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ATORY
ASTRONOMY
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LABORATORY ASTRONOMY
BY
ROBERT WHEELER WLLLSON, PH.D.
PROFESSOR OF ASTRONOMY IN HARVARD UNIVERSITY
GINN & COMPANY
BOSTON NEW YORK CHICAGO LONDON
COPYRIGHT, 1900, 1905
BY ROBERT W. WILLSON
ALL RIGHTS RESERVED
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PRIETORS BOSTON U.S.A.
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PREFACE
THE subjects treated in elementary text-books of astronomy which
are most difficult and discouraging to the beginner are those which
deal with the diurnal motion of the heavens and the apparent
motions of the sun, moon, and planets among the stars. A clear
conception of these fundamental facts is, however, necessary to
a proper understanding of many of the striking phenomena to
which the study of astronomy owes its hold upon the intellect and
the imagination.
No adequate notion of those subjects which involve the ideas of
force and mass can be given to the average student who has not
mastered the elements of mechanics ; but to explain the motions
of the heavenly bodies, the knowledge of a few principles of solid
geometry and of the properties of the ellipse will suffice, no
more, indeed, than may "be easily explained- in the pages of the
text-book itself.
Most of the difficulties which arise at the outset of the study may
be satisfactorily met by methods which require the student to make
and discuss simple observations and to solve simple problems. This
necessity is recognized in many recent text-book's which introduce
such methods to a greater or less extent, in all cases to great
advantage and in some with marked success. I have gathered in
this book some of those which I have found practicable, intending
that they should explain in natural sequence those phenomena which
depend on the diurnal motion, the moon's motion in her orbit and
the change in position of that orbit, the motion of the sun in the
ecliptic, and the geocentric motions of the planets.
The methods chosen may be carried out with fair-sized classes
and do not require a place of observation favored with an extensive
view of the heavens. The gnomon-pin, the hemisphere, the cross-
staff, a simple apparatus for measuring altitude and azimuth which
iii
iv PREFACE
may be converted into an equatorial by inclining it at the proper
angle, together with a few maps and diagrams, form an outfit so
inexpensive that it may be supplied to each pupil, and much work
may be done at home. It is obvious that the possibility thus offered
of utilizing favorable opportunities for observation is especially
valuable in a study which is so much dependent on the weather.
All members of the class, too, will be doing the same or similar
work at the same time, a principle of cardinal importance in
elementary laboratory work with large classes.
The meridian work of Chapter VI is added for the sake of logical
completeness, to explain the determination of the zero of right
ascensions, a subject which is usually neglected in the text-books
and would not be included in an ordinary course.
Nothing has been directly planned for teaching the names of the
constellations and the use of star maps. The work of Chapters II,
III, and IV, covering a period of some months, results in a very
good acquaintance with the principal stars and asterisms. It may
be assumed, too, that the teacher is familiar with the heavens and
will gather the class as early as possible to introduce them at least
to the polar constellations.
The book is intended primarily for teachers, but much of it is
suitable for use as a text-book, in spite of its rather condensed
form. It is meant to be used in connection with one of the many
admirable text-books on descriptive astronomy adapted to high-
school pupils.
The first six chapters were printed in 1900, and various changes
and additions might now be made, notably an improvement in the
protractor for laying off altitudes on the hemisphere, which is now
so constructed that it may be used as a ruler for the accurate draw-
ing of great circles. This permits a much simpler determination of
the pole of a small circle than that described in the first chapter.
ROBERT W. WILLSON
HARVARD UNIVERSITY
STUDENTS' ASTRONOMICAL LABORATORY
December, 1905
TABLE OF CONTENTS
CHAPTER I
THE SUN'S DIURNAL MOTION
PAGE
Path of the Shadow of a Pin-head cast by the Sun upon a Horizontal Plane 1
Altitude and Bearing .......... 4
Representation of the Celestial Sphere upon a Spherical Surface. . . 6
The Sun's Diurnal Path upon the Hemisphere is a Circle a Small Circle
except about March 20 and September 21 8
Determination of the Pole of the Circle ....... 9
Bearing of the Points of Sunrise and Sunset 11
The Meridian the Cardinal Points 11
Magnetic Decimation 12
Azimuth .............. 12
The Equinoctial ............ 14
Position of the Pole as seen from Different Places of Observation . . 15
Latitude equals Elevation of Pole 16
Hour-angle of the Sun .......... 17
Uniform Increase of the Sun's Hour-angle Apparent Solar Time . . 18
Declination of the Sun its Daily Change . . . . . .20
CHAPTER II
THE MOON'S PATH AMONG THE STARS
Position of the Moon by its Configuration with Neighboring Stars . .21
Plotting the Position of the Moon upon a Star Map .... 24
Position of the Moon by Measures of Distance from Neighboring Stars . 25
The Cross-staff 25
Length of the Month ' 29
Node of the Moon's Orbit 30
Errors of the Cross-staff . 31
VI TABLE OF CONTENTS
CHAPTER III
THE DIURNAL MOTION OF THE STARS
PAGE
Instrument for measuring Altitude and Azimuth 34
Adjustment of the Altazimuth .35
Determination of Meridian by Observations of the Sun .... 37
Determination of Apparent Noon by Equal Altitudes of the Sun . . 39
Meridian Mark 40
Selection of Stars Magnitudes . . . . . . . . .41
Plotting Diurnal Paths of Stars on the Hemisphere .... 42
Paths of Stars compared with that of the Sun .42
Drawing of Hemisphere with its Circles ....... 42
Rotation of the Sphere as a Whole 43
Declinations of Stars do not change like that of the Sun ... 43
Equable Description of Hour-angle by Stars ...... 43
Hour-angle and Declination fix the Position of a Heavenly Body as well
as Altitude and Azimuth Comparison of the Two Systems of
Coordinates ........... 44
Equatorial Instrument for measuring Hour-angle and Declination . . 45
Universal Equatorial Advantages of the Equatorial Mounting . . 45
CHAPTER IV
THE COMPLETE SPHERE OF THE HEAVENS
Rotation of the Heavens about an Axis passing through the Pole explains
Diurnal Motions of Sun, Moon, and Stars . . . . .47
Relative Position of Two Stars determined by their Declinations and the
Difference of their Hour-angles 48
Use of Equatorial to determine Positions of Stars . . . . .49
Use of a Timepiece to improve the Foregoing Method .... 50
Map of Stars by Comparison with a Fundamental Star . . . .53
Extension of Use of Timepiece to reduce Labor of Observation . . 54
The Vernal Equinox to replace the Fundamental Star Right Ascension 56
Sidereal Time Sidereal Clock 57
Right Ascension of a Star is the Sidereal Time of its Passage across the
Meridian 58
Right Ascension of any Body plus its Hour-angle at any Instant is Side-
real Time at that Instant ........ 58
Finding Stars by the Use of a Sidereal Clock and the Circles of the Equa-
torial Instrument .......... 59
The Clock Correction 60
List of Stars for determining Clock Error 61
TABLE OF CONTENTS vii
CHAPTER V
MOTION OF THE MOON AND SUN AMONG THE STARS
PAGE
Plotting Stars upon a Globe in their Proper Relative Positions . . 63
Plotting Positions of the Moon upon Map and Globe by Observations of
Declination, and Difference of Right Ascension, from Neighboring
Stars ' . . . . .64
Variable Rate of Motion of the Moon . . . . ' . . . 65
Variable Semi-diameter of the Moon ........ 65
Position of Greatest Semi-diameter and of Greatest Angular Motion . ( 65
Plotting Moon's Path on an Ecliptic Map 65
Observations of Sun's Place in Reference to a Fundamental Star by Equa-
torial and Sidereal Clock ........ 66
Sun's Place referred to Stars by Comparison with the Moon or Venus . 68
Plotting the Sun's Path upon the Globe the Ecliptic .... 70
CHAPTER VI
MERIDIAN OBSERVATIONS
Use of the Altazimuth or Equatorial in the Meridian 72
The Meridian Circle 73
Adjustments of the Meridian Circle ........ 74
Level ............. 74
Collimation ............. 78
Azimuth . ... 78
Determination of Declinations . . 80
Determination of the Polar Point . . . . . . . . 81
Absolute Determination of Declination ....... 81
Determination of the Equinox 83
Absolute Right Ascensions . ... . . . . . . .84
Autumnal Equinox of 1899 85
Autumnal Equinox of 1900 * . . .87
Length of the Year . 88
CHAPTER VII
THE NAUTICAL ALMANAC
Mean Time 91
The Equation of Time 92
Standard Time . 93
viii TABLE OF CONTENTS
PAGE
The Calendar Pages 94
Examination of the Several Columns ....... 99
Data for the Planets and Stars ........ 102
Comparison of Observations with the Ephemeris 102
Observations of the Moon with the Cross-staff ; Length of the Month . 103
Observation at Apparent Noon 104
Observations of the Planets. Observations of the Moon with Equatorial 105
Observations of the Sun's Place ........ 106
Determination of the Equinox ........ 107
CHAPTER VIII
THE CELESTIAL, GLOBE
Description of the Globe . . . , " . . . . . .108
Rectifying the Globe for a Given Place and Time . . . . . Ill
The Sun's Place on the Globe . . . : ,. . ... . .112
The Altitude Arc 113
Problems which do not require Rectification of the Globe . . . 114
Problems which require Rectification of the Globe for a Given Time . 117
Finding an Hour-angle by the Globe . . . . . . .119
Reduction to the Equator . . . . . . . . 121
CHAPTER IX
EXAMPLES OF THE USE OF THE GLOBE
Problems which require Rectification of the Globe for a Given Place . 122
Rising and Setting of Stars 122
Sunrise . . . . 124
Altitude and Azimuth ; Hour-angle ........ 125
Finding the Time from the Sun's Altitude 126
Identifying a Heavenly Body by its Altitude and Azimuth at a Given Time 129
Aspect of the Planets at a Given Time ....... 130
Rising and Setting of the Moon 131
Twilight 133
Orientation of Building by Sun Observation 134
Latitudes in which Southern Cross is Visible . . . ... 135
The Midnight Sun ; the Harvest Moon 136
Change of Azimuth at Rising and Setting 137
Graduating a Horizontal Sundial . . 137
Graduating a Vertical Sundial 138
Determining Path of Shadow by Globe 139
The Hour-index 141
TABLE OF CONTENTS IX
CHAPTER X
THE MOTION OF THE PLANETS
PAGE
Elliptic Orbits a Result of the Law of Gravitation ..... 143
Properties of the Ellipse 144
To draw an Ellipse from Given Data . . . ' ' '. . . .145
Mean and True Place of a Planet ; Equable Description of Areas . 146
The Equation of Center . . .148
Measurement of Angles in Radians . 149
The Diagram of Curtate Orbits .151
To find the Elements of an Orbit from the Diagram . . . . 154
Place of the Planet in its Orbit - . .156
To find the True Heliocentric Longitude of a Planet . . . . 157
To find the Heliocentric Latitude ........ 161
Geocentric Longitude of a Planet . . . . ... . . 161
The Sun's Longitude and the Equation of Time 162
Geocentric Latitude .......... 163
Perturbations; Precession 166
The Julian Day 167
Right Ascensions and Declinations of the Planets 167
Configurations of the Planets 168
The Path of Mars among the Stars in 1907 169
LABORATORY ASTRONOMY
PART I
CHAPTER I
THE DIURNAL MOTION OF THE SUN
THE most obvious and important astronomical phenomenon that
men observe is the succession o.f day and night, and the motion of
the sun which causes this succession is naturally the first object of
astronomical study. Every one knows that the sun rises in the east
and sets in the west, but very many educated people know little
more of the course of the sun than this. The first task of the
beginner in astronomy should be to observe, as carefully as possible,
the motion of the sun for a day. What is to be observed then?
A little thought shows that it can only be the direction in which
we have to look to see it at different times ; that is, toward what
point of the compass how far above the ground. All astronom- /
ical observation, indeed, comes down ultimately to this the direc- /
tion in which we see things. The strong light of the sun enables
us to make use of a very simple method depending on the principl
that the shadow of a body lies in the same straight line with
body and the source of light.
Path of the Shadow of a Pin-head. If we place a pin upright on
a horizontal plane in the sunlight and mark the position of the
shadow of its head at any time, we thus fix the position of the
sun at that time, since it is in the prolongation of the line drawn
from the shadow to the pin-head. In order to carry out systematic
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2 LABORATORY ASTRONOMY
observations by this method in such a form that the results may be
easily discussed, it will be convenient to have the following appa-
ratus : (1) A firm table in such a position as to receive sunlight for
as long a period as possible. It is better that it should be in the
open air, in which case it may be made by driving small posts into
FIG. i
the ground and securely fastening a stout plank about 18 inches
square as a top. (2) A board, 18 inches long and 8 inches broad,
furnished with leveling screws and smoothly covered with white
paper fastened down by (3) thumb tacks. (4) A level for leveling
the board. (5) A compass. (6) A glass plate, 6 inches long and
2 inches broad, along the median line of which a straight black
line is drawn. (7) A pin, 5 cm. long, with a spherical head and
an accurately turned base for setting it vertical. (8) A timepiece.
Draw a straight pencil line across the center of the paper as
FIG. 2
nearly as possible perpendicular to the length of the board. Place
the board upon the table and level approximately. Put the com-
pass on the middle of the pencil line and put the glass plate on the
compass with its central line over the center of the needle ; turn
the plate till its median line is parallel to the pencil line (Fig. 2),
THE DIURNAL MOTION OF THE SUN 3
and swing the whole board horizontally, till the needle is parallel to
the two lines, which are then said to be in the magnetic meridian.
Press the leveling screws firmly into the table, and thus make dents
by which the board may at any future time be placed in the same
position without the renewed use of the compass. Level the board
carefully, placing the level first east and west, then north and south.
Place the pin in the pencil line, in the center if the observation
is made between March 20 and September 20, but near the south-
ern edge of the board at any other time of the year, pressing it
firmly down till the base is close to the paper, so that the pin is
perpendicular to the paper. Mark with a hard pencil the estimated
center of the shadow of the pin-head, A (Fig. 3), noting the time by
the watch to the nearest minute, affix a number or letter, and affix
the same number to the recorded time of the observation in the
note-book. It is a good plan to use pencil for notes made while
FIG. 4
observing, and ink for computations or notes added afterward in
discussing them. Kepeat at hourly, or better half -hourly, intervals,
thus fixing a set of points (Fig. 4), through which a continuous
curve may be drawn showing the path of the shadow for several
hours. The same observation should be repeated two weeks later.
LABORATORY ASTRONOMY
ALTITUDE AND BEARING
By the foregoing process we obtain a diagram on which is shown
the position of the pin point, a magnetic meridian line through this
point, and a series of numbered points showing the position of the
shadow of the pin-head at different times ; the height of the pin is
known and also the fact that its head was in the same vertical line
with its point.
In the discussion of these results, it will be convenient to proceed
as follows :
Eemove the pin and draw with a hard pencil a fine line, AB
(Fig. 5), through the pinhole and the point marked at the first obser-
vation. This line is called a line of bearing, and the angle which
FIG. 5
it makes with the magnetic meridian is called the magnetic bearing
of the line. This angle, which may be directly measured on the
diagram by a protractor, fixes the position of the vertical plane which
contains the observed point and passes also through the center of
the pin-head and the sun. If this point bears KW. from the pin,
the sun evidently bears S.E.
Imagine a line, AC (Fig. 3), connecting the observed point with the
sun's center and passing also through the center of the pin-head.
The position of the sun in the vertical plane is evidently fixed by
this line. The angle between the line of bearing and this line, BA C,
is called the altitude of the sun ; it measures, by the ordinary con-
vention of solid geometry, the angle between the sun's direction
and the plane of the horizon.
THE DIURNAL MOTION OF THE SUN o
To determine this angle, lay off the line B'C' (Fig. 6), equal in
length to the pin, 5 cm., draw a perpendicular through E\ and by
means of a pair of compasses or scale laid
between the two points A and B (Fig. 5),
lay off the line A'B' on the perpendicular,
draw A'C'f and measure the angle B'A'C'
by a protractor. We now have the bearing
and altitude of the sun at the time of the
first observation, the bearing of the sun
from the pin being opposite to that of
the point from the pin. In like manner
the altitude and bearing are determined for
each observed point upon the path of the
shadow, and noted against the correspond-
ing time, in the note-book (to avoid con-
fusion, it is convenient to make a separate
figure for the morning and afternoon
observations, as shown in Fig. 6). We
have thus obtained a series of values
which will enable us to study more easily
the path of the sun upon the concave of
the sky.
Plotting the Sun's Path on a Spherical Surface. Probably the
most evident method of accomplishing this object would be to
construct a small concave portion of a sphere, as in the accom-
panying figure, which suggests how the position of the sun might
be referred to the inside of a glass shell.
6 LABORATORY ASTRONOMY
But the hollow surface offers difficulty in construction and
manipulation, and it requires but little stretch of the imagination
to pass to the convex surface as follows. The glass shell, as
seen from the other side, would appear thus :
FIG.
and we can more readily get at it to measure it, and moreover can
more easily recognize the properties of the lines which we shall
come to draw upon it, since we are used to looking upon spheres
from the outside rather than from the inside, except in the case of
the celestial sphere.
On both Figs. 7 and 8 is shown a group of dots which have
nearly the configuration of a group of stars conspicuous in the
southern heavens in midsummer and called the constellation of
Scorpio. It is evident that the constellation has the same shape in
both cases, except that in Fig. 8 it is turned right for left or semi-
inverted, as is the image of an object seen in a mirror. This prop-
erty obviously belongs to all figures drawn on the concave surface
as seen from the center, when they are looked at from the outside
directly toward the center.
So also the diurnal motion of the sun, which as we see it
from the center is from left to right, would be from right to
left as viewed from the outside of such a surface. This latter
is so slight an inconvenience that it is customary to represent
the motions of the heavenly bodies in the sky upon an opaque
globe, and to determine the angles which these bodies describe
about the center, by measuring the corresponding arcs upon the
convex surface.
THE DIURNAL MOTION OF THE SUN 7
Plotting on a Hemisphere. The apparatus required for plotting
the sun's path consists of : a hemisphere,, a, 4{- inches in diameter ;
a circular protractor, b, a quadrantal protractor, c, of 2^- inches
FIG. 9
radius, and a pair of compasses, d, whose legs may be bent and one
of which carries a hard pencil point.
Determine by trial with the compasses the center of the base of
the hemisphere, and mark two diameters by drawing straight lines
upon the base at right angles through the center. Prolong these by
marks about -J inch in length upon the convex surface. Place the
Fio. 10
hemisphere exactly central upon the circular protractor, by bring-
ing the marked ends of one of the diameters upon those divisions
of the protractor which are numbered and 180, and the other on
the divisions numbered 90 and 270. Determine and mark the
8
LABORATORY ASTRONOMY
highest point of the hemisphere by placing the quadrant with its
base upon the circular protractor, and its arc closely against the
sphere, and marking the end of the scale (Fig. 10). Eepeat this
with the arc in four positions, 90 apart on the base. The points
thus determined should coincide ; if they do not, estimate and mark
the center of the four points thus obtained. This point represents
the highest point of the dome of the heavens the point directly
overhead, called the zenith, and the zero and 180 points on the base
protractor may be taken as representing the south and north points
respectively of the magnetic meridian.
The Sun's Path a Circle. To plot the altitude and bearing of the
first observation, place the foot of the quadrant or altitude arc
close against the sphere, the foot of its graduated face on the
degree of the protractor which corresponds to the bearing. Mark
a fine point on the sphere at that degree of the altitude arc corre-
sponding to the altitude at the first observation. This point fixes
the direction in which the sun would have been seen from the center
of the hemisphere at the time of observation if the zero line had
been truly in the magnetic meridian. Proceed in the same manner
with the other observations of bearing and altitude, and thus obtain
FIG. 11
a series of points (Fig. 11), through which may be drawn a con-
tinuous line representing the sun's path upon that day.
It will appear at once that the arcs between the successive points
are of nearly equal length if the times of observation were equi-
distant, and otherwise are proportional to the intervals of time
THE DIURNAL MOTION OF THE SUN
between the corresponding observations a property which does
not at all belong to the shadow curve from which the points are
derived. We thus have a noteworthy simplification in referring
our observations to the sphere. It will also appear that a sheet of
FIG. 12
stiff paper or cardboard may be held edgewise between the hemi-
sphere and the eye, so as to cover all the points ; that is, they all
lie in the same plane. This fact shows that the sun's path is a
circle on the sphere. It is shown by the principles of solid geometry
that all sections of the sphere by a plane are circles. If the plane
of the circle passes through the center, it is the largest possible, its
radius being equal to that of the sphere ; it is then called a great
circle. Near the 20th of March and 22d of September it will be
found that the path of the shadow is nearly a straight line on the
diagram, and that the path of the sun is nearly a great circle ; that
is, the plane of this circle passes nearly through the center of the
sphere. In general, the shadow path is a curve, with its concave
side toward the pin in summer and its convex side toward it in
winter, while the path on the sphere is a small circle, that is, its
plane does not pass through the center of the sphere.
Determining the Pole of the Circle. It is proved by solid geometry
that all points of any circle on the sphere are equidistant from two
10
LABORATORY ASTRONOMY
points on the sphere, called the poles of the circle. It is important
to determine the pole of the sun's diurnal path.
Estimate as closely as possible the position on the sphere of a
point which is at the same distance from all the observed points of
the sun's path and open the compasses to nearly this distance. For
a closer approximation to the position of the pole, place the steel
point of the compasses at the point on the hemisphere correspond-
ing to the first observation, a, and with the other (pencil) point draw
a short arc, m (Fig. 12), near the estimated pole. Draw the arc n
from the point of the
last observation, <?, and
join these two arcs by
a third drawn from an
observed point, b, as
near as possible to the
middle of the path;
the pole of the sun's
diurnal circle will lie
nearly on the great
circle drawn from b to
the middle point o of
the arc last drawn.
Place the steel point
at o, and the pencil
point at b, and try the
distance of the pencil
FIG. is point from the sun's
path at either ex-
tremity. If the pencil point lies above (or below) the path at both
extremities, the compasses must be opened (or closed) slightly and
the assumed pole shifted directly away from (or toward) the middle
of the path.
The proper opening of the compasses is thus quickly determined
as well as a close approximation to the position of the pole. Place
the steel point at this new position, p, the pencil point at b, and
again test the extreme points. If the west end of the path is below
the pencil point (Fig. 13), the latter should be brought directly down
THE DIURNAL MOTION OF THE SUN 11
to the path by shifting the steel point on the sphere in the plane
of the compass legs, that is, along the great circle from p to s.
From the point thus found a circle can be described with the
compasses so as to pass approximately through all the observed
points ; that is, this point is the pole of the sun's path, and when
it is fixed as exactly as possible a circle is to be drawn from horizon
to horizon which will represent the sun's path from the point of
sunrise to that of sunset, and passing very nearly through all the
observed points. The bearing of the points of sunrise and sunset
may then be read off on the horizontal circle.
THE MERIDIAN
The pole as thus determined marks a very interesting and
important point in the heavens. We will draw a great circle
through the zenith and the pole. To do this, place the altitude arc
against the sphere, as if to measure the altitude of the pole ; and
FIG. 14
using it as a guide, draw the northern quadrant of the vertical
circle through the zenith and the pole. Note the bearing of this
vertical circle. Place the altitude arc at the opposite bearing, and
draw another or southern quadrant of the same great circle till it
meets the south horizon. This great circle (Fig. 14) is called the
meridian of the place of observation, and its plane is called the
plane of the meridian of the place of observation, sometimes
the true meridian, to distinguish it from the magnetic meridian.
12 LABORATORY ASTRONOMY
The line in which it cuts the base of the hemisphere represents the
meridian line or true meridian line, just as the line first drawn repre-
sents the line of the magnetic meridian. If the observations are made
in the United States, near a line drawn from Detroit to Savannah,
it will be found that the true meridian coincides very nearly with
the magnetic meridian. East of the line joining these cities, the
north end of the magnet points to the west of the true meridian by
the amounts given in the following table :
21 at the extreme N.E. boundary of Maine.
15 at Portland.
10 at Albany and New Haven.
5 at Washington and Buffalo.
While on the west the declination, as it is called, is to the east of the
true meridian.
5 at St. Louis and New Orleans.
10 at Omaha and El Paso.
15 at Deadwood and Los Angeles.
20 at Helena, Montana, and C. Blanco.
23 at the extreme N.W. boundary of the United States.
By drawing these lines on the map, as in Fig. 15, it is easy to
estimate the declinations at intermediate points within one or two
degrees, at the present time west declinations in the United States
are increasing and east declinations decreasing by about 1 in fifteen
years.
A great circle perpendicular to the meridian may be drawn by
placing the altitude protractor at readings 90 and 270 from the
meridian reading and drawing arcs to the zenith in each case.
This circle is the prime vertical, and intersects the horizon in the
east and west points ; thus all the cardinal points are fixed by the
meridian determined from our plotting of the sun's path.
Azimuth. Place the hemisphere upon the circular protractor in
such a position that the line of the true meridian on the hemisphere
coincides with the zero line of the protractor.
Place the altitude arc so as to measure the altitude at any part of
the sun's path west of the meridian (Fig. 16). The reading of the
foot of the arc will give the angle between the true meridian and
THE DIURNAL MOTION" OF THE SUN
13
the vertical plane containing the sun at that point of its diurnal
circle. This angle is its true bearing and differs from its magnetic
5
15
FIG. 15
bearing by the declination of the compass, being evidently less than
the magnetic bearing, if the decimation is west of north. It is also
called the azimuth of the sun's vertical circle, or, briefly, of the sun.
FIG. is
Formerly azimuth was usually reckoned from north through the
west or east, to 180 at the south point. It is now customary to
measure it from south through west up to 360, so that the azimuth
14 LABORATORY ASTRONOMY
of a body when east of the meridian lies between 180 and 360.
The present method is more convenient because the given angle
fixes the position of the vertical circle without the addition of the
letters E. and W. It is worthy of notice that with this notation
the azimuth of the sun as seen in northern latitudes outside of the
tropics always increases with the time ; and indeed this is true of
most of the bodies we shall have occasion to observe.
Now place the altitude quadrant so that its foot is at a point on
the circular protractor where the reading is 360 minus the azimuth
of the point just measured ; the sun at this point of its path is just as
far east of the meridian as it was west of the meridian at the point
last considered, and it will be found that the altitude of the two
points is the same. On the path shown in Fig. 16 the altitude is
45 at the points whose azimuths are 60 and 300 (60 E. of S.).
This fact, that equal altitudes of the sun correspond to equal
azimuths east and west of the true meridian, is an important one,
and will presently be made use of to enable us to determine the
position of the true meridian with a greater degree of precision.
THE EQUINOCTIAL
We shall find it convenient to draw upon the hemisphere another
line, which plays an important role in astronomy, the great circle
90 from the pole. Placing the steel point of the compasses at
the zenith, open the legs until the pencil point just comes to the
horizon plane where the spherical surface meets it, so that if it
were revolved about the zenith, the pencil point would move in
the horizon. The compass points now span an arc of 90 upon the
hemisphere. Place the steel point at the pole, and draw as much
of a great circle as can be described on the sphere above the horizon.
This will be just one-half of the great circle, and will cut the horizon
in the east and west points. The new circle is called the equinoctial
or celestial equator (Fig. 17).
We have seen that the path of the sun over the dome of the
heavens appears to be a small circle described from east to west
about a fixed point in the dome as a pole. The ancient explanation
of this fact was that the sun is fixed in a transparent spherical shell
THE DIURNAL MOTION OF THE SUN
15
of immense size revolving daily about an axis, the earth being a
plane in the center of unknown extent, but whose known regions
are so small compared to the shell that from points even widely
separated on the earth the appearance is the same ; just as the
FIG. 17
apparent direction and motion of the sun would be practically
the same on our hemisphere to a microscopic observer at the
center, and to another anywhere within one-hundredth of an inch
of the center. When observations were made, however, at points
some hundreds of miles apart on the same meridian, very per-
ceptible differences were found, whose nature will be understood
from a comparison of the hemisphere (Fig. 18 a), plotted from
FIG. 18
observations made Aug. 8, 1897, at a point in Canada, not far
from Quebec, with a second hemisphere (Fig. 18 &), on which is
shown the path of the sun on the same date derived from observa-
tion of the shadow of a pin-head at Polfos in Norway. It appears
on comparison that the distance of the pole above the north horizon
16 LABORATORY ASTRONOMY
is considerably greater in the latter, while the equator is just as
much nearer the southern horizon ; the sun is at the same distance
from the equator in each case. This fact cannot be explained on
the supposition that the horizon planes of the two places are the
same, for in that case we should have the spherical shell which
contains the sun revolving at the same time about two different
.fixed axes, which is impossible. It is not, however, improbable
that the earth's surface should be curved, if we can admit as
a possibility that the direction of gravity, which is perpendicular
to a horizontal plane, may be different at different places. That
the earth's surface in the east and west direction is curved, we
know; for men have traversed it from east to west and returned
to the starting point, so that we have good reason to believe that its
surface is everywhere curved. Long before this conclusive proof
was obtained, however, the globular form of the earth was inferred
on good grounds.
It was early suggested (regarding the fact that, if the sun is fixed
in a shell, that shell is of enormous size as compared with the earth)
that it is inherently more probable that the apparent motion of
the sun is due to a rotation of the spherical earth about an axis
passing through the earth's center and the poles of the sun's circle.
This argument is greatly strengthened when we investigate the
apparent motion of the stars in connection with their size and dis-
tance, and it is now beyond a doubt that this is the true explanation
of the apparent diurnal motion of the sun.
LATITUDE EQUALS ELEVATION OF THE POLE
This subject is treated in all text-books on descriptive astronomy,
and it is pointed out that the pole of the sun's path is the point
where the line of the earth's axis of rotation cuts the sky, and the
equinoctial or celestial equator is the great circle in which the plane
of the earth's equator cuts the sky. The fact is proved also that
the elevation of the pole above the horizon at any place is equal to
the latitude of the place.
This angle, as measured on the hemisphere shown in Fig. 18 a, is
47, and on the hemisphere of Fig. 18 b is 62. The latitudes of
THE DIURNAL MOTION OF THE SUN
17
Quebec and Polfos as determined by more accurate measures are
46 50' and 61 57'.
It is easy to see that the arc of the meridian from the zenith
to the equinoctial is also equal to the latitude, while the arc from
the south point of the horizon to the equator and that from the
zenith to the pole are each equal to 90 minus the latitude, or, as it
is usually called, the co-latitude.
It will be well here, as in all our measurements, to form some idea
of the accuracy of our results. As one degree on our hemisphere
is quite exactly equal to l mm , a quantity easily measured by ordi-
nary means, it is not difficult with ordinary care to determine the
FIG. 19
pole of the sun's path so closely that no observed point lies more
than a degree from the path. The pole is then fixed within one
degree unless the length of the path is very short ; usually if the
path is more than 90 in length the pole may be placed within less
than a degree of its true place and the latitude measured with an
error of less than one degree.
HOUR-ANGLE OF THE SUN
Open the dividers as before (see p. 14) so as to draw a great circle.
Place the steel point upon the place of the sun, S, on its diurnal
circle at the time of the last observation in the afternoon (Fig. 19),
and with the pencil point strike a small arc cutting the equator at Q.
18 LABORATOKY ASTRONOMY
Place the steel point where this arc cuts the equator, and draw a
great circle which will pass through the sun's place and the pole ;
notice that it also cuts the equator at right angles. Such a^ circle is
called an hour-circle. It is the intersection of the surface of the
sphere with a plane that passes through the poles and the place of
the sun. The number of degrees in the arc of the equator, included
between the meridian and the hour-circle which passes through the
sun, is called the hour-angle of the sun. By the ordinary convention
of solid geometry it measures the wedge angle between the plane of
the hour-circle and the plane of the meridian. If a book be placed
with its back in the line from the pole to the center of the sphere,
and with its title-page to the west, and the western cover opened
till it is in the plane of the hour-circle, while the title-page is in
the plane of the meridian, the wedge angle between the title-page
and the cover will be the hour-angle and will be measured by the
arc of the equator indicated above. It is reckoned as increasing
from the meridian towards the west in the direction in which the
cover is opened. If the hour-circle of the first morning observa-
tion is determined in the same way, the hour-angle measured in
the opposite direction from the meridian is sometimes called the
hour-angle east of the meridian ; but more commonly by astronomers
this value is subtracted from 360, and the angle thus obtained is
called the hour-angle, this being more convenient because the hour-
angle of the sun thus measured constantly increases with the time
as the sun pursues its course ; being at noon, 180 at midnight,
360 at the next noon, etc.
UNIFORM INCREASE OF HOUR-ANGLE
Let us now examine more carefully the truth of the surmise pre-
viously made, that the arc of the sun's path between two successive
observations is proportional to the interval of time between the
observations. Draw the hour-circles of the sun at each point of
observation (Fig. 20) ; measure the arc on the equator between the
first and the last hour-circles ; divide by the number of minutes
between the two times. This will give the average increase of
hour-angle per minute. Multiply this increase by the difference in
THE DIURNAL MOTION OF THE SUN
19
minutes of each of the observed times from the time of the first
observation, and compare with the progressive increase of the hour-
angle as measured off on the equator by means of the graduated
quadrant. They will be found to be nearly the same in each case.
It is thus shown that the hour-angle of the sun increases uniformly
with the time. The rate is nearly a quarter of a degree per minute,
since 360 are described in 24 hours. Notice that when the hour-
angle is zero, the actual time by the watch is not very far from 12
o'clock (in extreme cases it may be 45 minutes, if the clock is keep-
ing standard time), and that if the hour-angle in degrees (west of
the meridian) is divided by 15, the number of hours differs from the
FIG. 20
watch time just as much as the time of meridian passage differs
from 12 hours. In fact, the hour-angle of the sun measures what
is called apparent solar time, i.e., when H.A. = 15, it is 1 o'clock;
H. A. = 75, it is 5 o'clock j H.A. = 150, 10 o'clock, etc. ; those angles
east of the meridian lying between 180 and 360, i.e., between 12 h
and 24 h , so that 12 hours must be subtracted to give the correct hours
by the ordinary clock, which divides the day into two periods of 24
hours each ; for instance, if H.A. = 270, it is 18 h past noon or 6 A.M.
of the next day. Astronomical clocks usually show the hours con-
tinuously from to 24, thus avoiding the necessity of using A.M.
and P.M. to discriminate the period from noon to midnight and from
midnight to noon.
20 LABORATORY ASTRONOMY
DECLINATION OF THE SUN
The distance of the sun's path from the celestial equator, meas-
ured along the arc of an hour-circle, is called its declination, and
will be found appreciably the same at all points. It requires more
delicate observation than ours to find that it changes during the
few hours covered by our observation. If, however, the observa-
tion be repeated after an interval, say, of two weeks at any time
except for a month before or after the 20th of June or December,
it will be found that although the sun at the second observation
describes a circle, this circle is not in the same position with regard
to the equator that its declination has changed (between March
13 and 27, for instance, by about 5.5). The inference to be drawn
is that even during the period of our observation the sun's path is
not exactly parallel to the equator, although our observations are
not delicate enough to show that fact.
It is true in general, as in this case, that the first rude meas-
urements applied to the heavenly bodies give results which when
tested by those covering a longer time, or made with more delicate
instruments, are found to require correction.
CHAPTER II
THE MOON'S PATH AMONG THE STARS
NEXT to the diurnal motion of the sun the most conspicuous
phenomenon is the similar motion of the stars and the moon j this
will form the subject of a future chapter.
The study of the moon, however, discloses a new and interesting
motion of that body. It partakes indeed of the daily motion of
the heavenly bodies from east to west, but it moves less rapidly,
requiring nearly 25 hours to complete its circuit instead of 24, as
do the sun and stars, and returning to the meridian therefore
about an hour later on each successive night.
In consequence of this motion it continually changes its place
with reference to the stars, moving toward the east among them
so rapidly that the observation of a few hours is sufficient to show
the fact. At the same time its declination changes like that of the
sun, but much more rapidly.
We should begin early to study this motion, and it will be found
interesting to continue it at least for some months at the same time
that other observations are in progress a very few minutes each
evening will give in the course of time valuable results.
POSITION BY ALIGNMENT WITH STARS
The first method to be used consists in noting the moon's place
with reference to neighboring stars at different times. Some sort
of star map is necessary upon which the places of the moon may be
laid down so that its path among the stars may be studied. As
the configurations that offer themselves at different times are of
great variety, it will be well to give a few examples of actual
observations of the moon's place by this method.
Dec. 12, 1899, at X 12 h O m P.M., the moon was seen to be near
three unknown stars, making with them the following configuration,
21
22 LABORATORY ASTRONOMY
which was noted on a slip of paper as shown in Fig. 21. The
relative size of the stars is indicated by the size of the dots. (The
original papers on which the observations are made should be care-
ms -Dec 12* fr^ty preserved ; indeed, this should always be the
practice in all observations.)
At the same time, for purposes of identification,
it was noted that the group of stars formed, with
/ Capella and the brightest star in Orion, both of
A a mmeiricatfi urt wn ^ cn were known to the observer, a nearly equi-
lateral triangle. It was also noted that the moon
was about 6 from the farthest star, this being
estimated by comparison with the known distance between the
"pointers" in the "Dipper" (about 5). With these data it was
easily found by the map that these stars were the brightest stars
in Aries, and the moon was plotted in its proper place on the map
(page 24).
December 13, at 5 h 35 m P.M., the moon was (half its diameter)
below (south of) a line drawn from Aldebaran (identified by its
position with reference to Capella and Orion and by the letter V of
stars in which it lies, the Hyades) to the faintest of the three
reference stars of December 12. It was also about f west of a
line between two unknown stars identified later as Algol (equi-
distant from Capella and Aldebaran) and y Ceti (at first supposed
on reference to the map to be a Ceti,
Dec. 13 d 5*3S m <<w ^
but afterward correctly identified by
comparing the map with the heavens). \
The original observation is given
below (Fig. 22) of about one-half the
size of the drawing, all except the
underscored names being in pencil.
The underscored names are in ink and . \
made after the stars were identified.
This is a useful practice when addi-
tions are made to an original, so that subsequent work may not be
given the appearance of notes made at the time of observation. It
is well to give on the sketch map several stars in the neighborhood
of those used for alignment, to facilitate identification.
THE MOON'S PATH AMONG THE STARS 23
The alignment was tested by holding a straight stick at arm's
length parallel to the line joining the stars.
December 14, 6 h 30 m P.M. Moon on a line from Algol through
the Pleiades (known) about 2^ (5 diameters of moon) beyond the
latter, which were very faint in ag^^^ . *cya sec ^ d s^o^
the strong moonlight. No figure. \ v
December 15, 5 h 10 m P.M. Moon \ \
in a line between Capella and Aide- \ \
baran. Line from Pleiades to moon -_. . . \
filaun -, \
bisects line from Aldebaran to (3 \ \
Tauri (identified by relation to
Aldebaran and Capella).
9 h 25 m P.M. Moon in line from
3 Aurigse to Aldebaran (Fig. 23).
FIG. 23
(NOTE. Henceforth details of identification are omitted.)
December 16, 7 h 40 m P.M. Moon almost totally eclipsed 2J east
of line from ft Aurigae to y Orionis ; same distance from, ft Tauri as
Tauri (revised estimate about nearer ft Tauri
.\ December 18, 10 h 30 m P.M. Observation
\ snatched between clouds. Moon's western edge
tangent to line from a Geminorum to Procyon
and about 1 north of center of that line.
\ In the sketch maps above no great accuracy is
attempted in placing the stars, but in the final
" plotting on the map the directions of the notes
\ are carefully followed. The plotting should be
\ y done as soon as possible after the observation
is made, for even a hasty comparison with the
map will often show that stars have been mis-
identified or that there is some obvious error in
FlG 24 the notes, which may be rectified at once if there
is an opportunity to repeat the observation. Such
a case occurs in the observations of December 13 recorded above,
where y Ceti was mistaken for a.
24
LABORATORY ASTRONOMY
PLOTTING POSITIONS OF THE MOON ON A STAR MAP
Figure 25 shows the positions of the moon plotted from the fore-
going observations, together with the lines of construction from
which they were determined.
A drawing should be made of the shape of the illuminated portion
of the moon at each observation, and the direction among the stars
FIG. 25
of the line joining the points of the horns (cusps) for future study
of the cause of the moon's changes of phase.
If the star map accompanying this book is used, the identification
of the stars consists. in determining which of the dots represents
the star of reference ; the name may be determined by reference
to the list; thus the two stars near the line XXIV on the upper
portion of the map are "a Andromedee O h 5 m + 29" and "yPegasi
Qh g m _^_ 140 rpj^ mean j n g w hich attaches to these numbers is
given in Chapter IV. It is a good plan to keep a copy of the
map on which to note the names for reference as the stars are
learned ; most of the conspicuous ones will soon be remembered
as they are used.
THE MOON S PATH AMONG THE STARS
25
THE MOON'S PLACE FIXED BY ITS DISTANCE FROM
NEIGHBORING STARS
One month's observation by this method will show that the moon's
path is at all points near to the curved line drawn on the map,
which is called the ecliptic and which is explained on page 70.
To establish more accurately its relations to this line it will be
advisable in the later months to adopt a more accurate means of
observation, although when the moon is very near a bright star, its
position may be quite accurately fixed by the means that we have
indicated ; and if it chances to pass in front of a bright star and
produce an occupation, the moon's position is very accurately fixed
indeed, as accurately as by any method. But such opportunities
are rare, and for continuous accurate observation we should have a
means of measuring the distance of the moon from stars that are
at a considerable distance from it. An instrument sufficiently accu-
rate for our purpose is the cross-staff described below. It should be
mentioned that, on ac-
count of the distortion
of the map, the place of
the moon is usually more
accurately given by dis-
tances from the com-
parison stars than by
alignment. The sextant
may be used instead of
the cross-staff, but is
less convenient and also
more accurate than is
necessary.
The Cross-staff. The
cross-staff (Fig. 26) con-
sists of a straight graduated rod upon which slides a " transversal "
or " cross " perpendicular to the rod ; one end of the staff is placed
at the eye and the " cross " is moved to such a place that it just
fills the angle from one object to another ; its length is then the
chord of an arc equal to the angle between the objects as seen from
FIG. 26
26 LABORATOKY ASTRONOMY
that end of the staff at which the eye is placed. The figure, which
is taken from an old book on navigation, illustrates the use of this
instrument for measuring the sun's altitude above the sea horizon ;
the rod in the position shown indicates that the sun's altitude is
about 40.
Obviously a given position of the cross corresponds to a definite
angle at the end of the rod, and the rod may be graduated to give
this angle directly by inspection, or a table may be constructed by
which the angle corresponding to any division of the rod may be
found ; such a table is given on page 27. For our purpose an instru-
ment of convenient dimensions is made by using a cross 20 cm. in
length, sliding on a rod divided into millimeters (Fig. 27) ; this may
be used for measuring angles up to 30, which is enough for our
l.i.;.i.r.M.U.I*M
FIG. 27
purpose. The smallest angle that can be measured is about 12,
which corresponds to a chord of of the radius ; but by making a
part of the cross only 10 cm. long, as shown in the figure, we may
measure angles from 6 upwards, and for smaller angles may use
the thickness of the cross, which is 5 cm., and thus measure angles
as small as 3 ; the longer cross will not give good results above
30, as a slight variation of the eye from the exact end of the rod
makes a perceptible difference in the value of the angles greater
than 30.
Measures with the Cross-staff. As an example of the use of the
cross-staff, the following observations are given: They were made
with a staff about 3 feet in length, graduated by marking the point
for each degree at the proper distance in millimeters from the eye
end of the staff, as given by Table II on page 27. After the points
were marked a straight line was drawn through each entirely across
the rod, using the cross itself as a ruler ; graduations were thus
made on one side for use with the 20 cm. cross, on the other for the
THE MOON S PATH AMONG THE STARS
27
TABLE I ANGLE SUBTENDED BY CROSSES
TABLE II
Distance
from
Eye
LEXGTH OF CROSS
Distance
from
Eye
LENGTH OF CROSS
Angle
subtended by
20 cm. Cross
20cm.
10 cm.
5 cm.
20cm.
10 cm.
5 cm.
lOQcm
11. 4
5. 7
2. 9
62cm
18.3
9. 2
4. 6
99
11 .5
5 .8
2 .9
61
18.6
9.4
4 .7
98
11 .6
5.8
2 .9
60
18 .9
9.5
4 .8
97
11 .8
5 .9
3.0
59
19 .2
9 .7
4 .9
96
11 .9
6.0
3.0
58
19.6
9.9
4.9
95
12.0
6.0
3.0
57
19.9
10 .0
5 .0
94
12 .1
6 .1
3 .0
56
20.2
10 .2
5 .0
12
951mm
93
12 .3
6.2
3.1
55
20 .6
10 .4
5 .2
13
878
92
12 .4
6 .2
3 .1
54
21 .0
10 .6
5 .3
14
814
91
12 .5
6 .3
3.1
53
21 .4
10 .8
5 .4
15
760
90
12 .7
6.4
3.2
52
21 .8
11 .0
5 .5
16
711
89
12.8
6 .4
3 .2
51
22 .2
11 .2
5 .6
17
669
88
13 .0
6 .5
3.3
50
22 .6
11 .4
5 .7
18
631
87
13 .1
6.6
3 .3
49
23 .1
11 .6
5 .8
19
598
86
13.3
6 .7
3 .3
48
23 .5
11 .9
6.0
20
567
85
13 .4
6 .7
3.4
47
24.0
12 .1
6.1
21
540
84
13.6
6 .8
3 .4
46
24.5
12 .4
6 .2
22
514
83
13.7
6 .9
3.5
45
25 .1
12 .7
6 .4
23
491
82
13 .9
7.0
3.5
44
25.6
13 .0
6 .5
24
470
81
14.1
7 .1
3 .5
43
26 .2
13.3
6 .7
25
451
80
14 .3
7 .2
3.6
42
26 .8
13 .6
6.8
26
433
79
14 .4
7.2
3 .6
41
27 .4
13 .9
7 .0
27
416
78
14 .6
.7 .3
3 .7
40
28 .1
14.3
7 .2
28
401
77
14 .8
7 .4
3.7
39
28.8
14 .6
7 .3
29
387
76
15 .0
7 .5
3 .8
38
29.5
15 .0
7 .5
30
373
75
15 .2
7 .6
3.8
37
30.2
15 .4
7 .7
31
361
74
15 .4
7 .7
3 .9
36
31 .0
15.8
7 .9
32
349
73
15 .6
7 .8
3 .9
35
31 .9
16.3
8 .2
33
338
72
15 .8
7 .9
4 .0
34
32.8
16 .7
8.4
34
327
71
,16 .0
8 .1
4 .0
33
33.7
17 .2
8.7
35
317
70
16 .3
8 .2
4 .1
32
34.7
17.7
8 .9
36
308
69
16 .5
8.3
4 .2
31
35 .8
18.3
9.2
37
299
68
16 .7
8 .4
4 .2
30
36 .9
18 .9
9 .5
38
290
67
17 .0
8 .5
4.3
29
38.1
19.6
9 .9
39
282
66
17.2
8.7
4 .3
28
39.3
20 .2
10 .2
40
275
65
17 .5
8 .8
4 .4
27
40.6
21 .0
10.6
64
17 .7
8 .9
4.5
26
42 .1
21 .8
11 .0
63
18 .0
9.1
4 .5
25
43 .6
22 .6
11 .4
28 LABORATORY ASTRONOMY
10 cm. cross, and on one edge for the thickness of the cross. By
means of these graduations the angle subtended by the cross in any
position is read directly from the scale, quarters or thirds of a
degree being estimated and recorded in minutes of arc.
The observations are :
1900. January 2. 5 h 15 m .
Moon to e Pegasi, 35 45'
" " Altair, 26 30
" " Fomalhaut, 41 40
January 3. 6 h O m .
Moon to e Pegasi, 23 30'
" " Altair, 29 20
" " j8 Aquarii, 8 20
January 4. 5 h 20 m .
Moon to e Pegasi, 17 40'
" " |8 Aquarii, 8 30
" "5 Capricorni, 9 45
January 6. 5 h 50 m .
Moon to 7 Pegasi, 12 0'
" " a Pegasi, 16 40
" " e Pegasi, 33 30
January 7. 5 h 45 m .
Moon to 7 Pegasi, 9 40'
" " j8 Arietis, 19 45
" "a Andromedee, 21 15
" " j8 Ceti, 27 30
January 8. 6 h O m .
Moon to a Arietis, 11 0'
" " 7 Pegasi, 21 30
January 9. 10 h O m .
Moon to a Arietis, 9 45'
" " Alcyone, 16
" " a Ceti, 15 30
To represent these observations on the star map, open the com-
passes until the distance of the pencil point from the steel point is
equal to the measured distance making use for this purpose of
the scale of degrees in the margin, and then with the steel point
THE MOON'S PATH AMONG THE STARS
29
carefully centered on the comparison star, strike a short arc with
the pencil point near the estimated position of the moon ; the inter-
section of any two of these arcs fixes the position of the moon. If
the different stars give different points, those nearest the moon may
JO-
Pleiades
-20-
tyades
-w-
FlQ. 28
be assumed to give results nearer the truth. Fig. 28 shows the
positions of the moon January 6 to January 9 as plotted from the
above measures.
Length of the Month. If it happens that one of the positions ob-
served in the second month falls between the places obtained on two
successive days of the first month, or vice versa, a determination of
the moon's sidereal period may be made by interpolation. Thus, on
plotting the observation of December 12 (p. 22), which places the
moon between the two observations on January 8 d 6 h O m and Janu-
ary 9 d 10 h O m , its distance from the former is 6.0 and from the latter
10.0, while the interval is 28 h ; the moon's place on December 12 at
12 h O m is therefore the same as on January 8 at 6 h + X 28 h , or Jan-
uary 8 d 16 h .5, that is, January 9 at 4 h 30 m A.M., and the interval
between these two times is 27 d 4 h 30 m , which is the time required for
the moon to make a complete circuit among the stars or the length
30 LABORATORY ASTRONOMY
of the sidereal month. This is a fairly close approximation ; the
observation of December 12 having been made under favorable
circumstances, the configuration being well denned and the stars
near, so that the position on that date by alignment is nearly as
accurate as those determined by the measures on January 8 and 9.
After three months the moon comes nearly to the same position at
about the same time in the evening, so that it is convenient to deter-
mine its period without interpolation by observing the time when the
moon comes into the same star line as at the previous observation ;
moreover, the interval being three months, an error of an hour in
the observed interval causes an error of only 20 m in the length of
the month.
THE MOON'S NODE
When a sufficiently large number of observations have been plot-
ted to give a general idea of the moon's path among the stars, a
smooth curve is to be drawn as nearly as possible through all the
points and this curve should be compared with the ecliptic, as shown
on the map. Its greatest distance from the ecliptic and the place
where it crosses the ecliptic the position of the node should be
estimated with all possible precision. For this purpose, only the
more accurate positions obtained by the cross-staff should be used.
After a few observations of alignment are made, the student will
desire to use the more accurate method at once, but it is better to
have at least one month's observation by the first method (even if
the cross-staff is also used) for comparison with later observations by
alignment for the purpose of determining the length of the month,
as suggested above, without any instrumental aid whatever.
The records of the positions of the node should be preserved by
the teacher for comparison from year to year to show the motion
of this point along the ecliptic. The node, as determined by the
observations above given, was nearly at the point where the ecliptic
crosses the line from y Orionis to Capella. Observations made in
November, 1897, by the method of Chapter IV, gave its place on the
ecliptic at a point where the latter intersects a line drawn through
Castor and Pollux, thus indicating a motion of about 40 in the
interval
THE MOON'S PATH AMONG THE STARS 31
Observations made with the cross-staff are sufficiently accurate
to show that the motion of the moon is not uniform, but as the dis-
tortion of the map complicates the treatment of this subject, we
shall defer its consideration until the method of Chapter V has
been introduced.
It will be well, however, as soon as measures with the cross-staff
are begun, to devote a few minutes each evening to measures of the
moon's diameter with an instrument measuring to 10", such as a good
sextant ; or, better, a telescope provided with a micrometer, in order
to show the variations of the moon's apparent size at different parts
of its orbit. The relative distances of the moon from the earth as
inferred from these measures should be compared with the varia-
tions of her angular motion as read off from the chart ; although
on account of the distortion referred to above, it will not be possible
to show more than the fact that when the moon is nearest, her
angular motion about the earth is greatest, and vice versa.
The sextant or micrometer may henceforward be used also for
observations of the sun's diameter, which should be measured as
often as once a week for a considerable period.
When the moon's diameter is measured, a rough estimate of her
altitude should be made in order to make the correction for aug-
mentation in a future more accurate discussion of the measures for
determining the eccentricity of her orbit.
DETERMINING THE ERRORS OF THE CROSS-STAFF
Observations with the cross-staff are most easily made just before
the end of twilight or in full moonlight, so that the cross may be
seen dark against a dimly lighted background. When used for
measuring* the distance of stars in full darkness, it is convenient
to have a light so placed behind the observer that, while invisible
to him, it shall dimly illuminate the arms of the cross.
As the angles which are determined by the- cross-staff, especially if
large, are affected by the observer's habit of placing the eye too near
to or too far from the end of the staff, it is a good plan to measure
certain known distances and thus determine a set of corrections to
be applied, if necessary, to all measures made with that instrument.
32
LABORATORY ASTRONOMY
The following table gives the distances between certain stars always
conveniently placed for observation in the United States, together
with the results of measures made upon them with a cross-staff
held in the hands without support, and indicates fairly the accuracy
which may be obtained with this instrument. The back of the
observer was toward the window of a well-lighted room, and the
cross was plainly visible by this illumination.
STARS
TRUE
DIS-
TANCE
MEASURED DIS-
TANCES
MEAN
CORREC-
TION
aUrsse
Majoris to /3 Ursse Majoris
5. 4
5.8
5. 8
- 0.4
a "
i i il y i . it
10 .0
10 .5 10. 6 10. 6
10 .6
-0 .6
a "
it a f u u
15 .2
15 .6 15 .5 15 .7
15 .6
-0 .4
/3
" "f "
19 .9
20 .0 20 .3 20 .2
20 .2
-0 .3
a "
; t it a a
'1
25 .7
26 .6 26 .0 26 .1
26 .2
-0 .5
a "
" " Polaris
28 .5
29 .0 29 .2 28 .9
29 .0
-0 .5
ft "
u u u
33 .9
35 .1 34 .7 34 .4
34 .7
-0 .8
n "
u u t
41 .2
42 .2 42 .0 42 .0
42 .1
- .9
The measured distances are about one-half degree too large, and
if a correction of this amount is applied to all angles measured by
this instrument up to 30, the corrected values will seldom be so
much as half a degree in error, and the mean of three readings will
probably be correct within a quarter of a degree.
CHAPTER III
THE DIURNAL MOTION OF THE STARS
As the observations of the moon require but a few minutes each
evening, observations may be made on the same nights upon the
stars. The first object is to obtain the diurnal paths of some of the
brighter stars, and as they cast no shadow we must have recourse
to a new method of observation to determine their positions in the
sky at hourly intervals.
A simple apparatus for this purpose is represented in Fig. 29.
A paper circle is fastened to the leveling board used in the sun
FIG. 29
observations so that the zero of its graduation lies as nearly as
possible in the meridian, and a pin with its head removed is placed
upright through the center of the circle.
A carefully squared rectangular block about 10 inches by 8 inches
by 2 inches is placed against the pin so that the angle which its
face makes with the meridian may be read off upon the horizontal
33
34
LABORATORY ASTRONOMY
circle. A second paper circle is attached to the face of the block
with the zero of its graduations parallel to the lower edge ; a light
ruler is fastened to the block by a pin through the center of its
circle j the ruler may be pointed at any star by moving the block
about a vertical axis till its plane passes through the star, and then
moving the ruler in the vertical plane till it points at the star ; a
lantern is necessary for reading the circles and for illumination of
the block and ruler in full darkness ; it should be so shaded that
its direct light may not fall on the observer's eye. Sights attached
to the ruler make the observation slightly more accurate, but also
rather more difficult, and without them the ruler may be pointed
within half a degree, which is about as closely as the angles can be
determined by the circles.
THE ALTAZIMUTH
An inexpensive form of instrument for measuring altitude and
azimuth is shown in Fig. 30. Here the ruler provided with
sights A, B is movable about d, the center of the semicircle E.
This semicircle is movable about an axis
perpendicular to the horizontal circle F,
and its position on that circle is read
off by the pointer g, which reads zero
when the plane of E is in the meridian.
The circle F is mounted on a tripod
provided with leveling screws. If the
circle is so placed that the pointer reads
zero when the sight-bar is in the mag-
netic meridian, then its reading when
the sights are pointed at any star will
give the magnetic bearing of the star.
It will, however, be more convenient to
adjust the instrument so that the pointer
reads zero when the sight-bar is in the
true meridian.
To insure the verticality of the standard a level is attached to
the sight-bar, and by the leveling screws the instrument must be
FIG. 30
THE DIURNAL MOTION OF THE STARS
35
adjusted so that the circle E may be revolved without causing the
level bubble to move. (See page 36.)
A more convenient and not very expensive instrument is the
altazimuth or universal instrument shown in Fig. 31, which contains
some additional parts by the use of which it may be converted into
an equatorial instrument.
(See page 45.) It consists
of a horizontal plate carry-
ing a pointer and revolving
on an upright axis which
passes through the center
of a horizontal circle grad-
uated continuously from
to 360. The plate carries
a frame supporting the
axis of a graduated circle;
this axis is perpendicular
to the upright axis, and the
circle is graduated from
to 90 in opposite directions.
Attached to the
circle is a tele-
scope whose op-
tical axis is in
the plane of the
circle. The cir-
cle is read by
a pointer which
is fixed to the
frame carrying its axis and reads when the optical axis of the
telescope is perpendicular to the upright axis. A level is attached
to the telescope so that the bubble is in the center of its tube when
the telescope is horizontal. In what follows, all these adjustments
are supposed to be properly made by the maker.
FIG. 31
36
LABORATORY ASTRONOMY
ADJUSTMENT OF THE ALTAZIMUTH
If the altazimuth is so adjusted that the upright axis is exactly
vertical, and if we know the reading of the horizontal circle when
the vertical circle lies in the meridian, we may determine the
position of a heavenly body at any time by pointing the telescope
upon it and reading the two circles. The difference between the
reading of the horizontal circle and its meridian reading is the azi-
muth, and the reading of the vertical circle is the altitude of the
body. Before proceeding to the observation of stars, it will be well
to repeat our observations on the sun, using this instrument, and
making them in such a manner that we may at the same time get a
very exact determination of the meridian reading by the method
suggested on page 14.
Place the instrument upon the table used for the sun observation ;
bring the reading of each circle to ; and turn the whole instru-
ment in a horizontal plane until the telescope points approximately
south, using the meridian determination obtained from the shadow
observations. One leveling screw will then be nearly in the meridian
of the center of the instrument, while the two others will lie
in an east and west line. Bring the level bubble to the middle
of its tube by turning the north leveling screw; then set the
telescope pointing east; and "set" the level by turning the east
and west screws in opposite directions. Be careful to turn them
equally ; this can be done by taking one leveling screw between the
FIG. 32
finger and thumb of each hand, holding them, firmly, and turning
them in opposite directions by moving the elbows to or from the
body by the same amount. Turn the telescope north, and the bubble
THE DIURNAL MOTION OF THE STARS 37
should remain in place j if it does not, adjust the north screw. The
instrument is very easily and quickly adjusted by this method.
The upright axis is vertical when the telescope can be turned about
it into any position without displacing the bubble.
Determination of the Meridian and Time of Apparent Noon. After
completing the adjustment of the instrument, the reading of the circle
FIG. 33
when the telescope is in the meridian is determined as follows : Point
the telescope upon the sun approximately. Place a sheet of paper or
a card behind it, and turn the telescope about the vertical axis until
the shadow of the vertical circle is reduced to its smallest dimensions
and appears as a broad straight line. By moving the telescope
about the horizontal axis, bring the shadow of the tube to the form
of a circle ; in this circle will appear a blurred disk of light. Draw
the card about 10 inches back from the eyepiece, and pull out the
latter nearly -J- of an inch from its position when focused on distant
objects and the disk of light becomes nearly sharp ; complete the
focusing of this image of the sun by moving the card to or from
the eyepiece. The distance of the card and the drawing out of the
eyepiece should be such that the sun's image shall be about -J- to j-
of an inch in diameter. Now move the telescope until the image is
centered in the shadow of the telescope tube, note the time, and read
both circles ; this observation fixes the altitude and azimuth of the
38 LABORATORY ASTRONOMY
sun. For determining the meridian it is not necessary that the
time should be noted, but it will be convenient to use these obser-
vations for a repetition of the determination of the sun's path, deter-
mining the altitudes and azimuths by this more accurate method.
This observation should be made at least as early as 9 A.M.
Now increase the reading of the vertical circle to the next exact
number of degrees, and follow the sun by moving the telescope
about the vertical axis. After a few minutes the sun will be again
centered by this process. Note the time, and read the horizontal
circle. Increase the reading of the vertical circle again by one
degree to make another observation, and so on for half an hour.
Observations may be made at one-half degree intervals of altitude,
but those upon exact divisions will evidently be more accurate. If
circumstances admit, observations may be made, during the period
of two hours before and after noon, for the purpose of plotting the
sun's path ; but, owing to the slow change of altitude in that time,
the corresponding azimuths are not well determined, and they will
be nearly useless for placing the instrument in the meridian.
Some time in the afternoon, as the descending sun approaches the
altitude last observed in the forenoon, set the vertical circle upon the
reading corresponding to that observation, and repeat the series in
inverse order ; that is, decrease the readings of altitude by one degree
each time, and note the time and the reading of the horizontal circle
when the sun is in the axis of the telescope at each successive
altitude.
Since equal altitudes correspond to equal azimuths (see page 14),
east and west of the meridian, the difference of the horizontal read-
ings is twice the azimuth at either of the two corresponding obser-
vations (360 must be added to the western reading, if, as will
generally be the case, the point lies between the two readings).
Therefore, one-half this difference added to the lesser or subtracted
from the greater reading gives the meridian reading. The same
value is more easily found by taking half the sum of the two read-
ings. In the same way one-half the interval of time between the
two observations added to the time of the first reading gives the
watch time of the sun's meridian passage, or apparent noon, as it
is called.
THE DIURNAL MOTION OF THE STARS
39
Each pair of observations gives the value of the meridian reading
and of the watch time of apparent noon ; their accordance will give
an idea of the accuracy of the observations.
The following observations of the sun were made March 8, 1900,
with an instrument similar to that shown in Fig. 33.
TIME
ALTITUDE
HORIZONTAL
CIRCLE
TIME
ALTITUDE
HORIZONTAL
CIRCLE
1 8*
54
37"
27.
5
307. 6
9
2 h 32 m
10
31.0
47. 7
2 8
58
10
28 .
308 .4
10
2 36
30
30 .5
48 .7
3 9
1
42
28 .
5
SM9 .2
11
2 39
45
30 .0
49 .45
4 9
4
51
29 .
310 .0
12
2 43
27
29.5
50.35
5 9
9
5
29 .
5
310 .9
13
2 47
29 .0
51 .15
6 9
12
20
30.
311 .8
14
2 50
17
28 .5
51 .95
7 9
15
35
30 .
5
312 .6
15
2 54
7
28.0
52 .85
8 9
19
37
31 .
313 .45
16
2 57
33
27 .5
53 .6
The 1st and 16th of these observations give for the meridian
reading % [307.6 + (53.60 + 360)] = 360.60, and for the correspond-
ing watch time [8 54 37 + (2 57 33 + 12 h )] = ll h 56 m 5 s .
Taking the corresponding A.M. and P.M. observations in this man-
ner, we find for the eight pairs of observations above the following
values.
MERIDIAN READING
360. 6
360 .625
.575
.575
.625
.625
.65
.575
ALTITUDE
27. 5
28 .0
28 .5
29 .0
29.5
30 .0
30 .5
31 .0
mean
360
360
360
360
360
360
360.61
WATCH TIME OF NOON
ll h 56 1 " 5.QS
56 8.5
55 59.5
55 55.5
56 16.0
56 2.5
56 2.5
55 53.5
11 56 2.9
The agreement of these results is closer than will usually be
obtained, the observations being made by a skilled observer and the
angles carefully read by means of a pocket lens, which in many
cases enabled readings to be made to 0.05 ; any reading such as that
of the 8th observation, where the value was estimated to lie between
two tenths, being recorded as lying halfway between them. This
practice adds little to the accuracy if several observations are made,
and is not to be recommended to beginners.
40 LABORATORY ASTRONOMY
MERIDIAN MARK
It will be convenient to fix a meridian mark for future use. This
may be done by fixing the telescope at the meridian reading, turning
it down to the horizontal position, and placing some object (as a
stake) at as great a distance as possible, so that it may mark the
line of the axis of the telescope when in the meridian. A mark on
a fence or building will serve if at a greater distance than 50 feet,
though a still greater distance is desirable. For setting the tele-
scope upon the mark, it is convenient to have two wires crossing in
the center of the field of view, but the setting may be made within
0.l without this aid. Having established such a mark, set the
horizontal circle at 0, and move the whole base of the instrument
until the telescope points upon the meridian mark. Level carefully ;
then set the telescope again, if the operation of leveling has caused
it to move from the meridian mark ; level again, and by repeating
this process adjust the instrument so that it is level and that the
telescope is in the meridian. Then press hard on the leveling
screws, and make dents by which the instrument can be brought
into the same position at any future time.
After the A.M. and P.M. observations recorded above, the tele-
scope was pointed upon a meridian mark established by observations
made with the shadow of a pin, and the reading of the horizontal
circle was 359.8. The mark was then shifted about a foot toward
the west, and the telescope again pointed upon it. As the reading
pf the circle was then 3 60. 6, it may be assumed that the mark was
now very nearly in the meridian.
If circumstances are such that no point of reference in the meridian
is available, it will be necessary, after determining the meridian
readings by the sun, to set the telescope upon some well-defined object
in or near the horizontal plane and read the circle. The difference
between this reading and the meridian reading will be the azimuth
of the object. Set the pointer of the horizontal circle to this value,
and set the telescope upon the reference mark by moving the whole
base as before. If the pointer of the circle is now brought to 0, the
telescope will evidently be in the meridian ; and the position is to
be fixed by making dents with the leveling screws as before.
THE DIURNAL MOTION OF THE STARS 41
CHOICE OF STARS
We are now ready to begin observations of the stars.
The most familiar group of stars in the heavens is, no doubt, that
part of the Great Bear which is variously called the Dipper, Charles's
Wain, or the Plough.
At the beginning of October, at 8 o'clock in the evening, an
observer anywhere in the United States will see the Dipper at an
altitude between 10 and 30 above the N.W. horizon. Set the
telescope upon that star which is nearest the north point of the
horizon ; read both circles to determine its altitude and azimuth,
and note the time. Even if the telescope is provided with cross-
hairs, the illumination of the light of the sky will not be suffi-
cient to render them visible : but sufficient accuracy in pointing is
obtained by placing the star at the estimated center of the field.
Observe in succession the altitude and azimuth of the other six
stars forming the Dipper, noting the time in each case.
Using the Dipper as a starting point, we will now identify and
observe a few other stars.* The total length of the Dipper is about
25. Following approximately a line drawn joining the last two
stars of the handle of the Dipper, at a distance of about 30, we
come to a bright star of a strong red color, much the brightest in
that portion of the heavens ; this is Arcturus. Observe its altitude
and azimuth, and note the time as before. Almost directly over-
head, too high to be conveniently observed at this time, is a bril-
liant white star, Vega (a Lyrse). A little east of south from Vega,
at an altitude of about 60, is a group of three stars forming a line
about 5 in length. The central and brightest star of the three is
Altair (a Aquilse), and its position should be observed.
Diurnal Paths of the Stars. Proceed in this way for about an
hour, observing also, if time permits, the group of five stars whose
middle is at azimuth 220 and altitude 35. This is the constella-
tion of Cassiopeia. Another interesting asterism will be found
supposing that by this time it is 9 o'clock at azimuth 270 and
altitude 45, consisting of four stars of about equal magnitude,
* Many of the latest text-books on astronomy contain small star maps which
are valuable aids in the identification of the less conspicuous groups.
42 LABORATORY ASTROHOMY
placed at the corners of a quadrilateral whose sides are about 15 d
in length, and forming what is called the Square of Pegasus.
It is convenient as an aid in identification to note in each case the
magnitude of the star observed. As a rough standard of compari-
son, it may be remembered that the six bright stars of the Dipper
are of about the second magnitude; that at the junction of the
handle and bowl is of the fourth. The three stars in Aquila are of
the first, third, and fourth magnitudes. Vega and Arcturus are
each larger than an average first magnitude star. The brightest
stars in the constellation Cassiopeia and in the Square of Pegasus
range from the second to the third magnitude.
The little quadrilateral of fourth magnitude stars about 15 east
of Altair and known as Delphinus, or vulgarly as Job's Coffin, may
be observed.
At the expiration of an hour, set again upon the Dipper stars and
repeat the series, going through the same list in the same order. Arc-
turus will have sunk so low in a couple of hours as to be beyond the
reach of observation, even if the place of observation affords a clear
view of the horizon. Vega, however, will be less difficult to observe,
and may be now added to the list. We should not omit to make
an observation of the pole star, which, as its name indicates, may
be found near the pole and can be easily found, since the azimuth
of the pole is 180, and its altitude is equal to the latitude of the
place.
From the observed values of altitude and azimuth plot the suc-
cessive places on the hemisphere exactly as in the case of the sun,
and thus represent upon the hemisphere the paths of a number of
stars in various parts of the heavens. It will be found that these
paths are all circles of various dimensions, and that the circles are
all parallel to the equator, as determined from the sun observations,
that is, they have the same pole as the diurnal circles of the sun.
At this stage it is a good plan to devote some attention to the
representation of the various results as shown on the hemisphere,
by means of figures on a plane surface, that is, to make careful free-
hand drawings of the hemisphere and the circles which have now
been drawn upon it as seen from various points of view. This is
an important aid to the understanding of the diagrams by which it
THE DIURNAL MOTION OF THE STARS 43
is necessary to explain the statement and solution of astronomical
problems ; with this purpose in view the drawings should be lettered
and the definitions of the various points and lines written under
them.
ROTATION OF THE SPHERE AS A WHOLE
So far the result of our observations is to show that the heavenly
bodies appear to move as they would if they were all attached in
some way to the same spherical shell surrounding the earth, and
were carried about by a common revolution, as if the shell rotated
on a fixed axis, passing through the point of observation. The sun
may be conceived as carried by the same shell, but observations at
different dates show that its place on the shell must slowly change,
since its declination changes slightly from day to day.
If these observations on the stars are repeated ten days or one
hundred days later, we shall find that the declinations determined
from them are the same ; that is, the declinations of the diurnal
paths of the stars do not change like that of the sun. It will appear
also that, as in the case of the sun, equal arcs of the diurnal circle
and consequently equal hour-angles are described in equal times.
It follows from this, of course, that stars nearer the pole will appear
to move more slowly, since they describe paths which are shorter
when measured in degrees of a great circle, as may be shown by
measuring the diurnal circles on the hemisphere by a flexible milli-
meter scale, 1 mm. being equal to 1 of a great circle on our
hemisphere.
If the field of view of our telescope is 5, a star on the equinoc-
tial will be carried across its center by the diurnal motion in 20
minutes, while a star at a declination of 60 will remain in the field
for twice that time, since its diurnal circle is only half as large as
the equinoctial and an angular motion of 10 of its diurnal circle is
only 5 of great circle. Since the declinations of the stars do not
change, it is unnecessary to make our observations of the stars on
the same night ; or, rather, observations made on different nights
may be plotted as if made on the same night. We may thus obtain
extensions of the diurnal- circles by working early on one evening
and at later hours of the night on following occasions.
44 LABORATORY ASTRONOMY
POSITIONS FIXED BY HOUR-ANGLE AND DECLINATION;
THE EQUATORIAL
It is evident that we have, in the hour-angles and declinations
of the stars, another system of coordinates on the celestial sphere
by means of which their position may be fixed. The altitude and
azimuth refer the position of the star to the meridian and to the
horizon ; while the hour-angle and declination refer its position to
the meridian and the equator. We have hitherto found it more
convenient to deal with the first set of coordinates, but it is often
desirable to determine the hour-angle and declination of a body by
direct observation, and this may be done by means of an instrument
similar to the altazimuth but with the upright axis pointed to the
pole of the heavens, so that the horizontal circle lies in the plane of
the equator. With this instrument the angles read off on the circle
which is directly attached to the telescope measure distances along
the hour-circle, perpendicular to the equator, i.e., declinations, while
an angle read off on the other circle measures the angle between the
meridian and the hour-circle of the star at which the telescope points,
and is therefore the star's hour-angle. The two circles are there-
fore appropriately called the declination circle and the hour-circle
of the instrument. As these terms are used with another meaning
as applied to circles on the celestial sphere, it would seem that there
might be confusion from their use in this sense, but in practice it
is never doubtful whether " circle " means the graduated circle of
an instrument or a geometrical circle on the surface of the sphere.
It is here supposed that the instrument has been so adjusted
that both circles read when the telescope is in the plane of the
meridian and points at the equator. An instrument so mounted is
called an equatorial instrument. Our altazimuth is adapted to this
purpose by constructing the base so that it may be revolved about a
horizontal axis perpendicular to the plane in which the altitude
circle lies when the azimuth circle reads 0. If, then, it has been
placed in the meridian by the observation of equal altitudes as
before described, it may be inclined about this latter axis through
an angle equal to the complement of the latitude, and thus brought
into the proper position for observing declination and hour-angle
THE DIURNAL MOTION OF THE STARS
45
directly. An instrument so constructed is called a "universal"
equatorial. To adjust the universal equatorial so that the axis
points to the pole, adjust it as an altazimuth with both circles
reading and level it with the telescope in the meridian pointing
south. Depress the telescope till the reading of the vertical circle
equals the co-latitude. Tip the whole instrument so as to incline
the vertical axis to-
ward the north till
the bubble plays and
the telescope is hori-
zontal ; to do this the
vertical axis must
have been tipped
back through an
angle equal to the co-
latitude, and it will
be in proper adjust-
ment directed toward
a point in the merid-
ian whose altitude
is equal to the lati-
tude. (Fig. 34 shows
the instrument
adjusted for latitude
45.)
A notch should be cut in the iron arc at the bottom of the coun-
terpoise, into which the spring-catch may slip when the adjustment
is correct, so that the instrument may be quickly changed from one
position to the other. If the notch is not quite correctly placed,
the final adjustment may be made by a slight motion of the north
leveling screw to bring the level exactly into the horizontal position,
the vertical circle having been set to the co-latitude for this purpose.
The proper adjustment of the altazimuth is simpler, since it
depends only on the use of the level, while to place an equatorial
instrument in position we must know the latitude as well. On coin-
paring the two systems of coordinates, it is clear that, while the
altitude and azimuth both change continuously, but not uniformly
FIG. 34
46
LABORATORY ASTRONOMY
with the time, the hour-angle changes uniformly with the time,
and the declination remains the same. One advantage of the
latter system of coordinates is that in repeating our observations
on the same star after the lapse of an hour, we need only set
the declination circle to the previously observed declination, and
set the hour-circle at a reading obtained by adding to the former
setting the elapsed time in hours reduced to degrees by multiply-
ing by 15 ; we shall then pick up the star without difficulty. This
is an important aid in identifying stars, which has no counterpart
in the use of the altazimuth, and we shall henceforth use this
method of observation in preference to the other.
CHAPTEE IV
THE COMPLETE SPHERE OF THE HEAVENS
THE study of the motions of the sun, moon, and stars has thus,
far led to the conclusion that their courses above the plane of the
horizon can be perfectly represented by assuming the daily rotation
from east to west of a sphere to which they are attached, or a rota-
tion of the earth itself from west to east about an axis lying in the
meridian and inclined to the horizon at an angle equal to the latitude
of the place of observation, while the sun moves slowly to and from
the equator, and the moon, like the sun, changes its declination con-
tinually, and has also a motion toward the east on the sphere at a
rate of about 13 in each 24 hours. The combination of the two
motions of the moon causes it to describe a path which will be more
fully discussed later. We shall now begin to observe the sun, to
see if its motion among the stars resembles that of the moon in
having an east and west component in addition to its motion in
declination.
The motion of the moon can be directly referred to the stars, since
both are visible at the same time, although the illumination of the
dust of our atmosphere, by strong moonlight, cuts us off from
the use of the smaller stars, which cannot be seen except when
contrasted with a perfectly dark background.
The illumination produced by the sun, however, is so strong that
it completely blots out even the brightest stars, so that we cannot
apply either of the methods that we have employed in observing
the moon.
We are only able to see the stars, of course, when they are above
the plane of the horizon, but it is natural to suppose that they con-
tinue the same course below the horizon from their points of setting
to those of their rising. This inference is confirmed by the fact that
some of the bright stars which set within a few degrees of the north
point of the horizon, and which we infer complete their course below
47
48 LABORATORY ASTRONOMY
the horizon, may be seen actually to do so by an observer at a point
on the earth some degrees farther north, from which they may be
observed throughout the whole of their courses. In the case of the
sun, the following facts lead to the same conclusion. Immediately
after sunset a twilight glow is seen in the west whose intensity is
greatest at the point where the sun has just set. This glow appears to
pass along the horizon towards the north, and its point of greatest
intensity is observed to be directly over the position which the sun
would occupy in the continuation of its path below the horizon, on
the assumption that it continues to move uniformly in that path.
In high latitudes this change of position in the twilight arch can be
followed completely around from the point of sunset to the point of
sunrise, the highest point being due north at midnight. It is impos-
sible not to believe that the sun is actually there, though concealed
from our sight by the intervening earth. (Of course, too, it is now
generally known that in very high latitudes the sun at midsummer
is visible throughout its diurnal course.) As the sun sinks farther,
the light of the sky decreases, the brighter stars begin to appear,
and it is clearly impossible to resist the conclusion that they have
been in position during the daylight, but simply blotted out by the
overwhelming light of the sun.
OBSERVATIONS WITH THE EQUATORIAL
When we have fixed the idea that the heavenly sphere revolves
as a whole, carrying with it in a general sense all the bodies that we
observe, the next step is to devise some means of locating the dif-
ferent bodies in their proper relative positions on the sphere. For
this purpose the equatorial instrument furnishes us with an admi-
rable means of observation. The relative position of two stars is
completely fixed when we know the position of their parallels of
declination and their hour-circles, since the place of each star is
at the intersection of these two circles.
Since an observation with the equatorial gives directly the decli-
nation and hour-angle of a star, the method of fixing the relative
position of two stars, A and B, is as follows :
Point the telescope at A, and read the circles ; then set on B, and
THE COMPLETE SPHERE OF THE HEAVENS 49
read the circles ; then again on A, and read the circles, taking care
that the interval between the first and second observations shall be
as nearly as possible equal to the interval between the second and
third. Obviously the mean of the two readings of the hour-circle
at the pointings upon A gives the hour-angle of A at the time when
B was observed, since the star's hour-angle changes uniformly.
The difference between this mean and the reading of the hour-
circle when the pointing was made upon B is, therefore, the dif-
ference between the hour-angles of the stars at the time of that
observation ; and this fixes the relative position of their hour-circles,
since this difference is the arc of the equator included between
them ; their declinations are given by the readings of the declina-
tion circles, and thus the relative position of the two stars is
completely known.
As an illustration of this method, we may take the following
example :
With the telescope pointed at A, the readings of the hour-circle
and declination circle were 68.2 and 15.l, respectively. The
telescope was then pointed at B, and the circles read 85.9, 28.l,
and finally upon A, the readings being 69.l, 15.l ; the intervals
were nearly the same, as will usually be the case, unless there is
some difficulty in finding the second star. Of course the first star
can be re-found by the readings at the first observation ; indeed,
if the intervals are plainly unequal, a repetition of the observation
may always be made at equal intervals by setting the circles for
each star so that no time is lost in finding.
From the above observations we infer .that when the hour-angle
of B was 85. 9, that of A was 68. 65 ; and, therefore, that the hour-
circles of the two stars cut the equator at points 17.25 apart;
the hour-circle of B being to the west of that of A, so that B comes
to the meridian earlier, or "precedes" A.
It may be noted that the observations apparently occupied a little
less than 4 minutes, since in the whole interval the hour-angle of A
changed by 0.9.
50
LABORATORY ASTRONOMY
USE OF A CLOCK WITH THE EQUATORIAL
If the intervals between the observations are not exactly equal,
it will still be easy to fix the hour-angle of A at the time of the
observation on B if the ratio of the intervals is known; if, for
instance, the first observation of A gives an hour-angle of 25. 3, and
the later observation an hour-angle of 26.3, while the intervals
are l m between the first and second observations, and 3 m between
the second and third, the hour-angle of A at the second observa-
tion was obviously 25. 3 + 0.25. We may thus obtain by " inter-
polation " the hour-angle of A at any known fraction of the interval.
Plainly it is an advantage to note the time of each observation for
this purpose, as in the following observations, which were made
Feb. 5, 1900, for the purpose of determining the relative positions
of the stars forming the Square of Pegasus.
STAR
WATCH TIME
DECL. CIRCLE
HOUR-ClKCLE
1 7 Pegasi
7h 14m Os
+ 15. 2
66. 3
2 a Pegasi
15
. + 15 .2
83 .6
3 ft Pegasi
16 15
+ 28 .1
84 .1
4 a AndromedaB
17 10
+ 29 .0
68 .3
5 7 Pegasi
18 30
+ 15 .2
67 .6
6 7 Pegasi
21 30
+ 15 .1
(69 .2)
7 a AndromedaB
22 30
+ 29 .1
69 .6
8 ft Pegasi
23 30
+ 28 .1
85 .9
9 oJPegasi
24 20
+ 15 .3
86 .0
10 7 Pegasi
25 30
+ 15 .1
69 .1
11 7 Pegasi
27 30
+ 15 .1
69 .6
The observations here follow each other rapidly. They were made
by an experienced observer, and the arrangement of the stars is such
that, after setting y Pegasi, a Pegasi is brought into the field by
moving the telescope about the hour-axis only ; .we pass to ft Pegasi
by motion around the declination axis only, to a Andromedae by
motion about the hour-axis, and back to y Pegasi by rotation about
the declination axis ; so that the stars are found more quickly
than if both axes must be altered in position at each change ; in
observations 6 to 10 the series is observed in reversed order.
THE COMPLETE SPHERE OF THE HEAVENS 51
If the instrument was correctly adjusted, the declination of the
four stars was as follows : y Fegasi + 15.14, a Pegasi 15.25,
ft Pegasi 28.l, a Andromedse 29. 05, each being determined as the
mean of all the observations made upon the star.
The first advantage of the recorded times is to show that the
reading of the hour-circle in 6 was an error, probably for 68.2, as
we see by comparison with the other values of the hour-angle of
y Pegasi, which increase uniformly about 1 in each 4 minutes. It
will be better, however, to reject the observation entirely, as it is
not necessary to use it for the first set of observations 1 to 5, which
we will now discuss.
By interpolation between 1 and 5 we find that the hour-angle of
y Pegasi at 7 h 15 m s was f of 1.3 greater than 66.3, or 66.59;
at 7 h 16 m 15 s it was of 1.3 greater than 66.3, or 66.95; and at
7 h 17 m 10 s it was _ 8 _p_ O f i.3 i ess than 67.6, or 67.21. As the
hour-angles of the other stars were observed at these times, we
can at once find the differences of their hour-angles from that of
y Pegasi, which are as follows : a Pegasi, 17.01 ; ft Pegasi, 17.15 ;
a Andromedse, 1.09. All the hour-angles are greater than those of
y Pegasi, so that all the stars precede y Pegasi. By using all the
observations we may presumably obtain more accurate results, and
it will be well, as in all cases when a considerable number of
observations must be dealt with, to arrange the reductions in a
more systematic manner.
In the table on the following page the difference of hour-angle is
obtained by subtracting the observed hour-angle in each case from
the hour-angle of y Pegasi, so that its value is negative, if, as in
the results given above, the stars precede y Pegasi, and positive
if they follow it. An observation of Venus, 'made on the same
occasion, is added to the list, and an additional observation of
a Pegasi is included ; the erroneous observation of y Pegasi at
7 h 21 m 30 s is excluded.
The values of the hour-angle of y Pegasi at the successive times,
as given in column 6, are computed from the following considera-
tions, the proof of which is left to the student. If a quantity
changes uniformly, and its values at several different times are
known, the mean of these values is the same as the value which
LABORATORY ASTRONOMY
STAB
TIME
DECL.
H.A.
H.A. OF y PEG.
STAB FOLLOWS
y PEG.
1 Venus
7 h 12 m s
+ 4.0
7 5. 5
65. 86
- 9.64
2 a Peg.
13
+ 15 .1
83 .1
66 .10
- 17 .00
3 7 Peg.
14
+ 15 .2
66 .3
66 .35
+ .05
4 a Peg.
15
+ 15 .2
83 .6
66 .59
- 17 .01
5 /3 Peg.
16 15
+ 28 .1
84 .1
66 .89
- 17 .21
6 a Androm.
17 10
+ 29 .0
68 .3
67 .12
- 1 .18
7 7 Peg.
18 30
+ 15 .2
67 .6
67 .44
.16
8 a Androm.
22 30
+ 29 .1
69 .6
68 .44
- 1 .16
9 |8 Peg.
23 30
+ 28 .1
85 .9
68 .88
- 17 .22
10 a Peg.
24 30
+ 15 .3
86 .0
68 .93
- 17 .07
11 7 Peg.
25 30
+ 15 .1
69 .1
69 .17
12 7 Peg.
27 30
+ 15 .1
69 .6
69 .65
+ .05
the quantity has at the mean of the times. Using this principle,
we find the hour-angle of y Pegasi at 7 h 21 ra 22 s was 68.15.
Between observations 3 and 12 it changed 3.3 in 13J m , or 0.244
per minute. Assuming this rate of change, it is easy, though labori-
ous, to compute the hour-angle at any one of the given times ; for
example, at 7 h 12 m s the hour-angle was 68.15 - (9| times 0.244),
or 65.86. Labor will be saved by making a table of the values at
the even minutes by successive additions of 0.244, from which the
values at the observed times are rapidly interpolated. The sixth
column contains the number of degrees by which the hour-circle of
the star follows that of y Pegasi. The mean values for each star
obtained from this column are as follows.
STAB
DECL.
DIFF. H.A.
.7 Pegasi
Venus ........ .
+ 15. 15
- 4 .0
o.oo
- 9 .64
a Pegasi
+ 15 20
17 .03
j8 Pegasi
a Androm.
+ 28 .10
+ 29 .05
- 17 .22
1 .17
The true values of the declinations of these stars as determined by
many years of observations are for y Pegasi 14. 63, a Pegasi 14.67,
ft Pegasi 27.55, a AndromedsB 28.53. The values from our
THE COMPLETE SPHERE OF THE HEAVENS
53
observations are 15.15, 15.20, 28.10, 29.05, so that the latter
require corrections of 0.52, 0.53, 0.55, and 0.52, respec-
tively. This is due to a faulty adjustment of the instrument, but
the error from this cause evidently affects all the observations by
nearly the same amount, 0.53, so that the relative positions are
given quite accurately ; our observations placing the whole constel-
lation about y too far north.
Since Venus is in the near
neighborhood of y Pegasi, we
may assume that the observa-
tions of that planet are subject
to the same corrections, that
she preceded y Pegasi by 9.64,
and that her true declination
was - 4.0 - 0.53, or - 4.53.
The correction is applied alge-
braically with the same sign as
to the other stars, since it must
be so applied as to make the
true place farther south than
the observed place.
The places of the Square of Pegasus and the planet Venus, as
seen in the sky Feb. 5, 1900, are shown in Fig. 35.
Before plotting the stars on the hemisphere from the above
data, it must be prepared by drawing upon it in their proper posi-
tions the meridian, zenith, pole, and equator. Draw the hour-circle
of y Pegasi (see Fig. 19, p. 17) at the proper hour-angle from the
meridian, to give its position at the time of the last observation,
which may be determined by making it intersect the equator at the
proper point 69. 6 west of the meridian, and place the star upon it
at a distance from the equator equal to the observed declination,
15. 14. The hour-angle of a Pegasi should be drawn in the same
manner to cut the equator at 8 6. 66 from the meridian, and the
star placed upon it at the observed declination, 15. 20. Of course
on the scale of so small a hemisphere the nearest half degree is
sufficiently accurate. Bern ember that the configurations on the
hemisphere and on the map are semi-inverted.
QfAndrom.
fega,
rtfj
/
Venus
D -15 -30
FIG. 35
54 LABORATORY ASTRONOMY
CLOCK REGULATED TO SHOW THE HOUR-ANGLE OF THE
FUNDAMENTAL STAR
The method of calculating the hour-angles of y Pegasi in the last
example shows that if the reading of the watch can be relied upon,
the observations of that star need only be made at the beginning
and at the end of the period of observation, the hour-angle at any
time being determined by its uniform increase ; or even from a
single observation at the beginning of the period, since at the time
of observation of any star the hour-angle of y Pegasi can be
inferred from that at its first observation by adding the number of
degrees which it would have described in the time elapsed, obtained
by multiplying the number of hours by 15, or, what gives the same
results, dividing the minutes by 4. Moreover, if the rate of the
watch is such that it completes its 24 hours in the time in which
the stars complete their daily revolution, and if its hands are so set
as to read 12 hours when y Pegasi is on the meridian, the difference
of hour-angle at any time will be equal to the reading taken directly
from the hands of the watch reduced as above to degrees, for when
the star is on the meridian and its hour-angle therefore zero, the
watch marks O h O m s . Four minutes later by the watch the hour-
angle of the star has increased by the diurnal revolution to 1;
in four minutes more to 2 ; when the watch indicates 1 hour, the
star's hour-angle has increased to 15, and so on, till 24 hours have
elapsed, when the star will again be on the meridian and the cycle
recommences.
The rate of an ordinary watch is sufficiently near to that of the
stars to allow of its use for this purpose for periods of an hour
without causing any error in our observations.
In the use of this method we may regard the observation of the
fundamental or zero star as a means of finding out whether the
clock is set to the right time : thus, in the following set of obser-
vations the first observation gives the hour-angle of y Pegasi 67. 6
at 7 h 15 m 10 s , but as 67. 6 equals 4 h 30 m 24 s , we may regard the
clock as 2 h 44 m 46 s fast ; and by applying this correction to all the
observed times, may write down at once under the title " corrected
time " what the readings would have been if the clock had been set
THE COMPLETE SPHERE OF THE HEAVENS
55
to show hours, when the star's hour-angle was 0. Multiplying
these by 15 we have the hour-angle in degrees given in column 4.
The following observations were undertaken for determining the
configuration of the stars in Orion and its neighborhood, Feb. 6, 1900.
STAR
OBS. TIME
CORRECTED
TIME
H.A. OF
y PEG.
OBSERVED
H.A. OF STAR
DECL.
FOLLOWS
y PEG.
7 Pegasi
7h 15m IQs
4h 3()m 24
67. 6
67. 6
+ 15.5
a
7 20
4 35 14
68.8
348 .5
- 1 .4
80. 3
b
7 22
4 37 14
69 .3
349 .95
- .6
79 .35
c
7 23 30
4 38 44
69 .7
348.1
- 2 .2
81 .8
d
7 25 20
4 40 34
70.1
344 .8
+ 7 .1
85 .3
e
7 27 10
4 42 24
70 .6
353 .0
+ 6.1
77 .6
f
7 28 45
4 43 59
71 .0
347 .5
- 9 .9
83 .5
g
7 30 20
4 45 34
71 .4
356 .4
- 8 .2
75 .0
h
7 32
4 47 14
71 .8
351 .6
- 5 .5
80 .2
i
7 34
4 47 14
72 .3
334 .7
-16 .9
97 .6
j
7 35 45
4 50 59
72 .7
321 .3
+ 5 .05
111 .4
a
7 37 45
4 52 59
73.2
352 .9
- 1 .4
80 .3
7 Pegasi
7 39 50
4 55 4
73 .8
73.9
+ 15 .4
Moon
7 42
4 57 14
74 .3
27 .6
+ 20 .4
46.7
The results of columns 6 and 7 enable us to map the constellation
as in Fig. 36.
One or two constellations may be plotted in this manner both on
the map, which shows the constellation as seen in the sky, and on
, j A"
jjftgasi
Prt
cyon
Orion
, in"
*
"20*
Sir
ius
105 90* 75' 60 45 30 15 C
FIG. 36.
56 LABORATORY ASTRONOMY
the hemisphere, where it is semi-inverted. It will be advisable,
however, before much work has been done in this way, to introduce
a slight modification.
THE VERNAL EQUINOX RIGHT ASCENSION
The precession of the equinoxes causes a change in the position
of the equator, which slowly changes the declinations of all the
stars. For this reason it is found more convenient to select, instead
of y Pegasi as a zero star, the point upon the equator at which the
sun crosses it from south to north about March 21 of each year.
This point, which is called the vernal equinox, is not fixed, but
its motion, due to precession, is simpler than that of any star which
might be selected as a zero point ; it precedes the hour-circle of
y Pegasi at present by about 8 minutes of time, or 2 of arc, and it
was because of this proximity that we first selected that star.
Instead, therefore, of adjusting our clock so that it reads O h O m s
when y Pegasi is on the meridian, we set it to that time when the
vernal equinox is in that plane ; its readings then give the hour-angle
of the vernal equinox, and the difference between the hour-angles of
that point and of the star may be directly obtained from our obser-
vations. The distance by which a star follows the vernal equinox
is called its right ascension; more carefully defined, it is the arc
of the equator intercepted between the hour-circle of the star and
the hour-circle of the vernal equinox (which measures the wedge
angle between the planes of these circles) ; it is also the angle between
the tangents drawn to these two circles where they intersect at the
pole. Since any star which is east of the vernal equinox follows it,
the right ascensions of different stars increase toward the east, that
is, toward the left in the sky as we face south, but toward the right
on the solid hemisphere as we look down from the outside upon its
southern face.
Hereafter we shall fix the positions of the stars by their right
ascensions and declinations. We may make use of the observations
already reduced with very little additional labor. Since y Pegasi
follows the vernal equinox by 2, we need only add that amount to
the quantities given in column 7 on page 55 to know the right
THE COMPLETE SPHERE OF THE HEAVENS 57
ascension of the different stars. If we learn later that on Febru-
ary 6 the right ascension of y Pegasi was more exactly O h 8 m 5 8 .64,
we may further correct by adding 5 s , or even 5 8 .64, if the accuracy
of the observations warrants it. The method of determining the
exact position of the zero star with reference to the vernal equinox
is given in Chapter VI.
Formerly right ascensions were measured altogether in degrees,
but owing to the modern use of clocks, it has long been customary
to give them in hours ; for this reason the hour-circle of instruments
mounted as equatorials is graduated to read hours and minutes
directly. Since our universal equatorial is intended to serve also
as an altazimuth, its circles are both graduated to degrees.
SIDEREAL TIME
In the last section right ascension has been defined as the angle
between the hour-circle passing through a star and the great circle
passing through the pole and the vernal equinox. The latter circle
is called the equinoctial colure. We have also suggested the use
of a clock set to read O h O m s at the time when the vernal equinox
is on the meridian ; so that the hour-angle of the vernal equinox at
any time will be given directly by the reading of the face of the
watch in hours, minutes, and seconds, from which the angle in
degrees is found by multiplying by 15. A clock set in this manner,
and running at such a rate that it completes 24 hours in the time
that the star completes its revolution from any given hour-angle to
the same hour-angle again, is said to keep sidereal time. We
shall find later that a clock so regulated gains about 4 minutes a
day on a clock keeping mean time, thus gaining 24 hours on an
ordinary clock in the course of a year, and agreeing evidently with
a clock keeping apparent time, as defined on page 19, at that time
when the sun is at the vernal equinox and crosses the meridian at
the same time with the latter.
Let us suppose now that the vernal equinox has passed the
meridian by one hour, then its hour-angle is l h , or 15; and our
sidereal clock indicates exactly l h O m s . Any star which is at this
time on the meridian, that is, whose hour-angle is 0, must therefore
58
LABORATORY ASTRONOMY
follow the vernal equinox by l h , or 15, while at the same instant
the time by our sidereal clock is l h O m s . By our definition of
right ascension, since the star follows the vernal equinox by l h , its
right ascension is l h ; in this case, therefore, the right ascension of
the star in hours, minutes, and seconds has the same value as the
time given by the hands of the clock. In the same way, if the
vernal equinox has passed the meridian so far that its hour-angle is
2 h 15 m , the face of the clock will show 2 h 15 m j and any star then
upon the meridian follows the vernal equinox by 2 h 15 m . The same
relation holds here ; namely, that the right ascension of the star is
equal to the time by the sidereal clock when the star is upon the
meridian. This might have been given as a definition of the term
" right ascension " ; and, indeed, so closely are the two connected
in the mind of the practical astronomer that if the right ascension
of a star is given, he at once thinks of this number as representing
the time of its meridian passage.
RIGHT ASCENSION PLUS HOUR-ANGLE EQUALS
SIDEREAL TIME
We may here give an explanation of a general principle of very
frequent application, and of which this is simply a particular case.
Suppose the vernal equinox, represented by the symbol T (Fig. 37), to
have passed the meridian by 5 h 10 m .
Then a star, S, whose right ascen-
sion is 2 h 15 m , since it follows the
vernal equinox by that amount,
will have passed the meridian by
2 h 55 m ; and its hour-angle will be
2 h 55 m . The arc of the equator
between the meridian and the ver-
nal equinox may be considered as
made up of two parts : the right
ascension of the star, which is measured by the arc eastward
from the vernal equinox to the hour-circle of the star, and the
hour-angle of the star, which extends from the meridian westward
to the hour-circle of the star. Since this is true of any star, or,
FIG. 37
THE COMPLETE SPHERE OF THE HEAVENS 59
indeed, of any heavenly body, we may make the following general
statement : The right ascension of any body plus its hour-angle at
any instant will be equal to the sidereal time at that instant ; or,
as it is sometimes written : R.A. + H.A. = Sidereal Time. If the
body is a point on the meridian, its H.A. = zero ; hence the E/.A.
of a star on the meridian, or briefly, R.A. of the meridian = Sidereal
Time, as we have before shown.
From this relation we may most simply determine the right
ascension of any heavenly body by observing its hour-angle with
the equatorial instrument, and at the same time noting the sidereal
time, since R.A. = Sidereal Time H.A. It is by this method
that we shall now proceed to make a somewhat extended catalogue
of stars from which we may plot their positions upon the globe.
We will here notice some of the important uses to which this
principle may be put. If by any other means the right ascension
of a body is known, we may find its hour-angle at any given sidereal
time by the equation, Sidereal Time R.A. = H.A. This gives us
an easy way to point upon any object whose right ascension and
declination are known, if we have a clock keeping sidereal time ;
and this is the usual way in which the astronomer finds the objects
which he wishes to observe, since they are generally so faint that
they cannot be seen by the naked eye. For example, to point the tele-
scope at the great nebula in Orion, whose right ascension is 5 h 28 m ,
and declination 6 S., we first set the decimation circle to 6,
and if the sidereal time is 7 h 30 m we set the hour-circle to 2 h 2 m ,
then the telescope will be pointed upon the star. If the sidereal
time is 4 h 30 m , in which case the star evidently has not reached
the meridian by nearly an hour, we must add 24 hours to the sidereal
time ; then the expression, H.A. = Sidereal Time R.A. will
become H.A. = 28 h 30 m 5 h 28 m , or 23 h 2 m , the hour-angle being
reckoned, as before stated, from O h up to 24 h . If then the hour-
circle is brought to the reading 345 = 15 x 23^, we shall find
the star in the field.
60 LABORATORY ASTRONOMY
THE CLOCK CORRECTION
The same principle enables us to set our clock correctly to sidereal
time by observing the hour-angle of any star whose right ascension
is known. For example, the right ascension of Sirius being 6 h 40,
or 100, and its hour-angle being observed to be 330, or 22 h , the
sidereal time is R.A. -f H.A., that is, 430, or, subtracting 360, is
70, corresponding to 4 h 40 m ; and a clock may be set to agree; or,
by subtracting the time which it then indicates, we determine a
correction to be applied to its reading to give the true sidereal
time. If, for instance, at the observation above, the clock time is
4 h 41 m 10 s , the clock correction is l m 10 8 . In this case the clock
is l m 10 s fast, the time which it indicates is greater than the true
time, and its "error" is said to be -f- l m 10 s . On the other hand,
when the clock is slow the correction to true time is positive, while
the " error " is negative.
It is customary to observe this distinction between the terms
" error " and " correction " ; the former is the amount by which the
observed value of a quantity exceeds its true value, while the correc-
tion is the quantity which must be added to the observed to obtain the
true value. They are thus numerically equal but of opposite sign.
The error of the declination circle determined by the observations
of page 53 was -f- 0.53, while the correction was 0.53.
For the constantly occurring " clock correction," we shall use the
symbol A, the value of which is positive if the clock is slow and
negative if it is fast.
If, as is often desirable, we wish to observe a body of known right
ascension upon the meridian, we have only to observe it when the
time by the sidereal clock is equal to its right ascension.
We may of course find the right ascension of the moon by a direct
comparison with the neighboring stars, just as we have determined
the difference in right ascension of a Pegasi, from that of y Pegasi,
for the brighter stars can be easily observed at the same time as the
moon ; but no star is so bright that it can be readily observed by
our small instrument when the sun is above the horizon,* and we
have therefore no means of making a direct comparison between
* See, however, page 69.
THE COMPLETE SPHERE OF THE HEAVENS 61
a star and the sun. But by means of our clock and our new method
of observation this becomes easy ; and the sun is to be added to the
list of bodies whose right ascension we are to observe regularly. It
is only necessary that we should be provided with a clock which
keeps correct sidereal time. (See page 67.)
We have already spoken of the means of setting the clock;
now a few words as to how the regularity of its rate may be deter-
mined. It is only necessary to observe the watch time at which
any star is at a given hour-angle on successive nights. If the
rate of the clock is such that the interval between the observa-
tions is greater than 24 hours, the watch is gaining ; if the amount
is less than half a minute a day, the watch may be assumed for our
purposes to be keeping correct sidereal time, its actual error at any
time being checked, as before described, by the observation of the
hour-angle of bodies of known right ascension.
LIST OF STARS
Our first care will be to observe a number of bright stars not very
far from the equator which will serve for setting the clock or deter-
mining its error, selecting them so that several shall always be above
the horizon and may at any time be used for this purpose. Several
of those already observed will be found in the list given on the
following page, which contains the approximate places of a number
of conspicuous stars.
By repeated comparisons of these stars with each other and with
y Pegasi, their right ascensions may easily be fixed within 30 8 , and
they may then be used for determining the clock error at any time
when they are visible. The observations of each evening should be
reduced as soon as possible and maps made of the various constella-
tions similar to those of Figs. 35 and 36 ; it is, however, impossible
to represent any large portion of the sphere satisfactorily on a
plane surface, and, in order to have a proper idea of the relative
positions of the various constellations, we must plot our results on
a globe a proceeding still more necessary when we come to study
the motion of the sun and moon among the stars by the method of
the following chapter.
62
LABORATORY ASTRONOMY
A globe 6 inches in diameter is sufficiently large for our purpose ;
it. should be so mounted that it may be turned about its axis 011 a
firm support, and upon it should be traced 24 hour-circles 15 apart,
and small circles (parallels of declination) parallel to the equator
and 10 apart ; its surface should be smooth and white, and of such
a texture as to take a lead-pencfl mark easily, but permit of erasure.
TIME STARS
STAB
MAG.
K.A.
6
STAB
MAG.
R.A.
8
7 Pegasi
3
O h .l
+ 15
Denebola
2
ll h .7
+ 15
/S Ceti
2
.6
- 19
5 2 Corvi
3
12 .4
- 16
/3 Andromedse
2
1 .1
+ 35
Spica
1
13 .3
- 11
a Arietis
2
2 .0
+ 23
Arcturus
1
14 .2
+ 20
a Ceti
2
3 .0
+ 4
a 2 Librae
3
14 .8
+ 16
Alcyone
3
3 .7
+ 24
a Serpentis
3
15 .7
+ 7
Aldebaran
1
4 .5
+ 16
Autares
1
16 .4
-26
Capella
1
5 .2
+ 45
a Ophiuchi
2
17 .5
+ 13
Rigel
1
5 .2
- 8
7 2 Sagittarii
3
18 .0
- 30
e Orionis
2
5 .5
1
Vega
1
18 .6
+ 39
Betelgeuze
1
5 .8
+ 7
Altai r
1
19 .8
+ 9
Sirius
1
6 .7
- 17
a 2 Capricorni
4
20 .2
- 13
Castor
2
7 .5
+ 32
a Delphini
4
20 .6
+ 16
Procyon
1
7 .6
+ 5
e Pegasi
H
21 .7
+ 9
Pollux
1
7 .7
+ 28
a Aquarii
3
22 .0
- 1
a Hydrae
2
9 .4
- 8
a Pegasi
2i
23 .0
+ 15
Regulus
1
10 .1
+ 12
The number attached to the Greek letter indicates that the star to be
observed is the following of two neighboring stars.
CHAPTER V
MOTION OF THE MOON AND SUN AMONG THE STARS
FOR plotting the stars on the globe in their proper places, as given
by their right ascensions and declinations, it is convenient to have
the equator graduated into spaces of 10 m each ; this may be done
by laying the edge of a piece of paper along the equator, and mark-
ing off the points of intersection of the equator with two consecu-
tive hour-circles ; laying the paper upon a flat surface, bisect the
space between the two lines with the dividers, and trisect each of
these spaces by trial, testing the equality of the spacing by the
dividers ; this may be satis-
factorily done by two or three
trials, and the short scale thus
obtained may be easily trans-
ferred to the arcs on the equator
between each two hour-circles.
It may be found convenient to
bisect each of the spaces on the
scale, thus dividing the equator
into spaces of 5 m each.
A strip of parchment or
parchment paper about 8 inches
long and inch wide, of the
shape shown in Fig. 38, and
graduated to degrees, completes
the apparatus necessary for
plotting. The hole being
placed over the axis of the
globe, the graduated edge of
the strip may be made to coincide with the hour-circle of any star
by causing it to intersect the equator at a point corresponding to the
star's right ascension, taking care that the edge lies in a great circle
63
FlG 38
64 LABORATOKY ASTRONOMY
of the sphere ; the graduated edge gives at once the proper declina-
tion for plotting the star upon its hour-circle, and the point may be
marked with a well-sharpened, hard lead pencil ; the latter should
be carefully kept, and used for purposes of plotting only. With
this simple apparatus the stars may be rapidly and accurately
placed upon the globe.
An attempt should be made to represent the magnitudes of the
stars by the size of the dots which indicate their places.
THE MOON'S PATH ON THE SPHERE
The moon should be placed on the list of objects for regular
observation, the observations being made in precisely the same
manner as those of the stars, and its place should be plotted
upon the globe at each observation and marked by a number,
giving, the date of the month. This method of fixing the moon's
place is much more accurate than those made use of in Chapter II,
and, as the places are plotted upon a globe, we may study -to better
advantage those peculiarities of her motion which are masked by
the distortion of the map referred to in Chapter II.
The position of the node may now be fixed with such a degree of
accuracy that its regression is shown by the observations of two or
three months, if some care is taken to observe as nearly as possible at
the same altitude in the successive months, so that the corrections
for parallax may be nearly the same ; indeed, a very few months will
force upon the notice of the observer the fact that the moon's path
does not lie in one plane, just as observations a few days apart
show that the sun's diurnal path is not really a small circle lying-
in one plane.
We also study the variable motion of the moon by applying
dividers between the successive plotted places and then placing
the dividers against the parchment scale to measure the distance
in degrees traversed in the plane of the orbit. The scale must
lie along an hour-circle so as to conform to the curvature of the
sphere.
The average rate being about 13 a day, the points on the orbit
should be determined as nearly as possible at which the motion is
MOTION OF THE MOON AND SUN AMONG THE STARS 65
greater and less than this amount, and the point of most rapid
motion fixed as closely as possible ; this point is most simply fixed
by its distance an degrees from the ascending node of the moon's
orbit. Since the latter point, however, is continually changing,
it is customary to reckon the so-called " longitude in the orbit " of
the point by measuring from the vernal equinox along the ecliptic
to the node, and adding the angle measured along the orbit from
the node to the point.
The variations of the moon's angular diameter and the point of
the orbit where the diameter is greatest should be compared with
V777 VH Vf V
-20
-50
120
110
100
70 60 50
FIG. 39
30 20
the -results obtained from the investigation of the angular velocity
in the orbit, since we thus gain some knowledge of the moon's
relative distances from us at different points of its orbit, and of the
relation between its distance and its rate of motion about the earth.
The scale of the 6-inch globe is too small to do justice to the
accuracy of our observations, which are accurate to a quarter or a
tenth of a degree, and it will be interesting to plot these observations
66 LABORATORY ASTRONOMY
on a map constructed on a larger scale, and on a plan which
reduces the distortion to very 'small limits in the region of the
ecliptic ; such a map is shown in Fig. 39 on a reduced scale ; the
ecliptic is here taken as a straight horizontal line, as the equator
is in the star map previously used ; the latitude, or angular distance
of a point from the ecliptic measured on a great circle perpendicular
to the latter, serves as the coordinate corresponding to the declina-
tion on our former map, while right ascension is replaced by lon-
gitude, or distance along the ecliptic measured from the vernal
equinox up to 360. The same map will serve also for plotting
the paths of the planets in our later study.
For convenience in plotting, the parallels of declination and the
hour-circles are printed in broken lines upon the map. The obser-
vations of the moon shown in the figure are those of December,
1899, already plotted on the map of Fig. 25.
THE SUN'S PLACE AMONG THE STARS
By means of the equatorial we may also determine the place of
the sun among the stars, although the method of direct comparison
with stars we have used in the case of the moon is not applicable,
since the stars are not visible when the sun is above the horizon ;
the most obvious method which is capable of any degree of accuracy
involves the use of a clock regulated to sidereal time.
To determine the place of the sun, point upon it with the equa-
torial about two hours before sunset ; note the time, and read the
circles ; as soon as possible after sunset observe a star in the same
manner, with the instrument as near as may be to its position at
the sun observation. It is evident that if the circumstances were
fortunately such that the telescope did not have to be moved between
the observations, the difference in right ascension of the sun and the
star would be the difference in time noted by the sidereal clock,
while the declinations of the sun and star would be the same. The
nearer the star is to the position in which the sun was observed,,
the less will be the errors arising from imperfect adjustment and
orientation of the instrument; while the shorter the interval be-
tween the observations, the smaller will be the error due to the
MOTION OF THE MOON AND SUN AMONG THE STARS 67
uncertainty in the rate of the clock. As the condition of not
moving the telescope can seldom be fulfilled, however, we must
treat the observation as follows :
Let E.A., H.A., t, and A be the right ascension, hour-angle,
clock time, and clock correction at the time of the star observation,
and E.A.', H.A.', t', and A, the corresponding quantities at the
sun observation. The equation
E.A. + H.A. = Sidereal Time = t -f A
determines the value of A, which substituted in the equation
Sidereal Time = t 1 + A == E.A.' + H.A.'
determines the value of E.A.', the sun's right ascension at the
moment of observation.
The value of A#, as determined from the first equation, will be
negative if the clock is fast, and positive if the clock is slow ; and
it must always be applied to the observed time with the proper
sign. The declination of the sun is, of course, given directly by
the reading of the declination circle.
The following example illustrates the method :
March 29, 1899, an observation of the sun with an equatorial
telescope, and a clock keeping sidereal time, gave the following
values :
Observed time = 5 h 36 m 26 s ; H.A. = 75.7 = 5 h 2 m 48 8 ; 3 = + 4.l.
About an hour after sunset an observation of a Ceti was made
in nearly the same position of the instrument, which gave the
following values :
Observed time = 7 h 53 m 43 s ; H.A. = 74.l = 4 h 56 m 24 s ; 3 = + 4.2.
This latter gives, from the known right ascension of a Ceti,
2 h 57 m Q. + 4 h 56 m 24B = Sidereal Time = 7 h 53 m 43 s + A*,
and hence A = 19 s ; and, applying the same equation to the- sun
observation,
Sun's E.A. + 5 h 2 m 48 s = 5 h 36 m 26 s - 19 s = 5 h 36 m 7 s ;
hence the sun's right ascension at the time of the first observation
was O h 33 m 19 s . This is liable to an error equal to the uncertainty of
the circle readings, which may be at least one-twentieth of a degree,
68 LABORATORY ASTRONOMY
or 12 s of time, and to an error equal to the uncertainty of the gain
or loss of the clock during the interval of 2^- hours between the
two observations, probably five or ten seconds of time. We may
assume that the errors arising from defective adjustment of the
instrument were the same for both objects, and may be neglected,
since the position of the instrument was very nearly the same for
both observations.
DIFFERENTIAL OBSERVATIONS
The declination of a Ceti, as read from the circles, was +4.2,
while its known decimation was -f 3. 7. The correction necessary
to reduce the circle reading to the true value is, therefore, 0.5,
and, applying this quantity to the reading on the sun, we have for
the true value of the sun's declination + 4.l 0.5 = -f 3. 6. It is
worthy of note that the correction is about the same as that deter-
mined from the observations discussed on page 53, which were
made with the same instrument in nearly the same adjustment, but
from a different place of observation. These results indicate an
inherent defect in the instrument, which is at least in great part
neutralized by the method of observation. It is a very important
thing, even with the most delicate instruments, to avail ourselves of
methods which accomplish this object, and surprisingly good work
may be done with poor instruments by paying proper attention
to the details of observation for this purpose.
Methods by which an unknown body is thus compared with a
known body under circumstances as nearly alike as possible are
called "differential methods. 7 '
INDIRECT COMPARISON OF THE SUN WITH STARS
It is often possible to determine the difference of right ascension
of the sun and some well-known star by using the moon as an inter-
mediary, determining the difference of right ascension of the sun and
moon during the daytime and comparing the moon and a star as soon
as possible after sunset, the motion of the moon during the interval
being allowed for. The irregularity of the moon's motion may,
MOTION OF THE MOON AND SUN AMONG THE STARS 69
however, introduce a greater error than that arising from uncertainty
in the rate of the clock. A better method is offered on those not
infrequent occasions when the planet Venus is at its greatest
brilliancy, when it may be easily observed in full daylight ; the
motion of Venus in the interval is much smaller and more nearly
uniform, and, therefore, more accurately determined ; and by this
method the interval between the observations connecting the sun
with Venus and Venus with the star may be reduced to a very few
minutes, or even seconds, so that the error due to the clock may be
regarded as negligible.
The following observations illustrate the method.
1900
WATCH TIME
H.A.
6
April 19.3.
Procyon
8 h 17 m 45 s
15 4
+ 5 5
Venus .....
8 19 33
56 1
+ 26 1
Procyon . . . ...
Venus . - . . . . . .-
Procyon
8 21 45
8 23
8 24 53
16 .4
57 .0
17 2
+ 5 .55
+ 26 .05
+ 53
April 20.0.
Sun . .
1 28 45
358 5
+ 116
Venus ...
1 31 35
313 2
+ 25 35
Sun .
1 36 10
15
+ 11 6
Venus
Sun . .
1 38 33
1 41 21
315 .0
1 4
+ 25.3
+ 116
April 20. 3.
Procyon ....
9 31 27
33 25
+ 5 45
Venus
9 32 30
72 9
+ 25
Procyon
Venus ... ...
9 33 28
9 35
33.9
+ 5 .4
+ 25 9
Procyon
9 36
34 .25
+ 5.4
The observations April 19.3, that is, April 19 about 7 P.M., give for
the hour-angle of Venus 5 6. 55 at the watch time 8 h 21 m 17 s , and for
that of Procyon 16.33 at S h 21 m 28 s ; hence at 8 h 21 m Procyon
followed Venus 40.22.
In the same way we find that April 20.3 Procyon followed Venus
39.3, the change of the right ascension of Venus being 0.92 in 25.2
hours. A simple interpolation shows that April 20.0 Procyon
TO LABORATORY ASTRONOMY
followed Venus 39.59, and the observations at that time show that
Venus followed the sun 45.92, so that Procyon followed the sun
39.59 + 45. 92 = 85.51, and the difference of right ascension between
Procyon and the sun at noon on April 20 was, therefore, 5 h 42 m 2 s .
ADVANTAGES OF THE EQUATORIAL INSTRUMENT
Observation with, the equatorial we shall find especially useful
in getting exact positions of the moon, since it is available at any
time when the moon is above the horizon, and after sunset we can
always find some bright star sufficiently near to afford a fairly
accurate value of its place.
It is often inconvenient to observe the moon by the more accurate
method which is described in Chapter VI, that of meridian observa-
tions, which is confined to a short interval of one or two minutes
each day, and is often interfered with by clouds passing at the
critical moment, although nine-tenths of the whole day may be
suitable for observations made out of the meridian. Moreover,
until the moon is several days old, it is too faint for observation at
its meridian passage. It is, therefore, upon the equatorial that we
shall mainly rely for the determination of the moon's motion, as
well as for many observations of the planets out of the meridian.
Although it is far more convenient to find the right ascension and
declination of the sun by the method of the following chapter, at
least a few positions should be found by observations with the
equatorial and plotted on the globe. The result will be to show
that the path of the sun is very exactly a great circle fixed on the
sphere or so nearly fixed that some years of observation with the
most refined instruments are necessary to detect any change in its
position among the stars, although a much shorter time even would
serve to show the slow change of its intersection with the celestial
equator due to precession.
This great circle is called the ecliptic, and its position is shown on
the map which we have used for plotting our first moon observation.
Three months will give a sufficient arc of this circle to enable us
to determine with some accuracy its position with respect to the
equator, its inclination to the latter, and their points of intersection ;
MOTION OF THE MOON AND SUN AMONG THE STAKS 71
if possible, observations should, however, be continued throughout
the year which the sun requires to complete its circuit, so that the
variability of its motion may be observed, most of the work, how-
ever, being done with the meridian circle.
The sun's diameter should occasionally be measured to determine
the points at which it is nearest to and farthest from the earth.
CHAPTER VI
MERIDIAN OBSERVATIONS
WE have now arrived at a point where we can see what are the
desirable conditions for making observations as accurately as possible
of the position of a heavenly body. To adjust the equatorial instru-
ment so that its axis lies in the meridian and at the proper inclina-
tion, and to keep it so adjusted, is a matter of some difficulty. In
the last chapter we have shown how, by observing an unknown body
in a certain fixed position of the instrument, and later a body whose
right ascension and declination are known in as nearly as possible
the same position of the instrument, we lessen the effect of the
instrumental errors. We made our observation of the sun shortly
before sunset, so that the interval between this observation and that
of the comparison star should be as short as possible. If, however,
the rate of the clock can be relied upon, there is no reason why the
observation should not be made when the sun is on the meridian,
the interval of time required to connect it with stars in that case
being not necessarily more than eight or nine hours in the most
extreme case ; and the comparative ease with which an instrument
may be constructed so that it shall be at all observations exactly in
the meridian, and the possibility of constructing very accurate time-
pieces, has determined the use of such instruments for all the more
precise observations in astronomy, such as fix the positions of the
fundamental stars and the vernal equinox on the celestial sphere.
The equatorial instrument may be used for this purpose by clamp-
ing it in such a position that the reading of the hour-circle is 0, in
which case the declination axis is horizontal east and west, and
when the telescope is moved about its axis it always lies in the
plane of the meridian. If, with the instrument so adjusted, we
observe the sun at the time of its meridian passage, we may find
its declination by reading the declination circle, and its right ascen-
sion by noting the interval which elapses before the meridian transit
72
MERIDIAN OBSERVATIONS
73
of some known star after nightfall, free from any error involved in
reading the hour-circle. As before, a star should be chosen at
nearly the same declination, so that the interval of time may be
very nearly equal to the difference in right ascension between the
sun and the star, even if the instrument is not very exactly in the
meridian. Observation of several different stars will enable us to
determine whether the instrument actually does describe the plane
of the meridian as it is rotated about the horizontal axis (see Chapter
VIII) ; and by the observation of stars near the pole, as described
on page 81, we may determine whether the declination circle reads
exactly when the telescope points to the equator, as should be
the case.
THE MERIDIAN CIRCLE
An instrument which is to be used in this manner, however, is
not usually so constructed that it can be pointed at any point in the
heavens. Thus, it is un-
necessary that it should
consist of so many moving
parts as the equatorial in-
strument, and steadiness,
strength, and ease of ma-
nipulation are very much
increased by constructing
it as shown in Fig. 40,
which represents a very
small instrument built on
the plan of the meridian
circle of the fixed observ-
atory. The strong hori-
zontal axis revolves in
two Y's, which are set
in strong supports in an
east and west line. The
axis is enlarged towards
the center, and through the center passes at right angles the
telescope tube. The axis carries at one end a graduated circle
FIG. 40
74 LABORATOKY ASTRONOMY
perpendicular to the axis of rotation. If the axis of the telescope
is perpendicular to the axis of rotation, and if the latter is adjusted
horizontally east and west, the telescope may be brought into any
position of the meridian plane, but must always be directed to some
point of the latter. A pointer attached to the support marks the
zero of the vertical circle when the telescope points to the zenith,
and if the telescope be pointed to a star at the time of its meridian
passage, the angle as read off on the circle is the zenith distance of
the star ; while the time of the star's meridian passage by a clock
giving true sidereal time is its right ascension. If the latitude of
the place of observation is known, the star's decimation is deter-
mined by the fact that the zenith distance plus the declination of
any body equals the latitude (see page 81). At first the latitude
may be used as determined by the sun observation of Chapter I, or
from a good map showing the place of observation, but ultimately
its value should be determined with the meridian circle itself.
LEVEL ADJUSTMENT
We will now proceed to show how to make the necessary adjust-
ments for placing the telescope so that it may move in the plane of
the meridian.
Place the instrument on its pier and bring the Y's as nearly as
possible into an east and west line. If the pier is the same that
has been used in the previous work, this may be done by bringing
the telescope into the meridian which has been determined by the
method of equal altitudes.
The axis must first be brought into a horizontal line, making use
for this purpose of the striding level (Fig. 41), which is a necessary
auxiliary of this instrument. This is a glass tube nearly but not
quite cylindrical, ground inside to such a shape that a plane passing
through its axis, CD, cuts the wall in an arc, AB, of a circle whose
center is at 0. In this tube is hermetically sealed a very mobile
liquid in sufficient quantity nearly but not quite to fill it the
space remaining, called the " bubble," always occupying the top
of the tube. When CD is horizontal, the bubble rests in the
middle of the tube with its ends, of course, at equal distances from
MERIDIAK OBSERVATIONS
75
the middle; the tube is graduated so that this distance may be
measured, the numbering of the graduations usually increasing in
both directions from the center of the tube. If the radius of the
arc is 14.3 feet, a length of 3 inches of this arc will be equal to
about 1, since the arc of 1 in any circle is about of the radius ;
o7.o
1 inch of the arc will then be about 20', and 0.05 inch 1'. These
are about the actual values for the level used with the instrument
\1
\i
FIG. 41
shown in Fig. 40, the scale divisions being about fa of an inch
apart and therefore corresponding to an arc of 1'.
If the line CD is inclined at an angle of 1' to the horizontal
line by raising the end A, the center of curvature will be dis-
placed toward the left, and the level will have the same inclina-
tion as if the whole tube had been turned to the right about the
point through an angle of 1'; and the highest point of the arc,
which is always directly above 0, is now fa of an inch from the
middle toward A. Since the bubble always rests at the highest point
of the arc, it follows that its ends will each be moved toward A by
one division ; if, for instance, the readings of the ends are 5 and 5
when CD is horizontal, they will be 6 and 4 when CD is inclined
76 LABORATORY ASTRONOMY
by 1', and evidently 7 and 3 when CD is inclined 2', etc., the incli-
nation in minutes of arc being one-half the difference of the readings
of the ends of the bubble, or - if A and B represent the
2i
readings of the ends of the bubble in each case. If the reading of
B is greater, the end A is depressed by one-half the difference of
the readings ; and the above expression applies to both cases if we
agree that it shall always denote the elevation of A, a negative value
of - indicating depression of A.
REVERSAL OF THE LEVEL
The level tube is attached to a frame (Fig. 40) resting on two stiff
legs terminating in Y 's, which are of the same shape and size as
those in which the axis of the meridian circle rests, the axis of the
level tube being adjusted as nearly as possible parallel to the line
joining the Y's. It is difficult to insure this condition, but if it is
not exactly fulfilled, the horizontality of the axis may still be deter-
mined by placing the level on the axis, and determining the value
- , and then turning it end for end, and again reading the value ;
L
for if the end A is high by the same amount in each case, the axis
is obviously horizontal, and the measured angle of inclination is due
to the fact that the leg of the level adjacent to A is longer than the
other leg. The practical rule is to read the west and east ends in
TJ/- T^
each position. If these readings are W l E l W z E 2 , ^ is the
elevation of the west end according to the first observation, and
TT7" -Cl
2-jr - at the second. If the leg which is west at the first
observation is too long, the first observation gives a value for the
elevation of the west end too great, and the second a value too
small by the same amount ; and the average of the two values
TTT- 77 Tl^ rj
^ - and 2-jij - gives the true value of the inclination of the
axis.
MERIDIAN OBSERVATIONS 77
It is usual to write this ( w * +w *\ ( E i + E *) and to record
4
the observations in the following form :
W 1 E,
Subtract the second sum from the first and divide by 4. This
gives a positive value if the west end is high, and the axis may
be made horizontal by turning the leveling screw so as to make
the level bubble move through the proper number of divisions.
The level should be again determined in the same way, and the
axis is level when
The following record of level observation made Feb. 26.3, 1900,
conforms to the above scheme :
W E
1* 2*
-1
- \ division = 15"
The west end being too low, the screw was turned so as to raise it
enough to move the bubble J division toward the west, the level
remaining on the axis during the adjustment and watched as the
screw was turned ; the readings were then as follows :
2i If
If 2J
4 4
And the axis was truly level, since ( Wi 4- TF 2 ) (E l -{- E z ) = 0.
1
78 LABORATORY ASTRONOMY
COLLIMATION ADJUSTMENT
The line of collimation of the telescope is the line drawn from
the center of the lens to the wires that cross in the center of the
field. When the telescope is " pointed " or " set " upon a star, the
image of the star falls upon the point where these wires cross, and
when the instrument is correctly adjusted the line of collimation is
perpendicular to the axis of rotation, so that the line of collimation
cuts the celestial sphere in a great circle as the telescope turns upon
its axis.
To make this adjustment, point the telescope exactly upon any
well-defined distant point, the meridian mark will, of course, be
chosen if it has been located, then remove the axis from its Y's
and replace it after turning it end for end ; if the telescope is still
set on the mark in the second position, the adjustment is correct;
otherwise move the wire halfway toward the mark by means of
the screws a, a (Fig. 40). Set again upon the mark by moving
the screws in the eyepiece tube ; reverse the axis again, and thus
continue until the telescope points exactly upon the mark in both
positions of the axis.
If the adjustments for level and collimation are properly made,
the intersection of the wires in the center of the field of view will
appear to describe a vertical circle, that is, a great circle through
the zenith, as the instrument is turned on its axis. The final
adjustment consists in bringing this circle to coincide with the
meridian, but for this we must have recourse to observations of
stars.
AZIMUTH ADJUSTMENT
The simplest method is to observe the time of transit by a sidereal
clock of a circumpolar star at its upper transit, and again, 12 hours
later, at its transit below the pole ; if the interval is exactly 12
hours, the adjustment is correct ; if the interval is less than 12
hours, the telescope evidently points west of the pole, and the west
end of the rotation axis must be moved toward the north. This is
done by the screws a, a (Fig. 40), the fraction of a turn being noted j
MERIDIAN OBSERVATIONS 79
the observation is repeated upon the following night, and by com-
paring the change which has been produced by moving the screws,
the further alteration required is readily estimated. On Feb. 26,
1900, the lower transit of e Ursse Minoris was observed at 4 h 58 m
12 s , and the upper transit at 16 h 53 m 32 s ; the times were taken by
a sidereal clock and have been corrected for its error ; the interval
being ll h 55 m 40 s , it is evident that the telescope pointed to the
east of the meridian, the arc of the star's diurnal path between
the lower and upper transits lying to the east of the meridian and
being less than 12 h by 4 m 20 s or 260 s .
To correct the error, the west end of the axis was moved toward
the south by turning the adjusting screws through one-quarter of
a turn. On the following day the observations were repeated as
follows :
Feb. 27.25, lower transit 4 h 54 m 45 s ; Feb. 27.75, upper transit
16 h 54 m 28 s ; the eastern arc was still too small, but the error had
been reduced to 17 & , and required a further correction of ^ ? 7 5 of a
quarter turn of the screws, which were therefore turned through
about 6 in the same direction as before, and the instrument was
thus brought very closely into the meridian.
This method can only be used with small instruments when the
night is more than 12 hours long ; but it is the only independent
method ; it requires that the rate of the clock shall be known
between the two observations, and it requires observations at in-
convenient times. A more convenient method is always used in
practice, but requires an accurate knowledge of the right ascensions
of a considerable number of stars in the neighborhood of the pole.
It has been stated that it is often inconvenient to observe the moon
when on the meridian, but with this exception all the fundamental
observations of astronomy are now made with meridian instruments
on account of the simplicity and permanence of the necessary adjust-
ments. A body observed on the ineridian is also at its greatest
altitude and least affected by atmospheric disturbances, which often
interfere with the observation of bodies near the horizon.
80 LABORATORY ASTRONOMY
DETERMINATION OF DECLINATIONS WITH THE
MERIDIAN CIRCLE
The circle of the meridian instrument may be used to determine
the declination of a star in two ways, of which that now described
is perhaps the most obvious, but also the least convenient.
If the reading of the circle is known when the telescope is pointed
at the pole, the angle through which the telescope must be moved to
point upon any star, that is, the polar distance of the star, is the
difference between this value and the circle reading when the tele-
scope is pointed at the star ; this angle is 90 the star's declination;
if the star is on the equator, the angle is 90; and if the star is
south of the equator, the angle is greater than 90 by an amount
equal to the declination of the star ; if we consider the declination a
negative quantity for a star south of the equator, the value 90 8
represents the polar distance in all cases.
To determine the reading of the "polar point" we may set the
telescope upon a circumpolar star at its "upper culmination" and
read the circle, and again, 12 hours later, set on the same star at its
" lower culmination/' the mean of the two readings is the reading
of the polar point. The effect of refraction may be neglected with
our small instruments without causing an error of J^ of a degree
at any place in the United States if we restrict ourselves to stars
within 10 of the pole, or the circle readings may be corrected by
a refraction table. Immediately after making this determination it
is advisable to make a setting on the meridian mark and note the
reading ; this point may thereafter be used as a reference point from
which the reading of the polar point may be at any time determined
if the meridian mark has not in the mean time changed its position.
Better still, the observation of the polar point may be combined
with a determination of the circle reading when the telescope points
at the zenith, by one of the methods to be described later; the
difference of the readings in this case is obviously equal to the
co-latitude, and such an observation constitutes an "absolute deter-
mination of the latitude," that is, a determination made without
reference to observations made at any other place. When the lati-
tude has once been satisfactorily determined, the observations of
MERIDIAN OBSERVATIONS
81
the declinations of stars can be made to depend upon determinations
of the zenith point by means of the fact that for a body on the meridian
Declination = Latitude Zenith Distance,
latitude and declinations being reckoned positive northward from
the plane of the equator, and zenith distance positive southward
from the zenith. The proof of this relation is left to the student
as well as the interpretation of the result when the observation is
made at the transit below the pole.
At the time of observing the transits of e Ursse Minoris described
on page 79 the following readings of the circle were made when
the star was in the center of the field. Each of these observations
consists of two readings : one of the index A on the south end of
a horizontal bar fixed to the supports of the axis, and the other
of an index B at the other extremity of the bar, as nearly as
possible half a circumference from A. An angle given by the
mean of two readings made in this manner is free from the " error
of eccentricity," which affects readings by a single index in case
the center of the graduated circle does not exactly coincide with
the axis about which it is turned between the two observations.
DATE
A
B
MEAN
February 26.25 . . . .
55. 45
55. 35
55.40
26.75 ....
39 .95
39 .85
39 .90
27.25 ....
55 .45
55 .35
55 .40
27.75 ....
39 .95
39 .85
39 .90
Hence the reading when the instrument was pointed at the pole
was
55.40 + 39.90
= 47.65.
Evidently the polar distance of the star was
55 40 _ 39 90
7-75,
and its declination 82. 25; and we have thus obtained an "inde-
pendent " or " absolute " determination of the declination of c Ursae
Minoris ; that is, a determination independent of the work of other
observers, and only dependent on the accuracy of our circle and of
our observations.
82 LABOR ATOKY ASTRONOMY
The circle was known to be adjusted so that the reading of the
zenith was very exactly zero, hence the latitude of the place of
observation was 42.35. The exact agreement of these observations
indicates that the magnifying power of the telescope was such that
it could be set more accurately than the circle could be read, and
not that the results are reliable to a hundredth of a degree.
For convenience in recovering the zenith reading, in case the
adjustment of the circle should be disturbed, the zenith distance
of a meridian mark was measured repeatedly, the result showing
that its polar distance was 137. 47, and this was used to check the
polar reading in later observations upon stars when it was impossible
to get observations of the same star above and below the pole.
Another method of making absolute determinations of the latitude
with the meridian circle is to observe the zenith distance of the sun
at the solstices ; the mean of these values being the zenith distance
of the equator, which is equal to the latitude. This observation,
however, is subject to considerable uncertainty on account of the
difference in atmospheric conditions at the summer and winter
solstice, and to great inconvenience on account of the lapse of
time ; it is, however, of course, the means upon which we must
rely for the accurate determination of the obliquity of the ecliptic,
one of the fundamental quantities of astronomy.
For the use that we shall make of the meridian circle, it will
probably be most convenient to make a careful determination of
the polar distance of the meridian mark, and use this habitually
as a point for reference.
PROGRAM OF WORK WITH THE MERIDIAN CIRCLE
Work with the meridian circle should at first consist of reobser-
vation of all the stars which have been previously observed with
the equatorial, except those which are west of the meridian after
nightfall and cannot be observed for six months. Attention should
be given to gathering a list of stars within 15 or 20 of the pole
for the purpose of quickly setting the instrument in the meridian
by the methods of page 79. The sun should be observed at least
once a week and its place plotted on the globe, and many stars
MERIDIAN OBSERVATIONS 83
in the neighborhood of the moon's path to form a basis for finding
the moon's place by differential observations, of course, also the
moon itself, the planets and a comet, if any of sufficient bright-
ness appears. In this way, by observing a few stars each night,
a great amount of material may be stored for future use.
Especial attention should be given to getting a good number of
observations of stars near the equator, so that fairly accurate values
of their differences of right ascension may be obtained, and at the
first opportunity the absolute right ascension of one of their num-
ber must be determined in order that thus the places of all may be
known. The results may be best recorded by making a list of
their right ascensions referred to an assumed vernal equinox. Thus,
the observations discussed on page 52 show that a Pegasi precedes
y Pegasi by 17.03 = l h 8 m 7 s , or, in other words, follows it by
22 h 51 m 53 s ; and if the right ascension of y Pegasi referred to the
assumed equinox is O h 8 m , that of a Pegasi is 22 h 59 m 53 s . If
in the course of the year observation shows that the true right
ascension of y Pegasi is O h 8 m 5 s , it is evident that the true value
for a Pegasi is 22 h 59 m 58 s , and that the right ascension of all stars
referred to the assumed equinox by comparison with y Pegasi must
be increased by 5 s .
DETERMINATION OF THE EQUINOX
An opportunity for observing the absolute right ascension of the
zero star, which is often called a " determination of the equinox,"
occurs about the middle of March and September.
If the course is begun in September, it will be well to make this
determination with the help of more experienced observers, even
before the nature and object of the measures are understood.
The observation consists in determining the difference of right
ascension of some star from the sun at the instant when the latter
crosses the equator, for at that time it is either at the vernal or
autumnal equinox, and its right ascension is in the one case hours
and in the other 12 hours.
If a meridian observation of the sun's altitude shows that the sun
is exactly on the equator at meridian passage, and the time of transit
84 LABORATORY ASTRONOMY
is noted by a sidereal clock, and as soon as it is sufficiently dark the
transit of a star is observed, the difference of the times is the absolute
right ascension of the star if the observation is made at the vernal
equinox, or equals the right ascension of the star minus 12 h if the
observation is made at the autumnal equinox.
Inasmuch as the meridian of the observer will rarely be that
one on which the sun happens to be as it crosses the equator, we
must make observations on the day before and the day after the
equinox, thus getting the difference of right ascension of the star
from the sun at noon on both days. The declination of the sun
being also measured at these two times, a simple interpolation gives
the time at which the sun crossed the equator, and this time being
known, another simple interpolation between the differences of right
ascension at the two noons gives the difference of right ascension
of the sun and star at the time when the sun was at the equinox,
which is the star's absolute right ascension.
The first interpolation assumes that the sun's declination changes
uniformly with the time, and the second that its right ascension
changes uniformly with the time.
Observations should extend over a period of a week before and a
week after the equinox to test the truth of these assumptions.
In observing the sun, a shade of colored or smoked glass may be
placed over the eyepiece, or the eyepiece may be drawn out as in
the method of observation described on page 37, and the screen
held in such a position that the cross-wires are sharply focused
upon it. As the image of the sun enters the field it should be
adjusted by moving the telescope slightly north or south till the
horizontal wire passes through the center of the disk, and as the
latter advances, the time should be noted when the preceding and
following limbs cross the vertical wire, as well as the time when
the vertical wire bisects the disk ; at the instant of transit the disk
should be neatly divided into four equal divisions, a very small
deviation from this condition being quite perceptible to the eye.
MERIDIAK OBSERVATIONS
85
THE AUTUMNAL EQUINOX OF 1899
The following table gives the details of observations taken a*t
the autumnal equinox of 1899 for the purpose of determining the
equinox.
The latitude of the place of observation was 42.5, and the declina-
tions given in the last column are calculated by subtracting the zenith
distance in each case from this quantity, as explained on page 81.
DATE
OBJECT
TIME OF TRANSIT
ZEN. DIST.
DECL.
Sept. 22
Sun
12 h O m 2 s .O
S42.2
+ 0.3
77 Serpentis
18 18 22.6
45 .4
- 2.9
X Sagittarii . .
18 24 2.4
67 .95
-25.45
Vega ....
18 35 44.5
3 .87
+ 38 .63
Altair ....
19 48 7.6
33 .98
+ 8 .52
Sept. 23
Sun .
12 3 45.1
42 .62
.12
17 Serpentis
18 18 20.1
45 .4
- 2 .9
X Sagittarii . .
18 23 57.3
67 .97
- 25 .47
Vega ....
18 35 42.6
3.85
4- 38 .65
Altair ....
19 48 1.5
The intervals between the observed times of transit of each star
on the two different dates range from 23 h 59 m 53 8 .9 to 23 h 59 m 58M,
showing that the clock was losing about 4 s daily, a quantity so
small that for our purpose it may be neglected.
Observations of the sun made on different dates between Sep-
tember 18 and September 23, but not here recorded, showed that
its right ascension and declination were changing uniformly at the
rate of about 3 m 45 s and 0.39 per day. The table above shows
that from September 22 to September 23 the rates were 3 m 43 8 .1
(or, allowing for clock rate, about 3 m 39 s ) and 0.42 per day, and
the latter value we shall use to determine the time of the equinox,
as follows :
At noon September 22, or September 22 d .O, as it is expressed by
astronomers, the sun's declination was + 0.3, and September 23.0
86
LABORATORY ASTRONOMY
its declination was 0.12. Hence its declination was Septem-
ber 22f f, or September 22 d .714. It was at that time, as exactly as
our observations can show, at the autumnal equinox, and its right
ascension was 12 h O m s .
Since rj Serpentis followed it to the meridian 6 h 18 m 20 8 .6, that
quantity is the difference between the right ascension of the star
and that of the sun September 22.0. Similarly the difference of
right ascension of sun and star September 23.0 was 6 h 14 m 35 s .O ;
that is, it was 3 m 45 8 .6 less than at the previous date. Assuming
this change to be uniform, the difference of right ascension of sun
and star at the moment of the equinox on September 22 d .714 was
0.714 X 3 m 45 8 .6, or 2 m 41M less than on September 22.0; that is,
it was 6 h 15 m 39 s . 5, and since the right ascension of the sun Sep-
tember 22.714 was 12 h O m O 8 , the right ascension of t] Serpentis was
18 h 15 m 39 8 .5.
The following table gives the data from which the "absolute
right ascensions " of the four stars are thus determined. In the
last column are the declinations, which are the means obtained from
several observations between September 14 and September 23.
STAB
R.A. OF STAB MINUS R.A. OF SUN
STAB'S
R.A. DECL.
SEPT. 22.0
SEPT. 23.0
SEPT. 22.714
t\ Serpentis
X Sagittarii
Vega
Altair
6* 18m 2 (K6
6 24 0.4
6 35 42.5
7 48 5.6
6h i4m 35s. o
6 20 12.2
6 31 57.5
7 44 16.4
6* 15^ 39.5
6 21 17.4
6 33 1.8
7 45 21 .9
18h 15m 398.5
18 21 17.4
18 33 1.8
19 45 21 .9
- 2. 89
-25 .48
+38 .65
+ 8 .59
The measurements upon which the above results depend are of
two kinds : observed clock times, which are liable to errors of a
very few seconds, so that the differences of right ascension may be
assumed to be correct within perhaps 4 s ; and measures of the sun's
declination, which with the greatest care may be in error at least
0.05 on any given date.
It is quite wUhin the bounds of probability, for instance, that
the sun's declination was + 0.25 on September 22.0 and 0.17
MERIDIAN OBSERVATIONS
87
on September 23.0 ; and recomputing with these values, the date
of the equinox was September 22ff, or September 22 d .595, and
the right ascensions of the stars 18 h 16 m 6 8 .4, 18 h 21 m 44 8 .6,
18 h 33 28 S .6, 18 h 45 m 49 S .2 ; that is, the uncertainty of the equinox
is 0.12 days and of the right ascensions about 27 8 , although the
relative right ascension is altered only by a fraction of a second
in each case. It is thus evident that the accuracy of the right
ascensions depends chiefly upon the accuracy with which the sun's
declination can be measured.
THE AUTUMNAL EQUINOX OF 1900
In order to increase the accuracy of determination of declination,
a new circle reading to minutes of arc was substituted for that
used for the observations of the equinox in 1899, and the observa-
tions were repeated at the same place in 1900. The weather con-
ditions were unfavorable, so that only the following observations
could be made.
DATE
OBJECT
TIME OF
TRANSIT
ZEN.
DlST.
DECL.
Sept. 22
Sept. 23
Sun -
Vega . .
18
19
12
19
35
47
3
48
n 44s. 8
27 .0
49.0
1 .5
35.0
842
33
42
33
ll'.S
49.0
51.0
33.1
54.0
+
+ 38
+ 8
-
+ 8
18'. 5
41.0
39.0
3.1
36.0
Altair
Sun ......
Altair . ...
From these data, by the same method as before, the date of the
equinox is found to be September 22^f;|, or September 22.8565.
If each declination of the sun is accurate to 1', the result may be
in error by ^f ^ days, or about .09 day ; the actual error is probably
less than half this amount, and the concluded right ascensions
probably within 10 s of the true values.
The observed times of Altair on the two dates show that the
clock was gaining 46 s daily, since the true sidereal time of transit,
88
LABORATORY ASTRONOMY
being equal to the star's right ascension, is the same on both nights.
This rate is so large that it cannot be neglected as in the discus-
sion of the result for 1899.
If the clock correction A (see page 60) at the time of the sun's
transit, September 22, be assumed s and the gaining rate 46 s per
day, or 1 8 .916 per hour, the corrections for Vega and Altair Sep-
tember 22 were 12 8 .6 and 14.9, and for the sun and Altair
September 23 were 45.9 and 61 8 .0. The times obtained by
applying these corrections are said to be " corrected for rate of
the clock to the epoch September 22.0."
In this manner the times, as they would have been observed with
a clock having an exact sidereal rate, are found to be :
SEPTEMBER 22
SEPTEMBER 23
Time of transit
u u
u u
of the Sun . . .
" Vega ....
" Altair ....
Hh 58m 446.8
18 35 15 .4
19 47 34.1
I 2 h 2 m 15 8 .6
19 47 34 .0
Hence Altair followed the sun
September 22.0 7 h 48" 49 8 .3
23.0 7 45 18.4
22.856 7 45 48.8
and the right ascension of Altair was 19 h 45 m 48 8 .8 ; since Vega pre-
cedes Altair by l h 22 m 18 8 .7, its right ascension was 18 h 33 m 30 s . 1.
In 1899 the difference of right ascension of the two stars was
l h 22 m 20M, but the right ascensions of 1900 are greater by 28 8 .3
and 26 8 .7 than those of 1899.
If we assume the later determination to be absolutely correct,
we must regard the earlier as having placed the equinox farther
toward the east among the stars than its true place, so that right
ascensions referred to the equinox observed in 1899 are too small.
We may say that the observations of 1900 indicate a correction of
27 8 .5 to the " equinox of our little catalogue of four stars " ; that
is, a correction of -f 27 s . 5 to all their right ascensions as determined
in 1899.
MERIDIAN OBSERVATIONS 89
Applying these corrections, their right ascensions become for
t] Serpentis 18 h 16 ra 7 s .
X Sagittarii 18 21 44 .9
Vega 18 33 29 .3
Altair 19 45 49 .4
Since the later observations were made with an instrument
giving more accurate values of the declination, it is probable that
their results are more nearly correct. The clock rate was neglected
in the first observations, and the effects of precession, parallax, and
refraction in both series, following out the principle that no correc-
tions will be made until observations shall show their necessity.
The effect of refraction is to delay the autumnal equinox about
an hour, and hence to decrease the right ascensions of the stars
by about 10 s . At the vernal equinox, however, refraction hastens
the equinox an hour and increases the right ascensions by 10 s ;
its effect may be shown by observations at the two equinoxes of
the same year and eliminated by their combination. Parallax
hastens the autumnal and delays the vernal equinox by about 8 m ,
thus affecting right ascensions by a little more than I 8 , the mean
of observations at the two equinoxes being free from error from
this source. The effect of precession will be manifest in less
than ten years with an instrument like that used in the above
observations of 1900.
By comparing the equinox of September 22.714 0.12, 1899, and
September 22.856 .09, 1900, the length of the tropical year is
found to be 365.142, but may lie between 364.93 and 365.35 as far
as our observations can surely determine. Since refraction delays
the vernal and hastens the autumnal equinox by nearly the same
amount (about an hour) in each case, it has no effect upon the
length of the year. As the greatest error to be feared with our
improved instrument is less than 0.1 day, the length of ten or one
hundred years may be determined with less than twice that error,
in those periods the length of the year may be determined within
0.02 and .002 day, respectively.
With the best modern instrument used to the greatest advantage,
the sun's declination may be determined near the equinox within
90 LABORATORY ASTRONOMY
0".5, and hence the time of the equinox within 30 8 and right
ascensions within 8 .08. A single tropical year may be measured
with an error of less than l m .
We have now explained the methods by which it is possible to
fix the places of the sun, moon, and stars at different times and
thus to obtain data from which their apparent motions about the
earth may be studied and theories formed from which their future
places may be predicted. More or less complete accounts of these
theories are to be found in all works on descriptive astronomy,
and the predictions derived from them are published for three
years in advance by several governments for the use of navigators
and astronomers. Such .a publication is the American Ephemeris
and Nautical Almanac, of which it will be convenient to give some
account before taking up the motions of the planets.
The apparent motions of the planets are less simple than those
of the sun, moon, and stars, which at all times seem to move about
the earth as a center with approximately uniform velocities. The
planets, it is true, in the long run continually move like the sun
and moon around the heavenly sphere toward the east, but their
velocities are variable within wide limits and at certain times are
even reversed, so that they move in the opposite direction or
" retrograde " among the stars.
For this reason a longer period of observation is necessary to
determine their motions than can be given by the individual student.
We may, however, regard the nautical almanacs of past years as
predictions that have been verified, and they stand for us as an
accredited set of exceptionally accurate observations from which
we may draw material to combine with the results of our own
observations.
CHAPTEK VII
THE NAUTICAL ALMANAC
THE American Ephemeris and Nautical Almanac consists of two
parts, the Nautical Almanac proper, which is published separately
and contains data especially useful in navigation, and a second part,
which contains additional tables adapted to the use of astronomers.
The Nautical Almanac will suffice for most of our purposes, but the
complete work is convenient for a few references.
The tables contain data for the sun, moon, and planets, for suc-
cessive equidistant points of Greenwich mean time, so near together
that the values at any intermediate time may be obtained by inter-
polation with a degree of accuracy greater than can be obtained by
a single observation made with the most refined instruments. The
dates are given in astronomical time, each day beginning at noon
of the corresponding civil date.
At this point a few words are necessary in explanation of the term
" mean time."
We have already defined apparent solar time as the hour-angle of
the sun, and sidereal time as the hour-angle of the vernal equinox.
Owing to the fact that the sun moves at a varying angular rate and
in a path inclined to the equinoctial, the hour-angle of the sun does
not increase uniformly, and the hours of apparent time are, there-
fore, of unequal length.
We have not yet obtained material for a complete discussion of
the relation between apparent and mean solar time, and for this we
must refer to the text-books of descriptive astronomy. It will
be convenient to explain one simple statement of this relation which
is not always explicitly given.
The time required by the sun to complete its circuit of the heavens,
from one passage through the vernal equinox to another, is 365.2422
days. As it describes 360 of longitude in that time, its average
daily motion in longitude is 0. 98564 7.
91
92 LABORATORY ASTRONOMY
To establish, a convenient measure of time not greatly different
from apparent solar time, a fictitious body is imagined to start
with the sun at perihelion and to move along the ecliptic with a
uniform daily motion in longitude of 0.9S565. Its longitude at
any time is called, appropriately enough, the " mean longitude of
the sun."
When this body reaches the vernal equinox, a second fictitious
body, called the " mean sun," is supposed to start out from that
point eastward along the equator, moving with a uniform velocity
equal to the mean daily motion of the sun in the ecliptic.
The mean sun, therefore, continually increases its right ascension
by 0.98565 per day ; and since both fictitious suns are at the vernal
equinox in longitude zero at the same instant and move at the same
rate, one in the ecliptic and the other in the equator, it is obvious
that at all times the right ascension of the mean sun is equal to
the sun's mean longitude.
The hour-angle of the mean sun is equal to the mean solar time,
just as the hour-angle of the true sun is equal to the apparent solar
time.
A clock, properly regulated and set so that it shows O h O m s
at each successive passage of the mean sun over the meridian of
a given place, is said to keep the local mean time of that place.
When the hour-angle of the mean sun is 10, 20, 30, the local
mean time is O h 40 m , l h 20 m , 2 h , respectively.
It is of course true of the mean sun as of any other heavenly
body (see page 58) that its H.A. + R.A. = Sid. T. We may there-
fore write :
H.A. of mean sun -f K.A. of mean sun = Sid. T.
H.A. of sun -f K A. of sun = Sid. T.
And from these equations, remembering the definitions of mean
and apparent time, we derive the following :
Mean T. = App. T. + (E.A. of sun E.A. of mean sun).
The quantity in the parenthesis, which must be added to App. T.
to give the corresponding Mean T., is called the equation of time.
THE NAUTICAL ALMANAC 93
The equation of time is the difference between mean time and
apparent time, and when positive must be added to apparent time
to give the corresponding mean time, or subtracted from mean time
to find the corresponding apparent time.
Standard Time. It is now usual to regulate the clocks over large
sections of country to the mean time of a neighboring meridian.
Thus, clocks in the central part of the United States are set to
show O h O m s when the sun is in the meridian whose longitude
is 90 west of Greenwich, and they are said to keep Central
standard time ; which is, therefore, 6 hours slow of Greenwich
time. Other sections use the mean time of the 75th, 105th, and
120th meridians, 5, 7, and 8 hours slow of Greenwich, respectively.
More than one half the people of the United States use Central
standard time.
The fact that our watches are set to standard time is a convenience
in using the Almanac, since the watch time gives us Greenwich mean
time by applying so simple a correction, the minutes and seconds
being unchanged and the hours increased by a small constant number.
THE CALENDAR
About four-fifths of the Nautical Almanac consists of data regard-
ing the sun and moon, eighteen successive pages being devoted to
each month, and the corresponding pages of the different months
numbered with the Roman numerals from I to XVIII. These pages,
which form the Calendar, we will now consider in detail. The
reading matter of the Explanation which follows the tables should
be carefully read in connection with the following paragraphs :
reduced facsimiles of several pages are shown at page 176, to which
reference may be made.
The positions are given as they would appear to an observer at
the earth's center, and the times are, as stated at the head of each
page, Greenwich mean time. We pass at once to page II, which,
rather than the very similar page I, it will be always more con-
venient to use when, as in most of our observations, the Greenwich
time is known for which the data are required.
94 LABORATORY ASTRONOMY
Page II. The first and second columns give the day of the week
and month. The third column contains the sun's apparent right
ascension at Gr. Mean Noon, that is, its right ascension as affected
by the annual aberration (which makes it appear to be about 20"
behind its true place in its orbit) and measured from the actual
equinox of the date. Column 4 contains the hourly difference, or
the amount by which the right ascension is changing per hour.
To illustrate the use of this column, let it be required to find
the right ascension of the sun at the time of the first observa-
tion recorded on page 39 at 8 h 54 m 37 s A.M., Eastern standard time,
March 8, 1900.
We must first notice that the corresponding astronomical time,
which is reckoned from noon to noon, is 20 h 54 m 37 s after noon of
the preceding day, that is, the local date was March 7 d 20 h 54 m 37 s ;
adding 5 h to change E. Std. T. to G.M.T., we have March 7 d 25 h
54 ra 37 s , or March 8 d l h 54 m 37 s , G.M.T.
The sun's right ascension, March 8, at Greenwich mean noon, is
given as 23 h 13 m 57 8 .68. To this, since the sun's right ascension is
always increasing, must be added the change in l h 54 m 37 s (= l h .91),
the time elapsed since noon, which is obtained by multiplying the
hourly difference found in column 4 by 1.91 ; this gives the correc-
tion to be added to the tabular right ascension as 1.91 x 9 S .237,
or 17 8 .64, and the right ascension at the time of observation was
therefore 23 h 13 m 57 8 .68 + 17 8 .64, or 23 h 14 ra 15 S .32.
This simple process, which is fully illustrated in the Explanation,
will never give a value more than 8 .4 in error. A method of inter-
polation by which an accuracy of 8 .01 may be attained is given in
the Explanation. The error of the simple method arises from the
fact that the hourly difference is not constant, as will appear at
once from inspection of the values in the fourth column.
Columns 5 and 6 give the sun's apparent declination and its
hourly difference. The value at any time may be found by inter-
polation in the manner just explained.
North declinations are regarded as positive, and south decli-
nations negative, and in accordance with this convention the hourly
difference is marked + when the change of declination is toward
the north and when toward the south, so that the true declination
THE NAUTICAL ALMANAC
95
is found by applying the correction algebraically : thus, to find the
declinations at 4 P.M., G.M.T., on the following dates, we have :
1900
8 AT MEAN NOON
H. DlFF.
CORK. FOR 4 h
S AT 4> G.M.T.
Jan. 10
-21 59' 4".0
+ 22". 25
+ 4 x 22". 25 = + 89".0
- 21 57' 35".0
April 10
+ 7 53 3 .7
+ 55 .48
+ 4 x 55 .48 = + 221 .9
+ 7 56 45 .6
Aug. 10
+ 15 38 18 .2
-43 .73
-4 x43 .73 = -174 .9
+ 15 35 23 .3
Nov. 10
- 17 6 18 .2
-42 .31
-4 x42 .31 =-169 .2
- 17 9.7 .4
The error in a declination determined by a simple interpolation
from the preceding mean noon can never exceed 12". By the more
accurate method given in the Explanation, it is always less than 0".l.
To' make sure that the correction has been applied with the proper
sign, it is sufficient to notice that the computed value must lie
between the values for the including dates.
Columns 7 and 8 contain the equation of time and its hourly
difference. The correction to be applied is obtained, as in the pre-
ceding examples, by multiplying the hourly difference by the num-
ber of hours elapsed since Greenwich mean noon, and must either
be added or subtracted so as to give a value between the values of
the including dates.
The heading of the column indicates whether the equation of time
is to be added to or subtracted from mean time to give apparent
time. Of course when it is additive to mean time it must be sub-
tracted from apparent time to give mean time, as will appear on
comparing the corresponding column of page I.
Example. What is the equation of time January 10, 1900, at
3 h 45 m , Central standard time ?
The corresponding G.M.T. is 9 h 45 m = 9 h .75
Eq. of T. at Gr. Mean Noon . + 7 m 39 s . 87 H. Diff. = K014
Change in 9 h .75 = 9.75 x 1 8 .014 9 .88 x 9.75
Eq. of T. at 3 h 45 m , Cent. T. . +7 49.75 Corr. =9 8 .88
The correction 9 s . 88 is added because the value of the equation
January 11 is seen to be 8 m 3 8 .90, and the correction must be applied
so as to increase numerically the value on January 10.
96 LABORATORY ASTRONOMY
The ninth column contains the right ascension of the mean sun.
Since at mean noon the mean sun is on the meridian and since
(p. 59) the right ascension of a body which is on the meridian at
a given instant equals the sidereal time at that instant, the right
ascension of the mean sun at Greenwich mean noon equals the
Greenwich sidereal time at Greenwich mean noon, and this explains
the alternative heading which appears at the top of the column.
Since the right ascension of the mean sun increases uniformly,
the constant hourly difference requires no special column, but is
given at the foot of the page. For interpolation it is most con-
venient to use Table III, which occupies three of the last pages of
the Almanac, and gives directly the multiples of 9 8 .8565 by each hour
and minute up to 24 hours, thus saving the reduction of minutes to
decimals of an hour.
Example. Right ascension of mean sun, January 15, 1900, at
4 h 44 m 30 s .
R.A. mean sun, Gr. Mean Noon . ... . . . . . 19 h 37 m 55 s . 26
Add 4M4 m 30 s x 9 s . 8505 (Table III) . . . . . 46.74
E.A. meansunat4h44 m 30 8 19 38 42 .00
This is obviously the sidereal time of mean noon at a place in
longitude 4 h 44 m 30 8 west, and if desired a table of this quantity
may be computed for such a place by adding 46 8 .74 to the values
given each day in the Almanac for Greenwich.
Page I. The quantities on page I are only used for reducing
meridian observations of the sun, which are made, of course, at local
apparent noon. This page is convenient when the Greenwich mean
time has not been noted, for the time elapsed since the preceding
Greenwich apparent noon is equal to the west longitude of the
place of observation. This is the quantity, therefore, by which the
hourly difference must be multiplied to give the correction. An
example of the use of this page is given on page 104.
All the quantities given on page I may be found more easily from
page II if we know the G.M.T. for which they are required. The
only quantity for which we are obliged to consult page I is the
semi-diameter, and this never differs by so much as 0".01 from its
value at mean noon.
THE NAUTICAL ALMANAC 97
Page III. Column 2 gives the day of the year corresponding to
the given date, and is convenient for finding the number of days
intervening between dates. Thus, January 15, 1900, is the 15th
day of the year and September 25 is the 268th ; hence from noon,
January 15, to noon, September 25, is 268 15, or 253 days.
Column 3 contains the sun's longitude measured from the vernal
equinox of the given date. For some purposes it is more convenient
to measure from the mean equinox of the beginning of the fictitious
year, an epoch much used in astronomical calculations but of no
intrinsic interest. The minutes and seconds of the longitude as
thus measured are found in column 4. The longitude of column 3
is measured from the actual place of the equinox at the given date
as affected by precession and nutation.
Column 6 gives the sun's latitude, which is always nearly but
not exactly zero, as will be explained further on in this chapter.
Column 7 gives the logarithm of the earth's distance from the
sun in astronomical units. An astronomical unit is equal to the
semi-axis major of the earth's orbit, about 93,000,000 miles.
For those unacquainted with logarithms the following table will
make it easy to find by interpolation the approximate distance cor-
responding to a given logarithm.
Logarithm 9.9925000 corresponds to 0.9829 astronomical units.
" 9.9950000 " " 0.9886 " "
" 9.9975000 " " 0.9943 " "
" 0.0000000 " ' 1.0000 " "
" 0.0025000 " " 1.0058 " "
" 0.0050000 " " 1.0116 " "
" 0.0075000 " " 1.0174
Example. January 19, 1900, log radius vector = 9.99299, which
is very nearly ^ of the way from 9.9925 to 9.9950 ; hence on that
date the distance of the earth from the sun is of the way between
0.9829 and 0.9886, or 0.9840 astronomical units. The value can be
obtained within less than ^1^ of its amount without interpolation
by taking the nearest value of the logarithm given in the table.
Column 9 gives the mean time at which the vernal equinox is on
the meridian of Greenwich (when the number of hours is greater
than 12 the time is after midnight, and therefore during the morning
98 LABORATORY ASTRONOMY
hours of the next civil date). This quantity is sometimes used in
converting sidereal to mean time, but its use may be easily avoided
and is sufficiently treated in the Explanation.
Page IV. The quantities on page IV relate to the moon. They
are given for each 12 hours of Greenwich mean time, and seem to
call for no explanation, except perhaps the symbol 6, signifying
conjunction, which occurs once (and occasionally twice) upon each
page, on the day before or after that of new moon. Since successive
transits follow each other nearly 25 hours apart, in general one date
in each month would be left blank, the moon crossing the meridian
during the hour preceding noon of one date, and during the hour
following noon of the succeeding date. The symbol 6 occupies the
vacant space and marks the date of new moon.
Pages V to XII contain the right ascension and declination of the
moon for every hour of G.M.T., together with their differences for
each minute of time. The rapid motion of the moon makes it necessary
to give these quantities at shorter intervals than suffice for the sun,
in order that an equal accuracy may be attained in interpolation.
These are of course places as seen from the earth's center, and
it is to be remembered that at any point on the earth's surface the
moon may be displaced by parallax a little more than 1.
On page XII are given the exact dates to the nearest hour of
G.M.T. of the moon's phases and the times of perigee and apogee.
Pages XIII to XVIII contain tables of " lunar distances," that
is, distances for each three hours of Greenwich mean time between
the moon's center and certain bright stars and planets not far from
the plane of its motion ; the sun is included in the list, as the moon
is often visible in full daylight, so that its distance from the sun
may be easily measured.
This table is used in determining longitude ; the local time being
known, the G.M.T. may be found by the method of lunar distances,
as follows : The distance from moon to star or sun being measured
is found to lie between two distances given in the table ; the G.M.T.
of the observation then lies between the hours corresponding to the
two tabular distances, and its exact value may be determined by
interpolation. The difference between this time and the known
local time of the observation is the longitude.
THE NAUTICAL ALMANAC 99
The method requires accurate observations, and troublesome com-
putations are necessary to correct the measured distance for the
effects of refraction and parallax so as to find the distance from
moon to star as seen from the earth's center.
Data for the Planets, Eclipses. Following the calendar pages of
the Nautical Almanac are thirty pages giving the right ascension
and declination and the time of meridian passage of the five planets
which are visible to the naked eye, and three pages containing the
right ascensions and declinations of 150 of the brighter fixed stars.
A few pages are devoted to the eclipses of the year, with maps
from which may be obtained the approximate times of the successive
phases of the solar eclipses as seen from any given point of obser-
vation on the earth.
EXAMINATION OF THE SEVERAL COLUMNS
Having given this general summary of the contents of the tables,
we will now call attention to some of the interesting facts and rela-
tions that appear on running through the various columns throughout
the whole year.
The date of the solstices may be determined as the days on which
the sun's declination has its maximum northern and southern values.
The date of the equinoxes may be found, from either the right
ascension or declination columns, as the date on which the decli-
nation changes sign, and the right ascension is either O h or 12 h ; the
exact time may be found by interpolation. (See page 107.)
The number of days between the equinoxes may be determined
by using the column of days, page III. It will be found that the
sun is for some days more than half the year in that part of its
orbit which lies in the northern hemisphere.
The column of hourly difference shows that the declination is
changing slowly at the solstices and most rapidly at the equinoxes ;
moreover, the change at the latter dates is nearly uniform both in
right ascension and declination, as stated on page 85. If a right
triangle 'be drawn with the difference in right ascension for the
date of the equinox as base and difference in declination as alti-
tude, the angle between the base and the hypotenuse measured by
100 LABORATORY ASTRONOMY
a protractor will be found to be 23^. It obviously equals the angle
between the equator and the ecliptic.
Notice that the equation of time is the difference between right
ascension of mean and true sun, as stated on page 92, thus :
From the Almanac for 1900 (p. II), we have the following
values : January 21, Sun's B,. A. = 20 h 13 m 2 8 .79 ; K.A. Mean Sun
20 h l m 34 s . 61. Subtracting the latter from the former, we have for
the equation of time + ll m 28 8 .18. This is the value given on page II ;
the positive sign indicates that it is to be added to apparent time to
find mean time, or subtracted from mean time to find apparent time.
The dates on which the equation of time is and dates and values
of greatest and least equations should be noticed ; also that on those
dates for which the equation is the values of the sun's right ascen-
sion and declination, etc., on pages I and II, are the same, since
apparent noon and mean noon coincide. For 1900 the civil dates
are as follows :
EQ. OF T.
February 11 + 14 m 27 8 .28
April 15
May 15 - 3 m 49 8 .40
June 14
July 27 r . . . . ... .". + 6 m 17 8 .22
September 1
November 3 - 16 m 20 s . 40
December 25
The hourly difference of the right ascension of the mean sun has
the same integers as the mean daily motion of 'the sun in longitude,
0.98565 0.98565 X 3600"
0.98565 ; for .98565 per day = , or - , per
hour, and reducing this to seconds of time by dividing by 15, we
find the motion of the mean sun to be 9 8 .8565 per hour. This illus-
trates the fact that the mean motion of the sun in longitude (0. 98565
per day) is the same as that of the mean sun in right ascension
(9 8 .8565 per hour), page 92.
The column which gives the sun's latitude will repay an investi-
gation. It appears at a glance that there is a small but regular
change,, from south to north and return, with a period of about 27
or 28 days.
THE NAUTICAL ALMANAC 101
The principal cause of this is that it is not the earth, but the
center of gravity of the earth and moon, which describes an orbit
in the plane of the ecliptic ; and by the known properties of the
center of gravity, when the moon is above the ecliptic the earth
must be below. It is not very difficult to show that from this cause
the latitude may be 0".67 greater or less than when both bodies are
in the ecliptic, that is, when the moon is at one of her nodes.
The attractions of Venus and Jupiter also draw the earth out of the
ecliptic by an amount which may reach 0".5. In January, 1900, this
" planetary perturbation " was about + 0".13. The total range of lat-
itude during the month (see page 178) was from -f 0".68 to 0".48.
The moon was at her nodes January 12.33 and January 26.85.
From the radius vector column (p. Ill) we may find the sun's
distance at any date by the table on page 97. By comparing this
with the semi-diameter column (p. I), it is shown that the sun's
distance is inversely proportional to its angular semi-diameter.
Thus, January 1, 1904 :
Log r = 9.9926540, Dist. = 0.9832, Senii-diam. = 16' 17".90
and July 1, 1904 :
Log r = 0.0072095, Dist. = 1.0167, Semi-diam. = 15'45".67
and
0.9832 : 1.0167 = 945".67 : 977".90,
as appears on multiplying the means and extremes and comparing
the products.
The dates of the moon's perigee and apogee may be determined
from the greatest and least semi-diameter, page IV, column 2, or
from the greatest and least parallax in column 4. Since both semi-
diameter and parallax are inversely proportional to the moon's
distance from the earth, the latter may be determined by multiplying
the former by a constant quantity. This constant is 3.6625, and it
is not difficult to show that it is the ratio of the earth's equatorial
radius to that of the moon.
Compare the last two columns, noting that at new rnoon the moon
comes to the meridian with the sun at noon and that at full moon
(age 15 days) it comes to the meridian at midnight.
102 LABORATORY ASTRONOMY
TABLES OF THE PLANETS AND STARS
The data for the planets which follow the calendar pages illus-
trate many facts which are explained in the text-books on descriptive
astronomy.
Ketrograde motion, for example, is shown by negative hourly
differences in right ascension ; the stationary points occur on those
dates on which the hourly difference changes sign ; opposition takes
place when the time of transit is 12 h ; conjunction, when it is O h ;
the retrograde motion is a maximum at opposition.
By means of the right ascensions and declinations the path for
the year may be plotted on a star map, for which purpose an ecliptic
map (see page 65) is especially adapted.
The time of passing the node may be found from the point where
the path cuts the ecliptic, and the sidereal period from the interval
between two passages of the same node.
A series of Almanacs covering some years is useful in following
the outer planets as well as for comparison of the calendar pages to
show the repetition of the solar data after four years.
The table of star places contains columns of annual variation,
that is, the sum of the precession and proper motion (the latter
always a very small quantity), which are useful in showing the
effects of precession on the right ascensions and declinations of
stars- in different parts of the heavens. Compare in this respect
8 Draconis, (3 Ursae Minoris, Polaris, y Pegasi, y Geminorum, and
A. Sagitarii.
COMPARISONS OF OBSERVATIONS WITH THE EPHEMERIS
Many of the facts which we have obtained by observation in
former chapters may be found in the columns of the Almanac, and
after a thorough comprehension of the methods has been acquired
much time may be saved by employing these data ; but it is to be
remembered that facts thus obtained are not so thoroughly grasped
or so easily retained. With this caution, we may compare some of
the results of our previous work with the tables, to give an idea of
the methods of using the latter. Following are comparisons of a
THE NAUTICAL ALMANAC 103
few of the observations of the preceding chapters with the values
given by the Ephemeris :
Observations of the Moon. From careful measurement of the map
on page 29, the moon's declination on January 9, 1900, at 10
P.M., was + 19.3, and its right ascension was 2 h 38 m . The place of
observation was 4 h 44 m .5 west of Greenwich, and the time used was
Eastern standard time, which is 5 hours slow of Greenwich ; the
G.M.T. was therefore 15 h O m , at which time. the moon's declination
and right ascension are given in the Ephemeris (p. 180) as + 18 48'
and 2 h 39 m . The difference between the observed and calculated
places is about ^ -in declination and l m in right ascension, mainly
due to error of observation with the cross-staff.
Length of the Month. We may use the Ephemeris to find the
length of the month by seeking the next date at which the moon's
right ascension and declination are the same, which is February 5,
at about 21 hours, G.M.T., as will be seen from page VI for February.
This gives 27 d 6 h as the period of the moon's revolution among the
stars.
Passing to page V for December, we find that the right ascension
was again 2 h 39 m on December 3 at 19 hours, at which time the
declination was 17 19'. This shows that the moon's orbit had
shifted during this time so that it did not pass through exactly the
same points of the heavens in these two months, its December path
in the neighborhood of right ascension 2 h 39 m being l south of
the corresponding point of its path in January.
By column 2 of page III, January 9 is the 9th day of the year
and December 3 is the 337th ; hence the moon completed an integral
number of revolutions in 337 d 19 h - 9 d 15 h , or 328 d 4 h .
The -period having been determined as 27 days approximately
and 328 -s- 271 being nearly 12, it is evident that the number of
complete revolutions between these dates is 12. Dividing 328 d 4 h
by 12, we have 27 d 8 h as a closer approximation to the sidereal
month.
Taking the length of the successive months during the year, it is
interesting to note how very considerable is the difference in length
of the successive sidereal months due to the " perturbations " of
the moon's motion.
104 LABORATORY ASTRONOMY
Observations at Apparent Noon. The observations recorded on
page 39 were made at Cambridge, in longitude 4 h 44 m .5 west of
Greenwich, and the watch time of apparent noon was ll h 56 m 2 8 .9.
By the use of the Almanac, we find the correction of the watch
to standard time as follows :
Since the observation was made at local apparent noon, it will
be better to use page I of the Almanac, which gives for March 8,
at Greenwich apparent noon, equation of time ll m 1 8 .46, to be added
to apparent time, and hourly difference s . 61 9.
The time of observation was 4 h 44 m .5, or nearly 4 h .75 later, and
the change of the equation of time in this interval was 4.75 X s . 619
= 2 8 .93. As the equation of time was decreasing, its value at the
time of observation was 10 m 5S S .53. Since no sign is appended to
the hourly difference, we check this result by noting that it falls
between the values tabulated for March 8 and 9. Hence :
Camb. App. T. . . . 12 h O m s
Eq. of T. (add) 10 58 .53
Camb. M.T. 12 10 58.53
Corr. for Long 4 44 30
G. M.T. of observation 16 55 28.53
Subtracting 500
Eastern Std. T. of observation 11 55 28.53
Observed watch time . . . . ... . ... 11 56 2.9
Corr. of watch to Std. T. . . . . . .... . . . . - 34 .37
The correction for longitude to give G.M.T. is added, because at
any given instant the local time of any place is greater than that
of a place to the westward, since the sun passes its meridian earlier
and always has a greater hour-angle than at the western place.
Kemembering that Cambridge is 15 m 30 s east of the meridian
from which Eastern standard time is reckoned, we may find the
watch correction more simply, thus :
Camb. M.T 12 h 10 m 58 S .53
Reduction for Long, (subtract) - 15 30
Eastern Std. T 11 55 28 .53
Watch time 11 56 2 .9
At - 34 .37
Observations of the Planets. The data on page 52 show that on
February 5, 1900, at 7 h 12 m (the watch keeping Eastern standard
THE NAUTICAL ALMANAC
105
time), the right ascension of Venus was 9.64 = 38 m 33 8 .6 less than
that of y Pegasi, which from the Ephemeris was O h 8 m 5 S .69 ; hence
from this differential observation the right ascension of Venus was
23 h 29 m 32 s .09.
The G.M.T. of the observation was 12 h 12 m = 12 h .2.
The tables for Venus (p. 224) give :
H. DlFFS.
+ 11 s . 106 +77". 31
x 12.2 x 12 .2
135.49 943 .2
2 m 15 8 .49 15' 43". 2
The observation differs from the Ephemeris by l m 38 s in right
ascension and 8' in declination, although the method should give
angles within 0.2. The discrepancy is much greater than usually
occurs, and this observation of Venus is affected by some unexplained
error ; it depends on a single reading of the hour-angle. To exhibit
the usual accuracy, we may compare with the following observa-
tions, made February 6 :
FEBRUARY 5 R.A. OF VENUS
At Gr. M. noon . . 23 h 25 m 38 8 .14
Diff. for 12 h .2 . . +2 15.49
DECL.
-4 55' 46"
+ 15 43
At 12M2 m , G.M.T. . 2327 53.63
Observed values (p. 53) 23 29 32 .1
-4 40 3
-4 32
WATCH TIME
H.A.
DECL.
7 Pegasi . . . . .
Venus
7h 3m IQs
5 10
64. 6
74 .05
+ 15. 45
- 3.4
7 Pegasi ... . . .
7 10
65 .7
+ 15 .5
Hence Venus preceded y Pegasi 8.90 = 35 m 36 s , Decl. = 3.4
0.53 = 3.93 ; and since the right ascension of y Pegasi was
O h 8 m 6 s , our observation gives for the place of Venus at 12 h 5 m
G.M.T., E. A. = 23 h 32 m 30 s , and 8 = - 3 56'. The Ephemeris gives
E.A. = 23 h 32 m 17 S .2, and 8 = 4 9' 14".5.
Observations of the Moon's Place. The data given on page 55
show that on February 6, 1900, the moon followed y Pegasi 46.7
= 3 h 6 m 48 s . The right ascension of y Pegasi was O h 8 m 6 s ; hence
the moon's right ascension was 3 h 14 m 54 s , while its declination,
given directly by the circle, was + 20. 4. The Eastern standard
time was 7 h 42 m , corresponding to 12 h 42 m G.M.T.
106 LABORATORY ASTRONOMY
The Ephemeris gives :
MOON'S K.A. DECL. DIFFS. FOB l m
At 12 h G.M.T 3 h 14 m 34 8 +20 25' + 2 s . 33 + 6". 2
Diff. for 42 m + 1 38 + 4 s x 42 x 42
At time of observation . . 3 16 12 + 20 29 97 .9 260.4
Observed values 3 14 54 + 20 24 I m 37 8 .9 + 4' 20"
The agreement here is satisfactory considering that the moon is
more than 45 from the star with which it is compared. Part of
the difference is due to parallax.
Observations of the Sun's Place. By the observation treated on
page 67, the sun's right ascension and declination at 5 h 36 m 26 s ,
Cambridge sidereal time, March 29, 1899, by comparison with a Ceti,
were found to be O h 33 m 19 s and + 3.6. To compare this with the
Ephemeris of the sun, we must first find the Greenwich mean time
corresponding to 5 h 36 m 26 8 , Cambridge sidereal time. Heretofore
we have had given either local apparent time or standard time of
observations, and the Greenwich mean time has been found by adding
the equation of time and longitude in one case or an integral number
of hours in the other. In this case we have given the local sidereal
time, to find the corresponding Greenwich mean time.
The first step is to find the Greenwich sidereal time by adding the
longitude west of Greenwich, after which G.M.T. isfoundas follows :
Gr. Sid. T. = 5 h 36 m 26 s + 4 h 44 m 30 s . = 10 h 20 m 56 8
March 29, Gr. Sid. T. of Gr. M. noon ..... 30 37.57
Hence the sidereal interval elapsed since Gr. M. noon is 9 50 18 .43
And, by Table II, the quantity to be subtracted from
this to give the equivalent mean interval is ... 1 36 .71
Hence the corresponding mean time interval is . . . 9 48 41 .72
This is the mean time interval since Greenwich mean noon, which
of course is the required G.M.T.
We may now determine the sun's place at 9 h 48 m , or 9 h .8, G.M.T.,
by means of page II of the Ephemeris, as follows :
SUN'S R.A. DECL. H. DIFFS.
At Gr. M. noon . . . ,0 h 31 m 33* + 3 24'.4 -f 9 s .l + 58"
Diff. for 9 h .8 .... + 1 29 + 9 .5 x 9.8 x 9 .8
At time of observation . 33 2 -f 3 34 .9 89 568
Observed values (p. 67) . 33 19 + 3 36 l m 29 s 9'. 5
THE NAUTICAL ALMANAC
107
Determination of the Equinox. The following data from the Alma-
nacs of 1899 and 1900 may be compared with the results of page 89 :
AT GK.
APP. NOON
SUN'S DECL.
DJFF.
DATE OF EQUINOX BY
INTERPOLATION
1899.
1900.
Sept. 22.0
23.0
Sept. 23.0
24.0
+ 018 / 8". 7
-0 5 13 .9
+ 26 .4
-0 22 57 .5
23'22".6
23 23 .9
23' 22".6
0'26".4 ,
= Sept. 23.01880
23' 23". 9
The longitude of the place of observation was 4 h 48 m 40 8 W.
4 h 48 m 40 s 4.811H
24 h
24
days = O d .20046.
Hence the local dates of the equinoxes were September 22.57574,
1899, and September 22.81834, 1900, and the length of the tropical
year was 365.24260 days, as compared with the observed values
September 22.714, 1899,
September 22.856, 1900.
365.14 days.
Observations of Star Places. The right ascensions and declinations
of the stars given on pages 86 and 89 may be compared with the
mean places given in the Nautical Almanac for 1899 and 1900, or,
better, with the apparent places given in Part II of the American
Ephemeris. From the latter we find for September 22, 1900 :
E.A.
77 Serpentis 18 h 16 m 11 8 .4
X Sagittarii 18 21 51 .9
Vega "... 18 33 35.5
Altair . 19 45 57.9
DECL.
- 2 55'. 3
- 25 28 .5
+ 38 41 .8
+ 8 36.6
which are in close agreement with the results of observation.
CHAPTER VIII
THE CELESTIAL GLOBE
WHEN a globe such as that described on page 63 has had a num-
ber of constellations plotted on it in their proper positions, and the
sun's path added, showing the positions occupied by the sun at dif-
ferent times of the year, it becomes a very useful apparatus for
many purposes.
If, for instance, it is so placed that its axis points to the pole,
and is turned about the axis until the place of the sun as marked
on the globe for a certain date is on the under side and in a vertical
plane through the center, the sphere will represent the heavens as
seen at midnight on the given date.
When the globe has been so adjusted, if a straight line is drawn
from the center to any star on the surface of the globe, the prolon-
gation of this line will lead to the real star at the point which it
occupies on the sphere of the heavens. Thus used, such a globe is
helpful to a beginner in identifying the constellations. Obviously
the plane of the sun's path on the globe, if extended to the heavens,
will mark out the ecliptic, and all the hour-circles and parallels of
declination will mark the corresponding circles in the sky.
If the globe is turned slowly about its axis so that a point on
the equator moves from east to west through 15 per hour, we have
a sort of working model of the moving sphere of the heavens on
which we may measure off arcs and angles and thus solve approxi-
mately many problems suggesting themselves to one beginning to
study the apparent motions of the heavens. Such an apparatus has
from very early times been an important aid to astronomers and
students of astronomy, and no aid is so useful in arriving easily at
correct ideas on the subject. Especially was it useful and appropri-
ate in those days when the mechanism of the heavens was believed
to correspond closely to that of the model and the globe was regarded
as being a fair representation of their actual construction, in fact,
108
THE CELESTIAL GLOBE 109
a representation of the eighth or outer sphere which carried the
fixed stars, turning about a material axis somehow fixed in the
" Primum Mobile." The planets moving inside, each in its crystal
sphere, were treated by projecting them each on to its proper place
on the outside sphere for any particular time to solve a given prob-
lem. For the beginner, who stands to a certain extent in the place
of the early astronomers, it is still most important in studying many
problems. Usually the diagrams by which we illustrate our state-
ments of astronomical problems are drawn as if the celestial sphere
were seen from the outside as we see the globe. This is because
it is impossible to represent on a plane any large part of a spherical
surface as seen from the inside.
As usually constructed for demonstration and the solution of
problems, the celestial globe is made by building up layers of strong
paper laid in glue upon a solid wooden sphere so as to cover it with a
light but stiff shell, which is then cut through along a great circle,
so that the core may be taken out. The two halves of the shell
are fastened together by gluing on a strip of thin, strong cloth,
and after passing an axis of stiff wire through the center, several
layers of a mixture of glue and whiting are applied to the surface,
each being smoothed before drying. The whole is then turned so
as to form a very light and accurate spherical shell. Upon the
surface are pasted gores of paper, on which the circles and principal
stars are printed in such a manner as to lie in their proper places
on the globe. The outlines of the constellations are shown on the
plates, and the conventional figures which have been ascribed to
them. A small circular piece centered on the pole completes the
map. The figures are colored by hand, and the whole is then cov-
ered with a hard, transparent varnish.
Both equinoctial and ecliptic are graduated to degrees, and the
hours of right ascension on the former are marked by Koman
numerals, ' The places of the sun are usually indicated on the
ecliptic at dates five days apart. Since the circuit of the sun is
completed in 365J days, while the length of the year is sometimes
365 and sometimes 366 days, an average position of the sun must be
chosen, which is done with sufficient accuracy by plotting its place
for the second year after leap year.
110 LABORATORY ASTRONOMY
The axis of the globe is supported by a stiff brass circle, so that
the center of the sphere lies exactly in the plane of one of its
faces, and this face is graduated into degrees, one semicircle near
the outer edge from at either pole to 90 at the equator, and the
other semicircle near the inner edge from at the equator to 90
at either pole. The inner graduation is used for measuring the
angular distance from the equator to any point on the globe, that is,
the declination of any point. The graduation on the outer edge
is used for placing the axis at the proper angle to the horizon in
rectifying the globe, as explained on page 111. This graduated
circle which supports the axis is called the " brass meridian." It
is mounted in two slots in a somewhat larger wooden circle called
the "horizon," in such a manner that it is perpendicular to the
latter and that its center lies in the plane of the upper surface of
the wooden circle.
The horizon is graduated on its inner edge, and each quadrant
has two sets of numbers, one of which reads from at the prime
vertical to 90 at the meridian, and the other from at the
meridian to 90 at the prime vertical. These numbers serve for
the direct reading of amplitude and bearing respectively, which are
easily translated into azimuth, remembering that W. is 90, N. 180,
and E. 270, if azimuth is measured from the south point toward
the west from to 360. The brass meridian may be turned in its
own plane, sliding easily in the slots so that the axis of the globe
shall make any desired angle with the horizon.
If the globe is accurately made and mounted, its center will coin-
cide with the common center of the graduated face of the brass
meridian and the upper surface of the horizon, whatever may
be the inclination of the axis. No irregularities should appear
in the small space between these circles and the surface of the
globe when the latter is whirled rapidly on its axis. Some idea
of the^correct placing of the circles on the globe may be obtained
by noting whether all points of the equator and parallels come
under the proper divisions of the brass meridian, whether all points
of the equator pass through the east and west points of the horizon
90 from the graduated face of the brass meridian, and whether
the points of the equator which lie in the east and west points of
THE CELESTIAL GLOBE HI
the horizon are twelve hours apart whatever the inclination of
the axis.
It is desirable to have a means of fixing a point on the globe by
some mark that may be afterward removed without injuring the sur-
face. Gummed paper should not be used : small pieces of unglazed
paper when well moistened will adhere long enough for ordinary
purposes.
A good mark may be made with water-color paint mixed with
glycerine so as to be very thick and applied with a rubber point or
soft pen point. Such a mark may easily be removed with a moist-
ened finger even after several weeks.
Ink suitable for fountain pens is usually safe if removed within
an hour or two.
TO RECTIFY THE GLOBE
In order that the globe shall represent the heavens at any partic-
ular place, the axis must be inclined to the horizon by an angle
equal to the latitude. This may be accomplished by rotating the
brass meridian in its plane and measuring the angle of elevation of
the pole by the outside graduation, which reads from at the pole
to 90 at the equator. This process is called " rectifying " the globe
for a given place.
Having been rectified for a given place, the globe may be rectified
for a given time by bringing it to such a position that a line drawn
from its center to any star is parallel to the line drawn from the
given place to the actual place of the star in the heavens at the
given time. For this purpose, the pole being elevated to the proper
inclination, that is, the latitude, the whole apparatus is turned on
its base until the brass meridian is in the meridian of the place,
and the globe is turned on the polar axis until some one point is
known to be in the proper position ; then all points of the globe
will be in their proper positions.
The point chosen for this purpose will vary with circumstances.
If the local sidereal time is given, it is only necessary to place the
globe so that the hour-angle of the vernal equinox equals the given
sidereal time. (See page 57.) This is easily done by the graduation
112 LABORATORY ASTRONOMY
of the equator on the globe. When the hour-angle of the vernal
equinox is l h , 2 h , 3 h , the reading of the equinoctial under the brass
meridian is l h , 2 h , 3 h , etc., and the globe is therefore rectified to a
given sidereal time by turning it about the polar axis until the given
sidereal time is brought to the graduated face of the brass meridian.
The vernal equinox will then be at the proper hour-angle and all
points on the globe will be properly related to the corresponding
points on the sky.
If the apparent time is given, the globe may be rectified by the
following process. Mark the place of the sun in the ecliptic for
the given day. Bring this point to the meridian, which rectifies
the globe for apparent noon ; then, to rectify it for the given ap-
parent time, it is necessary to turn the globe until the hour-angle
of the sun is equal to the given apparent time. This may be done
by using the graduations of the equator as follows. Rectify for
apparent noon and read the hours and minutes of the graduation
on the equinoctial which comes under the brass meridian (this is
the sidereal time of apparent noon). Add to this reading the given
apparent time, and the sum will be the hours and minutes of the
equatorial graduation that must be brought to the meridian to
place the sun at the proper hour-angle.
If local mean time is given, the apparent time may be obtained
by applying the correction for the equation of time for the given
date, and the globe may then be rectified for apparent time, as
described in the last paragraph.
If, as will generally be the case, standard time is given, this may
be reduced to local mean time by applying the correction for longi-
tude, and we may then proceed as before.
We may here remark that in rectifying the globe for solar time
we make use of the sun's place as marked on the ecliptic for the
given date ; and that this place may be inaccurate by as much as
half a degree is obvious from the following consideration. Suppose
the place of the sun on the globe to be exact for any one year on
February 28. It will be exact on March 1 or about 1 in error,
according as the year has not or has the date February 29. The
following table of the sun's longitude shows more clearly the
nature of the facts.
THE CELESTIAL GLOBE
113
YEAR
FEBRUARY 20
MARCH 2
SEPTEMBER 23
1901
331. 2
34P.2
179. 8
1902
330 .9
341 .0
179 .5
1903 . . -V . . .
330 7
340 7
179 3
1904
330 .5
341 .5
180 .0
Average
330 .8
341 .1
179 .7
The values nearly repeat themselves after four years.
It is obvious that by assuming an average value of the longitude
for February 20, March 2, and September 23, we should sometimes
be in error by about in. the sun's place, though never more, and
by some such compromise the places must be selected for the posi-
tion of the sun upon a globe for general use. The error that thus
arises may amount to 2 m in the determination of the sun's right
ascension from the globe.
An indispensable attachment for the celestial globe is a thin
flexible strip of brass graduated to degrees and so constructed that
it may be attached to the brass meridian at its highest point by a
pivot, about which it can be turned so as to be brought to coincide
with any vertical circle ; its graduated edge may then be brought
over any point 'on the globe and the azimuth of the point fixed by
noting the place where the arc meets the graduations on the horizon.
The altitude of the point may be directly read on the flexible arc,
which is graduated from at the horizon to 90 at the place where
it is fixed to the brass meridian. The graduations are continued
below the horizon from to 18 for the purpose of determining
the end of twilight (page 133). The flexible arc is usually called
the "altitude arc."
The globe thus equipped may be used for the approximate solu-
tion of all problems which arise from the diurnal motion, some of
which we will now discuss. These approximate solutions are not
only sufficient for many purposes, but always indicate the proper
statement of the problem for purposes of computation, and serve
to detect gross errors in the numerical results.
114
LABORATORY ASTRONOMY
PROBLEMS WHICH DO NOT REQUIRE RECTIFICATION
OF THE GLOBE
Many problems are independent of the position of the observer
on the earth's surface, and for their solution it is immaterial at
what angle the polar axis is inclined. By bringing the axis to the
plane of the horizon, any star may be brought to view above the
horizon, but unless it is convenient to stand so that one can look
down upon the globe from above, it is often better to take a sitting
position and place the polar axis nearly vertical. In following the
solutions of the examples below, the accompanying figures serve to
show whether the globe has been brought to the proper position.
Problem 1. To find the right ascension and declination of a star.
Kotate the globe until the. star is in the plane of the brass
meridian ; note the hours, minutes, and seconds of that graduation
of the equinoctial which falls under the brass meridian. This is the
right ascension of the star. This value we
may call the "meridian reading" of the
equator and in future abbreviate to R.A.M.
(right ascension of the meridian). The
declination of the star equals that degree
of the graduation of the meridian under
which the star lies.
Example 1. The star rj Ursse Majoris in
the end of the Dipper handle is brought to
the brass meridian (Fig. 42) and is found
to lie halfway between the divisions 49 and
50 north of the equator ; the declination is
therefore + 49.5. The meridian reading
is 13 h 44 m , which is the star's right ascen-
sion. (For reading the declination the graduations on the inner
edge of the brass meridian must be used.)
Problem 2. Given the right ascension and declination of a star,
to find the star.
Rotate the globe until the meridian reading (R.A.M.) is equal to
the given right ascension, and under the brass meridian at the
given declination will be found the star.
FIG. 42. R.A.M. 13" 44'
Decl.
THE CELESTIAL GLOBE
115
FIG. 43. K.A.M. 19^ 46> ;
Decl. + 8
Example 2. The right ascension of a certain star is 19 h 46 m and
its declination + 8jL. What is the star ?
The division on the equator marked 19 h 46 m is brought to the
brass meridian (Fig. 43), and halfway between the graduations 8
and 9 on the meridian is found Altair, which
is the star sought.
Problem 3. To find the angular distance
between two stars.
Place the flexible quadrant along the sur-
face of the globe so that its graduated edge
passes through both stars, and read the
graduation at the points where it touches
each star ; the difference of the readings is
the angular distance between the stars. The
graduated edge should lie along the great
circle; as this is not always easy to adjust,
it is well to repeat the measure with the
quadrant in different adjustments and take
the smallest value obtained. An alternative method free from this
source of error is to adjust the points of a pair of compasses so that
they may just span the distance between the two stars. The com-
passes may then be applied to the globe
with one leg at the vernal equinox (0); the
other leg being brought to the equinoctial
its reading will give the angular distance
between the stars. To guard against defects
in the globe, the second point may be
brought to the ecliptic, and the reading
should be the same as on the equinoctial ;
if the readings differ, the mean of the values
should be taken.
In the use of the compasses care must
be taken not to scratch the surface of the
FIG. 44. Length of Dipper 26 g^^
Example 3. The following measures were made to determine the
distance between allrsse Majoris and TjUrsse Majoris. With the
flexible quadrant applied to the globe (Fig. 44) so as to lie as nearly
116 LABORATORY ASTRONOMY
as possible along the great circle between the stars, the readings
were:
ij URS.E MAJOKIS a UKS^E MAJOBIS DISTANCE
0.0 26.0 26.0
0.0 26.1 26.1
20.0 46.1 26.1
40.0 66.1 26.1
Here no difficulty was found in laying the arc along the great
circle, as the distance is not great, and the value is taken to be
26.l. Adjusting the points of a pair of compasses to the stars
and then placing the compasses with one point at the vernal equi-
nox, the other point was found to reach to 25. 6 of right ascension
on the equinoctial and to 25.6 of longitude on the ecliptic, which
gives the distance between the stars as 25.6.
Problem 4. To find the sun's longitude, rig/it ascension, and
.declination at a given date.
If the sun's place at different dates is marked on the ecliptic,
its longitude may be read off directly on the graduations of the
ecliptic. In all old globes, however, and in many modern ones the
ecliptic is not thus marked, and the place of the sun must be
found by determining the longitude by a table such as that given
on page 173, which is nearly correct for the first half of the present
century. A substitute for this table is generally to be found in
the form of two contiguous concentric circles on the horizon circle,
one graduated into degrees of longitude and the other into months
and days, so that the line for a given date in the outer circle is
found opposite the corresponding degree of the sun's longitude in
the inner circle. Commonly also the divisions both of this circle
and of the ecliptic are divided into groups of 30, each correspond-
ing roughly to one month of time. The 30 of Aries reach from the
first of Aries on March 20 to the first of Taurus on April 20, and so
on in the order of the signs. Thus, opposite May 6 is the fifteenth
degree of Taurus, corresponding to longitude 45 in the usual way
of reckoning ; opposite January 1 is the tenth degree of Capricor-
nus, nine complete signs and 10, or longitude 280. In the table
on page 173 the equivalents of the degrees of longitude are given
in signs and degrees.
THE CELESTIAL GLOBE 117
By whatever method the sun's place in the ecliptic is fixed,
its right ascension and declination are found by the method of
Problem 1.
Example 4- What are the sun's right ascension and declination
on April 20 ?
The longitude is found by the table to be 29.5, and on bringing
this point of the ecliptic to the meridian (Fig. 45) it is found to be
in declination -f-11^- , while the reading of
the meridian is l h 50 m . The sun's right
ascension is therefore l h 50 m and its decli-
nation is 11- north.
PROBLEMS WHICH REQUIRE RECTI-
FICATION OF THE GLOBE FOR A
GIVEN TIME
Such are problems which require a deter-
mination of the angle between the meridian
and some one of the hour-circles of the Fl( >- 45. Sun's E.A. i* 53 m ;
globe. They are independent of the latitude
of the place of observation, but depend upon the position of the
heavenly bodies with respect to the meridian. The brass meridian
being taken as the meridian of the place of observation, the only
quantities involved are differences of hour-angle and of right
ascension, and it will be advisable here to collect the following rela-
tions, which have already been explained.
All time is measured by the continually increasing hour-angle of
some point of the celestial sphere.
Local sidereal time (Camb. Sid. T.) is the hour-angle of the vernal
equinox.
Local apparent (solar) time (Camb. App. T.) is the hour-angle
of the sun.
Local mean (solar) time (Camb. M. T.) is the hour-angle of the
mean sun.
For example, at 21 h 20 m , Camb. Sid. T., the hour-angle of the
vernal equinox at Cambridge is 21 h 20 m ; at 10 h 30 m , Chicago apparent
118 LABORATORY ASTRONOMY
time, the hour-angle of the sun at Chicago is 10 h 30 m ; at 5 h 10 m ,
New York mean time, the hour-angle of the mean sun at New York
is 5 h 10 m .
The hour-angle is in all cases measured westward from the
observer's meridian up to 24 h .
Greenwich mean time (G.M.T.) is the hour-angle of the mean
sun measured from the meridian of Greenwich. When we say
that a place is a certain number of hours and minutes of longitude
west of Greenwich, we mean that the rotation of the earth brings
the sun to the meridian of the place just so many hours and minutes
after its arrival at the meridian of Greenwich. At local noon, then,
its hour-angle, reckoned from the Greenwich meridian, is equal to
the difference of longitude between the two meridians. As the sun
thereafter moves westward equally from the two meridians, Green-
wich time is always greater than that of any place west of it by
exactly the difference of their longitudes.
Therefore, to find the G.M.T. corresponding to a given local mean
time, we add to the latter the longitude (west) from Greenwich.
Standard time is directly obtained from G.M.T. by subtracting 4,
5, 6, 7, 8 hours, respectively, for Colonial, Eastern, Central, Moun-
tain, and Pacific time. Thus, the "reduction for longitude," so
called, from Cambridge mean time is + 4 h 44 m .5 to G.M.T. and
-f- 4 h 44 m .5 5 h to Eastern standard time ;
or, by a single operation, 15 m .5 directly to
Eastern time. The " reduction for longi-
tude" for San Francisco is -f 8 h 9 m .7 to
Greenwich and + 8 h 9 m .7 - 8 h = + 9 m .7 to
Pacific time. Problems, therefore, which
involve standard time require a knowledge
of the observer's longitude.
Problem 5. To rectify the globe for a
given sidereal time.
Eotate the globe till the E.A.M. equals
the given sidereal time. This brings the
FIG. 46. Sid. T. 7" 50- vernal equinox to an hour-angle equal to
the given sidereal time, and all points of the sphere into their
proper relation to the meridian.
THE CELESTIAL GLOBE 119
Example 5. To rectify the globe for 7 h 50 m sidereal time, rotate
the globe until E.A.M. is 7 h 50 m (Fig. 46).
Problem 6. The globe being rectified for a given sidereal time,
to determine the hour angle of a body.
Note the E.A.M. when the globe is in the given position; then
bring the body to the meridian and read its right ascension. Sub-
tract the latter reading from the former and the result is the hour-
angle of the body.
Since the reading of the meridian is always the sidereal time
(page 59), this process exemplifies the equation H.A. = Sid. T.
E.A. It is of course understood . that if in adding two times
or hour-angles the result is greater than twenty-four hours, that
amount is to be subtracted ; thus, an hour-angle of 35 h 25 m 10 8
corresponds to the same position as an hour-angle of ll h 25 m 10 s .
Also, if it is required to subtract a larger froni a smaller hour-
angle, the latter should be increased by twenty-four hours before
performing the subtraction : thus, 6 h 41 m
- ll h 17 m = 30 h 41 m - ll h 17 m = 19 h 24 m .
Example 6. What is the hour-angle of
Sirius at (a) 7 h 50 ra , sidereal time, and at
(b) 4 h 20 m , sidereal time ?
(a) Rectifying the globe, as in Problem
5, to 7 h 50 m Sid. T., the E.A.M. = 7 h 50 m .
Bringing Sirius to the meridian (Fig. 47),
E.A.M. = 6 h 41 m = E.A. of Sirius, as in
Problem 1. Hence H.A. of Sirius at
7 h 50 m Sid. T. = 7 h 50 m - 6 h 41 m = l h 9 m
(Fig. 46).
(b) Eectifying to 4 h 20 m Sid. T., E.A.M. FlG " 47 ' KA ' of Siriu8 ' 6 " 41m
= 4 h 20 m , and, as before, H.A. = 4 h 20 m - 6 h 41 m = 28 h 20 m -
6 h 41 m = 21 h 39 m .
Problem 7. The globe being rectified for a given apparent time, to
determine the hour-angle of a body.
Bring the sun's place to the meridian and take the E.A.M. (this
is the sun's right ascension, Problem 4). Eotate the globe through
an hour-angle equal to the given apparent time, and the sun is
brought to the required hour-angle ; the E.A.M. thus becomes H.A.
120 LABORATORY ASTRONOMY
of the sun -f- R.A. of the sun, and the globe is properly rectified
when this reading of the equator is brought under the meridian.
Since H.A. + E.A. = Sid. T., the rule may be given as follows :
Determine the sun's right ascension by the globe (Problem 4).
Add the given apparent time. The sum is the sidereal time.
For this sidereal time rectify the globe by Problem 5, and find the
hour-angle by Problem 6.
Example 7. What is the hour-angle of Sirius at 10 P.M., apparent
time, February 13 ?
Sun's R. A. by globe 21 h 50 m
App. T 10 0_
Sid. T 7 50
R.A. of Sirius by globe ' 6 41
H.A. of Sirius 1 9 (Fig. 46)
Problem 8. The globe being rectified for a given mean time, to
determine the hour-angle of a body.
Apply the equation of time (with the proper sign) to the given
mean time to find the corresponding apparent time, and with this
value rectify as in Problem 7.
Example 8. What is the hour-angle of Sirius at 5 A.M., local mean
time, July 10 ?
Equation of time + 5 m (add to App. T.)
July 10, 5 A.M = July 9 d 17 h O m
Eq. of T. (subtract) 5_
App. T. . . *** 16 55
Sun's R.A. by globe (add) . . . 7 20*
Sid. T 15
R.A. of Sirius (Problem 1) (subtract) 6 41
H.A. of Sirius 17 34
Problem 9. The globe being rectified for a given standard time, to
determine the hour-angle of a body.
Apply the reduction for longitude to find the corresponding mean
time and rectify as in Problem 8.
* The sun's place is marked on the globe for noon of the indicated date. It
is therefore more accurate in this problem to make use of the sun's place for
July 10 and in general for the nearest noon, which is always that of the civil
date.
THE CELESTIAL GLOBE
121
Example 9. At Chicago (longitude -f 5 h 50 m ) what is the hour-
angle of Sirius at 6.30 P.M., Central standard time, October 30?
Red. for Long. Chicago T. to Central T. - 10 m
Eq. of T. - 16 m (subtract from App. T.)
Central standard time 6 h 30 m
Red. of Long, to Chicago M. T.
Chicago M. T
Eq. of T. (add to M. T.) . .
+10
6 40
+16
App. T 6 56
Sun's R.A. by globe (add) 14 23
Chicago Sid. T 21 19~
R.A. of Sirius (subtract) 6 41
H.A. of Sirius (by Problem 5) 14 38~
Reduction to the Equator. In the solution of Example 4, page
117, it was shown that when the sun's longitude is 29.5 its R.A.
is l h 50 m , or 27.5.
The quantity which must be added to the longitude of a point
on the ecliptic to find its E.A. (in this case 2) is called the
" reduction to the equator " and is used in finding the equation of
time as explained in Chapter X. Its value for any given point of
the ecliptic may be found by the globe as in Example 4.
Following are the results :
LONGITUDE
and 180
10 190
20 200
30 210
40 220
50
60
70
80
90
230
240
250
260
270
RED. TO
EQUATOR
0.0
-0.8
-1 .5
-2 .1
-2 .4
-2 .4
-2 .2
- 1 .6
-0 .9
0.0
LONGITUDE
RED. TO
EQUATOR
90 and
270
o.o
100
280
+ 0.9
110
290
+ 1.6
120
300
+ 2 .2
130
310
+ 2.4
140
320
+ 2 .4
150
330
+ 2.1
160
340
+ 1.5
170
350
+ 0.8
180
360
0.0
CHAPTER IX
EXAMPLES OF THE USE OF THE GLOBE
MOST of the problems with which we have to deal require that
the observer's exact place on the earth shall be known, that is,
his latitude as well as his longitude ; and in order that they may
be solved it is necessary that the globe should be rectified to the
latitude by inclining the axis to the horizon by an angle equal to
the latitude.
This chapter contains some typical examples and the methods
by which they are solved, with references to the problems of the
preceding chapter.* Attention should be paid to the arrangement
of the solutions, and all numerical results should be fully labeled
so that it may be seen how they are obtained and combined. In all
the problems, unless otherwise stated, the globe must be rectified to
the latitude of Cambridge, 42.4 N. The longitude may be assumed
411 4^m wes f; o f Greenwich.
Example 10. At what sidereal time do
the Pleiades rise at Cambridge ?
Rectify the globe by raising the north
pole to such an angle that the graduation
42.4 on the outside edge of the brass merid-
ian coincides with the surface of the hori-
zon. Rotate the globe about the polar axis
until the Pleiades are in the plane of the
eastern horizon (Fig. 48). The R.A.M.
equals the sidereal time sought, 20 h 12 m .
This result is independent of the longitude.
The Pleiades rise at any place in latitude
42.4 K at 20 h 12 m of local sidereal time.
FIG. 48. Rising of Pleiades :
20h 12 Camb. Sid. T.
* These solutions were obtained with a not very accurate globe nine inches in
diameter. Better results may be obtained with a larger globe in good condition.
122
EXAMPLES OF THE USE OF THE GLOBE
123
Example 11. At what apparent time do the Pleiades rise at Cam-
bridge on October 30 ?
Determine the sidereal time, as in the last example, 20 h 12 m .
The sun's right ascension is determined to be 14 h 17 m by bringing
it to the meridian (Fig. 49), as in Prob-
lem 4, and the relation App. T. = Sid. T.
Sun's E.A. gives
20 h 12 m - 14 h 17 m = 5 h 55 m Camb. App. T.
Example 12. At what Cambridge mean
time do the Pleiades rise October 30 ? Eq.
of T. = - 16 m (subtract from App. T.).
The apparent time being 5 h 55 m by the
last example, the mean time is 5 h 55 m
16 m = 5 h 39 m .
Example 13. At what Eastern standard
time do the Pleiades rise at Cambridge
~ . , nr . n
October 30?
The arranement of the work is as follows :
FIG. 49. October 30: Sun's
Camb. Sid. T. by globe (Example 10) ... ..... 20 h 12 m
Sun's R.A. by globe (Problem 4) ......... 14 17
Camb. App. T. (Example 11) ........ . '. . ~6 55~
Eq. of T. by table ....... - . , ...... - 16
Camb. M. T. (Example 12) 5 39
Red. to E. Std. T. - 16
E. Std. T. of rising of Pleiades 5 23
Example 14- At what standard time do
the Pleiades set at Cambridge March 1 ?
Bringing the Pleiades to the western
horizon, we have, as in Example 13 :
Camb. Sid. T. by globe (Fig. 50) ... 11* 15 m
Sun's R.A. (Problem 4) 22 50
Camb. App. T :...... 12 25
Eq. of T. by table . . . . . . . . . + 13
Camb. M. T . ... . 12 38
Red. to E. Std. T -16
E. Std. T. of setting of Pleiades March 1 12 22
Example 15. What is the standard time
FIG. 50. Pleiades setting : . . ~ , . , ,_- + ~ n
Sid. T. iib i 5 m of sunrise at Cambridge on May 15 ?
124
LABORATORY ASTRONOMY
Mark the place of the sun on the ecliptic for May 15 and bring
this point to the plane of the eastern horizon (Fig. 51).
The R.A.M. gives the Camb. Sid. T. by
globe 20 h 28 m
Sun's R.A. (Problem 4) by globe . . . 3 28
Camb. App. T T? 00~
Eq. of T. by table _ -4
Camb. M. T 16 56
Red. for Long 16
Std. T. of sunrise May 15 ~IQ 40~
Or, May 16, 4 h 40 m A.M. But see the note to Prob-
lem 8. Since the place of the sun was taken for
May 15, the solution gives the time of sunrise for
that civil date.
FIG. 51. Sunrise May 15:
Sid. T. 20h 28">
Example 16. What is the azimuth of the
sun at Cambridge at sunrise June 21 ?
The sun's place for June 21, being brought
to the horizon as in the preceding problem, was found to be on the
division 59 of the graduation which reads from zero at the north
point of the horizon to 90 at the east point (Fig. 52) ; its bearing,
therefore, is N. 59 E., and its azimuth
reckoned from the south point is 180 +
59, or 239.
The graduation on the inner edge of the
horizon has a second set of numbers begin-
ning with at the east and west points
and running to 90 at the north and south
points. By means of this amplitudes may
be directly measured. The amplitude of
the sun in this case was E. 31 N.
Example 17. At Cambridge, September 10,
in the afternoon, the sun's altitude is 20.
What is its azimuth?
For the solution of this problem the alti-
tude arc must be applied to the brass
meridian, attaching the clamp so that the 90 mark of the gradua-
tion is as exactly as possible under the graduation 42.4 on the
inner edge of the brass meridian ; this is at the highest point
FIG. 52. Sunrise June 21 :
Sun's Bearing N. 59 E. ;
Az. 239
EXAMPLES OF THE USE OF THE GLOBE
125
of the globe, corresponding to the zenith of the sphere in latitude
42.4 north.
The longitude of the sun for September 10 being found, by the
circles printed on the horizon for this purpose, to be 17.7 in Virgo,
or five signs and 17.7 = 167.7, this point was brought into the
southwest quadrant halfway from the south
to the west point and the altitude arc made
to pass through it ; the altitude was seen to
be approximately 40. The foot of the arc
was then moved about 20 toward the west
point and the sun's place brought to it ; the
altitude was now about 30. The foot of
the arc was moved again about 20 farther
toward the west point and the sun's place
brought to it, the
sun's altitude being
about 15. The arc
FIG. 53. September 15 : Sun's was now moved
Alt. 20 ; Az. 77.5 , . . .
back a few degrees
toward the south and by a few trials a
position found (Fig. 53) such that the sun's
place coincided exactly with the division
marking an altitude of 20 ; the zero of the
graduated edge of the arc was then halfway
between 77 and 78 of the graduation on
the inner edge of the horizon circle. The
bearing was then S. 77.5 W. and the azi-
muth 77.5.
Example 18. At Cambridge Altair is east
of the meridian at an altitude of 30. Find its azimuth and
hour-angle and the sidereal time. Bringing the place of Altair
to 30 on the flexible arc, as described in the last problem, the
bearing is found to be S. 73 E. Hence the azimuth is 287. With
the same adjustment the E.A.M. is 15 h 56 m , which is the sidereal
time. By bringing Altair to the meridian, its right ascension is
found to be 19 h 43 m , and, by Problem 5, H.A. = 15 h 56 m - 19 b 43 m
= 20 h 13 m .
FIG. 54. Alt. of Altair 30 :
H.A. 20h 13-" ; Az. 287;
Sid. T. 15* 56*
126
LABORATORY ASTRONOMY
Example 19. On September 10, at Cambridge, in the forenoon,
the sun's altitude is 20. What is the local mean time ?
The sun's longitude being 167.7, as in
Example 17, its place is brought to 20 on
the flexible arc in the southeast quadrant
(at a bearing S. 78 E., with which compare
the result of Problem 17) and the problem
solved as follows :
Sun's forenoon Alt. 20
R.A.M. 6 h 42 m
Sun's R. A. (Problem 4) 11 13
App. T 19 29
Eq. of T. by table -3
Camb. M. T 19 26
FIG. 55. Sun's Alt. 20; Or - , 7 26A.M.
K.A.M. G> 42>
It would appear that our result means
7.26 A.M. of the following day. But it is to be remembered that we
have used the sun's place for September 10 (the places are marked
for noon), and our solution then applies more nearly to the morning
of that date. Example 19 is perhaps the most important that we
have solved, since it illustrates the method
by which the longitude is determined at
sea. The sun's altitude is measured by a
sextant and its hour-angle computed. From
the apparent time thus obtained the local
mean time is found as above and compared
with G-.M.T. kept by a chronometer.
Example 20. On July 10, at Cambridge,
what is the sun's hour-angle when it is in
the prime vertical ? What is the local
mean time ?
In the summer half of the year the sun
is in the prime vertical once in the fore-
noon and once in the afternoon, so that
there will be two solutions of the problem.
The place of the sun July 10 is found by the table to be in
longitude 107.7. The altitude arc being adjusted with its foot
FIG. 5G. Sun in Prime Verti-
cal : July 10, forenoon ;
R.A.M. 3h 3">
EXAMPLES OF THE USE OF THE GLOBE
127
at the east point of the horizon, the sun's place is brought to the
graduated edge of the are and R.A.M. noted. The altitude arc
being brought in the same way to coincide with the west quadrant
of the prime vertical, the sun's place is brought again to the gradu-
ated edge and R.A.M. noted. Then the sun's right ascension is
determined, and the results may be recorded and the computation
made in the following form :
Sun in prime vertical A.M.
R.A.M 3 h 3 m
Sun's R.A. by globe . '. .' . . . '. . . 7 20
App. T > . . . . . 19 43
Eq. of T. by table + 5
Local M. T. .
19 48
Ll h 36 m
7 20
4 16
+ 5
4 21
Or
7 48A.M., 4 21p.M.
Example 21. At Cambridge, at O h sidereal time, what bright
stars are seen near the meridian ? What are their declinations ?
Eectify the globe for latitude + 42.4. Eotate the globe until
the R.A.M. is O h , and the following stars
will be found near the meridian : y Pegasi,
Decl. + 14.0 ; a Andromedse, Decl. + 27.5;
ft Cassiopeise, Decl. + 58; Polaris, of course,
but too near the pole to be seen on the globe ;
y Ursse Majoris, Decl. 54 ; 8 Ursse Majoris,
Decl. 58. The two latter are below the
pole, and to determine their declinations
the globe must be rotated 180 to bring them
under the inner graduations of the meridian.
Notice that the four first stars lie along
the same hour-circle, which is 'the equinoc-
tial colure, in R.A. O h , and that this circle
is divided roughly by them into multiples
of 15, thus : Polaris to (3 Cassiopeise, 30 ;
ft Cassiopeise to a Andromedse, 30; a Andromedse to y Pegasi, 15.
By continuing the line of stars about 15 we arrive at Decl. = 0,
R.A. = 0, that is at the vernal equinox, which though marked by
no conspicuous star is easily fixed by this alignment.
FIG. 57. Stars on Meridian
at Cambridge at V* Side-
real Time
128
LABORATORY ASTRONOMY
Example 22. What is the standard time corresponding to O h of
sidereal time at Cambridge October 10 ?
The sidereal time being given, this problem is similar to Exam-
ples 13, 14, and 15, and illustrates the general process of passing
from sidereal to mean or standard time by means of the globe, thus :
Sid. T .............. ..... O h O m
Sun's R. A. by globe ...... - ' ....... 13 _ 4
App. T ................... 10 56
Eq. of T ............... ... - 13
Camb. M. T ................. 10 43
Red. for Long, to Std. T ...... *. . . .--.'. . - 16
Eastern standard time 10 27
Example 23. Find the altitude and azimuth of Arcturus at
8 P.M., standard time, at Cambridge, September 10.
This problem requires the globe to be
rectified for both latitude and time. The
latter adjustment is made as follows :
Std. T 8 h O m
Red. for Long + 16
Camb. M. T 8 16
Eq. of T. by table (add to M.T.) . . . +3
App. T 8 19
R.A. Sun by globe 11 15
Camb. Sid. T 19 34
Rectify for Cambridge, Lat. + 42.4.
Rotate the globe till the E.A.M. is 19 h 34 m .
FIG. 58. Arcturus: Septem-
ber 10, 8 P.M., E. std. T. ; Apply the altitude quadrant so as to pass
Ait. 20 ; AZ. 98 through Arcturus, and we find its altitude
19.5, and its bearing K 80.5 W.; hence its azimuth is 99.5.
Example 24. What constellation is rising in the east at 9 P.M.,
Eastern standard time, at Cambridge, November 10?
As in the preceding problem :
Std. T 9 h O m
Red. for Long, to Camb. M. T. . . . , + 16
Camb. M. T 9 16
Eq. of T. by table (subtract from App. T.) + 15
Camb. App. T. 9 31
R.A. of Sun by globe 16 3__
Sid. T. 34
EXAMPLES OF THE USE OF THE GLOBE
129
To rectify for time rotate the globe till the E.A.M. is O h 34 m .
It will be found that the constellation of Orion has just risen
above the eastern point of the horizon.
Compare the form of this solution with that
of Example 13, which is the inverse of this,
the rising of a star being given and the
standard time sought.
PROBLEMS INVOLVING THE USE OF
THE NAUTICAL ALMANAC
Example 25. At Cambridge, November 30,
1904, at 5 h 15 m P.M., standard time, a bright
star is seen due southwest about 10 above
,, , . , T ,. , . . ., , . FIG. 59. Orion rising: Cam-
the horizon. .No other stars being visible in bridge, November 10, 9 P.M.,
the twilight, it is desired to identify the star.
E. Std. T .
Red. for Long. .
Camb. M.T
Eq. of T. (subtract from A pp. T.) ....
App. T .- .
R.A. of Sun
Camb. Sid. T. .
Std. T.
5 h I 5
+ 16
5 31
+ 11
5
16
42
20
22 8
Rectifying for
Cambridge, Lat.-f-
42.4, and for 22 h
8 m Sid. T., it is
found, by means
of the altitude
arc (Fig. 60), that
there is no star
upon the globe at
the given altitude
and azimuth, the
nearest star being
FIG. 60. Star 10 above South- cr Centauri, which FIG. 61. Star brought to Merid-
re;^^: >uld *t t>e visi-
22" 7 ble at that altitude
ian : R.A 18> 56 ; Decl. -
130 LABORATORY ASTRONOMY
in twilight. The exact point being marked is brought to the
meridian and found to be in R.A. 18 h 56 m and Decl. 23^ (Fig.
61). The fact that its position is very near the ecliptic suggests
that it may be a planet, and on consulting the Almanac it is
found that on November 30 the right ascension of Venus is 19 h
4 m and its declination 24.7, or within about 2 of the observed
place.
Example 26. Which of the planets that are visible to the naked
eye are above the horizon at Cambridge at 8 P.M., standard time,
October 1, 1904 ?
From the Nautical Almanac are taken the following data for the
given date :
R.A. DECL.
Mercury ............ ll h 25 m + 5.l
Venus ............. 13 55 - 11 .4
Mars ............ . 10 10 + 12 .7
Jupiter ............ . 1 44 + 9 .1
Saturn ........... . . 21 10 - 17 .0
Marking these places upon the globe and
rectifying for the given place and time, it
is at once seen that the first three are below
the western horizon, while Jupiter is 20
above the east point of the horizon and
Saturn approaching the meridian at an
altitude of about 30.
Where only an approximate result is
desired, it will often be sufficient to neglect
the corrections for longitude and equation
of time, the sum of which at Cambridge
never amounts to much more than half an
FIG. 62. pjaneta, October i, hour! Thig of CQurse assumeg standard
time to equal apparent time. Thus, in this
problem we may bring the sun to the meridian and, noting E.A.M.
= 12 h 30 m and adding 8 h , we have 20 h 30 m ( 30 m ) as the E.A.M.
corresponding to 8 h apparent time. The general terms in which
the answer is given above will apply equally well, and some
time is saved where only the general aspect of the heavens is
required.
EXAMPLES OF THE USE OF THE GLOBE 131
Example 27. At what standard time does Jupiter set at Cam-
bridge December 25, 1904?
By the tables in the Nautical Almanac, we find that on the
given date the right ascension of Jupiter is l h 17 m and its declina-
tion -f 6.8. Marking this place on the globe and bringing it to
the western horizon, the E.A.M. is 7 h 38 m , which is the sidereal
time. Converting to standard time :
Sid. T 7 h 38 m
Sun's R.A. by globe 18 12
App. T 13 26
Eq. of T . .
Camb. M. T 13 26
Red. for Long - 16
Std. T 13 10
Or 1 10 A.M.
Example 28. At what time does the moon
rise at Cambridge December 25, 1904 ?
If the moon's position were known directly
from the Nautical Almanac, the solution of p IGi 63. j up iter setting :
this problem would be similar to the last: R - A - M - 7h 38ffi J E - std -
T. 13 h 10 m
but the moon's right ascension and declina-
tion are changing so rapidly that we must reach the result by approx-
imation. We may first assume the moon's place at rising to be
the same as at standard noon, December 25 (or 5 h , G.M.T.), and at
that time the Almanac gives the moon's right ascension 8 h 54 m ,
Decl. H- 14.9. Marking this place on the globe and bringing it
to the eastern horizon, we find E.A.M. = l h 56 m , and continue the
computation as in the second column of the table below. (See
Example 15.)
G.M.T. 5 h 12 h 28 m 12 h 45
Moon's Place 8 h 54, + 14. 9 9 h ll m , + 13.9 9 h 13 m , + 13.9
R.A.M. . . . l h 56 m 2 h 13 2 h 18 m
R.A. of Sun. . 18 12 18 12 18 12
App. T. ... 7 44 8
Eq. of T. . . . 0_
Camb. M. T. . 7 44
Red. for Long. . 16
E. Std. T. . 7 28 7 45 7 50
132
LABORATORY ASTRONOMY
FIG. 64. Moonrise at Cam-
bridge December 25, 1904 :
K.A.M. 2* 18
This gives as the approximate time of moonrise 7 h 28 m , E. Std. T.,
or 12 h 28 m , G.M.T., and finding the moon's place for this time,
R.A. 9 h ll m , Decl. + 13.9, we better our result by the computation
shown in the third column, which gives
7 h 45 m , E. Std. T., or 12 h 45 m , G.M.T. With
this value we find the moon's place 9 h 13 m ,
f 13.9, and compute as in the last column,
finding E. Std. T. = 7 h 50 m .
As this is within ten minutes of the time
for which the data were assumed, and since
in ten minutes the moon's right ascension,
as shown by the difference column, changes
by 24 s , a quantity too small to be surely
measured on an ordinary 10-inch globe,
we may regard the last solution as suffi-
ciently accurate.
It would appear that the two last results
should be in closer agreement, since the difference in the assumed
times is only seventeen minutes ; the two first measures, however,
were not made with care, as only approxi-
mate values were sought.
It is obviously an advantage to estimate
the approximate time of moonrise as closely
as possible before beginning the solution :
this may be done by noting the age of the
moon (page IV of the month) and remem-
bering that the moon rises and sets about
48 m , or O h .8, later each night than the night
before, and that at new moon sun and moon
rise and set together. Assuming that the
sun rises at 6 A.M. and sets at 6 P.M., stand-
ard time, we shall find an approximate value
of the standard time of moonrise or moonset
by adding to these times a number of hours
equal to eight-tenths of the moon's age in days. Thus, in the pre-
ceding problem, the moon's age being eighteen days on December
25, we add 0.8 x 18 h = 14 h .4 to 6 A.M. to find the time of moonrise ;
FIG. 65. Moonset at Cam-
bridge December 18, 1904 :
R.A.M. 9* 41"
EXAMPLES OF THE USE OF THE GLOBE
133
this gives 8 h .4 P.M. as the approximate time, which is within an hour
of the final result.
Example 29. Find the time at which the moon sets at Cam-
bridge December 18, 1904.
The moon's age is found by the Ephemeris to be eleven days ;
hence we add 9 h to 6 h P.M., and have as the approximate time of
moonset 15 h , corresponding to 20 h , G.M.T. We may record the
successive approximations as follows :
Assumed G.M.T.
Moon's R.A. and Decl.
FIRST
APPROXIMATION
20 h
SECOND
APPROXIMATION
2 h
R.A.M. at moonset
Sun's R.A. . .
App. T
Eq. of T. . . .
Red. for Long.
Std. T.
17
39m
46
15 53
- 4
-16
15 33
20 h 33 m
Qh 41m
17 46
15 55
-20
15 35
A single recomputation will always be sufficient if the moon's
place is first determined by computing from its age.
MISCELLANEOUS EXAMPLES
Example 30. Find the duration of twilight
at Cambridge March 1.
Evening twilight ends when the sun has
sunk so far below the horizon that his direct
rays can no longer fall upon and be reflected
by any particles in that portion of the atmos-
phere which lies above the plane of the hori-
zon. This is usually assumed to be the case
when the sun is 18 below the horizon.
Bringing the sun's place for March 1 to the
horizon, and then, by means of the extension of the altitude arc, to
point 18 below the horizon (Fig. 66), we have the following values
R.A.M. at sunset . . 4 h 20 m
R.A.M. at end of twilight 6 0_
Difference . 1 40
FIG. 66. End of Twilight at
Cambridge March 1
134
LABORATORY ASTRONOMY
which equals the change in the sun's hour-angle, or the time elapsed
between sunset and the end of twilight.
Example 31. At what hour, apparent time, does morning twi-
light begin at Cambridge June 21?
June 21. Sim's place 18 below E. horizon, R. A. M 20 h 8 m
Sun's R.A. by globe . . . 6
App. T 14 . 8
Or 28 A.M.
Example 32. At what point of the horizon does the first glim-
mer of dawn appear in latitude 42.4 on June 21?
Bringing the sun's place by trial to the altitude arc at a point
18 below the horizon (Fig. 67), the reading on the horizon at the
graduated edge of the altitude arc is E. 57
N. = Az. 213 ; and as this is the nearest
point of the horizon to the sun when it is
18 below the horizon, it is at this point or
a little to the south that the first light will
appear.
Example S3. How many hours can the
sun shine into north windows June 21 in
latitude 41?
By the method of Example 15, it is found
that the apparent times of sunrise and sun-
set on June 21 are 4 h 30 m A.M. and 7 h 30 m
FIG. 67. Dawn at Cambridge P.M., and by the method of Example 20,
june2i,at2fr8A.M.: Sun's that the sun is in the prime vertical at 7 h
Az 213
56 m A.M. and 4 h 4 m P.M. Hence from 4 h 30 m
to 7 h 56 m A.M. and from 4 h 4 m to 7 h 30 m P.M., a total of 6 h 52 m , the
sun shines on the north face of an east and west wall. The length
of the day is fifteen hours.
Example 84. August 20, in latitude 42, longitude 4 h 48 m , at
ten minutes past 10 A.M., Eastern standard time, the sun begins to
shine upon the front wall of a building. How does the building face ?
Since at the given time the sun is in the same vertical plane
with the front wall of the building, the problem requires us to
determine the direction of this plane by finding the sun's azimuth,
which may be done as follows :
EXAMPLES OF THE USE OF THE GLOBE 135
Rectifying for latitude 42^-, we have :
Std. T. 10 h 10 m A.M = 22 h 10 m
Red. for Long, (from E. Std. T.) + 12
Local M.T 22 22
Subtract Eq. of T. (additive to App. T.) . . . . ' . . - 3
App. T 22 19
Sun's R.A 10 1_
Sid. T 8 20
Eectifying for this time and bringing the altitude arc to the
sun's place for August 20, we find the sun's azimuth to be 315.
Hence the front wall is in a line from southeast to northwest, and
the building fronts southwest.
Example 35. What is the greatest north-
ern latitude in which all of the four bright
stars of the Southern Cross are visible?
What must be the time of year ?
Rectifying the globe for the equator, the
Southern Cross (about R.A. 12 h , Decl. - 60)
is brought to the meridian and the brass
meridian is moved in its own plane until
the lowest star is brought to the horizon at
its south point. The elevation of the pole
above the north horizon is then read on the
brass meridian and found to be 28, which FIG. 68. August 20: std.T.io*
is the required latitude. The star being still 10m ; R ; A - M - 8h ""J Sun ' 8
Az. 315
in the same position, the altitude arc is then
used to mark the points of the ecliptic which are 18 below the
horizon. These are found to be at points occupied by the sun
January 2 and May 25, and between these dates, therefore, the
whole cross may be above the horizon in latitude 28 in the full
darkness of night, the sun being below the twilight limit.
Example 36. What is the latest date at which we can see Sirius
in the evening twilight in latitude 42?
Sirius is visible when the sun is about 10 below the horizon, and
cannot be seen later than the day on which he sets at the instant
that the sun is 10 below the horizon.
Rectifying for 42 and bringing Sirius to the western horizon, we
find that the point of the ecliptic which is 10 below the horizon is
136
LABORATORY ASTRONOMY
the place occupied by the sun on May 15, which is, therefore, the
required date.
Example 37. Between what dates is the sun visible at midnight
at the North Cape, in latitude 70 north ?
Eectifying the globe for 70 north and rotating the globe slowly,
it is found that points on the ecliptic in longitudes 58 and 122 can
be brought exactly to the north point of the horizon ; any point
between these may be brought to the meridian below the pole and
above the horizon. The dates at which the sun occupies these posi-
tions are May 19 and July 25, and between these dates the sun will
always come to the meridian at midnight above the horizon.
Example 38. Illustrate the " harvest moon " by finding the time
of moonrise at Edinburgh, latitude 56, on successive dates about
the time of full moon, September 24, 1904.
As only approximate results are desired, we may take from the
Ephemeris the moon's place for 6 h P.M., G.M.T., and solve as follows:
1904
R.A.
DECL.
K.A.M. AT
MOONRISE
SUN'S ll.A.
APPARENT
TIME
September 22
22h 36 m
-8
17 h 22m
12 h O m
5 h 22 m
23
23 22
-4
17 42
12 4
5 38
24
7
- 1
18 9
12 7
2
25
52
+ 3
18 28
12 10
6 18
26
1 38
+ 7
18 51
12 14
6 37
And it appears that the moon rises about twenty minutes latter
each night than it did on the previous night.
Example 39. Find the time of moonrise at Edinburgh on succes-
sive nights at full moon, March 31, 1904.
We have, as in Example 38, the moon's place at 6 h P.M., G-.M.T. :
1904
R.A.
DECL.
R.A.M. AT
MOONRISE
SUN'S E.A.
APPARENT
TIME
March 30 . .
11* 56 m
+ 1 Q
5h 5 7 m
0^ 38
5 h IQm
31 . .
12 52
-4
7 20
41
6 39
April 1 . . .
13 49
-8
8 42
44
7 58
EXAMPLES OF THE USE OF THE GLOBE 137
Therefore the full moon at the time of the vernal equinox rises
about one hour and twenty minutes later each night. (Notice and
explain the difference in the accuracy attained in these two
examples.)
Example 40- Find the rate at which 8 Orionis is changing its
azimuth at rising and setting in latitude 42.
Rectifying for 42 and bringing 8 Orionis to the eastern horizon,
we find R.A.M. = 23 h 23 m ; Az. = 271. Increasing the hour-angle
half an hour by making R.A.M. = 23 h 53 m , we find, by the alti-
tude arc, Az. = 276. Bringing the star to the western horizon, we
have K.A.M. = ll h 24 m ; Az. = 89^. Decreasing the hour-angle
by making E.A.M. = 10 b 54 m , we find Az. = 84 half an hour
before setting. In both cases the diurnal rotation causes the
azimuth to increase at the rate of 5 in half an hour.
By solving the same problem for stars in various parts of the
heavens, as, for instance, Vega, y Pegasi, Airfares, and a Gruis, it
appears that stars of whatever declinations, when near the horizon,
are increasing their azimuths by about 10 per hour in latitude 42.
(This is the rate at which the plane of the pendulum appears to
revolve in Foucault's experiment.)
Example 41. To mark the hour-lines on a horizontal sundial for
use in latitude 42.
The gnomon of an ordinary sundial (Fig. 69) is directed toward the
pole, and its shadow at apparent noon falls upon the horizontal dial
on the line of XII hours, which, when
properly adjusted, lies in the direc-
tion of the meridian. The shadow
at that time is in a line drawn through
the foot of the gnomon toward azi-
muth 180. It always passes through
the intersection of the gnomon with
the dial and, continually shifting F- 69 - Horizontal sundial,
/ Latitude 42
toward the east, at any instant lies
in the plane containing the sun and the gnomon. This plane cuts
the celestial sphere in the sun's hour-circle. The shadow, therefore,
is a line which passes through the foot of the gnomon and whose
azimuth is that of the intersection of the sun's hour-circle with
138
LABORATORY ASTRONOMY
the plane of the horizon. For a given hour-angle the position of
this line will be the same whatever the position of the sun upon
its circle, and is therefore the same for a given apparent time
whatever the time of year.
We may find the azimuth of the intersection of a given hour-
circle with the horizon by means of the globe as follows. Rectify-
ing the globe for 42, the vernal equinox is brought to the meridian,
so that the equinoctial colure cuts the horizon at azimuth 180.
In this position R.A.M. is O h , and the azimuth of the shadow is
180. Increasing the hour-angle of the colure by successive incre-
ments of 15, we have the following values for the azimuths of
the hour-lines :
FOB THE P.M. HOURS:
I
II
III
IV
V
VI
VII
R.A.M.
1*
2
3
4
5
6
7
)URS:
AND SIMILARLY
FOR THE A.M. HOURS :
Azimuth of
Shadow
R.A.M.
Azimuth of
Shadow
190
XI
23
170
201
X
22
159
214
IX
21
146
230
VIII
20
130
249
VII
19
111
270
VI
18
90
291
V
17
69
If the hour-circles are shown for each 15 as on most modern
globes, it is sufficient to bring one hour-circle to the meridian and
note the points where the other circles cut the horizontal plane ;
Fig. 57 shows the globe rectified to 42 and O h Sid. T., and therefore
in position for reading the azimuths of the successive hour-lines
directly on the horizon.
Example 42. To mark the hour-lines of a vertical sundial for use
in latitude 42 N., the bearing of the plane being W. 24 S.
Here the shadow of the gnomon falls upon a vertical plane,
and the line for noon is a vertical line through the intersection
of the gnomon with the plane.
At any given hour after noon the shadow falls below the gnomon
and to the east of the XII line (Fig. 70), since it marks the inter-
section of the plane of the dial by the sun's hour-circle. It makes
an angle with the XII line which may be defined as the " nadir
EXAMPLES OF THE USE OF THE GLOBE
139
distance " of the line of intersection of the two planes, and this is
equal to the zenith distance of that part of the same line which
lies above the gnomon.
This problem therefore requires us to find
the zenith distance of the intersection of the
sun's hour-circle with the vertical plane for a
given hour-angle of the sun, and may be solved
with the globe as follows :
Rectify the globe for latitude 42, and adjust
the altitude arc to the zenith with its foot at
azimuth 66 on the horizon ; its plane then
corresponds to that of the dial.
Bringing the vernal equinox to the meridian, R.A.M. = O h , the
equinoctial colure intersects the altitude arc at zenith distance 0.
Increasing the hour-angle of the colure, as in Example 41, we have
successively
FIG. 70. Vertical Dial,
Latitude 42
HOUK-LlNE
I
II
III
IV
V
R.A.M.
lh
2
3
4
5
ZENITH DISTANCE
OF INTERSECTION
13
30
49
70
90
which gives the angles of the afternoon lines from the noon line.
Setting the arc at azimuth 246, we find in the same way
HOUR-LINE
XI
X
IX
VIII
VII
VI
V
R.A.M.
23*
22
21
20
19
18
17
ZENITH DISTANCE
OF INTERSECTION
11
22
33
44
56
70
90
which gives the morning lines. The A.M. and P.M. divisions will
not be symmetrical about the XII line unless the vertical plane
faces due south.
Example 4&- Find the path of the shadow of a pin head on a
horizontal plane at Cambridge March 21, from 8 A.M., apparent
time, to 5 P.M., apparent time.
140
LABORATORY ASTRONOMY
Rectifying the globe for latitude 42, bringing the sun's place to
hour-angles which correspond to the successive hours from 8 A.M.
to 5 P.M., and measuring its altitude and azimuth in each position
by the altitude arc, we have the following results :
APP. TIME
RA.M.
ALTITUDE
AZIMUTH
DISTANCE
8 h A.M.
20h
22
291
12.5cm.
9
21
32
303
8.1
10
22
40
318
6.0
11
23
46
337
4.9
Noon
48
4.5
l h P.M.
1
46
22
4.9
2
2
40
42
6.0
3
3
32
56
8.0
4
4
22
C8
12.6
5
5
11
80
26.5
To construct the curve we must know the length of the pin ;
assuming this to be 5 cm. long, a point on the paper is chosen to
represent the point vertically under the pin head, and through it
is drawn a line to represent the meridian, and other lines are
FIG. 71. Azimuth, of Shadow
drawn at the azimuths differing by 180 from those given in the
above table. (See Fig. 71.) The shadow path will cross these
lines at the corresponding hours.
To find the distance of any point of the shadow path from the
foot of the pin, we may reverse the process explained on page 5.
Drawing a line from C, the center of the base in Fig. 6, through
the divisions of the protractor corresponding to any one of the alti-
tudes of the above table and measuring the line A'B', we have the
distance in centimeters from the foot of the pin to the point where the
shadow falls on the corresponding azimuth line. The last column
of the above table gives the distances measured in this manner.
EXAMPLES OF THE USE OF THE GLOBE
Fig. 72 shows the shadow path as thus constructed, and it is
evidently a straight line. This will always be the case on the day
of the equinox, when the sun is in the equator and its diurnal path
is consequently a great circle.
FIG. 72. Path of Shadow
THE HOUR-INDEX
The globe is usually provided with an arrangement by means of
which approximate solutions may be made of problems involving
time without the use of the graduations of the equinoctial.
This process is so simple that its explanation might well have
preceded that of the method of finding the sun's hour-angle given
on page 112 and used in Problem 7. It is, however, very inaccurate,
and should only be chosen where an error of several minutes is
unimportant.
The most convenient form given to the attachment is that of a
small pointer fixed to the brass meridian in such a manner that it
revolves about the same center as the polar axis, but with sufficient
friction to keep it fixed in any position where it may be placed.
This pointer, or " hour-index," lies close to the surface of the
globe, which revolves freely under it. The end of the index lies
over a small circle on the globe, about 15 from the pole ; and this
circle is graduated into hours and quarters in two groups of 12
hours each, numbered in the same direction as the graduations of
the equinoctial.
The following example illustrates the use of the hour-index,
which in this case gives sufficiently good results with less trouble
than the method already explained.
Example 44. Find the apparent times, October 1, 1904, of rising
and setting of the planets whose places are given on page 130.
142 LABORATORY ASTRONOMY
Mark the places of the planets and of the sun ; bring the latter to
the meridian and set the hour-index to read XII noon. Eotate the
globe through any angle, and the reading of the index will equal
the hour-angle of the sun in its new position, and thus will give
directly the corresponding apparent time.
We may, therefore, rapidly determine the apparent time of rising
and setting of all the planets by bringing each in turn to the eastern
and western horizon and noting the reading of the hour-index.
The hour-index may be adjusted to give local mean time or
standard time directly by making it read the local mean time or
standard time of apparent noon when the sun is brought to the
meridian. Thus, for October 1, at Cambridge, longitude 4 h 44 m :
App. T. of App. noon 12 h O m
Eq. of T -10
Camb. M. T. of App. noon 11 50
Red. for Long 16
Std. T. of App. noon 11 34
And the index should be set to read ll h 34 m when the sun is on
the meridian, in order to give Eastern standard time.
CHAPTER X
THE MOTIONS OF THE PLANETS
IT has been the aim of the preceding chapters to show how the
diurnal motion and the motion of the sun and moon among the
stars may be studied in such a manner that the student shall acquire
and fix his knowledge in large part by his own observations.
There remains to be considered the motion of the planets, which
cannot be studied in the same way because they move so slowly
that a long time would be required to obtain a sumcient number
of observations on which to base a satisfactory theory. It is of
course desirable, however, during the continuance of the observa-
tions on the moon and stars to include the planets in order to
establish a few fundamental facts, such as that they never appear
far from the ecliptic and that in general they move from west to
east like the sun and moon, but that when opposite the sun, so
that they come to the meridian at midnight, they are moving from
east to west among the stars. Their places in the heavens should
be occasionally observed, for comparison with the places derived
from the theory which forms the subject of the present chapter.
In treating of this theory we shall first assemble the few prin-
ciples which have been shown to account for the observed motions,
and shall then show how these principles may be applied to the
graphical solution of problems involving the determination of the
place in the heavens of a planet as seen from the earth at any
given time. These problems serve to illustrate and explain the
phenomena resulting from the planetary motions, as the globe
problems of the preceding chapter serve for those resulting from
the diurnal rotation of the earth.
Results of the Law of Gravitation. In consequence of the attrac-
tion of the sun, each planet describes an ellipse, having the sun in
one focus ; this is "Kepler's first law." The mutual attractions of
the planets produce " perturbations " of their motion, but in no case
143
144
LABORATORY ASTRONOMY
are these perturbations sufficient to alter the place of the planet
by so much as one degree from its place as determined by the sun's
attraction. Jupiter may be displaced about 0.3 and Saturn nearly
0.8 ; but with this exception no displacement of a planet amounts
to J. The asteroids are subject to much greater perturbations.
The orbit of each planet is in a plane which remains nearly fixed,
and the planes of all the orbits are so nearly coincident with the
ecliptic that the projections of their paths on the ecliptic are no
more distorted than the roads of a moderately rugged country are
distorted in their representations on an ordinary plane map. This
fact makes it as easy to determine their motions by an accurate
map of their orbits on the plane of the ecliptic as to follow the
motion of a traveler over a well-charted country, when his point
of departure and rate of travel are known.
PROPERTIES OF THE ELLIPSE
An ellipse may be drawn by putting two pins upright in a board,
as in Fig. 73, laying a knotted loop of thread on the board so as to
include both pins, and then putting the point of a well-sharpened
pencil on the surface inside the loop. Let the pencil be moved out
FIG. 73. Drawing an Ellipse
so as to form the loop into a triangle, and then drawn along the
surface so as to pass successively through all the points which it
can reach without allowing the thread to become slack. The curve
which it follows will be an ellipse whose shape and size will depend
only on the distance between the pins and the size of the loop.
The form of the curve is shown in Fig. 74.
F 1 and F 2 are the foci, AB the major axis, and C, which bisects
both F^FZ and AB, is the center of the ellipse. PF-^ is the radius
THE MOTIONS OF THE PLANETS 145
vector from any point P to F lf and PF 2 the radius vector to F 2 .
They are usually represented by r and r 2 . r + r z is a constant for
all points of the ellipse, being always equal to the length of the
thread minus FiF z . For
the point A
and since from the sym-
metry of the curve 2?
AF l = BF 2 ,
A C is usually represented
by a, and CF-, or CF 2 by c.
J FIG. 74. Fundamental Points and Lines
Since 2 c equals the
distance between the foci, and 2 a + 2 c the length of the thread,
the shape and size of the ellipse are completely fixed by the values
of a and c. The ratio c/a is called the eccentricity and is repre-
sented by e ; it is always less than unity. The line along which
the major axis lies is called the line of apsides.
To draw a Given Ellipse. Let it be required to draw an ellipse
whose semi-major axis is one inch, and eccentricity i, with one
focus at the point F 1 of Fig. 75, and with its major axis inclined
30 to the horizontal.
Draw the line of apsides MN at the proper angle. Since e = J, we
locate C one-fourth of an inch from F l on the line of apsides.
Take F 2 at an equal distance beyond C, make the total length of
the thread 2 inches = 2 a -f- 2 c, and draw the ellipse as shown in
the figure.
The dotted line surrounding the ellipse is a circle drawn about
C as a center with a radius of one inch (equal to the semi major
axis). It is worthy of notice that the ellipse differs but little
from this circle, the greatest distance between the two being
about y^ of an inch. With a less eccentricity the agreement of
the two curves is closer. For e = 0.10 the difference is but .005
of the semi major axis, so that an ellipse of that eccentricity whose
semi major axis is two inches differs at no point more than T ^ of
146
LABORATORY ASTRONOMY
an inch from a circle struck about its center with a radius of two
inches. If the orbits of the planets are drawn with their true
eccentricities and with a line 0.01 inch in width, and in each case
a circle is struck with radius a about the center of the ellipse, and
having a width of .01 inch, no white space will be anywhere visi-
ble between the two lines unless the diameter of the circle is greater
Horizontal
FIG. 75. Ellipse drawn with Given Constants
than about 1 inch for Mercury, 4^ inches for Mars, 17 inches for
Jupiter, and 12^ inches for Saturn. For Venus and the earth the
circles may be several feet in diameter. The orbits may therefore
be represented by such circles with a considerable degree of
accuracy.
MEAN AND TRUE PLACE OF A PLANET
Having considered the geometrical properties of the planetary
orbits, it is next in order to inquire as to the law which regulates
the motions of the planets in their orbits.
Since the sun is at one focus of the orbit, the planet's distance
from the sun varies continually. It is nearest the sun at the peri-
helion point, which is at one extremity of the major axis. Aphelion
occurs at the opposite end of the major axis, and the planet is then
at its greatest distance.
THE MOTIONS OF THE PLANETS
147
Kepler's second law states that the planet moves in such a way that
its radius vector sweeps over equal areas in equal times. The appli-
cation of this principle will be evident from the following illustration.
Fig. 76 represents the orbit of Mercury in its true proportions.
The period of the revolution of the planet is eighty-eight days, in
which time the radius vector sweeps over the whole area of the
ellipse. To pass from perihelion to aphelion would require forty-
four days, or one-half the period, since the area described is one-
half the area of the whole ellipse. It is not difficult to fix very
nearly the point reached by the planet twenty-two days after pass-
ing through perihelion. It will then have accomplished a quarter
of a revolution, and be at
such a point P that the area
ASP is one-quarter of the
ellipse, or one-half of A PBS,
so that APS equals BPS.
It may be shown that this
point must be very nearly in
the line Pf drawn perpen-
dicular to the major axis
through f, the " empty "
focus of the orbit, as it is
sometimes called.
Assuming P to be on this
line, and drawing a perpen-
dicular Sk through the focus
occupied by the sun, and also the radius vector PS, we have from
the symmetry of the ellipse, Area ASk equal Area BfP, and the
triangle PkS evidently equals the triangle PfS. The difference of
the two areas ASP and BSP is therefore the segment of the ellipse
cut off by the chord Pk ; this segment is so very small that the
area ASP is very nearly equal to BSP.
The angle ASP through which the planet has moved about the
sun since perihelion is called its " true anomaly." In this case it
is about 110. We may now infer that the true anomalies of Mer-
cury 22, 44, 66, and 88 days after perihelion would be about 110,
180, 250, and 0, respectively.
FIG. 76. Equal Areas in the Ellipse
148
LABORATORY ASTRONOMY
It is convenient to refer the motion of the planet to that of a
hypothetical planet moving in the orbit in such a way as to be at
perihelion with the real planet and describe equal angles in equal
times ; thus the anomaly of the so-called " mean planet " after 22,
44, 66, and 88 days would be 90, 180, 270, and 360, respectively.
The Equation of Center. The quantity to be added to the anomaly
of the mean planet, or briefly, the " mean anomaly " of the planet, in
order to find its true anomaly, is called the "equation of center";
in the cases above given it is for the four positions 0, + 20, 0, and
20. It is always positive for values of the mean anomaly between
and 180, and negative
for values between 180
and 360. It appears from
Fig. 77, in which P and P'
mark the true and mean
places of the planet re-
spectively, that at all
points from perihelion A
to aphelion B, the true
anomaly ASP is greater
than the mean anomaly
A SP', while from aphelion
to perihelion ASP is less
than ASP'.
The value of the mean
anomaly being given for
any time, its value for any other time is easily found, since it
increases uniformly from to 360 in the time required for the
planet to make one revolution.
The mean anomaly being known, we may pass to the true anomaly
by means of a table of the equation of center (page 174), in which
the value of the latter is given for each degree or ten degrees of
the planet's mean anomaly.
The computation of these tables lies far beyond our scope, but it
is worth while to note that approximate values of the equation of
center may be found by a graphical method, which rests upon the
principle that in describing equal areas about one focus of an
FIG. 77
THE MOTIONS OF THE PLANETS
149
ellipse of small eccentricity, a planet describes very nearly equal
angles about the other focus.
If then the ellipse be carefully constructed on a large scale, say
with a major axis of ten inches, and through the empty focus lines
be drawn making angles of 10, 20, 30, etc., with the line of
F 2
FIG. 78
apsides, these lines will cut the ellipse at the places occupied
by the true planet when its mean anomalies are 10, 20, 30, etc.
Fig. 78 shows one-half of the orbit of Mercury divided into six
equal parts in this manner.
The true places being thus fixed, and lines drawn from each to
the sun, the true anomalies may be read off with a protractor ; and
by comparison with the mean anomalies the equation of center for
each ten degrees of mean anomaly may be determined.
MEASUREMENT OF ANGLES IN RADIANS
It has been assumed that the student is familiar with the ordi-
nary method of measuring angles in degrees. For some purposes
it is convenient to select a different unit, the u radian."
One radian is the angle subtended by an arc whose length
(measured by a flexible scale laid along the curve of the arc) is
equal to that of the radius. This angle measured in the ordinary
way is found to be 57.3 = 3438', or 206,265".
If the length of an arc a is known, and also the radius of the
circle r, the angle subtended by the arc is a/r (arc -^ radius) radians.
Thus in a circle two feet in diameter, an arc of one inch subtends
an angle of 1/12 radian, 6 inches of 0.5 radian, 1 foot of 1 radian,
etc. Since 1 radian equals 57. 3, an arc of one inch in the above circle
150 LABORATORY ASTRONOMY
subtends 1/12 x 57.3; and, in general, radians are transformed to
degrees, minutes, or seconds of arc by multiplying by 57.3, 3438,
and 206,265, respectively; and degrees, minutes, or seconds to
radians by dividing by 57.3, 3438, and 206,265, respectively.
The use of the radian is especially convenient in problems in-
volving an angle so small that the corresponding arc nearly equals
its chord or the perpendicular drawn from one extremity of the arc
to the radius drawn through its other extremity. The method is
illustrated by the following instances :
1. The moon's distance is 240,000 miles, and its angular diameter
is 31', or 31/3438 radian. Its diameter in miles is given by the
equation
24^00 = 38' HenC6 D = 2164 miles ' approximately.
2. The height of a tree is 30 feet, and the length of its shadow
is 150 feet. The altitude of the sun is
a/r = 30/150 = 0.2 radian = 11.46.
The true value obtained by trigonometrical computation is 11.54,
differing by .08, and this approximate method will give results
within 0.l so long as the angle does not exceed this value.
3. By means of a sextant the angle between the water line of a
distant war ship (Fig. 79) and the top of its military mast is found
.O-OiL-
FIG. 79
to be 17' 10". The height of the mast is known to be 120 feet.
Assuming this height to be equal to the arc subtended by the
measured angle, we have
17' 10" = 0.005 radian = ? = beight of ">Bt
r distance of ship
and the distance of the ship is about 8000 yards.
THE MOTIONS OF THE PLANETS 151
DIAGRAM OF CURTATE ORBITS
Fig. 80 represents a diagram of the orbits of the five inner
planets projected on the plane of the ecliptic, which serves to solve
many problems regarding the planetary motions. The diagram is
of convenient size for actual use, if its dimensions are such that
one astronomical unit equals about f of an inch.
In order to show how small is the distortion of the orbits as pro-
jected, we may compare the length of the radius vector to any
point in the orbit with that of its projection on the ecliptic, which
is called the " curtate " distance from the sun.
Even in the case of the orbit of Mercury, which has the greatest
inclination, the curtate distance differs from the true distance at
most by y^, in the case of Venus by less than ff J^, and in the
case of all the other planets by less than y^ 1 ^- If the scale of the
diagram is such that one astronomical unit equals 1| inches, no
radius vector drawn in any one of the " curtate " orbits will differ
from the corresponding radius vector drawn in the actual orbit by
so much as ^1^ of an inch ; and by referring to the data given on
page 146 it will be seen that on that scale the elliptic orbits may
be represented with considerable accuracy as circles.
The position of the line of apsides is fixed by the longitude of
perihelion, page 174; the distance c of the center of the ellipse
from the sun is found from the ratio c /a = e, and a circle struck
about the center with a radius a very closely represents the curtate
orbit ; the distances c and a are of course to be laid off from the
scale of astronomical units.
To draw such a diagram is a useful exercise, and by careful draw-
ing and erasure a single diagram may serve for many problems, but
it is convenient to have several printed copies when it is desired to
preserve the solutions.
It is also convenient to have diagrams on which an astronomical
unit equals 2j, j, and f inches, respectively, the first extending to the
orbit of Mars, the second to that of Jupiter, and the third to that
of Saturn. The larger scale should be used for problems referring
to Mercury and Venus, while the smaller scales are required for
the major planets.
152
LABORATORY ASTRONOMY
ELEMENTS OF THE SIX INNER PLANETS, JAN. I, 1900
Sj~M
S
|
e
t
y
h
MUD DUUnco
0.887
0.72S
1.000
1.524
6.203
9.639
Eoetttridt,
0.2056
0.0068
0.0168
0.0933
0.0482
0.0561
Inclination
7*0
3*.4
1*9
1*3
2*5
Longitad. of A.-
eeoduigNod.
17*1
76*.7
48*7
99*4
U2*7
Longitod. of Peri-
helion
76*
1304
101*2
S34!2
12*7
9o!tt
Meu Longitude,
Gr. Meu Nooa
182?22
344*33
100*67
294*27
238*13
266*61
Sidereal Period,
lieu Solar Dy>
87*9693
224-701
865*266
686*979
4332*58
10759*2
Mean dailj motion
4*09234
1*60213
0*98661
o!62403
0*08309
0?03346
EQUATION OF CENTER
Mm,
A "*"
A*.*
0*
0.0
0.0
0.0
0.0
0.0
0.0
860
10
5.4
0.1
tj
2.1
1.0
1.2
350
10.5
0.3
0.7
4.1
2.0
2.4
340
14.9
0.4
1.0
6.9
2.9
8.4
830
18.6
0.5
1.2
7.5
3.7
4.4
820
21.1
0.0
.5
8.8
4.4
6.2
310
22.8
0.7
.7
9.8
4.9
5.8
300
23.6
0.7
.8
10.4
5.3
6.
290
23.6
0.8
.9
10.7
5.5
6.
280
22.9
M
.9
10.6
u
6.
270
21.7
0.8
19
10.2
M
6.
260
19.9
0.7
.8
0.6
5.1
5.
250
15.3
o.n
1.4
7.6
4
4.7
230
12.5
0.5
1.2
6.8
S.
3.9
220
50
9.6
0.4
0.9
4.8
2
3.0
210
60
6.5
0.3
0.8
S.S
1.
2.1
200
70
3.3
0.1
0.3
1.7
0.
1.0
190
80
0.0
0.0
0.0
0.0
0.
0.0
180
FIG. 80. Diagram of Curtate Orbits
THE MOTIONS OF THE PLANETS 153
On the plan of each orbit the symbol of the planet is placed at
the perihelion point, whose position is thus approximately known
at a glance.
That part of the orbit which is above the plane of the ecliptic is
marked with a full line, and the part below is marked by a broken
line. The line of nodes is therefore determined as a line joining
the two points where the character of the line changes. This line,
of course, passes through the sun.
The inclinations of the orbit planes are shown by the triangles
which appear below the diagram, each marked by the symbol of
the planet to whose orbit it pertains. A scale of astronomical units
is printed at the bottom.
The attached tables (see page 174) give the values of the elements
of each orbit and certain other quantities which are required in
finding the place of the planet in its orbit at a given time.
Measurements may be made on the diagram between any two
points by laying a strip of paper with its straight edge through
the points, and marking the edge of the strip opposite each point.
By laying the straight edge along the scale the distance in astro-
nomical units is found. Instead of the paper strip a pair of com-
passes may be used.
The map shows the orbits as they would be seen from the north
side of the ecliptic, and the motions of the planets as thus seen are
always counter-clockwise about the sun. The plane of the map is
that of the ecliptic, and it is so oriented on the paper that hori-
zontal lines drawn from left to right would strike the celestial
sphere at the vernal equinox. Therefore the direction which on
an ordinary terrestrial map would be east on this map is toward
longitude zero ; up is toward longitude 90, down toward longitude
270, and the direction of any other line on the map is fixed by
determining the angle which it makes with the line drawn to the
vernal equinox. Thus, the line in Fig. 81 from E to M makes an
angle of 45 with the line SR, and is therefore directed ^toward
longitude 45, and EJ is directed toward longitude 260. By draw-
ing lines through the sun parallel to EM and EJ, respectively, the
longitude may be read off directly on the circle which bounds the
diagram.
154
LABORATORY ASTRONOMY
1 _L
1 Z 3 4 5 6
FIG. 81. Direction of a Line fixed by Longitude
To find the Elements of an Orbit. The elements of the planetary
orbits may be obtained from measurements on the diagram. These
elements are as follows :
a Semi-axis major of the ellipse or mean distance.
e Eccentricity of the ellipse = c/a, where c is distance of
focus from center.
TT Heliocentric longitude of perihelion.
Heliocentric longitude of node.
i Inclination of plane of orbit to plane of the ecliptic.
THE MOTIONS OF THE PLANETS 155
To find a draw a straight line from the perihelion point of the
orbit through the sun to cut the orbit at the aphelion point. This
is the line of apsides. Measure the distance from perihelion
to aphelion along the line of apsides in astronomical units. This
gives the major axis of the ellipse, one-half of which is the value
of a.
To find c, bisect the major axis and thus fix the center of the
ellipse. The distance from focus to center may then be measured
in astronomical units. This is the value of c; it is not regarded
as one of the elements, since it is fixed by the values a and e.
To find e, determine c/a from the above measurements.
To find TT, prolong the line of apsides through the perihelion
point ; the reading at the point where it cuts the graduated circle
is the longitude of perihelion.
To find Q, , prolong the line of nodes through the point where
the planet moving counter-clockwise passes from the dotted por-
tion of the orbit to the full line. The reading at the point where
this line cuts the graduated circle is the longitude of the ascending
node.
To find the inclination i, measure the angle of the proper triangle
by a protractor ; or, more accurately, measure the altitude h and
the base b of the triangle ; h/b is equal to the inclination in radians.
57.3 h/b = i in degrees.
The following measurements were made on the orbit of Jupiter :
Sun to perihelion ... ............. 4.96
Sun to aphelion ..... .... ....... 5.42
Major axis ...... ........ ..... 10.38
a Semi-axis a ................ 5.19
Center to perihelion ............. 5.19
Focus to perihelion ............. 4.96
c Center to focus c .... ...... .... 0.23
.
a 5.19
TT The line of apsides cuts the circle at 12. 7.
Q The line to ascending node cuts the circle at 99.4.
i The altitude of the triangle is 0.13 and the base 5.43; hence
i = h/b = 0.13/5.43 = 0.024 radian = 57.3 x 0.024 = 1.37.
156
LABORATORY ASTRONOMY
PLACE OF THE PLANET IN ITS ORBIT
If the heliocentric longitude of a planet is known, it may be
plotted at its proper place on the diagram by drawing a line from
the sun to that division of the graduated circle which indicates the
given longitude ; the intersection of this line with the orbit gives
the required place. When, for instance, the heliocentric longitude
of Jupiter is 280, the intersection falls very close to the descend-
ing node. In this particular case the place of the planet is com-
pletely known, since it is in the ecliptic. Usually the planet is
FIG. 82. The Z Coordinate
many millions of miles from the ecliptic, but its exact distance may
be easily found by the use of its inclination triangle.
This will appear by consideration of Fig. 82, which represents a
diagram in which the orbit of Jupiter has been cut through along
the heavy line, and the part of the orbit which is above the ecliptic
turned up around the line of nodes so as to be at the proper incli-
nation. The exact angle is insured by supporting it by wedges
having the proper angle.
THE MOTIONS OF THE PLANETS
157
The height of the planet at P above the plane of the ecliptic,
which we shall call its " Z coordinate," or simply Z, is evidently
the altitude of a right-angled triangle whose small angle is i (the
inclination of the orbit), and whose base is the line drawn from
the place of the planet on the diagram to the line of nodes. This
line (which practically equals the hypotenuse) we will call U.
To find Z, then, it is sufficient to measure U on the diagram and
to lay off the same distance along the horizontal side of the incli-
nation triangle. The vertical line drawn to the hypotenuse from
the point thus fixed gives the length of Z in astronomical units.
A far more accurate method is to make use of the obvious relation
Z/U = i in radians, or 57.3 Z/U = i in degrees. Thus, for Jupiter
Z = U x 1.3/57.3 = 0.023 U.
TO FIND THE TRUE HELIOCENTRIC LONGITUDE OF A
PLANET
To find the true position of any planet at a given time we must
first know its mean anomaly at that time, and then, by applying
the equation of center,
find the correspond-
ing value of the true
anomaly which enables
us to place the planet Eo
at the proper position
in its orbit.
Thus, if the earth's
mean anomaly is 70,
we find by the table,
page 174, that the equa-
tion of center is + 1.8,
and hence the true
anomaly is 71.8. Since the longitude of perihelion is 101.2, the
true heliocentric longitude is 101.2 + 71.8, or 173.0, and this
value enables us to plot the earth in its proper place on the
diagram, Fig. 83.
158 LABORATORY ASTRONOMY
We may find the mean anomaly if we know the number of days
elapsed since perihelion, and the mean daily motion along the orbit.
The fact that the planets move very nearly in the ecliptic, so that the
motion in the real and curtate orbit is very nearly the same, makes
it easier to proceed in a somewhat different manner, as follows :
In the Table of Elements appended to the chart is given the "mean
daily motion w (in heliocentric longitude), which is found by divid-
ing 360 by the period in days. This quantity enables us by a simple
multiplication to find the mean motion, or increase in heliocentric
longitude of the mean planet in any given number of days.
Knowing the mean (heliocentric) longitude at any given epoch,
the mean longitude at any later date is found by addition of the
mean motion in the elapsed time. The Table of Elements supplies
the necessary " longitude at the epoch " for Greenwich mean noon,
January 1, 1900.
We may summarize the process of finding the planet's true helio-
centric longitude as follows :
Let E be the longitude at the epoch,
t " " elapsed time in days,
/u, " " mean daily motion,
TT " " longitude of perihelion,
M " " mean anomaly,
v " " true anomaly,
/ " " true longitude (heliocentric).
First find the mean anomaly at the time t, as follows :
pi = Mean motion in elapsed time,
E -f- fA.t = Mean longitude at given date,
E + fjit TT = Mean anomaly.
With this value of the mean anomaly find the equation of center
by the table, and since
True anomaly = Mean anomaly -f Equation of center,
or v = E + pi TT + Equation of center,
and True longitude = v -f- TT, we have directly
True longitude = E + ^t + Equation of center.
The form of the computation is shown in the solution of the
following problem :
THE MOTIONS OF THE PLANETS 159
Find the true place of Mars and the earth May 8, 1905, at
Greenwich, midnight.
The elapsed time may be found as follows :
Gr. Mean Noon. Jan. 1, 1900, to Jan. 1, 1901 365 days
1902 365
1903 365
1904 365
1905 366
Jan. 1, 1905, to Feb. 1, 1905 31
Mar. 1, 1905 28
Apr. 1, 1905 31
May 1, 1905 30
Noon. May 1 to Midn., May 8, 1905 7.5
Elapsed time = 1953.5 days.
For Mars td = 0. 52403 x 1953.5 = 1023. 69.
For the earth & = 0.98561 x 1953.5 = 1925.39.
MARS EARTH
Mean longitude Jan. 1, 1900 = E 294. 27 100. 67
Mean motion 1953.5 days = nt 1023 .69 1925 .39
E + fit 1317 .96 2026 .06
Subtract complete revolutions ...... 1080 . 1800 .
Mean longitude May 8.5, 1905 ...... 237 .96 226 .06
Subtract longitude of perihelion ir [~ 334.2 101.2 ~|
Mean anomaly M |_ 263.8 124.9 J
Equation of center . . . . . . . . . . - 10.36 + 1.50
True longitude / = E + id + Equation of center 227.60 227.56
It will be noted that in each case the value of E + ^t has been
diminished by an integral number of revolutions : 3 x 360 for
Mars and '5 x 360 for the earth. It appears, also, that the num-
bers inclosed in brackets enter the computation only for the purpose
of obtaining the equation of center which is then applied directly
to the mean longitude following the equation
I = E + fjit + equation of center.
On plotting the planets it appears that Mars is exactly opposite
the sun, as indeed is evident from the fact that the earth and Mars
are in the same heliocentric longitude. The Ephemeris gives May 8,
8 P.M., G.M.T., as the time of opposition. The actual distance
between Mars and the earth, as measured on the diagram, is 0.56
astronomical units, or fifty-two million miles.
160
LABORATORY ASTRONOMY
The planet may be plotted with a very fine-pointed, hard pencil,
against the edge of a ruler passing through the sun and the point of
the graduated circle whose reading equals the planet's true helio-
centric longitude. It is quite an advantage to have the ruler of a
transparent substance in order that its edge may be correctly placed
on the graduations.
A better method, however, is to put a pin through the sun's place
firmly into the drawing board or table, and pass around the pin a
long loop of smooth black thread. The other end of the loop is
FIG. 84. Plotting with a Loop
held between the thumb and forefinger, with the threads slightly
separated (about ^ of an inch). The loop is then drawn taut, and
the middle of the white space between the threads may be bisected
by the proper point on the graduation ; the place of the planet is
then marked by putting the point of the pencil exactly midway
between the threads where they intersect the orbit (Fig. 84).
The planet having been placed in its true position on the orbit
by plotting it as above, so that its curtate radius vector is drawn
toward the true heliocentric longitude, its place is completely known
if we measure U and find Z, as on page 157. The usual method of
fixing the distance of the planet from the ecliptic is to give its
heliocentric latitude b, or angular distance from the ecliptic, which
may be found thus (Fig. 85) :
THE MOTIONS OF THE PLANETS
161
b = Z/r = angular distance (radians) of planet above ecliptic as
seen from the sun. Combining this with Z = U x i (radians), as
explained on page 157,
b (radians) = i (radians) ;
and turning each side of the equation
into degrees by multiplying by 57.3, we
have
(57.36)=- x (57.3 i),
or
The inclinations are so small that the
latitude is always well determined by
this method.
FIG. 85. Heliocentric Latitude
GEOCENTRIC POSITIONS
When a planet has been placed on the diagram by its heliocentric
coordinates, we may find its position as seen from the earth ; that
is, we may find the longitude and latitude of that point of the
celestial sphere upon which it is seen projected by an observer
upon the earth.
The line drawn from the earth to the planet is called the "line of
sight," and its projection on the ecliptic is the line from the earth
to the planet on the diagram. If this line is horizontal, it cuts
the celestial sphere at the vernal equinox, and the planet's geocen-
tric longitude is zero.
Geocentric Longitude. The angle between the (projected) line of
sight and the line drawn to the vernal equinox is the planet's geo-
centric longitude. It is equal to the angle between the line of sight
and the line drawn from the sun to the zero of the graduated circle.
This angle may be measured in several ways :
1. By prolonging the line of sight, if necessary, till it cuts the
line of equinoxes on the diagram, and measuring the angle with a
protractor.
162
LABORATORY ASTRONOMY
2. By drawing a line through the sun parallel to the line of
sight, and noting the point where it cuts the graduated circle.
3. The most accurate method is usually the following: Bring a
straight edge to pass accurately through the places of earth and
planet. Note the points of intersection with the graduated circle.
FIG. 86. Geocentric Longitude
Call the reading where the line of sight (from earth to planet)
cuts the circle A, and the other (opposite) reading B. Then the
geocentric longitude of the planet is
A+B
90, if A is less than
B ; and \- 90, if A is greater than B. This may be proved by
2
the theorem that the angle between two chords of a circle is meas-
ured by the half sum or half difference of the included angles,
according as they intersect inside or outside the circle.
Better than a straight edge is a fine line on a transparent ruler
(celluloid, glass, mica, tracing cloth), or a stretched thread laid over
the two points.
Fig. 86 illustrates the three methods, the heliocentric longitudes
of the earth and Venus being 150 and 90, respectively. The angle
at C measured by the protractor is 13, the line through S parallel
to AB cuts the graduated circle at 13.0, while the readings at A
and B are 20.0 and 186.0, so that - 90 = 13.0.
The Sun's Longitude and the Equation of Time. It is an important
fact that, since the line of sight to the sun is drawn to a point
THE MOTIONS OF THE PLANETS 163
whose heliocentric longitude is opposite to that of the earth, the
sun's geocentric longitude is always 180 + the earth's heliocentric
longitude.
The sun appears to move about the earth in an orbit whose ele-
ments are the same as those of the earth about the sun, except
that E and TT are each greater by 180.
The sun's mean longitude is therefore 280.67 -f- fit and its mean
anomaly is 280.67 -f /* 281.2, where t is. the number of days
since January 1, 1900, and /A is the earth's mean daily motion.
To find the sun's true longitude we add to the mean longitude
the equation of center taken from the table for the earth, and from
the true longitude we may find the R.A. by adding the reduction
to the equator (page 121). We may therefore write :
Sun's R.A. = Sun's mean longitude + Eq. center -f- Red. to equator.
Sun's R.A. Sun's mean longitude = Eq. center + Red. to equator.
And since the sun's mean longitude equals the R.A. of the mean
sun (page 92),
Sun's R.A. R.A. of mean sun = Eq. center -f- Red. to equator.
The first member of the last equation is the equation of time
, whose approximate value may thus be computed for any date :
Jan. 31, 1900. ^t = 30 x 0.9S56 = 29.57
E + pt = 280.67 + 29.57 = 310 .24
E + iLt-Tr = 310.24 - 281.2 = 29 .04
Equation of center = + .97 + 0.97
True longitude 311.21
Red. to equator + 2 .4
Equation of time + 3.37
or 13.5 minutes to be added to apparent time.
Geocentric Latitude. The geocentric latitude ft of the planet is
the angular distance of the planet from the ecliptic as seen from the
earth. It is found by the same method as that used for finding the
heliocentric latitude b. (See Fig. 87.)
Draw the line A from earth to planet on the diagram. Z/A
equals the angle (3 in radians, and Z U x i (in radians).
164 LABORATORY ASTRONOMY
Hence, by reasoning applied on page 161,
FlG. 87. Geocentric Latitude
The whole process of finding geocentric latitude and longitude
is illustrated in the following example :
To find the positions of the five inner planets at Greenwich
mean noon, July 6, 1907, the elapsed time from January 1,
1900, being 2742 days (see page 167).
$
9
d 1
3
E
182.22
344.33
100.67
294 27
238.13
Add fit ..'....
11221 .20
4393 .04
2702 .54
1436 .89
227.83
E 4- id
11403 42
4737 37
2803 21
1731 16
465 96
Subtr. complete revolutions
11160 .
4680 .
2520 .
1440 .
360.
E -f fit
243 .42
57 .37
283 .21
291 .16
105.96
Subtract TT
75 .9
130 .2
101 .2
334 .2
12.7
Mean anomaly ....
Eq. of center by table
Heliocentric longitude I .
167 .5
+ 4 .10
247 .52
287 .2
-0 .73
56 .64
182 .0
-0 .06
283 .15
317 .0
-7 .89
283 .27
93.3
+ 5.47
111.43
Plotting the planets on the diagram, we determine the geocentric
places by finding the following values :
THE MOTIONS OF THE PLANETS
165
9
*
i
From
diagram,
A . . . .
130. 4
80. 1
283. 2
111 2
u
u
B
302 3
266 9
103. 3
288 8
u
U
0.14
0.22
1.16
1.09
u
From
u
table
A
i
0.70
70
1.60
3. 4
0.40
1 9
6.26
1 3
A + J
3
90
126 35
83 5
283 25
110
2
U
X I
8 .
- 0.47
- 5. 51
+ 0.21
A
FIG. 88. Geocentric Places, July 6, 1907
166
LABORATORY ASTRONOMY
The signs attached to the latitudes are fixed by the fact that
Jupiter is in the full-line part of its orbit and therefore above the
ecliptic, while all the other planets are in the dotted parts of their
orbits and therefore in south latitudes.
Since the full line extends from the longitude of the ascending
node to that of the descending node, which is 180 greater, we may
also fix the sign of /3 by the following rule :
From the true heliocentric longitude subtract that of the ascend-
ing node ; if I Q> < 180, the latitude is positive ; if I & > 180,
the latitude is negative. Thus, in the above example :
$
9
cf
y
1
247. 5
56. 6
283. 3
111.4
Q
47 .1
75.7
48.7
100.1
I _ o
200 .4
340 .9
234 .6
11 .3
B .
nesr.
near.
nesr.
pos.
Perturbations. The longitudes above obtained are liable to an
error of more than a tenth of a degree if the elapsed time exceeds
a half century, and the perturbations which are neglected may add
somewhat to the error. The effect of the mutual perturbations of
Jupiter and Saturn may be approximately corrected by adding to
the mean longitudes the following quantities :
3
h
h
1800-1890
+ 0.3
1800-1840
-0.8
1940-1960
-0.4
1890-1950
+ .2
1840-1870
-0 .7
1960-1980
-0 .3
1950-1990
+ .1
1870-1910
-0 .6
1980-1990
-0 .2
1990-2000
.0
1910-1940
-0.5
1990-2000
-0 .1
Effect of Precession. The true longitudes found by the method
above described are referred to the equinox of 1900, the point from
which the mean longitude of the table is measured.
Since the vernal equinox moves along the ecliptic 50" per year
toward the west, or nearly 6' in seven years, the longitudes meas-
ured from the true equinox of 1907 will be about 0.l greater than
THE MOTIONS OF THE PLANETS 167
if measured from the equinox of 1900. This "reduction to the
equinox of date" is 50" x t, or 0.014 t, where t is the number of
years elapsed since 1900.
The Julian Day. The process of computing the elapsed time used
on page 159 is tedious and liable to error where the elapsed time
is considerable. Where the interval between distant dates is to be
accurately determined astronomers find it convenient to make use
of the number of each day in the Julian period. It is sufficient
here to say that January 1, 4713 B.C., was the first day of this
period, and the Ephemeris gives each year the number of the
Julian day for January 1 ; thus, the 1st of January, 1900, was
No. 2415021 in the cycle. To find the number for any given
date, we turn to page III of the corresponding month, add the
day of the year (taken from the second column), and subtract 1.
The table on page 175 gives for each year from 1800 to 2000 a
number one less than that of the Julian day corresponding to Jan-
uary 1 of the given year. The subsidiary table for months gives
for each month a number one less than the day of the year cor-
responding to the first of the given month.
It is easy to see that by adding together the year number, month
number, and day of the month, we get the corresponding Julian
day. Thus we compute the interval from January 1, 1900, to July 6,
1907, as follows :
Year number for 1900, 2415020 1907 2417576
Month number for January, _ July 181
Day of month, 1 6 6
Julian day, 2415021 2417763
2415021
Elapsed time, 2742
Right Ascensions and Declinations of the Planets. By means of the
geocentric latitudes and longitudes which we have thus determined
the planets may be placed in their respective positions upon the globe.
The proper longitude being found upon the ecliptic of the globe,
the latitude is laid off on a strip of paper by placing it along the
ecliptic and marking off the proper number of degrees along its
edge. The paper is then applied to the globe so as to mark off this
distance perpendicular to the ecliptic. The latitude is never so
168 LABORATORY ASTRONOMY
great as 8, so that no serious error in the place will occur if the
strip is not exactly perpendicular to the ecliptic.
The place of the planet being thus marked on the globe, its right
ascension and declination may be determined, and problems relat-
ing to its diurnal motion, such as its times of rising and setting,
may be solved by the methods of Chapters VIII and IX.
CONFIGURATIONS OF THE PLANETS
The elongation of a planet is its distance from the sun along the
ecliptic as seen from the earth. It is therefore equal to the differ-
ence of the geocentric longitudes of the sun and planet. The elonga-
tion is measured either way from the sun up to 180, at which
point the planet is at opposition, or opposite the sun. When the
elongation is zero the sun and planet are in the same longitude,
and the planet is in conjunction with the sun.
The symbols and <$ are used for opposition and conjunction,
respectively. When the longitude of the planet is greater than
that of the sun it is east of the latter, and follows it in its diurnal
revolution. It is therefore above the horizon at sunset and is
said to be an "evening star,' 7 since it is visible in the twilight
after sunset except when near conjunction. On the other hand, all
planets whose longitudes are less than that of the sun precede it,
and they will be above the horizon at sunrise and therefore visible
at dawn, except when very near conjunction. They are then " morn-
ing stars," just as stars in eastern elongation are evening stars.
The geocentric longitude of the sun, July 6, 1907, is 103.2 (since
the earth's heliocentric longitude is 283. 2, page 164). The longi-
tude of Jupiter being 110. 2, its elongation is about 7 east, and it
is an evening star, though too close to the sun to be visible ; it will
become a morning star about July 14.
The elongation of Mars is very nearly 180, and it is at opposi-
tion and becoming an evening star. The longitude of Venus is 84.6 ;
it is 18.6 west of the sun and is a morning star. On referring to
the diagram (Fig. 88), and remembering that it moves more rapidly
than the earth, it is evident that it is approaching conjunction
beyond the sun (" superior" conjunction), after which it will pass
THE MOTIONS OF THE PLANETS 169
to eastern elongations and be an evening star. Mercury's longitude
is 126 ; it is 23 east of the sun, and referring to the diagram, we
see that it is approaching conjunction between the earth and sun
(" inferior " conjunction), after which it will be a morning star.
The preceding principles enable us to find the place of a planet
at any given date, and thus to answer many of the questions which
continually suggest themselves to one interested in watching the
courses of the planets in the sky.
It is evident, for instance, from the problems solved on pages 159
and 164, that in 1907 the greater proximity of Mars to the earth
offers conditions for the study of its surface which are much more
favorable than those of the opposition of 1905.
The oppositions of Mars recur at an average interval of about
780 days, which is the synodic period of the earth and Mars, as
explained in the text-books of descriptive astronomy.
We may fix the dates of other oppositions approximately, as in
August, 1877, September, 1909, November, 1911, December, 1913,
etc., and by computing for the first and last days of those months a
closer approximation to the day of opposition may quickly be made,
and finally a careful computation for the exact date will fix the
time within a few hours. The geocentric place and the distance of
the planet may then be found.
It appears that favorable oppositions occur in the summer, and
that the planet is then quite a distance south of the equator, so
that it is far from the zenith of any northern observatory.
The satellites of Mars were discovered in 1877, and in the same
year an expedition was sent to the island of Ascension to observe
Mars for a determination of the solar parallax.
In conclusion we will consider the motion of Mars during the
summer of 1907, to illustrate the form which, the computation takes
when many places are to be found at comparatively short intervals.
We first carefully determine the mean longitudes of Mars and
the earth for March 22 to be 235.51 and 178.74, respectively, and
then easily form the second column of the following schedule by
successive additions of 10.48 and 19.71, the mean motions of the
two planets in twenty days.
170
LABORATORY ASTRONOMY
The third column is formed for Mars by writing the longitude
of perihelion 334. 2 on the upper edge of a slip of paper and
placing it under the numbers of the second column successively,
subtracting from each to find the corresponding mean anomaly.
The same result is more easily obtained by adding in the same
way 25.8 (360-334.2) to each number in the second column.
The third column is checked by noting that the differences of the
successive values are 10.48, which insures the accuracy of both
columns. The equation of center is taken from the table and
entered in the fourth column, and the true heliocentric longitude
found by adding corresponding numbers of the second and fourth
columns. The same process gives the earth's true heliocentric
longitude.
The labor is by no means proportionate to that required in
computing a single place, and the comparison of the successive
numbers of each column is an important aid in detecting errors.
MARS
THE EARTH
Date
E + fj.t
E + u.t-n
Eq. of
Center
I
E + nt
E+H'-TT
Eq. of
Center
I
Mar. 22
235. 51
261.3
- 10.3
225. 2
178. 74
77.5
+ 1.9
180. 6
April 11
245 .99
271 .8
-10 .6
235 .4
198 .45
97 .2
+ 1 .9
200 .3
May 1
256 .47
282 .3
-10 .6
245 .9
218 .16
117 .0
+ 1 -7
219 .9
21
266 .95
292 .8
-10 .3
256 .6
237 .87
136 .6
+ 1.3
239 .2
June 10
277 .43
303 .2
-9.5
267 .9
257 .58
156 .4
+ 0.7
258 .3
30
287 .91
313 .7
-8.3
279 .6
277 .29
176 .1
+ 0.1
277 .4
July 20
298 .39
324 .2
- 6 .8
291 .6
297 .00
195 .8
- .5
296 .5
Aug. 9
308 .87
334 .7
- 5 .0
303 .9
316 .71
215 .5
-1 .1
315 .6
29
319 .35
345 .1
- 3 .1
316 .3
336 .42
235 .2
-1 .5
334 .9
Sept. 18
329 .83
355 .6
-0.9
328 .9
356 .13
254 .9
-1.8
354 .3
Oct. 7
340 .31
6 .1
+ 1.3
311 .6
15 .84
274 .6
-1 .9
13 .9
The planets were plotted from the above data on a scale of 1.6
inches to the astronomical unit, the boundary circle being 9|
inches in diameter. The values of A, B, U, and A were determined
and the geocentric longitudes and latitudes A and ft found as in
the following table :
THE MOTIONS OF THE PLANETS
171
A
B
U
A
A.
/3
March 22 . .
244. 6
103. 7
+ 0.13
1.10
264. 1
+ 0.2
April 11 . .
254 .8
112 .6
-0.16
0.91
273 .7
-0 .3
May 1 . .
263 .9
119 .2
0.44
0.75
281 .5
- 1 .2
21 . .
271 .4
120 .9
0.69
0.60
286 .1
-2 .2
June 10 .
277 .8
117 .1
0.90
0.50
287.4
-3 .4
30 . .
282 .2
108 .0
1.10
0.44
285 .0
-4 .8
July 20 . .
285 .7
94.3
1.25
0.42
280 .0
-5 .7
Aug. 9 . .
289 .7
84 .9
1.36
0.47
277 .3
-5 .5
29 . .
296 .8
84.1
1.39
0.54
280.4
-4 .9
Sept. 18 . .
305 .0
88.2
1.36
0.64
286 .6
-4 .0
Oct. 7 . .
316 .6
98 .4
1.27
0.76
297 .5
-3 .2
310 300 290 280 270 260 250 240
FIG. 89. Path of Mars in the Summer of 1907
172 LABORATORY ASTRONOMY
In order to form an idea of the path described by the planet
among the stars, the positions may be plotted on an ecliptic map, as
in Fig. 89, which shows the form of the loop in the constellation
of Sagittarius.
During March the motion of the planet is eastward, or in the
direction of increasing longitudes, and is said to be " direct." The
rate of motion diminishes from one-half degree per day at the out-
set to half that amount in May, and soon after the beginning of
June the planet reaches its first " stationary point " and begins to
move slowly in the opposite direction in longitude, or "retrograde."
Its continuous motion in latitude toward the south prevents it from
exactly retracing its path and causes it to describe a "loop."
Its velocity in the retrograde arc increases to a maximum of
a quarter of a degree per day at opposition early in July, and
then decreases until the second stationary point is reached about
August 9, when the planet resumes its direct motion.
The exact dates of the stationary points may be found by com-
puting a few places in the neighborhood of June 10 and August 9.
The Ephemeris gives the dates as June 5 and August 8.
THE MOTIONS OF THE PLANETS
173
TABLE III AVERAGE VALUES OF THE SUN'S LONGITUDE AND
THE EQUATION OF TIME
LONGITUDE
LONGITUDE
MEAN LONGITUDE
EQ. OF TIME
Jan. 1 . . .
280. 3
>J 10. 3
280. 1
+ 3">. 5
11 . . .
290 .5
>? 20 .5
289 .9
+ 7 .9
21 ...
300 .7
XX 30 .7
299 .8
+ 11 .3
31 ...
310 .8
SZS 10.8
309 .6
+ 13 .6
Feb. 10 ...
320 .9
SXH 20 .9
319.5
+ 14 .4
20 ...
331 .1
X 1 .1
329 .4
+ 14 .0
Mar. 2 ...
341 .3
X 11.3
339 .5
+ 12 .4
12 ...
351 .4
X 21 .4
349 .3
+ 10 .0
22 .
1 .3
f> 1 .3
359 .2
+ 7 .1
April 1 ...
11 .2
| 11 .2
9 .0
+ 4 .1
11 .
21 .0
P 21 .0
18.9
+ 1 .2
21 .
30 .8
8 0.8
28 .7
- 1 .2
May 1 ...
40 .6
8 10.6
38 .6
- 2 .9
11 ...
50.3
8 20.3
48 .5
- 3 .7
21 ...
59 .9
8 29.9
58 .3
- 3 .6
31 . . .
69 .5
n 9 .5
68 .2
- 2 .6
June 10 ...
79 .0
n 19 .0
78 .0
- .9
20 ...
88 .6
n 28.6
87 .9
+ 1 .2
30 ...
98 .1
EB 8 .1
97 .7
+ 3 .3
July 10 . . .
107 .7
ZB 17 .7
107 .6
+ 5 .0
20 ...
117 .2
5 27 .2
117 .5
+ 6 .1
30 ...
126 .7
1 6.7
127 .3
+ 6 .2
Aug. 9 ...
136 .3
1 16 .3
137 .2
+ 5 .4
19 ...
145.9
1 25.9
147 .0
+ 3 .6
29 ...
155 .6
TTJ? 5 .6
156 .9
+ .9
Sept. 8 ...
165 .3
H 15 .3
166 .7
- 2 .3
18 .
175 .0
-n^ 25 .0
176 .6
- 5 .8
28 ...
184 .8
4 .8
186 .5
- 9 .2
Oct. 8 ...
194 .6
:: 14 .6
196 .3
-12 .3
18 . .
204 .5
24.5
206 .2
-14 .7
28 ...
214 .5
Til 4 .5
216 .0
-16 .1
Nov. 7 ...
224 .5
TH. 14 .5
225 .9
-16 .2
17 . .
234 .6
m. 24 .6
235 .7
- 15 .0
27 ...
244 .7
t 4.7
245 .6
-12 .4
Dec. 7 . . .
254 .8
t 14 .8
255 .4
- 8 .5
17 ...
265 .0
t 25 .0
265 .3
- 3 .9
27 ...
275 .2
>? 5.2
275 .2
+ 1 .0
Jan. 6 .
285 .4
V? 15.4
285 .0
+ 5 .7
174
LABORATORY ASTRONOMY
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Mean Longitude E
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Sidereal Period T
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THE MOTIONS OF THE PLANETS
175
TABLE VII THE JULIAN DAY
Add together the year number, the month number, and the day of the month.
1800 2378496
1843 2394201
1886 2409907
1929 2425612
1972 2441317
1801 78861
1844 94566
1887 10272
1930 25977
1973 41683
1802 79226
1845 94932
1888 10637
1931 26342
1974 42048
1803 79591
1846 95297
1889 11003
1932 26707
1975 42413
1804 79956
1847 95662
1890 11368
1933 27073
1976 42778
1805 80322
1848 96027
1891 11733
1934 27438
1977 43144
1806 80687
1849 96393
1892 12098
1935 27803
1978 43509
1807 81052
1850 96758
1893 12464
1936 28168
1979 43874
1808 81417
1851 97123
1894 12829
1937 28534
1980 44239
1809 81783
1852 97488
1895 13194
1938 28899
1981 44605
1810 82148
1853 97854
1896 13559
1939 29264
1982 44970
1811 82513
1854 98219
1897 13925
1940 29629
1983 45335
1812 82878
1855 98584
1898 14290
1941 29995
1984 45700
1813 83244
1856 98949
1899 14655
1942 30360
1985 46066
1814 83609
1857 99315
1900 15020
1943 30725
1986 46431
1815 83974
1858 99680
1901 15385
1944 31090
1987 46796
1816 84339
1859 2400045
1902 15750
1945 31456
1988 47161
1817 84705
1860 00410
1903 16115
1946 31821
1989 47527
1818 85070
1861 00776
1904 16480
1947 32186
1990 47892
1819 85435
1862 01141
1905 16846
1948 32551
1991 48257
1820 85800
1863 01506
1906 17211
1949 32917
1992 48622
1821 86166
1864 01871
1907 17576
1950 33282
1993 48988
1822 86531
1865 02237
1908 17941
1951 33647
1994 49353
1823 86896
1866 02602
1909 18307
1952 34012
1995 49718
1824 87261
1867 02967
1910 18672
1953 34378
1996 50083
1825 87627
1868 03332
1911 19037
1954 34743
1997 50449
1826 87992
1869 03698
1912 19402
1955 35108
1998 50814
1827 88357
1870 04063
1913 19768
1956 35473
1999 51179
1828 88722
1871 04428
1914 20133
1957 35839
2000 51544
1829 89088
1872 04793
1915 20498
1958 36204
Month Num.
1830 89453
1873 05159
1916 20863
1959 36569
Common
Lp.
1831 ' 89818
1874 05524
1917 21229
1960 36934
Jan.
~0
1832 90183
1875 05889
1918 21594
1961 37300
Feb. 31
31
1833 90549
1876 06254
1919 21959
1962 37665
March 59
60
1834 90914
1877 06620
1920 22324
1963 38030
April 90
91
1835 91279
1878 06985
1921 22690
1964 38395
May 120
121
1836 91644
1879 07350
1922 23055
1965 38761
June 151
152
1837 92010
1880 07715
1923 23420
1966 39126
July 181
182
1838 92375
1881 08081
1924 23785
1967 39491
Aug. 212
213
1839 92740
1882 08446
1925 24151
1968 39856
Sept. 243
244
1840 93105
1883 08811
1926 24516
1969 40222
Oct. 273
274
1841 93471
1884 09176
1927 24881
1970 40587
Nov. 304
305
1842 93836
1885 09542
1928 25246
1971 40952
Dec. 334
335
n.
MARCH, 1899.
AT GREENWICH MEAN NOON.
THE SUN'S
|
|
Equation of
Sidereal
0)
Time,
to be
Time,
or
1
3
Subtracted
Right Ascension
8
*J
Apparent
Diff. for
Apparent
Diff. for
from
Diff. for
of
S
Q
Right Ascension.
i Hour.
Declinatioa
i Hour.
Mean Time.
i Hour.
Mean .Sun.
Wed.
I
h m s
22 48 49.18
s
S. 7 33 9-5
+56.99
m s
12 31.64
8
0.496
h m 8
22 36 17-54
Thur.
2
22 52 33-57
9-34
7 10 18.4
57.26
12 19.48
O.5l6
22 40 14.09
Frid.
3
22 56 17.49
9-320
6 47 21.0
57-51
12 6.84
0.536
22 44 10.64
Sat.
4
23 o 0.95
9.301
6 24 17.9
+57-75
ii 53-75
0-555
22 48 7.20
SUN.
5
23 3 43-98
9.284
6 i 9.3
57-97
ii 40.23
0.572
22 52 3-75
Mon.
6
23 7 26.60
9.267
5 37 55-6
58.17
ii 26.29
0.589
22 56 0.30
Tues.
7
23 ii 8.82
9-251
5 H 37-3
+58.35
ii 11.96
O.6O5
22 59 56.86
Wed.
8
23 14 50.66
9.236
4 5i 14-8
58.52
10 57.25
O.62O
23 3 53-41
Thur.
9
23 18 32.14
9.221
4 27 48.4
58.67
10 42.17
0.635
23 7 49-96
Frid.
10
23 22 13.27
9.207
4 4 18.6
+58.80
10 26.75
0.649
23 II 46.52
Sat.
ii
23 25 54.08
9.194
3 4 45-8
58.92
IO II.OI
0.663
23 15 43-07
SUN.
12
23 29 34-57
9.181
3 17 10.3
59-02
9 54-95
0.675
23 19 39-62
Mon.
13
23 33 14.77
9.169
2 53 32-7
+59-10
9 38.60
0.687
23 23 36.17
Tues.
4
23 36 54-69
9.158
2 29 53.2
59-17
9 21.96
0.698
23 27 32.73
Wed.
23 4 34-35
9.147
2 6 12.3.
59-22
9 5-7
0.709
23 31 29.28
Thur.
16
23 44 13.76
9-137
I 42 30-4
+59.26
8 47-93
0.719
23 35 25-83
Frid.
17
23 47 52.95
9.128
I 18 47.8
59.28
8 30.56
0.728
23 39 22.38
Sat.
18
23 51 31.93
9.120
o 55 4-9
59.28
8 12.99
0.736
23 43 18.94-
SUN.
19
23 55 10-72
9-"3
31 22.2
+59-27
7 55-23
0-744
23 47 15-49
Mon.
20
23 -58 49.34
9.106
S. o 7 39.8
59-25
7 37-30
0.750
23 51 12.04
Tues.
21
2 27.82
9.100
N. o 16 1.7
59-21
7 19.22
0.756
23 55 8.60
Wed.
22
o 6 6.17
9.095
o 39 42.1
4.59.15
7 1.02
0. 7 6l
23 59 5-15
Thur.
23
o 9 44.41
9.092
i 3 21. o
59-o8
6 42.71
0.765
o 3 1.70
Frid.
2 4
o 13 22.57
9.089
I 26 58.1
59.00
6 24.32
0.767
o 6 58.26
Sat.
25
o 17 0.68
9.087
i 50 33-o
+58.90
6 5-87
0.769
o 10 54.81
SUN.
26
o 20 38.75
9.086
2 H 5'5
58.79
5 47-39
0.770
o 14 51.36
Mon.
27
o 24 16.81
9.086
2 37 35-i
58.67
5 28.90
0.770
o 18 47.91
Tues.
28
o 27 54.88
9.087
3 i 1.6
+58.53
5 10.41
0.769
o 22 44.47
Wed.
29
o 31 32.99
9.089
3 24 24.7
58.38
4 5i-97
0.767
o 26 41.02
Thur.
3
o 35 n. 16
9.092
3 47 44-o
58.22
4 33-59
0.764
o 30 37-57
Frid.
o 38 49.41
9.096
4 10 59.0
58.04
4 J 5- 2 9
O.760
o 34 34-12
Sat.
32
o 42 27.77
9.101
N. 4 34 9.7
+57-84
3 57.09
0.756
o 38 30.68
NOTE. The semidiam&er for mean noon may be assumed the same as that for apparent noon.
Diff. for i Hoar,
The sign + prefixed to the hourly change of declination indicates that south declinatioas are
+ 9 8 .856s.
decreasing, north declinations increasing.
(Table m.)
176
II.
JANUARY, 1900.
AT GREENWICH MEAN NOON.
THE SUN'S
i
4
Equation of
Sidereal
a
Time,
Time,
9
o
to be
or
*5
.c
Subtracted
Right Ascension
3
3
Apparent
Diff. for
Apparent
Diff for
from
Diff. for
of
8
fr
a
Right Ascension
i Hour.
Declination.
i Hour.
Mean Time.
i Hour.
Mean Sua
Mon.
i
h m s
18 46 23.63
s
11.045
S.23 I 23.1
+12.24
m s
3 40-17
1.190
h m s
18 42 43.46
Tues.
2
18 50 48.55
11.031
22 56 15.4
13-39
4 8 -53
1.175
1 8 46 40.02
Wed.
3
18 55 13.12
11.015
22 50 40.4
14-53
4 36.54
1.159
18 50 36.58
Thur.
4
18 59 37.28
10.998
22 44 38.1
+15.66
5 4-15
1.141
18 54 33-13
Frid.
5
19 4 1.04
10.980
22 38 8.7
16.78
5 31-35
1. 122
18 58 29.69
Sat.
6
19 8 24.33
10.961
22 31 12.6
17.89
5 ,58-08
I.I03
19 2 26.25
SUN.
7
19 12 47.13
10.940
22 23 49.8
+19.00
6 24.33
1.082
19 6 22.81
Mon.
8
19 17 9.41
10.918
22 l6 0.6
20.10
6 50.05
1. 060
19 10 19.36
Tues.
9
19 21 31.17
10.895
22 7 45-3
21. l8
7 15-25
1.037
19 14 15.92
Wed.
10
19 25 52.36
10.870
21 59 4-0
+22.25
7 39-87
I.OI4
19 18 12.48
Thur.
1 1
19 30 12.94
10.845
21 49 57.1
23-31
8 3-90
0.990
19 22 9.04
Frid.
12
19 34 32.93
10.819
21 40 24.8
24-37
8 27.34
0.964
19 26 5.59
Sat.
13
19 38 52.30
10.793
21 30 27.3
+25.41
8 50.14
0-937
19 30 2.15
SUN.
14
19 43 ii. 01
10.766
21 2O 4.9
26.43
9 12.30
0.910
19 33 58-71
Mon.
15
19 47 29.07
10.738
21 9 18.0
27-45
9 33-8i
0.882
19 37 55-26
Tues.
16
19 51 46.45
10.710
20 58 6-7
+28.46
9 54-63
0.854
19 41 51.82
Wed.
17
19 56 3.15
10.681
20 46 31.4
29.46
10 14.77
0.825
19 45 48.38
Thur.
18
20 .0 19.15
10.651
20 34 32-5
30.44
10 34.21
0-795
19 49 44.94
Frid.
19
20 4 34.42
10.621
20 22 IO.I
+31.41
10 52.93
0.765
19 53 41.49
Sat.
20
20 8 48.98
10.591
20 9 24.7
32.37
ii 10.94
0-735
19 57 38.05
SUN.
21
20 13 2.79
10.560
19 56 16.6
33-31
ii 28.18
0.704
20 I 34.61
Mon.
22
2O 17 15.86
16.529
19 42 46.1
+34-23
ii 44.70
0.672
20 5 31.16
Tues.
23
2O 21 28.17
10.497
19 28 53.5
35-14
12 0.45
0.641
20 9 27.72
Wed.
24
20 25 39.7?
10.465
19 14 39-3
36.04
12 15.44
0.609
20 13 24.28
Thur.
25
20 29 50.49
10.433
19 o 3.8
+36.92
12 29.66
0.577
20 17 20.83
Frid.
26
20 34 0.48
10.400
1 8 45 7.4
37-78
12 43.09
0.544
2O 21 17.39
Sat.
27
20 38 9.69
10.367
18 29 50.4
38.62
12 55.74
0.511
20 25 13.94
SUN.
28
20 42 18.10
10.334
18 14 13.3
+39-45
13 7-59
0.478
20 29 10.50
Mon.
29
20 46 25.71
10.300
17 58 16.5
40.27
13 18.64
0.444
20 33 7.06
Tues.
3
20 50 32.49
10.266
17 42 0.4
41.07
13 28.88
0.410
20 37 3.61
Wed.
31
20 54 38.46
10.232
17 25 25.4
41.85
13 38.29
0.376
20 41 0.17
Thur.
32
20 58 43.61
10.198
S. 17 8 31.9
+42.61
13 46.89
0.341
20 44 56.72
NOTE. The semidiameter for mean noon may be assumed the same as that for apparent noon.
Diff. for i Hour,
The sign + prefixed to the hourly change of declination indicates that south declinations are
decreasing
(Table III.)
177
JANUARY, 1900,
III.
AT GREENWICH MEAN NOON.
Mean Time
of
Sidereal Noon
Day of the Month.
I
3
1
s x
THE SUN'S
Logarithm
of the
Radius Vector
of the
Earth.
Diff. for
i Hour.
TRUE LONGITUDE.
DiflE. for
i Hour.
LATITUDE.
A
A'
I
2
3
2
3
152.96
152.96
152.96
4-0.26
0.40
0.50
9.9926699
9.9926694
9.9926706
- 0.6
+ O.I
0.8
h m s
5 16 24.56
5 12 28.65
5 8 32.74
280 40 8.7
28l 41 20.1
282 42 31.4
39 51-2
41 2.3
42 13.4
4
I
4
5
6
283 43 42.4
284 44 53.1
285 46 3.5
43 24.3
44 34.8
45 45-o
I52-95
152.94
152.92
4-0-59
0.65
0.67
9.9926734
9.9926780
9.9926845
+ 1.5
2-3
3-i
5 4 36-83
5 o 40.91
4 56 45-0
7
8
9
7
8
9
286 47 13.3
287 48 22.7
288 49 31.5
46 54-7
48 3-9
49 12.5
152.90
152.88
152.85
+ 0.65
O.6o
0.51
9.9926929
9.9927034
9.9927163
+ 3-9
4.9
5-9
4 52 49-og
4 48 53-i8
4 44 57-27
10
ii
12
10
ii
12
289 50 39.7
290 51 47.3
291 52 54.4
50 20.6
51 28.0
52 34-9
152.83
152.81
152.79
+ 0.42
0.30
0.17
9.9927317
9.9927495
9.9927699
+ 6.9
8.0
9.0
4 4 1 1-36
4 37 5-44
4 33 9-53
*3
H
15
13
H
15
292 54 0.8
293 55 6.7
294 56 1 2. 1
53 41-2
54 46-9
55 52.r
152.76
152.74
152.71
+ 0.04
O.og
O.2O
9.9927930
9.9928190
9.9928476
+ 10.1
W*3
12.5
4 29 13.62
4 25 17.71
4 21 21.80
16
17
18
16
17
18
295 57 16.9
296 58 21.2
297 59 25.0
56 56-7
58 0.9
59 4-5
152.69
152.67
152.65
-0.31
0-39
0.45
9.9928790
9.9929131
9.9929500
+13.6
14.8
15-9
4 17 25.89
4 J 3 29.98
4 9 34-07
19
20
21
19
20
21
299 o 28.3
300 1-31.2
301 2 3.3.5
o 7.7
i 10.4
2 12.6
152.63
152.61
152.59
0.48
0.48
0.46
9.9929894
9.9930314
9.9930759
+17.0
18.0
19.0
4 5 38.16
4 i 42.24
3 57 46.33
22
23
2 4
22
23
24
302 3 35-4
303 4 36-7
304 5 37-5
3 14-3
4 15-5
5 16.1
152.57
152.54
152.52
-0,41
0-34
0.23
9.9931227
9-993I7I9
9.9932232
+20. o
20.9
21.8
3 53 50-42
3 49 54-51
3 45 58.60
3
27
25
26
27
305 6 37.8
306 7 37-5
307 8 36.5
6 16.2
7 15-7
8 14.6
152.50
152.48
152.45
O.I I
-f- 0.02
0.15
9.9932765
9.9933316
9-9933886
4-22.6
23.4
24.1
3 42 2.69
3 38 6.78
3 34 10.87
28
29
3
31
28
29
3
31
308 9 34.8
309 10 32.4
310 ii 29.0
311 12 24.7
9 12.8
10 10.2
II 6.7
12 2.2
152.42
152.38
152.34
152.30
+ 0.29
0.44
o-55
0.63
9.9934471
9-993507I
9.9935684
9.9936309
+24.7
25.3
25.8
26.3
3 30 14-96
3 26 19.05
3 22 23.14
3 18 27.23
32
32
312 13 19.2
12 56.6
152.25
+ 0.68
9.9936948
+26.9
3 H 3I-32
NOTE. The numbers in column A correspond to the true equinox of the date; in column A' to the
Diff. for i Hour,
-9 a .82g6.
(Table II.)
178
IV.
JANUARY, 1900.
GREENWICH MEAN TIME.
THE MOON'S
J3
c
5
1
SEMIDIAMETER.
HORIZONTAL PARALLAX.
UPPER TRANSIT.
AGE.
>>
B
Noon.
Midnight.
Noon.
Diff. for
i Hour
Midnight.
Diff. for
i Hour.
Meridian of
Greenwich.
Diff. for
i Hour.
Noon.
i
, .,
.
,
b ra
m
d
I
16 19.6
16 23.4
59 48-5
+1-33
60 2.8
+ 1.05
6
29-5
2
16 26.4
16 28.3
60 13.6
0.75
60 2O.7
+0.44
o 57-9
2.43
O.g
3
16 29.2
16 29.2
60 24.1
+0.13
60 23.8
-0.17
i 55-o
2-33
1.9
4
16 28.1
16 26.2
60 20.0
-0.45
60 13.0
-0.70
2 49.6
2.22
2.9
5
16 23.6
16 20.2
60 3.2
0.92
59 50-9
I.IO
3 4*-9
2.14
3-9
6
16 16.3
16 12.0
59 36.7
1.25
59 20.9
1.36
4 32.8
2.IO
4-9
7
16 7.4
16 2.6
59 4-o
-1.44
58 46-4
-1.48
5 23.1
2.10
5-9
8
*5 57-7
15 52-8
58 28.4
1.50
58 10.2
1.50
6 13.9
2.1 3
6.9
9
15 47-9
15 43-i
57 52-2
1.49
57 34-5
1.46
7 5-6
2.18
7-9
10
15 38.4
15 33-8
57 17-2
-1.43
57 0.3
-1.38
7 58-5
2.22
8.9
ii
15 2 9-3
15 25.0
56 44.0
J-34
56 28.2
1.29
8 52.2
2.24
9.9
12
15 20.9
15 16.9
56 13-0
1.24
55 58.4
1.19
9 45-8
2.22
10.9
13
15 i3-i
15 9-5
55 44-4
-1.14
55 3i-i
-1.09
10 38.4
2.16
11.9
H
15 6.0
J5 2.7
55 18.3
1.03
55 6.2
0.98
ii 29.1
2.06
12.9
15
H 59-6
H 56.7
54 54-8
0.92
54 44- 2
0.85
12 17.2
1.95
13-9
16
H 54- 1
H 5i-7
54 34-5
-0.77
54 25-7
-0.68
13 2.7
1.8 5
14.9
*7
14 49.6
14 47.9
54 '8.1
-58
54 u-8
0.47
13 46.1
1.76
15-9
18
14 46.5
H 45-6
54 6.9
0.34
54 3-5
-O.2I
14 27.8
I.7I
16.9
19
14 45.2
14 45.2
54 i-9
0.06
54 2.1
+0.10
15 8. 5
1.69
17.9
20
H 45-9
14 47.1
54 4-4
+0.28
54 8.8
0.46
15 49-3
I. 7 I
18.9
21
14 48.9
H 5i-4
54 15-5
0.65
54 24.6
0.85
16 30.9
1.76
19.9
22
H 54-5
H 58.3
54 36-1
+1.06
54 50-1
+ 1.27
17 14-3
1.86
20.9
23
15 2.8
15 7-9
55 6.5
1.48
55 25.4
1.6 7
1 8 0.4
1.99
21.9
24
15 13-7
15 20.0
55 46-5
1.85
56 9-9
2. 02
18 49.8
2.14
22.9
25
15 26.9
15 34-2
56 35-i
+2.17
57 i-9
+2.29
19 43.0
2.29
23-9
26
15 4i-8
15 49-7
57 30-0
2.37
57 58.8
2.41
20 39-5
2.42
24.9
27
15 57-6
16 5.4
58 27.8
2.40
58 56-4
2.34
21 38.5
2.49
25.9
28
16 12.9
16 19.9
59 23-9
+2.22
59 49-6
+2.04
22 38.3
2.48
26.9
29
16 26.2
16 31.6
60 12.8
I.8l
60 32.8
1-52
23 37-3
2-43
27.9
30
16 36.0
16 39.3
60 49.0
I.lS
61 0.9
+0.80
6
28.9
31
16 41.3
16 41.9
61 8.2
+0.40
61 10.6
o.oo
o 34.5
2.34
0.4
32
16 41-3
16 39-4
61 8.3
-0.39
61 1.3
-0.75
I 29.8
2.26
1.4
179
JANUARY, 1900.
VII
GREENWICH MEAN TIME.
THE MOON'S RIGHT ASCENSION AND DECLINATION.
Hour.
Right
Ascension.
Diff for
i Minute.
Declination.
Diff. for
i Minute.
Hour.
Right
Ascension.
Diff. for
i Minute.
Declination.
Diff. for
I Minute.
TUESDAY 9.
THURSDAY n.
h m
2 5 1.09
2.2544 N.l6 38 10.6
9.442
o
h m 8
3 55 9-21
2.3237
N.22 4 39.6
3*944
i
2 7 16.41
8.2563
16 47 34.2
9-343
i
3 57 28.65
2.3242
22 8 32.5
3.818
2
2 9 3I-85
2.258?
16 56 51.8
9.243
2
3 59 48-12
2.3247
22 12 17.8
3*692
3
2 ii 47-39
2.2600
17 6 3.4
9.142
3
4 2 7-62
2.3252
22 15 55-5
3.566
4
2 14 3.05
2.2619
17 15 8.9
9.040
4
4 4 27.14
2.3254
22 19 25.8
3.440
5
2 16 18.82
2.2637
17 24 8.2
8.937
5
4. 6 46.67
2.3257
22 22 48.4
3.3I3
6
2 18 34.70
2.2656
17 33 1-4
8.835
6
9 6.22
2.3260
22 26 3.3
3.186
7
2 2O 50.69
2.2675
17 41 48.4
8.731
7
ii 25.79
2.3262
22 29 IO.7
3.059
8
2 23 6.8O
2.2693
17 50 29.1
8.627
8
13 45-36
2.3262
22 32 10.4
2.932
9
2 25 23.01
2.2712
17 59 3-6
8.521
9
16 4.93
2.3262
22 35 2.5
2.805
10
2 27 39.34
2.2730
18 7 31.6
8.414
10
18 24.50
2.3262
22 37 47.0
8.677
ii
2 29 55.77
2.2748
'8 15 53-3
8.307
ii
4 20 44.07
2.3261
22 40 23.8
2-549
12
2 32 12.32
2.2767
18 24 8.5
8.199
12
4 23 3-63
2.3259
22 42 52.9
2.422
*3
2 34 28.98
2.2785
18 32 17.2
8.091
13
4 25 23.18
2.3257
22 45 14.4
2.295
4
2 36 45-74
2.2802
18 40 19.4
7.982
M
4 27 42.71
2.3254
22 47 28.3
2.167
5
2 39 2.61
2.2820
18 48 15.1
7.873
15
4 30 2.23
2.3251
22 49 34-5
2.039
16
2 41 19.58
2.2837
18 56 4.2
7.76a
16
4 32 21.72
2.3246
22 51 33-o
I.9
7
2 43 36.66
2.2856
19 3 46-6
7.651
17
4 34 41.18
2.3241
22 53 23.9
1.784
18
2 45 53.85
2.2873
19 ii 22.4
7-539
18
4 37 0.61
2.3235
22 55 7-1
1.657
9
2 48 11.14
2.2890
19 18 51-4
7-427
19
4 39 20.00
2.3229
22 56 42.7
1.529
20
2 5 28.53
2.2907
19 26 13.7
7.315
20
4 41 39.36
2.3222
22 58 10.6
1.401
21
2 52 46.02
2.2923
19 33 29.2
7.201
21
4 43 58.67
2.32:4
22 59 30.8
1.273
22
2 55 3-6:
2.2940
19 40 37.8
7.086
22
4 46 17-93
2.3207
23 43-4
1. 147
23
2 57 21.30
2.2956
N.I9 47 39-5
6.971
23
4 48 37-15
2.3197
N.23 i 48.4
1.019
WEDNESDAY 10.
FRIDAY 12.
2 59 39.08
2.2972
N.I9 54 34-3
6.856
4 50 56-30
2.3187
N.23 2 45-7
0.892
i
3 i 56-96
2.2987
20 I 22.2
6.741
I
4 53 I5-40
2.3177
23 3 35-4
0.764
2
3 4 14-93
2.3003
20 8 3.2
6.624
2
4 55 34-43
2.3166
23 4 17-4
0.637
3
3 6 33.00
2,3018
20 I 4 37.1
6.507
3
4 57 53-39
s-3'54
23 4 51-9
0.511
4
3 8 5i-i5
2.3032
2O 21 4.0
6.389
4
5 o 12.28
-3Ma
23 5 18.7
0.384
5
3 'I 9-39
2.3047
2O 27 23.8
6.271
5
5 2 31.09
2.3128
23 5 38.o
0.258
6
3 13 27.72
2.3061
20 33 36-5
6.152
6
5 4 49-82
2.3115
23 5 49-7
0.132
7
3 15 46-12
2.3074
20 39 42.1
6.033
7
5 7 8.47
2.3102
23 5 53-8
+ 0.006
8
3 18 4.61
2.3087
20 45 40.5
5.913
8
5 9 27.04
2.3087
23 5 50-4
- 0.120
9
3 20 23.17
2.3100
20 51 31.7
5-793
9
5 " 45-51
2.3070
23 5 39-4
0.246
10
3 22 41.81
2.3113
20 57 15.7
5.672
10
5 M 3-88
.3053
23 5 20.9
0.371
ii
3 25 0.53
2.3125
21 2 52.4
5-55*
ii
5 16 22.15
2.3037
23 4 54-9
0.495
12
3 27 19.31
2.3136
21 8 21.9
5.43I
12
5 18 40.32
2.3019
23 4 21.5
0.619
13
3 29 38.16
2.3147
21 13 44.1
5.308
13
5 20 58.38
2.3001
23 3 40.6
0-744
14
3 3i 57.o8
2.3158
21 18 58.9
5.186
14
5 23 16.33
2.2982
23 2 52.2
0.868
15
3 34 16.06
2.3168
21 24 6.4
5.064
15
5 25 34.16
2.2962
23 * 56-4
0.992
16
3 36 35-io
2.3178
21 29 6.6
.941
16
5 27 51.87
2.2942
23 o 53.2
1.115
17
3 38 54-20
2.3187
21 33 59-3
.8.7
17
5 30 9-46
2.2921
22 59 42.6
1.237
18
3 4i 13-35
2.3196
21 38 44.6
.693
18
5 32 26.92
2.2899
22 58 24.7
1.360
19
3 43 32.55
2.3204
21 43 22.5
.569
19
5 34 44.25
8.2877
22 56 59-4
1.482
20
3 45 51.80
2.3212
21 .47 52.9
445
20
5 37 i-44
2.2853
22 55 26.8
1.604
21
3 48 ii. 10
2.3220
21 52 15.9
.320
21
5 39 18.49
2.2830
22 53 46-9
1.725
22
3 50 30-44
2.3226
21 56 31.3
.194
22
5 4i 35.40
2.2807
22 51 59.8
1.845
23
3 52 49.8i
2.3231
22 39.2
4-069
23
5 43 52-17
2.2782
22 50 5-5
1.965
24
3 55 9-21
2.3237
N.22 4 396
3-944
24
5 46 8.79
2.2757
N.22 48 4.0
2.085
180
VI.
FEBRUARY, 1900.
GREENWICH MEAN TIME.
THE MOON'S RIGHT ASCENSION AND DECLINATION.
Hour.
Right
Ascension.
Diff.for
t Minute.
Declination.
Diflf. for
i Minute.
Hour.
Right
Diff.for
i Minute.
Declination.
Diff.for
i Minute.
MONDAY 5.
WEDNESDAY 7.
i m s
s
O t n
h m s
8
O * M
*
51 20.62
a. 295*
N.is 26 59.8
10.261
o
3 42 30*05
8.3283
N.2I 30 28.1
4-685
i
53 38-36
2.2962
15 37 12.4
10. 159
i
3 44 49-75
2.3283
21 35 5-4
4.558
2
55 56-16
8.2971
15 47 18.9
10.057
2
3 47 9-45
2.3283
21 39 35-1
4*432
3
58 14.01
2.2981
15 57 19-3
9*954
3
3 49 29.15
2.3282
21 43 57-2
4.305
4
o 31-93
a. 2991
16 7 13.4
9.850
4
3 5i 48.84
2.3280
21 48 11.7
4*179
5
2 49.90
2.3001
16 17 1.3
9*745
5
3 54 8.51
3.3378
21 52 18.7
4.052
6
5 7-94
3.3012
16 26 42.8
9.639
6
3 56 28.17
2.3376
21 56 18.0
3.935
7
7 26.04
2.3021
16 36 18.0
9-533
7
3 58 47-82
3.3272
22 O 9.7
3.798
8
9 44.19
2.3030
16 45 46.8
9.427
8
4 * 7-44
2.3268
22 3 53.8
3.672
9
12 2.40
2.3040
16 55 9.2
9.319
9
4 3 27.04
2.3264
22 7 30.3
3.544
10
14 20.67
2.3050
17 4 25.1
9.210
10
4 5 46-61
3.3259
22 10 59.2
3.417
ii
16 39.00
2.3059
17 3 34-4
9.100
ii
4 8 6.15
2.3254
22 14 20.4
3.290
12
18 57.38
2.3069
17 22 37.1
8.990
12
4 10 25.66
2.3248
22 17 34.0
3.163
13
21 15.83
2.3080
17 3i 33-2
8.880
13
4 12 45.13
2.3242
22 20 39.9
3-035
14
23 34-34
3.3089
17 40 22.7
8.769
M
4 15 4-57
2.3236
22 23 38.2
8.908
15
25 52-90
2.3098
17 49 5-5
8.657
5
4 17 23.96
2. 3 228
22 26 28.9
.781
16
28 11.52
2.3108
17 57 4!-5
8.544
16
4 19 43-3-
2.3221
22 29 II.9
S.653
17
30 30.20
2.3117
18 6 10.8
8.431
17
4 22 2.6l
2.3212
22 31 47-3
8.537
18
32 48-93
3.3126
18 14 33.2
8.317
18
4 24 21.86
2.3204
22 34 15.1
8.400
19
35 7.7i
2.3135
18 22 48.8
8.202
19
4 26 41.06
2.3195
22 36 35.3
a.373
20
37 26.55
2.3144
18 30 57-5
8.087
20
4 29 0.20
8.3184
22 3 8 47 .8
8.145
21
39 45-44
2.3153
18 38 59-3
7.972
21
4 31 19.27
8.3173
22 40 52.7
3.018
22
42 4-39
2.3162
18 46 54.1
7.856
22
4 33 38-28
3.3163
22 42 50.O
1.892
2 3
44 23-39
2.3170
N.i8 54 42.1
3-739
23
4 35 57-23
2.3152
N.22 44 39.7
1.765
TUESDAY 6.
THURSDAY 8.
o
46 42.43
2.3177
tf.ig 2 22.9
7.622
o
4 38 16.10
2-3139
N.22 46 21.8
1.638
i
49 1-52
8.3186
19 9 56.7
7.504
i
4 40 34.90
2.3127
22 47 56.3
1.512
2
51 20.66
3.3194
19 17 23.4
7-386
2
4 42 53-62
2.31I3
22 49 23.2
1.386
3
53 39.85
2.3202
19 24 43.0
7-267
3
4 45 12.26
2.3100
22 50 42.6
1.260
4
55 59-oS
8.3208
19 3i 55-5
7.148
4
4 47 30-82
2.3086
22 51 54-4
1.134
5
58 18.35
2.3215
19 39 0.8
7.028
5
4 49 49-29
2.3071
22 52 58.7
1.008
6
3 o 37.66
2.3222
19 45 58.9
6.908
6
4 52 7-67
3.305J
22 53 55-4
o.88z
7
3 2 57.01
2.3228
19 52 49.8
6.787
7
4 54 25.95
3.3039
22 54 44.6
0-757
8
3 5 16.40
8.3234
*9 59 33-4
6.666
8
4 56 44-14
2.3023
22 55 26.3
0.632
9
3 7 35-82
2.3240
20 6 9.7
6.545
9
4 59 2.23
2.3007
22 56 0.5
0.508
10
3 9 55-28
2.3246
20 12 38.8
6.424
10
5 I 20.22
2.2989
22 56 27.3
0.384
ii
3 12 14.77
8.3251
20 19 0.6
6.303
ii
5 3 38-10
2.2971
22 56 46.6
0.260
12
3 14 34-29
2.3256
20 25 15.0
6.178
12
5 5 55-87
2.2932
22 56 58.5
0.136
13
3 16 53.84
2.3260
20 31 22.0
6.056
13
5 8 I3-52
2.2933
22 57 2.9
+ 0.012
14
3 9 i3-4i
2.3264
20 37 21.7
3*933
14
5 10 31.06
3.2913
22 56 59-9
0.1(1
15
3 21 33.01
2.3268
20 43 14.0
5.8o8
15
5 12 48-48
2.2893
22 56 49.6
0.233
16
3 23 52-63
2.3271
20 48 58.8
5-685
16
5 15 5-78
2.2872
22 56 31.9
0.357
17
3 26 12.26
2.3273
20 54 36.2
S.56I
J 7
5 17 22.95
2.2851
22 56 6.8
0*479
18
3 28 31.91
2.3277
21 O 6.1
3-436
18
5 19 39-99
2.2829
22 55 34-4
0.601
19
3 30 51-58
8.3279
21 5 28.5
5.3i*
19
5 21 56.90
2.2807
22 54 54-7
0.733
20
3 33 "-26
2.3281
21 10 43.5
3-187
20
5 24 13-67
8.2784
22 54 7-7
0.843
21
3 35 30-95
2.3282
21 15 50.9
5.061
21
5 26 30.31
2.2761
22 53 13-5
0.963
22
3 37 50-65
2.3283
21 20 50.8
4-936
22
5 28 46.80
2.2737
22 52 12. 1
1.084
23
24
3 40 10.35
3 42 30.05
2.3283
2.3283
21 25 43.2
N.2I 30 28.1
4.8u
4.685
23
24
5 3 3-i5
5 33 19.36
8.3713
8.2689
22 5 3-4
N.22 49 47.6
1.204
1.323
181
XVIII.
FEBRUARY, 1900.
GREENWICH MEAN TIME.
LUNAR DISTANCES.
**
P
Name and Direction
of Object
Midnight
P.L.
of
Diff.
XVk
P.L..
of
Diff.
XVIIIk
P.L.
of
Diff.
XXIk
P.L,
Diff.
18
Pollux W.
86 46 29
3081
88 15 2
3076
89 43 41
3069
91 12 28
3063
Regulus W.
49 49 o
3051
51 18 10
344
52 47 28
3037
54 16 55
3030
Antares E .
50 7 41
3038
48 38 15
3032
47 8 42
3026
45 39 2
3022
JUPITER E .
50 57 29
3062
49 28 33
3056
47 59 30
3050
46 30 19
3044
SATURN E .
74 45 5 1
3050
73 16 4
3044
71 47 22
3038
70 17 57
33i
a Aquilae E .
102 48 44
3500
101 28 20
3489
ioo 7 43
3479
98 46 55
3468
19
Pollux W.
98 38 27
3027
100 8 6
3020
K 37 54
301 r
I3 7 53
3003
Regulus W.
61 46 34
2989
63 17 i
2979
64 47 40
2969
66 18 31
2960
Antares E
38 8 59
2992
36 38 36
2985
35 8 4
2977
33 37 23
2971
JUPITER E .
39 2 19
3007
37 32 15
998
36 2 O
2989
34 3i 34
2980
SATURN E .
62. 48 39
2994
61 18 19
2985
59 47 48
2976
58 17 5
2967
a Aquilae E
92 o 7
3421
90 38 14
3412
89 16 ii
3403
87 53 58
3395
20
Regulus W.
73 55 57
2906
75 28 8
2894
77 o 34
2882
78 33 16
2870
JUPITER E .
26 56 24
2930
25 24 43
23 52 49
2909
22 2O 41
2898
SATURN E .
50 40 26
2914
49 8 25
2903
47 36 10
2891
4 6 3 39
2878
a Aquilae E .
81 o 35
3356
79 37 28
3350
78 14 14
3343
76 50 52
3337
SUN E.
109 33 56
3275
to8 9 15
3262
106 44 19
3249
105 19 8
3235
21
Regulus W.
86 20 53
2802
87 55 18
2788
89 30 i
2773
9i 5 4
2758
Spica W.
32 18 23
2793
33 53 o
2779
35 27 56
2763
37 3 12
2747
SATURN E .
38 16 56
36 42 42
2797
35 8 10
2782
33 33 18
2766
a Aquilae E .
69 52 25
3313
68 28 29
33"
67 4 30
3309
65 40 29
3307
SUN E.
98 9 o
3162
96 42 5
3U6
95 14 5 1
3130
93 47 18
3114
22
Spica W.
45 4 53
2666
46 42 19
2649
48 20 8
2631
49 5 8 21
2613
SATURN E .
25 33 48
2686
23 56 49
2669
22 19 28
2652
20 41 44
2634
Aquilae E .
58 40 32
3322
57 16 46
3330
55 53 9
334
54 29 44
3352
SUN E.
86 24 25
3028
84 54 47
3009
83 24 45
5989
81 54 19
2971
23
Spica W.
58 15 32
2522
59 56 H
2504
61 37 21
2485
63 18 55
2467
a Aquilae E .
47 37 27
3468
46 16 27
3505
44 56 8
3548
43 36 37
3598
SUN E.
74 16 17
2876
72 43 28
2856
71 10 13
2836
69 36 32
2817
24
Spica W.
7i 53 21
2373
73 37 34
2355
75 22 14
4335
77 7 22
2317
JUPITER W.
24 51 20
2407
26 34 45
2387
28 18 39
2367
30 3 i
2348
SUN E.
61 41 44
2719
60 5 29
2699
58 28 48
2680
56 5i 4i
2661
25
Spica W.
85 59 40
2228
87 47 26
2210
89 35 38
2194
91 24 15
2177
JUPITER W.
38 51 44
2256
40 38 49
2238
42 26 20
2220
44 *4 J 7
2203
SUN E .
48 39 50
2571
47 o 15
2554
45 20 17
2538
43 39 56
2522
26
Spica W.
TOO 33 22
2101
102 24 19
2087
104 15 38
2074
106 7 17
2061
Antares W.
55 5 57
2113
56 36 37
2098
58 47 39
208 5
60 39 2
2071
JUPITER W.
53 20 ii
2126
55 10 31
2112
57 i 12
2093
58 52 14
2085
SATURN W.
29 40 6
2120
3i 30 35
2105
33 21 26
2092
35 12 37
2079
SUN E.
35 13 9
2457
33 30 55
8446
31 48 26
2438
30 5 46
2433
27
JUPITER W.
68 12 4
2030
70 4 51
2021
7i 57 52
2013
73 5i 6
2005
SATURN W.
44" 33
2025
46 26 7
2017
4 8 19 15
2009
50 12 36
2000
SUN E.
21 31 15
2439
19 4 8 3 6
2456
18 6 21
2 4 8l
16 24 41
2520
182
MARCH, 1900.
AT GREENWICH APPARENT NOON.
THE SUN'S
j
8
|
Sidereal
Equation of
i
z.
Semi-
Time,
to be
1
1
diameter
Added to
3
"S
Apparent
Diff. for
Apparent
Diff. for
Semi-
Passing
Apparent
Diff. for
I
.1
Right Ascension.
i Honr.
Declination,
diameter.
Meridian,
Time.
i Hour.
Thur
I
h m s
22 47 57.02
s
9-37 1
S. 7 38 24.5
+56.96
16 9.26
65-37
m
12 34.70
.485
Frid
2
22 51 41.63
9-350
7 15 34-2
57,22
16 9.02
65-30
12 22.8O
.506
Sat.
3
22 55 25.75
9-330
6 52 37-9
57-47
16 8.77
65-23
12 10.41
.527
SUN.
4
22 59 9.38
9.310
6 29 35.9
+57-70
16 8.52
65.16
" 57-52
547
Mon.
5
23 2 52.53
9.291
6 6 28.7
57-91
16 8.27
65.09
II 44.16
.566
Tues
6
23 6 35.24
9.272
5 43 16.5
58.11
1 6 8.02
65.03
II 30.35
0.584
Wed.
7
23 10 17.52
9.254
5 *9 59-8
+58.29
16 7.77
64.97
II l6.I2
0.602
Thur.
8
2 3 13 59-37
9.237
4 56 39-0
58.46
16 7.52
64.91
II 1.46
0.619
Frid.
9
23 17 40.83
9.221
4 33 H-4
58.61
16 7.26
64.86
10 46.40
0.635
Sat.
10
23 21 21.91
9.206
4 9 46.5
+58.74
1 6 7.00
64.81
10 30.97
0.650
SUN.
ii
23 25 2.6 4
9.192
3 46 15-5
58.86
16 6.74
64.76
10 15.19
0.665
Mon.
12
23 28 43.04
9.178
3 22 41.9
58.96
1 6 6.48
64.71
9 59-07
0.679
Tues.
13
23 32 23.11
9.165
2 59 6.1
+59-05
16 6.22
64.66
9 42-65
0.691
Wed.
H
23 36 2.90
9.154
2 35 28.3
59.12
16 5.96
64.62
9 25.93
0.702
Thur
15
23 39 42.42
9.144
2 II 48.9
59.18
1 6 5.69
64.58
9 8.95
0.712
Frid
16
23 43 21.72
9-134
I 48 8.4
+59-23
16 5.42
64-55
8 51-73
0.721
Sat.
i7
23 47 0.80
9.125
I 2 4 27.0
59-26
16 5-15
64.52
8 34-31
0.730
SUN.
18
23 50 39.68
9.118
i o 45.0
59.27
*6 4.88
64-49
8 16.68
0.738
Mon.
19
23 54 18.38
9.113
o 37 2.8
+59-27
16 4.61
64.47
7 58-87
0.744
Tues.
20
2 3 57 5 6 -94
9.106
S. o 13 20.8
59.26
16 4.34
64-45
7 4-93
0.750
Wed.
31
o i 35-37
9.101
N. o 10 20.7
59.23
1 6 4.07
' 64.43
7 22.85
0-755
Thur.
22
o 5 I3-7I
9.098
o 34 i-3
+59-i8
16 3.79
6 4 . 4I
7 4-69
o-759
Frid.
23
o 8 51.97
9-095
o 57 40-8
59-12
16 3-51
64.40
6 46.44
0.762
Sat.
2 4
12 30.17
9.093
i 21 18.6
59-05
16 3.24
64'39
6 28.14
0.764
SUN.
25
o 16 8.33
9.092
i 44 54.6
+58.97
1 6 2.96
64.39
6 9.80
0.765
Mon.
26
o 19 46.47
3.092
2 8 28.1
58.86
16 2.69
64.38
5 5i-45
0.765
Tues.
27
o 23 24.61
9-093
2 3i 59-i
58.74
16 2.41
64.38
5 33-09
0.764
Wed.
28
o 27 2.78
9.094
2 55 27.0
+58.60
16 2.13
64.38
5 14-77
0.763
Thur.
29
o 30 41.00
9-096
3 18 51.5
58.44
16 1.86
64-39
4 56-48
0.761
Frid.
30
o 34 19.27
9-098
3 42 12.2
58.27
16 1.58
64.40
4 38.23
0-759
Sat.
31
o 37 57.60
9.101
4 5 28.7
58.09
16 1.31
64.41
4 20.05
0.756
SUN.
32
o 41 36.01
9.105
N. 4 28 40.7
+57.90
16 1.03
64.42
4 1^8
0.752
NOTE. The mean time of semidiameter passing may be found by subtracting o.i8 from the sidereal time.
The sign + prefixed to the hourly change of declination indicates that south declinations are decreasing; nonb
declinations, increasing.
183
II.
MARCH, 1900.
AT GREENWICH MEAN NOON.
Day of the Week.
Day of the Month.
THE SUN'S
Equation of
Time,
to be
Subtracted
from
Mean Time.
Diff. for
i Hour.
Sidereal
Time,
or
Right Ascension
Mean Sun.
Apparent
Right Ascensioa
Diff. for
i Hour.
Apparent
Declination.
Diff. for
i Hour.
Thur.
Frid.
Sat.
I
2
3
h m s
22 47 55.05
22 51 39.70
22 55 23.86
s
9-371
9-350
9-330
S. 7 38 36-4
7 15 46-0
6 52 49-6
57-22
57-47
12 34.80
12 22.90
12 10.51
0.485
0.506
0-527
22 35 20.24
22 39 16.80
22 43 13-35
SUN.
Mon.
Tues.
4
5
6
22 59 7-53
23 2 50.72
23 6 33.47
9.310
9.291
9.272
6 29 47.4
6 6 40.0
5 43 27.6
+57-70
57-91
58.11
II 44.27
1 1 30.46
0-547
0.566
0.584
22 47 9.90
22 5 I 6.45
22 55 3.01
Wed.
Thur.
Frid.
7
8
9
23 10 15.79
23 13 57-68
23 17 39-i8
9-254
9-237
9.221
5 20 10.7
4 56 49-7
4 33 24.9
+58.29
58.46
58.61
II 16.23
ii i-57
10 46.51
0.602
0.619
0.635
22 58 59-56
23 2 56.11
23 6 52.66
Sat.
SUN.
Mon.
10
ii
12
23 21 20.30
23 25 1.07
23 28 41.51
9.206
9.192
9.178
4 9 56.8
3 46 25.6
3 22 51.8
+58.74
58.86
58.96
10 31.08
10 15.30
9 59-i8
0.650
0.665
0.679
23 10 49.22
23 14 45-77
23 18 42.32
Tues.
Wed.
Thur.
13
H
23 32 21.63
23 36 1-47
23 39 41.04
9.165
9.154
9.144
2 59 15-7
2 35 37-7
2 II 58.0
+59-05
59.12
59.18
9 42.76
9 26.04
9 9.06
0.691
0.702
0.712
23 22 38.88
23 26 35-43
2 3 30 31-98
Frid.
Sat.
SUN.
16
17
18
23 43 20.38
23 46 59.50
23 50 38.42
9.134
9.125
9.118
I 48 17.2
1 24 35-5
i o 53.2
+59-23
59.26
59-27
8 51-84
8 34-4i
8 16.78
0.721
0.730
0.738
23 34 28.54
23 38 25.09
23 42 21.64
Mon.
Tues.
Wed.
19
20
21
23 54 17.17
23 57 55-77
o i 34.25
9.112
9.106
9.101
o 37 10.7
S. o 13 28.4
N. o 10 13.4
+59-27
59.26
59-23
7 58.98
7 22.95
0-744
0.750
0-755
23 46 18.19
23 50 14.74
23 54 "-So
Thur.
Frid.
Sat.
22
23
2 4
o 5 12.63
o 8 50.93
O 12 29.18
9.098
9-095
9-093
33 54-3
o 57 34-i
I 21 I2'.2
+59-I8
59.12
59.05
7 4-78
6 46-53
6 28.22
0-759
0.762
0.764
23 58 7-85
O 2 4.40
o 6 0.96
SUN.
Mon.
Tues,
25
26
27
o 16 7.39
o 19 45.58
o 23 23.77
9.092
9.092
9-093
i 44 48.5
2 8 22.3
2 31 53-6
+58.97
58.86
58.74
6 9.88
5 5i-5 2
5 33-i6
0.765
0.765
0.764
o 9 57-51
o 13 54.06
o 17 50.61
Wed.
Thur.
Frid.
Sat.
28
29
3
31
o 27 1.99
o 30 40.25
o 34 18.56
o 37 56.94
9.094
9.096
9.098
9.101
2 55 21.8
3 1 8 46-7
3 42 7-7
4 5 24-5
+58.60
58.44
58.27
58.09
5 14-83
4 56.54
4 38-29
4 20. 1 2
0.763
0.761
0.759
0.756
O 21 47.16
o 25 43.72
o 29 40.27
o 33 36.82
SUN.
3 2
o 41 35.40
9.105
N. 4 28 36.8
+57-90
4 2.03
0.752
o 37 33-37
NOTE. The semidiameter for mean noon may be assumed the same as that for apparent noon.
The sign + prefixed to the hourly change of declination indicates that south declinations are
decreasing ; north declinations, increasing.
Diff. for i Hour,
(Table III.)
184
II.
APRIL, 1900.
AT GREENWICH MEAN NOON.
Day of the Week.
|
i
THE SUN'S
Equation of
Time,
to be
Subtracted
Diff. for
Sidereal
Time,
Right Ascension
Mean Sun.
Apparent
Right Ascension.
Diff. for
i Hour.
Apparent
Declination.
Diff. for
j Hour.
Added to
Mean Time.
Mon.'
Tues.
2
3
h m s
o 41 35.40
o 45 13.96
o 48 52.63
9.105
9.109
9.114
N. 4 28 36.8
4 51 44.2
5 14 46-3
+57.90
57.69
57-47
4 2.03
3 44-04
3 26.15
0.752
0.748
0-743
h m s
o 37 33-37
o 41 29.93
o 45 26.48
Wed.
Thur.
Frid.
4
5
6
o 52 31.44
o 56 10.39
o 59 49.49
9.120
9.126
9-133
S 37 42-7
6 o 33.2
6 23 17.3
+57-23
56.97
56.70
3 8.40
2 50.8l
2 33.36
0-737
0.731
0.724
o 49 23.03
o 53 19-59
o 57 16.14
Sat.
SUN.
Mon.
7
8
9
i 3 28.79
i 7 8.27
i 10 47.97
9.141
9.150
9.160
6 45 54-7
7 8 25.1
7 30 48-2
+56.41
56.11
55.80
2 16.09
i 59-02
I 42.17
0.716
0.707
0.697
i i 12.69
i 5 9-24
i 9 5.8o
Tues.
Wed.
Thur.
10
ii
12
i 14 27.91
i 18 8.ii
I 21 48.58
9.170
9.181
9-193
7 53 3-7
8 15 ii. i
8 37 10.3
+55-48
55-15
54.80
I 25.57
I 9.21
53-13
0.687
0.676
0.664
1 *3 2.35
i 16 58.90
I 20 55.46
Frid.
Sat.
SUN.
Mon.
Tues.
Wed.
13
H
16
18
i 25 29.35
i 29 10.43
i 32 51-84
i 36 33.60
i 40 15.72
i 43 58.24
9.206
9.219
9-233
9.248
9.264
9.280
8 59 0.8
9 20 42.4
9 42 14-7
10 3 37.4
10 24 50.1
10 45 52.7
+54-43
54.05
+53-24
52.82
52-39
o 37-34
o 21.86
o 6.72
0.651
0.638
0.624
0.609
0-593
0.576
I 24 52.01
I 28 48.56
i 32 45-12
i 36 41-67
i 40 38.22
i 44 34.78
o 8.07
o 22.50
o 36.54
Thur.
Frid.
Sat.
19
20
21
i 47 41.17
i 51 24.51
i 55 8.29
9.297
9-3I5
9-334
ii 6 44.7
ii 27 25.9
ii 47 55.8
+51.94
51.48
51.01
o 50.16
I 3-38
I 16.15
0.559
0.541
0.523
i 48 31.33
i 52 27.88
i 56 24.44
SUN.
Mon.
Tues.
22
2 3
2 4
i 58 52.53
2 2 37.24
2 6 22.42
9-353
9-373
9-393
12 8 14.2
12 28 20.7
12 48 15.0
+50.52
50.02
49.50
I 28.46
I 40.32
I 51.68
0.504
0.484
0.464
2 20.99
2 4 J 7-55
2 8 14.10
Wed.
Thur.
Frid.
y
27
2 10 8.09
13 54.26
17 40.94
9-413
9-434
9-455
13 7 56.8
13 27 25.7
13 46 41.4
+48.97
48.43
47.87
2 2.56
2 12.95
2 22.82
0.443
0.422
0.401
2 12 10.65
2 16 7.21
2 20 3.76
Sat.
SUN.
Mon.
28
29
y-
21 28.13
25 15.84
29 4-07
9.476
9.498
9.520
14 5 43-5
14 24 31.7
14 43 5.6
+47.29
46.70
46.10
2 32.18
2 41.03
2 49.36
0.380
0.358
0.336
2 2 4 0.32
2 27 56.87
2 31 53-42
Tues.
3*
2 32 52.82
9-542
N.is i 25.0
+45-49
2 57.16
0.314
2 35 49-98
NOTE. The semidiameter for mean noon may be assumed the same as that for apparent noon.
The sign + prefixed to the hourly Change of declination indicates that north declinations are
increasing.
Diff. for i Hoar,
+ 9 '.8 5 6 5 .
(Table III.)
185
II.
AUGUST, 1900.
AT GREENWICH MEAN NOON.
THE SUN'S
Equation of
j
-a
Time,
Sidereal
1
S
to be
Subtracted
Time,
a
1
from
Right Ascension
g
Apparent
Diff. for
Apparent
Diff. for
Added to
Diff. for
of
>,
Right Ascension.
i Hour.
Declination.
i Hour.
Mean Time.
i Hour.
Mean Sun.
a
Q
Wed.
I
h m s,
8 44 41.30
s
9.716
N.i8 5 i.i
-37.61
6 8.12
s
0.140
h m s
8 38 33.18
Thur.
2
8 48 34.18
9.690
17 49 49-7
38.34
6 4.44
o.i 66
8 42 29.74
Frid.
3
8 52 26.44
9.664
17 34 21.0
39-05
6 0.15
0.191
8 46 26.29
Sat.
4
8 56 1 8.08
9-638
17 18 35.2
-39-75
5 55-23
0.217
8 50 22.85
SUN.
5
9 o 9.10
9.613
17 2 32.6
4-45
5 49-70
0.243
8 54 19-4
Mon.
6
9 3 59-5i
9.587
16 46 13.5
4I.I3
5 43-55
0.269
8 58 15.96
Tues.
7
9 7 49-3 1
9.562
16 29 38.3
-41.80
5 36-80
0.294
9 2 12.51
Wed.
8
9 " 38-5 1
9-537
1 6 12 47.1
42.46
5 29.44
0.319
9 6 9.07
Thur.
9
9 15 27.11
9-5I3
15 55 4-3
43.10
5 21.48
0-344
9 10 5.62
Frid.
10
9 19 15-12
9.489
15 38 18.2
-43-73
5 12.94
0.368
9 14 2.18
Sat.
1 1
9 23 2.56
9.465
15 20 41.0
44-35
5 3-82
0.392
9 i7 58.73
SUN.
12
9 26 49.43
9.442
15 2 49.1
44.96
4 54-14
0.415
9 21 55-29
Mon.
13
9 30 35-75
9.419
14 44 42.7
-45.56
4 43-91
0.438
9 25 51-84
Tues.
H
9 34 21.54
9-397
14 26 22.
46.15
4 33-14
0.460
9 29 48.40
Wed.
15
9 38 6.80
9-375
H 7 47-5
46.72
4 21.85
0.482
9 33 44-95
Thur.
16
9 4i 5*-55
9-354
13 48 59-4
-47-28
4 10.04
0.503
9 37 4*-50
Frid.
17
9 45 35-79
9-333
13 29 57.9
47.83
3 57-73
0.524
9 41 38.06
Sat.
18
9 49 *9-54
9.3I3
13 10 43.5
48.36
3 44-93
0-544
9 45 34-6i
SUN.
19
9 53 2.81
9-293
12 51 16.4
-48.88
3 31-64
0.564
9 49 31.17
Mon.
20
9 56 45.61
9.273
12 31 37.0
49-38
3 17-89
0.583
9 53 27.72.
Tues.
21
10 o 27.94
9.254
12 II 45.6
49.88
3 3-67
0.602
9 57 24.28
Wed.
22
10 4 9.83
9.235
II 51 42.6
-50.36
2 49.00
0.620
10 i 20.83
Thur.
23
10 7 51.27
9.217
II 31 28.2
50.83
2 33-89
0.638
10 5 17.38
Frid.
2 4
10 ii 32.28
9.199
11 II 2.8
51-28
2 18.34
0.656
10 9 13.94
Sat.
25
10 15 12.86
9.182
10 50 26.8
-51.72
2 2.37
0.674
10 13 10.49
SUN
26
10 18 53.03
9.165
10 29 40.5
52.14
i 45-99
0.691
10 17 7.04
Mon.
2 7
10 22 32.8l
9.149
10 8 44.3
52-55
i 29.21
0.707
10 21 3.60
Tues.
28
10 26 12.19
9-133
9 47 38-4
-52.94
i 12.04
0.723
10 25 0.15
Wed.
2 9
10 29 51.20
9.118
9 26 23.2
53-32
o 54.50
0.738
10 28 56.70
Thur.
30
10 33 29.86
9.104
9 4 59-i
53-69
o 36.60
0-753
10 32 53.26
Frid.
31
10 37 8.16
9.090
8 43 26.3
54.04
o 18.35
0.767
10 36 49.81
Sat.
32
10 40 46.13
9.076
N. 8 21 45.2
-54-38
o 0.23
0.781
10 40 46.36
NOTE. The semidiameter for mean noon may be assumed the same as that for apparent noon.
Diff. for i Hour,
The sign preBxed to the hourly change of declination indicates that north declinations are
+ 9'.8 5 6 5 .
decreasing.
(Table IIL)
186
SEPTEMBER, 1900
III.
AT GREENWICH MEAN NOON.
THE SUN'S
|
1
i
TRUE LONGITUDE.
Logarithm
of the
Radius Vector
Mean Time
5
i
Diff. for
LATITUDE.
of the
Diff. for
of
I
s
A
y
t Hour.
Earth.
I Hour.
Sidereal Noon
i
244
158 34 13-5
33 23.3
145-25
0.18
0.0038298
-44.4
h m s
13 17 2.70
2
245
159 32 20.2
31 29.9
145.31
0.06
0.0037217
44-9
13 13 6.80
3
246
160 30 28.3
29 37.9
M5-37
+ 0.07
0.0036125
43-4
13 9 10.89
4
247
161 28 37.8
27 47-3
M5-43
-f- O.2O
0.0035024
-45-9
13 5 I4-98
5
162 26 48.7
25 58-2
145.49
0.28
0.0033914
46.3
13 i 19.07
6
249
163 25 1.2
24 10.6
M5-55
o-35
0.0032799
46.6
12 57 23.17
7
250
l6 4 23 15.3
22 24.5
145-62
+ 0.40
0.0031678
-46.8
12 53 27.26
8
9
251
252
165 21 3I.O
1 66 19 48.4
2O 40.1
18 57-5
145.69
M5-77
0.42
0.40
0.0030554
0.0029427
46-9
47-o
12 49 31.35
12 45 35-45
10
253
167 18 7.8
17 16.7
145-85
+ 0.35
0.0028297
-47-1
12 41 39.54
ii
254
1 68 1 6 29.0
15 37-9
M5-93
0.27
0.0027166
47-2
12 37 43.64
12
255
169 14 52.3
14 i.o
146.01
0.15
0.0026032
47-3
12 33 47-73
13
256
170 13 17.6
12 26.3
146.10
+ 0.03
0.0024894
-47-5
12 29 51.82
H
257
171 ii 45.2
10 53-7
146.19
O.IO
0.0023751
47-7
12 25 55.91
15
258
172 10 14.9
9 23.4
146.28
0.23
O.OO226O3
48.0
12 22 O.OI
16
259
173 8 46.8
7 55-2
146-37
-0.35
0.0021448
-48-3
12 18 4.10
17
260
174 7 20.9
6 29.2
146.46
0.47
0.0020285
48.6
12 14 8.19
18
261
175 5 57-2
5 5-5
146.56
0.56
O.OOI9II4
49.0
12 10 12.29
19
262
176 4 35.8 v
3 43-9
146.65
0.63
0.0017934
-49-4
12 6 16.38
20
263
177 3 16.5
2 24.6
146.74
0.68
0.0016743
49.8
12 2 20.48
21
264
178 i 59-3
i 7-3
146.83
0.70
0.0015543
50.2
II 58 24.57
22
265
178 60 44.2
59 52-0
146.91
0.70
0.0014332
50.6
ii 54 28.66
23
266
179 59 31.0
58 38.8
147.00
0.66
O.OOI3II2
51.0
ii 50 32.76
24
267
1 80 58 20.0
57 27.7
147.08
0.61
O.OOII882
5'-4
ii 46 36.85
25
268
181 57 10.9
56 r8. 5
147.16
-0.53
0.0010642
-51.8
ii 42 40.94
26
269
182 56 3.7
55 "-3
147.24
0.44
0.0009394
52.1
ii 38 45.04
27
270
183 54 58-4
54 5-9
M7-32
0.33
0.0008137
52.4
ii 34 49-13
28
271
184 53 55-o
53 2.4
147-39
0.21
0.0006872
-52.7
ii 30 53.22
29
272
185 52 53-4
52 0.7
147-47
0.08
0.0005603
33-o
ii 26 57.32
30
273
186 51 53.6
51 0.8
147-54
+ 0.04
0.0004328
53-2
ii 23 1.41
31
274
187 50 55-5
50 2.6
147.62
4-0.17
0.0003049
-53-3
ii 19 5-5
NOTE. The numbers in column A correspond to the true equinox of the date ; in column A' to the
Diff. for t Hour.
OT829-6.
mean equinox of January o d .o.
(Table II.)
187
II.
NOVEMBER, 1900.
AT GREENWICH MEAN NOON.
THE SUN'S
Day of the Week.
8
?.
1
I
Equation of
Time,
to be
Added to
Mean Time.
Diff. for
Sidereal
Time,
Right Ascension
Mean Sun.
Apparent
Right Ascensioa
Diff. for
i Hour.
Apparent
Declination.
Diff. for
i Hour.
Thur
Frid.
Sat
i
2
3
h m s
14 24 57-33
14 28 52.67
14 32 48.80
9.790
9.823
9-856
S. I 4 22 58.5
I 4 4 2 8.4
15 I 4.0
-48.20
47.61
47.01
16 18.76
16 19.97
1 6 20.40
0.067
0.034
0.00 1
14 41 16.09
14 45 12.64
14 49 9.20
SUN.
Mon.
Tues.
4
5
6
14 36 45.72
14 40 43.46
14 44 42.01
9.889
9.923
9.958
15 19 44.8
15 38 10-5
15 56 20.6
-4 6 -39
45-75
45.09
16 20.03
16 18.85
16 16.85
0.032
0.065
0.100
H 53 5-75
14 57 2.31
15 o 58.86
Wed.
Thur.
Frid.
7
8
9
14 48 41.40
14 52 41.62
14 56 42.70
9-993
10.028
10.063
16 14 14.9
16 31 52.8
16 49 14.1
-44.42
43-73
43-03
16 14.02
16 10.35
16 5-83
0-135
0.170
0.206
15 4 55-42
15 8 51.97
15 12 48.53
Sat.
SUN.
Mon.
10
1 1
12
15 o 44.63
15 4 47-43
15 8 51.09
10.099
10.135
10.171
17 6 18.2
17 23 4.8
17 39 33-6
-42.31
41-57
40.82
16 0.45
15 54,2 1
15 47.10
0.242
0.278
0.314
15 16 45.08
15 20 41.64
15 24 38.19
Tues.
Wed.
Thur.
13
14
J 5
15 12 55.63
15 17 1.03
15 21 7.30
10.207
10.243
10.279
17 55 44-i
18 ii 35.9
18 27 8.6
-40.05
39-26
38.46
15 39-12
15 30.27
15 20.56
0.351
0.387
0.423
15 28 34.75
15 32 31-30
15 36 27.86
Frid.
Sat.
SUN.
16
17
18
15 25 14.43
15 29 22.42
15 33 31.26
10.315
10.351
10.386
18 42 21.8
18 57 15-1
19 ii 48.1
-37- 6 4
36.80
35-95
15 9.98
H 58.55
14 46.27
0.459
0.494
0.529
15 40 24.42
15 44 20.97
15 48 17.53
Mon.
Tues.
Wed.
19
20
21
15 37 40.94
15 41- 51.46
15 46 2.79
10.421
10.455
10.489
19 26 0.5
19 39 5^-8
19 53 21.8
-35.08
34.19
33-29
H 33-14
14 19.18
14 4.40
0.564
0.598
0.632
15 52 14.08
15 56 10.64
16 o 7.20
Thur.
Frid.
Sat.
22
23
24
15 50 14.94
15 54 27.87
15 58 41.59
10.522
10.555
10.587
20 6 29.8
20 19 15.8
20 3 I 39.2
-32.37
31-44
30.50
13 48.82
13 32-43
13 15-27
0.666
0.699
0-731
16 4 3-75
16 8 0.31
16 ii 56.86
SUN
Mon.
Tues.
25
26
27
16 2 56.07
16 7 11.29
16 ii 27.24
10.619
10.649
10.678
20 43 39.8
20 55 17.2
21 6 31.0
-29-54
28.57
27-58
12 57-35
12 38.69
12 19.30
0.762
0.792
0.822
16 15 53-42
16 19 49.98
16 23 46.53
Wed.
Thur.
Frid.
28
2 9
30
16 15 43.88
l6 20 1.22
16 24 19.22
10.707
10.736
10.764
21 17 2O-9
21 27 46.7
21 37 48.0
-26.58
25.56
24.54
II 59-21
II 38.43
II 16.99
0.851
0.880
0,907
16 27 43.09
1 6 31 39.65
16 35 36.20
Sat
31
16 28 37.86
10.790
S.2i 47 24.5
-23.51
10 54.90
0.933
16 39 32.76
NOTE. The semidiaraeter for mean noon may be assumed the same as that for apparent noon.
The sign prefixed to the hourly changs of declination indicates that soutn declinations are
Diff. for i Hour,
4-9'.8s65.
mcreasmg.
(Table III.)
188
VENUS, 1900.
GREENWICH MEAN TIME.
JANUARY.
FEBRUARY.
|
Apparent
Right
Ascension. .
Var. of
R.A.
for i
Hour.
Apparent
Declination.
Var. of
Decl.
for i
Hour.
Meridian
A
7.
Apparent
Right
Var. of
R.A.
Hour.
Apparent
Declination.
Var. of
Decl.
for i
Meridian
5
Passage.
Passage.
1
Noon.
Noon.
MM*.
Noon.
1
Noon.
Noon.
Noon.
Noon.
t
20 39 23.93
+12.819
-20 9 51.7
+46.64
156-8
I
23 746.06
+H.S33
-6 58 20.4
+75-82
2 22.9
2
20 44 30.90
12.763
19 5 55-3
48.06
i 57-9
2
23 12 .5.26
11.199
6 27 55.5
76.24
2 23.4
3
204936.51
12.706
1931 25.0
49-45
i 59.1
3
23 16 43-65
11.167
5 57 20.9
76.63
2 23.9
4
20 54 40.75
12.648
19 ii 21.7
50.82
2 O.2
4
23 21 II. 26
11.136
5 26 37.4
76.99
224.5
5
20 59 43-61
12.5
y>
18 50 46.0
52.15
2 1.3
5
23 25 38.14
11.106
4 55 45-8
77-31
2 25.0
6
21 4 45.08
+12.532
-18 29 38.8
+53-45
2 2.4
6
23 30 4-31
+11.077
-4 24 46.7
+77-60
225.5
7
21 94,5-15
12.474
18 8 0.8
54-72
2 3-5
7
23 34 29.81
11.050
3534J.I
77-86
2 25.9
8
21 14 43.82
12.4
5
174552.7
55-95
2 4-5
8
23 38 54-68
11.024
3 22 29.6
78.09
2 26.4
9
21 19 41.08
12-357
17 23 15-4
57-15
2 5-5
9
23 43 18.95
11.000
2 51 13.0
78.19
2 26.9
10
21 24 36.95
12.299
17 o 9.6
58-32
2 6. 5
10
2347 42.66
10.978
2 19 52.O
78.46
227.3
ii
21 2931.42
+12.241
-16 36 36.1
+59-46
2 7-5
ii
2352 5-85
+10.957
-I 48 27.4
+78.60
227-7
12
21 34 24.51
I2.I
3
16 12 35.7
60.57
2 8.4
12
23 56 28.56
10.937
I l6 59.8
78.70
2 28.2
13
21 39 16.22
12.1
f
1548 9.2
61.64
2 9-3
13
o o 50.82
10.919
o 45 29.9
78.77
2 28.6
14
2144 6.57
12.0
15 23 17.3
62.68
2 1O.2
M
o 5 12.68
10.903
-o 13 58.6
78.82
2 29.1
15
21 48 55.57
12.014
14 58 0.9
63.69
2 II. I
'3
o 934.17
10.889
+o 17 33.5
78.84
2 29.5
16
21 5343-25
+ 11.959
-14 32 20.7
+64-66
2 II.9
16
1355-34
+10.876
+049 5.7
+78.83
2 29.9
17
21 58 29.61
11.905
14 6 17.5
65.60
2 12.8
17
o 18 16.22
10.865
I 20 37.2
78.79
230.3
18
22 3 14.70
11.852
13 39 52.0
66.51
2 13.6
18
22 36.86
10.856
1 52 7.4
78.72
230.7
19
22 7 58.53
ii. 8
13 13 5-i
67-39
2 14.3
J9
o 26 57.29
10.848
2 23 35.6
78.62
2 3I.I
20
22 12 41.11
11.749
12 45 57.6
68.23
2 15.1
20
o 31 17.54
10.841
2 55 i.o
78.49
2 31-5
21
22 17 22.48
+11.699
-12 18 30.2
+69.04
2.5.8
21
o 35 37-66
+10.836
+3 26 23.0
+78.33
2 3'-9
22
22 22 2.67
11.650
ii 5043.7
69.83
2 16.6
22
o 39 57-69
10.833
3 57 4-8
78.14
2 32-3
23
22 26 41.70
11.602
II 22 38.8
70.58
2 17-3
23
o 44 17.66
10.832
4 28 53.8
77-92
2 32.6
24
22 31 19.60
11-556
10 54 16.6
71.29
2 l8.0
2 4
o 48 37.62
10.832
5 o 1.2
77-68
233-0
25
22 35 56-41
11.511
10 25 37.6
71-97
2 18.7
25
o 52 57-59
10.833
5 31 2.4
77.41
233-4
26
224032.15
+11.4
3
- 9 56 42-5
+72.62
2 19-3
26
o 57 17.62
+10.836
+6 i 56.6
+77."
233-8
27
22 45 6.86
11.426
927 32.1
73-24
2 19-9
2 7
i i 37-74
10.840
6 32 43.2
76.77
2 34-2
28
22 49 40.57
11.385
858 7-4
73-82
2 20.5
28
i 557-96
10.846
7 3 21.4
76.40
2 34-6
29
22 54 13-32
11.345
8 28 29.0
74-37
2 21. 1
29
i 10 18.33
10.853
7 33 50-5
76.01
2 35-o
3
225845.13
11.306
7 58 38-0
74.89
2 21-7
30
i 14 38.88
10.860
8 4 9.8
75-59
2 35-4
31
23 3 16.03
+ 11.269
- 7 28 34.9
+75-37
222-3
31
i 18 59.62
+10.869
+8 34 18.5
+75-14
235-8
32
23 746.06
+11.233
- 6 58 20.4
+75-82
2 22.9
32
i 23 20.59
+10.879
+9 4 16.0
+74-65
2 36.2
Day of the Month.
1st.
6th. llth. 16th
21st. 26th. 31st.
Day of the Month. 6th.
10th. 15th. 20th.
25th.
Semidiameter
Hor. Parallax
LS
593 6.03 6.14
6.15 6.25 6.35
'
6.24 6.37 6.50
6.47 6.59 6.73
Semidiameter . . 6.6
Hor. Parallax 6.8
3 6.78 6.95 7.15
1 703 7-19 7-37
7-31
7-57
i
NOTE.-
-The sign + indicates north declinations ; the sign indicates south declinations.
189
14 DAY USE
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ASTRON-MATH-STAT.
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MAY 2 3 1994
LD 21-1001
LD 21-40m-10,'65
(F7763slO)476
General Library
University of California
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M298787
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