ELEMENTS OF ASTRONOMY, WITH Numerous Examples anti Examination papers BY GEORGE W. PARKER, M.A. OF TRINIT-.y COLLEGE, DUBLIN FOURTH EDITION LONGMANS, QKEEN, AND CO 39, PATERNOSTER ROW, LONDON NEW YORK AND BOMBAY 1902 A3 PRINTED AT THE BY PONSONBY & WELDRICK. PREFACE. HPHE present volume is intended to meet the wants of those Students whose knowledge of Mathe- matics is limited to an acquaintance with the Elements of Euclid, Algebra, and Plane Trigono- metry. In a few cases easy formulae in Dynamics are introduced, but the articles containing these may, if necessary, be omitted without a breach in the continuity of the work. Many of the examples have been selected from papers set to third and fourth year Students of Trinity College, Dublin ; while a considerable num- ber have been chosen with a view to assist those reading for Degrees in the London and Royal Universities. The book forms, to some extent, a connecting link between the many popular works on Astronomy and more advanced treatises on the subject. The author, therefore, hopes that it may be found useful, not only by those for whom it has been specially 179755 VI PREFACE. written, but also by many others among the general public. The author is much indebted to MR. PIERS WARD, M.A., LL.B., for his kind assistance in read- ing the proof-sheets. 13, TBINITY COLLEGE, DUBLIN, , July 19th, 1894. PREFACE TO THE SECOND EDITION. IN this Second Edition the work has been carefully revised and some new matter added. The number of the examples has also been increased. The Author desires to express his thanks to several friends who have suggested many of the additions, and more especially to MR. RAYMOND WALTER A. SMITH (Sch.), B.A., to whom most of the corrections are due. 13, TRINITY COLLEGE, DUBLIN, March 13th, 1900. CONTENTS. CHAPTER I. PROPERTIES OF THE SPHERE. DEFINITIONS. PAGE Fundamental Definitions. Apparent diurnal Motion of the Heavens. The Ptolemaic and Copernican Systems. Changes in the Sun's Decimation, 1 CHAPTER II. THE EARTH. Altitude of the Celestial Pole. Length of a Degree of Latitude. Magnitude of the Earth. Proofs of the Earth's Rotation. Foucault's Experiment, 1 CHAPTER III. THE OBSERVATORY. The Transit Instrument. Various Errors nd Adjustments. The Meridian Circle. Regulation of the Clock. Equatorial and Micrometer. Alt- Azimuth Instrument, . 33 CHAPTER IV. ATMOSPHERIC REFRACTION. Effect on the Apparent Position of a Heavenly Body. Law of Refraction. The Constant Coefficient determined hy various methods. Oval appear- ances of Sun and Moon, 53 Vlil CONTENTS. CHAPTER V. THE SUN. PAGE The Sun's apparent Annual Path. Proofs of the Earth's Annual Motion. The Seasons. Heat from the Sun. Sun Spots. Eotation of Sun. Twilight. Its duration determined, 62 CHAPTER VI. THE MOTIONS OF THE PLANETS. THE SOLAE SYSTEM. Definitions. Phases and Brightness of the Planets. Periodic Times deter- mined. Kepler's Laws. Direct and Retrograde Motions. Rotations of Planets. Transits of Venus and Mercury. Comets and Meteoric Showers. Lengths of the Seasons. Eccentricity of Ellipse, . . 78 CHAPTER VII. PARALLAX. Law of Diurnal Parallax. Effect on the apparent position of a body. Horizontal Parallax of the Moon or a Planet determined. Finding the Sun's Parallax. Magnitudes of Moon, Sun, and Planets. Annual Parallax Bessel's Method. Annual Parallax of Jupiter, . . .114 CHAPTER VIII. DETERMINATION OF THE FIRST POINT OF AEIES. PRECESSION, NUTATION, AND ABERRATION. Flamsteed's Method for finding the Right Ascension of a Star. Precession of the Equinoxes its Period and Physical Cause. Nutation. Velocity of Light. Law of Aberration. General Effect of Aberration, . . 134 CHAPTER IX. THE MOON. The Moon's Phases. Determination of her Synodic Time and Sidereal Period. Metonic Cycle. The Moon Rotates round an axis. Librations. Harvest Moon explained geometrically. Revolution of the Moon's Nodes. Height of a Lunar Mountain. Physical State of the Moon, . . 149 CONTENTS. IX CHAPTER X. ECLIPSES. PAGE Causes of Lunar Eclipses. Breadth of the Earth's Shadow at the Moon. Solar Eclipses. Conditions for Eclipses. Ecliptic Limits. Saros of the Chaldeans, 169 CHAPTER XI. TIME. The Equation of Time. Its Causes. Unequal Lengths of Morning and Afternoon. Local Time. Mean and Sidereal Time. Reduction of Time. The Julian and Gregorian Calendars. The Sun-Dial, .... 184 CHAPTER XII. APPLICATION TO NAVIGATION. Hadley's Sextant. Latitude determined by various Methods. Mean Local Time calculated. Longitude by Chronometers. Lunar Method of determining the Longitude, 202 CHAPTER XIII. THE FIXED STAJBS. SPECTRUM ANALYSIS. Star Magnitudes. Clusters and Nebulae. Proper Motions. Double, Binary, and Variable Stars. The Solar Spectrum. Surface of the Sun and Solar Prominences. Spectra of Stars and Nebulae. . . . .214 CHAPTER XIV. MASSES OF THE HEAVENLY BODIES. The Mass of the Earth Maskelyne's Method and the Cavendish Experiment. Masses of the Sun and Planets. Masses of Binary Stars. Note on the Celestial Globe, 227 EXAMINATION PAPERS AND MISCELLANEOUS QUESTIONS, .... [l] INDEX, [17], b CHAPTER I. PROPERTIES OF THE SPHERE. DEFINITIONS. 1. Definition. A SPHERE is a solid bounded by one surface, such that all right lines drawn to that surface from a certain point within it are equal. That point is called the centre. A sphere might also be defined as being generated by causing a circle to revolve round one of its diameters. Thus if a circular hoop or ring be taken, and if, having fixed two diametrically opposite points in its circumference, we make it revolve round the diameter joining those points, we see that the circumference will trace out the surface of a sphere. 2. Every plane meeting a sphere cuts its surface in a circle. For, let DEF represent a plane section of a sphere. From 0. the centre of the sphere, let fall 00' perpendi- cular to the plane DEF\ take any point P in the circum- ference DEF. Now, since 00 is perpendicular to the plane DEF, it must be perpendicular to OP which lies in that plane ; therefore (EUCLID, i. 47), But R is of constant length, being the radius of the sphere, and p is constant ; therefore r is of constant length, and PROPERTIES OF THE SPHERE. [CHAP. I. therefore as P changes its position along the curve LEF, its distance from 0' is constant. Therefore DEFm\Lst be a circle. The reader can illustrate this experimentally by taking an apple as nearly spherical as possible, and, with a knife, cutting a section through. On examining the interior of the apple thus brought to view, he will find that the shape of the section is circular. 3. Definition. A great circle on the surface of a sphere is that whose plane passes through the centre of the sphere. Thus the circle AmnB* (fig. 2) is a great circle. A small circle is such that its plane does not pass through the centre of the sphere. Thus (fig. 1) DBF is a small circle. It is evident that all great circles on a sphere are equal in magnitude, but small circles are not, as they vary from being almost great circles till they dwindle down to mere points. Definition. If at the centre of a circle on a sphere a perpendicular be erected to its plane and produced out both ways, the two points in which it cuts the sphere are called the poles of that circle. Thus, if at the centre we erect PQ perpendicular to the plane of the great circle AB, the two points P, Q are the poles of the great circle AB. Definition. Great circles A which pass through the poles of another great circle are called secondaries to that great circle. Thus, if through P and Q (fig. 2) great circles PmQ 9 PnQ be drawn, these FIG 2. circles are secondaries to the great circle AB. CHAP. I.] DEFINITIONS. 3 N.B.A. great circle and its secondaries cut at right angles. Also the arcs of secondaries Pm and Pn drawn from P to AB are equal, and each is equal to a quadrant = 90. Apparent Diurnal Motion of the Heavens. Celestial Sphere. 4. To an observer situated in the middle of a level plane the appearance which the heavens present to him is that of a vast hollow dome of hemispherical shape, whose base appears to rest on the plane on which he stands, meeting it in a circle. This circle which bounds his view of the heavens is called the sensible or apparent horizon, the plane in which he stands being the plane of the sensible horizon. If now the time of observation be a cloudless night, there will be seen, apparently spangled over the concave surface of this hollow dome, a great number of shining bodies or specks of light called stars. By far the greater number of these bodies always main- tain nearly the same situation with respect to each other ; that is, if the angle which any two subtend at the eye of an observer be measured, it is found never to undergo any alteration in magnitude, except such changes as are of so minute a character as only to be observable after considerable intervals of time. These bodies are therefore called fixed stars. To even the most careless observer it will at once appear that all these fixed stars appear to move across the sky in a common direction without altering their positions relative to one another. Some rise in the eastern horizon, ascend in the sky, and, after describing arcs which seem circular, they sink below the western horizon, only to reappear again in the east on the next night at the same point as before, the whole revolution of each being completed in about 23 h 56 m 4 s . Many others make smaller circuits in the sky such that they never reach the horizon, and can therefore be seen throughout their whole course. The paths of these stars also seem circular, the time taken by each to complete a revolution B 2 4 PROPERTIES OF THE SPHERE. [CHAP. I, being about 23 h 56 m 4 s , the same as for those which rise and set. Moreover, their apparent diurnal paths all seem to be round one common point as pole. That point is called the celestial pole. Those stars which circle so closely to the pole that they do not rise and set are called circumpolar stars. In order to more accurately observe this apparent diurnal motion of the heavenly bodies, it is necessary to have a tele- scope suitably mounted, called an equatorial, such as is to be found in every observatory. This telescope, of which a full description will be given later on, can be directed to any part of the heavens, so that any star desired may be brought into the field of view. Moreover, by means of a clockwork arrangement, the field of view can be made to move uni- formly round the celestial pole in the same direction as the stars appear to move, completing its revolution in 23 h 56 m 4 s . It is now found that whatever star is chosen, once the revolving clockwork apparatus is set going, it is possible to keep it in the field of view throughout its whole course above the horizon. If the reader bear in mind that every point in the field of view of the telescope moves uniformly in a small circle round the celestial pole, and that the time of complet- ing a revolution is 23 h 56 m 4 s , he will at once arrive at the following conclusions : (1) The stars appear to move in small circles round the celestial pole. (2) This apparent motion is uniform. (3) The time occupied in completing one revolution is the same for all, viz. 23 h 56 m 4 s . 5. Celestial Sphere. If two observers be situated at diametrically opposite points of the earth, say in England and somewhere about New Zealand, each will see an appa- rently concave hemispherical surface of the heavens having a celestial pole. If, therefore, the observer could see CHAP. I.J DEFINITIONS. 5 Leavens at a glance, the appearance would be that of a com- plete sphere, on the concave surface of which all the heavenly bodies would seem to be situated, and revolving round two diametrically opposite points, the north and south celestial poles. This apparently spherical surface of the heavens is called the celestial sphere. It is usual, from a mathematical point of view, to regard the celestial sphere as a sphere of infinitely great radius compared with any distance on the earth, so that when we say that the earth occupies a position in the centre of this imaginary sphere, we mean that we may regard it as a mathematical point, i.e. of no dimensions. The Sim, Moon, and Planets. 6. The Sun. Let us now change the time of observation to the day-time, and see if the sun shares in the diurnal revolution common to the fixed stars. At first sight we might come to the conclusion, that his apparent diurnal motion is exactly the same ; -iS^rises in the east, describes an arc in the sky, sets in the west, and reappears again in the east next morning. But there is a difference ; the time taken by him to make a complete revolution is not 23 b 56 m 4 s , but 24 hours, or about 4 minutes longer than for the fixed stars. This can be verified roughly by waiting any day until one edge of the sun is in a line with a vertical wall, or, better still, with two vertical strings hung with weights at the ends of them. The interval which elapses until he gets into the same position next day is then noted, and is found to be about 24 hours ; whereas, as we have seen above, the same experiment applied to a fixed star-would give 23 h 56 m 4 s . The apparent position of the sun among the fixed stars cannot therefore be the same from day to day, but must be slowly shifting from west to east, so that he, as it were, hangs 6 PROPERTIES OF THE SPHERE. [CHAP. I, behind the fixed stars in his diurnal revolution from east to west, taking a slightly longer time to complete the circuit. If we could observe the fixed stars during the day-time with our naked eyes as can be done through an astronomical telescope, we could actually see the sun's slow change of position from west to east, and also that he returns to the same position among the fixed stars at the end of a period of time which is called a year. But we can demonstrate it without the aid of a telescope, thus : If we note each evening some group of stars which sets in the west some time after sunset, it will be seen, if tha observations be continued for some weeks, that the interval that elapses between sunset and the disappearance of the stars becomes shorter and shorter until eventually the stars set before the sun, and therefore cease to be visible in the early part of the night. If, however, we get up before dawn and look towards the eastern part of the sky we will see that these stars have risen before the sun. But if we continue our observations for 365 days, we shall find at the end of that time the sun in the same position relatively to the fixed stars as before, and the group in question will again be visible in the early part of the night. We thus say that the sun has two apparent motions : (1) A daily revolution from east to west in common with all the heavenly bodies. (2) A slow yearly revolution from west to east among the fixed stars. 7. The moon. Besides its apparent diurnal motion, the moon also appears to move from west to east among the fixed stars, but much more quickly than the sun, as it seems to complete a revolution relative to the sun and earth in a period of time which is called a month. CHAP. I.] DEFINITIONS. 7 8. The Planets. Besides the fixed stars, sun and moon, there are five other bodies visible to the naked eye whose apparent motions among the fixed stars are, as it were, so whimsical, that it is difficult to reduce them to any general laws. On this account they are called Planets or wandering stars. Sometimes they appear to move among the fixed stars in the same direction as the sun and moon, when their motion is said to be direct : sometimes in the opposite direction, when their motion is retrograde, and occasionally for a short time they appear stationary among the fixed stars. A fixed star to the naked eye burns with a twinkling light, a planet shines with a steady light. Also when a fixed star is examined through even the most powerful telescope it does not seem increased in size, the only difference being that its brilliancy is intensified. On the other hand, the disc of a planet appears enlarged when seen through a telescope. Four of the planets, Venus, Mars, Jupiter, and Saturn, are as bright or brighter than the most brilliant fixed star. If the planet appear shining in the south it is Mars, Jupiter, or Saturn. Yenus is an evening or a morning star ; in the former case it is seen in the west after sunset, in the latter case in the east before sunrise. The only other heavenly bodies visible at any time to the ordinary observer are comets and shooting stars or meteors, which complete the list of those bodies of which the follow- ing chapters give a more detailed description. 9. The Ptolemaic System. The different celestial phenomena mentioned above the diurnal revolution of all heavenly bodies, the yearly path of the sun, the monthly motion of the moon, and the apparently irregular courses taken by the planets were accounted for by Ptolemy, who lived in the second century after Christ, on apparently so 8 PROPERTIES OF THE SPHERE. [CHAP. I. satisfactory a basis that it was not until the sixteenth century that the true explanation was accepted. The whole celestial sphere was supposed to revolve round an axis passing through the north and south celestial poles, and called the celestial axis, the earth being in the centre. The sun, besides this daily revolution with the celestial sphere, was supposed to have a motion of its own in the opposite direction on the sphere describing a circle round the earth as centre once every year, the moon similarly com- pleting a circuit once every month. The retrograde and stationary stages in a planet's motion they accounted for in a rather ingenious way by supposing the planets to describe circles round the sun, which in its turn described a circle round the earth. The Copernican System. The true explanation, however, first given by Copernicus, and now so well known, shows that the diurnal revolution of the heavens is only apparent, and that it is really the earth which rotates in the opposite direction, from west to east, round an axis which, if produced out, would pass through the north and south celestial poles, thus causing the plane of the observer's horizon by its motion to uncover and bring into view new stars in the east, when these stars are said to rise, and to cover up and hide from view other stars in the west which are then said to set. Copernicus also referred the apparent yearly motion of the sun round the earth to a motion of the earth round the sun, and showed that the earth and planets form one system revolving round tho sun, from which they derive their light and heat. This explanation was at first received by astronomers with tho greatest suspicion, and it was only subsequent discoveries which placed it beyond any doubt ; we should therefore not take it for granted, but examine carefully the successive steps which led up to these conclusions. Although CHAP. I.] DEFINITIONS. 9 the diurnal revolution of the earth on its axis was not generally believed in until about three centuries ago, it is, however, not by any means a new idea. Cicero mentions that it was the opinion of Hicetas of Syracuse, who lived 400 years before Christ. Copernicus says that it was this statement of Cicero's which first led him to consider the earth's motion. 10. For the purpose of reference, the relative positions of the different fixed stars and the apparent path of the sun among them are mapped out on the surface of a globe, such that the arc joining the positions of any two stars on the surface of this miniature globe subtends the same angle at its centre as the two stars in question subtend at the eye of the observer. Such a celestial globe, the observer being supposed at its centre, would then serve as a representation of the appearance of the heavens. The reader should bear in mind that such a celestial globe only represents the angular distances of the heavenly bodies from one another, and not their distances from the earth : for the fixed stars are immensely further distant than the sun or planets, while being on the surface of a globe they are represented as being at the same distance from the observer. Definitions. (1) The great circle in which the plane of the horizon cuts the celestial sphere is called the celestial horizon. N.B. At sea, it is easy to observe the position of a heavenly body with respect to the horizon ; but on land, on account of the inequalities of the earth's surface, the horizon cannot be seen ; but we can determine it by taking a plane perpendicular to the direction in which a plumb line hangs, and note the position of the body with reference to that plane. Tfye surface of a small portion of liquid at rest, such as mercury, is also used by astronomers to determine the plane of the horizon. 10 PROPERTIES OF THE SPHERE. [CHAP. I. (2) If we imagine the direction of a plumb line produced upwards, that point in which it would cut the celestial sphere is called the zenith. (3) If we imagine the direction of the plumb line pro- duced downwards so as to cut the celestial sphere in the diametrically opposite point, that point is called the nadir. N.B. It is evident that the zenith and nadir are the two poles of the celestial horizon. (4) Celestial meridian. That great circle in the heavens drawn through the zenith and celestial pole is called the meridian. (5) Great circles drawn perpendicular to the horizon, i.e. secondaries to the horizon, are called verticals. That vertical drawn due east and west at right angles to the meridian is called the prime vertical. (6) The four points in which the meridian and prime vertical cut the horizon are called the four cardinal points the north, south, east, and west points. (7) The celestial equator is that great circle in the heavens whose plane is at right angles to the direction of the celes- tial pole. The north and south celestial poles are evidently the poles of the equator. The small circles described round the celestial pole by the stars in their apparent diurnal motion are all parallel to the celestial equator. It is evident since any two great circles bisect each other (having a common diameter) that one-half of the celestial equator is above the horizon and half below, so that if any star or other heavenly body be situated in the equator it will, in its diurnal revolution, remain equal times above and below the horizon, rising at the east point, and setting at the west point. CHAP. 1.] DEFINITIONS. 11 FIG. 3. DIAGRAM OF CELESTIAL SPHERE, THE OBSERVER BEING AT ABOUT THE LATITUDE OF DUBLIN. ASCB is the diurnal parallel of a star S, A being the point where the star rises, B where it sets, and C the point where it crosses the meridian. . . Observer. R . . North Point HmEn . . Horizon. jff . . South Point. Z . . Zenith. Wl East Point. P . . Qelestial Pole. n . . West Point. HZPR . . Meridian. 8 . . A Star. mQn . . Celestial Equator. ZSK . . Vertical through S. mZn . . Prime Vertical. LSPZ . . Hour angle of S. (8) Definition. The ecliptic is the apparent path, of the sun among the fixed stars in the course of a year. When this apparent annual path of the sun is traced out on the celestial sphere it is found that it can be represented by a great circle. Its name arises from the fact that if the moon in her monthly revolution happen to cross the plane of the ecliptic when it is full or new moon, there will be an eclipse, in the former case of the moon, and in the latter of the sun. Obliquity of Ecliptic to Equator. Equinoxes. 11. The angle at which the planes of the ecliptic and equator cut is about 23 28' which is called the obliquity of the ecliptic to the equator. These two great circles must 12 PROPERTIES OF THE SPHERE. [CHAP. I. intersect in two points, therefore on two days each year the sun is in the act of crossing the equator, and on those days his diurnal path almost coincides with the equator, rising due east and setting due west (fig. 3); one-half of his diurnal path is therefore above horizon and half below, and day and night are of equal duration all over the world. From this latter circumstance these two periods are called the Equinoxes, the two points of intersection of the ecliptic with the equator being called the equinoctial points. One of these points is called the first point of Aries (T), the other the first point of Libra (,) 9 because when these points were first named by the ancient astronomers they were in the constellations of Aries and Libra respectively. The sun is at the first point of Aries on the 21st March when crossing from the south to the north side of the equator ; this date is called the vernal or spring equinox, and is at Libra on the 23rd September, in his passage from the north to the south side of the equator, this date being the autumnal Equinox. The Signs of the Zodiac. 12. Ancient astronomers found by observation that the moon and planets were never at any time at a very great angular distance from the ecliptic; they therefore conceived an imaginary belt in the heavens extending for about 8 on either side of the ecliptic. Inside this space the moon and planets, and of course the sun were always to be found. They called this belt the zodiac from their imagining certain forms of animals situated within it, which they named the signs of the zodiac. There are twelve signs of the zodiac: these together with the symbols which represent them are as follows : Aries. Taurus. Gemini. Cancer. Leo. Virgo. T 8 n Si WSL Libra. Scorpio. Sagittarius. Capricornus. Aquarius. Pisces. -= m V? zz K CHAP. I.] DEFINITIONS. 13 Altitude, Azimuth. 13. The altitude of a heavenly body is its distance from the horizon measured on the arc perpendicular to the horizon drawn through the body (i. e. on the vertical drawn through the body). ' The azimuth of a body is the arc intercepted on the horizon between the foot of the vertical drawn through the body and the meridian. Thus (fig. 3) SK= altitude, and HK = azimuth of the star S. Of course it is immaterial whether we call RK or HK the azimuth of the star S, provided we mention whether we are measuring it from the north or south point. In northern latitudes the azimuth is generally measured east and west from the south point, and in southern latitudes from the north point. Thus if the arc HK= 30, the azimuth of 8 = 30 E. The arc SZis called the zenith distance of. the body, and is evidently the complement of the altitude. The position of a body on the celestial sphere with respect to the observer's horizon and meridian can be described by knowing its altitude and azimuth, but as the horizon of the observer is, owing to the earth's rotation, changing every instant, and, moreover, both the horizon and meridian are different for different places on the earth, therefore the altitude and azimuth of a heavenly body only describe its position at some particular instant and observed from a certain definite place on the earth. Declination and Right Ascension. 14. Instead of describing the position of a body with reference to the horizon we may refer it to the equator. The measurements by which its position is then indicated are 14 PROPERTIES OF THE SPHERE. [ CHAP ' * independent of the position of the observer on the earth, and do not change appreciably each instant, but, as we shall subsequently see, only after comparatively long periods of time. The declination of a heavenly body is its distance from the equator measured on an arc perpendicular to the equator drawn through the body. The right ascension is the arc of the equator intercepted between the firsLpwntfef Aries and the perpendicular to the equator drawn through the body. The right ascension is reckoned from T eastward from to 360. Thus (fig. 4) let JSQ re- present the equator, AB the ecliptic; draw a common secondary AQBP to both passing through P the pole of the equator (celestial pole) and P f the pole of the ecliptic. Then if S be the position of a heavenly body we have : FIG. 4. Sm = declination of body measured along PM. rm = right ascension of body measured along the equator. The arc SP is called the polar distance of the body and is evidently the complement of the declination. Celestial Latitude and Longitude. The position of a body may be indicated also with reference to the ecliptic. The latitude of a heavenly body is its distance from the ecliptic measured on a perpendicular arc to the ecliptic. The longitude is the arc of the ecliptic intercepted between CHAP. I.] DEFINITIONS. 15 the first point of Aries and a perpendicular arc to the ecliptic drawn through the body. Thus (fig. 4) Sn = latitude of S and T = longitude. The terms celestial latitude and longitude are applied to these measurements to distinguish them from terrestrial latitude and longitude with which they are not in any way connected. The longitudes of heavenly bodies are, like their right ascensions, measured from T eastward from to 360. Both the declinations and latitudes of heavenly bodies vary from to 90 on either side of the equator and ecliptic respectively. They are counted north or south according to whichever celestial pole happens to be on the same side of the great circle from which they are measured. \ Decimation Circles. Hour Angle. ^ Secondaries to the equator are called declination circles because it is on these circles that the declinations of heavenly bodies are measured. The angle which the declination circle through a star makes with the meridian is called the hour angle of the star, because when this angle is known we are able to calculate the time that must elapse before the star crosses the meridian or the time which has elapsed since it last crossed it, from the fact that the star completes a revolution of 360 round the celestial pole in 23* 56 m 4 8 . Thus (fig. 3) the angle SPZ = hour angle of star 8. Declination circles are on this account also called hour circles. Changes in the Sun's Declination during the year as he describes the Ecliptic. 15. At the spring equinox the declination of the sun is zero, he being at v (fig. 4). Each day, however, on account 16 PROPERTIES OF THE SPHERE. [CHAP. I. of his slow annual motion his declination increases until, some time about 21st June, he reaches his greatest declina- tion, viz. the arc BQ, (fig. 4). But BQ is the arc intercepted by the ecliptic and equator on their common secondary, and must therefore measure the angle between those great circles, which is 23 28'. Therefore the arc BQ = greatest declination of sun at mid- summer = 23 28' north. This period is called the summer solstice (sol, stare], because the sun before descending to = seems for some time to stand still. After midsummer the sun's declination gradually de- creases until at === (about 23rd September) it is again zero. Between 23rd September and the 21st of the following March the declination of the sun is south, reaching at mid-winter (21st December) a value AE which = 23 28' south. This period is called the winter solstice; therefore we have : The sun's declination at the vernal equinox = summer solstice = 2328'N, autumnal equinox = winter solstice =2328'S. During this period which is called a year the sun's right ascension and longitude increase from at vernal equinox to 360 immediately before the following vernal equinox, each being 90 on 21st June, 180 on 23rd September, and 270 on 21st December. It is hardly necessary to mention that the sun being in the ecliptic has his latitude always zero. N.B. When we speak of the sun's declination, &c., we mean that of the centre of the sun's disc. Tropics of Cancer and Capricorn. If on the celestial sphere we draw two small circles parallel to the equator, and distant from it 23 28' north I.] DEFINITIONS. 17 and south, these small circles will nearly coincide with the sun's apparent diurnal path on 21st June and 21st December. They are called the tropics because the sun seems to be on the point of turning at these periods. The northern circle is the Tropic of Cancer, the southern the Tropic of Capricorn. The Equinoctial Colure is the secondary to the equator passing through the equinoctial points. The Solstitial fjolure is the secondary to the equator passing through the solstices, and hence is also a secondary to the ecliptic. The altitude of a Star is greatest when on the Meridian. Let S represent the star in the meridian, and 8' its position at any other time. Join ZS' and PS'. Now since any two sides of a spherical triangle are together greater than the third, therefore we have j but PS' = PS, since a star always maintains the same dis- tance from the pole ; .-. ZS' + ZP>PS. Take away the common part ZP, therefore we get Z& c 18 PROPERTIES OF THE SPHERE. [cHAP. I. greater than ZS, that is, the zenith distance is least when on the meridian, and hence the meridian altitude is greatest. In the same way it can be shown that the depression of a body below the horizon is greatest when on the meridian. EXERCISES. \ 1. What are the altitude and hour angle of the zenith ? Ans. 90 : 0. 2. What are the declination and latitude of the celestial pole ? Ans. 90: 66 32' (90 -23 28') 3. How far is the pole of the ecliptic from the celestial pole, or, in other words, what is the magnitude of the arc PF in fig. 4 ? Ans. 23 28'. 4. What are the declination, right ascension, latitude, and longitude of ^ ? Ans. : 180 : : 180. 5. What point in the heavens has its declination, right ascension, latitude, nd longitude each equal to zero ? Ans. First point of Aries ( T ) . 6. If a certain star cross the meredian at 11 o'clock P.M. to-night, at what o'clock will it cross the meridian (1) to-morrow night; (2) 15 days hence, assuming the sun's change of right ascension throughout the year to be uniform ? See Arts. (5) and (6).) Ans, (I) About 10.56 P.M. (2) About 10 P.M. 7. At what hour will the same star cross the meridian a year hence ? Ans. 11 P.M. again. 8. A star is in the meridian 10 above the pole at midnight to-night, where will it be at midnight (1) six months hence ; (2) a year hence, sup- posing the sun's apparent motion in the ecliptic to be uniform ? 9. What is the sun's right ascension on 21st March, 21st 'June, 23rd Sep- ember, 21st December. Ans. : 90 : 180 : 270. 10. Calculate what would be the declination and right ascension of the sun on 21st April if the changes in these quantities were uniform throughout th year. Ans. 7 49' 20" N. : 30. 11. Making the same assumption as in the last question (1) Find at what ime the sun's right ascension should be 120 ; (2) at what time should his eclination be 15 38' 40" N. Ans. (I) 21st July. (2) 21st May or 21st July. N.B. The reader can, by reference to a celestial globe or the Nautical Almanac, see that the results obtained in Examples (10) and (11) are not the correct values of the right ascension and declination of the sun on the dates mentioned, which shows that the changes in these quantities throughout the year are not at all uniform. 12. What is the time of .sunrise and sunset at any place during the equi- noxes? A_ ns . About 6 A.M. and 6 P.M. 13. What is the hour angle of the sun at sunrise on 21st March ? Ans. 90. CHAPTER II. THE EARTH. 16. THAT the earth's shape is approximately spherical lias been known from the earliest times. It will not here be necessary to do more than mention the different reasons which lead us to this conclusion. They are : (1) The hull of a ship disappears first, which shows that the ship is sailing on a convex surface. (2) The outline of the earth's shadow, as seen on the surface of the moon during an eclipse, always seems an arc of a circle, and no body but a sphere can project a circular shadow in all positions. (3) The most conclusive proof, however, depends on the fact, which is found by observation, that equal distances gone over by the observer due north or south produce almost equal variations in the meridian altitude of any chosen star (or of the celestial pole). This could not happen except on the supposition that the earth is nearly spherical. Celestial Pole Constant in Direction. 17. The celestial pole being supposed to be situated at an indefinitely great distance away comp ared with any distance on the earth, therefore, as the observer changes his position on the earth's surface, the lines drawn from those posi- tions in the direction of the celestial pole are practically parallel. C2 20 THE EARTH. [CHAP. II. Earth's Axis. Terrestrial Equator. Terrestrial Latitude and Longitude. That diameter of the earth which is parallel to the constant direction of the celestial pole is called the earth's axis. The earth's axis cuts the surface of the earth in two points called the north and south poles of the earth. That great circle drawn round the earth whose plane is perpendicular to the earth's axis is called the terrestrial equator. Great circles drawn through the poles of the earth are called terrestrial meridians. Therefore, every place on the earth's surface may be supposed to have its meridian. The meridian of Greenwich is called the first meridian. The latitude of a place is its distance north or south of the equator measured on the meridian through the place. The longitude of a place is its distance east or west of the first meridian, and is measured by the number of degrees in the arc intercepted on the equator between the meridian of the place and the first meridian. All places situated on the same parallel to the equator have evidently the same latitude, and situated on the same meridian have the same longitude. Latitude is measured north and south from to 90, and longitude east and west from to 180. Corresponding to the Tropics of Cancer and Capricorn on the celestial sphere, we imagine two small circles on the earth parallel to the equator, one north the other south, and distant from it about 23 28' : these small circles are also called the Tropics of Cancer and Capricorn. The two small circles drawn round the north and south poles of the earth at DIRECTION OF POLE. CHAP. ,11.] ALTITUDE OF CELESTIAL POLE. 21 a distance of 23 28' are called the arctic and antarctic circles respectively. The portion of the earth's surface enclosed between the two tropics is called the torrid zone, between the tropics and the arctic and antarctic circles the temperate zones, and between the arctic and antarctic circles and the poles the frig id zones. 18. The altitude of the celestial pole at any place is equal to the latitude of the place. For let be the position of the observer ; EOQ, the meri- dian of the place, cutting the equator in E and Q. If OP represent the direction of the celestial pole as seen from 0> then the line CP drawn from C, the centre of the earth, in the direction of the celestial pole, will be parallel to OP (the pole being so far dis- tant). The horizon of the observer will be represented by a tangent plane OH drawn to the earth at 0. Then we have to prove that the angle 9 which is the altitude of the pole = the arc EO or the angle 0, which is FlG - 6< the latitude of the place. Since OP is parallel to CP, the angle a = the angle j3 ; but is the complement of a, and is the complement of j3 ; therefore 9 = 0, or altitude of pole = latitude of place. From this it follows that the change in the altitude of the pole must equal the change in the lati- tude of the observer as he proceeds north or south. 22 THE EARTH. [CHAP. II. Length of a degree of Latitude. Magnitude of Earth. Shape of Earth. 19. The measurement of the length of a degree of lati- tude on the earth is an operation of much practical difficulty. A position is chosen on the earth, and the altitude of the pole observed. Another station is chosen due north or south of the former position at such a distance from it that the altitude of the pole is increased or diminished by 1, as the case may be. The length of the arc of the meridian between the two stations is then measured, and is found to have a mean value of about 69 T V miles, which must be the length of a degree. The length of a degree has thus been calculated at about twenty different places on the earth, and the results have not been found to differ to any very great extent, which is confirmatory evidence of the earth's approximate spherical shape. It has, however, been found that the length of a degree near the poles is somewhat greater than near the equator, which shows that the curvature of the earth is not so great at the poles as at the equator, or, in other words, that the earth is slightly flattened at the poles. In fact the figure of the earth is what is called an oblate spheroid, differing but little from a sphere. The lengths of a degree of latitude at different parts of the earth have been found to be as fol ows : At the equator, . . 68704 miles. At latitude 20, . . 68-786 40% . . 68-993 60, , . 69-230 80, . . 69-386 i The length of a degree being about 69 T V miles, art CHAP. II.] MAGNITUDE OF EARTH. approximate value for the circumference and diameter of the earth can thus be found.* 1 = 69 T V miles ; /. 360 = somewhat under 25,000 miles = circum- ference of earth. Diameter of earth = ^ = somewhat under 8000 miles. The polar diameter of the earth is found to be about 26 miles shorter than the equatorial. Appearances of the Celestial Sphere due to Observer's Change of Place on Earth. 20. If the observer, starting from any place north of the equator, move due north along the meridian of the place, the celestial pole will appear to rise in the sky as his latitude increases (Art. 18). If he reach the north pole the celestial pole will appear right overhead in the zenith ; the celestial equator will therefore coincide with his horizon (see fig. 7). The apparent diurnal paths of the stars will appear as small circles parallel to the horizon ; therefore, all the stars visible will be circumpolar. Those stars, on the other hand, whose positions in the heavens are south of the celestial equator will never rise into view. Therefore, an observer at the north pole will never see more than half the heavens, no part of which, however, ever sinks below his horizon. Such a celestial sphere is called parallel sphere. Z-&. FIG. 7. Parallel Sphere, observer being at either pole of Earth. is called a The sun for half the year (21st March to * The above method was that employed by Eratosthenes (230 B.C.) in de- termining the magnitude of the earth, except that he measured the meridian altitude of the sun instead of the altitude of the pole. 24 THE EARTH. [CHAP. II. 23rd September) being north of the equator, will during this period appear above the observer's horizon. For these six months he will appear to make a circuit every 24 hours in the heavens, which would be parallel to the horizon but for his continual change of declination. His greatest altitude is reached on the 21st June, and is then 23 28'. For the remaining six months the sun keeps below the horizon, reaching a distance below it of 23 28' on 21st December. Therefore, at the north pole the day and night are each six months long. However, of the six months' night at the north pole a considerable portion is twilight. Observer at Equator. Let the observer now proceed southwards, and he will find that the celestial pole gradually falls in the sky until at the equator it will appear on the horizon coinciding with the north point, the south celestial pole coinciding with the south point. The celestial equator will therefore pass through the zenith and nadir, and, cutting the horizon at right angles, will coincide with the prime vertical (fig. 8). The apparent diurnal paths of the stars being parallel to the celestial equator will be bisected at right angles by the horizon ; the stars will therefore be an equal time above and below the horizon. There are therefore no stars cir- cumpolar, but every star in the heavens appears for nearly twelve hours above the horizon. FIG. 8. Right Sphere, observer being at Equator. It is evident, also, that day and night are equal through- out the year at the equator. CHAP. II.] PROOFS OF EARTH'S ROTATION. 25 Since the horizon bisects the diurnal paths of the stars at right angles, this sphere is called a right sphere. Similarly, in the southern hemisphere the south celestial pole will increase its altitude as the south latitude increases. Observer at about Latitude of Dublin. As the latitude of Dublin is about 53 20'N., the celestial pole will have an altitude PR = 53 20' (fig. 3). The apparent diurnal paths of the stars cut the horizon obliquely. Some stars are circumpolar, and some rise and set, while other stars whose decimations are south become visible for a small portion of their daily circuit. The sun's apparent diurnal path during the summer months may be represented by the circle A CB north of the equator, of which there is more than half above the horizon ; therefore during the summer we have a long period of daylight and a short night. Also in the winter, when the sun's declination is south, we have a short period oj daylight and a long night. The sphere in this position is called an oblique sphere. Diurnal Rotation of the Earth. 21. That the apparent diurnal motion of the heavens from east to west is really due to the earth rotating round an axis from west to east appears from the following con- siderations : (1) From simplicity. (2) From analogy. (3) From centrifugal force. (4) The experiment of letting a body fall from the top of a high tower. (5) Foucault's pendulum experiment. Under the first three of these headings are comprised the arguments which show that it is extremely probable that the 26 THE EARTH. |_ CHAP - n earth does rotate ; but (4) and (5) are experimental proofs of its rotation. From Simplicity. At the time of Copernicus, the only argument in favour of the earth's rotation was, that this was a much simpler, and therefore a much more probable, expla- nation than that all the stars and other heavenly bodies should be connected in such a complicated manner as to perform each its revolution round the celestial pole in the same time. From Analogy. However, the subsequent invention of telescopes (1609) supplied an additional argument. By the aid of the telescope we can see that many of the planets, as well as the sun and moon, are spherical bodies rotating about axes, from which we conclude that it is probable the earth also rotates. From Centrifugal Force. If the sun and planets, not to speak of the fixed stars, really described circles of such large radius in such a short period as one day, it would need an enormous attracting force acting towards the centres of those circles to keep them from flying off in a tangent. For we know from mechanics that if m be the mass of a body moving in a circle of radius r with a periodic time = T, the necessary force acting towards the centre to keep it in its circular path would be given by the formula : But here r would be very large and T very small, there- fore F would be enormously great. But there are no bodies we know whose attractions could be as great as this ; therefore the idea of the diurnal motion of the sun and planets, as well as the stars, is very improbable. CHAP. II.] . \ VERSITYl of J r >*^ PROOFS or EARTH'S ROTATION. 27 Q Experimental Proof from Falling Bodies. 22. Newton first suggested that if the earth rotate from west to east, a body, on being let fall from a considerable height above the earth's surface, should fall to the east of the vertical line. For, let P be the place from which the body is let drop, suppose the top of a tower (fig. 9) ; PAC the vertical through P passing through (7, the centre of the earth. Then, if PQ represent the arc de- scribed by the top of the tower while the body is falling, AB will represent the arc described by the base of the tower which, being less than PQ, shows that the base moves with a less velo- city than the top. But while the body is falling in the air it preserves the same velocity towards the east which it had at starting in common with the top of the tower, which, being slightly greater than that of the base of the tower, will cause the body to deviate slightly to the east. Thus, if we cut off AB' = PQ, B will represent the position of the base of the tower, and B' of the body, when it reaches the ground. If now we let a body fall from a high tower, and we find by actual measurement that daring its fall it has de- viated to the east of the vertical, the only cause we can give for this deviation is that the earth rotates from west to east. It is, however, very difficult to perform the experiment so as to give a decided result, the height of the tower being so small compared with the radius of the earth that FIG. 9. 28 THE EARTH. [CHAP. II. the deviation would be very slight. It has been tried at Boulogne and Hamburg, and the deviation was found to be one-third of an inch in a fall of 250 feet. Pendulum Experiments. 23. The experimental proof of the earth's rotation which is most striking is that first performed by Foucault in Paris in 1851, and very many times since by different observers. Before entering upon the details of the experiment, we will first suppose that the earth does rotate from west to east, and see what effect this rotation would have on a pendulum swinging at the north pole. We kn ow, from mechanics, that if a pendulum vibrate under the action of gravity alone, the plane of oscillation will remain fixed in space, for there is no force to make it deviate from that plane. Therefore, if it were possible to have a pendulum vibrating at the north pole, the observer and the plane in which he stands would be carried by the rota- tion of the earth round the fixed plane of the pendulum through 360 in 23 h 56 m 4 s . The observer, however, would be altogether unconscious of his own motion and that of the plane in which he stands, and it would appear to him as if the plane of the pendulum turned round in the opposite direction, making a complete circuit in 23 h 56 m 4 s (fig. 10). On the contrary, if a pendulum be set in motion at the equator, the plane of vibration, together with the observer and the surrounding surface of the earth will be carried 1 Equator \ } I FIG. 10. CHAP. II.] PENDULUM EXPERIMENTS. 29 FIG. II. bodily round in one common motion; therefore there will be no disturbance in the re- lative positions of the plane of the pendulum and the landmarks about it (fig. 11). At a place intermediate be- tween the equator and pole, the parts of the earth in the immediate neighbourhood of the pendulum which are near- est the equator will have a greater velocity towards the east than the parts nearest the pole ; therefore the plane in which the observer stands will really revolve beneath the pendulum, or, in other words, the plane of vibration of the pendulum will seem to revolve in the opposite direction with respect to the observer and surrounding landmarks. The time of the apparent revolution of the pendulum will get greater the nearer we approach the equator, until on the equator itself, as we have seen above, the plane of vibration does not seem to change at all. It is easy to prove, supposing that the earth does rotate, that at a place whose north or south lati- tude is X, the time of apparent revolution of the pendulum would be T cosec X, where T = time of revolution of the earth on its axis. For, let Om be the direc- tion of the observer, P the north or south pole of the earth, EQ the equator, and X = latitude of observer. 30 THE EARTH. [CHAP. II. Now, the earth revolves round OP through 360 in T SfiO units of time ; therefore it revolves through =- In 1 unit 360 of time. This angular velocity of =- per unit of time can be resolved into two components in direction at right angles to one another ; for we know, from dynamics, that rotations round axes can be resolved in exactly the same way as forces. Therefore, if we cut off ol to represent an QJA angular velocity of =- round OP, we find that this rotation is equivalent to a rotation represented by Om round the radius drawn to observer, and a rotation of On round a line at right angles to that radius ; but Om =01 cos (90 - \] = 01 sin X ; therefore to an observer at A the plane of a vibrating pen- 360 dulum will appear to revolve through sin X in 1 unit of time, and the time of making a complete revolution would therefore be SfiO T = * J v = -_ = Tcosec X = (23 h 56 m 4 s ) cosec X. <360 smX sinX Foucault's Experiment. 24. Foucault took a heavy iron ball and let it hang from the roof of the Pantheon by means of a wire about 200 feet long. A circular ridge of sand was placed in such a position that at every swing of the pendulum a pin attached to the lower part of the ball just scraped a mark in the sand. The ball was then drawn aside by means of a cord, and when at rest the cord was burnt off so that the pendulum should swing in as true a plane as possible. It was then observed that the marks made in the sand at each swing did not coincide, but that the plane of the pen- dulum seemed to be slowly turning round with a watch-hand rotation. What actually happened, however, was that the CHAP, ii.] FOUCAULT'S EXPERIMENT. 31 whole Pantheon, together with the observer and the circular ridge of sand, slowly rotated in the opposite direction. The wire was taken of this great length (200 feet) in order that the pendulum might move very slowly, thus meeting with very small resistance from the air, which enables it to keep up its motion for a long time. The reason a long wire ensures a long time of vibration is that the time of vibration is proportional to the square root of the length of the pendulum therefore the longer the pendulum is made the greater is the time of one vibration. If it were possible to keep the pendulum vibrating long enough at Paris to enable its plane to appear to make a complete revolution the time taken would be about 32 hours. We can account for this phenomenon on no other sup- position than that the earth revolves round an axis, and as the plane of the pendulum does not seem to change at the equator we know that the axis of revolution of the earth must be perpendicular to the equator. There are various other phenomena which can be ex- plained on the hypothesis of the rotation Qliha~eartk,- such as trade winds *mj_flprf.fl.iTi nnnst-flnf- nnrrAnfa in fh&jQP^" j also the revolution of cyclones which in the southern hemi- sphere move with a watch-hand rotation, and in the northern hemisphere in the opposite direction. EXAMPLES. 1. Find the lowest latitude at which it is possible to have a midnight sun. Ana. 66 32' north or south. 2. What circles on the earth correspond to the latitudes 66 32' north or south ? Ans. The arctic or antarctic circles. 3. What is the highest latitude north or south at which it is possible to see the sun in the zenith at noon ? Ans. 23 28'. 32 THE EARTH. [CHAP. II. 4. What is the latitude of a place at which the celestial equator and horizon coincide ? Ans. 90, at the poles. 5. What is the latitude of a place at which the ecliptic coincides with the horizon ? Ans. 66 32'. 6. Why is the sun never seen in the zenith at Dublin ? 7. If a pendulum be made to vibrate at a place whose latitude is 30, in what period of time will the plane of vibration appear to make a complete revo- lution ? Here T= (23 h 56 m ) cosec 30 = (23h 56 ra ) 2 = 47 h 52. 8. At what part of the earth would a body have no deviation towards the east when let drop from a height ? Am. At the poles. 9. If a person travelling eastward go round the world, he will at the end of his journey appear to have gained a day. On the other hand, if he travel west- ward, he will appear to lose a day. Explain this. 10. How far should a man travel northwards from the equator in order that the altitude of the pole might become 10 ? Assume the radius of the earth to be 4000 miles (J. S., T. C. D.). 2 x 3-14159 x 4000 Here 60 - ; iy , 2 x 8-14169 x 4000 x 10 B698 . 360 CHAPTER III. THE OBSERVATORY. Astronomical Clock. 25. WE have seen in Chapter I. that the apparent uniform revolution of the stars round the celestial pole is completed in about 4 minutes less time than the apparent diurnal revolution of the sun. This latter interval of time (more accurately its mean value throughout the year) is what is taken as the ordinary day, and is called a mean solar day. It is divided into 24 mean solar hours. On the other hand, the interval of time taken by the fixed stars to complete a revolution round the pole is called a sidereal day. The sidereal day is divided into 24 sidereal hours, which are reckoned from 1 to 24, therefore we have 24 sidereal hours = 23 h 56 m mean solar time. The astronomical clock is regulated so as to mark sidereal time, and as the sidereal day commences when the first point of Aries is on the meridian, the clock should then be set to mark O h O m s . It will then indicate the sidereal hours up to 24 when the next transit occurs. Definition. The sidereal time at any instant is the interval that has -elapsed since the preceding transit of the first point of Aries expressed in sidereal hours, minutes, &c. As the right ascensions of heavenly bodies are measured eastwards along the equator from the first point of Aries, it follows that those stars which have a small right ascension will cross the meridian before those stars whose right ascensions are greater. In fact, 360 of right ascension correspond to 24 sidereal hours or 15 to 1 sidereal hour. Eight ascensions T) 34 THE OBSERVATORY. [CHAP. in. may therefore be expressed in degrees or in time, the former being reduced to the latter by dividing by 15. Hence we might define the right ascension of a body as the sidereal time of its passage across the meridian. The hour angle of the first point of Aries (Art. 14) at any instant reduced to time (by dividing by 15) is evidently the sidereal time at that instant. The Transit Instrument, 26. The object of this instrument is to determine the exact instant at which a body crosses the meridian. It consists of a telescope rigidly fixed to a horizontal axis. At the extremities of this horizontal axis are two cylindrical pivots, of the same diameter, which move in sockets fixed on two piers of solid masonry. In order to diminish the pres- sure of the pivots on the sockets, and consequently the wear caused by friction, a great part of the weight of the telescope is balanced by two weights which are hung at the extremities of a pair of levers, the other extremities being attached to the cylindrical pivots (see fig. 13). In the plane of the principal focus of the object-glass is placed a framework of five, or seven, or a greater number of vertical wires or spider lines (see fig. 14) placed at equal intervals apart. These are intersected at right angles by two horizontal lines, to which the path of the image of a star or other body across the field of view will be almost parallel and in a position midway between them.* * In some portions of this Chapter, to avoid complexity, mention is only made of one horizontal line supposed to be situated midway between the two mentioned above. FIG. 13. CHAP.. III.] TRANSIT INSTRUMENT. 35 FIG. 14. Since the principal focus of the object-glass is in the same plane as that in which the lines are stretched, the image of a star under observation and the spider lines can he seen at the same time, and for purposes of adjustment the frame- work of lines admits of various move- ments by means of screws. When the instrument is used at night it is necessary to illuminate the spider lines. This is done by means of a lamp placed opposite one of the cylindrical pivots, the light from which by means of mirrors is re- flected down the tube on to the lines. It is the object of the observer that the telescope be so adjusted that the middle vertical spider line may coincide as nearly as possible with the meridian. The time at which the star crosses the meridian can therefore be estimated by ob- serving the instant, as indicated by the astronomical clock, at which it crosses this line. But as there is always a small error in noting the time of transit over one line, the seven are used in order that the observer may note the time at which the image crosses each of them, when the mean of these being taken will, in all probability, give a more accurate result than one observation could afford, as the observer may in some cases be too precipitate, in others too tardy, the positive and negative errors thus to some extent neutralizing one another. Line of Confutation. When the image of an object is formed at the principal focus of the object-glass of a telescope, then the direction in which it is viewed is the same as its true direction as seen by the naked eye ; this line along which the object is viewed is called the line of collimation, or line of sight. For practical purposes the line of collimation of a telescope may be defined as the line joining the optical centre of the object-glass with that point of the central vertical spider line midway between the two horizontal lines. T) 2 36 THE OBSERVATORY. [CHAP. III. Collimation, Level, and Deviation Errors, with the corresponding Adjustments. 27. Every transit instrument, to be perfectly adjusted, must satisfy the three following conditions : (1) The line of collimation should be perpendicular to the axis of rotation of the telescope. (2) The axis of rotation should be horizontal. (3) This horizontal axis should point due east and west, that is, the line of collimation should be due north and south. Corresponding to the above conditions we have there- fore in every instrument three errors (1) collimation error ; (2) level error ; (3) deviation error ; and to correct these we have three corresponding adjustments. Collimation Error* The error of collimation may be defined as the amount by which the angle between the line of collimation and tha axis of revolution of the telescope falls short of a right angle. Let XT (see fig. 15) represent the axis of revo- P \ lution of the telescope, and \ AB the line of collimation \ supposed not at right & angles to XT. Let the telescope be pointed at an object or mark on the earth placed at some distance away, and let P be a point of the object which coincides with the middle vertical spider line. The telescope Fm - 15 - is then reversed in its bearings, the right hand pivot being * Besides the above method of correcting the collimation error two other methods, which are more frequently used in observatories, are given in Arts. 36, 36A. CHAP. III.] TRANSIT INSTRUMENT. 37 placed in the left socket, and vice versa. If the point P still coincide with the central spider line, there is no collimation error. If not, the line of collimation, on the telescope being reversed, occupies the position A'B', making an equal angle with the perpendicular on the other side, the error of colli- mation being half the angle between AB and A'B'. To correct for this error the spider lines must be moved by means of screws until the central line coincides with the same point before and after reversing the telescope. When this adjustment has been made we know that the line of collimation sweeps out a great circle in the heavens. The object of the next two adjustments will be to insure that this great circle shall coincide with the meridian. Level Error. 28. This error is due to the axis of revolution not being horizontal. To correct for it we make use of a spirit-level which is long enough to reach from one extremity of the axis to the other. The level is first hung on to the axis by means of hooks, and the position of the bubble is noted by means of a scale attached. The level is then reversed and the reading of the scale again noted. If the bubble occupies the same position as before there is no level error. But if not, one end of the axis must be raised or lowered by means of a screw until the reading of the bubble is the mean of the two former readings. After correcting for this error we know that the line of collimation not only describes a great circle, but that this circle passes through the zenith, or in other words, is a vertical circle. Deviation Error or Error of Azimuth. 29. This error is due to the line of collimation not pointing due north and south. The middle vertical spider line will therefore not coincide with the meridian, but with a vertical THE OBSERVATORY. [CHAP. 111. such as ZX (see fig. 16). The error is detected by observing* the interval between the upper and lower transits of a circum- polar star (preferably the pole star), and again the interval between its lower and upper transits. These two intervals should be equal, as the meridian bi- sects the circle described by the star round the pole. But if not equal, the line of collimation cannot coin- cide with the meridian, the star appearing to transit at FIG. 16 m and n (fig. 16). To correct for this error one extremity of the axis must be moved horizontally by means of a screw, until the two intervals above mentioned are identical. Observing a Transit. Eye and Ear Method. 30. As the apparent motion of a star across the field of view is greatly magnified by the telescope, the star may appear at one side of a wire or spider line at the termination of one second, and at the other side before the end of the following second. An expert observer, however, can esti- mate to a small fraction of a second the instant of crossing the wire. When the star appears in the field of view he writes down the hour and minute from the clock, and then, without again looking up from his observation, keeps counting the seconds by the beats of the clock. The relative distances of the image of the star to the right and left of the wire at the end of two consecutive seconds enables him to determine the exact time of its passage across the wire. A similar observation is made for each of the seven wires. > This method is called the " eye and ear method." CHAP. III.] MERIDIAN CIRCLE. 39 The Chronograph. A more accurate method of observing a transit is now coming into general use, by means of the chronograph. The clock is so arranged that at every beat an electric circuit is broken, which causes a dot to be made on a sheet of paper wrapped round a uniformly-revolving cylinder. The cylin- der, besides revolving, let us say round a vertical axis, has a slow motion, either up or down in the direction of its length, so that the dots corresponding to seconds of time are arranged at equal intervals in a spiral or corkscrew curve on the cylinder. The observer at the instant of transit across a wire, presses a button, which causes a dot to be made on the cylinder in addition to those caused by the clock-beats. The position of this dot relatively to the two dots caused by the clock immediately before and after, enables him, by direct measurement, to determine the time of transit to a very small fraction of a second. Meridian Circle. 31. The meridian circle, or, as it sometimes called, the . 17. transit circle, consists of a transit instrument MN (fig. 17) 40 THE OBSERVATORY. [CHAP. in. such as we have already described, with the addition of a pair of graduated circles placed one on either side of the telescope. These circles are fixed with their planes at right angles to the horizontal axis, and revolve with the telescope. The rim of each circle is graduated by means of fine lines usually into intervals of 5'. As mechanical subdivision cannot go much further than this, the intermediate minutes and seconds are determined by means of a microscope. Usually, however, six microscopes placed at equal intervals are used, each of them being read off and the mean of them all taken. These are represented in fig. 17 by the letters A, B, C, &c. In addition to these there is a microscope of low magnifying power called the Pointer, which is used to read off the degrees and graduations corresponding to the intervals of 5'. These microscopes are all fixed, and therefore, as the circles revolve, the graduations pass across the field of view of each of them. The pointer microscope should read zero when the line of collimation points to the zenith. Heading Microscopes. In the focal plane of the object-glass of each of the six microscopes is fixed a small metal scale mn, cut into fine notches and called a comb. This scale and the image of the graduations on the circle are both seen together in the field of view as represented in fig. 18. There are 5 notches to each interval ab of the graduated circle, therefore each notch cor- responds to 1'. A small aperture p in the scale is placed to mark F IO . is. the central notch, so that when the pointer is opposite a graduation^ shall coincide with one also. A spider line xy, CHAP. III.] READING MICROSCOPE. 41 stretched across the field of view, can be moved parallel to itself by means of a micrometer screw, the head of which is divided into 60 equal parts. Five turns of the screw-head serve to bring the spider line through an interval ab of the graduated circle ; therefore we have 5 turns screw-head = 5' of graduated circle ; /. 1 turn = 1' 1 division _ 1 " Now, suppose we have to take the reading of the circle in any position. The pointer microscope gives the reading roughly in intervals of 5'. The odd minutes are given by the number of notches in the comb from the central notch p to the preceding graduation, say at c. The number of divisions through which the screw-head must be turned in order to bring the spider line into coincidence with c from the preced- ing notch will give the number of seconds. Error of Eccentricity avoided by taking the mean of two micro- scope readings by diametrically opposite points. 32. The error of eccentricity is caused by the graduated circle revolving round some poin^ not its centre. This is FIG. 19. entirely eliminated by taking the mean of the readings at 42 THE OBSERVATORY. [CHAP. III. diametrically opposite points. For let be the centre of the circle (fig. 19), 0' the point round which the circle revolves. Now let the circle revolve round 0' through an angle 0, so that the line AB occupies the position CD 9 then the angles a and /3 subtended at the centre by the arcs AC and BD will correspond to the two readings at opposite sides. We have to prove that the mean of a and ]3 will equal 0. By Euclid (i. 32) we have also a = + $ ; To find the Zenith Point on the Meridian Circle. 33. We have already stated that the pointer should read zero when the line of collimation points to the zenith. But the mean reading of the six microscopes is not generally zero at the same time. We have, therefore, in every transit circle to find the zenith point or the reading of the circles corresponding to the zenith. In order to find this point the telescope is directed verti- cally downwards to a basin of mercury placed beneath it. It is then moved until the fixed horizontal wire and its image reflected by the mercury are perfectly coincident. The line of collimation then points directly to the nadir. The reading of the pointer and microscopes, therefore, gives the nadir point, 180 from which is the zenith point. Another method of finding the zenith point is by observ- ing the pole star (which is chosen because on account of its slow motion it remains a long time in the field of view), and taking the reading of the circle when the star appears on the horizontal wire. The telescope is then depressed until the image of the star reflected from the surface of a basin of CHAP. III.] MERIDIAN ZENITH DISTANCE. 43 mercury coincides with the horizontal wire, and the reading again taken. As the telescope in the two instances must have been inclined at equal angles above and below the horizon, the mean of the two readings gives the horizontal point, 90 from which is the zenith point. It is also evident that half the difference of these two readings, corresponding to the direction of a star and of its image in a basin of mercury, is the meridian altitude of the star. The Polar Point on the Meridian Circle. 33A. To find the polar-point* i.e. the reading of the meridian circle when the telescope is pointed to the pole. A circumpolar star is observed in its upper and lower transits, and in each case the reading of the meridian circle is taken. z FIG. 19A. Thus, if m and n represent the positions of the star at its upper and lower culminations, we have Pm = Pn = polar distance of star = A. Let polar point = x. Then x + A = reading of circle when star is at n, and x- A = reading of circle when star is at m. Therefore the polar point x = half the sum of the two readings. 44 THE OBSERVATORY. [CHAP. III. Since PR = altitude of pole = latitude of place ; .*. ZP = the complement of the latitude = colatitude. The latitude of the observatory or place can now be found, for the difference between the zenith point and the polar point is the colatitude ZP which, subtracted from 90, gives the latitude. Meridian Zenith Distance. Meridian Altitude. Declination. 34. In order to measure the zenith distance of a star when in the meridian, the telescope is pointed to the star and the reading of the circle taken. The difference between this reading and the zenith reading gives the meridian zenith distance of the body, which must be corrected for refrac- tion and other errors. The meridian altitude is obtained by subtracting the Q FIG. 20. observed zenith distance from 90. Having obtained the meridian altitude of the star we can now, the latitude of the place being known, determine its declination. For let S be the position of a star in the meridian, then : S&= meridian altitude = a ; SE = decimation = S. CHAP. III.] ASTRONOMICAL CLOCK. 45 Also.Z5~ = colatitude ZP (both having a common comple- ment ZE). Now we have or colat + S = a. Similarly, if the star be at S', we have colat- S = a. Therefore colatitude declination = meridian altitude, the plus or minus sign being taken in the northern hemi- sphere, according as the star's declination is north or south.* From this equation, knowing the meridian altitude of the star and the latitude of the place, its declination is deter- mined. 35. Standard Stars. In the Nautical Almanac, which is published every year, is a list of stars whose right ascen- sions and declinations are recorded for each day. Their declinations are determined by the method we have just in- dicated, and a method of finding their right ascensions will be given later on (see Flamsteed's Method, Chapter VIII.) . These stars are called standard stars. The declination of any other star can be found by comparing the reading of the transit circle when the telescope is directed to the star with the corresponding reading for a standard star. The difference of the two readings being the difference of their declinations, the declination of the body in question can therefore be found. Regulation of the Clock. As the first point of Aries is an imaginary point in the sky, such that we cannot observe its passage across the meri- dian, we are not therefore able to tell by direct observation when to set the clock at O h O m s . But the time of transit of * In case the star transits between Z and P the equation becomes colat + S = 180 - a. 46 THE OBSERVATORY. [CHAP. III. a standard star is known from its right ascension, and the clock can therefore be set at correct sidereal time when one of these stars is observed in the meridian. The rate of the clock, i.e. the amount it gains or loses each day, can be determined by noting the interval between the transits of a fixed star on two successive nights. The interval should be 24 sidereal hours, from which the daily gain or loss of the clock is found. A good clock should be such that its rate of gain or loss is uniform. To find the Right Ascension of a Body. The clock being set correctly and its rate being known, then the sidereal time at which a body crosses the meridian is its right ascension, which can be reduced to degrees, minutes, and seconds by multiplying by 15. Collimating Telescopes. 36. In order to correct the error of collimation, we have seen that the axis of the telescope has to be reversed in its sockets, and the direction of a distant mark observed before and after reversal. This method has, however, now been superseded by the use of two small telescopes, called col- limating telescopes.^ fixed one to the north, and the other to the south side of the transit telescope. Each of these is furnished with cross wires, so that, on looking through one into the other, which can be managed by means of an opening in the tube of the large telescope, the images of the cross wires (illuminated) appear coincident. If now the cross wires of the transit instrument itself be so adjusted as to coincide with those of the north collimator, and it be found on rotating the telescope round that they are also coincident with those on the south, the line of collimation must be perpendicular to the horizontal axis. By this method the troublesome operation of reversing the axis of the telescope is avoided. CHAP. III.]: EQUATORIAL. MICROMETER. 47 36A. There is yet another method of determining the collimation error by pointing the telescope vertically down- wards towards a hasin of mercury. If the axis be perfectly horizontal and there be no collimation error, the spider lines should coincide with their image formed by reflection from the surface of the mercury ; for the rays of light divergiug from the spider lines (which are illuminated), after passing through the object glass, fall in parallel lines on the surface of the mercury, from which they are again reflected in parallel lines, which are converged back again to its focus by the object glass. If, therefore, the level error having been previously corrected, the real system of spider lines do not coincide with the reflected system, the difference, which may be measured by a micrometer, is twice the error of colli- mation. The Equatorial. 37. Most of the large telescopes in observatories are mounted equatorially. This arrangement consists in an axis AB (fig. 21) which points to the celestial pole, called FIG. 21. the polar axis. This polar axis turns in fixed bearings A and B attached to two fixed piers. The telescope can be turned 48 THE OBSERVATORY. [CHAP. III. round an axis (7, so as to be set at any angle to the polar axis. A clockwork apparatus is generally attached to the larger instruments, by means of which the polar axis is made to revolve uniformly in its bearings in the same direction as the diurnal motion of the heavens, the revolution being completed in 23 h 56 m , so that once the telescope is pointed at a star, and the clockwork apparatus set going, it is possible to keep that star in the field of view for a prolonged period. By combining the two motions which it is possible to give the equatorial, it can be pointed at any star in the heavens, which need not, as in the case of the transit in- strument, be in the meridian. It is therefore for obser- vation of bodies not in the meridian that this instrument is used. A graduated circle mn, whose plane is at right angles to the polar axis, serves to set the telescope at any required right ascension. It is called the hour circle. The axis C, round which the telescope turns, also carries a graduated circle, which is not drawn -on the figure. It is called the declination circle, as by means of it the telescope can be set at any required declination. Both circles are read off by pointer microscopes. The equatorial, on account of its high magnifying power, enables us to observe the nature of the moon and planets and other heavenly bodies. It is also used in stellar photography and in the spectroscopic analysis of the stars. Micrometers. 38. Every equatorial is furnished with a micrometer for measuring small angular distances such as the angle subtended at the observer by two neighbouring stars in the field of view of the telescope. The kind most commonly used is the parallel wire or spider line micrometer. It consists of a CHAP. III.] MICROMETERS. 49 rectangular framework (fig. 21 A) with a graduated screw- FlG. 2lA. head gg at each end ; bbb, ccc are two metal forks which slide within one another on which are fixed two parallel spider lines d and e ; two fine screws/,/ having milled heads g, g connected with graduated circles, are attached one to each fork, so that by turning these milled heads each fork can be drawn out or pushed in according to the direction in which the head is turned, and thus the spider lines can be brought as wide apart or placed as close together as we please. There is also a fixed transverse spider line k at right angles to d and e. The circumference of each of the circles in connexion with the screw- heads is divided into 100 equal parts. There is also a fine scale cut into notches, every fifth notch being cut deeper than the others as is seen in the above diagram ; the distance between two consecutive teeth being equal to* the interval between the threads of each of the screws, there- fore a complete revolution of one of the screw-heads just moves the corresponding spider line through a distance equal to the common interval between the teeth. In order to measure the angle subtended by two neighbour- ing stars at the observer, the micrometer is placed in the focal plane of the telescope and rotated until the fixed transverse spider line passes through the images of the two stars. The two parallel lines are then shifted by means of the screws until each coincides with an image of a star. The distance between the two wires can now be found by noting, by means of the teeth cut in the scale, how many turns must be given E 50 THE OBSERVATOHY. [CHAP. III. to the screw-head (or screw-heads) to make the parallel Hues coincide. The fractional parts of a turn can be read off on the graduated circle attached to each screw-head ; and the angular value of each turn being known, we are able to calcu- late the angle subtended by the stars. The micrometer also serves to measure the angular dia- meters of the sun, moon, or planets, one of the parallel lines being placed so as to touch one limb, and the other the dia- metrically opposite limb of the circular disc presented by the body, and the distance between them is measured as before. 38A. To find the angular value of each turn of the micrometer screw, a circumpolar star is chosen, preferably the pole star, for, on account of its very small distance from the pole, its motion is very slow, and can therefore be most accurately observed. The micrometer is then adjusted so that the diurnal motion of the star is along or parallel to the fixed spider line. The two movable lines are then separated by a certain number of turns of the screw, and the time taken by the image of the star to pass from one line to the other is noted, from which, knowing that the star describes 360 of a small circle in 24 sidereal hours, the angular value of the distance between the wires is easily found, and hence the angle, expressed in seconds of a small circle, corresponding to one turn is known : but the relative magnitude of the small circle described by the pole star to a great circle can be found since the declination of the star is known ; hence the number of seconds of the arc of a great circle corresponding to each turn is obtained. The Alt-Azimuth Instrument. The alt-azimuth may be described as an equatorial, of which the axis points to the zenith instead of the celestial pole. It admits of a double motion in altitude and azimuth, just as the equatorial does in right ascension and declination. Like the equatorial, it is used in ex-meridian observations. HAP. III.] EXAMPLES. 51 Given the zenith distances of a circumpolar star at its upper and lower transits to calculate the latitude of the place and the star's declination. Let the zenith distances* Zn and Zm (fig. 10A) be repre- sented by % and 2', also Polar distance Pm = Pn = A, and ZP = colat ; .% 2 = colat + A 2' = colat - A ; /. 2 + 2' = 2 colat, and 2 - 2' = 2 A ; , 2 + 2' , . . . A s + 2' .*. colat = jr ; hence lat. = 90 > 2 and polar distance A = ^- ; hence decln. S = 90 <& N.B. If the star in one of its transits souths, i.e. if it cross the meridian south of the zenith, its zenith distance at this transit is to be considered negative. EXAMPLES. 1. Supposing the earth to rotate with the same angular velocity as at present, but in the opposite direction, what would be the length of a mean solar day and the number of mean solar days in the year ? Am. 23 h 52 m ; 367. 2. How many sidereal days are there in the year ? Am. 366J. 3. What is the meridian altitude of the sun at Dublin on the 21st June, the latitude of Dublin being 53 20' ? Ans. 60 8'. Here colat 8 = (Art. 34) ; but 5 = 23 28' N. and colat = 90 - 53 20' = 36 40' j .'. 36 40' + 23 28' = o; .-. a = 60 8'. 4. "What is the meridian altitude of the sun at Dublin (1) during the winter solstice ; (2) at the equinoxes ? Ans. (1) 13 12'. (2) 36 40'. N.B. At winter solstice S = 23 28' S. (minus). * In all cases these zenith distances when measured by the meridian circle are to be corrected for refraction and other errors. E 2 52 THE OBSERVATORY. [CHAP. Ill- 1/5. The zenith distances of a circumpolar star as it crosses the meridian above and helow the pole are found, after correcting for refraction, &c., to he 13 7' 16" and 47 18' 26". Calculate from this the latitude of the place and the decli- nation of the star. (See Art. 38A). Am. 59 47' 9"; 72 54' 25". 6. The latitude of John o' Groat's house is 58 59' N. Find the sun's meridian altitudes at that place on midsummer and midwinter days, respectively. Ans. 54 29'; 7 3'. 7. Find the latitude of a place where the greatest elevation of the sun above the horizon at midsummer is 76 42'. Ans. 36 46'. ^ 8. The declination of Canopus is 52 38' S. ; if we travel southwards, where shall we first find it attain a meridian altitude of 10 ? Ans. 27 22' North latitude. 9. Find the declination of a star whose corrected meridian zenith distance, as observed at Dublin (lat. 53 20'), is 72- 18' 40". Ans. 18 58' 40" S. 10. What is the sun's midnight depression below the horizon at Dublin during midsummer and midwinter, respectively. Ans. 13 12'; 60 8'. 11 . The zenith distances of a star at lower and upper culminations are found, after correcting for refraction, &c., to be 76 4' and 2 52 S. respectively. Find the latitude of the place, and the declination of the star. N.B. Apply formulae in Art. 38A, but 2 52' being south is given a minus sign. Ans. 53 24'; 50 32'. >V 12. The declination of Vega (a Lyrae) is 38 41' N. ; does it cross the meridian of Dublin (lat. 53 20') north or south of the zenith ? Ans. Upper transit, 14 39' S. of zenith. Lower transit, 87 59' N. of zenith. ( 53 ) CHAPTER IV. ATMOSPHERIC REFRACTION. 39. WE have seen (Chapter in.) how the altitude of a star can be found by observation. This observed altitude, however, is liable to some error, owing to the rays from the star being bent in passing through the atmosphere before they reach the eye of the observer, thus leading him to think tli at the star is in a different direction than is really the case. This apparent displacement is due to the refracting power of the atmosphere. We know from op- tics that when a ray of light passes from a rarer to a denser me- dium, it is refracted or bent towards the perpendicular to the common surface of the two media. Thus A OB would represent the path of such a refracted ray, the angle i being the angle of incidence, r the angle of refrac- FIG. 22. tion, while i - r is the amount of the refraction. It is also a law of optics that the angles of incidence and refraction are such that their sines are in a constant ratio ; therefore, sin smr = a constant 54 ATMOSPHERIC REFRACTION. [CHAP. iv. Now the atmosphere is a gaseous fluid, subject to the action of gravity. Its density in its upper layers is very small ; but as we approach the earth its density increases as the weight of the superincumbent air on any given area increases. Therefore, when a ray of light from a star S strikes the atmosphere, we may suppose it in its passage to the earth to pass through an indefinite number of media each denser than the preceding, like a number of concentric spherical shells. The path AP of the ray through the atmosphere thus being continually bent will be curved. To an observer at P the star will appear in the direction PS', a tangent to the curve at the point P ; whereas its real direction, if there were no atmosphere to refract the ray, would be PS, a parallel drawn through P to AS ; for, being so far distant, the lines drawn from A and P to the star will be practically parallel. The angle SPS f between the apparent direction of the star and the direction in which it would appear if there were no atmosphere is called the refraction. The effect of refraction, therefore, on the position of a heavenly body, is to raise it in the sky, so as to increase its altitude and diminish its zenith distance. But as this CHAP. IV. J VARIATION IN DENSITY OF ATMOSPHERE. 55 apparent displacement takes place in a vertical plane, the azimuth of the body is not affected. Therefore, in all observations of the altitudes of heavenly bodies each apparent altitude must be diminished by the amount of the refraction, in order to get the true altitude. This correction is called the correction for refraction. The amount of refraction is greatest when the angle of incidence is greatest, i.e. when the body is on the horizon. The refraction is then called the horizontal refraction. The refraction is zero at the zenith, as the rays from a body situated right overhead strike the different layers of the atmosphere at right angles, and, therefore, do not get bent. The horizontal refraction is about 35'. Therefore, a body on the horizon will appear a little more than half a degree above the horizon. As the angular diameter of the sun is about 32', or a little more than half a degree, we are able to form some sort of idea as to how much a body on the horizon is displaced by refraction, by remembering that it is through an arc nearly equal to the breadth of the sun's disc. From this we conclude that when it appears to us that the sun is about to set he has in reality just set, and we would not see him at all were there no atmosphere to refract his rays. The amount of refraction is influenced by the changes in the pressure and temperature of the atmosphere. A rise in the barometer is accompanied by an increase in the amount of refraction, provided the altitude of the body remain the same. On the contrary, an increase of temperature produces a diminution of refraction under the same circumstances. In an observatory it is necessary, in estimating the error due to refraction, to take into account not only the zenith dis- tance of the body, but also the pressure and temperature of the atmosphere as indicated by the barometer and ther- mometer respectively. We have seen that the horizontal refraction is about 35' : therefore, how rapidly it decreases as the zenith distance decreases is seen from the fact that the 56 ATMOSPHERIC KEFRACTION. [CHAP. IV. refraction at an altitude of 45 has a mean value of only 58"-2. Law of Refraction. 40. As the height of the atmosphere is so very small compared with the radius of the earth, we may assume that the lines drawn from A and P (fig. 23) to the centre of the earth are parallel, or, in other words, that the surface of the earth is a horizontal plane, with an indefinite number of horizontal layers of atmosphere of gradually decreasing density resting on it. We can now very easily deduce a law according to which the refraction varies ; for the ray will get bent through the same amount if, instead of passing through a number of layers of varying density, we suppose it to pass through a homogeneous atmosphere of the same density throughout as the layer in contact with the earth, when we can imagine it to get bent once for all at its entrance into the atmosphere, and then proceed in a straight line to the observer. The refraction of a heavenly body, the temperature and pres- sure being constant, varies as the tangent of the apparent zenith distance. Let SAP represent the path of a ray from a star to an observer at P (fig. 24). The apparent direction of a star will then be P/S", the angle z being the apparent, and z + x the real zenith distance, while angle SAS f = amount of refraction = x. , T sine (angle of incidence) Now - } . j 5 -. ( = a constant = p. sine (angle of refraction) sin (z + x] , . . . or - = p, that is, sin (z + x) = p sin z ; sin z cos x + cos z sin x = p sin z. But x is a very small angle, and we know from trigo- nometry that the cosine of a very small angle is almost = 1, CHAP. IV.] LAW OF REFRACTION. 57 and as the perpendicular and arc almost coincide, its sine = its circular measure ; /. sin x = x (expressed in circular measure), and cos x = 1 ; z z Atmosphere Earth, FIG. 24. therefore the above equation becomes sin z + x cos z = /n sin z ; .*. x cos z = fi sin z - sin z = . sin 2 or - 1) sin z ; Let = (/* - 1) tan z. - 1 = JT; = IT tan z ; .*. # varies as tan s. This law has been found to be approximately true for zenith distances up to 75. Nearer the horizon the law does not hold, as the constitution of the different layers of the atmosphere will affect it. It is evident at once that the law 58 ATMOSPHERIC REFRACTION. [CHAP, iv could not hold at the horizon where the zenith distance = 90, for tan 90 = infinity. The amount of the refraction at any observed zenith dis- tance less than 75 can be found by substituting for tan ^its value, provided the value of the constant K be known, for finding which we give the following methods. 41. To find the constant coefficient of refraction when the latitude of the place is known. This is done by observing with the meridian circle the zenith distances of a circumpolar star as it crosses the meridian above and below the pole. Let m and n repre- sent the true positions of the star at its two culminations, then to the observer the star will appear at m f and n' as raised by refraction. Let the observed zenith distances Zm f and Zn'\)Q represented by 2 and 2' : FIG. 25, /. Zm = Zm' + the refraction = 2 + K tan 2, Zn = Zn' + the refraction = 2'+ JTtan 2' ; adding, we get Zm + Zn = 2 + si + K (tan 2 + tan 2'). But Zm + Zn = Z colat (Art. 38A) (for PR = latitude of place) = 180 - 2 lat ; /. 180 - 2 lat = 2 + 2' + K (tan 2 + tan 2') ; 180 - 2 lat - 2- 2' _ tan 2 + tan 2' But the latitude of the place is known, and 2 and si are the observed apparent zenith distances ; therefore K is determined. CHAP, iv.] BRADLEY'S METHOD. 59 42. Bradley's "Method. The coefficient of refraction can be found when the latitude of the place is not known by the method of Dr. Bradley, who, besides observing the zenith distances of a circumpolar star at its two culminations, measured the zenith distances of the sun when in the meri- dian at the summer and winter solstices, when the sun's declination is 23 28' north and 23 28' south respectively. Let these observed zenith distances be denoted by s and /. If now A and B (fig. 25) represent the real positions of the sun, the apparent positions when observed will appear raised to A' and B' ; .*. ZA = ZA' + refraction = s + K tan s, ZB =- ZB' + refraction = s' + K tan s' ; adding, we get ZA + ZB = s + s' + K (tan s + tans') ; but ZA + ZB = 2ZE as before, and arc ZE = PR (having a common complement ZP) = lat ; .*. 2 lat - s + s' + K (tan s + tan s'). We have also from observations on a circumpolar star, as in the last method, 180 - 2 lat = s + s' + K (tan s + tan s'). By adding these two equations, we eliminate the latitude thus : 180 = s + s' + s + s' + K {tan s + tan s' + tan s + tan s'j ; 180 - s - s' - s - s' K tan s + tan s' + tan s + tan / But s, s', s, and s' are observed; therefore K is deter- mined. The fact that the latitude need not be known is an advantage in Brad ley's method, but it takes six months to complete the observation. By these methods the constant of refraction has been estimated at about 58"-2 ; /. r = 58"-2 tan z. 60 ATMOSPHERIC REFRACTION. [CHAP. IV. 42 A. The coefficient of refraction may also be found and the latitude of the place determined at the same time, by observing the apparent zenith distances of two circumpolar stars at their transits above and below the pole. Thus we have for one star : 180 - 2 lat = z + z' + K (tan z + tan z') ; the second circumpolar star will similarly give 180 - 2 lat = 2i + zi + K (tan z l + tan 2/) ; /. z + z' + K (tan 2 + tan 2') = z l + Zi + K (tan % + tan 2/) ; .*. K (tan 2 + tan / - tan 21 - tan s/j = 2i + 2/ - 2 - /, , / r Zi -t Z l 3 S tan 2 + tan 2' - tan 21 - tan > > the value of K being thus found, the latitude may be determined by substitution in one of the above equations. 43. A curious effect of refraction is the oval shapes which the sun and moon appear to have when near the horizon. The reason of this phenomenon is, that the lower limb being nearer the horizon than the upper limb will be raised to a greater extent by refraction. The vertical diameter AB GD FIG. 26. will therefore appear shortened as A'B', while the horizontal diameter remains the same. This apparent diminution in the vertical diameter of the sun and moon when near the horizon amounts to about one-sixth part of the whole, or about 5'. CHAP. IV.] EXAMPLES. 61 EXAMPLES. 1 . The apparent zenith distance of a starts 30 : calculate the true zenith, assuming the coefficient of refraction to he 58"*2.J Here the refraction = 58" -2 tan 30. = 58"-2x _L = 33"-6; .. True zenith distance = 30 0' 33"-6. 2. The apparent altitude of a star is 30 ; calculate the true altitude, the coefficient of refraction heing 58"*2. Ans. 29 58' 19"-2. 3. An altitude of a star is observed, and found to he the angle whose sine is 1-3 ; calculate the true position of the star, assuming the amount of refraction at an altitude of 45 to he 58"-2 (J. S., T.C.D.). Here the refraction = jfiTtan Z, hut K= 58"-2 for tan 45 = 1 , and tan Z= cot (alt) = \* ; .-. refraction = ^x 58"-2 = 2' 19"- 7. Therefore the true altitude is 2' 19" 7 less than the observed altitude *' 4. The meridian altitudes of a circumpolar star are 20 and 30, and the corresponding corrections for refraction are 1' 40" and 1' 9" ; find the latitude of the place (Degree, T.C.D.). Ans. 24 58' 35"-5. 5. If a, a' he the true and apparent altitudes of a body affected by refrac- tion, prove the equation a = a! 58" -2 cot a'. '6. Find the latitude of a place at which the observed meridian zenith distances of a circumpolar star were 47 28' and 22 18', given that the tangents of these angles are 1*09 and *41 respectively, and taking the coefficient of refraction to be 58"-2. Here (Art. 41) 2 colat = Z + Z' + K (tan Z + tan Z'} ; or 2 colat = 47 28' + 22 18' + 58"-2 (1-09 + '41) = 69 46' + 58"-2 x 1-5 = 69 47' 27"-3 ; .-. colat = 34 53' 43"-6 ; .-. lat = 55 6' 16"-4. CHAPTER V. THE SUN. 44. To an inhabitant of the earth the sun is by far the most important of all the heavenly bodies. His rays supply light and heat not only to the earth, but to the other planets, and his attraction controls their motions, causing them -to describe their respective orbits. It is therefore hardly to be wondered at, that from the most ancient times a body of such splendour, whose influence on earthly affairs was so supreme, should have been an object of great awe and veneration. The sun is an intensely hot and luminous body, distant from the earth by about 92,700,000 miles. The angle which the diameter of his disc subtends at the earth, when measured by a micrometer, is found to have a mean value of about 32'. From this the sun's diameter in miles can be obtained, for 32' x 60 <]__ 206265" " 92,700,000* Prom which we get d the diameter of the sun to be about 860,000 miles, or about 110 times the earth's diameter. As the volumes of two spheres are to one another as the cubes of .their diameters, this would give vol. of sun = (110) 3 x vol. of earth = 1,331,000 x vol. of earth. So that if 1,331,000 spheres like the earth were massed CHAP. V.] DIURNAL AND ANNUAL MOTIONS OF THE SUN. 63 together into one sphere, the resulting volume would about equal that of the sun. The sun's density, however, owing to his physical state, is only about one-fourth that of the earth, from which we conclude that his mass is about 833,000 times the mass of the earth. The Sun's Apparent Diurnal and Annual Motions. 45. We have seen in Chapter I., that besides a compara- tively rapid diurnal motion from east to west which the sun has in common with all the other heavenly bodies, he seems to have a slow motion from west to east among the fixed stars at the rate of about 1 daily, so as to make a complete revolution each year. A mean daily change in right ascen- sion of 1 is equivalent to 4 minutes of time, for 15 corre- sponds to one hour. The mean solar day is thus 4 minutes longer than the sidereal day. We have seen, in Chapter II., that the apparent diurnal motion of the heavenly bodies is really due to a revolution of the earth on its axis, and it will presently be shown that the sun's apparent annual motion in the ecliptic is due to a motion of the earth in an orbit round the sun. On account of the sun's change of position among the fixed stars, the appearance which the heavens present to us each night at a certain fixed hour goes through a regular cycle of changes in the course of the year. For instance, stars and constellations which are visible at, say 11 o'clock at night during winter such as Sirius, Aldebaran, the Pleiades, and the constellation of Orion, will be below the horizon at the same hour in summer. The reason of this is evident if it is borne in mind that 11 P.M. means 11 hours after the sun has been in the meridian ; and therefore, when observed at the same hour each night, each star will have shifted with reference to both meridian and horizon. 64 THE SUN, [CHAP. v. To Trace the Annual Path of the Sun on the Celestial Sphere. 46. On account of the sun's brightness it is impossible, even in an observatory, to see those stars which are at all close to his disc, and therefore the sun's position with refe- rence to them cannot be directly measured. How, then, can the ecliptic be traced out ? To the ancient astronomers, who were without instruments of great accuracy, this was indeed a difficult problem. Hipparchus (160 B.C.) noted the sun's position relative to the moon during the daytime, and then, during the night, he determined the moon's position among the fixed stars, from which he deduced the position occupied by the sun. But in a modern observatory, by means of the transit circle and astronomical clock, we can find the right ascension and declination of the sun's centre, from which we are able to note on the celestial globe his position among the fixed stars. By repeating these observations at noon each day, his annual path can be traced out. When the ecliptic is thus mapped out on the celestial sphere it is found to be a great circle, that, is, its plane passes through the earth, which is situated at its centre. But let us not for a moment suppose that the sun's apparent yearly motion could be explained by supposing it to describe a circle round the earth as centre merely because the projection of that path on the imaginary celestial sphere is a circle. For if the sun were to move in a circle round the earth as centre, the angle subtended by the diameter of its disc should be always the same, that is, supposing that the sun itself does not undergo any change in volume. However, we find that this angle is not constant, but goes through a regular cycle of changes throughout the year, being greatest on the 31st December, when it is 32' 36", and least on 1st July, when it has a value 31' 32". From this it is seen that the sun is nearest the earth on CHAP, v.] EARTH'S ANNUAL MOTION. PROOFS. 65 31st December, and furthest away on 1st July, but that the difference is not very great. From this we may conclude that if the sun moves round the earth, his path must be nearly, but not quite, circular. Apparent Annual Motion of the Sun due to a Motion of the Earth. 47. As the apparent annual motion of the sun in the ecliptic, together with the changes in the seasons, could be explained on the supposition that the earth describes an annual orbit about the sun, we have therefore one or other of two alternatives to choose Either the sun revolves round the earth in an orbit nearly circular ; or The earth revolves round the sun in an orbit nearly circular. That the second explanation is the only admissible one appears from the following considerations : (1) It is known (Chapter VI.) that the planets, which are opaque bodies, receiving light and heat from the sun like the earth, revolve round the sun in orbits nearly circular. That some of these are much larger and some smaller than the earth ; some at greater and some at less distances from the sun; also the earth's periodic time (365| days), and its mean distance from the sun (92,000,000 miles) satisfy Kepler's 3rd Law (Chapter VI.), viz. that the squares of the periodic times of the planets vary as the cubes of their mean distances from the sun. We therefore argue from analogy that the earth, like the other planets, revolver round the sun. (2) We know from dynamical principles that the sun, earth, and planets, on account of their mutual attractions, must either come together or revolve round the common centre of gravity of the whole system. But the sun's mass F OF THE I W I \/ r D e i T \/ 66 THE SUN. [CHAP. v. being much greater than that of all the planets put together, the common centre of gravity of all is a point within the sun not far removed from his centre ; and round this point the earth and planets must revolve. (3) The aberration of the fixed stars (Chapter VIII.) cannot be explained on any other hypothesis except on the supposition that the earth moves round the sun. Parallelism of the Earth's Axis. 48. We find that although the earth revolves round the sun, the position of the celestial pole among the fixed stars remains very nearly constant throughout the year : we there- fore conclude that the axis of the earth is constant in direc- tion, i.e. remains parallel to itself, while the earth moves round the sun. Since the plane of the ecliptic, or, in other words, the plane of the earth's orbit, is inclined to the equator at an angle of 23 28', therefore the earth's axis, which is perpen- dicular to the equator, must be inclined to the plane of its orbit at an angle of 66 32', the complement of 23 28'. The Seasons. 49. The changes of the seasons are due to this constant obliquity of the earth's axis to the plane of its orbit (66 32'). Let fig. 27 represent the orbit of the earth round the sun. NS represents the axis of the earth, of which there are four parallel positions taken corresponding to the summer and winter solstices, and the autumnal and vernal equi- noxes. EQ represents the equator, ab and cd the arctic and antarctic circles, the centre of the earth, and H the sun. Position (1) Winter Solstice (left side of figure}. This represents the position of the earth when the north- ern portion of its axis is turned from the sun, i.e. when the L NOH is greatest, which happens at about 21st December, CHAP. V.] THE SEASONS. 67 when the sun is vertical to the Tropic of Capricorn mn. Since the L bOH = 90, the L NOH therefore = 90 + 23 28' = 11 3 28'. To an observer at the north pole -ZV'this period coincides with the middle of the long night lasting for six months, for it is evident, on looking at the figure, that the diurnal revolution of the earth on its axis could not bring any part not distant from N more than 23 28' into sunlight. (3) (2) . FIG. 27. If we draw a small circle round N at a distance from it of 23 28', just reaching the line of demarcation of light and darkness, this circle coincides with the arctic circle. The reverse is the case round the south pole, where this period corresponds to the middle of the long period of daylight. Similarly, at this period the sun will not set even at 12 o'clock, P.M., to any observer within the antarctic circle. Position (3) Summer Solstice (right side of figure). Here the conditions are reversed : the north pole of the earth is turned towards the sun, such that the L NOR has its least'value, viz. 90 - 23 28' = 66 32'. The sun in this case is vertical to the Tropic of Cancer xij. This period cor- responds to the middle of the six months' daylight at the north pole and six months' night at the south pole. F 2 68 THE SUN. [CHAP, v Positions (2) and (4). These two positions represent the earth at two inter- mediate periods when the plane of the equator passes through the sun, which therefore occupies a position on the celestial equator at one or other of the equinoctial points. Here the L NOH = 90 ; therefore the line of demarcation of light and darkness will pass through the north and south poles of the earth, and day and night are of equal duration all over the world. These two periods are therefore called the two equi- noxes, position (2) corresponding to the vernal, and (4) to the autumnal equinox. Amount of Heat received daily from the Sun. 50. The average amount of heat received from the sun each day in summer is greater than in winter. There are two reasons for this : (1) The sun remains a longer time above the horizon each day in summer than in winter ; and (2) he attains a greater meridian altitude in summer than in winter. But why should we get more heat from the sun when he has a great meridian altitude than when he is low FIG. 28. down near the horizon ? Could the explanation be that he is then nearer to us? No, for he is at practically the same distance from us at noonday when his rays are warm as at sunset, when he seems to give out very little heat ; and CHAP. V.] HEAT FROM THE SUN. 69 moreover lie is, as we have seen, nearer to us at midwinter than at midsummer. The explanation, however, depends on the fact that, when the sun has a great altitude in the sky, his rays strike the earth directly ; on the other hand, when low down near the horizon, they strike obliquely. Why the efficiency of his rays in warming the earth should be greater in the former case than in the latter appears at once from fig. 28. Let 8 represent a point on the sun, SAB and SAB' two cones having equal vertical angles at 8, the former striking the earth directly, the latter obliquely, so that to an observer on the earth situated inside the area AB the sun will appear high up in the heavens, and viewed from a point within A'B' he will appear quite close to the horizon. We may now assume, since the cones have equal vertical angles that equal quantities of heat radiate from 8 along the cones SAB and SA'B', and that therefore the areas AB and A'B' receive the same amount of heat, but the area A'B' being an oblique section of the cone is greater than AB, which is a direct section ; therefore, as the same amount is distributed over both, the quantity of heat per unit of area must be less inside A'B' than AB. This explanation accounts both for the fact that the aver- age amount of heat derived from the sun each day in summer is greater than in winter, and also that, other conditions being the same, the sun should feel hotter at noon on any day than at any other hour. This difference is still further increased owing to more heat being absorbed by the atmo- sphere when the sun is near the horizon, for the rays of the sun have a greater thickness of atmosphere to pass through when coming almost horizontally than vertically. From this we should expect that in northern latitudes June should be the hottest month of the year, and December the coldest. But we generally find that the mean tempera- ture is higher in August than in June, and lower in February than in December. The reason of this is, that during June 70 THE SUN. [CHAP. v. the earth has not had sufficient time to regain the heat lost during the winter ; but for some months after June, the earth gains more heat during the day than it loses at night ; there is, therefore, a continuous increase in the mean tempera- ture until the amounts of heat gained and lost during the twenty-four hours become equal. Again, for some time after the winter solstice, the amount of heat lost during the night exceeds that gained during the day ; therefore, during this period the earth is losing heat, the lowest mean tem- perature being in general registered when the gain and loss during the twenty-four hours become exactly equal. This is the explanation of the old saying : " as the day lengthens the cold strengthens." For the same reason, mid-day is not generally the warmest hour of the day, as there is in general a continuous gain in heat for some time into the afternoon : nor is the coldest period of the night generally reached for some hours after midnight. The mean temperature at any place is, however, greatly influenced by other conditions, such as prevailing winds, insular or continental position, proximity to the gulf-stream, height above sea-level, &c. Rotation of Sun. Sun Spots. 51. When the disc of the sun is observed through a telescope, dark spots are very often seen on its surface. These appear first at the eastern edge, move slowly across the bright face of the sun, and after disappearing behind the western edge, reappear again on the same side as before. Moreover, the times of appearance and disappearance are equal, each being about 13 J days. From these observations we are led to one or other of two conclusions (1) Either they are due to bodies revolving round the sun, so that they, coming between the sun and observer, appear as dark spots projected on the sun's surface ; or CHAP. V.] ROTATION OF SUN. SUN SPOTS. 71 (2) They are due to actual appearances on the surface of the sun itself, the sun rotating round an axis. That the first conclusion is in the highest degree improb- able appears at once. For let FCD (fig. 29) represent the supposed orbit of such a body round the sun. D J ^ r 5 FIG. 29. AP and BP are tangents drawn to the sun from the observer on the earth; these are almost parallel, as the observer is so far distant compared with the diameter of the sun. Now it is evident that the time during which a body moving in the orbit FCD would appear on the sun's surface would be while passing through the arc CD, the time of disappearance corresponding to the arc CFD ; therefore, assuming that the velocity of the body is uniform, the time of appearance would be much less than that of disappearance. But observation shows that these two periods are almost equal. From this we conclude that the sun rotates round an axis. The period of rotation is, however, somewhat less than the apparent period of revolution of the spots, as allow- ance must be made for the motion of the earth in its orbit. The period for the spots is about 27 days, while the sun rotates once in 25 1 days These spots are darker at the centre than round the margins. The dark central portion is called the umbra, surrounding which is the penumbra, of a somewhat lighter hue, apparently composed of radiating filaments. Apart 72 THE SUN. [CHAP, v altogether from the motion due to the sun's rotation they are observed to undergo changes in their size and shape, and after some weeks or months to disappear altogether. " The infe- rence from these various facts is irresistible." (I here quote Sir Bobert Ball.) " It tells us that the visible surface of the sun is not a solid mass is not even a liquid mass but that the sun, as far as we can see it, consists of matter in the gaseous or vaporous condition." " It often happens that a large spot divides into two or more smaller spots, and these parts have been sometimes seen to fly apart with a velocity, in some cases, of not less than 1000 miles an hour." In the case of some of the largest spots the umbra has been found to subtend an angle of 1' 30" at the eye of the observer, which would give a diameter of about 40,000 miles about five times greater than that of the earth. The Sun a Sphere. 52. We have seen that the sun rotates round an axis; we also know that the shape of the disc which he turns to the observer is always circular, for all the diameters, when measured in different directions with a micrometer, are found to be equal. The sun must therefore be a sphere, as no body rotating in the same way as the sun does could always present a circular margin unless it were spherical. Twilight. 53. After sunset a considerable time elapses before com- plete darkness sets in. We call this interval twilight. There is a corresponding interval before sunrise, which we call dawn. Twilight is caused by the diffused reflection of the sun's rays from the upper layers of the atmosphere. After the sun sets, when his rays, on account of the curvature of the earth, can no longer reach us, he still continues to illumine the atmosphere or particles suspended therein, which reflect the light down to us. CHAP. V.] TWILIGHT. 73 Twilight is considered at an end when minute stars of the sixth magnitude can be seen in the zenith. Of course, atmospheric conditions will alter considerably the interval of time after sunset which must elapse before this takes place ; but generally stars of the sixth magnitude appear when the perpendicular distance of the sun below the horizon exceeds 18. Therefore, twilight lasts until the perpendicular distance of the sun below the horizon exceeds 18. Twilight is shortest at the equator. The student will see this at once by referring to the diagram of the celestial sphere for an observer at the equator (Art. 20). The sun's diurnal path here cuts the horizon at right angles, and there- fore he takes a very short time to get 18 below the horizon. On the other hand, for an observer in the British Isles, the sun in setting cuts the horizon at an acute angle (which gets more acute the further north we go), and therefore he has to skim below the horizon a much greater distance, and for a much longer time, before his perpendicular distance below the horizon reaches 18. 54. Twilight at the North and South Poles. At the north pole we have seen (Chap. II.) that the sun remains for about six months below the horizon, from 23rd of Sep- tember until 21st of the following March. He is never, however, during that period, at a very great perpendicular distance below the horizon, the greatest depth being 23 28', which he reaches on 21st December. However, of this six months of so-called night a great portion is twilight, for it will not be altogether dark as long as the sun is within 18 of the horizon. That the period during which twilight lasts will be as great a portion of the six months as 18 is of 23 28' we can by no means say, for the sun's change in declination is not uniform. Assuming, however, that this is the case, we would have about four out of the six months during which twilight lasts, viz. two months after the 23rd 74 THE SUN. [CHAP, v September and two months before the 21st of the following March. Of course, the above will also apply to the south pole during the period when the sun is below the horizon, viz. from 21st March till the 23rd September. To find the duration of Twilight at the Equator during the Equinoxes. 55. During the Equinoxes the sun's diurnal path almost coincides with the celes- tial equator, which, for an observer at the earth's equator, cuts the horizon at right angles, passing through the zenith and nadir. Let S represent the sun at sunset (fig. 30). Let $' represent the sun at end of twilight. We have therefore to find the interval of time POLE N FIG. 30. corresponding to SS' or 18 of his daily course. But 360 correspond to 24 hours ; .'. As 360 : 18 : : 24 h : x, 18x24 360 ours = l h 12 m . 56. To calculate the duration of twilight at any place we have to solve two spherical 'triangles, the three sides being given. . For, let S represent the sun at sunset, let S' represent the sun at end of twilight. Join 8 and S' to zenith and pole by four arcs of great circles. CHAP. V.] TO FIND DURATION OF TWILIGHT. 75 Now, the sides of the A ZPS f are known ; for ZP = 90 - lat = colatitude; ZS' = 90 + 18 = 108, since S' is 18 below horizon, E - declination of sun. But the declination of the sun is known for each day in the year from the Nautical Almanac ; there- fore P/S' is known ; there- fore by solving we are able to calculate the Fl <>. 31 - L ZPS', which is the hour angle of the sun at end of twilight. Similarly, the sides of the A ZPS are known, and there- fore we can solve for the L ZPS, which is the hour angle of the sun when setting. Subtracting these two angles, we get the t_SPS', which measures the duration of twilight. Con- verting this into time at the rate of 360 to 24 hours, or 15 to 1 hour, gives the duration. It is obvious that the duration of twilight depends upon the latitude of the place and the declination of the sun, for these quantities alone are involved in the solution of the above spherical triangles. That is, it depends on the part of the earth at which the observer is situated, and, even in the same place, it varies according to the season of the year. 57. It is evident that twilight cannot last all night at or near the equator, the sun's diurnal path cutting the horizon at nearly a right angle. The question then arises as to what are the conditions which must hold, in order that twilight may last all night : Twilight will last all night at any place, provided the 76 THE SUN. [CHAP. v. latitude of the place plus the declination of the sun is not less than 72. z For let 8 represent the sun at midnight when in E meridian below the horizon, then PE = alt. of pole == lat' H of place = /, and 8Q = decl. of sun = S. Now PQ = 90; that is, ono FIG. 32. But if twilight just lasts all night, SR = 18 at mid- night ; .-. if /+ 72, twilight lasts all night. This rule holds when the latitude of the place and the declination of the sun are both north or both south. When the latitude is north and the declination of the sun south, or vice versa, then the condition becomes /- 8 not < 72 EXAMPLES. 1. What effect would be produced upon the seasons if the earth's axis were in the plane of the ecliptic or were perpendicular to it ? 2. If the declination of the sun be 10, find the lowest latitude at which twilight lasts all night. Here I + 8 = 72, or J+10=72; .-. I = 62. CHAP. V.] EXAMPLES. 77 3. Find the latitude of the place for which twilight just lasts all night when the sun's declination is 16 N. (Degree Exam., T.C.D.). Am. 56 N. 4. How does the duration of twilight at a given place alter with the season of the year ? (S. S., T.C.D.). Am. See Art. 56. 5. Determine the limits of the latitudes of places at which twilight lasts all night long, when the sun's declination is 10 15' N. Ans. At lat. 61 45' and places further north. 6. Find the decimation of the sun when twilight begins to last all night at Dublin (lat. 53 20'). Ans. 18 40' N. 7. Find the lowest latitude at which it is possible for twilight to last all night. Am. 48 32'. 8. Upon what does the duration of twilight depend ? Ans. The latitude of the place and the declination of the sun. 9. Can twilight last all night at Paris (lat. 48 50') ? (See question 7.) Ans. Yes, but only for several nights before and after the summer solstice. 10. Show how, by solving a spherical triangle, the time of sunset or sunrise can be calculated for any place at a given date. Ans. See Art. 56. 78 CHAPTER VI. THE MOTIONS OF THE PLANETS. THE SOLAR SYSTEM. 58. WE mentioned in a previous chapter that those planets which are visible to the naked eye and with which the ancients, who were not possessed of telescopes, were acquainted, are; Mercury, Venus, Mars, Jupiter, and Saturn. If the ordinary observer wish to find out whether a bright object in the sky be a planet or a fixed star he has only to note its position with reference to the neighbouring fixed stars ; for instance, it may happen to be in a line with two stars, or form an equilateral triangle with them. If, after several weeks, the body seems to have altered its posi- tion with reference to these fixed stars, it is probably one of the above planets. Since the invention of telescopes several other large planets, with some hundreds of very small ones, have been discovered. The names of the planets at present known are, in their order, from the sun outwards -^ ,, i Interior Planets. Jbjartn Inferior Planets I Mercury \ Venus ( Mars / The Asteroids Superior Planets 1 day ; 360 99 99 -jT*X9> 99 X fajZ' 9 .*. the L ESV is known and all the angles of the triangle SEV are known ; but SE B o*. the ratio of SJE to 8 Vis found. Similarly the ratio of the distances of a superior planet and the earth from the sun can he found, the proof being the same, the earth in this case gaming on the planet. Definition. The periodic time of a planet or, as it is often called, its sidereal period, is the time taken by the planet to make one revolution round the sun. The synodic period is the interval that elapses between two conjunctions of the same kind (both inferior or superior) or in the case of a superior planet, between two oppositions. To find the Periodic Time of a Planet when the Synodic Period is known. INFERIOR PLANET. 67o Let P = periodic time of planet expressed in days, E= earth T = synodic period ; SfiO v. = L moved through by planet in 1 day, and gr = earth 360 360 , , , - ,, c . -^ --- = angle gained by planet in 1 day, for the Jr MI inferior planet goes at the greater rate ; 88 THE PLANETS. SOLAR SYSTEM. [CHAP. VU 60 =7 360 but =7- = angle gained by planet in 1 day also ; 360 360 360 " P --. XP E T ' but E = 365-25 days, and T being known, P is determined. EXAMPLE. The interval between two inferior conjunctions of Mercury is 116 days : find its periodic time. 1_ I I P 365-25 ~ 116 ' nearly. Similarly for a superior planet we have the formula I_I = I E P " T' \ the earth going at the greater rate. Thus by noting the interval between two inferior conjunctions (or oppositions) of a planet we can calculate its periodic time, supposing its orbit and that of the earth circular. Kepler's Three Laws. 68. Kepler, the Danish astronomer, who lived at the commencement of the seventeenth century, first enunciated the following laws : i. Each planet moves in an elliptic orbit with the sun in one of the foci. ii. The straight line drawn from the sun to a planet (the planet's " radius vector ") sweeps out equal areas in equal times. in. The squares of the periodic times of the planets are to one another as the cubes of their mean distances from the sun. CHAP. VI.] KEPLER S LAWS. Definition. An ellipse is a plane figure bounded by one line called the circumference, such that the sum of the dis- tances of any point on that circumference from two fixed points within it is constant. Those two fixed points are called the foci. Thus (see fig. 39) if JPand F f be the two foci we have EP+F'P = FIG. 39. = a constant. The following is therefore a mechanical method of de- scribing an ellipse : Let two pins be taken and fixed into a plane board or table, say at F and F' (fig. 39), and round these let a loose endless string be thrown. If now a pencil be taken and, keeping the string tightly stretched, let it be carried round, occupying successively the points P, Q, R, &c., the curve traced out will be an ellipse, the two pins JFandP' being situated at the foci. Kepler's Second Law asserts that the line drawn from the sun to a planet sweeps out equal areas in equal times, that is, if the times of de- scribing the distances AB and PQ are equal, then the area 8AB = area SPQ. From this we can conclude that the nearer a planet ap- proaches the sun the greater must be its velocity, for if we regard the arcs AB and PQ as being described in a small unit of time, they, being small compared with the planet's distance from the sun, may be taken as straight lines. Now if the distance from S to AB be greater than from S to PQ, the FIG. 40. 90 THE PLANETS. SOLAR SYSTEM. [CHAP. VI. base PQ must in its turn be greater than the base AB in order that the two triangles may be equal, and therefore the velocity at PQ is greater than at AB. Corollary. As the earth is nearest the sun at mid- winter (Art. 46), we now see that its velocity at midwinter is greater than at any other part of its orbit. Verification of Kepler' 's Laws. 69. As regards the earth it can be seen by measurement of the sun's apparent diameter with the parallel wire or other micrometer that the orbit is not a true circle, and that there- fore the earth's distance from the sun is not constant, being greatest when the angle subtended by the sun's diameter is least. We can now therefore construct the curve which represents the earth's orbit, for if lines be drawn from a point S (fig. 40), the lengths of these lines being inversely proportional to the different angles which the sun's diameter subtends at the earth, measurements of which can be made daily, the extremities of these lines will be found to trace out an ellipse with S in one of the foci. Kepler determined the orbit of Mars before that of the earth had been ascertained. He determined the position of Mars relative to the earth and sun by a method somewhat similar to that in Art. 66, and by this means he arrived at the conclusion that the orbit of Mars was elliptic. The fact that he considered the orbit of the earth as circular in these calculations did not give rise to very serious error, as the eccentricity of the earth's orbit is very small and much less than that of the orbit of Mars. 70. Newton showed that Kepler's Third Law was a direct consequence of the law of universal gravitation which may be enunciated as follows : Every particle in the universe attracts every other particle with a force directly proportional to the mass of each, and inversely proportional to the square of their distance apart. CHAP, vi.] BODE'S LAW. 91 To deduce Kepler's Third Law from the Law of Gravitation. Let M = mass of the sun. Let r and r' be the distances of two planets from the sun whose periodic times are Tand T f respectively. Now by the law of gravitation the attractions of the sun at distances r and r from its centre are in the proportion M M r Z ' ^/2 ' But the centrifugal acceleration of a body moving in a circle of radius r is given by the equation t/ fJl% ' therefore assuming the orbits of the planets circular we have MM 4ir z r 4ir z r' ^ ' V* '' '' ~T r '' T /z ' Multiplying the extremes and means we get eventually r' r r * f'l ~ r '22 1Z ' 7^2 . 7^2 . . *,3 . A/3 * -* * i , which is Kepler's Third Law. Bode's Law. 71. There is a remarkable relation between the distances of the different planets from the sun which bears the name of the astronomer Bode. Write down the following numbers in which each after the first is doubled : 1 2 4 8 16 32 64 128 Now multiply by 3 and add 4 to each and we get 4 7 10 16 28 52 100 196 388 corresponding to Mercury, Venus, Earth, Mars, Asteroids, Jupiter, Saturn, Uranus, and Neptune. 92 THE PLANETS. SOLAR SYSTEM. [CHAP. VI. These numbers are approximately proportional to the distances of the different planets from the sun, that of the earth being 10. There is, however, a serious discrepancy in the case of Neptune, which is represented by the number 388, whereas to represent its actual distance it should 300*369. This law received a remarkable confirmation from the discovery of the Asteroids, which consist of a number of small planets whose orbits lie between the orbits of Mars and Jupiter. They are over 300 in number. Before their dis- covery there was no planet known whose distance from the sun corresponded to the number 28. However this number is found to approximately represent the mean distance of the different asteroids from the sun. Bode's Law can be expressed by means of the general formula D = 4 + 3 x 2 M ~ 1 , where D represents the distance of a planet from the sun, and n the number of the planet beginning with Venus. By giving to n the values 1, 2, 3, &c., the numbers corresponding to the distances of the different planets from the sun commencing with Venus, are found to be the same as those mentioned above. True Distance. Mercury, r ,< \, 4 = 4 3-871. Venus, ,t v 4 + 3x2 = 7 7'233. Earth, 4 + 3X2 1 = 10 10-000. Mars, . ' 4 + 3 x 2 2 = 16 15-237. Asteroids, 4 -1- 3 x 2 3 = 28 22 to 31. Jupiter, 4 + 3x2* = 52 52-028. Saturn, 4 + 3 x 2 5 = 100 95-388. Uranus, . 4 + 3 x 2 6 = 196 191-826. Neptune, . 4+3x2* = 388 300-369. Direct and Retrograde Motion. Stationary Points. 72. A planet's apparent motion is said to be direct when it seems to move in the same direction as the sun in the ecliptic, and retrograde when it appears to move in a contrary direction. In other words, its motion is direct when its longitude is increasing, and retrograde when diminishing. CHAP. VI.] DIRECT AND RETROGRADE MOTION. 93 As the earth E moves in its orbit in the direction indi- cated by the arrow (fig. 41) the sun appears to move as if the line E8 were rotating round E from right to left in a direction contrary to the hands of a watch. Therefore we will, for greater clearness, regard a contra- watch-hand rotation round E as direct, and a watch-hand rotation as \ retrograde, E being supposed fixed. An inferior planet moves with greater velocity than the earth (Art. 47) ; therefore when the planet is at inferior conjunction the extremity V of the line VE moving with greater velocity than E, the line will appear to revolve round FIG. 41. E like the hands of a watch, and therefore the apparent motion of a planet at inferior conjunction is retrograde. At the points of greatest elongation P and Q the planet's own velocity will produce no change in its direction, as it is along the lines EP and EQ, respectively, but the earth's motion will make the lines EP and EQ, appear to revolve in a direction contrary to that of the hands of a watch. Hence the planet's apparent motion at P and Q is direct. At any point on the 94 THE PLANETS. SOLAR SYSTEM. [CHAP. VI. arc P V'Q, the motion will appear direct, for both the planet's own velocity and that of the earth will combine to make the line joining them revolve round E in a direction contrary to that of the hands of a watch. Again, as the planet's motion appears retrograde at V r and direct at P and Q, it must pass through two points m and n, at which the retrograde motion is on the point of changing into direct or vice versa, and at which the planet does not seem to move. These two positions are called the stationary points. A superior planet on the other hand moves with a velo- city which is less than that of the earth ; therefore when the planet is in opposition at the point M (fig. 41) the line EM will appear to rotate round E like the hands of a watch. Hence the motion of a superior planet in opposition is retrograde. Again, when the planet is in quadrature at X and Y the velocity of the earth will have no effect, as it is in the direction of the line joining the observer to the planet. The planet's own motion, however, will cause EX or JEYto revolve round E in a contra- watch-hand direction. Hence the apparent motion of a superior planet in quadrature is direct. Also at any position along the arc JOf'P'both the planet's velocity and that of the earth combine to cause the line join- ing them to appear to revolve with a contra-watch-hand rotation, i.e. the planet's motion will be direct. As the planet appears retrograde at M and direct at JTand P, there will be two points p and q at which the retrograde motion is on the point of changing into the direct, and vice versa, these points being the two stationary points of the planet. Rotations of the Planets round their Axes. 73. We have already seen that the earth and sun rotate. It has also been shown by observing the markings and CHAP. VI.] ROTATIONS OF PLANETS. 95 on the surfaces of the planets that most of them, and probably all, rotate in the same manner. Mars rotates once in 24 h 37 m , so that a day in Mars is almost of the same length as a day on the earth. Jupiter takes 9 h 55 m , and Saturn 10 h 29 m . It is much more difficult to find the period of rotation of an inferior than of a superior planet ; for the latter when in opposition can be seen all night, whereas an inferior planet only appears as an evening or morning star, and observations can only be made at intervals of 24 hours. Now supposing that markings are observed on the surface of Venus after sunset, it is found that they occupy nearly the same position on the following night. From this we might be led to one or other of two conclusions (1) either Yenus makes a revo- lution on its axis in about 24 hours, or (2) it takes a very long period to complete a revolution so that the angle turned through in 24 hours would be very small. In either case it is evident the markings would not be much changed during 24 hours. Until quite recently it was believed that the first conclusion was true. From observations by Schrater the period for Yenus was believed to be 23 h 21 m , and for Mercury 24 h 5 m . Professor Schiaparelli, however, has recently con- tended that Mercury and Yenus take the same time to rotate on their axes as they do to revolve round the sun, the period for the former being 88 and the latter 224 days, and that therefore they turn always nearly the same face towards the sun, just as the moon does to the earth, large portions of each being in perpetual sunlight, and other portions always in darkness. As these planets are only to be seen close to the horizon after sunset or before sunrise the changes in the temperature and density of the lower strata of the atmosphere render it very difficult to observe the markings on their surface with sufficient accuracy to determine the exact truth ; but more recent observations would seem to show that Schiaparelli's conclusion is, at all events, false in the case of Yenus. 96 THE PLANETS. SOLAR SYSTEM. [CHAP. VI. To prove that the Velocities of Two Planets round the Sun are inversely as the square roots of their distances from the Sun. 74. For by Kepler's Third Law we have T z : r 2 ::r 3 :/ 3 ; but circumference of orbit = velocity x time. .-. 2wr = vT-, ., /2wrV /27T/V therefore we have ( - j : ( J : : r z : r z ; r' .. vi v' : : Corollary. Hence of two planets the nearer to the sun has the greater velocity. 75. We shall conclude this chapter with a brief review of the different bodies which constitute the solar system. Mercury ? . This is the nearest planet to the sun. Its diameter is about 3000 miles, being much smaller than that of the earth (e), whose diameter is 8000 miles. The orbit of Mercury is much more eccentric than those of the other principal planets, that is, it does not so nearly approach a circular shape. At one time the planet approaches to within 28,000,000 miles of the sun, and again in the opposite point of its orbit recedes to a distance of 43,000,000 miles. It is also distinguished by the great inclination of its orbit to the ecliptic, namely, about 7. Its periodic time about the sun is about 88 days. CHAP. VI.] TRANSITS OF VENUS. 97 Venus $ . Venus has a diameter almost equal to that of the earth. Its orhit also like that of the earth differs but little from a circle. The inclination of its orbit to the ecliptic is 3 23'. AVe have seen how Mercury and Venus being inferior planets ure only to be seen within certain angular distances on the east or west side of the sun, and are therefore morning or evening stars ; and also that their discs, when seen through a telescope, show phases like those of the moon. In both these planets it has been observed that the line of separation of the light from the dark portions is not continuous but notched, and also that the horns of the crescents they present are sometimes cut off abruptly. This is caused by mountains on their surfaces, which have been calculated to rise in both planets to heights considerably greater than those on the earth. The periodic time of Venus is ^24 -7 days. Transits of Venus and Mercury. 76. We have already seen that the transit of Venus or Mercury can only occur when the planet is in inferior con- junction at or near one of its nodes. If a transit of one of these planets occur at any time another transit at the same node will not occur until the earth and the planet shall have each made an exact number of revolutions. Now 8 revolutions of the earth expressed in days are almost equal to an exact number of revolutions of Venus,. viz. 13, there being only a difference of one day, for 8 x 365-242 = 2922 days nearly, and 13 x 224-7 - 2921-1 days. Hence, if a transit of Venus occur at any time there may. be another at the same node 8 years afterwards if one has not u 98 THE PLANETS. SOLAR SYSTEM. [CHAP. VI. already occurred 8 years before. There will not, however, be a transit 16 years afterwards, as, on account of the above difference of one day, the distance from the node when in conjunction will be too great In fact, a transit at the same node cannot in this case occur for another 235 years, whicli is the next number of years which corresponds to an exact number of revolutions of Yenus, for 235 x 365-242 = 85835 days nearly, and 382x2247 =85835 The first transit of Venus ever observed was that seen by Horrox, in 1639, which occurred at the ascending node. A transit at this node did not again occur for 235 years, viz. in 1874, and again in 1882. Transits at the descending node have been observed in the years 1761 and 1769, the next occurring in the year 2004. Transits of Mercury occur more frequently than those of Yenus, for its periodic time is such that it more frequently happens that an exact number of revolutions of the planet correspond to an exact number of years. Thus transits of Mercury at the same node may happen at intervals of 7, 13, 33, or 46 years. At present the earth in its orbital motion is opposite a node of Yenus on the 5th of June, and again on the 7th of December. Hence, for a very long period of time, transits of Yenus will occur in December and June. For a similar reason transits of Mercury will occur in May and November. Transits of Yenus are of great practical interest, as their observation furnishes the most accurate methods of deter- mining the sun's parallax and distance (Chapter VII.). Transits of Mercury cannot be used in the same manner, as its distance from the earth approaches too nearly that of the sun to give reliable results. Besides it moves too rapidly across the sun's disc to give time for accurate observations ; CHAP. VI.] MARS. 99 ?and also as its orbit does not so nearly approach a circular shape as that of Venus, the ratio of its distance from the sun to that of the earth cannot be so easily calculated. Mars $ . 77. Tho nearest of the superior planets is Mars. Its -distance from the sun varies from 127,000,000 to 153,000,000 miles, and therefore its orbit is much more eccentric than that of the earth. If the orbits of Mars and the earth were both circular the planet would be closest to us at opposition, its distance being then only the difference of the radii of the orbits. But the distance of Mars from the sun, as we have seen above, is very variable, and that of the earth from the sun changesfrom 90,500,000 miles at midwinter to 93,500,000 miles at midsummer. We can, therefore, see how some oppositions are much more favourable for observation than others. For, suppose during opposition that Mars were at its least distance from the sun, and the earth at its greatest distance, the planet would only be distant from us by 34,000,000 miles, and astronomers would then have the opportunity of viewing it under most favourable conditions. If we imagine the two points where Mars is nearest to and at its greatest distance from the sun to be joined to one another, it is found that the earth in its orbital motion passes close to this line on August 26, and again on February 22. On August 26 the earth passes that point of its orbit which j is in a line between the sun and the position which Mars I would occupy when closest to the sun (perihelion] ; and on February 22 it crosses the line between the sun and the point in which Mars would be situated when at its greatest distance from the sun (aphelion). Therefore if we regard the orbit of the earth as being circular (for it is much more nearly so than that of Mars), the nearer the date of an opposi- tion approaches to August 26, the more favourable are the conditions under which the planet can be observed, and the H2 100 THE PLANETS. SOLAR SYSTEM. [CHAP. VI- closer that date is to February 22 the more unfavourable i such an opposition for accurate observation. The periodic time of Mars is about 687 days, and the- inclination of its orbit to the ecliptic is about 2. Two small satellites of Mars were discovered by Mr. Hall, of Washington, during the opposition which occurred on epteinber 5, 1877, when the planet was very close to the earth, the date of tlie discovery being only ten days after the best date possible. They have been named Deimos and Phobos,* the former, which is the outer, completing a revolution round Mars in about 30 h 18 m , and the latter in 7 h 39 m . As Mars completes a revolution on its axis in 24 h 37 m , we have in Phobos an example of a satellite revolving round the primary planet much more quickly than the latter rotates on its axis, a case which is without a parallel in the solar system. At the beginning of this century a number of very small planets were discovered with orbits lying between Mars and Jupiter. They are called the Asteroids. There are at present considerably over 300, the smaller ones being only a few miles in diameter. The four largest are Vesta, Juno, Ceres, and Pallas. Their orbits are generally very eccentric, some of them being also inclined at considerable angles to the ecliptic. Jupiter if. 78. This is the largest of all the planets, its diameter being 11 times that of the earth. Its orbit is nearly circular like that of the earth, and is inclined to the ecliptic at an angle of about 1-^. When observed through a telescope, a number of bright belts or bands are seen encircling it parallel to its equator, which are probably belts of clouds or vapours * In Homer Deimos and Phobos are represented as the attendants of Mars. The passage in the Iliad which first suggested the names of the satellites he been thus construed by Professor Tyrrell : " Mars spa"ke and called Dismay and Rout To yoke his steeds, and he did on his harness sheen." II. 15, 119, 120. CHAP. Vic] SATURN. 101 in its atmosphere. It has five satellites or moons, which may he seen with a good field glass. The periodic times of these five moons, and their mean distances from Jupiter, satisfy Kepler's third law, which we have seen is true for the orbits of the planets round the sun. This is true of the satellites of all the planets. They are frequently eclipsed when they enter the shadow cast hy Jupiter on the side opposite the sun. An eclipse must not, however, be confounded with an occultation which happens when a satellite is in a line with Jupiter and the earth so as to be hidden from the observer's view. Again, a very curious phenomenon is observed when a satellite comes between Jupiter and the sun. The shadow cast by the satellite will then be observed as a dark spot moving across the face of Jupiter, which is indeed a wonder- ful sight, illustrating, as it does, the appearance which the arth would present if viewed from Mercury or Venus during a total eclipse of the sun by our satellite the moon. A transit of the satellite may also occur when the satellite is in a direct line between Jupiter and the earth. Saturn J? . 79. The orbit of Saturn is also very nearly circular. It is inclined to the ecliptic at an angle of about 2 j. At certain periods Saturn when viewed through a telescope presents a most wonderful appearance. It is surrounded by a series of circular rings which do not touch the surface of the planet ; indeed through the interval between the rings and the body of the planet fixed stars are sometimes seen. The plane of these rings is inclined at a constant angle of about 28 to the plane of Saturn's orbit, and, therefore, they being seen obliquely by us, will not appear circular but oval. They are supposed to be formed of immense numbers of small satellites. They become invisible (1) when the plane of the rings, when produced, passes through the earth, for being very thin when the edge is turned towards us, it is not possible to see 102 THE PLANETS. SOLAR SYSTEM. [CHAP. VJ. them ;e&0ept through the most powerful telescopes ; (2) when their plane passes through the sun, for they, deriving their light from the sun, have only their edge illuminated; (3) when their plane passes between the earth and the sun, for their dark surface being towards us it is not possible to* see them., Saturn has, besides, eight satellites, all situated external to the rings. The seven nearest move in orbits whose planes almost coincide with the plane of the rings, but that of the eighth is inclined to this plane at an angle of about 10. 7 FIG. 42. Phases of Saturn's Rings. Uranus y. Uranus was discovered in 1781 by Herschel. Its orbit is nearly circular, and inclined at a very small angle to the ecliptic. Four satellites have been discovered which revolve in orbits nearly perpendicular to the plane of the orbit of Uranus. Neptune ^ , The discovery of Neptune is one of the most brilliant in the history of Astronomy. It was found that the positions which it was calculated Uranus should occupy, after making allowance for all known disturbing forces, did not coincide with the observed positions. It was therefore thought that there must be some unknown planet whose attraction produced these disturbances. After the most laborious calculations the CHAP. VI.] COMETS. 103 position which this unknown body should occupy was deter- mined at almost the same time by Leverrier in France, and Adams in England, in the year 1846. One satellite of Neptune has been discovered. Comets. 80. The solar system includes a number of other bodies which differ widely from the planets, both in their physicaj state and in the nature of the orbits described by them round the sun. These bodies are called comets. Comets differ very much as regards their shape, and even the shape and size of the same comet may change considerably at different parts of its orbit ; but we generally find at one end a brilliant nucleus surrounded by nebulous matter stretching out into an elon- gated tail. The tails of some comets which have appeared have been of enormous dimensions; that of 1811 had a tail 23 in length, another in 1843 had a length of 40, while that of 1618 extended across the sky through an arc of 104. Comets generally appear suddenly in the sky, remaining visible for some weeks, or months, during which time they approach the sun with great velocity ; they then recede from it, and finally disappear from view. The mass and density of comets are extremely small ; it is even possible to see faint stars shining through them almost as if no material body were interposed between. By far the greater number of comets describe orbits of such great eccentricity that we may regard them as parabolas described round the sun as focus, the other focus being practically at an infinite distance. But there are a few comets whose orbits, although much more elongated than those of the planets, are sufficiently small to be contained within the solar system. The motions of these can be cal- culated and the dates of their return predicted from knowing 1 he magnitudes of the ellipses which they describe. These .ire called periodic comets. 104 THE PLANETS. SOLAR SYSTEM. [CHAP. VI. The orbits of comets, besides being much more eccentric, also differ from those of the planets in that they may be inclined at any angle to the plane of the ecliptic ; moreover, all the planets go round the sun in the same direction as the earth moves, whereas the motion of some comets is direct and of others retrograde. Periodic Comets. 81. H alley's Comet. Of the periodic comets, that known as Halley's is, perhaps, the most remarkable. It was observed that the comets which appeared in the years 1531, 1607, and 1682 were almost identical as regards the position of the nodes, the perihelion distance from the sun, the incli- nation of the orbit to the plane of the ecliptic, and certain other measurements, from which Halley concluded that they were really one and the same comet, having a periodic time about the sun of 75 years, and he therefore predicted its return in 1758. Clairaut, having calculated that, owing to perturbations caused by the attractions of Jupiter and Saturn, it would be retarded 518 days and 100 days respectively, predicted that it would be closest to the sun, or, in other words, at perihelion, at about the middle of April, 1759. No allowance was made for disturbances caused by the attractions of Uranus and Neptune, as these planets were not then discovered. It actually appeared at the end of 1758, and reached the perihelion at the middle of March, 1759. Halley's Comet has since appeared in 1835, and it may be again expected in 1910. One of the previous visits of Halley's Comet was on a very memorable occasion in the year 1066, the date of the Norman conquest ; the picture of this comet is depicted in the Bayeux tapestry. Encke's Comet. This periodic comet is also known to describe an elliptic path round the sun. At perihelion it is closer to the sun than Mercury ; and at aphelion, at its CHAP. VI.] COMETS. 105 greatest distance, it is not altogether as far from the sun as Jupiter; so that its orbit is well within the limits of the solar system. Its periodic time is about 3| years. The motion of this comet has been most carefully observed, and the perturbations in its movements due to the attractions of the earth and the other planets have been calculated. But it is found that after making allowance for all these disturb- ing forces, there is still a diminution in its periodic time of FIG. 43. Orbits of some Periodic Comets. 2J hours in each successive revolution. Encke accounted for this by supposing that there exists for a considerable distance round the sun a medium which, although of extreme tenuity, is still capable of offering sufficient resistance to the passage of a body of such small density as a comet as to appreciably diminish its periodic time. Non-periodic comets are much more numerous than peri- odic. To this class belonged the great comet of 1843, Donati's (jomet, which appeared in 1858, and the comet of 1881. 106 THE PLANETS. SOLAR SYSTEM. [eHAP. VI, Meteors or Shooting Stars. 82. In addition to the different members of the solar system which we have already enumerated there are an innumerable number of minute bodies which are called meteors. When these bodies, which move with great velocity, impinge on the earth's atmosphere in a direction opposite to that in which the earth is moving, the relative velocity is so large that the heat developed by the resistance of the air is sufficient to consume them, and they appear as a streak of light in the sky. Others, whose relative velocity is not so great, on rare occasions, fall to the earth unconsumed. These are called meteorites or meteoric stones. The heights of meteors have been found to vary from 16 to 160 miles. Although it is possible to see many stray meteors on almost any night, there are three periods in the year at which they occur in very considerable numbers, viz. August 9-11, November 12-14, and November 27-29. Radiant Point. During these August and November meteoric showers the apparent paths on the celestial sphere of most of the meteors seem to spring from one common point called a radiant point. This is merely an effect of perspective. For, as we will shortly see, all the meteors which compose the swarm through which the earth happens to be passing are, for the short period they are under observation, approxi- mately moving in parallel straight lines. If now we imagine planes drawn through these lines and the observer they will cut the celestial sphere in a number of great circles all having a common diameter, viz. the line drawn through the observer parallel to the common direction of the motion of the meteors. This line when produced will cut the celestial sphere in the two common points of intersection of all the circles, one of which is the radiant point. The radiant point for the August meteors is in the constellation of Perseus, and those for the two showers which take place in November are in .CHAP. VI. j COMETS AND METEORS. APSIDES. 107 the constellations of Leo and Andromeda respectively. Hence we have the three showers The Perseids, Aug. 9-11, radiant point in Perseus. The Leonids, Nov. 12-14, in Leo. The Andromedes, Nov. 27-29, in Andromeda. Connexion between Comets and Meteors. 83. The meteors which we see dashing into the atmo- sphere at about the 14th November are believed to t>e portions of a train of an innumerable number of minute bodies whose orbits are almost identical with that of Temple's Comet. The orbit of this comet actually cuts that of the earth, the earth arriving each year at the point of intersection on the 14th November. There are portions of this elliptic belt where these minute bodies are crowded together into groups; and on certain occasions, separated by long intervals of time, the earth passes through one of these groups or shoals, as, for instance, on November 13, 1866, when the appearance of the heavens, lit up by myriads of meteors, was of the most wonderful description. As the periodic time of this shoal is 33 years, it was fully expected that an equally brilliant display would be observed in November, 1899. But although a few individual members of the band were observed from several places on the earth, the result, for reasons which we can at present only con- jecture, was very disappointing. The showers known as the Andromedes and Perseids can be similarly accounted for, the former being due to the fact that the orbit of Biela's Comet cuts that of the earth at a point corresponding to November 27. 84. Line of Apsides. Those points where the earth, or a planet, in its orbit is nearest to and most remote from the sun (perihelion and aphelion) are called apsides. The earth 108 THE PLANETS. SOLAR SYSTEM. [CHAP. YI. is nearest the sun at the apse A (fig. 44) on the 31st December, and arrives at the opposite apse B on the 1st July, the line AB being called the line of Apsides. This line evidently coincides with the major axis of the ellipse. N.B. The sun's apparent diameter is greatest when the earth is at the apse A, and least when at B. Also as observed from two positions E and E f such that the L ASE = L ASE' the sun will have the same apparent diameter, for from the symmetry of the ellipse it is evident that the distances SE and 81? are equal. FIG. 44. To find the Direction of the Apse Line. It might at first suggest itself that the direction of the line of Apsides could easily be found by noting the position of the sun in the ecliptic when its apparent diameter is least or greatest. It is, however, very difficult to tell when this occurs, as the apparent diameter remains very nearly constant for some time before and after the earth's passage through the apse. The follow- ing is the method employed: When the earth is at E some considerable time before the apse is reached the sun's apparent diameter is measured and its position in the ecliptic noted by calculating its longi- tude. The longitude is again noted when its angular diameter CHAP. VI.] APSIDES OF EAKTH's ORBIT. 109 measures the same as before,, the earth being at E'. The mean of these two longitudes will give the point on the ecliptic occupied by the sun during the earth's passage through the apse, the line joining this point to the earth giving the direction of the line of apsides. Slow Motion of the Apse tine. By observing the position of the apse line for a number of years it is found that it has a slow direct motion in the plane of the ecliptic at the rate of 11*25" each year. 85. Length* of the Seasons. If P and Q represent the positions of the earth at the two solstices, a perpendicular ^ firing ///>//, Autumn / FIG. 45. XT erected at 8 to PQ (fig. 44) will give the positions X and Y of the earth at the vernal and autumnal equinoxes respectively. The orbit of the earth is thus divided into the fou, arcs XQ, QY, FP, and PX, corresponding to the four seasons, Spring, Summer, Autumn, and Winter, respectively. The four seasons are unequal in length, spring and summer lasting from 21st March till the 23rd September, being about 8 days longer than autumn and winter, which last from the 23rd September till the 21st of the following March. This inequality can very easily be explained from Kepler's Second Law, thus : Let AB represent the line of solstices (fig. 45), which for 110 THE PLANETS. SOLAR SYSTEM. [CHAP. VI. simplicity is supposed to coincide with the apse line, XY being the line of equinoxes, and CD the axis minor of the ellipse, i.e. perpendicular to AB erected at the centre of the ellipse Now since CD bisects the area of the ellipse we have area CBD = area CAD ; area CBD is > area XAY; .'. d fortiori area XB Y is > area XA T. But since equal areas are described in equal times, it follows that the combined length of spring and summer is greater than that of autumn and winter. ri The lengths of the four seasons are as follows : Spring. Summer. Autumn. "Winter. 92 d 20J h . 93 d 14J h . 89 d 18J h . 89 d OJ h . Eccentricity of the Earth's Orbit. 86. Definition. The ratio of the distance of the centre of the ellipse from the focus to the semiaxis major is called the eccentricity. Thus C\ Q Eccentricity E = - (fig. 45). (JA We can express the eccentricity of the earth's orbit in terms of the greatest and least apparent diameters of the sun. For, since 08-*(8B-8A) 9 ! _ SB-SA "' SB + SA But SA and SB are inversely proportional to the apparent diameters of the sun at A and B respectively. Therefore, if d and d' represent these diameters, we have _ d-d' Jo d + d' Therefore the eccentricity of the earth's orbit is equal to the CHAP. VI.] GENERAL EXAMPLES. Ill difference between the greatest and least apparent diameters of the sun divided by their sum. Example. The greatest apparent diameter of the sun being 32' 36" and the least 31' 32", calculate from this the eccentricity of the earth's orbit. d-d' 1956-1892 1 Here 1956 + 1892 60 GENERAL EXAMPLES. 1. A planet is found to have an elongation from the sun of 150. Is it an inferior or superior planet ? Ans. Superior (Corollary, Art. 59). 2. A planet is found to be in quadrature. Is it inferior or superior? Ans. Superior. 3. Two planets are observed through a telescope. One appears as a thin crescent, the other appears dichotomized. State whether they are inferior or superior planets. Ans. Both inferior (Art. 64). 4. If the exterior angle at a planet formed hy lines drawn from the earth and sun he 120, find what part of the hemisphere which is turned towards the earth is illuminated. Ans.. frd (Art. 61). 5. In question 4 find the ratio of the apparent hreadth of the visible illu- minated portion at its widest part to the apparent diameter of planet's complete disc (Art. 62). Here apparent hreadth _ r versin 120 _ r(l-cosl20) _ r (1 + 1) _ 3r _ 3 diameter of planet 2r 2r 2r ~ 4r ~ 4 6. Find what should he the radius of a planet's orbit in order that its greatest elongation from the sun, as seen from the earth, should be 30, assum- ing the distance of the earth from the sun as 92,000,000 miles. Ans. 46,000,000 miles. 7. Calculate approximately in miles per second the velocity of the earth in its orbit (J. S.,.T. C. D.). Ans. 18-3. 8. If there be 378 days between two successive oppositions of Saturn ; find the length of Saturn's year. (Degree, T. C. D.). Here -- and solving, we get P= 10828-6 days, = 29-6 vears. 112 THE PLANETS. SOLAR SYSTEM. [CHAP. VI. 9. The periodic time of Mercury being 88 days ; find the interval between two successive inferior conjunctions of this planet. 1 1 1 88" 366T5 "F cind solving, we get T = 115'9 days. 10. Assuming the mean distance of Venus to he -72, that of the earth being^ unity, apply Kepler's Laws to find the periodic time of Venus. Here, by Kepler's Third Law, we have T 2 : T' 2 : : r 3 : r' 3 , or T 2 : (365-25J 2 :: (-72) 3 : (1)3; .-. T= v/(365-25) 3 x (-72) 3 = 223 days nearly. 11. Assuming the distances of the different planets from the sun as given by Bode's Law, calculate from this the periodic time (1) of Mercury, (2) of Saturn. Am. (1) 90-11 days, (2) 11550-3 days. 12. Supposing a planet were to revolve round the sun at a distance of half a million miles, find what should be its periodic time. Am. 3| hours nearly. 13. Why do comets move with much greater velocity when at perihelion than at other parts of their eccentric orbits ? (Kepler's Second Law). 11. The two satellites of Mars have periodic times, which are about 30 hours and 7 hours respectively ; find the ratio of their mean distances from, Mars. Since Kepler's Laws apply to the motions of the satellites, we have : T 2 : T* : : t* : r'* ; that is, (30) 2 : (7|) 2 : : * : r*, or 4 2 : I 2 : : r 3 ; r' 3 ; .-. -J/16: 1 :: r : r'. Hence their mean distances from Mars are in the ratio of Z/IG to 1 or 2J/2 tol. 15. The velocity of Mercury in its orbit is 30 miles per second ; hence calcu- late the velocity of Saturn. Here, v : v' : : V^ : ^/r, or, 30 : v' : : \/100: v/4 (Bode's Law), or, 30 : v' : : 10 : 2 ; 10 v' = 60 ; .. v' = 6 miles per second. 16. Why was September 5, 1877, when the satellites of Mars were dis- covered, a date particularly favourable for observing that planet? (Art. 77.) CHAP. VI.] NAMES, PERIODS, ETC., OF THE PLANETS. 113 Arcs which they retrograde. o o o o o o o 2^-2 b co ^-i bi b cb i>- o o 1 o 1 o o o o t CO i * t-H O< O I-H Velocity in miles per second. He Hot H|n fN CO 55 f< -! 1 li 00 O CO Tt< CO COOJOO |ot-(fllCO ^H 1 -M 04 -* Tj< 1 1 11 Oft |>. 10 *O ^ O O) CO O) 1^- i 1 OO 00 O4 t- CD O CO CO O cbi>-o>ocbcMb^*o C/} xxc+e^o 5* c a* B H S Illllllll S n r^ CHAPTER VII. PARALLAX. 87. Definition. By the diurnal parallax of a heavenly body is meant the angle subtended at the body by that radius of the earth which is drawn to the observer. Thus, if Cbe the centre of the earth, the observer, the parallax of the body M is the angle subtended by CO at M, viz. the L p. The fixed stars are so very far away that we may regard the lines joining one of them to the observer and to the centre of the earth as being so nearly parallel that the parallax is practically zero. To illus- trate how small this angle becomes, let the reader take a marble one inch in diameter, and try to imagine what angle its radius could subtend at a point, say 1000 miles away. The most delicate instrument we possess would be unable to measure it ; and yet this angle is more than one hundred times as great as the angle which the radius of the earth could subtend at even the nearest fixed star. The planets, however, as well as the sun and moon, are comparatively so near us that this difference in the direction of the lines drawn from a point on the surface and from Fm. 46. [CHAP. VII.] EFFECT OF PARALLAX. 115 ithe centre of the earth to the planet is large enough to be (measured. Also the directions in which these bodies are observed Ifrom any two positions on the earth's surface are not exactly jthe same. All observers therefore, wherever situated, reduce ieir observations to what they would be if situated at the mtre of the earth. This reduction is what is called the \correction for parallax. The declinations, right ascensions, ;o., of bodies which we see noted in the Nautical Almanac :e those which they would have if seen from the centre of Ithe earth. Definition. The horizontal parallax is the parallax of body when on the horizon. Thus, if the body M be on FIG. 47. the horizon of the observer 0, the L P is the horizontal [parallax. 88. Tbe effect of Parallax on a heavenly body is depress it in the heavens. For, if be the position of the observer (fig. 46), then OZ, the production of the radius drawn to 0, is the direction of the zenith, and the L z is the zenith distar.ee of the body i2 116 PARALLAX. [CHAP. vii. M as seen from 0. Also the L z' would be its zenith distance if the observer were at the centre of the earth ; but LZ = LZ' + LP\ that is, apparent zenith distance = true zenith distance + parallax ; therefore, as seen from 0, the body appears lower down in the heavens than if seen from C. To find the parallax of a body for a given zenith distance. a = radius of earth (fig. 46), D - CM= distance of body. Since, from Trigonometry, we know that the sides of the A CO M are as the sines of the opposite angles; sinp a *' sin (180-*) "5 sinp a or -T -=T;; sm 2 D a . .v sin P=JJ sins; but p being in all cases a very small angle, therefore sin p = p (expressed in circular measure) ; When the body is on the horizon z - 90, and p becomes the horizontal parallax P ; therefore substituting, we have p = P sin z, or, parallax = horizontal parallax x sine of apparent zenith distance. CHAP. VII.] LAW OF PARALLAX. 117 Hence tbe parallax of a heavenly body varies as the sine of its apparent zenith distance. As sin s is a maximum when a = 90, we see that the parallax is a maximum when the body is on the horizon. EXAMPLES. 1. Supposing the sun's observed altitude to be 60 and the parallax 4"-4, find his true altitude. Here, since parallax depresses a body, true altitude = observed altitude + parallax : therefore true altitude = 60 + 4"-4 = 60 0' 4"-4. 2. Given the moon's horizontal parallax as being 57' 6", find its true altitude corresponding to an observed altitude of 60. Here p = Psin and s = 90 -60 = 30 ; .-. p = (57' 6") sin 30 = (57' 6") = 28' 33"; therefore true altitude = 60 28' 33". 3. The sun's horizontal parallax being 8"-8, find the true zenith distance corresponding to an observed zenith distance of 60. Here p = Psin z, or 9 = 2 therefore true zenith distance = 60 - 7"'6 = 59 59' 52"-4. Given the horizontal parallax of a body, to find its distance, and vice-versa. We have just seen that P = -=;, but P is expressed in circular measure. Hence, if expressed in seconds, we have : _ 206265" ~Z> 118 PARALLAX. [CHAP. Vi EXAMPLES. 1. Given that the moon's horizontal parallax is 57' 6" ; find its distance from the earth, the earth's radius heing 4000 miles. Am. About 240,000 miles. 2. The sun's horizontal parallax heing 8"- 8, find its distance from the earth. Ana. About 93,700,000 mile?. 3. The moon's distance heing 60 times the earth's radius, find the moon's horizontal parallax. P' a I Here 206265" D 60 ' .-. P=57'17". 89. The displacement of a heavenly body due to parallax like that from refraction is in the direction of the vertical drawn through the body. Hence the azimuth of a body is not affected by either parallax or refraction. "We have seen, (Art. 39) that refraction does not depend on the distance of the body from us, for the rays only get bent on their entrance into the atmosphere; the parallax, however, becomes less the greater the distance of the body, the moon's hori- zontal parallax being about 57', while that of the sun, which is much further away, is only about 8", and the fixed stars are so remote that their parallax is zero. All bodies ex- cept the moon are much more elevated by refraction than depressed by parallax. For instance, horizontal refraction amounts to about 34', whereas the sun when on the horizon , as we have seen above, is only depressed by parallax through about 8". For the moon, however, parallax is much greater than refraction ; hence the combined effect of both in this case produces a depression. To find the Angle which two distant places on the Earth's Surface,, nearly in the same Meridian, subtend at the Moon or a Planet. 90. Let A and B be two distant places on the earth, a& nearly as possible in the same meridian (Greenwich and the Cape of Good Hope are favourably situated for the purpose) ; CHAP. VII.] MOON S PARALLAX DETERMINED. 119 M represents the moon or planet when in the meridian of A and B. Let a fixed star be observed from A and S, in nearly the same part of the heavens as M 9 so that their right ascensions and declinations differ very slightly. The lines joining A and B to the star are nearly parallel, the star being so distant. FIG. 48. The angles a and ]3, the angular distances of the star from Jf, are measured at A and B respectively by means of micrometers; but and L $ = L a by parallel lines ; .-. z.0 = Za + /l|3, and a and /3 being known, 9 is determined. If the two places A and B are not in the same meridian, then the two observers, not making their measurements at the same time, a correction must be made for the small dis- tance moved by the moon or planet (owing to the orbital motion) in the interval between its passages over the two meridians. To find the Horizontal Parallax of the Moon or a Planet. 91. Two positions A and B (fig. 49) are chosen as before on the same meridian of the earth in the northern and southern hemispheres respectively. The meridian zenith 120 PARALLAX. [CHAP. vii. distances of the moon or planet M are then measured simultaneously by means of the meridian circle. Let these be z and 2', the lines O^and OZ' being the directions of the FIG. 49. zenith at A and B respectively ; also let P be the horizontal parallax. Now (Art. 88) we have p = -=r sin z = P sin 2, and .p' = -=r sins' = P sins'; .% p + jf = P (sin s + sin 2') ; sn z + sn s but p +p' is known, being angle subtended at M by A and B (Art. 90), and s and 2' being observed, the horizontal parallax P can be found. This method is free from any serious errors due to refraction, for in Art. 90 the moon and fixed star are nearly in the same position in the sky, and therefore almost equally affected by refraction, and therefore the value of p + p' is got with great accuracy. CHAP. VII.] SUN'S PARALLAX DETERMINED. 121 The above result for the horizontal parallax can be put in another form. For, draw the equator EQ, and let / and t be the latitudes of A and B respectively ; .-. L ZOQ = I, and L Z'OQ = I 9 but (Euclid, i. 32), L Z + L z = L ZOM + L Z M + L p + Z/, or z + z = I + I' + p + p' : .*. p+p' =z + z' - I- l f \ z'-l-l' . , .'. by substitution P = - sin z + sin z It is not possible to determine the sun's parallax in this manner, for, owing to the intensity of his rays the neighbour- ing stars cannot be observed. It can, however, be calculated indirectly. For, let the parallax of Mars when in opposition be observed by the above method, from which the distance of that planet from the earth can be found (Art. 88), this dis- tance is the difference of the distances (r and /) of Mars and the earth from the sun or r - r'. But the ratio of r to r' is known from Kepler's Third Law ; hence we can solve for / the distance of the earth from the sun, and the sun's parallax is determined by Art. 88. But the most accurate methods of obtaining the sun's parallax, and hence his distance from the earth, are from observations of the transit of Venus across his disc (Art. 76), as follows : Delisle's Method of finding the Sun's Parallax. 92. Two stations A and B (fig. 50) are chosen, both near the earth's equator, but separated as far apart as pos- sible, the circle AB being the equator of the earth. Let us now suppose, in order to simplify the explanation, that the sun and the orbit of Venus VV are in the plane of the equator AB. Draw tangents AS and BS to the sun. 122 PARALLAX. [CHAP. vii. The observer at A notes the instant at which internal contact takes place, which occurs when Venus is at F, touching the line AS internally ; a similar observation is made at B, internal contact occurring when Yenus is at F'. FIG. 50. The time of each observation is reduced to Greenwich time, which corrects for the difference in longitude of A and B. The difference of the two results will give the interval of time during which Yenus appears to move round the sun through the angle VSV. (The reader must remember that we are supposing the earth to be at rest, and that Yenus has an angular velocity round the sun equal to the excess of its real angular velocity over that of the earth.) But the rate at which the angle VS V is described is known, being 360 in each synodic period : we can therefore calculate the angle VS V or ASS.* Thus, knowing the angle which two distant places on the earth's surface subtend at the sun, we can calculate the sun's horizontal parallax as in Art. 91, and hence his distance from the earth. In actual practice a great many difficulties have to be met, the principal being the inclination of the orbit of Yenus to the ecliptic. The point S is here taken as practically coincident with the sun's centre. CHAP. VII.] STINTS PARALLAX DETERMINED. 123 In Delisle's method the longitudes of the places have to he very accurately known. In the following method, pro- posed hy Halley in 1716, it is not necessary to know the longitudes of the places, as only the duration of the transit is observed at each place, the fact that the clocks indicate different hours being of no consequence. Halley' s Method or the Method of Durations. 93. In this method the duration of the transit is observed from two places A and B on the earth, separated as far apart as possible, one in a high northern and the other in a high southern latitude so that there may be as large a dif- ference as possible in the observed length of time during which the transit lasts, as seen from the two places. B FIG. 51. Let V represent Yenus, the plane containing A, B, and V being the plane of the paper, while the reader must bear in mind that the plane of the circle, of which S is the centre, and which represents the sun's disc, is at right angles to this plane. To an observer at A, Venus, in her apparent motion in the direction indicated by the arrow, will appear to cross the sun's face in the direction cd 9 and to the observer at B in the direction ab, the time occupied being noted in each case. But the rate at which Venus appears to cross the sun's face can be calculated (Ex. 4, p. 125), being at the rate of 4" in each minute of time, we can therefore, by a statement in simple proportion, calculate the number of seconds in ab and cd 124 PARALLAX. [CHAP. VII. respectively, very much more accurately than if measured by a micrometer. Therefore the halves of these chords, viz. eb and fd are known in seconds. But the sun's semidiameter sb or sd is also known in seconds ; therefore we can find the number of seconds in se and sf, for we have approximately, by Euclid (I. 47), se* = sb*-be z ; also sf z =s(^-df z . Knowing se and s/, we find, by subtraction, the number of seconds in ef. Again, the number of miles in efcan be found from know- ing the number of miles between the two stations A and B, for, regarding the two triangles A VB and eVf as similar, we have But the ratio of F/to AV can be found (Art. 66), being 723 to 277 ; therefore ef in miles is known. Lastly, knowing efin miles and the angle in seconds subtended by it at the earth, the distance of the sun can be found thus : ef" efin miles 206265 = sun's distance' and consequently his parallax is determined. To find the Radius of the Moon in Miles. 94. Having shown how to determine the parallax of the FIG. 62. moon, sun, or a planet, we can calculate the radii of these CHAP. VII.] MAGNITUDE OF MOON OR PLANET DETERMINED. 125 bodies in miles by a comparison with the radius of the earth. Let r = radius of earth. / = radius of moon, or other body. P = moon's horizontal parallax = earth's angular semidiameter as seen from the moon. P* = moon's angular semidiameter. Now -3 = P (in circular measure), - = P (in circular measure) ; or (radius of earth) : (radius of moon) : : (moon's parallax) : (moon's semidiameter). EXAMPLES. 1. Taking the moon's horizontal parallax as 57', and its angular diameter as 32', find its radius in miles, assuming the earth's radius to he 4000 miles. Here moon's semidiameter = 16' ; .-. 4000 : / : : 57' : 16' ; .-. r' = 400 * 16 = 1123 miles. &7 2. The sun's horizontal parallax heing 8"-8, and his angular diameter 32', find his diameter in miles. Ans. 872,727 miles. 3. The synodic period of Venus heing 584 days, find the angle gained in each minute of time on the earth round the sun as centre. Ans. l"-54 per minute. 4. Find the angular velocity with which Venus crosses the sun's disc, assuming the distances of Venus and the earth from the sun are as 7 to 10, as given by Bode's Law. Since (fig. 50) 8V: VA : : 7 : 3. But /SThas ^.relative angular velocity round the sun of 1"'54 per minute (see Example 3) ; therefore, the relative angular velocity of AT round A is greater than this in the ratio of 7 : 3, which gives an approximate result of 3"-6 per minute, the true rate heing about 4" per minute. 126 PARALLAX. [CHAP, Annual Parallax. 95. We have already seen that no displacement of the observer due to a change of position on the earth's surface could apparently affect the direction of a fixed star. How- ever, as the earth in its annual motion describes an orbit of about 92 million miles radius round the sun, the different positions in space from which an observer views the fixed stars from time to time throughout the year must be sepa- rated from one another by very great distances indeed. For instance, any two diametrically opposite points in the orbit of the earth are separated by an interval of about 184 million miles, the earth proceeding from one of these points to the other in about six months. "We should therefore expect that, when viewed from two points separated by such a distance as this, the fixed stars should not occupy exactly the same position with respect to one another, those which are nearer the earth being more displaced than those further away. This is to a certain extent quite true ; but to such vast depths are the fixed stars sunk in space, that only in the case of some of those nearest to us can any appreciable displacement be detected ; or, in other words, a base line of 184 million miles is much too small a distance to take in an attempt to measure the distances of by far the greater number of these bodies. Owing to the small displacements in the apparent direc- tions of some of the fixed stars, due to the earth's changes of position throughout the year, we refer their directions on the celestial sphere to what they would have if viewed from the centre of the sun, which is fixed. This direction, as seen from the centre of the sun, being called the star's helio- centric direction, the correction, which must be applied to reduce the apparent or geocentric to the heliocentric direction being called the correction for annual parallax. /UNIVERSITY) V or J VJ^LIFORNV'W' CHAP. VII.] ANNUAL PARALLAX. 127 Definition. The annual parallax of a star is the angle subtended at the star by the line joining the earth and sun. Thus, if E represent the earth, E the sun, and 8 a star, the annual parallax of 8 is the angle subtended at 8 by EH or the angle/?. 96. The law according to which the annual parallax of a star should vary can be deduced by a method similar to that applied to the diurnal or geocentric parallax of the moon or planets. For sin p r smE = d' T .*. sin^=-sin^; but^> being small, sin^> =p (in circular measure), j0=-sin-E': a therefore, the annual parallax varies as the sine of the angular distance of the sun from the star. It is evident that the parallax of a star is a maximum when E = 90, which happens twice a year for each star. Let P represent this maximum value, and we have P = -sin 90=^; alsop = P sin E. Ct v N.B. G-enerally speaking, by a star's parallax is meant this maximum value of the parallax, unless otherwise stated. 128 PARALLAX. [CHAP. VII. As P is expressed in circular measure in the above formula; therefore, when expressed in seconds, we have P" r 206265 " d Thus, when the parallax of a star is known, we can deduce its distance from the solar system as r the distance of the earth from the sun is known (Art. 92). In the following questions, r may he taken as 92 million miles. EXAMPLES. The parallax of a Centauri being 0"-8, find its distance from the solar system. -8" _ 92000000 206265 " d ' 92000000 x 206265 .. d = - miles. *0 2. Supposing the parallax of a star to be 0"'2, find how long a ray of light would take to travel to the earth, being given the velocity of light as 190,000 miles per second. Ans. The effect of annual parallax on a star is to cause it to appear to move in a small ellipse throughout the year. 97. For as each displacement of the earth in its orhit produces a corresponding small displacement in the appa- rent position of the star in the sky, the star will therefore seem to describe a small yearly orbit round its heliocentric position (which is fixed) parallel to the plane of the earth's orbit. If now we assume the earth's orbit to be circular, we will consider the effect of parallax on a star, according as it is situated (1) near the pole of the ecliptic, (2) on the ecliptic, (3) at any point of the sky. (1) If a star be situated in the pole of the ecliptic, the plane of the small arc which we may suppose it to describe being at right angles to our line of sight will, when projected on the celestial sphere, still appear circular. CHAP. VII.] ANNUAL PARALLAX. BESSEL*& METHOD. 129 (2) If situated on the ecliptic, it will appear to move back and forward along the ecliptic in a straight line, this being explained by the fact that a circle seen edgeways appears as a line. (3) If situated in any other part of the sky, the apparent path throughout the year will appear as a diminutive ellipse, as a circle seen obliquely will appear elliptic. To determine the Annual Parallax of a Star Bessel's Method. 98. Bessel's method, otherwise called the differential method, consists in choosing a very faint star very close to the star whose parallax is sought. Being very faint, it is FIG. 54. presumably much further away than the star in question ; and we may, therefore, assume that its own parallax is so very small that any changes which take place throughout the year in the angular distance of the two stars from one another must be due almost entirely to the parallax of the near one. The actual measurement of these changes enables us to determine the annual parallax. N.B. The faint star is chosen very close to the star, 130 PARALLAX. L CHAP - V11 ' whose parallax we want, in order that both may be equally affected by errors due to refraction, aberration, &c., so that we have not to correct for these errors. The following will illustrate the method employed : Let A and B be two diametrically opposite points in the earth's orbit. AS and BS(ftg. 54) are the directions of the star S, as viewed from A and B\ and AS' and BS f the direc- tions of the faint star, these lines being taken as parallel, owing to the much greater distance of S'. A, B, S, and S' are supposed in the same plane. At A the observer measures, by means of the micrometer or heliometer, the angle a between S and $'; and again at B, six months afterwards, he measures the angle |3. But by Euclid (i. 32), we have but L = L |3 by parallel lines ; but a and |3 are known, therefore 6 is determined ; and 6 being the angle subtended by the diameter of the earth's orbit, is twice the annual parallax which can therefore be found. N.B. The reader can compare this method with that employed to find the angle subtended at the moon by two distant places on the earth's surface (Art. 90) by means of which the moon's diurnal or geocentric parallax was deter- mined. Strictly speaking, the lines AS' and BS' have some inclination to each other; therefore, the error to which Bessel's method is open is, that it determines not the parallax of the near star, but the difference in parallax of the two stars ; hence the parallax of a star thus deter- mined is always too small, but never too great, and CHAP. VII.] ANNUAL PARALLAX OF JUPITER. 131 therefore the distance of a star from us is found to be greater than is actually the case. Bessel, by this method, first measured the parallax, and hence the distance of 61 Cygni in the year 1838, and in the following year that of a Centauri was found. Instead of measurements with the micrometer, photo- graphy is now being very successfully employed in noting the changes in the angular distance of the two selected stars from one another. Absolute Method. 99. This method consists in measuring the star's right ascension and decimation when in the meridian at different times throughout the year, and after making, with as much accuracy as possible, all the corrections for precession, nutation, &c., the different results are compared together when any small differences which they may show give sufficient data to calculate the annual parallax of the star. EXAMPLES. 1. () Where must a star be situated so as to have no displacement due to parallax ; (b) where must it be situated so that the effect of parallax may be greatest. Ans. (a) In a line with earth and sun. (b) At an angular distance of 90 from sun. 2. If the parallax of 61 Cygni be 0"'5, find the parallax of a star which is ten times as far away from our solar system. Ans. 0"'05. 3. The parallax of a Centauri being 0"'75, compare its distance with that of 61 Cygni, whose parallax is 0"'5. Ans. (dist. a Centauri) : (dist. 61 Cygni) : : 2 : 3. To find the Annual Parallax of Jupiter. 100. As the distance of Jupiter at opposition is more than four times as great as that of the sun, its, diurnal parallax is therefore very small, so that it is not possible to observe K2 132 PARALLAX. [CHAP. VII. it with the same degree of accuracy as in the case of Mars (Art. 91). Its annual parallax, however, may be found thus : Let $, E, and J represent the sun, earth, and Jupiter, respectively when Jupiter is in quadrature, i.e. when the angle SEJ is a right angle. Again, let $, E f , J' be their positions when Jupiter is again in quadrature, the earth having moved through Eff and Jupiter through J>F. FIG. 55. The number of days between the two observations being known, we therefore know the LESE' described by the earth in that time. Similarly, we know the L JSJ'. There- fore, the /.a, which is half the difference of these two angles (the two triangles SEJ and SE'J' being equal in every respect), is known. But the angle a is the complement of the L 9. Therefore is found ; that is, the angle subtended at Jupiter by the radius of the earth's orbit is known. Again, n which determines the distance of Jupiter. This method also applies also to any planet outside the orbit of Jupiter. It can be easily shown that if T represent the synodic CHAP. VII.] EXAMPLE. 133 period of Jupiter, and Q the interval between its eastern and western quadratures, the annual parallax 90 C for ^^ = gained by earth on Jupiter in Q days ; 180Q , fl ,~ 9 .'. (fig. 55) 2a = = 180) / 2O\ .-. annual parallax 6 = 90 - -^-^ = 90f 1 -^ V EXAMPLE. The interval between eastern a-.id western quadratures of Jupiter is 175 days, and between two oppositions 400 days ; find the annual parallax of this planet. Am. 11 15'. ( 134 ) CHAPTER VIII. DETERMINATION OF THE FIRST POINT OF ARIES. PRECESSION, NUTATION, AND ABERRATION. 101. The first point of Aries being the zero point from which the right ascensions of all heavenly bodies are mea- sured, it is therefore necessary to know its position with reference to the fixed stars with very great accuracy. Once having fixed this point, and the astronomical clock being set at zero when it crosses the meridian, then the time at which any other star crosses the meridian will, on being reduced to degrees (by allowing 15 for each hour), give the right ascension of that star. It is evident that if we could find independently the right ascension of any one star, the position of the first point of Aries is immediately determined, and, conse- quently, the right ascensions of all other stars. The following method was first used by Flamsteed, the star selected being a Aquilse. Flamsteed 9 s method of finding the Right Ascension of a Star. Let a be the star whose right ascension is sought (fig. 56) : we have, therefore, to find X T (X being the foot of the declination circle drawn through a). The declination of the sun SM is measured (Art. 34) at noon on some day shortly after the vernal equinox, this being done by measuring his meridian zenith distance. At the same time the interval between his transit across the meridian and that of the star a is noted, this interval being the difference of their right ascensions MX, which we will denote by a. CHAP, viii.] FLAMSTEED'S METHOD. 135 Again, it can be ascertained at what time the sun shall have an equal declination shortly before the Autumnal Equinox by observing his meridian zenith distance at noon on successive days previous to September 23rd. But here we will have to do a little calculation ; for it is very impro- bable that the sun will have an equal declination exactly at noon on any one of these days ; but we can observe his declination at noon on two successive days, at which, in one case, it is greater, and, in the other case, less than SM ; and -Equator FIG. 56. then, if we assume that for short periods his changes in right ascension and declination are proportional to one another, we can, by a simple statement in proportion, calcu- late the exact time at which his declination SiN" shall be equal to SM. In this case also the difference between his right ascension and that of the star a is noted. This difference is NX, which we will call |3. It is evident that Let x - the required right ascension T X of star, fj. = right ascension of sun at S = MV ; /. 180 - fj. = right ascension of sun at Si = but XT - jffr - MX, or x - n = a ; also x- (lSO-fjL) =/3, or ^-180 + w = 6. 136 PRECESSION, NUTATION, ABERRATION. [CHAP. V11I- Thus we have two simultaneous equations, the unknown quantities being x and /x. Adding, we get but a and |3 being known, x is therefore determined. The above formula is open to some error, as during the interval between the two observations there is a slight increase in the right ascension of the star, owing to pre- cession. It can, however, be corrected as follows : Let p = the increase in the R.A. of a star dining the interval, and our two equations become x - p = a, and x+p- The uncorrected value of x gives, not the star's right ascension during the first observation near the vernal equinox, but its mean value between the two observations or the value it would have at about the 21st June. The advantages of Flamsteed's method are that it is not necessary to know exactly the sun's declination ; it is quite sufficient to observe when the declinations at the two observations are equal, so that any uncer- tainty in the latitude of the place due to instrumental errors, which will affect both observations equally, is of no consequence. Also, as the sun has nearly the same zenith distance during each observation, he will be equally affected by refraction and parallax, and hence these errors are avoided. CHAP. VIII.] OBLIQUITY OF EL1PTIC TO EQUATOR. 137 Determination of the Obliquity of the Ecliptic to the Equator. 102. This angle is measured by observing the meridian zenith distances of the sun at the summer and winter solstices. Let these be z and 2', and let the latitude of the place be /. Now if S be the position of the sun at one of the solstices, its declination SM is equal to the inclination w of the ecliptic to the equator (fig. 57), for the angle between two great circles is measured by the arc they intercept on a circle perpendicular to both. FIG. 57. But latitude = zenith distance declination* (Art. 34) ; .*. I = z + o> for summer solstice, also ? = z' - M for winter solstice. Subtracting, we get u> = / and, therefore, cu is found. In the above observations, it is not probable that the sun will be exactly at the solstitial point when in the meridian, but allowance can be made for his change of declination during the interval. * This equation is obviously identical with that given in Art. 34, viz. colat 8 = o. 138 PRECESSION, NUTATION, ABERRATION. [CHAP. VIII. Precession of the Equinoxes. 103. Eepeated observations of the right ascensions and decimations of the stars extending over a long period of time show us that the first point of Aries is not a fixed point in the sky, but has a very slow movement among the fixed stars along the ecliptic in a direction opposite to that of the yearly motion of the sun. This backward motion of Aries to meet the sun, in which the first point of Libra also takes part, causes the equinoxes, as it were, to precede their due time each year. Hence this slow movement is called the precession of the equinoxes. The rate of precession is 50"*24 in one year, or about 1 in 72 years. The time taken by Aries to complete one revolution of the heavens would, therefore, be about 26,000 years ; for 360 x 60 x 60 _ = ' JearS y ' Owing to precession, the longitude of each fixed star increases at the rate of 50"*24 each year. The right ascen- sions and declinations of the stars are also found to be slowly changing, but their latitudes remain almost constant. From this latter consideration we are led to the con- clusion that the ecliptic is very nearly fixed in the heavens, but that the equator must be slowly shifting on the ecliptic, thus causing their intersections T and === to move in the manner described above. This movement of the equator on the ecliptic is accom- panied by a corresponding gradual displacement of the celestial pole, which describes a circular path round the pole of the ecliptic at a distance from it of 23 28', one revolution being completed in 26,000 years. Hence, in some thousands of years the star which is now our pole star will be at a considerable distance from the celestial (HAP. VIII.] CAUSES OF P RECESSION. 139 pole. The bright star a Lyrse will, in about 10,000 years, be distant about 5 from the point round which the heavens will then seem to revolve, and will, like our present pole star, appear almost stationary in the sky. Physical Cause of Precession. 104. The precession of the equinoxes is almost entirely caused by the attraction of the moon and sun on the pro- tuberant portions of the earth at the equator^ If the earth's shape were perfectly spherical, the attractions of the sun and moon could each be represented by a single force passing through its centre, and would, therefore, not disturb the axis of rotation of the earth nor the plane of the equator. However, the shape of the earth is spheroidal, not unlike that of a sphere with an additional layer or belt of matter round the equatorial regions. In the adjoining figure, let 8 represent the sun, and PP f the axis of rotation of the earth (fig. 58). B 5^0. 58 Now, the attraction of the sun on the protuberant por- tions of the earth being greater on the nearer than on the more remote side, the resultant attraction will, therefore, be represented by a single force, OB acting at a point above the centre of gravity C, the effect of which would be to cause a disturbance of the axis of rotation of the earth. Qn first thoughts we might imagine that this would 140 PRECESSION, NUTATION, ABERRATION. [CHAP. VIII. result in a change in the plane of the equator, so as to make it eventually coincide with the ecliptic, and set the earth's axis at right angles to the plane of the ecliptic. And certainly this would be the case were it not that the earth is at the same time rotating rapidly round its axis, the resultant effect of these two rotations being that the axis of the earth is indeed disturbed, but in such a manner as not to alter its angle of inclination to the plane of the ecliptic. In fact, the axis of the earth has, as it were, a slow " wobbling " motion, so that the point in the heavens to which it is directed, viz. the celestial pole, describes the circle round the pole of the ecliptic, which we have pre- viously mentioned. FIG. 59. This motion of the earth's axis can be very well illus- trated by the "wobbling" of the axis of rotation of a spinning-top. The weight of the top acting vertically downwards tends to pull the axis of rotation AB away from the vertical ; but if the top be spinning sufficiently rapidly, it will not fall to the ground, but, as we all know, the axis of rotation describes a cone round the vertical AO, keeping at a constant angle to the ground in precisely the same CHAP. VIII. J NUTATION. 141 manner as in the case of the earth the celestial pole, which is the extremity of its axis, revolves round the pole of the ecliptic. The disturbing effect of the moon's attraction is more than twice as great as in the case of the sun, the ratio being as 7 : 3, the reason of this being on account of the greater proximity of the moon to the earth. In either case, the disturbance is greatest when the attracting body reaches its greatest north or south decli- nation, and is zero when the body is on the celestial equator. The precession caused by the sun and moon is some- times called the lunar-solar precession. It amounts to 50"*35 annually. This lias, however, to be diminished by a very small amount called the planetary precession, which, acting in the opposite direction, is found to be 0"-11 each year, and leaves an annual general precession of 50"'24. The planetary precession is caused by the action of the planets, which tends to disturb the earth's orbit, and therefore the plane of the ecliptic, producing at the same time a diminution of the obliquity of the ecliptic to the equator of about half a second each year. However, this change of obliquity will never exceed a certain fixed limit, viz. about 1 y on either side of the mean value. At present the vernal equinoctial point, though still retaining the name " First point of Aries," is not in the constellation of Aries, but, owing to precession, has shifted about 30 into the neighbouring constellation Pisces. Also the autumnal equinoctial point is not now in the constellation of Libra but in Yirgo. Nutation. 105. So far we have dealt with precession as if the celestial pole moved uniformly in a circle round the pole of the ecliptic, and this would certainly be the case if the disturbance due to the attractions of the sun and moon were 142 PRECESSION, NUTATION, ABERRATION. [CHAP. VIII constant ; but in consequence of a want of uniformity in this disturbance the celestial pole really describes a wavy path (see fig. 60). This nodding, as it were, of the celestial pole to and from the pole of the ecliptic is called nutation. The result is, that the precession is sometimes more and at other times less than its mean value, and there also results a small periodic increase and diminution of the obliquity of the ecliptic to the equator, according as the celestial pole P approaches or recedes from the pole of the ecliptic. FIG. 60. Nutation is almost altogether caused by the variable action of the moon depending on the position of the moon's nodes (the points where its path cuts the ecliptic) which make a complete revolution of the heavens in 18f years. The wavy motion of the celestial pole may be graphically represented in the following manner : Bound the mean position of the celestial pole as centre (fig. 60) describe a small ellipse ab, with a major axis ab = 18"-5 directed towards the pole of the ecliptic, and a minor axis 13"*7 along the circle mn ; then if we imagine the CHAP. VIII.] VELOCITY OF LIGHT. ABERRATION. 143 mean pole which is the centre of the ellipse, to move along the arc mn, the true pole P will move in the ellipse round it as centre, completing a revolution in 18f years. Bradley first discovered nutation by observing that, after correcting for aberration, &c., the apparent displacements in the fixed stars with reference to the equator and ecliptic could not be accounted for on the supposition of a uniform precession. The Velocity of Light. 106. That the propagation of light is not instantaneous was discovered by Eoemer, in 1675 from observing the eclipses of Jupiter's satellites. The times at which these eclipses should occur were predicted from a great number of previous obser- vations, and would therefore correspond to the mean distance of the planet from the earth. It was found, however, that when Jupiter was in opposition, or, in other words, nearest to the earth, the eclipses appeared to occur about eight minutes before the calculated time. On the other hand, when Jupiter was in superior conjunction or furthest from the earth the observed time was about eight minutes later than that pre- dicted ; from which it appeared that this difference of about sixteen minutes, or, more accurately, sixteen minutes and thirty- six seconds, was the time taken by a ray of light to move through the diameter of the earth's orbit. Taking this distance as 185,000,000 miles we get that the velocity of light is about 186,000 miles per second. As the velocity of the earth in its orbit is 18 J miles per second, we see that the velocity of light is about 10,000 times greater than that of the earth. We see from the above that light takes about 8 m 18 s to pass from the sun to the earth. This interval is sometimes called the equation of light. The velocity of light has since been measured directly by M. Fizeau, and afterwards by M. Foucault. 144 PRECESSION, NUTATION, ABERRATION. [CHAP. VIII. Aberration. 107. The Aberration of Light is the apparent displace- ment in the directions of the heavenly bodies due to a combination of the velocity of the earth with that of light. The velocity of the earth, although small compared with that of light, is still large enough to produce a sensible deflection in the direction of the rays of light coming to us, so that the direction in which we have to point a telescope in order to observe a star is not the same as if the earth were at rest. "We may illustrate the effect of aberration in the following manner: A man standing still in a shower of rain when the drops are falling vertically will, in order to shield him- self, hold an umbrella right over his head. But if he proceed to walk or run he will find that the drops seem to strike him in the face, so that he has to hold the umbrella before him. Also the more he increases the rate at which he is moving the greater will be the deflection in the direction of the rain drops. This deflection we might call the aberration of the rain. Effect of Aberration on a Star. 108. Let (fig. 61) be the position of the earth ; draw OA a tangent to the earth's orbit at 0, and cut off OA to represent v, the velocity of the earth. Then let OS be the direction of a star and, produce SO to B, so that OB may represent F", the velocity of light. Now in order to be able to consider the question as if the earth were at rest, let us apply a velocity equal and opposite to v to both the earth and to light. This leaves the relative motion unaltered. The point is thus reduced to a state of rest, while the light may be supposed to have two velocities 00 and OB which give a resultant velocity OD. The star will therefore appear in the direction OS' the production of OD and the angle SOS' or a, which measures the amount of the displacement, is called the aberration of the star. CHAP. VIII.l EFFECT OF ABERRATION. 145 Definitions. (1). The angle a between the real and apparent directions of the star is called the aberration. (2). The angle between the real direction of a star and the direction of the earth's motion is called the earth's ivay. Thus the L BOA or the L E is the earth's way. From the above it is seen that the effect of aberration is to displace a star in the direction of the earth's motion. As the direction of the earth's motion, being a tangent to its orbit, must be at right angles to the direction of the sun ; therefore, at any instant the earth seems to be moving to a point on the ecliptic 90 behind the sun ; and the displacement of each star in the heavens, owing to aberration, takes place along the great circle joining its position on the celestial sphere to this point. Since the sun's apparent motion in the ecliptic is from west to east, and as the longitudes of all heavenly bodies are measured from Aries in the same direction, hence the point on the ecliptic 90 behind the sun is that point whose longi- tute is less than that of the sun by 90. Thus if the sun's longitude be 120, all stars will aberrate towards that point of the ecliptic whose longitude is 30. FIG. 61. 146 PRECESSION, NUTATION, ABERRATION. [(3HAP. VIII, Aberration Varies as the sine of the Earth's Way. 109. We have, in the triangle OCD, _ amCOD~ CD' V sin a or . , sin j3 .-. sin a = .ZTsin ]3 ; but a being small, sin a = a (in circular measure), and /3 may be taken equal to the angle E or the earth's way, as they differ by a very small amount ; .*. = aberration = K sin E. K is called the coefficient of aberration, and, when expressed in circular measure, may be defined as the ratio of the velocity of the earth to that of light. If the aberration be expressed in seconds, we have 206265 = V * .-. " = 20"-6 sin E nearly. Therefore, the coefficient of aberration expressed in seconds is about 20"-6, a more accurate value being 20"*49. It is evident that the aberration is a maximum when the earth's way = 90 ; .-. maximum aberration = 20"-49 sin 90 = 20"'49. EXAMPLE. A star in the ecliptic has a longitude of 75, obtain the change in the position of the star owing to aberration, when the longitude of the sun is 135, assuming the constant of aberration to be 20"-49. Here the angular distance of star from sun = 135 - 75 = 60 ; .*. the earth's way = 30 since the direction of earth's motion is at right angles to direction ofsun; .-. a =JST sin ^ = 20" -49 sin 30 = 10"-245. 110. The effect of aberration is to cause each star to appear to describe a small ellipse round its true position in HAP. VIII.] LAW OF ABERRATION. 147 the course of a year. This can be shown in somewhat the same way as in the case of annual parallax. For we may regard the earth's orbit as being approximately a circle, and that its velocity throughout the year is uniform. We may therefore suppose each star to move in a circle, parallel to the earth's orbit, round its true position as centre. But when this imaginary circle is projected obliquely on the surface of the celestial sphere it becomes an ellipse of which the semi-axis major is parallel to the ecliptic, and equal to 20"-49 (the maximum aberration), the semi-axis minor being 20"*49 sin /, when / is the latitude of the star. Therefore, summing up, we have (1) Each star aberrates towards a point on the ecliptic 90 behind the sun. (2) The displacement varies as the sine of the earth's way. (3) A star situated at the pole of the ecliptic (that is, with latitude = 90) will, in the course of a year, appear to revolve round its true position in a circle whose angular radius is 20"*49. (4) A star situated on the ecliptic (that is, with zero lati- tude) will appear to oscillate through an arc on the ecliptic of 20" '49 on either side of its true position, the total annual displacement being 40"*9. (5) In general, a star whose latitude is I will, throughout the year, appear to describe a small ellipse round its true position as centre, the semi-axis major being 20"'49, and parallel to the ecliptic and the semi-axis minor 20"*49 sin I. The student will easily see that these results differ con- siderably from those obtained in the case of annual parallax, although they have some points of similarity. For the annual parallax of a star depends on its distance from us, whereas the constant of aberration is the same for all stars, irrespective of their distances. Also in the particular case, then a star is either in the same part of the celestial sphere L 2 148 PRECESSION, NUTATION, ABERRATION. [CHAP. VIII. as the sun or on the diametrically opposite point, the annual parallax is zero, but the aberration is a maximum. The displacement due to parallax takes place towards the sun, and that due to aberration towards a point on the ecliptic 90 behind the sun. 111. The aberration of a planet differs somewhat from that of a star, being due to two causes (1) That due to the velocity of the earth, and (2) to the velocity of the planet. If the planet's motion were equal to that of the earth, and in the same direction, there would be no aber- ration. In general, it is easy to calculate the aberration due to those two causes separately. As the velocity of the moon about the earth is very small compared with the velocity of light, we may regard the aberration due to this velocity as zero. Neither is there any aberration on account of the earth's orbital motion round the sun, for this motion is shared in by the moon. "We may, therefore, regard the moon as having practically no aberration. Discovery of Aberration. Bradley was first led to the discovery of aberration while attempting to find the annual parallax of y Draconis. Observing that the latitude of this star was subject to small annual variations for which he could not account by attributing them to any known cause, he was eventually led to adopt the above explanation. 112. Diurnal Aberration. Owing to the earth's rotation on its axis, a point on the equator turns through 25,000 miles in 23 h 56 m . This is at the rate of ^th of a mile per second, or g^th of the velocity of the earth in its orbit. Any other point on the earth not on the equator will have, of course, a less velocity than this. The aberration due to this motion is called diurnal aberration. It is, however, as we can easily see by com- paring the above velocity of rotation with that of light, so small as to bo almost inappreciable. ( 149 ) CHAPTER IX. THE MOON. 113. Next to the sun the moon is to us the most important of all the heavenly bodies. Besides its diurnal motion from east to west, which is imparted to it in common with all the other heavenly bodies in consequence of the rotation of the earth on its axis, it has, like the sun, a motion among the fixed stars in the opposite direction, making a complete revolution of the heavens in about 27 d 7 h 43 m . As the sun appears to make a complete revo- lution of the ecliptic in one year, we see that the moon's motion among the fixed stars is about thirteen times faster than that of the sun.. So rapid is this motion, that its change of position with respect to bright stars in its neigh- bourhood can be easily seen, even after as short an interval as two or three hours. The moon's path, on being mapped out on a celestial globe, is found to be represented by a great circle, cutting the ecliptic at an angle of 5 9', from which it follows that, like the planets, it is always to be found near the ecliptic, its north or south latitude never exceeding 5 9'. The moon's motion among the fixed stars is due to an orbital motion round the earth. In fact, the moon is the earth's satellite. We must not, however, suppose that its orbit round the earth is a circle, because the projection of this orbit on the celestial sphere, on being traced out, is represented by a great circle. No, for just as in the case of the sun, we find that the moon's distance from the earth is not constant. We are led to this conclusion by the fact 150 THE MOON. [CHAP, ix, that its angular diameter, on being measured at different times by means of a micrometer, is found to undergo periodic changes, which shows that its distance from the earth must be changing also, being least when the apparent diameter is greatest. Its greatest angular diameter is 33 J' ; least 291', and the mean 3iy, or a little more than half a degree. These changes in the apparent angular diameter lead us to the conclusion (1) that the moon's orbit round the earth is approximately elliptic with the centre of the earth situated in one of the foci, (2) the radius vector joining the centres of the earth and moon sweeps out equal areas in equal times. From this we might infer that the moon's motion among the fixed stars is not uniform. In fact, it varies from a maximum of 33' 40" per hour to a minimum of 27', its mean hourly velocity being 32' 56". So that we may say that the moon in its motion among the fixed stars moves through an arc equal to its own diameter in one hour. The mean distance of the moon from the earth is 238,000 miles, or about 60 times the earth's radius. As this distance is much less than the radius of the sun, which is 110 times the radius of the earth (Art. 44), we see that if the sun were placed with its centre at the centre of the earth its mass would extend considerably beyond the moon, a consideration which will perhaps enable the mind to form some idea of the magnitude of the body which forms the centre of our system. The Moon's Phases 114. One of the most interesting phenomena to be seen in the heavens is the series of changes which the visible portion of the moon's illuminated surface presents during its orbital motion about the earth. These appearances are called its phases. They prove that the moon is an opaque spherical body deriving its light from the sun. As only one CHAP. IX.] MOON'S PHASES. 151 hemisphere of the moon can be illuminated at once, viz. that half which is turned towards the sun, an observer will therefore see a variable amount of this bright surface depending on the relative positions of the sun, moon, and earth. Let ACMD (fig. 62) represent the orbit of the moon, E the earth, and S the direction of the sun. In the eight positions of the moon, which we have here depicted, the line mn, which is perpendicular to the direction of the sun, O FIG 62. separates the illuminated half of the moon from the unillu- minated half, and all the positions of mn are drawn as if parallel to one another, the sun being so far distant. The line ab may be taken as separating the half of the moon which is turned towards the observer from that which is turned away from him. When the moon is in conjunction at A, its dark hemi- sphere is turned towards the earth, and no portion is visible to the observer. It is then said to be new moon. 152 THE MOON. [CHAP. ix. Some four or five days afterwards, when the moon is at By the observer will see a small portion of the illuminated surface which will appear as a thin crescent in the sky, seen in the west after sunset. When the moon is at C, 90 from the sun ; that is, in quadrature, it will appear in the sky as a bright semicircle. This is said to be first quarter, and the moon is then said to be dichotomized. At D it is gibbous, and when in opposition at M, which occurs at about 15 days after conjunction, the whole of the illuminated hemisphere is turned towards the observer. The moon will then present a complete circular disc in the sky. This is said to be full moon. After full moon, these phases are repeated in reverse order, the moon being again in quadrature at G, which is called third quarter, and finally, conjunction is once more reached at A. When in conjunction and opposition, the moon is said to be in syzygy. Its elongation from the sun is then and 180, respectively. When in quadrature at first quarter its elongation is 90, and at third quarter 270. Definitions. (1). The time taken by the moon to make a complete revolution with reference to the fixed stars is called its periodic time or sidereal period. This period is 27 d 7 h 43 m . (2), The interval between two successive conjunctions or oppositions, or, in other words, the time taken to make a complete revolution with reference to the sun is called ,the synodic period or a lunation. This period is 29 days, or, more accurately, 29-5305887 days. It is obvious that if the sun had no apparent motion in the ecliptic, the synodic and sidereal periods would be exactly the same, so that the full moons would follow one another at intervals of 27 d 7 h , instead of 291 days. But while the CHAP. IX.] SYNODIC PERIOD DETERMINED. 153 moon is making a complete revolution round the earth, which it does in 27 d 7 h , the sun moves through an arc of about 27 on the ecliptic in the same direction (roughly at the rate of 1 daily), so that it takes the moon an additional two days to arrive at the same position relative to the sun and earth as when it started. In the above diagram illus- trating the phases of the moon we have, for the sake of simplicity of explanation, supposed the sun and earth fixed, and that the moon moves with its relative velocity with respect to the sun, completing the revolution in 29^- days. To determine the Moon's Synodic Period. 115. We know that when an eclipse of the moon takes place, the moon must be in opposition. Therefore, if we observe the exact interval of time that elapses between the middle of two eclipses, and divide by the number of luna- tions between them we get the length of a single lunation or synodic period. The mean length of a lunation can be calculated very accurately from the records of ancient eclipses. The earliest observations of eclipses of which there is an accurate account are those taken at Babylon in the years 720 and 719 B.C. The number of lunations between one of these eclipses and an eclipse at the present day being known, we are able to calculate the mean value of a lunation over a very long period of time. To find the Moon's Sidereal Period. 116. Knowing the moon's synodic period, we are able to calculate its sidereal period, or periodic time, in the same way as in the case of a planet (Art. 67). In fig. 63 E represents the earth, the inner circle being the orbit of the moon, and the outer circle the apparent orbit of the sun about the earth. A and B are the positions of the 154 THE MOON. [CHAP. ix. moon and sun at conjunction, and A' and B' their positions one day after conjunction. Let S = Period of sun's motion about earth = 365^ days. P = Moon's periodic time or sidereal period. L = Interval between two conjunctions = 2S-J-J- days. but FIG. 63. ' -p- = L described by moon in 1 day = L AEA', Q/2A ~- = L described by sun in 1 day = L 360 360 "" ~P --- 8~ = L day = = L gained by moon in 1 day, also: 360_360 _36Q P ' ~S~ ~L 1 or - 1 P 365-25 29-5 CHAP. IX.] METONIC CYCLE. 155 therefore, solving for P, we find the periodic time to be 27 d 7 h nearly. A more accurate value for the periodic time is 27 d 7 h 43 m 11 s , while that of a lunation is 29-5305887 days. Metonic Cycle. 117. Meton first discovered, B.C. 433, that 19 years expressed in days is an almost exact multiple of a lunation, for 365-25 x 19 = 6939-75 and 29-5305887 x 235 = 6939-688. So that in every 19 years there are almost exactly 235 luna- tions. Therefore, at the end of every 19 years the sun and moon, returning to the same positions with respect to the fixed stars, all the phases of the moon will occur again on the same days of the month as for the previous 19 years, the only difference being that they will occur about one hour sooner. This is called the Metonic Cycle. The discovery of the Metonic cycle was of considerable importance, as it afforded a ready method of predicting the dates of the full moons, etc., without the trouble of calculation. It has been much used in order to find the date on which Easter should fall in a given year, because this festival occurs on the Sunday following the first full moon after the 21st March. For this reason, the nineteen numbers, from 1 to 19, are called the golden numbers. The golden number, or the number in the Metonic Cycle, for any year, is the remainder got after dividing the year increased by unity by 19. Thus the golden number for 1901 is the remainder when 1902 is divided by 19, i. e. 2. Where zero is the remainder, then 19 is the golden number. Apparent area of illuminated Surface of Moon. 118. It can be shown in exactly the same way as in the case of a planet (Art. 62) that the apparent area of the bright portion turned towards the earth is proportional to 156 THE MOON. [CHAP. ix. the versed sine of the exterior angle subtended at the moon by the earth and sun. Thus, if (fig. 64) M represent the moon, E the earth, and 8 the sun, the external angle at the moon is the angle a ; therefore apparent area varies as versin a ; but (Euclid, I. 32), a = j3 + 0. Therefore the angle a is very nearly equal to j3, for the moon, being so near the earth, 6 is always a very small angle, being never more than 10'. Therefore the apparent area varies approximately as versin /3 where |3 is the angle of elongation of the moon from the sun. Of course this approximation is not true in the case of a planet, for its distance from the earth being so very much greater than that of the moon, we could by no means neglect the angle 0. FIG. 64. 119. Earth-shine. It is evident that if the earth were seen from the moon it would appear to pass through the same phases as the moon does when observed from the earth, but in inverse order. During new moon the earth, as seeu from the moon, would appear full. When the moon appears as a crescent the earth would appear gibbous, and vice versa. This accounts for the phenomenon which doubtless everyone has observed, that when the moon appears as a thin crescent in the sky the remainder of its surface can be seen shining with a dull grey light, caused by the earth-shine on the moon, which is reflected back again to the earth. CHAP. TX.] MOON'S ROTATION. 157 To find the Sun's distance by observing when the Moon is Dichotomized. 120. In Chapter VII. we described the different methods by which the sun's distance can be calculated. There is another method however, which, although not susceptible of the same degree of accuracy as those employed in modern times, is of great historical interest, as it was used by Aris- tarchus at Alexandria about 280 B.C., being the first attempt at determining the sun's distance. The angle of elongation j3 (fig. 64) of the moon from the sun is observed when the moon is dichotomized, or, in other words, when the L SMJE= 90. AT Q Now, COS = , and j3 being known, the ratio of the moon's distance to the sun's is known, from which, knowing that of the moon, the distance of the sun is determined. It is not possible to obtain accurate results by this method, as, owing to inequalities in the moon's surface, the line which separates the bright from the dark portion is, when seen through a telescope, very uneven, so that the observer is unable to tell the exact instant when the moon is dichoto- mized. Aristarchus deduced by this method that the sun was 19 times more distant than the moon, instead of 400 times, which modern observations give us. The Moon rotates round an Axis. 121 . It is a remarkable circumstance in connexion with the' moon that it always turns nearly the same face to the observer. The mountains and other markings which are to be seen on its surface are always to be found nearly in the same position with respect to the circumference of the moon's disc, and also 158 THE MOON. [CHAP. ix. relative to the plane of its orbit. From this circumstance we are led to conclude that (1) The moon revolves round an axis which is nearly perpendicular to the plane of its orbit, (2) The period of its rotation round its axis must be equal to the time of completing a revolution round the earth, viz. 27 d 7 h . On first thoughts it might appear to the reader as if the fact that the moon keeps the same face turned towards the earth proves that it has no rotation. The follow- ing illustration will serve to show how erroneous is such a conclusion : Let the reader place a lamp or other body in the middle of a room, and let him proceed to walk round it in a circle so as to keep his face turned towards it all the time. Now, let us suppose that at first his face is turned towards the north, and he will find, while he is completing a circuit, that he faces in turn towards all the points of the compass. He will be looking towards the south when he has moved through a semicircle, and will again face the north when he arrives at the point from which he started. In other words, in order to keep his face turned constantly towards the lamp he will have to rotate his body through 360 for every circuit he makes. So it is in the case of the moon's revolution round the earth. Moon's Librations Libration in Latitude. 122. The axis of rotation of the moon is not quite per- pendicular to the plane of its orbit, being inclined to it at an angle of 83J, or about 6^ to the perpendicular. Therefore while the moon revolves about the earth its north and south poles are alternately turned slightly towards or from the observer. At one part of its orbit we see about 6J C beyond the north pole, and at another time about 6| beyond the south pole. This phenomenon is called libration in latitude. CHAP. IX.] THE MOON'S ITERATIONS. 159 Libration in Longitude. We have seen that the period of the moon's rotation on its axis is equal to the time taken to go round the earth. But its motion round the earth is not uniform, as, owing to the elliptic form of its orbit, its distance from the earth is not constant. On the other hand its rotation on its axis is perfectly uniform. The consequence is, that although the two periods of making a complete revolution are the same for each, still at one time we are ahle to see a little more of the eastern side, and at another time a little more of the western side. This is called libration in longitude. Its maxi- mum amount is 7 45'. Diurnal Libration. There is also a diurnal libration which is really due to parallax. For, from the time the moon rises until it sets, the observer, on account of the rotation of the earth, has changed his point of observation, and therefore he does not in each case see exactly the same face. When the moon is rising in the east he sees a little more of its western side, and when setting in the west, a little more of its eastern side, than when it is high up in the sky crossing the meridian. The total effect of these librations is such that we are at different times able to see a total of about 59 per cent, of the moon's surface instead of about 50 per cent. Path of the Moon round the Sun. 123. We have seen that the moon's orbit relative to the earth is an ellipse, but, as it also follows the earth in its motion round the sun, we see that the path of our satellite round the sun is due to a combination of two motions, a monthly motion about the earth, and a yearly motion about the sun. If we neglect the small angle at which its orbit cuts the plane of the ecliptic, and assume them both in the same plane, the moon's path may be represented by the 160 THE MOON. [CHAP. ix. dotted line in the adjoining figure, going alternately inside and outside the orbit of the earth AEE', and crossing it about 25 times in the course of the year. M represents the moon, and E the earth at new moon, while H' and E' would be their positions at the next full moon after an interval of about a fortnight. It is to be observed that this path of the moon is always concave to the sun. . 65. More Moonlight in Winter than in Summer. The moon, when full, being in opposition, must be at almost the diametrically opposite point of the celestial sphere to that in which the sun is situated. Hence at full moon at midsummer the sun's declination being north the moon must have an equal southern declination, and therefore remains but a short time above the horizon (Art. 20). Again, at full moon during midwinter the conditions are reversed, the sun's decimation is south, and the moon's north ; hence we have the moon a long time above the horizon. This happens just when the days are shortest and we are most in need of light. Moon's Retardation. 124. The moon moves from west to east with reference to the sun through 360 every 29J days, or about 12J daily. Therefore its time of rising will be later and later each night by an interval whose mean value is about 50 minutes.* This retardation, as it is called, of moonrise is not by any means uniform throughout the year ; it may be as great as 1 hour 16 minutes or as small as 17 minutes. * Since 15 correspond to 1 hour, 12 are equivalent to 50 minutes. PHAP. IX.] HARVEST MOON. 161 Harvest Moon. 125. At the full moon nearest the autumnal equinox it :s found that the retardation in the time at which the moon rises is less than at any other full moon throughout the year Therefore for several nights in succession the moon will rise yery shortly after sunset, rising on the night of full moon, as it always does, at sunset. As this happens when the farmers are getting in their crops, thus enabling them to prolong their work into the night, it is called the Harvest Moon. FIG. 65A. In order to explain this phenomenon more clearly we shall suppose that the moon's path is along the ecliptic in- stead of being inclined to it at a small angle, and that it moves uniformly in the ecliptic at the rate of 13j daily ('360 in 27 d 7 h ), and we may remark that, owing to the ap- parent diurnal revolution of the heavenly bodies the angle at which the ecliptic cuts the horizon is continually changing, its greatest and least values being colat + 23 28' and colat - 23 28' respectively. The reason of this is that the pole of the ecliptic, which, in its diurnal motion, describes a small circle mn (fig. 65A) of angular radius 23 28' round M 162 THE MOON. [CHAP. ix. the celestial pole, is closest to the zenith at m when Zm = ZP- Pm = colat - 23 28', and at its greatest zenith distance at n when Zn = colat + 23 28'. Hence the angle between the ecliptic and horizon (being equal to the angular distance between their poles) must vary between the same limits, being least when r is rising at the east point X when the ecliptic KC passes between the horizon and equator, the order being horizon, ecliptic, equator ; and greatest when ^ is at X when the ecliptic takes the position jBT'O', the order being horizon, equator, ecliptic. During the full moon nearest the autumnal equinox the sun is in Libra, and the moon, being in opposition, is in Aries, crossing from the south to the north side of the equator ; the moon will therefore rise at X, when the ecliptic KC is at its smallest inclination (colat - 23 28') to the horizon. But after an interval of 23 h 56 m when, owing to the diurnal revolution of the celestial sphere, the point X returns to the same position as on the previous night, the moon will have moved about 13. Hence, if XM be cut off on the ecliptic equal to the moon's daily rate, and through M an arc of a small circle ML be drawn parallel to the equator, the moon, on the night following full moon, will rise at L, the amount of retardation being measured by the arc of the small circle LM or by the angle LPM. This retardation, expressed in time, is found to be in our latitudes only 18 J minutes, or 14j minutes with reference to the sun (allowing for the sun's daily retardation of 4 minutes). This phenomenon of course occurs each time the moon is in Aries or, in other words, every month ; but it is only during the harvest that the moon is in Aries and full at the same time. Similarly we might show that the reverse occurs when the sun is in Aries and the moon in Libra, the daily retardation being then a maximum. For when the moon is rising, the ecliptic takes the position K'C', cutting the horizon at the greatest angle possible (colat + 23 28') : if XM' be now cut CHAP. IX.] CHANGES IN THE MOON's NODES. 163 off equal to the moon's daily rate of motion, and a parallel 3fL' be drawn to the equator, the moon, on the following night, will rise at L', the retardation being measured by the arc of the small circle M' L' or by the angle M'PL' (the arc PM when produced passing through M.'). This, expressed in time, corresponds to a retardation of about l h 10 m with refe- rence to the fixed stars or l h 6 m with reference to the sun. The retardation is thus a maximum each month when the moon is in Libra, and therefore is a maximum during the full moon nearest the vernal equinox. It will, however, be a minimum for observers in the southern hemisphere, and to them the full moon at this period is a harvest moon. At the arctic circle there is even an acceleration in the time at which the moon rises on successive nights when the moon is in Aries. For, when Aries is rising, the ecliptic actually coincides with the horizon (Ex. 5, page 32), and the distance ML (fig. 65A) vanishes, and therefore the interval between two successive risings is only 23 h 56 m . So that, as measured by solar time, at the arctic circle, the moon after passing through Aries actually rises four minutes earlier than she did on the previous night. During the October full moon the same phenomena occur, but in a less marked degree. This moon is called the Hunter's Moon. Revolution of the Moon's Nodes. 126. The moon's nodes are not fixed points, but have a retrograde motion along the ecliptic at the rate of about 19 each year, completing a revolution in about 18 J years. This backward motion is similar to that of the equinoctial points v and =, but is very much more rapid, as the period for the precession of the equinoxes is about 26,000 years (Art. 103). The moon's motion is therefore very complicated, moving, as it appears to do, in a circle which is inclined to the ecliptic at an angle of 5, at the rate of rather more than half a M 2 164 THE MOON. [CHAP. ix. degree each hour, while the plane of this circle is carried backwards on the ecliptic at the rate of 19 each year, or about 8" an hour. 127. Synodic Revolution of the Moon's Modes. We have just seen that the sidereal period of the revolution of the moon's nodes is 18f years. The synodic period of revolution, i.e. the time taken to return from any position to the same position again with respect to the sun and earth, can now be calculated in the same way as that of a planet, f r ~ 360 = arc traversed by sun in 1 day, 365-25 360 and r-^r -r r= = arc traversed by node in 1 day; 18f x <365'25 360 360 360 , ' 365-25 + ^ = ~ (see Art 67) ' where T represents the synodic period. The plus sign is taken on the left-hand side of the equation, as the relative velocity of the sun and node is the sum of their angular velocities, the motion of the node being retrograde. On solving the above equation, T is found to be 346'62 days, which is the synodic period. The line of apsides of the moon's orbit is not fixed, but, like that of the earth's orbit, it has a slow progressive motion, making a complete revolution of the moon's orbit in about nine years, the period in the case of the earth being about 108,000 years. To find the Height of a Lunar Mountain. 128. It has been known, from the time of Galileo, that the surface of the moon is covered with mountains. Some of these mountains have been calculated to rise to heights of four or five miles above the surface of the sur- rounding plains, which shows that, considering the smaller size of the moon, its mountains are comparatively much CHAP. IX.] HEIGHT OF LUNAR MOUNTAIN. 165 more lofty than those of the earth. When the sun shines obliquely on the mountains, they cast long shadows over the surrounding plains on the side remote from the sun in exactly the same way as we are familiar with on the earth. Also, wherever there is a high mountain, its top- most peak catches the first rays of the rising and the last of the setting sun, when all the surrounding parts are still in complete darkness. Small points of light are for this reason sometimes seen on the dark portion of the moon's disc, separated by a measurable distance from the line of separation of light and darkness. 129. First Method. The method employed by Messrs. Beer and Madler in 1837 for finding the height of a lunar mountain consists in measuring, by means of a micrometer, the length of the shadow cast by the mountain when illu- minated by the sun's rays. By comparing this angular measurement with the angle subtended by the moon's diameter, the length of the shadow in miles can be found (for the moon's diameter in miles is known) ; from which, knowing the inclination of the sun's rays, the height of the mountain can be determined, just as the height of a tower on the earth can be found by knowing the length of its shadow and the altitude of the sun. In applying this method, allowance must be made for the effect of foreshort- ening, as the shadow being generally viewed obliquely, the micrometer measures merely the projection of its actual length on a plane perpendicular to the line of vision. 130. Second Method. This method consists in mea- suring, by means of a micrometer, the angular distance between the bright summit of the mountain-top appearing on the dark portion of the moon's disc and the line of separation of light and darkness. This measurement is made in a direction perpendicular to the line joining the extremities of the horns, and therefore parallel to the plane of the moon, earth, and sun. 166 THE MOON. [CHAP, ix* Let AB represent the line of separation of light and darkness on the moon, and P the top of a mountain when just illuminated by the ray PBS, the line PS being per- pendicular to AB, and touching the moon's surface at B. Let E be the earth, and ES' the direction of the sun, as Direction of Sun, FIG. 66. seen from E, which may be taken as parallel to PS, on account of the sun's great distance. The radius of the moon is denoted by r, and the height PC of. the mountain by h; then by Euclid (m. 36), or that is, (2r + h) 2rh + h* (see fig. 66) ; But h being very small compared with r, its square may be neglected ; /2 .-. 2rh = t z or 7* = AT Now the distance t is not measured directly ; what is actually measured being the projection of t on a plane CHAP. IX.] PHYSICAL STATE OF THE MOON. 167 perpendicular to the line of sight, viz. the perpendicular 8 let fall from P on BE (fig. 66). g But - = sin = sin (by parallel lines) ; But the angle is known, being the angle of elongation of the moon from the sun as seen from the earth ; therefore, t is known. Substituting this value of t, we have S 2 Height of mountain h = ?r r- 2r sm 2 ^ 131. Lunar Craters. Perhaps the most striking ob- jects to be seen in lunar landscapes are what are to all appearances enormous craters of what were once volcanoes, but which are now probably quite extinct. The typical lunar crater consists of an immense circular plain surrounded by a high wall or rampart. In the centre of the plain there generally rises a mountain, or sometimes more than one. Among the most characteristic of these craters are Tycho, having a diameter of fifty-four miles, and Archimedes, whose diameter is sixty miles. Another immense crater is Schickard, with a diameter of over 130 miles, and a sur- rounding wall, which, in some parts, attains a height of 10,000 feet. It has been pointed out that an observer situated in the centre of this walled space would think himself in the midst of a boundless desert, for, on account of the curvature of the moon's surface, the summit of the lofty surrounding wall would be altogether beneath his horizon. 132. Lunar Atmosphere. All observers of the moon have come to the conclusion that it either possesses no atmo- sphere at all, or, if any such gaseous covering exist, that it is of very extreme tenuity indeed. No change is observed in the intensity of the light from a fixed star as it approaches 168 THE MOON. [CHAP, ix the dark edge of the moon, such as there would he were there any appreciable thickness of atmosphere for its rays to penetrate. Also, when the moon passes hetween the observer and a fixed star, the observed time during which the occultation of the star lasts is found not to be less than the calculated time, as would be the case if the moon had an atmosphere of any considerable density; for the star would still be visible for some time after being actually covered by the moon, owing to its rays being refracted in their passage through the lunar atmosphere, if such existed, just as, owing to refraction by the earth's atmosphere, the sun remains visible to us for some time after he has sunk below our horizon. In addition to having no atmosphere, astronomers have not been able to detect water in any form on the moon's surface, which renders the existence of life, such as is known to us, altogether impossible. ( 169 ) CHAPTER X. ECLIPSES. 133. Eclipses are of two kinds, lunar and solar. Lunar Eclipses. A lunar eclipse is caused by the passage of the moon through the shadow of the earth. This can only happen when the earth is between the sun and moon, or, in other words, when the moon is in oppo- sition. If the plane of the moon's orbit coincided with the plane of the ecliptic instead of being inclined to it at an angle of about 5, we should have an eclipse of the moon at every opposition. However, on account of the above angle of inclination of its orbit, it generally happens that the moon, when in opposition, is either so far above or below the plane of the ecliptic that it fails to pass through the shadow of the earth. So we see that, in order that an eclipse may take place, the moon must be very nearly in the ecliptic, that is, at or near one of its nodes. Therefore, the conditions for a lunar eclipse are : (1) The moon must be in opposition, i.e. full. (2) It must be at, or near, one of its nodes. There are two kinds of lunar eclipse, total and partial. It is total when the whole surface of the moon passes through the shadow, and partial when only part of its surface is involved. 134. Let 8 and E (fig. 67) represent the centres of the sun and earth, respectively. Draw a pair of direct common tangents AB and CD to the sun and earth, meeting SE 170 ECLIPSES. [CHAP. x. produced in V, and a transverse pair AD and BC meeting SE in X. If these lines be now supposed to revolve round SE as axis they will generate cones, and there is thus a conical shadow BVD, having V as vertex, into which no direct ray from the sun can enter. This conical space is called the umbra. The spaces represented by VBL and VDN form what is called the penumbra, from which only part of the light of the sun is excluded. It is to be remembered that the passage of the moon through the penumbra does not give rise to any eclipse, but only to a diminution of brightness. FIG. 67. Thus the moon when at MI (fig. 67) receives light from portions of the sun next A, but rays from parts near C will not reach the moon, owing to the interposition of the earth ; consequently, the brightness of the moon is some- what diminished, the diminution being greater the nearer the moon approaches the edge of the umbra. An eclipse, properly so-called, however, does not commence until a por- tion of the moon's surface shall have entered the umbra. Phenomena due to Refraction. 135. As everyone who has seen a total eclipse is aware, the moon appears of a dull-red or brownish colour. It must, therefore, receive light from some source. That it is not due to earth-shine (Art. 119J is certain, for the moon CHAP. X.] BREADTH OF EARTH'S SHADOW. 171 being in opposition, the dark hemisphere of the earth is turned towards it. The phenomenon is caused by the refracting power of the earth's atmosphere, owing to which those rays from the sun, which nearly touch the earth, are bent round, and thus reach the moon's surface. Another curious phenomenon, due to refraction, is seen when an eclipse occurs at sunset or sunrise; for both the sun and moon being elevated by refraction, it is possible to see the moon eclipsed when the sun still appears shining in the heavens, a phenomenon which was observed in 1666, 1668, and 1750. N. B. Throughout the remainder of this Chapter we shall occasionally denote the sun by the symbol O, and the moon by @ . To find the Diameter of the Section of the Earth's Shadow where the Moon crosses it. 136. The angular diameter of the cross-section of the cone of shadow is represented by the arc MN. Let the semiangle MEV subtended by MN at the centre of the earth be a (fig. 68). FIG. 68. Let p = 0's hor. parallax = L EAX (fig. 68). / = C 's hor. parallax = L EMB or L EXB. s = angle subtended by O's semidiameter at E=LSEA. 6 = L E VB, the semiangle of cone of shadow. 172 ECLIPSES. [CHAP. x. Now by Euclid (i. 32) we have : a+0 = p'; .-. a=/-0. For the same reason = s - p ; .'. a = p' - (s - p) =p' + p - s. But p, p', and s are known ; therefore 2a, the angle sub- tended by MN at E, is determined. If the moon's horizontal parallax be taken as 57', the sun's as 8", and the sun's semidiameter as 16', the breadth of the shadow 2a, or 2 (p' + p - s) is found to be about 82'.* As the moon's angular diameter has a mean value of about half a degree, or 30', we see that the breadth of the section of the shadow at the distance of the moon is nearly three diameters of the moon ; and since the moon moves through an arc equal to its own diameter in about an hour (Art. 113), we see that when the moon passes through the axis of the shadow, that is, when the eclipse is central, it may remain totally eclipsed for about two hours. 137. In the above we see that the semiangle 9 of the cone = s - p = (G's semidiam.) - (O's hor. parallax). In the same way as the breadth of the section of the cone at MN has been found, it can be shown that the semidiameter of the section at XY 9 where the moon crosses it when in conjunction, is equal to p - p + s. For, by Euclid (i. 32), p' + 0=p' -p + s. To find the Length of the Earth's Shadow. 138. The distance EV (fig. 68) from the centre of the earth to the vertex of the cone is called the length or height of the earth's shadow. Its magnitude can now be found, * Or to be more accurate, the breadth of the shadow varies from a maximum of 89' 14" to a minimum value of 75' 38", the maximum value being reached when the moon is nearest the earth (perigee) and the earth at the same time farthest from the sun (aphelion), the minimum value being attained when these conditions are reversed. CHAP. X.] ECLIPSES OF THE SUN. 173 knowing the earth's radius and the semiangle of the cone. For, since the angle is so small (being equal to s - p), we may assume that r, the radius of the earth, coincides with an arc of the circle whose centre is Pand radius VE (fig. 68). Therefore it follows at once from circular measure that _?"_ JL- 206265" ~ EV" _ 206265" r _ 206265" r r~ "7'^T 7 "' Taking r, the radius of the earth, roughly as 4000 miles, s the semidiameter of the sun as 16' or 960", and p, the sun's parallax as 8", we have 206265 x 4000 EV -" miles = about 860,000 miles, or 215 times the earth's radius. Since the moon's distance from the earth is only about sixty times the earth's radius, we see that the moon's orbit extends for a much less distance from the earth than the length of the cone of shadow, and therefore a lunar eclipse must happen if the moon is at one of its nodes and full at the same time. 139. Solar Eclipses. An eclipse of the sun is caused by the interposition of the moon between the sun and the observer. As in the case of a lunar eclipse the moon must be nearly in the plane of the ecliptic. The two conditions for a solar eclipse are therefore : (1) The moon must be in conjunction, i.e. it must be new moon. (2) It must be at, or near, one of its nodes. In a lunar eclipse, the moon, on entering the umbra, loses its light, and consequently the eclipse is visible from any part of the hemisphere of the earth which is turned towards the moon. On the other hand, in the case of a solar eclipse, the 174 ECLIPSES. ("CHAP. x. light of the sun is merely hidden from the observer ; and the moon being much smaller than the earth, this shows that a solar eclipse can only be visible over a very limited area at the same time. There may be an eclipse of the sun visible from some portion of the earth if any part of the moon come within the arc XY (fig. 68) ; and there may be a lunar eclipse if it enter MN. Since the arc XYis greater than JfJVwe should expect that more solar eclipses should occur than lunar if we count the eclipses observed over the whole earth ; and this in fact is the case ; but, as we have just seen, an eclipse of the sun is only visible over a very limited area of the earth, and therefore it happens that there are more lunar than solar eclipses seen from any particular place. 140. There are three kinds of solar eclipses (1) total ; (2) annular ; and (3) partial. When the eclipse is total the whole of the sun's disc is hidden from view, whereas in the case of an annular eclipse only the central portion is darkened, with a bright ring surrounding it. In order to arrive at a clear idea as to how the moon, coming between the sun and the observer, can sometimes hide the whole of the sun's surface from view, and at other times only the central portion, it should be borne in mind that the moon's apparent angular diameter is not constant, for, on account of its elliptic orbit its distance from the earth is variable (Art. 113) ; the apparent diameter of the sun also varies, the mean values of both being very nearly equal. The moon's angular diameter varies from 33' 22" when nearest the earth (perigee) to 28' 48" when at its greatest distance (apogee) , and in the case of the sun the variation is from 32' 36" to 31' 32". When the moon's apparent diameter is greater than that of the sun, which occurs when it is closest to the earth, it will hide the whole of the sun's surface from the view of an observer situated on the line of centres of the two bodies, causing thus a total eclipse. When, CHAP. X.] LENGTH OF THE MOON'S SHADOW. 175 however, the moon's apparent diameter is less than that of the sun, as it is when at its greatest distance from the earth, it will, under the same conditions, hide only the central portion of the sun, giving rise to an annular eclipse. A very simple experiment will render the ahove expla- nation perfectly clear. If the reader take a coin, and, closing one eye, hold it in such a position before the other eye as to just completely hide the sun's surface from view, the position of the coin now is similar to that occupied by the moon when totally eclipsing the sun. If, however, the coin be removed to a greater distance from the eye, keeping its centre still in a direct line with that of the sun, it will be found, owing to the diminution in its apparent diameter, due to the increase of distance, that it only hides the central portion of the sun from view, thus illustrating how the moon, when farthest from the earth, causes an annular eclipse. When a partial eclipse takes place, only a portion of the sun's disc at one side becomes darkened, owing to the centres of the two bodies not being in a direct line with the observer. It is evident that all total and annular eclipses must begin and end as partial eclipses. To find the Length of the Cone of Shadow cast by the Moon. 141. Let S denote the centre of the sun, M of the moon, and R and r the radii of the sun and moon respectively (fig. 69). The apex of the shadow cast by the moon will be at where the common tangents CH and DFmeet. It is required to find the distance MO. Since the triangles OSC and OMH are similar we have, Euclid (vi. 4), OS JR ., OM+SM jR. therefore solving for OM and denoting SM by d we have 176 EGXIPSES. [CHAP. x. But r, the radius of the moon, is about 1076 miles, while d, the distance between the centres of the sun and moon varies from 11,717 to 11,713 times the earth's diameter. Substituting these values we find that OH varies from 28-94 to 28 -93 times the earth's diameter. FIG. 69. Since the distance from the moon's centre to the surface of the earth varies from 28 to 31 diameters of the earth, it follows that the observer may sometimes be situated at E (fig. 69) inside the cone of shadow, and at other times at E\ beyond the point 0, where the cone tapers to a vertex. In the former case a total eclipse takes place, the moon subtend- ing a greater angle than the sun, and in the latter case an annular eclipse occurs, the portion of the sun's surface hidden from view being represented by the inner circle QN (fig. 69j, marked off by tangents drawn from E to the surface of the moon, and produced out to meet the sun. To calculate the Conditions for a Lunar or Solar Eclipse. 142. Lunar Eclipse. Let (fig. 70) represent the centre of the section of the earth's shadow at the distance of the moon, and M the centre of the moon when touching the shadow externally. MN represents the apparent path of the moon, 2^0 the ecliptic, and N the position of the node. CHAP. X.] CONDITIONS FOR AN ECLIPSE. 177 Now it is evident that an eclipse of no portion of the moon's surface can take place unless the distance between the centres of the moon and shadow becomes less than MO. FIG. 70. But MO = (semidiam. of shadow) + (semidiam. of moon) m. But where a=p'+p-s (Art. 136); MO = p' + p-s + m, p = O's hor. parallax = 8", / = 's hor. parallax = 57', s = O's semidiam. = 16' (mean value), m = C's semidiam. = 15' (mean value). Therefore, we have MO = 57' + 8" - 16' + 15' = 56' (roughly). Similarly, for a total lunar eclipse the moon will, in the limiting position, touch the shadow internally, and we shall have MO = (radius of shadow) - (semidiam. of moon) = a- m *=p'+p-s-m = 26' (roughly). Therefore it is impossible for a lunar eclipse to occur if the distance between the centres of the moon and shadow exceeds 56', and for a total eclipse the distance cannot exceed 26'. 143. Solar Eclipse.- We have seen (Art. 137) that the angular radius of the section of the cone where the moon N 178 ECLIPSES. LCHAP. x. crosses it in conjunction at XF(fig. 68) isp'-p+ s; there- fore, it is evident that the limiting distance of the moon from the centre of the section for a partial eclipse of the sun is p r - p + s + m or about 88', the limiting distance for a total eclipse being p' -p +s - m or 58'. As the moon's orbit is inclined at such a small angle to the ecliptic (5), the distance MO must be nearly per- pendicular to the ecliptic, and therefore is almost equal to the latitude of the moon ; but, as the latitude of the moon varies from to 5, we see that an eclipse can only take place very near a node. 144. Definition. The greatest distance (measured along the ecliptic) of the moon from the node, when in opposition, at which an eclipse can happen is called the Lunar Ecliptic Limit. Thus, in fig. 71, the apparent path of the moon is represented by MN, the moon being taken just touching the shadow when nearly in opposition ; then the distance NO will represent the distance, measured along the ecliptic, of the moon from the node when in opposition (i.e. the pro- jection on the ecliptic of the moon's distance from the node when in opposition), and is therefore the ecliptic limit. To find the Lunar Ecliptic Limit. 145. In order to calculate the ecliptic limit NO, in the N FIG. 71. spherical triangle MON, the arc MO is known, being the sum of the semidiameters of the shadow and the moon (by CHAP. X.] ECLIPTIC LIMITS. 179 Art. 142, M = p' + p - s + m) ; the angle N, the inclination of the moon's orbit to the ecliptic is also known, being about 5, and the angle M is a right angle (since OM is the shortest distance from to MN) ; therefore, the arc NO can be calculated. Major and Minor Limits. The lunar ecliptic limit is not a constant quantity, as the parallax and semidiameter of both the sun and moon are variable. Moreover, the inclination of the moon's orbit varies from 5 20' to 4 57'. All these causes combine to produce considerable variations in the limit. When the moon is nearest the earth and the earth farthest from the sun, and at the same time the angle of inclination of the moon's orbit least, the circumstances are then most favourable for an eclipse, which may, therefore, take place at a greater distance from the node than at any other time. Under these circumstances the magnitude of ON is found to be 12 5', and is called the Major Ecliptic Limit. On the other hand, when the moon is farthest from the earth, the earth nearest the sun, and the angle at N (fig. 71) greatest, the circumstances are most unfavourable for an eclipse, and the moon must be much closer to the node than in the former case, in order that an eclipse should occur. Under these conditions ON is found to be 9 30', and is called the Minor Ecliptic Limit. When the distance of the moon from the node at oppo- sition is within the major limit the eclipse may take place, but within the minor limit it must take place. 146. Solar Ecliptic Limits. There is also- an ecliptic limit for the sun, viz. the greatest distance (measured along the ecliptic) of the moon from the node, when in conjunction, consistent with a solar eclipse. The maximum and mini- mum values are also called the major and minor limits ; the 180 ECLIPSES. [CHAP. x. former being 18 31' within which a solar eclipse may take place, and the latter 15 2 1/ within which it must occur. 147. In (Art. 126) it was seen that the moon's nodes have a retrograde motion on the ecliptic, making a complete revolution in 18-f years. From this it was proved (Art. 127), that the synodic period of revolution of the line of nodes is 346*62 days, or, in other words, we might say that the sun separates from the line of nodes through 360 in 346'62 days. Therefore, in one synodic lunar month of 29 J days, the sun separates from a node through an angle 30 38' =30 I nearly. 346-62 As the comparison of this result with the solar and lunar ecliptic limits enables us to calculate the frequency of eclipses, it is of importance that the student should remember the approximate values of the following quan- tities : Major. Minor. Lunar ecliptic limits, , , ; 12 9 J. Solar ecliptic limits, . 18| 15|. Eelative motion of sun = 30 | in each lunation. Period of sun's revolution = 346 days. Period from node to node = 173 days. Six lunations = 6 x 29 i =177 days. N.B. It is evident that during either a lunar or solar eclipse the distances of the sun and moon from the nearest node are nearly equal. To determine the Frequency of Eclipses. 148. Least Possible dumber. Let N and n repre- sent the moon's nodes (fig. 72). Cut off, on the ecliptic EC, distances NL, NL', nl, nl' each equal to the lunar ecliptic HAP. X.] FREQUENCY OF ECLIPSES. 181 limit ; and similarly NS, NS', ns, m' each equal to the solar ecliptic limit. Now as the sun moves with reference to the nodes through 30f in one synodic month, it follows that he will take more than a month in moving through the arc SS' or ss'; for the least value of these arcs, being double the sun's minor limit, is 31. Hence at least one new moon, and therefore one solar eclipse, must occur within each of these arcs. FIG. 72. On the other hand, the least value of LL' or U, being twice the moon's minor limit, is only 19, and the sun traverses each of these arcs in much less than a month (about 18 days) ; therefore, it is possible that there may be no full moon near either node, and therefore no lunar eclipse during the year. Hence the least possible number of eclipses in a year is two, both of the sun. 149. Greatest Possible Number. The sun takes 173 days to pass from N to n (fig. 72), or 4 days less than six lunations (177 days) ; therefore, when the moon happens to be full two days before the sun arrives at N, there will also be a full moon 2 days after his passage through n, thus rendering a lunar eclipse very close to each node a certainty. 182 ECLIPSES. [CHAP. x. But if a lunar eclipse occur 2 days before or after the sun's passage through a node, there may also be two solar eclipses near that node, viz. at the preceding and following new moon ; for, in half a lunation, or 14f days, the sun moves through 15-^; and even if to this we add the arc gone through in 2 days, the result is still well within the sun's major ecliptic limit. Therefore, there may be one lunar and two solar eclipses at each node in a period of 346 days. But if the eclipses within SS' (fig. 72) occur in January, there will be ample time before the year is completed for the sun to arrive a second time within SS'. There will now be another solar eclipse near S, followed by a lunar eclipse 6 days after the sun's passage through JV". There will not now, however, be a solar eclipse near /S', as the following new moon will take place outside the sun's major limit. In all, we have counted eight eclipses in 12J lunations or 368 days, viz. five of the sun and three of the moon. But all these eight eclipses cannot happen in a year (365 days) ; therefore one, either a solar or lunar, will have to be omitted. Hence the greatest possible number of eclipses in a year is seven, five of the sun and two of the moon, or four of the sun and three of the moon. In every 18 years there are generally 41 eclipses of the sun to 29 of the moon. Chaldean Saros. 150. The synodic period of the moon's nodes being 346-62 days, and a lunation being 29*53 days, we therefore have 19 synodic revolutions of node = 19 x 346-62 days = 6585 days ; also 223 lunations = 223 x 29-53 = 6585 days. Therefore, we see that after every period of 6585 days, which are equivalent to 18 years 11 days or 18 years CHAP. X.] SAROS OF JTHE CHALDEANS. 183 10 days, according as there are four or five leap years in the interval, the sun and moon will return to nearly the same positions relative to the nodes (each having made an exact numher of revolutions), and therefore the eclipses will repeat themselves in the following cycle in the same order as in the previous one. This period is called the Chaldean Saros, as, by means of it, the Chaldeans wero enabled to foretell the occurrence of eclipses. ( 184 ) CHAPTER XI. TIME. Mean and Apparent Time. Equation of Time. 151. We explained in Chapter III. the difference be- tween sidereal and solar time. The sidereal day is of constant length as the rotation of the earth on its axis is uniform. The length of the apparent solar day, however, is variable, as the sun's rate of change of right ascension is not uniform throughout the year. On account of this inequality a clock cannot be regulated to point to 12 o'clock j ust when the sun is in the meridian. Accordingly our clocks, instead of keeping apparent solar time keep mean solar time as indi- cated by the motion of an imaginary body called the mean sun, which is supposed to move uniformly in the equator at the same mean rate as that of the true sun in the ecliptic. Definition. A mean solar day is the interval between two successive transits of the mean sun across the meri- dian. As the mean sun changes its right ascension at a uniform rate we see that the length of a mean solar day is constant. The hour-angle of the mean sun at any instant measures the mean time at that instant, while that of the apparent or real sun gives the apparent time, or time as indicated by a sun- dial. 152. Definition. The equation of time is the difference between the mean and the apparent time. It is^coAinted positive when^the mean exceeds the apparent time, and CHAP. XI.] EQUATION OF TIME. 185 negative when the latter exceeds the former. Therefore we have (Mean Time) - (Apparent Time) = (Equation of Time), or (Clock Time) - (Dial Time) = (Equation of Time). As the true sun moves in the ecliptic and the mean sun in the equator, the motion of the former being variable, and of the latter, uniform, we see that the equation of time is due to two causes : (1) The variable motion of the true sun in the ecliptic owing to the eccentricity of the earth's orbit. (2) The obliquity of the ecliptic to the equator. We shall now consider each of these causes separately, and by combining their effects we shall be able to see how the equation of time varies throughout the year, when it reaches a maximum, and at what periods it vanishes. Equation of Time due to Unequal Motion of Sun. ' 153. On December 31st, at perihelion, or, in other words, when the earth is nearest the sun, its velocity is greatest (Art. 68), and therefore the rate at which the sun moves from west to east along the ecliptic will, at this period, be greater than the mean rate. As the earth turns on its axis from west to east this would cause the apparent or true solar days to exceed in length the mean solar days, and, if a sun-dial and a clock be started together at perihelion, the apparent time will gradually get behind mean time, so that the sun-dial will lose compared with the clock. This will continue for about three months until the rate of the sun in the ecliptic becomes equal to its mean rate. Hence that component of the equa- tion of time, due to the unequal motion of the sun, reaches at the end of March its greatest positive value, viz. about 7 minutes. The sun-dial will now begin to gain on the clock what it lost in the preceding three months, until aphelion (July 1st) is reached, when it will coincide with 186 TIME. [CHAP, xi the clock, and the equation of time, in so far as it is due to the unequal motion of the sun, vanishes. Similarly we can see that from aphelion to perihelion the equation of time, due to this cause, is negative, having a maximum value of - 7 minutes at the end of September. Equation of Time due to the Obliquity of the Ecliptic. 154. Even if the sun's motion along the ecliptic were uniform, his rate of change in right ascension would still be variable on account of the obliquity of the ecliptic to the equator. Let us now suppose that the true sun 8 and the mean sun /Si (fig. 73) start together at the vernal equinox r, the former moving in the ecliptic and the latter in the equator; they will again be together at the autumnal equinox =c=, and also their right ascensions will coincide at the two solstices. Hence that portion of the equation of time due solely to the obliquity of the ecliptic becomes zero four times each year, at the equinoxes and solstices. FIG. 73. Again, when the true sun is at S (fig. 73) his right ascension is r K (PSK being the arc passing through the CHAP. XI.] EQUATION OF TIME. 187 celestial pole and the sun's centre. But the position of the mean sun Si (as affected by the obliquity of the ecliptic alone) would be obtained by cutting off T Si = T $, Si falling to the east of K, since T S, being the hypotenuse of the right-angled spherical triangle T SK, is greater than T K. Therefore the true sun, being to the west of the mean sun, will each day cross the meridian first, the sun-dial being faster than the clock. Hence this portion of the equation of time is negative, its greatest value being about - 10 minutes. Similarly we may see that from solstice to equinox the dial will be slower than the clock, and the component of the equa- tion of time will be positive, having a maximum value of + 10 minutes. Combination of the Two Components. 155. Let X = that portion of equation of time due to the unequal motion of the sun. T = that due to obliquity of the ecliptic. If we summarize the above results we have (1) X vanishes twice each year, on December 31st and July 1st, and varies from a maximum of + 7 minutes at end of March to - 7 minutes at end of September. (See dotted curve, fig. 74.) (2) Y vanishes four times each year, at the equinoxes and solstices. From equinox to solstice T is negative, and from solstice to equinox, positive, varying from a maximum of + 10 minutes to - 10 minutes at intermediate points. (See continuous curve, fig. 74.) (3) The sum or difference of X and Y, according as they are of the same or opposite sign, gives the equation of time at any instant. (See curve, fig. 75.) " FnrH * \ {UNIVERSITY! 188 TIME. [CHAP. xi. The Equation of Time vanishes four times each Year. 156. We have seen that the equation of time is equal to the algebraic sum of X and F; the maxima values of T being + lOw, - 10m, + 10m, - 10w, which occur in the months February, May, August, November. Now as X has never a greater numerical value than 7 minutes it follows that, in the four months mentioned above, the equation of time (X + Y) must have the same sign as Y, whether X is positive or negative. Hence it follows that the equation of time has at least four changes of sign throughout the year, viz. : +> _. +> _ ? and therefore, on changing from positive to negative or vice versa, must at least on four occasions pass through a zero value.* The dates on which the equation of time vanishes are about April 16, June 15, September 1, and December 25. The greatest positive value is 14 m 28 s on February 11, and the greatest negative value 16 m 21 s on November 3. (See the curve in fig. 75.) 157. We can now represent graphically how the equation of time varies throughout the year, when it reaches a maxi- mum, and at what periods it vanishes. FIG. 74. These curves represent the variations in the two components of the equation of time. * That it does not vanish oftener than four times each year appears from, an examination of the curve in fig. 75. CHAP. XI.] MORNING AND AFTERNOON UNEQUAL. 189 In fig. 74 the dotted curve represents the component due to the unequal motion of the sun, and the continuous line that due to the obliquity of the ecliptic. In fig. 75 is a single curve representing the combined effect of the other two, the equation of time corresponding to any point on the curve being represented by the perpendicular distance of the point from the zero line. Thus the equation of time corresponding to p is represented by pm. All portions of the curve below the zero line represent negative values. The periods at which the equation of time vanishes are represented by the points where the curve cuts the zero line. FIG. 75. The curve represents the variations in the equation of time found by combining the two components whose curves are given in the preceding figure. Morning and Afternoon unequal in Length. 158. The interval of time from sunrise until the sun is in the meridian (apparent noon) is equal to the interval from apparent noon to sunset, neglecting the small change in the sun's declination throughout the day. But mean and apparent noon do not in general coincide ; therefore, morning and afternoon, as measured by our clocks, are not of equal length, the former being less (algebraically) than half the interval between sunrise and sunset by the equation of time, the latter being greater (algebraically) by the same amount. 190 TIME. [CHAP. xi. Hence the lengths of the morning and afternoon always differ by twice the equation of time, so that (length afternoon) - (length morning) = 2 (equation of time). Immediately after the winter solstice, the afternoons begin to lengthen, while the mornings still continue to get shorter. The explanation of this is simple. For the sun being at the winter solstice he, as it were, stands still for some days, during which time we may regard his declination as constant. The interval between apparent noon and sunset, therefore, remains constant. But as the equation of time is at this period increasing [see curve, fig- 75), mean noon precedes apparent noon by a greater amount each day. Hence the mean time of sunset increases and the afternoons get longer. Similarly, it may be shown that while the apparent time of sunrise remains the same, the mean time of sun- rise increases each day, and, consequently, the mornings are shortened. It is, however, to be borne in mind that very soon the increasing declination of the sun causes the mornings to lengthen as well as the afternoons. EXAMPLES. 1. Given mean time = 5 h 12 m 20 s p.m., and the equation of time =-f 5 m 25 s ; find apparent time. Am. 5 h 6 m 55 s p.m. 2. Given apparent time = 10 h 4 m 15 s a.m., on November 3, when the equation of time has its greatest negative value, viz. 16 m 21 s ; find mean time. Am. 9 h 47 m 54 s a.m. 3. Find th'e meantime of apparent noon in questions 1 and 2. Am. 1. 5 m 25 s p.m. 2. Il h 43 m 39 s a.m. 4. On Nov. 3, the sun-dial is 16 m 21 s faster than the clock. Given that the sun rose at 6 h 57 m a.m. ; find the time of sunset. Am. 4 h 30 m 18 s p.m. 5. Given that the sun rose on a certain date 6 h 54 m a.m., and set at 4 h 33 m p.m. ; find the equation of time. Am. - 16 m 30 s . 6. In question 5, by how much does the length of the morning exceed the length of the afternoon ? Am. 33 m . CHAP. XI.] IFFERENCE OF LOCAL TIMES. 191 Local Time. 159. As the earth rotates uniformly on its axis from west to east, it is evident that the further east a place is situated the sooner will the sun cross the meridian of that place, and, therefore, the later will be the local time. When it is noon at any place it will be 1 o'clock p.m. 15 to the eastward, and 11 o'clock a.m. 15 to the westward of that place. For : 360 correspond to 24 hours ; 15 to 1 hour. Given the longitudes of two places (A and B), and the time at one of them (A), to find the time at the other (). Rule. Divide the algebraic difference of the longitudes by 15, which gives the difference of the local times. Add this remit to, or subtract from , the given time at A, according as B is east or west of A, and the result will be the time at B. EXAMPLE. Find the time at New York (long. 74 10' W.) when it is 3 o'clock, P.M. at Dublin (long. 6 20' W.). Here the difference of longitudes = 67 50'. On dividing by 15, we get : Difference in times = 4 h 31 m 20". New York being west of Dublin, the time is earlier. There- fore, subtract 4 h 31 m 20 s from 3 p.m., that is, from 15 hours. H. M. S. 15 4 31 20 New York time = 10 28 40 a.m. Should one longitude be east and the other west, their algebraic difference will be got by adding the longitudes. 192 TIME. [CHAP. xi. -ZV..Z?. In the same way, when given the sidereal time at one of two places whose longitudes are known, the sidereal time at the other place can be found ; for the earth turns on its axis through 360 relative to the fixed stars in 24 sidereal hours, and therefore 15 relative to the fixed stars correspond to 1 sidereal hour. To reduce a given Interval of Mean Time to Sidereal Time, and vice versa. 160. There are 365^ mean solar days in the year, and 366|- sidereal days, the sun making one less diurnal revolu- tion than the fixed stars, on account of his annual motion in the ecliptic ; .*. 365| mean solar days = 366| sidereal days. Therefore, if m be any interval of mean time, and s the corresponding interval of sidereal time, we have 365|: 366J-: : m:s. v From which, if m be given, s can be found, and vice versa. EXAMPLES. 1. Express in sidereal time an interval of 16 h 15 m 23 s mean time. Am. 16 h 18 m 3 s . 2. Express in mean time an interval of 12 h 16 m 26 s sidereal time. Ans. 12 h 14 ra 16". N*B. We cannot convert, by means of the above formula, the actual mean time at any instant in sidfereal time, or vice versa. This is done as follows : Given the Sidereal Time at any instant at Greenwich, to find the Mean Time at that instant. 161. Let AQ (fig. 76) represent the meridian, r<2 the celestial equator, and m the mean sun. CHAP. XI.] Then 193 MEAN AND SIDEREAL TIME. Qr = sidereal time (expressed in arc), Qm = mean time (expressed in arc), m