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. Trans., 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 diameter, and probably partly solid, partly fluid, and partly gaseous. It is rotating about an axis inclined at about 11° or 12° to the normal to the ecliptic, with a period of from two to four hours and is revolving about the sun with a period not much shorter than our present year. The rapidity of the planet’s rotation causes so great 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 meteorites, which were once part of the primeval planet. Up to this point the history is largely speculative, for the conditions of instability of a rotating mass of fluid have not yet been fully investigated. 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. [Footnote 378-1] 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 of 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 acquire 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 earth on the tides raised in the moon, and this retardation takes place at a far greater rate than the similar retardation of the earth’s rotation. As soon as the moon rotates round her axis with twice the angular velocity with which she revolves in her orbit, the position of her axis of rotation (parallel with the earth’s axis) becomes dynamically unstable. The obliquity of the lunar equator to the plane of the orbit increases, attains a maximum, and then diminishes. Meanwhile the lunar axial rotation is being reduced towards identity with the orbital motion. Finally, her equator is nearly coincident with the plane of the orbit, and the attraction of the 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.

All this must have taken place early in the history of the earth, to which we now return. As the month increases in length the lunar orbit becomes eccentric, and the eccentricity reaches a maximum 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.

There are other collateral results which must arise from a supposed primitive viscosity or plasticity of the earth’s mass. For during this course of evolution the earth's mass must have suffered 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 increase 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 fluids; these are verae causae. Thus changes of the kind here discussed must be going on, and must have gone on in the past. And for is history of the earth and moon to be true throughout, it is only necessary to postulate a sufficient lapse of time, and that there is not enough matter diffused through space to materially resist the motions of the moon and earth in perhaps 200,000,000 years. It seems hardly too much to say that, granting these two postulates, 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 verae causae, 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.


Footnotes

378-1 See criticisms by Mr Nolan, Genesis of Moon, Melbourne, 1885; also Nature, 18th February 1886.






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