Encyclopædia Britannica, Ninth Edition/Tides/Chapter 10
X. Cosmogonic Speculations founded on Tidal Friction.
§ 50. History of the Earth and Moon.
We shall not attempt to discuss the mathematical methods by which the complete history of a planet, attended by one or more satellites, is to be traced. The laws indicated in the preceding sections show that there is such a problem, and that it may be solved, and we refer to Mr Darwin's papers for details (Phil. Trails., 1879-81). It may be interesting, however, to give the various results of the investigation in the form of a sketch of the possible evolution of the earth and moon, followed by remarks on the other planetary systems and on the solar system as a whole.
We begin with a planet not very much more than 8000 miles in Conjeediameter, and probably partly solid, partly fluid, and partly gaseous, tural It is rotating about an axis inclined at about 11 or 12 to the nor- genesisof mal to the ecliptic, with a period of from two to four hours, and is moon revolving about the sun with a period not much shorter than our from present year. The rapidity of the planet's rotation causes so great earth. a compression of its figure that it cannot continue to exist in an ellipsoidal form with stability; or else it is so nearly unstable that complete instability is induced by the solar tides. The planet then separates into two masses, the larger being the earth and the smaller the moon. It is not attempted to define the mode of separation, or to say whether the moon was initially a chain of meteorites. At any rate it must be assumed that the smaller mass became more or less conglomerated and finally fused into a spheroid, perhaps in consequence of impacts between its constituent mete orites, which were once part of the primeval planet. Up to this point the history is largely speculative, for the conditions of insta bility of a rotating mass of fluid have not yet been fully investigated. Earth and moon subject of investigation.We now have the earth and moon nearly in contact with one another, and rotating nearly as though they were parts of one rigid body.[1] This is the system which was the subject of dynamical investigation. As the two masses are not rigid, the attraction of each distorts the other; and, if they do not move rigorously with the same periodic time, each raises a tide in the other. Also the sun raises tides in both. In consequence of the frictional resistance to these tidal motions, such a system is dynamically unstable. If the moon had moved orbitally a little faster than the earth rotated, she must have fallen back into the earth; thus the existence of the moon compels us to believe that the equilibrium broke down by the moon revolving orbitally a little slower than the earth rotates. In consequence of the tidal friction the periodic times both of the moon (or the month) and of the earth's rotation (or the day) increase; but the month increases in length at a much greater rate than the day. At some early stage in the history of the system the moon was conglomerated into a spheroidal form, and acquired a rotation about an axis nearly parallel to that of the earth.
The axial rotation of the moon is retarded by the attraction of The;he earth on the tides raised in the moon, and this retardation takes moon, ilace at a far greater rate than the similar retardation of the earth's otation. As soon as the moon rotates round her axis with twice;he angular velocity with which she revolves in her orbit, the >osition of her axis of rotation (parallel with the earth's axis) >ecomes dynamically unstable. The obliquity of the lunar equator
- o the plane of the orbit increases, attains a maximum, and then diminishes. Meanwhile the lunar axial rotation is being reduced;owards identity with the orbital motion. Finally, her equator is nearly coincident with the plane of the orbit, and the attraction of he earth on a tide, which degenerates into a permanent ellipticity of the lunar equator, causes her always to show the same face to the earth.
The earth All this must have taken place early in the history of the earth, and lunar to which we now return. As the month increases in length the orbit. lunar orbit becomes eccentric, and the eccentricity reaches a maxi mum when the month occupies about a rotation and a half of the earth. The maximum of eccentricity is probably not large. After this the eccentricity diminishes. The plane of the lunar orbit is at first practically identical with the earth's equator, but as the moon recedes from the earth the sun's attraction begins to make itself felt. We must therefore introduce the conception of two ideal planes (here called the proper planes), to which the motion of the earth and moon must be referred. The lunar proper plane is at first inclined at a very small angle to the earth's proper plane, and the orbit and equator coincide with their respective proper planes. As soon as the earth rotates with twice the angular velocity with which the moon revolves in her orbit, a new instability sets in. The month is then about twelve of our present hours, and the day about six such hours in length. The inclinations of the lunar orbit and of the equator to their respective proper planes increase. That of the lunar orbit to its proper plane increases to a maximum of 6 or 7, and ever after diminishes, that of the equator to its proper plane increases to a maximum of about 2 45 . and ever after diminishes. The maximum inclination of the lunar orbit to its proper plane takes place when the day is a little less than nine of our present hours, and the month a little less than six of our present days. The maximum inclination of the equator to its proper plane takes place earlier than this. Whilst these changes have been going on the proper planes have been themselves changing in their positions relatively to one another and to the ecliptic. At first they were nearly coincident with one another and with the earth's equator, but they then open out, and the inclination of the lunar proper plane to the ecliptic continually diminishes, whilst that of the terrestrial proper plane continually increases. At some stage the earth became more rigid, and oceans were formed, so that oceanic tidal friction probably came to play a more important part than bodily tidal friction. If this be the case, the eccentricity of the orbit, after passing through a stationary phase, begins to increase again. We have now traced the system to a state in which the day and month are increasing, but at unequal rates, the inclination of the lunar proper plane to the ecliptic and of the orbit to the proper plane are diminishing, the inclination of the terrestrial proper plane to the ecliptic is increasing and of the equator to its proper plane is diminishing, and the eccentricity of the orbit is increasing. No new phase now supervenes and at length we have the system in its present configuration. The minimum time in which the changes from first to last can have taken place is 54,000,000 years.
Distor- There are other collateral results which must arise from a suption of posed primitive viscosity or plasticity of the earth's mass. For plastic during this course of evolution the earth's mass must have suffered planet, a screwing motion, so that the polar regions have travelled a little from west to east relatively to the equator. This affords a possible explanation of the north and south trend of our great continents. Also a large amount of heat has been generated by friction deep down in the earth; and some very small part of the observed in crease of temperature in underground borings may be attributable to this cause. The preceding history might vary a little in detail according to the degree of viscosity which we attribute to the earth's mass, and according as oceanic tidal friction is or is not, now and in the more recent past, a more powerful cause of change than bodily tidal friction. The argument reposes on the imperfect rigidity of solids and on the internal friction of semi-solids and The fluids; these are verse causes. Thus changes of the kind here distheory cussed must be going on, and must have gone on in the past. And postu- for this history of the earth and moon to be true throughout, it is lates suf- only necessary to postulate a sufficient lapse of time, and that there ficieut is not enough matter diffused through space to materially resist lapse of the motions of the moon and earth in perhaps 200,000,000 years, time. It seems hardly too much to say that, granting these two postu lates, and the existence of a primeval planet, such as that above described, a system would necessarily be developed which would bear a strong resemblance to our own. A theory, reposing on verse causse, which brings into quantitative correlation the lengths of the present day and month, the obliquity of the ecliptic, and the inclination and eccentricity of the lunar orbit should have claims to acceptance.
§ 51. The Other Planetary Subsystems.
Other If this has been the evolution of the earth and moon, a similar planet- process must have been going on elsewhere. So far we have only ary sub- considered a single satellite and the sun, but the theory may of systems, course be extended, with modifications, to planets attended by several satellites. We will now, therefore, consider some of the other members of the solar system. A large planet has much more energy of rotation to be destroyed, and moment of momentum to be redistributed, than a small one, and therefore a large planet ought to proceed in its evolution more slowly than a small one. Therefore we ought to find the larger planets less advanced than the smaller ones. The masses of such of the planets as have satel lites are, in terms of the earth's mass, as follows: Mars = |; Jupiter = 340; Saturn = 100; Uranus = 17; Neptune = 20.
Mars should therefore be furthest advanced in its evolution, and Mars, it is here alone in the whole system that we find a satellite moving orbitally faster than the planet rotates. This will also be the ultimate fate of our moon, because, after its orbital motion has been reduced to identity with that of the earth's rotation, solar tidal friction will further reduce the earth's angular velocity; the tidal reaction on the moon will then be reversed, and the moon's orbital velocity will increase and her distance from the earth diminish. But, since the moon's mass is very large, she must recede to an enormous distance from the earth before this reversal takes place. Now the satellites of Mars are very small, and therefore they need only recede a very short distance from the planet before the reversal of tidal reaction. The periodic time of the satellite Deimos is 30 h 18 m, and, as the period of rotation of Mars is 24 h 37 m, Deimos must be still receding from Mars, but very slowly. The periodic time of the satellite Phobos is 7 h 39 m; therefore it must be approaching Mars. It does not seem likely that it has ever been remote from the planet.[2] The eccentricities of the orbits of both satellites are small: that of Deimos is 0057 and that of Phobos "0066. If the viscosity of the planet be small, or if oceanic tidal friction be the principal cause of change, both eccentricities are diminishing; but, if the viscosity be large, both are increasing. As we have no means of knowing whether the eccentricities are increasing or diminishing, the larger eccentricity of the orbit of Phobos cannot be a fact of much importance either for or against the present views. But it must be admitted that it is a slightly unfavourable indication. The position of the proper plane of a satellite is determined by the periodic time of the satellite, the oblateness of the planet, and the sun's distance. The inclination of the orbit of a satellite to the proper plane is not determined by anything in the system. Hence it is only the inclination of the orbit which can afford any argument for or against the theory. The proper planes of both satellites are necessarily nearly coincident with the equator of the planet; but it is in accordance with the theory that the inclinations of the orbits to their respective proper planes should be small. Any change in the obliquity of the equator of Mars to the plane of his orbit must be entirely due to solar tides. The present obliquity is about 30, and this points also to an advanced stage of evolution, at least if the axis of the planet was primitively at all nearly per pendicular to the ecliptic.
We now come to the system of Jupiter. This, enormous planet Jupiter is still rotating in about ten hours; its axis is nearly perpendicular to the ecliptic; and three of its satellites revolve in seven days or less, whilst the fourth has a period of 16 d 16 h . This system is obviously far less advanced than our own. The inclinations of the proper planes to Jupiter's equator are necessarily small, but the inclinations of the orbits to the proper planes appear to be very interesting from a theoretical point of view. They are in the case of the first satellite 0", in the case of the second 27 50", in that of the third 12 20", and in that of the fourth 14 58". We have shown above that the orbit of a satellite is first coincident with its proper plane, and that the inclination afterwards rises to a maximum and finally declines. If then we may assume, as seems reasonable, that the satellites are in stages of evolution corresponding to their distances from the planet, these inclinations accord well with the theory. The eccentricities of the orbits of the two inner satellites are insensible, those of the outer two small. This does not tell strongly either for or against the theory, because the history of the eccentricity depends considerably on the nature of the friction to which the tides are subject. Yet it on the whole agrees with the theory that the eccentricity should be greater in the more remote satellites. It appears that the satel lites of Jupiter always present the same face to the planet, just as does our moon. This was to be expected.
The case of Saturn is not altogether so favourable to the theory. Saturn. The extremely rapid rotation, the ring, and the short periodic time of the inner satellites point to an early stage of development; whilst the longer periodic time of the three outer satellites and the high obliquity of the equator indicate a later stage. Perhaps both views may be more or less correct, for successive shedding of satellites would impart a modern appearance to the system. It has probably been previously remarked that the Saturnian system bears a strong analogy to the solar system, Titan being analogous to Jupiter, Hyperion and lapetus to Uranus and Neptune, and the inner satel lites to the inner planets. Thus anything which aids us in forming a theory of the one system will throw light on the other. The details of the Saturnian system seem to be more or less favourable to the theory. The proper planes of the orbits (except that of lapetus) are nearly in the plane of the ring, and the inclinations of all the orbits thereto appear not to be large. As the result of a careful series of observations made at Washington in 1873, Prof. Asaph Hall l finds that the eccentricities of the orbits of Mimas, Enceladus, Tethys, Dione, and Rhea are insensible, that of Titan is -0284, of Hyperion 1000, and that of lapetus -0278. The satel lite lapetus appears always to present the same face to the planet.
Uranus Concerning Uranus and Neptune there is not much to be said, and as their systems are veiy little known; but their masses are much Neptune, larger than that of the earth, and their satellites revolve with a short periodic time. The retrograde motion and high inclination of the satellites of Uranus are very remarkable. The theory of the inclination of the orbit has been based on an assrumed smallness of inclination, and it is not very easy to see to what results investi gation might lead if the inclination were large. It must be ad mitted, however, that the Uranian system points to the probability of the existence of a primitive planet, with retrograde rotation, or at least with a very large obliquity of equator.
It appears from this review that the other members of the solar system present some phenomena which are strikingly favourable to the tidal theory of evolution, and none which are absolutely con demnatory. We shall show in the following section that there are reasons why the tidal friction arising in the planetary systems cannot have had so much effect as in the case of the earth and moon. That the indications which we have just noted were not more marked, but yet seemed to exist, agrees well with this conclusion.
§ 52. Influence of Tidal Friction on the Evolution of the Solar System.
Solar According to the nebular hypothesis, the planets and the satellites system are portions detached from contracting nebulous masses. In the as a following discussion that hypothesis will be accepted in its main whole, outline, and we shall examine what modifications are necessitated by the influence of tidal friction. It may be shown that the reaction of the tides raised in the sun by the planets must have had a very small influence in changing the dimensions of the planetary orbits round the sun. From a consideration of numerical data with regard to the solar system and the planetary subsystems, it appears improbable that the planetary orbits have been sensibly enlarged by tidal friction since the origin of the several planets. But it is possible that some very small part of the eccentricities of Planet- the planetary orbits is due to this cause. From arguments similar ary sub- to those advanced with regard to the solar system as a whole, it systems, appears unlikely that the satellites of Mars, Jupiter, and Saturn originated very much nearer the present surfaces of the planets than we now observe them. But, the data being insufficient, we cannot feel sure that the alteration in the dimensions of the orbits of these satellites has not been considerable. It remains, however, nearly certain that they cannot have first originated almost in con tact with the present surfaces of the planets, in the same way as in the preceding sketch ( 50) has been shown to be probable with regard to the moon and earth. Numerical data concerning the distribution of moment of momentum in the several planetary sub-systems exhibit so striking a difference between the terres trial system and those of the other planets that we should from this alone have grounds for believing that the modes of evolution have been considerably different. The difference appears to lio in the genesis of the moon close to the present surface of tho planet, and we shall see below that solar tidal friction may be as signed as a reason to explain how it has happened that the terres trial planet had contracted to nearly its present dimensions before the genesis of a satellite, but that this was not the case with the exterior planets. The efficiency of solar tidal friction is very much greater in its action on the nearer planets than on the further ones. The time, however, during which solar tidal friction has been operating on the external planets is probably much longer than the period of its efficiency for the interior ones, and a series of numbers proportional to the total amount of rotation destroyed in the several planets would present a far less rapid decrease as we recede from the sun than numbers simply expressive of the efficiency of tidal friction at the several planets. Nevertheless it must be admitted that the effect produced by solar tidal friction on Jupiter and Saturn has not been nearly so great as on the interior planets. And, as already stated, it is very improbable that so large an amount of momentum should have been destroyed as to materially affect the orbits of the planets round the sun.
Distribu- We will now examine how the differences of distance from the tion of sun would be likely to affect the histories of the several planetary satellites masses. According to the nebular hypothesis, a planetary nebula amongst contracts, and rotates quicker as it contracts. The rapidity of the the revolution causes it to become unstable, or perhaps an equatorial planets, belt gradually detaches itself; it is immaterial which of these two really takes place. In either case the separation of that part of the mass which before the change had the greatest angular momentum permits the central portion to resume a planetary shape. The contraction and the increase of rotation proceed continually until another portion is detached, and so on. There thus recur at inter vals a series of epochs of instability or of abnormal change. Now
See Brit. Assoc. Report, 1886, p. 543.
tidal friction must diminish the rate of increase of rotation due to contraction, and therefore if tidal friction and contraction are at work together the epochs of instability must recur more rarely than if contraction alone acted. If the tidal retardation is suffi ciently great, the increase of rotation due to contraction will be so far counteracted as never to permit an epoch of instability to occur. Since the rate of retardation due to solar tidal friction decreases rapidly as we recede from the sun, these considerations accord with what we observe in the solar system. For Mercury and Venus have no satellites, and there is a progressive increase in the number of satellites as we recede from the sun. Moreover, the number of satellites is not directly connected with the mass of the planet, for Venus has nearly the same mass as the earth and has no satellite, and the earth has relatively by far the largest satellite of the whole system. Whether this be the true cause of the observed distribu tion of satellites amongst the planets or not, it is remarkable that the same cause also affords an explanation, as we shall now show, of that difference between the earth with the moon and the other planets with their satellites which has caused tidal friction to be the principal agent of change with the former but not with the Case of latter. In the case of the contracting terrestrial mass we may earth and suppose that there was for a long time nearly a balance between moon the retardation due to solar tidal friction and the acceleration different due to contraction, and that it was not until the planetary mass from had contracted to nearly its present dimensions that an- epoch others. of instability could occur. It may also be noted that if there be two equal planetary masses which generate satellites, but under very different conditions as to the degree of condensation of the masses, the two satellites will be likely to differ in mass; we cannot of course tell which of the two planets would generate the larger satellite. Thus, if the genesis of the moon was deferred until a late epoch in the history of the terrestrial mass, the mass of the moon relatively to the earth would be likely to differ from the mass of other satellites relatively to their planets. If the contraction of the planetary mass be almost completed before the genesis of the satellite, tidal friction, due jointly to the satellite and to the sun, will thereafter be the great cause of change in the system; and thus the hypothesis that it is the sole cause of change will give an approximately accurate explanation of the motion of the planet and satellite at any subsequent time. We have already seen that the theory that tidal friction has been the ruling power in the evolution of the earth and moon coordinates the present motions of the two bodies and carries us back to an initial state when the moon first had a separate existence as a satellite; and the initial configuration of the two bodies is such that we are led to believe that the moon is a portion of the primitive earth detached by rapid rotation or other causes. There seems to be some reason to suppose that the earliest form in which the moon had a separate existence was as a ring or chain of meteorites; but this condition precedes that to which the dynamical investigation leads back.
Let us now turn to the other planetary sub-systems. The satellites of the larger planets revolve with short periodic times; this admits of a simple explanation, for the smallness of their masses would have prevented tidal friction from being a very efficient cause of change in the dimensions of their orbits, and the largeness of the planet's masses would have caused them to proceed slowly in their evolution. If the planets be formed from chains of meteorites or of nebulous matter, their rotation has arisen from the excess of orbital momentum of the exterior over that of the interior matter. As we have no means of knowing how broad the chain may have been in any case, nor how much it may have closed in on the sun in course of concentration, we are unable to compute the primitive angular momentum of a planet. A rigorous method of comparison of the primitive rotations of the several planets is thus wanting. If, however, the planets were formed under similar conditions, then we should expect to find the exterior planets now rotating more rapidly than the interior ones. On making allowance for the differ ent degrees of concentration of the planets, this is the case. That the inner satellite of Mars revolves with a period of less than a third of the planet's rotation is perhaps the most remarkable fact Satelin the solar system. The theory of tidal friction explains this lites of perfectly; and this will be the ultimate fate of all satellites, be- Mars, cause the solar tidal friction retards the planetary rotation without directly affecting the satellite's orbital motion. Numerical com parison shows that the efficiency of solar tidal friction in retarding the terrestrial and martian rotations is of about the same degree of importance, notwithstanding the much greater distance of tho planet Mars. In the above discussion it will have been apparent that the earth and moon do actually differ from the other planets to such an extent as to permit tidal friction to have been the most important factor in their history.
By an examination of the probable effects of solar tidal friction Sumon a contracting planetary mass, we have been led to assign a rnary. cause for the observed distribution of satellites in the solar system, and this again has itself afforded an explanation of how it happened that the moon so originated that the tidal friction of the lunar tides in the earth should have been able to exercise so large an influence. We have endeavoured not only to set forth the influence which tidal friction may have, and probably has, had in the history of the system, if sufficient time be granted, but also to point out what effects it cannot have produced. These investigations afford no grounds for the rejection of the nebular hypothesis; but, while they present evidence in favour of the main outlines of that theory, they introduce modifications of considerable importance. Tidal friction is a cause of change of which Laplace's theory took no account; and, although the activity of that cause may be regarded as mainly belonging to a later period than the events described in the nebular hypothesis, yet it seems that its influence has been of great, and in one instance of even paramount importance in determining the present condition of the planets and their satellites. Throughout the whole of this discussion it has been supposed that sufficient time is at our disposal; Sir W. Thomson and others have, however, adduced reasoning which goes to show that the history of the solar system must be comprised within a period considerably less than a hundred million years.[3] Limitation of time. It would perhaps be premature to accept this as the final and definite conclusion of science. If, however, it be confirmed, we shall only be permitted to accept the doctrine that tidal friction has effected considerable modification in the configuration of the moon and earth, and must reject the earlier portion of the history sketched above. (G. H. D.)