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 cla.s.s 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.

[Ill.u.s.tration: 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.

[Ill.u.s.tration: 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...o...b..t. 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...o...b..t 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, a.s.serts 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 s.p.a.cing 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...o...b..t.

Representing the first of these by _a_ and the second by _T_, we have, as the mathematical equivalent of the law,

a^{3} T^{2} = 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.--Ura.n.u.s requires 84 years to make a revolution in its...o...b..t. 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 ill.u.s.tration 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 formulae 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 s.p.a.ce, _s_, fallen through, satisfy the relation

s = 1/2 gt^{2},

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

TABLE

_t_ _v_ _d_ _s_

0 0 0 0 1 32 16 16 2 64 48 64 3 96 80 144 4 128 112 256 5 160 144 400 etc. etc. etc. etc.

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?

[Ill.u.s.tration: GALILEO GALILEI (1564-1642).]

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.

[Ill.u.s.tration: 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 ill.u.s.trated 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 ma.s.ses, 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 ma.s.s 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.