The elements of theoretical and descriptive astronomy, for the use of colleges and academies

Front matter

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PLATE I,

COMPARATIVE MAGNITUDES OF THE PLANETS,

Preface

I have endeavored in this treatise to present the main facts and principles of Astronomy in a form adapted to the course of instruction in that science which is commonly given at colleges and the higher grades of academies. I have selected those topics which appear to me to be both the most important and the most interesting, and have arranged them in the order which experience has led me to believe to be the best.

I have endeavored, in the descriptive parts of the work, to give the latest information upon every topic which is introduced. The distances and the dimensions of the heavenly bodies, together with other numerical data which depend for their values upon the value of the solar parallax, are given to correspond with the new value which is now adopted in the American Ephemeris. The results of the observations made at the United States Naval Observatory upon the November showers of the last two years will be found in the Chapter- on Meteoric Bodies. Some description has also been given of the spectroscope, and of the many interesting discoveries made with it concerning the constitution of the heavenly bodies, and, very recently, concerning the real motions of the stars.

No clear conception of the processes by which most of the fundamental truths of Astronomy have been established can be attained without some knowledge of Mathematics. I have endeavored, however, to confine the theoretical discussions within the limits of that moderate mathematical know-

PREFACE.

ledge which may fairly be expected in those readers for whom the treatise has been prepared. Certain definitions and formula?, with which the student may possibly not be familiar, will be found in the Appendix; and, with the^aid of these, I believe that any one who has had the slightest mathematical education can read every portion of the work without difficulty.

The treatises which I have especially consulted in the preparation of this book are Professor William Chauvenet's admirable Manual of Spherical and Practical Astronomy, and an elaborate treatise on Descriptive Astronomy, by George F. Chambers, recently published in England. An interesting chronological history of Astronomy, taken from the latter work, will be found in the Appendix; and Plates II., III., and IV. are from the same source. Plate I. is from a French work by Camille Flammarion, entitled La Plurality des Mondes Habitis. I am indebted to my friend Professor George A. Osborne, of the Massachusetts Institute of Technology, for his kindness in revising my manuscript.

I trust that this treatise, although prepared especially as a text-book for students, may be of interest to others who may wish to know the general principles and the present state of the science of Astronomy.

C. J. W.

Harvard College,

Cambridge, Mass., March, 1872.

Contents

The Greek Alphabet Page 10

Chapter I

GENERAL PHENOMENA OF THE HEAVENS. DEFINITIONS. THE CELESTIAL SPHERE.

The heavenly bodies. Astronomy. Form of the earth. Diurnal motions of the heavenly bodies. Right, parallel, and oblique spheres. Definitions. Theorems. The astronomical triangle. Spherical coordinates. Vanishing lines and circles. Spherical projections 11

Chapter II

ASTRONOMICAL INSTRUMENTS. ERRORS.

The clock : its error and rate. The chronograph. The transit instrument: its construction, adjustment, and use. The meridian circle. The reading microscope. Fixed points. The mural circle. The altitude and azimuth instrument. Method of equal altitudes. The equatorial. The sextant. The artificial horizon. The vernier. Other astronomical instruments. Classes of errors 28

Chapter III

REFRACTION. PARALLAX. DIP OF THE HORIZON.

General laws of refraction. Astronomical refraction. Geocentric and heliocentric parallax. The dip of the horizon 51

Chapter IV

THE EARTH. ITS SIZE, FORM, AND ROTATION.

Measurement of arcs of the meridian by triangulation. Spheroidal form of the earth. Its dimensions. Its volume, density, and weight. Rotation of the earth. Change of weight in different latitudes. Centrifugal force. The trade-winds. Foucault's pendulum experiment. Linear velocity of rotation 59

D CONTENTS.

Chapter V. Latitude And Longitude

Four methods of finding the latitude of a place. Latitude at sea. Reduction of the latitude. Longitude. Greenwich time by chronometers, celestial phenomena, and lunar distances. Difference of longitude by electric and star signals. Longitude at sea. Comparison of the local times of different meridians Page 72

Chapter VI

THE SUN. THE EARTH'S ORBIT. THE SEASONS. TWILIGHT. THE ZODIACAL LIGHT.

The ecliptic. Distance of the sun from the earth determined by transits of Venus. Magnitude of the sun. The earth's orbit about the sun. The seasons. Twilight. Rotation of the sun, and its constitution. The zodiacal light 83

Chapter VII

SIDEREAL AND SOLAR TIME. EQUATION OF TIME. THE CALENDAR.

The sidereal and the solar year. Relation of sidereal and solar time. The equation of the centre. The equation of time. Astronomical and civil time. The calendar 100

Chapter VIII

UNIVERSAL GRAVITATION. PERTURBATIONS IN THE EARTH'S ORBIT. ABERRATION.

The law of universal gravitation. The mass of the sun. The earth's motion at perihelion and aphelion. Kepler's laws. Precession. Nutation. Change in the obliquity of the ecliptic. Advance of the line of apsides. Diurnal and annual aberration. Velocity of light. Aberration a proof of the earth's revolution 107

Chapter IX. The Moon

The orbit of the moon, and perturbations in it. Variation of the moon's meridian zenith distance. Distance, size, and mass of the moon. Augmentation of the semi-diameter. The phases of the moon. Sidereal and synodical periods. Retardation of the moon. The harvest moon. Rotation ; librations and other perturbations. The lunar cycle. General description of the moon 120

CONTENTS. i

CHAPTEE X.

LUNAR AND SOLAR ECLIPSES. OCCULTATIONS.

Lunar eclipses. The earth's shadow. Lunar ecliptic limits. Solar eclipses. The moon's shadow. Solar ecliptic limits. Cycle and number of eclipses. Occupations. Longitude by solar eclipses and occultations Page 135

CHAPTER XL

THE TIDES.

Cause of the tides. Effect of the moon's change in declination. General laws. Influence of the sun. Priming and lagging of tides. The establishment of a port. Cotidal lines. Height of tides. Tides in bays, rivers, &c. Four daily tides. Other phenomena 146

CHAPTER XII.

THE PLANETS AND THE PLANETOIDS. THE NEBULAR HYPOTHESIS.

Apparent motions of the planets. Heliocentric parallax. Orbits of the planets. Inferior planets. Direct and retrograde motion. Stationary points. Evening and morning stars. Elements of a planet's orbit. Heliocentric longitude of the node. Inclination of the orbit. Periodic time. Mercury. Venus. Transits of Venus. Superior planets: their periodic times and distances. Mars. The minor planets. Bode's law. Jupiter: its belts, satellites, and mass. Saturn and its rings. Disappearance of the rings. Uranus. Neptune. The nebular hypothesis 155

CHAPTER XIII.

COMETS AND METEORIC BODIES.

General description of comets. The tail. Elements of a comet's orbit. Number of comets and their orbits. Periodic times. Motion in their orbits. Mass and density. Periodic comets. Encke's comet. Winnecke's or Pons's comet. Brorsen's comet. Biela's comet, D' Arrest's comet. Faye's comet. Mechain's comet. Halley's comet. Remarkable comets of the present century. The great comet of 1811. The great comet of 1843. Donati's comet. The great comet of 1861. Meteoric bodies. Shooting stars. The November showers. Height and orbits of the meteors. Detonating meteors. Aerolites. Connection of comets and meteoric bodies 187

O CONTENTS.

CHAPTER XIV.

THE FIXED STARS. NEBULiE. MOTION OF THE SOLAR SYSTEM. REAL MOTIONS OF THE STARS.

Proper motions of the fixed stars. Magnitudes. Constellations. Constitution of the stars. Distance of the stars. Bessel's differential observations. Real magnitudes of the stars. Variable and temporary stars. Double and binary stars. Colored stars. Clusters. Resolvable and irresolvable nebulae. Annular, elliptic, spiral, and planetary nebulae. Nebulous stars. Double nebulae. The Magellanic clouds. Variation of brightness in nebulae. The milky way. Number of the stars. Motion of the solar system in space. Real motions of stars detected with the spectroscope Page 213

Appendix

Mathematical definitions, theorems, and formulae 242

Rirkwood's law 247

Chronological history of astronomy 248

Sketch of the history of navigation 255

Table I. — Elements of the planets, the sun, and the moon 256

" II.— The earth 257

" III.— The moon 257

" IV.— Elements of the satellites 258

" V. — The minorplanets 259

" VI. — Schwabe's observations of the solar spots 261

" VII. — Elements of the periodic comets 261

" VIII.— Transits of the inferior planets 262

" IX. — Stars whose parallax has been determined 262

" X.— The constellations.." 263

" XI. — Examples of variable stars 266

" XII. — Examples of binary stars 267

Index 269

THE GREEK ALPHABET.

The following table of the small letters of this alphabet is given for the use of those readers who are unacquainted with the Greek language.

a

Alpha.

V

Nu.

Beta.

i

Xi.

r

Gamma.

Omicron.

s

Delta.

It

Pi.

Epsllon.

p

Rho.

Zeta.

a

Sigma.

n

Eta.

*

Tau.

£or

Theta.

V

Upsilon.

i

Iota.

Phi.

K

Kappa.

Z

Chi.

A,

Lambda.

Psi.

t*

Mu.

Omega.

ASTRONOMY.

Chapter I

GENERAL PHENOMENA OF THE HEAVENS. DEFINITIONS.

1. The heavenly bodies are the sun, the planets, the satellites of the planets, the comets, the meteors, and the fixed stars.

The planets revolve about the sun in elliptical orbits, and the satellites revolve in similar orbits about the planets. The earth is a planet, as we shall see hereafter, and the moon is its satellite. The comets revolve about the sun in orbits which are either ellipses, parabolas, or hyperbolas. Comparatively little is known with any degree of certainty about the meteors ; but it is probable that they too revolve about the sun.

The sun, the planets, the satellites, and the comets constitute what is called the solar system. The fixed stars are bodies which lie outside of this system, and preserve almost precisely the same configuration from year to year.

The heavenly bodies may be considered to be projected upon the concave surface of a sphere of indefinite radius, the eye of the observer being at the centre of the sphere. This sphere is called the celestial sphere.

2. Astronomy is the science which treats of the heavenly bodies. It may be divided into Theoretical, Practical, and Descriptive Astronomy.

Theoretical Astronomy may be divided into Spherical and Physical Astronomy.

Spherical Astronomy treats of the heavenly bodies when considered to be projected upon the surface of the celestial sphere. It embraces those problems which arise from the apparent diur-

ll

FORM OF THE EARTH.

nal motion of the heavenly bodies, and also those which arise from any changes in the apparent positions of these bodies upon the surface of the celestial sphere.

Physical Astronomy treats of the causes of the motions of the heavenly bodies, and of the laws by which these motions are governed.

Practical Astronomy treats of the construction, adjustment, and use of astronomical instruments.

Descriptive Astronomy includes a general description of the heavenly bodies ; of their magnitudes, distances, motions, and configuration ; of their appearance *nd structure ; of, in short, every thing relating to these bodies which comes from observation or calculation.

3. Form of the Earth. — We may assume, at the outset, that the form of the earth is very nearly that of a sphere. The following are some of the reasons which may be given for such an assumption : —

(1.) If we stand upon the sea-shore, and watch a ship which is receding from the land, we shall find that the topmasts remain in sight after the hull has disappeared. If the surface of the sea were merely an extended plane, this would not happen ; for the topmasts, being smaller in dimensions than the hull, would

in that case disappear first.

The supposition that the surface of the sea is curved, however, fully accounts for this phenomenon, as may be seen in Fig. 1. Let the curve CBG represent a portion of the earth's surface, and let A be the position of the observer's eye : it is at once evident that no portion of the ship, S, will be visible which is situated below the line AH, drawn from A tangent to the earth's surface at C. The sftme figure also shows why it is that when a ship is approaching land any object on shore can be seen from the topmasts before it is seen from the deck.

(2.) At sea, the visible horizon everywhere appears to be a circle. This also is easily explained on the supposition that the

DIURNAL MOTION.

earth is spherical in form ; for if, in Fig. 1, the line A H is turned about the point A, and is continually tangent to the sphere, the points of tan gen cy, C, D, E, &c, will form the visible horizon of the spectator, and will evidently constitute a circle.

(3.) A lunar eclipse occurs when the earth is situated between the moon and the sun. Now the shadow which the earth at such a time casts upon the moon is invariably circular in form : and a body which in every position casts a circular shadow must be a sphere.

4. Diurnal motion of the heavenly bodies. — Two things will be noticed by an observer who watches the heavens during any clear night. The first is, that all the heavenly bodies, with the exception of the moon and the planets, retain constantly the same relative situation ; and the second is, that all these bodies, without any exception whatever, are continually changing their positions with reference to the horizon. Let us suppose the observer to be at some place in the Northern Hemisphere. A plane passed tangent to the earth's surface at his feet will be his

/ p

Fig. 2.

sensible horizon, and a second plane, parallel to this, passed through the centre of the earth, will be his rational horizon. If

14 DIUKNAL MOTION.

these twu planes be indefinitely extended in every direction, they will intersect the surface of the celestial sphere in two circles; but the radius of the celestial sphere is so immense in comparison with the radius of the earth, that these two circles will sensibly coincide, and will form one great circle of the sphere, called the celestial horizon. In other words, the earth, when compared with the celestial sphere, is to be regarded as only a point at its centre.

Let Fig. 2, then, represent the celestial sphere, at the centre of which, 0, the observer is stationed. Let the circle HESW be his celestial horizon, of which H is the north point, S the south, E the east, and W the west.

If he looks towards the southern point of the horizon, and watches the movements of some star which rises a little east of south, he will see that it rises above the horizon in a circular path for a little distance, attains its greatest elevation above the horizon when it bears directly south, and then descends and passes below the horizon a little west of south. In the figure, abe represents the path of such a star. If he notices a star which rises more to the eastward, as at d, for instance, he will see that it also passes from the east quarter of the horizon to the west in a circular path, attaining, however, a greater altitude above the horizon than that which the star a attained, and remaining above the horizon a longer time. A star which rises in the east point will set in the west point, and will remain twelve hours above the horizon. Turning his attention to some star which rises between the north and the east, as the star g, for instance, he will find that its movements are similar to the movements of the stars already noticed, and that it will remain above the horizon for more than twelve hours. Finally, if he turns towards the north, he will see certain stars, called circwnjiolar stars, which never pass below the horizon, but continually revolve about a fixed point in the heavens, very near to which point is a bright star called the Pole-star, which, to the naked eye, appears to be stationary, though observation shows that it also revolves about this same fixed point. In the figure, Im represents the orbit of a circum polar star.

If the same course of observations be repeated on the follow-

BIURNAL MOTION. T5

ing night, the observer will find that the situations of the stars with reference both to each other and to the horizon are the same that they were when he first began to examine them ; the motions which have already been described will be repeated, the circumpolar stars will still revolve about the same point in the heavens, and, in short, all the phenomena of which he took note will again be exhibited.

5. Inferences. — Three important truths are proved by a series of observations similar to that which we suppose to have been made.

(1.) The points at which the path of each star intersects the horizon remain unchanged from night to night, as long as the geographical position of the observer remains the same.

(2.) All the stars, whether they move in great or in small circles, make a complete revolution in identically the same interval of time; — that is to say, in twenty-four hours.

(3.) If we call that point about which the circumpolar stars appear to revolve the north pole of the heavens, and call the right line drawn from this point through the common centre of the earth and the celestial sphere the axis of the celestial sphere, the planes of the circles of all the stars are perpendicular to this axis.

Whether, then, the earth remains at rest, and the celestial sphere rotates about its axis, as above defined, or the celestial sphere remains at rest, and the earth rotates within it on an axis of its own, one thing is certain : the axis of rotation preserves in either case a constant direction. If the celestial sphere rotates about its axis, this axis always passes through the same points of the earth's surface ; and if the earth rotates within the celestial sphere, its axis of rotation is constantly directed to the same points on the surface of the celestial sphere.

We shall see hereafter the reasons which have led to the adoption of the theory that these apparent motions of the stars are really due to the rotation "of the earth upon its own axis. At present, for the sake of convenience in description, we shall consider the earth to be at rest, and shall speak of the apparent motions of the heavenly bodies as though they were real.

6. Farther observations. — Let us now suppose that the observer

16 DIURNAL MOTION.

leaves the place where he has hitherto been stationed, and travels in the direction of the point about which the circumpolar stars have appeared to revolve, and which we have called the north pole of the heavens. The general character of the phenomena which he observes will not be changed ; but he will notice a change in this respect : the elevation of the north pole above the horizon will continually increase as he travels towards it, and the planes of the circles of the stars, remaining constantly perpendicular to the axis of the celestial sphere, will become less and less inclined to the plane of the horizon. The consequence of this will be that stars which are near the southern point of the horizon will remain a shorter time above the horizon, and will finally cease to appear; while in the northern quarter of the heavens the number of stars which never pass below the horizon will continually increase. Finally, if we suppose the observer to go on until the north pole is directly above his head, the stars which he sees will neither set nor rise, but will continually move about the sphere in circles whose planes are parallel to the plane of the horizon. In such a situation, it is evident that half of the celestial sphere will be perpetually invisible to him. Referring to Fig. 2, such a state of things is represented by supposing the line OP to be moved up into coincidence with OZ, the planes of the circles abc, def, &c, still remaining perpendicular to OP. The stars a and d will lie continually below the horizon, and the stars g and I continually above it.

Such a sphere as this just now described, where the planes of revolution are parallel to the plane of the horizon, is called a parallel sphere.

If the observer, instead of travelling towards the north pole, travels directly from it, its elevation above the horizon will continually decrease, and the obliquity of the planes of the circles to the plane of the horizon will continually increase, until he will at length reach a point at which the north pole will lie in his horizon and the planes of the circles will be perpendicular to its plane. Referring again to Fig. 2, the line PO will in this case coincide with OH, and the arcs abc, def, &c, will have their planes perpendicular to the plane of the horizon. It is evident that at this point every star in the celestial sphere will come above the horizon once in twenty-four hours, and that half of every circle will lie above the horizon, and half below it.

The geographical position which the observer has now reached is some point on the earth's equator (Art. 7). Such a sphere as this, where the planes of the circles are perpendicular to the plane of the horizon, is called a right sphere. Besides the right and the parallel sphere, we have also the oblique sphere, where the planes of the circles are oblique to the plane of the horizon. Such a sphere is represented in Fig. 2.

If the observer travels still farther in the same direction, the north pole will sink below T his horizon, and the other extremity of the axis of the celestial sphere, called the south pole, will rise above it. There will be circumpolar stars revolving about this pole, and, in brief, all the phenomena which the observer noticed while travelling towards the north pole will be repeated as he travels towards the south pole.

DEFINITIONS.

7. We are now prepared to define certain points, angles, and circles on the earth and on the celestial sphere.

The axis of the celestial sphere is, as we have already seen, an imaginary line drawn from the north pole of the heavens through the common centre of the earth and of the celestial sphere, and produced until it again meets the surface of the celestial sphere in the south pole. The points where this axis meets the surface of the earth are called the north pole and the south pole of the earth.

That pole of the heavens which is above the horizon at any place is called the elevated pole at that place ; the other is called the depressed pole.

The axis of the earth is that diameter which passes through the poles of the earth.

The earth's equator is a great circle of the earth, whose plane is perpendicular to the axis. This circle is, of course, equidistant from the two poles, and divides the earth into two hemispheres. That hemisphere which contains the north pole is called the northern hemisphere, the other is called the southern hemisphere.

Parallels of latitude are small circles of the earth whose planes are perpendicular to the axis.

Terrestrial meridians are great circles of the earth passing through the poles.

The latitude of any place on the earth's surface is its angular distance from the plane of the equator. This angle is measured by the arc of the meridian included between the place and the equator. Latitude is reckoned, either north or south, from 0° to 90°.

The longitude of any place is the inclination of its own meridian to the meridian of some fixed station, and is measured by the arc of the equator included between these two meridians. Longitude is usually reckoned east or west of the fixed meridian, from 0° to 180°. The meridian of Greenwich, England, is most commonly taken for the fixed meridian, though the meridians of Washington, Paris, and other places are also taken for the same purpose. The fixed meridian is called the prime meridian.

The arc of a parallel of latitude included between any two meridians is the departure between those meridians for that latitude. It is evident that if the departure between any two meridians is taken on two different parallels of latitude, the departure at the greater latitude will be the smaller.

Upward motion at any place is motion from the centre of the earth. Downward motion is towards the same point.

The celestial horizon has already been defined (Art. 4). Lines drawn perpendicular to the plane of the horizon are called vertical lines. The vertical line at any place, if indefinitely prolonged, meets the celestial sphere in two points. The upper point is called the zenith, the low T er point the nadir.

The celestial meridian of any place is the great circle in which the plane of the terrestrial meridian of that place, when indefinitely produced, meets the surface of the celestial sphere. The axis of the sphere divides the celestial meridian into two semicircumferences : that which lies on the same side of the axis as the zenith is called the upper branch of the meridian, and the other is called the lower branch. The points where the celestial meridian and the celestial horizon intersect are called the north and the south point of the horizon, — the point which is the nearer to the north pole being the north point. The line in which the planes of these same two great circles intersect is called the meridian line.

Vertical circles are great circles of the sphere which pass through the zenith and nadir. That vertical circle the plane of which is perpendicular to the plane of the celestial meridian is called the prime vertical. The points in which the prime vertical cuts the horizon are called the east and the west point of the horizon; and the line which joins these two points is the east and west line.

The celestial equator, also called the equinoctial, is the great circle of the sphere in which the plane of the earths equator, indefinitely produced, meets the celestial sphere.

The altitude of a heavenly body is its angular distance above the plane of the celestial horizon, measured on a vertical circle passing through that body. The zenith distance of the body is its angular distance from the zenith, and is evidently the complement of the altitude.

The azimuth of a celestial body is the inclination of the vertical circle which passes through the body to the celestial meridian, and is measured by the arc of the celestial horizon included between this vertical circle and the celestial meridian. Azimuth may be reckoned from either the north or the south point of the horizon, and towards either the west or the east. Navigators usually reckon it from the north point in north latitude and the south point in south latitude, and east or west as the body is east or west of the meridian, thus restricting it numerically to values less than 180°. Astronomers, on the contrary, usually reckon from the south point, to the right hand, from 0° to 360°.

The amplitude of a heavenly body is its angular distance from the prime vertical when in the horizon. It is reckoned from the east point when the body is rising, and from the west point when the body is setting, towards the north or the south as the body is to the north or the south of the prime vertical.

The vernal equinox is a certain fibred point upon the equinoctial. It is also called the first point of Aries.

Hour circles, or circles of declination, are great circles of the sphere passing through the poles of the heavens.

The right ascension of a heavenly body is the inclination of its hour circle to the hour circle which passes through the vernal equinox ; or it is the arc of the equinoctial intercepted between these two hour circles. Right ascension is usually reckoned in hours, minutes, and seconds (an hour being taken equal to 15° of arc), and is always reckoned to the eastward, from Oh. to 24h.

The declination of a heavenly body is its angular distance from the plane of the celestial equator, measured on the circle of declination passing through the body. It is reckoned in degrees, minutes, &c, to the north and the south. The polar distance of a body is its angular distance from either pole, measured on its hour circle. Usually, however, when we speak of the polar distance of a body, we mean its angular distance from the elevated pole.

If, in Fig. 2, with the arc HP, which measures the altitude of the elevated pole, as a polar radius, we describe a circle about the pole as a centre, it is evident that the stars whose circles lie within this circle will never set. This circle is called the circle of perpetual apparition. It is equally evident that stars whose circles lie within a circle of the same magnitude, described about the depressed pole as a centre, will never come above the horizon. This circle is therefore called the circle of perpetual occultation.

The passage of a celestial body across the meridian is called its transit or culmination. When the body is within the circle of perpetual apparition, both transits occur above the horizon, one above the pole, the other below it. These are called the upper and the lower transit. For all bodies outside this circle, and not within the circle of perpetual occultation, the upper transit occurs above the horizon, the lower below it. For all bodies whatever, the upper transit occurs when the body crosses the upper branch of the meridian, and the lower transit when it crosses the low r er branch.