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ELEMENTARY
. MATHEMATICAL ASTRONOMY,
EXAMPLES AND EXAMINATION PAPERS.
C. W. 0. ^ABLOW, M.A., B.Sc.,
GOLD MEDALLIST IN MATHEMATICS AT LONDON M.A.,
SIXTH WRANGLER, AND FIRST CLASS FIRST DIVISION PART II. MATHEMATICAL
TRIPOS, CAMBRIDGE,
AND
GK H. BBYAN, D.So., M.A., F.E.S.,
SMITH'S PRIZEMAN, LATE FELLOW OK ST. PETER'S COLLEGE, CAMBRIDGE,
JOINT AUTHOR OF " COORDINATE GEOMETRY, PART I.," " THE TUTORIAL ALGEBRA,
ADVANCED COURSE," ETC.
Third Impression (Second Edition).
LONDON: W. B. OLIVE,
(University Correspondence College Press],
13 BOOKSELLEB.S Row, STKAND, W.C.
1900.
p
PREFACE TO THE FIRST EDITION.
FOR some time past it has been felt that a gap existed between
the many excellent popular and non-mathematical works on As-
tronomy, and the standard treatises on the subject, which involve
high mathematics. The present volume has been compiled with
the view of filling this gap, and of providing a suitable text-book
for such examinations as those for the B.A. and the B.Sc. degrees of
the University of London.
It has not been assumed that the reader's knowledge of mathe-
matics extends beyond the more rudimentary portions of Geometry,
Algebra, and Trigonometry. A knowledge of elementary Dynamics
will, however, be required in reading the last three chapters, but
all dynamical investigations have been left till the end of the book,
thus separating dynamical from descriptive Astronomy.
The principal properties of the Sphere required in Astronomy
have been collected in the Introductory Chapter ; and, as it is
impossible to understand Kepler's Laws without a slight knowledge
of the properties of the Ellipse, the more important of these have
been collected in the Appendix for the benefit of students who have
not read Conic Sections.
All the more important theorems have been carefully illustrated
by worked-out numerical examples, with the view of showing how
the various principles can be put to practical application. The
authors are of opinion that a far sounder knowledge of Astronomy
can be acquired with the help of such examples than by learning
the mere bookwork alone.
Feb. 1st, 1892.
PREFACE TO THE SECOND EDITION.
THE first edition of Mathematical Astronomy having run out of
print in less than eight months, we have hardly considered it
advisable to make many radical changes in the present edition.
We have, however, taken the opportunity of adding several notes at
the end, besides answers to the examples, which latter will, we
hope, prove of assistance, especially to private students ; our readers
will also notice that the book has been brought up to date by the
inclusion of the most recent discoveries. At the same time we
hope we have corrected all the misprints that are inseparable from
a first edition. Our best thanks are due to many of our readers for
their kind assistance in sending us corrections and suggestions.
Nov. 1st, 1892.
CONTENTS.
INTRODUCTORY CHAPTER.
PAOB
ON SPHERICAL GEOMETRY i
Definitions ii
Properties of Great and Small Circles iii
On Spherical Triangles v
CHAPTER I.
THE CELESTIAL SPHERE.
/Sect. I. Definitions Systems of Coordinates 1
II. The Diurnal Rotation of the Stars 13
III. The Sun's Annual Motion in the Ecliptic
The Moon's Motion Practical Applications 20
CHAPTER II.
THE OBSERVATORY.
Sect. I. Instruments adapted for Meridian Observations 35
II. Instruments adapted for Observations off the
Meridian 54
CHAPTER III.
THE EARTH.
Sect. I. Phenomena depending on Change of Position
on the Earth 63
II. Dip of the Horizon 73
III. Geodetic Measurements Figure of the Earth 77
CHAPTER IV.
THE SUN'S APPARENT MOTION IN THE ECLIPTIC.
Sect. I. The Seasons 87
II. The Ecliptic 99
III. The Earth's Orbit round the Sun 105
CHAPTER V.
ON TIME.
^/Sect. I. The Mean Sun and Equation of Time 115
II. The Sun-dial 125
III. Units of Time The Calendar 127
IV. Comparison of Mean and Sidereal Times 129
CONTENTS.
CHAPTER VI.
PACK
ATMOSPHERICAL REFRACTION AND TWILIGHT 140
CHAPTER VII.
THE DETERMINATION OF POSITION ON THE EARTH.
Sect. I. Instruments used in Navigation 153
^X, II. Finding the Latitude by Observation 102
^ HI. To find the Local Time by Observation 171
IV. Determination of the Meridian Line 175
CXJ, V. Longitude by Observation 177
VI. Captain Sumner's Method 187
CHAPTER VIII.
THK MOON.
Sect. I. Parallax The Moon's Distance and Dimensions 191
II. Synodic and Sidereal Months Moon's Phases
Mountains on the Moon 200
III. The Moon's Orbit and Rotation 209
CHAPTER IX.
ECLIPSES.
Sect. I. General Description of Eclipses 219
,, II. Determination of the Frequency of Eclipses 224
III. Occultations Places at which a Solar Eclipse
is visible 232
CHAPTER X.
THE PLANETS.
Sect. I. General Outline of the Solar System ... ... 238
II. Synodic and Sidereal Periods Description of
the Motion in Elongation of Planets, as
seen from the Earth Phases 244
III. Kepler's Laws of Planetary Motion 253
IV. Motion relative to Stars Stationary Points ... 258
V. Axial Rotations of Sun and Planets 264
CHAPTER XL
THE DISTANCES OF THE SUN AND STARS.
Sect. I. Introduction Determination of the Sun's
Parallax by Observations of a Superior
Planet at Opposition 267
II. Transits of Inferior Planets 271
,, III. Annual Parallax, and Distances of the Fixed
Stars 283
IV. The Aberration of Light ... 293
CONTENTS.
DYNAMICAL ASTRONOMY.
CHAPTER XII. PAOR
THE ROTATION OF THE EARTH 315
CHAPTER XIII.
THE LAW OP UNIVERSAL GRAVITATION.
Sect. I. The Earth's Orbital Motion Kepler's Laws
and their Consequences 337
II. Newton's Law of Gravitation Comparison of
the Masses of the Sun and Planets 352
III. The Earth's Mass and Density 362
CHAPTER XIV.
FURTHER APPLICATIONS OF THE LAW OF GRAVITATION.
Sect. I. The Moon's Mass Concavity of Lunar Orbit... 371
II. The Tides 375
,, III. Precession and Nutation 392
IV. Lunar and Planetary Perturbations 406
NOTES.
Diagram for Southern Hemisphere 421
The Photochronograph 421
Effects of Dip, &c., on Rising and Setting 422
APPENDIX.
Properties of the Ellipse 423
Table of Constants 426
ANSWERS TO EXAMPLES AND EXAMINATION QUESTIONS 428
INDEX 434
INTRODUCTORY CHAPTER,
ON SPHERICAL GEOMETRY.
Properties of the Sphere which will be referred to in the course of the
Text.
(1) A Sphere may be defined as a surface all points on which are
at the same distance from a certain fixed point. This point is the
Centre, and the constant distance is the Radius.
(2) The surface formed by the revolution of a semicircle about
its diameter is a sphere. For the centre of the semicircle is kept
fixed, and its distance from any point on the surface generated will
be equal to the radius of the semicircle.
FIG. 1.
(3) Let PqQP' be any position of the revolving semicircle whose
diameter PP' is fixed. Let OQ be the radius perpendicular to PP',
Cq any other line perpendicular to PP', meeting the semicircle in
q. (We may suppose these lines to be marked on a semicircular disc of
cardboard.) As the semicircle revolves, the lines OQ, Cgwill sweep
out planes perpendicular to PP', and the points Q, q will trace out
in these planes circles HQRK, hqrJc, of radii OQ, Cq respectively.
From this it may readily be seen that Every plane section of a
sphere is a circle,
4-STKON, 5
ii
ASTRONOMY.
Definitions.
(4) A great Circle of a sphere is the circle in which it is cut by
any plane passing through the centre (e.g., HQRK, PqQP' or PrRP ).
A small circle is the circle in which the sphere is cut by any plane
not passing through the centre (e.g., hqrk).
(5) The axis of a great or small circle is the diameter of the
sphere perpendicular to the plane of the circle. The poles of the
circle are the extremities of this diameter. (Thus, the line PP is
the axis, and P, P' are the poles of the circles HQK and hqJc).
(6) Secondaries to a circle of the sphere are great circles passing
through its poles. (Thus, PQP' and PRP" are secondaries of the
circles HQK, hqk).
FIG. 2.
(7) The angular distance between two points on a sphere is
measured by the arc of the great circle joining them, or by the angle
which this arc subtends at the centre of the sphere. Thus, the dis-
tance between Q and Bis measuredeither by the arc QE, or by the angle
QOR. Since the circular measure of L QOR = (arc Qft) -f- (radius of
sphere), it is usual to measure arcs of great circles by the angles
which they subtend at the centre. This remark does not apply to
small circles.
(8) The angle between two great circles is the angle between
their planes. Thus, the angle between the circles PQ, PR is the angle
between the planes PQP', 7'EP'. It is called "the angle QPR."
(9) A spherical triangle is a portion of the spherical surface
bounded by three arcs of gr.eat_circles. Thus, in Fig. 2, PQR is a
spherical triangle, but Pqr is not a spherical triangle, because qr is
not an arc of a great circle. We may, however, draw a great circle
passing through q and r, and thus form a spherical triangle Pqr.
SPHERICAL GEOMETRY. ill
Properties of Great and Small Circles.
(10) All points on a small circle are at a constant (angular))
distance from the pole.
For, as the generating semicircle revolves about PP 7 , carrying g
along the small circle hk, to r, the arc Pq = arc Pr, and Z POq = L POr.
The constant angular distance Pq is called the spherical, or
angular radius of the small circle. The pole P is analogous to the
centre of a circle in plane geometry.
(11) The spherical radius of a great circle is a quadrant, or,
All points on a great circle are distant 90 from its poles.
For, as Q, by revolving about PP', traces out the great circle
HQRK, we have L POQ = L POR = 90, and therefore, PQ, PE are
quadrants.
(12) Secondaries to any circle lie in planes perpendicular to
the plane of the circle.
For PP' is perpendicular to the planes of the circles HQK, liqk,
therefore any plane through PP / , such as PQP' or PEP', is also per-
pendicular to them.
(13) Circles which have the same axis and poles lie in parallel,
planes. For the planes HQK, hqk are parallel, both being perpen-
dicular to the axis PP'. Such circles are often called parallels.
(14) If any number of circles have a common diameter, their
poles all lie on the great circle to which they are secondaries, and
this great circle is a common secondary to the original circles.
For if OA is the axis of the circle PQP', then OA is perpendicular-
to POP'. Hence, if the circle PQP 7 revolves about PP', A traces out.
the great circle HQRK, of which P, P 7 are poles. We likewise see that
(15) If one great circle is a secondary to another, the latter is
also a secondary to the former.
This is otherwise evident, since their planes are perpendicular.
(16) The angle between two great circles is equal to
(i.) The angle between the tangents to them at their points
of intersection ;
(ii.) The arc which they intercept on a great circle to which
they are both secondaries ;
(iii.) The angular distance between their poles.
Let Ft, Pu be the tangents at P to the circles PQ, PE, and let A, B
bo the poles of the circles. If we suppose the semicircle PQP' to
revolve about PP' into the position PEP', the tangent at P will
revolve from Pt to Pu, the radius perpendicular to OP will revolve
from OQ to 07?, and the axis will revolve from OA to OB. All these
lines will revolve through an angle equal to the angle between
the planes PQP', PRP / , and this is the angle QPE between the
circles (Def. 8). BLenee,
le between circles PQ, PR = L tPu = L QOR
{y ASTEONO^TT.
(17) The arc of a small circle subtending a given angle at the
pole is proportional to the sine of the angular radius.
Let qr be the arc of the small circle hqrJc, subtending L qPr at P,
and let G be the centre of the circle. Evidently L qCr = L QOR
(since Cq, Gr are parallel to OQ, OB). Hence, the arcs qr, QR are
proportional to the radii Cq, OQ,
. arc qr = G = Gq_ = ghl pQq = gin p^
arc QR OQ Oq
But QR is the arc of a great circle subtending the same angle at the
pole P hence the arc qr is proportional to sin Pq, as was to be shown.
Since qQ = 90 - PQ, therefore sin Pq - cos gQ, so that the arc qr is
proportional to the cosine of the angular distance of the small circle
(jr from the parallel great circle QR.
FIG. 3.
FIG. 4.
(18) Comparison of Plane and Spherical Geometry.
It may be laid down as a general rule that great circles and small
circles on a sphere are analogous in their respective properties to
straight lines and circles in a plane. Thus, to join two points on a
sphere means to draw the great circle passing through them.
Secondaries to a great circle of the sphere are analogous to per-
pendiculars on a straight line. The distance of a point from any
great circle is the length of the arc of a secondary drawn from the
point to the circle. Thus, rR is the distance of the point r from the
great circle HQRK.
SPHEEICAL GEOMETftf. V
On Spherical Triangles*
(19) Parts of a Spherical Triangle. A spherical triangle, like a
plane triangle, has six parts, viz., its three sides and its three angles.
The sides are generally measured by the angles they subtend at the
centre of the sphere, so that the six parts are all expressed as angles.
Any three given parts suffice to determine a spherical triangle,
but there are certain " ambiguous cases " when the problem admits
of more than one solution. The formulge required in solving
spherical triangles form the subject of Spherical Trigonometry,
and are in every case different from the analogous f ormulaj in Plane
Trigonometry. There is this further difference, that a spherical
triangle is completely determined if its three angles are given.
Thus, two spherical triangles will, in general, be equal if they
have the following parts equal :
(i.) Three sides.
(ii.) Two sides andincluded angle.
(iii.) Two sides and one opposite
angle.
(iv.) Three angles,
(v.) Twoanglesandadjacentside.
(vi.) Two angles and one opposite
side.
Cases (iii.) and (vi.) may be ambiguous.
(20) Right-angled Triangles. If one of the angles is a right
angle, two of the remaining five parts will determine the triangle.
(21) Triangle with two right angles. The properties of a
spherical triangle, such as PQR, Fig. 3, in which one vertex P is
the pole of the opposite side QR, are worthy of notice. Here two
of the sides, PQ, PR, are quadrants, and two angles Q, R are right
angles. The third side QR is equal to the opposite angle QPR,
(22) Triangle with, three right angles. If, in addition, the angle
QPR is a right angle (Fig. 4), QR will be a quadrant. The triangle
PQR will, therefore, have all its angles right angles, and all its sides
quadrants, and each vertex will be the pole of the opposite side.
The planes of the great circles forming the sides, are three planes
through the centre mutually at right angles, and they divide the
surface of the sphere into eight of these triangles ; thus the area of
each triangle is one-eighth of the surface of the sphere.
(23) The three angles of a spherical triangle are together
greater than two right angles.
[For proof, see any text-book on Spherical Geometry.]
(24) If the sides of a spherical triangle, when expressed as angles,
are very small, so that its linear dimensions are very small com-
pared with the radius of the sphere, the triangle is very approxi-
mately a plane triangle.
Thus, although the Earth's surface is spherical, a triangle whose
sides are a few yards in length, if traced on the Earth, will not be
distinguishable from a plane triangle. If the sides are several
miles in length, the triangle will still be very nearly plane.
vi AJSTKONOMY.
(25) Any two sides 6f a spherical triangle are together
greater than the third side. For if we consider the plane angles
which the sides subtend at the centre of the sphere, any two of
these are together greater than the third, by Euclid XL, 20.
(26) The following application of (25) is of great use in astronomy,
and is analogous to Euclid III., 7, 8.
Let AHBK be any given great or small circle whose pole is P,
Zany other given point on the sphere, and let the great circle ZP
meet the given circle in the points A, B. Then A, B are the two
points on the given circle whose distances from Z are greatest and
least respectively.
For let H be any other point on the circle. Join ZH, HP.
Then, in spherical A ZPH, ZP + PH> ZH. But PH = PA ;
/. ZP + PA > ZH,
i.e., ZA>ZH.
Also, if Z is on the opposite side of the circle to P, then
ZII+PH>PZ', .:ZH + PB>PZ; .:ZH>PZ-PB,
i.e., ZH>ZB.
If Z' be a point on the same side of the circle as P, then PZ' + Z'H
>PH. But PH - PB. .'. PZ'-t Z'H^PB.
.-. Z'H>PB-PZ',
i.e., Z'H>Z'B, as before.
lie nee, A is further from Z, Z', and B is nearer to Z, Z', than any
other point on the circle.
(27) If H, K are the two points on the circle equidistant from Z,
the spherical triangles ZPH, ZPK have ZP common, ZH = ZK (by
hypothesis^), and PH = PK [by (10)], hence they are equal in all
respects ; thus L ZPH = L ZPK, and L PZH = L PZK.
Hence PH, PK are equally inclined to PB, as are also ZH, ZK.
Similar properties hold in the case of the point Z'. These pro-
perties are of frequent uw.
ASTRONOMY.
CHAPTEE I.
THE CELESTIAL SPHERE.
SECTION I. Definitions Systems of Co-ordinate*.
1 . Astronomy is the science which deals with the celestial
bodies. These comprise all the various bodies distributed
throughout the universe, such as the Earth (considered as a
whole), the Sun, the Planets, the Moon, the comets, the fixed
stars, and the nebulae. It is convenient to divide Astronomy
into three different branches.
The first may be called Descriptive Astronomy. It is
concerned with observing and recording the motions of the
various celestial bodies, and with applying the results of
such observations to predict their positions at any subsequent
time. It includes the determination of the distances, and the
measurement of the dimensions of the celestial bodies.
The second, or Gravitational Astronomy, is an appli-
cation of the principles of dynamics to account for the motions
of the celestial bodies. It includes the determination of their
masses.
The third, called Physical Astronomy, is concerned
with determining the nature, physical condition, temperature,
and chemical constitution of the celestial bodies.
The first branch has occupied the attention of astronomers
in all ages. The second owes its origin to the discoveries of
Sir Isaac Newton in the seventeenth century ; while the
third branch has been almost entirely built up in the present
century.
In this book we shall treat exclusively of Descriptive and
Gravitational Astronomy.
ASTRONOMY.
: -;2: :The ;C.elesti.al Sphere. On observing the stars it is
' not^ 'difficult to imagine that they are bright points dotted
about on the inside of a hollow spherical dome, whose centre
is at the eye of the observer. It is impossible to form any
direct conception of the distances of such remote bodies ; all
we can see is their relative directions. Moreover, mof-t
astronomical instruments are constructed to determine only
the directions of the celestial bodies. Hence it is important
to have a convenient mode of representing directions.
FIG. 6.
The way in which this is done is shown in Figure 6. Let
be the position of any observer, A, , C, &c., any stars or
other celestial bodies. About 0, as centre, describe a sphere
with any convenient length as radius, and let the lines joining
to the stars A, J3, C meet this sphere in a, ft, c respectively.
Then the points a, I, c will represent, on the sphere, the
directions of the stars A, H, C, for the lines joining these
points to will pass through the stars themselves. In this
manner we obtain, on the sphere, an exact representation of
the appearance of the heavens as seen from 0. Such a
sphere is called the Celestial Sphere.
This sphere may be taken as the dome upon which the stars
appear to lie. But it must be carefully borne in mind that
the stars do not actually lie on a sphere at all, and that they
are only so represented for the sake-of convenience.
THE CELESTIAL SPHERE.
3. Use of the Globes. The representation of directions
of stars by points on a sphere is well exemplified in the old-
fashioned star globes. Such a globe may be used as the
observer's celestial sphere ; but it must be remembered that
the directions of the stars are the lines joining the centre to
the corresponding points on the sphere ; for in every case the
observer is supposed to be at the centre of the celestial
sphere.
The properties given in the Introduction on Spherical Geo-
metry are applicable to the geometry of the celestial sphere.
A knowledge of thorn will be assumed in what follows.
4. Angular Distances and Angular Magnitudes.
Any plane through the observer will be represented on the
celestial sphere by a great circle. The arc of the great circle
a b (Fig. 6) represents the angle a 01 or A OB which the stars
A, subtend at 0. This angle is generally measured in
degrees, minutes, and seconds, and is called the angular
distance between the stars. This angular distance must
not be confused with their actual distance AB. In the same
way, when we are dealing with a body pf perceptible dimen-
sions, such as the Sun or Moon (DF, Fig. 6), we shall define
its angular diametsr as the angle DOF, subtended by a
diameter at the observer's eye. This angular diameter is
measured by the arc df of the celestial sphere, that is, by the
diameter of the projection of the body on the celestial sphere.
From the figure it is evident that
Od 01)'
Since DF is the actual linear diameter of the body, mea-
sured in units of length, the last relation shows us that the
angular diameter (df) of a body varies directly as its linear
diameter DF, and inversely as OD, the distance of the body
from the observer's eye.
As the eye can only judge of the dimensions of a body
from its angular magnitude, this result is illustrated by the
1'act that the nearer an object is to the eye the larger it looks,
and vice versd. Thus, if the distance of the object be doubled,
it will only look half as large, as may be easily verified.
4 ASTRONOMY.
5. The Directions of the Stars are very approxi-
mately independent of the Observer's Position on
the Earth.
This is simply a consequence of the enormously great dis-
tances of all the stars from the Earth. Thus,
let x (Fig. 7) denote any star or other celestial
body, S, JZtwo different positions o^ the observer.
If the distance SJ be only a very small fraction
of the distance Sx, the angle Ex 8 will be very
small, and this angle measures the difference be-
tween the directions of x as seen from ^and from 8.
In illustration, if we observe a group of objects
a mile or two off, and then walk a few feet in any
direction, we shall observe no perceptible change FIG. 7.
in the apparent directions or relative positions of the objects.
If Ex be drawn parallel to Sx, the angle xEx will be
equal to ExS, and will therefore be very small indeed.
Hence, Ex will very nearly coincide in direction with Ex'.
Thus, considering the vast distances of the stars, we see that
The lines joining a Star to different points of the
Earth may be considered as parallel.*
The stars will, therefore, always be represented by the
same points on a star globe, or celestial sphere, no matter
what be the position of the observer. The great use of the
celestial sphere in astronomy depends on this fact.
6. Motion of Meteors. The projection of bodies on the
celestial sphere is well illustrated by the apparent motion
of a swarm of meteors. Where such a swarm is moving
uniformly, all the meteors describe (approximately) parallel
straight lines. II we draw planes through these lines and
the observer, they will intersect in a common line, namely,
the line through the observer parallel to the direction of the
common motion of the meteors. The planes will, therefore,
cut the celestial sphere in great circles, having this line as
their common diameter. These great circles represent the
apparent paths >i (he meteors on the celestial sphere. The
paths appear, therefore, to radiate from a common point,
namely, one of the extremities of this diameter.
This point is called the Radiant, and by observing its
position the direction of motion of the meteors is determined.
* This is not true in the case of the Moon.
tHE CELESTIAL StHE&E. 6
7. Zenith and Nadir. Horizon. If, through the
observer, a line be drawn in the direction in which gravity
acts (i.e., the direction indicated by a plumb-line), it will meet
the celestial sphere in two points. One of these is vertically
above the observer, and is called the Zenith; the other is
vertically below the observer, and is called the Nadir. (Fig.
6, and Z, N, Fig. 8.)
If the plane through the observer parallel to the surface
of a liquid at rest be produced, it will cut the celestial
sphere in a great circle. This great circle is called the
Celestial Horizon. (Fig. 6, and sEnW, Fig. 8.)
It is proved in Hydrostatics that the surface of a liquid at
rest is a plane perpendicular to the direction of gravity.
Hence, the celestial horizon is the great circle whose poles
are the zenith and nadir. "We might have defined the
horizon by this property.
From the above definition, it is evident that, to an observer
whose eye is close to the surface of the ocean, the celestial
horizon forms the boundary of the visible portion of the
celestial sphere. On land, however, the boundary, or visible
horizon (as it is called), is always more or less irregular,
owing to trees, mountains, and other objects.
8. Diurnal Motion of the Stars. If we observe the
sky at different intervals during
the night, we shall find that the
stars always maintain the same
configurations relative to one
another, but that their actual
situations in the sky, relative to
the horizon, are continually
changing. Some stars will set
in the west, others will rise in
the east. One star which is
situated in the constellation called
the l< Little Bear," remains almost FlG - 8 -
fixed. This star is called Polaris, or the Pole Star. All the
other stars describe on the celestial sphere small circles
(Fig. 8) having a common pole P very near the Pole Star,
and the revolutions are performed in the same period of time,
namely, about 23 hours 56 minutes of our ordinary time.
6
ASTEONOMt.
9. Celestial Poles, Equator, and Meridian. The
common motion of the stars may most easily be conceived by
imagining them to be attached to the surface of a sphere
which is made to revolve uniformly about the diameter PP'.
The extremities of this diameter are called the Celestial
Poles. That pole, P, which is above the horizon in northern
latitudes is called the North Pole, the other, P\ is called
the South Pole.
The great circle, JEQR W, having these two points for its
poles, is called the Celestial Equator. It is, therefore, the
circle which would be traced out by the diurnal path of a
star distant 90 from either pole.
The Meridian is the great circle (PZP'N, Fig. 9) passing
through the zenith and nadir and the celestial poles. It cuts
both the horizon and equator at right angles [by Spher.
Geom. (12), since it passes through their poles].
THE CELESTIAL SPHEKE. 7
10. The Cardinal Points. The East and West
Points (J, W, Eig. 9) are the points of intersection of the
equator and horizon. The North and South Points
(&, S) are the intersections of the meridian with the horizon.
Verticals. rSecondaries to the horizon, i.e., great circles
through the zenith and nadir., are called Vertical Circles,
or, briefly, Verticals. Thus, the meridian is a vertical.
The Prime Vertical is the vertical circle (ZENTF) passing
through the east and west points.
Since P is the pole of the circle QERW, and ^is the pole
of nEsWy therefore E, W are the poles of the meridian
PZP'N. Hence the horizon, equator, and prime vertical
which pass through E, W, are all secondaries to the meridian ;
they therefore all cut the meridian at right angles.
11. Annual Motion of the Sun. The Ecliptic.
The Sun, while participating in the general diurnal rotation
of the heavens, possesses, in addition, an independent
motion of its own relative to the stars.
Imagine a star globe worked by clockwork so as to revolve
about an axis pointing to the celestial pole in the same peri-
odic time as the stars. On such a moving globe the directions
of the stars will always be represented by the same points.
During the daytime let the direction of the Sun be marked on
the globe, and let this process be repeated every day for a year.
We shall thus obtain on the globe a representation of the
Sun's path relative to the stars, and it will be found that
(i.) The Sun moves from west to east, and returns to the
same position among the stars in the period called a year ;
(ii.) The relative path on the celestial sphere is a great
circle, inclined to the equator at an angle of about 23 27f.
This great circle (CTL ===, Fig. 9) is called the Ecliptic.
"We may, therefore, briefly define the ecliptic as the great
circle which is the trace, on the celestial sphere, of the Sun's
annual path relative to the stars.
The intersections of the ecliptic and equator are called
Equinoctial Points. One of them is called the First
Point of Aries ; this is the point through which the Sun
passes when crossing from south to north of the equator, and
it is usually denoted by the symbol T The other is called
the First Point of Libra, and is denoted by the symbol =0=,
ASTKONOMY.
12. Coordinates. In Analytical Geometry, the position
of a point in a plane is denned by two coordinates. In like
manner, the position of a point on a sphere may be denned by
means of two coordinates. Thus, the position of a place on
the Earth is denned by the two coordinates, latitude and
longitude. For fixing the positions of celestial bodies, the
following different systems of coordinates are used.
13. Altitude or Zenith Distance and Azimuth. Let
Fig. 10 represent the celestial sphere, seen from overhead, and
lot x be any star. Draw the vertical circle ZxX. Then the
position of x may be defined by either of the following pairs
of coordinates, which are analogous to the Cartesian and
polar coordinates of a point in a plane respectively :
(a) The arc s X and the arc Xx ;
(b) The arc Zx and the angle sZx.
Practically, however, the two systems are equivalent ; for,
since Z is the pole of sX, ZX = 90, therefore
Zx = 90 xXj and angle sZx = arc sX,
FIG. 10.
The Altitude of a star (Xx} is its angular distance from
the horizon, measured along a vertical.
The Zenith Distance (abbreviation, Z.D.) is its angular
distance from the zenith (Zx) , or the complement of the altitude.
The Azimuth (sX or sZx) is the arc of the horizon inter-
cepted between the south point and the vertical of the star,
or the angle which the star's vertical makes with the meridian
THE CELESTIAL SPHERE. 9
*14. Points Of the Compass. In practical applications of Astro-
nomy to navigation, it is usual to measure the azimuth in "points"
and " quarter points " of the compass. The dial plate of a mariner's
compass is divided into 32 points, by repeatedly bisecting the right
angles formed by the directions of the four cardinal points. Thus
each point represents an angle of Hi degrees. The points are again
subdivided into " quarter points " of 2\ degrees. Starting from the
north and going round towards the east, the various points are denoted
as follows :
N., N. byB., N.N.E., N.E. by N., N.E., N.E. by E., E.N.E., E. by N.
E., E. byS., E.S.E., S.E. by E., S.E., S.E. by S., S.S.E., S. by E.
S., S. by W. S.S.W., S.W. by S., S.W., S.W: by W., W.S.W , W. by S.
W., W. by N., W.N.W. N.W. by W., N.W., N.W. by N., N.N.W., N. by W.
The quarter points are denoted thus : E.N.B. E. means one
quarter point to the eastward of E.N.E., that is, 6 points, or
70 18' 45", from the north point, taken in an easterly direction.
So, too, S.S.W. W. meafli 2J points, or 28 7' 30' , measured from
the south point westwards.
15. Polar Distance, or Declination, and Hour Angle.
From the pole P, draw through x the great circle PxM-, this
circle is a secondary to the equator EQ, W.
Then we may take for the coordinates of x the arc Px and
the angle sPx. Or we may take the arc x3f, which is the
complement of Px, and the arc QM, which = angle QPx.
The North Polar Distance of a star (abbreviation,
N.P.D.) is its angular distance (Pa;) from the celestial pole.
The Declination (abbreviation, Decl.) is the angular
distance from the equator (xM), measured along a secondary,
and is, therefore, the complement of the N.P.D.
The great circle PxM through the pole and the star is
called the star's Declination Circle.
The Hour Angle of the star (ZPx] is the angle which
the star's declination circle makes with the meridian.
The declination may be considered positive or negative,
according as the star is to the north or south of the equator,
but it is more usual to specify this by the letter N. or S., as
the case may be, and this is called the name of the declination.
The hour angle is generally measured from the meridian
towards the west, and is reckoned from to 360.
Either the declination and hour angle or the N.P.D. and
hour angle may be taken as the two coordinates of a star.
10
ASTBONOHY.
16. Declination and Right Ascension. The position
of a celestial body is, however, more frequently defined by
its declination and right ascension.
'The declination has been already defined, in 15, as the
angular distance of the star from the equator, measured along
a secondary. (xM, Fig. 11.)
The Right Ascension (E.A.) is the arc of the equator
intercepted between the foot of this secondary and the First
Point of Aries. Thus, T^, Fig. 11, is the E.A. of the star a:.
The E.A. of a star is always measured from T eastwards
reckoning from to 360. Thus the star w Piscium, whose
declination circle cuts the equator 1 34' 18" west of T, has
the E.A. 360 1 34' 18", or 358 25' 42".
FIG. 11.
17. Celestial Latitude and Longitude. The position
of a celestial body may also be referred to the ecliptic instead
of the equator.
The Celestial Latitude is the angular distance of the
tody from the ecliptic, measured along a secondary to the
ecliptic. (Hx, Pig. 11.)
The Celestial Longitude is the arc of the ecliptic inter-
cepted between this secondary and the first point of Aries,
measured eastwards from T- (T#, Pig. 11.)
tflE CELESTIAL SPHERE. ll
18. Latitude of the Observer. The celestial latitude
and longitude, defined in the last paragraph, must not be
confounded with the latitude and longitude of a place on the
Earth, as there is no connection whatever between them.
The Latitude of a place is the angular distance of its
zenith from the equator, measured along the meridian.
Thus, in Pig. 1 1 , ZQ, is the latitude of the observer.
Since PQ nZ 90 ; .-. ZQ = nP, or in other words,
The latitude of a place is the altitude of the Celestial Pole.
The complement of the latitude is called the Colatitude.
Hence, in Pig. 11, PZ is the colatitude of the observer,
and is the angular distance of the zenith from the pole.
In this book the latitude of an observer will generally be
denoted by the symbol /, and the colatitude by c.
The longitude of a place will be defined in Chapter III.
19. Obliquity of the Ecliptic. The inclination of the
ecliptic to the equator is called the Obliquity. In Pig. 11,
Q T C is the obliquity. As stated in 1 1 , this angle is about
23 27-'. We shall generally denote the obliquity by i.
20. Advantages of the Different Coordinate
Systems. The altitude and azimuth of a celestial body
indicate its position relative to objects on the Earth. Owing,
however, to the diurnal motion, they are constantly changing.
The N.P.D. and hour angle also serve to determine the
star's position relative to the earth, and have this further
advantage, that the N.P.D. is constant, while the hour angle
increases at a uniform rate.
Since the equator and first point of Aries partake of the
common diurnal motion of the stars, the declination and right
ascension of a star are constant. These coordinates are, there-
fore, the most suitable for tabulating the relative positions of
the various stars on the celestial sphere.
The celestial latitude and longitude of a celestial body are
also unaffected by the diurnal motion. They are most useful in
defining the positions of the Sun, Moon, planets, and comets,
for the first always moves in the ecliptic, while the paths
described by the others are always very near the ecliptic.
21. Recapitulation. Por the sake of convenient refer-
ence, we give on the next page a list of all the definitions of
this chapter, with references to Pigs. 11, 12.
ASTRON. c
12
ASTRONOMY.
GREAT CIRCLES.
Horizon, nEsW.
Equator, EQWR.
Meridian, ZsZ'n.
Prime Vertical, ZEZ'W.
THEIR POLES.
Zenith, Z-, Nadir, Z '.
North Pole, P ; South Pole, P.
East Point, E\ West Point, W.
NorthPoint, n ; South Point, s.
Ecliptic, T i:Z ; Equinoctial Points, T, =2=, viz. : Eirst
Point of Aries, T , and Eirst Point of Libra, b ; Yertical of
Star, ZxX-, Declination Circle of Star, Pxlf.
FIG. 12.
COORDINATES.
Altitude, Xx ; '")
or Zenith Distance, Zx. )
North Polar Distance, Px.
Declination, MX.
Celestial Latitude, Hx.
Azimuth, sX = sZx.
Hour Angle, QM = ZPx.
Bight Ascension, T ^
Celestial Longitude,
OTHER ANGLES. Obliquity of Ecliptic (t) CT Q-
Observer's Latitude (1) = ZQ = nP. Colatitude (c) = PZ.
Notice that the circles on the remote side of the celestial sphere
are dotted.
CELESTIAL SPHEKE. 13
SECTION II. The Diurnal Rotation of the Stars.
22. Sidereal Day and Sidereal Time. A Sidereal
Day is the period of a complete revolution of tlie stars about
the pole relative to the meridian and horizon. Like the
common day it is divided into 24 hours (h.), and these are
subdivided into 60 minutes (m.) of 60 seconds (s.) each.
The sidereal day commences at "Sidereal Noon," i.e., the
instant when the first point of Aries crosses the meridian.
The Astronomical Clock, which is the clock used in
observatories, indicates sidereal time. The hands should
indicate Oh. Om. Os. when the first point of Aries crosses the
meridian, and the hours are reckoned from Oh. up to 24h.,
when T again comes to the meridian and a new day begins.
From the facts stated in 8, it appears that the sidereal
day is about 4 minutes shorter than the ordinary day. The
stars are observed to revolve about the pole at a perfectly
uniform rate, so that the sidereal day is of invariable length,
and the angles described by any star about the pole are pro-
portional to the times of describing them. Thus, the hour
angle of a star (measured towards the west) is proportional
to the interval of sidereal time that has elapsed since the star
was on the meridian.
Now, in 24 sidereal hours the star comes round again to
the meridian, after a complete revolution, the hour angle
having increased from to 360. Hence the hour angle in-
creases at the rate of 15 per hour. Hence, also, it increases
15' per minute, or 15" per second.
The hour angle of a star is, for this reason, generally
measured by the number of hours, minutes, and seconds of
sidereal time taken to describe it. It is then said to be
expressed in time. Thus,
The hour angle of a star, when expressed in time*
is the interval of sidereal time that has elapsed
since the star was on the meridian.
In particular, since the instant when T is on the meridian
is the commencement of the sidereal day, we see that
The sidereal time is the hour angle of the first
point of Aries when expressed in time.
14 ASTHONOMY.
23. To reduce to angular measure any angle ex-
pressed in time. Multiply ~by 15. The hours, minutes, and
seconds of time will thus be reduced to degrees, minutes, and
seconds of angle.
Conversely, to reduce to time from angular measure
we must divide by 15, and for degrees, minutes, and seconds,
write hours, minutes, and seconds.
EXAMPLES. 1. To find, in angular measure, the hour angle of a
star at 15h. 21m. 50s. of sidereal time after its transit. The process
stands thus
15 21 50
230 27 30
/. the angular measure of the hour angle is 230 27' ?0"
2. To find the sidereal time required to describe 230 27' 30"
(converse of Ex. 1).
15 ) 230 27 30
15 21 50 ; .-. required time = 15h. 21m. 50s.
24. Transits. The passage of the star across the meri-
dian is called its Transit.
Let x be the position of any star in transit (Fig. 13).
The star's E.A. = T Q or rPQ = hour angle of T
= sidereal time expressed in angle.
Hence, the right ascension of a star, when ex-
pressed in time, is equal to the sidereal time of its
transit.
In practice the R.A. of a star is always expressed in time.
Thus, the R.A. of a Lyrse is given in the tables aa
18h. 33m. 14-8s., and not as 278 18' 42".
THE CELESTIAL SPHEEE. 15
Again, let 2 be the meridian zenith distance Zx, considered
positive if the -star transits north of the" zenith, d the star's
north declination Qx, and I the north latitude QZ. Wo
have evidently -
Qx = QZ+Zx;
d = i+*c
or (star's N. decl.)
= (lat. of observer) + (star's meridian Z.D.)
This formula will hold universally if declination, latitude,
and zenith distance are considered negative when south.
Hence the R. A. and decl. of a star maybe found by observing
its sidereal time of transit and its meridian Z.D., the latitude of
the observatory being known.
Conversely, if the R.A. and decl. of a star are known, we
can, by observing its time of transit and meridian Z.D., deter-
mine the sidereal time and the latitude of the observatory.
By finding the sidereal time we may set the astronomical
clock.
A star whose E.A. and decl. have been tabulated, is called
a known star.
In Chapter II. we shall describe an instrument called the
Transit Circle, which is adapted for observing the times of
transit and meridian zenith distances of celestial bodies.
25. General Relation between R.A. and hour
angle. Let x l (Fig. 13) be any star not on the meridian.
Then
z Qp Xl = L QPr- t rP^ = ^ QPr rM]
hence, if angles are expressed in time,
(star's hour angle) = (sidereal time) (star's H.A.).
Hence, given the 11. A. and decl. of a star, we can find its hour
angle and N.P.D. at any given sidereal time, and by this means
determine the star's position on the 'observer's celestial sphere.
Or we can construct the star's position thus On the equator,
in the westward direction from Q, measure off Q T equal to
the sidereal time (reckoning 15 to the hour). Prom T east-
wards, measure f M equal to the star's It. A.; and from 3f, in
the direction of the pole, measure off Mx l equal to the star's
declinatiqn. We thus find the star x r
1 6 ASTRONOMY.
*26. Transformations. If the R.A. and decl. of a star are
given, its celestial latitude and longitude may be found, and vice
versti ; but the calculations require spherical trigonometry. The
process is analogous to changing the direction of the axes through
an angle i, in plane coordinate geometry. Again, the Z.D. and
azimuth may be calculated from the N.F.D. and hour angle, by
solving the triangle ZPx^ We know the colatitude PZ, Px^ and
L ZPx t , and we have to determine Zxi and L QZx } (= ISO PZxJ.
In the last article we showed how to find the hour angle in
terms of the R.A., or vice versA, the sidereal time being known.
Hence we see that, given the coordinates of a star referred to one
system, its coordinates referred to any other of the systems can bo
calculated at any given instant of sidereal time.
27. Culmination and Southing of Stars. A celestial
body is said to culminate when its altitude is greatest or
least.
Since the fixed stars describe circles about the pole, it
readily follows, from Spherical Geometry (26), that a star
attains its greatest or least zenith distance when on the meridian,
and, therefore, that its culmination is the same as its transit.
This is not strictly the case with the Sun, because, owing to
its independent motion, its polar distance is not constant ;
hence it does not describe strictly a small circle about the pole.
When a star transits S. of the zenith it is said to south.
28. Circumpolar Stars. A Circumpolar Star at any
place is a star whose polar distance is less than the latitude
of the place. Its declination must, therefore, be greater
than the colatitude.
On the meridian let Px and Px' be measured, each equal to
the KP.D. of such a star (Fig. 14). Then x and x' will be
the positions of the star at its transits. Since Px < Pn, both
x' and x will be above n. Hence, during a sidereal day a cir-
cumpolar star will transit twice, once above the pole (at x)
and once below the pole (at x'), and both transits will be
visible. The two transits are distinguished as the upper
and lower culminations respectively, and they succeed one
another at intervals of 12 sidereal hours ( since xPx' = 180).
The altitude of the star is greatest at upper, and least at
lower culmination, as may easily be seen from Sph. Geom.
(26) by considering the zenith distances. Hence the altitude
is never less than nx, and the star is always above the horizon.
Since
THE CELESTIAL SPHEBE.
nx-nP=Px = Px = nPnaf,
17
that is,
The observer's latitude is half the sum of the
altitudes of a circumpolar star at upper and lower
culminations.
Also, Px \ (nx nx) ;
that is,
The Star's N.P.D. is half the difference of its
two meridian altitudes.
These results will require modification if the upper culmi-
nation takes place south of the zenith as at 8. The meridian
altitude will then be measured by sS, and not nS. Here,
nS = 180 sS, and we shall, therefore, have to replace the
altitude at upper culmination by its supplement.
South Circumpolar Stars. If the south polar dis-
tance of a star is less than the north latitude of the observer,
the star will always remain below the horizon, and will,
therefore, be invisible. Such a star is called a South Cir-
cumpolar Star.
EXAMPLE. The constellation of the Southern Cross ( Crux)
is invisible in Europe, for its declination is 62 30' S ; there-
fore its south polar distance is 27 30', and it will, therefore,
pot be visible in north latitudes higher than 27 30'.
18
ASTBONOMY.
29. Rising, Southing, and Setting of Stars. If the
N. and S. polar distances of a star are both greater than the
latitude, it will transit alternately above and below the
horizon. This shows that the star will be invisible during a
certain portion of its diurnal course. Astronomically, the
star is said to rise and set when it crosses the celestial
horizon.
Let J, V be the positions of any star when rising and setting
respectively.
FIG. 15.
In the spherical triangles Pnb,
PI = Pb' (each being the star's KP.D.),
right L Pnb = right L Pnb',
and Pn is common.
Hence the triangles are equal in all respects ; therefore
Z nPb = Z nPb',
and the supplements of these angles are also equal, that is,
L sPb = L sPb'.
But the angle sPb, when reduced to time, measures the
interval of time taken by the star to get from b to the meri-
dian, and sPV measures the time taken from the meridian to
b'. Hence,
The interval of time between rising and southing
is equal to the interval between southing and setting.
THE CELESTIAL SPHERE. 19
Thus, if , f are the times of rising and setting, and T the
time of transit, we have T t tfT.
The time of transit is the arithmetic mean between
the times of rising and setting.
In order to facilitate the calculations, tables have been constructed
giving the values of T t for different latitudes and declinations.
If the observer's latitude Pn and the star's polar distance Pb are
known, it is possible (by Spherical Trigonometry) to solve the right-
angled triangle PZm, and to calculate the angle nPb, and therefore
also the angle &Ps. This angle, when divided by 15, gives the time
T t. Moreover, the sidereal time of transit T is known, being equal
to the star's R.A. Hence the sidereal times of rising and setting can
be found.
If the star is on the equator, it will rise at E and set at W.
Since JSQWis a semicircle, exactly half the diurnal path will
be above the horizon, and the interval between rising and
setting will be 12 sidereal hours. If the star is to the north
of the equator, it will rise at some point b between E and ,
so that
L IPs > Z JEPs,
i.e., / bPs > 90,
and the star will he above the horizon for more than 12 hours.
Similarly, if the star is south of the equator, it will rise at a
point c between E and *, and will be above the horizon for
less than 12 hours.
Prom the equality of the triangles bPn, b'Pn (Pig. 15), we
also see that
nb = nb', and sb = sb'.
Hence the diameter (ns) of the celestial sphere, joining the
north and south points, bisects the arc (W) between the
directions of a star at rising and setting.
This gives us an easy method of roughly determining, by
observation, the directions of the cardinal points ; but, owing
to the usual irregularities in the visible horizon, the methoij
is not very exac.
20
ASTRONOMY.
SECTION III. The Sun's Annual Motion in the Ecliptic
The Moon's Motion Practical Applications.
30. The Sun's Motion in Longitude, Bight Ascen-
sion and Declination. In 11, we briefly described
the Sun's apparent motion in the heavens relative to the
fixed stars. "We defined a Year as the period of a complete
revolution, starting from and returning to any fixed point
on the celestial sphere. The Ecliptic was defined as the
great circle traced out by the Sun's path, and its points of
intersection with the Equator were termed the First Point
of Aries and First Point of Libra, or together, the
Equinoctial Points.
We shall now trace, by the aid of Pig. 16, the variations
in the Sun's coordinates during the course of a year, starting
with March 21st, when the Sun is in the first point of Aries.
We shall, as usual, denote the obliquity by i, so that
i = 23 27' nearly.
FIG. 16.
On March 21st the Sun crosses the equator, passing
through the first point of Aries (r). This is the Vernal
Equinox, and it is evident from the figure that
Sun's longitude = 0, B.A. = O, Decl. = 0.
Prom March 21st to June 2 1st the Sun's declination is
north, and is increasing.
THE CELESTIAL SPHEEE. 21
On June 21st the Sun has described an arc of 90 from r
on the ecliptic, and is at C (Fig. 16). This is called the
Summer Solstice. If we draw the declination circle
PCQ, the spherical triangle T OQ is of the kind described in
Sph. Geom. (21), and CP is a secondary to the ecliptic.
Hence (Sph. Geom. 26) the Sun's polar distance CP is a
minimum and therefore its decl. a maximum.
Also r Q = 90 and CQ = tCrQ = i. Hence
Sun's longitude = 90, B.A. = 90 - 6h.,
N. Decl. = /, (a maximum).
From June 21 to September 23 the Sun's declination is
still north, but is decreasing.
On September 23rd the Sun has described 180, and is
at the first point of Libra (=), the other extremity of the
common diameter of the ecliptic and equator. This is the
Autumnal Equinox, and we have
Sun's long. = 180, R.A. = 180 = 12h., Decl. = 0.
From Sept. 23 to Dec. 22 the Sun is south of the equator,
and its south declination is increasing.
On December 22ud the Sun has described 270 from T,
and is at L (Fig. 16). This is called the Winter Solstice.
We have t L = 90, and the triangle . RL has two right
angles at R, L (Sph. Geom. 21). The Sun's polar dis-
tance LP is a maximum (Sph. Geom. 26), and
*R = L = 90, LR = / L^R = i. Hence
Sun's longitude = 270, R.A. = 270 = 18h.,
S. Decl. = i, (a maximum).
From December 22 to March 21 the Sun's declination is
still south, but is decreasing.
Finally, on March 21, when the Sun has performed a com-
plete circuit of the ecliptic, we have .
Sun's long. = 360, B.A. = 360 = 24h., Decl. = 0.
The longitude and R.A. are again reckoned as zero, and
they, together with the declination, undergo the same cycle
of changes in the following year.
22
ASTEONOMT.
31. Sun's Variable Motion in R.A. We observe that
the Sun's right ascension is equal to its longitude four times
in the year, viz., at the two equinoxes and the two solstices.
At other times this is not the case.
For example, between the vernal equinox and summer
solstice we have T-3f< T$, .'. Sun's E.A. < longitude.
Hence, even if the Sun's motion in longitude be supposed
uniform, its R.A. will not increase quite uniformly. There
is a further cause of the want of uniformity, namely, that
the Sun's motion in longitude is not quite uniform ; but this
need not be considered in the present chapter.
32. Direct and Retrograde Motions. The direction
of the Sun's annual revolution relative to the stars, i.e., motion
from west through south to east, is called direct. The
opposite direction, that of the diurnal apparent motions of the
stars or revolution from east to west, is called retrograde.
The revolutions of all bodies forming the solar system,
with the exception of some comets and one or two small
satellites, are direct.
We shall see in Chapter III. that the apparent retrograde
diurnal motion may be accounted for by the direct rotation
pf the Earth about its polar axis,
THE CELESTIAL SPHERE. 23
33. Equinoctial and Solstitial Points Colures.
From 30 it appears that the Summer and Winter Solstices
may be defined as the times of the year when the Sun attains
its greatest north and south declinations respectively. The
corresponding positions of the Sun in the ecliptic ((7, Z,
Fig. 17) are called the Solstitial Points. In the same way
the Equinoctial Points (T, ) are the positions of the
Sun at the Vernal and Autumnal Equinoxes when its
declination is zero.
The declination circle PTP'^j passing through the equi-
noctial points, is called the Equinoctial Colure. The
declination circle PCP'L, passing through the solstitial points,
is called the Solstitial Colure. The latter passes through
the poles of the ecliptic (7T, K').
34. To find the Sun's Right Ascension and Decli-
nation. In the "Nautical Almanack,"* the Sun's R.A.
and declination at noon are tabulated for every day of the
year. Their hourly variations are also given in an adjoining
column. To find their values at any time of the day,
we only have to multiply the hourly variation by the
number of hours that have elapsed since the preceding noon,
and add to the value at that noon.
EXAMPLE. Tfl find the Sun's R.A. and decl. on September 4, 1891
at 5h. 18m. in^gjs^ afternoon. We find from the Almanack for 1891
under Septembers :
Sun's R.A. a*oon = lOli. 52m. 15s., hourly variation 9'04s.
N. Decl. at noon = 7 12' 12" 55'4"
(1) RA. at noon = lOh. 52m. 15s.
Increase in 5h. = 9'04s. x 5 = 45*2
18m. = 27
.-. R.A. at 5h. 18m. - lOh. 53m. 3s.
(2) From the Almanack, decl. is less on September 5, and is
therefore decreasing.
N. Decl at noon = 7 12' 12"
Decrease in 6h. = 55'4" x 5 = 4' 37" \ To be
18m. - 17") subtracted.
N. Decl. at 6h. 18m. = 7 T 18 '
* Also in " Whitaker's Almanack," which may be consulted with
advantage.
24 ASTRONOMY.
35. Rough Determination of the Sun's R.A. "We
can, without the "Nautical Almanack," find to within a
degree or two, the Sun's E. A. on any given date, as follow^ :
A year contains 365 days. In this period the Sun's E.A.
increases by 360. Hence its average rate of increase is very
nearly 30 per month, or 1 per day.
Knowing the Sun's E.A. at the nearest equinox or solstice,
we add 1 for every day later, or subtract 1 for every day
before that epoch. If the E.A. is required in time, we allow
for the increase at the rate of 2h. per month, or 4m. per day.
EXAMPLES. 1. To find the Sun's R.A. on January 1st. On
December 22nd the R.A. = 18h. Hence on January 1st, which is
ten days later, the Sun's R.A. = 18h. 40m.
2. To find on what date the Sun's R.A. is lOh. 36m. On Sep-
tember 23rd the R.A. is 12h. Also 12h.-10h. 36m. = 84m., and
the R.A. increases Sim. in 21 days. Hence the required date is 21
days before September 23, i.e., September 2nd,
36. Solar Time. Apparent Noon is the time of the
Sun's upper transit across the meridian, that is, in north
latitudes, the time when the Sun souths. Apparent Mid-
night is the time of the Sun's transit across the meridian
below the pole (and usually below the horizon).
An Apparent Solar Day is the interval between two
consecutive apparent noons, or two consecutive midnights.
Like the sidereal day, the solar day is divided into 24 hours,
which are again divided into 60 minutes of 60 seconds each.
For ordinary purposes the day is divided into two portions :
the morning, lasting from midnight to noon ; the evening,
from noon till midnight ; and in each portion times are
reckoned from Oh. (usually called 12h.) up to 12h. For
astronomical purposes we shall find it more convenient to
measure the solar time by the number of solar hours that
have elapsed since the preceding noon. Thus, 6.30 A.M. on
January 2nd will be reckoned, astronomically, as 18h. 30m.
on January 1st. On the other hand, 12.53 P.M. will be
reckoned as Oh. 53m., being 53 minutes past noon.
During a solar day the Sun's hour angle increases from
to 360. It therefore increases at the rate of 15 per hour.
Hence
The apparent solar time = the Sun's hour angle
expressed in time.
THE CELESTIAL SPHERE. 25
At noon the Sun is on the meridian. The sidereal time,
being the hour angle of T, is the same as the Sun's H.A., i.e.,
Sidereal time of apparent noon Sun's R. A. at noon.
At any other time, the difference between the sidereal and
solar times, being the difference between the hour angles of
T and the Sun, is equal to the Sun's E.A. Hence, as in
25, we have
(Sidereal time) (apparent solar time) = Sun's R.A.
If a and a + x are the right ascensions of the Sun at two
consecutive noons, then, since a whole day has elapsed between
the transits, the total sidereal interval is 24h. +#, and exceeds a
sidereal day by the amount x. But the interval is a solar day.
Hence, the solar day is longer than the sidereal
day, and the difference is equal to the sun's daily
motion in R.A.*
37. Morning and Evening Stars. Sunrise and
Sunset. "When a star rises shortly before the Sun, and in
the same part of the horizon, it is called a Morning Star.
Such a star is then only visible for a short time before sunrise.
When a star sets shortly after the Sun, and in the same part
of the horizon, it is called an Evening Star. It is then
only visible just after sunset.
It will be readily seen from a figure, that a star will be a
morning star if its decl. is nearly the same as the Sun's, while
its E/.A. is rather less. Similarly, a star will be an evening
star if its decl. is nearly the same as the Sun's, but its RA.
somewhat greater. Thus, as the Sun's R.A. increases, the
stars which are evening stars will become too near the Sun to
to be visible, and will subsequently reappear as morning stars.
The times of sunrise and sunset are calculated in the
manner described in 29. The hour angles of the Sun, when
crossing the eastern and western horizons, determine the
intervals of solar time between sunrise, apparent noon, and
sunset. The two intervals are equal, if the Sun's decl. be
supposed constant from sunrise to sunset a result very
approximately true, since the change of decl. is always very
small.
* Owing to the sun's variable motion in R. A., the apparent solar day is not quite
of constant length. In the present chapter, however, it may be regarded as
approximately constant.
26 ASTRONOMY.
38. The Gnomon. Determination of Obliquity of
Ecliptic. The Greek astronomers observed the Sun's
motion by means of the Gnomon, an instrument consisting
essentially of a vertical rod standing in the centre of a hori-
zontal floor. The direction of the shadow cast by the Sun
determined the Sun's azimuth, while the length of the shadow,
divided by the height of the rod, gave the tangent of the
Sun's zenith distance. To find the meridian line, a circle was
described about the rod as centre, and the directions of the
shadow were noted when its extremity just touched the circle
before and after noon. The sun's Z.D.'s at these two
instants being equal, their azimuths were evidently (Sph.
Geom. 27) equal and opposite, and the bisector of the angle
between the two directions was therefore the meridian line.
The Sun's meridian zenith distances were then observed
both at the summer solstice, when the Sun's IS", decl. is i and
meridian Z.D. least, and at the winter solstice, when the Sun's
S. decl. is i and meridian Z.D. greatest. Let these Z.D.'s be z l
and s 2 respectively, and let I be the latitude of the place of
observation. From 24, we readily see that
2 t = l-i, 2 2 = Z+t,
/: *=*(.+*,), * = i(v-i);.
thus determining both the latitude and the obliquity.
39. The Zodiac. The position of the ecliptic was defined
by the ancients by means of the constellations of the Zodiac,
which are twelve groups of stars, distributed at about equal
distances round a belt or zone, and extending about 8 on
each side of the ecliptic. The Sun and planets were observed
to remain always within this belt. The vernal and autumnal
equinoctial points were formerly situated in the constellations
of Aries and Libra, whence they were called the First Point
of Aries and the First Point of Libra. Their positions are very
slowly varying, but the old names are still retained. Thus,
the " First Point of Aries" is now situated in the constel-
lation Pisces.
The early astronomers probably determined the Sun's
annual path by observing the morning and evening stars.
After a year the same morning and evening stars would be
observed, and it would be concluded that the Sun performed
a complete revolution in the year.
THE CELESTIAL SPHEEE. 27
40. Motion of the Moon. The Moon describes among
the stars a great circle of the celestial sphere, inclined to
the ecliptic at an angle of about 5. The motion is direct,
and the period of a complete " sidereal " revolution is about
27 days.
In this time the Moon's celestial longitude increases by 360.
"When the Moon has the same longitude as the Sun, it is
said to be New Moon, and the period between consecutive
new Moons is called a Lunation. AVhen the Moon has
described 360 from new Moon, it will again be at the same
point among the stars ; but the Sun will have moved forward,
so that the Moon will have a little further to go before it
catches up the Sun again. Hence the lunation will be rather
longer than the period of a sidereal revolution, being about
29 \ days.
The Age of the Moon is the number of days which have
elapsed since the preceding new Moon. Since the Moon
separates 360 from the Sun in 29j days, it will separate at
the rate of about 12, or more accurately 12-|- , per day,
or 30' per hour. This enables us to calculate roughly the
Moon's angular distance from the Sun, when the age of the
Moon is given, and conversely, to determine the Moon's age
when its angular distance is given.
EXAMPLE. On September 23, 1891, the Moon is 20 days old.
To find roughly its angular distance from the Sun and its longitude
on that day.
(1) In one day the Moon separates 12^- from the Sun; therefore,
in 20 days it will have separated 20 x 121, or 244, and this is the
required angular distance from the Sun.
(2) On September 23 the Sun's longitude is 180 ; therefore the
Moon's longitude is 180 + 244 = 424 = 360 + 64, or 64.
This method only gives very rough results; for the Moon's
motion is far from uniform, and the variations seem very
irregular.
Moreover, the plane of the Moon's orbit is not fixed, but
its intersections with the ecliptic (called the Nodes) have a
retrograde motion of 19 per year. Hence, for rough pur-
poses, it is better to neglect the small inclination of the Moon's
orbit, and to consider the Moon in the ecliptic. If greater
accuracy be required, the Moon's decl. and R.A. may be
found from the Nautical Almanack.
28 ASTRONOMY.
41. Astronomical Diagrams and Practical Applica-
tions. We can now solve many problems connected with
the motion of the celestial bodies, such as determining the direc-
tion in which a given star will be seen from a given place, at
a given time, on a given date, or finding the time of day at
which a given star souths at a given time of year.
"We have, on the celestial sphere, certain circles, such as
the meridian, horizon, and prime vertical, also certain points,
such as the zenith and cardinal points, whose positions relative
to terrestrial objects always remain the same. Besides these,
we have the poles and equator, which remain fixed, with
reference loth to terrestrial objects and to the fixed stars.
"We have also certain points, such as the equinoctial points,
and certain circles, such as the ecliptic, which partake of
the diurnal motion of the stars, performing a retrograde
revolution about the pole once in a sidereal day. Lastly,
we have the Sun, which moves in the ecliptic, performing
one retrograde revolution relative to the meridian in a solar
day, or one direct revolution relative to the stars in a year,
and whose hour angle measures solar time.
In drawing a diagram of the celestial sphere, the positions
of the meridian, horizon, zenith, and cardinal points should
first be represented, usually in the positions shown in Pig.
18. Knowing the latitude nP of the place, we find the
pole P. The points Q, ft, where the equator cuts the meri-
dian, are found by making PQ = PR = 90 ; and the points
Q, Ii, with E, W, enable us to draw the equator.
We now have to find the equinoctial points. How to do
this depends on the data of the problem. Thus we may
have given
(i.) The sidereal time ;
(ii.) The hour angle of a star of known E.A. and decl ;
(iii.) The time of (solar) day and time of year.
In case (i.), the sidereal time multiplied by 15 gives, in
degrees, the hour angle (Qf) of the first point of Aries.
Measuring this angle from the meridian westwards, we find
Aries, and take Libra opposite to it. Any star of known
decl. and R.A. can be now found by taking on the equator
= star's R.A., and taking on MP, MX = star's decl.
THE CELESTIAL SPHERE.
29
The ecliptic may be drawn passing through Aries and
Libra, and inclined to the equator at an angle of about 23 \
(just over right angle). As we go round from west to east, or
in the direct sense, the ecliptic passes from south to north of
the equator at Aries ; this shows on which side to represent
the ecliptic. Knowing the time of year, we now find the
Sun (roughly) by supposing it to travel to or from the
nearest equinox or solstice about 1 per day from west to east.
Finally, if the Moon's age be given, we find the Moon by
measuring 12-i- per day, or 30' per hour eastwards from the
Sun.
P'
FIG. 18.
In case (ii.), we either know the hour angle, QMoi QPMof.
a known star (#), or, what is the same thing, the sidereal
interval since its transit ; or, in particular, it is given that the
star is on the meridian. Each of these data determines J/~,
the foot of the star's declination circle. From M we measure
westwards equal to the star's R.A. This finds Aries.
80 ASTRONOMY.
fn case (iii-)> the solar time multiplied by 15 gives the-
Hun's hour angle QPS in degrees. From the time of year
we can find the Sun's R.A., TJPS. From these we find
Q,PT and obtain the position of Aries just as in case (ii.)
It will be convenient to remember that azimuth and hour
angle are measured from the meridian westwards, while
right ascension and celestial longitude are measured from the
first point of Aries eastwards. Thus, since the Sun's diurnal
motion is retrograde, and its annual motion direct, the Sun's
azimuth, hour angle, R.A., and longitude are all increasing.
Most problems of this class depend for their solution chiefly
on the consideration of arcs measured along the equator, or
(what amounts to the same) angles measured at the pole.
In another class of problems depending on the relation be-
tween the latitude, a star's decl. and meridian altitude ( 24),
we have to deal with arcs measured along the meridian.
These two classes include nearly all problems on the celestial
sphere which do not require spherical trigonometry.
EXAMPLES.
1. To represent, in a diagram, the positions of the Sun and Moon,
and the star Herculis as seen by an observer in London on Aug. 19,
1891, at 8 p.m., the following data being given : Latitude of London-
= 51, Moon's age at noon on Aug. 19 = 14 days 19 hours, Moon's
latitude = 2 S., K.A. of (Herculia = 16h. 37m., decl. = 31 48' N.
The construction must be performed in the following order :
(i.) Draw the observer's celestial sphere, putting in the meridian,
horizon, zenith Z, and four cardinal points n, E, s, W.
(ii.) Indicate the position of the pole and equator. The observer' s-
latitude is 51. Make, therefore, nP = 51. P will be the pole. Take
PQ = PR = 90, and thus draw the equator, QERW.
(Hi.) Find the declination circle passing through the Sun. The-
time of day is 8 p.m. Therefore the Sun's hour angle is 8 x 15, or
120. On the equator measure QK = 120 westwards from the-
meridian. Then the Sun Q will lie on the declination circle PK.
Since QW = 90, we may find K by taking WK = 30 = $ WR.
(iv.) Find the first points of Aries and Libra. The date of obser-
vation is August 19. Now, on September 23 the Sun is at =2=. Also-
from August 19 to September 23 is 1 month 4 days. In this-
interval the Sun travels about 34 from west to east. Hence the
Sun is 34 west of rO=. And we must measure K* = 34 eastwards^
from 8, and thus find z.
The first point of Aries ( T ) is the opposite point on the equator..
THE CELESTIAL SPHERE.
31
(v.) We may now draw the ecliptic Cri^= passing through the
first points of Aries and Libra, and inclined to the equator at an
angle of about 23 (i.e., slightly over of a right angle). The Sun
is above the equator on August 19; hence the ecliptic cuts PK above
K. This shows on which side of the equator the ecliptic is to be
-drawn ; we might otherwise settle this point by remembering that
the ecliptic rises above the equator to the east of T .
The intersection of the ecliptic with PE determines Q, the position
of the Sun.
FIG. 19.
ascenfion is 16h. 37m., in time, = 249 15' in angular measure. On
the equator measure off T M = 249 15' in the direction west to east
(i.e., the direction of direct motion) from T ; we must, therefore,
take ^=M = 69 15'. On the declination circle HP, measure off
MX = 31 48' towards P. Then x is the required position of
Herculis.
(vii.) Find the Moon. At 8 p.m. the Moon's age is 14d. 19h + 8h.
= 15d. 3h. Hence, the Moon has separate/! from the Sun by
about 185 in the direction west to east. Measure off }) = 185
from west to east, and put in }) about 2 below the ecliptic. The
Moon's position is thus found.
32
ASTRONOMY.
a/-
2. To find (roughly) at what time of year the Star o Cygni
(R.A. = 20h. 38m., clecl. = 44 53' N.) souths at 7 p.m.
Let o be the position of the star on the meridian (Fig 20). At
7 p.m. the Sun's western hour angle (QS or QPS) = 7h. = 105.
Also TEQ, the Star's R.A. = 20h.
38m. Hence rRS, the Sun's R.A.
= 20h. 38m. - 7h. = 13h. 38m. ; or,
in angular measure, Sun's R.A.
= 204 30'. Now, on September 23,
Sun's R.A. = 180, and it increases at
about 1 per day. Hence the Sun's
R.A. will be 204 about 24 days later,
i.e., about October 17th.
3. At noon on the longest day (June
21) a vertical rod casts on a horizontal
plane a shadow whose length is equal p IG 20
to the height of the rod. To find
the latitude of the place and the Sun's altitude at midnight.
FIG. 21.
From the data, the Sun's Z.D. at noon, Z, evidently = 45.
Also, if QR be the equator, 0Q = Sun's decl. = i = 23 27' (approx.);
.-. latitude of place = ZQ = 45 + 23 27' = 68 27'.
If ' be the Sun's position at midnight,
P0' = PQ = 90-2.327' = G6 33'.
But Pn = lat. = 68 27'.
... Q' w = 68 27' -66 33' = 1 54';
and the Sun will be above the horizon at an alt. of 1 54' at
midnight.
THE CELESTIAL SPHERE.
EXAMPLES. I.
1. Why are the following definitions alone insufficient? Tlie zenith
and nadir are the poles of the horizon. The horizon is the great
circle of the celestial sphere whose plane is perpendicular to the
line joining the zenith and nadir.
2. The R.A. of an equatorial star is 270 ; determine approximately
the times at which this star rises and sets on the 21st June. In
what quarter of the heavens should we look for the star at mid-
night ?
3. Explain how to determine the position of the ecliptic relatively -
to an observer at a given hour on a given day. Indicate the position .
of the ecliptic relatively to an observer at Cambridge at 10 p.m. at
the autumnal equinox. (Lat. of Cambridge = 52 12' 51'6".)
VV!
i
4. Prove geometrically that the least of the angles subtended at
an observer by a given star and different points of the horizon
that which measures the star's altitude.
5. Show that in latitude 52 13' N. no circumpolar star when
southing can be within 75 34' of the horizon.
C. Represent in a figure the position of the ecliptic at sunrise on
March 21st as seen by an observer in latitude 45. Also in lati-
tude 67. ,
7. If the ecliptic were visible in the first part of the preceding
question, describe the variations which would take place during the
day in the positions of its points of intersection with the horizon.
8. Determine when the star whose declination is 30" N. and whose .
E.A. is 356 will cross the meridian at midnight.
9. The declination and R.A. of a given star are 22 N. and
6h. 20m. respectively. At what period of the year will it be (i.) a
morning, (ii.) an evening star ? In what part of the sky would you
then look for it ?
10. Find the Sun's R.A. (roughly) on January 25th, and thus de-
termine about whatxtime Aldebaran (R.A. 4h. 29m.) will cross the
meridian that night.
11. Where and at what time of the year would you look for
Fomalhaut ? (R.A. 22h. 51m., decl. 30. 16' S.)
12. At the summer solstice the meridian altitude of the Sun is
75. What is the latitude of the place ? What will be the meridian
altitude of the Sun at the equinoxes and at the winter solstice ?
~
34 ASTRONOMY.
EXAMINATION PAPER. I.
1. Explain how the directions of stars can be represented by
means of points on a sphere. Explain why the configurations of
the constellations do not depend on the position of the observer,
and why the angular distance of two different bodies on the celestial
sphere gives no idea of the actual distance between them.
2. Define the terms horizon, meridian, zenith, nadir, equator,
ecliptic, vertical, prime vertical, and represent their positions in a
figure.
3. Explain the use of coordinates in fixing the position of a body
on the celestial sphere, and define the terms altitude, azimuth t
polar distance, hour angle, right ascension, declination, longitude,
latitude. Which of these coordinates alwa3 T s remain constant for
the same star ?
4. Define the obliquity of the ecliptic and the latitude of the
observer. Give (roughly) the value of the obliquity, and of the latitude
of London. Indicate in a diagram of the celestial sphere twelve
different arcs and angles which are equal to the latitude of the
observer.
5. What is meant by a sidereal day and a sidereal hour ? How
could you find the length of a sidereal day without using a tele-
scope ? Why is sidereal time of such great use in connection with
astronomical observations ?
6. Show that the declination and right ascension of a celestial
body can be determined by meridian observations alone.
7. What is meant by a circumpolar star ? What is the limit of
declination for stars which are circumpolar in latitude 60 N. ?
Indicate in a diagram the belt of the celestial sphere containing all
the stars which rise and set.
8. Define the terms year, equinoxes, solstices, equinoctial and
solstitial points, equinoctial and solstitial colures. What are the
dates of the equinoxes and solstices, and what are the corresponding
values of the Sun's declination, longitude, and right ascension?
Find the Sun's greatest and least meridian altitudes at London.
9. Why is it that the interval between two transits of the Sun or
Moon is rather greater than a sidereal day ? Show how the Sun's
R.A. may be found (roughly) on any given date, and find it on
July 2nd, expressed in hours, minutes, and seconds.
10. Indicate (roughly) in a diagram the positions of the following
stars as seen in latitude 51 on July 2nd at 10 p.m, : Capella (R.A.
5h. 8m. 38s., decl. 45 53' 10" N.), a Lyras (R.A. 18h. 33m. 14s.,
decl. 38 40' 57" N.), a Scorpii (R.A. 16h. 22m. 43s., decl. 26 11'
22" S.), a Ursse Majoris (R.A. lOh. 57m. Os., dec!. 62 20' 22" N.)
CHAPTER II.
THE OBSERYATOHY.
SECTION I. Instruments adapted for Meridian Observations.
42. One of the most important problems of practical astro-
nomy is to determine, by observation, the right ascension and
declination of a celestial body. We have seen in Chapter I.
that these coordinates not only suffice to fix the position of a
star relative to neighbouring stars, but they also enable us to
find the direction in which the star may be seen from a given
place at a given time of day on a given date (41). More-
over, it is evident that by determining every day the decli-
nation and right ascension of the Sun, the Moon, or a planet,
the paths of these bodies relative to the stars can be mapped
out on the celestial sphere and their motions investigated.
In Section II. of the preceding chapter we showed that
the right ascension and declination of a star can be deter-
mined by observations made when the star is on the meridian.
We proved the following results :
The star's R.A. measured in time is equal to the time of
transit indicated by a sidereal clock ( 24).
The star's north decl. d can be found from z its meridian
zenith distance, and I the latitude of the observatory by the
iormula d = l+z,
where if the decl. is south d is negative, and if the star tran-
sits south of the zenith z is negative (24).
Lastly, I can be found by observing the altitudes of a
circumpolar star at its two culminations, and is therefore
known ( 28).
Hence the most essential requisites of an observatory must
include (i.) a clock to measure sidereal time, (ii.) a telescope
so fitted as to be always pointed in the meridian, provided
with graduated circles to measure its inclination to the ver-
tical, and with certain marks to fix the position of a star in
its field of view.
36
ASTRONOMY.
43. The Astronomical Clock is a clock regulated to
indicate sidereal time. It should be set to mark Oh. Om. Os.
at the time when the first point of Aries crosses the meridian.
It will therefore gain about 4 minutes per day
on an ordinary clock, or a whole day in the
course of a year ( 22, 36).
The clock is provided with a seconds hand, and
the pendulum beats once every second, produc-
ing audible "ticks"; hence an observer can
estimate times by counting the ticks, whilst he
is watching a star through a telescope.
The pendulum is a compensating pendu-
lum, or one whose period of oscillation is un-
affected by changes of temperature. The form
most commonly used is Graham's Mercurial
Pendulum, in which the bob carries two glass
cylinders containing mercury (Fig. 22). If the
temperature be raised, the effect of. the increase
in length of the pendulum rod is compensated
for by the mercury expanding and rising in the
cylinders. The same result is also effected in
Harrison's Gridiron Pendulum, described in
Wallace Stewart's Text-Boole of Heat, page 37.
The clock is sometimes regulated by placing
small shot in a cup attached to the pendulum.
FIG. 23.
THE OBSERVATORY. 37
44. The Astronomical Telescope (Fig. 23) consists
essentially of two convex lenses, or systems of lenses, and
0', fixed at opposite ends of a metal tube, and called the
object-glass and eye-piece respectively. The former lens
receives the rays of light from the stars or other distant objects,
and forms an inverted " image " (al) of the objects. The
centre of the round object-glass is. called its " optical
centre," and the image is produced as follows: Let AAA
be a pencil of rays from a distant star. By traversing the
object-glass these rays are refracted or bent towards the
middle ray A 0, which alone is unchanged in direction. The
rays all converge to a common point or "focus'' at a point a
in A produced, and, if received by the eye after passing #,
they would appear to emanate from a luminous point or
" image " of the star at a.
Similarly, the rays BBB, coming from another distant star,
will converge to a focus at a point b in BO produced, and
will give the effect of an " image" of the star at b. All
these images (a, b) lie in a certain plane FN, called the focal
plane of the object-glass, and they form a kind of picture or
image of such stars as are in the field of view.
The eye-piece 0' acts as a kind of magnifying glass, and
enlarges the image ab just as if it were a small object placed
in the focal plane FN. The figure shows how a second image
A'B' is formed by the direction of the pencils of light after
refraction through (/. This is the final image seen on looking
through the telescope. The eye must be placed in the plane
EE, so as to receive the pencils from A', B'.
If, now, a framework of fine wires or spider's threads
(Fig. 25) be stretched across the tube in the focal plane
FNj these wires, together with the image (#J), will be
equally magnified by the eye-piece. They will thus be
seen in focus simultaneously with the stars, and the field
of view will appear crossed by a series of perfectly distinct
lines, which will enable us to fix any star's position, and
thus determine its exact direction in space. Suppose, for
example, that we have two wires crossing one another at the
point F', and the telescope is so adjusted that the image of a
star coincides with F', then we know that the star lies in the
line joining F' to the optical centre of the object-glass.
00 ASTRONOMY.
45. The Transit Circle (Figs. 24, 26) is the instrument
used for determining both right ascension and declination. It
consists of a telescope, ST, attached perpendicularly to a
light, rigid axis, WPPE, hollow in the interior. The ex-
tremities of this axis are made in the form of cylindrical pivots,
E, W, which are capable of revolving freely in two fixed forks,
called Y's, from their shape. These Y's rest on piers of solid
stone, built on the firmest possible foundations, and they are
carefully fixed, so as always to keep the axis exactly hori-
zontal and pointing due east and west.
FIG. 24.
In order to dimini?0i the effect of friction in wearing away
the pivots, the axis is also partially supported at P, P upon
friction rollers (not represented in the figure) attached to a
THE OBSERVATORY. 3<>
system of levers ( Q, Q) and counterpoises (R, R) placed within
the piers. These support about four-fifths of the weight of
the telescope, leaving sufficient pressure on the Y's to ensure-
their keeping the axis fixed.
Within the telescope tube, in the focal plane of the object-
glass ( 44), is fixed a framework of cross wires, presenting^
the appearance shown in Fig. 25. Five, or sometimes seven,
wires appear vertical, and two appear horizontal. Of the
latter, one bisects the field of view ; the other is movable up
and down by means of a screw, whose head is divided by
graduation marks which indicate the position of the wire.
The line joining the optical centre of the object-glass to
the point of intersection of the middle vertical wire with the-
fixed horizontal wire is called the Line of
Colliinatiou. The wires should be so
adjusted that the line of colliination is per-
pendicular to the axis about which the
telescope turns. For this purpose the
framework carrying the wires can be moved
horizontally, by means of a screw, into the
right position. If the Y's have been accu-
rately fixed, then, as the telescope turns,
the line of collimation will always lie in the plane of the
meridian. Hence, when a star transits we shall, on looking
through the telescope, see it pass across the middle vertical,
wire.
Attached to the axis of the telescope, and turning with it,
are two wheels, or graduated circles, GH, having their
circumferences divided into degrees, and further subdivided
by fine lines at (usually) intervals of 5'. By means of these
graduations the inclination of the line of collimation to the
vertical is read off by aid of sevi ral fixed compound micro-
scopes, A, /, JB, pointed towards the circle. One of these
microscopes (7), called the Pointer or Index, is of
low magnifying power, and shows by inspection the number
of degrees and subdivisions in the mark of the circle, which
is opposite a wire bisecting its field of view. The pointer
should read zero when the line of collimation points to the
zenith, and the graduations increase as the telescope is.
turned northwards.
40
FIG. 26,
46. Beading Microscopes. In addition to the pointer
there are four (sometimes six) other microscopes, called
Reading Microscopes, arranged symmetrically round each
circle, as at ABCD (Fig. 26). These serve to determine the
number of minutes and seconds in the inclination of the tele-
scope, by means of the following arrangement. Inside the
tube of each microscope in the focal plane of its object-
glass* is fixed a graduated scale NL (Fig. 27) in the form of
a strip of metal with fine teeth or notches. This scale, and
the image of the telescope circle, formed by the object-glass of
the microscope, are simultaneously viewed by the eye-glass,
and present the appearance shown in Fig. 27.
FIG. 27.
A small hole O marks the middle notch, and 5 notches
correspond to a division of the telescope circle, hence the
number of notches from the hole to the next division of the
circle gives the number of minutes to be added to the pointer
reading.
* A compound microscope, like a telescope, consists of an object-
glass, which forms an image of an object, and an eye-piece which
enlarges this image. A scale or wires fixed in the plane of the
the image will, therefore, be seen in distinct focus, like the wires
in the telescope.
THE OBSERVATORY. 41
To read off the number of seconds, a pair of parallel
wires, Sit, are attached to a framework, and can be moved
across the field of view by means of a screw. One whole
turn takes the wires from one notch of the metal scale to
the next, i.e., over a space representing 1' on the telescope
circle ; and the head of the screw is divided into 60 parts,
each, therefore, representing V. The wires are adjusted
so that the graduation on the telescope circle appears midway
between them, and the reading of the screw-head then gives
the number of seconds. With practice, tenths of a second
can be estimated.
The four microscopes of one of the circles are all read, and
the best result is obtained by taking the mean of the readings.
47. Clamp and Tangent Screw. When it is required
to rotate the telescope of the transit circle very slowly, this
is done by means of the bar represented at LK in Fig. 24.
The telescope axis may be firmly clamped to this bar by
means of a clamp (not represented in the figure), which
grips the rim of one of the circles as in a vice. When this
has been done, the bar JTZ, and with it the telescope, may be
slowly turned by means of a horizontal screw at Z, called
the Tangent Screw, and provided with a long handle
attached to it by a " universal joint." This handle is held
by the observer, and he can thus turn the tangent screw
without ceasing to watch the stars.
48. Arrangements for Illumination. As most obser-
vations are conducted at night, the wires in the telescope and
the graduations of the circles must be illuminated. This is
done by a lamp placed exactly in front of one of the pivots,
the light from which is concentrated by means of a bull's-eye
lens in front and a mirror behind. Part of the rays are
reflected, by a complicated arrangement of mirrors and
prisms, so as to illuminate the parts of the graduated circle
viewed by the microscopes. The rest of the light passes
through a plate of red glass down the hollow axis to a ring-
shaped mirror, whence it is reflected up to the wires ; thus
the wires appear as dark lines on a dull red ground. There
is also another arrangement for illuminating the wires from in
front, if desired, so that they appear bright on a dark ground
42 ASTRONOMY.
49. Taking a Transit. Eye and Ear Method. If
a star is to be observed with the transit circle, its R.A.
and decl. must have been roughly estimated beforehand ;
hence, its meridian Z.D. [= (star's decl.) (observer's lat.)}
is known roughly. Before the star is expected to-
cross the meridian, the telescope is turned by hand
until the pointer indicates this roughly determined Z.D. ;
this adjustment is sufficiently accurate to ensure the-
star traversing the field of view. The telescope is then
clamped ( 47). The observer now " takes a second" from
the astronomical clock, i.e., he observes and writes down the-
hour and minute, observes the second, and begins counting
seconds by the clock's ticks. Thus, if he sees the time to be-
llh. 23ni. 29s., he writes down "llh. 23m.," and at the-
subsequent ticks he counts " 303132 33 " and so on ;
in this way he knows, during the rest of the observation, t he-
exact time at every clock -beat without looking at the clock.
The star soon approaches the first vertical wire, and passes
it, usually between two successive ticks. With practice, the
observer is able to estimate fractions of a second as follows :
Suppose the star crosses the wire between the 34th and 35th
tick. The positions of the star are noticed at tick 34 and at
tick 35, and by judging the ratio of their distances from the
wire on the two sides, the observer estimates the time of
crossing the wire by a simple proportion, and writes down,
this time, say 34'6. The estimate is difficult to make,,
because the two positions of the star are not visible simulta-
neously, and the star does not stop at them, but moves
continuously; hence to estimate tenths of seconds (as is
usually done) requires much training and practice.
Moreover, the observer must not lose count of the ticks of'
the clock, for when he has written down the instant of transit.
over the first wire the star will be nearing the second wire.*
The time of transit over the second vertical wire is now
estimated in the same way, and the process repeated at
each wire. The average of the times of crossing the five
or seven wires is taken as the time of transit ; in this way,
* In most instruments the wires are placed at such a distance-
that a star in the equator takes about 13 seconds from one wire-
to the next.
THE OBSERVATORY. 4$
the effect of small errors of observation will be much smaller
than if the transit over one wire only were observed.
This method of taking the time of transit is called the
" Eye and Ear Method."
While observing the transit, the observer turns the tele-
scope by means of the tangent screw, until the horizontal
wire bisects the image of the star ; during the rest of the
observation the star will appear to run along the horizontal
wire. After the observation, one of the circles is read by the
pointer and the four microscopes. If the circle reads 0' 0",
when the line of collirnation points to the zenith, the reading
for the star will determine its meridian Z.D., in other cases
we must subtract the zenith reading. Prom the meridian
Z.D. the declination can be found.
50. The Chronograph. To obviate the difficulty of
observing tr?nsits by the eye and ear method, an instrument
called the Chronograph is now frequently used. A cylin-
drical barrel, covered with prepared paper, is made to turn
slowly and uniformly by clockwork about an axle, on which
a screw is cut. In this way the barrel is made to move
forward in the direction of its axis, about one-tenth of an
inch in every revolution. The observer is furnished with a
key or button, which is in electric communication with a pen
or marker. At the instant when the star crosses one of the
vertical wires, the observer depresses the key, and a mark is
made upon the paper of the barrel. The astronomical clock,
also, has electric communication with the marker, and marks
the paper once every second, the beginning of a new minute
being indicated, in some instruments, by the omission of the
mark, in others, by a double mark. In this way, a record is
made of the times of transit over the wires, the marks being
arranged in a spiral, owing to the forward motion of the
barrel. The distance of the beginning of any transit-mark
from the previous second-mark can be measured at leisure
with very great accuracy, and the time of transit may thus
be readily calculated. Indeed, there is no difficulty in
recording, by this method, the transits of two, or even more,
near stars which are simultaneously in the field of view of
the telescope, for the transit-marks of the different stars can
be readily distinguished from one another afterwards.
ASTRON. E
44 ASTRONOMY.
51. Corrections. After the transit of a star has been
observed, certain corrections have to be allowed for in practice
before its true B. A. and decl. are obtained. These corrections,
which depend on errors of observation, may be conveniently
classified as follows :
(a) Corrections required for the Right Ascension :
1 . Error and rate of the astronomical clock.
2. Personal equation of the observer.
3. Errors of adjustment .of the transit circle, including
(a) Collimation error.
(5) Level error.
(c) Deviation error.
(d) Irregularities in the form of the pivots.
(e) Corrections for the " vertically" and " wire
intervals."
(5) Corrections required in finding the Declination :
1. Beading for zenith point, or for the nadir, hori-
zontal or polar point.
2. Errors of imperfect centering of the circles.
3. Errors of graduation.
4. Errors of " runs " in the reading microscopes.
Besides these corrections, which we now proceed to de-
scribe, there are others of a physical nature, such as refraction,
parallax, aberration, the description of which will be given
later. A correction is always regarded as positive when it
must be added to the ol served value of a quantity in order
to get the true value, negative if it has to be subtracted.
(a) CORRECTIONS REQUIRED FOR THE RIGHT ASCENSION.
. 52. Clock Error and Hate. A good astronomical clock
can generally be regulated so as not to gain or lose more than
about 2s. in a sidereal day. But to estimate times with
greater accuracy, it is necessary to apply a correction to the
time indicated, owing to the clock being either fast or slow.
The Error of a clock is the amount by which the clock is
sloiv when it indicates Oh. Om. Os. Thus, the error must be
added to the indicated time in order to obtain the correct
time. If the clock is fast, its error is negative.
The Rate of the clock is the increase of error during 24
hours. It is, therefore, the amount which the clock loses in
the 24 hours. If the clock gains, the rate is negative.
THE OBSERVATORY. 45
The rate of a clock is said to be uniform or constant
when the clock loses equal amounts in equal intervals of
time. In a good astronomical clock, the rate should remain
uniform for several weeks.
53. Correction for Error and Hate. If the error of a
clock and its rate (supposed uniform) are known, the correct
time can be readily found from the time shown by the clock.
The method will be made clear by the following example :
EXAMPLE. If the error of an astronomical clock be 2'52s., and its
rate be O44s., to find to the nearest hundreth of a second the correct
time of a transit, the observed time bythe clock being 19h.23m.25'44s.
Here in 24h. the clock loses 0'44s.
.-. in Ih. it loses -^ x 0'44s. = 0'0183s.
Hence, loss in 19h. = 0'0183s. x 19 = 0'348s.,
and loss in 23m. = O'OOTs.
At Oh. Om. Os. the clock error is = 2'52s. ;
/. at 19h. 23m. 25'44s., clock is too slow by 2'52s. +0'355s. = 2'88s.,
/. the correct time = 19h. 23m. 25'44s. + 2'88s.
= 19h. 23m. 28-32s.
54. Determination of Error and Rate of Clock.
The clock error is found by observing the transit of a known
star, i.e., a star whose R.A. and decl. are known.
If the clock were correct, the time of transit (when cor-
rected for all other errors) would be equal to the star's R.A.
(see 24). If this is not the case, we have evidently
(Clock error) = (Star's R.A.)
(observed time of transit).
This determines the clock error at the time of transit.
To find the rate, the transits of the same star are observed
on two consecutive nights.
Let t and t x be the observed times of transit ; then x is
the amount the clock has lost in 24 hours, i.e., the rate of the
clock. Therefore
(Bate of Clock) = (observed time of Isb transit)
(observed time of 2nd transit).
Having found the rate of the clock and its error at the
time of transit, the error at Oh. Om. Os. may be found by
subtracting the loss between Oh. Om. Os. and the transit.
Stars used in finding clock error arc known as "Clock
Stars."
46 ASTBONOMY.
55. Personal Equation is the error made by any par-
ticular observer in estimating the time of a transit.
Of two observers, one may habitually estimate the transit
too soon, another may estimate it too late, but experience
shows that the error made by each observer in taking times
of transit by the same method is approximately constant.
If all observations are made, by the same individual there
will be no need to take account of personal equation, because
the error made in taking a transit will be compensated by the
error made in observing the clock stars to set the clock. If
the two operations are performed by different observers, we
must allow for the difference of their personal equations.
Personal equation may be measured by an apparatus for
observing the transit of a fictitious star, .<?., a bright point
moved by clockwork ; in this case the actual time of its transit
is known, and can be compared with the observed time.
Personal equation is positive if the observer is too quick,
so that the correction must be added to the observed time to
get the true time, as in 51.
56. Errors of Adjustment of the Transit Circle.
If the transit circle is in perfect adjustment, the line of colli-
mation of the telescope must always lie in the plane of the
meridian. If not, we must correct for the small errors of
adjustment. The conditions required for perfect adjustment,
together with the corresponding corrections when these con-
ditions are not fulfilled, may be classified as follows :
(a) The line of collimation should be perpendicular to the
axis about which the telescope rotates. If not, the corre-
sponding correction is called Collimation Error.
(b) The axis of rotation must be horizontal. Level Error.
(0) The axis must point due east and west. Deviation
(or Azimuthal) Error.
(d) The pivots resting on the Y's must be truly turned,
and form parts of the same circular cylinder. Correction for
shape of pivots.
(e) The vertical wires in the transit must be truly vertical
(i.e., parallel to the meridian) and equidistant. Verticality
and Thread Intervals.
THE OBSERVATOBT. 47
*57. Collimation Error. We have seen ( 45) that the frame-
work carrying the vertical wires in the transit telescope can be
adjusted by a screw, so that collimation error can be corrected.
Suppose, for simplicity, that no other error is present. Then the
line of collimation will always make a constant small angle with the
meridian, and this angle will measure the collimation error.
To correct this error, two telescopes, called Collimators, are
pointed towards each other, one due north, the other due south of
the instrument (n, s, Fig. 26). Both contain adjustable " collimating
marks," formed by cross wires in their focal planes. The transit
telescope being first pointed vertically, and two apertures in the
side of its tube being uncovered, the observer looks through the
telescope s, and sees through the apertures into the telescope n.
He then brings the wires in s into coincidence with the images of
the wires in n ; he then knows (from the optical theory of the tele-
scope) that the lines of collimation of n, s are parallel. Suppose
(e.g.) that they make a small unknown angle x" W. of S., and E. of
N., respectively.
He now looks through the transit telescope into the collimator s.
He adjusts the middle vertical wire of the transit to coincide with
the image of the cross mark in s, reading the graduated screw by
which the adjustment is made. The line of collimation of the
transit is now x" we*t of the meridian. He points the telescope into
n, and similarly adjusts the wires : the line of collimation is now x"
east of the meridian. He now turns the adjusting screw to a reading
midway between the two observed readings ; the line of collimation
is then in the meridian, and collimation error has been removed.
*58. Level Error is measured by the inclination to the horizon of
the axis of rotation of the telecope. It causes the line of collima-
tion to trace out, on the celestial sphere, a great circle inclined to
the meridian at an angle equal to the level error.
Level error is found by pointing the telescope (corrected for
collimation error) downwards over a trough of mercury (N, Figs. 24,
26, 28).
An eye-piece is provided, called a " collimating eye-piece " (EF,
Fig. 28, p. 49), containing a plate of glass M, which reflects the
light from a lamp straight down the tube. The mercury will
form a reflected image of the telescope, which may be treated just
as it' it ",vere a real telescope or collimator ; the wires in the actual
telescope will appear bright, and those in the image will appear
dark. By the law of reflection, if the middle wire coincide with its
image, the line of collimation will be vertical, and (since there is no
collimation error) there will be no level error. If not, the wires
are moved by the screw until the vertical wire coincides with its
image. The observer reads the angle through which the screw has
been turned, and thus measures the level error. The wires are then
replaced (otherwise collimation error would be introduced) and
level error is corrected by adjusting the Y's ( 59).
48 ASTBOtfOMY.
*59. Deviation Error is measured by the small angle which the
axis of rotation of the telescope makes with the plane of the prime
vertical. It causes the line of collimation of an otherwise correctly
adjusted transit circle to describe a great circle through the zenith
whose inclination to the meridian is equal to the deviation error.
Deviation error can be discovered by observing the times of upper
and lower transit of a circumpolar star, such as the pole star.
Suppose (e.g.) that the telescope axis points slightly south of east;
then it is readily seen by a diagram that when the telescope is
pointed north of the zenith, the line of collimation will be slightly
east of the meridian. Then, at upper transit, if the observed cir-
cumpolar star is north of the zenith it will reach the middle wire
before reaching the meridian. At lower transit it will not reach the
wire till after passing the meridian. Hence, the time from upper to
lower transit will be rather greater than 12h., and the time from
lower to upper transit will be rather less than 12h. By observing
the difference of the intervals the deviation error can be found.
In many observatories, the Y's of the transit circle can be adjusted
by screws, one moving vertically, to correct for level error, the
other horizontally, to correct for deviation error.
When these errors are corrected, the cross wires of the collimators
are brought into coincidence with the middle wire of the telescope
when pointed horizontally.
*60. The correction for the shape of the pivots is rather compli-
cated, but, in a good instrument, it should be very small. When
the pivots are much worn by friction, they should be re-turned.
The errors may be measured by making a small mark on the end
of each pivot, and observing, by means of reading microscopes, the
motions of the marks as the instrument is slowly turned round. If
the pivots are true, the marks should remain fixed, or describe circles.
*61. Verticality of the Wires maybe tested by observing one of
the collimators, whose cross wires are adjusted as in 69. If the
cross wires always appear to intersect on the middle wire of the
transit when the instrument is turned through any small angle, we
know that the middle wire is vertical.
*62. Wire Intervals By "Equatorial Wire Intervals" are
meant the intervals of time taken by a star on the equator in pass-
ing from one vertical wire of the transit to the next.
If the intervals between successive wires are unequal, the mean
of the times of transit over the wires will not in general be the
same as the time of transit over the middle wire. We may imagine
a straight line so drawn across the field of view that the time of
transit across it is exactly equal to the mean of the times of transit
over the five or seven wires. This line is called the Mean of the Wires.
By carefully determining the equatorial wire intervals, the very
small interval between the transits over the mean of the wires and
over the middle wire can be found.
For a star not in the equator, the wire intervals are proportional
to the secant of the declination. This follows from Sph. Gcom. (17).
THE OBSERVATORY.
(6) CORRECTIONS REQUIRED IN FINDING THE DECLINA-
TION OF A STAR.
63. Zenith Point. In 45 we stated that the pointer
of the transit circle is usually adjusted to read 0' when
the line of collimation is pointed to the zenith. Eut it would
be very difficult to adjust the microscopes to give a mean
reading of exactly 0' 0" for the zt nith. Hence it is neces-
sary to determine the zenith point, or zenith reading, and
in calculating the meridian Z.D. of any star, this must be
subtracted from the reading for the star.
Let ^and -ZVbe the readings when the telescope is pointed
to the zenith and nadir, respectively, ZTand H' the readings
for the north and south points of the horizon ; then evidently,
Z=. H-90 = ^-180 = #"'-270.
Also, if x is the reading for the meridian transit of any star,
then star's meridian Z.D.= # Z, if north of the zenith,
or, = 360 (xZ\ if south of the zenith.
64. To find the Nadir Point, use is made of the Colli-
mating Eye Piece, already mentioned in 56, and
represented in Pig. 28. It consists of
two lenses J2, F, between which is a
plate of glass, l/~, inclined at an angle of
45 to the axis. This plate illuminates
the wires from above by partially re-
flecting the light from a lamp on them,
at the same time allowing them to be
seen through the eye-glass, E,
The telescope is pointed downwards
over the trough of mercury, N\ and
the rays of light from any one of the
wires, Q, will produce by reflection a
distinct image of the wire at q in the
focal plane. Ey turning the telescope
with the tangent screw, the fixed hori-
zontal wire may be made to coincide
with its image ; it will then be verti-
cally over the " optical centre" of the
object-glass ( 44). The line of colli-
mation will, therefore, point to the
nadir, and the nadir reading is given by
the pointer and microscopes. Subtracting
the zenith reading.
FIG. 28.
180, we have the
50
ASTRONOMt.
65. Determination of Horizontal Point. Method
of Double Observation. Both the horizontal reading and
the meridian altitude of a star can be determined by observ-
ing the star, both directly and by reflection, in a trough of
mercury placed in a suitable position (M, Pigs. 26, 29).
FIG. 29.
Fig. 29 illustrates the method of double observation. Let
PZ be the direction of the line of collimation corresponding
to the zero reading, PR the horizontal direction, PS and
HTP the directions of the star viewed directly and its image
viewed by reflection. The reading of the circle for the direct
observation is the angle ZPS, the reading for the reflection
is the angle ZPM.
Since the angles of reflection and incidence S'MZ', TMZ'
at the mercury are equal, and MS', PS are parallel, we have
evidently L SPH= S'MS' = TMJT = MPH-,
.-. star's altitude, SPH= f 8PM-
= \ (ZPN-ZPS)
= half the difference of the two readings.
Also : Horizontal reading, ZPH \ (ZPM+ZPS) ;
=: half the sum of the two readings.
Subtracting 90 from the north horizontal point, the zenith
point is found.
*66. In using this method with the transit circle of a fixed
observatory, the star will remain sufficiently long in the field of
view to allow of both observations being made at the same transit,
and the fact of the star not being quite on the meridian will not
THE OBSEBVATOBY. 51
affect the results perceptibly. But there will not be time to read
the circles by means of the four microscopes, between the two
observations. This difficulty is obviated by proceeding thus :
Before the first observation, point the telescope (by means of the
pointer) in such a direction that the reflection of the star in the
mercury will cross the field of view during fhe transit; for this
purpose the star's meridian altitude must be known approximately.
(Jlamp the telescope, and read the microscopes. When the star
appears in the field of view, adjust the moveable horizontal wire (by
means of its graduated screw) till it crosses the star, keeping the
telescope fixed. Now un clamp the telescope, and point it to the star
direct, turning it with the tangent screw until the moveable horizontal
wire again crosses the star. After the observation, read the graduated
screw of the horizontal wire, and also the pointer and microscopes.
Since the star is bisected by the same wire at each observation,
the difference in the readings gives the angle through which the
telescope was rotated, and this angle is evidently double the star's
altitude. Half the sum of the readings gives what would be the
reading if the moveable wire were pointed horizontally. This must
be corrected by adding the angular interval between the moveable
and fixed wires as determined from the graduated screw, and we
then have the reading for the horizon point when the fixed wire is used.
67. Polar Point. In order to find the declination of a
star by means of the transit circle, it is necessary to know
the reading when the telescope is pointed to the pole. This
may be found, just as in 28, by observing the upper and
lower transits of a circumpolar star. The mean of the two
readings gives the polar point.
The N.P.D. of any star is found by taking the difference
of the readings for the star and the polar point. The decli-
nation is, of course, the complement of the N.P.D.
We may also find declinations thus : Since angles are
measured from the zenith northwards, it is evident (by draw-
ing a figure or otherwise) that the reading for the point of
the equator above the horizon is given by
Equatorial point = (Polar point) +270.
Since the decl. is the angular distance from the equator, we have
(North Decl.) = (Reading for star) (Equatorial point).
If the star transits north of the zenith, its reading must be
increased by 360.
The latitude of the observatory is given by
Latitude = Altitude of pole
= (North horizontal point) (Polar point).
52 ASTEONOMT.
*68. Errors of Graduation. The operation of testing the accuracy
of the graduations on the circles of the transit circle is very long
and laborious. One of the two graduated circles is so attached to
its axis, so that it can be turned through any angle relative to the
telescope. Then, by reading the microscopes belonging to both
circles, every graduation on one circle is compared with every
graduation on the other circle, and any errors of graduation are thus
detected and measured. The effect of such errors is much reduced
by using all the four microscopes, and taking the mean of their
readings.
*69. Errors due to Imperfect Centering of the Circles. By
taking the mean of the microscope readings, all errors due to imper-
fect centering are eliminated. In proof, let us suppose that only
two microscopes (A, C, Fig. 26) are used, but that these are opposite
to one another. If the circle is truly centred, with its centre on
the line AC, the two readings will differ by 180. If, now, the gradu-
ated circle is displaced, without being rotated, till its centre is at a
distance h from AC, then the points of the scale, now under AC,
will be at distances h from the points formerly under AC, both being
displaced in the same direction. Hence, since both readings are
measured the same way round the circle, one will be increased
and the other will be decreased by the same angle. The arithmetic
mean of the two readings will, therefore, be unaltered by the dis-
placement of the centre, and will be independent of any small error
due to imperfect centering. The same is, of course, true of the
mean reading for the other pair of microscopes, B, D.
The error in centering may be discovered by taking the difference
of the readings of a pair of opposite microscopes. This difference
should be 180 ' if the circle is properly centred ; if not, the amount
by which it differs from 180 will determine how much the centre of
the circle is to one side or the other of the line joining the centres
of the pair of microscopes.
*70. Error of Runs. In the reading microscopes, one turn of the
micrometer screw should move the parallel wires over a space corre-
sponding to exactly 1' on the graduated circle, so that the wires
should be brought from one mark of the circle to the next by exactly
five turns of the screw. In practice it will probably be found that
rather more or rather less than five turns will be necessary. In this
case the readings of the teeth and of the micrometer screw-head will
differ slightly from true minutes and seconds of arc on the circle,
and a correction will be required. This error is called Error of
Runs.
*71. Collimation, Level and Deviation Errors have no appre-
ciable effect on observations for declination, provided that such
errors are small compared with the star's N.P.D. Hence, they may
be left out of account, except in observations of the Pole Star.
1BE OBSEKVATOKI. 53
72. General Remarks. We first described the Transit
Circle, and the methods of " taking a transit" ; we afterwards
described the corrections which must be applied to the results
of the observations in finding the right ascension and decli-
nation of a star. But in practical work the various errors
must be determined before any observation can be made.
Among these, collimation, level and deviation error, and the
nadir point should be found daily, as they may be affected by
heat or cold, or by shaking the instrument.
Clock error and rate are also determined daily by observing
certain " clock stars." The accuracy of the corrections may
be tested by observing various "known stars" of different
declinations. If the corrections have been accurately made,
the observed right ascensions and declinations should agree
with their values as given in astronomical tables.
Before determining clock error and rate by nieuns of a
11 clock star," the R.A. of one such star must be known.
Since the R.A. is measured from the first point of Aries, that
point must first be found. The method of finding it will be
described in Chap. IY.
73. Observations on the Sun, Moon, and Planets.
The positions of the Sun, Moon, and Planets are defined by
the coordinates of their centres. In finding these, the
angular diameters must be taken into account.
In observing the Moon or a planet, the fixed horizontal wire
is adjusted to touch the illuminated edge of its disc, and the
times at which its edge touches the vertical wires are ob-
served. To find the coordinates of the centre, a correction
is made for the angular semi-diameter of the body, which
must be determined independently. It must not be forgotten
that the image formed by the telescope is inverted.
In observing the Sun, the semi-diameter may be found
during the observation by adjusting the moveable horizontal
wire to touch one edge of the disc, while the fixed wire
touches the other edge. The reading of the micrometer
screw gives the Sun's angular diameter. In finding the time
of transit, the times of contact of the disc on arriving at and
leaving each wire are separately observed ; their arithmetic
mean for any wire is the time of transit of the centre.
54
ASTRONOMY.
SECTION II. Instruments adapted for Okxertalions off the
Meridian.
74. The Transit Circle can only be used to observe celestial
bodies during the short period before and after their transit
that they remain in the field of view. It is, therefore, un-
suited for continuous observation of a celestial body, such as
is required more particularly in Physical Astronomy. Eor
this purpose, a telescope must be mounted in such a way that
it can be pointed in any required direction, or moved so as to
keep the same body always in the field of view. There are
two such forms of mounting, and the telescopes thus mounted
are called the Altazimuth and the Equatorial.
FIG. 30.
75. The Altazimuth, In this instrument, a telescope,
ST, is supported so that it can turn freely about a horizontal
axis, CD, sometimes called the secondary axis. This
secondaiy axis, with the attached telescope, is capable of
turning about a fixed vertical axis, AB, sometimes called the
primary axis, which is supported at its upper and lower
ends as shown in the figure.
Both axes are provided with graduated circles, GIT,
TTTF. OBSERVATORY. 55
attached to, and turning with them. Each circle is read
by means of one or more "pointer" microscopes, M and TV.
There are also clamps, furnished with tangent screws, hy
means of which the circles may be fixed in any desired posi-
tion, or rotated slowly if required. At C is a counterpoise,
which balances the telescope and the circle 7F, and so
prevents their weight from bending the axis AB.
By rotating the whole instrument about the vertical axis
AS, the telescope can be brought to any required azimuth.
If now the circle GH\)Q clamped, the telescope can be turned
about CD to any required altitude. The microscope N
should indicate zero when the telescope is pointed in the
plane of the meridian, and the microscope M should indicate
zero when the telescope is horizontal. If now the telescope
be pointed so that a star is in the middle of its field of view,
the readings of the two microscopes TV, M will give the star's
azimuth and altitude respectively. The time of observation
being also known, the position of the star on the celestial
sphere is completely determined, and its R.A. and decl. can
be calculated if required. But for observations of this class,
the altazimuth is not nearly so reliable as the transit circle.
As the altazimuth possesses two independent motions, while
the transit circle possesses only one, the former instrument
is liable to a far greater number of errors of adjustment;
moreover, its telescope is far less firmly and rigidly supported,
and the instrument is therefore more liable to bend.
A large altazimuth in Greenwich Observatory is used for
observing the Moon's motion, when it is so near the Sun
that it cannot be accurately investiga 4 o 1 by meridian observa-
tions alone.
A portable telescope, mounted on a tripod stand, such as is
commonly used for observing the stars at night, is an altazi-
muth unprovided with graduated circles.
A Finder (F) is usually attached to a large altazimuth,
whose field of view is of small angular breadth. This is a
small telescope of lower magnify ing-power, with a larger field
of view, the centre of which is marked by cross wires. To
point the large telescope to any celestial body, the altazimuth
is so adjusted that the body is seen in the centre of the finder.
It will then be in the field of view of the large telescope.
56
ASTRONOMY.
76. The Equatorial (Fig. 31). If we suppose an alta-
zimuth inclined so that its primary axis, instead of being
vertical, is pointed in the direction of the pole, we shall have
an Equatorial. In this instrument the framework carrying
the telescope turns as a whole about about the primary axis
A JB, which is supported at A and J?, so as to point towards
the pole. Attached perpendicularly to this axis, and turning
with it, is a graduated circle, called the Hour Circle, which
read by a " pointer " microscope N.
The framework AB carries a secondaiy axis perpendi-
cular to the primary axis, and the telescope ST\& attached
perpendicularly to this secondary axis, about which it
is free to turn. The axis of the telescope carries another
graduated circle called the Declination Circle which is
read by the " pointer" microscope M.
FIG. 31.
The declination circle should read zero when the telescope
is pointed in the plane of the equator, and the hour circle
should read zero when the telescope is in the plane of the
meridian. If now the telescope is pointed towards any
celestial body, the readings of the two microscopes will
give, respectively, the declination and hour angle of the
body.
When it is required to observe the same body continuously
with the equatorial, the declination circle is clamped, and
the observer must slowly rotate the hour circle by hand, so
as to keep the body observed in the field of view.
THE OBSERVATORY. 57
In large instruments the hour circle can be attached to a
clamp which is worked by clockwork in such a manner that
the whole framework turns uniformly round the primary axis
AB once in a sidereal day. This motion will ensure that the
star under observation shall always remain in the centre of
the field of view.
The pointer-microscope of the hour circle may be made to
revolve with the clamp, and to mark zero when the telescope is
pointed towards the first point of Aries ; its reading will then
give the right ascension of any observed star. But the decli-
nation and right ascension cannot be determined with any
great degree of accuracy by reading the circles of the equa-
torial. There are the same difficulties as in the altazimuth ;
moreover, the primary axis, being inclined to the vertical, is
more liable to bend under the weight of the telescope.
The clockwork by which the equatorial is driven could not
be regulated by an ordinary pendulum, as this would make
the telescope move forward in a series of jerks, one at every
beat. For this reason, a conical pendulum revolving uniformly
must be used. The reader will find the principle of the
conical pendulum explained in most text-books on elementary
dynamics; a working example maybe seen in the "Watt's
Governor" of a steam-engine.
In most modern equatorials, the primary axis is not sup-
ported as in Fig. 31, but on a pillar just underneath the
secondary axis. The advantage is that the primary axis is
less liable to bend than when supported at its two ends A, B.
77. Uses of the Equatorial. Amongst these the fol-
lowing may be mentioned :
(i.) " Differential " observations, i.e., micrometric obser-
vations of the relative distances and positions of two near
stars simultaneously visible.
(ii.) Observations of the appearance, structure, and magni-
tude of the celestial bodies.
(iii.) Stellar photography,
(iv.) Spectroscopic analysis.
58
ASTRONOMY.
78. Micrometers. Any instrument used for measuring
the small angular distance between two bodies simultaneously
visible in the field of view of a telescope is called a
Micrometer. Thus the moveable horizontal wire in the
transit circle, with its graduated screw, is a micrometer, for
if the instrument be so adjusted that the fixed wire crosses
one star, while the moveable wire crosses another neighbouring
star, the distance between the wires, as read off on the screw
head, gives the difference of declination of the stars. The
moveable wire in the field of view of the reading microscope
is identical in principle with a micrometer.
79. The Screw and Position Micrometer (Fig. 32)
serves to fmdboth the angular distance bet ween two neighbour-
ing stars and the direction of the line joining them. It contains
a framework of wires placed
in the focal plane of the tele-
scope. Two of these wires
are parallel, and one of them
can be separated from the
other by turning a screw with
a graduated head. A third
wire, which we will call the
" transverse wire," is fixed
in the framework perpendi-
cular to the two former. The
whole apparatus, together
with the eye piece of the
telescope, can be rotated so
that the wires may appear in any required direction across the
field of view. A graduated circle, called the Position Circle,
is attached to the eye-piece, and measures the angle through
which it has thus been turned. Besides the wires, the frame-
work contains a transverse strip of metal marked with notches,
at distances apart corresponding to complete turns of the micro-
meter screw, an arrangement similar to that employed in the
reading microscope ( 45).
In observing two stars, the equatorial and micrometer are
so adjusted that one of the stars may appear at the inter-
section of the two fixed wires, while the other appears at the
intersection of the fixed and moveable wires.
FIG. 32.
THE OBSEBYATORY. 59
Hie number of notches of the scale, together -with the
reading of the screw-head, determine the distance "between
the images of the stars in turns and parts of a turn of the
screw-head. To find the angular distance "between the stars,
we only require to multiply by the known angular distance
corresponding to one turn of the screw.
The reading of the position circle determines the direction
of the small arc joining the stars. The position-circle should
read zero if the stars have the same R.A. Then the reading
in any other position will determine their position angle,
i.e., the angle which the line joining the stars makes with a
declination circle through one of the stars.
*80. Dollond's Heliometer is another form of micrometer, de-
pending on the principle that if the object-glass of an astronomical
telescope be cut across in two, each half will form an image of the
whole field of view, in the same way as if the lens were still com-
plete.f In the Heliometer one half of the object-glass can be made
to slide along the other by means of a graduated screw.
Fm. 33.
Suppose that we want to measure the angular diameter of the Sun
(8, Fig. 33). When the halves of the object-glass are together, so
that their optical centres coincide, one image of the Sun will be
formed. When the two halves are separated, two separate images
will be formed in the focal plane of the telescope, and will be seen
simultaneously. The half -lenses are separated, till the two images
touch, as db and be. Let 0, 0' be the optical centres of the two
halves of the objective. The distance 00' is read off on the screw-
head ; from this reading the Sun's angular diameter may be found.
For at b, the point of contact of the images, the half-lens forms
an image of the lower limb B, and the half -lens 0' forms an image of
the upper limb A. Hence, BOb and AO'b are straight lines, and ObO'
is the angular diameter Bb A. But the focal length 06 is known
Hence, if 00' is also known, the angular diameter 060' can be found.
t To show this, it is only necessary to cover up half the object-
glass of an astronomical telescope. (N.B. Not an opera-glass.)
60 ASTRONOMY.
In measuring the angular distance between two stars, the helio-
meter is adjusted so that the image of one star formed by one half-
lens coincides with the image of the other star formed by the
other half -lens 0'. The principle is the same as before.
*81. To find the angular distance corresponding to a revolution
of the micrometer screw, the simplest plan is to observe the Sun's
diameter, and to compare the reading with its known value. The
latter is given in the Nautical Almanack for every day at noon.
To test the zero reading of the position circle, the equatorial
is pointed to a star near the equator, and fixed, and the micrometer
is turned till the diurnal rotation causes the star to run along the
transverse wire. The circle should then read 90.
82. Stellar Photography. For photographic purposes,
the equatorial is driven by clockwork, carrying with it a
sensitized plate, on which an image of the heavens is projected.
In this way a photograph of part of the sky is obtained, and
on such a photograph the distances and relative positions of
the various stars, nebulaB, &c., can be accurately measured.
Moreover, by continuing the exposure sufficiently long, even
the faintest rays of light will produce an impression on the
photographic plate ; and it is thus possible to detect stars and
nebulaB which would be invisible to the eye.
*83. Spectrum Analysis. A description of the spectrum is given
in Wallace Stewart's Text-Book of Light, Chap. VIII., and the spec-
troscope is described in 91 of the same treatise.
A detailed account of the methods of spectrum analysis would be
out of place in this book, as the subject belongs to the domain of
Physical Astronomy. The general principle is this : We can, by
means of the spectroscope, analyse the constituent waves of the
light rays which reach us from the Sun and stars. We can compare
these constituents with those emitted or absorbed by the various
chemical elements in a state of vapour. Such comparisons enable
us to infer what chemical elements are present in different celestial
bodies.
84. Other Instruments. The instruments described in
this chapter are all such as are used in fixed observatori3S.
Besides these, certain portable instruments are used in astro-
nomical observations. Among the latter class the Zenith
Sector will be described in the next chapter, in connection
with the determination of the Earth's form and radius ; and
the Sextant and Chronometer will be explained in treating of
the methods of finding latitude and longitude at sea.
THE OBSEEYATOBY. 61
EXA.MPLES.II.
1. Describe the Altazimuth. Why is it not so well suited for
continuous observations as the equatorial, and, in particular, why is
it quite unsuitable for stellar photography?
2. Show that the altitude of a star is greatest when the star is on
the meridian.
3. From the result of Question 2, show how the meridian zenith
distance of a star might be found by observing its altitude with an
altazimuth.
4. How may we most easily set the astronomical clock ?
5. Show that the rate of a clock might be found by observations
on successive nights with any telescope provided with cross wires,
and pointed constantly in a fixed direction.
6. Distinguish, with examples, direct and retrograde angular
motion. Is R.A. measured direct or retrograde ?
7. Show that in latitude 45 the interval between the time of
any star's passing due east and its time of setting is constant.
8. Show that, if a transit circle be not centred truly, the con-
sequent error can be eliminated by taking the mean of the readings
of the microscopes.
9. In a double observation made with the transit circle, the
readings of the pointer directly and by reflection are 59 35' and
125 20' ; the means of the microscope readings are in the two cases
3' 42" and 1' 13''. The moveable wire reads t 2", and the reflected
star runs along the fixed horizontal wire. Find the zenith reading.
10. Explain how it is that photography has revealed the existence
of stars which are so faint as to be invisible.
11. Find the decl. of a Ophiuchi from the following observations,
made at Greenwich (lat. 51 28' 31" N.) -.Pointer reading 321 10',
microscope readings, 1' 2", 0' 50", 0' 40", 0' 58", the zenith reading
being 0' 16".
12. Find also the R.A. of a Ophiuchi. Given : Time by sidereal
clock = I7h. 29m., the numbers of seconds at the transits over the
five wires being 37'4s., 50'2s., 1m. 2'9s., 1m. 15'2s., 1m. 27'4s. Clock
error = 10'Gs. ; personal equation = + 0'4s.
62 ASTBON01TY.
EXAMINATION PAPER. II.
1. Classify the various observations which are taken in astro-
nomical investigations, and state the respective instruments which
may be used for those observations.
2. Define the right ascension and declination of a star, and describe
shortly the principles of the methods of finding them.
3. Describe how the time of transit of a star across each of the
five or seven wires of a transit instrument is observed, and explain
how the time of transit across the meridian is deduced. Define the
equatorial interval of two wires.
4. Describe the Reading Microscope, and show how the zenith
distance of a star may be found by direct observation with the
transit circle.
5. Enumerate the errors of a transit instrument, and explain how
level error may be measured and corrected.
6. Explain what is meant by collimation error, and draw a diagram
showing the circle traced out on the celestial sphere by the line of
collimation in an instrument which has a small collimation error
east of .the meridian. Is the correction, to be applied to the times
of transit, positive or negative in such a case ?
7. Describe the Equatorial, and explain the adjustments and
principal uses of the instrument.
8. Describe the Screw and Position Micrometer, and explain how
the value of a turn of the screw may be found.
9. What is meant by the error and rate of a clock, and the personal
equation of an observer? How are they usually found ?
10. On 1st March, 1872, the time of transit of j8 Librae, at Green-
wich, was observed to be 15h. 9m. 615s., and on the 3rd March the
observed time was 15h. 9m. 4'73s. The tabular R..A. of the star was
15h. 10m. 7'25s. Find the error and rate of the clock on 3rd March.
CHAPTER 111.
THE EARTH.
SECTION I. Phenomena depending on Change of Position on
the Earth.
85. Early Observations of the Earth's Form. One
of the first facts ascertained by the early Greek astronomers
was that the Earth's surface is globular in form. Even
Homer (B.C. 850 circ.) speaks of the sea as convex, and
Aristotle (B.C. 320) gives many reasons for believing the
Earth to be a sphere. Among these may be mentioned the
appearances presented when a ship disappears from view. If
the surface of the ocean were a plane, any person situated
above this plane would (if the air were sufficiently clear) see
the whole expanse of ocean extending to the furthermost
shores, with all the ships sailing on its surface. Instead of
this, it is observed that as a ship begins to sail away its
lowest part will, after a time, begin to sink below the appa-
rent boundary of the surface of the sea ; this sinking will
continue till only the masts are visible, and, finally, these
will disappear below the convex surface of the water between
the ship and the observer.
Another reason is suggested, by observing the stars. If
the Earth's surface were a plane, any star situated above the
plane would be seen simultaneously from all points of the
Earth, except where concealed by mountains or other
obstacles, and any star below the plane would be everywhere
simultaneously invisible. In reality, stars may be visible
from one place which are invisible from another ; and all the
appearances presented were found by the Greeks to agree
with what might be expected on a spherical Earth. Eratos-
thenes even made a calculation of the Earth's size from the
distance between Alexandria and Assouan and their latitudes
(91) deduced from the Sun's greatest meridian altitudes.
He found the circumference to be 250,000 stadia, or furlongs.
Lastly, the Earth's spherical form will account for the
circular form of the Earth's shadow in a lunar eclipse.
64
86. General Effects of Change of Position. In 5,
we showed that, owing to the great distance of the stars, they
are seen in the same direction whatever be the position of
the observer. In confirmation of this fact, it is found by
observation that the angular distance between any two stars
(after allowing for refraction) is observed to be independent
of the place of observation.
But the directions of the zenith and horizon vary with the
position of the observer. If we suppose the Earth spherical,
the vertical at any point on it will be the radius drawn from
the Earth's centre, while the plane of the horizon will be
a tangent plane to the Earth's surface ; both will depend
on the place. This circumstance accounts for the difference
in appearance of the heavens as seen simultaneously from
different places.
87. Earth's Rotation. The apparent rotation of the
heavens is accounted for by supposing that the stars are at
rest, and that the Earth rotates once in a sidereal day, from
west to east, about an axis parallel to the direction of the
celestial pole. The observer's zenith, horizon and meridian
turn about the pole from west to east, relatively to the stars,
and this causes the hour angles of the stars to increase by 360
in a sidereal day, in accordance with observation.
It is impossible to decide from observations of the stars
alone whether it is the Earth or the stars which rotate, just
as when two railway trains are side by side it is very difficult
for a passenger in one train, when observing the other, to
decide which train is in motion. That the Earth rotates has,
however, been conclusively proved by means of experiments,
which will be described when we come to treat of dynamical
astronomy.
88. Definitions. The Terrestrial Poles are the two
points in which the Earth's axis of rotation meets its surface.
The Terrestrial Equator is the great circle on the
Earth whose plane is perpendicular to the Earth's axis.
A Terrestrial Meridian is the section of the Earth's
surface by a plane passing through its axis. If we suppose
the Earth to be a sphere, a meridian will be a great circle
passing through the terrestrial poles.
1HE EAUTH. 65
89. Phenomena depending on Change of Latitude.
A ssuming the Earth to be spherical, let p Oqp'r be a meridian
section, C being the Earth's centre, p, p the poles, q, r points
on the equator. Then, if an observer is situated on the
meridian at 0, the direction of his celestial pole P will be
found by drawing .OP parallel to the Earth's axis^' Cp ( 87),
while his zenith Z will lie in GO produced.
Since OP is parallel to CpP lt therefore,
angle ZOP = OCp,
.'. altitude of pole at = WZOP = 90- OCp = qCO.
But the latitude of has been shown to be the altitude of
the pole ; therefore
The latitude of a place on the Earth is the angle
subtended at the Earth's centre by the arc of the
meridian drawn from the place to the equator.
Since the angle qCO is proportional to the arc qO,
The latitude of a place is proportional to its
distance from the equator.
Suppose the observer to go northwards along the meridian
from to 0', then, from what has just been shown, the
altitude of the pole increases from qCO to Z.qCO\ hence
The increase in the altitude of the pole (= /. OCO'}
is proportional to the arc 00', i.e., to the distance
travelled northwards.
66 AStfRONOMt.
90. Southern Latitudes. To an observer situated in tlio
southern hemisphere of the Earth, as at 0", the North Pole of
the heavens is below, and the South Pole, p" is above the
horizon. The South Latitude of the place is measured by
the altitude of the South Pole, p", and is equal to the
angle qCO".
At the terrestrial equator, the altitude of the pole is
zero ; hence the pole is on the horizon. At the terrestrial
North Pole p, the altitude of the celestial pole is 90, there-
fore the celestial pole coincides with the zenith. Hence,
also, an altazimuth, if taken to the North Pole, would there
become an equatorial.
PIG. 35.
At the Earth's North Pole, those stars are only visible
which are north of the equator, and they always remain
above the horizon. 1 travelling southwards, other stars,
whose declination is south, are seen in the south parts of the
celestial sphere, and on reaching the Earth's equator all the
stars will be above the horizon at some time or other, but the
Pole Star will only just rise above the horizon, near the
north point. After passing the equator, the Pole Star and
other stars near the North Pole disappear.
THE EABTfl. 6?
91. Radius of the Earth. The Earth's radius
may be found by measuring the distance between
two places on the same meridian, and finding their
difference of latitude.
Let the places of observation be 0, 0' (Fig. 35). Let the
latitudes qCO, qCO' be I and I' degrees respectively, and let
the length 00' = s. We have, supposing the Earth spherical,
ansle OCO' arc 00'
360 circumference of Earth '
Of*(\
.'. Earth's circumference = s x = ;
and Earth's radius = circumference = 180 .
2?T 7T / I
which determines the Earth's radius in terms of the data.
By observations of this kind the Earth's radius is found to
be very nearly 3,960 miles. For many purposes it will be
sufficiently approximate to take the radius as 4000 miles.
Its circumference is found by multiplying the radius by 27r,
and is about 24,900 miles, or,' roughly, 25,000 miles.
Conversely, knowing the Earth's radius, we can find the
length of the arc of the meridian corresponding to any
given difference of latitude.
92. Metre, Nautical Mile, Geographical Mile,
Fathom. The French Metre was originally defined as the
ten-millionth part of the length of a quadrant of the Earth's
meridian.
A Nautical mile is defined as the length of a minute of
arc of the meridian. Thus a quadrant of the meridian con-
tains 90 x 60, or 5,400 nautical miles, and the Earth's
circumference contains 21, GOO nautical miles.
A Fathom is the thousandth part of a nautical mile. It
contains almost exactly six feet.
A Geographical Mile is defined as the length of a minute
of arc measured, on the Earth's equator. Taking the Earth
as a sphere, the nautiral mile and geographical mile are equal.
68 ASTEOftOMT.
93. The "Knot." Use of the Log Line in Naviga-
tion. A nautical mile is sometimes called a knot. But the
Knot is more correctly the unit of velocity used in navigation,
being a velocity of one nautical mile per hour. Thus, a ship
sailing 12 knots travels at 12 nautical miles an hour.
The velocity of a ship is measured by means of the Log
Line. This consists of a "log," or float, attached to a cord
which can unwind freely from a small windlass. The log is
"heaved " or dropped into the sea, and allowed to remain at
rest, the cord being " paid out " as the ship moves away. By
measuring the length paid out in a given interval of time
(usually half a minute), the velocity of the ship may be
found. To facilitate the measurement, the line has knots
tied in it at such a distance apart that the number of knots
paid out in the interval of time is equal to the number of
nautical miles per hour at whioh the ship is sailing. It is
from these that the unit of velocity derives the name of knot.
Now one nautical mile per hour = nautical mile per
half-minute. Hence, for this interval, the knots should be
tied on the line at intervals of of a nautical mile apart.
94. From the definitions of 92, 93, it is easy to reduce
metres or nautical miles to ordinary foot and miles, and
conversely.
EXAMPLES.
1. To find the number of miles in an arc of 1.
An arc of 1 = circumference of Earth = 24900 = 69 , miles>
360 , 360
2. To find the number of feet in one fathom.
By Ex. 1, 60 nautical miles = 69 ordinary miles j i.e., 60,000
fathoms = 69jt x 5280 feet ;
/. 1 fathom = 69 * x 528 feet = 6'086 feet.
3. To express a metre in terms of a yard.
By definition, 40,000,000 metres = Earth's circumference =24,900
miles ;
.-. 1 metre = ^^S^Ur y ards = 1 '0956 yards.
Tttfe EARTH. 69
95. Terrestrial Longitude. The Longitude of a
place on the Earth is the angle between the terrestrial
meridian through that place, and a certain meridian fixed on
the Earth, and called the Prime Meridian.
Thus, in Eig. 36, if PEP' represents the prime meridian,
the longitude of any place q is measured by the angle RPq.
The longitude of q is also measured by R Q, the arc of the
equator intercepted between the meridian of the place and
the prime meridian.
FIG. 36.
Since the latitude of q is measured by the arc Qq, we see
that latitude and longitude are two coordinates denning the
position of a place on the Earth just as decl. and 11. A., or
celestial latitude and longitude define the position of a star.*
The choice of a prime meridian is purely a matter of con-
venience. The meridian of Greenwich Observatory is univer-
sally adopted by English-speaking nations. The Erench use
the meridian of Paris, and the University of Bolognahas recently
proposed the meridian of Jerusalem as the universal prime me-
ridian. Longitudes are measured both eastward and westward
from the prime meridian, from to 180, not from to 360.
*Note, however, that terrestrial latitude and longitude, being
referred to the equator, correspond more nearly to declination and
right ascension than to celestial latitude and longitude.
?0 ASTEOIfOMiT.
96. Phenomena depending on Change of Longitude.
(i.) Let q, r (Fig. 37) be two stations in the same latitude,
and let the longitude of q be L west of r, so that Z rPq = L.
As the Earth revolves about its axis at the rate of 360 per
sidereal day, or 15 per sidereal hour, the points q, r will
be carried forward in the direction of the arrow. After an
interval of -^ L sidereal hours, q will have revolved through
Z and will arrive at the position originally occupied by r.
Hence the appearance of the heavens to an observer at q will
be same as it was, -^ L sidereal hours previously, to an
observer at r. The stars will rise, south, and set -^ L hours
earlier at r than at q.
(ii.) If Aj B be two places in different latitudes, whose
difference of longitude is Z, the transits of a star at A and
B will take place when the meridian planes PAP' and
PBP' (which are evidently also the planes of the celestial
meridians of A, B respectively), pass through the direction of
the star. Hence, in this case also, the transits will occur
J-g- L hours earlier at B than at A.
Now an observer at B will set his sidereal clock to indicate
Oh. Om. Os. when T crosses the meridian of B. When T
transits at A, the clock at B will mark -fa L h., but an
observer at A will then set his clock at Oh. Om. Os. Hence,
if the two clocks be brought together and com] ared, the
clock from B will be -^ L h. faster than the clock from A.
This fact may be expressed briefly by saying that the
" local " sidereal time at B is T y h. faster than the local
sidereal time at A.
Since the Earth makes one revolution relative to the Sun
in a solar day, in like manner the local solar time at B
will be -jig-Z solar hours faster than the local solar time at A.
Therefore, whether the local times be sidereal or solar, we
have Longitude of A west of B = long, of B east of A
= 15 {(local time at .B) (local time at A)}.
In particular, Long, west of Greenwich
= 15 {(Greenwich time) (local time)}
= 15 (Greenwich time of local noon).
THE EARTH.
71
97. To find the length of any arc of a given
parallel of latitude, having given the difference of
longitude of its extremities.
[A small circle of the Earth parallel to the equator is
called a Parallel of Latitude.]
Let qr be the given arc of the parallel hqrk, I its latitude,
and let qPr, the difference of longitudes of q and r, be = Z.
Let a be the radius of the Earth.
If the meridians of q, r meet the terrestrial equator in
Q, R, we have, by Sph. Geom. (17),
arc qr = arc QR X sin Pq = arc QR x cos I.
But arc QR : circumference of Earth = Z : 360;
.-. arc QR = 27T0Z/360 =
180
/. arc qr =
iraL cos I
180
COROLLARY. Since V of arc of the equator measures a
geographical mile, it follows that
In latitude ?, the arc of a parallel corresponding to
1' difference of longitude is cos I geographical miles.
72 ASTRONOMY.
98. Changes of Latitude and Longitude due to a
Ship's Motion. Suppose a ship, in latitude I, to sail
m nautical miles in a direction A degrees west of north.
If m is small, we may easily see (by drawing a diagram)
that the ship would arrive at the same place by sailing
m cos .4 nautical miles due north, and then sailing msinA
nautical miles due west. Hence,
The ship's latitude will increase by m cos^4 minutes ( 92).
Its W. long, will increase by m sin^ sec I minutes ( 97, cor.).
NOTE. The shortest distance between two points on a sphere is
along a great circle. Hence, the shortest distance between two
places in the same latitude is less than the arc of the parallel joining
them (except at the equator). But the difference is imperceptible
when the arc is small.
99. To explain the Gain or Loss of a Day in going
round the World. If a traveller, starting from a place A,
go round the world eastward, and if, during the voyage, the
Earth revolves n times relative to the Sun, the traveller will
have performed one more revolution relative to the Earth in
the same direction, and therefore n + 1 revolutions relative to
the Sun. Hence, to a person remaining at -df, the voyage
will appear to have taken n days, while to the traveller,
n + 1 days will appear to have elapsed in other words, the
traveller will, apparently, have " gained a day."
But, as he goes eastward, he will find the local time con-
tinually getting faster, and he will have to move the hands
of his watch forward Ih. for every 15, or 4m. for every 1
of longitude. Thus, by the end of the voyage he will have
put his watch forward through 24h., and the day apparently
gained will be made up of the times apparently lost every
time the watch is put forward to local time.
Similarly, a traveller going round the world westward,
and starting and arriving back simultaneously with the first
traveller, will have made n 1 revolutions relative to the Sun,
instead of n. Hence, the journey will appear to have taken
n 1 days, and he will apparently have lost a day.
But, during the journey, he will have been continually
moving the hands of his watch backwards, so that the 24h.
apparently lost will be made up of the times apparently
gained each time the watch is put back to local time.
THE EAETH. 73
SECTION II. Dip of the Horizon
100. Definitions. Let be an observer situated above
the surface of the land or sea. Draw OT, OT tangents to the
surface. Then it is evident, from the figure, that only those
portions of the Earth's surface will be visible whose distance
from the observer is less than the length of the tangents
OT, OT.
FIG. 38.
The boundary of the portion of the Earth's surface visible
from any point is called the Offing or Visible Horizon.
Hence, if OA CB be the Earth's diameter through 0, and the
Earth be supposed spherical, the offing at is the small circle
TtT, formed by the revolution of T about OB, and having
for its pole the point A vertically underneath 0. If, however,
the Earth be not supposed spherical, the form of the offing
will, in general, be more or less oval, instead of circular.
Conversely, since it is observed that the " offing " at sea is
very approximately circular, whatever be the position of
the observer, it may be inferred that the Earth is approxi-
mately spherical.
The Dip of the Horizon at is the inclination to the
horizontal plane of a tangent from to the Earth's surface.
Hence, if HOH' be drawn horizontally (i.e., perpendicular
to OC\ the dip of the horizon will be the angle HOT.
74
ASTEONOMY.
101. To determine the Distance and Dip of the
Visible Horizon at a given height above the Earth.
Let h = A = given height of observer ;
a = CA = Earth's radius;
d OT = required distance of horizon ;
D = L HOT = required dip expressed in circular
D" the number of seconds in the dip D.
(i.) By Euclid III. 36, OT 2 = OA . OB
This determines d accurately. But in practical applications
h is always very small compared with 2a ; therefore A 2 may be
neglected in comparison with 2ah, and we have the approxi-
mate formula, rf 2 = 2ah .*. d = */ (2a7i).
(ii.) Since CTO is a right angle,
.-. z OCT= complement of L COT '= L TOR= D.
Therefore, D being expressed in circular measure, we hav<j
7)-
~
AT
radius CT
FIG. b9.
Now, in practical cases, where the dip is small, the -arc AT
will not differ perceptibly in length from the straight line OT.
We may, therefore, take arc AT= d ;
__ I2h
~ \ a'
THE EAETlt. 75
To reduce to seconds, we must multiply by 180 x 60 x GO/vr,
tbe number of seconds in a unit of circular measurement, and
we bave
, 180 X 60 X 60
/2h
V ft '
COROLLARY 1. Let #, h, d be measured in miles, and let
h' be tbe number of feet in tbe beigbt h.
Then h' = 52807& ; and taking tbe Eartb's radius a as 3960
miles, we bave
2x3960xA'
a very useful formula.
COROLLARY 2. Since tbe offing is a circle whose radius is
very approximately equal to OT QT d, we have
Area of Earth's surface visible from = nd 2 = lirah = f ?r7i'
in square miles.
*102. Accurate Determination of Dip. The use of approxi-
mations can be avoided by the exact formula :
toD-
which is adapted to logarithmic computation.
In this, as in the preceding formulae, no account has been taken
of the effect of refraction due to the atmosphere.
For this reason it is important to determine dip of the horizon
by practical observations. An instrument called the Dip Sector is
constructed for this purpose.
Tables have also been constructed, giving the dip of the horizon
as seen from different heights. They are of great use at sea,
where the altitude of a star is usually found by observing its angular
distances from the offing.
103. Disappearance of a Ship at Sea. Wben a ship
has passed the offing, the lower part will be the first to dis-
appear. Let A' 0' (Fig. 38) be the position of the ship ; let its
distance 0' be s, and let k = A 0' be tbe height above sea
level of the lowest portion just visible from 0. By the
approximate formula we have OT= ^/(2,a?i), 0'T= ,y/(2#)
This formula determines the distance s at which an object of
given height k disappears below the hori/on.
A.STKON. G
ASTRONOMY.
104. Effect of Dip on the Times of Rising and
Setting. To an observer on land, the offing is generally
more or less broken by irregularities of the Earth's surface.
At sea, however, the offing is well denned, and if the dip of
the horizon in seconds be D", the visible horizon, which
bounds the observer's view of the heavens, is represented on
the celestial sphere by a small circle parallel to the celestial
horizon, and at a distance D" below it (n'E's, Pig. 40).
Hence the stars appear to rise and set when they are at an
nngular distance D" below
the celestial horizon. Thus
they will rise sooner and set
later than they would if
there were no dip.
Taking the observer's lati-
tude to be I, let x', x be the
positions of a star of decli-
nation d, when rising across
the visible horizon n'E's and
the celestial horizon nEs
respectively. Draw x' ZTperpendicular to nEs, then x'H= D".
Then, if the star rise t seconds earlier at x' than at x, we have
15 t = Z x'Px (in seconds of angle)
= arc xx> = arc **'. (Sph. Geom., 17.)
sin xP cos d
But treating the small triangle x'xH&s plane (Sph. Geom., 24),
and remembering that Z Pxx = 90, we have
cos nxP '
.. t = If' sec d . sec nxP.
lo
Evidently the acceleration at rising = retardation at setting.
COROLLARY 1. To an observer at the Equator, P
coincides with w, .'. Z nxP = 0,
.-. the time of rising is accelerated by -^D" sec d seconds.
COROLLARY 2. If the star is on the equator, d = 0,
x coincides with E, and z nEP = nP = I,
.-. the acceleration = -&D" sec I seconds.
THE EARTH. 77
SECTION III. Geodetic Measurements Figure of the Earth.
105. Geodesy is the science connected with the accurate
measurement of arcs on the surface of the Earth. Such
measurements may be performed with either of the two
following objects :
(i.) The construction of maps.
(ii.) The determination of the Earth's form and magnitude.
Only the second application falls within the scope of this book.
10G. Alfred Russell Wallace's Method of Finding
the Earth's Radius. An approximate measure of the
Earth's radius can be readily found by means of the following
simple experiment, due to Mr. A. 11. Wallace.
FIG. 41.
Let Z, M, JV(Fig. 41) be the tops of three posts of the same
height set up in a line along the side of a straight canal.
Owing to the Earth's curvature the straight line LM will, if
produced, pass a little above N. Hence, in order to see Z, M
in a straight line, an observer at the post JV^will have to place
his eye at a point JST, a little above JV, and the height -ZTJV
may be measured. Let JL, .Of be also measured.
Since the posts are of equal height, Z, Jf, N will lie on a
circle concentric with, and almost coinciding with, the
Earth's surface. Let the vertical KN meet this circle again
in n. By Euclid III. 36,
KL . EM = EN. Kn; .-. Kn = KL . EMI EN,
and Radius of Earth = \ Kn (very approximately)
_ EL . EM
1EN
This method cannot be relied on where accuracy is required,
for the small height EN is very dim cult to measure, and a
very slight error in its measurement would affect the final
result considerably. Moreover the observations are consider-
ably affected by refraction.
78 ASTROXOMT.
107. Ordinary methods of Finding the Earth's
Radius. "Where greater accuracy is required, the radius of
the Earth is obtained by measuring the length of an arc of
the meridian and determining the difference of latitude of its
extremities; the radius may then be calculated as in 91.
The instruments required for the observations include
(i.) Measuring rods, such as the double bar ;
(ii.) A theodolite, for measuring angles ;
(iii.) A zenith sector.
108. Measurement of a Base Line. The first step is
to measure, with extreme accuracy, the length of the arc
joining two selected points, several miles apart, on a level
tract of country ; this line is called a Base Line. A series of
short upright posts are placed at equal distances apart along
the base line, and they are adjusted till their tops are seen
exactly in the same vertical plane, and are on the same level
as shown by a spirit level. Across these posts are laid
measuring rods of metal, whose length is very accurately
known, and these are also adjusted in a line, and made level
by the spirit level. These rods are not allowed to touch,
but the small distances between their ends are measured with
reading microscopes. In this way, a base line several miles
long can be measured correctly to within a small fraction
of an inch f
*109. The Double Bar.
If the measuring rods be made
of a single metal, their length i>. ^ iron I \j'
will vary with the tempera- }
ture. This disadvantage is, c 'l
however, sometimes obviated
by the use of the double bar
(Fig 42).
It consists of two bars, al, cd, one of iron, the other of brass.
These are joined together in the middle, and to their ends are
hinged perpendicular pointers eac, fbd of such length that
ea : ec = /& : fd
= coefficient of linear expansion of iron : that of brass,
= about 11 : 18.f
If the temperature be raised, the rods will expand, say to a'b',
c'd'. But aa' : cc' = ea : ec, therefore e, and similarly /, will remain
fixed. Hence the distance ef will be unaffected by the changes of
temperature. _
f Wallace Stewart's Heat, Table 22.
Brass
K
THE EARTH. 79
110. Triangnlation. When once a base line has been
measured, the distance between any two points on the Earth
can be determined by the measurement of angles alone. For,
calling the base line AB, let C be any object visible from
both A and B. If the angles CAB, CBA
be observed, we can solve the triangle H - G
ABC and determine the lengths of the ,+''*
sides CA, CB. Either of these sides, say ^S s
CA, may now be taken as the base of a new
triangle, whose vertex is another point, D.
Thus, by observing the angles of the tri-
angle A CD we can determine DA, DC in
terms of the known length of AC. Pro-
ceeding in this way, we may divide any
country into a network of triangles connect-
ing different places of observation A, B, (7, D,
and the distance between any two of the
places calculated, as well as the direction of ^C /
the line joining them. Finally, two stations ^'
(7, H are taken, which lie on the same meri-
dian, and the distance CU is calculated ; in IG '
this way it is possible to measure an arc of the meridian.
111. The Theodolite. The measurement of the angles
is far easier in practice than the measurement of a base line.
The instrument used for measuring angles is called a Theo-
dolite, and is really a portable form of altazimuth. It is
provided with spirit-levels, by means of which the instrument
fan be adjusted so that the horizontal circle is truly horizon-
tal, and the vertical axis, therefore, truly vertical; the
direction of the north point is usually found by means of a
compass needle. Most theodolites are only furnished with a
small arc of the vertical circle, sufficient for measuring the
altitude of one terrestrial object as seen from another.
By reading the horizontal circle of the theodolite, the azimuths
of B, C, as seen from A, are found. By using the difference of
azimuth instead of the angle ABC, it becomes unnecessary to take
account of the height of the various stations above the Earth. For
if A, B, C are replaced by any other points, A', B', C', at the sea
level, and vertically above or below A, B, G t the vertical planes
joining them will be unaltered in position, and therefore the
azimuths will also be unaffected.
80
ASTRONOMY.
112. Having thus found, with great accuracy, the length
of the arc joining two stations on the same meridian, it only
remains now to observe their difference of latitude.
The Zenith Sector is the most useful instrument for
this purpose. It consists essentially of a long telescope ST
(Eig. 44), mounted so as to turn about a horizontal axis, A,
near its object-glass ; this axis is adjusted to
point due east and west (as in the transit
circle). Attached to the lower end near the
eye piece is a graduated arc of a circle GH,
whose centre is at A. The line of collimation
of the telescope is indicated by cross-wires
placed in the field of view. A fine plumb-
line, AP, is attached to the axis A, and hangs
freely in front of the graduated arc. The
plumb-line should mark zero when the line of
collimation points to the zenith. When the
instrument is pointed to any star, the reading
opposite the plumb-line will be the star's zenith distance
This reading can be determined with great accuracy by
means of a reading microscope.
113. A star is selected which transits near the zenith*
and its meridian zenith distances are observed at the two
stations. Let these be s and z' degrees. Then if /, and /. 2
are the latitudes of the stations, and d the declination,
we have, by 24,
l'-l= (d-z')-(d-z) = z-z'.
Hence, if s is the measured length of the arc of the meri-
dian joining the stations, and r the radius of the Earth, 91
gives
18Q * _ 13 _
FIG. 44.
whence the Earth's radius is found.
* This position is chosen because the effects of atmospheric
refraction are least in the neighbourhood of the zenith,
THE EARTH. 81
114. Exact Figure of the Earth. If the Earth were
an exact sphere, the same value would be found for the
radius r in whatever latitude the observations were made.
But in reality the length of a degree of latitude, and therefore
also r, is found to be larger when the observation is made near
the poles than when made near the equator, and hence it is
inferred that the meridian curve is somewhat oval.
Let PQP'R represent the meridian curve, 00' two near
places of observation on it. Then, if 0J5Tand O'K be drawn
normal (i.e., perpendicular) to the Earth's surface at 0, 0',
they will be the directions of the plumb lines of the zenith
sectors at 0, 0'. Hence the observed difference of latitudes
or meridian altitudes at 0, 0' will give the angle OKO'.
Eegarding the small arc 00' as an arc of a circle whose
centre is JT, we shall have approximately,
Circular measure of OKO' = arc 00' -f- OJT,
arc 00' 180 s
_
circ. measure of OKO' TT I' V
and hence r, calculated as in 113, is the length OK.
The length OK is called the
radius of curvature of the arc,
and K is called the centre of
curvature ; they are respec-
tively the radius and centre of
the circle whose form most nearly
coincides with the meridian along
the arc 00'.
This radius of curvature OK
is not, in general, equal to C,
the distance from the centre of
the Earth, owing to the Earth FlG - 45 -
not being quite spherical.
As the result of numerous observations, the meridian curve
is found to be an ellipse (see Appendix), whose greatest
and least diameters, called the major and minor axes, are
the Earth's equatorial and polar diameters respectively. The
Earth's surface is the figure formed by making the ellipse
revolve about its minor axis POP'. This figure is called an
oblate spheroid.
ASTRONOMY.
115. To find the Equatorial and Polar Radii of Cur-
vature of the meridian curve, supposing 1 it to be an
ellipse. Let PQP'R be the ellipse. Let 2, 2i be the
lengths of its equatorial and polar diameters QCR, PCP'.
Let r v r z be the required radii of curvature at Q and P
respectively.
Take any point on the ellipse,
and let the normal at meet the
two axes in G and g respectively.
It is proved in treatises on
Conic Sections* that
OG : Og = CP* : C& = i 2 : a\
First take very near to Q.
Then OG will become equal to
the radius of curvature r^ ; also
Og will evidently become ulti-
mately equal to CQ or a.
Therefore, ^ : a = b* : a? ;
Next take very near to P.
to I and Og to r%.
Therefore, I : r 2 = W : 2 ;
Thus r x , r 2 are found in terms of a,
r =
Then G will become equal
r =
and I may be found ;
I r*r.
Conversely, if r, and r 2 are known,
for, by solving, we find a = %/(rfr
~We notice that since a > J, .*. r^ < r r
That the equatorial radius of curvature is less than the
polar is also evident from the shape of the curve. This, as the
figure shows, is most rounded at Q, It, and flattest or least
rounded at P, P'. Hence it will require a smaller circle to
fit the shape of the curve at the equator than at the poles.
116. Exact Dimensions of the Earth. The lengths of
the Earth's equatorial and polar semi-diameters, , i, are
a = 3963-296 miles, I = 3949'791 miles.
Thus, the Earth's equatorial semi-diameter exceeds its
polar semi-diameter by 13-505 miles.
* Appendix, Ellipse (9).
THE EAETH. 83
The mean radius of an oblate spheroid is the radius of a
sphere of equal volume, and is equal to ^/(a-1}. Thus, the
Earth's mean radius is approximately 3958-8 miles.
The ellipticity or compression (0) is the fraction
For the Earth, c = - nearly.
293
The eccentricity (e) is given by the relation
a~
Hence L l = s (I-* 2 ) = 8 (1 e)*;
.-. !-* = (! --<?)" = 1
Since c is small, 2 c 2, approx. ; .'. e* = 20, approx.,
which gives the Earth's eccentricity e '0826.
117. Geographical and Geocentric Latitude. The
Geographical Latitude of a place is the angle which the
normal to the Earth's surface at that place makes with the
plane of the equator. It is the latitude denned in 18,
Thus, L QGO (Fig. 46) is the geographical latitude of 0.
The Geocentric Latitude is the angle subtended at the
Earth's centre by the arc of the terrestrial meridian between
the place and the equator. Thus, / QCO is the geocentric
latitude of 0.
*118. Relations between the Geocentric and Geographical
Latitudes. Let / QGO = I, Z QGO = I'. Draw ONperp. to CQ.
Then GN : CN = OG : Og = 6 2 : a 2 ; .'. NO/CN = (NOJON) x (& 2 /o-) {
/. tan I' = tan I x & 2 /a 2 = (1- e 2 ) tan I.
We deduce also tan (l-l f ) = ^ S1 ^ 2 ^ 2 = ie"sin2l (approx.),
since e 2 is small.
84 ASTRONOMY.
EXAMPLES. III.
1. Show that the locus of points on the Earth's surface at which
the Sun rises at the same instant is half a great circle ; and state
the corresponding property possessed by the other half.
2. Find the least height of a mountain in Corsica in order that it
may be visible from the sea-level at Mentone, at a distance of 80
3. At the equator, in longitude L, a given vertical plane declines
a from the north towards the west ; find the latitude and longitude
of the places to whose horizon the given plane is parallel.
4. Prove that, at either equinox, in latitude I, a mountain whose
height is 1/n of the Earth's radius will catch the Sun's rays in the
morning , / hours bei'ore he rises on the plain at the base.
7T COSt Y n
5. Estimate to the nearest minute the value of this expression for
a mountain three miles high in latitude 45.
6. Find the distance of the horizon as seen from the top of a hill
1056 feet high.
7. Find, to the nearest mile, the radius of the Earth, supposing the
visual line of a telescope from the top of one post to the top of
another post two miles off, cuts a post, half way between, 8 inches
below the top, the posts standing at equal heights above the water
in a canal.
8. In Question 7, what would be the length of a nautical mile,
adopting the usual definition.
9. Supposing the Earth spherical, and of radius r, and neglecting
the refraction of the air, show that, if from the top of a mountain
of height a above the level of the sea, the summit of another
mountain is seen beyond the horizon of the sea, and at an elevation
e above the horizon, and if its distance be known to be D, its height is
approximately given by
a .ran. D (2-J*i
10. A railway train is moving north-east at 40 miles an hour in
latitude 60; find approximately, in numbers, the rate at which it is
phanging its longitude.
THE EARTH. 85
MISCELLANEOUS QUESTIONS.
1. Explain the different systems of coordinates by which a star's
position is fixed in thb hcnvenn.
2. Show, by a figure, where a star will be found at 9 p.m. on the
5th of June in latitude 50N., if the star's right ascension is 12 hours
and its declination 5 south.
3. Define dip, azimuth, culmination, circumpolar, zenith. Why
would it be insufficient to define the declination of a star as its
distance from the equator measured along a declination circle ?
4. Three stars, A, B, C, are on the same meridian at noon, JB being
on the equator, and A and C equidistant from B on either side.
Prove that the intervals between the setting-times of A and B and
J? and C are equal.
5. Show how to find approximately the Sun's R.A. at a given
date. Obtain its approximate value for March 1, August 10,
October 23, and January 15.
6. Describe the transit circle.
7. Define a morning and evening star. Show that on the 1st of
September a star, whose declination is 0, and R.A. llh. 28m., is an
evening star, but that it is a morning star three weeks Inter.
8. Assuming the Earth to be a sphere, show how its radius may
be practically measured.
9. Explain clearly the nature and uses of the zenith sector.
10. A, B, C are the tops of the masts of three ships in a line, and
are at equal heights above the sea- level, and is the centre of the
Earth. If the distance BC be x miles, and r is the Earth's radius
in miles, show that L BAC = \ L BOG ; and hence deduce that
zIU C= 18Qx6Qx6Q JL seconds.
TT 2r
Find this angle, having given so = 2, r = 3960, IT = 3f.
86 A.STEONOMT.
EXAMINATION PAPER. III.
1. Assuming the Earth to be a sphere, show that, as we travel from
the equator due north, our astronomical latitude (i.e., the altitude of
the Pole) will increase. Taking this increase as 1 for every
69 miles, find the circumference and the radius of the Earth.
2. Define the metre, the nautical mile, and the knot, and calculate
their values in feet and feet per second respectively, taking the
Earth's radius as 3960 miles.
3. How is the speed of a ship estimated ? Find, in feet, the dis-
tance apart of the knots on a log line, so constructed that the
number run out in half a minute measures the ship's velocity in
nautical miles per hour.
4. What are the difficulties in measuring an arc of the meridian
and how are they met ?
5. Find the Earth's radius in fathoms, and in metres. Express
the nautical mile in French units of length.
6. Obtain formulae for the distance of the visible horizon from a
place whose height is given. Deduce that, if the height h be
measured in inches, the distance in miles will be*/ , taking the
V 8
Earth's radjus as 3960 miles.
7. Define the dip of the horizon, and show how to find it. Prove
that the number of seconds in the dip is nearly 52 times the
distance in miles of the offing.
8. If A, B, and G be the tops of three equal posts arranged in
order two miles apart along a straight canal, show that the straight
line AB passes 5 feet 4 inches above C, and that AC passes 2 feet
8 inches below B.
9. Find the length of a given parallel of latitude intercepted
between two given circles of longitude.
10. Is the Earth an exact sphere ? Show that a degree of latitude
increases in length as we go northward. Distinguish a nautical
from a geographical mile.
CHAPTER IV.
THE SUN'S APPARENT MOTION IN THE ECLIPTIC.
SECTION I. The Seasons.
119. In Section III. of Chapter L* we described the Sun's
annual motion among the stars, and showed how, in con-
sequence of this motion, the Sun's right ascension increases
at an average rate of nearly 1 per day, while his declination
fluctuates between the values 23 27J' north, and 23 27J'
south of the equator. We shall now show how this annual
motion, combined with the diurnal rotation about the poles,
gives rise to the variations, both in the relative lengths of day
and night, and in the Sun's meridian altitude, during the
course of the year ; how these variations are modified by the
observer's position on the Earth ; and how they produce the
phenomena of summer and winter.
Although both the diurnal and annual apparent motions of
the Sun are known to be really due to the Earth's motion, it
will be convenient in this section to imagine the Earth to be
fixed, while the Sun and stars are moving ; thus the zenith,
pole, horizon, meridian, and equator will be considered fixed,
as they actually appear to be to an observer on the Earth.
As the change in the Sun's declination during a single day
is very small, the Sun's apparent path in the heavens from
morning till night is very approximately a small circle parallel
to the equator, and may be regarded as such for purposes of
explanation. The effects of the variation in the declination
will, however, become very apparent when we compare the
Sun's diurnal paths at different seasons of the year.
Throughout this section we shall denote the obliquity of
the ecliptic by ", the Sun's declination at any time by ^, his
zenith distance at noon by z, and the observer's latitude by I.
* The student will do well to revise Chapter I., Section III.,
before proceeding further.
88 ASTRONOMY.
120. Zones of the Earth. Definitions. From 24
it is evident that if the Sun passes through the zenith at
noon, d must = I.
But d lies between i (north) and t (south).
Therefore I must lie between the limits i N. and i S.
Thus, if the Sun be vertically overhead at some time in the
year, the latitude must not be greater than 23 27|' N. or S.
Again, from 28 we sec that the Sun, like a circumpolar
star, will remain above the horizon during the whole of its
revolution provided that 90^ < I.
This requires that I > 90- i.
Thus, if the Sun be visible all day long during a certain
period of the year, the latitude must be greater than 66 32^'
K. or S.
These circumstances have led to the following definitions.
The Tropics are the two parallels to the Earth's equator
in north and south latitude , or 23 27|-'. The northern
tropic is called the Tropic of Cancer, the southern the
Tropic of Capricorn.
The Arctic and Antarctic Circles are respectively the
parallels of north and south latitude 90 *, or 66 32f.
These four parallels divide the Earth's surface into five
regions or zones.
The portion between the tropics is called the Torrid Zone.
The portion between the tropic of Cancer and the arctic
circle is called the North Temperate Zone. The portion
between the tropic of Capricorn and the antarctic circle is
called the South Temperate Zone.
The portions north of the arctic circle, and south of the
antarctic circle are called the Frigid Zones, and are distin-
guished as the Arctic and Antarctic Zones.
121. Sun's Diurnal Path at Different Seasons and
Places. "We shall now describe the various appearances
presented by the Sun's diurnal motion at different times of
the year, beginning in each case with the vernal equinox.
We shall first suppose the observer at the Earth's equator,
and shall then, describe how the phenomena are modified as
he travels northward towards the pole.
SUN'S APPARENT MOTION IN THE ECLIPTIC.
89
122. At the Earth's equator, I = 0, and the poles of
of the celestial sphere are on the horizon (P, P', Fig. 47).
Hence, between sunrise and sunset, the Sun has always to
revolve about the poles through an angle 180, and the days
and nights are always equal, each being 12 hours long.
On March 21 the Sun is on the celestial equator, and it
describes the circle EZW, rising at the east point, passing
through the zenith at noon, and setting at the west point.
Between March 21 and Sept. 23, the Sun is north of
the celestial equator; it therefore rises north of E., transits
north of the zenith Z, and sets north of W. Its IS", meridian
zenith distance 2 is always equal to its !N". declination d
(since by 24, 2 d I and I = 0) .
Hence, from March 21 to June 21, z increases from to
i N. On June 21, z has its greatest JN". value f, and the
Sun describes the circle E'QW, where ZQ' = i.
From June 21 to Sept. 23, z decreases from i to 0.
On Sept. 23, the Sun again describes the great circle EQ W.
Between Sept. 23 and March 21, the Sun is south of the
equator, and therefore it transits south of the zenith. "We
now have z = d, both being S.
From Sept. 23 to Dec. 22, the Sun's south Z.D. at noon,
2, increases from to i.
On Dec. 22, 2 has its greatest value i (south) and the Sun
describes the circle E 'Q," W" where ZQ, " = i.
From Dec. 22 to March 21, 2 diminishes again from to 0.
On March 21, the Sun again describes the circle EQW, and
the same cycle of changes is repeated the following year.
90 ASlRONOM*.
123. In the Torrid Zone North of the Equator".
On March 21, the Sun describes the equator KQW (Fig,
48), rising at ^and setting at W. Here L ZPE L ZPW
90, and the day and night are each 12h. long. The
Sun transits S. of the zenith at Q, where ZQ = z =7.
From March 21 to June 21, d increases from to t, and
the Sun's diurnal path changes from EQVto E'QW.
The hour angles at rising and setting increase from ZPE
and ZPWiQ ZPE' and ZPW, respectively ; hence the days
increase and the nights decrease in length. The day is
longest on June 21, when the hour angle ZPE' is greatest.
The increase in the day is proportional to the angle EPE',
and is greater the greater the latitude I.
At first the Sun transits S. of the zenith, and z = ld.
"When d = , z = 0, and the Sun is directly overhead at noon.
After this, the Sun transits N. of the zenith, and z = d L
On June 21,2 attains its maximum N. value ZQ' = il.
From June 21 to Sept. 23, the phenomena occur in the
reverse order. The diurnal path changes gradually back to
EQW. The day diminishes to 12h. The Sun, which at first
continues to transit N". of the zenith, becomes once more ver-
tical at noon when d again = I, and then transits S. of the
zenith.
From Sept. 23 to Dec. 22, the Sun's path changes from
EQWto E"Q'W".
The eastern hour angle at sunrise decreases to ZPE"; thus
the days shorten and the nights lengthen. The day is
shortest on Dec. 22.
Also z increases from I to 1 -f i.
On Dec. 22, s attains the maximum value ZQ" = -f-, and
the Sun is then furthest from the zenith at noon.
From Dec. 22 to March 21, the length of the day increases
again to 12 hours, and the Sun's meridian zenith distance
decreases to z = L
124. On the Tropic of Cancer, I = i. The variations
in the lengths of day and night partake of the same general
character as in tbe Torrid Zone. But the Sun only just
reaches the zenith at noon once a year, namely, on the longest
day, June 21. At other times the Sun is south of the zenith
at noon, and z attains the maximum value 2* on December 22.
TIIE SUN'S APPABENT MOTION IN THE ECLIPTIC. 9l
Z Q' 2
P'
FIG. 49.
125. In the North Temperate Zone I > i but < 90 - i.
Here the variations in the lengths of day and night are
similar, hut more marked, owing to the greater latitude.
On March 21, the Sun describes the equator EQWR (Fig.
49), which is bisected by the horizon ; hence the day is 1 2h . long.
The length of the day increases from March 21 to June 21.
The day is longest on June 21, when the jSun describes
E'Q'WR', and the hour angles ZPE', ZPW are greatest.
The days diminish to 12h. on Sept. 23, when the Sun again
describes EQ, WE. The day is shortest on Dec. 22, when the
Sun describes E"Q!'W"R".
From Dec. 22 to March 21, the days increase in length, and
on March 21 the day is again 12 hours long.
The difference between the longest and shortest days is the
time taken by the Sun to describe the angles E'PE", W"PTP',
and is therefore
= iV ( ^ E'PE" + L W'PW} = A . / E'PE".
It will be seen that L E'PE" is greater in Fig. 49 than in
Fig. 48, thus the variations are more marked in the tem-
perate zone than in the torrid zone. The variations increase
as the latitude increases.
The Sun never readies the zenith' in the temperate zone,
but always transits south of the zenith. The Sun's zenith
distance at noon is least on June 21, when z = ZQ ' = li,
and is greatest on Dec. 22, when % = ZQ" = l+i. At the
equinoxes (March 21 and Sept. 23), z = ZQ = /.
ASTEON. H
92 ASTRONOMY.
126. On the Arctic Circle, I = 90 t. Hence on June
21, when the Sun's KP.D. = 90-*', the Sun at midnight
will only just graze the horizon at the north point without
actually setting. On Dec. 22 at noon, the Sun's Z.D. = 90,
and the Sun will just graze the horizon without actually
rising. As in the preceding case, the days increase from Dec. 22
to June 21, and decrease from June 21 to Dec. 22; on
March 21 and Sept. 23, the day and night are each 12h. long.
127. In the Arctic Zone we have l> 90- 1, and the
variations are somewhat different (Fig. 50).
On March 21, the Sun describes the circle EQW, and the
day is 12h. long.
As d increases, the days increase and the nights decrease,
and this continues until d = 90 I. When this happens,
the Sun at midnight only grazes the horizon at n.
Subsequently, while ^>90 I, the Sun remains above
the horizon during the whole of the day, circling about the
pole like a circumpolar star. This period is called the Per-
petual Day.
During the perpetual day, the Sun's path continues to rise
higher in the heavens every twenty -four hours until June 21,
when the Sun traces out the circle R' Q'. The Sun's least and
. greatest zenith distances will then be ZQ! = I i , and
ZR' 180 tZ respectively.
After June 21, the Sun's path will sink lower and lower.
When d is again 90 I the perpetual day will end.
Subsequently, the Sun will be below the horizon during
part of each day. The days will then gradually shorten and
the nights lengthen.
On Sept. 23, the Sun will again describe the circle EQ, W,
and the day and night will each be 12 hours long.
The days will continue to diminish till the Sun's south
declination d' 90 L When this happens the Sun at noon
will only just graze the horizon at s.
While d' >90 Z, the Sun remains continually below the
horizon. This period is called the Perpetual Night.
On Dec. 22 the Sun traces out the circle R"Q" below the
horizon.
When d' is again = 90 /, the perpetual night will end.
Subsequently, the day will gradually lengthen until
March 21, when it will again be 12 hours long.
THE SUN'S APPARENT MOTION IN THE ECLIPTIC. 98
Z P
FIG. 50.
Sun's altitude mil attain its greatest val ,' on June 21
when the Sun will trace out the circle QK '
21 ther night.
.
of the equator. In fact, if we consider two antipodal
or places at opposite ends of a diameter of the Earth
at one place will coincide with the night at the other
l , equat r and antarctic c Me,
the longest day, and June 21 the shortest.
Within the antarctic circle there will be perpetual day for
j certam penod before and after Dec. 22, and perpetual
for a certain period before and after June 21.
_ in _ _ _ _
r V OF THK
UNIVERSITY
94 ASTRONOMY.
The variations in the Sun's north zenith distance at noon
will be the same as the variations in the south zenith distance
in the corresponding north latitude six months earlier.*
130. The Seasons. Having thus described the variations
in the Sun's daily path at different times and places, we shall
now show how these variations account for the alternations
of heat and cold on the Earth.
Astronomically, the four seasons are denned as the portions
into which the year is divided by the equinoxes and the
solstices. Thus, in northern latitudes,
Spring commences at the Yernal Equinox (March 21),
Summer ,, ,, Summer Solstice (June 21),
Autumn ,, ,, Autumnal Equinox (Sept. 23),
Winter ,, Winter Solstice (Dec. 22).
It is obvious that the temperature at any place will depend
in a great measure upon the length of the day. While the
Sun is above the horizon, the Earth is receiving a considerable
portion of the heat of his rays, the remaining portion being
absorbed by the Earth's atmosphere through which the rays
have to pass. When the Sun is below the horizon, the
Earth's heat is radiating away into space, although the heated
atmosphere retards this radiation to a considerable extent.
Thus, on the whole, the Earth is most heated when the days
are longest, and conversely.
The variations in the Sun's meridian altitude have a still
greater influence on the temperature. When the Sun's rays
strike the surface of the Earth nearly perpendicularly, the
same pencil of rays will be spread over a smaller portion of the
surface than when the rays strike the surface at a considerable
angle ; hence the quantity of heat received on a square foot
of the surface will be greatest when the Sun is most nearly
vertical. By this mode of reasoning it is shown in Wallace
Stewart's Text-Boole of Light, 10, that the intensity oi
illumination of a surface is proportional to the cosine of the
angle of incidence, and the same argument holds good with
* The student will find it instructive to trace out fully the varia-
tions in S. latitudes corresponding to those described in 122-128.
See diagram, p. 421.
IN THE ECLIPTIC. 95
regard to radiant heat as well as light. Hence the Sun's heat-
ing power when ahove the horizon is always proportional to
the cosine of the Sun's zenith distance or the sine of its altitnde.
In this proof, however, the absorption of heat by the
Earth's atmosphere has been neglected. But when the
Sun's rays reach the Earth obliquely, they will have to pass
through a greater extent of the Earth's atmosphere, and
will, therefore, lose more heat than when they are nearly
vertical. This cause will still further increase the effect of
variations in the Sun's altitude in producing variations in the
temperature.
131. Between the Tropics the combination of the two
causes above described tends to produce high temperatures,
subject only to small variations during the year. The Sun's
meridian altitude is always very great, and the variations in the
lengths of day and night are small. If the latitude be north, the
Sun's heating power is greatest while the Sun transits north
of the zenith. During this period the Sun's meridian
altitude is least when the days are longest. Thus the effects
of the two causes in producing variations in the Sun's heat
counteract one another, to a certain extent, and give rise to
a period of nearly uniform but intense heat.
In the North Temperate Zone, the Sun is highest at
noon when the days are longest, and therefore both causes
combine to make the spring and summer seasons warmer
than autumn and winter. But the highest average tempera-
tures occur some time after the summer solstice, and the
lowest temperatures occur after the winter solstice ; for
the Earth is gaining heat most rapidly about the summer
solstice, and it continues to gain heat, but less rapidly, for
some time afterwards. Similarly, the Earth is losing heat
most rapidly at the winter solstice, and it continues to lose
heat, but less rapidly, for some time afterwards. Por this
reason, summer is warmer than spring, and winter is colder
than autumn. '
As we go northwards, the Sun's altitude at noon becomes
generally lower throughout the year, and the climate therefore
becomes colder. At the same time, the variations in the length
of the day become more marked, causing a greater fluctua-
tion of temperature between summer and winter.
96 ASTRONOMY.
Within the Arctic Circle there is a warm period during
the perpetual day, but the Sun's altitude is never sufficiently
great to cause very intense heat. During the perpetual night
the cold is extreme ; and the low altitude of the Sun, when
above the horizon at intermediate times, gives rise to a very
low average temperature during the year.
In the Southern Hemisphere the seasons are reversed ;
for, in south latitude I, when the Sun's south declination is d,
the same amount of heat will be received from the Sun as in
north latitude I, when his north declination is d. Hence, the
seasons corresponding to our spring, summer, autumn and
winter will begin respectively on September 23, December 22,
March 21, and June 21, and will be separated from the corre-
sponding seasons in north latitude by six months.
132. Other Causes affecting the Seasons and
Climate. It is found (as will be explained in the next
section) that the Sun's distance from the Earth is not quite
constant during the year. The Sun is nearest the Earth
about December 3 1 , and furthest away on July 1 (these are
the dates of perigee and apogee respectively) . As shown in
Wallace Stewart's Text-Book oj Light, 9, the intensity of
illumination, and therefore also of heating, due to the Sun's
rays, varies inversely as the square of the Sun's distance.
Hence the Earth receives, on the whole, more heat from the
Sun after the winter solstice than after the summer solstice.
This cause tends to make the winter milder and the summer
cooler in the northern hemisphere, and to make the summer
hotter, and the winter colder in the southern hemisphere.
The variations in the Sun's distance are, however, small,
and their effect on the seasons is more than counter-
acted by purely terrestrial causes arising from the unequal
distribution of land and water on the Earth. The sea has a
much greater capacity for heat than the rocks forming the
land ; it is not so readily heated or cooled. In the southern
hemisphere the sea greatly preponderates, the largest land-
surfaces being in the northern hemisphere. Hence, the
climate of the southern hemisphere is generally more equable,
and the seasons are not so marked as in the northern hemi-
sphere, quite in contradiction to what we should expect from
the astronomical causes.
THE SUN'S APPAKENT MOTION IN THE ECLIPTIC. 97
133. Times of Sunrise and Sunset. The times of
sunrise and sunset at Greenwich are given for every day
of the year in Whitaker > & and other almanacks. For any
other latitude, the Sun's declination must be found from the
almanack, the times of sunrise and sunset can then be found
by means of tables of double entry constructed for the pur-
pose (29). These are called ''Tables of Semidiurnal
and Seminocturnal Arcs." . They give, for different latitudes
and declinations, the interval between apparent noon
and sunset, .#., the apparent time of sunset, or half the
length of the day. Subtracting this from 12 hours, the
apparent time of sunrise is found, and is half the length
of the night.
If, as in 129, we consider two antipodal places A
and S, the planes of their horizons will be parallel, and the
Sun will be above the horizon at A when he is below the
horizon at J3, and vice versd. Hence, the apparent time of
sunrise (measured from noon) in N. latitude I will be the
apparent time of sunset (measured from midnight) in S.
latitude I on the same date.
For this reason the tables are usually constructed only for
N. latitudes. For S. latitudes they give the time of sunrise
instead of sunset.
The times found in this manner will be the local solar times.
To reduce to Greenwich solar time we must add or sub-
tract 4m. for each degree of longitude, according as the place
is W. or E. of Greenwich.
134. To find the length of the perpetual day and
night at places within the Arctic or Antarctic
Circles.
The perpetual day lasts while the Sun's declination at local
midnight is greater than the colatitude (or complement of the
latitude), during spring and summer. The perpetual night
lasts while the Sun's S. decl. at local noon is greater than the
colat. during autumn and winter. The Sun's decl. at Green-
wich noon being given for every day of the year, in the
Nautical Almanack, it is easy to find, to within a day,
the durations of the perpetual day and night in any given
latitude greater than 66 32|'.
98 ASTRONOMY.
135. To find the time the Sun takes to rise or
set. Let D" be the Sun's angular diameter, measured in
seconds. When the Sun begins to rise, his upper limb just
touches the horizon, and his centre is at a depth \D" below
the horizon. When the Sun has just finished rising, his
lower limb touches the horizon, and his centre is at an altitude
|_D" above the horizon. During the sunrise, the centre rises
through a vertical height D". The problem is closely similar
to that of 104, where the effect of dip is considered. Hence
if t seconds be the time taken in rising, d the declination of
the Sun's centre, and x the inclination to the vertical of the
Sun's path at rising (Hx'x or nxP, Tig. 40) we have
t = -jV D" sec d sec #,
= 4 sec d sec x x (O's angular diameter in minutes).
As in 104, this gives, for a place on the equator,
t -^D n sec d,
and at an equinox in latitude ?,
t = T V D" sec I.
EXAMPLE. At an equinox in latitude 60, the O's angular
diameter being 32',
the time taken to rise will be = 4 x 32 x sec 60 seconds
= 256s. = 4m. 16s.
136. Note. It may be mentioned that, owing to atmos-
pheric refraction, the Sun really appears to rise earlier and
set later than the times calculated by theory. As the pheno-
mena of refraction will be discussed more fully in Chapter
VI., it will be sufficient to mention here that the rays of light
from the Sun are bent to such an extent by the Earth's
atmosphere that the whole of the Sun's disc is visible when it
would just be entirely below the horizon if there were no
atmosphere.
Moreover, there is daylight, or rather twilight, for some
time after the Sun has vanished, so that what is commonly
called night does not begin for some time after sunset.
For the same reasons, the perpetual day at a place in the
arctic circle is lengthened, and the perpetual night shortened,
by several days.
The time taken in rising and setting is, however, prac-
tically UTI affected.
99
SECTION II. The Ecliptic.
137. The First Point of Aries. In determining the
right ascensions of stars, the first step must necessarily be
to find accurately the position of the first point of Aries, since
this point is taken as the origin from which R.A. is measured.
In other words, we must first find the R.A. of one star.
When this is known we can use that star as a " clock star,"
to determine the sidereal time and clock error ; and, these
being known, we can then find the R.A. of any other star, as
explained in Chapter II. But until the position of T has
been found, the methods of Chapter II. will only enable us
to find the difference of R.A. of two stars by observing the
difference of their times of transit, as indicated by the astro-
nomical clock, and will determine neither the sidereal time
nor the clock error, nor the R.A.'s of the stars.
138. First Method. The position of T may be found
thus : At the vernal equinox the Sun's declination changes
from south to north, or from negative to positive. Let the
Sun's declination be observed by the Transit Circle at the pre-
ceding and following noons, and let the observed values be
^and -f^ 2 (.*., ^ S., and d t IT.). Let t v 2 be the corre-
sponding times of transit of the Sun's centre, as observed by
the astronomical clock, and let T^ the time of transit of any
star, be also observed. Then,
T tfj = difference of R. A. of star and Sun at first noon,
Tt z = ,, ,, ,, at second noon.
Let T rfj = ^ and T t z = 2 . "We have
Increase in Sun's decl. in the day = d% ( d l ) = d z + d ly
,, ,, R.A. ,, = t t t l a l a. 2 ,
and both coordinates increase at an approximately uniform
rate during the day.
Therefore the Q's decl. will have increased from d l to
in a time d l /(d l + d^ of a day, and the corresponding increase
in R.A. will be
fa-oa) x dj(d l + d,\
The Sun is now at T, .' O's R.A. is now = 0. Hence,
The star's R.A. = a, - ^"^ *** + A
100 A.STEONOMT.
*139. Flamsteed's Method for finding the First
Point of Aries. The principle of the method now to be
described is as follows : Let 8 lt $ be two positions of the
Sun shortly after the vernal and before the autumnal equinox
respectively, and such that the declinations S l J/j and SM are
equal. Then the right-angled triangles r^/"A and ^MS
will be equal in all respects, and we shall therefore have
FIG. 52.
At noon, some day shortly after March 21, the Sun is
observed with the Transit Circle, say when at 8 V We thus
determine its meridian zenith distance z 15 and also the dif-
ference between the times of transit of the Sun and some fixed
star x, whose R.A. is required. This difference, which is the
difference of E.A. of the Sun and star, we shall call a r If
d l be the Sun's declination, and I the observer's latitude, we
shall have
= a
We now have to determine J/7V, the difference of R.A. of the
Sun and star shortly before September 23, when the Sun'g
declination SMis again equal to d r But the Sun can only
be observed with the Transit Circle at noon, and it is highly
improbable that the Sun's declination will again be exactly
equal to d 1 at noon on any day. We shall, however, find two
consecutive days in September on which the declinations at
noon, S 2 M 2 and $ 3 Jf 3 , are respectively greater and less than d^
THE SUN'S APPARENT MOTION^ Iff T3l2
Let 2 2 and 2 8 be the observed meridian zenith distances at
3 and S & ; d% and <? 8 the corresponding declinations S. 2 lf. 2 ,
(S' 3 I/3 ; 2 an ^ ^3 the observed arcs M^N and J/g-ZV, being the
differences of R. A. of the Sun and star on the two days.
During the day which elapses between the observations at
$a> $s> we ma y assume that the Sun's decl. and R.A. both
vary at a uniform rate, so that the change in the decl. is
always proportional to the corresponding change in R.A.*
Therefore,
and MN= M^N-M,M= a,-
Now we have shown that
-M 1 N= HN- ^
-12 hours;
= 6h. +
This determines T-ZV, the star's R. A., in terms of 15 a v 3 , the
observed differences between the times of transit of the Sun
and star, and d lt d^ d^ the Sun's declinations at the three
observations. But we need not even find the declinations, for
d l = l-z v d, = l-z v d s = l-% ;
therefore, substituting, we have
The star's R.A., r^= 6h.+f j ^-f^-^^ (a^-a,) } .
2 3~ 2 2
In applying either of the above methods to the numerical calcula-
tion of the right ascension of any star, it is advisable to follow the
various steps as we have described them, instead of merely sub-
stituting the numerical values of the data in the final formulas.
* In other words, we assume, as in Trigonometry, that tho
" principle of proportional parts " holds for the small variations in
decl. and E.A. during the day.
10 V 2 ASTRONOMY.
*140. The Advantages of Flamsteed's Method. Among these
the following may be mentioned.
1st. The method does not require a knowledge of the latitude, for
we do not require to find the Sun's declination. Hence, errors
arising from inaccurate determination of the latitude are avoided.
2nd. One great source of error in determining Z.D.'s is the refrac-
tion of the Earth's atmosphere. Since the Sun is observed each
time in the same part of the sky, z lt z 2 , 3 will be nearly equally
affected by refraction. Hence, the " principle of proportional
parts" will hold, so that the small differences in the true Z.D.'s
are proportional to the differences in the observed Z.D.'s. Hence
we may use the observed Z.D.'s uncorrected for refraction.
EXAMPLE.
To find the Right Ascension of Sirius and the clock errors in
March and Sept., 1891, from the following data, the rate of the clock
being supposed correct. (Decl. of Sirius = 16 34' 2" S.)
Mar. 25, 1891.
Sept. 18.
Sept. 19
Decl. of Sun at noon...
Time of transit of Sun
Time of transit oiSirius
1 48' 56"
Oh. 15m. 36s.
6h. 39m. 10s.
1 53' 0"
llh. 42m. 42s.
6h. 40m. 25s.
1 29' 43"
llh. 46m. 17s.
6h. 40m. 25s.
OnMar.25,(R.A.ofSmws)-(Sun'sR.A.)=6h 39m. 10s. -Oh. 15m. 36s.
=6h.23m.34s.
Hence, in angular measure, the difference of R.A. is about 96.
Draw the diagram as in Fig. 52, but make the angle SiPN = 96; iV
will therefore lie between M l and J5f 2 , instead of where represented.
Also, since Sirius is south of the equator, it should be represented
at a point x on PN produced through N. In this figure we shall have
8^ = 148'56"; MiN = 6h.39m.10s. -Oh.15m.36s. = 6h.23m.34a.
S 2 3f 2 = 153' 0"; NM. 2 = llh.42m.42s.-6h.40m.25s. = 5h. 2m. 17s.
= 129'43"; NM 3 = Ilh.46m.l7s.-6h.40m.25s. = 5h. 5m.52s.
Also, SM is by construction equal to S^M } .
Hence, applying the principle of proportional parts, we have
SoMg-giJf! = 4' 4" = 244
S. 2 M 2 -S 3 M 3 23' 17" 1397'
and M%M 3 = 3m. 35s. = 215s. ;
.-. M*M = 215 x 244/1397 = 37'5 seconds ;
.-. NM = 5h. 2m. 17s. + 37s. = 5h. 2m. 54s.
Now, NMt-NAI = NT -N~ = 2Nr -12h.
bonce, TN = 6}i. + %(NMi-NM) = 6h. + i(6h.23m.34s.-5h.2m.54s.)
= 6h. + (lh. 20m. 40s.) = 6h. 40m. 20s.
Thus the right ascension of Sirius = 6h, 40m. 20s.
Also, clock error in March = 6h.40m.20s.-6h.39m.10s. = + 1m. 10s.
Sept. = 6h.40m.20s.-6h.40m.25s. = 5s.
103
141. Precession of the Equinoxes. Thus far we have
treated the first point of Aries as being fixed, and this will
evidently be the case if the equator and ecliptic are fixed in
direction. But if the right ascensions of various stars are
observed over an interval of several years, it will be found
that the position of the first point of Aries is slowly changing,
and that it moves along the ecliptic in the retrograde direc-
tion at the rate of about 50-2" in a year. This motion is
called Precession of the Equinoxes, or, briefly, Precession.
Precession is found to be due almost entirely to gradual
changes in the direction of the plane of the equator, the
ecliptic remaining almost fixed among the stars. Its effect is
to produce a yearly increase of 50-2" in the celestial longi-
tudes of all stars, their latitudes being constant.
In a large number of years the effect of precession will be
considerable. Thus, T will perform a complete revolution
in the period
360x60x60 years, i.e., about 25,800 years.
o(j' 2i
At the present time the vernal equinoctial point has moved
right out of the constellation Aries into the adjoining con-
stellation Pisces. It still, however, retains the old name of
lt First Point of Aries." Similarly, the autumnal equinoctial
point is in the constellation Virgo, but it is still called the
" First Point of Libra."
The rate of precession can be found very accurately by
observations of the first point of Aries separated by a con-
siderable number of years. The larger the interval, the
larger is the change to be observed, and the less is the result
affected by instrumental errors.
*142. Correction for Precession in using Flamsteed's Method.
During the interval that elapses between the two observations in
Flamsteed's method, the right ascension of the observed star will have
increased slightly, owing to precession, and the E.A. given by the
formula will be the arithmetic mean of the E.A.'s at the times of
the two observations.f As the change in E.A. is very approximately
uniform, this mean will be the star's E.A. at a time exactly half
way between the two observations, i.e., at the summer solstice.
t This may be most readily seen by imagining the equator and
ecliptic to be at rest, and the change in E.A. to be due to motion of
the star.
104
ASTROJDMT.
143. Determination of Obliquity of Ecliptic. The
method now used for finding the obliquity of the ecliptic is
similar in principle to that of 38, hut the Sun's meridian
zenith distance is observed by means of the transit circle
instead of the gnomon.
The obliquity is equal to the Sun's greatest declination at
one of the solstices. Since observations with the Transit
Circle can only be performed at noon, while the maximum
declination will probably occur at some intermediate hour of
the day, it will be necessary, in exact determinations, to
make observations of the Sun's decl. for several days before
and after the solstice. Prom these it is possible to determine
the maximum decl. ; the method is, however, too complicated
to be described here. For rough purposes the Sun's greatest
noon decl. may be taken as the measure of the obliquity.
144. When the position of T has been determined, the
obliquity can also be found by a single observation of the
Sun's E-.A. and decl. For we thus find the two sides T-3/i
MS of the spherical triangle T^S, and these data are
sufficient to determine both the obliquity flfTS, and the
Sun's longitude T S.
THE SUN'S APPARENT MOTION IN THE ECLIPTIC. 105
SECTION III. The Earth's Orbit about the Sun.
145. Observations of the Sun's Relative Orbit. By
daily observations with the Transit Circle, the decl. and R. A.
of the Sun's centre at noon are found for every day of the
year. From these data the Sun's long, is calculated, as
in 144, by solving the spherical triangle T SM (Fig. 53).
If the obliquity of the ecliptic is also known, we have three
data, any two of which suffice to determine the long., T$.
Thus the accuracy of the observations can be tested, and the
Sun's motion at various times of the year can be accurately
determined.
Although the determination of the Sun's actual distance
from the Earth in miles is an operation of great difficulty, it
is easy to compare the Sun's distance from the Earth at dif-
ferent times of the year, for this distance is always inversely
proportional to the Sun's angular diameter. This property is
proved in 4, but numerous simple illustrations may also
be used to show that the angular diameter of any object varies
inversely with its distance (see 4).
The Sun's angular diameter may be readily observed by
means of the HeHometer ; or, if preferred, any other form of
micrometer may be used. The Sun's distances at two different
observations will be in the reciprocal ratio of the corresponding
angular diameters. Thus, by daily observation, the changes
in the Sun's distance during the year may be investigated.
If the circular measure of the Sun's angular diameter is
2r, then Trr 2 is called the Sun's apparent area. In fact,
this is the area of a disc which would look the same size as
the Sun if placed at unit distance from the eye.
EXAMPLE.
The Sun's angular diameter is 31' 32" at midsummer, and 32' 36"
at midwinter. To find the ratio of its distances from the Earth at
these times.
The distances being inversely proportional to the angular dia-
meters, we have
Dist. at midsummer = 82' 36" = 1956 = 489 _ , 1 ,
Dist. at midwinter 31' 32" 1892 473 "**
Hence the Sun is further at midsummer than at midwinter, in the
proportion of very nearly 81 to 30.
106 ASTRONOMY.
146. Kepler's First and Second Laws. We may no\\
construct a diagram of the Sun's relative orbit. Let E repre-
sent the position of the Earth, ET the direction of the first
point of Aries. Then, by making the angle TES equal to the
Sun's longitude at noon, and ES proportional to the Sun's
distance, we obtain a series of points S, S'... , 8 r .. , representing
the Sun's position in the plane of the ecliptic, as seen from
the Earth at noon on different days of the year. Draw the
curve passing through the points S, S'... , S r .. ; this curve will
represent the Sun's orbit relative to the Earth, and it will be
found that
I. The Sun's annual path is an ellipse, of which the
Earth is one focus.
II. The rate of motion is everywhere such that the
radius vector (i.e., the line joining the Earth to the
Sun) sweeps out equal areas in equal intervals of time.
These laws were discovered by Kepler for the motion of
Mars about the Sun, and he subsequently generalized them by
showing that the orbits of all the other planets, including the
Earth, obeyed the same laws. In their general form they are
known as Kepler's First and Second Laws. [See p. 253.]
FIG. 54.
147. Perigee and Apogee. When the Sun's distance
from the Earth is least, the Sun is said to be in perigee.
When the distance is greatest, the Sun is said to be in apogee
The positions of perigee and apogee are called the two
Apses of the orbit ; they are indicated at p, a in Fig. 54.
The line pEa joining them is the major axis of the ellipse
(Ellipse, 4), and is sometimes also called the apse line.
107
148. Verification of Kepler's First Law. The Sun's
angular diameter is observed to be greatest on Dec. 31,
and least on July 1 ; we therefore conclude that these are the
days on which the Sun passes through perigee and apogee
respectively. The positions of perigee and apogee being thus
found, the angle TEp is known, which is the long, of perigee.
From the winter solstice to perigee is about 10 days.
Hence, during this interval the Sun will have moved through
an angle of about 10 ;
.-. longitude of perigee = 270 + 10 = 280 roughly.
To verify that the orbit is an ellipse, it is now only neces-
sary to show that the relation connecting ES and the angle
pES is the same as that which holds in the case of the ellipse.
If the orbit is an ellipse of eccentricity <?, we must have
ES x (1 +0 cospJSS) = I (a constant). (Ellipse, 3.)
Therefore the Sun's angular diameter must be always pro-
portional to 1 + * wspES.
As the result of numerous observations, it is found that
this is actually the case, and the truth of Kepler's Eirst Law
for the Sun's orbit relative to the Earth is confirmed.
149. To find e, the eccentricity of the ellipse, the
best plan is to compare the greatest and least angular dia-
meters of the Sun, i.e., the diameters at perigee and apogee.
Since at these positions pES becomes and 180 respectively,
we have, from above,
ang. diam. at p : ang. diam. at a = IjEp : IjEa
= l+0cosO : 1+* cos 180 = \+e : l-e.
from which proportion e can be found.
Taking the angular diameters at perigee and apogee to be
3,2' 36" and 31' 32" (as in the Ex. of 145), the Sun's distances
at those times are in the ratio of 1956" : 1892", or 489 : 473 ;
1+g = 489 _ 489-473 _ 16 _ 8
' \-e 473 ~ 489+473 ~ 962 ~~ 481*
Hence e is very nearly equal to 1/60.
The Nautical Almanack contains a table giving the Sun's
angular diameter daily throughout the year. The average
angular diameter may be taken as 32' approximately.
Owing to the smallness of 0, the orbit is very nearly circular,
being, really, much more nearly so than is shown in Eig. 54
ASTEON. i
108 ASTEONOMY.
150. Verification of Kepler's Second Law. It is
found, as the result of observation, that the Sun's increase in
longitude in a day, at different times of year, is always pro-
portional to the square of the angular diameter, and is, there-
fore, inversely proportional to the square of the Sun's distance.
From this it may he deduced (as follows) that the area de-
scribed by the radius vector in one day is always constant.
PIG. 55.
Let SS' represent the small arc described by the Sun
in a day in any part of the orbit. Then the sector US 8' is
the area swept out by the radius vector. This sector does
not differ perceptibly from the triangle JESS' ; therefore, by
trigonometry,
area JS88' = %ES . US' . sin 8E8'.
Since the change in the Sun's distance in one day is imper-
ceptible, we may write JES for JES' in the above formula
without materially affecting the result ; also, since the angle
SES' is small, the sine of SES' is equal to the circular
measure of the angle SES'.
Therefore, area JESS' = \E& x L SES '.
But, by hypothesis, the change of longitude SES' varies
inversely as ES*, so that US' 2 x L SES' is constant ;
area ESS' is constant,
that is, the area described by the radius vector in a day is
constant. Thus, the area described in any number of days
is proportional to the number of days, and generally the areas
described in equal intervals of time are equal.
109
151. Deductions from Kepler's Second Law.
(i.) If the circular measure of the Sun's angular diameter
is 2r, then Trr 2 is the Sun's apparent area ( 145). Hence
the Sun's daily rate of change of longitude is proportional to
the apparent area of its disc.
(ii.) If T, K, :, L represent the Sun's positions at the
equinoxes and solstices, we have
Z rEK = z.KE = tEL LLEv = 90,
and it is readily seen from the figure that
area LET < area lEL < area TEE < area KE<,
and the lengths of the seasons, being proportional to these
areas, are unequal, their ascending order of magnitude being
Winter, Autumn, Spring, Summer.
Their lengths are, at the present time (1891), about
39d. 0|h., 89d. IS^h., 92d. 20h., 93d. 14|h.
(iii.) Since the intensity of the Sun's heat ( 131) and its
rate of motion in longitude both vary as the inverse square of
its distance, they are proportional to one another. Hence
the Earth, as a whole, receives equal amounts of heat while the
Sun describes equal angles. In particular, the total quantities
of heat received in the four seasons are equal.
(iv.) The Sun's longitude changes most rapidly on Decem-
ber 31, and least rapidly on July 1.
(v.) Since the apse line, or major axis, pSa, bisects the
ellipse, the time from perigee to apogee is equal to the time from
apogee to perigee.
*152. To find the Position of the Apse Line.
The Sun's distance remains very nearly constant for a short
time before and after perigee and apogee, hence it is difficult
to tell the exact instant when this distance is greatest or least.
For this reason, the following method is generally used :
The Sun's long, is observed at two points, S, S v before and
after the apse, when its angular diameters, or its rates of
motion in long., are found to be equal. Then ES = ES^
and the symmetry of the ellipse shows that JLpES = LpES^
and L aES = L aES r Hence the long, of the apse is the
arithmetic mean of the Sun's longitudes at the two observations.
153. Progressive Motion of Apse Line. Prom such
observations, extending over a long period of years, it is found
that the apse line is not fixed, but has a forward or direct
motion in the ecliptic plane of 1 1 "25" in a year.
1 10 ASTRONOMY.
154. The Sun's apparent annual motion may be
acco anted for by supposing the Earth to revolve
roun$ the Sun.
The annexed diagram will show how the Sun's annual
motion in the ecliptic, as well as the changes in the seasons,
may be accounted for on the theory that the Sun remains at
rest while the Earth describes an ellipse round it in the
course of the year in a plane inclined at an angle 23 27' to
the plane of the Earth's equator.
Mar 21
FIG. 56.
The distance of the nearest of the fixed stars is known to
be over 200,000 times as great as the Earth's distance from
the Sun. Hence, 5 shows that the directions of the fixed
stars will not change to any considerable extent, as the
Earth's position varies. "We shall, therefore, in the present
description, consider the directions of the stars to be fixed.
The directions of the various points and circles of the celestial
sphere, such as the first point of Aries, will also be fixed.
On March 21, the Earth is at JS lt and the Sun's direction
determines the direction of T, the First Point of Aries.
Ill
The Sun is vertical at a point Q on the equator, and as the
Earth revolves about its axis through P, all points on the
equator will come vertically under the Sun. There is night
all over the shaded portion of the Earth, day over the rest.
The great circle bounding the illuminated part passes through
the pole P, and, therefore, bisects the small circle traced out
by the daily rotation of any point on the Earth ; thus, the
day and night are everywhere equal. At the pole P the Sun
is just on the horizon.
On June 21, the Earth is at E^ and the Sun's longitude
TE^S = 90. The Sun is vertical at a point on the tropic
of Cancer. Since the arctic circle is entirely in the illumi-
nated part there is perpetual day over the whole arctic zone.
On September 23, the Earth is at E%, and the Sun's longi-
tude TE S S is 180. The Sun is once more vertical at a
point JR on the equator, and the day and night are everywhere
12 hours long, as they are at E r
On December 22, the Earth is at E, and the Sun's longi-
tude vEfi (measured in the direction of the arrow) is 270.
The Sun is now at its greatest angular distance south of the
equator, and overhead at a point on the tropic of Capricorn ;
this tropic is not represented, being on the under side of the
sphere. Since the arctic circle is entirely within the shaded
part there is perpetual night over the whole arctic zone.
155. New Definitions and Pacts. According to the
theory of the Earth's orbital motion, Kepler's First and
Second Laws must be re-stated thus for the Earth.
I. The Earth describes an ellipse, having the
Sun in one focus.
II. The radius vector joining the Earth and Sun
traces out equal areas in equal times about the Sun.
The ecliptic is now definedasthe great circle of the celestial
sphere, whose plane is parallel to that of the Earth's orbit.
The Earth is nearest the Sun on December 31, and is then
said to be in perihelion. The Earth is furthest from the
Sun on July 1, and is then said to be in aphelion. Thus,
when the Sun is in perigee the Earth is in perihelion, when
the Sun is in apogee the Earth is in aphelion. The positions
of perihelion and aphelion are indicated by the letters p, a in
Fig. 56. The line joining them is the apse line.
112 ASTROWOttt.
156. Geocentric and Heliocentric Latitude and
Longitude. Hitherto we have been dealing only with the
directions of the celestial bodies as seen from the Earth.
In dealing with the motion of the planets, it is more con-
venient, as a rule, to define their positions by the directions
in which they would be seen by an observer situated at the
centre of the Sun.
In every case, the direction of a celestial body may be
specified by the two coordinates, celestial latitude and longi-
tude, which measure respectively the arc of a secondary from
the body to the ecliptic and the arc of the ecliptic between
this secondary and the first point of Aries ( 17).
These coordinates are called the Geocentric Latitude
and Longitude when employed to define the body's geocen-
tric position, or position relative to the centre of the Earth.
The names Heliocentric Latitude and Longitude are
given to the corresponding coordinates when employed to
define the body's heliocentric position, or position relative
to the Sun's centre.
"When the distance of a fixed star is immeasurably great
compared with the radius of the Earth's orbit, its geocentric
and heliocentric directions coincide, and there is no difference
between the two sets of coordinates. There is a slight differ-
ence between the geocentric and heliocentric positions of a
few of the nearest fixed stars. But, in the case of the
planets, and of comets, the heliocentric latitude and
longitude differ entirely from the geocentric, and laborious
calculations are required to transform from one system of
coordinates to the other.
One fact may, however, be noted. The direction of the
Earth as seen from the Sun is always opposite to the direction
of the Sun as seen from the Earth. Hence,
The Earth's heliocentric longitude differs from the
Sun's geocentric longitude by 180.
This may be illustrated by referring to Pig. 56. We see
thatr&E^oyr/SLE^ 90, r&Ei = 180, TSJS, = 270;
thus, the Earth's longitude is on September 23, 90 on
December 22, 180 on March 21, and 270 on June 21.
THE SUN'S APPARENT MOTION IN THE ECLIPTIC. 113
EXAMPLES. IV.
1. Describe the phenomena of day and night at a pole of the
Earth.
2. Show how to find how long the midwinter Moon when full is
above the horizon at a place within the arctic circle of given
latitude.
3. Show that the ecliptic can never be perpendicular to the
horizon except at places between the tropics.
4. Show that for a place on the arctic circle the Sun always rises
at 18h. sidereal time from December 21 to June 20, and sets at the
same sidereal time from June 20 to December 21.
5. Find the angle between the ecliptic and the equator in order
that there should be no temperate zone, the torrid zone and the
frigid zone being contiguous.
6. Show how, by observations on the Sun, taken at an interval of
nearly six months, the astronomical clock may be set to indicate
Oh. Om. Os. when T is on the meridian.
7. On March 24, 1878, at noon, the Sun's declination was
1 29' 5*1", and the difference of right ascension of the Sun and a
star 6h. 1m. 34'45s. On September 18, 1878, at noon, the Sun's
declination was 1 49' 30'2", and it was distant from the star
5h. 27m. 32'97s. in right ascension. On September 19, 1878, at
noon, the Sun's declination was 1 26' 12'8", and it was distant from
the star 5h. 31m. 8'3s. in right ascension. Find the right ascension
of the star and that of the Sun at the first observation.
8. Describe the appearance presented to an observer in the Sun
of the parallels of latitude and the meridians of the Earth, any day
(i.) between the vernal equinox and the summer solstice,
(ii.) between the autumnal equinox and the winter solstice.
9. If a sunspot be situated near the edge of the Sun's disc,
describe how its position, relative to the horizon, will change between
sunrise and sunset.
10. Describe how the Sun's apparent velocity in the ecliptic
varies throughout the year; and give the dates of apogee and
perigee. Compare the daily motion in longitude at these dates,
having given that the eccentricity of the Earth's orbit is ^5.
114
ASTRONOMY,
EXAMINATION PAPEK. IV.
1. What is the astronomical reason for the Earth being divided
into torrid, temperate, and frigid zones ?
2. Assuming your latitude to be 52, show by a figure the daily
path of the Sun as seen by you on June 21, December 22, and
March 21 respectively.
3. Explain the causes of variation in the length of the day on the
Earth. Give the dates at which each season begins, and calculate
their lengths in days.
4. Discuss the variations in the length of the day at points within
the arctic circle ; and show how to find, by the Nautical Almanack,
the length of the perpetual day.
5. Prove that, in the course of the year, the Sun is as long above
the horizon at any place as below it.
6. Explain how it is that winter is colder than summer, although
the Sun is nearer.
7. Investigate Flamsteed's method of determining the first point
of Aries.
8. From the following observations calculate the Sun's R.A. on
March 30, 1872 :
Sun's
declination.
Sun crossed
meridian.
a Serpentis
crossed meridian.
March 30, 1872...
Sept. 11, 1872 ...
Sept. 12,1872 ...
4 (/ 8-1"
4 20' 58-8"
3 58' 3-0"
Oh. 1m. 4'47s.
Oh. 1m. 4-09a.
Oh. 1m. 4-07s.
15h. 1m. 5476s.
4h. 19m. ll-38s.
4h. 15m. 49-33s.
9. State Kepler's First Law for the orbit of the Earth relative to
the Sun, and explain how the eccentricity of the orbit can be found
by observations of the Sun's angular diameter.
10. State Kepler's Second Law, and find the relation between the
Sun's angular velocity and its apparent area.
CHAPTER V.
ON TIME.
SECTION I. The Mean Sun and Equation of Time.
157. Disadvantages of Sidereal and Apparent Solar
Time. In Chapter L, Sections II., III., we explained two
different ways of reckoning time. One of these, called
Sidereal Time, was denned by the diurnal motion of the first
point of Aries ; the other, called Apparent Solar Time, was
defined by the Sun's diurnal motion. We shall now show
that neither of these measures of time is suitable for every-
day use.
If we were to adopt sidereal time, the time of apparent
noon on any day of the year would be measured by the Sun's
K.A. on that day, and therefore would get later and later by
24h. during the course of the year.
Thus (0.^.), the time of noon would be Oh. on March 21,
6h. on June 21, 12h. on September 23, and 18h. on Decem-
ber 22, and the phenomena of day and night would bear no
constant relation to the time.
Apparent solar time is free from these disadvantages, but
it cannot be measured by a clock whose rate is uniform,
because the length of the solar day is not quite invariable.
In 36 we showed that the difference between a solar and
a sidereal day is equal to the Sun's daily increase in R.A.,
and in 31 we showed that this increase takes place at a rate
which is not quite the same at different times of the year.
Hence, the difference between a solar and a sidereal day is
not quite constant. But the length of a sidereal day is con-
stant ( 22). Hence the solar day is not quite constant,
and a clock cannot be regulated so as to always mark exactly
Oh. Om. Os. when the Sun crosses the meridian.
116 ASTRONOMY.
158. The Mean Sun. Definitions. To obviate these
disadvantages, another kind of time, called Mean Time, has
been introduced, and this is the time indicated by clocks, and
used for all ordinary purposes. Mean Time is defined by
means of what is CAiled the Mean S&o. This is not really
a Sun at all, but simply a point, which is imagined to move
round the equator on the celestial sphere.* The hour angle
of this moving point measures mean time, just as the hour
angle of T measures sidereal time ; and the mean Sun has to
satisfy the following requirements :
1st. It must never be very far from the Sun.
2nd. Its R.A. must increase uniformly during the year.
Now the inequalities in the motion in R.A., which render
the true Sun unsuitable as a timekeeper, are due to two
causes.
1st. The Sun does not move uniformly in the ecliptic, its
longitude increasing less rapidly in summer than in winter
( 151). t
2nd. Since the Sun moves in the ecliptic, and not in the
equator, its celestial longitude is in general different from its
R.A. ( 31 ). Hence, even if the Sun were to revolve uni-
formly, its R.A. would not increase uniformly.
In defining the mean Sun, or moving point which measures
mean time, these two causes of irregularity are obviated
separately as follows :
The Dynamical Mean Sun is defined to be a point which
coincides with the true Sun at perigee, and which moves
round the ecliptic in the same period (a year) as the true
Sun, but at a uniform rate.
Thus, in the dynamical mean Sun, irregularities due to
the Sun's unequal motion in longitude are removed, but those
due to the obliquity of the ecliptic still remain.
The Astronomical Mean Sun is defined to be a point
which moves round the equator in such a way that its R.A.
is always equal to the longitude of the dynamical mean Sun.
* The conception of the mean Sun as a moving point is important.
It would be physically impossible for a body to move in this manner.
Ott TIME. 117
Since the longitude of the dynamical mean Sun increases
uniformly, the R.A. of the astronomical mean Sun increases
uniformly. Hence the motion of the latter point does give
us a uniform measure of time.
The astronomical mean Sun is, therefore, the moving point
chosen in denning mean time. It is usually called simply
the Mean Sun.
159. Mean Noon and Mean Solar Time. Equation
of Time.
Mean Noon is defined as the time of transit of the mean
Sun.
A Mean Solar Day is the interval between two successive
mean noons. Like the apparent and sidereal days, it is
divided into 24 mean solar hours. During this interval, the
hour angle of the mean Sun increases from to 360.
Hence the mean solar time at any instant is measured by
the mean Sun's hour angle, converted into time at the rate
of Ih. per. 15, or 4m. per 1.
The Sun itself is frequently spoken of as the True Sun,
or Apparent Sun, to distinguish it from the mean Sun.
As explained in 36 the hour angle of the true Sun measures
the apparent solar time, and its time of transit is called
apparent noon.
The Equation of Time* is the name given to the amount
which must be added to the apparent time to obtain the mean
time.
Thus, the time indicated by a sun-dial ( 167) is determined
by the position of the shadow thrown by the true Sun, and is
the apparent solar time ; while a clock, which should go at a
uniform rate, is regulated to keep mean time. The equation
of time will then be defined by the relation,
(Time by clock ) = (Time by dial) + (Equation of time).
At apparent noon the sun-dial will indicate 12h., or, as it
is more conveniently reckoned, Oh. Hence,
Equation of time = Mean time of apparent noon.
* Thus, " equation of time " is not an equation at all in the
generally accepted sense of the word, but an interval of time (posi-
tive or negative).
118 ASTRONOMY.
The equation of time is positive if the Sun is " after the
clock," or the true Sun transits after the mean Sun. If the
Sun is "before the clock," or the true Sun transits first, the
equation of time is negative. The value of the equation of
time for every day in the year is given in most almanacks,
under the heading " Sun before clock," or " after clock."
160. The equation of time is divided into two parts. The
first, which is called the equation of time due to the eccen-
tricity, or to the unequal motion, is measured by the
difference between the hour angles of the true and dynamical
mean Suns. The second, or the equation due to the
obliquity, is measured by the difference of hour angle
between the dynamical and astronomical mean Suns.
161. Equation of Time due to Unequal Motion.
"We shall now trace the variations during the year of that
portion of the equation of time which is due to the Sun's
unequal motion in the ecliptic. We shall denote this portion
by^.
Let the true Sun be denoted by 8, and the dynamical mean
Sun (which moves in the ecliptic) by 8 r If angles are
measured in time, then
E l (hour angle of SJ (hour angle of S) = L SPS l ' t
.-. E l = (RA. of S) -(R.A. of 8J ;
since R.A. and hour angle are measured in opposite directions.
When the Sun is in perigee (p) (on December 31), /S t coin-
cides with S by definition ; .*. E l = O.
From perigee (p} to apogee (a), the Sun, has described 180,
and the time taken is ( 151, v.) half that of a complete
revolution. Hence, S l will also have described 180 ;
.*. at apogee (July 1), E l is again O.
Now ( .151, iv.) S is moving most rapidly at perigee, and
most slowly at apogee. Hence, after perigee, S will have got
ahead of S v and after apogee, S will have got behind 8 V
Thus : From perigee to apogee, E l is positive,
From apogee to perigee, E l is negative.
and E v vanishes twic a year, viz., at perigee and apogee.
ON TIME.
119
162. Equation of Time due to Obliquity. Let the
portion of the equation of time due to the obliquity be
denoted by E v
Take S 3 on the equator so that r 8 t = T 8 r Then 2
will be the astronomical mean Sun. Draw PS^, the
secondary to the equator through 8 r Then
EI = hour angle of $ 2 hour angle of ^
= L SJP&i (taken positive if 8 9 is west of SJ
= L rPS^ z rPS* = rM- rS, = r Jf- r ^,
all angles being supposed converted into time at the rate
of 15 to the hour.
At the vernal equinox,* when 8 l is at T , 2 will also be
of 1 */* /'' ~~ o
ai T , -^z u -
Between the vernal equinox and summer solstice, the angle
will be < 90, and, therefore, < T-WSiJ hence,
is negative.
* The vernal and autumnal equinoxes are, strictly, the times when
fif, and not 8 l} coincides with the equinoctial points, but, as Sj is
always near 8, the distinction need not be considered here. The
same remarks apply to the solstices.
120
ASTRONOMY.
At the summer solstice, 8 l is at C, and S 9 at Q, where
r Q = r C = 90. Hence (Sph. Geom., 21), r QC = 90;
and Mis also at Q ; .-. _# 2 o.
Between the summer solstice and autumnal equinox we
shall have M= < 8^. But rM^ = rS^ = 180;
.:TM>r8 l i .:rK>r8 t ; .-.Dispositive.
At the autumnal equinox, since fC = TQ==z 180,
S lt &J will both coincide with b; .. JEJ 8 = 0.
In a similar manner we may show that :
From the autumnal equinox to the winter solstice, U 3 is
negative.
At the winter solstice, 1 2 as O.
From the winter solstice to the vernal equinox, E% is
positive.
Collecting these results, we see that
(i.) From equinox to solstice /^ is negative.
(ii.) From solstice to equinox J 3 is positive.
(iii.) Es vanishes four times a year, viz., at the
equinoxes and solstices.
ON TIME. 121
163. Graphic Representation of Equation of Time.
The values of the equation of time at different seasons may
now be represented graphically by means of a curved line, in
which the abscissa of any point represents the time of year,
and the ordinate represents the corresponding value of the
equation of time.
In the accompanying figure (Fig. 59) the horizontal line
or axis from JE^ to E^ represents a year, the twelve divisions
representing the different months as indicated. The thin curve
represents the values of E^ the portion of the equation of
time due to the unequal motion ; this curve is obtained by
drawing ordinates perpendicular to the horizontal axis and
proportional to E r Where the curve is below the horizontal
line E i is negative.
FIG. 59.
The thick curved line is drawn in a similar manner, and
represents, on the same scale, the values of E v the equation
of time due to the obliquity.
In drawing the diagrams to scale, it is necessary to know
the maximum values of JEJ, E v These can be calculated,
but the calculations do not depend on elementary methods
alone. We shall therefore have to assume the following
facts :
The greatest value of E l is about 7 minutes.
Hence the greatest distances of the thin and thick curves
from the horizontal axis should be taken to be about 7 and 1
units of length respectively.
122
ASTRONOMY.
We may now draw the diagram representing JE, the total
equation of time. We have
Hence, at every point of the horizontal line we must erect
an prdinate whose length is equal to the algebraic sum of the
ordinates (taken with their proper sign) of the two curves
which represent E^ and E The extremities of these ordi-
nates will determine a new curve which represents E.
FlG. 60.
This curve is drawn separately in the annexed diagram
(Fig. 60). It cuts the horizontal axis in four points. At
these points the ordinate vanishes, and E is zero. Hence,
The Equation of Time vanishes four times a year.
164. Alternative Proof. But without representing the
values of the equation of time graphically, it can he readily
proved that E vanishes four times a year. The proof
depends on the fact stated in the last paragraph, that
The greatest equation of time due to the obliquity is greater
than the greatest equation due to the eccentricity.
Off TIME. 123
From 162 it is evident that JS t must attain its greatest
positive value some time between a solstice and the following
equinox, and its greatest negative value between an equinox
and the following solstice. These maxima occur, in fact, in
the months :
February, May, August, November.
Their values, with the proper signs, are respectively about
-flOm., 10m., 4- 10m., 10m.
Now, E l is never greater than the maximum value of 7m. ;
hence, whether E^ is positive or negative, the total equation,
E^ -f EX corresponding to either of these maxima, must have
the same sign as E Hence, in the year beginning and ending
with the date of the maximum value of E^ in February, E
will have the following signs alternately :
+ + +
Thus, 73 changes sign, and thereJx^e vanishes, four
times in the year.
165. Miscellaneous Remarks. "From Pig. 59 it will
be seen that the largest fluctuations in the equation of time
occur in the autumn and winter months ; during spring and
summer they are much smaller.
The days on which the equation of time vanishes are about
April 16, June 15, September 1, and December 25.
Between these days E increases numerically, and then
decreases, attaining a positive or negative value at some inter-
mediate time. These maxima are :
+ 14m. 28s. on February 11 ; 3m. 49s. on May 14 ;
-f-6m. 17s. on July 26 ; 16m. 21s. on November 3.
166. Inequality in the Lengths of Morning and
Afternoon. If we neglect the small changes in the Sun's
declination during the day, the interval from sunrise to
apparent noon is equal to the interval from apparent noon to
sunset ( 37). But by morning and afternoon are meant the
intervals between sunrise and mean noon, and between mean
noon and sunset respectively. Hence, unless mean and appa-
rent noon coincide, i.e., unless the equation of time vanishes,
the morning and afternoon will not be equal in length.
4-STKOF, K
124 ASTBONOMT.
Let r, * be the mean times of sunrise and sunset, E the
equation of time. Then
12h. r = interval from sunrise to mean noon.
But apparent noon occurs later than mean noon by E\
.'. 12h. r-\-E interval from sunrise to apparent noon.
Similarly, sJE= interval from apparent noon to sunset;
.-. 12h.-r+.# =*-.#,
or r + s= 12h. + 2^,
so that the sum of the times of sunrise and sunset
exceeds 12 hours by twice the equation of time.
The length of the morning is 12h. r, and that of the
afternoon is . Now the last relation gives
2J0 = -(12-r);
.-. 2 (equation of time)
= (length of afternoon) (length of morning).
About the shortest day (December 22) the curve represent-
ing the equation of time is going upwards, hence E is
increasing. But the length of day is changing very slowly
(because it is a minimum), hence, for a few days, the half
length, jE", may be regarded as constant. Hence, must
increase, and, therefore, the mean time of sunset is later
each day. Similarly, it may be shown that sunrise is also
later. The afternoons, therefore, begin to lengthen, while
the mornings continue to shorten.
Similarly, about June 21, the afternoons continue to
lengthen after the longest day, although the mornings are
already shortening.
EXAMPLE. On Nov. 1, the sun-dial is 16m. 20s. before the clock.
Given that the Sun rose at 6h. 54m., find the time of sunset.
Time from sunrise to mean noon = 12h. 6h. 54m. = 5h. 6m.
apparent noon to mean noon = Oh. 16m. 20s.
sunrise to apparent noon = 4h. 49m. 40s.
apparent noon to sunset = 4h. 49m. 40s.
mean noon to sunset
= 4h. 49m. 40s. - 16m. 20s. = 4h. 33m. 20s.
Hence, the time of sunset was 4h. 33m., correct to the nearest
minute.
ON TIME. J 25
SECTION II. The Sun-dial
essentially of a rod or flat
the direction of the celestial pole The shadow
Pia. 61.
The plane through OA, the edge of the style, and throueh
the edge of the shadow, evidently passes through the Sun
also it passes through the celestial pole, therefore it will meet
' "Declination ce
shln t plane, which is the plane of the
known t,? p P arent f 00 ?' and whose position is supposed
123' oli e A 1 Y- rder , to graduate the p late for *~
1, ^, do clock, it is only necessary to determine the posi-
esof no,-' " - c "' w
l ' 1 ' ' , 45 . ' &0 '' Wlth the meridian plane. Since
be thn - S P erour ' te e P^e* wffl
the planes bounding the shadow at 1, 2 3 o'clock
rcspecfavely. If we join the points 0,, On. dm *'"&* , hese
,
lines of shadow in the plane of the
m * * he cirram ^ of the dial-plate
126 ASTRONOMY.
168. Geometrical Method of Graduating the Dial-
plate. To find the planes OA i., OAn., &c., suppose a plane
AKR drawn through A perpendicular to OA, meeting the
plane of the dial-plate in KR and the meridian plane in A~s.ii.
If, in this plane, we take the angles xn.-4i., i.^n., n.^tm.,
&c., each = 15, the points i., n., m...., &c., will evidently
determine the directions of the shadow at 1, 2, 3,... o'clock
respectively.
FIG. 62
But in practice it is much more convenient to perform the
construction in the plane of the dial itself. Imagine the
plane AKR of Fig. 62 turned about the line KR till it is
brought into the plane of the dial, the point A of the plane
being brought to U (Fig. 62). Then, by making the angles
xii. 7i., i. ^n., n..Z7"m., &c., each = 15, we shall obtain the
same series of points i., n., m. as before.
If the dial -plate is horizontal, and I is the latitude of the
place (xn. OA), we have evidently therefore the following
construction :
On the meridian line, measure xn. = OA sec I, and
xn. U = xn. A = Oxn. sin I. Draw -ZTxn. R perpendicular
to OU. Make the angles xn.ZTi., i.Z7n., n.JTni., &c.,
each = 15, taking i., n., m., &c., on KR. Join 0i., On.,
0m., &c., and let the joining lines meet the circumference
of the dial in 1, 2, 3, &c. These will be the required
points of graduation for 1, 2, 3,... o'clock respectively.
127
SECTION III. Units of Time The Calendar.
169. Tropical, Sidereal, and Anomalistic Years.
Hitherto we have defined a year as the period of a complete
revolution of the Sun in the ecliptic. In order to give a
more accurate definition, however, it is necessary to specify
the starting point from which the revolution is measured.
"We are thus led to three different kinds of years.
A Tropical Year is the period between two successive
vernal equinoxes, or the time taken by the Sun to perform a
complete revolution relative to the first point of Aries.
The length of the tropical year in mean solar time is very
approximately 365d. 5h. 48m. 45 -5 Is. at the present time.
For many purposes it may be taken as 365 days.
A Sidereal Year is the period of a complete revolution
of the Sun, starting from and returning to the secondary to
the ecliptic through some fixed star. Thus, after a sidereal
year the Sun will have returned to exactly the same position
among the constellations.
If T were a fixed point among the stars, the sidereal and
tropical year would be exactly of the same length. But T
has an annual retrograde motion of 50-22" among the stars
( 141). Consequently, the tropical year is rather shorter
than the sidereal.
An Anomalistic Year is the period of the Sun's revo-
lution relative to the apse line in other words, the interval
between successive passages through perigee.
Owing to the progressive motion of the apse line, the positions
of perigee and apogee move forward in the ecliptic at the rate
of 11-25" per annum ( 153). Hence the anomalistic year is
rather longer than the sidereal.
It is easy to compare the lengths of the sidereal, tropical,
and anomalistic years. For, relative to the stars,
In the sidereal year the Sun describes 360,
In the tropical year it describes 360 - 50-22",
In the anomalistic year it describes 360 -f 11-25" ;
.'. (Sidereal year) : (tropical year) : (anomalistic year)
= 36O : 360 50-22": 360+11'25".
From this proportion it will be found that the sidereal year
is about 20 m. longer than the tropical, and 4 Jin. shorter than
the anomalistic.
128 AST&ONOMf.
170. The Civil Tear. For ordinary purposes, it is
important that the year shall possess the following qualifications :
1 st. It must contain an exact (not a fractional) number of days.
2nd. It must mark the recurrence of the seasons.
Now the tropical year marks the recurrence of the seasons,
but its length is not an exact number of days, being, as we
have seen, about 365d. 5h. 48m. 45 -5 Is. To obviate this
disadvantage, the civil year has been introduced. Its length
is sometime? 365, and sometimes 366 days, but its average
length is almost exactly equal to that of the tropical year.
Taking an ordinary civil year as 365d., four such years
will be less than four tropical years by 23h. 15m. 2'04s. ? or
nearly a day. To compensate foi this dilierence, every fourth
civil year is made to contain 366 days, instead of 365, and is
called a leap year. For convenience, the leap years are chosen
to be those years the number of which is divisible by 4, such as
1892, 1896.
The introduction of a leap year once in every four years
is due to Julius Caesar, and the calendar constructed on this
principle is called the Julian Calendar.
Now three ordinary years and one leap year exceed four
tropical years by 24h. 23h. 15m. 2'04s., t.g.!44m. 57j96a^
Thus, 400 years of the Julian Calendar will exceed 400
tropical years by (44m. 57'96s.) x 100, i.e., by 3d._2h^56m.36s.
To compensate for this difference, Pope ~Gregory~XlII.
arranged that three days should be omitted in every 400 years.
This correction is called the Gregorian correction and is
made as follows : Every year whose number is a multiple 0/100 is
taken to be an ordinary year of 365 days, instead of being a leap
year of 366, unless the number of the century is divisible by 4;
in that case the year is a leap year.
EXAMPLES. (i.) 1892 is divisible by 4, .*. the year 1892 is a
leap year, (ii.) 1900 is a multiple of 100, and 19 is not divisible
by 4, .'. 190O is not a leap year. (Hi.) 2000 : the number of tho
century is 20, and is divisible by 4, .'. 2OOO is a leap year.
The Gregorian correction still leaves a small difference
between the tropical year and the average length of the civil
year, amounting to only Id. 5h. 26m. in 4,000 years.
171. A Synodic Year is a period of 12 lunar months,
being nearly 355 days. The name is, however, rarely used.
OK TIME. 129
SECTION IV. Comparison of Mean and Sidereal Times.
172. Relation between Units. One of the most
important problems in practical astronomy is to find the
sidereal time at any given instant of mean solar time, and
conversely, to find the mean time at any given instant of
sidereal time. Before doing this it is necessary to compare
the lengths of the mean and sidereal days.
We have seen ( 169) that a tropical year contains abont
365| mean solar days. In this period both the true and
mean Sun describe one complete revolution, or 360 from
west to east relative to T ; or, what is the same thing, T
describes one revolution from east to west relative to the
mean Sun. But the mean Sun performs 365 revolutions
from east to west relative to the meridian at any place.
Therefore T performs one more revolution, i.e., 366 revo-
lutions, relative to the meridian.
Now, a sidereal day and a mean solar day have been defined
( 22, 159) as the periods of revolution of the mean Sun
and of T relative to the meridian ;
.-. SOS} mean solar days = 366| sidereal days.
Prom this relation we have,
One mean solar day = ( 1 + ~ ) sidereal days
\ ODO^-/
= (1 + '002738) sidereal days
= 24h. 3m. 56'5s. sidereal time
= 1 sidereal day + 4m. 4s. nearly;
.. one mean solar hour = Ih. + 10s. s. sidereal time,
and 6m. of mean solar time = Gin. + Is. sidereal time nearly.
In like manner we have
One sidereal day = ( 1 ) mean solar days
\ o66f/
-- (1 -002730) mean days
= 23h. 56m. 4' Is. mean time
= 1 mean day 4m. -f 4s. nearly ;
.*. one sidereal hour = Ih. 10s. +-J-S. of mean time,
and 6m. sidereal time = 6m. Is.meansolartimenearly.
L
130 tf __ |4
173. From the results of the last paragraph we have the
following approximate rules :
(i.) To reduce a given interval of mean time to
sidereal time, add 10s. for every hour, and Is. for every
6m. in the given interval. For every minute so added, sub-
tract Is.
(ii.) To reduce a given interval of sidereal time
to mean time, subtract 10s. for every hour, and Is. for
every 6m. in the given interval. Then add Is. for everr
minute so subtracted.
EXAMPLE 1. Express in sidereal time an interval of 13h. 23m. 25s.
mean time.
The calculation stands as follows : H. M. s.^
Mean solar interval =13 23 25
Add 10s. per hour on 13h.... ' 2 10
Is. per 6m. on 23m. ... ... ... 4
13 25 39
Subtract Is. per 1m. on 2m. 13'8s. ... 2
.-. Required sidereal interval = 13 25 37
EXAMPLE 2. Find the mean solar interval corresponding to
14h. 45m. 53s. of sidereal time.
The calculation stands as follows : H. M. s.
Given sidereal interval ... ... ... ... = 14 45 53
Subtract 10s. per hour on 14h. = 2m. 20s. \ o 28
Is. per 6m. on 46m. (nearly )= 8s. /
14 43 25
Add Is. per 1m. on 2m. 28s. .... ... 3
.. Required interval of mean time =14 43 28
If accuracy to within a few seconds is not required, the
second correction of Is. per 1m. may be omitted. On the
other hand, if the interval consists of a considerable number
of days, or if accuracy to the decimal of a second is needed,
the results found by the rules will no longer be correct.
"We must, instead, add 1/365^- of the given mean solar interval
to get the sidereal interval, or subtract 1/366 J of the given
sidereal to get the mean solar interval.
In order to still further simplify the calculations, tables
have been constructed ; in most cases, these give the quantity
to be added or subtracted according as we are changing from
mean to sidereal, or from sidereal to mean time.
Off TIME. 131
174. To find the sidereal time at a given instant
of mean solar time on a given date at Greenwich.
The Nautical Almanack* gives the sidereal time of mean
noon at Greenwich on every day of the year.
Now the given mean time represents the number of hours,
minutes, and seconds which have elapsed since mean noon,
expressed in mean time. Convert this interval into sidereal
time ; we then have the sidereal interval which has elapsed
since mean noon. Add this to the sidereal time of mean
noon ; the result is the sidereal time required.
Thus, let m he the mean time at the given instant, mea-
sured from the preceding mean noon,
* the sidereal time of mean noon from the Nautical Almanack,
and let k = l/365 ; so that l+ is the ratio of a mean solar
unit to the corresponding sidereal unit.
Then, from mean noon to given instant,
Interval in mean time = m
.*. interval in sidereal time = m+lan
But, at mean noon, sidereal time = s
.*. at given instant,
required sidereal time, s-=s Q +m+km.
If the result he greater than 24h., we must subtract 24h., for
times are always measured from Oh. up to 24h.
EXAMPLE. Find the sidereal time corresponding to 8h. 15m. 40s.
P.M. on Dec. 20, given that the sidereal time of mean noon was
I7h. 55m. 8s.
From mean noon to the given instant, the interval in mean time
is 8h. 15m. 40s.
Converting this interval to sidereal time, by the method of 173,
we have Mean solar interval = 8h. 15m. 40s.
Add 10s. per hour on 8h. 1m. 20s.
Add Is. per 6m. on 15m. 40s. 3s.
8h. 17m. 3s.
Subtract Is. per 1m. on 1m. 23s. ls._
.*. Sidereal interval since mean noon = 8h. 17m. 2s.
But sidereal time of mean noon = I7h. 55m. 8s.
.*. Sidereal time at instant required = 26h. 12m. 10s.
Or, deducting 24h., sidereal time is = 2h. 12m. 10s.
* Or Whitaker's Almanack, which may be used if the Nautical is
not at hand.
132 ASTEONOMf.
175. To find the mean solar time corresponding to
a given instant of sidereal time at Greenwich.
Subtract the sidereal time of mean noon from the given
sidereal time ; this gives the interval which has elapsed since
mean noon, expressed in sidereal time. Convert this interval
into mean time ; the result is the mean time required.
Let It = 1/366 J ; so that 1 k' is the ratio of a sidereal to a
mean solar unit.
Let the given sidereal time = ,
and let the sidereal time of the preceding mean noon = ;
Then, from mean noon to given instant,
Interval in sidereal time = s S Q
.'. interval in mean time = (s s ) '(s s ).
.'. required mean time m = (s * ) M> ).
If s be less than , we must add 24h. to s in order that the
times s, s may be reckoned from the same transit of T
EXAMPLE. Find the solar time corresponding to 16h. 3m. 42s.
sidereal time on May 6, 1891, sidereal time at mean noon being
2h. 52m. 17s.
Sidereal interval since mean noon
= 16h. 3m. 42s. -2h. 52m. 17s. = 13h. llm. 25s.
/. Mean solar interval ( 173)
= 13h. llm. 25s. -2m. 10s. 2s. + 2s. = 13h. 9m. 15s.
Hence, 13h. 9m. 15s. is the mean time; which, in our usual
reckoning, would be called Ih. 9m. 15s., on the morning of May 6
( 36). The sidereal timo was also 16h. 3m. 42s. a sidereal day
or 23h. 56m. 4s. previously, i.e., Ih. 13m. 11s. a.m. on the morning
of May 5. %
176. To find the mean time corresponding to a
given instant of sidereal time at Greenwich (alterna-
tive method). The Nautical Almanack also contains the
mean time of tl Sidereal Noon," i.e., the mean time when T
is on the meridian, and when the sidereal clock marks
Oh. Om. Os. Let this be m , and let s be the given sidereal
time, k' the factor l/366 as before. Then
From sidereal noon to given instant, sidereal interval = s ;
.-. ,, ,, mean solar,, = s-k's.
But, at sidereal noon, mean time = m ;
/. at given instant,
The required mean time = m Q +s Jc's.
Oft TIME. 133
177. To find the sidereal time from the mean solar,
or the mean time from the sidereal, in any given
longitude. If the longitude is not that of Greenwich, the
ahove methods will require a slight modification, because the
sidereal time of mean noon and mean time of sidereal noon
are tabulated for Greenwich.
In such cases, the safest plan is as follows : Find the
Greenwich time corresponding to the given local time ( 96).
Convert this Greenwich time from mean to sidereal, or sidereal
to mean, as the case may be, and then find the corresponding
local time again.
Let the longitude be L west of Greenwich (Z being nega-
tive if the longitude is east),
lot m l be the mean and s l the sidereal local time,
m, s the corresponding times at Greenwich,
and let /c, &', w> , have the same meanings as in 172-4.
By 96 we have, whether the times be local or sidereal,
(Greenwich time) (local time in long. Z W.) = T 1 - F Zh.
= 4Z m. Therefore, s s l ^L = m m r
(i.) If m l is given and s l is required, we have (in hours),
By 174, 8 = 8H +
s l = srfa =
(ii.) If s 1 is given and m l is required, we have
By 175, 176, m = (-* ) - #(-) or = i +-#,
i.e. m *-*~^ s -* > + L
EXAMPLE. Find the solar time when the local sidereal time is
5h. 17m. 32s. on March 21, the place of observation being Moscow
(long. 37 34' 15" B.) ; given that sidereal time of mean noon was
23h. 54m. 52s. at Greenwich.
Eeduced to time ( 23), 37 34' 15" is 2h. 30m. 17s.
/. Greenwich sidereal time at instant required
= 5h. 17m. 32s. -2h. 30m. 17s. = 2h. 47m. 15a.
Sidereal interval since Greenwich noon
= 2h. 47m. 15s. + 24h -23h. 54m. 52s. = 2h. 52m. 23s.
.'. Greenwich mean time = 2h. 52m. 23s. -20s. -9s. = 2h. 51m. 54s.
.'. Moscow mean time = 2h. 51m. 54s. + 2h. 30m. 17s. - 5h. 22m. 11s
134 AStKONOiTT.
178. Equinoctial Time. For the purpose of comparing
the times of observations made at different places on the
Earth, another kind of time has been introduced.
The Equinoctial Time at any instant is the interval of
time that has elapsed since the preceding vernal equinox,
measured in mean solar units.
The advantage of equinoctial time is that it is independent
of the observer's position on the Earth, since the instant when
the Sun passes through T is a perfectly definite instant of
time, and is independent of the place of observation. On the
other hand, mean time and sidereal time, being measured
from the transits of the mean Sun and of T across the
meridian, depend on the position of the meridian that is, on
the longitude of the observer.
The chief disadvantage of equinoctial time is that since the
tropical year contains 365d. 5h. 48m. 46s., and not exactly
365 days, the vernal equinox will occur 5h. 48m. 46s. later
in the day every year, so that at the end of each tropical year
the equinoctial clock will have to be put back 5h. 48m. 46s.
Hence also the same equinoctial time will represent a different
time of day on the same date in different years.
The disadvantages of using local time are obviated in Great
Britain by the universal use of " Greenwich Mean Time."
179. Practical Applications. In 41 we showed how
to determine roughly the time of night at which a given star
would transit on a given day of the year. "With the intro-
duction of mean time, in the present chapter, we are in a
position to obtain a more accurate solution of the problem.
Por the R.A. of any star (expressed in time) is its sidereal
time of transit. If this be given, we only have to find the
corresponding mean time ; this will be the required time
of transit, as indicated by an ordinary clock.
In the calculations required in converting the time from
one measure to the other, it is advisable not to quote the
formula? of 174-177, but to go through the various steps
one by one.
If neither the sidereal time of mean noon nor the mean
time of sidereal noon is given, we must fall back on the
rough method of 35.
ON TIME. 135
EXAMPLES.
1. Find the solar time at 5h. 29m. 28s. sidereal time on July 1, 1891 ;
mean time of sidereal noon being 17h. 20m. 8s.
Sidereal interval from sidereal noon to the given instant = 5h.29m.28s.
.-. Mean solar interval = 5h. 29m. 28s. 50s. 5s. + Is. = 5h.28m.34s.
i.e., Mean solar time = 5h. 28m. 34s. + I7h. 20m. 83. =22h. 48m. 42s. ;
or, lOh. 48m. 42s. A.M., July 2.
It was also 5h. 29m. 28s., a sidereal day or 23h. 56m. 4s. pre-
viously, i.e., lOh. 52m. 38s. a.m. July 1. ^
2. To find the mean time of transit of Aldelaran at Greenwich on
December 12, 1891. Given H M s
R.A. of Aldelaran = 4 29 40 ;
Sidereal time of noon, December 12, 1891 = 17 23 56.
Since the star's R.A. is less than the sidereal time of noon, we
must increase the former by 24h., in order that both may be mea-
sured from the same " sidereal noon." H. M. s.
Sidereal time of transit + 24h. = 28 29 40
Subtract noon = 17 23 56
/. Sidereal interval from noon to transit =11 5 44
To convert into mean solar units, subtract 1 49
.'. Mean Solar interval from noon to transit =11 3 55
.'. Aldelaran transits at llh. 3m. 55s. mean time. 7~&~*ii-
3. To find the (local) sidereal time at New York at 9h. 25m. 31g.
(local mean time) on the morning of September 1, 1891.
Longitude of New York = 74 W.
Sidereal time of mean noon at Greenwich, Sept. 1 = lOh. 42m. 24s.
The given local mean time is measured from midnight, therefore
we must take the time measured from noon as H. M. s.
August 31, 1891. = 21 25 31
Add for 74 west longitude reduced to time = 4 56
.*. Greenwich mean time is, August 31, 26 21 31
or, September 1, 2 21 31
To convert this interval to sidereal units, add 24
/. Sidereal time elapsed since Greenwich noon = 2 21 55
But at Greenwich noon, sidereal time (by data) = 10 42 24
.-. Sidereal time at Greenwich is 13 4 19
Subtract for 74 west longitude, 4 56
.-. Sidereal Time at New York = 8 18 9
136 ASTRONOMY.
4. To find the Paris mean time of transit of Regulus at Nice on
December 26, 1891. H. M. s.
Longitude of Paris = 2 21' E.E.A. of Regulus =10 2 34
Nice = 7 18' E.
Sidereal time at Greenwich noon = 18 18 48
Here local sidereal time of transit at Nice 10 2 34
Subtract east longitude of Nice, T 18', in time 29 12
V. Greenwich sid. time of transit at Nice -I- 24h. C3 33 22
Subtract Greenwich sidereal time at noon, 18 18 48
/. Sidereal interval since Greenwich noon 15 14 34
To convert to mean solar units, subtract 2 30
.'. Greenwich mean time = 15 12 4
Add east longitude of Paris, expressed in time = 9 24
.-. Paris mean time of transit = 15 21 28
That is, 3h. 21m. 28s. in the morning on December 27.
5. Find the E.A. of the Sun at true noon on October 8, 1891, given
that the equation of time for that day is 12m. 24s., and that the
sidereal time of mean noon on March 21 was 23h. 54m. 52s.
Mean solar interval from mean noon March 21 to mean noon Oct. 8
= 201 days.
Mean solar interval from mean noon to apparent noon on Oct. 8
= -12m. 24s.
/. interval from mean noon on March 21 to apparent noon on Oct. 8
= 201d.-12m. 24s.
Now, in 365 days the mean Sun's E.A. increases 24h., and the
increase takes place quite uniformly.
.'. increase in mean Sun's E.A. in 201 days H. M. s.
= 24h.x 201 -=-365| = 13 12 27
Add mean Sun's E.A. on March 21
( = sidereal time of mean noon) = 23 54 52
.'. mean Sun's E.A. at mean noon Oct. 8 = 37 7 19
or, subtracting 24h., =13 7 19
Subtract change of E.A. in 12m. 24s. = 2
.'. mean Sun's E.A. at apparent noon Oct. 8 =13 7 17
But true Sun's E.A. mean Sun's E.A
= equation of time = 12 24
/. True Sun's E.A. at apparent noon Oct. 8 = J2h. 54m. 53s,
ON TIME. 137
EXAMPLES. -V.
1; To what angles do Sidereal Time, Solar Time, and Mean Time
correspond on the celestial sphere ? Are these angles measured
direct or retrograde ?
2. Draw a diagram of the Equation of Time, on the supposition
that perihelion coincides with the vernal equinox.
3. On May 14 the morning is 7'8 minutes longer than the after-
noon : find the equation of time on that day.
4. On a sun-dial placed on a vertical wall facing south, the
position of the end of the shadow of a gnomon at mean noon is
marked on every day of the year. Show that the curve passing
through these points is something like an inverted figure of eight.
5. Why are not the graduations of a level dial uniform ? Show
that they will be so if the dial be fixed perpendicular to the index.
6. Show that if every 5th year were to contain 366 days, every
25th year 367 days, and every 450th year 368 days, the average
length of the civil year would be almost exactly equal to that of the
tropical year. How many centuries would have to elapse before the
difference would amount to a day ?
7. Give explicit directions for pointing an equatorial telescope to
a star of R.A. 22h., declination 37 N., in latitude 50 N., longitude
25 E., at lOh. Greenwich mean time, when the true Sun's E.A. is
14h. 47m. 17s., and the equation of time is 16m. 14s.
8. If the mean time of transit of the first point of Aries be
9h. 41m. 24*4s., find the time of the year, and the sidereal time of
an observation on the same day at Ih. 22m. 13'5s.
9. At Greenwich, the equation of time at apparent noon to-day is
- 3m. 39'42s., and at apparent noon to-morrow it will be 3m. 35'39s.
Prove that the mean solar time at New York corresponding to ap-
parent time 9 A.M. there this morning is 8h. 56m. 2O9s., having given
that the longitude of New York is 74 I' W.
10. Find the sidereal time at apparent noon on Sept. 30, 1878, at
Louisville ( long. 85 30' W.) having given the following from the
Nautical Almanack :
At mean noon.
Sun's apparent right
ascension.
Sept. 30. 12h. 26m. 23'16s.
Oct. 1. 12h. 30m. 0'51s.
Equation of time
to be added to mean time.
10m. 0'77s.
10m. 19-98s,
138 ASTBONOMT.
MISCELLANEOUS QUESTIONS.
1. Explain how to determine the position of the ecliptic relatively
to an observer in S. latitude at a given time on a given day.
2. Indicate the position of the ecliptic relatively to an observer
at Cape Town (lat. 33 56' 3'5" S.) at noon on August 3.
3. Explain why a day seems to be gained or lost by sailing round
the world. State which way round a day seems to be lost, and give
the reason why.
4. If the inclination of the ecliptic to the equator were 60, instead
of 23 27^', describe what would be the variations in the seasons to
an observer in latitude 45, illustrating your description with a
diagram.
5. Describe the changes of position in the point of the Sun's
rising at different times of the year, and at different points on the
Earth's surface.
6. If the equator and ecliptic were coincident, what kind of curve
would be described in space by a point on the Earth's surface, say
at the equator, daring the course of the year ?
7. Examine when that part of the equation of time due to the
eccentricity of the Earth's orbit is positive.
8. On September 22, 1861, the times of transit of a Lyrse and of
the Sun's centre over the meridian of Greenwich were observed to
be 18h. 32m. 51'3s. and 12h. Om. 23'3s. by a sidereal clock whose
rate was correct. Given that the R. A. of a Lyrae was 18h. 31m. 43'9s.,
find the Sun's B..A. and the error of the clock.
9. Define mean time and sidereal time, and compare the lengths
of the mean second and the sidereal second.
10. If a, a' are the hour angles in degrees of the Sun at Greenwich,
at t and t' hours mean time, show that the equations of time at the
preceding and following mean noons, expressed in fractions of an
hour, are respectively
a't-at' 21 .X(24-Q-a(24 f)
oir TIME.
EXAMINATION PAPER. v.
1. Define tiie dynamical mean Sun and the mean Sun, stating at
what points they have the same R.A., and when the former coin-
cides with the true Sun. Show that the mean Sun has a uniform
diurnal motion, and state how it measures mean time.
2. Define the equation of time. Of what two parts is it generally
taken to consist? State when each of these parts vanishes, is
positive, or negative. Give roughly their maximum values, and
sketch curves showing their variations graphically.
3. Show that the equation of time vanishes four times a year.
4. If, on a certain day, the sun-dial be 10 minutes before the clock,
what is the value of the equation of time on that day ? Will the
forenoon of that day or the afternoon be longer, and by how much ?
5. Define the terms solar day, mean solar day, sidereal day.
What is the approximate difference and the exact ratio of the
second and third ?
6. Define the terms civil year, anomalistic year, equinoctial
time. Why was this last introduced ?
7. Show how to express mean solar time in terms of sidereal
time, and vice versd.
8. If the mean Sun's R.A. at mean noon at Greenwich on June 1
be 4h. 36m. 54s., find the sidereal time corresponding to 2h. 35m. 45s.
mean time (1) at Greenwich, (2) at a place in longitude 25 E.
9. On what day of the year will a sidereal clock indicate lOh. 20m.
at 4 P.M. ?
.10. In what years during the present century have there been
five Sundays in February ? When will it next happen ?
ASTBON. L
CHAPTER VI.
ATMOSPHERICAL KEFBACTION AND TWILIGHT.
180. Laws of Refraction. It is a fundamental prin-
ciple of Optics that a ray of light travels in a straight line,
so long as its course lies in the same homogeneous medium ;
but when a ray passes from one medium into another, or
from one stratum of a medium
into another stratum of dif-
ferent density, it, in general,
undergoes a change of direction
at their surface of separation.
This change of direction is
called Refraction.*
Letarayof light S0(Fig. 64)
pass at from one medium into
another, the two media being
separated by the plane surface
AB, and let OT be the direc-
tion of the ray after refraction
in the second medium. Draw
ZOZ' the normal or perpendicular to the plane AB at 0.
Then the three laws of refraction may be stated as follows :
I. Thfrancident and refracted rays SO, OTand the normal
ZOZ 1 all lie in one Diane.
. .
is a constant quantity, being tlie same for all directions of the
rays, so long as the two media are the same.]
This constant ratio is called the relative index of
refraction of the two media, and is usually denoted by the
Greek letter fi.
* For a fuller description, see Stewart's Light, Chap. YI.
f The value of the ratio varies slightly for rays of different colours,
but with this we are not concerned in the present chapter.
ATMOSPHERICAL REFRACTION AND TWILIGHT. 141
Thus, if TO be produced backwards to S',
sin ZOS = /i sin Z' OT = p sin ZOS r ,
The angles ZOS and Z' OT are usually called the angle of
incidence and the angle of refraction respectively.
III. When light passes from a rarer fo a denser medium, the
angle of incidence is greater than the angle of refraction.
Since Z ZOS> L Z'OT, sin ZOS > sinZ'OT and /. /i > 1.
181. General Description of Atmospherical Refrac-
tion. If the Earth had no ,
atmosphere, the rays of light
proceeding from a celestial
body would travel in straight
Lines right up to the obser-
ver's eye or telescope, and we
should see the body in its
actual direction.
But when a ray Sa (Fig. 65)
meets the uppermost layer
AA' of the Earth's atmo-
sphere, it is refracted or bent
out of its course, and its direc- FlG -
tion changed to aft. On passing into a denser stratum of aiv
at BB', it is further bent into the direction be, and so on ;
thus, on reaching the observer, the ray is travelling in
a direction OT, different from its original direction, but
(by Law I.) in the same vertical plane.
The body is, therefore, seen in the direction OS',
although its real direction is aS or OS. Also, since the
successive horizontal layers of air A A', BB', CC', ...
are of increasing density, the effect of refraction is to
bend the ray towards the perpendicular to the surfaces of
separation, that is, towards the vertical.
Hence : The apparent altitudes of the stars are
increased by refraction.
In reality, the density of the atmosphere increases gradually
as we approach the Earth, instead of changing abruptly at
the planes A A', BB', .... Consequently, the ray, instead of
describing the polygonal path Sabc 0, describes a curved path,
but the general effect is the same.
142
ASTRONOMY.
182. Law of Successive Refractions. Let there be
any number of different media, separated by parallel planes
JLA', ', CO', HH' (Fig. 66), and let Sale OT represent
the path of a ray as refracted at the various surfaces. Then
it is a result of experiment that the final direction S'T
of the ray is parallel to what it would have been if the ray
had been refracted directly from the first into the last medium
without traversing the intervening media.
Thus, if a ray SO, drawn parallel to Sa, were to pass
directly from the first medium to the last by a single refrac-
tion at 0, its refracted direction would be the same as that
actually taken by the ray Sa, and would coincide with OT.
S'
FIG. 66.
FIQ. 67.
183. The Formula for Astronomical Refraction.
"We shall now apply the above laws to determine the change
in the apparent direction of a star produced by refraction.
Since the height of the atmosphere is only a small fraction
of the Earth's radius, it is sufficient for most purposes
of approximation to regard the Earth as flat, and the surfaces
of equal density in the atmosphere as parallel planes. "With
this assumption, the effect of refraction is exactly the same
( 182) as if the rays were refracted directly into the lowest
stratum of the atmosphere, without traversing the intervening
strata.
ATMOSPHERICAL EEFEACTION AND TWILIGHT. 143
Let OS (Fig. 67) be the true direction of a star or other
celestial body. Then, before reaching the atmosphere, the
rays from the star travel in the direction SO. Let their
direction after refraction be S'OT, then OS' is the
apparent direction in which the star will be seen, and the
angle SOS' is the apparent change in direction due to
refraction. The normal OZ points towards the zenith.
Hence ZOS is the star's true zenith distance, and ZOS 1
or Z'OT is its apparent zenith distance, and the first and
third laws of refraction show that the star's apparent direction
is displaced towards the zenith.
Let L ZOS' = 8, tS'OS = u, and .-. L ZOS = z + u ;
and let /j be the index of refraction.
By the second law of refraction,
sin (a -f u) = p sin s.
sin 2 cos w-j-cos s sin u = yu sin z.
Now the refraction u is in general very small. Hence,' if
u be measured in circular measure, we know by Trigonometry
that sin u = , and cos u = 1 very approximately. Therefore
we have
sin 2 -f- u cos 8 = (j. sin 8 ;
Let U be the amount of refraction in circular measure
when the zenith distance is 45. Putting s 45, we have
.-. u =
Thus the amount of refraction is proportional to
the tangent of the apparent zenith distance.
The last result does not depend on the fact that the refrac-
tion is measured in circular measure. Hence, if w", U" be
the numbers of seconds in u, U, we have
u" = U" tan a.
The quantity U" is called the coefficient of refraction.
Since U is the circular measure of Z7", we have
V" = 180X60X6 . V= 206265 (,,-1),
7T
whence, if U" is known, p cm be found, and conversely,
144 ASTEONOMT.
184. Observations on the preceding Formula. In
the last formula u" represents the correction which must be
added to the apparent or observed zenith distance in order to
obtain the true zenith distance. By the first law, the azimuth
of a celestial body is unaltered by refraction.
Thus the time of transit of a star across the meridian, or
across any other vertical circle, is unaltered by refraction.
In using the transit circle, there will, therefore, be no cor-
rection for observations of right ascension, but in finding the
declination the observed meridian Z.D. will require to be
increased by U" tan z.
A star in the zenith is unaffected by refraction, and the
correction increases as the zenith distance increases. When
a star is near the horizon, the formula u" = U" tan z fails,
since it makes u" = co, when 2 = 90. In this case u is no
longer a small angle, so that we are not justified in putting
sin u = u and cos u = 1. But there is a more important reason
why the formula fails at low altitudes, namely, that the rays
of light have to traverse such a length of the Earth's atmo-
sphere that we can no longer regard the strata of equal density
as bounded by parallel planes. In this case, it is necessary to
take into account the roundness of the Earth in order to obtain
any approach to accurate results.
For zenith distances less than 75, the formula is found to
give fairly satisfactory results ; for greater zenith distances it
makes the correction too large-.
The coefficient of refraction U" is found to be about 57",
when the height of the barometer is 29 -6 inches and the
temperature is 50. But the index of refraction depends on
the density of the air, and this again depends on the pressure
and temperature. Hence, where accurate corrections for
refraction are required, the height of the barometer and
thermometer must be read. Any want of uniformity in the
strata of equal density, or any uncertainty in determining
the temperature, will introduce a source of error ; hence it is
desirable that the corrections shall be as small as possible.
For this reason observations made near the zenith are always
the most reliable,
ATMOSPHERICAL EEFEACTION AND TWILIGHT.
145
*185. Cassini's Formula. The law of refraction was also investi-
gated by Dominique Cassini on the hypothesis that the atmosphere
is spherical but homogeneous throughout ; in this way he obtained
the approximate formula
u = (ju1) tans (1 n sec 2 s),
where n is the ratio of the height of the homogeneous atmosphere
to the radius of the Earth.
Cassini's formula may be proved as follows : Let SO'O be the
path of a ray of light from a star 8.
By hypothesis this ray undergoes a
single refraction on entering the homo-
geneous atmosphere at 0'. Let be
the position of the observer, G the
centre of the Earth. Produce 00'
to 8', CO to Z, and GO' to Z'. Let
u = L SOS' (in circular measure),
Then, by 183, if u is small, we have
u = (jti 1) tans';
but here z' is not the apparent zenith
distance, so that we must express tan z'
in terms of tan z.
Draw CT perpendicular to O'O pro-
duced, and O'N perpendicular to COZ.
Then O'T tans' = TG = OTtansj
tan_s_ = OT = 1 + ^0
tans'" Or OT
OIV sec z __ , O# gec 2
00
= 1 + '
FIG. 68.
00 cos z
But ON is very approximately the height of the homogeneous
atmosphere OH, and is therefore = n . OG ;
tan 2 . , tans
tans'
= 1 + n sec 2 z ;
tans'
1 + Ti sec 2 z
whence, by substituting in the formula, we have
, .. , tan z
1 4 n sec 2 z
= (fj. 1) tans {l wsec 2 s + 'n- 2 sec 4 s n 3 sec 6 s, &c.}
Now n is very small ; we may therefore neglect its square and higher
powers; hence we obtain approximately
u= (/* !) tan z (1Ti sec- s),
which is Cassini's formula.
If the value of n be properly chosen, Cassini's formula is found
to give very good results for all zenith distances up to 80.
146 . A8TBONOMY.
186. To determine the Coefficient of Refraction
from Meridian Observations. Assuming the " tangent
law," u= 7"tanz, the coefficient of refraction U may be
found from observations of circumpolar stars as follows.
Let Z D z 2 , the apparent zenith distances of a circumpolar
itar, be observed at upper and lower culminations respectively.
Then the true zenith distances will be
%i -f 7~tan z l and z 2 + 7"tan z 2 .
Now, the observer's latitude is half the sum of the meri-
dian altitudes at the two culminations ( 28), hence if I be
the latitude, we have
or 90-? = i(2 1 + 2 2 ) + J^(tanz 1 +tanE 2 ) ...... (i.).
Now let a second circumpolar star be observed. Let its
apparent zenith distances at upper and lower culminations be
z' and % '. Then we obtain in like manner
90-Z = i (a'+a") + i T(tan z' + tan 2") ...... (ii.).
Eliminating I from (i.) and (ii.) by subtraction, we have
(tan Zj + tan z 2 ) (tan z' + tan z")*
If the two stars have the same declination, we shall have
Zj = z and z 2 = z", and the above formula will fail. Hence
it is important that the two observed stars should differ con-
siderably in declination ; the best results are obtained by
selecting one star veiy near the pole (e.g-, the Pole Star) and
the other about 30 from the pole.
187. Alternative Method (Bradley's). Instead of
using a second circumpolar star, Bradley observed the Sun's
apparent Z.D.'sat noon at the two solstices. Let these be Z v Z y
By 38, since the true Z.D.'s are
Z^ + 7tan Z t and Z^ + U tan Z v
Z^ + Z7tan Z l = l- i, Z^ + Vtan Z z = l + i; (i = obliquity.)
.-. 2Z = ' 1 + 4+7(tan;+tan 2 ) ......... (iii.).
Eliminating ?from (i.), (iii.), we have
^hence 7" is found.
UNIVERSITY
TWILIGHT,. 147
188. Other Methods of finding the Refraction.
Suppose that at a station on the Earth's equator, either a
star on the celestial equator, or the Sun at an equinox, is
observed during the day. Its diurnal path from east to west
passes through the zenith, and during the course of the
day its true zenith distance will change uniformly at the
rate of 15 per hour. Thus the true Z.D. at anytime is
known. Let the apparent Z.D. be observed with an altazi-
muth. The difference between the observed and the calcu-
lated Z.D. is the displacement of the body due to refraction.
By this method we find the corrections for refraction at
different zenith distances without making any assumptions
regarding the law of refraction.
Except at stations on the Earth's equator, it is not possible
to observe the refraction at different zenith distances in such
a simple manner. Nevertheless, methods more or less similar
can be employed. For this purpose the zenith distances of a
known star are observed at different times. The true zenith
distance at the time of each observation can be calculated
from the known R.A. and declination ( 26). Hence
the refraction for different zenith distances of the star
can be determined. This method is very useful for verifying
the law of refraction after the star's declination and the
observer's latitude have been found with tolerable accuracy.
Moreover, it can be employed to find the corrections for
refraction at low altitudes when the " tangent law " ceases
to give approximate results.
189. Tables of Mean Refraction. From the results
of such observations tables of mean refraction have been con-
structed by Bessel,* and are now used universally. These
are calculated for temperature 50 and height of barometer
29*6 inches ; they give the refraction for every 5' of altitude
up to 10, for larger intervals at altitudes between 10
and 54, and for every 1 at altitudes varying from 54 to
90. Other tables give the "Correction for Mean Refraction,"
which must be added to or subtracted from the mean refrac-
tion given in the first table in allowing for differences in the
temperature and barometric pressure. The corrections for
temperature and pressure are applied separately.
* See any book of Mathematical Tables, such as Chambers'^.
148 ASTBOffOMY.
190. Effects of Refraction on Rising and Setting.
At the horizon the mean refraction is about 33' ; con-
sequently a celestial body appears to rise or set when it
it is 33' below the horizon. Thus, the effect of refraction is
to accelerate the time of rising, and to retard, by an equal
amount, the time of setting of a celestial body. In particular,
the Sun, whose angular diameter is 32', appears to be
just above the horizon when it is really just below.
The acceleration in the time of rising due to refraction can
be investigated in exactly the same way as the acceleration
due to dip ( 104). If u" denotes the refraction at the hori-
zon in seconds, d the declination, x the inclination to the
vertical of the direction in which the body rises, the accelera-
tion in the time of rising in seconds
= u" sec x sec d.
lo
Taking the horizontal refraction as 33', or 1980", and
putting x = 0, d = 0, we see that at the Earth's equator at
an equinox, the time of sunrise is accelerated by about
2m. 12s. owing to refraction,
"When the Sun or Moon is near the horizon, it appears
distorted into a somewhat oval shape. This effect is due to
refraction. The whole disc is raised by refraction, but the
refraction increases as the altitude diminishes ; so that the
lower limb is raised more than the upper limb, and the
vertical diameter appears contracted. The horizontal dia-
meter is unaffected by refraction, since its two extremities
are simply raised. Hence, the disc appears somewhat flat-
tened or elliptical, instead of truly circular.
According to the tables of mean refraction, the refraction
on the horizon is 33', while at an altitude 30', the refraction
is only 28' 23", and at 35' it is 27' 41". Hence, taking the
Sun's or Moon's diameter as 32', the lower limb when on the
horizon is raised about 5' more than the upper. The con-
traction of the vertical diameter, therefore, amounts to 5',
i.e., about one-sixth of the diameter itself, so that the appa-
rent vertical and horizontal angular diameters are approxi-
mately in the ratio of 5 to 6.
ATMOSPHERICAL REFRACTION AND TWILIGHT. 149
191. Illusory Variations in Size of Sun and Moon.
The Sun and Moon generally seem to look larger when
low down than when high up in the sky. This is, however,
merely a false impression formed by the observer, and is not
in accordance with measurements of the angular diameter
made with a micrometer. When near the horizon, tho
eye is apt to estimate the size and distance of the Sun and
Moon by comparing them with the neighbouring terres-
trial objects (trees, hills, &c.). When the bodies are at
a considerable altitude no such comparison is possible, and a
different estimate of their size is instinctively formed.
192. Effect of Refraction on Dip, and Distance of
the Horizon. Since refraction increases as we approach
the Earth, its effect is always to bend the path of a ray of
light into a curve which is concave downwards (Fig. 69).
FIG. 69.
Let be any point above the Earth's surface, and let T'
be the curved path of the ray of light which touches the Earth
at T' and passes through 0. Then OT' is the distance of
the visible horizon. Draw the straight tangent OT, then
OT would be the distance of the visible horizon if there
V^ere no refraction; hence, it is evident from the figure that
The Distance of the horizon is increased by
refraction.
Draw OT", the tangent at to the curved path OT', then
OT" is the apparent direction of the horizon. Hence, from
the figure we see that
The Dip of the horizon is diminished by refraction.
Both dip and distance are still approximately proportional
to the square root of the height of the observer.
150 ASTBONOMY.
193. Effect of Refraction on Lunar Eclipses and
on Lunar Occupations. In a total eclipse the Moon's
disc is never perfectly dark, but appears of a dull red colour.
This effect is due to refraction. The Earth coming between
the Sun and Moon prevents the Sun's direct rays from reach-
ing the Moon, but those rays which nearly graze the Earth's
surface are bent round by the refraction of the Earth's
atmosphere, and thus reach the Moon's disc.
From observing the " occultations " of stars when the
unilluminated portion of the Moon passes in front of them,
we are enabled to infer that the Moon does not possess an
atmosphere similar to that of our Earth. For the directions
of stars would be displaced by the refraction of such an
atmosphere just before disappearing behind the disc, and just
after the occultation ; and no such effect has been observed.
194. Twilight. The phenomenon of twilight is also due
to the Earth's atmosphere, and is explained as follows :
After the Sun has set, its rays still continue to fall on the
atmosphere above the Earth, and of the light thus received
a considerable portion is reflected or scattered in various
directions. This scattered light is what we call twilight,
and it illuminates the Earth for a considerable time after
sunset. Moreover, some of the scattered light is transmitted
to other particles of the atmosphere further away from the
Sun, and these reflect the rays a second time ; the result of
these second reflections is to further increase the duration of
twilight. Twilight is said to end when this scattered light has
entirely disappeared, or has, at least, become imperceptible.
From numerous observations, twilight is found to end when
the Sun is at a depth of about 18 below the horizon.
If the Sun does not descend more than 18 below the
horizon, there will be twilight all night.
Let I = latitude, d = Sun's declination, then it is easily
seen by a figure that the Sun's depth below the horizon
at midnight = 90 dl.
This depth is less than 18, if I > 72 <?.
But the greatest value of d is , or nearly 23J (mid-
summer). Hence, there is twilight all the night about
midsummer, at any place whose latitude I is not less than
72-23j, or48i.
ATMOSPHERICAL KfcFKACTiON AND TWILIGHT, 151
EXAMPLES. YI.
1. What would IDG the effect of refraction on terrestrial objects as
seen by a fish under water ?
2. For stars near the zenith show that the refraction is approxi-
mately proportional to the zenith distance, and that the number of
seconds in the refraction is equal to the number of degrees In the
zenith distance. (Take coefficient of refraction = 57".)
3. From the summit of a mountain 2400 feet above the level of
the sea, it is just possible to see the summit of another, of height
3450 feet, at a distance of 143 miles. Find approximately the radius
of the Earth, assuming that the effect of refraction is to alter the
distance of the visible horizon in the ratio 12 : 13.
4. Trace the changes in the apparent declination of a star due to
refraction in the course of a day, at a place in latitude 45 N"., the
actual declination being 50 N.
5. At Greenwich (latitude 51 28' 31" N.) the star o Oygni was
observed to transit 6 34' 57" south of the zenith. Find the
star's declination, employing the results of Question 2.
6. Prove that if the declination of a star observed off the meridian
is unaffected by refraction, the star culminates between the pole
and the zenith, and that the azimuth of the star from the north
is a maximum at the instant considered.
7. Show how the duration of twilight gives a measure of the
height of the atmosphere.
8. What is the lowest latitude in the arctic circle at which there
is no twilight at midwinter, and what is the corresponding distance
from the North Pole in miles ?
152 ASTRONOMt.
EXAMINATION PAPEB.-VL
1. What effect Las refraction on the apparent position of a star ?
Show that the greater the altitude of the star the less it is displaced
by refraction, and that a star in the zenith is not displaced at all.
2. Prove (stating what optical laws are assumed) that, if the
Earth and the layers of the atmosphere be supposed flat, the
amount of refraction depends solely on the temperature and pressure
at the Earth's surface.
3. Prove the formula for refraction, r = (/j. 1) tan z. Is this
formula universally applicable ? Give the reason for your answer.
4. Given that the optical coefficient of refraction of air (u)
= r0003, find the astronomical coefficient of refraction (U) in
seconds.
5. What is the refraction error? How may we approximately
determine the correction for refraction from observations made
on the transits of circumpolar stars ?
6. Show how the constant of refraction (on the usual assumption
that the refraction is proportional to the tangent of the zenith
distance) might be determined by observing the two meridian alti-
tudes of a circumpolar star whose declination is known.
7. Assuming the tangent formulas for refraction, find the latitude
of a place at which the upper and lower meridian altitudes of a cir-
cumpolar star were 30 and 60 ( ^ 3 = T732), the coefficient of
refraction being 57".
8. Why is the Moon seen throughout a total eclipse ?
9. In the Scientific American, June 18, 1887, it was stated by the
editor that " The atmosphere by its refraction acts as a lens, pro-
ducing an apparent increase in the diameter (of the Sun and Moon)
near the horizon. When we consider that the atmosphere, as seen
from the surface of the globe, is a section of a vast lens whose radius
is the semi-diameter of the Earth, it is reasonable to assume a small
increase in the size of the objects seen through it, and a still greater
increase when seen in the obliquity of the horizon." Why is the
above statement altogether incorrect?
10. Find the duration of twilight at the equator at an equinox.
CHAPTER VII.
THE DETERMINATION OF POSITION ON THE
EARTH.
SECTION I. Instruments used in Navigation.
195. Among the different uses to which Astronomy has
been put, perhaps the most important of all is its application
to finding the geographical latitude and longitude of any
place on the Earth from observations of celestial bodies. Such
observations may be made for either of the following purposes :
1 . The determination of the exact latitude and longitude
of an observatory. These must be known accurately before
the coordinates of a star can be found or observations taken
at different observatories can be compared.
2. The construction of maps. The geographical latitude
and longitude of a place form a system of coordinates which
enable us to represent its exact position on a map.
3. The determination of the exact position of a ship in
mid-ocean. This is the most useful application of all ; on a
long sea voyage it is necessary to calculate daily the ship's
latitude and longitude correct to within a mile or so.
Now, owing to the motion and rocking of a ship, all the
astronomical instruments hitherto described are useless at
sea. The mariner is therefore obliged to have recourse to
others which are unaffected by the unsteadiness of the vessel.
The two instruments best fulfilling this condition are the
Sextant and the Chronometer, which we shall now describe.
154
196. The Sextant. The use of the Sextant is to measure
the angular distance between two objects by observing them
both simultaneously. It consists of a brass framework form-
ing a sector CDE graduated along the circular arc or limb
DE\ the angle DCE is usually about 60 or rather more.
To the centre C of the arc is fixed an arm BI, capable of
turning about C, and which carries the small mirror B, called
the index glass. Another small mirror A, called the
horizon-glass, is fixed to the arm CD, making an angle of
about 60 with BD. Of this mirror half the back is usually
silvered, the other half being transparent. Finally, at T is
fixed a telescope, pointed towards A in such a manner as to
receive the rays of light from the mirror B after reflec-
tion at A (Pigs. 70, 71).
FIG. 70.
On looking through the telescope T we shall see two sets
of images, for objects at -ZTwill be seen directly through the
unsilvered part of the mirror A, while objects at S will be
seen after two reflections at the mirrors B and A. The
mirror is so near the object glass of the telescope as to ba
quite out of focus ; hence these two sets of images will not
appear separate, but will overlap one another.
THE DETERMINATION OF POSITION ON THE EARTH. 155
The arm BI carries at / an index mark or pointer by which.
its position can be read off on the graduated scale DE. The
pointer should read zero when the mirrors A, B are parallel
(as in the position B'E, Fig. 70). When this is the case, the
two images of any very distant object H will coincide. For
when a ray of light is reflected in succession at two parallel
mirrors, its final direction is parallel to its initial direction.*
Hence if H' CA T represents the path of a ray of light from
the object H, as reflected in succession at B' and A, the por-
tion AT is parallel to H'C, and therefore coincides with the
ray HAT, by which the object is seen directly.
Now let it be required to find the angular distance between
the two objects .ZTand 8. To do this, the mirror B is rotated
by means of the arm BI until the image of S (formed by
the two reflections) is seen to coincide with H. The angle
EC1, through which the mirror B has been turned from its
original position, is then half the required angular distance
between H, S.
For draw CN', CN perpendicular to the two positions
JB'j B of the mirror respectively. Since in reflection at a
plane mirror the angles of incidence and reflection are equal,
and .-. ACH'= 2
also L NCS = A CN and .-. L A CS 2 z A ON.
Hence /.ACS- ACH'= 2(^ACJV- LACN'\
i.e., LH'CS = 2. LN'CN
= 2 . L ECI-,
or the angular distance between the objects is double the
angle ECL
On the scale ED, every half -degree is marked as 1. The
reading of the pointer / will therefore give double the angle
ECI, and this is the angular distance required.
The coincidence of the two images in the field of view of
the sextant will not be affected by any small displacement of
the instrument in its own plane. This peculiarity renders
the sextant particularly useful on board ship, where it is
impossible to hold the instrument perfectly steady.
* See Stewart's Text-Book of Light, Chap. IV.
ASTRON. M
156 ASTRONOMY.
197. Shades, Clamp and Tangent Screw, Beading
Glass, Vernier.
For viewing the Sun, the sextant is provided with
shades. These consist simply of plates of glass blackened
for the purpose of reducing the great intensity of the Sun's
rays. There are two sets of shades, G, 6r, hinged to the
frame CE in such positions that one set can be inserted
between A and (7, to deaden the rays from S, while the other
set can be turned behind A to deaden the rays from H.
They are called respectively the " index shades" and
"horizon shades."
FIG. 71.
The arm or index bar B C is furnished with a clamp, by
means of which it can be clamped at any desired part of the
graduated limb DJE. When this has been done the arm can
be moved slowly by means of a tangent screw -ZT, and in
this way can be adjusted with great precision.
The arc D12is usually graduated to divisions of 10',* and
is used by means of the lens Jf, called the " reading glass."
But the index bar also carries a scale V called a Vernier
( 198) which, sliding beside the scale on the limb, enables
us to read off observations to within 10".
*0f course these divisions are only 5' apart, but in what follows
we shall speak of half-minutes as minutes.
THE DETEKMINATION OF POSITION ON THE EABTH. 157
198. The Vernier is a scale the distance between whose gradua-
tions is 10' 10", i.e., 9' 50", or 10" less than the distance between
the graduations on the limb. These graduations are marked 0",
10", 20", &c., being measured in the same direction as on the limb.
For example, let us suppose the zero point on the vernier is between
the marks 26 20' and 26 30' on the limb. We take the reading by
the limb as 26 20 7 . We then look along the vernier scale until we
find that one of the marks on it exactly coincides ivith one of the marks
on the limb. Suppose that this is the 25th graduation from tne
zero point of the vernier, i.e., the point marked 4' 10". We add
this 4' 10" to the 26 20' read on the limb, and the sum gives the
correct reading, namely, 26 24' 10".
The principle is as follows. Let us denote by P the mark which
coincides on the two scales.
Then from zero of vernier scale to P is 25 divisions of vernier,
i.e., an arc of 25 x (10' -10").
Also from 26 20' of scale on limb to P is 25 divisions of limb, i.e.>
an arc of 25 x 10'.
.*. from 26 20' on limb to of vernier, represents an arc of
25 x 10' -25 x (10' -10") ; i.e., 25 x 10", or 4' 10".
Hence the zero mark of the vernier scale is at a distance 26 20'
+ 4' 10" from the zero on the limb, and the reading is 26 24' 10". f
199. The Errors of the Sextant need not be described in detail.
If the sextant does not read zero when the two mirrors are parallel,
it is said to have an Index Error, and a constant correction for
index error must be added to all readings made with the instrument.
There are also errors due to eccentricity or want of coincidence
between the centre about which the index bar turns, and the
centre of the limb, errors of graduation, &c.
200. To determine the Index Error of the Sextant, In all goou
sextants the graduated limb is continued backwards for about 5
behind the zero point. This portion of the limb is called the "arc
of excess," and is used for finding the index error, as follows. The
Sun or full Moon is observed ; the two images of its disc are
brought into contact. Let e be the index-error, r the sextant reading,
D the angular diameter of the disc, then we have evidently D = r + e.
Now let the index bar be moved along the arc of excess until the
images again touch, the image which was before uppermost being
undermost. If the reading on the arc of excess be r', we have
now D = r' + e, or D = / e.
Hence, 2e = r'r.
f The simpler forms of mercurial barometer are provided with a
vernier by means of which the height of the mercury is read off to
the nearest hundredth of an inch. The student will find it of great
assistance to carefully examine the vernier in such an instrument
158 ASTEONOMY.
201. To take altitudes at Sea by the Sextant.
The principal use of the sextant is for finding altitudes.
Now the altitude of a star is its distance from the nearest
point of the celestial horizon. To find this, the sextant is so
adjusted that the reflected image of the star appears to lie on
the ofiing or visible horizon ; when the plane of the sextant
is slightly turned, the image of the star should just graze the
horizon without going below it. The sextant reading then
gives the star's angular distance from the nearest point of the
"offing." Subtract the dip of the horizon and the correc-
tion for refraction, both of which are given in books of
mathematical tables. The star's true altitude is thus
obtained.
202. To take the Altitude of the Sun or Moon.
In observing the Sun's altitude, the " index " shades must be
turned into position between the two mirrors, and the instru-
ment adjusted so that the Sun's lower limb appears just to
graze the horizon. The reading of the sextant, when
corrected for dip and refraction, gives the altitude of the
Sun's lower limb. Add the Sun's angular semi-diameter,
which is given in the Nautical Almanack ; the altitude of the
Sun's centre is then obtained.
Both the Sun's altitude and its angular diameter may be
obtained by observing the altitudes of the upper and lower
limbs. The difference of the two corrected readings gives the
Sun's angular diameter, and half the sum of the readings
gives the altitude of the Sun's centre.
If this method is used, allowance must be made for the
change in the Sun's altitude between the observations. For
this purpose, three observations must be made. First take
the altitude of the Sun's lower limb, then of the upper limb,
and lastly, again of the lower limb. Also note the time
of each observation. The difference between the first and
third readings determines the Sun's motion in altitude ; from
this, by a simple proportion, the change in altitude between
the first and second observations is found, and thus the alti-
tude of the lower limb at the second observation is known.
We can now find the Sun's angular diameter, and the altitude
of its centre at the second observation.
THE DETERMINATION OP POSITION ON THE EAETH. 159
Let , = time of 1st observation, when a = alt. of lower limb ;
3 = time of 2nd observation, when I = alt. of upper limb ;
# 3 = time of 3rd observation, when a' = alt. of lower limb ;
Then in time t s t v the alt. of lower limb increases a' a.
.'. in time 2 ^ it increases (a a) x f ~.
h~h
Hence if 2 denote the alt. of lower limb at second observation,
tn - t j 69 - TI
This finds # 2 , and we then have
Sun's angular diameter = l a. r
Alt. of Sun's centre at second observation =
In taking the altitude of the Moon, the altitude of the
illuminated limb must be observed, an'l the angular semi-
diameter, as given in the " Nautical Almanac," must be
added or subtracted, according as the lower or upper limb is
illuminated.
203. Artificial Horizon for Land Observations.
Owing to the absence of a well-defined offing on land, an
artificial horizon must be used. This is simply a shallow
dish of mercury, protected in some manner from the disturbing
effect of the wind. The sextant is used to observe the
angular distance between a star and its image as reflected in
the mercury. Half this angular distance is the star's apparent
altitude ; correcting this for refraction, the true altitude is
obtained (<?/. 65).
As the limb of the sextant is generally an arc of not more
than 70, the instrument will not measure angular distances
of more than 140, and it can, therefore, only be used with an
artificial horizon for altitudes of under 70. For greater
altitudes the zenith sector must be used.
At sea, where altitudes are measured from the offing, this
objection does not apply. On account of the motion of the
vessel an artificial horizon is useless; hence, no observations
can be taken when the offing is ill- defined, which fre-
quently happens, especially at night. The mariner is,
for this reason, chiefly dependent upon observations of the
Sun and Moon, and such stars of the first magnitude, or
planets, as are visible about dusk.
160 ASTRONOMY.
204. The Chronometer is the form of timepiece used on
board ship, and in all observations in which clocks are un-
available, owing to their want of portability. In principle,
the chronometer is simply a large and very accurately con-
structed watch ; its rate of motion being controlled, not by a
pendulum, but by a balance-wheel, which oscillates to and
fro under the influence of a steel hair-spring. In order that
the chronometer may go at a uniform rate, the balance-wheel
is constructed in such a manner that its time of oscillation is
unaffected by changes of temperature. If the wheel were
made of one continuous piece of metal, any increase of tem-
perature would cause the whole to expand, and the couple
exerted by the spring would not reverse its motion so readily,
so that the time of oscillation would be increased. To
FIG. 72.
obviate this, the rim of the wheel is made in several (generally
three) disconnected arcs, each being formed of steel within
and of brass without. When the temperature rises, the sup-
porting arms or spokes expand, pushing the arcs outward;
but in each arc the outer half of brass expands more than the
inner half of steel, and this causes it to curl inwards,
bringing the extremity actually nearer the centre than it was
before. The arcs carry small screw weights, and by adjusting
these nearer to or further from the supports, the compensa-
tion can be arranged with great accuracy.*
* The student who has read a little Rigid Dynamics will notice
that the compensation must be so arranged that the " moment of
inertia " of the balance-wheel is unaffected by the temperature.
ttE DETERMINATION OF POSITION ON THE EALITII. 161
Another peculiarity of the chronometer consists in the
"detached escapement." The action of the main spring,
while keeping up the oscillations, must not affect their
periodic time, and to secure this condition the escapement is
so arranged that the balance wheel is only acted on during a
very small portion of each oscillation.
The chronometer is usually suspended in a framework, in
such a manner that when the vessel rolls the instrument
always swings into a horizontal position ; the framework
also serves to protect it from violent shaking.
205. Error and Bate of the Chronometer. A chrono-
meter is constructed to keep Greenwich mean solar time. As
in the case of the astronomical clock, the amount that a chrono-
meter is slow when it indicates noon is called its error, and
the amount which it loses in 24 hours is called its rate. If
the chronometer is fast, the error is negative ; if it gains, the
rate is negative.
The essential qualification of a good chronometer is that
its rate must be quite uniform. It is not necessary that the
rate shall be zero, provided that its amount is known, since
a correction can easily be applied to obtain the correct
time from the chronometer reading. During sea voyages
extending over a large number of days, the correction for rate
may become considerable, and there is no very satisfactory
method of finding the chronometer error at sea ; for this
reason the instrument is rated, i.e., has its rate determined
by comparisons with a standard clock, whenever the ship is in
port. Moreover, many ships carry several chronometers, which
serve to check each other ; if the rate of one should vary slightly,
this change would be detected by comparison with the others.
Many of the best chronometers used in the !N~avy arid
elsewhere are tested at the Greenwich Observatory. They
are there kept in a special room, in which they can be
subjected to artificial variations of temperature, with a view
of ascertaining whether the compensation for temperature is
perfect or not. The chronometers are compared daily with
the standard clock. The process of rating is performed by
two assistants who have acquired the power of counting the
beats of the clock while reading off the errors of one chrono-
meter after another. In this manner, about a hundred
chronometers can be rated in half an hour.
1 62 ASTBONOMY.
SECTION II. Finding the Latitude by Observation.
206. The methods of finding latitude may be conveniently
classified as follows :
A. Meridian Observations.
(1) By a single meridian altitude of the Sun or a known
star.
(2) By meridian altitudes of two stars, one north and one
south of the zenith, taken with the sextant.
(3) By two observations of a circumpolar star.
B. Observations not made on the Meridian.
(" Ex-meridian Observations")
(4) By a single observed altitude, the local time being known.
(4A) By " circum-meridian altitudes."
(4s) By observing the altitude of the Pole Star.
(5) By observations of two altitudes.
(6) By the Prime Vertical instrument.
We now proceed to examine the various methods in detail,
but it must be premised that the " ex-meridian " methods
cannot be thoroughly explained without spherical trigo-
nometry.
207. Latitude by a Single Meridian Altitude. Let
S (Pig. 73) represent the position of the Sun or a star of
known declination when southing.
Let the meridian altitude sS be observed, and let it be = a\
also let % be the meridian Z.D. ZS, so that z = 90- a. Let
tfbe the known K". decl. QS, and / the required N. latitude QZ.
THE DETEKMINATION OF POSITION ON THE EARTH. 166
EXAMPLE.
On April 11, 1891, in longitude 80 12' E. (roughly) with an
artificial horizon, the meridian reading of the sextant for the Sun's
lower limb was observed to be 107 59' 48". Barometer 307 inches,
thermometer 72. Find the latitude, having given the following
data :
Q i II
's (Sun's) decl. at Greenwich noon, Ap. 11 =8 19 4 "I From
Hourly variation of decl = 55'1 \ Nautical
O's semi-diameter = 15 59 J Almanack.
Mean refraction at altitude 54 = 41 \ from
Correction for barometer = +1 I m ^
for thermometer = 2 J
The calculation is best arranged as follows :
O I II
(i.) Double observed alt. of lower limb = 107 59 48
.'. observed alt = 535954
Corrected refraction at this alt.
(which is nearly 54) = 40 (-)
.'. true alt. of lower limb = 53 59 14
Aug. semi- diam = 1559( + )
Merid. alt. O's centre =, 54 15 13
Subtract from 90
Mcrid. ZJD. of Q's centre = 35 44 47 S (i.)
M. 8.
(ii.) Long. 8 12' E. in time ... = 32 48
.'. tune of observation = 32 48 before Greenwich noon.
O / II
's decl. at Greenwich noon April 11 = 8 19 4 N. (increasing).
Variation in 30m. before noon ... = 27 ( )
2m. 48s. (about) ... = 3(-)
.'. 's decl. at time of observation ... = 8 18 34 N.
Add O's merid. Z.D. from (i.) = 35 44 47 S.
Required north latitude = 44 3' 21".
166 ASTKONOMT.
211. To find the latitude by sextant observations
of the meridian altitudes of two stars which culmi-
nate on opposite sides of the zenith. This is really only
a modification of the first method. Two stars of known
declination are selected which culminate, one south and the
other north of the zenith, at very nearly the same altitude.
The latitude is calculated independently from ohservations of
the meridian altitudes of either star, and the mean of the two
results is taken as the correct latitude.
This method possesses the following advantages :
1st. There is no need to correct the observed altitudes for
dip of the horizon ;
2nd. The result is unaffected by any constant instrumental
errors (index error, &c.) which affect both altitudes equally;
3rd. The correction for refraction is reduced to a minimum,
or even entirely eliminated, if the altitudes are almost equal.
.For let rfj, d. 2 be the north declinations of the two stars ;
Zj (south) and z 2 (north) their true meridian Z.D.'s ;
#! and # 2 their observed meridian altitudes ;
t and u. 2 the corrections for refraction;
D the dip of the horizon ;
e the correction for constant instrumental errors.
For true meridian altitudes of the two stars we have
90 z l a^e D-u v 90 s 2 = a ^eDu y
The two observations give, therefore, for the latitude (by 204)
I ^+ 2l = ^ + 90- 0! -<? + ./) + !,
1= d 2 -% = d 3 9Q+a z + eD-u 2 .
Therefore, taking the mean of the two results,
a result involving no corrections beyond the difference of
refractions, w 2 u r
Moreover, if the altitudes a^ and # 2 are greater than 45,
and their difference (0 a 0,) is less than a degree, then
\ ( 2 - MJ) is < 1", and therefore the refraction correction
may be entirely neglected.
THE DETERMINATION OP POSITION ON THE EARTH. 167
212. Latitude by Circumpolars. This method has
already been mentioned in 28, but we will here repeat
the investigation for convenience.
Let x, x 1 (Fig. 74) represent the positions of a circumpolai
star at its upper and lower transits. Let its meridian
altitudes nx and nx 1 be observed, and let their corrected
values be a^ and tf a respectively. Since
Px = star's N.P.D. = P^
or
In this formula no knowledge of the star's declination is
required, but the observed altitudes require to be corrected
for refraction, dip, &c.
The circumpolar method is most useful in determining the
latitude of a fixed observatory, because this must be done
before the declination of any star can be determined. The
transit circle is used to determine the meridian altitudes at
the two culminations.
By observing two or more circumpolars the correction for
refraction may be found, as in 186, and the observed alti-
tudes may then be corrected for refraction.
168 ASTRONOMY.
As the declinations of a large number of stars are given
in astronomical tables, the circumpolar method is never
used at sea. It would possess no advantage, and would have
the disadvantage of requiring a correction for the change in
the ship's place between the two culminations.
EXAMPLES.
1. The observed meridian altitude of /3 Ceti (decl. 18 36' 44'5" S.)
is 36 43' 12", and that of a Ursse Minoris (decl. 88 41' 53'1" N.) at its
upper culmination is 30 9' 57", both altitudes being measured from
the " offing," and the dip being unknown. Find the latitude, given
Refraction at alt. 36 = 1' 20" ; at alt. 37 = 1' 17".
This is an example of the method of 211. The calculation stands
thus :
Ceti (south).
36 43' 12"
- 1 18
a
Observed altitudes
Refraction corrections
Corrected Altitudes
Zenith Distances
Declinations
Ursse Minoris (north).
36 9' 57"
- 1 19-5
36 41 54
90
36 8 37-5
90
+ 53 18 6 S.
-18 36 44-5 S.
-53 51 22-5 N.
+ 88 41 53-1 N.
34 41 21-5 N. Calculated Latitudes 34 50 30 6 N.
Thus, lat. by star north of zenith = 34 50 7 30'6" N.
south = 34 41 21-5 N.
2)69 31 52-1
Mean latitude = 34 45' 56" N?
Here, owing to dip, one of the calculated latitudes is 4' 34'6" too
great, and the other is 4' 34'5" too small, but the mean of the two
results is the correct latitude.
2. The observed altitudes of /3 Ursa? Minoris at lower and upper
culmination are 29 58' 16" and 60 45' 3". Find approximately the
latitude, assuming the coefficient of refraction to be 57".
By the " tangent formula," refraction at altitude 30 (approx.)
= 57" tan 60 = 57" x A/3 = 57" x 1-732 = 1' 39".
Refraction at alt. 60 = 57" tan 30 = 57" x A/3/3 = 1' 39"-=-3 - 33".
Hence truealt. at lower culmination = 29 58' 15" 1' 39" = 29 56' 36"
upper =60 45 3 - 33" =60 44 30
2) 90 41 6
.-. Required North Latitude= 45 20' 33"
THE DETERMINATION OF POSITION ON THE EARTH. 1 GO
LATITUDE BY EX-MERIDIAN OBSERVATIONS.
213. To find the latitude "by a single altitude, the
local time being known. If the local time be known, a
single altitude of the Sun or a known star is sufficient to
determine the latitude.
For let S be the observed body, ^the zenith, P the pole.f
Then in the spherical triangle PZS, the known local time
enables us to find the hour angle ZPS. For, if the Sun be
observed, its hour angle ZPS
= 15 x (apparent local time)
= 15 x (mean local time equation of time) ;
and if a star be observed, its hour angle ZPS
= 15 x (local sidereal time star's R. A.).
Also ZS = observed body's Z.D. = 90 (observed altitude) ;
PS= ,, N.P.D. = 90 -(known dccl.).
Hence, ZS, PS, and the angle ZPS are known. These
data completely fix the spherical triangle ZPS, and from
them ZP can be found by Spherical Trigonometry.
Hence the latitude is found, being = 90 ZP.
*214. By Circum-meridian Altitudes. This is a par-
ticular case of the method last described. In attempting to
find the latitude by meridian observations, it may happen
that passing clouds prevent the body from being observed at
the instant of transit. In this case the latitude can be found
from the observed altitude when very near the meridian.
The hour angle ZPS is then small, and the difference
between the observed and meridian altitudes is also
small. This difference is called the " Itoduction," and is
found by approximate methods.
The best results are obtained by taking a number of alti-
tudes of the body before and after passing the meridian.
*215. By a Single Altitude of the Pole Star. The
N.P.D. of Polaris is only about 1 16'. Hence, if its alti-
tude is observed at any time, the latitude may be found by
adding to, or subtracting from, this altitude, a small correc-
tion, never greater than about 1 16J'.
iThe student will have no difficulty in illustrating 213-216
with diagrams. For 213, Fig 75 may be copied.
1 70 ASTRONOMY.
This correction consists of three parts, which are given by
three tables in the Nautical Almanack. The first two cor-
rections depend on the sidereal time, and on the observed
altitude ; the third is due to variations in the R.A. and
N.P.D. of Polaris, due to precession ( 141), etc.
*216. Latitude by observation of Two Altitudes. By observing
the altitudes of two known stars, both the latitude and the local
sidereal time can be found.
The same method can be employed to determine the latitude by
two observations of the Sun's altitude, separated by a known interval
of time.
The necessary calculations are very complicated, involving
Spherical Trigonometry, and they cannot be materially simplified
even by the use of tables.
A very useful geometrical construction, enabling us, from the two
observed altitudes, to indicate the exact position of a ship on a
globe without calculation, will be detailed in Section VI. of this
chapter.
217. Latitude by the Prime Vertical Instrument.
The latitude of a fixed observatory may be found by means
of an instrument similar to the Transit Circle, but whose
telescope turns in the plane of the prime vertical instead of
the meridian. A star will cross the middle wire of such an
instrument when its direction is either due east or west ;
the times of the two transits are observed. Let S, S' be the
positions of a known star at its eastern and western transits,
Z the zenith. P the pole. The sidereal interval between the
two transits determines the angle SPS', and this is evidently
twice the angle ZPS. Hence z ZPS is known. Also PS, the
star's N.P.D., is known, and PZS is a right angle. Therefore,
the spherical triangle ZPS is completely determined, and the
colatitude ZP can be found.
The times of the transits are unaffected by refraction, and
this fact constitutes the principal advantage of the method.
The observations may be performed by an altazimuth, whose
horizontal circle is clamped so that the telescope moves in
the prime vertical. The instrument must be so adjusted that
the interval of time between the first transit and culmination
is equal to the interval between culmination and the second
transit. The culmination must be observed with a Transit
Circle.
THE DETERMINATION OF POSITION ON THE EARTH. 171
SECTION III. To find the Local Time by Observation.
218. In determining the longitude of a place on the Earth,
the first step is to find the local time by observations of the
hour angle of a known celestial body. If the time indicated
by a chronometer or clock at the instant of observation be
also noted, we shall find the difference between the true local
time and the indicated time. This difference is the error of
the clock on local time.
In 167 we described one instrument for observing local
time the Sun-dial. This cannot, however, be used except
for very rough observations, as the boundary of the shadow
cast by the style is not sufficiently well defined to admit of accu-
rate measurements. Moreover the Sun-dial is not portable.
For this reason the local time is usually found by one or
other of the following methods :
1st. By meridian observations.
2nd. By equal altitudes.
3rd. By a single altitude, the latitude being known.
4th. By observation of two altitudes.
219. Local Time by Meridian Observations. In a
fixed observatory, the local sidereal time is found by means of
the Transit Circle, as explained in 24, 54. The transit of
a known star is observed ; the local sidereal time of transit is
equal to the star's E.A., and is therefore known.
Or by observing the transit of the Sun's centre, the time
of apparent local noon may be found. The equation of time
is the mean time of apparent noon, and is given in the
" Nautical Almanack " ; ht-ncc the local mean time is found.
These methods are not available at sea, as the Transit
Circle cannot be used. It might be thought that we could
use a soxtant to ascertain the instant when the body's altitude
is greatest, bat, for a short interval before and after the transit,
the altitude remains very nearly constant ; it is therefore
impossible to tell with any degree of accuracy when it is
a maximum.
On the other hand, a slight error in the time of observation
does not affect the altitude perceptibly, so that the meridian
altitude may be observed with great accuracy, as in 208.
ASTRON. N
172
ASTRONOMY.
220. Method of Equal Altitudes. "When it is required
to find the local time from observations taken with a sextant,
the simplest method is as follows : Observe the altitude of
any celestial body some time before it culminates. After the
body has passed the meridian, observe the instant of time
when its altitude is again the same as it was at the first
observation. Half the sum of the times of the two observa-
tions gives the time of transit.
FIG. 75.
For let S, S' be the two observed positions of the body, Z
the zenith, and P the pole.
The altitudes of SX, S'X' being equal, the zenith distances
are equal ;
.-. ZS = ZS'.
Also PS = PS',
and the spherical triangles ZPS, ZPS' have ZP in common.
.-. tSPZ= Z ZPS'.
Now let ^ and 2 be the times of the two observations,
t the time of transit.
Then tt^ is the time taken to describe the angle SPZ-,
a t ,, ,, ,, ,, ZPS'.
Since the two angles are equal,
.-. t-t^ t^-t-,
.-. * = }(, +3).
From the time of transit the local time can be found, as in
the last article.
THE DETERMINATION OF POSITION ON THE EARTH. 173
221. In observing the Equal Altitudes with a
Sextant, the following method is used : At the first ob-
servation clamp the index bar at an altitude slightly greater
than that of the body. Continue to observe the body as it
rises, till its image is in contact with the horizon, and note
the instant of time (^) at which this happens. Keep the
index bar clamped until the second observation ; commence
observing the body again just before it has reached the same
altitude again, and note the instant of time ( 2 ) when its
image is again in contact with the horizon. The two observed
times (t v tf s ) are the times of equal altitude.
If an artificial horizon le used, we must observe the two
instants of time (t v 2 ) when the two images are in contact.
222. Equation of Equal Altitudes. If the Sun be the
observed body, its declination will, in general, change
slightly between the two observations ; hence PS will not be
exactly equal to PS', and the angles SPZ, ZPS' will not be
quite equal. For this reason a small correction must be
applied, in order to allow for the effect of the change of
declination. This correction is called the Equation of
Equal Altitudes, and may be found from tables which
have been calculated for the purpose.
At Sea allowance must also be made for the change of
position of the ship between the two observations, and thk
correction is also effected by means of tables.
223. The method of Equal Altitudes possesses the
following advantages :
1st. The results are unaffected by errors of graduation of
the sextant, for the actual readings are not required.
2nd. The semi-diameter of the observed body need not be
known.
3rd. The observed altitudes, being equal, are equally affected
by refraction, and no refraction correction need therefore be
made.
4th. The dip of the horizon need not bo known, provided
that it is the same at both observations.
1 74 ASTBONOMY.
224. With, a Gnomon, the time of apparent noon can be
roughly found in a very simple manner. A rod is fixed
vertically in a horizontal plane, and on the latter are
drawn several circles, concentric with the base of the rod.
Let the times be observed, before and after noon, when
the extremity of the shadow cast by the rod just touches
one of these circles. At these two instants the Sun's alti-
tudes are, of course, equal, and therefore the time of apparent
noon is the arithmetical mean between the observed times.
EXAMPLE. The shadow of a vertical stick at Land's End (long.
5 40' W.) is observed to have the same length at 9h. 27m. A.M. and
3h. 1m. 40s. P.M., Greenwich time. Find the equation of time on
the day of observation.
Greenwich mean time of local apparent noon is
i { 9h. 27m. Os. + 3h. 1m. 40s. 12h. } = 14m. 20s.
But, by 96, Greenwich mean time of local mean noon = 22m. 40s.
.'. Eqn. of time = local mean time of apparent noon = 8m. 20s.
*225. The Latitude may also be found by the method of
equal altitudes, though the calculations require Spherical
Trigonometry. For this purpose, the altitude at either
observation must be read off' on the sextant, and corrected for
refraction, dip, &c. The zenith distance SZ is therefore
known. The angle SPZis also known, being half the angle
described in the interval t^ t lt and PS, being the comple-
ment of the declination, is also known. The spherical triangle
ZP8 is therefore completely determined, and ZP, which is
the complement of the latitude, can be found.
226. Local Time by a Single Altitude, the Latitude
being known. This is the converse of the method for
finding the latitude described in 213. If the altitude of a
known body, S, be observed in known latitude, we know
ZS, SP, PZ, which are the complements of the observed
altitude, the declination, and the latitude respectively ; hence
the hour angle SPZ, and therefore also the local time, may
be found.
*227. Local Time by Two Altitudes. The method of 216
determines, not only the latitude, but also the hour angles of the
bodies at the two observations, and these determine the local time,
The method of equal altitudes is in reality only a particular case
THE DETERMINATION OF POSITION ON THE EARTH.
175
SECTION IY. Determination of tTie Meridian Line.
228. Before setting up a transit circle or equatorial in a
fixed observatory, it is necessary to know with considerable
accuracy the direction of the meridian line, i.e., the line
joining the north and south points of the horizon. At sea,
the directions of the cardinal points are determined by a
mariner's compass ; but here, too, it is of great use, on long
voyages, to determine the variation of the compass, or
the deviation of the magnetic needle from the meridian line.
This deviation is different at different parts of the Earth.
There are three ways of finding the meridian line : first,
by two observations of a celestial body at equal alti-
tudes ; second, by a single observation of the azimuth ; third,
by one or more observations of the Pole Star.
229. By Equal Altitudes. When a body has equal
altitudes before and after culmination, the corresponding
azimuths are equal and oppo-
site.
For if 8, S' denote the two
positions of tho body, the tri-
angles ZPS, ZPS' are equal in
all respects ;
.-. Z PZS = Z PZS' and
/. Z sZS = z sZS'.
230. At Sea, the Sun's azi-
muth, or compass bearing, may
be observed when rising and
when setting; the meridian FlG -
line bisects the angle between the two directions ( 29).
231. On Land, we may observe tho directions of the
shadow cast by a vertical rod on a horizontal plane when it
has equal lengths ; for this purpose we mark the points at
which the end of the shadow just touches a circle concentric
with the base of the rod (of. 224). Bisecting the angle
between the two directions, the north and south points are found.
If greater accuracy is required, an altazimuth may be used.
The readings of the horizontal circle arc taken when tho
altitudes of a star are equal ; the meridian reading is the
176 VSTBONOMT.
arithmetical mean of the two readings. While observing th
equal altitudes, the vertical circle must be kept clamped.
*232. By a Single -Observation. If the direction o
the vertical plane through a single celestial body S b
observed at any instant, the direction of the meridian lin
may be found by means of Spherical Trigonometry.
For if any three parts of the triangle ZPS are known, th
triangle is completely determined, and the angle PZS can b
found.
The azimuth sZS = 180 PZS, and is then known
hence the meridian line ZS is found.
Now the sides PS, ZS, ZP are the complements of th
declination, the altitude, and the latitude ; and the hour angl
ZPS is known, if the local time be known. Any three o
these data are sufficient to determine the angle PZS.
Thus, for example, the Sun's direction, either at sunrise fl-
at sunset, determines the meridian line, if either the loca
time or the latitude is known.
233. By Observations of the Pole Star. The direc
tion of the meridian may be very accurately determined b
observations of the star Polaris. If the azimuthal readings o
this star be observed at the two instants when it i
furthest from the meridian, east and west, respectively
the reading for the meridian is half their sum. Th
observations maybe made with an altazimuth. The azimut
at either observation is a maximum, and it remains ver
nearly constant for a short interval before and after attainin
its maximum. Hence, a slight error in the time of observe
tion will not perceptibly affect the azimuth. The sam
method is applicable to any star which culminates betwee
the pole and the zenith.
The most accurate method is, however, that employed i
finding the deviation error of the Transit Circle ( 59
If the telescope always moves in the plane of the meridiai
the interval from upper to lower culmination, and tb
interval from lower to upper culmination, will both I
exactly twelve sidereal hours. If not, the small amount b
which the vertical plane swept out by the telescope is east c
west of the meridian, can be found by observing the amouni
by which the two intervals are greater and less than 12h.
THE DETERMINATION OF POSITION ON THE EARTH. 177
SECTION Y. Longitude ly Observation.
234. In Section III. of the present chapter we showed
how the local time can he found hy observing \he celestial
bodies. "When this has been done, the longitude of the place
of observation may be fonnd by comparing the observed local
time with the corresponding Greenwich time.
For in 96 we showed that if the longitude of a place west
of Greenwich be Z, then
(Greenwich time) (local time) = -^L h. = 4Z m. ;
whence, knowing the difference of the two times, L may be
found.
The methods of finding Greenwich mean time, and henco
longitude, may be classified as follows :
A. Methods available at Sea.
(1) By the chronometer.
(2) By the method of lunar distances.
(3) By celestial signals.
B. Methods suitable for Land Observations.
(4) By repeated transmission of chronometers.
(5) By the chronograph.
(6) By terrestrial signals.
(7) By Moon culminating stars or by the Moon's meridian
altitude.
235. Longitude by the Chronometer. By reading
the chronometer used on board ship, and making the necessary
corrections for error and rate, the Greenwich mean time at
any instant may be found. If, then, the local mean time is
determined by observing the Sun, or one of the other
celestial bodies, and the observations are timed by the chrono-
meter, the difference between the local and Greenwich mean
times will be found, and this determines the ship's longitude
measured from Greenwich.
EXAMPLE 1. At apparent noon a chronometer indicates
19h. 33m. 25.s., Greenwich mean time, and the equation of time is
- 2m. Is. To find the longitude.
Here the local mean time is 2m. Is.
.'. Greenwich mean time local mean time ... = 191i. 35m. 26s.
Mult, by 15, we have long. W. of Greenwich ... = 293 51' 30"
or sub. from 360, long. E. of Greenwich =66 8' 30"
178 ASTROtfOMT.
EXAMPLE 2. Find the longitude, from the following data :
Sun's computed hour angle = 75E. Time by chronometer = 23h.7m.31s.
Equation of time = + 3m. 55s. Correction for error and rate, Im.lSs.
(i.) Here 0's hour angle in time = 5h. before noon
/. apparent local time = 19h.0m. Os.
Equation of time = 3 55
/. mean local time =19h.3m.55s.
(ii.) Observed time = 23h. 7m. 31s.
Correction = 1 18
Greenwich time = 23 6 13
19 3 55
W. Long, in time =42 18
15
.-. required long ... = 60 34' 30" W.
EXAMPLE 3. On June 29, from a ship in the North Atlantic
Ocean, the Sun was observed to have equal altitudes when the
chronometer indicated llh.27m. 26s. and 6h. 48m. 32s. At noon on
June 25, the chronometer was 3s. too fast, and it gains 8s. a day.
Tho equation of time on June 29 at 3 p.m. was -f 2m. 58s. To
find the ship's longitude.
The process stands as follows : H. M. s.
Chronometer time of first observation =11 27 26
second observation + 12h. ., =18 48 32
2)30 15 58
15 7 59
Hence the chronometer time of local apparent noon =3 7 59
Correction forchronometer error June 25 = 3s. \
rate in 4 days^= 32s. j- = 36
,,3 hours =- Is.J
/. Greenwich time of local apparent noon =3 7 23
Subtract equation of time (since mean noon occurs
first) = -2 58
/. Greenwich time of local mean noon =3 4 25
15
/. longitude west of Greenwich ... ... ... ... = 46 6'
THE DETERMINATION OF POSITION ON THE EARTH. 179
236. Method of Lunar Distances. If from any cause
the ship's chronometer should stop, or its indications should
become unreliable, the Greenwich time may be found by
observations of lunar distances. In this method the Moon,
by its rapid motion among the stars, takes the place of a
chronometer, its position relative to the neighbouring stars
determining the Greenwich time. The Moon moves through
360 in 27 days ; hence it travels at the relative rate of about
33' per hour, or rather over 1" in every 2s., and this motion
is sufficiently rapid to render it available as a timekeeper.
For this purpose, tables of lunar distances are given in the
Nautical Almanack. These tables give the angular distances
of the Moon's centre from the Sun or from such bright stars or
planets as are in its neighbourhood, calculated for every third
hour of Greenwich mean time, and for every day of the year.
The angular distance of the Moon's bright limb from one
of the given stars may be observed by means of a sextant.
By adding or subtracting the Moon's semi-diameter, as given
in the Nautical Almanack, and correcting as explained below,
the angular distance of its centre may be found. During the
interval of three hours between the times given in the
Nautical Almanack, the angular distance changes at an
approximately uniform rate, and therefore the Greenwich
time of the observation may be computed by proportional parts.
237. Clearing the Distance. One of the great draw-
backs of the lunar method consists in the laborious calculations
necessary for what is called "clearing the distance." The
angular distance between the Moon and the star will be
affected by refraction, and this alone requires a correction to
be applied to the observed lunar distance ; but there is another
correction, for what is called parallax, which is equally
important. This latter correction depends on the fact
that the Moon's distance from the Earth is only about 60
times the Earth's radius, and at this comparatively small
distance the direction of the Moon cannot be considered as
independent of the observer's position on the Earth, as has
been done with the fixed stars* (5).
* Indeed, if a star happens to be behind the Moon's disc, it may
sometimes appear on opposite sides of the Moon to two observers at
nearly opposite points on the Earth.
180 ASTRONOMY.
For this reason, the lunar distances of a star, as tabulated
in the Nautical Almanack, are the angles which the Moon
and star subtend at the centre of the Earth. They are,
therefore, sometimes called the geocentric lunar distances.
Hence it is necessary to calculate the Moon's geocentric
position from that observed, before the Greenwich time of the
observation can be determined.
The correction for parallax, will be dealt with more fully
in the next chapter. Suffice it to mention here that the
parallax, like the refraction correction, depends only on
the Moon's zenith distance, and therefore, the only data
needed for clearing the distance are the altitudes of the two
bodies at the time of observation. The calculations are then
greatly simplified by the use of tables.
238. Advantages and Disadvantages of the Lunar
Method. The method of lunar distances was introduced at
a time when chronometers were very imperfectly constructed,
and could not be relied on during a moderate voyage. At the
present time, owing to the high degree of accuracy attained
in the construction of chronometers, combined with the
reduction in the length of sea voyages since the introduction
of steam, the lunar method has been almost entirely super-
seded by the use of chronometers. It is still used, however,
for the occasional corre-ction of a chronometer if the voyage
be extremely long ; and explorers rely upon it mainly.
The principal disadvantages of using lunar distances are :
1st. The calculations necessary for clearing the distance
are very tedious, and not such as could be performed readily
by a seaman possessing little or no knowledge of mathematics.
Moreover, the corrections are often considerable.
2nd. A slight error in the observed lunar distance would
introduce a considerable error in the estimated longitude.
The best sextants are only divided to every 10", and an error
of 10" in the observed lunar distance would introduce an error
of 20s. in the computed Greenwich time. This would give,
in the longitude, an error of 5', or of 5 geographical miles at
the equator. Even this degree of accuracy would be difficult
to attain in practice, while the rate of a well-constructed
chronometer can be depended upon to within Is. per day.
THE DETERMINATION OP POSITION ON THE EARTH. 181
EXAMPLE. On 'Nov. 14, the cleared angular distance of the
Moon's centre from Aldebaran was found to be 32 44' 52". Find
the Greenwich time, having given the following data :
ANGULAR DISTANCE OF THE MOON FROM Aldebaran.
Date.
Position of Star.
6 P.M.
9 P.M.
Midnight.
Nov. 14.
East.
33 32' 57'
31 44' 14"
29 55' 32"
The calculation stands as follows :
Ang. dist. at 6 P.M. =33 32' 57"
at observation = 32 44 52
Decrease since 6 P.M. = 48 5
Ang. dist. at 6 P.M. = 33 32' 57"
at 9 P.M. = 31 44 14
Decrease in 3 h. = 1 48 43
/. In 3h. the Moon's angular distance from Aldebaran decreases
148'43", or 6523";
.*. the time in which it decreases 48' 5", or 2885", ia
- 3h. x _ lh. 19m. 37s.
. Greenwich time of observation = 6h. + lh. 19m. 37s.
= 7h. 19m. 37s.
239. Longitude by Celestial Signals. The eclipses of
Jupiter's satellites begin and terminate at times which can
be calculated beforehand ; it would, therefore, appear
possible to ascertain the Greenwich time by observing the
instants at which a satellite disappears into, or emerges
from, the shadow cast by the planet. .But, as the dis-
appearance and emergence take place gradually, it is im-
possible to employ this method with accuracy to the
determination of longitude. The same objection applies still
more forcibly in the case of eclipses of the Moon.
By observing the occultations of stars behind the disc of
the Moon, we have another way of determining the Greenwich
time and finding the longitude. This is merely a particular
case of the method of lunar distances, since at the instant of
disappearance, the star's apparent (unconnected) distance from
the Moon's centre is equal to the Moon's semi-diameter.
182 ASTRONOMY.
METHODS OF FINDING LONGITUDE ON LAND.
240. Longitude by repeated transmission of Chro-
nometers. The chronometer method of comparing longitudes
can be employed with far greater accuracy on land, on account
of the possibility of taking repeated journeys to and fro
in order to effect the comparison of the local times. The
rate of the chronometer is determined by observing its
error at the first station, both before and after taking it
to the second.
Suppose, for example, that it is required to find the differ-
ence of longitude between two stations, A and B. A chrono-
meter is compared with the standard clock at A, and its
error is noted. It is then carried to JB, and its indications are
compared with those of a clock regulated to keep local time.
It is then again brought back to A, and compared a
second time with the standard clock. The increase in the
chronometer error during the whole interval serves to
determine the rate of the chronometer. Wo can now
correct for error and rate the time indicated by the
chronometer at A, and thus determine the difference
between the local times at A and . By converting
this difference into angular measure at the rate of 15
to the hour, the required difference of longitude of the two
stations is determined.
It is probable that the rate of the chronometer may not be
the same while it is being shaken about on its journey
as while it is at rest. This difference of rate may be
allowed for by comparing the chronometer with the local
clock soon after arrival, and again before departing. The
total loss while at rest is thus found, and by subtract-
ing we have the total loss during the two journeys. The
only assumption which it is necessary to make is that
the rate is the same on the outward journey as on the
return journey.
In order to obtain a result as free from error as possible,
a number of journeys to and fro are performed, and several
chronometers are used on each journey. The most accurate
result is found by taking the mean of the calculated values
for the difference of longitude.
THE DETEllMINATIOff OF POSITION OK THE EAUTH. 183
EXAMPLES.
At I7h. by a chronometer, the Greenwich mean time was found
fco be 16h. 59m. 57'2s. It was taken to a place A, and indicated 4h.,
when the local mean time was 3h. 47m. 46'9s. ; and when it indicated
llh., the Greenwich time was llh. Om. 9'7s. To find the longitude
of A in time and in angle.
Here, at I7h., the chronometer error by Greenwich time was 2'8s.
24 + llh. +9'7s
.'. in 18h. the chronometer lost 12'5s. ;
/. the loss in llh. = x 12'5s. = 7'64s. nearly ;
18
.'. the Greenwich time, when the chronometer indicated 4h., was
= 4h.-2'8s. + 7'64s. = 4h. Om. 4'84s.,
and the local time at the same instant was = 3h. 47m. 46'9s.
/. required longitude = 12m. 17'9s. W. = 3 4' 28" W.
2. As a ship starts from Liverpool, its chronometer indicates Oh.,
and is correct by Greenwich mean time. After 16 days, as it reaches
Quebec, the chronometer indicates 7h. Om. 23s., and Quebec time is
2h. 5m. 42s. Nearly seven days afterwards, the ship departs at
Quebec noon, the chronometer then reading 4h. 54m. 39s. ; and when
it reaches Liverpool, after a voyage of just over fourteen days, it is
found to be 17s. slow by Greenwich mean time. Find the longitude
of Quebec.
By Quebec time, the ship stayed in port 7d. 2h. 5m. 42s.
= 6d. 21h. 54m. 18s.
By chronometer, the ship stayed in port 7d.4h.54in.39s. 7h.0m.23s.
= 6d. 21h. 54m. 16s.
.-. in 7 days in port, chronometer lost 2s.
But in 37 days altogether, 17s.
/. in 30 days at sea, 15s.
/. in 16 days, from Liverpool to Quebec, it lost 8s.
But chronometer time on arrival was 7h. Om. 23s.
.'. Greenwich time was 7h. Om. 31s.
And local time was 2h. 5m. 42s.
The difference = longitude of Quebec (in time) = 4h. 54m. 49s.
/. Longitude of Quebec (in angle) = 73 42' 15" W.
184 ASTRONOMY.
241. Longitude by the Chronograph. When two
observatories are in telegraphic communication, the local
time maybe readily signalled from one to the other by means
of the electric current, and the difference between the longi-
tudes thus determined.
This method is employed in connection with the chrono-
graphic method of recording transits, the chronographs being
connected by the telegraph line, so that a transit is recorded
nearly simultaneously at both stations.
Let us call the two stations A and B. When the star
crosses the meridian at A, the observer presses the button of
his chronograph. Let t% be the times of transit at A as
thus recorded at A and B respectively. When the same star
crosses the meridian at B, the times of transit are again
recorded at A and B. Let these recorded times be T^ and
T 2 respectively.
The transmission of the signal from one station to the
other is not quite instantaneous, because a small interval of
time must always elapse before the current has attained
sufficient strength to make the signal at the distant station.
Let this interval be x. Then the transit at A will be recorded
too late at B by the amount x, and the transit at B will be
recorded too late at A by the same amount x.
When this correction is applied, the true times of the two
transits, as determined by the chronograph record at A, will
be j and T^x. Hence, if L denote the difference of longi-
tude in time measured westwards from A to , the chrono-
graph record at A gives
L T^-x-t r
Again, the true times of the two transits, as determined by
the chronograph record at B, will be 2 x and T y Hence
the chronograph record at B gives
z = r t -(v-*)==zi
By addition, we have
2L=T l -t l + T,-t s ; .-. L = \(T^t^T,-t,\
a result which does not involve x.
Thus we see that, by using both chronograph records, and
taking the mean of the separately calculated differences of
longitude, the corrections due to the time occupied by the
passage of the signals are entirely eliminated.
THE DETEKMINATION OF POSITION ON THE EAETH. 185
*242. Elimination of Personal Equation. In the
above investigation we have taken no account of the personal
equations of the two observers. But if e is the correction for
personal equation of the observer at A, and E is that of the
observer at B, the observed times t lt 2 must both be increased
by e, and 2^, jF 2 must both be increased by E. Introducing
these corrections, the formula gives
To eliminate the corrections, let the two observers change
places, and repeat the operations, and let the new recorded
times of transit be denoted by accented letters. The cor-
rection E must now be applied to the times /, 2 ', and the
correction e must be applied to T^ and T 2 '. Therefore
By again taking the mean of the two results we get
a result in which personal equation is eliminated.
243. Longitude "by Terrestrial Signals. Before the
introduction of the electric telegraph and the chronometer,
other signals had to be used. Among such signals may
be mentioned flashes of light and rockets visible simul-
taneously from two stations at a considerable distance apart.
The heliograph, in which signals are transmitted by flashes
of reflected sunlight, forms another means of determining
differences of longitude between two stations visible one from
the other ; and this method is still often found very useful
in surveying a country. A flash of lightning and the bursting
of a meteor have also occasionally been used, but they are
far too uncertain in their occurrence to be of much value.
The local time of the signal is noted at each place, and the
difference of these times gives the difference of longitudes.
The signals must in every case be seen, not heard, as an ex-
plosion, even if audible at two distant stations, would not be
heard simultaneously at both, owing to the comparatively small
velocity of sound. Where the distance between the two stations
is great, a chain of intermediate stations must be established,
and the local time of each station compared with that of the
next ; this method was used in most of the earliest determina-
tions of longitude. Now such methods are entirely superseded
by the use of the chronometer and the electric telegraph.
186
ASTRONOMY.
244. Longitude by Moon culminating Stars. Here,
as in the method of lunar distances, the Moon's position
determines the Greenwich time, but instead of observing the
Moon's angular distance from a neighbouring star, we
observe the difference of right ascension between the
Moon and the star by taking their times of transit with a
tran.-it circle.
The method is not available at sea, because transits cannot
be taken with a sextant. It can be used to determine, by
means of a portable transit circle, the longitude of a tem-
porary observatory set up in a country where there is no
means of telegraphic communication with the outer world.
Its great advantage over the method of lunar distances is that
it does not involve the laborious, process of " clearing the
distance," because the times of passage across the meridian
are unaffected by parallax and refraction.
The necessary data for the calculations are given in the
Nautical Almanack. The time of transit of the star deter-
mines the local sidereal time at the place, and when the
observatory clock is thus corrected, the time of the Moon's
transit is its R.A. The tables in the Nautical Almanack give
the Moon's R.A. at the time of its transit at Greenwich.
The increase of R.A. is proportional to the time which elapsed
between the transits at Greenwich and at the place of
observation, and hence the Greenwich time of the local
transit is known. Hence, the longitude may be found.
*245. Longitude by Meridian Altitude of the Moon. Another
method of finding the longitude is sometimes used, namely to find
the Greenwich time by observations of the Moon's declination. For
this purpose, the Moon's meridian altitude is observed with a
transit circle and its declination deduced ( 24). The Nautical
Almanack contains the Moon's declination for every 3h. of Green-
wich time ; from this the Greenwich time of observation may be
found by proportional parts. But the method is difficult to
employ, because the observations are affected by the same
sources of error, arising from parallax and refraction, as in the
method of lunar distances, and there is also a correction for dip in
observations made at sea. Moreover, the Moon's daily motion in
declination is so small (the greatest variation being about 5 per
day), that a slight error in the computed declination would very
considerably affect the calculated value of the longitude.
THE DETERMINATION OF POSITION ON THE EARTH. 187
SECTION VI. Captain Sumner's Method.
246. "We shall now show that, by taking two altitudes of
the Sun with a sextant, and noting the Greenwich times of
observation with a chronometer, we can construct a ship's
position on a terrestrial globe geometrically.
The Sub-Solar Point. We can at once find the position
on the terrestrial globe of a place at which the Sun is in the
zenith on a given day, at a given instant of Greenwich time.
For, evidently, the latitude of the place is equal to the Sun's
declination, and is, therefore, known ; while the longitude
west of Greenwich is equal to the Greenwich apparent time,
which may be found by subtracting the equation of time from
the mean time. The place is called the Sub-Solar Point.
The Circle of Position. Assuming the Earth to be
spherical, the Sun's Z.D. at any place is equal to the angular
distance of the place from the sub-solar point. (For it is
evidently the angle between the directions of the zeniths at
the given place and at the sub-solar point.) Hence, the
places at which the Sun has a given Z.I), all lie on a small
circle of the terrestrial globe, whose pole is at the sub -solar
point, and whose angular radius is equal to the Sun's Z.D.
This circle is the circle of position.
Geometrical Construction for the Position of the
Ship. If, then, two altitudes of the Sun be observed,
and the Greenwich times noted with a chronometer, we can
find the sub-solar points, and thus construct the circles of
position, and we know that the ship lies on each circle. The
ship must, therefore, be at one of the two points in which
the two circles cut. To decide which is the actual position,
the Sun's azimuth must be very roughly estimated at the
two observations. On the globe it will be easy to see at which
of the two places the Sun had the observed azimuths. Thus
the ship' s exact position on the globe is found . It is easy to allow
for the ship's motion between the observations.
If two stars are observed, the two substellar points (or
places at which the stars are in the zenith) can be con-
structed. For the latitude of either is equal to the corres-
ponding star's decl., and its longitude is equal to the star's
hour angle at Greenwich = sidereal time star's R.A.
The ship's place can now be found by drawing the circles
of position as before.
ASTRON. o
188 ASTRONOMY.
EXAMPLES. VII.
1. At noon on the longest day a circumpolar star is passing over
the observer's meridian, and its zenith distance is the same as that
of the Sun's centre ; at midnight it just grazes the horizon. Find
the latitude.
2. On January 2, 1881, on a ship in the North Atlantic in longi-
tude 48 W., it was observed that the Sun's meridian altitude was
15 21' 45". The Sun's declination at noon at Greenwich on the
same day was 22 54' 33", and the hourly variation 13'78". Find
the ship's latitude.
3. Show how to find the latitude by observing the difference of
the meridian zenith distances of two known stars which cross the
meridian on opposite sides of the zenith at nearly equal distances
from it. Explain whether the stars chosen should be near to or
remote from the zenith. Give also the advantages and disadvan-
tages of this method of finding the latitude, as compared with the
method of circumpolars.
4. On a certain day the observed meridian altitude of a Cassiopeia
(decimation 55 49' ll'l" N.) was 85 10' 18". The eye of the
observer was 18 feet above the horizon, and the error for refraction
for the altitude of the star is 5" ; determine the latitude.
5. The deck of a ship (stationary) is 25 feet from the sea, and the
dip of the horizon at 1 foot is 1' ; if the two meridian altitudes of a
circumpolar star from the sea horizon be 60 2' and 29 58', find the
latitude.
6. At the winter solstice the meridian altitude of the Sun is 15.
What is the latitude of the place ? What will be the meridian
height of the Sun at the equinoxes and at the summer solstice ?
7. Describe the altazimuth, and show how it can be used to find
the time of apparent noon and the azimuth of the meridian by the
method of equal altitudes.
8. A vertical rod is fixed exactly in the centre of a circular foun-
tain basin, and it is observed that on the 25th of July the extremity
of the shadow exactly reaches the margin of the water at lOh. 7m.
A.M., and at 2h. 25m. P.M. The equation of time on that day is
+ 6m. What is the error, compared with local time, of the watch
by which these observations were taken ?
9. In the railway station at Ventimiglia is a clock one face of
which indicates Paris time, the other Eoman time. It is observed
that, when the former indicates 12h. 39m. 4s., the latter indicates
Ih. 19m. 40s. The longitude of Paris being 2 21' E., find the i
longitude of Borne,
THE DETERMINATION OP POSITION ON THE EARTH. 189
10. In Question 9, what is the corresponding local time at Venti-
miglia, the longitude being 7 35' E. ?
11. A chronometer is set by the standard clock at Greenwich at
6 A.M. It is then taken to Shepton Mallet, and indicates noon when
the local time is llh. 49m. 50s. The chronometer is then brought
back to Greenwich, and indicates 9 P.M., when the correct time is
8h. 59m. 55s. Find the longitude of Shepton, supposing the chrono-
meter rate uniform.
12. In applying the lunar method, find the error in the calculated
longitude of the observer due to an error of 1' in the tables of the
Moon's longitude.
13. Amerigo Vespucci is said to have found his longitude in lati-
tude 10 N. in the following manner. At 7.30 P.M. the Moon was
1 E. of Mars, at midnight the Moon was 5^ E. of Mars. The
Nuremberg time of conjunction of the Moon and Mars was midnight.
Hence he calculated that his longitude was 82 W. of Nuremberg.
Discuss the accuracy of the method, and point out the necessary
corrections.
14. A chronometer whose rate is uniform is found at Greenwich
to have an error of Sj hours when the time which it indicates is t\. It
is then taken to a place A t and when it indicates t 2 it is found that
the excess of the observed local time of the place A over 2 is 5 2 hours.
It is now again brought back to Greenwich, and the chronometer
time and error are observed to be t z and S 3 hours respectively.
Prove that the longitude of A east of Greenwich is
15 (So^ + Mi + S^-Ms-^i-W/^-*,) degrees.
15. The sidereal times of transit of a certain star across the
meridian of an observatory A, as recorded at A, and by a telegraphic
signal at B, are t\, t. 2 respectively. The sidereal times of transit of
the same star across the meridian of -B, recorded by telegraphic
signal at A, and at B, are T 1? T 2 respectively. If the signals take
the same time to travel in either direction, show that the difference
of the longitudes of B and A in angular measure
16. The altitudes of two known stars are observed at a given
instant of time. Show how to find on a terrestrial globe the places
at which the stars are vertically overhead, and give a geometrical
construction for the place of observation.
17. In Question 16, find the condition that there should be two,
one, or no possible positions of a ship at which the altitudes of the
known stars have certain given values.
18. If longitude is found by lunar distances, and latitude by
meridian altitudes, find the latitude in which an error of 1' in the
sextant reading will introduce the same error in both observations
if estimated not in angle, but in miles on the Earth's surface.
190 ASTBONOMY.
EXAMINATION PAPER, VII.
1. Give a description of the Sextant, and explain how to use it
for taking altitudes (1) at sea, (2) on land.
2. How does a Chronometer differ from an ordinary watch ?
What are its error and rate ?
3. Prove that a single meridian altitude of a star, whose declina-
tion is known, will determine the latitude. Why is a zenith sector
sometimes preferred to a transit circle for this purpose ?
4. Show how the latitude is determinable by two meridian obser-
vations of a circumpolar star. Why is this method not generally
applicable on board ship ?
5. Show how to find the latitude of a place (1) by observing the
Sun's altitude at a given time ; (2) by the Prime Vertical Instru-
ment.
6. Describe the method of equal altitudes for finding the time of
transit of a celestial body. If the times be observed by the ship's
chronometer, show how to find the longitude.
7. What methods are available for the determination of Greenwich
time at sea ? Describe the method of taking lunar distances.
8. How is the difference of longitude determined by electric
telegraph ? Explain how the personal equation and the time of
transmission of the signal are eliminated.
9. Contrast the method of Moon-culminating Stars with that of
Lunar Distances in respect of the instruments employed, and of the
intricacy of the calculations involved. What other celestial signals
have been proposed, and what is their disadvantage ?
10. Knowing the Greenwich time, show how to construct graphi- j
cally on a globe the position of the ship without any calculation
whatever.
CHAPTER VIII.
THE MOON.
SECTION I. Parallax The Moon's Distance and Dimensions.
247. Definitions. By tlie Parallax of a celestial body
is meant the angle between the straight lines joining it to
two different places of observation.
In 5 we stated that the fixed stars are seen in the same
direction from all parts on the Earth ; hence such stars have
no appreciable parallax. The Moon, Sun, and planets, on
the other hand, are at a (comparatively) much smaller dis-
tance from the Earth, and their parallax is a measurable
quantity. The distance of the Moon from the Earth's centre
is about 60 times the radius of the Earth. The effects of
parallax in connection with the method of Lunar Distances
have already been mentioned (237).
To avoid the necessity of specifying the place of observa-
tion, the direction of the Moon or any other celestial body is
always referred to the centre of the Earth. The direction
of a line joining the body to the Earth's centre is called the
body's geocentric direction. The angle between the geo-
centric direction and the direction of the body relative to
any given observatoiy is called the body's Geocentric
Parallax, or more shortly, its Parallax. Thus the
geocentric parallax is the angle subtended at the body by
the radius of the Earth through the point of observation.
The Horizontal Parallax is the geocentric parallax of
a body when on the horizon of the place of observation.*
192
248. General Effects of Geocentric Parallax.
Assuming the Earth, to be spherical, let C (Fig. 77)
be the Earth's centre, the place of observation, and M the
centre of the Moon or other observed body. Then the angle
MC is the geocentric parallax of M.
Produce CO to Z\ then OZ is the direction of the zenith
at 0, and ZOM is therefore the zenith distance of M as seen
from (corrected of course for refraction). Now
L ZOM = L ZCM+ L OMC ;
therefore the apparent zenith distance of M is increased by
the amount of the geocentric parallax. Conversely to find
L ZCM we must subtract the parallax OMC from the
observed zenith distance ZOM.
The azimuth is unaltered by parallax, because OM } CM
lie in the same plane through OZ.
FIG. 77.
249. To find the Correction for Geocentric Parallax.
In Fig. 77, let
a = CO J^vth's ladiuA,
d = CM = Moon's (or other body's) geocentric distance,
s = ZOM = observed zenith distance of M,
p = OMC = parallax of M.
By Trigonometry, since the sides of A OMC are propor-
tional to the sines of the opposite angles,
sin CMO _ CO
' ' sin COM CM'
THE MOON. 193
sin a
ji,
that is , .
sins a
Therefore sinp = sin .
Cv
Let P be the horizontal parallax of Jf. Then, when
2 = 90, p = P, and therefore the last formula gives
sin P= sin 90 = .
rt d
Hence, by substitution,
sinp = sin P . sin s.
This formula is exact. But the angles p and P are in
every case very small, and therefore their sines are very
approximately equal to their circular measures. Hence we
have the approximate formula
p = JP . sin z,
or, The parallax of a celestial body varies as the
sine of its apparent zenith distance.
The last formula holds good no matter what be the unit of
angular measurement. Thus if p", P" denote the numbers
of seconds in p, P respectively, we have, by reducing to
seconds, p" = P" sin 2.
EXAMPLES.
1. Supposing the Sun's horizontal parallax to be 8'S", to find the
correction for parallax when the Sun's altitude is 60.
Here z = 90 -60 = 30, P" = 8'8", and therefore
p" = P" sin 30 - 8-8" x i - 4'4".
2. To find the corrections for the Moon's parallax for altitudes of
30 and 45, the Moon's horizontal parallax being 57'.
In the two cases wo have respectively z = 60 and z = 45, and
the corresponding corrections are
p" = 57' sin 60 = 57' x x /3 = 28' 30" x ^/3
= 1710" x 1-7320 = 2961-7"= 49' 21-7",
and p" = 57' sin 45 = 57 X x ^2 = 28'' 30" x v/2
= 1710" x 1-4142 - 2418-3"= 40' 18'3".
*94 ASTRONOMY.
250. Relation between the Horizontal Parallax
and Distance of a Celestial Body. In the last paragraph
we showed that sin P = .
a
This formula may be proved independently by drawing
MA to touch the Earth at A. M is on the horizon at A ;
the Z CMA is therefore the horizontal parallax P, and we
have immediately
sin P = sin CMA = CA/CM= /d.
Since P is small, we have approximately
Circular measure of jP = a/d.
and therefore in seconds
p ,, = 180X60X60 ^ = 206265
if d d
which shows that, The horizontal parallax of a body
varies inversely as its distance from the Earth.
FIG. 78.
If we know the Earth's radius a and the distarce d, the
last formula enables us to calculate the horizontal j ara lax
P". Conversely, if we know the horizontal parallax of a body
we can calculate its distance.
EXAMPLE 1. Given that the Moon's distance is 60 times the
Earth's radius, to find the Moon's horizontal parallax.
We have - = ;
d 60
circular measure of P = ^r approximately,
bu
"NTow the unit of circular measure = 57'2957 ;
P (in angular measure) = x 57 - 2957 = 57'2957'
= 57' 17-7",
and this is tho required horizontal parallax.
THE MOON. 195
EXAMPLE 2. Given that the Sun's parallax* is 8'8", to find
the Sun's distance, the Earth's radius being 3,960 miles.
The circular measure of 8*8" is = ** ,
and, by the formula, we have, for the Sun's distance in miles,
d = a = 3960x180x60x60
circ. meas. of P 8'8 x TT
Taking TT =3f , and calculating the result correct to the first three
significant figures, we find the Sun's distance d
= 92,8OO,OOO miles approximately.
It would be useless to carry the calculations beyond the third
figure, for, of course, the values of the Earth's radius and Sun's
parallax are only approximate; moreover, we should have to use
the more accurate value of TT, viz., 3'141592
251. Comparison between Parallax and Refraction.
It will be noticed that while parallax and refraction both
produce displacements of the apparent position of a body along
a vertical circle, the displacement due to parallax is directed
away from the zenith, and is always proportional to the sine
of the zenith distance, while that due to refraction is directed
towards the zenith, and is proportional to the tangent of the
zenith distance, provided the altitude is not small. Also the
correction for parallax is inversely proportional to the distance
of the body, and is imperceptible, except in the case of mem-
bers of our solar system ; while the correction for refraction
is independent of the body's distance, and depends only on
the condition of the atmosphere.
The Moon's horizontal parallax is about 57', while the
horizontal refraction is only 33'. Hence, by the combined
effects of parallax and refraction, the Moon's apparent
altitude is diminished, or its Z.D. increased. The time of
rising is, therefore, on the whole retarded, and the time of
setting accelerated. The effect of parallax on the times of
rising and setting may be investigated by the methods of
104, 190.
For all other bodies, including the nearest planets, the
correction for refraction far outweighs that due to parallax.
* When astronomers speak of the parallax of the Sun, Moon, or
a planet, without further specifying the observation, the horinontal
parallax is always to be understood.
6 ASTRONOMY.
252. To find the Moon's Parallax by Meridian
Observations. The Moon's parallax may be conveniently
determined as follows. Let A and B be two observatories
situated on the same meridian, one north, the other sonth of
the equator. Let M denote the Moon's centre, and let x be
a star having no appreciable parallax, whose R.A. is approxi-
mately equal to that of the Moon, their declinations being
also nearly equal.
Let the Moon's meridian zenith distances ZAM and Z'BM
be observed with the transit circles at A and B, and let xA M
and xBM, the differences of the meridian Z.D.'s of the Moon
and star at the two stations, be also observed.
Let 2, = / ZAM, 2 2 = L Z'BM.
a, = L xAM, 3 = L xBM.
P = Moon's required horizontal parallax.
By 249, we have, approximately,
/ A M C = P sin 54, Z BMC = P sin s 2 .
FIG. 79.
.-. Z AMB = P (sin 2, + sin 2 3 ) (i.).
Moreover, if MX be drawn parallel to Ax or Bx,
Z XMA = Z MAx = a, ;
Z XMB Z J/#r = fl a ;
.-. z ^J/7? = !-, '. (ii.),
From'(i.) and (ii.),
P (sin , + sin 2 2 ) = ^j 2 ;
P = ^i- 3 .
siu 2, -{-sins.,
whence the Moon's parallax, P, may be found.
tSE Moott. 197
253. If the two observatories are not on the same meridian,
allowance must be made for the change in the Moon's
declination between the two observations. Let the stations
be denoted by A, JJ, and let S' be the place on the meridian
of A, which has the same latitude as B. Then, if the Moon's
meridian Z.D. be observed at B, we can, by adding or sub-
tracting the change of declination during the interval, find
what would be the meridian Z.D. if observed from B'.
Moreover, the star's meridian Z.D. is the same both at B and
at B'. Hence it is easy to calculate what would be the angles
at B' corresponding to the observed angles at B. From the
former, and the observed angles at A, we find the parallax
P, as before.
To ensure the greatest accuracy, it is advisable that the
difference of longitude of the two stations should be so small
that the correction for the Moon's motion in declination
is trifling. It is necessary, however, that 0, # 2 should
be large ; for this reason the stations should be chosen
one as far north and the other as far south of the equator as
possible. The observatories at Greenwich and the Cape of
Good Hope have been found most suitable.
The principal advantage of the above method is that the
probable errors arising from any uncertainty in the corrections
for refraction are diminished as far as possible.
For, since the Moon and observed star have nearly the
same declination, the corrections for refraction to be applied
to #!, # 2 , their small differences of Z.D., are very small indeed.
The errors are not of so much moment in the denominator
sin z l -f- sin z 2 , as the latter is not itself a small quantity.
From such observations, the mean horizontal parallax of
the Moon has been found to be 57' 2 '70 7".
This value corresponds to a mean distance of 60-27 times
the equatorial radius of the Earth, or 238,840 miles. The
distance and parallax of the Moon are not, however, quite
constant ; their greatest and least values are in the ratio of
(roughly) 19 : 17. For rough calculations, the Moon's
distance may be taken as 60 times the Earth's radius.
Neither this method nor the next ( 254) gives accurate
resulls for the Sun, for the brilliancy of the rays renders all
stars in its neighbourhood invisible
198 ASTKONOMY.
254. To find the Parallax of a rlanet from Observa-
tions made at a Single Observatory. The parallax of
Mars, when nearest the Earth, has also been determined by
the following method, depending on the Earth's rotation.
Since the apparent altitude of a body is always diminished
by parallax, it can easily be seen by a figure, that, shortly after
a planet has risen, its R.A. and longitude appear greater
than their geocentric values (the planet being displaced east-
wards), while shortly before setting they appear less
than their geocentric values (the displacement being west-
wards). The planet's position, relative to certain fixed stars,
is observed soon after rising and before setting by means of
an equatorial furnished with a micrometer or heliometer.
The observed change of position is due partly to parallax
and partly to the planet's motion relative to the Earth's
centre during the interval between the observations, which
produces displacements far greater than those due to
parallax. But by repeating the observations on successive
days, the planet's rate of motion can be accurately
determined, and the displacements due to parallax can thus
be separated from those due to relative motion. Refraction
need not be allowed for ; because it affects those stars with
which the planet is compared, as well as the planet itself.
This method can be used for the Moon, but the Moon's
motion is so rapid that the calculations are more complicated.
*255. Effect of the Earth's Ellipticity. The effect of parallax
is made rather more complicated by the spheroidal form of the
Earth. For, by 249, the magnitude of the horizontal parallax at
any place depends on its distance from the Earth's centre, and since
this distance is not the same for all places on the Earth, the horizontal
parallax is not everywhere the same. Again, the direction in which
the body is displaced is away from the line (produced) joining the
centre of the Earth with the observer ( 248). But this line does not
pass exactly through the zenith ( 117). Hence the displacement
is not in general along a vertical, so that the azimuth as well as
altitude is very slightly altered by parallax.
250. The Equatorial Horizontal Parallax is the geo-
centric parallax of a body seen on the horizon of a place at the
Earth's equator. It is generally adopted as the measure of
the parallax of a celestial body. Its sine is equal to
(Earth's equatorial radius)/(body's geocentric distance).
THE M001T. 199
257. Relation between Parallax and Angular
Diameter. In Fig. 80 it will be seen that the angle CM A,
which measures the parallax of M, also measures the Earth's
angular semi-diameter as it would appear from M. Thus,
the Moon's parallax is the angular semi-diameter of the Earth
as it would appear if observed from the Moon.
FIG. 80.
258. To Find the Moon's Diameter. Let 0, c be the
radii of the Earth and Moon respectively, measured in miles,
d the distance between their centres, Pthe Moon's horizontal
parallax, m the Moon's angular semi-diameter as it would
appear if seen from the Earth's centre. Then, from Fig. 80,
.. c : a = sin m : sin P = m : P approximately ;
i.e. (rad. of Moon) : (rad. of Earth)
= ( C 's ang. semi-diam.) : ( <T 's hor. parx.).
Hence, knowing the Moon's horizontal parallax and its
angular diameter, the Moon's radius can be found.
The Moon's mean angular diameter 2m is observed to be
about 31' 5". From this the Moon's actual diameter is readily
found to be about 2160 miles, or T 3 T of the Earth's diameter.
The surfaces of spheres are proportional to the squares, and the
volumes to the cubes of their radii. Hence the Moon's superficial
area is about -jf T , or $, and its volume about T f| T , or -^ of that
of the Earth.
EXAMPLE. To find the Moon's diameter in miles, given
< 's angular diameter = 31' 7",
([ 's equatorial horizontal parallax = 57' 2",
Earth's equatorial radius = 3963 miles.
.-. <t 's diameter 2c = a x ^ = 3963 x ^J^. = 3963 x -- = 2162.
Thus the Moon's diameter is 2162 miles.
200 ASTEONOMT.
SECTION II. Synodic and Sidereal Months Moon's Phases
Mountains on the Moon.
259. Definitions. In 40 we defined the lunation as
the period between consecutive new Moons, and showed that
it was rather longer than the period of the Moon's revolution
relative to the stars. We shall now require the following
additional definitions, most of which apply also to the planets.
The elongation of the Moon or planet is the difference
between its celestial longitude and that of the Sun. If the
body were to move in the ecliptic its elongation would be its
angular distance from the Sun.
The Moon or planet is said to be in conjunction when it
has the same longitude as the Sun, so that its elongation is
zero. The Moon is in conjunction at new Moon ( 40). The
body is in opposition when its elongation is 180. In both
positions it is said to be in syzygy. The body is said to be
in quadrature when its elongation is either 90 or 270.
The period between consecutive conjunctions is called the
synodic period of the Moon or planet. The Moon's
synodic period is, therefore, the same as a lunation; it is
also called a Synodic Month. In this period the Moon's
elongation increases by 360, the motion being direct.
The period of revolution relative to the stars is called the
sidereal period ; that of the Moon, the Sidereal Mouth.
The average length of the Calendar Month in common
use is slightly in excess of the synodic month (cf. 171).
260. Relation between the Sidereal and Synodic
Months.
Let the number of days in a year be F, in a sidereal month
J/j and in a synodic month S.
In M days the Moon's longitude increases 360 ;
.'. in 1 day the Moon's longitude increases 360/J/.
Similarly in 1 day the Sun's longitude increases 360/F,
and the Moon's elongation increases 360/&
Now, from the definition,
(Moon's elongation) = (Moon's long.) (Sun's long.),
and their daily rates of increase must be connected by the
same relation :
THE MOON. 201
360^360 360.
8 " M Y '
=- or =
8 " M ' M~~
1
sider. month synod, month, year
EXAMPLE. Find (roughly) the length of the sidereal month, given
that the synodic month (8) = 29|d., and the year (Y) = 365id.
Here we have i = i+ JL.
To simplify the calculations, we put the relation into the form
= 29 5 - 29-5 x = 29-5 - 2'20 = 27'3.
1579
Hence the sidereal month is very nearly 27 J days.
261. To determine the Moon's Synodic Period.
An eclipse of the Sun can only happen at conjunction, and
an eclipse of the Moon at opposition, and the middle of the
eclipse determines the exact instant of conjunction or oppo-
sition, as the case may be. Hence, by observing the exact
interval of time between the middle of two eclipses, and
counting the number of lunations between them, the length
of a single lunation, or synodic period, can be found with
great accuracy expressed in mean solar units of time.
The records of ancient eclipses enable us to find a still closer
approximation to the mean length of the lunation. From
modern observations, the length of a lunation has been found
with sufficient accuracy to enable us to tell the exact number
of lunations between these ancient eclipses and a recent lunar
eclipse (this number being, of course, a whole number], ]3y
dividing the known interval in days by this number, the
mean length of the synodic period during the interval can be
accurately found. At the present time the length of a
lunation is 29-5305887 days, or 29d. 12h. 44m. 2'7s. nearly.
Prom this the length of the Moon's sidereal period is cal-
culated, as in 260, and found to be 27d. 7h. 43m- 11 -5s.
nearly
202 ASTBOUOMY.
262. Phases of the Moon. The acccompanying dia-
grams will show how the phases of the Moon are accounted
for on the hypothesis that the Moon is an opaque body
illuminated by the Sun. In the upper figure the central
globe represents the Earth, the others represent the Moon in
different parts of its orbit, while the Sun is supposed to be at
a great distance away to the right of the figure.* The half
of the Moon that is turned towards the Sun is illumi-
nated, the other half 'being dark. The Moon's appearance
depends on the relative proportions of the illuminated and
darkened portions that are turned towards the Earth.
FIG. 81.
The lower figures, 0, b, c, d, e, /, g, h, represent the appear-
ances of the Moon relative to the ecliptic, as seen from the
Earth when in the positions represented by the corresponding
letters in the upper figure.
* The Sun's distance is about 390 times the Moon's. If the
former be represented by an inch, the latter will be represented by
about 11 yards.
TKb MOON. 203
At A, a the Moon is in conjunction, and only the dark
part is towards the Earth. This is called M"ew Moon.
At B, b a portion of the bright part is visible as a crescent
at the western side of the disc. The Moon's appearance is
known as horned. The points or extremities of the horns
are called the cusps.
At 7, c the Moon's elongation is 90, and the western half
of the disc, or visible portion, is illuminated, the eastern half
being dark. The Moon is then said to be dichotomized.
This is called the First Quarter. The Moon's age is about
7| days.
At D, d more than half the disc is illuminated. The
Moon's appearance is then described as gibbous.
At E) e the Moon is in opposition. The whole of the disc
is illuminated. This is called Pull Moon. The Moon's age
is about 15 days.
At F, f a portion of the disc at the western side is dark.
The Moon is again gibbous, but the bright part is turned in
the opposite direction to that which it has at D, d.
At 6r, g the Moon's elongation is 270. The eastern half
of the disc is illuminated, and the western half is dark. The
Moon is again dichotomized. This is called the Last
Quarter. The Moon's age is about 22 days.
At -ZZ", h only a small crescent in the eastern portion is still
illuminated. The Moon is now again horned, but the horns
are in the opposite direction to those in , I.
Finally, the Moon comes round to conjunction again at A,
and the whole of the part towards the Earth is dark.
From new to full Moon, the visible illuminated portion
increases , and the Moon is said to be waxing 1 . From full to
new, the illuminated portion decreases, and the Moon is said
to be waning.
It will be noticed from a comparison of the figures that
the illuminated portion of the visible disc is always that
nearest the Sun. Moreover, its area is greater the greater
the Moon's elongation.*
* The phases of the Moon may be readily illustrated experi-
mentally, by taking an opaque ball, or an orange, and holding it in
different directions relative to the light from the Sun or a gas-
burner.
ASTEON. P
201 ASTRONOMY.
263. Relation between Phase and Elongation. Let
M (Fig. 82) "be the centre of the Moon, MS the direction of
the Sun, E'ME that of the Earth. Draw the great circles
AMB perpendicular to ME, and CMD perpendicular to MS ;
the former is the boundary of the part of the Moon turned
towards the Earth, and 'the latter is the boundary of the
illuminated portion. Hence the visible bright portion is the
lune A MC. The angle of the lune, L AMC, is equal to
Z E'MS (Sph. Greom. 16). The area of a spherical lune is
proportional to its angle. Hence,
area of visible illuminated part _ Z AMC _ E'MS
area of hemisphere 180 ' 180
180 ~
180
FIG. 83.
But this does not give the " apparent area " of the bright
part. For, as in 145, the apparent area of a body is the
area of the disc formed by projecting the body on the celestial
sphere. If IT denote the projection of the point C on the
plane AMB (so that CN is perpendicular to BA\ the arc
A C will be seen in perspective as a line of length AN, and
the bright part will be seen as a plane lune (Fig. 83), whose
boundary POP' optically forms the half of an ellipse whose
major axis is PP' t and minor axis
THE MOON. 205
It may be shown that
area of half-ellipse POP' : area of semicircle PAP 1
and .-. area APCP' : area APBP' = AN~ AB
= 1 - cos AMC : 2 = 1 cos E'MS \ 2.
Hence the apparent area of the bright part is proportional to
FIG. 84.
m le 83E ' diffcrs from the Moo *'s elongation
r \r C Smal ]. anglG ESK (Fig. 84); i.e., the Bangle
which the Moon's distance subtends at the Sun. This angle
is very small, being always less than 10'. Hence the
area 01 the phase is very approximately proportional to
1 cos(Moon's elongation).
264. Determination of the Sun's Distance by Aris-
tarchus. From observing the Moon's elongation when
dichotomized Aristarchus (B.C. 270 circ.) made a computation
ol the bun s distance in the following manner. When the
^^/dichotomized, L SME = 90, the Moon's elongation
t SEM= 90- z ESM, and cos SZM= JEM/US. Hence
by observing the angle SEN, the ratio of the Sun's distance
to the Moon's was computed.
But this method is incapable of giving reliable results, owing
to the impossibility of finding the exact instant when the Moon
is dichotomized. The Moon's surface is rough, and covered
with mountains, and the tops of these catch the light before
the lower parts, while throwing a shadow on the portions
behind them. Hence the boundaiy of the bright part is
always jagged, and is never a straight line, as it would be at
the quarters, if the surface of the Moon were perfectly smooth.
In tact, Aristarchus estimated the Sun's distance as only
about 19 times that of the Moon, whereas they are really in
the proportion of nearly 400 to 1.
206 ASTRONOMY.
265. Earth-Shine on the Moon. Phases of the
Earth. When the Moon is nearly new, the tmilluminated
portion of its surface is distinctly visible as a disc of a dull-
red colour. This appearance is due to the light reflected
from the Earth as " Earth-shine," which illuminates the
Moon in just the same way that the moonshine illuminates
the Earth at full Moon. From 258, the Earth's superficial
area is greater than the Moon's in the proportion of about
40 : 3. Consequently the Earth-shine on the Moon is more
than 13 times as bright as the moonshine on the Earth.
The Earth, as seen from the Moon, would appear to pass
through phases similar to those of the Moon, as seen from
the Earth. The Earth's and Moon's phases are evidently
supplementary. Thus, when the Moon is new the Earth
would appear full, and vice-versd ; when the Moon is in the
first quarter, the Earth would appear in the last quarter.
Owing, however, to twilight, the boundary of the Earth's
illuminated portion would not be so well denned as in the
case of the Moon ; there would be a gradual shading off from
light to darkness, extending over a belt of breadth 18 on
beyond the bright part. The entire absence of twilight on
the Moon is one of the strongest evidences against the exist-
ence of a lunar atmosphere similar to that of our Earth.
266. Appearance of Moon relative to the Horizon.
"We are now in a position to represent, in a diagram, the
Moon's position and appearance relative to the horizon at a
given time of day and year when the Moon's age is given.
The ecliptic having been found, as explained in 41,
the age of the Moon determines the Moon's elongation,
as in 40. Measuring this angle along the ecliptic, we find
the Moon's position roughly j for the Moon is never very far
from the ecliptic (cf. 40). The elongation also determines
the phase, and enables us to indicate the appearance of
the disc. The bright side or limb is always turned towards
the Sun. The cusps, therefore, point in the reverse direction,
and the line joining them is perpendicular to the ecliptic.
We can also trace the changes in the direction of the
Moon's horns relative to the horizon, between its time of
rising and setting.
THE MOON. 207
Take, for example, the case when the Moon is a few (say
three) days old. The Moon is then a little east of the Sun ;
therefore the bright limb is at the western side of the disc,
and the horns point eastward. Hence, at rising, the horns
are pointed downwards, and at setting they are pointed
upwards (Fig. 85).
n
FIG. 85. FIG. 86.
When the Moon is waning, the reverse will be the case
(Fig. 86).
267 Heights of Lunar Llouutains. We stated m
264 that the Moon's surface is covered with mountains, and
that in consequence the bounding line between the illumi-
nated and dark portions of the disc is always jagged and
irregular , while the mountains themselves throw their
shadows on the portions of the surface behind them. These
circumstances have led to the two following different ways
of measuring the height of the lunar mountains
First Method. If a tower is standing in the middle
of a perfectly level plain, it is evident from trigono~
nietry that the length of the shadow, multiplied by the
tangent of the Sun's altitude, gives the height of the tower.
The same will be true in the case of the shadow cast by a
mountain, provided we measure the length of the shadow
from a point vertically underneath the summit. Now, in
the case of the Moon it is possible, from knowing the Moon's
age, to calculate exactly what would be the altitude of the
Sun as it would be seen from any point of the lunar surface.
The apparent length of the shadows of the mountains can be
measured, in angular measure, by means of a micrometer ;
from this their actual length can be calculated, allowance
being, of course, made for the fact that we are not looking
vertically down on the shadows, and hence they appear fore-
shortened. In this way. the height of the mountains can be
found.
208 ASTKONOMY.
The principal disadvantage of this method is, that if the
surface of the Moon surrounding the mountain should be less
flat than it has been estimated, there will be a corresponding
error in the height of the mountain. In particular, it would
be impossible to apply the method to find the heights of
mountains closely crowded together.
268. Second Method. In treating of the Earth in
104, we showed that one effect of the dip of the horizon is
to accelerate the times of rising, and to retard the times of
setting of the Sun and stars. We also showed how to calcu-
late the amount of the acceleration if the dip be known.
Conversely, if the acceleration in the time of rising be known,
the dip of the horizon can be calculated, and from this the
height of the observer above the general level of the Earth
may be found.
Tfow precisely the same method may be applied to measure
the heights of lunar mountains. When the Moon is waxing
the Sun is gradually rising over those parts of the Moon's
surface which are turned towards the Earth. The tops of
the mountains catch the rays before the lower parts, and,
therefore, stand out bright against the dark background of
the unilluminated parts below. Similarly, when the Moon
is waning, the summits of the mountains remain as bright
specks after the lower portions are plunged in shadow. By
noticing the exact instant at which the Sun's rays begin or
oeasc to illuminate the summit, this acceleration or retarda-
tion, due to dip, may be calculated, and the height of the
mountain determined.
If the Moon's surface around the mountain is fairly level,
the distance of the mountain from the illuminated portion at
the instant of disappearance determines the distance of the
visible horizon as seen from the mountain. This distance can
be calculated from measurements made with a micrometer
(proper allowance being made for foreshortening if the moun-
tain is not in the centre of the disc).
Hence the height (h) of the mountain may be calculated
by the formula of 101 (i.), viz., h = d^/Za, where d is the
estimated distance of the horizon, and a the Moon's radius,
THE MOON. 209
SECTION III. The Moon's Orbit and Rotation.
269. The Moon's Orbit about the Earth can be inves-
tigated by a method precisely similar to that employed in the
case of the Sun (see 145). The Moon's E.A. and decl.
may be observed daily by the Transit Circle. The observed
decl. must be corrected for refraction and parallax (neither of
which affect the R.A., since the observations are made on the
meridian}. "We thus find the positions of the Moon on the
celestial sphere relative to the Earth's centre for eveiy day
at the instant of its transit across the meridian of the obser-
vatory,
Instead of observing the Moon's parallax daily, the Moon's
distances from the Earth's centre on different days, may be
compared by measuring the Moon's angular diameters, with
the heliometer. Here, however, another correction for
parallax is required. For the observed angular diameters
are inversely proportional to the corresponding distances of
the Moon from the observer, and not from the centre of the
Earth.
This correction is by no means inconsiderable. Thus, for
example, if the Moon be vertically overhead, its distances from
the observer and from the Earth's centre will differ by the
Earth's radius, i.e., by about -^ of the latter distance, and
its angular diameter will, therefore, be increased in the pro-
portion of about 60 to 59.
Having thus determined the direction and distance of the
Moon's centre, relative to the Earth's centre, for every clay in
the month, the Moon's orbit may be traced out in just the same
way as the Sun's orbit was traced out in 146. It is thus
found that the motion obeys approximately the following
laws :
(i.) The Moon's orbit lies in a plane through the Earth's
centre, inclined to the plane of the ecliptic at an angle of about
5 8'.
(ii.) The orbit is an ellipse, having the- Earth's centre in
one focus, the eccentricity of the ellipse being about .
18
(iii.) The radius vector joining the Earth's and Moon\
centres traces out equal areas in equal intervals of time.
210 ASTRONOMY.
The period of revolution is, of course, the sidereal lunar
month, as denned in Section II., namely, about 27^ days.
The laws which govern the Moon's motion are thus iden-
tical with Kepler's laws for the Earth's orbital motion round
the Sun ( 155).
270. The Eccentricity of the Moon's Orbit is found
by comparing the Moon's greatest and least distances, which
are inversely proportional to its least and greatest (geocentric)
angular diameters respectively. The latter are in the ratio
of about 17 to 19, and it is inferred that the eccentricity is
'about (19-17)/(19 + 17) 5 or ^ (ef. 149).
The terms perigee, apogee, apse line are used in
the same sense as in 147. Perigee and apogee are the
points in the orbit at which the Moon is nearest to and
furthest from the Earth respectively. Both are called the
apses or apsides, the line joining them being called the apse
line, apsidal line or line of apsides, according to choice. It is
the major axis of the orbit.
As in 151, it follows that the Moon's angular motion in
its orbit is swiftest at perigee, and slowest at apogee.
271. Nodes. The points in which the Moon's orbit, or its
projection on the celestial sphere, cuts the ecliptic are called
the Moon's Nodes (ef. 40). The line joining them is
called the Nodal Line. It is the line of intersection of the
planes of the Moon's orbit and ecliptic. That node through
which the Moon passes in crossing from south to north of the
ecliptic is distinguished as the ascending node, the other is
distinguished as the descending node.
272. Perturbations. As the result of observations extend-
ing over a large number of lunar months, it is found that the
Moon docs not describe exactly the same ellipse over and
over again, and that, therefore, the laws stated in 269 are
only approximate. The actual motion can, however, be
represented by supposing the Moon to revolve in an ellipse,
the positions and dimensions of which are very slowly vary-
ing. This mode of representing the motion may be illustrated
by imagining a bead to revolve on a smooth elliptic wire
which is very slowly moved about and deformed.
THE MOON. 211
The complete investigation of these small changes or
perturbations, as they are called, belongs to the domain of
Gravitational Astronomy. It will be necessary here to
enumerate the chief perturbations, on account of the important
part they play in determining the circumstances of eclipses.
273. Retrograde Motion of the Moon's Nodes. The
Moon's nodes are not fixed, but have a retrograde motion
along the ecliptic of about 19 in a year. This phenomenon
closely resembles the retrograde motion of T (Precession,
141), but is far more rapid. Its effect is to carry the line
of nodes, with the plane of the Moon's orbit, slowly round
the ecliptic, performing a complete revolution in 6793-391
days, or rather over 18-6 years.
One result of this nodal motion is that the angle of inclination
of the Moon's orbit to the equator is subject to periodic
variations. When the Moon's ascending node coincides with the
first point of Aries, the angle between the Moon's orbit and the
equator will be the difference of the angles they make with the
ecliptic, i.e. about 23 28' - 5 8' or 18 20'. When, on the contrary,
the ascending node coincides with the first point of Libra, the angle
between the orbit and the equator will be the sum of the angles they
make with the ecliptic, i.e., 23 28' + 5 8' or 28 36'. The period of
fluctuation is the time of revolution of the Moon's nodes relative to
the first point of Aries, and is a few days (nearly five) greater than
their sidereal period of revolution, on account of precession.
274. Progressive Motion of Apse Line. The line oi
apsides is not fixed, but has a direct motion in the plane
of the Moon's orbit, performing a complete revolution in
3232-575 days, or about nine years. A similar progressive
motion of the apse line of the Earth's orbit about the Sun
was mentioned in 153. The latter motion is, however,
much less rapid, its period being about 108,000 years.
275. Other Perturbations. The inclination of the
Moon's orbit to the ecliptic is not quite constant. It is
subject to small periodic variations, its greatest and least
values being 5 13' and 5 3'.
In addition there are variations in the eccentricity of the
orbit, in the rates of motion of the nodes, and in the length
of the sidereal period. All of these render the accurate
investigation of the Moon's orbit one of the most complicated
problems of Astronomy.
212 A8TEONOMT.
276. The Moon's Rotation. It is a remarkable fact
that the Moon always turns the same side of its surface to th
Earth. Whether we examine the markings on its surface
with the naked eye, or resolve them into mountains and
streaks with a telescope, they always appear very nearly the
same, although their illumination, of course, varies with the
phase.
From this it is evident that the Moon rotates upon its axis
in the same "sidereal" period as it takes to describe its
orbit about the Earth, i.e., once in a sidereal month. It
might, at a first glance, appear as if the Moon had no rota-
tion, but such is not the case. To explain this, let us consider
the phenomena which would be presented to an observer if
situated on the Moon in the centre of the portion turned
towards the Earth.
The Earth would always appear directly overhead, i.e., in
the observer's zenith. But as the Moon describes its orbit
about the Earth, the direction of tlie line joining the Earth
and Moon revolves through 360, relative to the fixed stars,
in a sidereal month. Hence the direction of the observer's
zenith on the Moon must also revolve through 360 in a
sidereal month, and therefore the Moon must rotate on its
axis in this period.
The Moon would be said to describe its orbit without
rotation, if the same points on its surface were to remain
always directed towards the same fixed stars. "Were this the
case, different parts of the surface would become turned to-
wards the Earth as the Earth's direction changed, and this is
not what actually occurs.
It thus appears that, to an observer on the Moon, the
directions of the stars relative to the horizon would appear
to revolve through 360 once in a sidereal lunar month.
Thus, the sidereal month is the period corresponding to the
sidereal day of an observer on the Earth. In a similar way,
the Sun's direction would appear to revolve through 360 in
a synodic month. This, therefore, is the period corresponding
to the solar day on the Earth, as is otherwise evident from
the fact that the Moon's phases determine the alternations of
light and darkness on the Moon's surface, and that they
repeat themselves once in every synodic month.
THE MOON. 213
277. Libratipns of the Moon. Libration in Lati-
tude. If the axis about which the Moon rotates were per-
pendicular to the plane of the Moon's orbit, we should not be
able to see any of the surface beyond the two poles ('.*., ex-
tremities of the axis of rotation). In reality, however, the
Moon's axis, instead of being exactly perpendicular to its
orbit, is inclined at an angle of about 6| to the perpendicular,
just as the Earth's axis of rotation makes an angle of about
23 28' with a perpendicular to the ecliptic. The conse-
quence is that during the Moon's revolution the Moon's north
and south poles are alternately turned a little towards and a
little away from the Earth ; thus, in one part of the orbit we
see the Moon's surface to an angular distance of 6 44' beyond
its north pole, in the opposite part we see 6 44' beyond the
southpolc. This phenomenon is called the Moon's libration in
latitude. It makes the Moon's poles appear to nocl, oscillat-
ing to and fro once in every revolution relative to the nodes.
Libration in latitude may be conveniently illustrated by
the corresponding phenomenon in the case of the Earth's
motion round the Sun, as represented in Fig. 56 ( 154). At
the summer solstice the whole of the Arctic circle is illumi-
nated by the Sun's rays, and therefore an observer on the
Sun (if such could exist) would see the Earth's surface for a
distance of 23 28' beyond the north pole. Similarly, at the
winter solstice an observer on the Sun would see the whole
of the Antarctic circle, and a portion of the Earth's surface
extending 23 28' beyond the south pole.
278. Libration in Longitude. Owing to the elliptical
form of the orbit, the Moon's angular velocity about the
Earth is not quite uniform, being least at apogee and greatest
at perigee. But the Moon rotates about its polar axis with
perfectly uniform angular velocity equal to the average
angular velocity of the orbital motion (so that the periods
of rotation and of orbital motion are equal).
Thus, at apogee the angular velocity of rotation is slightly
greater than that of the orbital motion, and is, therefore,
greater than that required to keep the same part of the
Moon's surface always turned towards the Earth. In con-
sequence, the Moon will appear to gradually turn round, so
as to show a little more of the eastern side of its surface.
214 ASTRONOMY.
At perigee, the angular velocity of rotation is less than
that of the orbital motion, and is, therefore, not quite suffi-
cient to keep the same part of the Moon's surface always
turned towards the Earth. In consequence we shall begin to
see a little further round the western side of the Moon's disc.
This phenomenon is called libration in longitude. Its
maximum amount is 7 45' ; thus, during each revolution of
the Moon relative to the apse line, we alternately see 7 45'
of arc further round the eastern and western sides of the disc
than we should otherwise.
279. Diurnal Libration. The phenomenon known as
diurnal libration is really only an effect of parallax. If
the Moon were vertically overhead, and if we were to travel
eastwards, we should, of course, begin to see a little further
round the eastern side of the Moon's surface. If we were to
travel westwards we should begin to see a little further round
the western side. Now, the rotation of the Earth carries the
observer round from west to east. Hence, when the Moon is
rising wo see a little further round its western side, and
when setting we see a little further round its eastern side,
than we should from a point vertically underneath the Moon.
Similarly an observer in the northern hemisphere would
always see rather more of the Moon's northern portion, and
an observer in the southern hemisphere would see rather more
of the southern portion than an observer at the equator.
The greatest amount of the diurnal libration is equal to the
Moon's horizontal parallax, and is therefore about 57'. We
see 57' round the Moon's western corner when rising, and 57'
round the eastern corner when setting.
An observer at any given instant sees not quite half
(49-998 per cent.) the Moon's surface. The visible portion is
bounded by a cone through the observer's eye enveloping the
Moon, and is less than a hemisphere by a belt of breadth equal
to the Moon's angular semi-diameter, i.e., about 16'.
280. General Effects of Libration. In consequence
of the three librations, about 59 per cent, of the Moon's sur-
face is visible from the Earth at some time or other, instead of
rather under 50 (49-998) per cent., as would be the case if
there were no libration. At the same time only about 41 per
cent, of the surface is always visible from the Earth. The
remainder is sometimes visible, sometimes invisible.
THE MOON. 215
To an observer on the surface of the Moon the result of libra-
tion in latitude and longitude would be that the Earth, instead
of remaining stationary in the sky, would appear to perform small
oscillations about its mean position. It would really appear to de-
scribe a series of ellipses. The motion of the different parts of the
Earth across its disc in the course of the Earth's diurnal revolution
would be the only phenomenon resulting from the cause which pro-
duces diurnal libration.
281. Metonic Cycle. A problem of great historic interest in the
study of the lunar motions is the finding of a method of ready pre-
diction of the Moon's phases. From the earliest times there have
been religious festivals regulated (as Easter still is) by the Moon's
phases; but the direct calculation, from first principles, of the phase
for a given day would be long and tedious.
This difficulty was overcome by the discovery of the EO-called
Metonic Cycle by Meton and Euctemon, B.C. 433. They found that
after a cycle of nineteen years the new and full Moons recurred on the
same days of the year. To show this it is necessary to prove that
nineteen years is nearly an exact multiple of the synodic month.
Now, 1 tropical year = 365'2422 days ; .'.19 years = 6939'60 days,
and 1 synodic month = 29'5306 days ; /. 285months = 6939'69days;
.'. 19 years differs from 235 lunations by '09 days, i.e., 2h. 10m. nearly.
If we define the Golden Number of a year as the remainder when
(1 + the number of the year A.D.) is divided by 19, and the Epact as
the Moon's age on the 1st of January, we see that two years which
have the same Golden Number have corresponding lunar phases on
the same days, and in particular have the same epact.
Hence, the Golden Number of the year 1 B.C. (which might be
more consistently called A.D.) is evidently 1 ; and it happens that
that year had new Moon on January 1, and, therefore, its epact is
zero. But twelve lunar months contain 354'37 days, and fall short
of the average year (365'25 days) by 10'88 days, which is nearly
5^ lunations. Hence, the epact is greater by -^ of a lunation each
year ; and since whole months are not counted in estimating the
Moon's age, it is (in months) the fractional part of
|i (Golden Number -1};
or, in days, the remainder when 11 {Golden No. 1 j is divided by 30.
Thus the Golden Number of 1892 is the remainder when 1893 is
divided by 19, i.e., 12. Hence, the epact is the remainder when
11 {12 1} is divided by 30, i.e., 1 ; hence, the Moon is one day old
on January 1, 1892, and new on December 31, 1891.
In the epact, fractions of a day are never reckoned. Owing to
the extra day in leap year, the rule is sometimes a day wrong; .but
it is near enough for fixing the ecclesiastical calendar.
ASTROXOMT.
282. Harvest Moon. The full Moon which occurs
nearest the autumnal equinox is called the Harvest Moon.
Owing to the Moon's direct motion in its orbit the time of
moonrise always occurs later and later every day, but in the
case of the harvest Moon the daily retardation is less than
in the case of any other full Moon, as we shall now show.
To simplify our rough explanations we suppose the Moon to
be moving in the ecliptic.
The Moon's E.A. determines the time at which the Moon
crosses the meridian (cf. 24). In consequence of the
orbital motion the R. A. increases continuously, just as in the
case of the Sun ( 30), only the increase is more rapid (360
per month instead of per year). Therefore the Moon transits
later and later every night.
When the Moon is in the first point of Aries it is passing
from south to north of the equator, and its declination is
increasing most rapidly. Now, the arguments of 123-125
are applicable to the Moon as well as the Sun, and they show
that, as the declination increases, there is, in north latitudes,
a corresponding increase in the length of time that the
Moon is above the horizon. The effect of this increase is to
lengthen the interval from the Moon's rising to its transit ;
this lengthening tends to counterbalance, more or less, the
retardation in the time of transit, thus reducing the retarda-
tion in the time of moonrise to a minimum.
Similarly it may be shown that whenever the Moon passes
the first point of Libra, the daily retardation of moonrise will
be a maximum, while that of the time of setting will be a
minimum. These phenomena, therefore, recur once each
lunar month.
Now, at harvest time the Sun is near ; hence, when the
Moon is near T it is full ; and the minimum retardation of the
Moon's rising, therefore, takes place at full Moon. And since
the Moon is then opposite the Sun, it rises at sunset. Both
these causes make the phenomenon more conspicuous in itself
than at other times, and as the continuance of light is useful
to the farmers when gathering in their harvest, the name
Harvest Moon has been applied.
At the following full Moon the phenomena are similar but
less marked. But as it is now the hunting season, the Moon
is called the " Hunter's Moon."
TOE MOON. 217
EXAMPLES. VIII.
1. If a, a 1 be the true and apparent altitudes of a body affected
by parallax, prove the equation a = a' + P cos a'.
2. If the Sun's parallax be 8'80", find the Sun's distance.
3. If in our latitude, on March 21, the Moon is in its first quarter,
about what time may it be looked for on the meridian, and how
long does it remain above the horizon ?
4. Show that from a study of the Moon's phases we can infer the
Sun to be much more distant than the Moon. Prove that if the
synodic period wore 30 days, and the Sun only twice as distant as
the Moon, the Moon would be dichotomized after only 5 days
instead of 7.
5. Taking the usual values of the Sun's and the Moon's distances,
calculate, roughly, the mean value of the angle ESM when the Moon
is dichotomized.
6. Under what conditions is the line of cusps perpendicular to
the horizon ? Consider specially the appearance to an observer on
the Arctic circle.
7. There was an eclipse of the Moon on Jan. 28, 1888, central at
11.10 in the evening. What is the Moon's age on May 21 of that
year?
8. Find approximately the position and appearance of the Moon,
relatively to the horizon, in latitude 50 N., in the middle of Novem-
ber at 10 P.M., when it is ten days old
9. At a place in the temperate zone can the Sun or the Moon be
longer above the horizon ?
10. What would be the effect on the Harvest Moon (i.) if the
polar axis of the Earth were perpendicular to the ecliptic, or (ii.) if
the Moon were to move in the ecliptic ?
218 ASTRONOMY.
EXAMINATION PAPER. VIII.
1. What is parallax, and under what conditions is the parallax of
a heavenly body greatest ? Show by some simple illustrations that
as the distance of an object increases, its parallax lessens.
2. Prove the formula sin p = sin P sin z, where P is the Moon's
horizontal parallax, and p its parallax when its zenith distance
is z.
3. How is the distance of the Moon determined by observations
made in the plane of the meridian? Why cannot the Sun's parallax
be accurately determined in this way ?
4. Show that we can calculate the Moon's sidereal period given
its synodic period and the length of the year. Find it, given that
these are 29^ and 365J: days respectively.
5. Describe the phases of the Moon, and find an expression for
the phase when the Moon is at a given elongation. Show how ao
observation of the Moon, when at its first quarter, would help ua to
find the ratio of the distances of the Moon and the Sun.
6. Describe some methods for determining the heights of lunar
mountains.
7. Describe the phenomena of the Moon's motion. Given that
the Moon moves in a plane inclined at 5 to the ecliptic, find the
lowest north latitude of a place where the full Moon can never rise
at the summer solstice.
8. Explain (and illustrate by figures) how it is that we see more
than half the Moon's surface, and define the terms node, phase,
libration.
9. Describe the general appearance presented by the solar system
to an observer situated at the centre of the Moon's hemisphere
turned towards the Earth. When would the Earth be partially
eclipsed to such an observer ?
10. Explain the phenomenon called the Harvest Moon, and show
that from a similar cause the daily retaliation in the sidereal time
of sunrise is least at the vernal equinox.
CHAPTER IX,
ECLIPSES.
SECTION I. General Description of Eclipses.
283. Eclipses are of two kinds, lunar and solar. If at
full Moon the centres of the Sun, Earth, and Moon arc very
nearly in a straight line, the Earth, acting as a screen, will
stop the Sun's rays from reaching the Moon, and the Moon
will, therefore, be either wholly or partially darkened. This
phenomenon is called a Lunar Eclipse.
On the other hand, if the three centres are nearly in a
straight line when the Moon is new, the Moon, by coming
between the Earth and the Sun, will cut off the whole or a
portion of the Sun's rays from certain parts of the Earth's
surface. In such parts the Earth will be darkened, and the
Sun will appear either wholly or partially hidden. This
phenomenon is a Solar Eclipse.
If the Moon were to move exactly in the ecliptic we
should have an eclipse of the Moon at every opposition, and
an eclipse of the Sun at every conjunction, for at either
epoch the centres of the Earth, Sun, and Moon would be in
an exact straight line. In consequence, however, of the
Moon's orbit being inclined to the ecliptic at an angle of
about 5i, the Moon at " syzygy " (conjunction or opposition)
is generally so far on the north or south side of the ecliptic
that no eclipse takes place. An eclipse only occurs when
the Moon at syzygy is very near the ecliptic, and,
therefore, not far from the line of nodes ( 271).
ASTBOtf. Q
220 * ASTRONOMY.
284. Different Kinds of Lunar Eclipse. Eclipses of
the Moon are of two kinds, total and partial. Let S, E be
the centres of the Sun and Earth respectively. Draw the
common tangents ^LSFand A'B'Vio the two glohes, meet-
ing on SE produced in F, and draw also the other pair of
tangents AB'K', A'BK cutting at Z7, between S and J If
the figure be supposed to revolve about &#, the tangents
will generate cones, enveloping the Sun and Earth, and
having their vertices at U and F. The space BVB , inside
the inner cone, is called the umbra ; the space between the
inner and outer cone is called the penumbra.* The
character of the lunar eclipse will vary according to the
following conditions :
FIG. 87.
(i.) If at opposition, the Moon falls entirely within the
umbra or inner cone B FZ?', as at J/J, no portion of the Moon's
surface then receives any direct rays from the Sun, and the
Moon is therefore plunged in darkness (except for the light
which reaches it after refraction by the Earth's atmosphere,
as explained in 193). The eclipse is then said to be total.
(ii.) If the Moon falls partly within and partly without
the umbra B VB\ as at J/~ 2 , the portion within the umbra
receives no light from the Sun, and is, therefore, obscured,
while the remaining portion receives light from part of the
Sun's surface about A, and is, therefore, partially illuminated.
The eclipse is then said to be partial.
*For further description of the formation of the umbra and
penumbra, see Wallace Stewart's Text-Book of Lie/lit, 5,
ECLIPSES. 221
(iii.) If the Moon falls entirely within the "penumbra,"
or outer cone, as at M z , it receives the Sun's rays from A,
but not from A'. There is no true eclipse, but only a
diminution of brightness (sometimes called a " penumbral
eclipse").
A lunar eclipse is visible simultaneously from all places on
that hemisphere of the Earth over which the Moon is above
the horizon at the time of its occurrence.
!N"ear the boundary of the hemisphere there are two strips
in the form of lunes, comprising those places respectively at
which the Moon sets and rises during the eclipse j at such
places only its beginning or end is seen.
285. Phenomena of a Total Eclipse of the Moon.
As the Moon gradually moves towards opposition, the first
appearance noticeable is the slight darkening of the Moon's
surface as it enters the penumbra. This darkening increases
very gradually as the Moon approaches the umbra, or true
shadow. At "First Contact" a portion of the Moon
enters the umbra, and the eclipse is then seen as a partial
eclipse, the dark portion being bounded by the circular arc
formed by the boundary of the umbra. As the Moon
advances, the dark portion increases till the whole of the
Moon is within the umbra, and the eclipse is total. "When
the Moon begins to emerge at the other side of the umbra,
the eclipse again becomes partial, and continues so until
"Last Contact," when the Moon has entirely emerged
from the umbra, after which the Moon gradually gets brighter
and brighter till it finally leaves the penumbra.
In the case of a partial eclipse, the umbra merely appears
to pass over a portion of the Moon's disc, which portion is
greatest at the middle of the eclipse*
286. Effects of Refraction on Lunar Eclipses. In
193 it was stated that, owing to atmospheric refraction, the
Moon's disc appears of a dull-red colour during the totality
of the eclipse. A still more curious phenomenon is noticed
when an eclipse occurs at sunset or sunrise. The refraction
at the horizon increases the apparent altitudes of the Sun and
Moon in the heavens, so that both appear above the horizon
when they are just below. Hence a total eclipse of the
Moon is sometimes seen when the Sun is shining.
222
ASTRONOMY.
287. Different Kinds of Solar Eclipse. An eclipse of
the Sun may be either total, annular, or partial. To
explain the difference between the first two kinds of eclipse,
let us suppose that the observer is situated exactly in the
line of centres of the Sun and new Moon, so that both bodies
appear in the same direction. Then, if the Moon's angular
diameter is greater than the Sun's, the whole of the Sun will
be concealed by the Moon ; the eclipse is then said to be
total. If, on the other hand, the Sun has the greater
angular diameter, the Moon will conceal only the central
portion of the Sun's disc, leaving a bright ring visible all
round; under such circumstances, the eclipse is said to be
annular. Lastly, if the observer is not exactly in the line
of centres, the Moon .may cover up a segment at one side of
the Sun's disc ; the eclipse is then partial.
Now, the Moon's angular diameter varies, according to the
distance of the Moon, from 28' 48" at apogee to 33' 22" at
perigee, the corresponding limits for the Sun's diameter being
31' 32" at apogee, and 32' 36" at perigee. Hence, both total
and annular eclipses of the Sun are possible. Thus, when
the Sun is in apogee and the Moon in perigee an eclipse must
be either total or partial ; when the Sun is in perigee and
the Moon in apogee, an eclipse must be annular or partial.
FIG. 88.
288. Circumstances of a Solar Eclipse. Fig. 88
shows the different circumstances under which a solar eclipse
is seen from different parts of the Earth. Draw the common
tangents CDQ, C'D'Q, CRU, C'RD to the Sun and Moon,
forming the enveloping cones DQD' and fRg\ these consti-
tute respectively the boundaries of the umbra and penumbra
of the Moon's shadow. First let the umbra DQD* meet the
Earth's surface (E^ before coming to a point at Q, the curve
223
of intersection being de. Also let the penumbra fRg meet
the Earth's surface in the curve fg. Then from anyplace on
the Earth within the space de the Sun appears totally eclipsed.
At a place elsewhere within the penumbra fg, the Sun appears
partially eclipsed, a portion only being obscured by the Moon.
Next let the umbra DQD' come to a point Q before
reaching the Earth E y Then, if the cone of the umbra be
produced to meet the Earth in d'e', an observer anywhere
within the space d'e' sees the eclipse as an annular eclipse.
At any place elsewhere within the penumbra /y, the eclipse
appears partial, as before. At parts of the Earth which fall
without the penumbra there is no eclipse. Hence a solar
eclipse is only visible over a part of the Earth's surface,
and its circumstances are different at different places.
As the Sun and Moon move forward in their relative orbits,
and the Earth revolves on its axis, the two cones of the
Moon's shadow travel over the Earth, and the eclipse becomes
visible from different places in succession The inner cone
traces out on the Earth a very narrow belt, over which
the eclipse is seen as a total or annular eclipse, according
to circumstances. The outer cone, or penumbra, sweeps out
a far broader belt, including that part of the Earth's surface
where the eclipse is visible as a partial eclipse.
A total or annular eclipse of the Sun, like a total eclipse
of the Moon, always begins and ends as a partial eclipse, the
totality or annular condition only lasting for a short period
about the middle of the eclipse. The maximum duration
of totality at the Equator is just under eight minutes.
In the case of an annular eclipse, there are two internal,
as well as two external, contacts, and the eclipse remains
annular during the interval between the internal contacts.
This may sometimes be rather more than twelve minutes.
Owing to the limited area of the belt over which a solar
eclipse is visible, the chance that any eclipse may be visible
at any given place is far smaller than in the case of a lunar
eclipse. The chance of an eclipse being total at any place is
very small indeed. The last eclipse visible as a total eclipse
in England occurred in 1724 ; the next will take place on
June 29th, 1927. One or more partial eclipses are visible at
Greenwich in nearly every year.
224
ASTROBOMf.
SECTION IT. Determination of the Frequency of Eclipses.
289. To Find the Limits of the Moon's geocentric
position consistent with a Solar or Lunar Eclipse.
In Fig. 89, let the plane of the paper represent any piano
through the Sun's and Moon's centres; and let ABV and
A 'B' ^"represent the common tangents bounding the cone of
the Earth's true shadow. Let A UB' be the other common
tangent, which goes (nearly) through B 1 ; and let the line SE,
joining the centres of the Sun and Earth, meet the common
tangents in Fand U. Let T, t, t' be those points on AB Fand
AB' whose distance from E is equal to that of the Moon.
FIG. 89.
Then, if J/i, M. 2 denote the positions of the Moon's centre,
when touching the cone B V externally and internally at T,
it is evident that a lunar eclipse occurs whenever the full
Moon is nearer the line of centres than N r Hence, if m
denote the Moon's angular semi-diameter TEM^ the Moon's
angular distance from JSVmnst be less than VEM or VET+ m.
Similarly, the lunar eclipse is total when the Moon is not
further from the line of centres than Jf 2 ; for this the Moon's
(geocentric) angular distance from the line of centres must
be not greater than VEN^ qr VET-m.
Let m v m 2 be the centres of the Moon at internal and
external contact with ^47? near t. There is evidently a solar
eclipse visible at some point of the Earth's surface (such as
Z?) as a partial eclipse, if the Moon's angular distance from
the Sun is less than SEm^ or SEt+m.
Supposing the Moon's distance to be such that its angular
radius is less than that of the Sun, there is an annular
eclipse whenever the Moon lies wholly within the cone A VA',
as at m v This requires the Moon's geocentric angular dis-
tance from the Sun to be less than SEm v or SEtm.
ECLIPSES. 225
If, however, the Moon is so near that its angular radius
is greater than that of the Sun, the angle it subtends is
greater than ABA', and therefore there is a total
eclipse at B whenever the edge of the Moon reaches the
internal tangent A'B. Taking m s to represent the corre-
sponding position of the Moon when touching the other
tangent AB' at t' (for the sake of clearness in the figure),
we see that, in order that there may be a total eclipse
fomewhere on the Earth's surface, the geocentric angular
distance between the Moon's and Sun's centres must be less
than SEm B or SEt' + m.
Now, as the cone A VA' tapers to a point at V, the breadth
of its cross section is greater near m v m v m s than near M ly M 9 ,
and when the Moon is in syzygy, its angular distance from EV
or ES = its latitude. Hence the limits of latitude aro greater
for a solar than for a lunar eclipse, and therefore the proba-
bility of the occurrence of a solar eclipse is greater than the
probability of a lunar eclipse. This explains why, on the
whole, solar eclipses are more frequent than lunar.
*290. We shall now calculate the angles VEM^ VEM. 2 ,
SEm^ , SEm^ SEm y Let p, P denote the horizontal parallaxes .
of the Moon and Sun respectively; m, s their respeetive
angular semi-diameters (Fig. 89). We have s= Z SEA,
p-A BTE = z BtE= Z B't'E, P = z BAE=* Z B'AE,
and m = Z TJEM l = Z TEM Z = z tEm l = Z tEm^ = Z t'Em y
For the lunar eclipses we have, from the triangle TEA,
Z ETJB+ Z EAB = 180- z TEA = Z VET+ Z SEA
.-. VET = LETB+ LEAB- LSEA=p+P-s-,
.-. z VEMi = ^ VET+ z TEM^ = p+ps+m ;
and z rJSMt = Z VET- z TEM 9 =p+P-s-m ;
For the solar eclipses we have, from the triangle tEA,
Z EtB z EAB = z tEA - Z SEt z SEA
.-. z SEin^ = p P + s + in ,
and z SEni^ =p JP + s m,
Lastly, from the triangle t'EA we have
Z Et'B'- /.EAB' z AEt' = Z AES+ Z SEf.
:. L SEt 1 = Z B't'E- z B'AE- Z ^4^>S = p-P-s.
. . z >S^i, = p -P s -i- w-
226
[As an examplo, the student may show that the greatest
latitudes the Moon can have, in ordei that it may bo partially
or wholly within the penumbra at opposition arc p + s + P-f m
and p + s + P m respectively.]
*291. Greatesb Latitudes of the Moon at Syzygy.
Since S and V are in the ecliptic, it follows that when
the Moon is in conjunction or opposition, the plane of the
paper in !Fig. 89 is perpendicular to the ecliptic. Therefore
the angles VEM^ VEM. 2 measure the Moon's latitude at con-
junction, and SJEm v SEm. 2 , SEm 3 measure its latitude at
opposition in the positions represented. The above expres-
sions are, therefore, the greatest possible latitudes at syzygy
consistent with eclipses of the kinds named.
Now, taking the mean values we have, roughly,
s=l6'- m=l5'; p = 57' ; P = 0' 8".
Substituting these values, and collecting the results, we have,
roughly, the following limits for the Moon's geocentric lati-
tude, or angular distance from the line of centres :
(1) For a lunar eclipse, VEM^ =p+ P-s+m = 56';
(2) Fora total lunar eclipse, VEM^p + Psm = 26';
(3) For a solar eclipse, SEm^ =zp P+s + m 88';
(4) For an annular eclipse, SEm z = pP+s m = 58'.
Lastly, taking the Sun at apogee, and the Moon at perigee,
we have, m = 17' and s = 16' nearly, whence we have, in the
most favourable case,
(40) For a total solar eclipse, SEm^ -=p P 8+ m = 58'.
292. Ecliptic Limits. From the last results it appears
that a lunar eclipse cannot occur unless at the time of oppo-
sition the Moon's latitude is less than about 56', and that a
solar eclipse cannot occur unless at conjunction the Moon's
latitude is less than about 88'. Now the Moon's latitude
depends on its position in its orbit relatively to the line of
nodes ; hence there will be corresponding limits to the Moon's
distance from the node consistent with the occurrence of
eclipses. These limits are called the Ecliptic Limits.
*The ecliptic limits may be computed as follows : Let the
geocentric direction of the Moon's centre be represented on
the celestial sphere by Jf. Let JV represent the node,
ECLIPSES. 227
secondary to the ecliptic. [The ecliptic limit, strictly speak-
ing, means the limit of NH measured along the ecliptic, and
not that of NMJ\
Now the limit of latitude MH lias been calculated in the
last paragraph for the different cases. Let this be denoted by
I. Also let I be the inclination of the Moon's orbit to the
ecliptic. Then in the spherical triangle NHM, right-angled
at H t we have HM = I, and z JINM= /; both of these are
known, hence NIL can be calculated.
FIG. 90.
For rough purposes it will be sufficient either to treat the
small triangle ITNMas a plane triangle (Sph. Geom. 24), or
to regard Mil as approximately the arc of a small circle,
whose pole is JV. The first method gives
.-. NH= I cot /.
Or, adopting the second method, we have (Sph. Geom. 17)
I = MH= z MNITx sin NH = /sin Nil-,
.-. sin NH =1/1,
whence the ecliptic limit Nil is found.
EXAMPLES.
1. To find the Lunar Ecliptic Limit. For a lunar eclipse we have,
by 291, I = 56'. Also, 1-5 roughly.
Ilence
= sin 11 (from table of natural sines)
and the lunar ecliptic limit is about 11.
2. To find the Solar Ecliptic Limit. For a solar eclipse we have
I = 88'. Hence, taking I = 5 as before, we have
= sin 17, roughly,
and the solar ecliptic limit is about 17.
228 ASTRONOMY.
293. Major and Minor Ecliptic Limits. Owing to
the variations in the distances of the Sun and Moon their
parallaxes and angular semi-diameters are not quite constant.
Hence the exact limits of the Moon's latitude /, as calculated
by the method of 291, are subject to small variations.
This alone would render the ecliptic limits variable. But
there is another cause of variation in the ecliptic limits,
arising from the fact that Z the inclination of the Moon's
orbit, is also variable, its greatest and least values being about
5 19' and 4 57'.
The greatest and least values of the limits for each kind of
eclipse are called the Major and Minor Ecliptic Limits.
For an eclipse of the Moon the major and minor ecliptic
limits have been calculated to be about 12 5' and 9 30' re-
spectively at the present time. For an eclipse of the Sun the
limits are 18 31' and 15 21' respectively.
Thus a lunar eclipse may take place if the Moon, when
full, is within 12 5' of a node; and a lunar eclipse must
take place if the full Moon is within 9 30' of a node.
Similarly, a solar eclipse may take place if the Moon, when
new, is within 18 31', and a solar eclipse must take place if
the new Moon is within 15 21' of a node.
The mean values of the lunar and solar ecliptic limits are
now 10 47' and 16 56'. But the eccentricity of the Earth's
orbit is very slowly decreasing ; consequently the major limits
are smaller and the minor limits larger than they were, say,
a thousand years ago.
294. Synodic Revolution of the Moon's Nodes. An
eclipse is thus only possible at a time when the Sun is within
a certain angular distance of the Moon's nodes. Hence the
period of revolution of the Moon's nodes, relative to the Sun,
marks the recurrence of the intervals of time during which
eclipses are possible. This period is called the period of a
synodic revolution of the nodes.
In 273 it was stated that the Moon's nodes have a retro-
grade motion of about 19 per annum, more exactly 19 21'.
In one year (365d.) the Sun, therefore, separates from a node
by 360+ 19 21' or 379*35, hence it separates 360 in
(360 x 365)/379-35 days, or about 346'62d. This, then,
is the period of a synodic revolution of the node.
EClttSES. 229
In a synodic lunar month (29| days), the Sun separates
from the line of nodes by an angle
379jx29j-h365, or 30 36',
a result which will be required in the next paragraph.
295. To find the Greatest and Least number of
Eclipses possible in a Year. Let the circle in Fig. 9*
represent the ecliptic, and let JVJ n be the Moon's nodes.
Take the arcs NL, NL', til, nl' each equal to the lunar eclip-
tic limit, and NS, JVb", ns, ns' each equal to the solar ecliptic
limit. Then the least value of S3' or ss' is twice the minor
solar ecliptic limit, and is 30 42', and this is greater than
30 36', the distance traversed by the Sun relative to the
nodes between two new Moons. Hence, at least one new
Moon must occur while the Sun is travel-
ling over the arc SS', and two may occur.
Therefore there must be one, and there may
le two eclipses of the Sun, while the Sun
is in the neighbourhood of a node.
Again, the greatest value of LL', IV is
double the major lunar ecliptic limit, and
is, therefore, 24 10'. This is consider-
ably less than the space passed over by
the Sun relative to the nodes between Fio.TJi.
two full Moons. Hence, there cannot
be more than one full Moon while the Sun is in the arc LL\
and there may be none. Therefore there cannot le more than
one eclipse of the Moon while the Sun is in the neighbourhood of
a node, and there may be none at all.
296. The case most favourable to the occurrence of
eclipses is that in which the Moon is new just after the Sun
has come within the solar ecliptic limits, i.e., near S. There
will then be an eclipse of the Sun.
When the Moon is full (about 14| clays later) the Sun will
be near N, at a point within the lunar ecliptic limits ; there
will therefore be an eclipse of the Moon.
At the following new Moon the Sun will not have reached
S'-, and there will be a second eclipse of the Sun.
In six lunations from the first eclipse the Sun will have
travelled through just over 180, and will be within the space
ss', near s ; there will therefore be a third eclipse of the Sun.
2,30 ASTROttOMt.
At the next full Moon the Sun will be near , and there
will be a second eclipse of the Moon.
The Sun may just fall within the space 88 near *' at the
next new Moon ; there will then be & fourth eclipse of the Sun.
In twelve lunations from the first eclipse, the Sun will
have described about 368, and will, therefore, be about 8
beyond its first position, and well within the limits ss' ; there
will, therefore, be & fifth eclipse of the Sun.
About 14f days later, at full Moon, the Sun will be well
within the lunar ecliptic limits LL, and there will be a third
eclipse of the Moon.
All these eclipses occur in 12J lunations, i.e., 369 days, or
a year and four days. "We cannot, therefore, have all the
eight eclipses in one year, but
There may be as many as seven eclipses in a year, namely,
either five solar and tivo lunar, or four solar and three lunar.
297. The most unfavourable case is that in which the
Moon is full just before the Sun reachesthe ecliptic limits at L.
At new Moon the Sun will be near N, and there will be
one solar eclipse.
At the next full Moon the Sun will have passed L', so that
there will be no lunar eclipse. After
six .lunations the Sun will not have
arrived at I.
At the next new Moon the Sun will
be within the ecliptic limits, and there
will be a second solar i-flijM-e.
At the next full Moon the Sun will be
again just beyond I', and at 12 lunations
from first full Moon, the Sun may again
not have quite reached L.
At 12 lunations there will be a third solar eclipse.
The interval between the first and third eclipses will be 12
lunations, or about 354 days. If, therefore, the first eclipse
occurs after the llth day of the year, i.e., January 11, the
third will not occur till the following year. Therefore,
The least possible number of eclipses in a year is two. These
must both le solar eclipses.
ECLIPSES. 231
298. The Saros of the Chaldeans. The period of a
synodic revolution of the nodes is ( 294) approximately
346-62 days. Hence,
19 synodic revolutions of the node take 6585-78 days.
Also 223 lunar months = 6585*32 days.
It follows that after 6585 days, or 18 years 11 days, the
Moon's nodes will have performed 19 revolutions relative to
the Sun, and the Moon will have performed 223 revolutions
almost exactly. Hence the Sun and Moon will occupy almost
exactly the same position relative to the nodes at the end of
this period as at the beginning, and eclipses will therefore
recur after this interval.
The period was discovered by observation by the Chaldean
astronomers, who called it the Saros. By a knowledge of it
they were usually able to predict eclipses. Indeed, in the
records of eclipses handed down to us in the form of cuneiform
inscriptions, they invariably stated whether the circumstances
accorded with prediction by the Saros or not.
A "synodic revolution of the Moon's apsides," or the
period in which the Sun performs a complete revolution
relative to the Moon's apse line, occupies 411-74 days.
Hence sixteen such revolutions occupy 6587*87 days, or
about two days longer than the Saros. Therefore the Moon's
line of apsides also returns to very nearly the same
position relative to the Sun and Moon. Hence, the
solar eclipses, as they recur, will be nearly of the same
kind (total or annular) in each Saros. The whole number
of eclipses in a Saros is about 70. The average of all
eclipses from B.C. 1207 to A.D. 2162 shows that there are
20 solar eclipses to 13 lunar.
The present values of the mean solar and lunar ecliptic limits,
16 56', and 10 47', are in the ratio of 31 : 18 very nearly.
This ratio gives, on the whole, a higher average proportion
of solar eclipses to lunar than that given above. It must,
however, be remembered that all the angles used in calcu-
lating the limits are subject to gradual changes. Con-
sequently the numbers of eclipses in that period aro subject to
very gradual variation ; after a large number of Saroses have
recurred, the order of eclipses in each will have changed,
232 ASTRONOMY.
*SECTION III. Occultations Places at which a Solar Eclipse
is visible.
299. Occultations. When the Moon's disc passes in front
of a star or planet, the Moon is said to occult it.
An occultation evidently takes place whenever the ap-
parent angular distance of the Moon's centre from the star
becomes less than the Moon's angular semi- diameter. As the
apparent position of the Moon is affected by parallax, the cir-
cumstances of an occultation are different at different
places on the Earth's surface.
FIG.
Let m denote Moon's angular semi-diameter, p its horizontal
parallax. In the figure, let E and M be the centres of the
Earth and Moon, and let s C, sC' represent the parallel rays
coming from a star, and grazing the Moon's disc. These
rays cut the Earth's surface along a curve 00*, and it
is evident that only to observers at points within this curve
is the star hidden by the Moon's disc. Let EC, Es, EM,
EC' cut the Earth's surface in c, x, m, c' ; the rays EC, EC'
cut the Earth's surface in a small circle cc, whose angular
radius mEc = MEC = m. Let d be the geocentric angular
distance SEM between the Moon's centre and the star.
Then the angle ECO = angle subtended by the Earth's
radius EO at C ;
= parallax of C when viewed from ;
= ^sin COZ(\ 249);
= p sin OEx (by parallels).
But ECO = CEs ;
= angle subtended by ex ;
.% sin OEx = aD " Ic **'.
P
ECLIPSES. 233
Hence we have the following construction for the curve
separating those points on the Earth's surface at which the
occultation is visible at a given instant from those at which
the star is not occulted. Taking the sublunar point m as
pole, describe a circle cc on the terrestrial globe, with the
Moon's angular semi-diameter (m) as radius. Through the sub-
stellar point x draw any great circle, cutting this small circle
in any point c. Measure along it an arc c such that sin c
is always the same multiple f ) of me. The locus of the
points 0, thus determined, is the curve required.
Half of the circle cc' consists of points under the advancing
limb of the Moon; hence, over the portion of the curve 00'
corresponding to this half -circle, the occultation is just
beginning. At points on the other half of cc the Moon's
limb is receding ; hence over the other portion of 0' the
star is reappearing from behind the Moon's disc.
Since the greatest and least values of ex in any position
are d+m and d m, it is evident that the greatest value of d
for which an occultation can take is when
d m=p; d=m+p.
300. Occnltation of a Planet. If s be a planet, the
lines Es, Os can no longer be regarded as rigorously parallel;
but the angle between them, Es 0,
= angle subtended at s by the Earth's radius EO
= parallactic correction at ( 248)
= P sin ZOs ( 249) = P sin OEx very nearly.
As before, EGO - p sin OEx. But ECO = EsO + CEs;
.-. p sin OEx = P sin OEx + ex ; sin OEx = -^.
pP
With this exception, the construction is the same as for a
star.
If the planet be so large that we must take account of its
angular diameter, the method of the next paragraph must be
used.
234 ASTRONOMY.
301. Eclipse of the Sun. There is a total eclipse of the
Sun, provided the Moon's disc completely covers the Sun's;
this occurs if the Moon's angular semi-diameter (m) is larger
than the Sun's (s), and the apparent angular distance
between the Sun's and Moon's centres (as seen from any
point at which the eclipse is visible) is less than m s.
ilcnce, if the Moon's angular semi-diameter were reduced
to m s, the Sun's centre would then be occulted. Hence
the points 0, whose locus encloses the places from which
the eclipse is visible, can be found as follows :
With centre m the sublunar point, and angular radius
m s, describe a circle. Through the subsolar point x draw
any arc of a great circle xc, cutting the circle in 0, and
take 0, on xc produced, such that
xo
p-p
For an annular eclipse m < *, and the apparent angular
distance between the centres is s m\ hence the same con-
struction is followed, save that s m is the angular radius of
the small circle first described. For a partial solar eclipse,
the angular radius is s + m.
When a planet has a sensible disc, the beginning of its
occupation may be compared to a partial eclipse of the Sun ;
and the planet is entirely occulted when the conditions are
satisfied corresponding to those for a total eclipse.
EXAMPLE. Supposing the centres of the Earth, Moon, and Sun to
be in a straight line and the Moon's and Sun's semi-diameters to be
exactly 17' and 16', to find the angular radii of the circles on the
Earth over which the eclipse is total and partial respectively, taking
the relative horizontal parallax as 57'.
At those points at which the eclipse is total, the apparent angular
distance between the centres, as displaced by parallax, must be not
greater than 17' - 16', or 1'. Hence, since the centres are in a line,
with the Earth's centre, the parallactic displacement must be not
greater than 1'. Hence, if z be the Sun's zenith distance at the
boundary, then 57' sin z = 1' ; .*. sin z = ?, or approximately cir-
cular measure of z = -^j-. But a radian contains about 57 ; .*. -gV of
a radian = 1 approx. Hence the eclipse is total over a circle of
angular radius 1 about the sub-solar point.
Similarly, the eclipse is partial if 57' sin z < 16' + 17', or 33', or
sin < ff, or '58. From a table of natural sines, we find that
sin- 1 '58 = 35^ roughly ; therefore the angular radius is 35 3 .
ECLIPSES. 235
EXAMPLES ON ECLIPSES GENERALLY.
1. To find (roughly) the maximum duration of an eclipse of tho
Moon, and the maximum duration of totality.
From 291 we see that a lunar eclipse will continue as long as the
Moon's angular distance from the line of centres of the Earth and
Sun is less than 58', and the eclipse will continue total while the
angular distance is less than 26'. Hence, the maximum duration of
the eclipse is the time taken by the Moon to describe 2 x 58', or 116',
and the maximum duration of totality is the time taken to describe
2 x 26', or 52'.
Now the Moon describes 360 (relative to the direction of the Sun)
in the synodic month, 29 -r days. Therefore, the times taken to
describe 116' and 52'' respectively are
29^x116 , 29jj<52 d
360x60 360x60 ay8> ,.
i.e. 3h. 48m. and Ih. 42m.,
and these are the maximum durations of the eclipse and of totality.
The eclipse of Nov. 15, 1891, lasted 3h. 28m., and was total for Ih. 23m.
2. To calculate roughly the velocity with which the Moon's
shadow travels over the Earth. (Sun's distance = 93,000,000 miles.)
The radius of the Moon's orbit being about 240,000 miles, its cir-
cumference is about 1,508,000 miles. Relative to tho line of centres,
the Moon describes the circumference in a synodic month, i.e., about
29 days. Hence its relative velocity is about 1,508,000 -f- 29, or
51,000 miles per day, i.e., 2,100 miles per hour. If q 'denote the
point where the middle of the shadow reaches the Earth (Fig. 88),
and if the Earth's surface at q is perpendicular to Sq, wo have
velocity of q : vel. of M = Sq : 8M
= 93,000,000 : 93,000,000-240,000 - 1-0026 nearly.
Hence the velocity of the shadow at q = vel. of M very nearly
= 2,100 miles an hour.
To find the velocity of the shadow relative to places on the Earth,
we must subtract the velocity of the Earth's diurnal motion. This,
at the Earth's equator, is about 1,040 miles an hour. Hence, if the
Earth's surface and the shadow are moving in the same direction,
the relative velocity is about 1,060 miles an hour.
3. To find the maximum duration of totality of the eclipse of
the example on page 234, neglecting the obliquity of the ecliptic.
The angular radius of the shadow being 1, or about 69 miles, its
diameter is 139 miles. The obliquity of the ecliptic being neglected,
the eclipse is central at a point on the equator, and the shadow and
the Earth are therefore moving in the same direction with relative
velocity 1,060 miles an hour (by Question 2). The greatest duration
of totality is the time taken by the shadow to travel over a distance
equal to its diameter, i.e., 139 miles, and is therefore 139 x 60/1000
minutes, i.e., 7'9 minutes (roughly).
ASTKON. E
236 ASTEONOMT.
EXAMPLES. IX.
1. If a total lunar eclipse occur at the summer solstice, and at
the middle of the eclipse the Moon is seen in the zenith, find the
latitude of the place of observation.
2. If there is a total eclipse of the Moon on March. 21, will the
year b.e favourable for observing the phenomenon of the Harvest
Moon?
3. Having given the dimensions and distances of the Sun and Moon,
show how to find the diameter of the umbra where it meets the
Earth's surface.
4. Calculate (roughly) the totality of a solar eclipse, viewed from
the Equator at the Equinox, supposing
Moon's diameter 2,1GO miles, Sun's diameter 400 times Moon's ;
Distance of Moon from Earth 222,000 miles ;
Distance of Sun from Earth 92,000,000 miles.
6. If 8 is the semi-diameter of the Sun, and p, P the horizontal
parallaxes of the Sun and the Moon at the time of a lunar eclipse, show
that to an observer on the Earth the angular radius of the Earth's
shadow at the distance of the Moon is P + p S, and that of the
penumbra P + p + S. Determine, also, the length of the shadow.
6. If the distance of the Moon from the centre of the Earth is
taken to be 60 times the Earth's radius, the angular diameter of the
Sun to be half a degree, and the synodic period of the Sun and
Moon to be 30 days, show that the greatest time which can be
occupied by the centre of the Moon in passing through the umbra
of the Earth's shadow is about three hours, and explain how this
method might be employed to find th<? Sun's parallax.
7. If the distance of the Moon were diminished to 30 times the
Earth's radius, what would be the time occupied in passing through
the shadow ?
8. Determine what length of the axis of the Earth's shadow is
absolutely dark, having given that the horizontal refraction is about
35' j and account for the copper colour often seen on the Moon
when eclipsed.
9. What kind -of eclipse is most suitable for the determination of
longitude, and why?
10. What would be the greatest possible inclination of the plane
of the Moon's orbit to the ecliptic, that there might be a partial
eclipse at each conjunction?
(The greatest distance of the Moon - 60 x Earth's radius.)
ECLIPSES. 237
EXAMINATION PAPER. IX.
1. What is the cause of eclipses of the Sun, and of the Moon ?
Why is a solar eclipse visible over so small a portion, and a lunar
eclipse over so large a portion of the Earth ?
2. Account for the phenomenon called a Lunar Eclipse. Show
that it begins and ends at the same instant at all places from which
it is visible.
3. Explain briefly the manner in which a solar eclipse passes over
the Earth.
4. Explain clearly how an annular eclipse of the Sun is produced.
Why are there no annular eclipses of the Moon ? Explain why
solar eclipses are sometimes total and sometimes annular.
5. Explain why, though there are, on the whole, more eclipses of
the Sun than of the Moon, many more of the latter than of the
former are visible at Greenwich.
6. Define umbra and pemnnbra. Calculate the lengths of the
cones of shadow (umbra) cast by the Earth and Moon, and find the
breadth of the Earth's umbra at the distance of the Moon.
7. Define and roughly calculate the solar and lunar ecliptic limits.
What is the greatest number of lunar eclipses which can occur in a
year ? What is the least number of solar eclipses which can occur
in the same interval ?
8. What is the Saros ? State its length, and why it has to be
an exact multiple of the synodic period of the Moon and nearly
a multiple of that of the node.
9. Do occultations of a star by the Moon occur at the same
instant at all observatories ?
10. Show how to find at what point (if any) of the Earth's
surface a solar eclipse will be central.
CHAPTER X.
THE PLANETS.
SECTION I. General Outline of the Solar System.
302. The name planet, or " wanderer," was applied by
the Greeks to designate all those celestial bodies, except
comets and meteors, which changed their position relative to
the stars, independently of the diurnal motion ; these included
the Sun and Moon. At present, however, only those bodies
are called planets which move in orbits about the Sun. The
Sun itself is considered to be a star, while the Earth is
classed among the planets, and the Moon, which follows the
Earth in its annual path, and has an orbital motion about
the Earth, is described, along with similar bodies which
revolve about other planets, as a satellite or secondary.
303. The Sun, 0, is distinguished by its immense size and
mass. It forms the centre of the solar system, for, in spite
of the great distances of some of the furthest planets, the
centre of mass of the whole system always lies very near the
Sun. The Sun resembles the other fixed stars in being self-
luminous.
Its diameter is 110 times that of the Earth, or nearly
twice as great as the diameter of the Moon's orbit about the
Earth.
Erom observing the apparent motion of the spots or cavities
which are usually seen on the Sun's disc, it is inferred that
the Sun rotates on its axis in the sidereal period of about
25 days.
THE PLANETS. 239
304. Bode's Law. The distances of the planets from the
Sun have been observed to be approximately connected by a
remarkable law known as Bode's Law. This law is purely
empirical, that is, it is merely a result of observation, and it
has not as yet been proved to be a consequence of any known
physical principle. Moreover, it is only roughly true, giving,
as it does, a result far too great for the furthest planet
Neptune.
The law is given by the following rule : Write down the
series of numbers
0, 3, 6, 12, 24, 48, 96, 192, 384,
each number (after the second) being double the previous one.
Now add 4 to every term ; thus we obtain
4, 7, 10, 16, 28, 52, 100, 196, 388.
These numbers represent fairly closely the relative distances
of the various planets from the Sun, the distance of the Earth
(the third in the series) being taken as 10.
The planets all revolve round the Sun in the same direction
as the Earth. Their motion is, therefore, direct.
305. Mercury, > ^ s the planet nearest the Sun, its dis-
tance on the above scale being represented by 4. It is
characterized by its small size, the great eccentricity of its
elliptical orbit, amounting to about 3-, and the great inclina-
tion of the orbit to the ecliptic, namely, about 7. The
sidereal period of revolution round the Sun is about 88 of our
days.
Thus, Mercury's greatest and least distances from the Sun
ore in the ratio of 1+i : 1 i (cf. 149),
or 3:2.
Professor Schiaparelli, of Milan, has found that Mercury
rotates on its axis once in a sidereal period of revolution ;
consequently it always turns nearly the same face to the
Sun, like the Moon does to the Earth ( 276).
Owing, however, to the great 'eccentricity of the orbit, the
" libration in longitude " is much greater than that of the
Moon, amounting to 47. Consequently, rather over one
quarter of the whole surface is turned alternately towards
and away from the Sun, three-eighths is always illuminated,
and three-eighths is always dork.
240 ASTKONOMY.
306. Venus, ? , is the next planet, its mean distance
from the Sun being represented by about 7 (really 7-2). Its
orbit is very nearly circular, arid is inclined to the ecliptic at
an angle of about 3 23'.
Yenus revolves about the Sun in a period of 224 days.
307. The Earth, , comesnext, its mean distance being re-
presented by 10, audits orbit very nearly circular (eccentricity
^i_). Its period of revolution in the ecliptic is 365|- days,
and its period of rotation is a sidereal day, or 23h. 56m. mean
time. It is the nearest planet to the Sun having a satellite
(the Moon, ([ ), which revolves about it in 27|- days.
308. Mars, <?, is at a mean distance represented roughly
by 16, or more accurately by 15 '2. Its orbit is inclined at
less than 2 to the ecliptic, and is an ellipse of eccentricity
about T a T . It revolves about the Sun in a sidereal period of
about 686 days, and rotates on its axis in about 24h. 37m.
Mars has two very small satellites, which revolve about it
in the periods 7J and 30^ hours, roughly. The appearance
which would be presented by the inner satellite, if observed
from Mars, is rather interesting. As it revolves much faster
than Mars, it would be seen to rise in the west and set in the
cast twice during the night. The outer satellite would appear
to revolve slowly in the opposite direction from east to west.
The inner satellite is eclipsed often at opposition, and would
appear to transit the Sun's disc often at conjunction.
309. The Asteroids. The next conspicuous planet,
Jupiter, is at a distance represented by 52 ; but, according
to Bode's law, there should be a planet at the distance 28. It
was for a long time thought that no planet existed at this
distance, but the gap was filled, at the beginning of the cen-
tury, by the discovery of a number of small planets, to which
the name of Asteroids, or Minor Planets, was given.
Since that time a few new asteroids have been discovered
almost every year, the total number found up to October 15,
1891, being 321. It is probable that this number will be
very largely increased by stellar photography.
The largest asteroid, Vesta, is just visible to the naked
eye when in opposition ; and the length of its diameter is
THE PLANETS. 24!
between -1 and -2 of that of the Moon. Among the others
Juno, Ceres, Pallas, and Astreea are the most con-
spicuous telescopic objects. Many of the smaller asteroids
are less than ten miles in diameter, and are probably simply
masses of rock flying round and round the Sun.
The periodic times of revolution of the asteroids vary con-
siderably, but their average is about 1,600 days. The orbits
are in many cases very oval, the eccentricity of one (Poly-
hymnia) being over i, and they are often inclined at consider-
able angles to the ecliptic, the inclination in the case of Pallas
amounting to nearly 35, while that of Juno is 13.
The planets outside the asteroid belt are distinguished from
those hitherto described by their far greater dimensions and
masses, and by their smaller densities. In this respect they
resemble the Sun. They are also supposed to be at high
temperatures, though not hot enough to emit light.
310. Jupiter, I/, is at a mean distance almost exactly re-
presented by 52. It revolves round the Sun in a period of
twelve years, in an orbit nearly circular and inclined at only
1J to the ecliptic.
The diameter of Jupiter is about eleven times that of the
Earth, and through a telescope the disc is seen to be encircled
with a series of belts or streaks parallel to its equator. On
account of their variability, these are supposed to be due to
lelts of clouds in the atmosphere of the planet.
Jupiter is now known to have five satellites. The four
outer ones arc interesting as being the first celestial bodies
discovered with the telescope by its inventor Galileo (A.D.
1610). A fairly powerful opera glass will just show them.
The outermost of all revolves in an ellipse of considerable
eccentricity inclined to the ecliptic plane at about 8, its
period being about lOd. 17h. The three next revolve in
orbits nearly circular, and in the ecliptic, in periods of 7d. 4h.,
3d. 13h., and Id. 18jh. The fifth or innermost satellite
has only just been discovered (1892) by Mr. Barnard with
the great Lick telescope ; it revolves in a period of nearly
12h., at a mean distance of 70,000 miles from the surface, or
113,000 miles from the centre of Jupiter. Jupiter's satellites
are frequently eclipsed by passing into the shadow cast by Jupi-
ter, or occulted whenJupitercom.es between them and the Earth,
242 ASTRONOMY.
311.. Saturn, J? , is at a mean distance from the Sun of 95 J,
taking the Earth's distance as 10. This is rather less than
the distance given by Bode's Law. The periodic time of revo-
lution is 29-^ years. The orbit is nearly circular, and inclined
to the ecliptic at an angle of 2J.
Saturn's rings are among the most wonderful objects
revealed by the telescope. They appear to be three flat
annular discs of extreme thinness, lying in a plane .inclined,
to the ecliptic at an angle of about 28, and extcndingjto a
distance rather greater than the radius of the planet ; the
middle ring is by far the brightest, whilejthe inner ring is
very_f aint. When the Earth is in the plane of the rings they
are seen edgewise, and, owing to their very small thickness,
they then become invisible except in the best telescopes.
It is probable that the rings consist of a large number _oi
small satellites or meteors. It is certain that they do not
consist of a continuous ma?s of solid or liquid matter. The
surface of the planet itself is encircled with belts similar to
those on Jupiter.
In addition to the rings, Saturn has at least eight satellites,
all situated outside the rings. The seven nearest move in
planes nearly coinciding with that of the rings, while the
orbit of the eighth is inclined to it at an angle of 10. The
sixth satellite is by far the largest, having a probable dia-
meter not far short of that of the planet Mars. The seventh
has been observed, like our moon, always to turn the same*
side towards the planet. The distances of the satellites from
Saturn range from 3 to 60 times the planet's semi-diameter,
and the corresponding periods range from 22^ h. to 79 d.
312. Uranus, IjJ, at mean distance 192, revolves in an
approximately circular orbit, nearly coinciding with the
ecliptic, in a period of 84 years. It was discovered in 1781
by Sir William Herschel, who named it the Georgium Sidus
in honour of the king.
Uranus is attended by four satellites at least, and these
possess the remarkable peculiarity of revolving, in a plant 1
nearly perpendicular to the ecliptic and in a retrograde
direction. In fact, the plane of their orbits makes an angle
of 82 with the ecliptic. Their periods are 2|d., 4d., 8|d..
and 13|d. roughly.
THE PLANETS.
243
313. Neptune, ^. The position of this planet was pre-
dicted in 1846 almost simultaneously by Adams and Leverrier,
from., the observed effects of its attraction on the orbital
motion of Uranus. It was first actually seen by Galle, of
Berlin, in September, 1846, very close to the position which
had been computed beforehand. It has a mean distance
300 (being considerably less than that which it would have
according to Eode's Law), and it revolves in its orbit in about
164 years.
^Neptune has one satellite moving in a retrograde direction
in a plane inclined to the ecliptic at about 35.
The discovery of Neptune will be treated more fully in the
chapter on Perturbations.
314. Tabular View of the Solar System. For con-
venient reference, the mean distances of the planets, measured
in terms of the Earth's mean distance as the unit, and their
periodic times, are given below, together with the inclina-
tions and eccentricities of the orbits, and the numbers of
their satellites.
Name of
Planet.
Mean Dist.
of Planet.
of Earth.'
Periodic Time.
Inclination
of Orbit.
Eccen-
tricity
of
Orbil.
No. of
Satel-
lite*
Mercury, $
0-38
days =
88
= years
0-24
o /
7
206
Venus, ?
0-72
224
0-62
3 23
007
Earth,
1-00
335
100
017
1
Mars, 3
1-52
687
1-88
1 51
093
2
Ceres, (?)
2-77
1,681
4-60
10 37
076
Jupiter, "4
Saturn, \i
Uranus, fl
5-20
9-54
19-18
4,332
10,759
30,687
11-86
29-46
84-02
1 19
2 30
46
048
056
046
I 8 & 3
I rings
4
Neptune, ttf
30-05
60,181
164-78
1 47
009
1
244 ASTHONOMY.
SECTION 11. Synodic and Sidereal Periods Description of
Motion in Elongation of Planets as seen from the Earth
Phases.
315. Inferior and Superior Planets. Definitions.
In describing the motions of the planets relative to the
Earth, it is convenient to divide the planets into two classes,
inferior and superior planets.
An inferior planet is one which is nearer to the Sun than
the Earth ; Mercury and Venus are the two inferior planets.
A superior planet is one which is further from the Sun
than the Earth : all the planets except Mercury and Yenus
are superior.
The angle of elongation is the difference between the
geocentric (156) longitude of the planet and that of the
Sun. It has the same meaning as in the case of the Moon
(259).
We shall now describe the changes in elongation of the
inferior and superior planets, as seen from the Earth. It
appears from the preceding section that
(i.) The planets all revolve round the Sun in the same
direction ;
(ii.) The planets which are nearer the Sun travel at a
greater speed than those which arc more remote.
The second fact can be easily verified from comparing the
distances and periods of the planets given in the previous
section. Even if we take into account the fact that the more
distant ones ha\c further to travel, we shall still find that
they take longer to travel over the same distance.
In order to further simplify the descriptions we shall
assume that the planets all revolve uniformly in circles, about
the Sun as centre, in the plane of the ecliptic. These
assumptions are only roughly true, on account of the small
eccentricities of the orbits and their small inclinations to
the ecliptic ; hence our results will only agree roughly with
observation.
316. Changes in Elongation of an Inferior Planet.
Let E be the Earth, Pan inferior planet moving in the orbit
AUBU' about S the Sun. Since $F revolves more rapidly
about S than SE, the motion of V relative to .#, as it would
appear from S, is direct,
THE PLANETS.
245
S V separates from SJS at a rate which is the difference
of the rates at which J, V revolve in their orbits. The
changes in the positions of the planet relative to the Sun
are therefore the same as if E were at rest and V re-
volved with an angular velocity equal to the excess of the
angular velocity of the planet oVer that of the Earth.
Let the line E& meet the orbit of V in A and B. "When
V is at A or B it has the same longitude as S, and if the
planet actually moved in the ecliptic it would be in front of
the Sun at A, behind the Sun at B. In reality, owing
to the inclination of the orbits, this but rarely happens.
At A, the planet is said to be in inferior conjunction
with the Sun ; it has the same longitude and is nearer the
Earth. At B the planet is said to be in superior conjunc-
tion with the Sun ; it has the same longitude but is further
away. If we consider the appearances which would be pre-
sented on the Sun, the planet is in " heliocentric conjunction "
with the Earth at A and in " heliocentric opposition " at B.
After inferior conjunction at A, the pianet is seen on the
westward side of the Sun, as at V r The elongation SEV
gradually increases till the planet reaches a point 7 such
that EU\$ a tangent to the orbit. The planet is then at its
greatest elongation, the angle SEU being a maximum.
Subsequently, as at V tt the elongation diminishes, and the
planet approaches the Sun, until superior conjunction occurs,
as at B. The planet then separates from the Sun, reappear-
ing on the opposite (eastern) side, as at F~ 3 , attains its maxi-
mum elongation at Z7 7 , and finally comes round again to
inferior conjunction at A.
246 AsriiONOMi.
The time between two consecutive conjunctions of the
same kind (superior or inferior) is called the synodic period
of the planet (cf. 259), and is the period in which
SV separates from SB through 360.
317. To find (roughly) the Ratio of the Distance
from the Sun of an Inferior Planet to that of the
Earth, it is only necessary to observe the planet's greatest
elongation. F.or if 7", E (Fig- 95) represent the planet and
Eartli at the instant of greatest elongation, the angle EUS is
a right angle, and therefore
that is,
Distance of planet . .
Distance of Earth = S1UC f greatest elo ** at "> n -
This method is, however, much modified by the fact that
the real orbits are not circles, but ellipses.
EXAMPLE 1. Given that the greatest elongation of Yenus is 45,
find its distance from the Sun, that of the Earth being 93,000,000
miles.
Here distance of Venus = 93,000,000 sin 45 = 93,000,000 x V^
= 93,000,000 x -70711 = 65,760,000 miles.
EXAMPLE 2. Taking the Earth's distance as unity, to find the
distance of Mercury, having given that Mercury's greatest elonga-
tion is 22i.
The distance of Mercury = Ixsin 22 = ^{1(1 -cos 45)}
= ^(2-^/2) = -38268.
THE PLAN
318. Changes in Elongation of a Superior Planet.
Let us now compare the apparent motion of the superior planet
7with that of Sun. Since it revolves about the Sun in the
same direction as the Earth does, but more slowly, the line
SJwill move, relative to SE, in the opposite or retrograde
direction. Hence, in considering the changes in the position
of the planet relative to the Sun, we may regard SE as a
fixed line, and J must then revolve about 8 in the circle
ARBTwifh a retrograde motion, i.e., in the same direction
as the hands of a watch.*
At A the planet is in opposition with the Sun, and its
elongation is 180. At B it is in conjunction, and its
elongation is 0. If, however, we were to refer the directions
of the Earth and planet to the Sun, the planet would be in
heliocentric conjunction with the Earth at A, and in helio-
centric opposition at B.
The planet is nearest the Earth at A, and since its orbital
Telocity is constant, its relative angular velocity is then
greatest, and the elongation SEJ is decreasing at its most
rapid rate. As the planet moves round from opposition A to
FIG. 96.
conjunction B, the elongation SEJ decreases continuously
from 180 to 0.
At R the elongation is 90, and the planet is said to be in
quadrature.
* As a simple illustration, both the hour and minute hands of a
watoh revolve in the same directions, but the minute hand goe*
faster and leaves the hour hand behind. Hence the hour hand
separates from the minute hand in the opposite direction to that in
which both tare moving.
248
ASTRONOMY".
At conjunction, J?, the elongation is 0; and we may also
consider it to be 360. As the planet revolves from B to A,
the elongation (measured round in the direction BRA} de-
creases from 360 to 180.
At T the elongation is 270, and the planet is again seid
to be in quadrature.
At A the elongation is again 180, the planet being once
more in opposition. After this the elongation decreases from
180 to as before, as the planet's relative position changes
from A through R to B.
The cycle of changes recurs in the synodic period, i.e.,
the period between two successive conjunctions or oppositions.
"We see that the elongation decreases continually from 360
to as the planet revolves from conjunction round to con-
junction, and there is no greatest elongation.
FIG. 97.
319. To compare (roughly) the Distance of a
Superior Planet with that of the Earth. Here there
is no greatest elongation, and therefore we must resort to
another method.
Let the planet's elongation SEJ (Fig. 97) be observed at
any instant, the interval of time which has elapsed since the
planet was in opposition being also observed. Let this
interval be , and let 8 denote the length of the planet's
synodic period. Then, in time S the angle JSE increases
from to 360 j therefore, if we assume the change to take
place uniformly, the angle JSE at time t after conjunction
is = 860 x tlS
THE PLANETS. 249
Hence, JSE is known. Also JJES has been observed, and
SJE (= 18QJJSSJSE) is therefore also known.
Therefore we have, by plane trigonometry,
Distance of Planet _ SJ _ sin SEJ
Distance of Earth " 8E sin SJE
which determines the ratio of the distances required.
This method is also applicable to the inferior planets. It
is, however, not exact, owing to the fact that the planetary
motions are not really uniform (see 327).
*320. It U not necessary to observe the instant of conjunction or
opposition. If 8 is known, t\ro observations of the elongation and
the elapsed time are sufficient to determine the ratio of the distances.
The requisite formulea are more complicated, but they only involve
plane trigonometry. We, therefore, leave their investigation as an
exercise to the more advanced student.
EXAMPLE. To calculate the distance of Saturn in terms of that
of the Earth, having given that 94 days after opposition the elonga-
tion of Saturn was 84 17', and that the synodic period is 376 days.
Given also tan 5 43' = !.
Let the Sun, Earth, and Saturn be denoted by fif, E, J. In 376
days / J8E increases from to 360.
.*. in 94 days after opposition L JSE = 90 j
also, by hypothesis, L JES = 84 17'.
Distance of Saturn = SJ = ^ flj?/ = ^ 840 1? ,,
Distance of Earth SE
-r cot 6 43' = ~ = 10.
Therefore the distance of Saturn, as calculated from the given
data is 10 times that of the Earth.
321. The synodic period of an inferior planet may
be found very readily by determining the time between two
transits of the planet across the Sun's disc and counting the
number of revolutions in the interval.
For a superior planet this is not possible, and we
must, instead, find the interval between two epochs at which
the planet has the same elongation.
250 ASTRONOMY.
322. Relations between the Synodic and Sidereal
Periods. The relation between the synodic and sidereal
periods is almost exactly the same as in the case of the Moon,
the only difference being that the planets revolve about the
Sun and not about the Earth.
The sidereal period of a planet is the time of the planet's
revolution in its orbit about the Sun relative to the stars.
The synodic period is the interval between two conjunc-
tions with the Earth relative to the Sun. It is the time in
which the planet makes one whole revolution as compared
with the line joining the Earth to the Sun.
Let S be the planet's synodic period,
P its sidereal period,
Yihc length of a year, that is, the Earth's sidereal period,
all the periods being supposed measured in days.
Then, in one clay,
the angle described by the planet about the Sun 360/P,
the angle described by the Earth = 360/F,
and the angle through which their heliocentric
directions have separated = 860//S.
If the planet be inferior, it revolves more rapidly
than the Earth, and 360/ represents the angle gained by
the planet in one day.
360 360 360
If the planet be superior, it revolves more slowly
than the Earth, and 360/ is the angle gained by iheJZarth
i-i one day.
360 _ 360 360
~^~ ~Y~ ~ ;
or _i=i-JL.
3 1 P
From these relations, the sidereal period can be found if
the synodic period is known, and vice vtrsd.
THE PLANETS.
323. Phases of the Planets. As the planets derive
their light from the Sun, they must, like the Moon, pass
through different phases depending on the proportion of their
illuminated surface which is turned towards the Earth.
Phases of an Inferior Planet. An inferior planet V
will evidently be new at inferior conjunction A, dichotomized
like the Moon at its third "quarter at greatest elongation 7",
full at superior conjunction B, dichotomised like the Moon at
first quarter when it again comes to greatest elongation at
IT. Thus, like the Moon, it will undergo all the possible
different phases in the course of a synodic revolution.
There is, however, one important difference. As the
planet revolves from A to B its distance from the Earth
increases, and its angular diameter therefore decreases. Thus
the planet appears largest when new and smallest when full,
and the variations in the planet's brightness due to the differ-
ences of phase arc, to a great extent, counterbalanced by the
changes in the planet's distance. For this reason, Venus
alters very little in its brightness (as seen by the naked eye)
during the course of its synodical revolution.
FIG. 98.
The phase is determined by the angle 8 VE, and this is
the angle of elongation of the Earth as it would appear from
the planet. The illuminated portion of the visible surface
of the planet at V is proportional to 180-F.#, and the
proportion of the apparent area of the disc which is illumi-
nated varies as 1 + cos S VE or 2 cos 2 \ S VE. ( Cf. 263).
The phases of Venus are easily seen through a telescope.
ASTROX.
252 ASTRONOMY.
324. Phases of a Superior Planet. For a superior
planet J the angle SJE never exceeds a certain value. It is
greatest when SJEJ = 90, being then the greatest elongation
of the Earth as it would appear from the planet. Hence tho
planet is always nearly full, being only slightly gibbous, and
the phase is most marked at quadrature.
FIG. 90.
The gibbosity of Mars, though small, is readily visible at
quadrature, about one-eighth of the planet's disc being
obscured. The other superior planets are, however, at a
distance from the Sun so much greater than that of the Earth
that they always appear very approximately full.
325. The "Phases" of Saturn's Kings are due to an entirely
different cause. The plane of the rings, like the plane of the Earth's
equator, is fixed indirection, and inclined to the ecliptic at an angle
of about 28. Hence, during the course of the planet's sidereal
revolution, the Sun passes alternately to the north and south side?
of the rings (just as in the phenomena of the seasons on our Earth,
the Sun is alternately N. and S. of the equator). The Earth, which,
relatively to Saturn, is a small distance from the Sun, also passes
alterrately to the north and south sides of the rings, and we see the
rings first on one side and then on the other. At the instant of
transition the rings are seen edgewise, and are almost invisible.
Unless Saturn is in opposition at this instant, the Sun and Earth,
do not cross the plane of the rings simultaneously, and between
their passages there is a B'.iort interval during which the Sun and
Earth are on opposite sides of the plane; and the unilluminaled
side of the rings is turned towards the Earth. The last " dis-
appearances" of the rings occurred in Sept., 1891 May, 1892, but
they occur twice in each sidereal period, or once about every 15 years.
Other interesting appearances are presented by the shadows
thrown by the planet on the rings and by the rings o^ *he planet.
THE PLANETS. 253
SECTION III. Kepler's Laws of Planetary Motion.
326. Kepler's Three Laws. We have already seen
that the orbits of most of the planets are nearly circular, their
-distances from the Sun being nearly constant and their
motions being nearly uniform. A far closer approximation
to the truth is the hypothesis held for a long time by Tycho
Brahe and other astronomers, namely, that each planet re-
solved in a circle whose centre was at a small distance from
the Sun, and described equal angles in equal intervals of
time about a point found by drawing a straight line from the
Sun's centre to the centre of the circle and producing it for
nn equal distance beyond the latter point.
The true laws which govern the motion of the planets were
discovered by the Danish astronomer Kepler, in connection
with his great work on the planet Mars (De Motibus Stellae
Jfartis). After nine years' incessant labour the first and second
of the following laws were discovered, and shortly afterwards
the third.
I. Every planet moves in an ellipse, with the Sun
in one of the foci.
II. The straight line drawn from the centre of
the Sun to the centre of the planet (the planet's
"radius vector") sweeps out equal areas in equal
times.
III. The squares of the periodic times of the
several planets are proportional to the cubes of their
mean distances from the Sun.
These laws are known as Kepler's Three Laws. We
Tiave already proved that the first two laws hold in the
case of the Earth. The third law is also found to hold good
for the Earth as well as the other planets, and this fact alone
.affords strong evidence that the Earth is a planet
254 ASTRONOMT.
By the mean distance of a planet is meant the arith-
metic mean between the planet's greatest and least dis-
tances from the Sun. If p, a (Fig. 100) be the planet' im-
positions at perihelion and aphelion (i.e., when nearest and
furthest from the Sun respectively), the planet's mean
distance = (Sp + Sa) = \pa = \ (major axis of ellipse
described) (147).
The periodic times are, of course, the sidereal periods.
Hence the third law is a relation between the sidereal periods-
and the major axes of the orbits.
FIG. 100.
327. Verification of Kepler's First and Second
Laws. We will now roughly sketch the principle of the-
methods by which Kepler determined the orbit of Mars, and
thus proved his Eirst and Second Laws. A verification of
the laws in the case of the Earth. has already been given, and
we have shown ( 145) how to determine exactly the position
of the Earth at any given time ; we may regard this, there-
fore, as known. We may also suppose the length of the
sidereal period of Mars to be known, for the average length of
the synodic period may be found, as in 261, and the sidereal
period may be deduced by the formula of 322.
Let the direction of the planet be observed when it is at
any point J/ in its orbit, the Earth's position being E. When
the planet has returned again to Jf after a sidereal revolution,
the Earth will not have returned to the same place in it*
'J HE PLAXETS.
255
orbit but will be in a different position, say F Let no
the planet's new direction FM be observed *
the "rie k ^T g F hG ^\ m0ti< 2' We knmv ^> ^ nd
tne angle -^A Prom th
the ieF 0< 2' We nmv ^> ^ nd
tne angle -^A Prom the observations of the two directions
of J/ we know the angles SEM ^\ SFM. These "* G
sufficient to enable us to solve the quadrilateral
Via. 101.
. .
We can thus determine SM and the angle .2SJ/; whence
the dis ance and d lre ction of M from the Suu am found?
Similarly, any other position of Mars in its orbit can be found
by two observations o the planet's sidereal period separated
by the interval of the planet's sidereal revolution. In
s way, by a senes of observations of Mars, extending ovei
dS
Clirccti011
d daity '
* For simplicity we suppose Mars to move in thelcliptic plane
The methods require some modification when the inclination of the
orbits 1S taken into account, but the general principle is the same.
r A
256 ASTEOXOMY.
b28. Verification of Kepler's Third Law. Kepler '&
Third Law can le verified much more easily, especially if we
make the approximate assumption that the planets revolve
uniformly in circles about the Sun as centre. The sidereal
periods of the different planets can be found by observing the
average length of the synodic period (the actual length of
any synodic period is not quite constant, owing to the planet
not revolving with exactly uniform velocity) and applying
the equations of 322. The distance of the planet may be
compared with that of the Earth, either by observing the
greatest elongation (317) in the case of an inferior planet,
or by the method of 319. It is then easy to verify the
relation between the mean distances and periodic times of
the several planets.
In the table of 314, the student will have little
difficulty in verifying (especially if a table of logarithms-
be employed) that the square of the ratio of the periodic
time of the planet to the year (or periodic time of the
Earth) is in every case equal to the cube of the ratio of
the planet's mean distance to that of the Earth.* The data
being only approximate, however, the law can only be veri-
fied as approximately true, although it is in reality accurate.
Owing to the importance of Kepler's Third Law, we append
the following examples as illustrations.
EXAMPLES.
1. Given that the mean distance of Mars is 1'52 times that of the
Earth, to find the sidereal period of Mars.
Let T be the sidereal period of Mars in days. Then, by Kepler'a
Third Law,
/. T - 305^ x A/(3-511S) - 305 L x T874 = 684'5.
Hence, from the given data, the period of Mars is T874 of a year,
or 684-5 days.
Had we taken the more accurate value of the relative distance,
viz., l - 5237, we should have found for the period the correct value,
namely, 687 days.
* In other words, 2 log (period in years) = 3 log (distance in terms-
of Earth's distance).
THE PLANETS. 257
2. The synodic period of Jupiter being 399 days, to find its distance
from the Sun, having given that the Earth's mean distance is 92
million miles.
Let T be the sidereal period of Jupiter. Then, by 322,
JL = _1 1 33f
T 365^ 399 36o x 399'
.
= 11-82, or nearly 12 years.
Let a be the distance of Jupiter in millions of miles. Then by
Kepler's Third Law,
I SL V = ( Y - 144
\ 92 ; IT I
/. a =92 x3/(l44) = 92x5-24 = 482;
that is, Jupiter's distance is 482 millions of miles.
By taking T - 11'82 and the Earth's distance as 92'04, we should
have found the more accurate value 477'6 for Jupiter's distance in
millions of miles.
329. Satellites. The motions of the satellites about any
planet are found to obey the same laws as those which Kepler
investigated for the orbits of the planets. For example, the
Moon's orbit about the Earth is an ellipse, and (except so far
as affected by perturbations) satisfies both of Kepler's First
and Second Laws. When a number of satellites are revolv-
ing round a common primary (i.e., planet) as is the case with
Jupiter, the squares of their periodic times are found, in
every case, to be proportional to the cubes of their mean
distances from the planet.*
EXAMPLE. To compare (roughly) the mean distances of its two
satellites from Mars. The periodic times are 30^ h. and7|h. respec-
tively, and these are in the ratio (nearly) of 4 to 1.
Hence the mean distances are as 4^ : 1, or %/W : 1.
Now, 2-yi6 = s/128 = 5 very nearly (since 5 :J - 125). Hence,
the mean distances are very nearly in the ratio of 5 to 2.
* Of course the relation docs not hold between the periodic times
and mean distances of satellites revolving round different planets,
nor between those of a satellite and those of a planet.
258 ASTRONOMY.
SECTION IV. Motions Relative to Stars Stationary Points.
330. Direct and Retrograde Motion. We have
described ( 316-318) the motion of a planet relative to the
Sun. In considering- its motion relative to the stars we must
take account of the Earth's motion.
FIG. 103.
An inferior planet moves more swiftly than the Earth.
Hence at inferior conjunction the line ^^(Fig. 102) joining
them is moving in the direction of the hands of a watch. The
planet therefore appears to move retrograde. At greatest elonga-
tion ( 7, U') the planet's own motion is in the line joining it to
the Earth, and hence produces no change in its direction ;
but the Earth's direct motion causes the line EU or EU' ta
turn about U QT U' with a rotation contrary to that of the
hands of a watch; and therefore the apparent motion is
direct. Over the whole portion UBU' of the relative orbit
both the Earth's motion and the planet's combine to make the
planet's apparent motion direct. There must, therefore, be
two positions, M between A and U and N between U' and
A, at which the motion is checked and reversed. At these
two positions the planet is said to be stationary.
A superior planet moves slower than the Earth ; hence at
opposition the line EA (Fig. 103) joining them is turning in
the direction of the hands of a watch. The planet therefore
appears to move retrograde. At quadrature (2t, T) the Earth is
moving along RET] hence its motion produces no change in
the planet's direction. Hence the planet's direct motion about
THE PLANETS.
259
the Sun makes its apparent motion also direct. In all parts
of the arc RBT the orbital velocities of Earth and planet
conspire to produce direct motion. Hence the planet is
stationary at If, between A and H, and at JV between
In both cases the longitude increases from J/ to JV and
decreases from Nto J/; hence it is a maximum at JV and a
minimum at M. After a complete synodic revolution the
planet's elongation is the same as at the beginning, and the
Sun's longitude has been increased ; therefore the planet's
longitude has also increased. Hence the direct preponderates
over the retrograde motion.
FIG. 105.
FIG. 104.
331. Alternative explanation. We may also proceed
;as follows. Let E, J represent two planets at heliocentric
conjunction. Let E^ E^ E z , ..., J l} J^ 7 8 , ..., be their
successive positions after a series of equal intervals. To find
the apparent motion of /among the stars, as seen from J2,
take any point E, and let E\, E'l, JS3, ... (Fig. 105) be
parallel respectively to E^'E^ E./^ .... Then the
points 1, 2, 3, ... represent 7's direction as seen from ^at a
'series of equal intervals, starting from opposition.
260
ASTKONOMY.
Again, if Jl, <72, <73 be taken parallel to
(Pig. 108), the points 1, 2 now represent j's direction as
seen from J.
We observe from Figs. 107, 108 that the relative motion is
retrograde from 1 to 2, and becomes direct near 3. At the
instant at which this takes place, either planet must be
stationary, relative to the other. Since J 4 E 4 is nearly a tan-
gent to JS's orbit, E is near its greatest elongation, and J
is near quadrature at the positions 4 ; hence, E appears
stationary from /between inferior conjunction and greatest
elongation ; and J appears stationary between opposition and
quadrature.
FIG. 107.
We notice that <71, J2, . . . are parallel to E 1, E<2, but
measured in opposite directions, showing that the motion of
E relative to J is the same (direct, stationary, or retrograde)
as that of / relative to E.
THE PLANETS. 261
332. Effects of Motion in Latitude. Hitherto we
have supposed the planet to move in the ecliptic. When,
however, the small inclination of the orbit to the ecliptic is
taken into account, it is evident that the planet's latitude is
subject to periodic fluctuations.
The points of intersection of the planet's orbit with the
ecliptic are (as in the case of the Moon) called the Nodes.
Whenever the planet is at a node its latitude is zero; and
this happens twice in every sidereal period of revolution.
A planet is stationary when its longitude is a maximum or
mininium, but unless its latitude should happen to be a
maximum at the same time, the planet does not remain
actually at rest. When the change from direct to retrograde
motion, and vice versa, is combined with the variations in lati-
tude, the effect is to make the planet describe a zigzag curve,
sometimes containing one or two loops, called " loops of
retrogression." This is readily verified by observation.
t Ecliptic
FIG. 109.
Fig. 109 is an example of the path of Venus in the neigh-
bourhood of its stationary points, the numbers representing
its positions at a series of intervals of ten days. Here,,
the planet is stationary close to the node JV, between 4
and 5, and it describes a loop in the neighbourhood of the-
stationary point near 9, where its motion changes from re-
trograde to direct.
The student will find it an instructive exercise to trace out
the path of any planet in the neighbourhood of its retrograde
motion, using the values of its decl. and R.A., at intervals of
a few days, as tabulated in the Nautical or
Almanack.
262
ASTRONOMY.
333. To find the condition that two planets may be
stationary as seen from one another, assuming the
orbits circular and in one plane. Let P, Q be the
positions of the planets at any instant ; P', Q' their position?
after a very short interval of time.
Then, if PQ and P'Q' are parallel, the direction of either
planet, as seen from the other, is the same at the beginning
and end of the interval ; that is, P is stationary as seen from
Q, and Q is stationary as seen from P.
Let u, v represent the orbital velocities of the planets P, Q ;
a, b the radii $P, SQ respectively.
FIG. 110.
Draw P'J/, Q'N perpendicular to PQ. Then, in the
stationary position, we must have P'M = Q'N.
But PP', QQ', being the arcs described by the two planets
in the same interval, are proportional to the velocities u, v.
Therefore P'M, Q'N are proportional to the component
velocities of the planets perpendicular to PQ. These com-
ponent velocities must, therefore, be equal, and we have
u sin P'PM= v sin Q'QN.
"Whence, since P'P is perpendicular to SP and Q'Qto SQ,
u cos SPQ = v cos SQN = vcos SQP (i.),
and this is the condition that the planets may be stationary
relative to one another.
THE PLANETS. 263
*334. To find the angle between the radii vectores in the station-
ary position, and the period during which a planet's motion is
retrograde. By projecting SQ, QP on SP, we have
a = 6 cos PSQ + PQ cos SPQ.
Similarly b = a cos PSQ + PQ cos SQP.
.-. cos SPQ : cos SQP = a - b cos PSQ : b - a cos PSQ.
Whence, by (i.), u (a-b cos PSQ) +v (b-acosPSQ) = 0;
= "" '" . (ii.).
ac -\-lni
By means of Kepler's Third Law, we can express the ratio of u to
v in terms of a and b. For if T l} To denote the periodic times, then.
evidently uT v = 2-, vT 2 = 2.T& ;
.-. u : v = aT 2 : Z-7',.
But T, I T, = ^ ; bl ;
Substituting in (ii.), we have
cos PSQ = ^
[From this result it may be easily deduced that
tan t PSQ = ( 1=? /SQ ) *
\l + cosPSQ/
In the above investigation PSQ is .the angle through which SQ 1
separates from SP between heliocentric conjunction and the station-
ary point. Hence, since L PSQ increases from to 360 in the
synodic period S, the time taken from conjunction to the stationary
- 4. /P-SO
point =Sx ^T'
If L PSQi = L PSQ, there is another stationary point before con-
junction, when the planets are in the relative positions P, Q. Hence,
the interval between the two stationary positions is twice the time
taken by the planets to separate through /PSQ, and is therefore
This represents the interval during which the motion of either
planet, as seen from the other, is retrograde. During the remainder
of the synodic period the motion is direct, and the time of direct
motion is therefore
264 ASTROXOMY.
SECTION Y. Axial Rotations of Sun and Planets.
335. The Period of Eotation of the Sun can be found by
observing the passage of sunspots across the disc. These spots, by
the way, are very easily exhibited with any small telescope by
focussing an image of the Sun on to a piece of white paper placed
.a few inches in front of the eye-glass for to look straight at the
Sun would cause blindness. As the Sun's axis of rotation is nearly
perpendicular to the ecliptic, the rotation of the spots is seen in
perspective, and makes them appear to move nearly in straight
lines across the disc. From this observed apparent motion (as
projected on the celestial sphere in a manner similar to that
explained in 263) their actual motion in circles about the Sun's
axis is readily determined. For example, if a spot move^ from the
-centre of the disc to the middle point of its radius, we may readily
see that the angle turned through = sin" 1 -| = 3C.
The spots are observed to return to the same position in about
.27| days, and this is their synodic period of rotation relative to the
Earth. Call it 8, and let T be the time of a sidereal rotation, T the
length of the year. Then, as in the case of an inferior planet
( 322), we may show that
J_ = 1 j_ m 1 = 1 t 1
8 ~* T - Y ' T ~ 27i "*" 365^ ;
whence the true period of rotation T = 25| days (roughly).
It has been observed that spots near the Sun's equator rotate
rather faster than those near the poles. This proves the Sun's surface
-to be in a fluid condition, for no rigid body could rotate in this way.
336. Periods of Rotation of Planets. The rotation period of a
.superior planet is easily found by observing the motions of the
markings across its di-c near opposition, allowance being made for
the motions of the Earth and planet. The surface of Mars has well-
-defined markings, which give the period 24h. 37m. The principal
mark on Jupiter is a great red spot amid his southern belts, which
rotates in the period of 9h. 56m. Saturn rotates in lOh. 14m.
For an inferior planet, the period is more difficult to observe.
There is still some uncertainty as to whether Venus rotates in about
;23h. 21m., or whether, like Mercury, it always turns the same face
to the Sun. There are no well-defined markings, and, as the
greatest elongation is only 45, Venus can only be seen for part of
the night as an evening or morning star, and in the most favourable
positions only a portion of the disc is illuminated. Moreover,
refraction, modified by air-currents, prevents the planet from being
seen distinctly when near the horizon. If the same markings are
:Been on the disc of a planet on consecutive nights, they may either
hare remained turned towards the Earth, or they may have rotated
through 360 during the day ; hence the difficulty of deciding between
the two alternative hypotheses. Before the researches of Schiapa-
j-elli ( 305), it was believed that Mercury also rotated in about 24h.
THE PLANETS. 265
EXAMPLES. X.
1. The Earth revolves round the Sun in 365'25 days, and Venus
in 224'7 days. Find the time between two successive conjunctions
of Venus.
2. If Venus and the Sun rise in succession at the same point of
the horizon on the 1st of June, determine roughly Venus' elongation.
3. Find tbe ratio of the apparent areas of the illuminated portions
of the disc of Venus when dichotomized and when full, taking
Venus' distance from the Sun to be T 8 T of that of the Earth.
4. Mars rotates on his axis once in 24 hours, and the periods of
the sidereal revolutions of his two satellites are 1\ hours and 3O
hours respectively. Find the time between consecutive transits
over the meridian of any place on Mars of the two satellites
respectively.
5. A small satellite is eclipsed at every opposition. Find an
. expression for the greatest inclination which its orbit can have to
the plane of the ecliptic.
6. If the periodic time of Saturn be 30 years, and the mean dis-
tance of Neptune 2,760 millions of miles, find (roughly) the mean
distance of Saturn and the periodic time of Neptune. (Earth's mean
distance is 92 millions of miles.)
7. If the synodic period of revolution of an inferior planet were a.
year, what would be its sidereal period, and what would be its mean
distance from the Sun according to Kepler's Third Law ?
8. Jupiter's solar distance is 5'2 times the Earth's solar distance ' t
find the length of time between two conjunctions of the Earth and.
Jupiter.
9. Saturn's mean distance from the Sun is nine times the Earth's
mean distance. Find how long the motion is retrograde, having
given cos" 1 \ = 65.
10. Show that if the planets further from the Sun were to move
-with greater velocity in their orbits than the nearer ones, there
would be no stationary points, the relative motion among the stars
"being always direct. AVhat would be the corresponding phenomenon,
if the velocities of two planets were equal ?
266 ASTRONOMY.
EXAMINATION PAPER. X.
1. Explain the apparent motion of a superior planet. Illustrate*
by figures.
2. Describe the apparent course among the stars of an inferior
planet as seen from the Earth, and the changes in appearance which
the planet undergoes.
3. Define the sidereal and synodic period of a superior or inferior
planet, and find the relation between them. Calculate the synodic
period of a superior planet whose period of revolution is thirty
years.
4. How is it that Venus alters so little in apparent magnitude (as-
seen by the naked eye) in her journey round the Sun ? Why does-
not Jupiter exhibit any perceptible phases ?
5. State Bode's Law connecting the mean distances of the various;
planets from the Sun.
6. Prove that the time of most rapid approach of an inferior
planet to the Earth is when its elongation is greatest, and that the-'
velocity of approach is then that under which it would describe its-
orbit in the synodic period of the Earth and the planet. Give the-
corresponding results for a superior planet. (The orbits are to be-
taken circular and in the same plane.)
7. AVhat is meant by stationary points in the apparent motion of a
planet ? Prove that, if a planet Q is stationary as seen from P>
then P will be stationary as seen from Q.
8. State Kepler's Three Laws, and, assuming the orbits of the-
Earth and Venus to be circular, show how the Third Law might be
verified by observations of the greatest elongation and synodic
period of Venus.
9. Find the periods during which Venus is an evening star and a
morning star respectively, being given that the mean distance of
Venus from the Sun is '72 of that of the Earth.
10. Having given that there will be a full Moon on the 5th of June,
that Mercury and Venus are both evening stars near their greatest
elongations, that Mars changed from an evening to a morning star-
about the vernal equinox, and that Jupiter was in opposition to the
Sun on April 21st, draw a figure of the configuration of these
heavenly bodies on May 1st. (All these bodies may be supposed to
move in one plane.)
CHAPTER XI.
THE DISTANCES OF THE SUN AND STARS.
SECTION I. Introduction Determination of the Surfs Parallax
by Observations of a Superior Planet at Opposition*
337. In Chapter VIII. , Section I., we explained the nature
of the correction known as parallax, and showed how to find
the distance of a celestial body from the Earth in terms of its
parallax. We also described two methods of finding the
parallax of the Moon or of a planet in opposition the first
by meridian observations at two stations, one in the northern
and the other in the southern hemisphere ( 252) ; the second
by micrometric observations made at a single observatory
shortly after the time of rising and shortly before the time
of setting of the planet or observed body ( 254).
In both methods the position of the body is compared with
that of neighbouring stars. This is impossible in the case of
the Sun, for the intensity of the Sun's rays necessitates the
use of darkened glasses in observations of the Sun, and these
render all near stars invisible.
Of course the star could theoretically be dispensed with in the
method of 252, but only (as there explained) at a great sacrifice
of accuracy ; and if a star is used which crosses the meridian at
night, the temperature of the air has changed considerably, and the
corrections for refraction are therefore quite different, besides
which other errors are introduced by the change of temperature
of the instrument.
* The student will find it of great advantag3 to revise Section I.
of Chapter VIII. before commencing the present Section.
A.STKON". T
268 ASTRONOMY.
In 264 we described a method, due to Aristarchus, in
which the ratio of the Sun's to the Moon's distance was
determined by observing the Moon's elongation when dicho-
tomized, but this method was rejected, owing to the irregular
boundary of the illuminated part of the disc, and the con-
sequent impossibility of observing the instant of dichotomy.
338. Classification of Methods. The principal prac-
ticable methods of finding the Sun's distance may be con-
veniently classified as follows :
A. Geometrical Methods.
(1) By observations of the parallax of a superior planet at
opposition (Section I.).
(2) By observations of a transit of the inferior planet
Venus (Section II.).
B. Optical Methods (Section IV.).
(3) By the eclipses of Jupiter's satellites (Roemer's Method).
(4) By the aberration of light.
C. Gravitational Methods (Chapter XIV., Section IV.).
(5) By perturbations of Venus or Mars.
(6) By lunar and solar inequalities.
339. To find the Sun's Parallax by Observation of
the Parallax of Mars. By observing the parallax of
Mars when in opposition, the Sun's parallax can readily be
found. For the observed parallax determines the distance of
Mars from the Earth, and this is the difference of the dis-
tances of the Sun from the Earth and Mars respectively.
The ratio of their mean distances may be found, if we assume
Kepler's Third Law ( 326), by comparing the sidereal period
of Mars with the sidereal year, and is therefore known.
Hence the distance of either planet from the Sun may readily
be found, and the Sun's parallax thus determined.
The parallax of Mars in opposition may be observed by
cither of the methods described in Chapter VIII., Section I.
The method of 252 (by meridian observations at two
stations) was employed by E. J. Stone in 1865. The observa-
tions were made at Greenwich and at the Cape, and the Sun's
parallax was computed as 8 -943". The method of 254 (by
observations at a single observatory) was employed by Gill
at Ascension Island in 1879, and the result was 8-783",
THE DISTANCES OF THE SUN AND STAES. 269
EXAMPLE.
If the parallax of Mars when in opposition be 14", to find the
Sun's parallax, assuming the distances of the Sun from the Earth
and Mars to be in the ratio of 10 : 16.
The distance of the Earth from Mars in opposition is the difference
of the Sun's distances from the two planets. Hence
Distance of Earth from Mars I Distance of Earth from Sun
= 16 - 10 : 10 = 3 : 5.
But the parallax of a body is inversely proportional to its dis-
tance ( 250).
.'. Parallax of Sun : Parallax of Mars = 3:5;
.'. Sun's parallax = 8 * 14// = 8'4".
5
*340. Effect of Eccentricities of Orbits. Owing to the eccen-
tricities of the orbits of the Earth and Mars, their distances from the
Sun when in opposition will not in general be equal to their mean
distances, and therefore their ratio will differ from that given by
Kepler's Third Law. But, by the method of 145, the Earth's dis-
tance at any time may be compared with its mean distance, and
similarly, since the eccentricity of the orbit of Mars and the position
of its apse line are known, it is easy to determine the ratio of Mars'
distance at opposition to its mean distance, and thus to compare its
distance with that of the Earth.
341. Sun's Parallax by Observations on the Aste-
roids and on Venus. The Sun's parallax may also be
found by observing the parallax of one of the asteroids when
in opposition, the method being identical with that employed
in the case of Mars. In this way Galle, by meridian obser-
vations of the parallax of Flora at opposition in 1873, com-
puted the Sun's parallax at 8'873", and Lindsay and Gill, by
observing the parallax of Juno in 1877, found the value
8-765".
The next planet, Jupiter, is too distant to be utilized in
this way. Its parallax at opposition is less than a quarter of
the Sun's parallax, and is too small to be observed with
sufficient accuracy.
The Sun's parallax might also be found by an observation of
Venus near its greatest elongation. The ratio of its distance
to the Sun's might be calculated and its parallax found by the
method of 252, and that of the Sun deduced. The method
of 254 could not be employed, because one of the observa-
tions would have to be made in full sunshine.
270 ASTLONOMT.
EXAMPLES.
1. Having given that the greatest possible parallax of Mars when in
opposition is 21'OS", to find the Sun's mean parallax, the eccentri-
cities of the orbits of the Earth and Mars being ^ and T y respec-
tively, and the periodic time of Mars being 1'88 of a year.
The parallax of Mars is greatest when Mars is nearest the Earth ;
hence the greatest possible value occurs when, at opposition, Mars is
in perihelion and the Earth is at aphelion.
Let r, r' denote the mean distances of the Earth and Mars from
the Sun respectively. By Kepler's Third Law we have
r' 3 _ (1-88) 2 . . r' n .J , ~ 9q
;. 3 - -p > - =
(The calculation is most easily performed with a table of logarithms.)
But since the Earth is in aphelion, its distance from the Sun at
the time of observation is greater than its mean distance by Jj,
and is therefore
= r (1 + e^) = 1-017 r.
Also the distance of Mars from the Sun at perihelion
= r'(l-TV) = (l-iV)x 1-523 r
= (1-523- -090) r- l'433r.
Hence the least distance of Mars from the Earth at opposition
= -416 r.
Therefore, since r is the Sun's mean distance from the Earth, we
have
Observed parallax of Mars : mean parallax of Sun = 1 : '416;
/. Sun's mean parallax = 21-08" x -416 = 8'77".
2. To find the Earth's moan distance from the Sun, and its dis-
tances at perihelion and aphelion, taking the Sun's parallax as 8"79".
If a denote the Earth's equatorial radius, we have, approximately,
r = __ a __ = <* - ^ a x 2Q6 ' 265
sin 8*79" circ. meas. of 8'79" 879 *
Taking a = 3963'3, this gives
r (Earth's mean solar distance) = 93,002,000 miles,
correct to the nearest thousand miles.
Also, perihelion distance from Sun = 93,002,000 x (1-^)
= 93,002,000-1,550,000 - 91,452,000 miles,
and aphelion distance = 93,002.000 x (1 + -L)
= 93,002,000 + 3,550,000 = 94,552,000 miles.
DISTANCES OF THE SUN AND STARS. 271
SECTION II. Transits of Inferior Planets.
342. When Yenus is very near the ecliptic at inferior con-
junction, it passes in front of the Sun's disc, appearing like
a black dot on the Sun. Now the circumstances of such a
transit are different at different places, for although both
the Sun and planet are displaced by parallax, their displace-
ments arc different, and their relative directions are therefore
not the same. Now the ratio of the parallaxes of the Sun
and planet at conjunction can be calculated from comparing
their periodic times, or from the ratio of their distances, as
determined by observations of the planet's greatest elonga-
tion or otherwise. Hence, by comparing the circumstances
of the transit at different places, it becomes possible to deter-
mine the parallaxes of both the Sun and planet.
The various methods of finding the Sun's parallax from
observing transits of Venus may be classified as follows :
(i.) By simultaneous observations of the relative position
of the planet at different stations, either by micrometric mea-
surements, or from photographs.
(ii.) Delislis method, by comparing the times of the begin-
ning or end of the transit at stations in different longitudes.
(iii.) Halley's method, by comparing the durations of the
transit at stations in different latitudes.
Of these methods Halley's is the earliest, Delisle's the next.
343. First Method. Let P and p be the horizontal
parallaxes of the Sun and of Yenus respectively at the time of
transit. Then, at a place where the planet's zenith distance
is z, its direction is depressed by parallax through an angle
p sin z ( 249) ; also the Sun is depressed through P sin z*
Hence the planet appears to be brought nearer to the Sun's
lower limb by an angle (pP) sin %.
If, now, the positions of the planet relative to the Sun's disc
be simultaneously observed at any two or more different
places, and the Sun's zenith distances be also determined,
the difference of parallaxes p P can be readily found.
Thus, if one of the stations be chosen where the Sun is
* Strictly speaking, this should be P sin z,, where z\ is the Z.D. of
the Sun's centre, but z l is very nearly equal to 2, and no sensible
error is introduced by taking z instead of K\.
272 ASTBONOMt.
vertical, and another where the Sun is on the horizon, the
relative displacement will be zero at the former station, and
p -P at the latter. Hence, the two directions of the planet
relative to the Sun will he inclined at an angle p P. If
two stations are at opposite ends of a diameter of the Earth,
the angular distance between the relative positions will be
'2 (pP). Hence, in either case, 2? P can be readily found.
Let now / and r denote the distances of Yenus and the
Earth from the Sun respectively. Then, if The the ratio of
the sidereal period of Yenus to a year, we have, by Kepler's
Third Law (assuming the orbits circular),
r'/r = T\
whence the ratio of r' to r is found. Also, since Yenus is in
conjunction, its distance from the Earth is = rr'. There-
fore p : P= r : r r' t
and ^_=n' = JL-l.
p-P r V
Whence, since the ratio of r to r is known, and P p has
been observed, the Sun's horizontal parallax Pmay be found.
We have roughly (by Bode's Law) r' =T*O r i an ^ therefore
Hence the displacement of Yenus on the Sun's disc at a place
where its zenith distance is 2, is about | P sin 2.
The apparent position of Yenus on the Sun's disc may be
observed either by measuring the planet's distance from the
edge of the disc with a micrometer or heliometer, or by taking
a photograph of the Sun. But the photographic method,
though easier, does not give such accurate results.
For, to obtain P correct to O'Ol", it would be necessary to find
2(p-P) correct to ^-xO'Ol", or about 0'05". Since the Sun's dia-
meter is 32', the greatest possible difference of positions would be
20 x 32 x 60 ' r 37400'
of the Sun's diameter. It is difficult to obtain a good photograph
of the Sun more than 4J inches in diameter, and it would, therefore,
be necessary to measure the planet's position correct to - & zoo f an
inch, a degree of accuracy unattainable in practice. The slightest
distortion or imperfection in the photographic plate would render
the observations worthless.
THE DISTANCES OF THE SUN AND STARS. 273
344. Delisle's Method. In this method, the Sun's
parallax is determined by observing the difference between
the times at which the transit begins or ends at different
places. Let A, B be two stations near the Earth's equator in
widely different longitudes, say at the ends of the diameter
of the Earth, and in the plane containing UV, the path of
Yenus' relative motion. Draw AUL and BVL, touching
the Sun in L and cutting the path of Yenus in &, V. Then,
when Yenus reaches U the transit begins at A, the planet
appearing to enter the Sun's disc at L t and when Yenus is at
V the transit begins at B. In the interval between the
times of commencement of the transit as seen from A and B,
the planet moves through the angle ULVor ALB about the
Sun relative to the Earth, and this angle, being the angle sub-
tended at the Sun by the Earth's diameter AB, is twice the
Sun 1 9 parallax.
FIG. 111.
But the rate of relative angular motion of Yenus is known,
being 360 in a synodic period. Hence the angle TJLV,
described in the observed interval, is known, and the Sun's
parallax is thus found.
In a similar way, the Sun's parallax may be determined by <
observing the interval between the times at which the transit
ends at two stations A, B. We should have to draw two
tangents from A, B to the opposite side of the Sun (M ). As
before, the angle described by Yenus in the observed interval
is twice the Sun's parallax.
274 ASTRONOMY.
In employing Delisle's method, the observed times ol
ingress or egress must be the Greenwich times, or must be
reckoned from an epoch common to both observers. For this
reason the difference of longitudes of the two stations must
be accurately known. In the following example the ob-
served interval 690s. corresponds to 8-86" of parallax, and it
follows that an error of Is. in the estimated interval would
give rise to an error of just over 0-01" in the computed
parallax. Hence if the interval of time be estimated correct
to the nearest second, the parallax will be correct to two
decimals of a second.
In practice it would be dim cult to make observations from
the extremity of a diameter of the Earth, but the method is
readily modified so as to be applicable when the stations are
not so favourably situated.
EXAMPLE.
Given that the synodic period is 584 days, and that the difference
between the times of ending of a transit, as seen from opposite ends
of a diameter of the Earth, is llm. 30s., to find the Sun's parallax.
In 584 days Venus revolves through 360 about the Sun relative
to the Earth ; therefore its angular motion per minute
360 x 60 x 60 . ,
= - seconds = T541".
584 x 24 x 60
Therefore in 11 Jm. Venus describes an angle T541" x 11^ = 1772".
This angle is twice the Sun's parallax ;
.'. Sun's parallax = 8'86".
345. Halley's Method. The method now to be de-
scribed was invented by Dr. Halley in 1716, and was first
put into use at the transits in 1761 and 1769. In Halley's
method the times of duration of the transits are observed
from two stations A, B, one in north and the other in south
latitude, in a plane as nearly as possible perpendicular to the
ecliptic, or, more strictly, to the relative path of Venus. Take
this plane as the plane of the paper in Fig. 112, and suppose
also (for the purpose of simplifying the explanation) that
-4, B are at the ends of a diameter of the Earth. Let LM
be the diameter of the Sun's disc perpendicular to the line of
centres, and let the directions of Yenus A V, BV, when pro-
duced, meet the disc in a, I. Then a, b are the relative posi-
tions of Yenus as seen at conjunction from A and 7>.
THE DISTANCES OP TRK SUN A\0 STARS.
275
In Fig. 113 the Sun's disc is represented as seen from the
Earth ; a, I are the positions of Venus as seen on the disc
from A, B, projected on L1I, in Fig. 112, and PQR, PQR'
are the apparent paths of Venus as it appears to cross the
disc at B and A respectively.
As in 343, the angular measure of the arc db or QQ!
measures the sum of the displacement of Venus due to relative
parallax at A and B, and this, in the circumstances here
considered, is twice the difference of the parallaxes of the
Sun and Venus.
Now the observed times of duration of the transit ut A
and B are the times taken to describe the chords P' Q-R' and
PQR respectively. Knowing the synodic period of Venus
and the ratio of its distances from the Sun and Earth, the
rate at which Venus travels across the Sun's face can be
found. Hence, the angular lengths of the chords PQR,
P'Q'R' can be found. Also the Sun's angular diameter ZJ/
is known. Hence the angular distances OQ, OQ', QQ f can
be calculated, for we have (very approximately)
whence the
and QQ' = OQ'-OQ..
Hence QQ' is known, and therefore the difference of
parallaxes of Venus and the Sun is found ;
Sun's parallax may bd found as in 343.
276 ASTRONOMY.
*34C. Or if AB be known in miles, the length cf ab in miles can
be found from the proportion ab : AB = Va : VA, and then, the angle
aAb being known (being the angular measure of QQ')> we can find
the Sun's distance in miles, for we have
circular measure of L aAb = t ; whence
aA
Sun's distance Aa (in miles) = lep S th ab (in mi]es > .
circular measure of L aAb
The working of Halley's method will be made much
clearer by a careful study of the following numerical
examples. The student should copy Pigs. 112 and 113.
EXAMPLES.
1. To find the angular rate at which Venus moves across the
Sun's disc.
Let 8, E, V denote the Sun, Earth, and Venus respectively
(Fig. 112).
From the example of 344, 8V separates from 8E with relative
angular velocity, about 8, of T54" per minute, or 1' 32'4" per hour.
But Venus is nearer the Earth than the Sun in the ratio 28 : 72
(roughly). And we have
angular velocity of EV I ang. vel. of 8V
Therefore EV separates from E8 with angular velocity
= ^ x 1' 32-4" per hour = 3' 57'6" per hour
= V per minute very nearly.
2. Neglecting the motion of the observatory due to the Earth's
rotation, find the position on the Sun's disc of the chord PR, tra-
versed by the planet, in order that the trsviisit may take four hours.
Draw the figures as in 345.
In four hours Venus moves 4x3' 58' , or very nearly 16' relative
to the Sun (by Ex. 1) ; /. the chord PR must measure 10'. Hence
PR is equal to the Sun's angular semi-diameter OP.
Therefore, PR is a side of a regular inscribed hexagon in the Sun,
and L MOP = 30.
3. If, at A, B. at opposite ends of a diameter of the Earth perpen-
dicular to the piano of the ecliptic, the durations of transit are
3h. 21in. and 4h. respectively, to find the Sun's parallax.
tHB DISTANCES OP THE SUN AND STARS. 2^7
Here tlie arc PR takes 39m. longer to describe than P*R'. Hence
it is longer by 39 x 4", or 156". Draw R'K perpendicular to PR.
Then, KR = ^PR-Pit) = x 156" * 78".
Now, by Example 2,
Z Jf OR = 60?
And JBE', being very small, is approximately a straight line perpen-
dicular to OR ; .'. R'RK = 30 approximately. Hence
Q'Q = R'K = RKtan 30 = RK^/% = ^V3" = 45" nearly.
But
angular measure of Q'Q : twice Sun's parallax
= 8V:EV= 18:7;
.*. twice Sun's parallax = 45'' x T ^ = 17'50";
.'. Sun's parallax = 8*75".
4. A transit of Venus was observed from two stations selected as
favourably as possible, one in N. the other in S. latitude, the zenith
distances of the planet being 53 8' (sin 53 8' = '8) and 30
respectively. Given that the times occupied by the planet in pass-
ing across the disc were 4h. 52m. and 4h. 30m., to find the Sun's
parallax, assuming the distances of Venus and the Earth from the
Sun to be in the ratio of 18 : 25 and neglecting the rotation of the
Earth.
Venus moves nearly 4" per minute relative to the Sun; hence in
4h. 30m. it moves through 18'.
In 4h. 52m. it moves through 19 7 28" j
/. in Fig. 113, P'Q' = 18' x = 9',
PQ = 19' 28" x * = 9-73',
and the Sun's semi-diameter SP ~ 16' nearly;
.'. SQ =Vsp-'-pQ2 = v/256- 94-67 - 12W;
SQ' = v/SP' 2 -P'Q 2 = v/256 -81 = 13-23 7 ;
/. QQ' = -53 7 = 31-8".
Now, if A and B be well chosen, QQ' is the sum of the relative
displacements of Venus at the two stations. Let P be the Sun's
parallax, p that of Venus ; then we have
QQ' = (2>-P)( s in 2 + sin z'} = (p - P) x (sin 30 + sin 53 8')
1*3
Again, P:p= :
/. P = 24-5" x T 7 ? = 9'5".
Hence, with the given data, the Sun's parallax is 9'5".
278 \STRONOMY.
347. Difficulties of Observing the Duration of a
Transit. In Examples 3, 4, above, the observed differences
of duration were 39m. and 20m. respectively. An error of
one second in the estimated durations of transit would give
rise to an error of less than O'l per cent., and if we could be
sure of observing the durations to within a second, the Sun's
parallax could be found correct to two decimal places. But
in practice it is extremely difficult to estimate the times of
beginning and ending of a transit, even to the nearest second.
For in the first place, Venus, when seen through the telescope,
is not a mere point, but a disc of finite dimensions, its angular
diameter at conjunction being about 67", or one-thirtieth of
the diameter of the Sun. Hence its passage across the edge
of the disc from external to internal contact occupies an
interval which is never less than about 17s. (See Example
on page 279.)
est
FIG. 114.
Now, it is impossible to observe the first external contact ( U}
of Venus with the Sun, because the planet is invisible until
it has cut off a perceptible portion from the edge of the Sun's
disc, and by that time it has advanced considerably beyond
the point of contact. The last external contact ( F') at the end of
the transit is also difficult (though rather less so) to observe,
for a similar reason.
For this reason, the internal contacts U', V. are alone
observed, and a correction is applied for the angular semi-
diameter of Venus.
But in observing the first internal contact U\ when the
planet's disc separates from the edge of the Sun, another
difficulty, in the form of an optical illusion, makes itself
manifest.
THE DISTANCES OF THE SUN AND STARS. 279
Instead of remaining truly circular, the planet's disc appears
to become elongated towards the edge of the Sun, and remains
for some time connected with the edge by a narrow neck
called the " black drop." This breaks suddenly at last, but
not until the planet has separated some distance from the
Sun's edge.* Even if the "black drop" be remedied, the
atmosphere surrounding the planet Venus renders the con-
tacts uncertain and ill-defined.
It is worthy of notire that in Dclisle's method the times of
ingress and egress at both stations are equally affected by the
"black drop" appearance, and therefore it has no effect on
the computation, provided that both observers take the same
stage of the phenomenon for the observed time of ingress.
EXAMPLE.
Having given that the angular diameter of Venus at conjunction
is 67", to find the interval between external and internal contact (i.)
when Venus passes across the centre of the Sun's disc, (ii.) in the
circumstances of Example 2, 346.
(i.) Between external and internal contacts the planet moves
through a distance equal to its angular diameter; therefore, since its
rate of motion is 4" per second, the time occupied = 67 -f 4s. = 17s.
very nearly.
(ii.) Here the planet is 67" nearer the centre at internal than at
external contact. Now the planet's direction of motion UV is
inclined at angle 60 to the radius through the centre of the disc (Fig.
114). Hence the planet's .component relative velocity along the
radius is 4" cos 60 per second, and therefore the interval required,
in seconds,
67 = 67
4 cos 60 2
= 33-5s.
348. Recent Determinations of the Parallax of
the Sun. Professor Arthur Auwers, the well-known Berlin
astronomer, has recently (December 11, 1891) completed the
calculations based on the observations in Germany of the
transit of Venus in 1882. He finds that the parallax of the Sun
is 8 800 seconds, with an error of 0*03 of a second at most.
From the old observations of the transits of 1761 and 1709,
Prof. Newcomb has lately computed the parallax at 8'79".
* The " black drop " may be illustrated by holding two globes in
the sunshine, at different distances from a white screen, and moving
them until their shadows nearly touch.
280 ASTRONOMY.
349. Advantages and Disadvantages of H alley's
and Delisle's Methods. In Halley's method the observed
data are the intervals of time occupied by Yenus in crossing
the Sun's disc at the two stations. It is not necessary to
know the actual times of the transit ; hence neither the
Greenwich time nor the longitude of the observatories need
be known. In Delisle's method it is essential that the
Greenwich times of the observations should be known with
great accuracy, but it is not necessary to observe both the
beginning and end of the transit at the two stations. Still,
if these be both observed, we have two independent data
for calculating the parallax, which afford some test of the
accuracy of the computations.
On the other hand, Delisle's method possesses the advan-
tage that the places of observation mut be near the Earth's
equator, and it may therefore be possible to select the stations
nearly at opposite ends of a diameter of the Earth, and thus
to get the greatest effect of parallax, while in Halley's
method it is necessary that the stations shall be in as high
latitudes as possible, and, owing to the practical difficulties
of taking observations near the poles, the greatest effect
of parallax cannot be utilized.
Delisle's method is most easily employed if the transit is
nearly central, i.e., if Venus passes nearly across the centre
of the Sun's disc. This condition is fatal to the success of
Halley's method ; here the best results are obtained when
Yenus transits near the edje of the disc.
For in Fig. 113 (page 27ij) we have
OQ' 2 -OQ 2 = QP 2 -Q'P' 2 ,
PR-P'R' QP+Q^
2 OQ + OQ'
Hence the effect on QQ' of a small error in the computed length of
PR or PR' will be least when QP + Q'P' is smallest and OQ+O'Q' is
largest, a condition satisfied when the transit takes place near the
edge M of the disc.
On the other hand, for a nearly central transit, OQ, O'Q' would be
email, and very slight errors in the estimated lengths of PR, P'R'
would produce such large errors in the computed displacement
QQ' as to render the method practically worthless.
The transits of 1874 and 1882 were both favourable to the
use of ll'illey's method.
THE DISTANCES OP THE SUN AND Sf AKS. 2S1
*350. To determine the frequency of Transits of
Venus. Since the Sun's angular semi-diameter is about 16',
a transit of Yenus only occurs when the angular distance
between the centres of the Sun and Yenus, as seen from some
place on the Earth, is 1 6'. Hence, neglecting the effects of
the relative parallax (P-p =. 23" by Ex. 3. 346, and this
is small compared with 16'), Yenus must be at an angular dis-
tance (SEV] < 16' from the ecliptic at the time of conjunc-
tion. Hence the planet's heliocentric latitude JSSVmust be
less than 16' xEVjSV, that is l6'x T 7 ? , or about 6'. Now
the orbit of Yenus is inclined to the ecliptic at about 3 23',
or 203'. Hence, by a method similar to that of 292, we see
that the planet must be at a distance from the node of not
more than about sin'^f-g = sin' 1 -^ (roughly) = 142', in
order that a transit may take place. The smallncss of
this limit alone shows that transits of Yenus are of rare
occurrence.
Now, a synodic period of Yenus contains about 584 days,
that is, 1-599, or, more accurately, 1-598662 of a year.
Hence five synodic revolutions occupy almost exactly eight
years, the difference only amounting to T -^ of a year. This
difference corresponds to an arc of f~f , or 2 24' on the
ecliptic. This arc is much less than the doulle arc 3 24'
within which transits take place. Hence it frequently
happens that, eight years after one transit has taken place,
the Sun and Yenus arc again at conjunction within the
necessary limits, and another transit occurs near the same
node. But after sixteen years, conjunction will occur at 4 48'
from its first position ; this is greater than 3 24' ; hence
there cannot be more than t\vo transits near the same node at
intervals of eight years. And if a transit should be central,
occurring almost exactly at the node, the conjunctions occurring
eight years before and after would fall outside the required
limits, and no second transit would then take place in eight
years.
Again, it maybe shown that 1-598662x147 = 235-003.
Hence 147 synodic periods of Yenus occupy almost exactly
235 years, the difference being only '003 of a year. Thus a
transit of Yenus may recur at the same node at an interval
of 235 years. And it is possible to prove that thore is no
282 ASTRONOMY.
intermediate interval between. 8 and 235 years at "which
transits recur at the same node.
If the orbits of the Earth and Venus were circular, a transit
at one node would be followed by one at the opposite node in
11 3^ or 121* years. For
1-598662x71 = 113+'005; 1-598662x76 = 121*--002.
But this result is modified by the eccentricities of the orbits
(which now cause a difference of nearly a day in the times
taken by the Earth to describe the two halves into which its
orbit is divided by the line of nodes).
In reality it is found that the intervals between transits of
Venus occur at present in the following order :
8, 105*; 8, 121*; 8, 105*; 8, 121*.
Transits have occurred, and are about to occur, in 1761, 1769,
1874, 1882, 2004, 2012 (the thick and thin type being
used to distinguish the two different nodes).
. Transits of Mercury occur much more frequently
than transits of Venus. For although the orbit of Mercury
is inclined to the ecliptic at about twice as great an angle as
that of Venus, this cause is more than compensated for by
the greater proximity of the planet to the Sun ; and since
the synodic period of Mercury is only about % of that of
Venus, conjunctions occur five times as often, so that we
should ceteris paribus expect five times as many transits. By
a method similar to that employed for Venus it is found that
transits occur at the same node at intervals of 7, 13, 33, or
-46 years. The next transit will occur in 1894.
Although transits of Mercury thus occur far more often
than transits of Venus, they cannot be used to determine the
Sun's parallax with such accuracy, for Mercury is so near the
Sun that the parallaxes of the two bodies are more nearly
equal, and the planet's relative displacement is therefore
much smaller than that of Venus. Moreover, Mercury moves
much more rapidly across the Sun's disc, giving less time for
accurate observations ; besides which, owing to the great
eccentricity of the orbit, the ratio of Mercury's to the Earth's
distance from the Sun cannot be so exactly computed.
THE DISTANCES OF THE SUN AND STAES. 283
SECTION III. Annual Parallax, and Distances of the.
Fixed Stars.
352. Annual Parallax, Definition. By Annual
Parallax is meant the angle between the directions of a
star as seen from different positions of the Earth in its annual
orbit round the Sun.
We haye several times ( 5, 247) mentioned that the fixed
stars have no appreciable geocentric parallax. Their distances
from the Earth are so great that the angle subtended at one
of them by a diameter of the Earth is far too small to be
observable even with the most accurately constructed instru-
ments. But the diameter of the Earth's annual orbit is
about 23,400 times as great as the Earth' R diameter, or about
186 million miles (twice the Sun's distance), and this
diameter subtends, at certain of the nearest fixed stars, an
angle sufficiently great to be measurable, sometimes amounting
to between \" and 2".
Now, the Earth, by its annual motion, passes in six months
from one end to the other of a diameter of its
orbit ; hence, by observing the same star at an
interval of six months, its displacement due to
annual parallax can be measured.
Since the Sun is fixed, the position of a star on
the celestial sphere is correctedfor annual parallax
by referring its direction to the centre of the Sun;
this is called the star's heliocentric direction, as
in 156.
The correction for annual parallax is
the angle between the geocentric and heliocen-
tric directions of a star. Let S be the Sun, E the Earth, x
the star (Fig. 115). Then Ex is the apparent or geocentric
direction of the star, Sx its heliocentric direction, and z ExS
is the correction for annual parallax. This angle is also equal
to xEx ! where Ex 1 is parallel to Sx.
We notice that the correction for annual parallax (ExS)
is the angular distance of the Earth from the Sun as they would
appear if seen ly an observer on the star.
ASTRON. u
284 ASTBONOMY.
353. To find the Correction for Annual Parallax.
Let r = JES = radius of Earth's orbit.
= Sx = distance of star.
E = L SEx = angular distance of star from Sun.
p = z ExS = annual parallax of star.
By trigonometry we have in the triangle SEx
smSJEx Sx '
whence* sinp= sin.E ..................... (i.).
Hence the parallactic correction p is greatest when
E = 90. This happens twice a year, and the corresponding
positions of the Earth in its orbit are evidently the inter-
sections of the ecliptic with a plane drawn through S per-
pendicular to Sx. Let this greatest value of p be denoted
by P, then P is called the star's annual parallax, or
simply the star's parallax. j
Putting E= 90 in (i.), we have
and therefore sin^? = sin P . sin E.
* Notice the close similarity between the present investigation
and that of 249.
f There is no risk of confusion in the use of the term parallax
alone, because a star has no geocentric parallax. The " parallax "
of a body means its equatorial horizontal parallax if the body belongs
to the solar system. If not, its " parallax " is its annual parallax.
THE DISTANCES OF THE SUN AND STAES. 285
But the angles P, p are always very small ; therefore their
sines are very approximately equal to their circular measures.
Thus we have approximately
_P (in circular measure) = ~ ,
d
# = JPsin E;
and, if P", p" denote the numbers of seconds in P, p,
P" = 180x60x60 -L = 206,265 r (approximately))
v d> d
and p" = P" sin E.
354. Relation between the Parallax and Distance
of a Star. If a star's parallax be known, its distance from
the Sun is given by the formula
fll = 180X60X60 JL =ao6266 r
TT d d
whence d = S r = 206,265 ,
where r is the Sun's distance from the Earth.
For most purposes r may be taken as 93 million miles.
EXAMPLES.
1. The parallax of Castor is 0'2" ; to find its distance. We have
d = 206265 *- = 206,265x93,000,000
P" 0-2
= 5 x 206,265 x 93,000,000
= 95,900,000,000,000, or 959 x 10 11 miles
approximately. It would be useless to attempt to calculate more
figures of the result with the given data, which are only approximate.
It is most convenient (besides being shorter) to write the result in
the second form.
2. To find the distance of a Centauri (i.) in terms of the Sun's
distance, (ii.) in miles, taking its parallax to be 0750".
Here d = . r = 275,000r
"75
275 x 10 3 x 93 x 10 fi = 25,575 x 10 9
256 x 10 11 miles approximately.
286 ASTRONOMY.
355. General Effects of Parallax. Since Ex is
parallel to Sz, it is in the same plane as US and JSx.
Hence the lines ES, Ex, Ex' cut the celestial sphere of E at
points 8, x, a? , lying in one great circle, and we have the
two following laws :
(i.) Parallax displaces the apparent position of a star from
its heliocentric position in the direction of the Sun.
(ii.) The parallactic displacement of any star at different times
varies as the sine of its angular distance from the Sun.
FIG. 117. FIG. 118.
Let Fig. 118 represent the observer's celestial sphere,
S the Sun. Let x be the apparent or geocentric position
of the star, whose parallax is P. Draw the great circle
Sx and produce it to # , making
XX Q = P sin Sx.
Then # represents the star's heliocentric position, and this
is its position as corrected for annual parallax.
Conversely, if the star's heliocentric position # is given,
we may obtain its geocentric or apparent position x by join-
ing tf $, and on it taking
xp = P sin 8x = P sin Sx Q very approximately
(for the difference between P sin Sx and P sin Sx is
exceedingly small, and may be neglected).
The terms Parallax in Latitude and Parallax in
Longitude are used to designate the corrections for parallax
which must be applied to the celestial latitude and longitude
of a star respectively. Similarly, the parallax in decl. and
parallax in R. A. denote the corresponding corrections for
the decl. and E-.A.
THE DISTANCES OP THE SUN AND STARS. 28?
356. To show that any star, owing to parallax,
appears to describe an ellipse.
In Fig. 117, Ex' is parallel to the star's heliocentric direc-
tion; therefore, x is fixed, relative to the Earth. Moreover,
x'x = ES. Hence, as the Sun 8 appears to revolve about
the Earth in a year, the star x will appear as though it
revolved in an equal orbit about its heliocentric position x ', in
a plane parallel to the ecliptic.
FIG. 119.
Let the circle MN(Fig. 119) represent this path, which the
star x appearsto describe in consequence of parallax. This circle
is viewed obliquely, owing to its plane not being in general
perpendicular to Ex'\ hence, if mn denote its projection on the
celestial sphere, the laws of perspective show that mn is an
ellipse. (Appendix, 12.) This small ellipse is the curve
described by the star on the celestial sphere during the year.
Particular Cases. A star in the ecliptic moves as if it
revolved about its mean position in a circle in the ecliptic
plane, hence its projection on the celestial sphere oscillates
to and fro in a straight line (more accurately a small arc of a
great circle) of length 2P.
For a star in the poie of the ecliptic the circle MN is per-
pendicular to Ed) hence Ex describes a right cone, and the
projection x describes on the celestial sphere a circle, of
angular radius P, about the pole K.
If the eccentricity of the Earth's orbit be taken into account, the
curve MN will be an ellipse instead of a circle, but its projection
mn will still be an ellipse.
288
ASTRONOMY.
357. Major and Minor Axes of the Ellipse. We
shall now prove the following properties of the small ellipse
described during the course of the year by a star whose
parallax is P, and celestial latitude I.
(i.) (A) The length of the semi-axis major is P.
(B) The major axis is parallel to the ecliptic.
(c) When the star is displaced along the major axis it
has no parallax in latitude.
(D) At these times the Sun's longitude differs from the
star's by 90.
(ii.) (A) The length of the semi-axis minor is Psinl.
(B) The minor axis is perpendicular to the ecliptic.
(c) When the star is displaced along the minor axis it
has no parallax in longitude.
(D) At these times the Sun's longitude is either equal to
the star's, or differs from it ly 180.
On the celestial sphere let a? denote the heliocentric
position of the star, ABAB' the ecliptic, JTits pole,
the secondary to the ecliptic through the star.
Then, if S is the Sun, the star X Q is displaced to #, where
x = P sin xS.
THE DISTANCES OF THE SUN AND STAES. 289
(i.) The displacement is greatest when sin x Q S is greatest,
and this happens when
Bin# /S=: 1, # =90.
If, therefore, we take A, A' on the ecliptic so that
A) A' are the corresponding positions of the Sun.
JTow A, A are the poles of BKB* (Sph. Geom., 11, 14,
15), and therefore the great circle Ax^A is a secondary to
BKB'. Hence, if a, a' denote the displaced positions of the
star, aa' is perpendicular to JK7?, and is therefore, parallel to
the ecliptic.
Also, x Q a = x$ = P sin 90 = P ;
therefore the semi-axis major of the ellipse is P.
Since AB = AB = 90, the star's longitude ( r B) differs
from the Sun's longitude at A or A by 90.
And since the star is displaced parallel to the ecliptic, its
latitude, or angular distance from the ecliptic, is unaltered,
and therefore the parallax in latitude is zero.
(ii.) The parallactic displacement is least when sin xJ3 is
least, and this happens when S is at B. For (Sph. Geom.,
26) B is the point on the ecliptic nearest to # . Also, since
sin xj = sin (180 # #) = sin x^B,
it follows that the parallactic displacement is also least when
S is at B'.
If, therefore, b, V be the extremities of the minor axis, the
arc lit is along JO?, and is therefore perpendicular to the
ecliptic.
Also, xjb = x$ = P sin x B = P sin / ;
therefore the semi-axis minor is P sin I.
When the Sun is at B, it has the same longitude as the
star ; when at ^, the longitudes differ by 180.
And since the star is displaced in a direction perpendicular
to the ecliptic, its longitude TB is unaltered; therefore the
parallax in longitude is zero,
290 ASTRONOMY.
The parallax in latitude is evidently equal to the apparent
angular displacement of the star resolved parallel to x Q K, and
its maximum value is xjb, or xj)'. The parallax in longitude
is not equal to the star's angular displacement perpendicular
to Jx Q , but to the change of longitude thence resulting, and
this is measured by the angle xKx^. Hence, in Tig. 120,
(i.) The maximum parallax in latitude = x Q b = I* sin .
(ii.) The maximum parallax in longitude = L x^Ka
= x^Kd = x Q a/sin Kx Q (Sph. Geom. l7)=P/eos x Q B
= P sec L
358. To determine the Annual Parallax of any
Star, the following methods have been employed :
(i.) The absolute method, by the Transit Circle ;
(ii.) Bessel's, or the differential method, by the micrometer
or heliometer j
(iii.) The photographic method.
The absolute method consists simply in observing with
the Transit Circle the apparent decl. and R.A. of a star at
different times in the year. From the small variations in
these coordinates it is possible to find the star's parallax.
Although this method has been successfully employed, it
possesses many disadvantages. For the observations are con-
siderably affected by errors of adjustment of the Transit
Circle and by refraction. Moreover, several other causes
give rise to variations in the star's apparent decl. and R.A.
during the year. These include aberration (vide Section I V.) ,
precession ( 141), and nutation, all of which produce dis-
placements much larger than those due to parallax.
In 372 we shall see that when either the latitude or longi-
tude is most affected by parallax it is unaffected by aberra-
tion. Hence the best plan is to find the changes in these
coordinates when they are respectively most affected by
parallax. These changes are P sin I and Psec I ( 357) and
from them P may be found.
359. Bessel's Method consists in observing with a micro-
meter (^79) or heliometer ( 80) the variations in the
angular distance and relative position of two optically near
stars during the course of a year.
THE DISTANCES OF THE STJN AND STARS. 291
The stars, being nearly in the same direction, are very
nearly equally affected by refraction, and we may also men-
tion that the same is true of aberration, precession and
nutation. These corrections do not therefore sensibly affect
the relative angular distance and positions of the stars. On
the other hand, the two stars may be at very different dis-
tances from the Earth ; if so, they are differently displaced
by parallax, and their angular distance and position undergo
variations depending on their relative parallax or difference of
parallax. Hence, by observing these variations during the
year the difference of parallax can be found.
This method does not determine the actual parallax of
either star. But if one of the observed stars is very bright
and the other is very faint, it is reasonable to assume that
the former is comparatively near the Earth, while the latter
is at such a great distance away that its parallax is insensible.
Under such circumstances the observed relative parallax is
the parallax of the bright star alone. By making compari-
sons between the bright star and several different faint stars
in its neighbourhood, this point may be settled.
If a considerable discrepancy .is found in the observed
relative parallaxes, one or more of the comparison stars must
themselves have appreciable parallaxes, but since the vast
majority of stars in any neighbourhood are too distant to have
a parallax, we shall be able to find the parallax not only of
the star originally observed, but of that with which we had
first compared it.
The parallax of a star can never be negative ; if the relative
parallax should be found to be negative, we should infer that
the comparison star has the greater parallax, and is therefore
nearer the Earth.
360. The Photographic Method is identical in prin-
ciple with the last, but instead of observing the relative
distances of different stars with a micrometer, portions of the
heavens are photographed at different seasons, and the dis-
placements due to parallax are measured at leisure by
comparing the positions of any star on the different plates.
This method has been used by Dr. Pritchard, of Oxford, and
possesses the advantages of great accuracy, combined with
convenience.
292 ASTRONOMY.
361. Parallaxes of certain Fixed Stars. The nearest
stars are a Centauri, with a parallax of 0-75", and 61 Cygni,
with parallax 0-54". Among others, the following may he
mentioned: a Lyra, 0-18", Sirim, 0-2", Arcturm, 0-1 3",
Polaris, 0-07", a Aquilce, 0-19". Of these, 61 Cygni is hy
no means bright ; and a companion star to Sirius is invisible
in all but two or three of the best telescopes. So it is not an
invariable rule that faint stars are most distant, and have no
appreciable parallax ; it is, however, true in the great majority
of cases.*
362. Proper Motions. Binary Stars. Many stars,
instead of being fixed in space, are gradually changing their
positions. They are then said to have a proper motion.
This motion may partly belong to the star, but is also partly
an apparent motion, due to the fact that the solar system is
itself moving through space in the direction of a point in the
constellation Hercules. The displacement due to this cause
can be allowed for approximately.
Many of these motions, like that of our own Sun, are apparently
progressive ; i.e., the star moves with constant velocity and
in the same direction. Others are orbital, i.e., the star
revolves about some other star, or (more accurately) two
stars revolve about their common centre of mass. Such a
system of stars is called a Binary Star. It is usually seen by
the naked eye as a single heavenly body, its components
being too near to be distinguished. Frequently a system of
stars has itself a progressive motion; and sometimes an
apparently progressive motion may really be an orbital one,
with a period so long that the path has not sensibly diverged
from a straight line during the short period for which stellar
motions have been watched.
A progressive or orbital motion cannot be confounded with
the displacement due to annual parallax, for the former is
always in the same direction, and the latter has a period dif-
fering from a year, while parallax always produces an annual
variation.
* These figures can only be regarded as very rough approxima-
tions, for considerable discrepancies exist between the values foun4
by different methods.
THE DISTANCES OP THE SUN AND STAES. 293
SECTION IV. The Aberration of Light.
363. Velocity of Light. "We now come to certain
methods of finding the Sun's distance which depend on the
fact that light is propagated through space with a large but
measurable velocity.
The velocity of light has been measured by laboratory
experiments in two different ways, invented by two French
physicists, Fizeau and Foucault. For the description of these
the reader is referred to "Wallace Stewart's Text Book of Light,
Chapter IX.* The experiments give the velocity of light in
air ; the velocity in vacuo can be obtained by multiplying
this by the index of refraction of air.f The latter quantity
may be found either by direct experiment or from the coeffi-
cient of astronomical refraction (see 183).
In 1876, Cornu, by employing Fizeau's method, found the
velocity of light in vacuo to be 300,400,000 metres per second.
Still more recently, Michelson, by a modification of Foucault's
method, has found the velocity to be 299,860,000 metres,
or 186,330 miles per second ; this may be taken as the
most probable value.
364. Roemer's Method. The Equation of Light.
In the last chapter we stated that Jupiter has four satellites,
which revolve very nearly in the plane of the planet's orbit.
Consequently a satellite passes through the shadow cast by
Jupiter once in nearly every revolution, and is then eclipsed,
as is our Moon in a lunar eclipse.
Since the orbits and periods of the satellites have been
accurately observed, it is possible to predict the recurrence
of the eclipses, so that when one eclipse has been observed
the times at which subsequent eclipses will begin and end
can be computed.
]S"ow, the Danish astronomer Roemer in 1675 observed a
remarkable discrepancy between the predicted and the
observed times of eclipses. If of two eclipses one happens
when Jupiter is near opposition, and the other happens near
the planet's superior conjunction, the observed interval
* The student will find it useful to read this chapter before com-
mencing the present section,
t Stewart's Light, 41.
294
ASTROXOMT.
between the former and the latter is always greater than the
computed interval ; similarly the observed interval between
an eclipse near superior conjunction and the next eclipse
near opposition is always less than the computed interval.
The eclipses at conjunction arc thus always retarded, relatively
to those at opposition, by an interval of time which is observed
to be about 16m. 40s. As explained by Roemer, this apparent
retardation is due to the fact that light travels from Jupiter
to the Earth with finite velocity, and therefore takes 1 6m.
40s. longer to reach the Earth when the planet is furthest
away at superior conjunction (B} than when the planet is
nearest the Earth at opposition (A).
The relative retardation is the difference between the times
taken by the light to travel over the distances AE and BE.
But BE- AE = 2S. Therefore the retardation is twice the
time taken by the light to travel from the Sun to the Earth.
Taking the retardation as 16m. 40s., we see that light takes
8m. 20s. to travel from the Sun to the Earth.
This interval is sometimes called the " equation of light?'
If we know the equation of light and the velocity of light,
\ve may calculate the Sun's distance. Conversely, if the
Sun's distance and the equation of light are known, the
velocity of light can be determined.
Knowing the Sun's distance, the Sun's parallax can be
computed, as in Chapter VIII., Section I. The present
method differs from those described in Sections I., II., in
that it gives the distance instead of the parallax of the Sun.
THE DISTANCES OF TIFE SUN AND STARS. 295
EXAMPLE 1. To find the Sun's distance, having given that the
velocity of light is 186,330 miles per second, and that eclipses of
Jupiter's satellites which occur when the planet is furthest from
the Earth, are retarded 16m. 40s. relatively to those which occur
when the planet is nearest.
Here the time taken by light to pass over a diameter of the
Earth's orbit is 16m. 40s. ; therefore light travels from the Sun to
the Earth in 8m. 20s., or 500 seconds.
/. the Sun's distance = 186.330 x 500 miles
= 93,165,000 miles.
EXAMPLE 2. Taking the value of the Sun's distance calculated in
the preceding example, the Sun's parallax will be found to be
about 8-78".
365. The Aberration of Light is a displacement of the
apparent directions of stars, due to the effect of the Earth's
motion on the direction of the relative velocity with which
their light approaches the earth.
The rays of light emanating from a star travel in straight
lines through space* with a velocity of about 186,330 miles
per second. We see the star when the rays reach our eye,
and the appearance presented to us depends solely on how
the rays are travelling at that instant. If the Earth were
at rest, and there were no refraction, we should see the star
in its true direction, "because the light would be travelling
towards our eyes in a straight line from the star. But in
every case the direction in which a star is seen is the direction
of approach of the light-rays from the star at the instant of
their reaching the eye.
Now the velocity of approach is the relative velocity of the
light with respect to the observer. If the observer is in
motion, this relative velocity is partly due to the motion
of the light and partly due to the motion of the ob-
server. If the observer happens to be travelling towards
or away from the source of light, the only effect of
his motion will be to increase or decrease the velocity of
approach of the light, without altering its direction, but if he
be moving in any other direction, his own motion will alter
the direction of the relative velocity of approach, and will
therefore alter the direction in which the star is seen.
* Of course the rays are refracted when they reach the Earth's
atmosphere, but the effects of refraction can be allowed for separately.
296 A8TEONOMY.
Suppose the light to be travelling from a distant star x
in the direction xO. Let T"be the velocity of light, and let
it be represented by the length M 0. Suppose also that an
observer is travelling along the direction NO with velocity w,
represented by the straight line NO. Then, if we regard
as a fixed point, the light is approaching with velocity re-
presented by MO. Also since the observer is approaching
with velocity represented by NO, the point is approaching
the observer jVwith an equal and opposite velocity repre-
sented therefore by ON. Hence the whole relative velocity
with which the light is travelling towards the observer is
the resultant of the velocities represented by M and ON.
By the Triangle of Velocities this resultant velocity is repre-
sented in magnitude and direction by MN. Hence MN
represents the direction of approach of the light towards the
observer's eye. Therefore when the observer has reached
the star is seen in the direction Ox' drawn parallel to NM,
although its real direction is Ox,
In consequence, the star appears to be displaced from its
true position x to the position x'. This displacement is
called the aberration of the star, and its amount is, of
course, measured by the angle xOx. This angle is sometimes
called the angle of aberration or the aberration error.
366. Illustrations of Eelative Velocity and Aberration. The
following simple illustrations may possibly assist the reader in
understanding more thoroughly how aberration is produced.
(1) Suppose a shower of rain-drops to be falling perfectly
vertically, with a velocity, say, of 40 feet per second. Then,
if a man walk through the shower, say with a velocity of 4 feet
THE DISTANCES OF THE SUN AND STABS. 297
per second, the drops will appear to be coining towards him, and
therefore to be falling in a direction inclined to the vertical. Here
the man is moving towards the drops with a horizontal velocity of
4 feet per second, and therefore the drops appear to be coming
towards the man with an equal and opposite horizontal velocity of
4 feet per second.
Their whole relative velocity is the resultant of this horizontal
velocity and the vertical velocity of 40 feet per second with which
the drops are approaching the ground. By the rule for the compo-
sition of velocities, this lelative velocity makes an angle tan~ l -$ or
tan* 1 '1 with the vertical. Hence the man's own motion causes an")
apparent displacement of the direction of the rain from the vertical Y"
through an angle tan' 1 g l. This angle corresponds to the angle ofJ
aberration in the case of light.
(2) Suppose a ship is sailing due south, and that the wind is blow-
ing from due west with an equal velocity. Then to a person on the
ship the wind will appear to be blowing from the south-west, its
southerly component being due to the motion of the ship, which is
approaching the south. In this case the ship's velocity causes the
wind to apparently change from west to south-west, i.e., to turn
through 45. We might, therefore, consistently say that the
" angle of aberration " of the wind was 45.
367. Annual and Diurnal Aberration. A point on
the Earth's surface is moving through space with a velocity
compounded of
(i.) The orbital velocity of the Earth in the ecliptic about
the Sun ;
(ii.) The velocity due to Earth's rotation about the poles.
These give rise to two different kinds of aberration, known
respectively as annual and diurnal aberration. Now the
Earth's orbital velocity is about 2?r x 93,000,000 miles per
annum, or rather over 18 miles per second, while the
velocity due to the Earth's rotation at the equator is roughly
2^x4000 miles per day, or 0*3 miles per second. The
former velocity is about T oi^o ^ * ne velocity of light, and
therefore the annual aberration is a small though measurable
angle. The latter velocity is only -fa as great ; hence the
diurnal aberration is much smaller and less important. For
this reason the term " aberration" always signifies annual
aberration, unless the word "diurnal" is also used. We shall
now consider the effects of annual aberration, leaving diurnal
aberration till the end of this section.
298 * ASTRONOMY.
368. To determine the correction for aberration
on the position of a Star. Let Ox be the actual direction
of a star x seen from the Earth at ; U the direction of the
Earth's orbital motion at the time of observation. On Ox
take OM representing on any scale the velocity of light, and
draw MY parallel to 017, and representing on the same scale
the velocity of the Earth. Then YO represents the relative
velocity of the light in magnitude and direction, so that OYx
is the direction in which the star x is seen (Mg. 123).
[For if ON be drawn parallel and equal to YM, the parallelogram
of velocities MNOT shows that 21/0, the actual velocity of the light-
rays in space is the resultant of the two velocities TO and NO, or
YO and MY, and therefore YO is the required relative velocity.]
FIG. 124.
Since Ox, Ox, and U all lie in one plane, it follows, by
representing their directions on the celestial sphere, that a
star is displaced by aberration along the great circle joining its
true place to the point on the celestial sphere towards which the
Earth is moving.
The displacement xOx is called the star's aberration
error. Let it be denoted by y, and let
u = NO = velocity of Earth,
V MO = velocity of light.
Then the triangle OM Y gives
sinMOY_ MY _ u t
sin M YO ~~ MO ~~ V*
or sin y = -^ sin HYO = ~ sin
THE DISTANCES OF THE SUN AND STARS. 299
The aberration error y is, therefore, greatest when UOx'
90. Let its value, then, be k. Putting UOx = 90, we
have sin H =s
and .*. sin y = sin k sin UOoc.
The angle UOx is called the Earth's Way of the star,
and k is called the Coefficient of Aberration. Since a and
k are both small, we have, approximately
y = k sin (Earth's way),
k (in circular measure) = u/ V\
and, therefore, if y", k" denote the number of seconds in y, k
respectively
?/' = k" sin (Earth's way),
, 180x60x60 u
~iT ' v
= 206,265 x.
velocity of light
369. General effect of Aberration on the Celestial
Sphere. Neglecting the eccentricity of the Earth's orbit,
the direction of motion of the Earth, in the ecliptic
plane, is always perpendicular to the radius vector drawn to
the Sun. Hence, on the celestial sphere, the point 7", towards
which the Earth is moving, is on the ecliptic, at an angular
distance 90 behind the Sun. This point is sometimes called
the apex of the Earth's Way.
Let x' denote the observed position of the star. Draw the
great circle x' 7, and produce it to a point x , such that
xx' = k sin x' U.
Then x represents the star's true position, corrected fV
aberration.
Conversely, if we are given the true position x, we can find
the apparent position x' by joining #7 and taking
xx' = k sinxV,
for it is quite sufficiently approximate to use k sin xU instead
of k sin x'U.
A3TROX. X
300 ASTRONOMT.
"We thus have the following laws :
(i.) Aberration produces displacement in the apparent
position of a star towards a point U on the ecliptic, distant 90
behind the Sun.
(ii.) The amount of the displacement varies as the sine of the
Earth's Way of the star, i.e., the star's angular distance
from the point U.
FIG. 125.
FIG. 123.
370. Comparison between Aberration and Annual
Parallax. The student will not fail to notice the close
analogy between the corrections for aberration and annual
parallax.
The point 7 for the former corresponds to the point S for
the latter, in determining the direction and magnitude of the
displacement. In fact, the aberration error of a star is exact///
the same as its parallactic correction would be three months earlier
(when the Sun was at U) if the star's annual parallax ice re k.
There is, however, this important difference that the annual
parallax depends on a star's distance, whilst the constant of
aberration k is the same for all stars.
For k depends only on the ratio of the Earth's velocity to
the velocity of light, and not on the star's distance. The
value of k in seconds is about 20'492" ; for rough purposes it
mnv he taken as 20'5".
THE DISTANCES OF THE SUN AND STARS. 301
371. To show that the aberration curve of a star is
an ellipse. This result, which follows immediately from the
analogy between aberration and parallax, may be proved inde-
pendently as follows: On Ox (Fig. 125), the true direction
of a star #, take Ox to represent the velocity of light, and
xM to represent the Earth's velocity. Then M meets the
celestial sphere in m, the star's apparent position.
As the Earth's direction of motion in the ecliptic varies,
while its velocity remains constant, Jfdescribes a circle, about
x as centre in a plane parallel to the ecliptic plane. The
projection of this circle on the celestial sphere is an ellipse
(cf. 356), and this is the curve traced out by a star during
the year in consequence of aberration.
Particular Cases. A star in the ecliptic oscillates to
and fro in a straight line, or more accurately an arc of a great
circle of length 2/j. A star at the pole of the ecliptic revolves
in a small circle of radius k (cf. 356).
372. Major and Minor Axes of the Aberration
Ullipse. By writing Z7"for S and k for P in the investiga-
tion of 357, we obtain the analogous results relating to the
ellipse described by a star in consequence of aberration,
namely :
(i.) (A) The length of the semi-axis major is k.
(B) The major axis of the ellipse is parallel to the ecliptic.
(c) When the star is displaced along the major axis it has no
aberration in latitude.
(D) At these times the Surfs longitude is either equal to the
.star's, or differs from it by 180.*
(ii.) (A) The length, of the semi-axis minor is k sin I.
(B) The minor axis is perpendicular to the ecliptic.
(c) When the star is displaced along the minor axis, it has no
aberration in longitude.
(D) At these times the Sun's longitude differs from the star's
.by 90.
COROLLARY. The maximum aberration in longitude = k sec?
(cf. 357, ii.).
* Note that (i., r) and (ii., D) are the reverse of the corresponding
-properties in 357.
302 ASTRONOMY.
*373. Effect of Eccentricity of Earth's Orbit. Owing to the
elliptic form of the Earth's orbit the Earth's velocity is not quite
uniform, and therefore the coefficient of aberration is subject to
small variations during the year. The earth's velocity is greatest
at perihelion and least at aphelion. The angular velocities at those
times are inversely proportional to the squares of the corresponding
distances from the Sun, but the actual (linear) velocities are in-
versely proportional to the distances themselves, and these are in
the ratio of l-e : 1 + e, or 1 --fa : 1 + ^ ( 149). Since the coeffi-
cient of aberration is proportional to the Earth's velocity, its
greatest and least values are therefore in the ratio of 61 : 59, and
are respectively and of its mean value.
Moreover, the direction of the Earth's motion is not always
exactly perpendicular to the line joining it to the Sun, hence the
" apex of the Earth's way," towards which a star is displaced, may
be distant a little more or a little less than 90 from the Suu at
different seasons.
The aberration curve is still an ellipse. The student who
has read the more advanced parts of particle dynamics may know
that the curve MN, tracecfout by M, is in this case the "hodograph "
of the Earth's orbital motion. It is also known, in the case of
elliptic motion, such as the Earth's, that this hodograph is a circle,
whose centre does not, however, quite coincide with x. Hence the
aberration-curve hTc is an ellipse.
374. Discovery of Aberration. Aberration was dis-
covered by Bradley, in 1725, in the course of a series of
observations made with a zenith sector on the star y Draconis
for the purpose of discovering its annual parallax. The star's
latitude was observed to undergo small periodic variations
during the course of the year, and these differed from the
variations due to annual parallax in the fact that the dis-
placement in latitude was greatest when the Sun's longitude
differed from that of the stars ly 90 ; that is, at the time when
the parallax in latitude should be zero ( 357, i., c.). The fact
that the phenomenon recurred annually led Bradley to suppose
that it was intimately connected with the Earth's motion
about the Sun, and he was thus led to adopt the explanation
which we have given above, It will be seen that the pecu-
liarity which led Bradley to discard annual parallax as an
explanation is quite in harmony with the results of 372.
375. To Determine the Constant of Aberration by
Observation. The constant k can best be found by observ-
ing different stars with a zenith sector or transit circle, as in.,
the direct method of finding a star's parallax ( 358).
THE DISTANCES OF THE SUN AND STAUS.
303
The differential method of 359 cannot be used, because
the coefficient of aberration is the same for all stars. But
aberration is much larger than parallax (the coefficient of
aberration being 20-49", while the greatest stellar parallax
is < I"), and can therefore be found directly with greater
accuracy. Of course it is necessary to make corrections for
refraction and precession. The former correction is the most
liable to uncertainty, as it varies slightly according to atmo-
spheric conditions. But, as all stars have the same constant
of aberration, a star may be selected which transits near the
zenith, and is therefore but little affected by refraction.
This condition was secured by Bradley when he observed
the star y Draconis. The star is very favourable in another
respect, for its longitude is very nearly 270. It therefore
lies very nearly in the " solstitial colure," its declination
circle passing nearly through the pole^f the ecliptic.
J,
At the vernal equinox, the star's longitude is less than the
Sim's by 90, and it is therefore displaced away from the
poles of the ecliptic and equator through a distance k" sin ?,
its decimation being therefore decreased by k" sin I. At the
autumnal equinox its declination is increased by k" sin L
Hence the difference of the apparent declinations = 2k' 1 sin ?,
and this is also the difference of the star's apparent meridian
zenith distances. By observing these, k" may be found,
7 being of course known.
The value of k" is very approximately 20 - 493".
304 A.STBONOMY.
376. Relation between the Coefficient of Aberration
and the Equation of Light. "We have seen (368) that
JU 180x60x60 u r x
T~ 'T '. ................. W '
where V is the coefficient of aberration in seconds, u the
velocity of the Earth, V that of light, hoth of which we will
suppose measured in miles per second.
Now let r represent the radius of the Earth's orbit (sup-
posed circular) in miles. Then in one sidereal year, or 365 J
days, the Earth travels round its orbit through a distance
2irr miles. Hence the Earth's velocity in miles per second is
365ix 24x60x60
Substituting in (i.), we have
jfc" - 15 JL
365 V
But r/ Fis the time taken by the light to travel from the
Sun to the Earth, measured in seconds, or the " equation of
light." Hence,
The coefficient of aberration in seconds
= --- x number of seconds taken by Sun's light to
365 4 reach Earth.
Thus, by observing the retardation of the eclipses of
Jupiter's satellites at superior conjunction, the coefficient of
aberration can be found independently of the methods of
375, the number of days (365^) in the sidereal year being
of course known.
The close agreement between the values found thus and
by direct observation affords the strongest evidence in support
of Bradley 's explanation of aberration.
EXAMPLE. To find the coefficient of aberration in seconds, having
given that light takes 8m. 20s. to travel from the Sun to the Earth.
Here the required coefficient of aberration
,// 15 x 500 7oOO
THE DISTANCES OF THE SUN AKD STARS. 305
377. To find the time taken by the light from a
star to reach the Earth. It is sometimes convenient to
estimate the distance of a star by the number of years which
the light from it takes to reach the Earth. This may he
determined from a knowledge of the star's parallax, and of
the coefficient of aberration, without knowing either the Snn's
distance or the velocity of light.
Let the parallax of a star he = P" in seconds = P radians,
and let the coefficient of aberration = k" seconds = k radians.
Then, if r, d be the Earth's and star's distances from the
Sun, we have
p _ r 7 _ velocity of Earth
d ' " velocity of light "
Now, in one year, the Earth travels over a distance 2?rr ;
27Tf
.-. in one year light travels a distance ;
/J
.-. the number of years taken by light to travel from the
star (distance d) to the Earth
' \ k ~ 27TT ~~ 27TP 27TP"'
The distance travelled by light in a year is sometimes
called a " light-year." Hence,
The product of a star's parallax and its distance in light-
years is equal to the coefficient of aberration divided by 2?r.
EXAMPLES. 1. To find how long the light would take to reach U8
from a star having a parallax Ol".
The required time, in years,
1 fc" 10x20-49x7
== = --- approximately
STT 0-1 2 x 22
= 32-6.
2. To find the time taken by the light from the nearest star,
a Centauri, taking its parallax as 075".
The parallax is 7'5 times that of the star in the last question,
therefore its distance is 10/75 as great, and the time taken by the
light = ^ = 4-35 years.
7'5
306 ASTRONOMY.
378. Relation between the Coefficient of Aber-
ration, the Sun's Parallax, and the Velocity of Light.
-It follows from 376 that if the coefficient of aberration
k" be determined by observation, the fraction rjV is also
known, independently of observations of the eclipses of
Jupiter's satellites. And if F, the velocity of light, be deter-
mined experimentally by the method of Foucault or Fizeau,
the Sun's distance r can be found. Thus the Sun's parallax
can be calculated from the coefficient of aberration and the
velocity of light. And generally, if, of the four quantities,
Sun's parallax, coefficient of aberration, velocity of light, and
length of sidereal year in days, any three are observed, the
value of the fourth may be deduced from them.
In this manner Foucault, by his determination of the
velocity of light, in 1862, found the Sun's parallax to be 8'86".
Cornu, by experiments in 1874 and 1877, combined with the
values for k" determined by Struve, obtained the values
8-83" and 8*80" respectively. Hichelson's experiments make
the parallax 8- 793".
EXAMPLE. If the velocity of light = 186,000 miles per second
and the Earth's radius (a) = 3,960 miles, to prove that the product
of the Sun's parallax and the coefficient of aberration, both measured
in seconds, is 180'35.
The Sun's parallax P" = 18 * G0 * 60
n- r
15 r GO
P " fc" = 18 x G0 x ( > x 60 a 200205 x GO 3000
14G1* V 14G1 186000
= 180-35.
379. Planetary Aberration. The direction of any
planet is affected by aberration, which is due partly to the
motion of the Earth, and partly to that of the planet itself.
For, during the time occupied by the light in travelling
from a planet to the Earth, the planet itself will have moved
from the position which it occupied when the light left it.
We shall, however, show that the direction in which a
planet is seen at any instant was the actual direction of the
planet relative to the Earth at the instant previously when the
lujlt left the planet.
THE DISTANCES OF THE SUN AND STARS. 307
Let t be the time required by the light to travel from
the planet to the Earth. Let P, Q be the positions of the
planet and Earth at any instant ; P', Q' their positions after
an interval t.
The light which leaves the planet when at P reaches the
Earth when it has arrived at Q' ; the direction of the actual
motion of the light is, therefore, along PQ. Eut PQ' and
Q Q' are the spaces passed over by the light and the Earth*
FIG. 128.
respectively in the time t (and QQ' is so small an arc that it
may be regarded as a straight line). Therefore
QQ' : PQ' = velocity of Earth : velocity of light.
Hence it follows from 368 that the line PQ represents the
direction of relative velocity of the light with respect to the
Earth. Therefore, when the Earth is at Q' the planet is seen
in a direction parallel to PQ, and its apparent direction
is exactly what its real direction was at a time t previously.
The same is true in the case of the Sun or a comet, or
any other body, provided that the time taken by the light
from the body to reach the Earth is so small that the Earth's
motion doe. not change sensibly in direction in the interval.
The aberration of the planet at any instant is the angle
between the apparent direction PQ and the actual direction
P'Q.
308 ASTRONOMY.
EXAMPLE. To find the effect of aberration on the positions of (i.)
the Sun, (ii.) Saturn in opposition, taking its distance from the
Sun to be 9^ times the Earth's.
(i.) The light takes 8m. 20s. to travel from the Sun to the Earth
therefore the Sun's apparent coordinates at any instant are its actual
coordinates 8m. 20s. previously. Thus, its apparent decl. and R.A.
at noon are its true decl. and R.A. at 23h. 51m. 40s., or llh. 51m.
,40s. A.M.
Now the Sun describes 360 in longitude in 365 days. Hence, in,
500 seconds it describes 20'492", and the Sun's aberration in longi-
tude is 20'492". This is otherwise evident from the fact that the-
Earth's way of the Sun is 90 ; and it is at rest, consequently its
aberration = fc.
(ii.) The distance of Saturn from the Earth at opposition is.
= 9| 1, or 8 times the Sun's distance. Light travels over this
distance in 8m. 20s. x 8 = 500x8|s. = Ih. 10m. 50s. Therefore,
the apparent coordinates are the actual coordinates Ih. 10m. 50s.
previously.
Thus the observed decl. and R.A. at midnight (12h. 0*m. Os.) are the-
true decl. and R.A. at lOh. 49m. 10s.
380. Diurnal Aberration is due to the effect of the
Earth's diurnal rotation about the poles on the relative velo-
city of light.
As the Earth revolves from west to east, the portion of the-
motion of an ohserver due to this diurnal rotation is in the-
direction of the east point E of the horizon.
The effect of diurnal aberration can thus be investigated
by methods precisely similar to those of 368, Staking the-
place of U.*
Hence, every star x is displaced by diurnal aberration,
towards the east point E. And if x' be its displaced position,,
then
the displacement xx' = A sin x E,
where
,-, i n . volocitv of observer
Circular measure of A = ^-- .
velocity ot light
* The student will find it useful to go through the various steps-
of 368-371, considering the diurnal motion.
THE DISTANCES OF THE STTN AND STARS. 309"
Taking a for the Earth's radius, V for the velocity of light,
let the observer's latitude be I.
In a sidereal day (86164-1 mean seconds) the Earth's
rotation carries the observer round a small circle, whose dis-
tance from the Earth's axis is a cos , and whose circumference
is, therefore, 'lira cos I. Hence, the observer's velocity
= miles per second :
86164-1 ( F
1-irtt COS /
.*. circular measure of A =
86164-1 x V 1
.'. A" (number of seconds in A]
__ 180x60x60 x 2Tra cos/
TT 86164-1 7'
Ida cos I , t
= approximately.
Thus, the coefficient of diurnal aberration varies as the
cosine of the latitude. If K" denote the coefficient of
diurnal aberration at the equator in seconds, we therefore,,
have
K" = - l5 * 39fi3 = n-32"
V 186,000
A" = K" cos I = O'32' cos L
* Effect of Diurnal Aberration on Meridian Observations.
The correction for diurnal aberration is greatest when the star
is 90 from the east point, i.e., is on the meridian. In this case,
the displacement is perpendicular to the meridian, and is equal
to A".
The star's meridian altitude is thus unaffected, but its time of
transit is somewhat retarded at upper culmination, and (for a cir-
cumpolar star) accelerated at lower culmination, since the star
appears on the meridian, when it is really A" west of the meridian-
The effect of diurnal aberration on the time of transit is thus equi-
valent to that of a small collimation error A" in the Transit Circle.
For a star on the equator, seen from the Earth's equator, the-
retardation of the time of transit would be -^ K " seconds, = -g^
of a second nearly, and it would be difficult to observe such a small
interval.
310 ASTRONOMY.
381. To determine the Coefficient of Diurnal Aber-
ration "by Observations of the Azimuths of Stars
when on the Horizon.
When a star is rising or setting it is evidently displaced by
diurnal aberration along the horizon towards the east point.
Consider two stars, one of which rises S. of E., and the other
"N. of E. It is evident that their rising points are drawn
towards one another. But the stars set S. of W. and "N.
of W., and their displacements are still towards the E.
point ; hence, their setting points are separated away from
one another. And, if the stars, at rising and setting, be
carefully observed with an altazimuth, the difference between
their azimuths at setting will exceed that between their
azimuths at rising by an amount proportional to the diurnal
aberration. From this, the coefficient of diurnal aberration
may be found.
The azimuths are unaltered by refraction ( 184), but the times
of rising and setting are slightly altered by refraction. If the co-
efficient of refraction be the same at both observations, however,
the acceleration in rising will be equal to the retardation at setting,
.and the refraction will increase the azimuths at rising and setting
by the same amount ; thus the data will be unaffected. If the tem-
perature of the air has changed considerably between rising and
setting, it is only necessary to make the observations at equal
intervals before and after the stars transit.
.382. Relation between the Coefficients of Aberra-
tion and the Sun's Parallax. We have evidently
7T" _ velocity of diurnal motion at equator
k" velocity of Earth's orbital motion
But the velocities in miles, per sidereal day, are 2-n-a and
This gives the coefficient of diurnal aberration at the equa-
tor in terms of the coefficient of annual aberration and the
Sun's parallax. Conversely, if it were possible to observe
the coefficient of diurnal aberration accurately, we should
thus have another way of finding the Sun's parallax.
But the smallness of the diurnal aberration renders it im-
possible to obtain good results by this method.
THE DISTANCES OF THE SUN AND STARS. 311
EXAMPLES. XI.
1. Prove that cosec 876" = 23546 approximately, and thence that
the distance of the Sun is nearly 81 million geographical miles, the
angle 8' 76" being the Sun's parallax, and a geographical mile sub-
tending 1' at the Earth's centre.
2. Find the Sun's diameter in miles, taking the Sun's parallax as
8'8", its angular diameter as 32', and the Earth's radius as 3,960
miles.
3. A spot at the centre of the Sun's disc is observed to subtend
an angle of 5". What is its absolute diameter?
4. Show, by means of a diagram, that the general effect of the
Earth's diurnal rotation is to shorten the duration of a transit of
Venus, and that this circumstance might be used to find the Sun's
parallax.
5. Supposing the equator, ecliptic, and orbit of Venus all to lie in
one plane, and that a transit of Venus would last eight hours, at a
point on the Earth's equator, if the Earth were without rotation ;
show that, if the Sun is vertically overhead at the middle of the
transit, the duration is diminished by about 9m. 55?s. owing to the
Earth's rotation, taking the Sun's parallax to be 8'8", and the syn-
odic period of Venus to be 586 clays.
6. If the annual parallax be 2", determine the distance of the star,
taking the Sun's distance to be 90,000,000 miles. Hence, deduce
the distance of a star whose pamllax is 0'2".
7. Find, roughly, the distance of a star whose parallax is 0'5",
given that the Sun's parallax is 9", and the Earth's radius is 4000
miles.
8. The parallax of 61 Cyyni is O'o", and its proper motion, per-
pendicular to the line of sight, is 5" a year; compare its velocity in
that direction with that of the Earth in its orbit round the Sun.
9. Account for the following phenomena : (i.) all stars in the
ecliptic oscillate in a straight line about their mean places in the
course of the year ; (ii.) two very near stars in the ecliptic appear to
Approach and recede from one another in the course of the year.
10. Suppose the velocity of light to be the same as the velocity of
the Earth round the Sun. Discuss the effect on the Pole Star as-
seen by an observer at the North Pole throughout the year.
312 ASTRONOMY.
11. Sound travels with a velocity 1,100 feet per second. Deter-
mine the aberration produced in the apparent direction of sound to
a person in a railway train travelling at sixty miles an hour, if the
source of sound be exactly in front of one of the windows of the
carriage.
12. Show that, in consequence of aberration, the fixed stars
whose latitude is I appear to describe ellipses whose eccentricity
is cos I.
13. How must a star be situated so as to have no displacement
-due to (i.) aberration, (ii.) parallax? "Where must a star be so that
the effect may be the greatest ?
14. On what stars is the effect of aberration or parallax to make
them appear to describe (i.) circles, (ii.) straight lines?
15. Show that the effect of annual parallax on the position of a
fitar may be represented by imagining the star to move in an orbit
equal and parallel to the Earth's orbit, and that the effect of aber-
ration may be represented by imagining it to revolve in a circle
whose radius is equal to the distance traversed by the Earth while
the light is travelling from the star.
16. Supposing the star 17 Virginia to be situated (as it nearly is)
at the first point of Libra, find the direction and magnitude of its
displacement due to aberration about the 21st day of every month
of the year, taking the coefficient of aberration to be 20*5". When
is its aberration greatest ?
17. At the solstices show that a star on the equator has no aber-
ration in declination. If its R.A. be 22h., show that its time of
transit is retarded at the summer and accelerated at the winter
solstice by "68 of a second.
18. If the coefficient of aberration be 20", and an error of 2,000
miles a second be made in determining the-velocity of light, find, in
miles, the consequent error in the value of the Sun's mean distance
as computed from these data.
19. Show that when a planet is stationary its position is unaffected
by aberration.
20. Taking the Earth's radius as 4,000, velocity of light 186,000
miles per second, show that the coefficient of diurnal aberration at
the equator is about one-third of a second.
THE DISTANCES OF THE SILX AND STARS. 313
MISCELLANEOUS QUESTIONS.
1. Explain the following terms: asteroid, libration, lunation
parallax, perihelion, planet's elongation, right ascension, synodical
period, gyxygies. ztn : .th.
2. Given that the R.A. of Orion's belt is 80, show by a figure its
position at different hours of the night about March 21 and
September 23.
3. Prove that the number of minutes in the dip is equal to the
number of nautical miles in the distance of the visible horizon.
4. Show how to determine the latitude of a place by meridional
observations on a circumpolar star, taking into account the refraction
5. Show how to find longitude from lunar distances. The cleared
lunar distance of a star at 8h. 30m. local mean time is 150'45", and
the tabular distances are 150'0" at 6h. and 151'30' / at 9h. of Green-
wich mean time. Find the longitude.
6. At what time of the year can the waning moon best be seen ?
7. On July 21 at 2 A.M. the Moon is on the meridian. What is
the age of the Moon ? Indicate the position on the celestial sphere
of a star whose declination is and whose R.A. is 30.
8. Taking the distance of Venus from the Sun to be f of that of
the Earth, find the ratio of the planet's angular diameters at superior
and inferior conjunction and greatest elongation, and draw a series
of diagrams showing the changes in the planet's appearance during
a synodic period, as seen through a telescope under the same
magnifying power.
9. Defining a lunar day as the interval between two consecutive
transits of the Moon across the meridian, find its mean length in
(i.) mean solar, and (ii.) sidereal units.
10. At what season is the aberration of a star least whose R.A. is
$0 and whose declination is 60 ?
11. Show that the constant of aberration can be determined by
observation of Jupiter's satellites, without a knowledge of the radius
of the Earth's orbit.
12. How is it possible to calculate separately the aberration the
constant of aberration being supposed unknown annual parallax,
and proper motion of a star, from a long series of observations of the
apparent place of a star ?
3H ASTRONOMY.
EXAMINATION PAPEK. XL
1. Why is the method for finding the Moon's parallax not available-
in the case of the Sun? Show how the determination of the
parallax of Mars leads to the determination of the Sun's parallax.
2. Show how the Sun's parallax can be found by comparing the
times of commencement or of termination of a transit of Venus at two-
stations not far from the Earth's equator.
3. Show how the Sun's parallax can be found by comparing the-
durations of a transit of Venus at two stations in high N. and S.
latitudes. Why is this method not available when the transit is-
central ?
4. Distinguish between solar and stellar parallax. Towards what
point does a star seem to be displaced by heliocentric parallax ?
Find an expression for the displacement.
5. Describe Bessel's method of determining the annual parallax
of a fixed star.
6. How might the Sun's parallax be determined by observations
of the eclipses of Jupiter's satellites?
7. Explain the aberration of light, and investigate the direction-
and magnitude of the displacement which it produces on the-
apparent position of a star.
8. Show that owing to aberration a star in the pole of the ecliptic
appears to describe a circle, and that a star in the ecliptic appears-,
to oscillate to and fro in a straight line during the course of the year^
9. Show how the velocity of light may be determined from the
aberration of.a star when the Sun's mean distance is known.
10. Investigate the general effects of diurnal aberration due to-
the Earth's rotation about its axis. In what direction nre stars-
displaced by diurnal aberration ? Show that the coefficient of
diurnal aberration at a place in latitude I is K cos I, where K is the-
coefficient at the equator.
DYNAMICAL ASTRONOMY,
CHAPTER XII.
THE ROTATION OF THE EARTH.
383. Introductory. In the preceding chapters we have
shown how the motions of the celestial bodies can be determined
by actual ob crvation, and we have also explained certain
resulting phenomena. But no use has yet been made of the
principles of dynamics ; consequently we have been unable
to investigate the causes of the various motions. In par-
ticular, while we have assumed that the diurnal rotation of the
stars is an appearance due to the Earth's rotation, we have
not as yet given any definite proof that this is the only pos-
sible explanation.
The ancient Greeks accounted for the motions of the solar
system by means of the Theory of Epicycles, according to
which each planet moved as if it were at the end of a system
of jointed rods rotating with uniform but different angular
velocities. Suppose AB, BO, CD to be three rods jointed
together at II, C. Let A be fixed ; let AB revolve uniformly
about A ; let BC revolve with a different angular velocity
about B ; and let CD revolve with another different angular
velocity about C. Then, by properly choosing the lengths and
angular velocities of the rods, the motion of J), relative to A,
may be made nearly to represent the motion, relative to the
Earth, of a planet.
Copernicus (A.D. 1500 eirc.) was the first astronomer who
explained the motions of the solar system on tho theory that
the diurnal motion is due to the Earth's rotation, and that
the Earth is one of the planets which revolve round the Sun.
This theory was adopted by Kepler (A.D. 1609 circ.) whose
laws of planetary motion have already been mentioned ( S26).
A.STRON. Y
316 ASTRONOMY.
These laws were, however, unexplained until their true cause
was found by Newton (A.B. 1687) by his discovery of the
law of gravitation.
384. Arguments in Favour of the Earth's Rota-
tion. Without appealing to dynamical principles, the pro-
bability of the Earth's rotation about its axis (87) may be
inferred from the following considerations :
(i.) If the Earth were at rest, we should have to imagine
the Sun and stars to be revolving about it with inconceivably
great velocities. If the Earth rotates, the velocity of a point
on its equator is somewhere about 1,050 miles an hour. But
since the Sun's distance is about 24,000 times the Earth's
radius, the alternative hypothesis would require the Sun a
body whose diameter is nearly 110 times as great as that of
the Earth to be moving with a velocity 24,000 times as
great, or about 25,000,000 miles an hour; while most of the
fixed stars are at such distances from the Earth that they
would have to move with velocities vastly greater than the
velocity of light. It is inconceivable that such should
be the case.
(ii.) The diurnal rotations all take place about the pole,
and are all performed in the same period a sidereal day.
This uniformity is a natural consequence of the Earth's rota-
tion, but it' the Earth were at rest, it could only be explained
by supposing the stars to be rigidly connected in some manner
or other. Were such a connection to exist it would be difficult
to explain the proper motions of certain fixed stars, and the
independent motions of the Sun, Moon, and planets.
(iii.) By observing the motion of the spots on the Sun at
different intervals, it is found that the Sun rotates on its axis.
Moreover, similar rotations may be observed in the planets ;
thus, Mars is known to rotate in a period of nearly 24
hours. There is, therefore, nothing unreasonable in suppos-
ing that the Earth also rotates once in a sidereal day.
(iv.) The phenomenon of diurnal aberration affords a proof
of the Earth's rotation. Were it not for the difficulty of its
observation, this proof alone would be conclusive.
We may mention that diurnal parallax ^ould be equally well
accounted for if the celestial bodies revolved round the Earth; not
so, however, diurnal aberration,
THE B.OTATIOTT OF THE EARTH. 317
385. Dynamical Proofs of the Earth's Rotation.
The following is a list of the methods by which the Earth's
rotation is proved from dynamical considerations :
(1) The eastward deviation of falling bodies.
(2) Eoucault's pendulum experiment.
(3) Foucault's experiments with a gyroscope.
(4) Experiments on the deviation of projectiles.
(5) Observations of ocean currents and trade winds.
(6) Experiments on the differences in the acceleration of
gravity in different latitudes, due to the Earth's centrifugal
force, as observed by counting the oscillations of a pendulum ;
combined with
(7) Observations of the figure of the Earth.
386. The Eastward Deviation of Falling Bodies.
If the Earth is rotating p.Jbout its polar axis, those points
which are furthest from the Earth's axis move with greater
velocity than those which are nearer the axis. Hence the
top of a high tower moves with slightly greater velocity than
the base. If, then, a stone be dropped from the top of the
tower, its eastward horizontal velocity, due to the Earth's
rotation, is greater than that of the Earth below, and it falls
to the east of the vertical through its point of projection.
The same is true when a body is dropped down a mine. This
eastward deviation, though small, has been observed, and
affords a proof of the Earth's rotation.
Consider, for example, a tower of height h at the equator. If a be
the Earth's equatorial radius, the base travels over a distance 2ira in
a sidereal clay, owing to the Earth's rotation, while the top of the
tower describes 2ir(a + h) per sidereal day. Thus, the velocity at
the top exceeds that at the bottom by 2irh per sidereal day.
If h be measured in feet, the difference of velocities is irh/'SQOO indies
per sidereal second, and is sufficiently great to cause a small but
perceptible deviation when a body is let fall from a high tower.
The earliest experiments were too rough to show this deviation,
and were, therefore, used as evidence against, instead of for, the
Earth's rotation. The deviation can only be observed in experi-
ments conducted with very great care, and it is very difficult to
measure. Its amount is largely modified by the resistance of the
air and other causes, and therefore differs considerably from that
by theory.
318
ASTRONOMY.
387. Foucault's Pendulum Experiment. In 1 85 1 , M .
Foucault invented an experiment by which the Earth's rota-
tion is very clearly shown. A pendulum is formed of a large
metal ball suspended by a fine wire from the roof of a high
building, and is set in motion by being drawn on one side and
suddenly released ; it then oscillates to and fro in a vertical
plane. If now the pendulum be sufficiently long and heavy
to continue vibrating for a considerable length of time, the
plane of oscillation is observed to very gradually change its
direction relative to the surrounding objects, by turning
slowly round from left to right at a place in the northern
hemisphere, or in the reverse direction in the southern. If
the experiment is performed in latitude ?, the plane of
oscillation appears to rotate through 15 x sin I in a sidereal
hour, 360 sin lin a sidereal day, or 360 in cosec I sidereal
days. This apparent rotation is accounted for by the Earth's
rotation, as follows.
(i.) Let us first imagine the experiment to be performed at
the north pole of the Earth. Let the pendulum AB be
vibrating about A in the arc BB' in
the plane of the paper. The only forces
acting on the bob are the tension of
the string BA and the weight of the
bob acting vertically downwards ; both
are in the plane of the paper. The
Earth's rotation about its axis CA pro-
duces no forces on the bob. Hence
there is nothing whatever to alter the
direction of the plane of oscillation ;
this plane therefore remains fixed in
space. But the Earth is not fixed in
space ; it turns from west to east, making
a complete direct revolution in a sidereal
day. Hence the plane of the pendulum's oscillation appears,
to an observer not conscious of his own motion, as though it
rotated once in a sidereal day, in the reverse or retrograde
direction (east to west). If, however, he were to compare
the plane of oscillation not with the Earth but with the
stars, whose directions are actually fixed in space, he would
FIG. 129.
THE ROTATION OF THE EARTH.
319
see that it always retained the same position relatively to
them.
Since, then, the pendulum at the pole of the Earth appears to
follow the stars, it evidently appears to rotate in the same
direction as the hands of a watch at the north pole, and in
the direction opposite to the hands of a watch at the south
pole.
7
(ii.) Next suppose the experiment performed at the Earth's
equator. If the bob be set swinging in
the plane of the equator, take this as the
plane of the paper (Fig. 130). The
direction of the vertical AQC is now
rotating about an axis through C per-
pendicular to the plane of the paper ;
hence it always remains in that plane.
Hence there is nothing whatever to
turn the plane of oscillation of the pen-
dulum out of the plane of the Earth's
equator. It therefore continues always
to pass through the east and west points,
and there is no apparent rotation of the
plane of oscillation.
FIG. 130.
If the pendulum do not swing in the plane of the equator,
the explanation is much more complicated. As the Earth
rotates, the direction of gravity performs a direct revolution
in a sidereal day. Hence, relative to the point of support,
gravity is gradually and continuously turning the bob west-
wards, in such a way as to keep its mean position always
pointed towards the centre of the Earth. When the bob is
south of its position of equilibrium, this westward bias tends
to turn the plane of oscillation in the clockwise direction,
but when the bob is north of the mean position, the west-
ward bias has an equal tendency to turn the plane in the
reverse direction. Consequently the two effects counter-
act one another, and therefore produce no apparent
rotation of the plane of oscillation relative to surrounding
objects.
320
ASTRONOMY.
Qu
(in.} Lastly, consider the case of an observer in latitude
I (Fig. 131). Let w denote the
angular velocity with which the
Earth is rotating about its polar axis
CP. It is a well-known theorem
in Rigid Dynamics that an angular
velocity of rotation about any line
maybe resolved into components about
any two other lines, by the parallelo-
gram law, in just the same way as a
linear velocity or a force along that
line; this theorem is called the
Parallelogram of Angular Velocities.
Applying it to the angular velocity
n about CP, we may resolve it into
two components
FIG 131.
and
n cosPCO or n sin I about CO,
n sin PGO or n cos I about a line CO' perpendicular to CO,
and we may consider the effects of the two angular velocities
separately.
As in case (i.), the component nsin I causes the Earth to
turn about CO, without altering the direction in space of the
plane of oscillation ; this plane, therefore, appears to rotate
relatively in the reverse or retrograde direction, with
angular velocity n sin I. As in case (ii.)> the angular velocity
n cos I about CO' produces no apparent rotation of the plane
of oscillation relative to the Earth. Hence the plane of oscilla-
tion appears to revolve, relative to the Earth, with retrograde
angular velocity n sin I.
But the angular velocity n = 15 per sidereal hour
= 360 per sidereal day.
Therefore the plane of oscillation turns through
15 sin I per sidereal hour = 360 sin I per sidereal day,
360
fcnd its period of rotation = , -
n sin I
= cosec I sidereal days.
THE IIOTATI02? OF THE EARTH. 321
388. The Gyroscope or Gyrostat is another apparatus
used by Foucault to prove the Earth's rotation. It is simply
a large spinning-top, or, more correctly, a heavy revolving
wheel IT (Fig. 132), whose axis of rotation AB is supported
by a framework, so that it can turn about its centre of gravity
in any manner. Thus, by turning the wheel and the inner
frame A CBD about the bearings CD, and then turning the
outer frame DECF about the bearings EF, the axis AB (like
the telescope in an altazimuth or equatorial) can be pointed
in any desired direction. The three axes A B, CD, EF all
pass through the centre of gravity of the top ; hence its weight
is entirely supported, and does not tend to turn it in any
way; and the bearings A, B, C, D, E, JPare very light, and
so constructed that their friction may be as small as possible.
The top may be spun by a string in the usual way, and it
continues to spin for a long time.
FIG. 132.
When a symmetrical body, such as the wheel H, is revolv-
ing rapidly about its axis of figure, and is not acted on by
any force or couple, it is evident that no change of motion
can take place, and therefore the axis of rotation AB must
remain fixed in direction. This is the case with the gyro-
scope, for, from the mode in which the weight of the wheel
is supported, there is no force tending to turn it round.
When the experiment is performed it is observed that the
axis AB follows the stars in their diurnal motion ; if pointed
to any star, it always continues to point to that star, its posi-
tion relative to the Earth changing with that of the star.
Hence it is inferred that the directions of the stars arc fixed
in space, and that the diurnal motion is not due to them, but
to the rotation of the Earth.
322 ASTllOXOMY.
389. If while the gyroscope is spinning rapidly any attempt be made
to alter the direction of the axis of rotation AB by pushing it in any
direction, a very great resistance will be experienced, and the axis
will only move with great difficulty. This shows that the small
friction at the pivots CD, EF can have but little effect in turning
the axis of the top, and therefore the gyroscope spins as if it were
practically free, as long as its angular velocity remains considerable.
The following additional experiments with the gyroscope can be
also used to prove the Earth's rotation.
Experiment 1. Let the hoop CEDF be steadily rotated about the
line EF. The line AB is no longer free to take up any position, for
the pivots and D obviously force it always to be in a plane through
EF and perpendicular to plane CEDF. Hence the axis of rotation
is no longer able to maintain always the same position, unless that
position coincides with EF. The result
is that the axis gradually turns about
CD till it does coincide with EF, the di-
rection of rotation of the wheel being
the same as that in which frame is forced
to revolve. It will then have no further
tendency to change its place. Of course
we suppose the hoop turned so quickly
that the effect of the slow motion of the
Earth is imperceptible.
Experiment 2. We may now repeat
Experiment 1, using the Earth's rota-
tion. Let the framework CEDFbe fixed
in a horizontal position, the line CD
being held pointed due east and west. The axis AB is then
free to turn in the plane of the meridian. Now, owing to the
Earth's rotation, the framework carrying CD is turning about the
Earth's polar axis, and this causes the top to turn till its axis points
to il\e celestial poles. The result of experiment agrees with
theory, thus affording a further proof of the Earth's rotation about
the poles.
Experiment 3. Let the framework CEDF be clamped in a vertical
plane. The axis AB can then turn in a horizontal plane, but it cannot
point to the pole. It will, however, try to point in a direction
differing as little as possible from the direction of the Earth's axis,
and will therefore turn till it points due north and south. This has
also been verified by actual observation.
Experiments 3 and 2, if performed with a sufficiently perfect
gyroscope, would enable us to find the north point, and then to find
the celestial pole, and thus determine the latitude without observing
any stars. By means of Foucault's pendulum experiment we could
also (theoretically) determine the latitude,.
THE ROTATION OF THE EARTH. 323
390. The Deviation of Projectiles. If we suppose a
cannon ball to be fired in any direction, say from the Earth's
North Pole, the ball will travel with uniform horizontal velocity
in a vertical plane. But, as the Earth rotates from right to left,
the object at which the ball was aimed will be carried round
to the left of the plane of projection, and therefore the ball
will appear to deviate to the right of its mark. At the South
Pole the reverse would be the case, because in consequence
of the direction of the vertical being reversed, the Earth would
revolve from left to right ; hence the ball would deviate to
the left of its mark. At the equator no such effect would
occur.
The deviation, like that in Foucault's pendulum, depends on
the Earth's component angular velocity about a vertical axis
at the place of observation, and this component, in latitude I,
is n sin ( 387, iii.). Now the Earth rotates about the poles
through 15" per sidereal second. Hence, if t be the time of
flight measured in sidereal seconds, the deviation is
= nt sin I = 15". t sin ,
and it is necessary to aim at an angle 15". t sin I to the left of
the target in N. lat. /, or 15". t sin/ to the right in S. lat. I.
The formula is sufficiently approximate even if t be measured
in solar seconds. It is necessary to allow for this deviation
in gunnery thus affording another proof of the Earth's
rotation.
391. The Trade Winds are due to a similar cause. The
currents of air travelling towards the hotter parts of the
Earth at the equator, like the projectiles, undergo a deviation
towards the right in the northern hemisphere, and towards the
left in the southern. This deviation changes their directions
from north and south to north-east and south-east respectively.
In a similar manner the Earth's rotation causes a deviation
in the ocean currents, making them revolve in a direction
opposite to that of the Earth's rotation, which is "counter
clockwise " in the N. and " clockwise " in the S. hemisphere.
The rotatory motion of the wind in cyclones is also due to
the Earth's rotation.
324 ASTEONOMY.
392. Centrifugal Force. If a body of mass m is revolving
in a circle of radius r with uniform velocity v under the action
of any forces, it is known that the body has an acceleration v*/r
towards the centre of the circle.* Hence the forces must
have a resultant mv*/r acting towards the centre, and they
would be balanced by a force mv 2 /r acting in the reverse
direction, i.e., outwards from the centre. This force is called
the centrifugal force.
Thus, in consequence of its acceleration, the body appears to
exert a centrifugal force outwards. If it be attached to the
centre of the circle by a string, the pull in the string is mv*/r.
If m be measured in pounds, r in feet, and v in feet per
second, then mv^/r represents the centrifugal force in poundals.
Similarly, in the centimetre-gramme-second system of units,
mv*/r is the centrifugal force in dynes.
If n represent the body's angular velocity in radians per
second, v = nr, and the centrifugal force is therefore mn*r.
393. General Effects of the Earth's Centrifugal
Force. If the Earth were at rest the weight of a body
would be entirely due to the Earth's attraction. But in con-
sequence of the diurnal rotation the apparent weight is the
resultant of the Earth's attraction and the centrifugal force.
Let QOR represent a meridian section of the Earth
(Fig. 134). Consider a body of mass m supported at any
point on the Earth's surface. Since the Earth is nearly,
but not quite, spherical, the force ^ of the Earth's attraction
on a unit mass is not directed exactly to the Earth's centre,
but along a line OK. But, owing to the body's central
acceleration along OM, the force which it exerts on the
support is not quite equal to the Earth's attraction mg^
but is compounded of mg Q acting along OJT, and the centri-
fugal force m . ri* . MO acting along M 0.
On KQ, take a point G such that
KG
* See any book on Dynamics.
THE ROTATION OF THE EARTH.
325
then, by the triangle of force?, OG is the direction of the re-
sultant force exerted by the body on its support, and this
force is the apparent weight of the body. Hence, also OG
represents the apparent direction of gravity, or the verti-
cal as indicated by a plumb-line. Producing GO, KO to Z,
Z", we see that the effect of centrifugal force is to displace the
vertical from Z" towards the nearest pole (P).
The angle ZGQ measures the (geographical) latitude of
the place, and is greater than Z ' KQ, which would measure
the latitude if the Earth were at rest. Hence the apparent
latitude of any place is increased ly centrifugal force.
I
FIG. 134.
Again, if the apparent weight be denoted by mg, we have,
by the triangle of forces,
g :y = GO: KQ-,
now from the figure it is evident that G < IL(), and there-
fore g < # . Hence the apparent iveight of a body is diminished
by centrifugal force.
394. Effect on the Acceleration of a Falling Body.
If a body is falling freely towards the Earth near 0, the
whole acceleration of its motion in space is due to the Earth's
attraction, and is # , along OK. But the Earth at has
itself an acceleration ri*OM to wards 31. Hence the accelera-
tion of the body relative to the Earth is the resultant of </
along (7, and w 2 . M along J/0, and is therefore g along
G. Hence the body approaches the Earth with acceleration
g along OG. Therefore its relative acceleration is the accele-
ration dve to its apparent weight, that is, to the resultant of
the Earths attraction and centrifugal force.
326 ASTEONOMT:
395. To find the loss of weight of a body at the
equator, due to centrifugal force. At the equator
centrifugal force is directly opposed to gravity ; hence, if a
denote the Earth's radius CQ,
ff = &*-***>
Now we have roughly
ff = 32-18 feet per second per second,
a = 3963 miles = 3903 x 5280 feet,
and n = 2?r radians per sidereal day
= radians per mean solar second.
Hence A = 3963x5280x4.' = .
86164 x 86164
and therefore - = = nearly.
ff 32-18 28J
Hence </ = f/ -JL </ ,
or the effect of the Earth's rotation is to decrease the weight of a
body by about of the whole.
289
For rough calculations it would be sufficient to take g = 32'2,
a = 3960 miles, and to neglect the difference between a solar and a
sidereal day. This would give -fa, as bei'ore.
396. To find approximately the loss of weight of a
body and the deviation of the vertical due to centri-
fugal force in any given latitude.
Let 1= QGO = astronomical latitude of 0; D GOK
= ZOZ" = deviation of vertical from direction of Earth's
attraction, or increase of latitude due to centrifugal force.
We have OM = CO cos COM
= a COB I approximately ;
where a is the Earth's radius, since the Earth is very nearly
spherical, and Z COM is therefore very nearly equal to the
latitude 1. Therefore centrifugal force per unit mass at
= n a . ON = n* . a cos I = ^ ^ cos I (from 395).
THE ROTATION OF TFE EARTH.
327
Resolving along 06r, we have, if y be the Earth's attraction
per unit mass at 0*,
g = 0Q cos J)n* . OlTcos I
= f7 cos 2 1 approximately
(since D is small, and .*. cos D = 1 nearly).
Hence, in latitude I, the Earth's rotation dim : nishes theweiyld
of a body ly approximately cos 2 1 of itself.
289
Resolving perpendicular to G,
we have
g~ sin D w 2 M sin I = ;
tfa cos I sin I
.'. sin J> = -
1 sin 2?
~ 289 2
Since d is small, this gives
approximately
1 sin 21
circular measure ot d = -^- -
FIG. 135.
" (number of seconds in
180x60x60
289X27T
206265
sin 21
578
sin 21 = 357" sin
Hence the deviation D = 5' 57". sin 21, and this is the in-
crease of latitude due to centrifugal force.
COROLLAEY. The deviation of the vertical due to centri-
fugal force is greatest in latitude 45 (v sin 2? = 1), and is
there 5' 57".
* Since the Earth is not quite spherical, g is not the same at as
at the equator. The difference may be neglected, however, when
multiplied by the small constant jy.
328 ASTRONOMY.
397. Figure of the Earth. In 114 we stated that the
form of the Earth has been observed to be an oblate spheroid.
Now it has been proved mathematically that a mass of
gravitating liquid- when rotating takes the form of an oblate
spheroid whose least diameter is along its axis of rotation.
Thus the Earth's form may be accounted for on the theory that
the Earth's surface was formerly in a fluid or molten state,
and that it then assumed its present form, owing to its diurnal
rotation. We thus have another argument in favour of the
Earth's rotation ; but it is only fair to say that this theory
of the Earth's origin has not been satisfactorily demonstrated.
It accounts satisfactorily, however, for the form of the
surface of the ocean.
This theory may be illustrated by the following general considera-
tions. When a mass of liquid is acted on by no forces beyond the
attractions of its particles, it is easy to realize that the whole is in
equilibrium in a spherical form, being then perfectly symmetrical.
If, however, the fluid be rotating about the axis PGP', the centri-
fugal force tends to pull the liquid away from this axis and towards
the equatorial plane. The liquid would, therefore, fly right off, but
its attraction is always trying to pull it back to the spherical form.
Hence, the only effect of centrifugal force (which, for the Earth, is
small compared with gravity) is to distort the liquid from its spheri-
cal form by pulling it out towards the equator ; and it is therefore
reasonable to suppose that the fluid will assume a more or less oblate
figure, whose equatorial is greater than its polar diameter.
It may also be remarked that the form assumed by the liquid
would be such that the effective force of gravity (i.e., the resultant
of the attraction and centrifugal force) on the surface would every-
where be perpendicular (i.e., normal) to the surface.
*398. Gravitational Observations. If the Earth were a sphere,
its attraction g would everywhere tend to its centre, and would be
of the same intensity at all points on its surface, while the variations
in g, the apparent intensity of gravity, would be entirely due to the
Earth's centrifugal force, its value in latitude I being proportional
to 1 -^-g cos- 1 ( 396). By comparing the values of g at different
places, we should then be able to demonstrate the Earth's centri-
fugal force, and hence prove its rotation. But, owing to the Earth's
ellipticity, its attraction gr does not pass through the centre, except
at the poles and equator, and its intensity is not everywhere con-
stant. It is, therefore, important to determine experimentally the
values of g at different stations. By allowing for centrifugal force, the
corresponding values of the Earth's attraction g can be found, and
the variations in its intensity at different places afford a measure of
THE ROTATION OF THE EARTH. 329
tlie amount by which the Earth differs from a sphere. We thus
have a gravitational method of finding the Earth's ellipticity.
But the Earth's ellipticity can also be determined by direct obser-
vation, as explained in Chapter III., Section III. The agreement
between the results thus independently obtained furnishes another
proof of the Earth's rotation.
In consequence of the EarthVellipticity it is found (by observa-
tion) that the difference in the intensity of gravity between the polo
and equator is increased from ^-g- to -3^-5 of the whole.
399. To compare the Intensity of Gravity at different places.
The intensity of gravity may be measured by the force with which
a body of unit mass is drawn towards the Earth. This cannot be
measured by weighing a body with a common balance, because the
weights of the body and of the counterpoise, by means of which it is
weighed, are equally affected by variations in the intensity of gravity,
and two bodies of equal mass will, therefore, balance one another
when placed in the scale pans, no matter what be the intensity of
gravity. In fact, by weighing a body with weights in the ordinary
way, we determine only its mass, and not the absolute force with
which it is drawn to the Earth.
We might determine the intensity of gravity by means of a
" spring balance," for the elasticity of the spring does not depend on
the intensity of gravity, and therefore the extension of the spring
gives an absolute measure of the force with which the body is drawn
towards the Earth. If the apparatus were to support a mass of one
pound, first at the equator and then at the pole, the force on it
would be greater at the latter place by about - l ^, and this spring
would thcro be extended about -j--^ more. It would be very difficult
to construct a spring balance sufficiently sensitive to show such a
small relative difference of weight, but it has been done.
Aticood's machine might be used to find g, but this method is not
capable of giving very accurate results.
The most accurate method of finding g is by timing the oscillations
of a pendulum of known length.
[* A theoretical simple pendulum, consisting of a mere heavy par-
ticle of no dimensions, suspended by a thread without weight, is of
course impossible to realize in practice, but the difficulty is over-
come by the use of a pendulum called Captain Rater's Reversible
Pendulum. This pendulum is a bar which can be made to swing
ab ut either of two knife-blades fixed ?.t opposite sides of, but un-
equal distances from, its centre of gravity, and it is so loaded that
the periods of oscillation, when suspended from either knife-edge,
are equal. It is then known that the pendulum will swing about
either knife-edge in just the same manner as if it were a simple
pendulum whose whole mass was concentrated at the other knife-
edge. The distance between the knife-edges is, therefore, to be
regarded as the length of the pendulum.']
330
400. Oscillation? of a Simple Pendulum. In a simple
pendulum, formed of a small heavy particle suspended by a
fine light thread of length I, the period of a complete oscillation
to and fro is
the time of a single swing or " leat " being of course half of
this.
Hence by observing the time of oscillation t and measuring
the length I, the intensity of gravity g can be found.
By the " seconds pendulum " is meant a pendulum in
which one beat occupies one second, hence a complete
oscillation occupies two seconds.
EXAMPLK. Having given that the length of the seconds pendulum
is 99'39 centimetres, to find g in centimetres per second per second.
t = 2nVZ/7 = 2 seconds, and I = 99'39 cm.,
.-. g =^i = 99-39 x (3-1416) 2 = 981.
It is often necessary to compare the lengths of two
pendulums whose periods of oscillation are very nearly equal,
to find the effect of small changes in the length of a pendulum
due to variations in temperature, or, in comparing the intensity
of gravity at different places, to find the effect of a small
alteration in the value of g on the period of oscillation and on
the number of oscillations in a given interval. If the differ-
ences are small, the calculations may be much simplified by
means of the following methods of approximation.*
401. To find the change in the time of oscillation of
a pendulum, and in the number of oscillations in a
given interval, due to a small variation in its length
or in the intensity of gravity.
If t be the time of a complete oscillation of a pendulum of
length J, we have, by 400,
? = 47T 2 - (i).
* The same results can of course be obtained by means of the
differential calculus.
THE HOTATIOX OF THE EARTH. 331
(i.) Suppose the length increased to 1' 9 and let t' be the new
period of oscillation. We have
t* = 47T 3 -.
g
Therefore, by division,
*"_r
*T*T
and therefore also
I'*-?-,, ..t + t I'-l
~1T ~ t} ~T
These formulae are exact. But if I' is very nearly equal to ,
t' is very nearly equal to t, and therefore, putting t + t'= 2t,
we have approximately
Q t'-t_l'-l
T T'
whence, if t, I be known, the change V t, consequent on the
increase of length I' I, may be readily found approximately
without the labour of extracting any square roots.
(ii.) Suppose the intensity of gravity increased to g', the
length I being unaltered, and let t' be the new period. Since
we have, by division,
and therefore also
But, if , g are very nearly equal to t', /, this gives
approximately
2 """ = - .
* f/
ASTEON. Z
332 ASTEONOMf.
(iii.) If I and g both vary, becoming V and g', we have, in
like manner
Therefore also
I 9
or approximately, if the variations are small,
2 *-* = r ~ ? - ff '- ff
t I <J '
showing that the effects of the two variations may be con-
sidered separately.
(iv.) If n, ri be the number of complete oscillations of the
pendulum in a given interval T, and if, in consequence of the
change, this number be altered to w', we have
nt = nt'=T,
n _ t
IT-T',
n '- n t t'
whence - = -7-
n t
If t' is very nearly equal to t, this gives approximately
which determines the number of beats gained by the pendulum
in the time T, in consequence of the variations, the original
number n being supposed known.
EXAMPLE. To find the number of oscillations gained or lost in an
hour by the pendulum of the Example of 400, supposing (i.) its
length increased to 1 metre; (ii.) the acceleration of gravity in-
creased to 982 ; (iii.) both changes made simultaneously.
(i.) The pendulum beats seconds ; therefore it performs 3600 half
oscillations or 1800 whole oscillations in an hour. Also V = lOO'OO
l'-l = 0-61, g'-g = Q,
THE ROTATION OF TUB EARTH. 333
Hence, if n' be the new number of oscillations in an hour,
'-1800 = _0^1 = _ 9^1, ) = _ OjGl
1800 21 21' 200*
.-. n' -1800 = -9 x -61 = -5'49.
Hence the pendulum loses nearly 5 oscillations in an hour, and the
number of oscillations is therefore 1794$.
... . TT tt'-1800 g'-g 982-981 1
-~ * = - = ~ ;
'- 1800 = =-9 = 1 nearly.
Hence the pendulum gains 1 oscillation in an hour, the total
number of oscillations being 1801.
(iii.) Since from the first cause the pendulum loses 5J oscillations
and from the second it gains 1 oscillation, therefore on the whole it
loses 5^ 1 or 4^ oscillations per hour. It therefore performs 1795^
oscillations or 3591 beats per hour.
402. To compare the times of oscillations of two
pendulums whose periods are very nearly equal.
If two pendulums of nearly equal periods are simultaneously
started swinging in the same direction, the one whose period
is a little the shortest will soon begin to swing before the
other. After some time it will gain a half oscillation, and
the pendulums will then be swinging in opposite directions.
After another equal interval, the quicker pendulum will have
gained one whole oscillation on the slower, and both will
be again swinging together in the same direction. Similarly,
every time the quicker pendulum has gained an exact number
of complete oscillations on the slower, both will be swinging
together in the same direction. Thus, the number of coinci-
dences, or the number of times that the two pendulums are
together, in any interval, is equal to the number of complete
oscillations (to and fro) gained by the quicker pendulum over
the slower, i.e., the difference between the numbers of com-
plete oscillations performed by the two pendulums.
Thus, if n, n 1 be the number of oscillations of the slower
and faster pendulums in any given interval, then n' n is the
the number of oscillations gained by the latter, and is, there-
fore, the number of " coincidences." If either of the num-
bers n, n' is known, we can, by counting the coincidences,
find the other number.
334 ASTBONOMY.
403. To find g, the acceleration of gravity, the
simplest plan is to use a Captain Kater's pendulum, the beat
of which is very nearly one second. By counting the
"coincidences" of the pendulum with the pendulum of a
clock regulated to beat seconds during, say, an hour (as
shown by the clock) the exact time of oscillation can be
found. Moreover, from the number of beats gained or lost,
and the observed length of the pendulum, we may calculate
the amount by which the length must be increased or decreased
in order to make the pendulum beat seconds. The length of
the seconds pendulum is thus known, and the value of g can
be found.
The reason for using two pendulums is that it would be extremely
difficult to measure the length of the pendulum of the clock, and it
would be equally difficult to find the period of oscillation of a pen-
dulum without comparing it with that of a clock, whose rate can be
regulated daily by astronomical observations.
404. To compare the value of g at two different
stations, the simplest plan is to determine the number of
seconds gained or lost in a day by a clock after it has been
taken from one station to the other, the length of the pen-
dulum remaining the same. If n, ri be the number of
seconds marked by the clock in a day at the two places, we
have exactly =--,
*' 9
or approximately,
n g
whence the ratio of g' to g may be found.
Here there is no necessity to use a Captain Kater's pendulum,
because the length of the pendulum is not required ; hence the
ordinary compensating pendulum of the clock answers the purpose.
If a non-compensating pendulum were used, it would be necessary I
to make allowance for any change in the length of the pendulum \
due to variations ID temperature.
THE ROTATION OF THE EARTH. 335
EXAMPLES. XII.
iat af tPr n CaUl -rf Pen ^ lu bein S sefc vibrating in latitude 30, show
at atter one sidereal day it is ao-ain vihr.r,nr ,v __ '_i7_.
that af r n-rf n S se vrang n
K?5 ^^
place m the northern hemisphere the pendulum which
the same direction as the hands of a watch will have
apparent angular velocity, and will gain two complete
e
sthrhsro p enuum
the North p g nt A *l C n8ld f' '? the first P lace > the Phenomena at
so^^mf 80 deSCnbeth P ondin g phenomena in the
.
3. If a railway is laid along a meridian, and a train is travelling
anTast^To " ^^^ P ole > "yeBtigate whether it wHIexert
an eastward or a westward thrust on the rails, and why
4. A bullet is fired in N. latitude 45, with a velocity of 1 600 frpf
5 Ihn ft ' f f?i I how many feet it will deviate to the right.
a bodv 7ti * 6 Ear S W6re t0 r tate ^venteen times as fist
a body at the equator would have no weight.
6. If the Earth were a homogeneous sphere rotating so fast thaf
^ f * he last q uest! . show that the Earth's
336 ASTRONOMY.
EXAMINATION PAPEB. XII.
1. Give reasons for supposing that the diurnal rotation of th(
heavens is only an appearance caused by a real rotation of the
Earth. Name methods by which it has been claimed that this ii
proved.
2. Describe the gyroscope experiment, and the gyroscope.
3. Give any theoretical methods of determining latitude withou
observing a heavenly body.
4. Describe Foucault's experiment for exhibiting the Earth'
rotation ; and find the time of the complete rotation of the plane c
vibration of a simple pendulum fieely suspended in latitude 60.
5. Having given that the Earth's circumference is 40,000 kilc
metres, find the acceleration of a body at the equator due to th
Earth's rotation in centime bres per second per second, and takinj
<7 , the acceleration of gravity, to be 981 of these units, deduce t
ratio of centrifugal force to gravity at the equator.
6. What is meant by the vertical at any point of the Earth
surface ? Supposing the Earth to be a uniform sphere revolvir
round a diameter, calculate the deflection of the vertical from t
normal to the surface.
7. State what argument is drawn from the Earth's form to suppo
the hypothesis of its rotation.
8. Why is it that the intensity of gravity is less at the equal
than in higher latitudes ? Show that the alteration in the appare
weight of a body due to centrifugal force varies nearly as cos
where I is the latitude, and state the ratio of centrifugal force
gravity at the equator.
9. If a body is weighed by a spring balance in London and
Quito, a difference of weight is observed. Why is this not observed"
an ordinary pair of scales be used ?
10. Show that an increase in the intensity of gravity will cat
a pendulum to swing more rapidly, and vice vers&. If the accele:
tion of gravity be increased by the small fraction l/r of its vali
show that a pendulum will gain one complete oscillation in every
CHAPTER XIII*
THE LAW OF UNIVERSAL GRAVITATION.
SECTION I.^T/ie Earttis Orlital Motion Kepler's Laws and
their Consequences.
405. Evidence in favour of the Earth's Annual
Motion round the Sun. The theory that the Earth is a
planet, and revolves round the Sun, was propounded by
Copernicus (circ. 1530) and received its most convincing
proof, over 150 years later from Newton (A.D. 1687), who
accounted for the motions of the Earth and planets as a
consequence of the law of universal gravitation. This proof
is based on dynamical principles ; but the following arguments,
based on other considerations, afford independent evidence in
favour of the theory that the Earth revolves round the Sun
rather than the Sun round the Earth.
(i.) The Sun's diameter is 110 times that of the Earth's,
and it is much easier to believe that the smaller body revolves
round the larger, than that the larger body revolves round
the smaller.
If the dynamical laws of motion be assumed, it is impossible to
gee how the larger body could revolve round the smaller, unless
either its mass and. therefore its density were very small indeed,
or the smaller one were rigidly fixed iu some way.
(ii.) The stationary points, and alternately direct and retro-
grade motions of the planets, are easily accounted for on the
theory that the Earth and planets revolve round the Sun
(Chap. X.) in orbits very nearly circular, and it would be
impossible to give such a simple explanation of these motions
on any other theory. It is true that we might suppose, with
Tycho Erahe (circ. 1600), that the planets revolve round the
Sun as a centre, while that body has an orbital motion round
the Earth, but this explanation would be more complicated
than that which assumes the Sun to be at rest. And it would
be hard to explain how such huge bodies as Jupiter and
Saturn could be brought to describe such complex paths.
33$ ASTRONOMY.
(iii.) As seen through a telescope, Venus and Mars are
found to be very similar to the Earth in their physical charac-
teristics, and their phases show that, like the Earth and
Moon, they are not self-luminous. It is, therefore, only
natural to suppose that their property of revolving round the
Sun is shared by the Earth. Moreover, the Earth's relative
distance from the Sun agrees fairly closely with that given
by Bode's law ; hence there is a strong analogy between the
Earth and the planets.
(iv.) The orbital motion of the Earth is in strict accordance
with Kepler's Laws of Planetary Motion. In particular, the
relation between the mean distances and periodic times given
by Kepler's Third Law ( 326) is satisfied in the case of the
Earth's orbit.
Moreover, a similar relation is observed to hold between the
periodic times of Jupiter's satellites and their mean distances
from Jupiter. Hence it is probable that the Earth and
planets form, like Jupiter's satellites, one system revolving
about a common centre. But it is improbable that the Sun
and Moon should both revolve about the Earth, for their
distances from it and their periods are not connected by this
relation.
(v.) The changes in the relative positions of two stars
during the year in consequence of annual parallax can only
be accounted for on the hypothesis either of the Earth's
orbital motion, or of a highly improbable rigid connection
between all the nearer stars and the Sun, compelling them
all to execute an annual orbit of the same size and position.
(vi.) The aberration of light affords the most convincing
proof of all. In particular, the relation between the coefficient
of aberration and the retardation of the eclipses of Jupiter's
satellites has been fully verified by actual observations, and
affords incontestible evidence that the phenomenon is actually
due to the finite velocity of light, as explained in Chapter XI.
And the alternative hypothesis which would account for
annual parallax would not give rise to aberration, but would
produce an entirely different phenomenon. Hence the evi-
dence derived from the aberration of light, unlike the previous
evidence, furnishes a conclusive proof, and not merely an
argument, in favour of the Earth's orbital motion.
THE LAW OF UNIVERSAL GRAVITATION. 339
406. NEWTON'S THEORETICAL DEDUCTIONS
FROM KEPLER'S LAWS.
Kepler's Three Laws of planetary motion naturally suggest
the following questions :
(1) What makes the planets move in ellipses ?
(2) Why does the radius vector from the Sun to any planet
trace out equal areas in equal times ?
(3) Why are the squares of the periodic times proportional
to the cubes of the mean distances from the Sun ?
These questions were first answered by Newton about 1687,
or nearly sixty years after the death of Kepler. The theore-
tical interpretation of the Second Law necessarily precedes
that of the first; accordingly we now repeat the laws in their
new order, together with Newton's interpretations of them.
Kepler's Second Law. The radius vector joining
each planet to the Sun moves in a plane describing
equal areas in equal times.
NEWTON'S DEDUCTION. The force under which a
planet describes its orbit always acts along the
radius vector in the direction of the Sun's centre.
Kepler's First Law. The planets move in ellipses,
having the Sun in one focus.
NEWTON'S DEDUCTION. The force on any planet
varies inversely as the square of its distance
from the Sun.
Kepler's Third Law. The squares of the periodic
times of the several planets are proportional to the
cubes of their mean distances from the San.
NEWTON'S DEDUCTION. The forces on different
planets vary directly as their masses, and inversely
as the squares of their distances from the Sun,
or, in other words, the accelerations of different
planets, due to the Sun's attraction, vary inversely
as the squares of their distances from the Sun.
340 ASTRONOMY.
.If, as we have every reason for believing, the planets are
material bodies, Newton's laws of motion show that they
cannot move as they do unless they are acted on by some
force, otherwise they would either be at rest or move uni-
formly in a straight line. Kepler's Second Law then enables
us to determine the direction of this force, his First Law
enables us to compare the force at different parts of the same
orbit, and his Third Law enables us to compare the forces on
different planets.
407. We have seen that the orbits of most of the planets are
nearly circular, the eccentricities being small, except in the
case of Mercury. Before proceeding to the general discussion
of the dynamical interpretation of Kepler's Laws, it will be
convenient therefore to consider the case where the orbits are
supposed circular, having the Sun for centre. Kepler's
Second Law shows that under such circumstances the planets
will describe their orbits uniformly, and it hence follows that
the acceleration of a planet has no component in the direction
of motion, but is directed exactly towards the centre of the
Sun. The law of force can now be deduced very simply, as
follows :
KEPLER'S THIRD LAW FOE, CIRCULAR
ORBITS.
408. To compare the Sun's attractions on different
Planets, assuming that the orbits are circular and
that the squares of the periodic times are propor-
tional to the cubes of the radii.
Suppose a planet of mass J/is moving with velocity v in a
circle of radius r. Let T be the periodic time, P the force to
the centre. . .', .
Since the normal acceleration in a circular orbit is 2 /r,
therefore *
In the period T 7 the planet describes the circumference lira ;
.-. vT= 2vr.
Substituting for v, we have
P iii^: _ ^L *****
jf ~- r 2 -yT
TfLE LAW OF UNIVERSAL GRAVITATION. $4l
Let M 1 be the mass of another planet revolving in a cir-
cular orbit of radius r', T its periodic time, P' the force of
the Sun's attraction ; then we have in like manner
p ,_ JT 4*V
- ^ x r * '
By Kepler's Third Law,
r- r
Therefore the forces on different planets vary
directly as their masses and inversely as the squares
of their distances from the Sun.
COEOLLAEY 1. Let P = CM/r* ; then C is called the abso-
lute intensity of the Sun's attraction, and we see that
The absolute intensity of the Sun's attraction is the
same for all planets.
For c
The constant C evidently represents the force with which
the Sun would attract a unit mass at unit distance, or the
acceleration which the Sun would produce at unit distance.
COROLLAS r 2. If another body be revolving in an orbit of
radius / in a period T\ under a different central force, which
produces an acceleration C"// 2 at distance r', we have
tT=l=! and (7=
.-. C'T : CT 2 = i : r 8 ,
a formula which enables us to compare the absolute intensities
of two different centres of force, which attract inversely as
the squares of the distances, when the periodic times and
distances of two bodies revolving about them are known.
342 ASTRONOMY.
409. To compare the velocities and angular velo-
cities of two planets moving in circular orbits. If
v, v are the velocities, n, ri the angular velocities (in radians
per unit time), we have
Also v = rw, v = rn' ;
.*. v:v'= r~*: r'~*.
EXAMPLES.
1. If the Earth's period were doubled, to find what would be its
new distance from the Sun.
If r, r' be the old and new distances, Kepler's Third Law gives
r' 3 : r 3 = 2 2 : 12;
/. S = r x */4 = 92,000,000 x T587
= 146,000,000 miles.
2. If the Earth's velocity were doubled, its orbit remaining cir-
cular, to find its new distance.
Here r' : r = v 2 : t/ 2 = 1 : 4 ;
... r'=ir = 23,000,000 miles.
3. If the Earth's angular velocity were doubled, to find its new
distance.
The new angular velocity being double the old, the new period
would be half the old, and therefore
r' 3 :*- 3 =()': I';
/. r' = r x */i = r/ V4 = 92,000,000 -f- T587
= 92,000,000 x -63 = 58,000,000 miles.
4. To find what would be the coefficient of aberration to an
observer situated on Venus.
The coefficient of aberration (in circular measure) is the ratio of
the observer's velocity to the velocity of light ; hence, if fc, k' are the
coefficients on the Earth and Venus,
_ = t/ r^ = \r_ /100
k v r-* V r' V 72 ;
.. k' = 20-493" x A/(l-38*) = 20'493" x M785
= 24-151".
THE LAW OF UNIVERSAL GRAVITATION. 343
We shall now prove Newton's deductions from Kepler's
Laws, for the general case of elliptic orbits, employing, how-
ever, different and simpler proofs to those used by Newton.
410. Areal Velocity. definition. If a point P is
moving in any path MPK about a centre S, the rate of
increase of the area of the sector MSP, bounded by the fixed
line SM and the radius vector SP, is called the areal
velocity of P about the point S.
If the radius vector SP describes equal areas in equal
times, in accordance with Kepler's Second Law, the areal
velocity of P about S is of course constant, and is then
measured by the area of the sector described in a unit of time.
If the rate of description of areas is not constant, we
must, in measuring the areal velocity at any point, pursue a
similar course to that adopted in measuring variable velocity
at any instant, as follows :
FIG. 136.
If the radius vector describes the sector PSP' in the inter-
val of time t, then the average areal velocity over the
arc PP' is measured by the ratio
area PSP'
time t
(Thus the average areal velocity is the areal velocity with
which a radius vector, sweeping out equal areas in equal
times, would describe the sector PSP' in the same time t.)
The areal velocity at a point P is the limiting value of
the average areal velocity over the arc PP when this arc
is infinitisimally small.
344 ASTRONOMY.
411. Relation between the Areal Velocity and the
Actual (linear) Velocity. Let PP' be the small arc
described by a body in any small interval of time t. Let
be the actual or linear velocity of the body, h its areal velocity.
Since the arc PP is supposed small, we have
PP'=vt,
area PSP'=M.
Draw S Y perpendicular on the chord PP' produced. Then
&PSP'= | (base) x (altitude)
or
FIG. 137.
But when the arc PP' is infinitesimally small, PFis the
tangent at P, and SYis therefore the perpendicular from S
on the tangent at P. If this perpendicular be denoted
we have therefore
or (areal vel. about S)
= J (velocity) x (perp. from S on tangent).
COROLLARY. Por planets moving in circular orbits
of radii r, r, h = |IT, and h'= \v'r r .
But v I v = r"* : r x ~ J ;
A:A'=r:r';
hence the areal velocity of a planet moving in a circular orbit
is proportional to the square root of the radius, . .
THE LAW OP UNIVERSAL GRAVITATION. 345
412. PROPOSITION I. If a particle moves in such a
manner that its areal velocity about a fixed point is
constant, to prove that the resultant force on the
particle is always directed towards the fixed point.
[Newton's Deduction from Kepler's Second Law.]
Let a body be moving in the curve PQ in such a way that
its areal velocity about S remains constant. Let v, v' be
the velocities at P, Q, and let PF, QY, the corresponding
directions of motion, intersect in R. Drop SY, S Y perpen-
dicular on PF, QY.
Since the areal velocities at P and Q are equal,
.-. v.SY=v'. SY.
But SY= Rsin&KF,
SY = SItsmSltY.
.-. v$wSRY=v sin
FIG. 138.
i.e.) Component velocity at P perpendicular to BR
= component vel. at Q pcrp. to SB.
Therefore, as the particle moves from P to Q, its velocity
perpendicular to JRS is unaltered, and therefore the total
change of velocity is parallel to ItS.
This is true whether the arc PQ be large or small. But if
the arc PQ be taken infinitesimally small, the average rate
of change of velocity over PQ, measures the acceleration at
P, and P coincides with P.
Therefore the direction of the acceleration of the particle
nt any point of its path always passes through S, and there-
fore the force acting on the particle also always passes
through S.
346 ASTEONOMT.
413. Conversely, if the force on the particle always passes through 8,
the areal velocity about 8 remains constant. For in passing from P
to Q, the direction of motion is changed from PR to EQ, and the
same change of velocity could therefore be produced by a suitable
single blow or instantaneous impulse acting at R. And since the
force at every point of PQ always passes through 8, this equivalent
impulse must evidently also pass through 8 ; it must therefore act
along RS. Hence the velocity perpendicular to R8 is unaltered by
the whole impulse, and is the same at P as at Q j therefore
FIG. 139.
v sin 8RT = v' sin 8RT j
therefore v .SY = v' .SY 1 ;
therefore areal vel. at P = areal vel. at Q.
414. PROPOSITION II. A particle describes an ellipse
under a force directed to wards the focus ; to show that
the force varies inversely as the square of the dis-
tance.
[Newton's Deduction from Kepler's First Law.]
If h is the constant areal velocity, we have, by (i.),
We will now express the kinetic energy of the particle in
terms of r, its distance from the focus. Let its mass be M.
In the Appendix (Ellipse 11) it is proved that for the
ellipse whose major and minor axes are 20, 2J,
m, j. 2 4# 4tfa / 2 1 \
Therefore t? 8 = - = ^- ( --- J.
jt? 2 i 2 \ r a r
and kinetic energy at distance r
THE LAW OP T7NIVEESAI, GBAVITATION. 347
If v is the velocity at distance r', we have, similarly,
and therefore, for the increase of kinetic energy,
(in.).
FIG. 140.
Now the increase of kinetic energy is equal to the work
done by the impressed force in bringing the particle from
distance r to distance r. The resolved part of the displace-
ment in the direction of the force is rr'. Hence if P'
denote the average value of the force between the distances
r and r', we have
Work done = P'
(r-r'} = JJf^-^W = ~^(~ - -M
b*
rr
rr
Put / = r ; then the average force P' becomes the actual
force P at distance r. Therefore
A. j- j. \
orce at distance r) = 2
This is proportional to 1/r 2 .
Therefore the force varies inversely as the square of the dis-
tance.
\STRON. 2 A
348 ASTRONOMY.
415. PROPOSITION III. Having given that the squares
of the periodic times of the planets are proportional
to the cubes of the semi-axes major of their orbits, to
compare the forces acting on different planets.
[Newton's Deduction from Kepler's Third Law.]
Let T be the periodic time of any planet; then, by
hypothesis, the ratio
is the same for all planets.
In the last proposition (vi.) we showed that the force at
distance r is given by
-p _
Let this be put = Jf(7/r 2 , where C is some constant ; then
4h~a , .. N
= ........................ ( yu -)-
Now in the period T the radius vector sweeps out the area
of the ellipse, and this area is nab (Appendix, Ellipse 13).
Hence, since the areal velocity is h, we have
hT= irab.
Substituting the value of h from this equation in (vii.), we
have
But a*/T 2 is the same for all the planets ; therefore C is con-
stant for all the planets, and since the force
it follows that
The forces on different planets are proportional to tlieir
masses divided by the squares of their distances from the Sun.
Or, as in 408, Cor. 1,
27)e absolute intensity of the Sun's attraction ( C) is the same
for all the planets.
CoROLLAjtY. Let accented letters refer to the orbit of
another particle revolving round a different centre of force of
intensity C'. Then, by (viii.),
FC: T'*C r = a s : a'\
THE LAW OP UNIVERSAL GRAVITATION. 349
416. Other Consequences of Kepler's Laws.
(i.) In 150 we showed that, in consequence of Kepler's
Second Law being satisfied by the Earth in its annual orbit,
the Sun's apparent motion in longitude is inversely propor-
tional to the square of the Earth's distance from it. Since
the areal velocity of any planet about the Sun always remains
constant^ it may be shown in like manner that its angular
velocity is inversely proportional to the square of its distance
from the Sun.
FIG. 141.
For, if the planet's radius vector revolves from SP to SP
in the time t, and if the arc PP' is very small, we have
area SPP' = SP* x Z PSP' ( 150),
the angle being measured in radians ;
area SPP' _ i o p2 v tPSP
___ = f * __,
i.e., (areal velocity of P) = %SP* x (angular velocity of P),
provided that the angular velocity is measured in radians per
unit of time.
If n denote the angular velocity, h the areal velocity, and r
the distance SP, we have therefore
And since h is constant, n is inversely proportional to r.
* (ii.) If the mass of the planet is M, its momentum is Mv along
PY, and the moment of this momentum about 8 is
= Mv x 8T = Mvp = 2hM. ( 411.)
This is the planet's angular momentum, and is constant, since 7i in-
constant.
350
ASTRONOMY.
*417. Having given, in magnitude and direction, the velocity of a
planet at any point of its orbit, to construct the ellipse described
under the Sun's attraction.
Let the attraction at distance r be 0/r 2 per unit mass, where C is
given. Suppose that at the point
P of the orbit the planet is moving
with velocity v in the direction
PT. We have
v x ST = 2h, which determines h.
Also, from (vii.),
G = 47i 2 a/6 2 .
Substituting in (ii.),
*-c(- 1)...(,).
Hence, by considering the planet
at P, we have
SP
Now v, G, and SP are known ; hence the last equation determines
the semi-axis major a. If r = SP, we have
2a
2C-ru 2 '
Let H be the other focus of the ellipse. Then it is known
(Ellipse 8) that HP, SP make equal angles with PT. Also SP + HP
= 2a. Hence, we can construct the position of If by making
/ TPI = / TPS, and producing IP to a point H such that
PH = 2a-SP.
The ellipse can now be constructed as in Appendix (Ellipse 2).
COROLLARY 1. Since SP + HP = 2a, equation (x.) gives
SP.a
COROLLARY 2. Substituting for h in terms of G, we see from
equation (iv.) that the work done when the body moves from dis-
tance r to distance / is
*jfafJL-4.
* This result is also proved independently in many treatises on
dynamics, but a fuller investigation would be out of place here.
THE LAW OF UNIVERSAL GBAVITATION. 351
Hence the work done by a mass M in falling from distance 2a to
distance r is
= MQ (--- = iMu 2 byfxi.
\ r 2a/
kinetic energy of the planet when at distance r.
Therefore, if a circle be described about the centre of force 8, with
radius equal to the major axis 2a, the velocity at any point of the
orbit is that which the planet would acquire by falling freely from
the circle to that point under the action of the attracting force.
COROLLARY 3. If the planet be revolving in a circle, r a, and
therefore v 2 = C/r = C/a, as in 408.
COROLLARY 4. If v 3 = 2C/r, (x.) gives I/a = 0; /. a = oo.
Hence the velocity is that acquired by falling from an infinite
distance. In this case, the orbit is not an ellipse, but a parabola, a
conic section satisfying the " focus and directrix " definition of
Appendix (1), but having its eccentricity equal to unity.
If v z > 2C/r, the velocity is greater than that due to falling from
infinity, a comes out negative, and the orbit is a hyperbola, a conic
section satisfying the focus and directrix definition, but having its
eccentricity e greater than unity.
A few cornel- have been observed to describe parabolas and hyper-
bolas .-ibout the Sun. In such a case the motion is not periodic; the
comet gradually moves away TO an infinite distance, and is lost for
ever, unless the attraction of some other heavenly body should
happen to divert its course, and send it back into the solar system.
EXAMPLE. To find how long the Earth would take to fall into
the Sun if its velocity were suddenly destroyed.
If the Earth's velocity were very nearly, but not quite destroyed,
it would describe a very narrow ellipse, very nearly coinciding with
the straight line joining the point of projection to the Sun. The
major axis of this ellipse would be very nearly equal to the Earth's
initial distance from the Sun, and therefore the Earth would have
very nearly gone half round the narrow ellipse when it would
collide with the surface of the Sun.
Hence, if r denote the Earth's distance from the Sun, the semi-
rnajor axis of the narrow ellipse is \r, and the periodic time in this
ellipse would be ()* years. The Earth would therefore collide
with the Sun in
2 x (1)^ years = years = - years
= x 1-4142 days = 64 days nearly.
8
352 ASTRONOMY.
SECTION II. Newton's Law of Gravitation Comparison of
the Masses of the Sun and Planets.
418. In the last section we showed that the Sun attracts any
planet of mass M at distance r with a force CM/r 1 , where C
is a constant. If we assume the truth of Newton's Third
Law of Motion (i.e., that action and reaction are equal and
opposite), the planet must also attract the Sun with an equal
and opposite force CM/r 2 . Since in the former case the
force is proportional to the mass of the attracted body, and in
the latter to the mass of the attracting body, it is reasonable
to suppose that the attraction between two bodies is propor-
tional to the mass of each.
Moreover, the motions of the various satellites, such as the
Moon, confirm the theory that they revolve in their orbits
under the attraction of their respective primary planets.
From evidence of this character Newton, after many years of
careful investigation, enunciated his Law of Universal
Gravitation, which he stated thus :
Every particle in the universe attracts every other
particle with a force proportional to the quantities
of matter in each, and inversely proportional to the
square of the distance between them.
By quantity of matter is, of course, meant mass, and the
word attracts implies that the force between two particles
acts in the straight line joining them and tends to bring them
together.
If M, M' be the masses of two particles, and r the distance
between them, the law asserts that either particle is acted on
by a force, directed towards the other, of magnitude
where k has the same value for all bodies in the universe.
The constant is called the constant of gravitation.
*419. Astronomical Unit of Mass. Taking any fundamental
units of length and time, it is possible to choose the unit of mass
such that fc = 1. This unit is called the astronomical unit of mass.
Hence, if M, M ' are expressed in astronomical units, the force
between the particles is equal to MH'jr". It is, however, usually
more convenient to keep the unit of mass arbitrary, and to retain
the constant fc.
THE LAW OF UNIVERSAL GRAVITATION. 353
420. Remarks on the Law of Gravitation. Newton's
Law states that not only do the Sun, 'the planets and their
satellites, and the stars, mutually attract one another, but
every pound of matter on one celestial body attracts every
other pound of matter, on either the same or another body.
But it is well-known that two spheres attract one another
in just the Fame way as if the whole of the mass of either
were concentrated at its centre, provided that the spheres
are either homogeneous or made up of concentric spherical
layers, each of uniform density. Since the Sun and
planets are very nearly spherical, and their dimensions are
very small compared with their distances, we see that their
attractions may be very approximately found by regarding
them as mere particles, instead of taking separate account of
the individual particles forming them.
Moreover, every planet is attracted by every other planet,
as well as by the Sun. But the mass of the Sun, and con-
sequently its attraction, is so much greater than that of any
other member of the solar system, that the planetary motions
are only very slightly influenced by the mutual attractions.
Kepler's Laws, therefore, still hold approximately, but the
orbits are subject to small and slow changes or perturbations.
The Moon , on the other hand, is far nearer to the Earth
than to the Sun ; hence the Moon's orbital motion is mainly
due to the Earth's attraction. The chief effect of the Sun's
attraction on the Earth and Moon is to cause them together
to describe the annual orbit ; but it also produces pertur-
bations or disturbances in the Moon's relative orbit ( 272)
with which we are not here concerned.
The fixed stars also attract one another and attract the
solar system, which in its turn attracts the stars. The
proper motions of stars are probably due to this cause ;
but when we consider the vast distances of the stars, and
remember that the attraction varies inversely as the square
of the distance, it is evident that the relative accelerations
are mostly too feeble to have produced any sensible changes of
motion within historic times, and that countless ages must
elapse before such changes can be discerned.
354 ASffcONOMf*
42 1 . Correction of Kepler's Third Law. Prom the fact
that a planet attracts the Sun with a force equal to that with
the Sun attracts the planets, it may he shown that Kepler's
Third Law cannot he strictly true, as a consequence of the
law of gravitation. Not only will the planet move under
the Sun's attraction, but the Sun will also move under the
planet's attraction. Eut since the forces on the two hodics
are equal, while the mass of the Sun is very great compared
with the mass of any planet, it follows that the acceleration
of the Sun is very small compared with that of the planet,
and hence the Sun remains very nearly at rest.
We may, however, obtain a modification of Kepler's Third
Law, in which the planet's reciprocal attraction is allowed
for as follows : ,,
Let S, M be the masses of the Sim and planet; then the
attraction betweeen them is
This attraction, acting on the mass JJ/of the planet, produces
an acceleration of the planet towards the Sun equal to
The corresponding attraction on the mass 8 of the Sun pro-
duces an acceleration, in the reverse direction, of
Hence the whole acceleration of the planet relative to the
Sun is iM,
yd
instead of kS/r z , as it would be if the Sun were at rest.
Hence the absolute intensity of the planet's acceleration
towards the Sun is k (S + M), and this depends on the values
of both M and 8. Let now T be the periodic time, r the
planet's mean distance from the Sun, or the semi-axis major
of the relative orbit ; then, by 408 (for a circular orbit), or
415 (for an elliptical orbit),
1HE LAW OF UNIVERSAL GRAVITATION. 355
If M 1 be the mass of another planet, we have in like manner
for its orbit Jc (8 +11') T l = 4^ r'\
Therefore T 2 (8+3T) : T l (8 + M 1 ) = r 3 : r*,
the correct relation between the periods and mean distances.
It is known that different planets have different masses.
Hence, the fact that Kepler's Third Law is approximately
true shows that the masses of the planets are small compared
with that of the Sun.
422. Motion relative to Centre of Mass. The
mutual attractions of the Sun and planet have no influence
on the position of the centre of mass (commonly called the
" centre of gravity ") of the solar system ; hence, in consider-
ing the relative motions, that point may be treated as fixed.
It is known from general dynamical principles that when a
system of bodies are under the influence of their mutual
reactions or attractions alone, the centre of mass of the whole
system is not accelerated. But it may be interesting to prove
independently that when two bodies, such as the Sun and a
planet, attract one another, they both revolve about their
centre of mass.
Let us suppose (to take a simple case) the relative
orbit circular and of radius (P=) r, the angular velocity
being n. Then, if G be the point about which the planet
(P) and Sun (S) revolve, individually, we have
n*xGP = acccl. of planet = kS/r* ;
w 2 x GS = acccl. of Sun =
Hence MxGP= Sx GS ;
which relation shows that G is the common centre of mass,
as was to be proved.
In the case of three or more bodies, such as the Sun and
pLinets, the centre of mass is still the common centre about
which they revolve, but the corresponding investigation is
more difficult, owing to the effect of the mutual attractions
of the planets in producing perturbations.
It may be mentioned that the mass of the Sun is so large,
compared with those of the planets, that, although the further
planets arc so very distant, the centre of mass of the whole
solar system always lies very near the Sun.
356 ASTRONOMY.
423. Verification of the Theory of Gravitation for
the Earth and Moon. Before considering the motions of
the planets about the Sun, Newton investigated the orbital
motion of the Moon about the Earth, with the view of dis-
covering whether the Earth's attractive force, which retains
the Moon in its orbit, is the same force as that which pro-
duces the phenomenon of gravity at the Earth's surface.
If we assume that the force varies inversely at the square
of the distance, and that the Moon's distance is 60 times the
Earth's radius, the acceleration of gravity at the Moon should
be (-frV) 2 g, where g is the acceleration of gravity on the
Earth's surface.
But the acceleration g = 32-2 feet per sec. per sec. ;
.. accel. at Moon's distance = 32-2/3600 feet per sec. per sec.
= 32-2 feet per min. per min.
From the length of the lunar month and the Moon's dis-
tance in miles, Newton calculated what must be the normal
acceleration -of the Moon in its orbit. At the time of his first
investigation (1666) the Earth's radius and the Moon's dis-
tance were but imperfectly known, and the Moon's normal
acceleration, as thus computed, came out only about 27 feet
per minute per minute. Some fifteen years later, the Earth's
radius, and consequently the Moon's distance, had been
measured with much greater accuracy, and, working with the
new values, Newton found that the Moon's normal accelera-
tion to the Earth agreed with that given by his theory.
Taking the lunar sidereal month as 27 -3 days, the Earth's
radius as 3960 miles, and the radius of the Moon's orbit as
60 times the Earth's radius, the angular velocity (n) of the
Moon, in radians, per minute is
27T
27-3x24x60'
The Moon's distance in feet (d) = 3960 X 60 x 5280.
Hence the Moon's normal acceleration (tfd) in feet per
minute per minute
= 3 150 XJL XJ280 x 47T 2 _ 2xll0 2 X7r 2
(27'3) 2 x 24 2 x 60 2 = (27-3) 2 x 10
= 32 approximately,
thus agreeing with that given by the law of gravitation.
THE LAW OF UNIVERSAL GRAVITATION. 357
EXAMPLE. Having given that a body at the Earth's equator loses
1/289 of its weight in consequence of centrifugal force,
(i.) To calculate the period in which a projectile could revolve in
a circular orbit, close to, but without touching the Earth, and
(ii.) To deduce the Moon's distance.
(i.) The centrifugal force on the body would have to be equal to
its weight, and would therefore have to be 289 times as great as that
at the Earth's equator.
Hence the projectile would have to move -v/289, or 17 times as
fast as a point on the Earth's equator, and would therefore have to
perform 17 revolutions per day.*
Therefore the period of revolution = j\- of a day.
(ii.) Assuming the law of gravitation, the periodic times and dis-
tances of the projectile and Moon must be connected by Kepler's
Third Law. Hence, taking the Moon's sidereal period as 27| days,
we have, if a = Earth's rad., d = Moon's dist.,
.-. d z = a? x (17 x 27^) 2 = a 3 {^^} 2 = a 3 . 215915'i ;
.-. d = a x 3/215915-1 = 59'99;
.'. distance of Moon = 60 x Earth's radius almost exactly.
424. Effect of Moon's Attraction. Moon's Mass.
If we take account of the Moon's attraction on the Earth we
must introduce a correction analogous to that made in Kepler's
Third Law (421). If J/, m are the masses of the Earth
and Moon, the whole relative acceleration is k(lf+m)l$,
instead of kM/d*. But, if g n is the acceleration of gravity on
the Earth's surface, ^ =
and, if I 7 is the length of the sidereal month, then, by 421,
jJf-fW/nj
w-jr
* 1H "^ :=r ^^'
This formula might be used (and has been used by Airy)
to find m/M, the ratio of the Moon's to the Earth's mass, in
terms of the observed values of a, d, g^ T. It is not, how-
ever, a very accurate method, owing to the smallness of M/Jf.
* Relative to the Earth it would perform 16 or 18 revolutions per
day, according to whether it was revolving in the same or the
opposite direction to the Earth.
358 ASTRONOMY.
425. To find the ratio of the Sun's Mass to that
of the Earth.
Let /S, M, m be the masses of the Sun, Earth, and Moon,
d, r the distances of the Moon and Sun from the Earth, T, Y
the lengths of the sidereal lunar month and year respectively.
Then, if k be the gravitation constant, the Earth's attraction
on the Moon is = klfm/d', and its intensity is kM.
The Sun's attraction on the Earth is = kSM/r 1 , and its
intensity is kS.
Therefore, by 415, Corollary,
kM . T 2 = 47rd 3 , kS . F 2 = 47rV ;
whence the ratio of the Sun's to the Earth's mass may be
found.
If we take account of the attraction of the smaller body
on the larger, the whole acceleration of the Earth, relative to
the Sun, is k (S + M+m)/^ (since the Sun is attracted by
the Moon as well as the Earth), and that of the Moon, relative
to the Earth, is k (M+m)/eP. Hence the corrected or more
exact formula is
Since the Moon's mass is about -fa of that of the Earth,
the first or approximate formula can only be used if the cal-
culations arc not carried beyond two significant figures.
In this manner it is found that the Sun's mass is about
331,100 times that of the Earth.
EXAMPLES.
1. To compare, roughly, the masses of the Earth and Sun, taking
the Sun's distance to be 390 times the Moon's, and the number of
sidereal months in the year to be 13.
We have 8:M = ~:l^ t
. mass of Sun 390? _ 2 _
mass of Earth 18* " l51 ' 000 -
To the degree of accuracy possible by this method, the Sun's
mass is therefore 350,000 times that of the Earth.
THE LAW OP UNIVERSAL GRAVITATION. 359
2. To find the ratio of the masses, taking the Moon's mass as Jj
of the Earth's, and the number of sidereal months in the year as 13^.
390 3 _ 390 3 x 3 2 _ 5338710 --,.._
oooooy ;
.'. 8 = 333668 (M + m) = 333668 (1 + ff T ) 31 = 337,787 M.
426. To determine the mass of a planet which has
one or more satellites.
The method of the last paragraph is obviously applicable
to the case of any planet which has a satellite. We require
to know the mean distance and the periodic time of the
satellite. The former may be easily found by observing the
maximum angular distance of the satellite from its primary,
the distance of the planet from the Earth at the time of
observation having been previously computed. The periodic
time of the satellite may also be easily observed.
Let M' , ml be the masses of the planet and satellite, d'
their distance apart, r' their distance from the Sun, T' the
period of revolution of the satellite, Y' the planet's period of
revolution round the Sun. Using unaccented letters to re-
present the corresponding quantities for the Earth and Moon
we have, roughly,
tE __ M'T" 2 __ SY'' 2 = SY 2 _ 31T*
k " d'* ' r* r & d* '
or, more accurately,
(S+M+m'^Y*
k ' ~~d'~
_ (S+M+m) Y'
whence the mass of the planet, or, more correctly, the sum of
the masses of the planet and satellite, may be determined in
terms of the mass of the Sun, or the sum of the masses of the
Earth and Moon. We do not require to know the periodic
time and mean distance of the planet from the Sun, since the
above expressions enable us to express the required mass,
M' + m' 9 in terms of the year and mean distance of the
Earth, or in terms of the lunar month and the mean distance
of the Moon.
360
EXAMPLE. To find the mass of Uranus in terms of that of the
Sun, having given that its satellite Titania revolves in a period of
8 days 17 hours at a distance from the planet = '003 times the
distance of tho Earth from the Sun.
Let M be the mass of Uranus, then we have
d 3 . r 3
and, by Kepler's Third Law, r*/Y' 2 is the same for Uranus as for the
Earth. Hence
M'8= C' 003 ) 3 . I 3
(8d. 17h.) a ' (365d.6h.) 2?
*L = I 3 Y /365d.6h.
8 UOOO/ \8d.l7h.
= ?7_ x /8766\ 2
~ 10 9 \ 209 /
Thus, the mass of Uranus is to that of the Sun in the ratio of
1 to 21,Q53.
*427. The Masses of Mercury and Venus (which have no satellites)
could theoretically be found by determining their mean distances
from the Sun by direct observation, and comparing them with
those calculated from their periodic times by Kepler's Third Law.
For, if 3T is the mass of such a planet, we have
(S+_J)_r 2 = (8 + M + m) T-
~
This enables ns to find the stun of the masses of the Sun and
planet, and, the Sun's mass being known, the planet's mass could
be found.
This method is, however, worthless, because the masses of Mercury
nd Venus are only about TOO^WO ^ *injW of that of the Sun >
and in order to calculate one significant figure of the fraction M'/S
it would be necessary to know all the data correct to about seven
significant figures, a degree of ^Qewracy unattainable in practice.
For this reason it is necesgar,y ?to Calculate the masses of these
planets by means of the pefltwrbatiqns t they produce on one another
and on the Earthy these p*L r All r ) ia ^9 I n j' vW^ ^ e discussed in the next
chapter.
T1TR LAW OP TTNTVERSAL GRAVITATION". 36 1
428. Centre of Mass of the Solar System. When
the masses of the various planets have been found in terms
of the Sun's mass, the position of the centre of mass of the
system can be found for any given configuration, and can
thus be shown to always lie very near the Sun.
EXAMPLES.
1. To find the distance of the centre of mass of the Earth and
Sun from the centre of the Sun.
Here the mass of the Sun is 331,100 times the Earth's mass, and
the distance between their centres is about 92,000,000 miles. Hence,
the centre of mass of the two is at a distance from the Sun's centre
of about 92,000,000 =278mil
331,100 + 1
2. To find the centre of mass of Uranus and the Sun, and to show
that it lies within the Sun.
The distance of Uranus from the Sun is 19'2 times the Earth's
distance, and its mass is 1/21053 of the Sun's. Hence the C,M- ia
at a distance from the Sun's centre of
92,000,000 x 19-2
21053 + 1
The Sun's semi-diameter is 433,200 miles ; hence the centre of mass
of the Sun and Uranus is at a distance from the Sun's centre of
rather less than the radius.
3. In the case of Jupiter, the mean distance is 5'2 times that of
the Earth, and the mass is 1/1050 of that of the Sun j hence the
C.M. is at a distance
5-2 x 92,000,000
1050 + 1
This is just greater than the Sun's radius (433,200), showing that
the centre of mass lies just without the Sun's surface.
362 ASTBONOMY.
SECTION III. The Earth's Mass and Density.
429. The so-called " Weight of the Earth " really
means the Earth's mass, and the operation called " weighing
the Earth," in some of the older text-books, means finding the
mass of the Earth. In the last section we explained how to
compare the masses of the Sun and certain planets with that
of the Earth, and in the next chapter we shall give methods
applicable to a planet having no satellites. But before the
masses can be expressed in pounds or tons it is necessary to
determine the Earth's mass in these units. The methods of
doing this all depend on comparing the Earth's attraction
with that of a body of known mass and distance ; and the only
difficulty lies in determining the \atter attraction, since the
force between two bodies of ordinary dimensions is always
extremely small. The following methods have been used.
The first two are by far the best.
(1) By the "Cavendish Experiment," or the balance.
(2) By observations of the influence of tides in estuaries.
(3) By the "Mountain" method.
(4) By pendulum experiments in mines.
430. The " Cavendish Experiment " owes its name to
its having been first used to determine the Earth's mass by
Cavendish, about the year 1798. The essential principle of
the method consists in comparing the attractions of two heavy
balls of known size and weight with the Earth's attraction.
Since the attraction of a sphere at any point is proportional
directly to the mass of the sphere and inversely to the square
of the distance from its centre, it is evident that by comparing
the attractions of different spheres such as the Earth and the
experimental ball of metal we can find the ratio of their
masses.
The comparison is effected by means of a torsion "balance.
Two equal small balls A, B are fixed to the ends of a light
beam suspended from its middle point by means of a slender
vertical thread or "torsion fibre" (in his recent experiments,
Professor C. V. Boys has used a fine fibre of spun quartz), so
as to be capable of twisting about in a horizontal plane
(the plane of the paper in Fig. 143). Two heavy metal balls
C, D, are brought near the small balls A, (as shown in the
THE LAW OP UNIVERSAL GRAVITATION. 363
figure), and their attraction causes the beam to turn about 0,
say from its original position of rest XX' to the position AB.
As the beam turns the fibre twists ; this twisting is resisted by
the elasticity of the fibre, which produces a couple, propor-
tional to the angle of twist XOA, tending to untwist it again.
Let us call this couple /x L XOA, where / is a constant
depending on the fibre, called its " torsional rigidity"
The beam AB assumes a position of equilibrium when the
moments about of the attractions of the large spheres (7, D
on the balls A, B, just balance the "untwisting couple"
/x Z XOA. The angle XOA being measured, and the
dimensions of the apparatus being supposed known, the
attractions of the spheres can now be determined in terms of
the torsional rigidity.
FIG. 143.
The value of /is found in terms of absolute units of couple
by observing the time of a small oscillation of the beam when
the balls A, B have been removed. [The beam will then
swing backwards and forwards like the balance wheel of a
chronometer (204). The greater the torsional rigidity, the
more frequently will it reverse the motion of the beam, and
the more frequent will be the oscillations.*]
Hence finally the attractions between the known masses
C, D and A, B are found in terms of known units of force,
and by comparing these attractions with that of gravity the
Earth's mass is found.
* The student who has read Kigid Dynamics should work out the
formula.
ASTRON. 2 B
364 ASTRONOMY.
In practice, instead of measuring the angle XOA, the masses C, I)
are subsequently placed on the reverse side of the beam, say with
their centres at c, fl, and they now deflect the beam in the reverse
direction, say to ab. The angle measured is the whole angle aOA,
and this angle is ticice the angle XOA, if the positions CD and cd
are symmetrically arranged with respect to the line XOX'.
In the earlier experiments the beam AB was six feet long, and the
masses C, D were balls of lead a foot in diameter. Quite recently,
however, Professor C V. Boys, by the use of a quartz fibre for the
suspending thread, has performed the experiment on a much smaller
scale, the whole apparatus being only a few inches in size and being
highly sensitive. He uses cylinders instead of spheres for the
attracting bodies, and this introduces extra complications in the
calculations.
Although the above description shows the general principle of the
method, many further precautions are required to ensure accuracy.
A full description of these would be out of place here.
431. The common balance has also been used to deter-
mine the Earth's mass. In this case the differences of weight
of a body are observed when a large attracting mass is placed
successively above and below the scale-pan containing it.
EXAMPLE. To find the Earth's mass in tons, having given that the
attraction of a leaden ball, weighing 3 cwt., on a body placed at a
distance of 6 inches from its centre is '0000000432 of the weight of
the body.
Let M be the mass of the Earth in tons.
The mass of the ball in tons is = /.
The Earth's radius in feet = 3960 x 5280 = 20,900,000 roughly ;
and the distance of the body from the ball in feet = ^.
Hence, since the attractions of the Earth and ball are proportional
directly to the masses and inversely to the squares of the distances
from their centres,
-0000000432 ; 1 = _
'
' 2 (20,900,000)
(20,900,000^ x J_ = 3 x 209- x 10*0
x -0000000432 5 x 432 x 10 - 10
5 432 2160
= 6067 x 10 1S .
Hence the mass of the Earth is (roughly) 6067 million billion
tons.
THE LAW OF UNIVERSAL GRAVITATION. 365
432. To determine the Earth's Mass by observa-
tions of the Attraction of Tides in Estuaries. A
method which admits of very great accuracy is that in
which the mass of the Earth is found hy comparing it with
that of the water brought by the tide into an estuary. Con-
sider an observatory situated (like Edinburgh Observatory)
due south of an arm of the sea, whose general direction is
east and west. The direction of its zenith, as shown either
by a plummet or by the normal to the surface of a "bowl of
mercury, is not the same at high tide as at low, because the
additional mass of water at high tide produces an attraction
which deflects the plummet and the nadir point northward,
and hence displaces the zenith towards the south. Hence
the latitude of the observatory is less at high tide than at
low ; and the difference is a measurable quantity. The great
advantage of this method is that the mass which deflects the
plumb-line can be measured with great certainty ; for the
density of the sea-water is exactly known (and, unlike that of
the rocks in the next methods, is uniform throughout) and
the shape and height of the layer of water brought in are
known from the ordnance maps, and the tide measurements
at the port.
*433. In the Pendulum Method the values of g r the
acceleration of gravity, are compared by comparing the oscil-
lations of two pendulums at the top and bottom of a deep
mine. The difference of the two values is due to the attrac-
tion of that portion of the Earth which is above the bottom
of the mine ; this exerts a downward pull on the upper pen-
dulum, and an upward pull on the lower one. If the Earth
were homogeneous throughout, the values of g at the top and
bottom would be directly proportional to the corresponding
distances from the Earth's centre. If this is not observed to
be the case, the discrepancy enables us to find the ratio of the
density of the Earth to that of the rocks in the neighbourhood
of the mine. If the latter density is known, the Earth's
density can be found, and knowing its volume, its mass can
be computed. But this method is very liable to considerable
errors, arising from imperfect knowledge of the density of the
rocks overlying the mine.
366 ASTEONOMT.
*434. In the Mountain Method the Earth's attraction is com*
pared with that of a mountain projecting above its surface. Suppose
a mountain range, such as Schiehallien in Scotland, running due E.
and W. ; then at a place at its foot on the S. side the attraction of
the mountain will pull the plummet of a plumb line towards the N.,
and at a place on the N. side the mountain will pull the plummet to
the S. Hence the Z.D. of a star, as observed by means of zenith
sectors, will be different at the two sides, and from this difference
the ratio of the Earth's to the mountain's attraction may be found.
In order to deduce the Earth's density it is then necessary to
determine accurately the dimensions and density of the mountain.
This renders the method very inexact, for it is impossible to find
with certainty the density of the rocks throughout every part of the
mountain.
435. Determination of Densities. Gravity on the
Surface of the Sun and Planets. When the mass and
volume of a celestial body have been computed, its average
density can, of course, be readily found. By dividing the
mass in pounds by the volume in cubic feet, we find the
average mass per cubic foot, and since we know that the
mass of a cubic foot of water is about 62 J Ibs., it is easy to
compare the average density with that of water. The deter-
mination of densities is particularly interesting, on account of
the evidence it furnishes regarding the physical condition of
the members of the solar system. The Earth's density is
about 5*58.
Prom knowing the ratios of the mass and diameter of the
Sun or a planet to that of the Earth, we can compare the
intensity of its attraction at a point on its surface with the
intensity of gravity on the Earth.
It may be noticed that attraction of a sphere at its surface is pro-
portional to the product of the density and the radius.
For the attraction is proportional to mass -*- (radius) 2 , and the
mass is proportional to the density x (radius) 3 ; .*. the attraction
at the surface is proportional to the density x radius.
EXAMPLES.
1. To find the Earth's average density and mass, having given
that the attraction of a ball of lead a foot in diameter, on a particle
placed close to its surface, is less than the Earth's attraction in the
proportion of 1 : 20,500,000, and that the density of lead is 11'4 times
that of water.
THE LAW OF UNIVERSAL GRAVITATION. 367
Let D be the average density of the Earth. Then, since the radii
of the Earth and the leaden ball are | and 20,900,000 feet respectively,
and the attractions at their surfaces are proportional to thei*
densities multiplied by their radii,
/. 1 : 20,500,000 = ll'4xi : -Dx 20,900,000;
/. average density of Earth. D = 5'7x|^ = 5-6.
Hence the average mass of a cubic foot of the material forming
the Earth is 5 - 6 x 62'5 pounds. But the Earth is a sphere of volume
|TT (20,900,000) 3 cubic feet.
Hence the mass of the Earth, with these data,
= -I* x 209 3 x 10 15 x 5-6 x 62-5 pounds
= 1H38 x 10 22 pounds = 597 x 10 19 tons.
2. To calculate the mean density of the Sun from the following
data:
Mass of O = 330,000 . (mass of $) ;
Density of = 5 '58 ;
Q's parallax = 8'8"j Q's angular semi-diameter = 16'.
The radii of the Sun and Earth being in the ratio of the Sun's
angular semi-diameter to its parallax ( 258), we have
Q's radius 16' 960 inQ-1
- = - = - = iuy j. j
(p's radius 8'S 8'8
/. volume of Sun = (109'1) 3 . (vol. of Earth)
= 1,298,000 . (vol. of Earth) roughly.
But mass of Sun = 330,000. (mass of Earth) ;
. density of Sun ^ 330 = _1_ nearl
density of Earth 1298 3'9
/. density of Sun = 1'4.
3. To find the number of poundals in the weight of a pound at the
surface of Jupiter, taking the planet's radius as 43,200 miles and
density 1^ times that of water.
Taking the Earth's radius as 3960 miles and density as 5'58, we
have
(gravity at surface of Jupiter) : (gravity on Earth)
= 1-33 x 43,200 : 5-58 x 3960.
But at the Earth's surface the weight of a pound
= 32-2 poundals ;
therefore on the surface of Jupiter the weight of a pound
= 83-7 poundals.
368 ASTRONOMY.
EXAMPLES. XIII.
1. Taking Neptune's period as SO years, and the Earth's velocity
as 91 miles per second, find the orbital velocity of Neptune.
2. If we suppose the Moon to be 61 times as far from the Earth's
centre as we are, find how far the Earth's attraction can pull the
Moon from rest in a minute.
3. If the Earth possessed a satellite revolving at a distance of only
6,000 miles from the Earth's surface, what would be approximately
its periodic time, assuming the Earth to be a sphere of 4,000 miles
radius ?
4. Assuming the distance between the Earth's centre and the
Moon's to be 240,000 miles, and the period of the Moon's revolution
28 days, find how long the month would be if the distance were
only 80,000 miles.
5. Calculate the mass of the Sun in terms of that of Mars, given
that the Earth's mean distance and period are 92 x 10 6 miles and
365i days, and the mean distance and period of the outer satellite
of Mars are 14,650 miles and Id. 6h. 18m.
6. Show that the periodic time of an asteroid is 3| years, having
given that its mean distance is 2'305 times that of the Earth.
7. Show that we could find the Sun's mass in terms of the Earth's,
from exact observation of the periods and mean distances of the
Earth and an asteroid, by the error produced in Kepler's Third Law
in consequence of the Earth's mass.
8. Show that an increase of 10 per cent, in the Earth's distance
from the Sun would increase the length of the year by 56' 14 days.
9. The masses of the Earth and Jupiter are approximately
"iuroVoT) an d ToW respectively of the Sun's mass, and their distances
from the Sun are as 1 : 5. Show that Kepler's Laws would give the
periodic time of Jupiter too great by more than 2 days.
10. Prove that the mass of the Sun is 2 x 10 27 tons, given that
the mean acceleration of gravity on the Earth's surface is 9'81
metres per second per second, the mean density of the Earth is
5*67, the Sun's mean distance T5 x 10 s kilometres, a quadrant of the
Earth's circumference 10,000 kilometres, and taking a metre cube of
water to be a ton.
THE LAW OF UNIVERSAL GRAVITATION. 369
11. Having given that the constant of aberration for the Earth
is 20'49", and that the distance of Jupiter from the Sun is 5'2 times
the distance of the Earth from the Sun, calculate the constant of
aberration for Jupiter.
12. If the mass of Jupiter is T oVo f t ne m ass of the Sun, show
that the change in the constant of aberration caused by taking into
account the mass of Jupiter is 004" nearly (see Question 11).
13. Find the centre of mass of Jupiter and the Sun. Hence find
the centre of mass of Jupiter, the Sun, and Earth, (1) when Jupiter
is in conjunction, (2) when in opposition. (Sun's mass = 1,048
times Jupiter's = 332,000 times Earth's. Jupiter's mean distance
= 480,000,000 miles ; Earth's = 93,000,000 miles.)
14. If the intensity of gravity at the Earth's surface be 32 - 185
feet per second per second, what will be its value when we ascend
in a balloon to a height of 10,000 feet ? (Take Earth's radius = 4,000
miles and neglect centrifugal force.) Would the intensity be the
same on the top of a mountain 10,000 feet high ? If not, why not ?
15. Show how by comparing the number of oscillations of a
pendulum at the top and bottom of a mountain of known density,
the Earth's mass could be found.
16. How would the tides in the Thames affect the determination
of meridian altitudes at Greenwich observatory theoretically ?
. 17. If the mean diameter of Jupiter be 86,000 miles, and his mass
315 times that of the Earth, find the average density of Jupiter.
18. If the Sun's diameter be 109 times that of the Earth, his mass
330,000 times greater, and if an article weighing one pound on the
Earth were removed to the Sun's surface, find in poundals what its
weight would be there.
19. Taking the Moon's mass as ^ that of the Earth, show that
the attraction which the Moon exerts upon bodies at its surface is
only about l-5th that of gravity at the Earth's surface.
20. If the Earth were suddenly arrested in its course at an
eclipse of the Sun, what kind of orbit would the Moon begin ^0
describe ?
370 ASTRONOMY.
EXAMINATION PAPER. XIII.
1. State reasons for supposing that the Earth moves round the
Sun, and not the Sun round the Earth.
2. State Kepler's Laws, and give Newton's deductions therefrom.
.3. If the Sun attracts the Earth, why does not the Earth fall into
the Sun ?
4. Show that the angular velocitiesof two planets are as the cubes
of their linear velocities.
5. State Newton's Law of Gravitation, and prove Kepler's Third
Law from it for the case of circular orbits, taking the planets small.
6. Explain clearly (and illustrate by figures or otherwise) what
is meant by a force varying inversely as the square of the distance.
7. Are Kepler's Laws perfectly correct ? Give the reason for your
answer. What is the correct form of the Third Law if the masses
of the planets are supposed appreciable as compared with the mass
of the Sun ?
8. How can the mass of Jupiter be found ?
9. Show that if a body describes equal areas in equal times about
a point, it must be acted on by a force to that point.
10. Find the law of force to the focus under which a body will
describe an ellipse ; and if C be the acceleration produced by the
force at unit distance, T the periodic time, and 2a the major axis
of the ellipse, find the relation between 0, a, T.
CHAPTEE XIV,
FURTHER APPLICATIONS OF THE LAW OF
GRAVITATION.
SECTION 1. The Hoori 's Mass Concavity of Lunar Orbit.
436. The Earth's Displacement due to the Moon.
In Section II. of the last chapter we saw that when two
bodies are under their mutual attraction they revolve about
their common centre of mass. Thus, instead of the Moon
revolving about the Earth in a period of 27-^ days, both
bodies revolve about their centre of mass in this period,
although from the Moon's smaller size its motion is more
marked.
In this case both the Earth and Moon are under the
attraction of a third body the Sun which causes them
together to describe the annual orbit. But the Sun's dis-
tance is so great compared with the distance apart of the
Earth and Moon, that its attraction is very nearly the same,
both in intensity and direction, on both bodies. To a first
approximation, therefore, the resultant attraction of the Sun
is the same as if the masses of both the Earth and Moon were
collected at their common centre of mass. Hence it is strictly
the centre of mass of the Earth and Moon, and not the centre
of the Earth, which revolves in an ellipse about the Sun with
uniform areal velocity, in accordance with the laws stated in
155. And, owing to the revolution of the Moon, the Earth's
centre revolves round this point once in a sidereal month,
threading its way alternately in and out of the ellipse
described, and being alternately before and behind its mean
position.
372
ASTRONOMY.
This displacement of the Earth has been used for finding
the Moon's mass in terms of tho Earth's, by determining the
common centre of mass of the Earth and Moon, as follows.
FIG. H4.
Let EV J/,, 6 r 1 (Fig. 144) be the positions of the centres of the
Earth and Moon, and their centre of mass, at the Moon's last
quarter, JE* 2 , M v 2 and E# M# G 3 their positions at new Moon
and at first quarter respectively, S the Sun's centre.
Then, at last quarter, E^ is behind # and the Sun's longi-
tude, as seen from E^ is less than it would be as seen from G l
by the angle E^SG^. At first quarter, JE 9 is in front of G 3 ,
and therefore the Sun's longitude is greater at E% than at G s
by the angle G^SE y If, then, the observed coordinates of
the Sun be compared with those calculated on the supposition
that the Earth moves uniformly (i.e., with uniform areal
velocity), its longitude will be found to be decreased at last
quarter and increased at first quarter.
From observing these displacements the Moon's mass may
be found. For, knowing the angle of displacement E l SG l
and the Sun's distance, the length E l G l may be found. Also
the Moon's distance E^ is known. And, since G l is the
FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 373
centre of mass of the Earth and Moon,
mass of Moon : mass of Earth = E l G l : G^J/j;
whence the mass of the Moon can be found.
The Sun's displacement at the quarters could he found by
meridian observations of the Sun's R.A. with a transit circle.
The displacement of the Earth will also give rise to an
apparent displacement, having a period of about one month,
in the position of any near planet ; this could be detected
by observations on Mars, when in opposition, similar to those
used in finding solar parallax ( 339).
From this and other methods it is found that the mass of
the Moon is about 1/81 of that of the Earth. The Moon's
density, as thus deduced, is about 3 '44, or of that of the
Earth.
EXAMPLE. To compare the masses of the Moon and Earth, having
given that the Sun's displacement in longitude at the Moon's
quadratures is equal in f of the Sun's parallax.
Since L E i SG i = f the angle subtended by Earth's radius at S,
therefore E& = | (Earth's radius).
But E^Mi = 60 (Earth's radius) ;
.'. EjJf, = 80. #!(?,;
.'. GjMi = 79..E? 1 G 1 ,
and mass of Moon : mass of Earth =E^G^ : O i M l = 1 : 79;
/. the Moon's mass = 1/79 of the Earth's mass.
437. Application to Determination of Solar Paral-
lax. If the Moon's mass be found by any other method, the
above phenomena give us a means of finding the Sim's
parallax and distance. For we then know E^ G l : G^^ and
therefore E& and the angle E.SG, is found by observation.
But the exact ratio of IS 1 SG 1 to the parallax is known, for it
is equal to that of ^ G l to tbe Earth's radius ; hence the Sun's
parallax and distance can be found. Since the Moon's mass
can be found with extreme accuracy by many different
methods, this method is quite as accurate as many that have
been used for finding the solar parallax.
*438 Concavity of the Moon's Path about the Sun. - The Moon,
by its monthly orbital motion about the Earth, threads its way alter-
nately inside and outside of the ellipse which the centre of mass of
the Earth and Moon describes in its annual orbit about the Sun,
,374 ASTRONOMY.
Hence the path described by the Moon in the course of the year is
a wavy curve, forming a series of about thirteen undulations about
the ellipse. It might be thought that these undulations turned
alternately their concave and convex side towards the Sun, but the
Moon's path is really always concave ; that is, it always bends
towards the Sun, as shown in Fig. 145, which shows how the path
passes to the inside of the ellipse without becoming convex.
To show this it is necessary to prove that the Moon is always
being accelerated towards the Sun. Let n, n' be the angular velo-
cities of the Moon about the Earth and the Earth about, the Sun
respectively. Then, when the Moon is new, as at M 2 (Fig. 145), its
acceleration towards G 2 , relative to G 2 , is n 2 . MG 2 . But (? 2 has a
normal acceleration n'' 2 G^S 'towards 8. Hetice the resultant accelera-
tion of the Moon Jf 2 towards 8 is n'-G. 2 S-
FIG. 145.
Now, there are about 13 sidereal months in the year ; therefore
TO = 13X- Also E^S is nearly 400 times E 2 M 2 , and therefore G 2 S is
slightly over 400 times GM- 2 . Therefore roughly
n'"G z S : n*M 2 Gz = 400 : 182 ; .'. ri-G^S > n 2 (? 2 3f 2 .
Thus, the resultant acceleration of M is directed towards, not away
from 8, even at Jf 2 , where the acceleration, relative to (? 2 . is directly
opposed to that of G 2 . Therefore the Moon's path is constantly
being bent (or deflected from the tangent at M 2 ) in the direction of
the Sun, and is concave towards the Sun.
*439. Alternate Concavity and Convexity of the Path, of a
Point on the Earth.. In consequence of the Earth's diurnal rota-
tion, combined with its annual motion, a point on the Earth's equator
describes a wavy curve forming 365 undulations about the path
described by the Earth's centre. In this case, however, it may be
easily shown in the same way that the acceleration of the point
towards the Earth's centre is greater than the acceleration of the
Earth's centre towards the Sun. The path is, therefore, not
always concave to the Sun, being bent away from the Sun in
the neighbourhood of the points where the two component accelera-
tions act in opposite directions.
FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 375
SECTION II. The Tides.
In the last section we investigated the displacements due to
the -Moon's attraction on the Earth as a whole. We shall
now consider the effects arising from the fact that the Moon's
attractive force is not quite the same either in magnitude or
direction at different parts of the Earth, and shall show how
the small differences in the attraction give rise to the tides.
440. The Moon's or Sun's Disturbing Force. Let C,
Jf be the centres of the Earth and Moon ; AC A' the Earth's
diameter through M ; B, B' points on the Earth such that
M C = MB = MB'. Let 3/, m denote the masses of the
Earth and Moon, a the Earth's radius, d the Moon's distance.
The resultant attraction of the Moon on the Earth as a
whole is &Mm/ CM*, and the Earth is therefore moving with
acceleration km/ CM* towards the common centre of mass of
the Earth and Moon, as shown in 422, 424.
FIG. 146.
(i.) N"ow at the sublunar point A the Moon's attraction on
unit mass is km/AM 2 and is greater than that at C (since
AM < CM). Hence the Moon tends to accelerate A more
than C and thus to draw a body at A away from the Earth,
with relative acceleration F, where
l CA
d*(d-aj 2 d* (1-a/dy
Since a/d is a small fraction, we have, to a first approximation,
376
ASTRONOMY.
(ii.) At A the Moon's attraction per unit mass is km/A'M' 2 ,
and is less than that at (7, since AM > CM. Hence the Moon
tends to accelerate C more than A, and thus to draw the
Earth away from A with relative acceleration F', where
= Jan
To a first approximation, therefore,
(l+a/d
Thus a body either at A or A tends to separate from the
Earth, as if acted on by a force away from C, of magnitude
approximately = 2kina/d* per unit mass.
FIG. 147.
(iii.) Consider now the effect of the Moon's attraction on a
body at B. This produces a force per unit mass of
which may be resolved into components
and
Since we have taken 7>3f = CM, the first component is
equal to km/ CM 3 ; that is, to the force at C. This component
therefore tends to make a body at B move with the rest of
the Earth, and produces no relative acceleration. Therefore
the Moon tends to draw a body at B towards the Earth with
relative acceleration /, represented by the second component ;
thus
FUKTHER APPLICATIONS OF THE LAW OP GRAVITATION. 377
The point B is approximately the end of the diameter BCB
perpendicular to AC (since BM, CM, B'M are nearly parallel
in the neighbourhood of the Earth).
Hence the relative acceleration at B is approximately per-
pendicular to CM, and its magnitude
f=km=km .
d it'
Similarly at B' the Moon tends to draw a body towards C,
with relative acceleration /= kma/d*.
At either of these points, B, B', therefore, a body tends to
approach the Earth, as if acted on by a force towards the Earth's
centre, of magnitude kma/d 3 per unit mass. Generally, the
Moon's attraction at any point tends to accelerate a body,
relatively to the Earth, as if it were acted on by a force depend-
ing on the difference in magnitude and direction between the
Moon's attractions at that point and at the Earth's centre.
This apparent force is called the Moon's disturbing
force or tide-generating force. AVc sec that the dis-
turbingforce produces a pull along^L4' and a squeeze along////.
A similar consequence arises from the attraction of the Sun.
The Sun's actual attraction on the Earth as a whole keeps the
Earth in its annual orbit, but the variations in the attraction
at different points give rise to an apparent distribution of
force on the Earth which is the Sun's disturbing force or
tide-generating force.
441. To find approximately the Moon's or Sun's
Disturbing Force at any point.
Let be any point of the Earth. Draw ON perpen-
dicular on CM. " Then the difference of the Moon's attractions
at and N tends to accelerate towards JV, with a relative
acceleration 1cm . NO/d* [by 440 (iii.)]- Also, the difference
of the attractions at N, C tends to accelerate TV away from C
with a relative acceleration 2km. CN/d 3 [by 440 (i.)].
The whole acceleration of 0, relative to C, is compounded
of these two relative accelerations. Therefore, if X. Foe the
components of the disturbing force at in the directions
CN, NO,
ON v ^
.- , Y= K
378 ASTRONOMY.
442. Hence the following geometrical construction ;
On CN produced take a point IT such that
Tlicn the line OH represents the disturbing force at in
direction, and its magnitude is
v 7 on
F = 1cm. -.
d*
The Sun's tide-raising force may be found exactly in the
same way. The force is everywhere directed towards a point
on the diameter of the Earth through the Sun, found by a
similar construction to the above. And if r, S denote the
Sun's distance and mass, the force is proportional to S/r s
instead of mj&.
In all these investigations we see that the tide-raising force
due to an attracting body is proportional directly to its mass
and inversely to the cube (not the square) of its distance.
From this it is easy to compare the tide-raising forces due
to different bodies acting at different distances.
EXAMPLES.
1. To compare the tide-raising forces due to the Sun and Moon.
The masses of the Sun and Moon are respectively 331,000 and
gL times the Earth's mass. Also, the Sun's distance is about 390
times the Moon's.
.*. Sun's tide-raising force : Moon's tide-raising force
= 33 : 73 nearly = 3:7 nearly.
Thus the Sun's tide-raising force is about'three-sevenths of that
of the Moon.
2. To find what would be the change in the Moon's tide-raising
force if the Moon's distance were doubled and its mass were in-
creased sixfold.
If /,/'betheold and new tide-raising forces at corresponding points,
/'/= f = 4 f
J ' J 2 3 ' l a> 4
Therefore the tide-raising force would have three-quarters of its
present value.
3. To compare the Moon's tide-raising forces at perigee and
apogee.
The greatest and least distances of the Moon being in the ratio of
FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 379
l + TT to ! Te> or 19 to 17 ( 270), the tide-raising power at perigee
is greater than at apogee in the ratio of 19 3 : 17 3 or 6859 : 4913, or
roughly 7 : 5.
4. To compare the maximum and minimum values of the Sun's
tide-raising force.
The eccentricity of the Earth's orbit being ^ these are in the
ratio of (1 + sV) 3 : (1 g^) 3 , or approximately 1 + ^ : I--&, or
21 : 19. As before, the force is greatest at perigee and least at
apogee.
Moon
443. The Equilibrium Theory of the Tides. Let
us imagine the Earth to be a solid sphere covered with an
ocean of uniform depth. If we plot out the disturbing forces
at different points of the Earth by the construction of 442,
we shall find the distribution represented in Eig. 148, the
lines representing the forces both in magnitude and direction.
Here the disturbing force tends to raise the ocean at the
sub-lunar point A and at the opposite point A, and to de-
press it at the points B, B'. At intermediate points it tends
to draw the water away from B and B\ towards A and A'.
Hence the surface of the ocean will assume an oval form,
as represented by the thick line in Eig. 148, and there will
be high water at the sublunar point A and the opposite point
A, low water along the circle of the Earth BB', distant 90
from the sublunar point. Thus we have the same tides
occurring simultaneously at opposite sides of the Earth.
It may be shown that the oval curve aba'b' is an ellipse
whose major axis is aa'. The surface of the ocean, therefore,
assumes the form of the figure produced by revolving this
ellipse about its major axis. This figure is called a prolate
spheroid, and is thus distinguished from an oblate spheroid,
which is formed by revolution about the minor axis.
2c
ASTRON.
380 ASTEONOMT,
But though this is the form which the ocean would assume
if it were at rest, a stricter mathematical investigation shows
that the Earth's rotation would cause the surface of the sea
to assume a very different form.
In fact, if the Earth were covered over with a sufficiently
shallow ocean of uniform depth, and rotating, we should really
have low tide very near the sublunar point A and its anti-
podal point A', and high tide at the two points on the Earth's
equator distant 90 from the Moon (Fig. 149).
If the Moon were to move in the equator, the equilibrium
theory would always give low ^ water at the poles. This
phenomenon is uninnue'nced by the Earth's rotation, and since
the Moon is never more than about 28 from the equator, we
see that the Moon's tide-raising force has the general effect
of drawing some of the ocean from the poles towards the
equator.
*444. A few other consequences of the equilibrium theory may
also be enumerated. (1) According to it the height of the tides, or
the difference of height between high and low water at any place, is
directly proportional to the tide-generating force, and consequently,
with the results of Example 1 of 442, the heights of the solar and
lunar tides are in the proportion of 3 to 7. (2) Since the distortion
of the mass of liquid is resisted by gravity, the height of the tide
depends on the ratio of the tide-producing force to gravity, and
therefore is inversely proportional to the intensity of gravity, and
therefore to the density of the Earth ; if the density were halved,
the height of the tides would be doubled. (3) If the diameter of
the Earth were doubled, its density remaining the same, the inten-
sity of gravity and the tide-producing force would both be doubled,
since both are proportional to the Earth's radius. This would cause
the ocean to assume the same shape as before, only all its dimensions
would be doubled. f Consequently the height of the tide would also
be doubled, and it thus appears that the height of the tide is pro-
portional to the Earth's radius.
We thus have the means of comparing the tides which would be
produced on different celestial bodies, for the above properties show
that the height of tide is proportional to ma/Dd?, where a and D are
the radius and density of the body under consideration, TO, d the
mass and distance of the disturbing body.
*445. Canal Theory of the Tides. As an illustration,
let us consider what would happen in a circular canal, not
extremely deep, supposed to extend round the equator of a re-
f Of course this is not a very strict proof.
FTTRTHEB APPLICATIONS OF THE LAW OF GBAYITATION. 381
volving globe. Then, in Fig. 149, it is clear that the direction
of the disturbing force would, if it acted alone, cause the
water in the quadrants AB and AB' to flow towards A ;
and, in the quadrants A'B and AB', towards A. Hence this
force acts in the same direction as the Earth's rotation in the
quadrants B'A and BA, and in the opposite direction in AB
and AB'. Hence, as the water is carried from A to B, it is
constantly being retarded, from B to A it is accelerated,
from A to B' it is retarded, and from B' to A it is again
accelerated, the average velocity being, of course, that of
the Earth's rotation. Hence the velocity is least at B and
', and greatest at A and A'.
Moon
Now, it is easy to see that when water moves steadily
in a uniform canal it must be shallow where it is swift and
deep where it is slow. For, if we consider any portion
of the canal, say AB, the quantity that flows in at one end
A is equal to the quantity that flows out at the other
end B. But it is evident that if the depth of the canal
were doubled at any point without altering the velocity of
the liquid, twice as much liquid would flow through the
canal ; consequently, in order that the amount which flows
through might be the same as before, we should have to halve
the velocity of the liquid. This shows that where the canal
is deepest the water must be travelling most slowly. Con-
versely, where the velocity is least the depth must be greatest,
and where the velocity is greatest the depth must be least.
Hence the depth is least at A and A', and greatest at B and
J?, just the opposite to what we should have expected from
the equilibrium theory.
882 ASTRONOMY.
In a canal constructed round any parallel of latitude the
same would be the case ; and hence, if we could imagine a
uniform ocean replaced by a series of such parallel canals,
low tide would occur at every place when the Moon was in
the meridian.
This theory (due to Newton), though sounder than Laplace's
equilibrium theory, is still not quite mathematically correct.
The true explanation of the tides, even in an ocean of uniform
depth, is far more complicated, and quite beyond the scope
of this book.
446. Lunar Day and Lunar Time. According to
either hypothesis, the recurrence of high and low water depends
on the Moon's motion relative to the meridian ; hence, in
investigating this, it is convenient to introduce another kind
of time, depending on the Moon's diurnal motion.
The lunar day is the interval between two consecutive
upper transits of the Moon across the meridian.
In a lunation, or 29 mean solar days, the Moon performs
one direct revolution relative to the Sun, and therefore per-
forms one retrograde revolution less relative to the meridian.
Thus 29J mean days = 28J lunar days ; whence the mean
length of a lunar day
= O + BT) mean 8 lar days = 24h. 50m. 32s. nearly.
The lunar time is measured by the Moon's hour an^le,
converted into hours, minutes, and seconds, at the rate of ^15
to the hour.
ttjItTHEB, APPLICATIONS OP THE LAW OF GRAVITATION. 383
*447. Semi-diurnal, Diurnal and Fortnightly Tides.
It has been found convenient to regard the tides produced
by the Moon's disturbing force as divided into three parts,
whose periods .are half a day, a day and a fortnight, the
" day " being the lunar day of the last paragraph.
If we adopt the equilibrium theory as a working hypothesis,
the lunar tide must be highest when the Moon is nearest to
the zenith or nadir. Hence high tide takes place at the
Moon's upper and lower transits, when its zenith distance and
nadir distance are least respectively. But, for a place in N.
lat. (Fig. 150) when the Moon's declination is K, it describes a
small circle Q,'R\ and its least zenith distance ZQ 'is less than its
least nadir distance NR ; hence the two tides are unequal
in height. This phenomenon can be represented by supposing
a diurnal tide, high only once in a lunar clay, combined
with a semi-diurnal tide, high twice in this period.
Again, the Moon's meridian Z.D. and N.D. go through a
complete cycle of changes, owing to the change of the Moon's
declination, whose period is a month. But after half a month,
the Moon's declination will have the same value but opposite
sign, and hence the diurnal circles Q[R' , Q,"R" ', equidistant
from the equator Q,R, are described at intervals of a fortnight.
But NJR"= ZQ', ZQ"=: NR' ; hence the two tides have the
same heights. This can be represented by supposing a fort-
nightly tide of the proper height combined with the
diurnal and semi-diurnal ones.
In just the same way the smaller tides caused by the Sun
may be artificially represented by combining a diurnal
and semi-diurnal tide (the solar day being used) and a
six-monthly tide.
448. Spring and Neap Tides. Priming andLagging
We have hitherto considered chiefly the tides due to the
action of the Moon. In reality, however, the tides are due
to the combined action of the Sun and Moon, the tide-raising
forces due to these bodies being in the proportion of about
3 to 7 (Ex. 1, 442). We shall make the assumption that the
height of the tide at any place is the algebraic sum of the
heights of the tides which would be produced at that place by
the Sun and Moon separately.
384 ASTRONOMY.
/ At new or full Moon the Sun is nearly in the line AA\
(, and the tide-raising powers of the Sun and Moon both act in
)the same direction, and tend to draw the water from B^B' to
i A^^AL^ hence the whole tide is that due to the sum of the
) separate disturbing forces of the Sun and Moon. The tides
/ are then most marked, the height of high water and.depth_pf
I low water being at their maximum. Such tides are called
Spring Tides. We notice that the height of the spring
tide = 1 +f or \- ^ * na * f tnc lunar tide alone.
Moon
At the Moon's first or last quarter the Sun is in a
line BB' perpendicular to A A'. Hence the Sun tends to draw
the water away from A, A' to B, B >', while the Moon tends to
draw the water in the opposite direction. The Moon's action
being the greater, preponderates, but the Sun's action
diminishes the tides as much as possible. The variations
are therefore at their minimum, although high water still
occurs at the same time as it would if the Sun were absent.
These tides are called Neap Tides. The^ndght of the
neap, _ tide is the difference of the heights of the lunar and
solar tides, and is therefore f of that of the lunar tide.
Hence spring tides and neap tides are in the ratio of
(roughly) 10 to 4.
For any intermediate phase of the Moon, the Sun's action
is somewhat different.
Between new Moon and first quarter, the Sun is over
a point S l behind A. Here the Moon tends to draw the
water towards A, A', and the Sun tends to draw the water
towards S l and the antipodal point $ s . Therefore the com-
bined action tends to draw the water towards two points Q, Q'
APPLICATIONS OF THE LAW OF GRAVITATION". 385
between A and S l and between A and S. 6 respectively, whose
longitudes are rather less than those of A and A respectively.
The resulting position of high water is therefore displaced to
the west, and the high water occurs earlier than it would if
due to the Moon's influence alone. The tides are then said
to prime.
Between first quarter and full Moon the Sun is over
a point $ 2 between ' and A, and the combined action of the
Sun and Moon tends to draw the water towards two points
jR, R', whose longitudes are slightly greater than those of
A, A. The resulting high tides are therefore displaced east-
wards, and occur later than they would if the Sun were
absent. The tides are then said to lag.
Between full Moon and last quarter the Sun is over
some point $ 8 between and A', but the antipodal point S l
is between A and B' ; hence the tide primes.
Between last quarter and new Moon, when the Sun
is at a point S between B and A, it is evident in like manner
that the tide lags.
Hence Spring Tides occur at the syzygies (conjunction
and opposition).
Neap Tides occur at the quadratures.
From syzygy to quadrature, the tide primes.
From quadrature to syzygy, the tide lags.
The heights of the spring and neap tides vary with the varying
distances of the Sun and Moon from the Earth. Spring tides are
the highest possible when both the Sun and Moon are in perigee,
while neap tides are the most marked when the Moon is in apogee
but the Sun is in perigee (because the Sun then pulls against the
Moon with the greatest power, as far as the Sun's action is con-
cerned). Both the spring and neap tides, and also the priming and
lagging, are on the whole most marked when the Sun is near per ^ee,
i.e.. about January.
It may be here stated, without proof, that, taking the Sun s am
Moon's tide-raising forces to be in the proportion ot o 7, ti
maximum interval of priming or lagging is found \
61 minutes.
386 ASTRONOMY.
449. Establishment of the Port. Both the equilibrium
and canal theories completely fail to represent the actual tides
on the sea, owing to the irregular distribution of land and
water on the Earth, combined with the varying depth of the
ocean. These circumstances render the prediction of tides
by calculation one of the most complicated problems of prac-
tical astronomy, and the computations have to be based largely
on previous observations. In consequence of the barriers
offered to the passage of tidal waves by large continents,
lunar high tide does not occur either when the Me on crosses
the meridian, as it would on the equilibrium theory, or
when the Moon's hour angle is 90, as it would on the canal
theory. But this continental retardation causes the high
tide to occur later than it would on the equilibrium theory,
by an interval which is constant for any given place. This
interval, reckoned inlunar hours, is called the Establishment
of the Port for the place considered. Thus the establish-
ment of the port at London Bridge is Ih. 58m., so that lunar
high water occurs Ih. 58m. after the Moon's transit, i.e.,
when the Moon's hour angle, reckoned in time, is Ih. 58m.
The same causes affect the solar tide as the lunar, hence
the Sun's hour angle (or the local apparent time) at the solar
high tide is also equal to the establishment of the port.
The actual high tide, being due to the Sun and Moon con-
jointly, is earlier or later than the lunar tide by the amount
of priming or lagging. By adding a correction' for this to
the establishment of the port, the lunar time of high water
may be found for any phase of the Moon ; and we notice in
particular that at the Moon's four quarters (syzygies and
quadratures), the lunar time of high water is equal to the
establishment of the port. And, knowing the lunar time of
high water, the corresponding mean time can be found, for
(mean solar time) (lunar time)
= (mean 0's hour angle) ( ([ 's hour angle)
= ( d 's R.A.) -(mean Q's R.A.)
[since R.A. and hour angle are measured in opposite directions].
Now the Moon's R.A. is given in the Nautical Almanack
for every hour of every day in the year. Also the mean Sun's
R.A. at noon is the sidereal time of mean noon, and is given
FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 387
in the Nautical Almanack. Hence the mean Sun's R.A.
[which = (sidereal time) - (mean time)] is easily found for
any intermediate time.
Hence the mean time of high water can be readily found.
The establishments of different ports, and the times of high
water at London Bridge, are given in the Nautical Almanack.
*450. If only a very rough calculation is required, we may proceed
as in 35, 40. We assume the Moon's R.A. to increase uniformly ;
we shall then have
( <[ 's R.A.) - ( 0's R.A.) = ( O elongation) ;
.'. (solar time) = (lunar time) + ( <t 's elongation).
Knowing the Moon's age, its elongation may be found, as in 40,
and this must be converted into time, at the rate of Ih. to 15. We
then have (time of high water)
= (establishment) + (amount of lag.) + ( C 's elongation in time)
EXAMPLE. To find, roughly, the time of high water at the Moon's
first quarter, at London Bridge.
Here there is no priming or lagging. Hence the lunar time, or
's hour angle at high water, is equal to the establishment, or Ih.
58m. Also the Moon's elongation is 90. Hence the Sun's hour
angle, in time, = Ih. 58m. + 6h ., and high water occurs about 7h. 58m.
*451. Tidal Constants. The excess of the establishment of the
port at any place, over that at London Bridge, expressed in mean
time, is sometimes called the Tidal Constant of that place.
If we assume the amount of priming or lagging to be the same
at both places, the tidal constant is the difference between the times
of high water at London Bridge and the given place. Hence,
knowing the tidal constant and the time of high water at London
Bridge, the time at any other place can be found.
Tables of tidal constants, and of the heights of the spring and neap
tides at different places, are given in Whitakers Almanack.
EXAMPLE. To find the times of high water at Cardiff and Ports-
mouth on January 25, 1892, the tide intervals from London Bridge
being +4h. 58m. and 2h. 17m. From the Almanack we find
times of high water at London Bridge are
Jan. 24. . Jan. 25.
9h. 15m. aft. 9h. 53m. morn., lOh. 31m. aft.
Add 4h. 58m. 4h. 58m. 4h. 58m.
/. Times at Cardiff are
(Jan. 25) 2h. 13m. morn. 2h. 51m. aft.
Again, subtract from first line 2h. 17m. 2h. 17m.
times at Portsmouth are (Jan. 25) 7h. 36m. morn., 8h. 14m. aft.
388
ASTROXOJltf.
452. The Masses of the Sun and Moon can be com-
pared by observing the relative heights of the solar and lunar
tide, the relative distances of the Sun and Moon being known.
Or, if the ratio of the masses be supposed known, the dis-
tances could be compared by this method. In this manner
Newton (A.D. 1687) found the masses of the Moon and Earth
to be in the proportion of 1 : 40 nearly. D. Bernouilli
(1738) found 1 : 70, and Lubbock (1862) found 1 : 67-3.
The two last make the Moon's mass a little too great. Newton
makes it double what it ought to be.
FIG. 152.
453. Effects of Tidal Friction. Retardation of
Earth's Rotation. Acceleration of Moon's Orbital
Motion. All liquids possess a certain kind of friction,
known as u viscosity," which tends to resist their motion
when they are changing their form, and to convert part of
their kinetic energy into heat. Owing to this friction between
the Earth and the oceans, the Earth, in its diurnal rotation,
tends to carry the tidal wave round slightly in front of the
point underneath the Moon, taking the positions of high water
forward from the line JI'CMto AC A. The Moon, on the
contrary, tends to draw the water back from A : A', the dis-
turbing forces AH, A' II' forming a couple, which is resisted
only by the Earth's friction. Hence the ocean exerts an
equal frictional couple on the Earth, and this couple tends to
diminish the angular velocity of the Earth's diurnal rotation,
and thus increase its period.
Therefore tidal friction tends to gradually lengthen the day.
APPLICATIONS OF THE JAW OF GBAVlTATlOtf. 389
But if the Moon exerts a couple on the Earth, tending to
retard it, the Earth must exert an equal and opposite couple
on the Moon, tending to accelerate it. That it really does so
is manifest from Fig. 152. The portion of the ocean heaped
up at A, being nearer the Moon, exerts a greater attraction than
that at A', in addition to which the angle CM A is very
slightly greater than GNA '. Hence the resultant of the
attractions of equal masses of water at A and A acts on M
in a direction slightly in front of M C, and tends to pull the
Moon forward. This tends to increase the Moon's areal
velocity. (Compare 413.) Since the areal velocity of a
body revolving in a circle varies as the square root of the
radius ( 411, Cor.), the Moon's distance must be gradually
increased by this means, and hence also its periodic time.*
Therefore tidal friction tends to increase the Moon's distance
and to lengthen the month.
Still the final effect of tidal friction must be to equalize
the lengths of the day and lunar month. The angular
velocities of the Earth and Moon "both decrease, but the
effect of the couple, in producing retardation, is far more
considerable on the Earth than on the Moon.
The student who has not read Rigid Dynamics may illustrate this
statement by the comparative ease with which a small top can be
spun with the fingers, and the great difficulty of imparting an equal
angular velocity to the same body by whirling it round in a circle
at the end of a string of considerable length. The top represents
the Earth, and the body on the long string the Moon.
In Rigid Dynamics it is shown that when a system of bodies are
revolving under their mutual reactions, their angular momentum, or
moment of momentum about their centre of mass, remains constant.
Hence the decrease in the Earth's angular momentum is equal to
the increase in that of the Moon. Now the angular momentum
of a particle revolving in an orbit is twice the product of its mass
into its areal velocity, and this is also approximately true of the
Moon. Hence, since the Moon's distance from the common centre
of mass is far greater (about sixty times as great) than the distance
of any point on the Earth from its axis of rotation, it is evident
that the same change in angular momentum produces far more effect
on the angular velocity of the Earth than on that of the Moon.
* This increase of the distance more than counterbalances the
tendency to increase the Moon's actual velocity. For the actnai
velocity is inversely proportional to the square root of the distance
( 409), and therefore diminishes as the distance increases,
Similarly, the angular velocity is decreased.
390 ASTfcONOMt.
It thus appears that, after the lapse of probably many
millions of years, tidal friction will equalize the periods of
rotation of the Earth and Moon, and the day and month will be
of equal length, each being probably about 1,400 hours long.
The Earth will then always turn the same face towards the
Moon, just as the Moon now does towards the Earth ; hence
there will be no lunar tides, and the retardation due to lunar
tidal friction will no longer exist.
The solar tides will, however, still continue to exist, pro-
vided that there is any water left on the Earth. The effect
of solar tidal friction will be to retard the Earth's rotation,
thus further lengthening the day ; and this again will retard
the Moon's orbital motion, and diminish its areal velocity.
The Moon will, therefore, approach the Earth, and will
ultimately fall into the Earth ; and finally, the Earth will
always turn the same face towards the Sun, so that there
will always be day over one hemisphere and night over the
other.
This theory of the probable future history of the Earth is
due to Professor G. H. Darwin. It is certain that the effect
of tidal friction on the Earth's rotation must be very small ;
hence a very long period must necessarily elapse before any
perceptible increase in the length of the day can be detected.
The records of history afford no data sufficiently accurate to
furnish conclusive evidence of such a lengthening, but there
are some grounds for believing that the sidereal day is in-
creasing in length by about *006 of a second in 1,000 years.
Moreover, the Earth is gradually cooling, and consequently
is shrinking ; and this shrinkage, by bringing the particles of
the Earth nearer to the axis, causes an increase of the angular
velocity of rotation.* It is quite possible that an increase of
this nature is at the present time either wholly or partially
counteracting the retardation due to tidal friction.
* For, according to the principles of Rigid Dynamics, the angular
momentum of the Earth = (its angular velocity) x (its moment of
inertia). And if the angular momentum remains constant, and the
moment of inertia decreases through shrinkage, the angular velocity
must increase.
FUHTHER APPLICATIONS OF THE LAW OF GEAVITATIOfl . 391
454. The Moon's Form and Rotation. The theory of
tidal friction affords a simple explanation of how it is that the
Moon always turns the same face to the Earth. Remember-
ing that the Earth's mass is 81 times the Moon's, but that its
radius is about four times as great, the Earth's tide-raising
force at a point on the Moon would be about 81/4, or over
twenty times as great as the Moon's on the Earth. Although
there are now no oceans on the Moon, still we have some
evidence that water may once have existed on its surface.
Furthermore, the large volcanic craters with which its sur-
face is dotted prove that the Moon was at one time filled
with molten lava, and that it was probably wholly in a liquid
or viscous state at an earlier period of its history. At that
time the huge tides on the Moon, ever following the Earth,
must, by their friction, have gradually equalized the Moon's
period of rotation with its period of revolution about the
Earth, in just the same way as if the Moon were surrounded
by a friction belt attached to the Earth. This continued till
the Moon always turned the same face to the Earth.
If the Moon was then not quite solid, the Earth's tide-
raising force, which had then become constant, must have
drawn it out into the form demanded by the equilibrium
theory, namely, to a first approximation, a prolate spheroid,
with its longest diameter pointed towards the Earth.
It may easily be seen, from the expressions in 440, that th-
tide-raising force of a body is slightly greater at the poinl
just under it than at the opposite point (when we do not
only consider approximate values). Hence the Moon is not
quite spheroidal, but is more drawn out on the side toward^
the Earth than on the remote side. Its form is, therefore,
that of an egg, the small end being towards the Earth. This
result of theory cannot, of course, be confirmed by direct
observation, the remote side being invisible ; butHansen, by
the theory of perturbations, has shown that _ the Moon's
centre of mass is further from the Earth than its centre of
figure, thus furnishing independent evidence in favour of
the theory.
392 ASTBONOMY.
*455. Application to Solar System. Since the Sun's
tide-raising force on different planets varies inversely as the
cube of their distance, the solar tides are far greater on the
nearer planets than on those more remote. It is, therefore,
quite natural to suppose that the effects of tidal friction may
have produced such a great retardation in the rotations of
Mercury, and possibly also Venus, that one or both of these
bodies already turn the same face towards the Sun, while the
Earth, and the remoter planets, must necessarily take a much
longer time to undergo the necessary retardation, and it
would be very unnatural to expect JS~eptune, for example,
always to turn the same face to the Sun. Thus Professor
Schiaparelli's recent researches on the rotations of Mercury
and Yenus are in support of the theory of tidal friction.
SECTION III. Precession and Nutation.
456. In 141 we stated that the plane of the Earth's
equator is not fixed in space, but that its intersections with
the ecliptic have a slow retrograde motion. This phenome-
non, which is known as Precession, is due to the fact that the
Earth is not quite spherical, and that, in consequence of its
spheroidal form, the Sun's and Moon's attractions exert a dis-
turbing couple on it.
457. The Sun's and Moon's Disturbing Couples
on the Earth.
Let the plane of the paper in Fig. 153 contain the Earth's
polar axis PP', and the Moon's centre M, say at the time
when the Moon's south declination is greatest.
Inside the Earth inscribe a sphere PAP' A', touching its
surface at the poles. Then we may (for the sake of illustra-
tion) regard the protuberant portion of the Earth outside this
sphere as a kind of tide firmly fixed to the Earth, and the argu-
ments of the last section ( 453) show that the variations in the
Moon's attraction at different points give rise to a distribution
of disturbing force identical with the tide -raising force, tending
to draw this protuberant part with its longest diameter QR
pointing towards the Moon. The Moon's attraction on the
matter inside the inscribed sphere- passes exactly through the
FTJBTHEB APPLICATIONS OP THE LAW OP GRAVITATION. 393
Earth's centre 0, and produces no such couple ; but the dis-
turbing forces at A, A', which are represented by AH, A' IT,
form a couple on the protuberant parts, A Q, A'R, tending
to turn the diameter A A towards CM. The same is true of
the disturbing forces at any other pair of opposite points of
the Earth in the quadrants HCK, H'CK'. Of course there
are couples in the two other quadrants tending in the reverse
direction, but they have less matter to act on, and are there-
fore insufficient to balance the former couples.
FIG. 153.
When the Moon is at the opposite point of its orbit, i.e., at
its greatest N. declination, it is again in the line CH', and
again tends to draw the Earth's equatorial plane towards the
line HH'. For any intermediate position of the Moon the
couple is smaller, and it vanishes when the Moon is on the
equator ; still, on the whole, the Moon's disturbing force always
tends to draw the plane of the Earth? s equator towards the plane
of the Moon's orbit.
Similarly, the Sun's disturbing force always tends to draw
the plane of the Earth's equator towards the ecliptic.
Since the Moon's nodes are rotating ( 273), the plane of the
Moon's orbit is not fixed ; but it is inclined to the ecliptic at
a small angle (5), while the plane of the equator is inclined
to the ecliptic at a much larger angle (23|). The average
effect of the Moon's disturbing couple is thus to pull the
Earth's equator towards the plane of the ecliptic. This ten-
dency is increased by the Sun's disturbing couple ; and the
two are proportional to the Sun's and Moon's tide-producing
forces, i.e., as 3 : 7 roughly. For this reason, the resulting
phenomenon is sometimes called luni-solar precession.
394 ASTBONOMT.
*458. Effect of the Couple on the Earth's Axis.
If the Earth were without rotation, the tendency of this
couple would be to bring the plane of the equator into coinci-
dence with the ecliptic, with the result that the equator
would oscillate from side to side of the ecliptic, like a pendu-
lum under gravity. But the rapid diurnal motion of the
Earth entirely alters the phenomena.
Let CR be a semi-diameter of the Earth, perpendicular to
CP and CM. The precessional couple would, alone, produce a
slow rotation in the direction PQM; i.e., about CR. If
now the Earth's rotation be represented in magnitude and
direction by CP, measured along the Earth's axis, this addi-
tional rotation must be represented by a very short length
CR', measured along CR.
FIG. 154.
Take PP', equal and parallel to CR' ; then, since PP' is
very small, CP' is of almost exactly the same length as CP.
But angular velocities, and momenta about lines which repre-
sent them in magnitude, are compounded by the same law as
forces, velocities, &c. \ef. 387 (iii.)] along the same lines of
corresponding magnitudes.
Hence, the resultant axis of rotation is shifted from CP to
CP', in a direction perpendicular to the plane of the acting
couple.
A full explanation of what follows would be impossible
without a close acquaintance with Rigid Dynamics. But it is
evident that a body flattened at the poles will spin more
readily about the line CP than about any other line drawn in
its substance. Hence it is easy to understand that the polar
axis CPis itself deflected towards CP', and thus moves per-
pendicular to the acting couple.
FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 395
This motion can be illustrated by that of a rapidly spinning
top, or of a gyroscope, the phenomena of which can readily be
investigated by experiment.
459. Precession of a Spinning Toy. Experiment 1.
Let a top be set spinning rapidly about its extremity, in the oppo-
site direction to the hands of a watch, as seen from above, the top
being supported at a point on its axis below its centre of gravity.
The weight of the top, acting vertically through the centre of gravity,
tends to upset the top by pulling its axis out of the vertical. But if
the top is spinning sufficiently rapidly, we know that it will not fall,
the only effect of gravity being to make it " reel," i.e., to cause its
axis of rotation to describe a cone about the vertical through the
point of support, revolving slowly in the counter-clockwise direction.
This slow revolution may be called the precession of the top, and
the experiment shows that when a top is acted on by a couple (such
as that due to its weight) tending to pull its axis away from the
vertical, it processes in the same direction in which it is spinning.
Experiment 2. Now suppose the top suspended from its upper
extremity, being thus supported above its centre of gravity. The
couple due to the weight and the reaction of the support, now tends
to draw the axis of the top towards the vertical. In this case the
axis of the top will be found to slowly describe a cone in the opposite
direction ; that is, the top now precesses in the opposite direction to
that in which it is spinning.
Experiment 3. Suppose the top supported as in Experiment 1.
If we give the top a push away from the vertical, its axis will not
move in this direction, but its processional motion will increase. If
we give a push in the direction of precession, its axis will approach
the vertical. If we push the axis in the direction of the vertical, it
will not move towards the vertical, but its rate of processional motion
will be increased, i.e., the top will acquire an additional increased
precessional motion. If we push it in the direction opposite to
that of precession, the axis will begin to move away from the vertical.
In every case the axis of the top moves in a direction perpendicular
to the direction of the force acting on it, and therefore a couple
acting on a very rapidly spinning top produces displacement of the
axis in a plane perpendicular to the plane of the couple.*
[If we push the top by pressing the side of a pencil against its
axis, it thus always moves in the direction in which the axis would
roll along the side of the pencil. Of course the displacement of the
axis is not due to rolling, as may easily be shown by repeating tl
same experiment with a gyroscope, this time pushing one of the
hoops carrying the top instead of touching the top itself ; here no
such rolling is possible.]
* These experiments may easily be performed by the reader with
any good-sized top.
ASTBOX. 2 D
396 ASTB-ONOMt.
r'r
FIG. 155.
460. Precession of the Earth's Axis. On the celestial
sphere, let P, K he the poles of the equator and ecliptic
respectively. The Sun's disturbing couple and the mean
couple due to the Moon tend to pull the Earth's equator
towards the ecliptic, or to pull the polar axis P towards the
axis of the ecliptic K. Hence the Earth "behaves like a top
suspended from above its centre of gravity, and the polar
axis slowly describes a cone about the axis of the ecliptic,
revolving in the opposite direction to that of the Earth's
rotation, i.e., in the retrograde direction.* The pole P there-
fore slowly describes a small circle PP' about It, the pole of
the ecliptic, with angular radius P7T, equal to the obliquity
of the ecliptic, i.e., 23 27'. As the pole revolves from P to
P' it carries the equator from r Q^ to f '$=', thus carry-
ing the equinoctial points T and b slowly backwards along
the ecliptic. The average angle T T ', or P/fP'f , described in
a year, is 50-2", and P therefore performs a complete revolu-
tion about -5Tin 25,800 years ( 141).
* Sec also Fig. 154. If K be pole of ecliptic (CK nearly perpen-
dicular to CM) it is evident that as P travels towards P' it moves
in the retrograde direction about It".
t PT andETT are each 90; .'. T is. pole of arc KP-, .'. LrKP is
a right angle. Similarly, T 'KP' is a right angle ;
/. PKP' = L TKr' = arc T T 7 ,
since T T'^ is a great circle, whose pole is K.
FURTHER APPLICATIONS OP THE LAW of GRAVITATION. 397
The position of the ecliptic is not affected by precession.
Hence the celestial latitude xH of any star x remains constant,
and its celestial longitude T Jf increases by the amount of pre-
cession r T', that is, at the rate of '50-2" per year.
A star's declination and right ascension are, however, con-
tinually changing. This change is, of course, due to the
motion of the equator, and not of the star. Thus, as P moves
to P', the KP.D. of the star x decreases from Px to P'x, and
its 11. A. changes from TPx to T'P'x. (The circles
T-P, T'F, %P, %P' are not represented, in order not to com-
plicate the figure unnecessarily. The reader should draw a
figure, inserting them.)
The declinations of some stars are increasing, of others
decreasing.
461. To apply the Corrections for Precession.
The changes in the decl. and R.A. of a star in one year are
always small, except in the case of the Pole Star, which is
so near the pole that a slight displacement of the pole pro-
duces a great change in the R.A. With this exception, the
rates of change of the decl. and R.A. of a star remain sensibly
constant for a considerable period. Hence, if the coordi-
nates are observed on any given date, and their rates of
variation are known, their values at any other date may be
found by adding or subtracting corrections obtained by mul-
tiplying these rates of variation by the elapsed time.
The rates of variation may be regarded as constant so long
as the interval of time is small compared with the period of
rotation of the pole. They are therefore sensibly uniform for
several years.
The most convenient plan, in correcting for precession, is to
calculate the right ascensions and declinations of all stars for
the same date or epoch.
For this purpose, the time of the vernal equinox in the year
1900 is now frequently chosen as the standard epoch of refer-
ence. "When the R.A. and decl. of a star are known, their
rates of variation can be calculated by Spherical Trigonometry
in terms of the known rate of precession, and the correction
can then be applied.
398 A.STBONOMY.
It would, of course, be possible to proceed somewhat differ-
ently, namely, from the decl. and R.A. to find tbe star's lat.
and long. Tbe long, could tben be increased by tbe amount
of tbe precession, namely, 50 -2" x (tbe number of years
elapsed) ; and from tbe new lat. and long, tbe new decl. and
R.A. could be found ; but tbe calculations would be longer.
For the purpose of facilitating observations of time, latitude
and longitude, and instrumental errors, tbe declinations and
right ascensions of certain bright stars are calculated at
intervals of ten days in tbe Nautical Almanack ; these stars
are the clock stars of 54.
The effects of aberration, as well as of precession and nuta-
tion, are taken into account, the tabulated coordinates being
those of tbe apparent and not tbe true positions of the star.
Such stars can therefore be used to determine clock error and
other errors, without applying any further correction.
462. Various Effects of Precession.
Since the R.A. and decl. of a star depend only on tbe
relative positions of the star and equator, their variations due
to precession are just the same as they would be if the equator
and ecliptic were fixed, and tbe stars bad a direct motion of
rotation, of 50*2" per annum, about the pole of the ecliptic.
If we make this supposition, tbe stars will describe circles
about JTin a period of 25,800 years.
(i.) If a star's distance Ex from the pole of the ecliptic is
less than the obliquity t, or its latitude (I) greater than
90 t, it will describe a circle ax^'x^ (Fig. 156), of radius
90 Z, not enclosing the pole P, and its greatest and least
N.P.D. will be
Pa' = f+(90-f), Pa = t-(90-Z).
Also the star's R.A. will fluctuate between the values
rPx, and rP# 2 . Now r is the pole of PK; hence EPr
is a right angle, and rPK= 270; therefore the maximum
and minimum R.A. are 270 + JTPx v and 270 KPx r
FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 399
(ii.) If, on the other hand, the star's latitude is <90' *',
it will describe a circle byb', enclosing the pole P. Its
greatest and least N.P.D. will be
PV = (90 - /) + , Pb = (90- /) - i.
The star's E.A. will continually increase from to 360.
In either case the star's N.P.D. will increase as its longitude
increases from 90 (at a or b) to 270 (at a' or J'), and will
decrease over the other half of the path.
The Pole Star will, after a time, move away from the pole,
and its place will be then occupied in succession by other
stars whose latitude is very nearly = 90 i = 66 33'. If
I, L be the latitude and longitude of such a star, it will be
nearest the pole in an interval of (90 -Z)-^ 50 -2" years, and
its N.P.D. will then be (90- J)~-
That precession has shifted the equinoctial points from the
constellations Aries and Libra, into Pisces and Virgo, has
already been mentioned. Since there are twelve signs ot the
zodiac, the equinoctial points shift from one " sign ' into the
next in 25,800/12 years, *.*., about 2,150 years,
400 ISTfiONOMT.
463. Effects on the Climate of the Earth's Hemi-
spheres. We have seen ( 132) that the fact of the Earth
being in perihelion near the winter solstices renders the
climate of the Earth's northern hemisphere more equable, but
makes the seasons more marked in southern hemisphere.
Owing, however, to precession, combined with the progres-
sive motion of the apse line ( 153), the reverse will be the
case in 180x60x6 or 10,545 years. The summer in the
50-22 + 11-25'
northern hemisphere will then be hotter, but shorter, and the
winter colder and longer. On the whole, the climate will be
colder, as the Earth's radiation will be more rapid during the
heat of summer, and therefore a larger proportion of the heat
received from the Sun will be lost before the winter.
In a recent paper, Sir Robert Ball has shown that the ice
ages, of which we have geological evidence, can probably be
accounted for in this manner. The eccentricity of the Earth's
orbit is not constant, but is changing very slowly, and is
decreasing at the present time. "When the orbit had its
greatest eccentricity and the winter solstice coincided with
aphelion, the autumn and winter were 199 days long, spring
and summer being only 166 days long. At this time the
climate of the northern hemisphere must have been so exceed-
ingly cold that the whole of northern Europe, including
Germany and Switzerland, was ice-bound. When aphelion
coincided with the summer solstice a similar effect took place
in the southern hemisphere, but the northern hemisphere was
warmer and more genial than it is now, spring and summer
being 199 days long, and autumn and winter only 166 days
long. Thus, at the time of greatest eccentricity 'there must
have been long ages of arctic climate, oscillating from one
hemisphere to the other and bask in a period of 10,500 years,
alternating with more equable, and, perhaps, almost tropical
climates.
464. Nutation of the Earth's Axis. In treating of
precession, we have supposed the Earth's poles to describe
small circles uniformly about the poles of the ecliptic. This
PT7ETHEE APPLICATIONS OF THE LAW OP GRAVITATION. 401
they would do if the Sun's and Moon's disturbing couples on
the Earth were always constant in magnitude, and always
tended to pull the Earth's poles directly towards the poles of
the ecliptic. But the couples, so far from being constant
are subject to periodic variations, in consequence of which
the Earth's poles really describe a wavy curve (shown in
-Fig. 157), threading alternately in and out of the small
circle which would be described under precession alone if the
couple were constant. This- phenomenon is called Nutation,
because it causes the Earth's poles to nod to and from the
pole of the ecliptic.
Nutation is really compounded of several independent
periodic motions of the Earth's axis; the most important of
these is known as Lunar Nutation, and has for its period
the time of a sidereal revolution of the Moon's nodes, i.e.,
about 18 years 220 days. The effect of lunar nutation may
be represented by imagining the pole P to revolve in a small
ellipse about its mean position jp as centre, in the above period,
in the retrograde direction, while p revolves about JT, the
pole of the ecliptic, with the uniform angular velocity of
precession of 50'2" per annum. The major and minor axes
of the little ellipse are along and perpendicular to Kp re-
spectively, their semi-lengths being pa 9" and pb = 6 -8"
respectively. The angle pKb = bp/smJp (Sph. Georn. 17)
^ 6-8"cosec 23 27' = 1 7' 1" nearly.
402 ASTRONOMY.
465. General Effects of Lunar Nutation. In con-
sequence of lunar nutation, the obliquity of the ecliptic is
subject to periodic variations. For this obliquity is equal
to the arc KP, and as P revolves about its mean position
from one end to the other of the major axis of the little
ellipse, the arc KP becomes alternately greater and less than
its mean value p, by 9". Thus the greatest and least values
of the obliquity of the ecliptic differ by 18", and the obliquity
fluctuates between the values 23 27' 20" and 23 27' 2" once
in about 18f years.
FIG. 158.
Again, when the pole is at an extremity of the minor axis
i, it has regreded further than its mean position p by the
angle pKb, which we have seen is about 17-1". Hence, also,
the first point of Aries has regreded 17'1" further than it
would have gone had its motion been uniform. Similarly, at
V it has regreded 17-1" less than it wt'uld have done if
moving uniformly. Hence the first point of Aries oscillates
to and fro about its mean position through an arc of 34' 2" in
the period of 18-| years, while its mean position moves through
an angle 18f X 50-2", or about 15' 37".
The angular distance between the true and mean positions
of the first point of Aries is called the Equation of the
Equinoxes. It is, of course, equal to the angle pKP.
Nutation does not affect the position of the ecliptic ; hence
the latitudes of stars are unaltered by it. Their apparent
longitudes are, however, increased by the equation of the
equinoxes. Both this cause and the varying obliquity of
the ecliptic produce variations in a star's R.A. and decl.
FURTHER APPLICATIONS OP TUB LAW OP GRAVITATION. 403
466. Discovery of Nutation. Nutation was discovered
by Bradley soon after his discovery of aberration, while con-
tinuing his observations on the star y Draconis and on a small
star in the constellation Camelopardus, by its effect on the
declinations of these stars. The peculiarity which led him
to separate nutation from aberration was their difference of
period. The period of the former phenomenon is about 19
years, while that of the aberration displacement is only a
year. Had the observed variations in declination been due
to aberration alone, the declination would always have had
the same apparent value at the same time of year, but such
was not the case.
Newton had, sixty years previously (1687), proved the
existence of nutation from theory, but had supposed that its
effects would be inappreciable.
467. To correct for Nutation, the coordinates of a star
are always referred to. the mean position of the ecliptic, i.e.,
the position which the ecliptic would occupy if its pole were
at p, the centre of the little ellipse. Hence, since the
apparent decl. and RA. of a star x are measured by 90 Px
andr^ (= 270 + 7TP^), the corrected decl. and K.A. are
90 jfcc and 270 + JTpx. If the star's position is specified
by its celestial latitude and longitude, the only correction
required is to increase the longitude by the equation of the
equinoxes.
*468. Bessel's Day Numbers. If the declinations and right ascen-
sion of stars have been tabulated for a certain date, their apparent
values for any other date, as affected by precession, nutation, and
aberration, can be found by adding certain small corrections to the
tabulated values, and it is found that these may be put into the
form Change of R.A. = Aa + Bb + Cc + Dd,
Change of decl. - Aa' + Bb' + Cc' -I- Dd',
where A, B, 0, D are constants, whose values depend only on the
date, and are the same for all stars ; while a, 6, c, d, a', &, c, d
depend only on the coordinates of the star, being alway* constant for
the same star, and independent of the time of observation.
The four quantities A, B, C, D are called Bessel's Day Numbers,
and their logarithms are given in the Nautical Almanack for every
day of the year. The logarithms of the eight constants a, 6, c,d,
a', b', c', d', have been tabulated for many thousands of stars m tt
star catalogues of the Boyal Astronomical Society.
404 ASTRONOMY.
469. Physical Cause of Nutation. If the Moon were
to move exactly in the ecliptic, the average couples exerted
by the Moon as well as the Sun would both tend to pull the
Earth's pole directly towards K, the pole of the ecliptic.
But the Moon's orbit is inclined to the ecliptic at an angle
of 5 ; hence, if L be its pole, KL = 5, and the Moon's
average disturbing couple tends to pull the pole P towards
L instead of K. When we consider the Sun's action also,
the resultant of the two couples tends to pull the pole towards
a point H which is intermediate between K and Z, but
nearer to L (because the Moon's disturbing couple is about
2J times the Sun's). Hence the pole P moves off in a direc-
tion perpendicular to HP, and not to KP. In consequence
of the rotation of the Moon's nodes, Z, and therefore also ZT,
revolves in a small circle about P in the period of 18| years
(see Fig. 159).
Let Z t , Z 2 , Z 3 , Z 4 , Z 6 be the positions of Z, and P v P 2 , P 3 ,
P 4 , P 5 the positions of P, when the angle PKL is 0, 90,
180, 270, 360 respectively, H v H^ the positions of ZT cor-
responding to Z 2 , Z 4 . Then at P l and P 8 the couple is
directed towards JT, and therefore P is then moving perpen-
dicular to KP. At P 2 the couple is directed towards J7 2 , an ^
the pole P 2 moves perpendicularly to JToPg, thuspassingfromthe
inside to the outside of the small circle described by its mean
position. Similarly, at P 4 the pole, by moving perpendicularly
to II fv passes back from the outside to the inside of the
small circle which it would describe if the couple were
always directed towards K. Thus the wavy form of the
curve described by P is accounted for. And since the whole
space Pj-ffPg or Zj J5fZ 5 , traversed in a revolution of Z, is very
small, the period of oscillation is almost exactly that of
revolution of the Moon's nodes.
Again, the Moon's couple depends on the angular distance
PZ, and is greater the greater this distance (as may easily be
seen by 457). Hence the resultant couple, and therefore
also the precessional motion, is least at P l and greatest at P 3 .
This accounts for the variable rate of motion of P, which
gives rise to the equation of the equinoxes.
FUTITHEE APPLICATIONS OF THE LAW OF GRAVITATION. 405
FIG. 159.
FIG. 160.
* 470. Solar and Monthly Nutations. The variations in the inten-
sity of the Sun's and Moon's disturbing couples during their orbital
revolutions give rise to two other kinds of nutation. Let us first
consider the variations in the Sun's disturbing couple, which pro-
duce Solar Nutation. It appears from 457, that the couple vanishes
when the Sun is on the equator, and that it is greater the greater
the Sun's declination. Also it is readily evident from Fig. 153 that
the couple in general acts in a plane through the Sun and the
Earth's poles, tending to turn the poles more nearly perpendicular
to the direction of the Sun. This shows that the couple is not
really directed towards the pole of the ecliptic (though this is its
average direction for the year) except at the solstices (Fig. 160).
Now at the vernal equinox, when the Sun is at T , the couple
vanishes, and therefore the Earth's tendency to precession, due to
the Sun, vanishes. Between the vernal equinox and the summer
solstice, when the Sun is at Sj, the couple is along SjP away from
Si, and this tends to make the pole precess along PG' perpendicu-
larly to SiP. At the summer solstice the couple along CP is a
maximum, and tends to produce precession along PG 5 perpendicular
to KP. At 8-2 the couple along S 2 P tends to make the pole precess
in the direction PG". At the autumnal equinox, ^, the couple, and
therefore the velocity of solar precession, vanishes. AtS a the Sun's
declination is negative, and the couple tends to draw P towards S 3 ;
hence the Earth again tends to precess along PG'. At the winter
solstice the direction of precession is again along PG, and the pro-
cessional velocity again a maximum. Finally, at 84 the direction
of precession is again along PG".
406 ASTBONOMY.
Hence the variations in the Sun'a declination cause the pole to
thread its way in and out of the circle it would describe under
uniform precession once every six months, and to cause the velocity
of revolution about K to fluctuate in the same period. This gives
rise to the nutation known as Solar Nutation, whose period is half a
tropical year. In the case of the Moon the corresponding
phenomenon is known as Monthly Nutation, and its period is half
a month ; the explanation is exactly the same.
The variations in the obliquity of the ecliptic due to these two
causes are small, because, owing to the comparatively small period in
which they recur, the pole has not time to oscillate to and from K
to any great extent. Moreover, the couple, and therefore the rate of
motion of P, decreases as the inclination of PG' to PO increases.
When the Sun is at T or ^ the displacement, if it existed, would be
along PK, in the most advantageous direction for producing nutation,
but at this instant the couple vanishes.
The solar nutation only displaces the pole about 1'2" to or from
K, and the displacement due to monthly nutation is imperceptible.
The effects on the equation of the equinoxes are more apparent.
Under the Sun's action alone, the pole would come to rest twice a
year, viz., at the equinoxes, and under the Moon's action its rate of
motion would vanish twice a month, viz., when the Moon crossed
the equator. At all other times the couples tend to produce retro-
grade never direct motion of the pole about K. Hence the pre-
cessional motion can never vanish unless the Sun and Moon should
happen to cross the equator simultaneously.
SECTION IY. Lunar and Planetary Perturbations.
471. In consequence of the universality of gravitation,
every body in the solar system has its motion more or less
disturbed by the attraction of every other body. Kepler's
Laws (with the modification of the Third Law given in 421 )
would only be strictly true if each planet were attracted
solely by the Sun, and each satellite described its relative
orbit solely under the attraction of its primary. Hence the
fact that these laws very nearly agree with the results of
observation shows that the mutual attractions of the planets
are small compared with that which the Sun exerts on each
of them, and that, in the orbital motion of a satellite, by far
the greater part of the relative acceleration is due to the
attraction of the primary.
FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 407
472. Lunar Perturbations- "We have seen, in Section
I., that tEe Moon's motion consists of two component parts, a
monthly orbital motion relative to the Earth or, more strictly,
relative to the centre of mass of the Earth and Moon and
the annual orbital motion of this centre of mass in an ellipse
about the Sun. If the acceleration of the Sun's attraction
were the same in magnitude and direction at the Moon as at
the Earth, it would be exactly the acceleration required to
produce the latter component, and the relative orbit about
the Earth would be determined by the Earth's attraction
alone. This is very nearly the case, owing to the great dis-
tance of the Sun. But the small differences of the
accelerations caused by the Sun's attraction on the Earth and
Moon tend to modify the relative motion of these two bodies,
by giving rise to perturbations ( 272). The relative
accelerations thus produced may be represented by a distri-
bution of disturbing force due to the Sun, just in the same
way that the relative accelerations of the oceans, which cause
the tides, are determined by distributions of disturbing force
due to the Sun and Moon. And since the Sun's distance is
nearly 400 times the Moon's, the expressions for ^the dis-
turbing force, corresponding to those investigated in 441,
are sufficiently approximate to account for the more impor-
tant lunar perturbations.
FIG. 161.
Let S, E, M denote the centres of the Sun, Earth, and
Moon. Drop M K perpendicular on ES, and on JSf produced
take KH 2EK. Then, if S denote the mass and r the dis-
tance of the Sun, the Sun's disturbing force produces at M a
relative acceleration along M H of magnitude JcS . MH/r*,
its components being Jc.S. MR ft* along MK and 2k . S . EK/r*
parallel to EK.
This force tends to accelerate the Moon towards the Earth
at quadrature (3f,), and away from the Earth at conjunction
and opposition (M w 3f 4 ) . At any other position it accelerates
408 ASTRONOMY.
the Moon towards a point (.fi^) in the line JZS, and thus
makes the Moon tend to approach the Sun, if its elongation
(Af^ZftS) is less than 90; but it accelerates the Moon
towards a point (7T 3 ) away from the Sun if its angle of
elongation from the Sun be obtuse.
473. The Rotation of the Moon's Nodes. Let CL
represent the ecliptic, JV^J/jJVi' the great circle which the
Moon would appear to describe on the celestial sphere if there
were no disturbing force acting upon it, and let 77", between
JVi and JV/on the ecliptic, represent either the Sun's position
on the celestial sphere or that of the point antipodal to it.
Then the reasoning of the last paragraph shows that the dis-
turbing force acts in the plane HEM V and therefore has a
component at M l directed along the tangent to the great
circle
Now let us suppose that the Moon is revolving under the
Earth's attraction alone, but that on arriving at J/j it is
acted on by a sudden impulse or blow directed towards H.
Clearly the effect of such an impulse is to bend the direction
of motion inward, from Mffl to J^JV^and the Moon will then
begin to describe a great circle Jfj.$i', which, if produced both
ways, will intercept the ecliptic at 'points N v N behind JVj,
JV/. The inclination of the orbit to the ecliptic will also be
diminished slightly if M l is within 90 of ^ ; for the exterior
angle MN^H > MN^ since the sides of the triangle
J/iJVyV 2 are each less than 90. But when the Moon comes
to My let another impulse act towards H. This will deflect
the direction of motion from M^N^ to M^, and the Moon
will now begin to describe the great circle NJil^N^ whose
nodes Ny N^ are still further behind their initial positions.
The inclination of the orbit to the ecliptic will, however, be
increased this time.
It is easy to see that the same general effect takes place
when the Moon is acted on by a continuous force, always
FURTHER AT-PLICATIOKS OP THE LAW OP GRAVITATION. 409
tending towards the ecliptic, instead of a series of impulses.
Such a force continuously deflects the Moon's direction of
motion, and draws the Moon down so that it returns to
the ecliptic more quickly than it would otherwise. Hence
the Moon, after leaving one node, arrives at the next hefore
is has quite described 180, and the result is an apparent
retrograde (never direct] motion of the nodes, combined with
periodic, hut small, fluctuations in the inclination of the
orbit.
*474. The retrograde motion of the Moon's nodes is, in some respects,
analogous to the precession of the equinoxes, and, although the analogy
is somewhat imperfect, the former phenomenon gives an illustration
of the way in which the latter is produced. If the Earth had a
string of satellites, like Saturn's rings, chosely packed together in a
circle in the plane of the equator, the Sun's disturbing force, ever
ac derating them towards the ecliptic, would, as in the case of the
Moon, cause a retrograde motion of the points of intersection of all
of their paths with the ecliptic, and this would give the appearance
of a kind of retrograde precession of the plane of the rings. If the
particles, instead of being separate, were united into a solid ring, the
general phenomena would be the same. And it is not unnatural to
expect that what occurs in a simple ring should also occur, to a
greater or less degree, in the case of other bodies that are somewhat
flattened out perpendicularly to their axis of rotation, such as the
Earth, thus accounting for the precession of the equinoxes. (Of
course this is only an illustration, not a rigorous proof; in fact, if
the Earth were qiiite spherical it would behave very differently.)
FIG. 163.
*475 Perturbations due to Average Value of Eadial Disturbing
Force -Let d be the Moon's distance. Then, when the Moon is m
conjunction or opposition, the Sun's disturbing force acts away from
the Earth, and is of magnitude 2k8d/* (Fig 163) When the
Moon is in quadrature the disturbing force acts towards the B.
but is only half as great. Hence, on the average, the disturbing
force tends to pull the Moon away from the Earth.
In consequence, the Moon's average centrifugal force must be
rather less than it would be at the same distance from the Eartl ,f
there were no disturbing force, and the effect of this is^ to make the
month a little longer than it would be otherwise for the same c
tance of the Moon.
410 ASTRONOMY.
Moreover, the disturbing force increases as the Moon's distance
increases, but the Earth's attraction diminishes, being proportional
to the inverse square of the distance ; this has the effect of making
the whole average acceleration along the radius vector decrease more
rapidly as the distance increases than it would according to the law
of inverse squares. The result of this cause is the progressive
motion of the apse line. It is difficult to explain this in a simple
manner, but the following arguments may give some idea of how
the effect takes place. At apogee the Moon's average acceleration
is less, and at perigee it is greater than if it followed the law of
inverse squares and had the same mean value. Hence, when the
Moon's distance is greatest, as at apogee, the Earth does not pull
the Moon back so quickly, and it takes longer to come back to its
least distance, so that it does not reach perigee till it has revolved
through a little more than 180. Similarly, at perigee the greater
average acceleration to the Earth does not allow the Moon to fly
out again quite so quickly, and it does not reach apogee till it has
described rather more than 180. Hence, in each case, the line of
apsides moves forward on the whole.
*476. Variation, Evection, Annual Equation, Parallactic
Inequality. When the Moon is nearer than the Earth to the Sun
(M } , Fig. 162), the Moon is more attracted than the Earth, and
therefore the disturbing force is towards the Sun ( 472). Its
effect is, therefore, to accelerate the Moon from last quarter to con-
junction, and to retard it from conjunction to first quarter. When
the Moon is more distant than the Earth from the Sun (A/ 3 , Fig. 163),
it is less attracted than the Earth, and therefore the disturbing
force is away from the Sun. Thus the Moon is accelerated from
first quarter to full Moon, and retarded from full Moon to last
quarter.f Hence we see that the Moon's motion in each case
must be swiftest at conjunction and opposition, and slowest at the
quadratures. This phenomenon is known as the Variation.
The force towards the Earth is greatest at the quadratures, and
least at the conjunction and opposition, since at the former the Sun
pulls the Moon towards, and at the latter away from the Earth.
Either cause tends to make the orbit more curved at the quadratures
and less curved at the syzygies. For, if v is the velocity, R the
radius of curvature, then v*]R = normal acceleration. Hence R is
greatest, and the orbit therefore least curved, when v is greatest,
and the normal acceleration is least. The effect of this cause would
be to distort the orbit, if it were a circle, into a slightly oval curve,
which would be most flattened, and therefore narrowest (compare
f These retardations and accelerations are closely analogous to
those of the water in an equatorial canal ( 445).
-PUHTHEB APPLICATIONS OF THE LAW OF GRAVITATION. 411
arguments of 114, 115), at the points towards and opposite the
Sun ; most rounded, and therefore broadest, at the points distant
90 from the Sun.
Of course the Moon's undisturbed orbit is not really circular, but
elliptic, and far more elliptic than the oval into which a circular
orbit would be thus distorted. But a distortion still takes place, and
gives rise to periodical changes in the eccentricity, depending on the
position of the apse line, and known as evection.
The Sun's disturbing force is greatest when the Sun is nearest,
and least when the Sun is furthest. These fluctuations, between
perihelion and aphelion, give rise to another perturbation, called the
whose most noticeable effect consists in the con-
sequent variations in the length of the month ( 475).
If, instead of resorting to a first approximation, we employ more
accurate expressions for the Sun's disturbing force on the Moon, it
is evident that this force is greater when the Moon is near con-
junction than at the corresponding position near opposition ; just
as the disturbing force which produces the tides is really greater
under the Moon than at the opposite point. Hence the Moon is
mere disturbed from last quarter through new Moon to first quarter
than from first quarter through full Moon to last quarter. Hence
the time of first quarter is slightly accelerated, and that of last
quarter retarded. This is called the Moon's Parallactic Inequality.
Its amount is proportional to TcScP/r*, instead of fcSeF/r^like the
other perturbations). For many reasons this perturbation is of
considerable use in determinations of the Sun's mass and distance.
477. Planetary Perturbations. The Sun's mass is so
great, compared with the masses of the planets, that the
orbital motion of one planet about the Sun is but slightly
affected by the attraction of any other planet. The mutual
attractions of the planets, and their actions on the Sun, give
rise to small planetary perturbations, which cause each
planet to diverge slowly from its elliptical orbit, besides
accelerating or retarding its motion.
Since the orbital motions of the planets are all usually
referred to the Sun as their common centre or " origin," nnd
not to the centre of mass of the solar system, the perturba-
tions of one planet, due to a second, depend, not on the actual
acceleration produced by the latter, but on the differences of
the accelerations which it produces on the former planet and on
the Sun.
As in the case of the Moon, the force which produces this
difference of accelerations is called the disturbing force.
ASTRON. 2 E
412 ASTRONOMt.
*478. Geometrical Construction for the Disturbing Force. The
approximate expressions, investigated in 472, for _the Sun s dis-
turbing force on the Moon, are inapplicable to the disturbing force
of one planet on another, because the distance of the distorting body
from the Sun is no longer very large, compared with that of the
disturbed body. We must, therefore, adopt the following con-
struction (Fig. 164) :
Let P, Q be two planets, of masses Jf, If ; 8 the Sun. Then the
planet P produces an acceleration fcM/PQ 2 on Q along QP, and an
acceleration IcM/PS* on 8 along SP. To find the acceleration of Q,
relative to 8, due to this cause, take a point T on PQ such that
PT : P3 = PS 2 : PQ 2 . Then the accelerations of 8, Q, due to P, are
fcjf . SP//SP 3 and TcM . TP/SP 3 respectively. Hence, by the triangle of
accelerations, the acceleration of Q, relative to 8, is represented^ in
magnitude and direction by fclf .TS/SP 3 . Therefore the disturbing
force per unit mass on Q, due to P, is parallel to T/8, and of magni-
tude TeM. TS/SP*.
FIG. 164.
Similarly, if we take a point 2* on QP such that QT : Q8
= QS 2 : QP 2 , the disturbing force per unit mass on P, due to Q, is
parallel to T'8, and is of magnitude JcM' . T'S/SQ 3 .
The disturbing force on Q, due to P, and that on P, due to Q, are
not equal and opposite, because they depend on the planets' attrac-
tions on 8, as well as on their mutual attractions.
When PQ = P/8, the points Q, T evidently coincide, and the dis-
turbing force on Q is along the radius vector QS. When PQ < P8,
PT>PQ, so that the disturbing force on Q tends to pull Q about 8
(as in Fig. 164) towards P, and when PQ>PS, the disturbing force
tends to push Q about 8 away from P.
JTTETHER APPLICATIONS OF THE LAW OP GBAVITATION. 413
f 7 f ' * h i? distlu * bin S f on P is along PS.
1*2 PUU P about 5 toward Q. an * when
push P about fl owa^/ from Q.
*479. Periodic Perturbations on an Interior Planet.-Let us con-
sider, m the first place, the perturbations produced by one planet
E on another planet F, whose orbit is nearer the Sun- as for
example, the perturbations produced by the Earth on Venus by
Jupiter or Mars on the Earth, or by Neptune on Uranus.
Let A, B be the positions of the planet, relative to E, when in
heliocentri^ conjunction and opposition respectively; U, IT points
on the relative orbit such that EU = EU' = E8. (These points are
near but not quite coincident with the positions of greatest elonga-
tion. ) Then, if we only consider the component relative acceleration
of F perpendicular to the radius vector Fflf, this vanishes when the
planet is at U or 17, as shown in the last paragraph.
FIG. 165.
The tangential acceleration also vanishes at A and B. Over the
arc U'AU the relative acceleration is towards E, therefore the planet's
orbital velocity is accelerated from U' to A j similarly it is retarded
from A to U.
Again, at a point F 2 on the arc UBU', the relative acceleration is
away from the Earth, and this accelerates the planet's orbital
velocity between U and B, and retards it between B and U'.
It follows that V is moving most swiftly at A and B, and most
slowly at U and U'. Hence, if we neglect the eccentricity of the
orbit, we see that the planet, after passing A, will shoot ahead of the
position it would occupy if moving uniformly; thus the disturbing
force displaces the planet forwards during its path from A to near
U. Somewhere near U, when the planet is moving with its least
velocity, it begins to lag behind the position it would occupy if
moving uniformly; thus from near Uto B the disturbing force dis-
places the planet backwards. Similarly, it may be seen that from B
to near U' the planet is displaced forwards, and from near U ' to A
it is displaced backwards.
414
ASTRONOMY.
The principal effect of the component of the disturbing force
along the radius vector, is to cause rotation of the planet's apsides,
as in the case of the Moon. The direction of their rotation depends
on the direction of the force, and is not always direct. The eccen-
tricity of the orbit is also affected by this cause, as in the phenome-
non of lunar evection, and the periodic time is slightly changed.
Owing to the inclination of the planes of the orbits of E, V, the
attraction of E, in general, gives rise to a small component perpen-
dicular to the plane of F's orbit, which is always directed towards
the plane of E's orbit. This component produces rotation of the line
of nodes, or line of intersection of the planes of the two orbits. This
rotation is always in the retrograde direction, and is to be explained
in exactly the same way as the rotation of the Moon's nodes.
It is thus a remarkable fact that since all the bodies in the solar
system (except the satellites of Uranus and Neptune) rotate in the
direct direction, all the planes of rotation and revolution, and all
their lines of intersection (i.e., the lines of nodes, and the lines of
equinoxes) in the whole solar system, with the above exceptions,
have a retrograde motion.
A
FIG. 166.
*480. Periodic Perturbations of an Exterior Planet. The accele-
rations and retardations produced by a planet E on one J, whose
orbit is more remote from the Sun, during the course of a synodic
period, may be investigated in a similar manner to the corre-
sponding perturbations of an interior planet, assuming the orbits to
be nearly circular.
If SJ is less than 2SE there are two points if, N on the relative
orbit at which EM = EN = E8. At these points the disturbing
torce is purely radial, and it appears, as before, that the planet J is
lerated from heliocentric conjunction A to M, and from helio-
Ce ^ r i C T PPOSition B to N ' retarded from N to A, and from M to B.
c, ' then E8<EA > hence the attraction of E is greater on
the hun than on J, and the disturbing force therefore always accele-
rates the planet J towards B. Thus the planet's orbital velocity
increases from A to B, and decreases from B to A, and it is greatest at
a and least at A. Therefore from B to A the planet is displaced in
FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 415
advance of its mean position, and from A to B falls behind its mean
position.
The effects of the radial and orthogonal components of the dis-
turbing force in altering the period and causing rotation of the apse
line, and regression of the nodes, can be investigated in the same
way for a superior as for an inferior planet.
*481. Inequalities of Long Period. If the orbits of the planets
were circular (except for the effects of perturbations), and in the
same plane, their mutual perturbations would be strictly periodic,
and would recur once in every synodic period. Owing, however, to
the inclinations and eccentricities of the orbits, this is not the case.
The mutual attractions of the planets produce small changes in the
eccentricities and inclinations, and even in their periodic times,
which depend on the positions of conjunction and opposition relative
to the lines of nodes and apses. Neglecting the motion of these
latter lines, the perturbations would only be strictly periodic if the
periodic times of two planets were commensurable ; the period of
recurrence being the least common multiple of the periods of the
two planets. But when the periodic times of two planets are nearly
but not quite in the proportion of two small whole numbers, inequali-
ties of long period are produced, whose effects may, in the course
of time, become considerable.
Thus, for example, the periodic times of Jupiter and Saturn are
very nearly but not quite in the proportion of 2 to 5. If the propor-
tionality were exact, then 5 revolutions of Jupiter would take the
same time as 2 revolutions of Saturn 5 and, since Jupiter would thus
gain three revolutions on Saturn, the interval would contain 3
synodic periods. Thus, after 3 synodic periods had elapsed from
conjunction, another conjunction would occur at exactly the same
place in the two orbits, and the perturbations would be strictly
periodical.
But, in reality, the proportionality of periods is not exact; the
positions of every third conjunction are very slowly revolving in
the direct direction. They perform a complete revolution in
2,640 years. But there are three points on the orbits at^which con-
junctions occur, and these are distant very nearly 120 from one
another. It follows that when the positions of conjunction have
revolved through 120, they will again occur at the same points on
the orbits, and the perturbations will again be of the same kind as
initially. The time required for this is one-third the above period,
or 880 years, and consequently Jupiter and Saturn are subject to
lono-- period inequalities which recur only once in 880 years.
Again, the periodic times of Venus and the Earth are nearly in
the proportion of 8 to 13 ; consequently 5 conjunctions of Venus occur
in almost exactly 8 years, thus giving rise to perturbations havinj
a period of 8 years. But the proportion is not exact, and, consequently,
there are other mutual perturbations having a very long period.
416 ASTBOtfOMY.
One of the most important secular perturbations is the alternate
increase and decrease in the eccentricity of the Earth's orbit. This,
at the present time, is becoming gradually more and more circular,
but in about 24,000 years the eccentricity will be a minimum, and
will then once more begin to increase. The effects of this cause on
the climate of the Earth's two hemispheres have already been
considered ( 463).
482. Gravitational Methods of Finding the Sun's
Distance. The Earth's perturbations on Mars and Venus
furnish a good method of finding the Sun's distance. For
the magnitude of these perturbations depends on the ratio of
the Earth's mass, or rather the sum of the masses of the
Earth and Moon (since both are instrumental in producing
the perturbations), to the Sun's mass. Hence, if S, M, m
denote the masses of the Sun, Earth, and Moon, it is possible,
from observations of these perturbations, to find the ratio of
(M+m) : S.
But, if r, d be the distances of the Sun and Moon from the
Earth, T and Fthe length of the sidereal lunar month and
year, we have, by Kepler's corrected Third Law,
(M+m) T* : (S+M+m) Y z = d 3 : r 8 ;
whence the ratio of r to d is known. If, now, the Moon's
distance d be determined by observation in any of the ways
described in Chapter VIII., or by the gravitational method
of 423, the Sun's distance r may be immediately found.
This method was used by Leverrier in 1872. From obser-
vations of certain perturbations of Venus he found the values
8-853" and 8-859" for the Sun's parallax, while the rotation
of the apse line of Mars gave the value 8'866".
The perturbations of Encke's comet were used in a similar
way by Von Asten, in 1876, to find the Sun's parallax, the
value thus obtained being rather greater, viz., 9-009".
The lunar perturbations also furnish data for determining
the Sun's distance, the principal of these being the parallactic
inequality of the Moon ( 476). Several computations of the
Sun's parallax have thus been made, the results being 8 -6"
by Laplace in 1804, 8-95" by Leverrier in 1858, 8-838" by
Newcomb in 1867. See also 437 for the determination of
the parallax from the apparent monthly displacement of the
Sun.
FUETHEB APPLICATIONS OF THE LAW OF GRAVITATION. 417
483. Determination of Masses. The mass of any
planet which is not furnished with a satellite can be deter
mined in terms of the Sun's mass by means of the perturba-
tions it produces on the orbits of other planets. The
amount of these perturbations is always proportional to the
disturbing force, and this again is proportional to the mass
of the disturbing planet. In this manner the mass of Yenus
has been found to be about 1/400,000 of the Sun's mass, and
that of Mercury about 1/5,000,000.
484. The Discovery of Neptune. The narrative of the
discovery of Neptune is one of the most striking and remark-
able in the annals of theoretical astronomy, and forms a fitting
conclusion to this chapter.
In 1795, or about 14 years after its discovery, the planet
Uranus was observed to deviate slightly from its predicted
position, the_ observed longitude becoming slightly greater
than that given by theory. The_ discrepancy increased till
1822, when Uranus appeared to undergo a retardation, and
to again approach its predicted position. About 1830 the
observed and computed longitudes of the planet were equal,
but the retardation still continued, and by 1845 Uranus had
fallen behind its computed position by nearly 2'.
As early as 1821, Alexis Bouvard pointed out that these
discrepancies indicated the existence of a planet exterior to
Uranus, but the matter remained in abeyance until 1846,
when the late Mr. (afterwards Prof.) Adams, in Cambridge,
and M. Leverrier, in Paris, independently and almost simul-
taneously, undertook the problem of determining the position,
orfcit, and mass oi_an unknown planet which would give rise
to the observed perturbations. Adams was undoubtedly the
first by a few months in performing the computations, but
the actual search for the planet at the observatory of Cam-
bridge was delayed from pressure of other work. Meanwhile
Leverrier sent the results of his calculations to Dr. Galle,
of Berlin, who, within a few hours of receiving them, turned
his telescope towards the place predicted for the planet, and
found it within about 52' of that place. Subsequent exami-
nation of star charts showed that the planet had been pre-
viously observed on several occasions, but had always been
mistaken for a fixed star.
418 ASTRONOMY.
It will be seen from 479 that the acceleration of Uranus
up to 1822, and its subsequent retardation, are ^at once
accounted for by supposing an exterior planet to be in helio-
centric conjunction with the Sun about the year 1822. But
Adams and Leverricr sought for far more accurate details
concerning the planet. At the same time the data afforded
by the observed perturbations of Uranus were insufficient to
determine all the unknown elements of the new planet's
orbit, and therefore the problem admitted of any number of
possible solutions. In other words, any number of different
planets could have produced the observed perturbations.
To render the problem less indeterminate, however, both
astronomers assumed that the disturbing body moved nearly
in the plane of the ecliptic and in a nearly circular orbit,
that its distance and period were connected by Kepler's
Third Law, and that its distance from the Sun followed
Bode's Law. The latter assumption led to considerable
errors, including an erroneous estimation of the planet's
period by Kepler's Third Law. For when Neptune was
observed, its distance was found to be only 30 '04 times the
Earth's distance, instead of 38-8 times, as it would have been
according to Bode's Law. Nevertheless, the actual planet
was subsequently found to fully account for all the observed
perturbations of Uranus.
The discovery of Neptune affords most powerful evidence
of the truth of the Law of Gravitation, and so indeed does
the theory of perturbations generally. The fact that the
planetary motions are observed to agree closely with theory,
that computations of astronomical constants (such as the
Sun's and Moon's distances), based upon gravitational methods,
agree so closely with those obtained by other methods, when
possible errors of observation are taken into account, affords
an indisputable proof that the resultant acceleration of any
body in the solar system can always be resolved into com-
ponents directed to the various other bodies, each__cpmponent
boingju:oportional directly to the mass and inversely to the
sijuare of the distance of the corresponding body. Such a
truth cannot be regarded as a fortuitous coincidence ; it can
only be explained by supposing every body in the universe
to attract every other body in accordance with Newton's Law
of Universal Gravitation.
FUKTHEK APPLICATIONS OF THE LAW Qlf OKAV1TATION.
EXAMPLES. XIY.
1. If the Sun's parallax be 8'80", and the Sun's displacement at
first quarter of Moon 6'52", calculate the mass of the Moon, the
Earth's radius being taken as 3,963 miles.
2. Supposing the Moon's distance to be 60 of the Earth's radii,
and the Sun's distance to be 400 times that of the Moon, while his
mass is 25,600,000 times the Moon's mass, compare the effects of
the Sun and Moon in creating a tide at the equator, in the event of
a total eclipse occurring at the equinox.
3. If the Earth and Moon were only half their present distance
from the Sun, what difference would this make to the tides ? Cal-
culate roughly what the proportion between the Sun's tide-raisng
power and the Moon's would then be, assuming the Moon's distance
from -the Earth remained the same as at present.
4. Taking the Moon's mass as -^ of the Earth's, and its distance
as 60 times the Earth's radius, show that the Moon's tide-raising
force increases the intensity of gravity by 1/17,280,000 when the
Moon is on the horizon, and that it decreases the intensity of gravity
by 1/8,640,000 when the Moon is in the zenith.
5. Compare the heights of the solar tides on the Earth and on
Mercury, taking the density of Mercury to be twice that of the
Earth, its diameter "38 of the Earth's diameter, and its solar distance
38 of the Earth's solar distance.
6. Explain how the pushing forward of the Moon by the tidal
wave enlarges the Moon's orbit.
7. Show that, owing to precession, the right ascension of a star
at a greater distance than 23| from the pole of the ecliptic will
undergo all possible changes, but that a star at a less distance than
23^ will always have a right ascension greater than twelve hours.
8. Prove that for a short time precession does not alter the decli-
nations of stars whose right ascensions are 6h., or 18h.
9. Exhibit in a diagram the position of the pole star (R.A.
= Ih. 20m., decl. = 88 40 7 ) relative to the poles of the equator and
ecliptic, and hence show that owing to precession its R.A. is increas-
ing rapidly, but that its polar distance is decreasing.
10. Describe the disturbing effects of Neptune on Uranus for a
short time before and after heliocentric conjunction, pointing out
when Uranus is displaced in the direct, and when in the retrograde
direction.
420 A.STRONOMr.
EXAMINATION PAPER. XIV.
1. Show that the Moon's orbit is everywhere concave to the Sun.
2. Show that the tide-raising force of a heavenly body is nearly
proportional to its (mass) -f- (distance) 3 .
3. How is it that we have tides on opposite sides of the Earth at
once ?
4. Explain the production of the tides on the equilibrium theory.
5. Define the terms spring tide, neap tide, priming and lagging ,
establishment of the port, lunar time.
6. What is meant by the expression " Luni-solar Precession" ?
Describe the action of the Sun and of the Moon in causing the
Precession.
7. Give a general description of Precession. Does precession
change the position of (a) the equator, (6) the ecliptic among the
stars ?
8. Describe nutation. What is the cause of Lunar Nutation?
What is meant by the equation of the equinoxes ?
9. Give a brief account of the discovery of Neptune.
10. Explain how the retrograde motion of the Moon's nodes is
caused bv the Sun's attraction on the Earth and Moon.
NOTE I.
DIAGRAM FOE SOUTH LATITUDES.
In order to familiarize the student with astronomical
diagrams drawn under different conditions, we subjoina/ftgure
showing the principal circles of the celestial splndre of an
observer in South latitude 45 at about 19h. of JSereal time
(QWRr = 270+15 = 19h.). The figure Jnows also the
Sun's daily paths at t^e" solstices ; also the arcs T^R^= QM,
and MX, which measure the E.A. and N. decl. of the star x.
R"
fi
NADIR
FIG. 169.
N.POLE
NOTE II.
THE PHOTOCHEONOGRAPH.
Quite recently photography has been applied to recording
transits, as an alternative for the methods explained in
Chap. II., 49, 50. The image of the observed star is
422 ASTRONOMY.
jjf ejected on a sensitized plate placed in front of the transit
circle, and, owing to the diurnal motion, it moves horizontally
across the plate. The plate is made to oscillate slightly in a
vertical direction, by means of clockwork, say once in a
second, and this motion, combined with the horizontal motion
of the image, causes it to describe a zigzag or wavy streak on
the plate. The star's position at each second is indicated by
the undulations, and the position of these is capable of being
measured with great exactness.
NOTE III.
NOTE ON 104.
It may be proved, by Spherical Trigonometry, that
sin nP = sin xP sin nxP, or sin I = cos d sin nxP ,
cos 2 d cos 2 nxP = cos 2 d cos 2 d sin 2 nxP = cos 2 d sin 2 1
= cos (d + 1} cos (d I) ;
acceleration t = -
15
U"
15 </{ cos (<M-Z) cos (<?/)}
The same formula is applicable to 135, 190.
sees.
APPENDIX.
PROPERTIES OF THE ELLIPSE.
For the benefit of'those readers who have not studied Conic
Sections, we subjoin a list of those properties of the ellipse which
are of astronomical importance. The proofs are given in books on
Conic Sections.
B
APSE
FIG. 168.
1. DEFINITION. A conic section is a curve such that the distance
of every point on it from a certain fixed point is proportional to its
perpendicular distance from a certain fixed straight line.
The fixed point is called the focus, the fixed line is called the
directrix, and the constant ratio of distances is called the eccen-
tricity.
If this constant ratio or eccentricity is less than unity, the curve
is called an ellipse. In this case the curve assumes the form of a
closed oval, as shown in the figure.
If 8 is the focus, and if from A, P, L, P r , A', &c., any points on
the curve, perpendiculars AX, PM, &c., be drawn on the directrix,
and if the eccentricity be e, the definition requires that
= 8 A _ 8P_ = SL_ = SP' = -8 A' = .
AX ~ PM LK P'M' A'X
and that e is less than unity.
The other conic sections, the parabola and hyperbola, are defined
by the same property, save that in the former e = 1, and in the
latter e > 1 ; but they are of little astronomical importance, except
as representing the paths described by non-periodic comets.
424
ASTRONOMY.
2. An ellipse has two foci (each focus having a corresponding
directrix), and the sum of the distances of any point from the two
foci is constant.
Thus in Fig 169, 8, Hare the two foci, and the sum 8P + PH is
the same for all positions of P on the curve.
From this property an ellipse may easily be drawn. For, let two
small pins be fixed at 8 and H, and let a loop of string SPH be passed
over them and round a pencil-point P ; then, if the pencil be moved so
as to keep the string tight, its point P will trace out an ellipse.
For SP + PH +H8 = constant, and /. SP + PR = constant.
3. For all positions of P on the ellipse, SP is inversely propor-
tional to 1 -t e cos ASP, so that
SP(l + ecos^4SP) = I = constant,
e being the eccentricity and 8A the line through 8 perpendicular
to the directrix.
>r
4. The line joining the two foci is perpendicular to the directrices.
The portion of this line (AA'), bounded by the curve, is called the
major axis or axis major. Its middle point C is called the centre,
and the^ curve is symmetrical about this point.
The line BCB', drawn through the centre perpendicular to AC A'
and terminated by the curve, is called the minor axis or axis minor.
The lengths of the major and minor axes are usually denoted by
2a and 26 respectively.
5. The extremities A, A' of the major axis are called the apses
or apsides. Since, by (2), 8P + HP is constant, therefore, taking
P at A or A', SP + HP = SA + HA = 8 A' + HA'
= $(SA + HA + SA' + HA') evidently
= AA' = 2a.
Taking P at B, SB + HB = 2a ;
.'. SB (evidently) = HB = a = CA.
PBOPERTIES OF THE ELLIPSE. 425
6. The eccentricity e = 08/CA ; /. CS = e . CA, and
52 = 02 = SB 2 - CiS 2 (Euc. I. 47) = a*-a 2 e 2 = a 2 (1-e 2 ) j
Hence also
S4= (L4-CS = a(l-e) and 8 A' = CA'
7. The latus rectum is the chord LSL' drawn through the focus
perpendicular to the major axis AA'. Its length is 21, where
I = a (Ift-e 2 ). Also I is the constant of (3), for when P coincides
with L, ASP = 90; .'. cos ASL = 0, and 8L = I. [Fig. 168.]
8. The tangent fPT and normal PGg, at P, bisect respectively
the exterior and interior angles (SPI, 8PH) formed by the lines
JSP, HP.
9. If the normal meets the major and minor axes in G, g,
PO : Pg = OB 2 : CA 2 (= 6 2 : a 2 ).
10. If ST, drawn perpendicular on the tangent at P, meets HP
produced in I, then evidently SP IP ;
.'. HI = SP + HP = 2a [by (2)].
If HT' is the other focal perpendicular on the tangent, it is known
that rectangle 8T . HT' = constant = 6 2 .
11. Relation between the focal radius SP and the focal perpen-
dicular on the tangent ST.
Let SP = r, 8T = p.
Then cos TIP = cos T8P = pfr.
By Trigonometry,
= I8* + IH--2. 18. IH. cos SIHi
4aV = 4j> 2 + 4a 2 - 8pa x p/r ;
2 (1-e 2 ) 2a
This may also be proved from the similarity of the triangles
8PT } HPT', which gives 8T : HT' = SP : HP ;
/. ST 2 : ST.HT = )S(P : .HP and ST.HF = b 2 (10) j
.\ p 2 : b 2 =r : 2a-r.
12. If a circular cone (i.e., either a right or oblique cone on a
circular base) is cut in two by a plane not intersecting its base, the
curve of section is an ellipse. More generally, the form of a circle
represented in perspective, or the oval shadow cast by a spherical
globe or a circular disc on any plane, are ellipses. A circle is a
particular form of ellipse for the case where 6 = a and /. e = 0.
13. The area of the ellipse is -nab.
426 ASTRONOMY.
TABLE OF ASTRONOMICAL CONSTANTS.
(Approximate values, calculated, when variable, for the Spring
Equinox, A.D. 1900.)
THE CELESTIAL SPHERE.
Latitude of London (Greenwich Observatory), 51 28' 31",
Cambridge Observatory, 52 12' 51".
Obliquity of Ecliptic, 23 27' 8",
OPTICAL CONSTANTS.
Coefficient of Astronomical Eefraction, 57".
Horizontal Eefraction, 33'.
Coefficient of Aberration, 20'493".
Velocity of Light in miles per second, 186,330.
metres 299,860,000.
Equation of Light, 8m. 18s
TIME CONSTANTS.
Sidereal Day in mean solar units = 1 l/366days = 23h. 56m.4'ls.
Mean Solar Day in sidereal units = 1 + 1/365| days = 24h. 3m. 56'5s.
Year, Tropical, in mean time, 365d. 5h. 48m. 45'51s.
Sidereal, 365d. 6h. 9m. 8'97s.
Anomalistic, 365d. 6h. 13m. 48'09s.
Civil, if the number of the year is not divisible by 4,
or if it be divisible by 100, bnt not by 400, 365 days.
In other cases, 366
Month, Sidereal, 27'32166d. = 27d. 7h. 43m. 11 '4s.
Synodic, 29'53059d. = 29d. 12h. 44m. 30s.
Metonic Cycle, 235 Synodic Months = 6939'69d
= 19 tropical years (all but 2 hours).
Period of Botatiou of Moon's Nodes (Sidereal), 6793'391d. = 18'60yr.
(Synodic), 346'644d.
= 346d. 14Jh.
Apsides (Sidereal), 3232'575d. = 8'85yr,
(Synodic), 411'74d.
Saros 223 Synodic Months = 6585'29d. = 18*0906 yr,
= 18 yr. 10 or 11 days.
= 19 Synodic periods of Moon's Nodes (very nearly/
= 16 Apsides (nearly).
Equation of Time, Maximum due to Eccentricity, 7m.
Obliquity, 10m,
TABLE OF ASTRONOMICAL CONSTANTS.
427
Equatorial Circumference,
THE EARTH.
Equatorial Eadius, 3963-296 miles.
Polar 3949-791
Mean 3959"!
22,902
360 x 60 = 21,600 geographical miles.
4 x 1Q7 = 40,000,000 metres.
Ellipticity or Compression, l-f-293.
Eccentricity, '0826.
Density (Water = 1), 5'58.
Mass, 6067 x 10 18 tons.
Mean Acceleration of Gravity in ft. per sec. per sec., 32-18.
Eatio of C entrif ugal Force to Gravity at Equator, 1 -. 289.
Eccentricity of its Orbit, l-s-60.
Annual Progressive Motion of Apse Line, H'25".
Eetrograde Motion of Equinoxes (Precession), 50-22".
Period of Precession, 25,695 years.
Nutation, 18'6
Greatest change in Obliquity due to Nutation, 9'23".
Equation of Equinoxes, 15' 37".
THE SUN.
Mean Parallax, 8'80".
Angular Semi-diameter, 16' 1".
Distance in miles, 92,800,000.
Diameter in miles, 866,400.
in Earth's radii, 109.
Density in terms of Earth's, ^.
(taking water a& lj, 1 J 4.
Mass in terms' of Earth's, 324,439.
Period of Axial Eotation, 25d. 5h. 37m.
THE MOON.
Mean Parallax, 57' 2707".
Angular Semi-diameter, 15' 34".
Distance in miles, 238,840,
in Earth's radii, 60'27.
in terms of Sun's distance, 1/389.
Diameter in miles, 2,162.
in terms of Earth's, 3/11.
Density in terms of Earth's, '61.
(taking water as 1), 3'4.
Mass, in terms of Earth's, 1/81.
Eccentricity of Orbit, 1/18.
Inclination of Orbit to Ecliptic, 5 8'.
Ecliptic Limits, Lunar, 12 5' and 9 30'.
Solar, 18 31' and 15 21'.
Tide-raising force in terms of Sun's, 7/3.
ASTRON. 2 F
ANSWERS.
NOTE. Where only rough values of the astronomical data are
given in the questions, the answers can only be regarded as rough
approximations, not as highly accurate results. It is impossible to
calculate results correctly to a greater number of significant figures
than are given in the data employed, and any extra figures so
calculated will necessarily be incorrect. As the use of working
examples is to learn astronomy rather than arithmetic, it is ad-
visable to supply from memory the rough values of such astronomi-
cal constants as are not given in the questions. These values will
thus be remembered more easily than if the more accurate values
were taken from the tables on pages 426, 427, though reference to
the latter should be made until the student is familiar with them.
I. EXAMPLES (p. 33).
1. Only their relative positions are stated; these do not completely
fix them.
2. 6 P.M., 6 A.M.; on the meridian. 8. On September 19.
9. (i.) Early in July ; (ii.) middle of June the Sun passes it about
June 26.
10. 304 = 20h. 16m.; at 8h. 13m. P.M.
11. Near the S. horizon about 10 P.M. early in October.
12. 38 27', 51 33', 28 5', or if Sun transits N. of zenith 8 27',
81 33', 58 5'.
I. EXAMINATION PAPER (p. 34).
7. 30. 8. 61 58' 37", 15 4' 21". 9. 6h. 43m. 16s. (roughly).
10. The figure should make Capella slightly W. of N., altitude about
15; o Lyras a little S.E. of zenith, altitude about 75;
a Scorpii slightly W. of S., altitude about 12; o Ursse Hajoris
N.W., altitude about 60.
ANSWERS. 429
II. EXAMPLES (p. 61).
. Direct. 7. Interval = 12 sidereal hours. 9. 2 3 29' 58'5".
11. 12 39' 9". 12. I7h. 29m. 52'42s.
II. EXAMINATION PAPER (p. 62).
6. Positive. 1O. lrn.2'52s., + 0718.
III. EXAMPLES (p. 84).
2. 4,267ft.
3. aN., L-90W. and o S., L + 90W., if L = W. longitude
given place.
5. 13m. 6. 39-8 miles. 7. 3960.
8. 6084ft. 10. 49' 6" per hour.
MISCELLANEOUS QUESTIONS (p. 85).
2. N.P.D. = 85, hour angle = 30 W.
3. Because declination circle has not been defined.
5. 22h. 40m., 9h. 20m., 14h. Om., 19h. 36m.. 10. 52".
' V
III. 1 EXAMINATION PAPER (p. 86).
1. 24,840 miles, 3,953 miles.
2. 3-285 ft., 6,084 ft., T69ft. per second. 3. 507 ft.
5. 3,437,700 fathoms, 6,366,200 metres (roughly), 1,851-851 metres.
9. See 97, cor.
IV. EXAMPLES (p. 113).
5. 45. 7. Star, 6h. 15m. 26'35s. ; Sun, Oh. 13m. 51'90s.
10. 3481 : 3721, or 29 : 31 nearly.
IV. EXAMINATION PAPER (p. 114).
3. See 130, 151.
3. Oh. 36m. 21'26s. (Note that the clock has a losing rate of
3m. 22'05s. on sidereal time ; it gives solar time approxi-
mately.)
V. EXAMPLES (p. 137).
1. Retrograde. 3. 3'9m. 6. 347 centuries exactly.
7. Star's hour angle = 4h. llm. 3s., N.P.D. = 53.
8. October 28, 15h. 39m. 27'32s.
1O. 12h. 27m. 13'26s. at Louisville = 18h. 9m. 13'26 at Greenwich.
430 ASTRONOMY.
MISCELLANEOUS QUESTIONS (p. 138).
3. Eastward. 5. Use Figs. 47, 50. 6. See 439.
7. See 161. 8. llh. 59m. 15'9s. ; - 1m. 7'4s.
9. 366-25 : 365'25 or 1465 : 1461.
V. EXAMINATION PAPER (p. 139).
4. - 10m. ; morning 20m. longer. 5. See 172.
8. (i.) 7h. 13m. 5s. ; (ii.) 7h. 12m. 48s. 9. June 26.
10. 1824, 1852, 1880, 1920.
VI. EXAMPLES (p. 151).
3. 3,963 miles.
4. From 50 9' 47" to 49 59' 55" (refraction at altitude 5 - 9' 47"
by tables).
5. 44 53' 28". 8. 84 33' ; 377 miles or 327 nautical miles.
VI. EXAMINATION PAPER (p. 152).
4. 462". 7. 44 58' 54". 1O. Ih. 12m.
VII. EXAMPLES (p. 188).
1. 37 49'. 2. 51 44' 26-09*.
4. 50 54' 58'6" or 60 43' 23'6" according aa star transits N. or S.
of zenith.
5. 44 55', or, if corrected for refraction (cf. Ex. 2, p. 168),
44 53' 54".
6. 51 33', 38 27', 61 54'. 8. - 10m. } i.e., 10m. fast.
9. 12 30'. 1O. Ih.Om. 11. 2 32'. 12. 27'.
13. See 237. 18. Lat. = cos- l -fr = 87 54' nearly.
VIII. EXAMPLES (p. 217).
2. 92,819,000 (see Ex. 2, p. 195).
3. At 6 p.m. ; about same length as Midsummer Sun, i.e., 16|h.
4. See 261. 5. 8' 48". 6. Use 266.
7. lOd. 4h. at noon.
8. Gibbous, bright limb turned slightly below direction of W.
Hour angle = 30, decl. = 0.
10. (i.) No harvest moon ; (ii.) Phenomena practically unaltered.
ANSWERS, 431
VIIL EXAMINATION PAI^ER (p. 218).
4. See 260. 7. 71 33". 9. When we have a solar eclipse.
IX. EXAMPLES (p. 236).
1. 23|S.
2. Favourable if moon passes from N. to S. at ecliptic on March 21.
4. 4m. 38s. 5. Length = (Earth's radius) -~ sin (8 P).
7. 6h. 32m. if month unaltered; or, by 329, a lunation = about
10 days, and then time = 2h. 10m.
8. 40 Earth's radii = 158,000 miles (roughly).
9. Total Solar. 1O. 128' (cf. 291).
IX. EXAMINATION PAPER (p. 237).
6. 850,000, 230,000, and 5,800 miles (roughly).
7. See 292, 295-297. 9. No.
10. In Fig. 93 take M on xm produced, such that sin xM = #m/(p - P).
X. EXAMPLES (p. 265).
1. 291'96 days, or, if conjunctions are of the same kind, 583'92 days.
2. 40. 3. 19 : 6, or nearly 3:1. 4. IQi^h., 120h.
5. p + P s with notation of 290. 6. 888 million miles, 164 yrs.
7. 6 months ; Vi or '63 of Earth's mean distance. 8. 398 days.
9. f of a year = 137 days.
10. Stationary at heliocentric conjunction only, never retrograde.
X. EXAMINATION PAPER (p. 266).
3. i_i_ years = 378 days.
4. See 323, 324. The alterations in Venus's brightening are
really not inconsiderable (see Ex. 3, p. 205).
6. Most rapid approach at quadrature j velocity that with which the
Earth would describe its orbit in synodic period.
9. 287 days.
10. Draw the circular orbits about , radii 4, 7, 10, 16, 52 ( 304).
The heliocentric longitudes (measured from Q T ) are roughly
as follows: $153, ? 175, 0220, <? 20, ^211. The
C should be drawn close to at an elongation C 90
at first quarter.
432
XI. EXAMPLES (p. 311).
2. 432,000 miles. 3. 2,250 miles.
6. 9,282,000 and 92,820,000 million miles respectively.
7. 37'8 billion miles - 378 x 10 n miles.
8. 5 : TT = 1'6 : 1 roughly.
10. It will always appear half-way between its actual direction and
a point on the ecliptic 90 behind Sun. Path is roughly a
small circle of angular radius 45.
11. 4 35'.
13. (i.) On ecliptic 90 from Sun. (ii.) In same or opposite direc-
tion to Sun. Effects greatest along great circles distant 90
from these points.
14. (i.) At either pole of ecliptic, (ii.) In ecliptic.
16. Jan. 21, 10'25" Eastwards; Feb., 17-75" E. ; Mar., 20'50" E. ;
April, 17-75" E. ; May, 10'25" E. ; June, 0" ; July, 10'25"
Westwards; Aug.,l7'75"W.j Sept.,20'50"W.; Oct., 1775" W. ;
Nov., 10-25" VV. ; Dec., 0".
18. 973,800 miles.
MISCELLANEOUS QUESTIONS (p. 313).
5. 15 E. 6. In the autumn.
7. 17d. 5h. ; star is on equator, hour angle 60 E. 8. 1 : v7 : 7.
9. 24h. 50m. 30s. mean units = 24h. 54m. 35s. sidereal units.
10. At the equinoxes. 11. See 376.
XII. EXAMPLES (p. 335).
1. 12 Ve sidereal hours = 16h. 58m. 5s. sidereal time.
2. Pendulum revolving in direction of hands of watch will have
less velocity in S. hemisphere.
7. Increased (i.) 59 54' 51"; (ii.) 60 15' 27". 12. 109 Ibs.
XII. EXAMINATION PAPER (p. 336).
3. By observing deviation of a projectile ( 390), or by 387 or 389.
4. 16 V3 = 27-7157 sidereal hours = Id. 3h. 33m. mean time.
5. 3*368 cm. per sec. per sec. ; ,$ T . 9. See 3PO.
ANSWERS. 433
XIII. EXAMPLES (p. 368).
1. 2-97 miles per sec. 2. 15 ft., or, if g = 32'2, 15'576ft.
3. 5h.35m. 4. 5-39 days. 5.2,959,000. 11. 8'98".
13. The distances from the centre of the Sun are 457,579 miles,
457,579 -H 278 milesj and 457,579 - 281 miles ; but these results
can only be considered as approximate.
14. 32'155 greater, owing to attraction of mountain.
17. '253 of Earth's density ; T415, taking water = 1.
18. 894 poundals.
20. At first a hyperbola under the Earth's attraction. After going
some distance this attraction would become insensible, and
the Moon would describe an ellipse about the Sun rather
more eccentric than the Earth's present orbit.
XIV. EXAMPLES (p. 419).
3. 24 : 7, by Ex. 1, 442 Cor., or 16 : 5, using result of last example.
5. Tide on Mercury is higher in proportion 1 : '2888, or 45 : 13, or
7 : 2 nearly.
1O. Direct shortly before, retrograde shortly after.
XIV. EXAMINATION PAPER (p. 420),
7. (a) Yes; (6) No.
UNIVERSITY
INDEX.
(The numbers refer to the pages throughout.)
Aberration of Light, 295; cor-
rection for aberration deter-
mined, 298 ; its general effect
on the celestial sphere, 299;
jomparison with annual paral-
lax, 300 ; to show that the
aberration curve of a star is an
ellipse, 301 ; its discovery by
Bradley, 302 ; the constant de-
termined by observation, 302 ;
relation between the coefficient
of aberration and the equation
of light, 304 ; relation between
the coefficient of aberration
and the Sun's parallax, 310.
diurnal, 308 ; its effect on
meridian observations, 309 ;
determination of its coefficient
by observations of the azimuths
of stars on the horizon, 310.
planetary, 306.
Altazimuth, 54.
Altitude, 8.
Angular diameter, 8.
distance, 3.
measure, its conversion to
time, 14.
velocities of planets, to com-
pare, 342.
Annual equation, 411.
Anomalistic year, 127.
Aphelion, 111.
Apogee, 106, 210.
Apparent area, 105, 109;
Moon's phase, 204.
midnight, 24.
motion of a planet, 258.
noon, 24.
solar day, 24.
solar time, its disadvantages
115.
Apparent Sun, 117.
.pse, 106.
Moon's, 210, 410.
line, 106, 111, 210; deter-
mination of its position, 109;
its progressive motion, 109, 211,
414.
Arctic and Antarctic circles, 88.
Areal velocity, 343 ; relation be-
tween areal velocity and actual
(linear) velocity, 344.
Aries, first point of, 7j to find,
99, 100; retrograde motion of
(see Precession}.
ARISTARCHUS : his method of
finding the Sun's distance, 205.
Asteroids, 240.
Astronomical clock, 13, 36.
diagrams : their practical
application, 28.
telescope, 36.
terms, table of, 12.
Astronomy defined, 1 ; its prac-
tical uses, 153.
Descriptive, Gravitational,
and Physical, defined, 1.
Autumnal equinox, 21.
Azimuth, 8.
Bar, double, 78.
Base line, measurement of, 78. ^
BESSEL : his method of determin-
ing the annual parallax of a star,
290 ; his day numbers, 403.
Binary stars, 292.
Black drop, 279.
BODE'S Law, 239.
BRADLEY : his discovery of aber-
ration, 302 ; his discovery of
nutation, 403; his determina-
tion of refraction, 146.
INDEX.
Calendar, Julian, 128; Gregorian
correction, 128.
month, 200.
ardinal points, 7.
CASSINI: his formula of refrac
tion, 145.
CAVENDISH: his experiment for
finding the Earth's mass, 362.
Celestial equator, 6.
horizon, 5.
latitude, 10.
longitude, 10.
meridian, 6.
poles, G.
sphere, 2.
entre of mass, 355.
'Centrifugal force, 324; its effects
on the acceleration of falling
bodies, 325 ; loss of weight of
a body due to it, 326
'Ceres, 241.
Chronograph, 43 ; photo-, 421
Chronometer, 160; its error and
rate, 161.
Circle, of position, 187 ; transit, 38.
Circumpolar stars, 16; determi
nation of latitude by, 167.
Civil Year, 128.
Clock, astronomical, 13, 36.
error and rate, 44, 45.
stars, 45, 398.
'Colatitude, 11.
Collimating Eyepiece, 49.
Collimation, error, 46.
line of, 39.
Colures, 23.
Compass, points of, 9.
Conjunctions, 200, 245.
Coordinates : their use explained,
8 ; advantages of the different
systems, 11 ; table of, 12
transformation of, 16.
Culmination, 16.
485
Day and night, relative lengths,
89 92.
lunar, 382.
mean, 117.
numbers, Bessel's, 403.
perpetual, 92.
sidereal, 13.
Declination, 9, 10; name of, 9-
expressed in terms of latitude
and meridian Z. D., 15 ; deter-
mination of the Sun's, 23-
method of observing, 51.
Declination Circle, 9, 56.
DELISLE : his method of deter-
mining the Sun's parallax, 271
Density of a heavenly body its
determination, 366.
Dip of horizon; defined, 73- its
determination, 74, 75; its effect
on tho times of rising and
setting, 76, 422.
Direct motion, 22.
Disappearance of a ship at sea 75
Diurnal motion of the stars 5.
aberration, 308
Double bar, 78.
Day, apparent solar, 24; explana-
tion of gam or loss of a clay in
going round the world, 72.
Earth : early observations of its
form, 63; general effects of
change of position on it 64-
its rotation, 64; measurement
of its radius, 67; A. R. Wai-
Jace's method of finding its
radius, 77; ordinary methods
of finding its radius, 78; its
exact form, 81 ; determination
of its equatorial and polar
n,^; its exact dimensions,
; its mean radius, 83; its
ellipticity or compression, 83
its eccentricity, 88 , its zones)
88 ; determination of the eccen-
tricity of its orbit, 107- its
phases, 206; its place in the
solar system, 240; its rotation,
315; arguments in favour of
its rotation, 316; dynamical
proofs of its rotation, 317
436
INDEX.
Earth (continued) : general
effects of its centrifugal force,
324; its figure, 328; evidence
in favour of its annual motion
round the Sun, 337; verification
of the law of gravitation, 356 ;
its so-called "weight," 362 ; the
Cavendish experiment, 362 ;
the mountain method of finding
its mass, 366 ; its mass deter-
mined by the common balance,
364; its mass determined by
observations of the attraction
of tides in estuaries, 365 ; the
pendulum method of finding its
mass, 365 ; its displacement
due to the Moon, 371 ; its rota-
tion retarded by tidal friction,
388 ; precession of its axis, 396 ;
nutation of its axis, 400.
Earth's way, 299.
Eclipses, 219 et seqq. ; different
kinds of lunar E., 220 ; effects
of refraction on lunar E., 150,
221 ; different kinds of solar E.,
222 ; determination of greatest
or least number possible in a
year, 229 ; of Jupiter's satellites,
241 ; their retardation, 293.
Ecliptic, 7, 20, 99, 111 ; its obli-
quity, 11 ; determination of its
obliquity, 26, 104.
Ecliptic limits, 226, 228.
Ellipse, properties of, 423.
Elongation, 200, 244 ; changes of
E. of planet, 244, 246.
Equation, Annual, 411.
of equinoxes, 402.
of light, 293 ; its relation to
the coefficient of aberration,
304.
of time, 117 ; due to un-
equal motion, 118 ; due to ob-
liquity, 119; its graphical
representation, 121 ; it vanishes
1 four times a year, 122 ; its
maximum values. 123 ; its de-
termination, ]24.
Equation, personal, 46.
Equator, celestial, 6.
terrestrial, 64.
Equatorial, 56 ; its use, 57.
Equinoctial colure, 23.
points, 7, 20, 23.
time, 134.
Equinoxes, 20, 21, 23 ; precession
of, 103.
Evection, 411.
Fathom, 67.
First point of Aries, 7, 20; its
determination, 100.
First point of Libra, 7, 20.
FLAMSTEED : his method of deter-
mining the first point of Aries,
100; advantages of the method,
102.
FOUCAULT : his pendulum experi-
ment, 318 ; his gyroscope, 321 ;
his determination of the velo-
city of light, 293.
Full Moon, 203.
Geocentric latitude, 83, 112.
longitude, 112.
lunar distances, 180.
parallax : its general effects >
192; correction for, 192.
Geodesy, 77.
Geographical latitude, 83.
mile, 67.
Gibbosity of Mars, 252.
Gibbous Moon, 203.
Globes : their use, 3.
Gnomon, 25, 125.
Golden Number, 215.
Gravitation : Newton's law of>
352 ; remarks, 353 ; verification
for the Earth and Moon, 356.
Gravity: to compare its intensity
at different places, 329, 334;
to find its value, 334.
GREGORY, Pope : his correction
of the Julian Calendar, 128.
Gyroscope or Gyrostat, 321, 395.
INDEX.
43?
HALLET : his method of deter-
mining the Sun's parallax by ob-
serving a transit of Venus, 271.
Harvest Moon, 216.
Heliocentric latitude, 112.
longitude, 112.
Heliometer, 59.
Horizon, celestial, 5; artificial,159,
visible, 5, 73-76.
- dip of, 73.
Horizontal parallax, 191.
point, 50.
Hour angle, 9 ; expressed in time,
13; its connection with right
ascension, 15.
circle, 56.
Instruments for meridian obser-
vations, 35 ; for ex-meridian
observations, 54; for geodesy,
78-80 ; for navigation, 153.
Introductory Chapter on Spheri-
cal Geometry, i.-vi.
JULIUS C^SAR : his calendar, 128.
Juno, 241,269.
Jupiter, 241 ; its satellites, 241.
KATER'S reversible pendulum, 329.
KEPLER : his laws of planetary
motion, 106, 111, 253 ; verifica-
tion of his first law, 107, 254 ;
verification of the second law,
108, 254 ; deductions from the
second law, 109; verification
of the third law, 256 ; Newton's
deductions from his laws, 339,
345, 346, 348 ; his third law for
circular orbits, 340 ; correction
of the third law, 354.
Knot, 68.
Known star, 15, 45.
Lagging of the tides, 383-5.
Latitude of a place defined, 10;
phenomena depending on
change of latitude, 65 ; change
due to ship's motion, 72.
Latitude (continued) : determi-
nation by meridian observa-
tions, 162 ; determination by
ex-meridian observations, 169.
celestial, 10.
geocentric, 83, 112.
geographical, 83.
heliocentric, 112.
parallel of, 71 ; length of
any arc of a given parallel, 71-
Leap year, 128.
Libra, first point of, 7.
Light, refraction of, 140 ; its velo-
city, 293; aberration of, 295
to find the time taken by the
light from a star to the Earth,.
305.
Light-year, 305.
Local time: its determination,.
171.
Log-line : its use in navigation,.
68.
Longitude, celestial, 10.
geocentric, 112.
heliocentric, 112.
terrestrial, 69 ; phenomena
depending on change of terres-
trial longitude, 70 ; change due-
to ship's motion, 72; its deter-
mination at sea, 177; the
method of lunar distances, 179;.
clearing the distance, 179 ; its
determination by celestial
signals, 181 ; its determination,
on land, 182 ; its determination
by transmission of chronome-
ters, 182; by chronograph, 184^
by terrestrial signals, 185; by
Moon-culminating stars, 186;.
bv Captain Sumner's method,.
187,
Loop of retrogression, 261.
Lunar distances, determination of
longitude by, 179.
geocentric, 180.
mountains: determination of
their height, 207.
Lunation. 27.
488
INDEX.
Mars, 240 ; Kepler's observations
on Mars, 254 ; its parallax used
to determine that of the Sun,
268.
Mass, astronomical unit of, 352.
Mean noon, 117.
solar day, 117.
solar time, 117 ; its deter-
mination at a given instant of
sidereal time, 132.
Sun, 116, 117.
time, 116.
Mercury, 239 ; its period of rota-
tion, 264 ; frequency of its
transits, 282; its mass, 360, 417.
Meridian, celestial, 6.
line : its determination, 175.
prime, 69.
terrestrial, 64.
it Meteors : their motion, 4.
Metonic cycle, 215.
Metre, 67.
Micrometers, 58.
Midnight, apparent, 24.
Mile, geographical, 67.
nautical, 67.
Moon : its motion, 27 ; its age, 27 ;
itsposition denned by its centre,
53; illusory variations in its
size, 149 ; method of taking its
altitude by the sextant, 158;
determination of its parallax,
196 ; its distance, 197 ; its dia-
meter determined, 199; its
elongation, 200 ; determination
of its synodic period, 201 ; its
phases, 202; relation between
phase and elongation, 204 ; its
use in finding the Sun's dis-
tance, 205; its appearance
i-elative to the horizon, 206;
determination of the height of
lunar mountains, 207 ; its orbit
about the Earth, 209; eccen-
tricity of its orbit, 210; its
nodes, 210; its perturbations,
210, 407 ; retrograde motion
of its nodes, 211, 408, 409.
Moon (continued) : progressive
motion of its apse line, 211, 410 ;
its rotation, 212 ; its librations,
213; general effects of libra-
tion, 214 ; its eclipses, 219-221 ;
determination of its geocentric
distance consistent with an
eclipse, 224; its greatest lati-
tude at syzygy consistent with
an eclipse, 226; synodic revo-
lution of its nodes, 228; its
occupations, 232 ; verification
of the law of gravitation, 356;
effect of its attraction, 357 ; its
mass, 357 ; concavity of its path
about the Sun, 374 ; its disturb-
ing or tide-generating force,
375, 377 ; its orbital motion
accelerated by tidal friction,
388 ; its form and rotation,
391 ; its disturbing couple on
the Earth, 392 ; the rotation of
its nodes, 408; its other in-
equalities, 410, 411.
Nadir, 5.
point, determination of, 49.
Nautical mile, 67.
Neptune, 243 ; its discovery, 417.
New Moon, 27.
NEWTON, Sir ISAAC : his deduc-
tions from Kepler's laws, 339,
345, 346, 348 ; his law of uni-
versal gravitation, 352.
Nodes, 27, 210 ; their retrograde
motion, 211.
North polar distance of a circuin-
polar star, 17.
Number of eclipses in year, 229.
Nutation, lunar, 401 ; its general
effects, 402 ; its discovery, 403 ;
to correct for, 403 ; its physical
causes, 404.
monthly, 406,
solar, 405.
Obliquity of ecliptic, 11 j its de
termination, 26.
INDEX.
439
Observatory, 35.
Occultafcions, 232.
Offing, 73.
Opposition, 200.
Parallactic inequality, 411.
Parallax, 179, 191; geocentric
parallax, 191; horizontal paral-
lax, 191 ; general effects of
and correction for geocentric
parallax, 192 ; relation between
horizontal parallax and dis-
tance of celestial body, 194;
compared with refraction, 195;
parallax of Moon determined,
196 ; parallax of planet deter-
mined, 198 ; relation between
parallax and angular diameter,
199 ; determination of the
Sun's parallax, 268 et seqq.;
annual parallax denned, 283;
to find the correction for
annual parallax, 284; relation
between the parallax and dis-
tance of a star, 285 ; its general
effects on the position of a star,
286; determination of the an-
nual parallax of a star, 290.
Pendulum, Foucault's, 318 ; Cap-
tain Kater's reversible, 329 j
oscillations of a simple pen-
dulum, 330; to find the change
in the time of oscillation due
to a variation in its length or
in the intensity of gravity,
330; to compare the times of
oscillation of two pendulums of
nearly equal periods, 333 ; pen-
dulum method of finding the
Earth's mass, 365.
Perigee, 106, 210.
Perihelion, 111.
Perpetual day : determination of
its length, 97.
Personal equation, 46.
Phases of Moon, 202 ; of planet,
251, 252.
Perturbations, lunar, 210, 407;
rotation of nodes, 408 ; due to
average value of radial disturb-
ing force, 409; variation, evec-
tion, annual equation and
parallactic inequality, 410, 411.
planetary, 411 ; periodical,
413, 414 ; inequalities of long
period, 415 ; secular, 416.
Photography, stellar, 60, 421.
Planet : its position defined by
centre, 53; determination of
its parallax, 198 ; its occulta-
tion, 235 ; definition, 238 ; in-
ferior and superior planets,
244 ; changes in elongation of
a inferior planet, 244 ; to find
the ratio of the distance from
the Sun of an inferior planet
to that of the Earth, 246;
changes in elongation of a
superior planet, 247; to com-
pare the distance from the Sun
of a superior planet with that
of the Earth, 248 ; determina-
tion of the synodic period of an
inferior planet, 249; relation
between the synodic and side-
real periods of a planet, 250 ;
phases of the planets, 251, 252 ;
motions relative to stars, 258 ;
transits of inferior planets,
271; its aberration, 306, 307;
to compare the velocities and
angular velocities of two planets
moving in circular orbits, 342 ;
having given the velocity of a
planet at any point of its orbit,
^ to construct the ellipse de-
scribed under the Sun's attrac-
tion, 350 ; to find the mass of a
planet which has one or more
satellites, 359; its perturba-
tions, 411 j masses determined,
417.
Points of the compass, 9*
Polar distance, 9-
point : its determination, 51.
440
INDEX.
Pole, celestial, 6.
terrestrial, 64.
Port, establishment of the, 386.
Precession of the equinoxes, 103,
392.
Earth's axis, 396.
a spinning-top, 395.
luni-solar, 393 ; to apply the
corrections for, 397 ; various
effects of, 398; its effects on
the climate of the Earth's
hemispheres, 400.
Prime vertical, 7.
Prime vertical instrument : deter-
mination of latitude by its use,
170.
Priming of the tides, 383-5.
-Quadrature, 200.
Hadiant, 4.
Eeading microscope, 40.
Refraction, 140; laws of R.,
140; relative index of E., 140;
general description of atmo-
spherical R., 141 ; its effect
on the apparent altitude of a
star, 141 ; law of successive
R., 142; formula for astro-
nomical R., 142 ; Cassini's for-
mula, 145; coefficient found
by meridian observations, 146 ;
other methods of determination,
147 ; its effects on rising and
setting, 148; effects on dip and
distance of horizon, 149 ; effects
on lunar eclipses and occulta-
tions, 150, 221 ; comparison of
R. with parallax, 195.
Retrograde motion, 22, 258.
Right ascension, 10 ; expressed in
time, 14 ; connection with hour
angle, 15.
ROEMER : his method of finding
the velocity of light, 293.
dotation of "Earth, 64, 315 ; of
Moon, 212; of Moon'snodes,211,
408 ; of Sun and planets, 264.
Saros of the Chaldeans, 231.
Satellite, defined, 238 ; their obe-
dience to Kepler's laws, 257.
Saturn, 242 ; phases of its rings,
252.
Seasons, 94 ; effect of the length
of day on temperature, 94 ;
other causes affecting tempera-
ture, 94; unequal length of, 109.
Secondary, iii., 238.
Sextant, 154 ; its errors, 157 ;
determination of theindex error,
157 ; method of taking altitudes
at sea, 158 ; method of taking
altitudes of Sun or Moon, 158.
Sidereal day, 13.
month, 200 ; its relation to
the synodic month, 200.
noon, 13.
period, 200, 250.
time, 13, 25 ; its disadvan-
tages, 115 ; its determination
at a given instant of mean solar
time, 131 ; its determination at
Greenwich or in any longitude,
133.
year, 127.
Solar day, apparent, 24.
system, tabular view of, 243
its centre of mass, 361.
time, 24 ; its disadvantages,
115.
Solstices, 21, 23.
Solstitial colure, 23.
points, 23.
Southing of stars, 16.
Spectrum analysis, 60.'
Stars : independence of their di-
rections relative to observer's
position on the Earth, 4 ; their
diurnal motion, 5, 13; culmi-
nation, 16 ; southing, 16 ; cir-
cumpolar stars, 16 ; rising and
setting, 18 ; time of transit, 19 ;
to show that a star appears to
describe an ellipse, owing to
parallax, 287 ; owing to aber-
ration, 301.
INDEX.
441
Stars, morning and evening, 25.
Stationary points, 258 ; their de-
termination, 262, 263.
Sub-solar point, 187.
Summer solstice, 21.
and winter, causes of, 94.
SUMNER, Captain : his method of
finding longitude, 187.
Sundial, 125 ; geometrical method
of graduation, 126.
Sun : its annual motion, 7 ; its
annual motion in the ecliptic,
20 ; its motion in longitude,
right ascension and declination,
20, 21 ; its variable motion in
right ascension, 22 ; determi-
nation of its right ascension and
declination, 23, 24 ; its position
defined by its centre, 53 ; its
diurnal path at different sea-
sons and places, 88 ; to find
length of time of sunrise or
sunset, 98 ; observations of its
relative orbit, 105 ; its apparent
area, 105, 109 ; its apparent
annual motion accounted for,
110; illusory variations in size,
149 ; method of finding its alti-
tude by the sextant, 158 ; diffi-
culty of finding its parallax, 197;
its distance determined by
Aristarchus, 205 ; solar eclipses,
219, 222, 234; description, 238 ;
its period of rotation, 264 ; de-
termination of its distance from
the Earth, 268 et seqq. ; its paral-
lax determined by observation
of the parallax of Mars, 268;
parallax by observations on the
asteroids and Venus, 269; paral-
lax determined by observations
-of the transit of Venus, 271 et
$eqq.; advantages and disadvan-
tages of Halley's and Delisle's
methods, 280 ; relation between
coefficient of aberration, Sun's
parallax, and velocity of light,
306.
Sun (continued) : to find the ratio
of its mass to the Earth's, 358 :
gravity on its surface, 366 ; its
parallax determined by observa-
tions of lunar and solar displace-
ments of the Earth, 373; its
disturbing or tide-generating
force, 375, 377 ; its mass com-
pared with that of the Moon,
from observations of the rela-
tive heights of the solar and
lunar tides, 388 ; its disturbing
couple on the Earth, 392; gravi-
tational methods of finding its
distance, 416.
Synodic month, 200.
period, 200, 250.
Syzygy, 200.
Telescope, astronomical, 37.
Terrestrial equator, 64.
longitude, 69.
meridian, 64.
pole, 64.
Theodolite, 79.
Tidal constants, 387.
friction, 388 ; application to
the solar system, 392.
Tides, 375 ; equilibrium theory of
their formation, 379; canal
theory, 380; semi - diurnal,
diurnal, and fortnightly tides
due to the Moon, 383; semi-diur-
nal, diurnal, and six-monthly
tides due to the Sun, 383; spring
and neap tides, 383_j their
priming and lagging, 383-
385 ; establishments of ports
386.
Time: its reduction to circular
measure, 14 ; relation between
the different units, 129, 134.
equinoctial, 134.
local : its determination by
method of equal altitudes, 171,
172.
lunar, 382.
Trade winds, 323.
442
LtfDEX.
Transit, 14 ; eye and ear method
of taking transits, 42; of Venus,
271-282"; of Mercury, 282.
circle, 38 ; corrections re-
quired for right ascension, 44 ;
corrections required for decli-
nation, 49.
Triangulation, 79.
Tropics, 88.
Tropical year, 127.
True Sun, 117.
Uranus, 242.
Variation, 410.
Venus, 240; its period of rota-
tion, 264; observations of its
transit used to determine the
Sun's parallax, 271 ; determi-
nation of the frequency of its
transits, 281; its mass, 360,
417.
Velocity, angular, 342.
area!, 343.
of light, 293.
Velocities of planets compared,
342.
Vernier, 157.
Vernal equinox, 20,
Vertical, 7.
circle, 7.
prime, 7.
Vesta, 2-40.
WALLACE, ALFRED RUSSELL : hi
method of finding the Earth's-
radius, 77.
Waning and waxing Moons, 203.
Winter solstice, 21.
Year, 20.
anomalistic, 127.
civil, 128.
' leap, 128.
sidereal, 127.
synodic, 128.
tropical, 127.
Zenith, 5.
distance, 8.
point, 51.
sector, 8&
Zodiac, 25.
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THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO SO CENTS ON THE FOURTH
DAY AND TO $1.OO ON THE SEVENTH DAY
OVERDUE.
UNIVERSITY .QF.CALIFORNIA LIBRARY