Fig. 16.—The orbits of Jupiter and Saturn.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 everyplanet 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.16and17, 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.—InFig. 16the 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 theirorbits 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.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 inopposition—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 inconjunctionwith 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 fromFig. 16only 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 thepoint 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:
A. D.Mercury.Venus.Earth.Mars.Period88.0 days.224.7 days.365.25 days.687.1 days.1900Feb. 18th.Jan. 11th.Sept. 23d.April 28th.1901Feb. 5th.April 5th.Sept. 23d....1902Jan. 23d.June 29th.Sept. 23d.March 16th.1903April 8th.Feb. 8th.Sept. 23d....1904March 25th.May 3d.Sept. 23d.Feb. 1st.1905March 12th.July 26th.Sept. 23d.Dec. 19th.1906Feb. 27th.March 8th.Sept. 23d....1907Feb. 14th.May 31st.Sept. 23d.Nov. 6th.1908Feb. 1st.Jan. 11th.Sept. 23d....1909Jan. 18th.April 4th.Sept. 23d.Sept. 23d.1910Jan. 5th.June 28th.Sept. 23d....
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 revolutionsin 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 eitherFig. 16orFig. 17any 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 starnow? 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.16and17are 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.
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).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 situatedupon and moved along a circular orbit, called the deferent, within which the earth was placed, as is shown inFig. 18, where the small circle is the epicycle, the large circle is the deferent,Pis the planet, andEthe 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.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. InFig. 17the dot, inside the orbit of Mercury, which is markeda, shows the position of the vacant focus of the orbit of Mars, and the dotbis 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 aradius vector. The point midway between the pins is thecenterof the ellipse, and the distance of either pin from the center measures theeccentricityof 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. InFig. 17it 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 ofthe 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 inFig. 17was 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 byaand the second byT, 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=T2/3, 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? (SeeChapter IIIfor their periodic times.) Wouldit 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 (1727A. 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 ofmotion; 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 markedvin 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 markedd, 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, markedsin the table, is found by taking the sum of all the preceding values ofd. The velocity, 32 feet per second, which measures the change of motion in each second, also measures theaccelerating forcewhich produces this motion, and it is usually represented in formulæ by the letterg. Let the student show from the numbers inthe table that the accelerating force, the time,t, during which it operates, and the space,s, fallen through, satisfy the relation
s= 1/2gt2,
which is usually called the law of falling bodies. How does the table show thatgis equal to 32?
tvds0000132161626448643968014441281122565160144400etc.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?
GALILEO GALILEI (1564-1642).GALILEO GALILEI (1564-1642).
Fig. 19shows the path through the air of a ball which has been struck by a bat at the pointA, and started off in the directionA Bwith 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 lineA Bat the uniform rate of 200 feet per second, and at the end of the fourth second it should be 800 feet fromA, 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 lineA C. No matter how far the ball may go before striking the ground, it can not get back to the pointA, and the curveA Ctherefore can not be a part of a circle, since that curve returns into itself. It is, in fact, a part of aparabola, 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.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 standsponsor for them, even though they are justified by a rigorous interpretation of the Newtonian law.
The wordmass, in the statement of the law of gravitation, means the quantity of matter contained in the body, and if we represent by the lettersm'andm''the respective quantities of matter contained in the two bodies whose distance from each other isr, 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 coefficientkis 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 massesm',m''are very great that the force of gravity becomes large, and the weight of bodies at thesurface 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.—InFig. 20Srepresents the sun andPa planet or other celestial body, which for the moment is moving along the straight lineP 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 lineP 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 theperihelion.
Without any formal application of mathematics we may readily see that the swifter the motion of the body atPthe 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 orbitsP 1andP 2.
On the other hand,P 5andP 6represent orbits in which the velocity atPwas 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 atPwere reduced to nothing at all?
What would be the effect if the body starting atPmoved directly away from1?
Fig. 20.—Different kinds of orbits.Fig. 20.—Different kinds of orbits.
The student should not fail to observe that the sun's attraction tends to pull the body atPforward along its path, and therefore increases its velocity, and that this influence continues until the planet reaches perihelion, at which point it attains its greatest velocity, and the force of the sun's attraction is wholly expended in changing the direction of its motion. After the planet has passed perihelion the sun begins to pull backward and to retard the motion in just the same measure that before perihelion passage it increased it, so that the two halves of the orbit on opposite sides of a line drawn from the perihelion through the sun are exactly alike. We may here note the explanation of Kepler's second law: when the planet is near the sun it moves faster, and the radius vector changes its direction more rapidly than when the planet is remote from the sun on account of the greater force with which it is attracted, and the exact relation between the rates at which the radius vectorturns in different parts of the orbit, as given by the second law, depends upon the changes in this force.
When the velocity is not too great, the sun's backward pull, after a planet has passed perihelion, finally overcomes it and turns the planet toward the sun again, in such a way that it comes back to the pointP, moving in the same direction and with the same speed as before—i. e., it has gone around the sun in an orbit likeP 6orP 4, an ellipse, along which it will continue to move ever after. But we must not fail to note that this return into the same orbit is a consequence of the last line in the statement of the law of gravitation (p. 54), and that, if the magnitude of this force were inversely as the cube of the distance or any other proportion than the square, the orbit would be something very different. If the velocity is too great for the sun's attraction to overcome, the orbit will be a hyperbola, likeP 2, along which the body will move away never to return, while a velocity just at the limit of what the sun can control gives an orbit likeP 3, a parabola, along which the body moves withparabolic velocity, which is ever diminishing as the body gets farther from the sun, but is always just sufficient to keep it from returning. If the earth's velocity could be increased 41 per cent, from 19 up to 27 miles per second, it would have parabolic velocity, and would quit the sun's company.
The summation of the whole matter is that the orbit in which a body moves around the sun, or past the sun, depends upon its velocity and if this velocity and the direction of the motion at any one point in the orbit are known the whole orbit is determined by them, and the position of the planet in its orbit for past as well as future times can be determined through the application of Newton's laws; and the same is true for any other heavenly body—moon, comet, meteor, etc. It is in this way that astronomers are able to predict, years in advance, in what particular part of the sky a given planet will appear at a given time.
It is sometimes a source of wonder that the planets move in ellipses instead of circles, but it is easily seen fromFig. 20that the planet,P, could not by any possibility move in a circle, since the direction of its motion atPis not at right angles with the line joining it to the sun as it must be in a circular orbit, and even if it were perpendicular to the radius vector the planet must needs have exactly the right velocity given to it at this point, since either more or less speed would change the circle into an ellipse. In order to produce circular motion there must be a balancing of conditions as nice as is required to make a pin stand upon its point, and the really surprising thing is that the orbits of the planets should be so nearly circular as they are. If the orbit of the earth were drawn accurately to scale, the untrained eye would not detect the slightest deviation from a true circle, and even the orbit of Mercury (Fig. 17), which is much more eccentric than that of the earth, might almost pass for a circle.
Fig. 21. An impossible orbit.Fig. 21.An impossible orbit.
The orbitP 2, which lies between the parabola and the straight line, is called in geometry a hyperbola, and Newton succeeded in proving from the law of gravitation that a body might move under the sun's attraction in a hyperbola as well as in a parabola or ellipse; but it must move in some one of these curves; no other orbit is possible.[A]Thus it would not be possible for a body moving under the law of gravitation to describe about the sun any such orbit as is shown inFig. 21. If the body passes a second time through any point of its orbit, such asPin the figure, then it must retrace, time after time, the whole path that it firsttraversed in getting fromParound toPagain—i. e., the orbit must be an ellipse.
Newton also proved that Kepler's three laws are mere corollaries from the law of gravitation, and that to be strictly correct the third law must be slightly altered so as to take into account the masses of the planets. These are, however, so small in comparison with that of the sun, that the correction is of comparatively little moment.
39.Perturbations.—In what precedes we have considered the motion of a planet under the influence of no other force than the sun's attraction, while in fact, as the law of gravitation asserts, every other body in the universe is in some measure attracting it and changing its motion. The resulting disturbances in the motion of the attracted body are calledperturbations, but for the most part these are insignificant, because the bodies by whose disturbing attractions they are caused are either very small or very remote, and it is only when our moving planet,P, comes under the influence of some great disturbing power like Jupiter or one of the other planets that the perturbations caused by their influence need to be taken into account.
The problem of the motion of three bodies—sun, Jupiter, planet—which must then be dealt with is vastly more complicated than that which we have considered, and the ablest mathematicians and astronomers have not been able to furnish a complete solution for it, although they have worked upon the problem for two centuries, and have developed an immense amount of detailed information concerning it.