A Text-Book of Astronomy

by George C. Comstock

Despite these objections and others equally serious that have been raised, the nebular theory agrees with the facts of Nature at so many points that astronomers upon the whole are strongly inclined to accept its major outlines as being at least an approximation to the course of development actually followed by the solar system; but at some points—e. g., the formation of planets and satellites through the casting off of nebulous rings—the objections are so many and strong as to call for revision and possibly serious modification of the theory.

One proposed modification, much discussed in recent years, consists in substituting for the primitive gaseous nebula imagined by Laplace, a very diffuse cloud of meteoric matter which in the course of its development would become transformed into the gaseous state by rising temperature. From this point of view much of the meteoric dust still scattered throughout the solar system may be only the fragments left over in fashioning the sun and planets. Chamberlin and Moulton, who have recently given much attention to this subject, in dissenting from some of Laplace's views, consider that the primitive nebulous condition must have been one in which the matter of the system was "so brought together as to give low mass, high momentum, and irregular distribution to the outer part, and high mass, low momentum, and sphericity to the central part," and they suggest a possible oblique collision of a small nebula with the outer parts of a large one.

232. Bode's law.—We should not leave the theory of Laplace without noting the light it casts upon one point otherwise obscure—the meaning of Bode's law ([§ 134]). This law, stated in mathematical form, makes a geometrical series, and similar geometrical series apply to the distances of the satellites of Jupiter and Saturn from these planets. Now, Roche has shown by the application of physical laws to the shrinkage of a gaseous body that its radius at any time may be expressed by means of a certain mathematical formula very similar to Bode's law, save that it involves the amount of time that has elapsed since the beginning of the shrinking process. By comparing this formula with the one corresponding to Bode's law he reaches the conclusion that the peculiar spacing of the planets expressed by that law means that they were formed at successive equal intervals of time—i. e., that Mars is as much older than the earth as the earth is older than Venus, etc. The failure of Bode's law in the case of Neptune would then imply that the interval of time between the formation of Neptune and Uranus was shorter than that which has prevailed for the other planets. But too much stress should not be placed upon this conclusion. So long as the manner in which the planets came into being continues an open question, conclusions about their time of birth must remain of doubtful validity.

233. Tidal friction between earth and moon.—An important addition to theories of development within the solar system has been worked out by Prof. G. H. Darwin, who, starting with certain very simple assumptions as to the present condition of things in earth and moon, derives from these, by a strict process of mathematical reasoning, far-reaching conclusions of great interest and importance. The key to these conclusions lies in recognition of the fact that through the influence of the tides ([§ 42]) there is now in progress and has been in progress for a very long time, a gradual transfer of motion (moment of momentum) from the earth to the moon. The earth's motion of rotation is being slowly destroyed by the friction of the tides, as the motion of a bicycle is destroyed by the friction of a brake, and, in consequence of this slowing down, the moon is pushed farther and farther away from the earth, so that it now moves in a larger orbit than it had some millions of years ago.

[Fig. 24] has been used to illustrate the action of the moon in raising tides upon the earth, but in accordance with the third law of motion ([§ 36]) this action must be accompanied by an equal and contrary reaction whose nature may readily be seen from the same figure. The moon moves about its orbit from west to east and the earth rotates about its axis in the same direction, as shown by the curved arrow in the figure. The tidal wave, I, therefore points a little in advance of the moon's position in its orbit and by its attraction must tend to pull the moon ahead in its orbital motion a little faster than it would move if the whole substance of the earth were placed inside the sphere represented by the broken circle in the figure. It is true that the tidal wave at I'' pulls back and tends to neutralize the effect of the wave at I, but on the whole the tidal wave nearer the moon has the stronger influence, and the moon on the whole moves a very little faster, and by virtue of this added impetus draws continually a little farther away from the earth than it would if there were no tides.

234. Consequences of tidal friction upon the earth.—This process of moving the moon away from the earth is a cumulative one, going on century after century, and with reference to it the moon's orbit must be described not as a circle or ellipse, or any other curve which returns into itself, but as a spiral, like the balance spring of a watch, each of whose coils is a little larger than the preceding one, although this excess is, to be sure, very small, because the tides themselves are small and the tidal influence feeble when compared with the whole attraction of the earth for the moon. But, given time enough, even this small force may accomplish great results, and something like 100,000,000 years of past opportunity would have sufficed for the tidal forces to move the moon from close proximity with the earth out to its present position.

For millions of years to come, if moon and earth endure so long, the distance between them must go on increasing, although at an ever slower rate, since the farther away the moon goes the smaller will be the tides and the slower the working out of their results. On the other hand, when the moon was nearer the earth than now, tidal influences must have been greater and their effects more rapidly produced than at the present time, particularly if, as seems probable, at some past epoch the earth was hot and plastic like Jupiter and Saturn. Then, instead of tides in the water of the sea, such as we now have, the whole substance of the earth would respond to the moon's attraction in bodily tides of semi-fluid matter not only higher, but with greater internal friction of their molecules one upon another, and correspondingly greater effect in checking the earth's rotation.

But, whether the tide be a bodily one or confined to the waters of the sea, so long as the moon causes it to flow there will be a certain amount of friction which will affect the earth much as a brake affects a revolving wheel, slowing down its motion, and producing thus a longer day as well as a longer month on account of the moon's increased distance. Slowing down the earth's rotation is the direct action of the moon upon the earth. Pushing the moon away is the form in which the earth's equal and contrary reaction manifests itself.

235. Consequences of tidal friction upon the moon.—When the moon was plastic the earth must have raised in it a bodily tide manifold greater than the lunar tides upon the earth, and, as we have seen in [Chapter IX], this tide has long since worn out the greater part of the moon's rotation and brought our satellite to the condition in which it presents always the same face toward the earth.

These two processes, slowing down the rotation and pushing away the disturbing body, are inseparable—one requires the other; and it is worth noting in this connection that when for any reason the tide ceases to flow, and the tidal wave takes up a permanent position, as it has in the moon ([§ 99]), its work is ended, for when there is no motion of the wave there can be no friction to further reduce the rate of rotation of the one body, and no reaction to that friction to push away the other. But this permanent and stationary tidal wave in the moon, or elsewhere, means that the satellite presents always the same face toward its planet, moving once about its orbit in the time required for one revolution upon its axis, and the tide raised by the moon upon the earth tends to produce here the result long since achieved in our satellite, to make our day and month of equal length, and to make the earth turn always the same side toward the moon. But the moon's tidal force is small compared with that of the earth, and has a vastly greater momentum to overcome, so that its work upon the earth is not yet complete. According to Thomson and Tait, the moon must be pushed off another hundred thousand miles, and the day lengthened out by tidal influence to seven of our present weeks before the day and the lunar month are made of equal length, and the moon thereby permanently hidden from one hemisphere of the earth.