A Text-Book of Astronomy

by George C. Comstock

236. The earth-moon system.—Retracing into the past the course of development of the earth and moon, it is possible to reach back by means of the mathematical theory of tidal friction to a time at which these bodies were much nearer to each other than now, but it has not been found possible to trace out the mode of their separation from one body into two, as is supposed in the nebular theory. In the earliest part of their history accessible to mathematical analysis they are distinct bodies at some considerable distance from each other, with the earth rotating about an axis more nearly perpendicular to the moon's orbit and to the ecliptic than is now the case. Starting from such a condition, the lunar tides, according to Darwin, have been instrumental in tipping the earth's rotation axis into its present oblique position, and in determining the eccentricity of the moon's orbit and its position with respect to the ecliptic as well as the present length of day and month.

237. Tidal friction upon the planets.—The satellites of the outer planets are equally subject to influences of this kind, and there appears to be independent evidence that some of them, at least, turn always the same face toward their respective planets, indicating that the work of tidal friction has here been accomplished. We saw in [Chapter XI] that it is at present an open question whether the inner planets, Venus and Mercury, do not always turn the same face toward the sun, their day and year being of equal length. In addition to the direct observational evidence upon this point, Schiaparelli has sought to show by an appeal to tidal theory that such is probably the case, at least for Mercury, since the tidal forces which tend to bring about this result in that planet are about as great as the forces which have certainly produced it in the case of the moon and Saturn's satellite, Japetus. The same line of reasoning would show that every satellite in the solar system, save possibly the newly discovered ninth satellite of Saturn, must, as a consequence of tidal friction, turn always the same face toward its planet.

238. The solar tide.—The sun also raises tides in the earth, and their influence must be similar in character to that of the lunar tides, checking the rotation of the earth and thrusting earth and sun apart, although quantitatively these effects are small compared with those of the moon. They must, however, continue so long as the solar tide lasts, possibly until the day and year are made of equal length—i. e., they may continue long after the lunar tidal influence has ceased to push earth and moon apart. Should this be the case, a curious inverse effect will be produced. The day being then longer than the month, the moon will again raise a tide in the earth which will run around it from west to east, opposite to the course of the present tide, thus tending to accelerate the earth's rotation, and by its reaction to bring the moon back toward the earth again, and ultimately to fall upon it.

We may note that an effect of this kind must be in progress now between Mars and its inner satellite, Phobos, whose time of orbital revolution is only one third of a Martian day. It seems probable that this satellite is in the last stages of its existence as an independent body, and must ultimately fall into Mars.

239. Roche's limit.—In looking forward to such a catastrophe, however, due regard must be paid to a dynamical principle of a different character. The moon can never be precipitated upon the earth entire, since before it reaches us it will have been torn asunder by the excess of the earth's attraction for the near side of its satellite over that which it exerts upon the far side. As the result of Roche's mathematical analysis we are able to assign a limiting distance between any planet and its satellite within which the satellite, if it turns always the same face toward the planet, can not come without being broken into fragments. If we represent the radius of the planet by r, and the quotient obtained by dividing the density of the planet by the density of the satellite by q, then

Roche's limit = 2.44 r ∛ q.

Thus in the case of earth and moon we find from the densities given in [§ 95], q = 1.65, and with r = 3,963 miles we obtain 11,400 miles as the nearest approach which the moon could make to the earth without being broken up by the difference of the earth's attractions for its opposite sides.

We must observe, however, that Roche's limit takes no account of molecular forces, the adhesion of one molecule to another, by virtue of which a stick or stone resists fracture, but is concerned only with the gravitative forces by which the molecules are attracted toward the moon's center and toward the earth. Within a stone or rock of moderate size these gravitative forces are insignificant, and cohesion is the chief factor in preserving its integrity, but in a large body like the moon, the case is just reversed, cohesion plays a small part and gravitation a large one in holding the body together. We may conclude, therefore, that at a proper distance these forces are capable of breaking up the moon, or any other large body, into fragments of a size such that molecular cohesion instead of gravitation is the chief agent in preserving them from further disintegration.

240. Saturn's rings.—Saturn's rings are of peculiar interest in this connection. The outer edge of the ring system lies just inside of Roche's limit for this planet, and we have already seen that the rings are composed of small fragments independent of each other. Whatever may have been the process by which the nine satellites of Saturn came into existence, we have in Roche's limit the explanation why the material of the ring was not worked up into satellites; the forces exerted by Saturn would tear into pieces any considerable satellite thus formed and equally would prevent the formation of one from raw material.

Saturn's rings present the only case within the solar system where matter is known to be revolving about a planet at a distance less than Roche's limit, and it is an interesting question whether these rings can remain as a permanent part of the planet's system or are only a temporary feature. The drawings of Saturn made two centuries ago agree among themselves in representing the rings as larger than they now appear, and there is some reason to suppose that as a consequence of mutual disturbances—collisions—their momentum is being slowly wasted so that ultimately they must be precipitated into the planet. But the direct evidence of such a progress that can be drawn from present data is too scanty to justify positive conclusions in the matter. On the other hand, Nolan suggests that in the outer parts of the ring small satellites might be formed whose tidal influence upon Saturn would suffice to push them away from the ring beyond Roche's limit, and that the very small inner satellites of Saturn may have been thus formed at the expense of the ring.