TWENTIETH CENTURY TEXT-BOOKS

EDITED BY
A. F. NIGHTINGALE, Ph.D., LL.D.
FORMERLY SUPERINTENDENT OF HIGH SCHOOLS, CHICAGO

TWENTIETH CENTURY TEXT-BOOKS

A TEXT-BOOK OF ASTRONOMY

BY

GEORGE C. COMSTOCK

DIRECTOR OF THE WASHBURN OBSERVATORY AND
PROFESSOR OF ASTRONOMY IN THE
UNIVERSITY OF WISCONSIN

NEW YORK
D. APPLETON AND COMPANY
1903

Copyright, 1901

By D. APPLETON AND COMPANY

PREFACE

The present work is not a compendium of astronomy or an outline course of popular reading in that science. It has been prepared as a text-book, and the author has purposely omitted from it much matter interesting as well as important to a complete view of the science, and has endeavored to concentrate attention upon those parts of the subject that possess special educational value. From this point of view matter which permits of experimental treatment with simple apparatus is of peculiar value and is given a prominence in the text beyond its just due in a well-balanced exposition of the elements of astronomy, while topics, such as the results of spectrum analysis, which depend upon elaborate apparatus, are in the experimental part of the work accorded much less space than their intrinsic importance would justify.

Teacher and student are alike urged to magnify the observational side of the subject and to strive to obtain in their work the maximum degree of precision of which their apparatus is capable. The instruments required are few and easily obtained. With exception of a watch and a protractor, all of the apparatus needed may be built by any one of fair mechanical talent who will follow the illustrations and descriptions of the text. In order that proper opportunity for observations may be had, the study should be pursued during the milder portion of the year, between April and November in northern latitudes, using clear weather for a direct study of the sky and cloudy days for book work.

The illustrations contained in the present work are worthy of as careful study as is the text, and many of them are intended as an aid to experimental work and accurate measurement, e. g., the star maps, the diagrams of the planetary orbits, pictures of the moon, sun, etc. If the school possesses a projection lantern, a set of astronomical slides to be used in connection with it may be made of great advantage, if the pictures are studied as an auxiliary to Nature. Mere display and scenic effect are of little value.

A brief bibliography of popular literature upon astronomy may be found at the end of this book, and it will be well if at least a part of these works can be placed in the school library and systematically used for supplementary reading. An added interest may be given to the study if one or more of the popular periodicals which deal with astronomy are taken regularly by the school and kept within easy reach of the students. From time to time the teacher may well assign topics treated in these periodicals to be read by individual students and presented to the class in the form of an essay.

The author is under obligations to many of his professional friends who have contributed illustrative matter for his text, and his thanks are in an especial manner due to the editors of the Astrophysical Journal, Astronomy and Astrophysics, and Popular Astronomy for permission to reproduce here plates which have appeared in those periodicals, and to Dr. Charles Boynton, who has kindly read and criticised the proofs.

George C. Comstock.

University of Wisconsin, February, 1901.

CONTENTS

LIST OF LITHOGRAPHIC PLATES

LIST OF FULL-PAGE ILLUSTRATIONS

ASTRONOMY

CHAPTER I

DIFFERENT KINDS OF MEASUREMENT

1. Accurate measurement.—Accurate measurement is the foundation of exact science, and at the very beginning of his study in astronomy the student should learn something of the astronomer's kind of measurement. He should practice measuring the stars with all possible care, and should seek to attain the most accurate results of which his instruments and apparatus are capable. The ordinary affairs of life furnish abundant illustration of some of these measurements, such as finding the length of a board in inches or the weight of a load of coal in pounds and measurements of both length and weight are of importance in astronomy, but of far greater astronomical importance than these are the measurement of angles and the measurement of time. A kitchen clock or a cheap watch is usually thought of as a machine to tell the "time of day," but it may be used to time a horse or a bicycler upon a race course, and then it becomes an instrument to measure the amount of time required for covering the length of the course. Astronomers use a clock in both of these ways—to tell the time at which something happens or is done, and to measure the amount of time required for something; and in using a clock for either purpose the student should learn to take the time from it to the nearest second or better, if it has a seconds hand, or to a small fraction of a minute, by estimating the position of the minute hand between the minute marks on the dial. Estimate the fraction in tenths of a minute, not in halves or quarters.

Exercise 1.—If several watches are available, let one person tap sharply upon a desk with a pencil and let each of the others note the time by the minute hand to the nearest tenth of a minute and record the observations as follows:

The letters h and m are used as abbreviations for hour and minute. The first and second columns of the table are the record made by one student, and second and third the record made by another. After all the observations have been made and recorded they should be brought together and compared by taking the differences between the times recorded for each tap, as is shown in the last column. This difference shows how much faster one watch is than the other, and the agreement or disagreement of these differences shows the degree of accuracy of the observations. Keep up this practice until tenths of a minute can be estimated with fair precision.

2. Angles and their use.—An angle is the amount of opening or difference of direction between two lines that cross each other. At twelve o'clock the hour and minute hand of a watch point in the same direction and the angle between them is zero. At one o'clock the minute hand is again at XII, but the hour hand has moved to I, one twelfth part of the circumference of the dial, and the angle between the hands is one twelfth of a circumference. It is customary to imagine the circumference of a dial to be cut up into 360 equal parts—i. e., each minute space of an ordinary dial to be subdivided into six equal parts, each of which is called a degree, and the measurement of an angle consists in finding how many of these degrees are included in the opening between its sides. At one o'clock the angle between the hands of a watch is thirty degrees, which is usually written 30°, at three o'clock it is 90°, at six o'clock 180°, etc.

A watch may be used to measure angles. How? But a more convenient instrument is the protractor, which is shown in [Fig. 1], applied to the angle A B C and showing that A B C = 85° as nearly as the protractor scale can be read.

The student should have and use a protractor, such as is furnished with this book, for the numerous exercises which are to follow.

Fig. 1.—A protractor.

Exercise 2.—Draw neatly a triangle with sides about 100 millimeters long, measure each of its angles and take their sum. No matter what may be the shape of the triangle, this sum should be very nearly 180°—exactly 180° if the work were perfect—but perfection can seldom be attained and one of the first lessons to be learned in any science which deals with measurement is, that however careful we may be in our work some minute error will cling to it and our results can be only approximately correct. This, however, should not be taken as an excuse for careless work, but rather as a stimulus to extra effort in order that the unavoidable errors may be made as small as possible. In the present case the measured angles may be improved a little by adding (algebraically) to each of them one third of the amount by which their sum falls short of 180°, as in the following example:

This process is in very common use among astronomers, and is called "adjusting" the observations.

Fig. 2.—Triangulation.

3. Triangles.—The instruments used by astronomers for the measurement of angles are usually provided with a telescope, which may be pointed at different objects, and with a scale, like that of the protractor, to measure the angle through which the telescope is turned in passing from one object to another. In this way it is possible to measure the angle between lines drawn from the instrument to two distant objects, such as two church steeples or the sun and moon, and this is usually called the angle between the objects. By measuring angles in this way it is possible to determine the distance to an inaccessible point, as shown in [Fig. 2]. A surveyor at A desires to know the distance to C, on the opposite side of a river which he can not cross. He measures with a tape line along his own side of the stream the distance A B = 100 yards and then, with a suitable instrument, measures the angle at A between the points C and B, and the angle at B between C and A, finding B A C = 73.4°, A B C = 49.3°. To determine the distance A C he draws upon paper a line 100 millimeters long, and marks the ends a and b; with a protractor he constructs at a the angle b a c = 73.4°, and at b the angle a b c = 49.3°, and marks by c the point where the two lines thus drawn meet. With the millimeter scale he now measures the distance a c = 90.2 millimeters, which determines the distance A C across the river to be 90.2 yards, since the triangle on paper has been made similar to the one across the river, and millimeters on the one correspond to yards on the other. What is the proposition of geometry upon which this depends? The measured distance A B in the surveyor's problem is called a base line.

Exercise 3.—With a foot rule and a protractor measure a base line and the angles necessary to determine the length of the schoolroom. After the length has been thus found, measure it directly with the foot rule and compare the measured length with the one found from the angles. If any part of the work has been carelessly done, the student need not expect the results to agree.

Fig. 3.—Finding the moon's distance from the earth.

In the same manner, by sighting at the moon from widely different parts of the earth, as in [Fig. 3], the moon's distance from us is found to be about a quarter of a million miles. What is the base line in this case?

4. The horizon—altitudes.—In their observations astronomers and sailors make much use of the plane of the horizon, and practically any flat and level surface, such as that of a smooth pond, may be regarded as a part of this plane and used as such. A very common observation relating to the plane of the horizon is called "taking the sun's altitude," and consists in measuring the angle between the sun's rays and the plane of the horizon upon which they fall. This angle between a line and a plane appears slightly different from the angle between two lines, but is really the same thing, since it means the angle between the sun's rays and a line drawn in the plane of the horizon toward the point directly under the sun. Compare this with the definition given in the geographies, "The latitude of a point on the earth's surface is its angular distance north or south of the equator," and note that the latitude is the angle between the plane of the equator and a line drawn from the earth's center to the given point on its surface.

A convenient method of obtaining a part of the plane of the horizon for use in observation is as follows: Place a slate or a pane of glass upon a table in the sunshine. Slightly moisten its whole surface and then pour a little more water upon it near the center. If the water runs toward one side, thrust the edge of a thin wooden wedge under this side and block it up until the water shows no tendency to run one way rather than another; it is then level and a part of the plane of the horizon. Get several wedges ready before commencing the experiment. After they have been properly placed, drive a pin or tack behind each one so that it may not slip.

5. Taking the sun's altitude. Exercise 4.—Prepare a piece of board 20 centimeters, or more, square, planed smooth on one face and one edge. Drive a pin perpendicularly into the face of the board, near the middle of the planed edge. Set the board on edge on the horizon plane and turn it edgewise toward the sun so that a shadow of the pin is cast on the plane. Stick another pin into the board, near its upper edge, so that its shadow shall fall exactly upon the shadow of the first pin, and with a watch or clock observe the time at which the two shadows coincide. Without lifting the board from the plane, turn it around so that the opposite edge is directed toward the sun and set a third pin just as the second one was placed, and again take the time. Remove the pins and draw fine pencil lines, connecting the holes, as shown in [Fig. 4], and with the protractor measure the angle thus marked. The student who has studied elementary geometry should be able to demonstrate that at the mean of the two recorded times the sun's altitude was equal to one half of the angle measured in the figure.

Fig. 4.—Taking the sun's altitude.

When the board is turned edgewise toward the sun so that its shadow is as thin as possible, rule a pencil line alongside it on the horizon plane. The angle which this line makes with a line pointing due south is called the sun's azimuth. When the sun is south, its azimuth is zero; when west, it is 90°; when east, 270°, etc.

Exercise 5.—Let a number of different students take the sun's altitude during both the morning and afternoon session and note the time of each observation, to the nearest minute. Verify the setting of the plane of the horizon from time to time, to make sure that no change has occurred in it.

6. Graphical representations.—Make a graph (drawing) of all the observations, similar to [Fig. 5], and find by bisecting a set of chords g to g, e to e, d to d, drawn parallel to B B, the time at which the sun's altitude was greatest. In [Fig. 5] we see from the intersection of M M with B B that this time was 11h. 50m.

The method of graphs which is here introduced is of great importance in physical science, and the student should carefully observe in [Fig. 5] that the line B B is a scale of times, which may be made long or short, provided only the intervals between consecutive hours 9 to 10, 10 to 11, 11 to 12, etc., are equal. The distance of each little circle from B B is taken proportional to the sun's altitude, and may be upon any desired scale—e. g., a millimeter to a degree—provided the same scale is used for all observations. Each circle is placed accurately over that part of the base line which corresponds to the time at which the altitude was taken. Square ruled paper is very convenient, although not necessary, for such diagrams. It is especially to be noted that from the few observations which are represented in the figure a smooth curve has been drawn through the circles which represent the sun's altitude, and this curve shows the altitude of the sun at every moment between 9 A. M. and 3 P. M. In [Fig. 5] the sun's altitude at noon was 57°. What was it at half past two?

Fig. 5.—A graph of the sun's altitude.

7. Diameter of a distant object.—By sighting over a protractor, measure the angle between imaginary lines drawn from it to the opposite sides of a window. Carry the protractor farther away from the window and repeat the experiment, to see how much the angle changes. The angle thus measured is called "the angle subtended" by the window at the place where the measurement was made. If this place was squarely in front of the window we may draw upon paper an angle equal to the measured one and lay off from the vertex along its sides a distance proportional to the distance of the window—e. g., a millimeter for each centimeter of real distance. If a cross line be now drawn connecting the points thus found, its length will be proportional to the width of the window, and the width may be read off to scale, a centimeter for every millimeter in the length of the cross line.

The astronomer who measures with an appropriate instrument the angle subtended by the moon may in an entirely similar manner find the moon's diameter and has, in fact, found it to be 2,163 miles. Can the same method be used to find the diameter of the sun? A planet? The earth?

CHAPTER II

THE STARS AND THEIR DIURNAL MOTION

8. The stars.—From the very beginning of his study in astronomy, and as frequently as possible, the student should practice watching the stars by night, to become acquainted with the constellations and their movements. As an introduction to this study he may face toward the north, and compare the stars which he sees in that part of the sky with the map of the northern heavens, given on [Plate I], opposite [page 124]. Turn the map around, upside down if necessary, until the stars upon it match the brighter ones in the sky. Note how the stars are grouped in such conspicuous constellations as the Big Dipper (Ursa Major), the Little Dipper (Ursa Minor), and Cassiopeia. These three constellations should be learned so that they can be recognized at any time.

The names of the stars.—Facing the star map is a key which contains the names of the more important constellations and the names of the brighter stars in their constellations. These names are for the most part a Greek letter prefixed to the genitive case of the Latin name of the constellation. (See the Greek alphabet printed at the end of the book.)

9. Magnitudes of the stars.—Nearly nineteen centuries ago St. Paul noted that "one star differeth from another star in glory," and no more apt words can be found to mark the difference of brightness which the stars present. Even prior to St. Paul's day the ancient Greek astronomers had divided the stars in respect of brightness into six groups, which the modern astronomers still use, calling each group a magnitude. Thus a few of the brightest stars are said to be of the first magnitude, the great mass of faint ones which are just visible to the unaided eye are said to be of the sixth magnitude, and intermediate degrees of brilliancy are represented by the intermediate magnitudes, second, third, fourth, and fifth. The student must not be misled by the word magnitude. It has no reference to the size of the stars, but only to their brightness, and on the star maps of this book the larger and smaller circles by which the stars are represented indicate only the brightness of the stars according to the system of magnitudes. Following the indications of these maps, the student should, in learning the principal stars and constellations, learn also to recognize how bright is a star of the second, fourth, or other magnitude.

10. Observing the stars.—Find on the map and in the sky the stars α Ursæ Minoris, α Ursæ Majoris, β Ursæ Majoris. What geometrical figure will fit on to these stars? In addition to its regular name, α Ursæ Minoris is frequently called by the special name Polaris, or the pole star. Why are the other two stars called "the Pointers"? What letter of the alphabet do the five bright stars in Cassiopeia suggest?

Exercise 6.—Stand in such a position that Polaris is just hidden behind the corner of a building or some other vertical line, and mark upon the key map as accurately as possible the position of this line with respect to the other stars, showing which stars are to the right and which are to the left of it. Record the time (date, hour, and minute) at which this observation was made. An hour or two later repeat the observation at the same place, draw the line and note the time, and you will find that the line last drawn upon the map does not agree with the first one. The stars have changed their positions, and with respect to the vertical line the Pointers are now in a different direction from Polaris. Measure with a protractor the angle between the two lines drawn in the map, and use this angle and the recorded times of the observation to find how many degrees per hour this direction is changing. It should be about 15° per hour. If the observation were repeated 12 hours after the first recorded time, what would be the position of the vertical line among the stars? What would it be 24 hours later? A week later? Repeat the observation on the next clear night, and allowing for the number of whole revolutions made by the stars between the two dates, again determine from the time interval a more accurate value of the rate at which the stars move.

The motion of the stars which the student has here detected is called their "diurnal" motion. What is the significance of the word diurnal?

In the preceding paragraph there is introduced a method of great importance in astronomical practice—i. e., determining something—in this case the rate per hour, from observations separated by a long interval of time, in order to get a more accurate value than could be found from a short interval. Why is it more accurate? To determine the rate at which the planet Mars rotates about its axis, astronomers use observations separated by an interval of more than 200 years, during which the planet made more than 75,000 revolutions upon its axis. If we were to write out in algebraic form an equation for determining the length of one revolution of Mars about its axis, the large number, 75,000, would appear in the equation as a divisor, and in the final result would greatly reduce whatever errors existed in the observations employed.

Repeat [Exercise 6] night after night, and note whether the stars come back to the same position at the same hour and minute every night.

Fig. 6.

Fig. 7.

11. The plumb-line apparatus.—This experiment, and many others, may be conveniently and accurately made with no other apparatus than a plumb line, and a device for sighting past it. In Figs. [6] and [7] there is shown a simple form of such apparatus, consisting essentially of a board which rests in a horizontal position upon the points of three screws that pass through it. This board carries a small box, to one side of which is nailed in vertical position another board 5 or 6 feet long to carry the plumb line. This consists of a wire or fish line with any heavy weight—e. g., a brick or flatiron—tied to its lower end and immersed in a vessel of water placed inside the box, so as to check any swinging motion of the weight. In the cover of the box is a small hole through which the wire passes, and by turning the screws in the baseboard the apparatus may be readily leveled, so that the wire shall swing freely in the center of the hole without touching the cover of the box. Guy wires, shown in the figure, are applied so as to stiffen the whole apparatus. A board with a screw eye at each end may be pivoted to the upright, as in [Fig. 6], for measuring altitudes; or to the box, as in [Fig. 7], for observing the time at which a star in its diurnal motion passes through the plane determined by the plumb line and the center of the screw eye through which the observer looks.

The whole apparatus may be constructed by any person of ordinary mechanical skill at a very small cost, and it or something equivalent should be provided for every class beginning observational astronomy. To use the apparatus for the experiment of [§ 10], it should be leveled, and the board with the screw eyes, attached as in [Fig. 7], should be turned until the observer, looking through the screw eye, sees Polaris exactly behind the wire. Use a bicycle lamp to illumine the wire by night. The apparatus is now adjusted, and the observer has only to wait for the stars which he desires to observe, and to note by his watch the time at which they pass behind the wire. It will be seen that the wire takes the place of the vertical edge of the building, and that the board with the screw eyes is introduced solely to keep the observer in the right place relative to the wire.

12. A sidereal clock.—Clocks are sometimes so made and regulated that they show always the same hour and minute when the stars come back to the same place, and such a timepiece is called a sidereal clock—i. e., a star-time clock. Would such a clock gain or lose in comparison with an ordinary watch? Could an ordinary watch be turned into a sidereal watch by moving the regulator?

13. Photographing the stars.—Exercise 7.—For any student who uses a camera. Upon some clear and moonless night point the camera, properly focused, at Polaris, and expose a plate for three or four hours. Upon developing the plate you should find a series of circular trails such as are shown in [Fig. 8], only longer. Each one of these is produced by a star moving slowly over the plate, in consequence of its changing position in the sky. The center indicated by these curved trails is called the pole of the heavens. It is that part of the sky toward which is pointed the axis about which the earth rotates, and the motion of the stars around the center is only an apparent motion due to the rotation of the earth which daily carries the observer and his camera around this axis while the stars stand still, just as trees and fences and telegraph poles stand still, although to the passenger upon a railway train they appear to be in rapid motion. So far as simple observations are concerned, there is no method by which the pupil can tell for himself that the motion of the stars is an apparent rather than a real one, and, following the custom of astronomers, we shall habitually speak as if it were a real movement of the stars. How long was the plate exposed in photographing [Fig. 8]?

14. Finding the stars.—On [Plate I], opposite [page 124], the pole of the heavens is at the center of the map, near Polaris, and the heavy trail near the center of [Fig. 8] is made by Polaris. See if you can identify from the map any of the stars whose trails show in the photograph. The brighter the star the bolder and heavier its trail.

Find from the map and locate in the sky the two bright stars Capella and Vega, which are on opposite sides of Polaris and nearly equidistant from it. Do these stars share in the motion around the pole? Are they visible on every clear night, and all night?

Observe other bright stars farther from Polaris than are Vega and Capella and note their movement. Do they move like the sun and moon? Do they rise and set?

In what part of the sky do the stars move most rapidly, near the pole or far from it?

How long does it take the fastest moving stars to make the circuit of the sky and come back to the same place? How long does it take the slow stars?

15. Rising and setting of the stars.—A study of the sky along the lines indicated in these questions will show that there is a considerable part of it surrounding the pole whose stars are visible on every clear night. The same star is sometimes high in the sky, sometimes low, sometimes to the east of the pole and at other times west of it, but is always above the horizon. Such stars are said to be circumpolar. A little farther from the pole each star, when at the lowest point of its circular path, dips for a time below the horizon and is lost to view, and the farther it is away from the pole the longer does it remain invisible, until, in the case of stars 90° away from the pole, we find them hidden below the horizon for twelve hours out of every twenty-four (see [Fig. 9]). The sun is such a star, and in its rising and setting acts precisely as does every other star at a similar distance from the pole—only, as we shall find later, each star keeps always at (nearly) the same distance from the pole, while the sun in the course of a year changes its distance from the pole very greatly, and thus changes the amount of time it spends above and below the horizon, producing in this way the long days of summer and the short ones of winter.

How much time do stars which are more than 90° from the pole spend above the horizon?

We say in common speech that the sun rises in the east, but this is strictly true only at the time when it is 90° distant from the pole—i. e., in March and September. At other seasons it rises north or south of east according as its distance from the pole is less or greater than 90°, and the same is true for the stars.

16. The geography of the sky.—Find from a map the latitude and longitude of your schoolhouse. Find on the map the place whose latitude is 39° and longitude 77° west of the meridian of Greenwich. Is there any other place in the world which has the same latitude and longitude as your schoolhouse?

The places of the stars in the sky are located in exactly the manner which is illustrated by these geographical questions, only different names are used. Instead of latitude the astronomer says declination, in place of longitude he says right ascension, in place of meridian he says hour circle, but he means by these new names the same ideas that the geographer expresses by the old ones.

Imagine the earth swollen up until it fills the whole sky; the earth's equator would meet the sky along a line (a great circle) everywhere 90° distant from the pole, and this line is called the celestial equator. Trace its position along the middle of the map opposite [page 190] and notice near what stars it runs. Every meridian of the swollen earth would touch the sky along an hour circle—i. e., a great circle passing through the pole and therefore perpendicular to the equator. Note that in the map one of these hour circles is marked 0. It plays the same part in measuring right ascensions as does the meridian of Greenwich in measuring longitudes; it is the beginning, from which they are reckoned. Note also, at the extreme left end of the map, the four bright stars in the form of a square, one side of which is parallel and close to the hour circle, which is marked 0. This is familiarly called the Great Square in Pegasus, and may be found high up in the southern sky whenever the Big Dipper lies below the pole. Why can it not be seen when Ursa Major is above the pole?

Astronomers use the right ascensions of the stars not only to tell in what part of the sky the star is placed, but also in time reckonings, to regulate their sidereal clocks, and with regard to this use they find it convenient to express right ascension not in degrees but in hours, 24 of which fill up the circuit of the sky and each of which is equal to 15° of arc, 24 × 15 = 360. The right ascension of Capella is 5h. 9m. = 77.2°, but the student should accustom himself to using it in hours and minutes as given and not to change it into degrees. He should also note that some stars lie on the side of the celestial equator toward Polaris, and others are on the opposite side, so that the astronomer has to distinguish between north declinations and south declinations, just as the geographer distinguishes between north latitudes and south latitudes. This is done by the use of the + and - signs, a + denoting that the star lies north of the celestial equator, i. e., toward Polaris.

Find on [Plate II], opposite [page 190], the Pleiades (Plēadēs), R. A. = 3h. 42m., Dec. = +23.8°. Why do they not show on [Plate I], opposite [page 124]? In what direction are they from Polaris? This is one of the finest star clusters in the sky, but it needs a telescope to bring out its richness. See how many stars you can count in it with the naked eye, and afterward examine it with an opera glass. Compare what you see with [Fig. 10]. Find Antares, R. A. = 16h. 23m. Dec. = -26.2°. How far is it, in degrees, from the pole? Is it visible in your sky? If so, what is its color?

Find the R. A. and Dec. of α Ursæ Majoris; of β Ursæ Majoris; of Polaris. Find the Northern Crown, Corona Borealis, R. A. = 15h. 30m., Dec. = +27.0°; the Beehive, Præsepe, R. A. = 8h. 33m., Dec. = +20.4°.

These should be looked up, not only on the map, but also in the sky.

17. Reference lines and circles.—As the stars move across the sky in their diurnal motion, they carry the framework of hour circles and equator with them, so that the right ascension and declination of each star remain unchanged by this motion, just as longitudes and latitudes remain unchanged by the earth's rotation. They are the same when a star is rising and when it is setting; when it is above the pole and when it is below it. During each day the hour circle of every star in the heavens passes overhead, and at the moment when any particular hour circle is exactly overhead all the stars which lie upon it are said to be "on the meridian"—i. e., at that particular moment they stand directly over the observer's geographical meridian and upon the corresponding celestial meridian.

An eye placed at the center of the earth and capable of looking through its solid substance would see your geographical meridian against the background of the sky exactly covering your celestial meridian and passing from one pole through your zenith to the other pole. In [Fig. 11] the inner circle represents the terrestrial meridian of a certain place, O, as seen from the center of the earth, C, and the outer circle represents the celestial meridian of O as seen from C, only we must imagine, what can not be shown on the figure, that the outer circle is so large that the inner one shrinks to a mere point in comparison with it. If C P represents the direction in which the earth's axis passes through the center, then C E at right angles to it must be the direction of the equator which we suppose to be turned edgewise toward us; and if C O is the direction of some particular point on the earth's surface, then Z directly overhead is called the zenith of that point, upon the celestial sphere. The line C H represents a direction parallel to the horizon plane at O, and H C P is the angle which the axis of the earth makes with this horizon plane. The arc O E measures the latitude of O, and the arc Z E measures the declination of Z, and since by elementary geometry each of these arcs contains the same number of degrees as the angle E C Z, we have the

Fig. 11.—Reference lines and circles.

Theorem.—The latitude of any place is equal to the declination of its zenith.

Corollary.—Any star whose declination is equal to your latitude will once in each day pass through your zenith.

18. Latitude.—From the construction of the figure

from which we find by subtraction and transposition

∠ E C Z = ∠ H C P

and this gives the further

Theorem.—The latitude of any place is equal to the elevation of the pole above its horizon plane.

An observer who travels north or south over the earth changes his latitude, and therefore changes the angle between his horizon plane and the axis of the earth. What effect will this have upon the position of stars in his sky? If you were to go to the earth's equator, in what part of the sky would you look for Polaris? Can Polaris be seen from Australia? From South America? If you were to go from Minnesota to Texas, in what respect would the appearance of stars in the northern sky be changed? How would the appearance of stars in the southern sky be changed?

Fig. 12.—Diurnal path of Polaris.

Exercise 8.—Determine your latitude by taking the altitude of Polaris when it is at some one of the four points of its diurnal path, shown in [Fig. 12]. When it is at 1 it is said to be at upper culmination, and the star ζ Ursæ Majoris in the handle of the Big Dipper will be directly below it. When at 2 it is at western elongation, and the star Castor is near the meridian. When it is at 3 it is at lower culmination, and the star Spica is on the meridian. When it is at 4 it is at eastern elongation, and Altair is near the meridian. All of these stars are conspicuous ones, which the student should find upon the map and learn to recognize in the sky. The altitude observed at either 2 or 4 may be considered equal to the latitude of the place, but the altitude observed when Polaris is at the positions marked 1 and 3 must be corrected for the star's distance from the pole, which may be assumed equal to 1.3°.

The plumb-line apparatus described at [page 12] is shown in [Fig. 6] slightly modified, so as to adapt it to measuring the altitudes of stars. Note that the board with the screw eye at one end has been transferred from the box to the vertical standard, and has a screw eye at each end. When the apparatus has been properly leveled, so that the plumb line hangs at the middle of the hole in the box cover, the board is to be pointed at the star by sighting through the centers of the two screw eyes, and a pencil line is to be ruled along its edge upon the face of the vertical standard. After this has been done turn the apparatus halfway around so that what was the north side now points south, level it again and revolve the board about the screw which holds it to the vertical standard, until the screw eyes again point to the star. Rule another line along the same edge of the board as before and with a protractor measure the angle between these lines. Use a bicycle lamp if you need artificial light for your work. The student who has studied plane geometry should be able to prove that one half of the angle between these lines is equal to the altitude of the star.

After you have determined your latitude from Polaris, compare the result with your position as shown upon the best map available. With a little practice and considerable care the latitude may be thus determined within one tenth of a degree, which is equivalent to about 7 miles. If you go 10 miles north or south from your first station you should find the pole higher up or lower down in the sky by an amount which can be measured with your apparatus.

19. The meridian line.—To establish a true north and south line upon the ground, use the apparatus as described at [page 13], and when Polaris is at upper or lower culmination drive into the ground two stakes in line with the star and the plumb line. Such a meridian line is of great convenience in observing the stars and should be laid out and permanently marked in some convenient open space from which, if possible, all parts of the sky are visible. June and November are convenient months for this exercise, since Polaris then comes to culmination early in the evening.

20. Time.—What is the time at which school begins in the morning? What do you mean by "the time"?

The sidereal time at any moment is the right ascension of the hour circle which at that moment coincides with the meridian. When the hour circle passing through Sirius coincides with the meridian, the sidereal time is 6h. 40m., since that is the right ascension of Sirius, and in astronomical language Sirius is "on the meridian" at 6h. 40m. sidereal time. As may be seen from the map, this 6h. 40m. is the right ascension of Sirius, and if a clock be set to indicate 6h. 40m. when Sirius crosses the meridian, it will show sidereal time. If the clock is properly regulated, every other star in the heavens will come to the meridian at the moment when the time shown by the clock is equal to the right ascension of the star. A clock properly regulated for this purpose will gain about four minutes per day in comparison with ordinary clocks, and when so regulated it is called a sidereal clock. The student should be provided with such a clock for his future work, but one such clock will serve for several persons, and a nutmeg clock or a watch of the cheapest kind is quite sufficient.

Exercise 9.—Set such a clock to sidereal time by means of the transit of a star over your meridian. For this experiment it is presupposed that a meridian line has been marked out on the ground as in [§ 19], and the simplest mode of performing the experiment required is for the observer, having chosen a suitable star in the southern part of the sky, to place his eye accurately over the northern end of the meridian line and to estimate as nearly as possible the beginning and end of the period during which the star appears to stand exactly above the southern end of the line. The middle of this period may be taken as the time at which the star crossed the meridian and at this moment the sidereal time is equal to the right ascension of the star. The difference between this right ascension and the observed middle instant is the error of the clock or the amount by which its hands must be set back or forward in order to indicate true sidereal time.

A more accurate mode of performing the experiment consists in using the plumb-line apparatus carefully adjusted, as in [Fig. 7], so that the line joining the wire to the center of the screw eye shall be parallel to the meridian line. Observe the time by the clock at which the star disappears behind the wire as seen through the center of the screw eye. If the star is too high up in the sky for convenient observation, place a mirror, face up, just north of the screw eye and observe star, wire and screw eye by reflection in it.

The numerical right ascension of the observed star is needed for this experiment, and it may be measured from the star map, but it will usually be best to observe one of the stars of the table at the end of the book, and to obtain its right ascension as follows: The table gives the right ascension and declination of each star as they were at the beginning of the year 1900, but on account of the precession (see [Chapter V]), these numbers all change slowly with the lapse of time, and on the average the right ascension of each star of the table must be increased by one twentieth of a minute for each year after 1900—i. e., in 1910 the right ascension of the first star of the table will be 0h. 38.6m. + (10/20)m. = 0h. 39.1m. The declinations also change slightly, but as they are only intended to help in finding the star on the star maps, their change may be ignored.

Having set the clock approximately to sidereal time, observe one or two more stars in the same way as above. The difference between the observed time and the right ascension, if any is found, is the "correction" of the clock. This correction ought not to exceed a minute if due care has been taken in the several operations prescribed. The relation of the clock to the right ascension of the stars is expressed in the following equation, with which the student should become thoroughly familiar:

A = T ± U

T stands for the time by the clock at which the star crossed the meridian. A is the right ascension of the star, and U is the correction of the clock. Use the + sign in the equation whenever the clock is too slow, and the - sign when it is too fast. U may be found from this equation when A and T are given, or A may be found when T and U are given. It is in this way that astronomers measure the right ascensions of the stars and planets.

Determine U from each star you have observed, and note how the several results agree one with another.

21. Definitions.—To define a thing or an idea is to give a description sufficient to identify it and distinguish it from every other possible thing or idea. If a definition does not come up to this standard it is insufficient. Anything beyond this requirement is certainly useless and probably mischievous.

Let the student define the following geographical terms, and let him also criticise the definitions offered by his fellow-students: Equator, poles, meridian, latitude, longitude, north, south, east, west.

Compare the following astronomical definitions with your geographical definitions, and criticise them in the same way. If you are not able to improve upon them, commit them to memory:

The Poles of the heavens are those points in the sky toward which the earth's axis points. How many are there? The one near Polaris is called the north pole.

The Celestial Equator is a great circle of the sky distant 90° from the poles.

The Zenith is that point of the sky, overhead, toward which a plumb line points. Why is the word overhead placed in the definition? Is there more than one zenith?

The Horizon is a great circle of the sky 90° distant from the zenith.

An Hour Circle is any great circle of the sky which passes through the poles. Every star has its own hour circle.

The Meridian is that hour circle which passes through the zenith.

A Vertical Circle is any great circle that passes through the zenith. Is the meridian a vertical circle?

The Declination of a star is its angular distance north or south of the celestial equator.

The Right Ascension of a star is the angle included between its hour circle and the hour circle of a certain point on the equator which is called the Vernal Equinox. From spherical geometry we learn that this angle is to be measured either at the pole where the two hour circles intersect, as is done in the star map opposite [page 124], or along the equator, as is done in the map opposite page 190. Right ascension is always measured from the vernal equinox in the direction opposite to that in which the stars appear to travel in their diurnal motion—i. e., from west toward east.

The Altitude of a star is its angular distance above the horizon.

The Azimuth of a star is the angle between the meridian and the vertical circle passing through the star. A star due south has an azimuth of 0°. Due west, 90°. Due north, 180°. Due east, 270°.

What is the azimuth of Polaris in degrees?

What is the azimuth of the sun at sunrise? At sunset? At noon? Are these azimuths the same on different days?

The Hour Angle of a star is the angle between its hour circle and the meridian. It is measured from the meridian in the direction in which the stars appear to travel in their diurnal motion—i. e., from east toward west.

What is the hour angle of the sun at noon? What is the hour angle of Polaris when it is at the lowest point in its daily motion?

22. Exercises.—The student must not be satisfied with merely learning these definitions. He must learn to see these points and lines in his mind as if they were visibly painted upon the sky. To this end it will help him to note that the poles, the zenith, the meridian, the horizon, and the equator seem to stand still in the sky, always in the same place with respect to the observer, while the hour circles and the vernal equinox move with the stars and keep the same place among them. Does the apparent motion of a star change its declination or right ascension? What is the hour angle of the sun when it has the greatest altitude? Will your answer to the preceding question be true for a star? What is the altitude of the sun after sunset? In what direction is the north pole from the zenith? From the vernal equinox? Where are the points in which the meridian and equator respectively intersect the horizon?

CHAPTER III

FIXED AND WANDERING STARS

23. Star maps.—Select from the map some conspicuous constellation that will be conveniently placed for observation in the evening, and make on a large scale a copy of all the stars of the constellation that are shown upon the map. At night compare this copy with the sky, and mark in upon your paper all the stars of the constellation which are not already there. Both the original drawing and the additions made to it by night should be carefully done, and for the latter purpose what is called the method of allineations may be used with advantage—i. e., the new star is in line with two already on the drawing and is midway between them, or it makes an equilateral triangle with two others, or a square with three others, etc.

A series of maps of the more prominent constellations, such as Ursa Major, Cassiopea, Pegasus, Taurus, Orion, Gemini, Canis Major, Leo, Corvus, Bootes, Virgo, Hercules, Lyra, Aquila, Scorpius, should be constructed in this manner upon a uniform scale and preserved as a part of the student's work. Let the magnitude of the stars be represented on the maps as accurately as may be, and note the peculiarity of color which some stars present. For the most part their color is a very pale yellow, but occasionally one may be found of a decidedly ruddy hue—e. g., Aldebaran or Antares. Such a star map, not quite complete, is shown in [Fig. 13].

So, too, a sharp eye may detect that some stars do not remain always of the same magnitude, but change their brightness from night to night, and this not on account of cloud or mist in the atmosphere, but from something in the star itself. Algol is one of the most conspicuous of these variable stars, as they are called.

Fig. 13.—Star map of the region about Orion.

24. The moon's motion among the stars.—Whenever the moon is visible note its position among the stars by allineations, and plot it on the key map opposite [page 190]. Keep a record of the day and hour corresponding to each such observation. You will find, if the work is correctly done, that the positions of the moon all fall near the curved line shown on the map. This line is called the ecliptic.

After several such observations have been made and plotted, find by measurement from the map how many degrees per day the moon moves. How long would it require to make the circuit of the heavens and come back to the starting point?

On each night when you observe the moon, make on a separate piece of paper a drawing of it about 10 centimeters in diameter and show in the drawing every feature of the moon's face which you can see—e. g., the shape of the illuminated surface (phase); the direction among the stars of the line joining the horns; any spots which you can see upon the moon's face, etc. An opera glass will prove of great assistance in this work.

Use your drawings and the positions of the moon plotted upon the map to answer the following questions: Does the direction of the line joining the horns have any special relation to the ecliptic? Does the amount of illuminated surface of the moon have any relation to the moon's angular distance from the sun? Does it have any relation to the time at which the moon sets? Do the spots on the moon when visible remain always in the same place? Do they come and go? Do they change their position with relation to each other? Can you determine from these spots that the moon rotates about an axis, as the earth does? In what direction does its axis point? How long does it take to make one revolution about the axis? Is there any day and night upon the moon?

Each of these questions can be correctly answered from the student's own observations without recourse to any book.

25. The sun and its motion.—Examine the face of the sun through a smoked glass to see if there is anything there that you can sketch.

By day as well as by night the sky is studded with stars, only they can not be seen by day on account of the overwhelming glare of sunlight, but the position of the sun among the stars may be found quite as accurately as was that of the moon, by observing from day to day its right ascension and declination, and this should be practiced at noon on clear days by different members of the class.

Exercise 10.—The right ascension of the sun may be found by observing with the sidereal clock the time of its transit over the meridian. Use the equation in [§ 20], and substitute in place of U the value of the clock correction found from observations of stars on a preceding or following night. If the clock gains or loses with respect to sidereal time, take this into account in the value of U.

Exercise 11.—To determine the sun's declination, measure its altitude at the time it crosses the meridian. Use either the method of [Exercise 4], or that used with Polaris in [Exercise 8]. The student should be able to show from [Fig. 11] that the declination is equal to the sum of the altitude and the latitude of the place diminished by 90°, or in an equation

Declination = Altitude + Latitude - 90°.

If the declination as found from this equation is a negative number it indicates that the sun is on the south side of the equator.

The right ascension and declination of the sun as observed on each day should be plotted on the map and the date, written opposite it. If the work has been correctly done, the plotted points should fall upon the curved line (ecliptic) which runs lengthwise of the map. This line, in fact, represents the sun's path among the stars.

Note that the hours of right ascension increase from 0 up to 24, while the numbers on the clock dial go only from 0 to 12, and then repeat 0 to 12 again during the same day. When the sidereal time is 13 hours, 14 hours, etc., the clock will indicate 1 hour, 2 hours, etc., and 12 hours must then be added to the time shown on the dial.

If observations of the sun's right ascension and declination are made in the latter part of either March or September the student will find that the sun crosses the equator at these times, and he should determine from his observations, as accurately as possible, the date and hour of this crossing and the point on the equator at which the sun crosses it. These points are called the equinoxes, Vernal Equinox and Autumnal Equinox for the spring and autumn crossings respectively, and the student will recall that the vernal equinox is the point from which right ascensions are measured. Its position among the stars is found by astronomers from observations like those above described, only made with much more elaborate apparatus.

Similar observations made in June and December show that the sun's midday altitude is about 47° greater in summer than in winter. They show also that the sun is as far north of the equator in June as he is south of it in December, from which it is easily inferred that his path, the ecliptic, is inclined to the equator at an angle of 23°.5, one half of 47°. This angle is called the obliquity of the ecliptic. The student may recall that in the geographies the torrid zone is said to extend 23°.5 on either side of the earth's equator. Is there any connection between these limits and the obliquity of the ecliptic? Would it be correct to define the torrid zone as that part of the earth's surface within which the sun may at some season of the year pass through the zenith?

Exercise 12.—After a half dozen observations of the sun have been plotted upon the map, find by measurement the rate, in degrees per day, at which the sun moves along the ecliptic. How many days will be required for it to move completely around the ecliptic from vernal equinox back to vernal equinox again? Accurate observations with the elaborate apparatus used by professional astronomers show that this period, which is called a tropical year, is 365 days 5 hours 48 minutes 46 seconds. Is this the same as the ordinary year of our calendars?

26. The planets.—Any one who has watched the sky and who has made the drawings prescribed in this chapter can hardly fail to have found in the course of his observations some bright stars not set down on the printed star maps, and to have found also that these stars do not remain fixed in position among their fellows, but wander about from one constellation to another. Observe the motion of one of these planets from night to night and plot its positions on the star map, precisely as was done for the moon. What kind of path does it follow?

Both the ancient Greeks and the modern Germans have called these bodies wandering stars, and in English we name them planets, which is simply the Greek word for wanderer, bent to our use. Besides the sun and moon there are in the heavens five planets easily visible to the naked eye and, as we shall see later, a great number of smaller ones visible only in the telescope. More than 2,000 years ago astronomers began observing the motion of sun, moon, and planets among the stars, and endeavored to account for these motions by the theory that each wandering star moved in an orbit about the earth. Classical and mediæval literature are permeated with this idea, which was displaced only after a long struggle begun by Copernicus (1543 A. D.), who taught that the moon alone of these bodies revolves about the earth, while the earth and the other planets revolve around the sun. The ecliptic is the intersection of the plane of the earth's orbit with the sky, and the sun appears to move along the ecliptic because, as the earth moves around its orbit, the sun is always seen projected against the opposite side of it. The moon and planets all appear to move near the ecliptic because the planes of their orbits nearly coincide with the plane of the earth's orbit, and a narrow strip on either side of the ecliptic, following its course completely around the sky, is called the zodiac, a word which may be regarded as the name of a narrow street (16° wide) within which all the wanderings of the visible planets are confined and outside of which they never venture. Indeed, Mars is the only planet which ever approaches the edge of the street, the others traveling near the middle of the road.

Fig. 14.—The apparent motion of a planet.

27. A typical case of planetary motion.—The Copernican theory, enormously extended and developed through the Newtonian law of gravitation (see [Chapter IV]), has completely supplanted the older Ptolemaic doctrine, and an illustration of the simple manner in which it accounts for the apparently complicated motions of a planet among the stars is found in Figs. [14] and [15], the first of which represents the apparent motion of the planet Mars through the constellations Aries and Pisces during the latter part of the year 1894, while the second shows the true motions of Mars and the earth in their orbits about the sun during the same period. The straight line in [Fig. 14], with cross ruling upon it, is a part of the ecliptic, and the numbers placed opposite it represent the distance, in degrees, from the vernal equinox. In [Fig. 15] the straight line represents the direction from the sun toward the vernal equinox, and the angle which this line makes with the line joining earth and sun is called the earth's longitude. The imaginary line joining the earth and sun is called the earth's radius vector, and the pupil should note that the longitude and length of the radius vector taken together show the direction and distance of the earth from the sun—i. e., they fix the relative positions of the two bodies. The same is nearly true for Mars and would be wholly true if the orbit of Mars lay in the same plane with that of the earth. How does [Fig. 14] show that the orbit of Mars does not lie exactly in the same plane with the orbit of the earth?

Exercise 13.—Find from [Fig. 15] what ought to have been the apparent course of Mars among the stars during the period shown in the two figures, and compare what you find with [Fig. 14]. The apparent position of Mars among the stars is merely its direction from the earth, and this direction is represented in [Fig. 14] by the distance of the planet from the ecliptic and by its longitude.

Fig. 15.—The real motion of a planet.

The longitude of Mars for each date can be found from [Fig. 15] by measuring the angle between the straight line S V and the line drawn from the earth to Mars. Thus for October 12th we may find with the protractor that the angle between the line S V and the line joining the earth to Mars is a little more than 30°, and in [Fig. 14] the position of Mars for this date is shown nearly opposite the cross line corresponding to 30° on the ecliptic. Just how far below the ecliptic this position of Mars should fall can not be told from [Fig. 15], which from necessity is constructed as if the orbits of Mars and the earth lay in the same plane, and Mars in this case would always appear to stand exactly on the ecliptic and to oscillate back and forth as shown in [Fig. 14], but without the up-and-down motion there shown. In this way plot in [Fig. 14] the longitudes of Mars as seen from the earth for other dates and observe how the forward motion of the two planets in their orbits accounts for the apparently capricious motion of Mars to and fro among the stars.

Fig. 16.—The orbits of Jupiter and Saturn.

28. The orbits of the planets.—Each planet, great or small, moves in its own appropriate orbit about the sun, and the exact determination of these orbits, their sizes, shapes, positions, etc., has been one of the great problems of astronomy for more than 2,000 years, in which successive generations of astronomers have striven to push to a still higher degree of accuracy the knowledge attained by their predecessors. Without attempting to enter into the details of this problem we may say, generally, that every planet moves in a plane passing through the sun, and for the six planets visible to the naked eye these planes nearly coincide, so that the six orbits may all be shown without much error as lying in the flat surface of one map. It is, however, more convenient to use two maps, such as Figs. [16] and [17], one of which shows the group of planets, Mercury, Venus, the earth, and Mars, which are near the sun, and on this account are sometimes called the inner planets, while the other shows the more distant planets, Jupiter and Saturn, together with the earth, whose orbit is thus made to serve as a connecting link between the two diagrams. These diagrams are accurately drawn to scale, and are intended to be used by the student for accurate measurement in connection with the exercises and problems which follow.

In addition to the six planets shown in the figures the solar system contains two large planets and several hundred small ones, for the most part invisible to the naked eye, which are omitted in order to avoid confusing the diagrams.

29. Jupiter and Saturn.—In [Fig. 16] the sun at the center is encircled by the orbits of the three planets, and inclosing all of these is a circular border showing the directions from the sun of the constellations which lie along the zodiac. The student must note carefully that it is only the directions of these constellations that are correctly shown, and that in order to show them at all they have been placed very much too close to the sun. The cross lines extending from the orbit of the earth toward the sun with Roman numerals opposite them show the positions of the earth in its orbit on the first day of January (I), first day of February (II), etc., and the similar lines attached to the orbits of Jupiter and Saturn with Arabic numerals show the positions of those planets on the first day of January of each year indicated, so that the figure serves to show not only the orbits of the planets, but their actual positions in their orbits for something more than the first decade of the twentieth century.

The line drawn from the sun toward the right of the figure shows the direction to the vernal equinox. It forms one side of the angle which measures a planet's longitude.

Fig. 17.—The orbits of the inner planets.

Exercise 14.—Measure with your protractor the longitude of the earth on January 1st. Is this longitude the same in all years? Measure the longitude of Jupiter on January 1, 1900; on July 1, 1900; on September 25, 1906.

Draw neatly on the map a pencil line connecting the position of the earth for January 1, 1900, with the position of Jupiter for the same date, and produce the line beyond Jupiter until it meets the circle of the constellations. This line represents the direction of Jupiter from the earth, and points toward the constellation in which the planet appears at that date. But this representation of the place of Jupiter in the sky is not a very accurate one, since on the scale of the diagram the stars are in fact more than 100,000 times as far off as they are shown in the figure, and the pencil mark does not meet the line of constellations at the same intersection it would have if this line were pushed back to its true position. To remedy this defect we must draw another line from the sun parallel to the one first drawn, and its intersection with the constellations will give very approximately the true position of Jupiter in the sky.

Exercise 15.—Find the present positions of Jupiter and Saturn, and look them up in the sky by means of your star maps. The planets will appear in the indicated constellations as very bright stars not shown on the map.

Which of the planets, Jupiter and Saturn, changes its direction from the sun more rapidly? Which travels the greater number of miles per day? When will Jupiter and Saturn be in the same constellation? Does the earth move faster or slower than Jupiter?

The distance of Jupiter or Saturn from the earth at any time may be readily obtained from the figure. Thus, by direct measurement with the millimeter scale we find for January 1, 1900, the distance of Jupiter from the earth is 6.1 times the distance of the sun from the earth, and this may be turned into miles by multiplying it by 93,000,000, which is approximately the distance of the sun from the earth. For most purposes it is quite as well to dispense with this multiplication and call the distance 6.1 astronomical units, remembering that the astronomical unit is the distance of the sun from the earth.

Exercise 16.—What is Jupiter's distance from the earth at its nearest approach? What is the greatest distance it ever attains? Is Jupiter's least distance from the earth greater or less than its least distance from Saturn?

On what day in the year 1906 will the earth be on line between Jupiter and the sun? On this day Jupiter is said to be in opposition—i. e., the planet and the sun are on opposite sides of the earth, and Jupiter then comes to the meridian of any and every place at midnight. When the sun is between the earth and Jupiter (at what date in 1906?) the planet is said to be in conjunction with the sun, and of course passes the meridian with the sun at noon. Can you determine from the figure the time at which Jupiter comes to the meridian at other dates than opposition and conjunction? Can you determine when it is visible in the evening hours? Tell from the figure what constellation is on the meridian at midnight on January 1st. Will it be the same constellation in every year?

30. Mercury, Venus, and Mars.—[Fig. 17], which represents the orbits of the inner planets, differs from [Fig. 16] only in the method of fixing the positions of the planets in their orbits at any given date. The motion of these planets is so rapid, on account of their proximity to the sun, that it would not do to mark their positions as was done for Jupiter and Saturn, and with the exception of the earth they do not always return to the same place on the same day in each year. It is therefore necessary to adopt a slightly different method, as follows: The straight line extending from the sun toward the vernal equinox, V, is called the prime radius, and we know from past observations that the earth in its motion around the sun crosses this line on September 23d in each year, and to fix the earth's position for September 23d in the diagram we have only to take the point at which the prime radius intersects the earth's orbit. A month later, on October 23d, the earth will no longer be at this point, but will have moved on along its orbit to the point marked 30 (thirty days after September 23d). Sixty days after September 23d it will be at the point marked 60, etc., and for any date we have only to find the number of days intervening between it and the preceding September 23d, and this number will show at once the position of the earth in its orbit. Thus for the date July 4, 1900, we find

1900, July 4 - 1899, September 23 = 284 days,

and the little circle marked upon the earth's orbit between the numbers 270 and 300 shows the position of the earth on that date.

In what constellation was the sun on July 4, 1900? What zodiacal constellation came to the meridian at midnight on that date? What other constellations came to the meridian at the same time?

The positions of the other planets in their orbits are found in the same manner, save that they do not cross the prime radius on the same date in each year, and the times at which they do cross it must be taken from the following table:

Table of Epochs

The first line of figures in this table shows the number of days that each of these planets requires to make a complete revolution about the sun, and it appears from these numbers that Mercury makes about four revolutions in its orbit per year, and therefore crosses the prime radius four times in each year, while the other planets are decidedly slower in their movements. The following lines of the table show for each year the date at which each planet first crossed the prime radius in that year; the dates of subsequent crossings in any year can be found by adding once, twice, or three times the period to the given date, and the table may be extended to later years, if need be, by continuously adding multiples of the period. In the case of Mars it appears that there is only about one year out of two in which this planet crosses the prime radius.

After the date at which the planet crosses the prime radius has been determined its position for any required date is found exactly as in the case of the earth, and the constellation in which the planet will appear from the earth is found as explained above in connection with Jupiter and Saturn.

The broken lines in the figure represent the construction for finding the places in the sky occupied by Mercury, Venus, and Mars on July 4, 1900. Let the student make a similar construction and find the positions of these planets at the present time. Look them up in the sky and see if they are where your work puts them.

31. Exercises.—The "evening star" is a term loosely applied to any planet which is visible in the western sky soon after sunset. It is easy to see that such a planet must be farther toward the east in the sky than is the sun, and in either [Fig. 16] or [Fig. 17] any planet which viewed from the position of the earth lies to the left of the sun and not more than 50° away from it will be an evening star. If to the right of the sun it is a morning star, and may be seen in the eastern sky shortly before sunrise.

What planet is the evening star now? Is there more than one evening star at a time? What is the morning star now?

Do Mercury, Venus, or Mars ever appear in opposition? What is the maximum angular distance from the sun at which Venus can ever be seen? Why is Mercury a more difficult planet to see than Venus? In what month of the year does Mars come nearest to the earth? Will it always be brighter in this month than in any other? Which of all the planets comes nearest to the earth?

The earth always comes to the same longitude on the same day of each year. Why is not this true of the other planets?

The student should remember that in one respect Figs. [16] and [17] are not altogether correct representations, since they show the orbits as all lying in the same plane. If this were strictly true, every planet would move, like the sun, always along the ecliptic; but in fact all of the orbits are tilted a little out of the plane of the ecliptic and every planet in its motion deviates a little from the ecliptic, first to one side then to the other; but not even Mars, which is the most erratic in this respect, ever gets more than eight degrees away from the ecliptic, and for the most part all of them are much closer to the ecliptic than this limit.

CHAPTER IV

CELESTIAL MECHANICS

32. The beginnings of celestial mechanics.—From the earliest dawn of civilization, long before the beginnings of written history, the motions of sun and moon and planets among the stars from constellation to constellation had commanded the attention of thinking men, particularly of the class of priests. The religions of which they were the guardians and teachers stood in closest relations with the movements of the stars, and their own power and influence were increased by a knowledge of them.

ISAAC NEWTON (1643-1727).

Out of these professional needs, as well as from a spirit of scientific research, there grew up and flourished for many centuries a study of the motions of the planets, simple and crude at first, because the observations that could then be made were at best but rough ones, but growing more accurate and more complex as the development of the mechanic arts put better and more precise instruments into the hands of astronomers and enabled them to observe with increasing accuracy the movements of these bodies. It was early seen that while for the most part the planets, including the sun and moon, traveled through the constellations from west to east, some of them sometimes reversed their motion and for a time traveled in the opposite way. This clearly can not be explained by the simple theory which had early been adopted that a planet moves always in the same direction around a circular orbit having the earth at its center, and so it was said to move around in a small circular orbit, called an epicycle, whose center was situated upon and moved along a circular orbit, called the deferent, within which the earth was placed, as is shown in [Fig. 18], where the small circle is the epicycle, the large circle is the deferent, P is the planet, and E the earth. When this proved inadequate to account for the really complicated movements of the planets, another epicycle was put on top of the first one, and then another and another, until the supposed system became so complicated that Copernicus, a Polish astronomer, repudiated its fundamental theorem and taught that the motions of the planets take place in circles around the sun instead of about the earth, and that the earth itself is only one of the planets moving around the sun in its own appropriate orbit and itself largely responsible for the seemingly erratic movements of the other planets, since from day to day we see them and observe their positions from different points of view.

Fig. 18.—Epicycle and deferent.

33. Kepler's laws.—Two generations later came Kepler with his three famous laws of planetary motion:

I. Every planet moves in an ellipse which has the sun at one of its foci.

II. The radius vector of each planet moves over equal areas in equal times.

III. The squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun.

These laws are the crowning glory, not only of Kepler's career, but of all astronomical discovery from the beginning up to his time, and they well deserve careful study and explanation, although more modern progress has shown that they are only approximately true.

Exercise 17.—Drive two pins into a smooth board an inch apart and fasten to them the ends of a string a foot long. Take up the slack of the string with the point of a lead pencil and, keeping the string drawn taut, move the pencil point over the board into every possible position. The curve thus traced will be an ellipse having the pins at the two points which are called its foci.

In the case of the planetary orbits one focus of the ellipse is vacant, and, in accordance with the first law, the center of the sun is at the other focus. In [Fig. 17] the dot, inside the orbit of Mercury, which is marked a, shows the position of the vacant focus of the orbit of Mars, and the dot b is the vacant focus of Mercury's orbit. The orbits of Venus and the earth are so nearly circular that their vacant foci lie very close to the sun and are not marked in the figure. The line drawn from the sun to any point of the orbit (the string from pin to pencil point) is a radius vector. The point midway between the pins is the center of the ellipse, and the distance of either pin from the center measures the eccentricity of the ellipse.

Draw several ellipses with the same length of string, but with the pins at different distances apart, and note that the greater the eccentricity the flatter is the ellipse, but that all of them have the same length.

If both pins were driven into the same hole, what kind of an ellipse would you get?

The Second Law was worked out by Kepler as his answer to a problem suggested by the first law. In [Fig. 17] it is apparent from a mere inspection of the orbit of Mercury that this planet travels much faster on one side of its orbit than on the other, the distance covered in ten days between the numbers 10 and 20 being more than fifty per cent greater than that between 50 and 60. The same difference is found, though usually in less degree, for every other planet, and Kepler's problem was to discover a means by which to mark upon the orbit the figures showing the positions of the planet at the end of equal intervals of time. His solution of this problem, contained in the second law, asserts that if we draw radii vectors from the sun to each of the marked points taken at equal time intervals around the orbit, then the area of the sector formed by two adjacent radii vectores and the arc included between them is equal to the area of each and every other such sector, the short radii vectores being spread apart so as to include a long arc between them while the long radii vectores have a short arc. In Kepler's form of stating the law the radius vector is supposed to travel with the planet and in each day to sweep over the same fractional part of the total area of the orbit. The spacing of the numbers in [Fig. 17] was done by means of this law.

For the proper understanding of Kepler's Third Law we must note that the "mean distance" which appears in it is one half of the long diameter of the orbit and that the "periodic time" means the number of days or years required by the planet to make a complete circuit in its orbit. Representing the first of these by a and the second by T, we have, as the mathematical equivalent of the law,

a³ ÷ T² = C

where the quotient, C, is a number which, as Kepler found, is the same for every planet of the solar system. If we take the mean distance of the earth from the sun as the unit of distance, and the year as the unit of time, we shall find by applying the equation to the earth's motion, C = 1. Applying this value to any other planet we shall find in the same units, a = T^2/3, by means of which we may determine the distance of any planet from the sun when its periodic time, T, has been learned from observation.

Exercise 18.—Uranus requires 84 years to make a revolution in its orbit. What is its mean distance from the sun? What are the mean distances of Mercury, Venus, and Mars? (See [Chapter III] for their periodic times.) Would it be possible for two planets at different distances from the sun to move around their orbits in the same time?

A circle is an ellipse in which the two foci have been brought together. Would Kepler's laws hold true for such an orbit?

34. Newton's laws of motion.—Kepler studied and described the motion of the planets. Newton, three generations later (1727 A. D.), studied and described the mechanism which controls that motion. To Kepler and his age the heavens were supernatural, while to Newton and his successors they are a part of Nature, governed by the same laws which obtain upon the earth, and we turn to the ordinary things of everyday life as the foundation of celestial mechanics.

Every one who has ridden a bicycle knows that he can coast farther upon a level road if it is smooth than if it is rough; but however smooth and hard the road may be and however fast the wheel may have been started, it is sooner or later stopped by the resistance which the road and the air offer to its motion, and when once stopped or checked it can be started again only by applying fresh power. We have here a familiar illustration of what is called

The first law of motion.—"Every body continues in its state of rest or of uniform motion in a straight line except in so far as it may be compelled by force to change that state." A gust of wind, a stone, a careless movement of the rider may turn the bicycle to the right or the left, but unless some disturbing force is applied it will go straight ahead, and if all resistance to its motion could be removed it would go always at the speed given it by the last power applied, swerving neither to the one hand nor the other.

When a slow rider increases his speed we recognize at once that he has applied additional power to the wheel, and when this speed is slackened it equally shows that force has been applied against the motion. It is force alone which can produce a change in either velocity or direction of motion; but simple as this law now appears it required the genius of Galileo to discover it and of Newton to give it the form in which it is stated above.

35. The second law of motion, which is also due to Galileo and Newton, is:

"Change of motion is proportional to force applied and takes place in the direction of the straight line in which the force acts." Suppose a man to fall from a balloon at some great elevation in the air; his own weight is the force which pulls him down, and that force operating at every instant is sufficient to give him at the end of the first second of his fall a downward velocity of 32 feet per second—i. e., it has changed his state from rest, to motion at this rate, and the motion is toward the earth because the force acts in that direction. During the next second the ceaseless operation of this force will have the same effect as in the first second and will add another 32 feet to his velocity, so that two seconds from the time he commenced to fall he will be moving at the rate of 64 feet per second, etc. The column of figures marked v in the table below shows what his velocity will be at the end of subsequent seconds. The changing velocity here shown is the change of motion to which the law refers, and the velocity is proportional to the time shown in the first column of the table, because the amount of force exerted in this case is proportional to the time during which it operated. The distance through which the man will fall in each second is shown in the column marked d, and is found by taking the average of his velocity at the beginning and end of this second, and the total distance through which he has fallen at the end of each second, marked s in the table, is found by taking the sum of all the preceding values of d. The velocity, 32 feet per second, which measures the change of motion in each second, also measures the accelerating force which produces this motion, and it is usually represented in formulæ by the letter g. Let the student show from the numbers in the table that the accelerating force, the time, t, during which it operates, and the space, s, fallen through, satisfy the relation

s = 1/2 gt²,

which is usually called the law of falling bodies. How does the table show that g is equal to 32?

Table

If the balloon were half a mile high how long would it take to fall to the ground? What would be the velocity just before reaching the ground?

[Fig. 19] shows the path through the air of a ball which has been struck by a bat at the point A, and started off in the direction A B with a velocity of 200 feet per second. In accordance with the first law of motion, if it were acted upon by no other force than the impulse given by the bat, it should travel along the straight line A B at the uniform rate of 200 feet per second, and at the end of the fourth second it should be 800 feet from A, at the point marked 4, but during these four seconds its weight has caused it to fall 256 feet, and its actual position, 4', is 256 feet below the point 4. In this way we find its position at the end of each second, 1', 2', 3', 4', etc., and drawing a line through these points we shall find the actual path of the ball under the influence of the two forces to be the curved line A C. No matter how far the ball may go before striking the ground, it can not get back to the point A, and the curve A C therefore can not be a part of a circle, since that curve returns into itself. It is, in fact, a part of a parabola, which, as we shall see later, is a kind of orbit in which comets and some other heavenly bodies move. A skyrocket moves in the same kind of a path, and so does a stone, a bullet, or any other object hurled through the air.

Fig. 19.—The path of a ball.

36. The third law of motion.—"To every action there is always an equal and contrary reaction; or the mutual actions of any two bodies are always equal and oppositely directed." This is well illustrated in the case of a man climbing a rope hand over hand. The direct force or action which he exerts is a downward pull upon the rope, and it is the reaction of the rope to this pull which lifts him along it. We shall find in a later chapter a curious application of this law to the history of the earth and moon.

It is the great glory of Sir Isaac Newton that he first of all men recognized that these simple laws of motion hold true in the heavens as well as upon the earth; that the complicated motion of a planet, a comet, or a star is determined in accordance with these laws by the forces which act upon the bodies, and that these forces are essentially the same as that which we call weight. The formal statement of the principle last named is included in—

37. Newton's law of gravitation.—"Every particle of matter in the universe attracts every other particle with a force whose direction is that of a line joining the two, and whose magnitude is directly as the product of their masses, and inversely as the square of their distance from each other." We know that we ourselves and the things about us are pulled toward the earth by a force (weight) which is called, in the Latin that Newton wrote, gravitas, and the word marks well the true significance of the law of gravitation. Newton did not discover a new force in the heavens, but he extended an old and familiar one from a limited terrestrial sphere of action to an unlimited and celestial one, and furnished a precise statement of the way in which the force operates. Whether a body be hot or cold, wet or dry, solid, liquid, or gaseous, is of no account in determining the force which it exerts, since this depends solely upon mass and distance.

The student should perhaps be warned against straining too far the language which it is customary to employ in this connection. The law of gravitation is certainly a far-reaching one, and it may operate in every remotest corner of the universe precisely as stated above, but additional information about those corners would be welcome to supplement our rather scanty stock of knowledge concerning what happens there. We may not controvert the words of a popular preacher who says, "When I lift my hand I move the stars in Ursa Major," but we should not wish to stand sponsor for them, even though they are justified by a rigorous interpretation of the Newtonian law.

The word mass, in the statement of the law of gravitation, means the quantity of matter contained in the body, and if we represent by the letters m' and m'' the respective quantities of matter contained in the two bodies whose distance from each other is r, we shall have, in accordance with the law of gravitation, the following mathematical expression for the force, F, which acts between them:

F = k (m'm'')/r².

This equation, which is the general mathematical expression for the law of gravitation, may be made to yield some curious results. Thus, if we select two bullets, each having a mass of 1 gram, and place them so that their centers are 1 centimeter apart, the above expression for the force exerted between them becomes

F = k {(1 × 1)/1²} = k,

from which it appears that the coefficient k is the force exerted between these bodies. This is called the gravitation constant, and it evidently furnishes a measure of the specific intensity with which one particle of matter attracts another. Elaborate experiments which have been made to determine the amount of this force show that it is surprisingly small, for in the case of the two bullets whose mass of 1 gram each is supposed to be concentrated into an indefinitely small space, gravity would have to operate between them continuously for more than forty minutes in order to pull them together, although they were separated by only 1 centimeter to start with, and nothing save their own inertia opposed their movements. It is only when one or both of the masses m', m'' are very great that the force of gravity becomes large, and the weight of bodies at the surface of the earth is considerable because of the great quantity of matter which goes to make up the earth. Many of the heavenly bodies are much more massive than the earth, as the mathematical astronomers have found by applying the law of gravitation to determine numerically their masses, or, in more popular language, to "weigh" them.

The student should observe that the two terms mass and weight are not synonymous; mass is defined above as the quantity of matter contained in a body, while weight is the force with which the earth attracts that body, and in accordance with the law of gravitation its weight depends upon its distance from the center of the earth, while its mass is quite independent of its position with respect to the earth.

By the third law of motion the earth is pulled toward a falling body just as strongly as the body is pulled toward the earth—i. e., by a force equal to the weight of the body. How much does the earth rise toward the body?

38. The motion of a planet.—In [Fig. 20] S represents the sun and P a planet or other celestial body, which for the moment is moving along the straight line P 1. In accordance with the first law of motion it would continue to move along this line with uniform velocity if no external force acted upon it; but such a force, the sun's attraction, is acting, and by virtue of this attraction the body is pulled aside from the line P 1.

Knowing the velocity and direction of the body's motion and the force with which the sun attracts it, the mathematician is able to apply Newton's laws of motion so as to determine the path of the body, and a few of the possible orbits are shown in the figure where the short cross stroke marks the point of each orbit which is nearest to the sun. This point is called the perihelion.

Without any formal application of mathematics we may readily see that the swifter the motion of the body at P the shorter will be the time during which it is subjected to the sun's attraction at close range, and therefore the force exerted by the sun, and the resulting change of motion, will be small, as in the orbits P 1 and P 2.

On the other hand, P 5 and P 6 represent orbits in which the velocity at P was comparatively small, and the resulting change of motion greater than would be possible for a more swiftly moving body.

What would be the orbit if the velocity at P were reduced to nothing at all?