Chapter 6

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 as a solid body, each bending the same face towards the other; but as the moon retreated, and as tides began to be raised on the earth, the length of the day began to increase, as did also the length of the month. We know, however, that the month increased more rapidly than the day, so that a time was reached when the month was twice as long as the day; and still both periods kept on increasing, but not at equal rates, for in progress of time the month grew so much more rapidly than the day, that many days had to elapse while the moon accomplished a single revolution. It is, however, only necessary for us to note those stages of the mighty progress which correspond to special events. The first of such stages was attained when the month assumed its maximum ratio to the day. At this time, the month was about twenty-nine days, and the epoch appears to have occurred at a comparatively recent date if we use such standards of time as tidal evolution requires; though measured by historical standards, the epoch is of incalculable antiquity. I cannot impress upon you too often the enormous magnitude of the period of time which these phenomena have required for their evolution. Professor Darwin's theory affords but little information on this point, and the utmost we can do is to assign a minor limit to the period through which tidal evolution has been in progress. It is certain that the birth of the moon must have occurred at least fifty million years ago, but probably the true period is enormously greater than this. If indeed we choose to add a cipher or two to the figure just printed, I do not think there is anything which could tell us that we have over-estimated the mark. Therefore, when I speak of the epoch in which the month possessed the greatest number of days as a recent one, it must be understood that I am merely speaking of events in relation to the order of tidal evolution. Viewed from this standpoint, we can show that the epoch is a recent one in the following manner. At present the month consists of a little more than twenty-seven days, but at this maximum period to which I have referred the month was about twenty-nine days; from that it began to decline, and the decline cannot have proceeded very far, for even still there are only two days less in the month than at the time when the month had the greatest number of days. It thus follows that the present epoch--the human epoch, as we may call it--in the history of the earth has fallen at a time when the progress of tidal evolution is about half-way between the initial and the final stage. I do not mean half-way in the sense of actual measurement of years; indeed, from this point it would seem that we cannot yet be nearly half-way, for, vast as are the periods of time that have elapsed since the moon first took its departure from the earth, they fall far short of that awful period of time which will intervene between the present moment and the hour when the next critical state of earth-moon history shall have been attained. In that state the day is destined once again to be equal to the month, just as was the case in the initial stage. The half-way stage will therefore in one sense be that in which the proportion of the month to the day culminates. This is the stage which we have but lately passed; and thus it is that at present we may be said to be almost half-way through the progress of tidal evolution.

My narrative of the earth-moon evolution must from this point forward cease to be retrospective. Having begun at that critical moment when the month and day were first equal, we have traced the progress of events to the present hour. What we have now to say is therefore a forecast of events yet to come. So far as we can tell, no agent is likely to interfere with the gradual evolution caused by the tides, which dynamical principles have disclosed to us. As the years roll on, or perhaps, I should rather say, as thousands of years and millions of years roll on, the day will continue to elongate, or the earth to rotate more slowly on its axis. But countless ages must elapse before another critical stage of the history shall be reached. It is needless for me to ponder over the tedious process by which this interesting epoch is reached. I shall rather sketch what the actual condition of our system will be when that moment shall have arrived. The day will then have expanded from the present familiar twenty-four hours up to a day more than twice, more than five, even more than fifty times its present duration. In round numbers, we may say that this great day will occupy one thousand four hundred of our ordinary hours. To realize the critical nature of the situation then arrived at, we must follow the corresponding evolution through which the moon passes. From its present distance of two hundred and forty thousand miles, the moon will describe an ever-enlarging orbit; and as it does so the duration of the month will also increase, until at last a point will be reached when the month has become more than double its present length, and has attained the particular value of one thousand four hundred hours. We are specially to observe that this one-thousand four-hundred-hour month will be exactly reached when the day has also expanded to one thousand four hundred hours; and the essence of this critical condition, which may be regarded as a significant point of tidal evolution, is that the day and the month have again become equal. The day and the month were equal at the beginning, the day and the month will be equal at the end. Yet how wide is the difference between the beginning and the end. The day or the month at the end is some hundreds of times as long as the month or the day at the beginning.

I have already fully explained how, in any stage of the evolutionary progress in which the day and the month became equal, the energy of the system attained a maximum or a minimum value. At the beginning the energy was a maximum; at the end the energy will be a minimum. The most important consequences follow from this consideration. I have already shown that a condition of maximum energy corresponded to dynamic instability. Thus we saw that the earth-moon history could not have commenced without the intervention of some influence other than tides at the beginning. Now let us learn what the similar doctrine has to tell us with regard to the end. The condition then arrived at is one of dynamical stability; for suppose that the system were to receive a slight alteration, by which the moon went out a little further, and thus described a larger orbit, and so performed more than its share of the moment of spin. Then the earth would have to do a little less spinning, because, under all circumstances, the total quantity of spin must be preserved unaltered. But the energy being at a minimum, such a small displacement must of course produce a state of things in which the energy would be increased. Or if we conceived the moon to come in towards the earth, the moon would then contribute less to the total moment of momentum. It would therefore be incumbent on the earth to do more; and accordingly the velocity of the earth's rotation would be augmented. But this arrangement also could only be produced by the addition of some fresh energy to the system, because the position from which the system is supposed to have been disturbed is one of minimum energy.

No disturbance of the system from this final position is therefore conceivable, unless some energy can be communicated to it. But this will demonstrate the utter incompetency of the tides to shift the system by a hair's breadth from this position; for it is of the essence of the tides to waste energy by friction. And the transformations of the system which the tides have caused are invariably characterized by a decline of energy, the movements being otherwise arranged so that the total moment of momentum shall be preserved intact. Note, how far we were justified in speaking of this condition as a final one. It is final so far as the lunar tides are concerned; and were the system to be screened from all outer interference, this accommodation between the earth and the moon would be eternal.

There is indeed another way of demonstrating that a condition of the system in which the day has assumed equality with the month must necessarily be one of dynamical equilibrium. We have shown that the energy which the tides demand is derived not from the mere fact that there are high tides and low tides, but from the circumstance that these tides do rise and fall; that in falling and rising they do produce currents; and it is these currents which generate the friction by which the earth's velocity is slowly abated, its energy wasted, and no doubt ultimately dissipated as heat. If therefore we can make the ebbing and the flowing of the tides to cease, then our argument will disappear. Thus suppose, for the sake of illustration, that at a moment when the tides happened to be at high water in the Thames, such a change took place in the behaviour of the moon that the water always remained full in the Thames, and at every other spot on the earth remained fixed at the exact height which it possessed at this particular moment. There would be no more tidal friction, and therefore the system would cease to course through that series of changes which the existence of tidal friction necessitates.

But if the tide is always to be full in the Thames, then the moon must be always in the same position with respect to the meridian, that is, the moon must always be fixed in the heavens over London. In fact, the moon must then revolve around the earth just as fast as London does--the month must have the same length as the day. The earth must then show the same face constantly to the moon, just as the moon always does show the same face towards the earth; the two globes will in fact revolve as if they were connected with invisible bonds, which united them into a single rigid body.

We need therefore feel no surprise at the cessation of the progress of tidal evolution when the month and the day are equal, for then the movement of moon-raised tides has ceased. No doubt the same may be said of the state at the beginning of the history, when the day and the month had the brief and equal duration of a few hours. While the equality of the two periods lasted there could be no tides, and therefore no progress in the direction of tidal evolution. There is, however, the profound difference of stability and instability between the two cases; the most insignificant disturbance of the system at the initial stage was sufficient to precipitate the revolving moon from its condition of dynamical equilibrium, and to start the course of tidal evolution in full vigour. If, however, any trifling derangement should take place in the last condition of the system, so that the month and the day departed slightly from equality, there would instantly be an ebbing and a flowing of the tides; and the friction generated by these tides would operate to restore the equality because this condition is one of dynamical stability.

It will thus be seen with what justice we can look forward to the day and month each of fourteen hundred hours as a finale to the progress of the luni-tidal evolution. Throughout the whole of this marvellous series of changes it is always necessary to remember the one constant and invariable element--the moment of momentum of the system which tides cannot alter. Whatever else the friction can have done, however fearful may have been the loss of energy by the system, the moment of momentum which the system had at the beginning it preserves unto the end. This it is which chiefly gives us the numerical data on which we have to rely for the quantitative features of tidal evolution.

We have made so many demands in the course of these lectures on the capacity of tidal friction to accomplish startling phenomena in the evolution of the earth-moon system, that it is well for us to seek for any evidence that may otherwise be obtainable as to the capacity of tides for the accomplishment of gigantic operations. I do not say that there is any doubt which requires to be dispelled by such evidence, for as to the general outlines of the doctrine of tidal evolution which has been here sketched out there can be no reasonable ground for mistrust; but nevertheless it is always desirable to widen our comprehension of any natural phenomena by observing collateral facts. Now there is one branch of tidal action to which I have as yet only in the most incidental way referred. We have been speaking of the tides in the earth which are made to ebb and flow by the action of the moon; we have now to consider the tides in the moon, which are there excited by the action of the earth. For between these two bodies there is a reciprocity of tidal-making energy--each of them is competent to raise tides in the other. As the moon is so small in comparison with the earth, and as the tides on the moon are of but little significance in the progress of tidal evolution, it has been permissible for us to omit them from our former discussion. But it is these tides on the moon which will afford us a striking illustration of the competency of tides for stupendous tasks. The moon presents a monument to show what tides are able to accomplish.

I must first, however, explain a difficulty which is almost sure to suggest itself when we speak of tides on the moon. I shall be told that the moon contains no water on its surface, and how then, it will be said, can tides ebb and flow where there is no sea to be disturbed? There are two answers to this difficulty; it is no doubt true that the moon seems at present entirely devoid of water in so far as its surface is exposed to us, but it is by no means certain that the moon was always in this destitute condition. There are very large features marked on its map as "seas"; these regions are of a darker hue than the rest of the moon's surface, they are large objects often many hundreds of miles in diameter, and they form, in fact, those dark patches on the brilliant surface which are conspicuous to the unaided eye, and are represented in Fig. 3. Viewed in a telescope these so-called seas, while clearly possessing no water at the present time, are yet widely different from the general aspect of the moon's surface. It has often been supposed that great oceans once filled these basins, and a plausible explanation has even been offered as to how the waters they once contained could have vanished. It has been thought that as the mineral substances deep in the interior of our satellite assumed the crystalline form during the progress of cooling, the demand for water of crystallization required for incorporation with the minerals was so great that the oceans of the moon became entirely absorbed. It is, however, unnecessary for our present argument that this theory should be correct. Even if there never was a drop of water found on our satellite, the tides in its molten materials would be quite sufficient for our purpose; anything that tides could accomplish would be done more speedily by vast tides of flowing lava than by merely oceanic tides.

There can be no doubt that tides raised on the moon by the earth would be greater than the tides raised on the earth by the moon. The question is, however, not a very simple one, for it depends on the masses of both bodies as well as on their relative dimensions. In so far as the masses are concerned, the earth being more than eighty times as heavy as the moon, the tides would on this account be vastly larger on the moon than on the earth. On the other hand, the moon's diameter being much less than that of the earth, the efficiency of a tide-producing body in its action on the moon would be less than that of the same body at the same distance in its action on the earth; but the diminution of the tides from this cause would be not so great as their increase from the former cause, and therefore the net result would be to exhibit much greater tides on the moon than on the earth.