Professor A. 0. Leuschner 1868-1953

Gift of Dr.Erida Leuschner Rei chert

Great Double Telescope, Visual and Photographic, of the Potsdam Astrophysical Observatory. (See Sec. 51.)

MANUAL OF ASTRONOMY

A TEXT-BOOK

BY

CHARLES A. (YOUNG, PH.D., LL.D.

LATE PROFESSOR OF ASTRONOMY IN PRINCETON UNIVERSITY

AUTHOR OF " THE SUN " AND OF A SERIES OF

ASTRONOMICAL TEXT-BOOKS

GINN AND COMPANY

BOSTON • NEW YORK • CHICAGO • LONDON ATLANTA • DALLAS • COLUMBUS • SAN FRANCISCO

ENTERED AT STATIONERS' HAT/T,

COPYRIGHT, 1902 BY CHARLES A. YOUNG

ALL RIGHTS RESERVED 224.1

GIFT

(gftc athenaeum jgregg

G1NN AND COMPANY • PRO- PRIETORS • BOSTON • U.S.A.

PEEFACE

THE present volume has been prepared in response to a rather pressing demand for a text-book intermediate between the author's Elements of Astronomy and his Greneral Astronomy. The latter is found by many teachers to be too large for convenient use in the time at their disposal, while the former is not quite sufficiently extended for their purpose.

The material of the new book has naturally been derived largely from its predecessors; but everything has been care- fully worked over, rearranged and rewritten where necessary, and changed and added to in order to bring it thoroughly up to date.

The writer is under great obligations to many persons who have assisted him in various ways, especially to Professor Anne S. Young, of the astronomical department in Mt. Holyoke College, who has carefully read and corrected all the proof. He is greatly indebted also to D. Appleton & Co. for permission to use illustrations from The Sun, to Warner & Swasey for photographs of astronomical instruments, and to numerous other friends who have kindly furnished material for engravings. Among these may be mentioned specially Pro- fessor Pickering of the Harvard Observatory, the lamented Keeler, and Professors Campbell and Hussey of the Lick

i"

Ss880816

iv PREFACE

Observatory, and Professors Hale, Frost, and Barnard of the Yerkes, besides several others to whom acknowledgment is made in the text.

The volume speaks for itself as to the skillful care of printers and publishers in securing the most perfect mechan-

ical execution.

C. A. YOUNG

PRINCETON, N.J.

PREFACE TO ISSUE OF 1912

IN the present issue a number of more or less important errata have been corrected, and various changes and additions nave been made, required by the recent rapid progress of

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MAY, 1912

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TABLE OF CONTENTS

PAGES

INTRODUCTION 1-5

CHAPTER I. — PRELIMINARY CONSIDERATIONS AND DEFINI- TIONS — Fundamental Notions and Definitions — Astronom- ical Coordinates and th« " Doctrine of the Sphere " - - The Celestial Globe — Exercises 6-31

CHAPTER II. — ASTRONOMICAL INSTRUMENTS — Telescopes, and their Accessories and Mountings — Timekeepers and Chronographs — The Transit-Instrument — The Prime Ver- tical Instrument — The Almucantar — The Meridian-Circle and Universal Instrument — The Micrometer — The Heliom- eter — The Sextant — Exercises 32-65

CHAPTER III. — CORRECTIONS TO ASTRONOMICAL OBSER- VATIONS — Dip of the Horizon — Parallax — Semidiameter

— Refraction — Twinkling or Scintillation — Twilight — Exercises 66-75

CHAPTER IV. — FUNDAMENTAL PROBLEMS OF PRACTICAL ASTRONOMY — Latitude — Time — Longitude — Azimuth

— The Right Ascension and Declination of a Heavenly

Body — Exercises 76-104

CHAPTER V. — THE EARTH AS AN ASTRONOMICAL BODY — Its Form, Rotation, and Dimensions — Mass, Weight, and Gravitation — The Earth's Mass and Density — Exercises . 105-135 CHAPTER VI. — THE ORBITAL MOTION OF THE EARTH - The Apparent Motion of the Sun and the Ecliptic — The Orbital Motion of the Earth — Precession and Nutation — Aberration — The Equation of Time — The Seasons and the

Calendar — Exercises 136-165

CHAPTER VII. — THE MOON — The Moon's Orbital Motion and the Month — Distance, Dimensions, Mass, Density, and Force of Gravity — Rotation and Librations — Phases — Light and Heat — Physical Condition — Telescopic Aspect and Peculiarities of the Lunar Surface . ... 166-194

vi TABLE OF CONTENTS

PAGES

CHAPTER VIII. — THE SUN — Its Distance, Dimensions, Mass, and Density — Its Rotation and Equatorial Accel- eration— Methods of studying its Surface — The Photo- sphere— Sun-Spots — Their Nature, Dimensions, Develop- ment, and Motions — Their Distribution and Periodicity — Sun-Spot Theories 195-216

CHAPTER IX. — THE SUN (Continued) — The Spectroscope, the Solar Spectrum, and the Chemical Constitution of the Sun — The Doppler-Fizeau Principle — The Chromo- sphere and Prominences — The Corona — The Sun's Light — Measurement of the Intensity of the Sun's Heat — Theory of its Maintenance — The Age and Duration of the Sun — Summary as to the Constitution of the Sun — Exercises 217-260

CHAPTER X. — ECLIPSES — Form and Dimensions of Shad- ows — Eclipses of the Moon — Solar Eclipses — Total, Annular, and Partial — Ecliptic Limits and Number of Eclipses in a Year — Recurrence of Eclipses and the Saros

— Occupations 261-274

CHAPTER XL — CELESTIAL MECHANICS — The Laws of

Central Force — Circular Motion — Kepler's Laws, and New- ton's Verification of the Theory of Gravitation — The Conic Sections — The Problem of Two Bodies — The Parabolic Velocity — Exercises — The Problem of Three Bodies and Perturbations — The Tides 275-310

CHAPTER XII. — THE PLANETS IN GENERAL — Bode's Law — The Apparent Motions of the Planets — The Elements of their Orbits — Determination of Periods and Distances — Perturbations, Stability of the System — Data referring to the Planets themselves — Determination of Diameter, Mass, Rotation, Surface Peculiarities, Atmosphere, etc. — Herschel's Illustration of the Scale of the System — Exercises . . . 311-345

CHAPTER XIII. — THE TERRESTRIAL AND MINOR PLANETS

— Mercury, Venus, and Mars — The Asteroids — Intra- mercurial Planets — Zodiacal Light 346-381

CHAPTER XIV. — THE MAJOR PLANETS — Jupiter : its Sat- ellite System; the Equation of Light, and the Distance of the Sun — Saturn : its Rings and Satellites — Uranus : its Discovery, Peculiarities, and Satellites — Neptune : its Dis- covery, Peculiarities, and Satellite — Exercises 382-408

TABLE OF CONTENTS vii

PAGES

CHAPTER XV METHODS OF DETERMINING THE PARAL- LAX AND DISTANCE OF THE SUN — Importance and Diffi- culty of the Problem — Historical — Classification of Methods

— Geometrical Methods — Oppositions of Mars and certain Asteroids, and Transits of Venus — Gravitational Methods . 409-421

CHAPTER XVI. — COMETS — Their Number, Designation, and Orbits — Their Constituent Parts and Appearance — Their Spectra — Physical Constitution and Behavior — Danger from Comets — Exercises 422-454

CHAPTER XVII. — METEORS AND SHOOTING-STARS — Aero- lites : their Fall and Physical Characteristics ; Cause of Light and Heat ; Probable Origin — Shooting-Stars : their Number, Velocity, and Length of Path — Meteoric Showers : the Radiant ; Connection between Comets and Meteors — Exercises 455-476

CHAPTER XVIII. — THE STARS — Their Nature, Number, and Designation — Star-Catalogues and Charts — The Photo- graphic Campaigns — Proper Motions, Radial Motions, and the Motion of the Sun in Space — Stellar Parallax — Exercises 477-505

CHAPTER XIX. — THE LIGHT OF THE STARS — Magnitudes and Brightness — Color and Heat — Spectra — Variable Stars — Exercises . . . 506-536

CHAPTER XX STELLAR SYSTEMS, CLUSTERS, AND NEBULA

— Double and Multiple Stars — Binaries — Spectroscopic Binaries — Clusters — Nebulse — The Stellar Universe — Cosmogony — Exercises 537-573

APPENDIX. — Transformation of Astronomical Coordinates — Projection and Calculation of a Lunar Eclipse — Greek Alphabet and Miscellaneous Symbols — Dimensions of the Terrestrial Spheroid — Time Constants and other Astro- nomical Constants 574-582

TABLE I. Principal Elements of the Solar System . . 583

TABLE II. The Satellites of the Solar System .... 584-585 TABLE III. Comets of which Returns have been observed 586

TABLE IV. Stellar Parallaxes, Distances, and Motions . 587

TABLE V. Radial Velocities of Stars 588

TABLE VI. Variable Stars 589

TABLE VII. Orbits of Binary Stars 590

TABLE VIII. Mean Refraction 591

INDEX ... , 592-611

ADDENDA TO MANUAL OF ASTRONOMY

Addendum A. SEC. 54. — In the instrument described in this section there is a considerable loss of light from the two reflections. A much simpler form with only one reflection, and with most of the advantages of the Coude, is now in use at Cambridge (England) for celestial photography. But it commands only the region between Declinations + 75° and — 30°.

Addendum B. SEC. 415. — Mr. Lowell has recently published an elabo- rate mathematical investigation of the temperature of Mars, with the fol- lowing results : mean temperature of the planet, 48° F. ; boiling point of water, 111° F.; density of air at the planet's surface, T\ of earth's, corre- sponding to a barometric height of 2J inches. The mean temperature ot the earth is usually taken as 60° F.

An expedition sent by him to northern Chili in charge of Professor Todd is reported to have obtained photographs of the planet showing many of the canals, and some of them double.

Addendum C. SECS. 418-421. — The present rate of asteroid discovery makes it impossible to keep up with it in a text-book. In 1906 more than 100 were found, and each succeeding year has added a large number to the list. Most of them are so faint as to be observable only by photog- raphy. Rev. J.H.Metcalf of Taunton, Massachusetts, has lately made a very effective modification of the Heidelberg method. While the telescope fol- lows the .stars by its driving-clock, the photographic plate receives a slight sliding motion, the same as that of an average asteroid in the region under observation. The image of a planet, if one happens to be present, remains therefore stationary on the plate, or nearly so, during the entire exposure, arid is many times more intense than if it had been allowed to trail. The stars do the trailing, and are easily recognized as such.

When first announced each new object is designated provisionally by two letters in an alphabetical arrangement : thus Eros was for a time known as « DQ," and already « 1907 ZZ " has been discovered. It is pro- posed to continue the same system, always, however, prefixing the year. When sufficient observations have been obtained to determine the planet's orbit, and its non-identity with any previously known, the Director of the Recheninstitut at Berlin assigns a permanent "number," and the

ADDENDA TO MANUAL OF ASTRONOMY

discoverer, if he chooses, gives it a name. Tn August, 1911, 714 asteroids had thus received " numbers," though many remained unnamed.

Among the recently discovered planets, TG (588), 1906 VY, and 1907 XM (all discovered at Heidelberg) are of special interest. They form a class by themselves, their orbits not differing greatly from that of Jupiter in size and period. They have already received the names of Achilles, Patroclus, and Hector. Their exact orbits are still only roughly deter- mined, but it is clear that the problem of their motion is peculiarly inter- esting, since they appear to present approximately the special case long ago pointed out by Lagrange, of a planet keeping permanently equidistant from the sun and Jupiter. 1908 CS, Nestor, belongs also to this group.

The asteroid Occlo (475), discovered in 1901, has an eccentricity of 0.38, even greater than that of ^thra. Planet 1906 WD has the enor- mous inclination of 48°.

Addendum D. SEC. 543. — Our determinations of the "Solar Apex" all depend on the assumption that the stars whose motions form the basis of the calculation are moving indiscriminately in all directions, so that in the general sum the motions balance. If this is not the case, — i.e., if there is any predominant drift, — the computed position of the Apex will be affected; and as this exact balance seldom holds good, different sets of stars generally give somewhat different results.

The recent investigations of Kapteyn on what he calls "star streaming " have excited great interest among astronomers. They seem to show that extensive systematic drifts actually exist, and that the nearer stars (those which have a measurable proper motion) mainly belong to two great systems, — one, the more numerous, drifting towards the region of Orion, the other streaming in the opposite direction.

MANUAL OF ASTEONOMY

INTRODUCTION

1. Astronomy is the science which treats of the heavenly bodies, as is indicated by the derivation of its name (aarpov VQIJLOS). It considers :

(1) Their motions, both real and apparent, and the laws which govern those motions.

(2) Their forms, dimensions, and masses.

(3) Their nature, constitution, and physical condition.

(4) The effects which they produce upon each other by their attractions, radiations, or any other ascertainable influence.

The earth is an immense ball, about 8000 miles in diameter, composed of rock and water, and covered with a thin envelope of air and cloud. Whirling on its axis, it rushes through empty space with a speed fifty times as great as that of the swiftest shot. On its surface we are wholly unconscious of the motion, because of its perfect steadiness.

As we look up at night we see in all directions the countless The off-look stars ; and conspicuous among them, and looking like stars, from the though very different in their real nature, are scattered a few planets. Here and there appear faintly shining clouds of light, like the so-called Milky Way and the nebulse, and perhaps now and then a comet. Most striking of all, if she happens to be in the heavens at the time, though really the most insignificant of all, is the moon. By day the sun alone is visible, flooding the air with its light and hiding the other bodies from the unaided eye, but not all of them from the telescope.

i

MANUAL OF ASTRONOMY

Branches of astronomy.

2. The Heavenly Bodies. — The bodies thus seen from the earth are the heavenly bodies. For the most part they are globes like the earth, whirling on their axes, and moving swiftly, though at such distances from us that their motions can be detected only by careful observation.

They may be classified as follows: First, the solar system proper, composed of the sun, the planets which revolve around it, and the satellites which attend the planets in their motion. The moon thus accompanies the earth. The distances between these bodies are enormous as compared with the size of the earth; and the sun, which rules them all, is a body of almost inconceivable magnitude.

Secondly, we have the comets and the meteors, which, while they acknowledge the sun's dominion, move in orbits of a dif- ferent shape and are bodies of a very different character.

Thirdly, we have the stars, at distances from us immensely greater than even those which separate the planets. The visible stars are suns, bodies like our own sun in nature, and like it, self-luminous, while the planets and their satellites shine only by reflected sunlight. The telescope reveals millions of stars invisible to the naked eye, and there are others, possibly thou- sands of them, that are dark and do not shine, but manifest their existence by effects upon their neighbors.

Finally, there are the nebulce, of which we know very little except that they are cloudlike masses of shining matter, and belong to the region of the stars.

3, Branches of Astronomy. — Astronomy is divided into many branches, some of which generally recognized are the following :

(1) Descriptive Astronomy, This, as its name implies, is merely an orderly statement of astronomical facts and principles.

(2) Spherical Astronomy. This, discarding all considerations of absolute dimensions and distances, treats the heavenly bodies simply as objects on the surface of the celestial sphere ; it deals

INTRODUCTION 3

only with angles and directions, and, strictly regarded, is merely spherical trigonometry applied to astronomy.

(3) Practical Astronomy. This treats of the instruments, the methods of observation, and the processes of calculation by which astronomical facts are ascertained. It is quite as much an art as a science.

(4) Theoretical Astronomy. This deals with the calculation of orbits and ephemerides, including the effect of perturbations.

(5) Astronomical Mechanics. This is simply the application of mechanical principles to explain astronomical facts, chiefly the planetary and lunar motions. It is sometimes called " gravita- tional astronomy," because, with few exceptions, gravitation is the only force sensibly concerned in the motions of the heavenly bodies.

Until about 1860 this branch of the science was generally Abandon- designated " physical astronomy," but the term is now objection- ^n* ^ able because of late it has been used by some writers to denote a " physical very different and comparatively new branch of the science, viz. : astron>-

(6) Astronomical Physics, or Astro-Physics. This treats of the physical characteristics of the heavenly bodies, their bright- ness and spectroscopic peculiarities, their temperature and radi- ation, the nature and condition of their atmospheres and surfaces, and all phenomena which indicate or depend on their physical condition. It is sometimes called The New Astronomy.

The above branches are not distinct and separate, but overlap in all directions. Valuable works exist, however, bearing the different titles indicated above, and it is important for the student to know what subjects he may expect to find discussed in each, although they do not distribute the science between them in any strictly logical and mutually exclusive manner.

4. Rank of Astronomy among the Sciences. — Astronomy is the oldest of the natural sciences. Obviously, in the very infancy of the race the rising and setting of the sun, the phases of the moon, and the progress of the seasons must have

MANUAL OF ASTRONOMY

Astronomy still pro- gressive.

Use in navi- gation and geodesy.

Use in regu- lation of time.

Chief value purely intel- lectual.

compelled the attention of even the most unobservant. Nearly the earliest of all existing records relate to astronomical subjects, such as eclipses and the positions of the planets.

As astronomy is the oldest of the sciences, so also it is one of the most perfect and complete, though not in the sense that it has reached a maturity which admits no further development, for in fact it was never more vigorously alive or growing faster than at present. In certain aspects astronomy is also the noblest of the sisterhood, being the most " unselfish " of them all, cultivated not so much for material profit as for pure love of learning.

5. Utility. — But although bearing less directly upon the material interests of life than the more modern sciences of physics and chemistry, it is really of high utility.

It is by means of astronomy that the latitude and longitude of points upon the earth's surface are determined, and by such determinations alone is it possible to conduct extensive naviga- tion. Moreover, all the operations of surveying upon a large scale, such as the determination of international boundaries, depend more or less upon astronomical observations.

The same is true of all operations which, like the railway service, require an accurate knowledge and observance of the time ; for our fundamental timekeeper is the daily revolu- tion of the heavens, as determined by the astronomer's transit instrument.

At present, however, the end and object of astronomical study is chiefly knowledge, pure and simple. It is not likely that great inventions and new arts will grow out of its prin- ciples, such as are continually arising from chemical, physical, and biological studies ; but it would be rash to say that such outgrowths are impossible.

The student of astronomy must, therefore, expect his chief profit to be intellectual, — in the widening of the range of thought and conception, in the pleasure attending the discovery

INTRODUCTION 5

of simple law working out the most far-reaching results, in the delight over the beauty and order revealed by the telescope and spectroscope in systems otherwise invisible, in the recogni- tion of the essential unity of the material universe and of the kinship between his own mind and the Infinite Reason.

In ancient time it was believed that human affairs of every kind, the welfare of nations, and the life history of individuals, were controlled, or at least prefigured, by the motions of the stars and planets ; so that from the study of the heavens it ought to be possible to predict futurity. The pseudo-science of astrology, based upon this belief, supplied the motives that Astrology led to most of the astronomical observations of the ancients. As modern a pseudo- chemistry had its origin in alchemy, so astrology was the progenitor of astronomy, and it is remarkable how persistent a hold this baseless delusion still retains upon the credulous.

6. Place in Education. — Apart from the utility of astronomy in the ordinary sense of the word, the study of the science is of high value as an intellectual training. No other so operates Educational to give us, on the one hand, just views of our real insignificance value- in the universe of space, matter, and time, or to teach us, on the other hand, the dignity of the human intellect as being the offspring, and measurably the counterpart, of the Divine, — able in a sense to comprehend the universe and understand its plan and meaning.

The study of the science cultivates nearly every faculty of the mind ; the memory, the reasoning power, and the imagina- tion all receive from it special exercise and development. By the precise and mathematical character of many of its discussions it enforces exactness of thought and expression, and corrects the vague indefiniteness which is apt to be the result of purely literary training ; while, on the other hand, by the beauty and grandeur of the subjects which it presents, it stimulates the imagination and gratifies the poetic sense.

NOTE. — The occasional references to "Physics" refer to Gage's Principles of Physics (Goodspeed's revision).

CHAPTER I PRELIMINARY CONSIDERATIONS AND DEFINITIONS

Fundamental Notions and Definitions— Astronomical Coordinates and the "Doctrine of the Sphere " — The Celestial Globe

ASTRONOMY, like all the other sciences, has a terminology of its own, and uses technical terms in the description of its facts and phenomena. In a popular work it would be proper to avoid such terms as far as possible, even at the expense of circumlocutions and occasional ambiguity; but in a text-book it is desirable that the student should be introduced to the most important of them at the very outset and be made sufficiently familiar with them to use them intelligently and accurately .

7. The Celestial Sphere.1 — The sky appears like a hollow vault, to which the stars seem to be attached, like gilded nail- heads upon the inner surface of a dome. We cannot judge of the distance to this surface from the eye further than to perceive that it must be very far away ; it is therefore natural and extremely convenient to regard the distance of the sky The celestial as everywhere the same and unlimited. The celestial sphere, sphere ag ^ jg ca}ie(j is conceived of as so enormous that the whole

conceived

as infinite, material universe of stars and planets lies in its center like a few grains of sand in the middle of the dome of the Capitol. Its diameter, in technical language, is taken as mathematically infinite, i.e., greater than any assignable quantity.

Since the radius of the sphere is thus infinite, it follows that all the lines of any set of parallels will appear, if produced

1 The study of the celestial sphere and its circles is greatly aided by the use of a globe or armillary sphere. Without some such apparatus it is rather difficult for a beginner to get clear ideas upon the subject.

6

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 7

indefinitely, to pierce it at a single point, the vanishing point of perspective, or the point at infinity of projective geometry. However far apart the lines may be and whatever, therefore, may be the distances in miles between the points at which they pierce the surface of the celestial sphere, yet, seen by the observer at its infinitely distant center, the angular distance between those points is utterly insensible, and they coalesce into one. Thus the axis of the earth and all lines parallel to it pierce the heavens at one point, the celestial pole; and the plane of the earth's equator, keep- ing parallel to itself during her annual circuit around the sun, marks out only one celestial equa- tor in the sky.

8. The place of a heavenly body is simply the point where a line drawn from the observer through the body in question and continued onward pierces the celestial sphere. It depends solely upon the direc- tion of the body and has nothing

to do with its distance. Thus, in Fig. 1 A, B, C, etc., are the apparent places of a, b, c, etc., the observer being at 0. Objects that are nearly in line with each other, as h, i, Jc, will appear close together in the sky, however great the real distance between them. The moon, for instance, often looks to us very near a star, which is always at an immeasurable distance beyond her.

9. Linear and Angular Dimensions and Measurement. — Linear dimensions are such as can be expressed in linear units; i.e., in miles, feet, or inches ; kilometers, meters, or millimeters. Angular dimensions are expressed in angular units ; i.e., in degrees, minutes, and seconds, or sometimes in radians, the radian being the angle which is measured by an arc equal in length to the radius, determined by dividing the circumference by 2 TT.

Apparent convergence of parallels to a single point on the celestial sphere.

FIG. 1

Place of a heavenly body de- pends solely on its direc- tion from observer.

Value of the radian in

minutes, and seconds.

MANUAL OF ASTRONOMY

Angular units used

ments on

celestial

sphere.

The radian, therefore, equals 57°.29 (i.e., 360° H- 2 TT), or 3437'.75 (i.e., 21600' -f- 2 TT), or 206264".8 (i.e., 1 296000" -*- 27r).

Hence, to reduce to seconds of arc an angle expressed in radians, we must multiply its value in radians by 206264-8 ; a relation of which we shall make frequent use.

Obviously, angular units alone can properly be used in describ- ing apparent distances in the sky. One cannot say correctly ing measure- that the two stars known as "-the pointers" are so many feet apart ; their distance is about five degrees.

It is very important that the student of astronomy should accustom himself as soon as possible to estimate celestial meas- ures in angular units. A little practice soon makes it easy, although the beginner is apt to be embarrassed by the fact that the sky appears to the eye to be not a true hemisphere, but a flattened vault, so that all estimates of angular distances for objects near the horizon are apt to be exaggerated. The moon when rising or setting looks to most persons much larger than when overhead, and the " Dipper-bowl " when underneath the pole seems to cover a much larger area than when above it.

Apparent This illusion (for it is merely an illusion), which makes the sun and

enlargement heavenly bodies when near the horizon appear larger than when high up

of sun and jn ^Q gj^ |g prokakiy due to the fact that in the latter case we have no

the horizon intervening objects by which to estimate the distance, and it therefore is

judged to be smaller than at the horizon. If we look at the sun or moon

when near the horizon through a lightly smoked glass which cuts off the

view of the landscape, the object immediately shrinks to its ordinary size.

Relation between distance, radius, and angular semi- diameter of a globe.

10. Relation between the Distance and Apparent Size of an Object. — Suppose a globe having a (linear) radius BC equal to r. As seen from the point A (Fig. 2) its apparent (i.e., angular) semidiameter will be BAC or s, its distance being AC or R.

We have immediately, from trigonometry, since B is a right angle, sin s = r /R, whence also r = R X ?in s, and R = r •*- sin s.

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 9

If, as is usual in astronomy, the diameter of an object is small as compared with its distance, so that sin s practically

FIG. 2

equals s itself, we may write s = r/R, which gives s in radians (not in degrees or seconds). If we wish to have it in the ordinary angular units,

8° = 57.3r/^; or «' = 3437.7 r/JR; or s" = 206264.8 r/R} also R = 206264.8 r/«"; andr = ^«"/206264.8;

where s° means s in degrees ; s', in minutes of arc ; s", in seconds of arc.

In either form of the equation we see that the apparent diameter varies directly as the linear diameter and inversely as the distance.

In the case of the moon, R = about 239000 miles ; and r, 1081 miles. Hence s (in radians) = ^tf-o"o = jj-y °f a radian, which is about 933", — a little more than one fourth of a degree.

It may be mentioned here as a rather curious fact that to most persons Apparent the moon, when at a considerable altitude, appears about a foot in diam- distance of eter ; — at least, this seems to be the average estimate. This implies that *~e surface the surface of the sky appears to them only about 110 feet away, since tial spnere that is the distance at which a disk one foot in diameter would have an angular diameter of Tyff of a radian, or £°.

Probably this is connected with the physiological fact that our muscular sense enables us to judge moderate distances pretty fairly up to 80 or 100 feet, through the "binocular parallax" or convergence of the eyes upon the object looked at. Beyond that distance the convergence is too slight to be perceived. It would seem that we instinctively estimate the moon's distance as small as our senses will permit when there are no intervening objects which compel our judgment to put her further off.

10

MANUAL OF ASTRONOMY

POINTS AND CIRCLES OF EEFERENCE AND SYSTEMS OF COORDINATES

In order to be able to describe intelligently the position of a heavenly body in the sky, it is convenient to suppose the inner surface of the celestial sphere to be marked off by circles traced upon it, — imaginary circles, of course, like the meridians and parallels of latitude upon the surface of the earth.

Three distinct systems of such circles are made use of in astronomy, each of which has its own peculiar adaptation for its special purposes.

A. SYSTEM DEPENDING ON THE DIRECTION OF GRAVITY AT THE POINT WHERE THE OBSERVER STANDS

11. The Zenith and Nadir. — If we suspend a plumb-line, and imagine the line extended upward to the sky, it will pierce the celestial sphere at a point directly overhead, known as the Astronomical Zenith, or the Zenith simply, unless some other qualifier is annexed.

As will be seen later (Sec. 130, b), the plumb-line does not point exactly to the center of the earth, because the earth rotates on its axis and is not strictly spherical. If an imagi- nary line be drawn from the center of the earth upward through the observer, and produced to the celestial sphere, it marks a different point, known as the geocentric zenith, which is never very far from the astronomical zenith, but must not be confounded with it.

For most purposes the astronomical zenith is the better practical point of reference, because its position can be deter- mined directly by observation, which is not the case with the other.

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 11

The Nadir is the point opposite to the zenith, directly under foot in the invisible part of the celestial sphere.

Both " zenith " and " nadir " are derived from the Arabic, as are many other astronomical terms. It is a reminiscence of the centuries when the Arabs were the chief cultivators of science.

12. The Horizon. — If now we imagine a great circle drawn The horizon completely around the celestial sphere half-way between the defined< zenith and nadir, and therefore 90° from each of them, it will

be the Horizon (pronounced ho-ri'-zon, not hor'-i-zon).

Since the surface of still water is always perpendicular to the direction of gravity, we may also define the horizon as the great circle in which a plane tangent to a surface of still water at the place of observation cuts the celestial sphere.

Many writers distinguish between the sensible and rational Unneces- horizons, — the former being denned by a horizontal plane drawn sary distinc-

tion foptwppn

through the observer's eye, while the latter is defined by a plane sensible and parallel to this, but drawn through the center of the earth, rational These two planes, however, though 4000 miles apart, coalesce upon the infinite celestial sphere into a single great circle 90° from both zenith and nadir, agreeing with the first definition given above. The distinction is unnecessary.

13. Visible Horizon. — The word " horizon " (from the Greek) The visible means literally "the boundary" — that is, the limit of the land- horizon- scape, where sky meets earth or sea; and this boundary line

is known in astronomy as the visible horizon. On land it is of no astronomical importance, being irregular; but at sea it is practically a true circle, nearly coinciding with the horizon above defined, but a little below it. When the observer's eye is at the water-level, the coincidence is exact ; but if he is at an elevation above the surface, the line of sight drawn from his eye tangent to the water inclines or dips down, on account of the curvature of the earth, by a small angle known as the dip of the horizon, to be discussed further on (Sec. 77).

12

MANUAL OF ASTRONOMY

14. Vertical Circles ; the Meridian and the Prime Vertical. - Vertical circles are great circles drawn from the zenith at right angles to the horizon, and therefore passing through the nadir also. Their number is indefinite.

That particular vertical circle which passes north and south through the pole, to be defined hereafter, is known as the Celes- tial Meridian, and is evidently the circle traced upon the celestial sphere by the plane of the terrestrial meridian upon which the observer is located. The vertical circle at right angles to the meridian is called the Prime Vertical. The points where

FIG. 3. — The Horizon and Vertical Circles

0, the place of the observer. OZ, the observer's vertical. Z, the zenith ; P, the pole. SWNE, the horizon. SZPN, the meridian. EZW, the prime vertical.

M, some star.

ZMH, arc of the star's vertical circle.

TMR, the star's almucantar.

Angle TZM, or arc SH, star's azimuth.

Arc HM, star's altitude.

Arc ZM, star's zenith-distance.

the meridian intersects the horizon are the north and south points; and the east and west points are midway between them. These are known as the Cardinal Points.

The parallels of altitude, or almucantars, are small circles of the celestial sphere drawn parallel to the horizon, sometimes called circles of equal altitude.

15. Altitude and Zenith-Distance. — The Altitude of a heav- enly body is its angular elevation above the horizon, i.e., the number of degrees between it and the horizon, measured on a vertical circle passing through the object. In Fig. 3 the

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 13

vertical circle ZMH passes through the body M. The arc MH is the altitude of M, and the arc ZM (the complement of MH) is its zenith-distance.

16. Azimuth. — The Azimuth (an Arabic word) of a heavenly Azimuth body is the same as its "bearing" in surveying; measured, defined- however, from the true meridian and not from the magnetic.

It may be defined as the angle formed at the zenith between the meridian and the vertical circle which passes through the object ; or, what comes to the same thing, it is the arc of the horizon intercepted between the south point and the foot of this circle.

In Fig. 3 SZM is the azimuth of M, as is also the arc SH, which measures this angle. The distance of H from the east or west point of the horizon is called the amplitude of the body, Amplitude but the term is seldom used except in describing the point defined- where the sun or moon rises or sets.

There are various ways of reckoning azimuth. Formerly Method of it was usually expressed in the same way as the " bearing " in reckoning surveying; i.e., so many degrees east or west of north or south. In the figure, the azimuth of M thus expressed is about S. 50° E. The more usual way at present, however, is to reckon it from the south point clear around through the west to the point of beginning, so that the arc SWNKEH would be the azimuth of Jf, — about 310°.

17. Altitude and azimuth are for many purposes in con- inconven- venient, because they continually change for a celestial object. lence of altl" It is desirable, therefore, in defining the place of a body in the azimuth, heavens, to use a different way which shall be free from this objection ; and this can be done by taking as the fundamental

line of our system, not the direction of gravity, which is differ- ent at any two different points on the earth's surface and is continually changing as the earth revolves, but the direction of the earths axis, which is practically constant.

14

MAKUAL OF ASTRONOMY

B. SYSTEM DEPENDING UPON THE DIRECTION OF THE EARTH'S AXIS OF ROTATION

Apparent 18. The Apparent Diurnal Rotation of the Heavens. — If on

theTe'avens some clear evening in the early autumn, say about eight o'clock on the 22d of September, we face the north, we shall find the

FIG. 4. — The Northern Circumpolar Constellations

appearance of that part of the heavens directly before us sub- stantially as shown in Fig. 4. In the north is the constellation of the Great Bear (Ursa Major), characterized by the conspicu- ous group of seven stars, known as the Great Dipper, which lies with its handle sloping upward to the west. The two

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 15

easternmost stars of the four which form its bowl are called the pointers, because they point to the pole-star, — a solitary star The pole- not quite half-way from the horizon to the zenith (in the latitude star and the

J pointers.

of New York), and about as bright as the brighter of the two pointers. It is often called Polaris.

High up on the opposite side of the pole-star from the Great Dipper, and at nearly the same distance, is an irregular zigzag of five stars, each about as bright as the pole-star itself. This is the constellation of Cassiopeia.

If now we watch these stars for only a few hours, we shall find that while all the configura- tions remain unaltered, their places in the sky are slowly changing. The Dipper slides down- ward towards the north, so that by eleven o'clock the pointers are directly under Polaris. Cassio- peia still keeps oppo-

site, however, rising towards the zenith; and if we were to continue to watch them the whole night, we should find that all the stars appear to be moving in circles around a point near the pole-star, revolving in the opposite direction to the hands of a watch (as we look up towards the north), with a steady motion which takes them completely around once a day, or, to be exact, once in the sidereal day, consisting of 23h56m48.l of ordinary time. They behave just as if they were attached to the inner surface of a huge revolving sphere.

Instead of watching the stars with the eye, the same result can be still better reached by photography. A camera is pointed

16

MANUAL OF ASTRONOMY

Polar star trails.

up towards the pole-star and remains firmly fixed while the stars, by their diurnal motion, impress their " trails " upon the plate. Fig. 5 is copied from a negative made by the author with an exposure of about three hours.

If instead of looking towards the north we now look south- ward, we shall find that there also the stars appear to move in the same kind of way. All that are not too near the pole-star rise somewhere in the eastern horizon, ascend not vertically but obliquely to the meridian, and descend obliquely to their setting at points on the western horizon. The motion is always in an arc of the circle, called the star's diurnal circle, the size of which depends upon the star's distance from the pole. Moreover, all these arcs are strictly parallel.

The ancients accounted for these obvious facts by supposing the stars actually fixed upon a real material sphere, really turning daily in the manner indicated. According to this view there must therefore be upon the sphere two opposite, pivotal points which remain at rest, and these are the poles.

19. Definition of the Poles. — The Celestial Poles, or Poles of Rotation (when it is necessary, as sometimes happens, to dis- tinguish between these poles and the poles of the ecliptic), may therefore be defined as those two points in the sky where a star would have no diurnal motion. The exact position of either pole may be determined with proper instruments by finding the center of the small diurnal circle described by some star near it, as for instance by the pole-star.

Since the two poles are diametrically opposite in the sky, only one of them is usually visible from a given place ; observers north of the equator see only the north pole, and vice versa in the southern hemisphere.

Knowing as we now do that the apparent revolution of the celestial sphere is due to the real rotation of the earth on its axis, we may also define the poles as the two points where the earths axis of rotation (or any set of lines parallel to it), produced indefinitely, would pierce the celestial sphere.

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 17

20. The Celestial Equator, or Equinoctial, and Hour-Circles. —

The Celestial Equator is the great circle of the celestial sphere, drawn half-way between the poles (therefore everywhere 90° from each of them), and is the great circle in which the plane of the earth's equator cuts the celestial sphere (Fig. 6). It is often called the Equinoctial. Small circles drawn parallel to the equinoctial, like the parallels of latitude on the earth, are called parallels of declination, a star's parallel of declination being iden- tical with its diurnal circle.

The great circles of the celestial sphere which pass through the poles, like the meridians on the earth, and are therefore perpendicular to the celestial equator, are called Hour- Circles. On celestial globes twenty-four of them are usually drawn, corresponding one to each of the twenty-four hours, but the real number is indefinite ; an hour-circle can be drawn through any star. That particular hour-circle which at any moment passes through the zenith of the observer coincides with the celestial meridian, already defined.

21. Declination and Hour Angle The Declination of a star

is its distance in degrees north or south of the celestial equator ; -f- if north, — if south. It corresponds precisely with the lati- tude of a place on the earth's surface, but cannot be called celestial latitude, because the term has been preoccupied by an entirely different quantity to be defined later (Sec. 27).

The Hour Angle of a star at any moment is the angle at the pole between the celestial meridian and the hour-circle of the star. In Fig. 7, for the body m it is the angle mPZ, or the arc QY.

FIG. 6. — The Plane of the Earth's Equator produced to cut the Celes- tial Sphere

Definition of the celestial equator.

Parallels of declination identical with diurna\ circles.

Hour-circles defined. The meridian as an hour-circle.

Declination defined.

Hour angle defined.

18

MANUAL OF ASTRONOMY

Relation of units of

time to units

of angle.

This angle, or arc, may of course be measured like any other, in degrees, but since it depends upon the time which has elapsed since the body was last on the meridian, it is more usual to measure it in hours, minutes, and seconds of time. The hour -g then equivalent to -£T of a circumference, or 15°, and the

- .

minute and second of time to 15' and 15" of arc, respectively. Thus, an hour angle of 4h2m38 equals 60° 30' 45".

FIG. 7. — Hour-Circles, etc.

O, place of the observer ; Z, his zenith.

SENW, the horizon.

POP', the axis of the celestial sphere.

P and P', the two poles of the heavens.

EQWT, the celestial equator, or equinoc- tial.

X, the vernal equinox, or " first of Aries."

PXP', the equinoctial colure, or zero hour- circle.

m, some star.

Ym, the star's declination ; Pm, its north- polar distance.

Angle mPR = arc QY, the star's (eastern) hour angle ; = 24h minus star's western hour angle.

Angle X Pm = arc X Y, star's right ascension. Sidereal time at the moment = 24>> minus angle XPQ.

The position of the body m (Fig. 7) is, then, perfectly denned by saying that its declination is +25° and its hour angle 40° east (or simply 320°, if we choose, as is usual, to reckon completely around in the direction of the diurnal motion). Instead of 40 degrees, we might say 2h40m of time east, or simply 21b20ra to correspond to the 320°.

22, The declination of a star, omitting certain minutiae for the present, remains practically unaltered even for years, but

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 19

the hour angle changes continually and uniformly at the rate of The hour 15° for every sidereal hour. This unfits it for use in ephemeri- angle

* changes

des or star-catalogues. We must substitute for the meridian continually some other hour-circle passing through a well-defined point ™ithtne

9 _ tiniG •

which participates in the diurnal rotation and so retains an unchanging position relative to the stars. Such a point, selected by astronomers nearly two thousand years ago, is the so-called Vernal Equinox, or First of Aries.

23. The Ecliptic, Equinoxes, Solstices, and Colures. — The sun, moon, and planets, though apparently carried by the diurnal revolution of the celestial sphere, are not, like the stars, apparently fixed upon it, but move over its surface like glow- worms creeping on a whirling globe. In the course of a year, as will be explained later (Sec. 156), the sun makes a complete circuit of the heavens, traveling among the stars in a great circle called the Ecliptic. The ecliptic.

The ecliptic cuts the celestial equator in two opposite points at an angle of about 23i°. These points are the equinoxes. The Vernal Equinox, or First of Aries (symbol <Y>), is the point The vernal where the sun crosses from the south to the north side of the e(1U11 equator, on or about the 21st of March. The other is the autumnal equinox.

The summer and winter Solstices are points on the ecliptic, The sol- midway between the two equinoxes and 90° from each, where s the sun attains its maximum declination of + 23£° and — 23£° . in summer and winter, respectively.

The hour-circles drawn from the pole (of rotation) through the equinoxes and solstices are called the equinoctial and

solstitial Colures. The colurea

Neglecting for the present the gradual effect of pre- cession (Sec. 165), these points and circles are fixed with reference to the stars, and form a framework by which the places of celestial objects may be conveniently defined and catalogued.

20

MANUAL OF ASTRONOMY

Position of the vernal equinox.

Definitions of right ascension.

The sidereal day.

Sidereal time.

Definition of sidereal time.

No conspicuous star marks the position of the vernal equinox ; but a line drawn from the pole-star through ft Cassiopeioe and continued 90° from the pole will strike very near it.

24. Right Ascension. — The Right Ascension of a star may now be denned as the angle made at the celestial pole between the hour-circle of the star and the hour-circle which passes through the vernal equinox (called the equinoctial colure), or as the arc of the celestial equator intercepted between the vernal equinox and the point where the star's hour-circle cuts the equator. Right ascension is reckoned always eastward from the equinox, completely around the circle, and may be expressed either in degrees or in time units. A star one degree west of the equinox has a right ascension of 359°, or 23h56m.

Evidently the diurnal motion does not affect the right ascen- sion of a star, but, like the declination, it remains practically unchanged for years. In Fig. 7. (Sec. 21), if X be the vernal equinox, the right ascension of m is the angle XPm, or the aro XY measured from X eastward.

25. Sidereal Day and Sidereal Time. — The sidereal day is the interval of time between two successive passages of a fixed star over a given meridian, and at any place it begins at the moment when the vernal equinox is on the meridian; it is about four minutes shorter than the solar day, and like it is divided into twenty-four (sidereal) hours with corresponding sidereal minutes and seconds, all shorter than the corresponding solar units.

The sidereal time at any moment is the time shown by a clock so set and regulated as to show zero hours, zero minutes, and zero seconds at the moment when the vernal equinox crosses the meridian. It is the hour angle of the vernal equinox, or, what is the same thing, the right ascension of the observer's meridian.

26. Observatory Definition of Right Ascension. — The right ascension of a star may now be correctly, and for observatory

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 21

purposes, most conveniently defined as the sidereal time at the Observatory moment when the star is crossing the observer's meridian. Since definitlon of

. right ascen-

the sidereal clock indicates zero hours at sidereal noon, i.e., at sion. the moment when the vernal equinox is on the meridian, its face at any other time shows the hour angle of the equinox ; and this is what has just been defined as the right ascension of all stars which may then happen to be on the meridian (common to them all since they all lie on the same hour-circle).

0. SYSTEM DETERMINED BY THE PLANE OF THE EARTH'S ORBIT

27. Celestial Latitude and Longitude — The ancient astrono- Definition mers confined their observations mostlv to the sun, moon, and of celestial

J latitude and

planets, which are never far from the ecliptic, and for this longitude, reason the ecliptic (which is simply the trace of the plane of the earth's orbit upon the celestial sphere) was for them a more convenient circle of reference than the equator, — especially as they had no accurate clocks. According to their terminology, Latitude (celestial) is the angular distance of a heavenly body north or south of the ecliptic ; Longitude (celestial) is the arc of the ecliptic intercepted between the vernal equinox (°f ) and the foot of a circle drawn from the pole of the ecliptic to the ecliptic through the object. Longitude, like right ascension, is always reckoned eastward from the equinox.

Circles drawn from the poles of the ecliptic perpendicular to the ecliptic are called secondaries to the ecliptic, — by some Secondaries writers " ecliptic meridians," and on- some celestial globes are to the

ecliptic.

drawn instead of hour-circles.

The poles of the ecliptic are the points 90° distant from the Poles of the ecliptic. The position of the north ecliptic pole is shown in ecllPtlc- Fig. 4. It is on the solstitial colure, about 231-0 distant from the pole of rotation, in declination 66J-0 and right ascension 18h. It is marked by no conspicuous star.

22

MANUAL OF ASTRONOMY

It is unfortunate, or at least confusing to beginners, that celes- tial latitude and longitude should not correspond with the ter- restrial quantities that bear the same name. Great care must be taken to observe the distinction.

The gravity system of coordinates.

The two systems which de- pend upon the rotation of the earth.

28. Recapitulation. — The direction of gravity at the point where the observer happens to stand determines the zenith and nadir, the horizon and the almucantars, or parallels of altitude, and all the vertical circles. One of the verticals, the meridian, is singled out from the rest by the circumstance that it is the projection of the observer's terrestrial meridian upon the celestial sphere and passes through the pole, marking the north and south points where it cuts the horizon. Altitude and azimuth, or their complements, zenith-distance and amplitude, define the position of a body by reference to the horizon and meridian.

This set of points and circles shifts its position among the stars with every change in the place of the observer and every moment of time. Each place and hour has its own zenith, its own horizon, and its own meridian.

In a similar way, the direction of the earths axis, which is independent of the observer's place on the earth, determines the pole (of rotation), the equator, parallels of declination, and the hour-circles. Two of these hour-circles are singled out as reference lines: one of them is the hour-circle which at any moment passes through the zenith and coincides with the merid- ian, — a purely local reference line ; the other, the equinoctial colure, which passes through the vernal equinox, a point chosen from its relation to the sun's annual motion.

Declination and hour angle define the place of a star with reference to the equator and meridian, while declination and right ascension refer it to the equator arid vernal equinox. The latter are the coordinates usually given in star-catalogues and almanacs for the purpose of defining the position of stars and planets, and they correspond exactly to latitude and longitude on

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 23

the earth, by means of which geographical positions are desig- nated. °f in the sky takes the place of Greenwich on the earth.

Finally, the earth's orbit gives us the great circle of the sky The ecliptic known as the ecliptic ; and celestial latitude and longitude define system- the position of a star with reference to the ecliptic and the ver- nal equinox (°p ). For most purposes this pair of coordinates is practically less convenient than right ascension and declination ; but it came into use centuries earlier, and has advantages in dealing with the planets and the moon.

29, The scheme given below presents in tabular form the relations of the four different systems to each other. In each case one of the two coordinates is measured along a primary circle, from a point selected as the origin, to a point where a secondary circle cuts it, drawn through the object perpendicular to the primary. The second coordinate is the angular distance of the object from the primary circle measured along this secondary.

SYS- TEM	PRIMARY CIR- CLE, HOW DETERMINED	PRIMARY CIRCLE	ORIGIN	SECONDARY CIRCLE	COORDI- NATES	USUAL SYMBOL X £i ffi H
A	Direction of gravity	Horizon	South point on horizon	Vertical cir- cle of star	Azimuth Altitude	c (A) <w
B«	" 1	Rotation of earth	Celestial equator	Foot of the meridian on equator	Hour-circle of star	Hour angle Declination	<** (5)
2	Rotation of earth	Celestial equator	The vernal equinox (°f)	Hour-circle of star	Right ascen- sion Declination	(«) (5)
C	Plane of earth's orbit	Ecliptic	The vernal equinox (°f3)	Secondary to ecliptic through star	Longitude Latitude	(X) 08)'

Tabular exhibit of the four systems of coordinates

30. Relation of the Coordinates on the Sphere. — Fig. 8 shows how these Diagram

coordinates are related to each other. The reader is supposed to be showing the

looking down on the celestial sphere from above, the circle SENWA being relation of the horizon.

the systems.

24

MANUAL OF ASTRONOMY

Z is the zenith ; P, the north pole (of rotation) ; P', the pole of the ecliptic ; °f , the vernal equinox, and £^, the autumnal ; S, E, N, W are the cardinal points of the horizon. The oval W^MQCE-^R is the celestial equator, and the narrower one, °fLB±±K, is the ecliptic. The angle B°fC, measured by the arcs EC and PP', is the obliquity of the ecliptic, for which the usual symbol is e or c.

O is some celestial object. Then the arc A 0 (projected as a straight line) is its altitude and the angle OZS its azimuth. OM is its declination

FIG. 8. —Relation of the Different Coordinates

and OPQ, its hour angle. °fPMis its right ascension = arc °f M. OL is its latitude and °f P'L (- arc °f L) is its longitude. °f P is 90° of the equi- noctial colure and P'PBC is part of the solstitial colure. The angles °f> P'B and TPC" are each 90°.

For methods and formulae by which either set of coordinates may be "transformed" into one of the others, see Sees. 700 and 701 (Appendix).

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 25

31. The Astronomical Triangle. — The triangle PZO (pole- zenith-object) (Fig. 8) is often called the astronomical triangle because so many problems, especially of nautical astronomy, depend on its solution. Its sides and angles are all named, — PZ is the colatitude of the observer, ZO is the zenith-distance of the object, and OP is its north polar distance, or complement of its declination. The angle P is the hour angle of the object, the angle Z is the supplement of its azimuth, and, finally, the angle at 0 is called the parallac-

tic angle, because it enters into the calculations of the effects of parallax and re- fraction upon the right ascension and declination of a body. Any three of the parts being given the others can, of course, be found.

32. Relation of the Place of the Celestial Pole to the Observer's Latitude. --If

an observer were at the north pole of the earth, it is clear that the pole-star would be very near his zenith, while it would be at his horizon if he were at the equator. The place of the pole in the sky, therefore, depends entirely on the observer's latitude, and in this very simple way the altitude of the pole (its height in degrees above the horizon) is always equal to the latitude of the observer. This will be clear from Fig. 9. The latitude (astronomical) of a place may be defined as the angle between the direction of gravity at that place and the plane of the earths equator, — the angle ONQ in Fig. 9. If at 0 we draw HH1 per- pendicular to ON, it will be a level line, and will lie in the plane of the horizon. From 0 also draw OP" parallel to CPf, the earth's axis. OP" and CPf, being parallel, will both be directed

The "astro- nomical triangle."

FIG. 9. — Relation of Latitude to the Elevation of the Pole

Position of the pole in the sky.

The altitude of the pole equals the observer's latitude.

26 MANUAI, OF ASTRONOMY

to their "vanishing point" in the celestial sphere (Sec. 7),

which is the celestial pole. The angle H'OP" is therefore the

altitude of the pole as seen at 0 ; and it obviously equals ONQ.

This fundamental relation, that the altitude of the pole is identical

with the observer's latitude, cannot be too strongly emphasized.

Aspect of 33. The Right Sphere. — If the observer is situated at the

the heavens eartn's equator, that is, in latitude zero, the pole will be in his

as seen from

the earth's horizon and the celestial equator will be a vertical circle, coin- equator, ciding with the prime vertical (Sec. 14). All heavenly bodies will rise and set vertically, and their diurnal circles will all be bisected by the horizon, so that they will be twelve hours above and twelve hours below it ; and the length of the night will always equal that of the day (neglecting refraction, Sec. 82). This aspect of the heavens is called the right sphere.

It is worth noting that for an observer exactly at the north pole the definitions of meridian and azimuth break down, since at that point the zenith coincides with the pole. Facing which direction he will, he is still looking directly south. If he change his place a few steps, how- ever, his zenith will move, and everything will become definite again.

Aspect of 34. The Parallel Sphere. — If the observer is at the pole of

the heavens the earth where hig latitude is 990 the celestial pole will be

as seen from

the pole. at his zenith and the equator will coincide with the horizon. If at the north pole, all the stars north of the celestial equator will remain permanently above the horizon, never rising nor falling, but sailing around the sky on almucantars, or parallels of altitude. The stars in the southern hemisphere, on the other hand, will never rise to view.

Since the sun and moon move among the stars in such a way that during half of the time they are north of the equator and half the time south of it, they will be half the time above

The six the horizon and half the time below it, at least approximately, *7 S*nce ^S statement nee(ls to be slightly modified to allow for the effect of refraction. The moon will be visible for about a fort- night each month and the sun for about six months each year.

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 27

35. The Oblique Sphere. — At any station between the poles and the equator the pole will be elevated above the horizon, and the stars will rise and set in oblique circles, as shown in Fig. 10. Those whose distance from the elevated pole is less than PN (the latitude of the observer) will of course never set, remaining perpetually visible. The radius of this circle of per- petual apparition, as it is called (the shaded cap around P in the figure), is obviously just equal to the height of the pole, becoming larger as the latitude increases. On the other hand, stars within the same distance

of the depressed pole will lie in the circle of perpetual occul- tation, and will never rise above the horizon. A star exactly on the celestial equa- tor will have its diurnal circle bisected by the horizon and will be above the horizon twelve hours. A star north of the equator, if the north pole is the elevated one, will have more than half its diur- nal circle above the horizon and will be visible for more than twelve hours each day; as, for instance, a star at A, rising at B and setting at B1.

Whenever the sun is north of the celestial equator, the day will therefore be longer than the night for all stations in north- ern latitude ; how much longer will depend on the latitude of the place and the sun's distance from the equator, i.e., its declination.

36. The Midnight Sun. — If the latitude of the observer is such that PN in the figure is greater than the sun's polar distance or codeclination at the time when the sun is far- thest north (about 66 J°), the sun will come into the circle of

FIG. 10. — The Oblique Sphere

The mid- night sun.

28

MANUAL OF ASTRONOMY

When the

sun shines into north windows.

perpetual apparition and will make a complete circuit of the heavens without setting, until its polar distance again becomes less than PN. This happens near the summer solstice at the North Cape and at all stations within the Arctic circle.

Whenever the sun is north of the equator it will in all north latitudes rise at a point north of east, as B in the figure, and will continue to shine upon every vertical surface that faces the north, until, as it ascends, it crosses the prime vertical EZW at some point V.

In the latitude of New York, the sun on the longest days of summer is south of the prime vertical only about eight hours of the whole fifteen during which it is above the horizon. During seven hours of the day it shines into north windows.

The celestial globe.

Its horizon and circles upon it.

The merid- ian ring, its graduation and clamp.

A celestial globe will be of great assistance in studying these diurnal phenomena. By means of this it can at once be seen what stars never set, which ones never rise, and during what part of the twenty-four hours a heavenly body at a known declination is above or below the horizon.

37. The Celestial Globe. — The celestial globe is a ball, usually of papier-mach£, upon which are drawn the circles of the celestial sphere and a map of the stars. It is mounted in a framework which represents the horizon and the meridian, in the manner shown by Fig. 11.

The horizon, HH' in the figure, is usually a wooden ring three or four inches wide, directly supported by the pedestal. It carries upon its upper surface at the inner edge a circle marked with degrees for measuring the azimuth of any heavenly body, and outside this the so-called "zodiacal circles," which give the sun's longitude and the equation of time (Sees. 99 and 174) for every day of the year.

The meridian ring, MM', is a circular ring of metal which carries the bearings of the axis on which the globe revolves. Things are so arranged, or ought to be, that the mathematical axis of the globe is exactly in the same plane as the graduated face of the ring, which is divided into degrees and fractions of a degree, with zero at the equator. The meridian ring fits into two notches in the horizon circle and is held underneath the globe

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 29

by a support with a clamp, which enables us to fix it securely in any desired position, the mathematical center of the globe being precisely in the planes both of the meridian ring and the horizon.

The hour index on the globe here figured is a pointer like the hour-hand of a clock, so attached to the meridian ring at the pole that it can be turned around the axis with stiffish friction, but will retain its position unchanged when the globe is made to turn under it. It points out the time on a small time-circle graduated usually to hours and quarters printed on the surface of the globe.

The surface of the globe is marked first with the celestial equator (Sec. 20), next with the ecliptic (Sec. 23), crossing the equator at an angle of 23^° (at X in the figure), and each of these circles is divided into degrees and fractions. The equinoctial and solstitial colures (Sec. 23) are also al- ways represented. As to the other circles, usage differs. The ordinary way at present is to mark the globe with twenty-four hour-circles, fif- teen degrees apart (the colures being two of them), and with parallels of declination ten degrees apart.

On the surface of the globe are plotted the positions of the stars and the outlines of the constellations.

38. To rectify a globe, — that is, to set it so as to show the aspect of the heavens at any given time, —

(1) Elevate the north pole of the globe to an angle equal to the observer's latitude by means of the graduation on the meridian ring, and clamp the ring securely.

(2) Look up the day of the month on the horizon of the globe and opposite to the day find, on the longitude circle, the sun's longitude for that day.

(3) On the ecliptic (on the surface of the globe) find the degree of longi- tude thus indicated and bring it to the graduated face of the meridian ring.

FIG. 11. — The Celestial Globe

The hour index.

Circles drawn on surface of globe.

30

MANUAL OF ASTRONOMY

The globe is then set to correspond to (apparent) noon of the day in question. (It may be well to mark the place of the sun temporarily with a bit of moist paper applied at the proper place in the ecliptic ; it can easily be wiped off after using.)

(4) Holding the globe fast, so as to keep the place of the sun on the meridian, turn the hour index until it shows on the graduated time-circle the local mean time of apparent noon, i.e., 12h ± the equation of time given for the day on the horizon ring. (If standard time is used, the hour index must be set to the standard time of apparent noon.)

(5) Finally, turn the globe until the hour for which it is to be set is brought to the meridian, as indicated on the hour index. The globe will then show the true aspect of the heavens.

The positions of the moon and planets are not given by this operation, since they have no fixed places in the sky and therefore cannot be put upon the globe by the maker. If one wants them represented, he must look up their right ascensions and declinations for the day in some almanac and mark the places on the globe with bits of wax or paper.

EXERCISES

1. What point in the celestial sphere has both its right ascension and declination zero ? ^"jf^

2. What are the celestial latitude and longitude of this poin^t? "Q

3. What are the hour angle' and azimuth of the zenith ? hyW

4. At what points does the celestial equator cut the horizon?^

5. What angle does the celestial equator make with the horizon at these points, as seen by an obsgryerAin latitude 40°? J Dl

6. What if his latitude is 10°f 20°? 50°V '60°^°

7. When the vernal equinox °f° is rising on the eastern horizon, what angle does the ecliptic make with the horizon at that point for an observer in latitude 40°? ^^3"

8. What angle when setting ?

9. What is the angle between the ecliptic and horizon when the autumnal equinox is rising, and when setting?

10. Name the fourteen principal points on the celestial sphere (zenith, poles, equinoxes, etc.).

11. What important circles on the celestial sphere have no correlatives on the surface of the earth ?

PRELIMINARY CONSIDERATIONS AND DEFINITIONS 31

12. What are the approximate right ascension and declination (a and 8) of the sun on March 21 and September 22?

13. What is the sun's altitude at noon on March 21 for an observer in latitude 42°? C/C

14. How far is the sun from the zenith at noon on March 21, as seen at Pulkowa, latitude 60°? How far at noon on June 21 ?

15. On March 21, one hour after sunset, whereabouts in the sky would be a star having a right ascension of 7 hours and declination of 40°, the observer being in latitude 40° ?

16. If a star rises to-night at 10 o'clock, at what time (approximately) will it rise 30 days hence ?

17. When the right ascension of the sun is 6 hours, what are its longi- tude (A) and latitude (0) ?

18. What, when its a is 12 hours?

19. What are the latitude and longitude of the north pole of rotation ?

20. What are the right ascension and declination of the north pole of the ecliptic ?

NOTE. — None of the above exercises require any calculation beyond a simple addition or subtraction.

21. What are the longitude and declination of the sun when its right

ascension is 3 hours ?

An§ ( Long. = 47° 27' 59".

< Dec. = 17° 03' 08".

NOTE. — This requires the solution of the spherical right angle triangle, in which the base is the given a (=45°), the angle adjacent is e (23° IT), and the parts to be found are the hypotenuse X and the other leg opposite c, which is 5.

CHAPTER II ASTRONOMICAL INSTRUMENTS

Telescopes, and their Accessories and Mountings — Timekeepers and Chronographs

— The Transit-Instrument — The Prime Vertical Instrument — The Almucantar

— The Meridian-Circle and Universal Instrument — The Micrometer — The Heliometer — The Sextant

39. Astronomical observations are of various kinds : some- times we desire to ascertain the apparent distance between two bodies ; sometimes the position which the body occupies at a given time, or the time at which it arrives at a given circle of the sky, — usually the meridian. Sometimes we wish merely to examine its surface, to measure its light, or to investigate its spectrum; and for all these purposes special instruments have been devised. We propose in this chapter to describe a few of the most important at present in use.

40. Telescopes in General. -- Telescopes are of two kinds, refracting and reflecting. The former were first invented and are much more used, but the largest instruments which

Fundamental have ever been made are reflectors. In both the fundamental Principle is identical. The large lens, or mirror, — the "object- ive " of the instrument — forms at its focus a " real " image of the object looked at, and this image is then examined and magnified by the eyepiece, which" in principle is only a magnify ing-glass.

Essential 41. The Simple Refracting Telescope. — This consists essen-

thTrefrac^ ^ally, as shown in Fig. 12, of two convex lenses, one the object- ing tele- glass A, of large size and long focus ; the other, the eyepiece scope. ^ o;f short focus ; the two being set at a distance nearly equal to

the sum of their focal lengths. Recalling the optical principles

ASTRONOMICAL INSTRUMENTS

33

of the formation of images by lenses,1 we see that if the

instrument is pointed toward the moon, for instance, all the

rays that strike the object-glass from the top of the object will

come to a focus at a, while those from the bottom will come to

a focus at &, and similarly with rays from other points on the

surface of the moon. We shall therefore get in the "focal Real image

plane " of the object-glass a small inverted " real " image of the

moon, so that if a photographic plate is inserted in the focal

plane at ab and properly exposed, we shall get a picture of the

object.

The size of the picture will depend upon the apparent Size of the angular diameter of the object and the distance of the image ab from the object-glass, and is determined by the condition

FIG. 12. — The Simple Refracting Telescope

that, as seen from point 0 (the optical center of the object-glass), the object and its image subtend equal angles, since rays which pass through the point 0 suffer no sensible deviation.

If the focal length of the lens A is 10 feet, then the image of the moon formed by it will appear, when viewed from a distance of 10 feet, just as large as the moon itself ; from a distance of 1 foot, the image will, of course, appear ten times as large

With such an object-glass, therefore, even without an eyepiece, one can see the mountains of the moon and satellites of Jupiter by simply putting the eye in the line of the rays, at a distance of 10 or 12 inches back of the eyepiece hole, the eyepiece itself having been, of course, removed. •

1 In this explanation we use the approximate theory of lenses (in which their thickness is neglected), as given in the elementary text-books. The more exact theory would require some slight modification in statements, but none of substantial importance.

34 MANUAL OF ASTRONOMY

42. Magnifying Power. — If we use the naked eye, one cannot, unless near-sighted, see the image distinctly from a distance much less than 10 inches; but if we use a magnifying- lens of 1-inch focus, we can view it from a distance of only an inch, and it will look correspondingly larger. Without stopping to demonstrate the principle, the magnifying power is simply equal to the quotient obtained by dividing the focal length of the object-glass by that of the eye-lens ; or, as a formula,

Formula for

themagni- M — F/f\ that is, Od/cd in the figure.

f ying power.

If, for example, the focal length of the object-glass be 4 feet and that of the eye-lens one quarter of an inch, then

M= 48 •*• J =4x48 = 192.

A magnifying power of unity, however, is often spoken of as " no magni- fying power at all," since the image appears of the same size as the object.

The magnifying power of the telescope is changed at pleasure by simply changing the eyepiece (see Sec. 47).

Light-gath- 43. Light-Gathering Power of the Telescope and Brightness

ering power Of the Image This depends not upon the focal length of the

to the square object-glass, but upon its diameter; or, more strictly, its area.

of the diam- If we estimate the diameter of the pupil of the eye at one fifth

object-glass °^ an ^ncn' then (neglecting the loss in transmission through

the lenses) a telescope 1 inch in diameter collects into the

image of a star twenty-five times as much light as the naked

eye receives; and the great Yerkes telescope of 40 inches

in diameter gathers 40000 times as much, or about 35000

after allowing for the losses. The amount of light collected is

proportional to the square of the diameter of the object-glass.

The apparent brightness of an object which, like the moon or a planet, shows a disk, is not, however, increased in any such ratio, because the light gathered by the object-glass is spread out by the magnifying power of the eyepiece. In fact, it can be demonstrated that no optical arrangement whatever can show

ASTRONOMICAL INSTRUMENTS 35

an extended surface brighter than it appears to the naked eye. No optical But the total quantity of light in the image of the object greatly arranse- exceeds that which is available for vision with the naked eye, increase the and objects which, like the stars, are mere luminous points, intrinsic have their brightness immensely increased, so that with the tele- Of j^** scope millions otherwise invisible are brought to light. With the extended telescope, also, the brighter stars are easily seen in the daytime. s

44. The Achromatic Telescope. — A single lens cannot bring Chromatic the rays which emanate from a single point in the obiect to anv aberration

. *f J of a single-

exact locus, since the rays 01 different color (wave-length) are lens object- differently refracted, the blue more than the green, and this glass- more than the red. In consequence of this so-called " chromatic aberration," the simple refract- ing telescope is a very poor instrument.1 r7ZT^ u Lutrow

About 1760 it was discov- ered in England that by making FIG. 13. — Different Forms of the the object-glass of two or more

lenses of different kinds of glass the chromatic aberration can be nearly corrected. Object-glasses so made — no others are now in common use — are called achromatic, and they fulfil The achro- with reasonable approximation, though not perfectly, the con- matiolens- dition of distinctness ; namely, that the rays which emanate from any single point in the object should be collected to a single point in the image. In practice, only two lenses are ordinarily used in the construction of an astronomical object- glass, — a convex of crown-glass, and a concave of flint-glass, the curves of the two lenses and the distances between them being so chosen as to give the best possible correction of the

1 By making the telescope extremely long in proportion to its diameter, the distinctness of the image is considerably improved, and in the middle of the seventeenth century instruments more than 200 feet in length were used by Cassini and others. Saturn's rings and several of his satellites were discovered by Huyghens and Cassini with instruments of this kind.

36

MANUAL OF ASTRONOMY

Imperfect achroma- tism of object-

More perfect object lenses from new kinds of glass.

The spuri- ous disk of a star.

spherical aberration as well as of the chromatic. Many forms of object-glass are made, three of which are shown in Fig. 13.

45. Secondary Spectrum. — It is not possible to obtain a perfect correction of color with the only kinds of glass which were available until very recently. Ordinary achromatic lenses, even the best of them, show around every bright object a strong purple halo, due to red and blue rays which are both brought to a focus further from the object-glass than are the yellow and green. This halo seriously injures the definition and makes it difficult to see small stars very near a bright one. It is specially obnoxious in large instruments.

Much is hoped from the new varieties of glass now being made at Jena in Germany. Several telescopes of considerable size have already been constructed, of which the lenses are practically aplanatic; that is, sensibly free from both spherical and chromatic aberration. Possibly a new era in telescope making is opening with the new century.

46. Diffraction and Spurious Disks. — Even if a lens were absolutely perfect as regards the correction of aberrations, it would still be unable to fulfil strictly the condition of distinctness.

Since light consists of waves of finite length, the image of a luminous point can never be also a point, but necessarily, on account of " diffraction," consists of a central disk of finite diameter, surrounded by a series of " interference " rings ; and the image of a line is a streak and not a line. The diameter of the " spurious disk " of a star, as it is called, varies inversely with the diameter of the object-glass ; the larger the telescope, the smaller the image of a star with a given magnifying power.

With a good 41-inch telescope and a power of about 120, the image of a small star, when the air is perfectly steady (which unfortunately seldom happens), is a clean, round disk, about 1" in diameter, with a bright ring around it, separated from the disk by a dark space about as wide as the disk. With a 9-inch

ASTRONOMICAL INSTRUMENTS 37

instrument the disk has a diameter of 0".5, — just half as great ; with the Yerkes telescope, about 0".ll. The angular diameter of a star disk in a telescope the aperture of which is a inches is, therefore, given by the following formula, due to Dawes :

Formula for

f/n — * - diameter of

a spurious

disk. If the magnifying power is too great (more than about sixty

to the inch of aperture), the disk of a star will become ill-defined at the edge ; so that there is very little use with most objects in pushing the magnifying power any higher.

This effect of "diffraction" has much to do with the supe- riority of large instruments in showing minute details; no increase of magnifying power on a small telescope can exhibit Superiority the object as sharply as the same power on a large one, pro- c vided, of course, that the object-glasses are equally good in work- glasses in manship and that the atmospheric conditions are satisfactory, defining

DOW6T»

(But a given amount of atmospheric disturbance injures the per- formance of a large telescope much more than that of a small one.)

47. Eyepieces, or " Oculars. " — For some purposes the simple convex lens is the best eyepiece possible ; but it performs well only for a small object, like a close double star, exactly in the center of the field of view. Generally, therefore, we employ eyepieces composed of two or more lenses, which give a larger field of view than a single lens and define fairly well over the whole extent of the field. They fall into two general classes, the positive arid the negative.

The positive eyepieces are much more generally useful. They Positive act as simple magnifying-glasses and can be taken out of the eyePieces- telescope and used as hand magnifiers if desired. The image of the object formed by the object-glass lies outside of this kind of eyepiece, between it and the object-glass.

In the negative eyepieces, on the other hand, the rays from Negative the object are intercepted by the so-called " field lens " before eyePleces-

38 MANUAL OF ASTRONOMY

reaching the focus, and the image is formed inside the eye- piece. It cannot therefore be used as a hand magnifier.

Fig. 14 shows the two most usual forms of eyepiece, and also the " solid eyepiece " constructed by Steinheil ; but there are a multitude of various kinds. All these eyepieces show the object inverted, which is of no importance in astronomical observations.

Steinheil 'Monocentric' (Positive)

Huyghenian (Negative)

FIG. 14. — Various Forms of Telescope Eyepiece

It is evident that in an achromatic telescope the objec1>glass is by far the most important and expensive member of the instrument. It costs, according to size, from $100 up to $65000, while the eyepieces cost only from $2 to $25 apiece, and every telescope of any pretension possesses a considerable stock, of various magnifying powers.

48. Reticle. — If the telescope is to be used for pointing The reticle, upon an object, it must be provided with a " reticle" of some sort. The simplest is a frame with two spider-lines stretched across it at right angles to each other, their intersection being the point of reference. This reticle is placed, not at or near the object-glass, as often supposed, but in the focal plane, as ab in Fig. 12 (Sec. 41). Of course, positive eyepieces only can be used in connection with such a reticle, though in sextant telescopes a negative eyepiece is sometimes used with a pair of cross-wires placed between the two lenses of the eyepiece. Sometimes a glass plate with fine lines ruled upon it is used instead of spider-lines. In order to make the lines of the

ASTRONOMICAL INSTRUMENTS 39

reticle visible at night, a faint light is reflected into the instru- ment by some one of various arrangements devised for the purpose.

49. The Reflecting Telescope. — About 16 TO, when the chro- Thereflect-

matic aberration of refractors first came to be understood (in mstele~

x scope.

consequence of Newton's discovery of the decomposition of light), the reflecting telescope was invented. For nearly one hundred and fifty years it held its place as the chief instru- ment for star-gazing. There are several varieties, differing in the way in which the image formed by the mirror is brought

Gregorian

Cassegrainian

Newtonian FIG. 15. — Reflecting Telescopes

within reach of the magnifying eyepiece. Fig. 15 illustrates

three of the most common forms. The Newtonian is most Various

used, but one t)r two large instruments are of the Cassegrainian forms of the

form, which is exactly like the Gregorian shown in the figure

(now almost obsolete), with the exception that the small mirror

is convex instead of concave.

In the Herschelian, or "front view" form, the large mirror is slightly inclined, throwing the rays to the edge of the open end of the tube, so that the secondary mirror is dispensed with, and the observer stands with his back to the object. This is practicable only with very large instruments, since the head

40

MANUAL OF ASTRONOMY

Mirrors of silver on glass.

Superiority of the refracting telescope over the reflecting.

Certain advantages of the reflector.

Large refractors

of the observer partly obstructs the light; the image also is somewhat distorted, and at present this construction is never used.

Until about 1870, the large mirror (technically speculum) was always made of speculum-metal, a composition of copper and tin. It is now usually made of glass, silvered on the front surface by a chemical process. When new, these silvered films reflect much more light than the old speculum-metal ; they tar- nish rather easily, but fortunately can be easily renewed.

50. Relative Advantages of Reflectors and Refractors. — There is much earnest discussion on this point, each form of instru- ment having its earnest partisans. On the whole, however, the refractor is usually better. Up to a certain limit, never yet reached, it gives more light than a reflector of the same size, defines better under all ordinary conditions, has a wider field of view, is more manageable and convenient, and more permanent; the speculum of a reflector usually needs to be resilvered every few years, while a carefully used object-glass never deteriorates.

The reflector is of course far less expensive than a refractor of the same size, and its absolute achromatism is a great advan- tage in certain lines of work, photographic and spectroscopic.

For a fuller discussion of the matter, see General Astronomy.

51. Large Telescopes. — The largest refractors1 at preseiit*(1909) exist- ing are those of the Yerkes Observatory (40 inches in diameter and 65 feet long), and the telescope of the Lick Observatory, which has an aperture of 36 inches and a focal length of 56 feet. There are about fourteen others which have apertures not less than 2 feet. The object lenses of more than half of these instruments, including both of the largest, were made (that is, ground and figured) in this country by the Clarks of Cambridgeport. The glass itself was made by various firms in Europe.

1 No account is taken in this reckoning of the great 48-inch telescope of the Paris Exposition. It is not certain as yet how it will turn out from an astro- nomical point of view.

ASTRONOMICAL INSTRUMENTS

41

The frontispiece is the great Potsdam double telescope, — two mounted together, — one 31^ inches in diameter for photography, the other 20 inches in diameter for visual observations ; the focal length of both is about 43 feet. It was erected in 1899.

At the head of the reflectors stands the enormous instrument of Lord Rosse of Birr Castle, 6 feet in diameter and 60 feet long, made in 1842, and still used occasionally. One still larger, 100 inches in diameter, Large is planned for the Carnegie Solar Observatory on Mt. Wilson, where reflectors, a 5-foot reflector was mounted in 1908. Another 5-foot reflector1 was made by Mr. Common in England in 1889. There are also four or five 4 -foot telescopes, of which Herschel's (erected in 1789, but long ago dismantled) was the first.

At the Lick Observatory is the 3-foot in- strument (made by Mr. Common and pre- sented to the observatory by Mr. Crossley) with which Keeler made his wonderful pho- tographs of nebulae, some of which are figured <&<y //^ ) ) ^\ s> C in the last chapter of this book. Another of 2 -foot aperture is mounted at the Yerkes Observatory, and there is a new 40-inch in- strument at Flagstaff, Arizona.

52. Mounting of a Telescope. — A

telescope, however excellent optically, is of little scientific use unless firmly and conveniently mounted.2

At present nearly all but small portable instruments are mounted as Equatorial*. resents the arrangement schematically. Its essential feature is that the "principal axis" — the one which moves in fixed bear- ings attached to the pier and is called the polar axis — is inclined so as to point towards the celestial pole. The graduated circle H attached to it is therefore parallel to the celestial equator,

FIG. 16. — The Equatorial (Schematic)

The

•IT,. w r> equatorial

Fig. 16 rep- £oimting.

1 Acquired and mounted at the Harvard College Observatory in 1905.

2 We may add that it must be mounted where it can be pointed directly at the stars, without any intervening window-glass between it and the object.

42

MANUAL OF ASTRONOMY

Advantages of equa- torial mounting.

Permits use of clock- work.

Makes it easy to find objects too faint to be

Use of equatorial in determin- ing position of planets or qpmets.

and is usually called the hour-circle of the instrument, — sometimes the right-ascension circle. At the upper extremity of the polar axis a sleeve is fastened, which carries the declination axis D passing through it. To one end of this declination axis is attached the telescope tube T, and at the other end the declination circle (7, and a counterpoise if necessary.

53. The advantages of the equatorial mounting are very great. In the first place, when the telescope is once pointed upon an object it is not necessary to turn the declination axis at all in order to keep the object in view, but only to turn the polar axis with a perfectly uniform motion, which can be, and usually is, given by clockwork (not shown in the figure).

In the next place, it is very easy to find an object, even if invisible to the eye (like a faint comet, or a star in the day- time), provided we know its right ascension and declination and have the sidereal time, — a sidereal clock or chronometer being an indispensable accessory of the equatorial. We set the declination circle by its vernier to the declination of the object and then turn the polar axis until the hour-circle shows the proper hour angle, which is simply the difference between the right ascension of the object and the sidereal time at the moment. When the telescope has been so set the object will be found in the field of view, provided a low-power eyepiece is used. On account of refraction the setting does not direct the instrument precisely to the apparent place of the object, but only very near it.

The equatorial does not give very accurate positions of heavenly bodies by means of the direct readings of its circles, but it can be used as explained later in Sec. 117 to determine with great precision the difference between the position of a known star and that of a comet or planet; and this answers the purpose as well as a direct determination.

ASTRONOMICAL INSTRUMENTS

43

The frontispiece shows the equatorial mounting of the great Potsdam tel- escope. Fig. 173 (Sec. 536) represents another form of equatorial mounting, adopted for several of the instruments of the photographic campaign. Lord Rosse's great reflector is not mounted equatorially, nor was HerschePs 4-foot reflector, but nearly all the other reflectors referred to above are equatorials.

54. Other Mountings. — With very large telescopes this mounting becomes unwieldy, notwithstanding the ingenious electrical and other arrangements by which the observer at

FIG. 17. —The Equatorial Coude

the eyepiece is enabled to control its motions. The enor- mous rotating dome — that of the Yerkes Observatory is 90 feet in diameter — and the requisite elevating floor are also extremely expensive, so that at present there is among astron- omers a tendency to adopt plans by which the telescope may be fixed in its position, while the light is brought to the eye- piece by one or more reflections from plane mirrors.

Fig. 17 represents the smaller equatorial coude, or "elbowed equato- rial," of the Paris Observatory. A silvered mirror at an angle of 45° in

44

MANUAL OF ASTRONOMY

The

equatorial

coude.

Importance

of Huy-

ghens'

invention

of the

pendulum

clock.

the box in front of the object-glass, and another one in the cube at the center of the instrument, effect the necessary changes in the direction of the ray. The observer sits motionless, under cover, at the eyepiece, looking downward towards the south, at an angle equal to the latitude of the place. A much larger, similar instrument, since mounted at the same observatory, has an aperture of 24 inches and a focal length of about 60 feet. Three or four instruments of this sort are now in use.1

Another arrangement is to place the telescope horizontally, pointing towards the south, and to direct the light from the object into it by reflection from the mirror of a so-called siderostat This is a simple plane mirror larger than the object-glass, properly mounted and driven by clockwork so as to send the reflected rays horizontally always in the same direction, and having connections by which its motions can be con- trolled from the eye end of the telescope. The great telescope of the Paris Exposition of 1900 was arranged in this way.

The ccelostat is a slightly different arrangement, in which the plane mirror, mounted upon a polar axis, revolves at half the diurnal rate, and the telescope, while retaining one fixed position for a body in a given declination, has to change its position to observe bodies in a different declination. There are still other forms in which a large reflector is used to give the rays a convenient direction.

But the use of the mirror or mirrors involves considerable loss of light ; and what is worse, if the mirror is large it is extremely difficult to figure the surface with the requisite accuracy, and to prevent slight distortions by variations of temperature and changes of position. As a consequence, definition is seldom as satisfactory as with telescopes pointed directlv to the heavens ; still, in certain operations of astronomical photography, the siderostat and coelostat are extremely useful.

55. Timekeepers and Recorders. — Obviously a good clock or chronometer is an essential instrument of the observatory. The invention of the pendulum clock by Huyghens in 1657 was almost as important to the advancement of astronomy as that of the telescope by Galileo ; and the improvement of the clock and chronometer through the invention of temperature compensation by Harrison and Graham in the eighteenth cen- tury is fully comparable with the improvement of the telescope by the achromatic object-glass.

1 See Addendum A, at beginning of book.

ASTRONOMICAL INSTRUMENTS

45

The astronomical clock differs from any other clock only in The astro- being made with extreme care and in having a pendulum so nom/cal constructed that its rate will not be sensibly affected by changes of temperature. The mercurial pen- dulum is most common, but other forms are also used. (See Fig. 18.)

The pendulum usually beats seconds (rarely half seconds), and the clock face ordinarily has its second-hand, minute-hand, and hour-hand each moving on a separate center, the hour- hand making its revolution not in twelve hours, as in an ordinary clock, but in twenty- four, the hours being numbered accordingly.

In cases where the extremest accuracy of performance is required, the clock is placed in an underground chamber, where the tem- perature varies only slightly or not at all, and is besides inclosed in an air-tight case, within which the air is kept at a uniform pressure, since changes in the density of the air slightly affect the swing of the pendulum. Usually a clock loses about one quarter of a second a day for a rise of one inch in the barometer.

Finally, also, the astronomical clock is usu- ally fitted with some arrangement for making or breaking an electric circuit at every second or every other second, so that its beats can be communicated tele- The break- graphically to all parts of the observatory. The minute is usually circult- marked either by the omission of a second or by a double tick.

56. Error and Rate. — The error or correction of a clock is the Error and amount which must be added (algebraically) to its face indication rate' to give the true time ; + when slow, — when fast. The rate is the amount it loses or gains daily ; + when losing, — when gaining. Sometimes the hourly rate is given instead of the.

FIG. 18

Compensation Pen- dulums

I. Graham's Pendulum 2. Zinc-Steel Pendulum

46 MANUAL OF ASTRONOMY

daily. The error is adjusted by simply setting the hands ; the rate by raising or lowering the pendulum bob, or for delicate final adjustment without stopping the clock, by adding or removing small pieces of metal on the cover of the cylindrical vessel which usually constitutes the pendulum bob.

Perfection in an astronomical clock consists in its maintain- ing a constant rate, i.e., in gaining or losing precisely the same amount each day; for convenience the rate should be small, and is usually kept less than half a second daily. But this is a mere matter of adjustment.

57. The Chronometer. — The pendulum clock not being port- able, it is necessary to provide timekeepers that are so. The chronometer is merely a carefully made watch with a balance- wheel compensated to run, as nearly as possible, at the same rate in different temperatures, and with a peculiar escapement, which, though 'unsuited to ordinary usage, gives better results than any other when treated carefully.

The box chronometer used on shipboard is usually about twice the diameter of a common pocket watch, and is mounted on " gimbals " so as to remain horizontal at all times, notwith- standing the motion of the vessel. It usually beats half seconds.

It is not possible to secure in the chronometer balance as perfect a temperature correction as in the pendulum, and for this and other reasons the best chronometers cannot quite com- pete with the best clocks in precision; but they are sufficiently accurate for most purposes, and of course are vastly more con- venient for field operations, while at sea they are simply indis- pensable. Never turn the hands of a chronometer backward ; it may ruin the escapement.

58. Eye-and-Ear Method of Observation. — The old-fashioned method of time observation consisted simply in noting by " eye and ear " the moment (in seconds and tenths of a second) when the phenomenon occurred ; as, for instance, when a star passed some wire of the reticle. The tenths, of course, are merely

ASTRONOMICAL INSTRUMENTS

47

estimated, but the skilful observer seldom errs by a whole tenth in his estimation. Skill and accuracy in this method are acquired only by long practice.

59. Telegraphic Method ; the Chronograph At present observation

such observations are usually made by the help of electricitv. by means of

J J J thechrono-

50s.

x

9 h. 35 m. oo.o s. graph.

FIG. 19. —A Chronograph Record

FIG. 20. — A Chronograph By Warner & Swasey

The clock is so arranged that at every beat (or every other beat) of the pendulum an electric circuit is made or broken for an instant, and this causes a sudden sideways jerk in the

48 MANUAL OF ASTRONOMY

armature of an electromagnet, like that of a telegraph sounder. This armature carries a fountain-pen, which writes upon a sheet of paper wrapped around a cylinder six or seven inches in diameter, which cylinder itself is turned uniformly by clock- work once a minute ; at the same time the pen carriage is drawn slowly along, so that the marks on the paper form a continuous helix, graduated into second or two-second spaces by the clock beats. When taken from the cylinder, the paper presents the appearance of an ordinary page crossed by parallel lines spaced off into two-second lengths, as shown in Fig. 19, which is part of an actual record.

Fig. 20 represents a chronograph of the usual American form.

The observer, at the moment when a star crosses the wire, presses a "key" which he holds in his hand, and thus inter- polates a mark of his own among the clock beats on the sheet ; as, for instance, at X and Y in the figure. Since the beginning of each minute is indicated on the sheet in some way by the mechanism which produces the clock beats, it is very easy to read the time of X and Y by applying a suitable scale, the beginning of the mark made by the key being the moment of observation.

In the figure the initial minute marked when the chronograph was started happened to be 9h35m, the zero in the case of this clock being indicated by a double beat. The signal at X, therefore, was made at 9h35m558.45, and that of Y at 9h36m588.63. The "rattle" just pre- ceding X was the signal that a star was approaching the transit wire.

In European observatories the record is usually made by a more simple but less convenient apparatus upon a long fillet or ribbon of paper drawn slowly along. At a few observatories in this country a more complicated printing chronograph, invented by Professor Hough of the Dearborn Observa- tory, is used. By this the minutes, seconds, and hundredths of a second are actually printed upon the fillet in type,' like the record of sales on a stock telegraph.

60. Meridian Observations. — A large proportion of all astro- nomical observations for determining the positions of the

ASTRONOMICAL INSTRUMENTS

49

heavenly bodies are made when the body is crossing the meridian or is very near it. At that time the effects of refraction and parallax (to be discussed later) are a minimum, and as they act only vertically they do not affect the time when a body crosses the meridian nor, consequently, its observed right ascension. In any other part of the sky both these coordinates are affected, and the calculation of the correction requires the computation of the uparallactic angle" in the astronomical triangle (Sec. 31).

61. The transit-instrument is the instrument used in connec- tion with a sidereal clock or chronometer, and often with a chronograph, to observe the time of a star's transit, or passage across the meridian. If the " error " of the sidereal clock at the moment is known and allowed for, the corrected time of the observation will be the right ascension of the star (Sec. 26).

Vice versa, if the right ascen- sion is known, the error of the clock will be the difference be- tween the right ascension of the object and the time observed.

The instrument (Fig. 21) con- sists essentially of a telescope carrying at the eye end a reticle and mounted on a stiff axis that turns in V-shaped bearings called "Y's," which can have their position adjusted so as to make the axis exactly perpendicular to the .meridian. A delicate spirit-level, which can be placed upon the pivots of the axis to measure any slight deviation from horizontality, is an essential accessory ; and it is practically necessary to have a small graduated circle attached to the instrument, in order to set it at the proper elevation for the star which is to be observed.

Advantage of observa- tions on the meridian.

FIG. 21. — The Transit-Instrument

The transit- instrument.

The level, setting- circle, and reversing apparatus.

50

MANUAL OF ASTRONOMY

It is desirable, also, that the instrument should have a reversing apparatus by which the axis may be easily lifted and safely reversed in the Y's without jar or shock.

The reticle usually contains from five to fifteen "vertical wires " crossed by two horizontal ones. Fig. 22 shows the reticle of a small transit intended for observations by " eye and ear." When the chronograph is to be used, the wires are much more numerous and placed nearer together.

In order to make the wires visible at night the field must be illuminated. For this purpose one of the pivots of the instrument is pierced (some- times both of them), so that the light from a lamp will shine through the axis upon a small reflector placed in the central cube of the instrument, where the axis and the tube are joined. This sends sufficient light towards the eye to illuminate the field, while it does not cut off any considerable portion of the rays from the object.

FIG. 22. — Reticle of the Transit-Instrument

The observation consists in noting the instant, as shown by the clock or chronometer, in hours, minutes, seconds, and tenths of a second, at which the star crosses each wire of the reticle.

62. The instrument, must be thoroughly rigid, without any loose joints or shakiness, especially in the mounting of the object- glass and reticle. Moreover, the two pivots should be of the same diameter, accurately round, without taper, and precisely in line with each other ; in other words, they must be portions of one and the same geometrical cylinder. To fulfil this condition taxes the highest skill of the mechanician.

When exactly adjusted, the middle wire of such an instru- ment always precisely coincides with the meridian, however the instrument may be turned on its axis; and the sidereal time when a star crosses that wire is therefore the star's right ascension.

ASTRONOMICAL INSTRUMENTS

51

Another form of the instrument now much used is often called the broken transit, of which Fig. 23 is a representation. A reflector (usually a right-angled prism) in the central cube of the instrument directs the rays horizontally through one end of the axis where the eyepiece is

FIG. 23. — A Broken Transit By Warner & Swasey

placed, so that whatever may be the elevation of the star the observer looks straight forward horizontally, without needing to change his position. The instrument is very convenient, but is usually subject to rather a large error, due to flexure of the axis, which, even if it exists, produces no such effect in transits of ordinary form. The error is, however, easily determined and allowed for if the axis is not too slender.

MANUAL OF ASTRONOMY

Necessary adjust- ments.

Tests of adjust- ments.

Non-perma- nence of adjust- ments.

63. Adjustments of the Transit. — These are four in number-,

(1) The reticle must be exactly in the focal plane of the object-glass and the middle wire accurately vertical.

(2) The line of collimation (i.e., the line which joins the optical center of the object-glass to the middle wire) must be exactly perpendicular to the axis of rotation. This may be tested by pointing on a distant mark and then reversing the instrument. The middle wire must still bisect the mark after the reversal. If not, the reticle must be adjusted by the screws provided for the purpose.

(3) The axis must be level. This adjustment is made mechan- ically by the help of the spirit-level. One of the Y's has a screw by which it can be slightly raised or lowered, as may be necessary.

(4) The azimuth of the axis must be exactly 90° ; i.e., the axis must point exactly east and west. This adjustment is made by means of star observations, with the help of the side- real clock.

Without going into detail, we may say that if the instrument is correctly adjusted, the time occupied by a star near the pole, in passing from its transit across the middle wire above the pole to its next transit below the pole, must be exactly twelve sidereal hours. Moreover, if two stars are observed, one near the pole and another near the equator, the difference between their times of transit ought to be precisely equal to their differ- ence of right ascension. By utilizing these principles the astronomer can determine the error of azimuth adjustment and correct it.

But it is to be remembered that no adjustments, however carefully made, will be absolutely exact or remain permanently correct, on account of changes in temperature which affect the instrument and the pier on which it is mounted. In cases where extreme accuracy of results is required, the slight errors which remain after the most careful adjustment must be

ASTRONOMICAL INSTRUMENTS 53

determined from the observations themselves by means of the little discrepancies between the results obtained from stars at different distances from the pole. The methods to be used are taught in practical astronomy.

64. Personal Equation. — It is found that skilled observers Personal are in the habit of noting the passage of a star across the e(iuatlon- transit wire slightly too late or too early by an amount which is different for each observer, but nearly constant for each. This is called the observer's personal equation, and in some cases for eye-and-ear observation is as much as half a second, In the telegraphic method it is much less, seldom exceeding Os..l. It is an extremely troublesome error, because it varies with the nature and brightness of the object and with the observer's position and physical condition.

Various devices have been proposed for dealing with it ; either by Mechanical measuring its amount, or by eliminating it by means of some apparatus method of

which reduces the observation to the accurate bisection of the star disk, ^e m^ ri made to appear to be at rest by a clockwork motion given to the eyepiece, equation and carrying with it a " micrometer wire " which is under the control of the observer. When the bisection is satisfactory he touches a key which instantly stops the motion and registers the time upon the chronograph ; afterwards, at his leisure, he measures the distance of his micrometer wire from the central wire of the reticle. In this way the disturbing effect of the star's motion is eliminated.

65. The Photochronograph. — Another method, and one of the most Photo- promising, is by means of photography. The eyepiece of the transit is graphic removed, and a small photographic plate, about as large as a microscope ° sei slide, is placed just back of the reticle, so arranged in the frame which holds it that it can move up and down slightly under the action of an electromagnet connected with the standard-clock circuit. When a star impresses its "trail" on the plate, the trail is broken every second (or every other second) by the clock, like the marks on a chronograph sheet, so that it consists of a row of small dashes. The image of the reticle wires is also imprinted upon the plate by holding a small lamp for an instant in front of the object-glass.

During the passage of the star some particular second is marked on the plate by cutting off the clock circuit for two or three seconds, or by

54

MANUAL OF ASTRONOMY

The prime

vertical

instrument.

Its use.

The almu- cantar and its use.

making a rattle, allowing the beats to resume their regular course at some instant recorded in the note-book. After the plate is developed, its inspec- tion and measurement under a microscope will show at what second and fraction of a second the star passed each reticle wire. But this part of the operation is laborious. On the other hand, the expensive and troublesome chronograph is dispensed with.

66. The Prime Vertical Instrument. — For certain purposes a transit-instrument, provided with an apparatus for rapid reversal, is turned quarter way round and mounted with its axis north and south, so that the plane of rotation lies east and west instead of in the meridian. It is then called the " prime vertical instrument." It may be used for determining the lati- tude of the observer, the precise declination of such stars as cross the meridian between the zenith and equator, and any minute change due to "aberration" and to slight movements of the terrestrial pole. (See Sec. 94.)

The observation consists in noting the instant when the star crosses (obliquely) the middle wire of the reticle.

67. The Almucantar. — This is an instrument invented about 1885 by Dr. S. C. Chandler of Cambridge, U.S., for the pur- pose of observing the time at which stars cross, not the meridian or any vertical circle, but some given parallel of altitude, usually the " almucantar " of the pole. From such observations can be determined with great accuracy the error of the clock, the decli- nation of the stars observed, or the latitude of the observer.

It consists of a firm base carrying a tank containing mercury, on which swims a float which carries the observing telescope, its inclination being preserved absolutely constant by the prin- ciple of flotation. This dispenses with the necessity of using spirit-levels (which are always more or less unsatisfactory) for determining the inclination.

The telescope is sometimes placed horizontally on the float, while a mirror in front of its object-glass brings down the rays of the star. Two such instruments of considerable size have been built since 1899 and give prom- ising results, — one at Durham, England, the other at Cleveland, Ohio.

ASTRONOMICAL INSTRUMENTS

55

68. The Meridian-Circle This is a transit-instrument of The merid-

large size and most careful construction, with the addition of a ian~circle :

essentials of

large graduated circle attached to the axis and turning with it. its construc- tion.

FIG. 24. —Meridian-Circle in United States Naval Observatory, Washington By Warner & Swasey

The utmost resources of mechanical art are expended in gradu- ating this circle with precision. The divisions are now usually made either two minutes or five minutes of arc, and the farther

56

MANUAL OF ASTRONOMY

Its zero points.

Determina- tion of the polar point.

Determina- tion of the nadir point.

subdivision is effected by so-called "reading microscopes," four of which at least are always used in the case of a large instrument. (For a description of the reading microscope, the reader is referred to Creneral Astronomy, Art. 64, or to Camp- bell's Practical Astronomy.) By means of these microscopes the "reading of the circle" is made in degrees, minutes, sec- onds, and tenths of a second of arc, the tenths being obtained by estimation.

On a circle 2 feet in diameter 1" of arc is only about T7^ part of an inch ; an error of that amount is now very seldom made by reputable constructors in placing a graduation line, or by a good observer in reading the instrument with the microscope.

Fig. 24 represents the new meridian-circle of the United States Naval Observatory at Washington, with a 6-inch telescope and circles about 27 inches in diameter.

69. Zero Points. — The instrument is used to measure the altitude or else the polar distance of a heavenly body at the time when it is .crossing the meridian. As a preliminary we must determine some zero point upon the circle, — the nadir point or horizontal point, if we wish to measure altitudes or zenith-distances ; the polar point or equator point, if polar dis- tances or declinations. The polar point is determined by taking the circle reading for some star near the pole when it crosses the meridian above the pole, and then doing the same thing again twelve hours later when it crosses it below. The mean of the two readings corrected for refraction will be the reading which the circle would give when the telescope is pointed exactly to the pole, — technically, the polar point. The equator point is, of course, 90° from this.

The nadir point is the reading of the circle when the tele- scope is pointed vertically downward. It is determined by the reading of the circle when the instrument is so set that the horizontal wire of the reticle coincides with its own image formed by a reflection from a basin of mercury placed on the

ASTRONOMICAL INSTRUMENTS

57

pier below the instrument. To make this reflected image visible it is necessary to illuminate the reticle by light thrown towards the object-glass from behind the wires, — the ordinary The colii illuminatiori used during observation comes from the opposite direction. This peculiar illumination is effected by what is known as the "collimating eye- piece." A thin glass plate inserted at an angle of 45° between the lenses of a Ramsden eyepiece throws down sufficient light, ad- mitted through a hole in the side of the eyepiece, and yet permits the observer to see the wires and their reflected image. The zenith point is, of course, just 180° from the nadir point thus determined.

Obviously, the meridian-circle can be used simply as a transit, so that with this instrument and a clock the observer is in a position to determine both the right ascension and declination of any heavenly body that can be seen when it crosses the meridian.

FIG. 25. — A 5-inch Altazimuth By Warner & Swasey

Extra- meridian observa- tions.

70. Extra-Meridian Observa- tions. — Many objects, however, are not visible when they cross the meridian; a comet, for in- stance, or a planet, may be in such a part of the heavens that it transits only by daylight. To observe such objects we may employ a so-called universal instru- ment, or astronomical theodolite, which is simply an instru- Theuniver- ment with both horizontal and vertical circles like a large salmstrn-

ment, or

surveyor's theodolite and is also called an altazimuth. By means altazimuth of this the altitude and azimuth of an object may be measured,

58

MANUAL OF ASTRONOMY

Extra- meridian observa- tions with the equa- torial.

and, if the time is given, from these the right ascension and declination can be deduced.

Fig. 25 shows the 5-inch altazimuth of the Washington Observatory.

More often, however, observations for the positions of bodies not on the meridian are made with the equatorial telescope

already described, with which the difference between the right ascension and declination of the observed body and that of some star in its neighbor- hood is determined by means of a micrometer or, at present, often by photography.

71. The Micrometer. - There are various forms of micrometers, the most common and generally useful being that

(A) r^QBJ — — ^lyilpW known as the filar-position mi-

crometer, shown in Figs. 26 A

FIG. 26. — The Filar-Position Micrometer

and B. It is a comparatively small instrument which is attached at the eye end of the telescope. It usually contains a set of fixed wires, two or three of them parallel to each other (only one, e, is shown in B, which represents the internal construction

ASTRONOMICAL INSTRUMENTS

59

of the instrument), crossed at right angles by a single line or set of lines. Under the plate which carries the fixed threads lies a fork moved by a carefully made screw with a graduated head, and this fork carries one or more wires parallel to the first set, so that the distance between the wires e and d (Fig. 26 B) can be varied at pleasure and read off by means of the screw- head graduation.

The box containing the wires is so arranged that it can itself be rotated around the op- tical axis of the telescope and set in any desired " position " ; for example, so that the movable wire d shall be parallel to the celestial equator when the position circle F should read 90°. When so set that the movable wire points from one star to another in the field of view, the " po- sition angle " (see Fig. 191, Sec. 585) can be read off on the circle F.

With such a micrometer we can measure at once the distance in seconds of arc between any two stars which are near enough to be distinctly seen in the same field of its use and view, and can determine the position angle of the line joining them. The available range in a small telescope may reach 30'. In large telescopes, which with the same eyepieces give much higher magnifying powers, the range is correspondingly less, — not more than from 5' to 10'. When the distance between the objects exceeds 2' or 3', the filar micrometer becomes difficult

FIG. 27. — Position Micrometer By Warner & Swasey

60 MANUAL OF ASTRONOMY

to use and inaccurate, because the observer cannot see both objects distinctly at the same time.

Fig. 27 is a complete micrometer, fitted with electric illumination.

The heiiom- 72. The Heliometer. — For the measurement of larger dis- eter:itscon- Dances not exceeding two or three degrees the heliometer is used. This is a complete equatorially mounted telescope with its object-glass (usually from 4 to 8 inches aperture) diametrically divided into two halves which can be made to slide past each other for 3 or 4 inches (Fig. 28), the distance being measured on a delicate scale read by long microscopes A\ Ao A2 which come down to the end of the instru- x ment. The telescope tube can be rotated j in its cradle so as to make the line of division of the lenses lie in any desired position.

When the objectrglass scale is at zero, MI MO Ma the two half lenses act as a single lens ^i S0 S* and each object in the field of view pre-

sents a single image, as Sn and Mn in

FIG. 28. — The Heliometer

the figure. But as soon as one of the

semi-lenses is pushed past the other, two images of each object appear, and the distance and direction between them can be varied at pleasure by sliding the lenses and rotating the tube.

The distance between any two different objects is measured by making their images coincide (as, for instance, M± with $0, or £2 with MQ), and the observer does not have to " look two ways at once," nor is he obliged to trust to the stability of his instru- ment or the accuracy of the clockwork motion.

On the whole, the heliometer stands at the head of astro- nomical instruments for the precision of its results and is employed in the most delicate investigations, like those upon solar and stellar parallax (Sees. 467 and 550). But it is a

ASTRONOMICAL INSTRUMENTS 61

very complicated and costly instrument, and extremely laborious The rank

to «Se' heHoLte,

The only one in the United States at present is the 6-inch among

instrument at the Yale University Observatory. astronomi-

At present, however, such measurements of the distance of ments. an object from neighboring stars are very generally effected by

means of photography. Photographs of the field of view con- Observa-

taininor the object are made and afterwards measured, and in tlonsby

means of

this case the limits of distance between the object and the stars photog- to which it is referred can be very much increased without raPhy- lessening the accuracy of the determination.

73. The Sextant. — All the instruments so far mentioned, The sextant: except the chronometer, require some firmly fixed support, and the mstru~ are therefore absolutely useless at sea. The sextant is the only mariner, one upon which the mariner can rely. By means of it he can measure the angular distance between two points (as, for instance, between the sun and visible horizon), not by pointing first to one and afterwards to the other, but by sighting them its peculiar both simultaneously and in apparent coincidence, a " double advantage image " measurement, in which respect the sextant is analo- instruments, gous to the heliometer. A skilful observer can make the measurement accurately even when he has no stable footing.

Fig. 29 represents the instrument. Its graduated limb is usually, as its name implies, about a sixth of a complete circle, with a radius of from 5 to 8 inches. It is graduated in itscon- half degrees (which are, however, numbered as whole degrees) structlon- and so can measure any angle not much exceeding 120°. The index arm, or " alidade " (MN in the figure), is pivoted at the center of the arc and carries a "vernier," which slides along the limb and can be fixed at any point by a clamp, with an attached tangent screw T. The reading of this vernier gives the angle measured by the instrument; the best instruments read to 10" only, because it is impracticable to use a telescope with very much magnifying power.

62

MANUAL OF ASTRONOMY

Just over the center of the arc the index-mirror M, about 2 inches by li in size, is fastened to the index arm, moving with it and keeping always perpendicular to the plane of the limb. At H the horizon-glass, about an inch wide and about twice the height of the index-glass, is secured to the frame of the instrument in such a position that when the vernier reads zero the index-mirror and horizon-glass will be parallel to each

M

IS'

FIG. 29. — The Sextant

other. Only half of the horizon-glass is silvered, the upper half being left transparent. E is a small telescope screwed to the frame and directed towards the horizon-glass.

If the vernier stands near, but not exactly at, zero, an observer looking into the telescope will see together in the field of view two separate images of the object towards which the telescope is directed; and if he slides the vernier, he will see that one of the images remains fixed while the other moves. The fixed image is formed by the rays which reach the object-glass directly through the unsilvered half of the horizon-glass ; the movable

ASTRONOMICAL INSTRUMENTS 63

image, on the other hand, is produced by rays which have Double suffered two reflections, having been reflected from the index- ima&e

& formed by

mirror to the horizon-glass and again reflected a second time sextant, from the lower, silvered half of the horizon-glass. When the two mirrors are parallel the two images coincide, provided the object is at a considerable distance.

If the vernier does not stand at or near zero, an observer Angle looking at an object directly through the horizon-glass will see not only that object, but also, in the same telescopic field of whose view, whatever other object is so situated as to send its rays imagescoin-

. . , cide equals

to the telescope by reflection from the mirrors ; and the reading half the of the vernier will give the angle at the instrument between the two ansle objects whose images thus coincide, — the angles between the mirrors planes of the two mirrors being, as easily proved, just half the angle between the two objects, and the half degrees on the limb being numbered as whole ones.

74. The principal use of the instrument is in measuring the altitude of the sun. At sea the observer usually proceeds as follows: first, setting Method of the index, loosely clamped, near zero and holding the sextant in his right observation hand with its plane vertical, he points the telescope towards the sun ; then at sea" he slides the vernier along the arc with his left hand until he brings the reflected image of the sun down to the horizon, all the time keeping it in view in the telescope; finally, tightening the clamp and using the tangent screw, he makes the lower edge or limb of the sun just graze the horizon as he swings the sun's image back and forth by a slight motion of the instru- ment — it would be impossible on board ship to hold the image in contact with the horizon, and is not necessary. As soon as the contact is satis- factory he marks the time and afterwards reads the angle. The reading of the vernier after due corrections (see next chapter) gives the sun's true altitude at the moment.

On land we have recourse to an " artificial horizon." This is a shallow Artificial basin of mercury covered with a roof of glass plates having their surfaces horizon used accurately plane and parallel. In this case we measure the angle between ° the sun and its image reflected in the mercury. The reading of the instru- ment corrected for index error then gives twice the sun's apparent altitude.

The skilful use of the sextant requires considerable dexterity, and from the low power of the telescope the angles measured are less precise than

64

MANUAL OF ASTRONOMY

those determined by large fixed instruments, but the portability of the instrument and its applicability at sea render it invaluable. It was invented in practical form by Godfrey of Philadelphia, in 1730, though Newton, as was discovered by Halley, had really struck upon the same idea long before.

Demonstra- 75. The principle that the angle between the objects whose images tion of the coincide in the sextant is twice the angle between the mirrors (or between ^cTsextant tne*r normals) is easily demonstrated as follows :

The ray SM (Fig. 30) coming from an object, after reflection first at M (the index-mirror) and then at H (the horizon-glass), is made to coincide with the ray OH coming from the horizon.

From the law of reflection, we have the two angles SMP and PMH

equal to each other, each being x. In the same way the two angles marked y are equal. From the geometric principle that the angle SMH, exterior to the triangle HME, is equal to the sum of the oppo- site interior angles at H and E, we get E = 2 x — 2 y. Similarly, from the tri- angle HM Q, Q = x - y ; whence E = 2 Q = 2 Q'.

76. With the instruments above described all the fundamental obser- vations required in the investiga- tions of spherical and theoretical astronomy can be supplied, the sex- tant and chronometer being, however, the only ones available in nautical astronomy.

Astrophysical studies require numerous physical instruments of an entirely different character, — spectroscopes, photometers, heat-measuring instruments, and various kinds of photographic apparatus. These will be considered later, as occasion arises.

FIG. 30. — Principle of the Sextant

ASTRONOMICAL INSTRUMENTS 65

EXERCISES

1. If a firefly were to alight on the object-glass of a telescope, what would be the appearance to an observer looking through the instrument ? Would he think he saw a comet ?

2. When a person is looking through a telescope, if you hold your finger in front of the object-glass, will he see it?

t/ 3. If half the object-glass of a telescope pointed at the moon is covered, how will it affect the appearance of the moon as seen by the observer ? ]/ 4. If a certain eyepiece gives a magnifying power of 60 when used with a telescope of 5 feet focal length, what power will it give on a tele- scope of 30 feet focal length ?

5. What is theoretically the angular distance between the centers of two star disks which are just barely separated by a telescope of 24 inches aperture (Sec. 46) ?

V 6. Why is it important that the two pivots of a transit-instrument should be of exactly the same diameter?

7. If the wires of a micrometer (Fig. 26) are so set that, used with a telescope of 10 feet focal length, a star moving along the right-ascension wire will occupy 15 seconds in passing from d to e, how long will it take when the micrometer is transferred to a telescope of 50 feet focus ?

8. If the threads of a micrometer screw are 7V of an inch apart, what is the angular value of one revolution of the screw when the micrometer is attached to a telescope of 30 feet focal length ?

9. Does changing the eyepiece of a telescope for the purpose of altering the magnifying power affect the value of the revolution of the microscope screw V

CHAPTER III

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS

Dip of the Horizon — Parallax— Semidiameter — Refraction — Twinkling or Scintil-

lation—Twilight

OBSERVATIONS as actually made always require corrections before they can be used in deducing results. Those that depend on the errors or maladjustment of the instrument itself will not be considered here, but only such as are due to other causes external to the instrument and the observer.

77. Dip of the Horizon -- In observations of the altitude of a heavenly body at sea, where the sextant measurement is made from the visible horizon, or sea-line, it is necessary to take into account the depres- sion of the visible below the true astro- nomical horizon by a small angle called the dip. The amount of this dip depends upon the observer's altitude above the sea-level. In Fig. 31 C is the center of the earth, AB a portion of its level surface, and 0 the eye of the observer at an elevation h above A. The line drawn perpendicular to OC is truly horizontal (regarding the earth as spherical), while the tangent OB

is the line drawn from 0 to B, the visible horizon. The angle HOB is the dip, and is obviously equal to OCB. From the triangle OCB we have

FIG. 31. — Dip of the Horizon

cos OCB = CB/ CO = designating the dip by A.

= cos A,

66

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS 67

The formula in this shape is inconvenient, because it deter- Formui»for mines a small angle by means of its cosine. But since 1 — cos A = 2 sin2 £ A, we easily obtain the following :

h)

Or, since A is always a small angle, and neglecting h in the denominator of the fraction as being insignificant compared with jft, we get

This gives with quite sufficient accuracy the true depression of the sea horizon as it would be if the line of sight were straight. But this is not the case, owing to refraction of the rays in pass- ing through the air, and the amount of this refraction is very uncertain and variable. Ordinarily the dip is diminished about one eighth of the amount computed by the formula.

An approximate formula, obtained by substituting the radius of the earth (20 890000 feet) and reducing, gives A' (i.e., in

minutes of arc) = 3438 (Sec. 9), whence A'

mula for

= V h (feet) (nearly) ; or, in words, the dip in minutes of arc dip.

equals the square root of the observer's elevation in feet; i.e., the dip is 1' at an elevation of 1 foot, 5' at an elevation of 25 feet, 10' at an elevation of 100 feet, etc.

This result is generally about five per cent too large, taking into account refraction ; but it is near enough for most practical purposes, since at sea the observer is seldom as much as 50 feet above the sea-level and cannot, with a sextant, measure altitudes more closely than to the nearest quarter of a minute.

The formula A' = V3 h (meters) agrees still more nearly with the actual value.

68

MANUAL OF ASTRONOMY

The distance OB of the sea horizon is easily seen, from Fig. 31, to be

3 h

Formula for

distance of R tan A. An approximate formula is, distance in miles =

sea horizon.

This, however, takes no account of refraction, and the actual distance is always greater.

General definition of parallax.

Annual or

heliocentric

parallax.

Diurnal or geocentric parallax.

Horizontal parallax.

78. Parallax (Fig. 32). — In general the word "parallax" means the difference between th^ direction of a heavenly body as seen by the observer and as seen from some standard point of reference.

The annual or heliocentric parallax of a star is the difference

of the star's direction as seen from the earth and from the sun. With this we have nothing to do for the present.

The diurnal or geocentric paral- lax of the sun, moon, or a planet is the difference of its direction as seen from the center of the earth and from the observer's sta- tion on the earth's surface ; or, what comes to the same thing, it is the angle at the body made by two lines drawn from it, one to the observer, the other to the center of the earth. In Fig. 32 the parallax of the body P is the angle OP C, which equals xOP, and is the difference between ZOP and ZCP. Obviously this parallax is zero for a body directly overhead at Z, and a maximum for a body rising at H. Moreover, and this is to be specially noted, this paral- lax of a body at the horizon — the horizontal parallax — is simply t he angular semidiameter of the earth as seen from the body. When we say that the moon's horizontal parallax is 57', it is equivalent to saying that, seen from the moon, the earth has an apparent diameter of 114'.

FIG. 32. — Parallax

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS 69

79. Law of the Parallax. — From the triangle OOP we have

PC : OC = sin COP : sin CPO, or, R : r = sin f : sin p (since C OP is the supplement of ?).

This gives Formula

Sin^ = -Sm?, (a) embodying

jg the laws

of diurnal

or, since p is always a small angle, parallax.

p" = 206265" - sin £ (b)

£kt

When a body is at the horizon its zenith-distance is 90° and sin f=l. Hence, the horizontal parallax, II", of the body is given by the formula

sinn = -, or II" = 206265-, (<?); and /' = II" sin ?. (d) ./? It

Or, in words, the parallax at any altitude equals the horizontal parallax multiplied by the sine of the apparent zenith-distance.

From equation (<?) we have also, for finding R, the distance Relation

Of the body, between dis-

tance of a

*= or £ = , (e} body and its

Sin II II" parallax.

a relation of great importance as determining the distance of a heavenly body when its parallax is known.

80. Equatorial Parallax. — Owing to the " ellipticity," or " oblateness," of the earth, the horizontal parallax of a body varies slightly at different places, being a maximum at the equator, where the distance of an observer from the earth's center is greatest. It is agreed to take as the standard the equatorial-horizontal-parallax, i.e., the earth's equatorial semi- Equatorial diameter in seconds as seen from the body. parallax.

If the earth were exactly spherical, the parallax would act in an exactly vertical plane and would simply diminish the altitude of the body without in the least affecting its azimuth.

70

MANUAL OF ASTRONOMY

Really, however, it acts along great circles drawn from the geo- centric zenith to the geocentric nadir (Sec. 11), and these circles are not identical with the vertical circles nor exactly normal to the horizon. For this reason the azimuth of the moon, which has a parallax of about a degree, is sensibly affected. The calculation of the parallax corrections to observations of the moon's right ascension and declination is also modified and greatly complicated. (See Campbell's Practical Astronomy, Sec. 26.)

In the calculation of the parallax of all other bodies it is sufficient to regard the earth as spherical.

81. Semidiameter. — In the case of the sun or moon the edge, or limb, of the object is usually observed, and to get the true position of its center the angular semidiameter must be added or subtracted. For all objects except the moon this may be taken directly from the ephemerides, but the moon's appar- ent diameter increases slightly with its altitude, being about FIT Par^ °r about 30", greater when in the zenith than at the horizon, because at the zenith it is about 4000 miles, or ^ part of its whole distance from the center of the earth, nearer than at the horizon. At any observed zenith-distance, OP (Fig. 32), the apparent or "augmented" semidiameter («'), as seen from 0, is greater than the semidiameter (*) given in the ephemeris as seen from C, in the ratio of PC to PO. From the triangle P OC we obtain, therefore,

sf : s::PC:PO: : sinPOC: sinPCO: : sin f : sin (£— p) (f being the apparent zenith-distance).

Whence

sn

sin(f—

This "augmentation" of the moon's diameter, amounting to about 30" near the zenith, has, of course, nothing whatever to do with the opti- cal illusion already referred to which makes the moon seem larger when near the horizon.

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS 71

82. Refraction. — As the rays of light from a star enter our atmosphere, unless they strike perpendicularly they are bent downwards by refraction and follow a curved path, as illus- trated in Fig. 33.

Since the object is seen in the direction from which the rays enter the eye, the effect is to make the apparent altitude of the object greater than the true.

Refraction, like parallax, is zero at the zenith and a maxi- mum at the horizon, where under average conditions it lifts an object about 35', leaving the azimuth, however, unchanged. But the law of refraction is very different from that of parallax.

Its amount depends upon the den- sity of the air (which is determined by the barometric pressure and tem- perature) as well as the altitude of the object, but is independent of its distance.

The theory of refraction is too

Astro- nomical refraction.

Its effect to increase the apparent altitude of a body.

7?'

FIG. 33.— Atmospheric Refraction

complicated to be discussed here, and

the reader is referred to Campbell's or Chauvenet's Practical

Astronomy.

The computation of the correction when precision is required is made by means of elaborate tables provided for the purpose and given in works on practical astronomy, the data being the observed altitude of the object, the temperature, and the height of the barometer. Increase of atmospheric pressure slightly increases the refraction, and increase of temperature diminishes it.

For altitudes exceeding 25° the following approximate for- mula, corresponding to a temperature of zero Centigrade (32° Fahrenheit) and a barometric pressure of 30 inches, may be used, and will generally give results correct within a few seconds, viz., r" = 60". 7 tan ?, in which £ is the apparent zenith- distance.

Affected by temperature and baro- metric pressure.

Approxi- mate for- mulae for bodies above 15° altitude.

72

MANUAL OF ASTRONOMY

Refraction table in Appendix.

Effect of refraction to increase length of the day at expense of the night.

The following formula (due to Professor Comstock) is a little more complicated, but much more accurate, viz.,

r» =

9835 460 +

tan

in which b is the height of the barometer in inches and t is the temperature on Fahrenheit's scale. For altitudes above 15° this formula will seldom be over V in error.

The little Table VIII (Appendix) gives by inspection pretty accurately the refraction under the circumstances stated in its heading ; and by applying the approximate corrections for barometer and thermometer indicated in the note below it, the results will seldom be more than 2" in error.

It is hardly necessary to add that this refraction correction, required by most astronomical observations of position, is very troublesome, and usually involves more or less uncertainty and error from the continually changing and unknown condi- tion of the atmosphere along the path followed by the rays of light.

For methods by which the amount of the refraction is deter- mined by observation, the reader is referred to works on practical astronomy, or to the author's General Astronomy, Art. 94.

83, Effect of Refraction near the Horizon. — The horizontal refraction, ranging as it does from 32' to 40', according to meteorological conditions, is always somewhat greater than the diameter of either the sun or the moon. At the moment, therefore, when the sun's lower limb appears to be just rising or setting, the whole disk is really below the" plane of the hori- zon ; and the time of sunrise in our latitudes is thus accelerated from two to four minutes, according to the inclination of the sun's diurnal circle to the horizon, which varies with the time of the year. Of course, sunset is delayed by the same amount, and thus at both ends the day is lengthened at the expense of the night.

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS 73

Near the horizon the refraction changes very rapidly ; while Effect of

under ordinary summer temperature it is about 35' at the hori- refraction

zon, it is only 29' at an elevation of half a degree, so that as form of the

the sun or moon rises the bottom of the disk is lifted 6' more disks of sun

than the top and the vertical diameter is thus made apparently when °°"y

about one-fifth part shorter than the horizontal. This quite near the notably distorts the disk into the form of an oval flattened on the under side. In cold weather the effect is much more marked.

Two other semi-astronomical effects, the twinkling of the stars and twilight, are due to the action of our atmosphere, and may be treated in this connection, though in no other way con- nected with the principal subject of the chapter.

84. Twinkling or Scintillation of the Stars. — This is a purely Scintilla- atmospheric phenomenon, usually conspicuous near the horizon, t where it is often accompanied by marked changes of color, phenome- Near the zenith it generally disappears, and at other altitudes non' it differs greatly on different nights. As a rule only the stars twinkle strongly ; the planets, Mercury excepted, usually shine with an almost steady light.

Authorities differ as to the details of explanation, but prob- ably scintillation is mainly due to two cooperating causes, both depending on the fact that the air is generally full of streaks and wavelets of unequal density carried by the wind.

(1) Light coming through such a medium is concentrated in Unequal some places and diverted from others by simple refraction, like refractlons

by drifting

light from an electric lamp shining through an ordinary window- atmospheric pane upon the opposite wall. If the light of a star were strong wavelets of enough, a white surface illuminated by it would be covered by density, bright and dark mottlings, drifting with the wind ; and as such mottlings pass the eye the star appears to fade and brighten by turns. Looked at in the telescope, it also "dances," being slightly displaced back and forth by the irregular refraction.

74

MANUAL OF ASTRONOMY

Supplemen- tary action of optical " inter- ference."

Effect upon the spec- trum of a star.

Why planets do not twinkle.

(2) The other cause of twinkling is optical interference. Pen- cils of light coming from a star (optically a mere luminous point] reach the observer's eye by routes differing only slightly, and are just in a condition to "interfere." The result is the temporary destruction of rays of certain wave-lengths and the reinforcement of others. Accordingly, the "spectrum" (Sec. 569) of a twinkling star is traversed by dark bands in the different colors, oscillating back and forth, but, on the whole, when the star is rising, progressing from the blue towards the red, and vice versa when the star is near the setting.

The planets do not twinkle, because they are not luminous points, but have disks made up of a congeries of such points ; while each point twinkles like a star, the twinklings do not synchronize with each other, and so the general sum of light remains practically uniform. When very near the horizon, how- ever, the irregular refraction is sometimes sufficiently violent to make them dance and change color. Since the disk of Mercury is very small, and the planet is never seen except near the horizon, it usually behaves like a star.

85. Twilight. — This is caused by the reflection of sunlight from the upper portion of the earth's atmosphere, perhaps from the air itself, perhaps from the minute solid particles in the air, — authorities differ. After the sun has set, its rays, passing over the observer's head, still continue to shine through the air above him, and twilight continues as long as any portion of the illuminated air remains in sight from where he stands. It is con- sidered to end when stars of the sixth magnitude become visible near the zenith, which does not occur until the sun is about 18° below the horizon; but this varies considerably for different places, according to the purity of the air.

Duration of The length of time required by the sun after setting to reach this depth

twilight. varies with the season and with the observer's latitude. In latitude 40° it

is about ninety minutes on March 1 and October 12, but more than two

hours at the summer solstice. In latitudes above 50°, when the days are

Cause of twilight.

CORRECTIONS TO ASTRONOMICAL OBSERVATIONS 75

longest, twilight never disappears even at midnight. On the mountains of Peru, on the other hand, it is said never to last more than half an hour, probably because the upper air in that region is practically clear from dust particles.

From the fact that twilight lasts until the sun is 1 8° below the horizon, Height of the height of the twilight-producing atmosphere can easily be computed, the earth's and comes out about 50 miles. This, however, is not the real limit of a the atmosphere. The phenomena of meteors show that at an elevation of 100 miles there is still air enough to resist their motion and cause their incandescence.

Soon after the sun has set, the twilight bow appears rising in the east, — The twilight a dark blue segment, bounded by a faintly reddish arc. It is the shadow b°w. of the earth upon the air, and as it rises the arc becomes rapidly diffuse and indistinct and is lost long before it reaches the zenith.

EXERCISES

V 1. What is the approximate dip of the horizon from a hill 900 feet high (Sec. 77)? Jo'/*«c

(/ 2. How high must a mountain be in order that the dip of the horizon from its summit may be 2°? /fy y r*

3. What is the distance of the horizon in miles, as seen from the summit of this mountain (Sec. 77) ?

\/ 4. Assuming the horizontal parallax of the sun at 8". 8, what is the horizontal parallax of Mars when nearest us, at a distance of 0.378 astro- nomical units? (The astronomical unit is the distance from the earth to the sun.)

5. What is the greatest apparent diameter of the earth as seen from Mars?

v 6. What is the horizontal parallax of Jupiter when at a distance of 6 astronomical units?

/ 7. Does atmospheric refraction increase or decrease the apparent size of the sun's disk when it is near the horizon?

8. What is the lowest latitude where twilight can last all night? Can it do so at New York? at London? at Edinburgh?

CHAPTER IV

Funda- mental problems of observation.

Definitions of astronom- ical latitude.

FUNDAMENTAL PROBLEMS OF PRACTICAL ASTRONOMY

Latitude — Time — Longitude — Azimuth — The Eight Ascension and Declination of

a Heavenly Body

86. There are certain problems of practical astronomy which are encountered at the very threshold of all investigations respecting the heavenly bodies, the earth included. The student must know how to determine his position on the surface of the earth, that is, his latitude and longitude ; how to ascertain the exact time at which an observation is made; and how to observe the precise position of a heavenly body and fix its right ascen- sion and declination.

87. Definitions of the Observer's Latitude. — In geography the latitude of a place is usually defined simply as its distance north or south of the equator, measured in degrees. This is not explicit enough unless it is stated how the degrees them- selves are to be measured. If the earth were a perfect sphere there would be no difficulty, but since the earth is sensibly flattened at its poles the geographical degrees have somewhat different lengths in different parts of the earth. The funda- mental definition of astronomical latitude has already been given (Sec. 32) as the angle between the direction of gravity where the observer stands and the plane of the equator. The angle between gravity and the earth's axis is the colatitude of the place. Other equivalent definitions of the latitude are the altitude of the pole and the declination of the zenith, which is the same as the altitude of the pole, as is clear from Fig. 34, where ZQ obviously equals NP.

76

PROBLEMS OF PRACTICAL ASTRONOMY

77

The problem, then, is to determine by observation of the heav- enly bodies either the angle of elevation of the celestial pole, or the distance in degrees between the zenith and the celestial equator.

88. First Method: by Observation of Circumpolar Stars. - The most obvious method (already referred to) is by observing with a suitable instrument the altitude of some star near the pole at the moment .when it is crossing the meridian above the pole, and again twelve sidereal hours later when it is once more on the meridian but below the pole. In the first case its altitude is the greatest possible; in the second, the least. The mean of the two altitudes (each corrected for atmospheric refrac- tion) is the altitude of the pole or the latitude of the observer.

The method has the great advantage that it is an independent one; that is, the observer is not obliged to depend upon his predecessors for any of his data. But the method fails for stations very near the equator, because there the pole is so near the horizon that the neces- sary observations cannot be made.

At an observatory the observations are usually made with the meridian- circle, and the mean of a

great number of observations is necessary in order to elimi- nate the slight errors in the computed refraction corrections due to varying atmospheric conditions. Where the meridian- circle is not available, the observations may also be made with a sextant or theodolite, but the results are much less precise.

89. Second Method: by the Meridian Altitude or Zenith- Distance of a Body whose Declination is accurately known.— In Fig. 34 the circle MQPN is the meridian, Q and P being

Latitude by observation of circum- polar stars-

Advantages and disad- vantages of the method.

FIG. 34

T8

MANUAL OF ASTRONOMY

Latitude by meridian altitude of object of known declination.

Formula for latitude in this case.

Advantages and disad- vantages of this method.

Latitude at sea by observation of the sun.

respectively the equator and the pole and Z the zenith. QZ is the declination of the zenith, or the latitude of the observer. If, when the star s crosses the meridian, we observe its zenith- distance, fa (Zs in the figure), its declination, Qs or Sg being known, then evidently QZ equals Qs plus %Z\ that is, the latitude equals the declination of the star plus its zenith-distance. If the star were at s', south of the equator, the same equation would still hold algebraically, because the declination Qs' is then a negative quantity ; and if the star were at n between the zenith and pole, its north zenith-distance, fw, would be negative. In all cases, therefore, we may write <j> = 8 + £.

If we use the meridian-circle in making our observations, we can always select stars that pass near the zenith, where the refraction is small, which is in itself a great advantage. Moreover, we can select them in such a way that some will be as much north of the zenith as others are south, and this will practically eliminate even the slight refraction errors that remain. On the other hand, in using this method we have to obtain our star declinations from the catalogues made by previous observers, so that the method is not an " independent " one.

90. At sea the latitude is usually obtained by observing with the sextant the sun's maximum altitude, which occurs, of course, at noon. Since at sea one seldom knows beforehand precisely the moment of local noon, the observer takes care to begin his observations some minutes earlier, repeating his measure of the sun's altitude every minute or two. At first the altitude will keep increasing, but immediately after noon occurs it will begin to decrease. The observer uses, therefore, the maximum1 altitude obtained, which, corrected for refraction, parallax, semidiameter, and dip of the horizon, will give him the true meridian altitude of the sun. The Nautical Almanac gives him its declination.

10n account of the sun's motion in declination and the northward or southward motion of the ship itself, the sun's maximum altitude is usually attained not precisely on the meridian, but a short time earlier or later. This requires a slight correction to the deduced latitude, the calculation of which is explained in books on navigation.

PROBLEMS OF PRACTICAL ASTRONOMY 79

91. Third Method: by Circummeridian Altitudes. — If the Latitude by observer knows his time with reasonable accuracy, he can obtain c

his latitude from observations of the altitude of a heavenly body altitudes. made when it is near the meridian with practically the same precision as at the moment of meridian passage. It lies beyond our scope to discuss the method of reduction, which is explained, with the necessary tables, in all works on practical astronomy.

The great advantage of the method is that the observer is not restricted to a single observation at each meridian passage of the sun or of the selected star, but can utilize the half-hours preceding and following that moment. The meridian-circle, of course, cannot be used. Usually the sextant, or a so-called "universal instrument" (Sec. 70), is employed.

92, Fourth Method1 : by the Zenith-Telescope. — The essential Latitude by

characteristic of the method is the measurement with a microm- the zemth~

telescope — eter of the d iffe r ence between the nearly equal zenith-distances the most

of two stars which pass the meridian within a few minutes accurate of each other, one north and the other south of the zenith, and not very far from it; such pairs of stars can now always be found in our star-catalogues.

A special instrument, known as the zenith-telescope, is gen- erally employed, though a simple transit-instrument, provided with reversing apparatus, a delicate level attached to the tele* scope, and a declination micrometer is now often used.

Fig. 35 shows the very complete zenith-telescope of the Flower Observa- tory near Philadelphia.

At the Georgetown Observatory a photographic zenith-telescope is used, having a photographic plate in place of the eyepiece.

The telescope is set at the proper altitude for the star which Method of first comes to the meridian and the " latitude level," as it is c called, — which is attached to the telescope — is set horizontal ;

1 Known as the "American method," because first practically introduced by Captain Talcott, of the United States Engineers, in a boundary survey in 1845. It is now very generally adopted and considered the best.

80

MANUAL OF ASTRONOMY

Advantage in dispens- ing with a graduated circle.

as the star passes through the field of view its distance north or south of the central horizontal wire is measured by the micrometer. The instrument is then reversed so that the tele- scope points towards the north (if it was south before), and the telescope so readjusted, if neces- sary, that the level is again horizontal, — taking great care, however, not to disturb the angle between the level and the telescope itself. The telescope is then evidently elevated at exactly the same angle as before, but on the opposite side of the zenith. As the second star passes through the field, we measure with the micrometer its distance north or south of the central wire. The compari- son of the two measures gives the difference of the two zenith- distances with great accuracy and without the necessity of depending upon any graduated circle.

In field operations like those of geodesy this is an enormous advantage, both as regards the portability of the instrument and the attainable precision of results.

FIG. 35. — A Zenith-Telescope By Warner & Swasey

PROBLEMS OF PRACTICAL ASTRONOMY 81

From Fig. 34 we have

for star south of zenith, </> = Sg -+- £ ; for star north of zenith, <£ = 8n — fn.

Adding the two equations and dividing by 2, we have

— (TA 2 /

— (T Formula for

the latitude.

The star-catalogue gives us the declinations of the two stars (&,+&n) ; and the difference of the zenith-distances (fg — fj is determined by the micrometer measures.

When the method was first introduced it was difficult to find pairs of stars whose declination was known with sufficient pre- cision. At present our star-catalogues are so extensive and exact that this difficulty has practically disappeared.

Refraction is almost eliminated, because the two stars of each Refraction pair are at very nearly the same zenith-distance. eliminated.

Evidently the accuracy depends ultimately upon the exactness with which the level measures the slight but inevitable difference between the inclinations of the instrument when pointed on the two stars.

In Dr. Chandler's Almucantar (Sec. 67) the telescope preserves its constant declination automatically, by being mounted upon a base which floats in mercury, thus dispensing with the level.

There are numerous other methods for obtaining the latitude. In Chauvenet's Practical Astronomy over forty are given, some of which can fairly compete in precision with those named above.

93, The Gnomon. — The ancients could not use any of the Ancient preceding methods for finding the latitude. They were, how- method of

J determining

ever, able to make a very respectable approximation by means the latitude of the simplest of all astronomical instruments, the gnomon. bythe This is merely a vertical shaft or column of known height erected on a perfectly horizontal plane, and the observation

82

MANUAL OF ASTRONOMY

Climate."

consists in noting the length of the shadow cast at noon at certain times of the year. Suppose, for instance, that on the day of the summer solstice, at noon, the length of the shadow is AC (Fig. 36). The height AB being given, we can easily compute in the right-angled triangle the angle ABC, which equals SBZ, the sun's zenith-distance when farthest north.

Again, observe the length AD of the shadow at noon of the shortest day in winter and compute the angle ABD, which is the sun's corresponding zenith-distance when farthest south. Now, since the sun travels equal distances north and south of the

celestial equator, the mean of the two zenith-distances will give the angular dis- tance between the equator and the zenith, i.e., the declination of the zenith, which is the latitude of the place.

The method is an inde- pendent one, like that of the observation of circum- polar stars, requiring no data except those which the observer determines for himself. It does not admit of much accuracy, however, since the penumbra at the end of the shadow makes it impossible to measure its length very precisely.

It should be noted that the ancients, instead of designating the position of a place by means of its latitude, used its climate; the climate (from /cAtjua) being the slope of the plane of the celestial equator, the angle AEB, which is the colatitude.

For the use of the gnomon in determining the obliquity of the ecliptic and the length of the year, see Sees. 164 (2) and 182. Many of the Egyp- tian obelisks are known to have been used for astronomical observations, and perhaps were erected mainly for that purpose.

FIG. 36. — Latitude by the Gnomon

PROBLEMS OF PRACTICAL ASTRONOMY

83

94. Variation of Latitude and Motion of the Poles of the Earth.

— It has long been doubted whether latitudes are strictly con- stant. They cannot be so if the axis of the earth shifts its position within the globe. Some have supposed that in the past there have been great changes of this kind, seeking thus to explain certain geological epochs, as, for instance, the glacial and the carboniferous. But thus far no evidence of any consid- erable displacement has appeared, nor is there any satisfactory proof of certain slow, continuous " secular " changes, which have been strongly sus- -< pected.

Theoretically, how- -°"10' ever, any alteration in the arrangement of the matter of the earth,

- +0*10

+ 0?20

•0?20.-

+ OJ30

FIG. 37

by elevation, subsi- dence, transportation, or denudation, must +°':2°t- necessarily disturb the axis and change the latitudes to some ex- tent. The question

is merely whether our observations can be made sufficiently accurate to detect the change. Since 1889 the limit has been reached, and we now have conclusive proof of such effects.

The first satisfactory evidence of the fact was obtained at Berlin by Kiistner, and at other German stations in 1888 and 1889, and the result has since been abundantly confirmed by observations at many other stations. Moreover, Dr. S. C. Chandler of Cambridge, U.S., by a brilliant and laborious series of investigations, finds the same variations clearly exhibited in almost every extended body of reliable observations made since

No evidence of any considerable changes in the position of the earth's axis.

Minute changes theoretically must occur.

First obser- vational evidence obtained in 1888.

84

MANUAL OF ASTRONOMY

Nature of the periodic

motion of

the pole.

1750. From the whole mass of evidence he concludes that the movement of the pole at present is composed of two motions, — one an annual revolution in an ellipse about 30 £QQ^ iong but varying in width and position, the other a revo-

J ...

lution in a circle about 26 feet in diameter and having a period of about 4-28 days, — both revolutions being counter-clockwise. The resultant motion presents a very irregular appearance and changes greatly from year to year.

Fig. 37 represents the actual motion from 1890 to 1898 as deduced by Albrecht from all available observations.

The annual component of this polar motion is very likely due to meteoro- logical causes which follow the seasons, such as the deposit of rain, snow, and ice. The explanation of the 428-day component is not yet entirely clear, and its discussion would take us too far.

It is likely also that irregular disturbances, due to various causes — for instance, perhaps, earthquakes — may modify the regular periodic motions.

Time de- fined as measured duration.

Determina- tion of time.

The three kinds of time.

95. Different Kinds of Time. — Time is usually defined as measured duration. From the beginning the apparent diurnal rotation of the heavens has been accepted as the standard unit, and to it we refer all artificial measures of time, such as clocks and watches.

In practice the accurate determination of time consists in finding the Hour Angle (Sec. 21) of the object or point which has been selected to mark the beginning of the day by its passage across the meridian.

In astronomy three kinds of time are now recognized: side- real time, apparent solar time, and mean solar time, — the last being the tijne of civil life and ordinary business, while the first is used for astronomical purposes exclusively. Apparent solar time (formerly called true time) has now practically fallen out of use, except in countries where watches and clocks are scarce or unknown and sun-dials are the ordinary timekeepers.

PROBLEMS OF PRACTICAL ASTRONOMY 85

96. Sidereal Time. — The celestial object which determines Sidereal sidereal time by its position in the sky at any moment is, it time~ the

J t hour angle

will be remembered, the vernal equinox or first of Aries (symbol, Of the vernal °f ), i.e., the point where the sun crosses the celestial equator e(iuinox' in the spring, about March 21 every year.

As already stated (Sec. 25), the local sidereal day1 begins at the moment when the first of Aries crosses the observer's meridian, and the sidereal time at any moment is the hour angle of the vernal equinox; i.e., it is the time marked by a clock so set and adjusted as to show sidereal noon (OhOmOs) at each transit of the first of Aries.

The equinoctial point is, of course, invisible ; but its posi- tion among the stars is always known, so that its hour angle at any moment can be determined by observing the stars.

97, Apparent Solar Time. — Just as sidereal time is the hour Apparent angle of the vernal equinox, so apparent solar time at any s<jlartime- moment is the hour angle of the sun. It is the time shown by angle of the the sun-dial, and its noon occurs at the moment when the sun's sun: identi-

cal with sun

center crosses the meridian. dial time.

On account of the earth's orbital motion (explained more fully in Chapter VI), the sun appears to move eastward along the ecliptic, completing its circuit in a year. Each noon, therefore, it occupies a place among the stars about a degree farther east than it did the noon before, and so comes to the meridian about four minutes later, if time is reckoned by a sidereal clock. In other words, the solar day is about four minutes longer than the sidereal, the difference amounting to exactly one day each year, which contains 366 J sidereal days.

But the sun's eastward motion is not uniform, for several

1 On account of the precession of the equinoxes (to be discussed later), the sidereal day thus denned is slightly shorter than it would be if denned as the interval between successive transits of some given star ; the difference being a little less than Tfo. of a second, or one day in 25800 years,— too little to be worth taking into account in any ordinary calculation.

86

MANUAL OF ASTRONOMY

Apparent solar days vary in length. Hence ap- parent time is unsatisfac- tory.

reasons, and the apparent solar days therefore vary in length. December 23, for instance, is about fifty-one seconds longer from sun-dial noon to noon again (by a sidereal clock) than Sep- tember 16. For this reason apparent solar or sun-dial time is unsatisfactory for scientific use and cannot be kept by any simple mechanical arrangement in clocks and watches. At present it is practically discarded in favor of mean solar time.

98, Mean Solar Time A fictitious sun is, therefore, imagined,

moving uniformly eastward in the celestial equator and complet- ing its annual course in exactly the same time as that in which the actual sun makes the circuit of the ecliptic. This fictitious sun is made the timekeeper for mean solar time. It is mean noon when its center crosses the meridian, and at any moment the hour angle of the fictitious sun is the mean time for that moment. The mean solar days are, therefore, all of exactly the same length and equal to the length of the average apparent solar day, the mean solar day being longer than the sidereal by 3m55s.91 (mean solar minutes and seconds) and the sidereal day shorter than the solar by 3m568.55 (sidereal minutes and seconds).

99. Sidereal time will not answer for business purposes, because its noon (the transit of the vernal equinox) occurs at all hours of the day and night in different seasons of the year : on September 22, for instance, it comes at midnight. Apparent solar time is unsatisfactory because of the variation in the length of its days and hours. Yet we have to live by the sun: its rising and setting, daylight and night, control our actions.

Mean solar time furnishes a satisfactory compromise. It has a time unit which is invariable, and it can be kept by clocks and watches, while it agrees nearly enough with sun-dial time for convenience. It is the time now used for all purposes except in some kinds of astronomical work.

The difference between apparent time and mean time (never amounting to more than about a quarter of an hour) is called

PROBLEMS OF PRACTICAL ASTRONOMY 87

the equation of time and will be discussed hereafter in connec- Equation of

tion with the earth's orbital motion (Sec. 174). time-

Since there are 365.2421 solar days in a year (Sec. 182) and Relation

one more sidereal day, we have the following fundamental rela- between the

^ ... number of

tion: — the number of sidereal seconds in any time interval : the sidereal and

number of mean solar seconds in the same interval : : 366.2421 : mean solar

secondsina

given time

From this it follows at once that to reduce a solar time interval. interval to sidereal, we must divide the number of seconds it Reduction of contains by 365.2421, and add the quotient to the number of asolartime

17 interval to

solar seconds. To reduce a sidereal interval to solar, divide by sidereal, and 366.2421, and subtract the quotient from the number of sidereal vice versa- seconds.

The Nautical Almanac gives the sidereal time of mean solar noon for every day of the year, with tables by means of which mean solar time can be accurately deduced from the correspond- ing sidereal time, or vice versa, by a very brief1 calculation.

100, The Civil Day and the Astronomical Day. — The astro- Theastro- nomical day begins at mean noon; the civil day, twelve hours nomical and

civil cljiv^

earlier at midnight. Astronomical mean time is reckoned around through the whole twenty-four hours instead of being counted in two series of twelve hours each: thus, 10 A.M. of Wednesday, February 27, civil reckoning, is Tuesday, February 26, 22 o'clock, by astronomical reckoning. This must be borne in mind in using the Almanac.2

1 The approximate relation between sidereal time and mean solar time is very simple. Assuming that on March 22 the two times agree, after that day the sidereal time gains two hours each month. On April 22, therefore, the sidereal clock is two hours in advance, on June 22, six hours in advance, and so on. On account of the differing length of months, this reckoning is slightly errone- ous in some parts of the year, but is usually correct within four or five minutes. March 22 is taken as the starting-point because it distributes the errors better than the 21st. For the odd days the gain may be taken as four minutes daily.

2 The astronomical day is made to begin at noon because astronomers are "night-birds," and would find it inconvenient to have to change dates at midnight in the middle of their work.

88

MANUAL OF ASTRONOMY

Determina- tion of time consists in ascertaining the error of a time- piece.

Determina- tion of time by the transit- instrument.

Almanac

stars.

Necessary to observe a number of stars in order to attain high precision.

DETERMINATION OF TIME

In practice the problem of determining time always consists in ascertaining the error or correction of a timepiece, i.e., the amount by which the clock or chronometer is faster or slower than the time it ought to indicate.

101. Determination of Time by the Transit-Instrument. — The method most employed by astronomers is by observations with the transit-instrument (Sec. 61). We observe the time shown by the sidereal clock at which a star of known right ascension crosses each wire of the reticle. The mean is taken as the instant of crossing the instrumental meridian, and when the instrument is in perfect adjustment the difference between the star's right ascension and the observed clock time will be the clock's "error"; or, as a formula, A£ = a — £, — At being the usual symbol for the clock error, and t the observed time.

The Almanac supplies a list of several hundred stars whose right ascension and declination are accurately given for every tenth day of the year, so that the observer at night has no diffi- culty in finding a suitable star at almost any time. In the day- time he is, of course, limited to the brighter stars.

The observation of a single star with an instrument in ordi- nary adjustment will usually give the error of the clock within half a second; but it is much better and usual to observe a number of stars, reversing the instrument upon its Y's once at least during the operation. This will enable him to determine and allow for the faults of instrumental adjustment, so that with a good instrument a skilled observer can thus determine his clock error within about a thirtieth of a second of time, provided proper correction is applied for his " personal equa- tion" (Sec. 64).

If instead of observing a star we observe the sun with this instrument, the time as shown by the mean solar clock ought to be twelve hours plus or minus the equation of time as given in

PROBLEMS OF PRACTICAL ASTRONOMY 89

the Almanac. But for various reasons transit observations of Soiartime the sun are less accurate than those of the stars, and it is far usually now

deduced

better to deduce the mean solar time from the sidereal time by from means of the almanac data. sidereal.

102. The Method of Equal Altitudes. — If we observe the time shown by the chronometer or the clock when a star attains a Method of certain altitude and then the time when it attains the same equal

altitudes.

altitude on the other side of the meridian, the mean of the two times will be the time of the star's transit across the meridian, provided, of course, that the chronometer runs uniformly during the interval.

We may also use stars of slightly differing declination, one Modification on one side of the meridian, and the other observed a few ofthe

method in

minutes later on the other side; and by a somewhat tedious observing calculation it is possible to determine the error of the clock starsof with practically the same accuracy as if both observations had different been made on the same star, and much more quickly. declination.

If we observe the sun in this manner in the morning, and again in the afternoon, the moment of apparent noon will seldom Correction be exactly half-way between the two observed times, and proper re(iulred m correction must be made for the sun's slight motion in decli- time from nation during1 the interval, — a correction easilv computed bv e<iual alti~

, , , , . ° , , ^, J tudesofthe

tables furnished for the purpose. sun

The advantage of this method is that the errors of gradua- tion of the instrument have no effect, nor is it necessary for the observer to know his latitude except approximately.

On the other hand, there is, of course, danger that the second observation may be interfered with by clouds. Moreover, both observations must be made at the same place.

103. Marine Method: by a Single Altitude of the Sun, the Observation Observer's Latitude being known. — Since neither of the preced- todetermine

time at sea-

ing methods can be used at sea, the following is the method usually practised. The altitude of the sun, at some time when it is rapidly rising or falling (i.e., not near noon), is measured

90

MANUAL OF ASTRONOMY

Computa- tion of the time from the obser- vation.

Time when observation should be made.

with the sextant, and the corresponding time shown by the

chronometer accurately noted.

We then compute the hour angle of the sun, P, from the

triangle PZS (Fig. 38), and this hour angle, corrected for the

equation of time, gives the mean solar time at the observed

moment. The difference between this time and that shown

by the chronometer is the error of the chronometer on local

time.

In the triangle ZPS (which is the same as SPO in Fig. 8) all

three of the sides are given : PZ is the complement of the

latitude c/>, which is sup- posed to be known ; PS is the complement of the sun's declination 8, which is found in the Almanac, as is also the equation of time; while ZS or f is given by observation, being the complement of the sun's altitude as measured by the

sextant and corrected for dip, semidiameter, refraction, and

parallax. The formula ordinarily used is

EH

FIG. 38. — Determination of Time by the Sun's Altitude.

8in !

= J s *

~ 8)] sin j [g - 0 -

cos

cos 8

In order to insure accuracy it is desirable that the sun should be on the prime vertical, or as near it as practicable. It should NOT be near the meridian, for at that time the sun is rising or falling very slowly, and the slightest error in the measured altitude would make an enormous difference in the computed hour angle. If the sun is exactly east or west at the time of observation, an error of even several minutes of arc in the assumed latitude produces no sensible effect upon the result.

PROBLEMS OF PRACTICAL ASTRONOMY 91

The disadvantage of the method is that any error of gradua- Disadvan- tion of the sextant vitiates the result, and no sextant is perfect. tage of the

method and

But with ordinary care and good instruments the sea-captain is limit of able to get his time correct within three or four seconds. accuracy.

When a number of altitude observations have been made for time, and it is desired to reduce them separately, so as to test their agreement and determine their probable error, there is an advantage in using the formula

cos £ ,>

cos P = - — _ — tan d> tan d,

cos $ cos 8

employing the " Gaussian logarithms " in the computation. The second term of the formula and the denominator of the first term remain constant through the whole series, saving much labor in reduction.

104. To compute the Time of Sunrise or Sunset. — To solve Calculation this problem we have precisely the same data as in finding the of time of time by a single altitude of the sun. The zenith-distance of the sunset. sun's center at the moment when its upper edge is rising equals 90° 51', — made up of 90° plus 16' (the mean semidiameter of the sun) plus 35' (the mean refraction at the horizon). The resulting hour angle, corrected for the equation of time, gives the mean local time at which the sun's upper limb reaches the horizon under average circumstances of temperature and baro- metric pressure. If the sun rises or sets over the sea horizon and the observer's eye is at any considerable elevation above sea-level, the dip of the horizon must also be added to 90° 51; before making the computation.

The beginning and end of twilight may be computed in the same way by merely substituting 108°, i.e., 90° + 18°, for 90° 51'.

DETEEMINATION OF LONGITUDE

Having now the means of finding the true local time at any place, we can take up the problem of the longitude, the most important of all the economic problems of astronomy. The great observatories. a.t Greenwich and Paris were established expressly

92

MANUAL OF ASTRONOMY

Definition of longitude.

Difference of longitude equals dif- ference of local times.

The knot of the problem.

Telegraphic method.

Details of process.

for the purpose of furnishing the observations which could be utilized for its accurate determination at sea.

105, The longitude of a place on the earth may be defined as the angle at the pole of the earth between the standard meridian and the meridian of the place; and this angle is measured by, and equal to, the arc of the equator intercepted between the two meridians.

As to the standard meridian there is some variation of usage. At sea nearly all nations at present reckon from the meridian of Greenwich, except the French, who insist on Paris.

Since the earth turns 011 its axis at a uniform rate, the angle at the pole is strictly proportional to the time required for the earth to turn through that angle ; so that longitude may be, and now usually is, expressed in time units, — i.e., in hours, minutes, and seconds, rather than degrees, etc., — and is simply the differ- ence between the local times at G-reenwich and at the place where the longitude is to be determined.

Since the observer can determine his own local time by the methods already given, the knot of the problem is to find the Greenwich local time corresponding to his own, without leaving his place.

106. First Method: by Telegraphic Comparison between his Own Clock and that of Some Station whose Longitude from Green- wich is known. — The difference between the two clocks will be the difference of longitude between the two stations after the proper corrections for clock errors, personal equation, and time occupied by the transmission of the electric signals have been applied or eliminated.

The process usually employed is as follows : The observers, after ascer- taining that they both have clear weather, proceed early in the evening to determine the local time at each station by an extensive series of star observations with the transit-instrument. Then at an hour agreed upon the observer at the eastern station, A, " switches his clock " into the tele- graphic circuit, so that its beats are communicated along the line and received upon the chronograph of the western station. After the eastern clock has thus sent its signals, say for two minutes, it is " switched out "

PKOBLEMS OF PRACTICAL ASTRONOMY 93

and the western observer puts his clock into the circuit, so that its beats are received upon the eastern chronograph. Sometimes the signals are communicated both ways simultaneously, so that the beats of both clocks appear upon both chronograph sheets at the same time. The operation is closed by another series of transit observations by each observer.

We have now upon each chronometer sheet an accurate comparison of the two clocks, showing the amount by which the western clock is slow of the eastern, and if the transmission of electric signals were instantaneous, the difference shown upon the two chronometer sheets would be identical on both. Practically, however, there will always be a discrepancy of some Elimination hundredths of a second, amounting to twice the time occupied in the trans- of error due mission of the signals; but the mean of the two differences after correcting to tr^ns~ for the carefully determined clock errors will be the true difference of longi- time and tude between the places. Especial care must be taken to determine with personal accuracy the personal equations of the observers, or else to eliminate them, which equation, may be done by causing the observers to change places.

In cases where the highest accuracy is required, it is customary to make observations of this kind on not less than five or six evenings.

The astronomical difference of longitude between two places Limit of

can thus be determined within about ^ of a second of time, i.e., 8 within about 20 feet in the latitude of the United States.

107. Second Method : by the Chronometer. — This method is The chrono-

available at sea. The chronometer is set to indicate Greenwich metric

, , . , . 55 i • i method of

time before the ship leaves port, its " rate having been care- determining fully determined by observation for several days. In order to longitude find the longitude by the chronometer, the sailor must determine its " error " upon local time by an observation of the altitude of the sun when near the prime vertical (Sec. 103). If the chro- nometer indicates true Greenwich time, its "error" deduced from the observation will be the longitude. Usually, however, the indication of the chronometer face must be corrected for the gain or loss of the chronometer since leaving port, in order to give the true Greenwich time at the moment.

Chronometers are only imperfect instruments, and it is Chr0nome- important, therefore, that several of them should be carried by ters needed the vessel to check each other. This requires three at least, g^ other

94

MANUAL OF ASTRONOMY

Failure of the method for long voyages.

Lunar method, the moon being regarded as a clock hand showing Greenwich time.

Lunar methods available on land.

Lunar distances at sea.

because if only two chronometers are carried, and they disagree, there is nothing to indicate which is the delinquent.

Moreover, in the course of months, chronometers generally change their rates progressively, so that they cannot be depended on for very long intervals of time ; and the error accumulates much more rapidly than in proportion to the time. If, there- fore, a ship is to be at sea more than three or four months without making port, the method becomes untrustworthy. For voyages of less than a month it is now practically all that could be desired.

108. Third Method : by the Moon regarded as a Clock Hand, with Stars for Dial Figures. — Before the days of reliable chro- nometers, navigators and astronomers were generally obliged to depend upon the moon for their Greenwich time. The laws of her motion are now fairly well known, so that the right ascension and declination of the moon are now computed and published in the Nautical Almanac, three years in advance, for every Greenwich hour of every day in the year. It is therefore possible to deduce the Greenwich time at any moment when the moon is visible by making some observation which will accurately determine her place among the stars.

On land it may be :

(a) The direct transit-instrument observation of her right ascension as she crosses the meridian.

(6) The observation at the moment when she occults a star (incomparably the most accurate of all lunar methods) or makes contact with the sun in a solar eclipse.

(c) The observation of the moon's azimuth with the universal instrument at an accurately determined time.

At sea the only practicable observation is to measure with a sextant a lunar distance, i.e., the distance of the moon from some star or planet nearly in her path.

Since, however, the almanac place of the moon is the place she would apparently occupy if seen from the center of the earth, most lunar

PROBLEMS OF PRACTICAL ASTRONOMY 95

observations require complicated and laborious reductions before they can Inferiority

be used for longitude. Moreover, the motion of the moon is so slow (she of the lunar

requires a month to make the circuit of 360°) that any error in the methods>

observation of her place produces nearly thirty times as great an error in ° the corresponding Greenwich time and the deduced longitude. It is as if one should try to read accurate time from a watch that had only an hour- hand.

109. Other Methods: Eclipses of the Moon and Jupiter's Satellites. — Longitude A rough longitude can be obtained from the observation of these eclipses, by eclipses since they occur at the same moment of absolute time wherever observed. of the moon By comparing the local times of observation with the Greenwich time T ., ,

eJU.pl LOT S

obtained by correspondence after the event, or from the Almanac, the satellites, difference of longitude at once comes out. The difficulty with this method is that the eclipses are gradual phenomena, presenting no well-marked instant for observation.

On the same principle artificial signals, such as flashes of powder and Longitude explosion of rockets, can be used between two stations so situated that by artificial both can see the flashes. Early in the century the difference of longitude S1^na • between the Black Sea and the Atlantic was determined by means of a chain of such signal stations on the mountain tops ; so also, later, the differ- ence of longitude between the eastern and western extremities of the northern boundary of Mexico. This method is now superseded by the telegraph.

110. Local and Standard Time. — Until recently it has been Local and always customary to use local time, each city determining its J^da" own time by its own observations. Before the days of the telegraph, and while traveling was comparatively slow and infre- quent, this was best. At present it has been found better for

many reasons to give up the system of local times in favor of a system of standard time. This facilitates all railway and tele- Advantages graphic business in a remarkable degree, and makes it practi- °.f standard cally easy for every one to keep accurate time, since it can be daily wired from some observatory (as Washington) to every telegraph office in the country. According to the system now American established in North America, there are five such standard times systems of

standard

in use, — the colonial, the eastern, the central, the mountain, time, and the Pacific, — which are slower than Greenwich time by

96

MANUAL OF ASTRONOMY

exactly four, five, six, seven, and eight hours, respectively. The minutes and seconds are everywhere identical.

At most places only one of these standard times is employed ; but in cities where different systems join each other, as, for instance, at Atlanta and Pittsburg, two standard times are in use, differing from each other by exactly one hour, and from the local time by about half an hour. In some such places the local time also maintains itself.

This system is now adopted in nearly all civilized countries, though with a half-hour modification in certain cases. Everywhere except in America the standard time is fast of Greenwich time. In Continental Europe, Russia excepted, it is one hour fast ; in Cape Colony, one and one-half hours ; in India, five and one-half hours ; in Burma, six and one-half hours ; in West Australia, eight hours ; in South Australia and Japan, nine hours ; in Eastern Australia, ten hours ; and in New Zealand, eleven and one-half hours.

In order to determine the standard time by observation it is only necessary to determine the local time by one of the methods given, correcting it by first adding the observer's longi- tude west from Greenwich, and then deducting the necessary integral number of hours.

111. Where the Day begins. — It is evident that if a traveler were to start from Greenwich on Monday noon and were by some means able to travel westward along the parallel of lati- tude as fast as the earth turns eastward beneath his feet, he would keep the sun exactly upon the meridian all day long and have continual noon. But what noon? It was Monday noon when he started, and when he gets back to London twenty-four hours later he will find it to be Tuesday noon there. Yet it has been noon all the time. When did Monday noon become Tuesday noon ?

It is agreed among mariners to make the change of date at the 180th meridian from G-reenwich, which passes over the Pacific hardly anywhere touching the land.

PROBLEMS OF PRACTICAL ASTRONOMY 97

Ships crossing this line from the east skip one day in so Loss or gain doing1. If it is Monday afternoon when a ship reaches the of a day by

vessels pass- line, it becomes Tuesday afternoon the moment she passes it, ing the

the intervening twenty-four hours being dropped from the date-line. reckoning on the log-book. Vice versa, when a vessel crosses the line from the western side, it counts the same day twice over, passing from Tuesday back to Monday, and having to do Tuesday over again.

There is considerable irregularity in the date actually used on the The date- different islands in the Pacific, as will be seen by looking at the so-called ^ne* date-line as given in the Century Atlas of the World. Those islands which received their earliest European inhabitants via the Cape of Good Hope have adopted the Asiatic date, even if they really lie east of the 180th meridian ; while those that were first approached via the American side have the American date. When Alaska was transferred from Russia to the United States, it was necessary to drop one day of the week from the official dates.

PLACE OF A SHIP AT SEA

112. Determination of the Position of a Ship. — The determi- nation of the place of a ship at sea is commercially of such importance that, notwithstanding a little repetition, we collect here the different methods available for the purpose. They are necessarily such that the requisite observations can be made with the sextant and chronometer, the only instruments available on shipboard.

The latitude is usually obtained by observations of the sun's Determine altitude at noon, according to the method explained in Sec. 90. tionof a

The longitude is usually found by determining the error upon tude and local time of the chronometer, which carries Greenwich time. longitude. (See Sees. 103 and 10T.)

In case of long voyages, or when the chronometer has for any reason failed, the longitude may also be obtained by meas- uring lunar distances and comparing them with the data of the Nautical Almanac.

98

MANUAL OF ASTRONOMY

Sumner's method.

The circle of position. Its center and radius at any time.

Position of ship deter- mined by the inter- section of two circles of position.

Practical application of the method.

These methods require separate observations for the latitude and for the longitude.

113. Sumner's Method. — At present a method known as Sumners Method, because first proposed by Captain Sumner of Boston, in 1843, has come largely into use. It is based on the principle that any single observation of the sun's altitude, giving, of course, its zenith-distance at the time, determines the so-called circle of position on which the ship is situated. The center of this circle of position on the earth's surface is the point directly under the sun at the moment of observation. The longitude of this point is the Greenwich apparent time at the moment of observation as determined by the chronometer, and its latitude is the sun's declination. The radius of the circle of position (reckoned in degrees of a great circle from this center) is the observed zenith-distance of the sun.

A second observation made some hours later will give a second circle of position, and if the ship has not moved mean- while the intersection of the two circles will give the place of the ship.

The circles intersect at two points, of course, but at which one the ship is situated is never doubtful, because the approxi- mate azimuth of the sun, observed simply as a compass bearing, tells roughly on what part of the circle the ship is placed. If, for instance, the sun is in the southeast at the first observa- tion, the ship must be on the northwestern part of the corre- sponding circle of position.

If the ship has moved between the two observations, as of course is usual, its motion as determined by log and compass can be allowed for with very little difficulty.

114. Usually the matter is treated as- follows : The latitude of the vessel is practically always known within a degree or so, from the " dead reckoning " since the last observation. Suppose the latitude is known to be about 51° ; then, from the first (morning) observation of the sun's altitude and the chronometer 'time, the navigator computes the longitude,

PROBLEMS OF PRACTICAL ASTRONOMY

99

Longitude West from Greenwich

assuming the latitude to be 52°, and finds it to be, say, 40° 52'. Again, assuming the latitude to be 50°, he gets 43° 20', and marks the two com- puted longitudes at A and JB on the chart (Fig. 39). A line drawn through these points will be very nearly a part of the vessel's circle of position at the time of that observation.

From the second (afternoon) observation the points C and D are computed in the same way, giving a piece of the second circle of position.

Suppose now that in the interval the ship has moved 60 miles on a course north 60° west. From the points A and B lay off 60 miles on the chart in the proper direction to the points a and 6, and join ab by a line. 5', the in- Lat. tersection of this line with the line CD, will be the position of the ship at the time of the second observation with all the approxi- mation necessary for the navigator's pur- pose ; and if we reckon back 60 miles from £', we shall find S, the ship's position in the morning. There are,

however, extended tables which greatly reduce the labor of computations and make the result more accurate than that derived from the chart.

The peculiar advantage of the method is that each observa- tion is used for all it is worth, giving accurately the position its peculiar of a line upon which the vessel is somewhere situated, and advantase- approximately (by the sun's azimuth) its position on that line. Very often this knowledge is all that the navigator needs to give him the knowledge of his distance from land, even when he fails in getting the second observation necessary to deter- mine his precise location. Everything, however, depends upon Must have the correctness of the Grreenwich time given by the chronometer, Greenwich just as in the ordinary method of longitude determination.

4	5° 4	4° 4	3° 4	2° 4	1° 4	0°
,at. 5o°	C		^/^ / "^ / /	^	/	toO
	\	/	'	/	4	52
	\	\/		/
11°		/\~-	^ o /	/		M
	/	\	\7*
	/		/\
50°	'"b ~	!\*zl		\D		eno
»"	(XT 2*	56m 2*	52m 2'	48™ 2"	44" 2*	40"

FIG. 39. — Sumner's Method

100

MANUAL OF ASTRONOMY

Determina- tion of

of pole-star.

115, Determination of "Azimuth." — A problem, important, though not so often encountered as that of latitude and longitude determinations, .is that of determining the " azimuth," or " true bearing," of a line upon the earth's surface.

With a theodolite having an accurately graduated horizontal circle the observer points alternately upon the pole-star and upon a distant signal erected for the purpose at a distance of say half a mile or more, — usually an " artificial star " consisting of a small hole in a plate of metal, with a lantern behind it. At each pointing he notes the time by a sidereal chronometer. The theodolite must be carefully adjusted for collimation,

and especial pains must be taken to have the axis of the telescope perfectly level.

The next morning by daylight the observer measures the angle or angles between the night signal and the objects whose azimuth is required.

If the pole-star were exactly at the pole, the mere difference between the two readings of the circle, obtained when the telescope is pointed on the star and on the signal, would directly give the azimuth of the signal. As this is not the case, the azimuth of the star must be computed for the moment of each observation, which is easily done, as the right ascension and declination of the star are given in the Almanac for every day of the year.

Referring to Fig. 40, N being the north point of the horizon, P the pole, and NZ the meridian, we see that PS is the polar distance of the star, or complement of its declination, the side PZ is the complement of the observer's latitude, while the angle at P is the hour angle of the star, i.e., the difference between the right ascension of the star and the sidereal time of observation. This hour angle must, of course, be reduced to degrees before making the computation. We thus have two sides of the triangle,