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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 PP7, 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 PQP7 revolves about PP', A traces out. the great circle HQRK, of which P, P7 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 0 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 0 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 0 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 0 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 0° 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 0° 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 0° 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 xl (Fig. 13) be any star not on the meridian. Then
z QpXl = 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 Mxl equal to the star's declinatiqn. We thus find the star xr
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 ZPxt, and we have to determine Zxi and L QZx} (= ISO0— 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 = nP—naf,
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 — tf—T.
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 0° 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 zl and s2 respectively, and let I be the latitude of the place of observation. From § 24, we readily see that 2t = l-i, 22 = 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-|-0, 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-0 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 0 }) = 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 pIG 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°-2.3°27' = G60 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, azimutht polar distance, hour angle, right ascension, declination, longitude, latitude. Which of these coordinates alwa3Ts 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, 0 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 0 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 0 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 0 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 " 30—31—32 — 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' 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° 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' 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°— (x—Z\ 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 investiga4 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 0 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 0 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° 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 CpPlt therefore, angle ZOP = OCp,
.'. altitude of pole at 0 = W°—ZOP = 90°- OCp = qCO. But the latitude of 0 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 0 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. 0 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 yards = 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 0° to 180°, not from 0° 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 Ty£ 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 0 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 0 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 0 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 0 is the inclination to the horizontal plane of a tangent from 0 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 0 = 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, OT2 = OA . OB
This determines d accurately. But in practical applications h is always very small compared with 2a ; therefore A2 may be neglected in comparison with 2ah, and we have the approxi- mate formula, rf2 = 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 0 = nd2 = 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 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 J£L, .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 ^Ss 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, Gt 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 0 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, supposing1 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 rv rz be the required radii of curvature at Q and P respectively.
Take any point 0 on the ellipse, and let the normal at 0 meet the two axes in G and g respectively.
It is proved in treatises on Conic Sections* that OG : Og = CP* : C& = i2 : a\
First take 0 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 0 very near to P. to I and Og to r%.
Therefore, I : r2 = W : «2; Thus rx, r2 are found in terms of a,
r =
Then 0 G will become equal
r =
and I may be found ; I — r*r.
Conversely, if r, and r2 are known, for, by solving, we find a = %/(rfr
~We notice that since a > J, .*. r^ < rr
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 Ll = 0s (I-*2) = 08(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 = 62 : a2; .'. NO/CN = (NOJON) x (&2/o-) { /. tan I' = tan I x &2/a2 = (1- e2) tan I.
We deduce also tan (l-lf) = — ^ S1^ 2^2 = ie"sin2l (approx.), since e2 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 50°N., 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 0 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
zIUC=18Qx6Qx6QJL 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 0 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 0 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 0 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 = l—d.
"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' = i—l.
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 ' = l—i, 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°— t—Z 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 cMe,
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 — -^Dn sec d, and at an equinox in latitude ?,
t = TV 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