Astronomy for Everybody/Part 4/Chapter 11

No work of the human intellect farther transcends what would seem possible to the ordinary thinker than the mathematical demonstrations of the motions of the heavenly bodies under the influence of their mutual gravitation. We have learned something of the orbits of the planets round the sun; but the following of the orbit is not the fundamental law of the planet's motion; the latter is determined by gravitation alone. This law, as stated by Newton, is so comprehensive that nothing can be added. The law that every particle of matter in the universe attracts every other particle, with a force which varies inversely as the square of the distance between them, is the only law of nature which, so far as we know, is absolutely universal and invariable in its action. All the other processes of nature are in some way varied or modified by heat and cold, by time or place, by the presence or absence of other bodies. But no operation that man has ever been able to perform on matter changes its gravitation in the slightest. Two bodies gravitate by exactly the same amount, no matter what we do with them, no matter what obstacles we interpose between them, no matter how fast they move. All other natural forces admit of being investigated, but gravitation does not. Philosophers have attempted to explain it, or to ​find some cause for it, but nothing has yet been added to our knowledge by these attempts.

The motions of the planets are governed by their gravitation. Were there only a single planet moving round the sun it would be acted on by no force but the sun's attraction. By purely mathematical calculation it is shown that such a planet would describe an ellipse, having the sun in one focus. It would keep going round and round in this ellipse forever. But in accordance with the law, the planets must gravitate towards each other. This mutual gravitation is far less than that of the sun, because in our solar system the planets are of much smaller mass than the central body. In consequence of this mutual attraction the planets deviate from the ellipse. Their orbits are very nearly, but not exactly, ellipses. Still, the problem of their motion is one of pure mathematical demonstration. It has occupied the ablest mathematicians of the world since the time of Newton. Every generation has studied and added to the work of the preceding one. One hundred years after Newton, Laplace and Lagrange showed that the ellipses near which the planets move gradually change their form and position. These changes can be calculated thousands, tens of thousands, or even hundreds of thousands of years in advance. Thus it is known that the eccentricity of the earth's orbit round the sun is now slightly diminishing, and that it will continue to diminish for about forty thousand years. Then it will increase so that in the course of many thousands more of years it will be greater than it now is. The same is true of all ​the planets. Their orbits gradually change their form back and forth through tens of thousands of years, like "great clocks of eternity which count off ages as ours count off seconds." The ordinary reader would be justified in some incredulity as to the correctness of these predictions for thousands of years to come, were it not for the striking precision with which the motions of the planets are actually predicted by the mathematical astronomer. This precision is reached by solving the very difficult problem of determining the effect of each planet on the motions of all the other planets. We might predict the motions of these bodies by assuming that each of them moves round the sun in a fixed ellipse, which, as I have just said, would be the case if it were not attracted by any other body. Our predictions would then, from time to time, be in error by amounts which might amount to large fractions of a degree; perhaps, in the course of a long time, to even more. To form an idea of this error we may say that one degree is the angle at which we see the breadth of an ordinary window frame at the distance of a hundred yards. The planet might then be predicted as in a line with one side of such a frame when in reality it would be at the other side or in the middle of the window.

But, taking account of the attraction of all the other planets, the prediction is so exact that the refined observations of astronomy hardly show any appreciable deviation. If we should mark on the side of a distant house a row of a hundred points, each apparently as far from the other as the average error of these predictions, the ​whole row would seem to the naked eye as a single point. The history of the discovery of Neptune, which was mentioned in the preceding chapter, affords the most striking example that we possess of the certainty of these predictions.

I shall now endeavour to give the reader some idea of the manner in which the mathematical astronomer reaches these wonderful results. To make them, he must, of course, know the pull each planet exerts upon the others. This is proportional to what the physicist and astronomer call the mass of the attracting planet. This word means quantity of matter, and around us on the surface of the earth, it has nearly the same meaning as the word weight. We may therefore say that, when the astronomer determines the mass of a planet, he is weighing it. He does this on the same principle by which the butcher weighs a ham in the spring balance. When the butcher picks the ham up he feels a pull of the ham toward the earth. When he hangs it on the hook, this pull is transferred from his hand to the spring of the balance. The stronger the pull the farther the spring is pulled down. What he reads on the scale is the strength of the pull. You know that this pull is simply the attraction of the earth on the ham. But, by a universal law of force, the ham attracts the earth exactly as much as the earth does the ham. So what the butcher really does is to find how much or how strongly the ham attracts the earth, and he calls that pull the weight of the ham. On the same ​principle, the astronomer finds the weight of a body by finding how strong is its attractive pull on some other body.

In applying this principle to the heavenly bodies, you meet at once a difficulty that looks insurmountable. You cannot get up to the heavenly bodies to do your weighing; how then will you measure their pull? I must begin the answer to this question by explaining more exactly the difference between the weight of a body and its mass. The weight of objects is not the same all over the world; a thing which weighs thirty pounds in New York would weigh an ounce more than thirty pounds in a spring balance in Greenland, and nearly an ounce less at the equator. This is because the earth is not a perfect sphere, but a little flattened. Thus weight varies with the place. If a ham weighing thirty pounds were taken up to the moon and weighed there, the pull would only be five pounds, because the moon is so much smaller and lighter than the earth. But there would be just as much ham on the moon as on the earth. There would be another weight of the ham on the planet Mars, and yet another on the sun, where it would weigh some eight hundred pounds. Hence, the astronomer does not speak of the weight of a planet, because that would depend on the place where it was weighed; but he speaks of the mass of the planet, which means how much planet there is, no matter where you might weigh it.

At the same time we might, without any inexactness, agree that the mass of a heavenly body should be fixed by the weight it would have at some place agreed upon, say ​New York. As we could not even imagine a planet at New York, because it may be larger than the earth itself, what we are to imagine is this: Suppose the planet could be divided into a million million million equal parts, and one of these parts brought to New York and weighed. We could easily find its weight in pounds or tons. Then multiply this weight by a million million million and we shall have a weight of the planet. This would be what the astronomers might take as the mass of the planet.

With these explanations, let us see how the weight of the earth is found. The principle we apply is that round bodies of the same specific gravity attract small objects on their surface with a force proportional to the diameter of the attracting body. For example, a body two feet in diameter attracts twice as strongly as one of a foot, one of three feet three times as strongly, and so on. Now, our earth is about forty million feet in diameter; that is, ten million times four feet. It follows that if we made a little model of the earth four feet in diameter, having the average specific gravity of the earth, it would attract a particle with one tenmillionth part of the attraction of the earth. We have shown in our chapter on the earth how the attraction of such a model has actually been measured, with the result of showing that the total mass of the earth is five and one half times that of an equal bulk of water. Thus this mass becomes a known quantity.

We come now to the planets. I have said that the mass or weight of a heavenly body is determined by its attraction on some other body. There are two ways in ​which the attraction of a planet may be measured. One is by its attraction on the planets next to it, causing them to deviate from the orbits in which they would move if left to themselves. By measuring the deviations, we can determine the amount of the pull, and hence the mass of the planet.

The reader will readily understand that the mathematical processes necessary to get a result in this way must be very delicate and complicated. A much simpler method can be used in the case of those planets which have satellites revolving round them, because the attraction of the planet can be determined by the motions of the satellite. The first law of motion teaches us that a body in motion, if acted on by no force, will move in a straight line. Hence, if we see a body moving in a curve, we know that it is acted on by a force in the direction toward which the motion curves. A familiar example is that of a stone thrown from the hand. If the stone were not attracted by the earth it would go on forever in the line of throw, and leave the earth entirely. But under the attraction of the earth it is drawn down and down, as it travels onward, until finally it reaches the ground. The faster the stone is thrown, of course, the farther it will go, and the greater will be the sweep of the curve of its path. If it were a cannon ball, the first part of the curve would be nearly a right line. If we could fire a cannon ball horizontally from the top of a high mountain with a velocity of five miles a second, and if it were not resisted by the air, the curvature of the path would be equal to that of the surface of our earth, and so the ​ball would never reach the earth, but would revolve round it like a little satellite in an orbit of its own. Could this be done the astronomer would be able, knowing the velocity of the ball, to calculate the attraction of the earth. The moon is a satellite, moving like such a ball, and an observer on Mars would be able, by measuring the orbit of the moon, to determine the attraction of the earth as well as we determine it by actually observing the motion of falling bodies around us.

Thus it is that when a planet like Mars or Jupiter has satellites revolving around it, astronomers on the earth can observe the attraction of the planet on its satellites and thus determine its mass. The rule for doing this is very simple. The cube of the distance between the planet and satellite is divided by the square of the time of revolution. The quotient is a number which is proportional to the mass of the planet. The rule applies to the motion of the moon round the earth and of the planets round the sun. If we divide the cube of the earth's distance from the sun, say ninety-three millions of miles, by the square of three hundred and sixty-five and a quarter, the days in a year, we shall get a certain quotient. Let us call this number the sun-quotient. Then, if we divide the cube of the moon's distance from the earth by the square of its time of revolution, we shall get another quotient, which we may call the earth-quotient. The sun-quotient will come out about three hundred and thirty thousand times as large as the earth-quotient. Hence it is concluded that the mass of the sun is three hundred and thirty thousand ​times that of the earth; that it would take this number of earths to make a body as heavy as the sun.

I give this calculation to illustrate the principle; it must not be supposed that the astronomer proceeds exactly in this way and has only this simple calculation to make. In the case of the moon and earth, the motion and distance of the former vary in consequence of the attraction of the sun, so that their actual distance apart is a changing quantity. So what the astronomer actually does is to find the attraction of the earth by observing the length of a pendulum which beats seconds in various latitudes. Then by very delicate mathematical processes he can find with great exactness what would be the time of revolution of a small satellite at any given distance from the earth, and thus can get the earth-quotient.

But, as I have already pointed out, we must, in the case of the planets, find the quotient in question by means of the satellites; and it happens, fortunately, that the motions of these bodies are much less changed by the attraction of the sun than is the motion of the moon. Thus, when we make the computation for the outer satellite of Mars, we find the quotient to be ${\displaystyle \scriptstyle {\frac {1}{3,093,500}}}$ that of the sun-quotient. Hence we conclude that the mass of Mars is ${\displaystyle \scriptstyle {\frac {1}{3,098,500}}}$ that of the sun. By the corresponding quotient, the mass of Jupiter is found to be about ${\displaystyle \scriptstyle {\frac {1}{1,047}}}$ that of the sun; Saturn, ${\displaystyle \scriptstyle {\frac {1}{3,500}}}$; Uranus, ${\displaystyle \scriptstyle {\frac {1}{22,700}}}$; Neptune, ${\displaystyle \scriptstyle {\frac {1}{19,500}}}$.

I have set forth only the great principle on which the astronomer has proceeded for the purpose in question. The law of gravitation is at the bottom of all his work. ​The effects of this law require mathematical processes which it has taken two hundred years to bring to their present state, and which are still far from perfect. The measurement of the distance of a satellite is not a job to be done in an evening; it requires patient labor extending through months and years, and then is not as exact as the astronomer would wish. He does the best he can and must be satisfied with the result until he can devise an improvement on his work, which he is always trying to do with varying success.