation. Equating these quantities gives : or : (8) In this equation, r is the moon's distance from the earth, which we here suppose expressed in feet; and t is the moon's sidereal period, in seconds of time. Let us then calculate g, and ascertain whether it agrees with its known value derived by physicists from laboratory experiments. The moon's sidereal period is 27 d 7 h 43 m 11.5 s , or 2360591.5 seconds. The moon's distance, r, is 238,840 miles, or 1,261,075,200 feet. The earth's radius is 3858.8 miles, or 20,902,464 feet. The value of TT is 3.1416. Making the calcu- lation by means of logarithms, the above data give, by the aid of equation (8) : g = 32.5, which is in very close accord with the value of g found directly by experiment in the physical laboratory. It is a most astounding thing that a series of quantities can thus be brought together, as it were, from various parts of the solar system : the moon's distance determined by astronomic observations at Greenwich and the Cape of Good Hope (p. 395) ; the earth's radius by triangu- lation measures (p. 97) ; the moon's period by noting the interval between distant full-moons (p. 162), it is astounding that these heterogeneous quantities, thus determined by direct observation, can be combined by a simple formula based on that wonderful law 404 APPENDIX of Newton, and made to produce the identical value for g which we obtain by terrestrial laboratory observations, quite without using astronomic material. There could not be a more striking proof of the unity of science ; nor can any doubt remain that the same force of gravity which controls experiments on the earth, also controls the moon's orbital motion. Note 26. Planet's Mass (p. 205). Let us suppose once more that orbits are all circular. Consider- ing the satellite orbit, we found, when discussing Newton's test of the law of gravitation by means of the moon, that the accelera- tion due to the attractive force toward the center of the orbit may be represented by the equation (p. 403) : where r is now the radius of the satellite's orbit in miles, and t its period of revolution. In this equation, / is due to a continuously acting attractive force toward the planet situated in the center of the orbit, supposed circular. It is easy to obtain another expression for this force. We have at once, from Newton's law of gravitation (p. 376), that the attrac- tion existing between the planet and the satellite is proportional to the product of their masses, and inversely proportional to the square of the orbital radius. If we let M indicate the planet's mass, and m that of the satellite, this force is : n Mm frr-'j r 2 where G is a constant depending on the units adopted for linear distances, etc. Now this Newtonian force produces the acceleration / in the planetary mass m ; and since force is measured by the acceleration produced, multiplied by the mass moved, it follows that : f r J = tr r 2 405 ASTRONOMY and this is the acceleration due to the attraction of the planet on the satellite. In an exactly similar way, we can show that the satellite pro- duces an acceleration of the planet equal to : so that the total acceleration existing between the two bodies is : If we now equate this value of the acceleration to that given in the equation for /, we have : = r r 2 t 2 or : Let us next apply this equation to two planets, each having a satellite, and indicate by subscript numbers quantities belonging to the two planets. We thus easily obtain the proportion : or: With the help of this general proportion, we can now find the planet's mass as compared with that of the earth. We need only let the subscript 1 refer to the earth and moon, and the subscript 2 to the planet and satellite. Then everything is known in the proportion except M 2 + -w 2 , if we have determined by direct observation the distance and period of the satellite with respect to its planet. It is to be noted that this method gives only the sum of the masses of the planet and its satellite, not the mass of the planet alone. But this is of minor importance, since the satellites 406 APPENDIX are almost always very small compared with their planets : and, in any case, it is the combined mass of the system, including both planet and satellite, that we really need to know. For it is this combined mass which pulls upon other bodies in space ; and it is the pulling force upon such other bodies which must be used in any further calculations relating to orbits, etc. When a planet has no satellite, as in the case of Venus and Mer- cury, we cannot employ the above simple and accurate method. We must then have recourse to a mathematical discussion of the slight perturbations the planet produces in the observed motions of other bodies in the solar system. These perturbations, of course, depend on the planet's mass, being greater for a massive planet than for a small one ; and consequently the planetary masses must admit of numerical evaluation from the observed perturbative effects they produce. Unfortunately our knowledge of the mass of Mercury is still incomplete ; that of Venus, however, is known with some precision. The mass of a planet once determined, it is easy to calculate the force of gravity on the planet's surface, its Superficial Gravity, as it is called. If we designate by P the planet's radius in terms of the earth's radius, and by g the planetary superficial gravity, analogous to the customary designation of the force of gravity on our earth's surface, we have at once, from Newton's law of gravi- tation : a- M 9~pi> where M is the planet's mass in terms of the earth's mass. To ascertain the planet's density in comparison with that of our earth, we proceed thus : We know, in general : Mass = Volume X Density. Therefore we have for the earth's mass M e : M e = FA, where A e represents the terrestrial density, and V, the earth's volume. And for the planet we have : M P = FA, 407 ASTRONOMY Consequently : A P F P = M v A.F. M.' But, again using P to indicate the planet's radius : 17 L _ D3 F. Therefore, if we take the mass of the earth as unity : If we wish the actual specific gravity of the planet, compared with water, we must substitute for the A e the value 5.53, as de- termined by means of the Cavendish experiment (p. 110). Note 27. Synodic and Sidereal Periods (p. 209). Let us indicate by / sld and E siA the sidereal periods of Jupiter and the earth, each expressed in mean solar days. E M , for in- stance, is then 365J, approximately. Then, regarding both orbits as circular, and the motions uniform, the earth in one day will pass 360 over a fraction of its total orbit represented by ; and Jupiter 360 will pass over a fraction represented by These two fractions Aid are not equal : if we take the difference : 360 360 this quantity will be the angle by which the earth and Jupiter fail to lie in a straight line, as seen from the sun at the end of one day after the beginning of Jupiter's synodic year (see Fig. 55, p. 208). This quantity is therefore, by definition, Jupiter's daily synodic motion. But if Jupiter's entire synodic period be represented 360 by J sy n, its daily synodic motion will also be Equating this " syn with the above value of the same quantity, we have : 360 360 360 . T ~ 1? T ' /syn ^sid "sid 408 APPENDIX or: 1 1 1 ^syn -^sid /sid By means of this equation, Jupiter's synodic period may be calculated from his sidereal period, and vice versa; for E M is known to be 365J days. Note 28. Periods of Inferior Planet (p. 210). The synodic motion, as in the case of a superior planet, again depends on the earth's orbital motion as well as on that of the planet. As before, the daily sidereal motions of Venus and the S60 S60 earth may be represented by and The difference will be Vsid #sid the daily synodic motion of Venus, supposed seen from the sun. This quantity is : 360 360 F 8ld E m ' Thus the formula for the daily synodic motion of an inferior planet is perfectly analogous to that for a superior planet, except that the terms are now interchanged. This is of course due to the fact that the superior planet has a slower angular motion around the sun than the earth, while the inferior planet has a faster angular motion. But, as before, if F 8yn be the synodic o/^r\o period of Venus, the daily synodic motion will be - ; and we 'syn have : 360 360 360 il It follows that for any planet whatever the reciprocal of the synodic period is always equal to the difference between the reciprocals of the planet's sidereal period and the earth's, sidereal period of 365J days. Note 29. Table of Periods (p. 211). It will be of interest to calculate some of the numbers in the table (p. 211) by means of the formulas in Notes 27 and 28. We find: 409 ASTRONOMY RECIPROCAL OF SIDEREAL PERIOD RECIPROCAL OF EARTH'S SIDEREAL PERIOD DIFFERENCE SYNODIC PERIOD Mercury .011364 002738 008626 116 .001456 002738 001282 780 Uranus 000033 002738 002705 370 In computing the numbers in this table, all periods have been reduced to days ; and the numbers in the last column are recipro- cals of those in the column headed " Difference." It is at once clear from this little calculation how the peculiarities of the table of periods arise. As the sidereal periods of the outer planets increase, the reciprocals of these periods must diminish. Consequently, these reciprocals must gradually approach zero, and the numbers in the column " Difference " must approach the value .002738. So the numbers in the final column of synodic periods must approach the value 365 J, or the earth's period. This is just what we should expect. For the outermost planets remain practically stationary for many days among the fixed stars, and must therefore have a conjunction every time the earth goes around its orbit, or very nearly so. The effect of their own slow orbital motion on their synodic motion is necessarily very slight. Note 30. Greatest Elongation, Mercury and Venus (p. 212). In Fig. 58 (p. 212) the triangle SEV is right-angled at V. We can therefore calculate the angle SEV, which is the required greatest elongation angle, by means of the formula : sin SEV sin of greatest elongation SE' distance of planet from sun distance of earth from sun Let us make the calculation for Mercury. The orbit of this planet is more flattened than any other in the solar system : the approximate distance of Mercury from the sun varies from 28.5 410 APPENDIX to 43.5 million miles. Obviously, the greatest elongation will be larger if it happens when the planet is in that part of its orbit which is farthest from the sun. We shall therefore make the calculation twice, using the two values just given for the distance from Mercury to the sun. We have : LEAST GREATEST Distance of Mercury 285 435 Distance of earth 930 930 1 4548 1 6385 1 9685 1 9685 Log sin greatest elongation 9 4863 9 6700 Greatest elongation 1751' 2753' From this calculation we see that Mercury can never attain an angular distance from the sun greater than 28, as seen projected on the sky from the earth ; and ordinarily its greatest elongation will be much less than 28. Note 31. Temperature of Mars (p. 226). The distance from Mars to the sun is about 1J times that from the earth to the sun. Therefore, if we assume the heat radiated by the sun to diminish with the square of the distance, Mars 1 receives only as much heat as the earth, or f as much. We (1.5)' may also assume that, on the average, all planets radiate annually the same amount of heat they receive ; otherwise they would be- come continuously hotter or colder. Now we have a law of physics known as Stefan's law, which gives us an estimate of the quantity of heat a body will radiate at different temperatures. According to this law, calling the quantity of radiated heat Q, and the tem- perature F (Fahrenheit), we have: for the earth, Q e = (458 + F e ) 4 , for Mars, Q m = (458 + ^J 4 . But if each planet radiates as much heat as it receives, Q. 4 411 ASTRONOMY Therefore : (458 + F e Y = 9 (458 + F m y 4' Now for the average temperature of the earth, we may put F. = 60. Therefore : 458 + F m = -V$ (518) = 0.82 X 518 = 425. So that : F m = - 33 Fahrenheit. This result is of course uncertain, because we cannot be sure that Stefan's law is really reliable in the case of Mars and the earth. It has been tested in the laboratory only, and for a black body radiating its heat freely. Note 32. Saturn's Ring (p. 245). We have already found (p. 402) a formula for the accelera- tion toward the center of an orbit. It is : /=< But, according to Newton's law, / is inversely proportional to r 2 ; so that F 2 must be inversely proportional to r. Therefore if the rings are really a mass of satellites, the squares of their linear velocities are inversely proportional to their distances from the planet. In other words, the outside of the ring should revolve more slowly than the inside. The outside radius of the ring has been measured by the usual methods (p. 203) to be 86,500 miles ; the inner, 55,700. The square roots of these numbers are in the ratio of 1 to 1.24 ; while the ob- served linear velocities are in the ratio of 1 to 1.25. There is therefore a surprisingly close agreement ; and there can be no doubt that the various parts of the rings rotate in accordance with Kepler's harmonic law, and' are composed of satellite swarms. Note 33. Halley's Transit of Venus Method (p. 269). We must first show how to calculate the length of the chord in seconds of arc. In Fig. 119, let S, Vi, and E be the positions of the 412 APPENDIX sun, Venus, and the earth at the moment of inferior conjunc- tion. Let P be the synodic period (p. 208) of Venus, in days. Then Venus gains a whole revolution of 360 on the earth in P days, from the defini- tion of the synodic period. In one day Venus gains ?S. Therefore, if we let s ==== $; ^^^ FIG. 119. Halley's Method. the arc ViV 2 represent the synodic gain of Venus on the earth in a day, as seen from the sun, we have: Angle S = But in the plane triangle SEVi, we have : sin S:sinE = V 2 E:V Z S, since the sines of the angles of any plane triangle are proportional to the opposite sides. Therefore: sin = M sin S . V 2 E V Sf But the ratio ~ is known from the known relative lengths of the radii of the two orbits belonging to Venus and the earth (cf. p. 262). The angle S being also known, as has just been shown, it follows that we can calculate the angle E, which is the angular distance through which Venus advances across the face of the sun in one day, as seen from the earth. This angle is transformed into seconds of arc ; and the observers having found the fraction of a day required by Venus to traverse the observed chords, we find at once by proportion the lengths of the chords in seconds of arc. As soon as the lengths of the two chords SP and sp (Fig. 73, p. 269) thus become known in seconds of arc, the further proceedings are simple. For the angular semi-diameter, or radius, of the sun's disk is of course known also in seconds of arc (p. 118); conse- quently, it is possible to calculate the distances Sa and Sb (Fig. 73) in seconds of arc, and also their difference ab. We also know (Fig. 73) the ratio of the lines VA and Fa, because we know 413 ASTRONOMY the relative distances of Venus and the earth from the sun. Va is 0.723 if Aa is 1.000. Therefore : Va : VA = 723 : 277 ; and ab, in miles, is ff-f AB, provided, of course, that the distance AB is perpendicular to the plane of Venus' orbit. If not, it is easy to calculate the necessary correction. Now, knowing ab, on the sun, both in miles and in seconds of arc as seen from the earth, we easily FIG. 120. Haiiey's Method. obtain the distance of the sun. The simple Fig. 120 shows how this is done. Calling r the radius of the earth's orbit, or the distance from the earth to the sun, we have, from the right-angled triangle Eab, in which the line ab is on the sun, as usual : tan a6(seconds of arc) = or "(miles) ' (miles) = afrMM . tan ab (seconds) Note 34. Solar Parallax from the Aberration of Light (p. 271). Let us study somewhat in detail the action of light aberration. In Fig. 121, suppose that an observer at t has his telescope pointed in the direction tT ; that the earth, carry- ing the observer and telescope, is for the moment moving in its annual orbit in the direction tt r , with the velocity v miles per second. Now suppose light from a star at S reaches T at the moment when the telescope is in the position tT. And suppose this light travels with a velocity of V miles per second in the direction ST. Now indicate by the angle tTt' '. Then we may say, as it were, 414 FIG. 121. Solar Parallax from Aberration of Light. APPENDIX that if the velocities v and V are properly proportioned to fit the angle a, the light will "stay in the telescope tube" while the tube is moving from tT to t'T 1 '. We shall then have : tan = |. This equation signifies that a star at S will really appear pro- jected on the sky in the direction t'T'. In other words, the aberra- tion of light displaces the apparent position of the star on the sky through the angle a. And there is no difficulty in measuring this angle a: for the displacement of the star is always in the direction of the earth's motion, here tt'. And as that motion takes place in a nearly circular orbit, the displacement a must be in opposite directions at intervals of half a year (cf. p. 137). For the earth's orbital motion is, of course, reversed in direction at opposite points of the orbit. We have therefore merely to determine by observation the apparent declination of a star on the sky at intervals of six months. If a suitably located star is selected, the declination will be found to vary by twice the angle a ; about 41" of arc- From this we easily compute the solar distance. For the velocity of light, V, is known from laboratory experiments. With V and a both known, we can compute v with the equation just ob- tained, and v is the earth's linear velocity in its orbit. Thus it has been found that v is about 18.5 miles per second. This we have now to multiply by the number of seconds in a year, to get the linear circumference of the earth's orbit. Finally, dividing by 2 TT, we have the orbital radius, or the solar distance. Note 35. Sun's Mass (p. 291). To ascertain this quantity, we resume the formula which ex- presses the acceleration which the sun gives a planet. It is (p. 402): s = Y1 (velocity of planet in orbit) 2 _ r radius of planetary orbit In the case of the earth r is 93,000,000 miles. Assuming the orbit approximately circular, we can find its circumference by the formula Circumference = 2 irr ; 415 ASTRONOMY and this being divided by the number of seconds in a sidereal year, we find V, the linear orbital velocity of the earth, in miles per second. It is approximately 18J. Now, calculating/, we find : / = 0.233 inch. If we now let g, as usual, represent the constant of terrestrial grav- ity, we may write a simple proportion by the aid of Newton's law : /.. _ sun's mass . earth's mass (sun's distance) 2 ' (earth's radius) 2 This proportion is a direct consequence of Newton's law, which makes attractive forces proportional to masses, and inversely proportional to squares of distances. The earth's radius becomes the distance for terrestrial gravity