Fig. 22.—A planet subject to great perturbations by Jupiter.
40. Weighing the planets.—Although these perturbations can not be considered dangerous, they are interesting since they furnish a method for weighing the planets which produce them. From the law of gravitation we learn that the ability of a planet to produce perturbations depends directly upon its mass, since the force F which it exerts contains this mass, m', as a factor. So, too, the divisor r2 in the expression for the force shows that the distance between the disturbing and disturbed bodies is a matter of great consequence, for the smaller the distance the greater the force. When, therefore, the mass of a planet such as Jupiter is to be determined from the perturbations it produces, it is customary to select some such opportunity as is presented in [Fig. 22], where one of the small planets, called asteroids, is represented as moving in a very eccentric orbit, which at one point approaches close to the orbit of Jupiter, and at another place comes near to the orbit of the earth. For the most part Jupiter will not exert any very great disturbing influence upon a planet moving in such an orbit as this, since it is only at rare intervals that the asteroid and Jupiter approach so close to each other, as is shown in the figure. The time during which the asteroid is little affected by the attraction of Jupiter is used to study the motion given to it by the sun's attraction—that is, to determine carefully the undisturbed orbit in which it moves; but there comes a time at which the asteroid passes close to Jupiter, as shown in the figure, and the orbital motion which the sun imparts to it will then be greatly disturbed, and when the planet next comes round to the part of its orbit near the earth the effect of these disturbances upon its apparent position in the sky will be exaggerated by its close proximity to the earth. If now the astronomer observes the actual position of the asteroid in the sky, its right ascension and declination, and compares these with the position assigned to the planet by the law of gravitation when the attraction of Jupiter is ignored, the differences between the observed right ascensions and declinations and those computed upon the theory of undisturbed motion will measure the influence that Jupiter has had upon the asteroid, and the amount by which Jupiter has shifted it, compared with the amount by which the sun has moved it—that is, with the motion in its orbit—furnishes the mass of Jupiter expressed as a fractional part of the mass of the sun.
There has been determined in this manner the mass of every planet in the solar system which is large enough to produce any appreciable perturbation, and all these masses prove to be exceedingly small fractions of the mass of the sun, as may be seen from the following table, in which is given opposite the name of each planet the number by which the mass of the sun must be divided in order to get the mass of the planet:
| Mercury | 7,000,000 (?) |
| Venus | 408,000 |
| Earth | 329,000 |
| Mars | 3,093,500 |
| Jupiter | 1,047.4 |
| Saturn | 3,502 |
| Uranus | 22,800 |
| Neptune | 19,700 |
It is to be especially noted that the mass given for each planet includes the mass of all the satellites which attend it, since their influence was felt in the perturbations from which the mass was derived. Thus the mass assigned to the earth is the combined mass of earth and moon.
41. Discovery of Neptune.—The most famous example of perturbations is found in connection with the discovery, in the year 1846, of Neptune, the outermost planet of the solar system. For many years the motion of Uranus, his next neighbor, had proved a puzzle to astronomers. In accordance with Kepler's first law this planet should move in an ellipse having the sun at one of its foci, but no ellipse could be found which exactly fitted its observed path among the stars, although, to be sure, the misfit was not very pronounced. Astronomers surmised that the small deviations of Uranus from the best path which theory combined with observation could assign, were due to perturbations in its motion caused by an unknown planet more remote from the sun—a thing easy to conjecture but hard to prove, and harder still to find the unknown disturber. But almost simultaneously two young men, Adams in England and Le Verrier in France, attacked the problem quite independently of each other, and carried it to a successful solution, showing that if the irregularities in the motion of Uranus were indeed caused by an unknown planet, then that planet must, in September, 1846, be in the direction of the constellation Aquarius; and there it was found on September 23d by the astronomers of the Berlin Observatory whom Le Verrier had invited to search for it, and found within a degree of the exact point which the law of gravitation in his hands had assigned to it.
This working backward from the perturbations experienced by Uranus to the cause which produced them is justly regarded as one of the greatest scientific achievements of the human intellect, and it is worthy of note that we are approaching the time at which it may be repeated, for Neptune now behaves much as did Uranus three quarters of a century ago, and the most plausible explanation which can be offered for these anomalies in its path is that the bounds of the solar system must be again enlarged to include another disturbing planet.
42. The shape of a planet.—There is an effect of gravitation not yet touched upon, which is of considerable interest and wide application in astronomy—viz., its influence in determining the shape of the heavenly bodies. The earth is a globe because every part of it is drawn toward the center by the attraction of the other parts, and if this attraction on its surface were everywhere of equal force the material of the earth would be crushed by it into a truly spherical form, no matter what may have been the shape in which it was originally made. But such is not the real condition of the earth, for its diurnal rotation develops in every particle of its body a force which is sometimes called centrifugal, but which is really nothing more than the inertia of its particles, which tend at every moment to keep unchanged the direction of their motion and which thus resist the attraction that pulls them into a circular path marked out by the earth's rotation, just as a stone tied at the end of a string and swung swiftly in a circle pulls upon the string and opposes the constraint which keeps it moving in a circle. A few experiments with such a stone will show that the faster it goes the harder does it pull upon the string, and the same is true of each particle of the earth, the swiftly moving ones near the equator having a greater centrifugal force than the slow ones near the poles. At the equator the centrifugal force is directly opposed to the force of gravity, and in effect diminishes it, so that, comparatively, there is an excess of gravity at the poles which compresses the earth along its axis and causes it to bulge out at the equator until a balance is thus restored. As we have learned from the study of geography, in the case of the earth, this compression amounts to about 27 miles, but in the larger planets, Jupiter and Saturn, it is much greater, amounting to several thousand miles.
But rotation is not the only influence that tends to pull a planet out of shape. The attraction which the earth exerts upon the moon is stronger on the near side and weaker on the far side of our satellite than at its center, and this difference of attraction tends to warp the moon, as is illustrated in [Fig. 23] where 1, 2, and 3 represent pieces of iron of equal mass placed in line on a table near a horseshoe magnet, H. Each piece of iron is attracted by the magnet and is held back by a weight to which it is fastened by means of a cord running over a pulley, P, at the edge of the table. These weights are all to be supposed equally heavy and each of them pulls upon its piece of iron with a force just sufficient to balance the attraction of the magnet for the middle piece, No. 2. It is clear that under this arrangement No. 2 will move neither to the right nor to the left, since the forces exerted upon it by the magnet and the weight just balance each other. Upon No. 1, however, the magnet pulls harder than upon No. 2, because it is nearer and its pull therefore more than balances the force exerted by the weight, so that No. 1 will be pulled away from No. 2 and will stretch the elastic cords, which are represented by the lines joining 1 and 2, until their tension, together with the force exerted by the weight, just balances the attraction of the magnet. For No. 3, the force exerted by the magnet is less than that of the weight, and it will also be pulled away from No. 2 until its elastic cords are stretched to the proper tension. The net result is that the three blocks which, without the magnet's influence, would be held close together by the elastic cords, are pulled apart by this outside force as far as the resistance of the cords will permit.