Time and Tide by Sir Robert Stawell Ball (read with me txt) 📖

attained at the moment when the alteration is zero. It therefore follows, that the critical periods of the system will arise when the relative rotation is zero, that is, when the earth's rotation on its axis is performed with a velocity equal to that with which the moon revolves around the earth. This is truly a singular condition of the earth-moon system; the moon in such a case would revolve around the earth as if the two bodies were bound together by rigid bonds into what was practically a single solid body. At the present moment no doubt to some extent this condition is realized, because the moon always turns the same face to the earth (a point on which we shall have something to say later on); but in the original condition of the earth-moon system, the earth would also constantly direct the same face to the moon, a condition of things which is now very far from being realized.

It can be shown from the mathematical nature of the problem that there are four states of the earth-moon system in which this condition may be realized, and which are also compatible with the conservation of the moment of momentum. We can express what this condition implies in a somewhat more simple manner. Let us understand by the _day_ the period of the earth's rotation on its axis, whatever that may be, and let us understand by the _month_ the period of revolution of the moon around the earth, whatever value it may have; then the condition of maximum or minimum energy is attained when the day and the month have become equal to each other. Of the four occasions mathematically possible in which the day and the month can be equal, there are only two which at present need engage our attention--one of these occurred near the beginning of the earth and moon's history, the other remains to be approached in the immeasurably remote future. The two remaining solutions are futile, being what the mathematician would describe as imaginary.

There is a fundamental difference between the dynamical conditions in these critical epochs--in one of them the energy of the system has attained a maximum value, and in the other the energy of the system is at a minimum value. It is impossible to over-estimate the significance of these two states of the system.

I may recall the fundamental notion which every one has learned in mechanics, as to the difference between stable and unstable equilibrium. The conceivable possibility of making an egg stand on its end is a practical impossibility, because nature does not like unstable equilibrium, and a body departs therefrom on the least disturbance; on the other hand, stable equilibrium is the position in which nature tends to place everything. A log of wood floating on a river might conceivably float in a vertical position with its end up out of the water, but you never could succeed in so balancing it, because no matter how carefully you adjusted the log, it would almost instantly turn over when you left it free; on the other hand, when the log floats naturally on the water it assumes a horizontal position, to which, when momentarily displaced therefrom, it will return if permitted to do so. We have here an illustration of the contrast between stable and unstable equilibrium. It will be found generally that a body is in equilibrium when its centre of gravity is at its highest point or at its lowest point; there is, however, this important difference, that when the centre of gravity is highest the equilibrium is unstable, and when the centre of gravity is lowest the equilibrium is stable. The potential energy of an egg poised on its end in unstable equilibrium is greater than when it lies on its side in stable equilibrium. In fact, energy must be expended to raise the egg from the horizontal position to the vertical; while, on the other hand, work could conceivably be done by the egg when it passes from the vertical position to the horizontal. Speaking generally, we may say that the stable position indicates low energy, while a redundancy of that valuable agent is suggestive of instability.

We may apply similar principles to the consideration of the earth-moon system. It is true that we have here a series of dynamical phenomena, while the illustrations I have given of stable and unstable equilibrium relate only to statical problems; but we can have dynamical stability and dynamical instability, just as we can have stable and unstable equilibrium. Dynamical instability corresponds with the maximum of energy, and dynamical stability to the minimum of energy.

At that primitive epoch, when the energy of the earth-moon system was a maximum, the condition was one of dynamical instability; it was impossible that it should last. But now mark how truly critical an occurrence this must have been in the history of the earth-moon system, for have I not already explained that it is a necessary condition of the progress of tidal evolution that the energy of the system should be always declining? But here our retrospect has conducted us back to a most eventful crisis, in which the energy was a maximum, and therefore cannot have been immediately preceded by a state in which the energy was greater still; it is therefore impossible for the tidal evolution to have produced this state of things; some other influence must have been in operation at this beginning of the earth-moon system.

Thus there can be hardly a doubt that immediately preceding the critical epoch the moon originated from the earth in the way we have described. Note also that this condition, being one of maximum energy, was necessarily of dynamical instability, it could not last; the moon must adopt either of two courses--it must tumble back on the earth, or it must start outwards. Now which course was the moon to adopt? The case is analogous to that of an egg standing on its end--it will inevitably tumble one way or the other. Some infinitesimal cause will produce a tendency towards one side, and to that side accordingly the egg will fall. The earth-moon system was similarly in an unstable state, an infinitesimal cause might conceivably decide the fate of the system. We are necessarily in ignorance of what the determining cause might have been, but the effect it produced is perfectly clear; the moon did not again return to its mother earth, but set out on that mighty career which is in progress to-day.

Let it be noted that these critical epochs in the earth-moon history arise when and only when there is an absolute identity between the length of the month and the length of the day. It may be proper therefore that I should provide a demonstration of the fact, that the identity between these two periods must necessarily have occurred at a very early period in the evolution.

The law of Kepler, which asserts that the square of the periodic time is proportioned to the cube of the mean distance, is in its ordinary application confined to a comparison between the revolutions of the several planets about the sun. The periodic time of each planet is connected with its average distance by this law; but there is another application of Kepler's law which gives us information of the distance and the period of the moon in former stages of the earth-moon history. Although the actual path of the moon is of course an ellipse, yet that ellipse is troubled, as is well known, by many disturbing forces, and from this cause alone the actual path of the moon is far from being any of those simple curves with which we are so well acquainted. Even were the earth and the moon absolutely rigid particles, perturbations would work all sorts of small changes in the pliant curve. The phenomena of tidal evolution impart an additional element of complexity into the actual shape of the moon's path. We now see that the ellipse is not merely subject to incessant deflections of a periodic nature, it also undergoes a gradual contraction as we look back through time past; but we may, with all needful accuracy for our present purpose, think of the path of the moon as a circle, only we must attribute to that circle a continuous contraction of its radius the further and the further we look back. The alteration in the radius will be even so slow, that the moon will accomplish thousands of revolutions around the earth without any appreciable alteration in the average distance of the two bodies. We can therefore think of the moon as revolving at every epoch in a circle of special radius, and as accomplishing that revolution in a special time. With this understanding we can now apply Kepler's law to the several stages of the moon's past history. The periodic time of each revolution, and the mean distance at which that revolution was performed, will be always connected together by the formula of Kepler. Thus to take an instance in the very remote past. Let us suppose that the moon was at one hundred and twenty thousand miles instead of two hundred and forty thousand, that is, at half its present distance. Applying the law of Kepler, we see that the time of revolution must then have been only about ten days instead of the twenty-seven it is now. Still further, let us suppose that the moon revolves in an orbit with one-tenth of the diameter it has at present, then the cube of 10 being 1000, and the square root of 1000 being 31.6, it follows that the month must have been less than the thirty-first part of what it is at present, that is, it must have been considerably less than one of our present days. Thus you see the month is growing shorter and shorter the further we look back, the day is also growing shorter and shorter; but still I think we can show that there must have been a time when the month will have been at least as short as the day. For let us take the most extreme case in which the moon shall have made the closest possible approximation to the earth. Two globes in contact will have a distance between their centres which is equal to the sum of their radii. Take the earth as having a radius of four thousand miles, and the moon a radius of one thousand miles, the two centres must at their shortest distance be five thousand miles apart, that is, the moon must then be at the forty-eighth part of its present distance from the earth. Now the cube of 48 is 110,592, and the square root of 110,592 is nearly 333, therefore the length of the month will be one-three hundred and thirty-third part of the duration of the month at present; in other words, the moon must revolve around the earth in a period of somewhat about two hours. It seems impossible that the day can ever have been as brief as this. We have therefore proved that, in the course of its contracting duration, the moon must have overtaken the contracting day, and that therefore there must have been a time when the moon was in the vicinity of the earth, and having a day and month of equal period. Thus we have shown that the critical condition of dynamical instability must have occurred in the early period of the earth-moon history, if the agents then in operation were those which we now know. The further development of the subject must be postponed until the next lecture.


LECTURE II.

Starting from that fitting commencement of earth-moon history which the critical epoch affords, we shall now describe the dynamical phenomena as the tidal evolution progressed. The moon and the earth initially moved