While the doctrine of tidal retardation is theoretically sound, and while the relations of the moon to the earth have probably been appreciably affected by tidal action, geological evidence indicates that it has not been sufficiently effective in producing crustal deformations to be clearly detected by its own distinctive results. This may be due (1) to the fact that there are compensating agencies that tend to acceleration of rotation, and (2) to the probable fact that the central rigidity of the earth is too high to give a very effective body-tide. Hence the process of retardation may have been too slow to have been geologically appreciable in the known period. The recent estimates of the effective rigidity of the earth are greater than former ones, and they may need to be modified yet further in the same direction.
Distribution of rigidity.—An important consideration in this connection is the distribution of interior rigidity. It is certain that the rigidity of the outermost part, taken as a mass, is somewhat less than that of rock of an average surface type, for it is fissured, and there is no reason to suppose that the rigidity of the rock next below the fissure zone rises at once to the rigidity of steel, and hence if the average rigidity of the whole earth is equal to that of steel, a portion of the interior must have a rigidity much higher than steel. There is probably some law of increase from surface to center, and there are theoretical grounds for thinking that it is in some way connected with the laws of pressure, density, compressibility, and temperature. All of these factors probably affect rigidity, but in different ways. The modulus of rigidity of steel is about 770 × 106 grms. per sq. cm. Milne and Gray[269] found that of granite to be 128 × 106. The ratio of the rigidity of steel to that of rock is, therefore, about 6 : 1. If it be assumed that the rigidity increases in depth directly as the density, the rigidity will nowhere reach that of steel, being only about two-thirds as much at the center. If it be assumed that the rigidity increases as the squares of the density ratios, the following values are obtained:
| Distances from center in terms of radius. | Densities under Laplace’s law. | Density ratios. | Density ratios squared. | Deduced rigidities. |
|---|---|---|---|---|
| 1.00 | 2.8 | 1 | 1 | 0.16 Steel |
| .75 | 5.7 | 2 | 4 | 0.6 ” |
| .50 | 8.39 | 3 | 9 | 1.5 ” |
| .25 | 10.27 | 3.7 | 13.7 | 2.3 ” |
| .00 | 10.95 | 3.9 | 15.2 | 2.5 ” |
These values seem fairly consistent with the apparent requirements of the case.
If the distribution of rigidity were of this nature, the average rigidity would be much less than that of steel, for more than half the volume lies in the outer division, between 1.00 and .75 radius, and yet the effective resistance to tidal deformation would be high, for, according to G. H. Darwin,[270] the tidal stress-differences are eight times as great in the center as at the surface. The rigidity would, therefore, be distributed so as to be much more effective in resistance than if it were uniform. The suggestion arises here that the tidal stresses and other analogous stresses arising from astronomical sources may be in themselves the causes of some such distribution of rigidity as this. The tidal stresses are rhythmical and give rise to a kind of kneading of the body of the earth, small in measure to be sure, but persistent and rapidly recurrent. Since these stress-differences at the center are eight times those at the surface, and since also the gravitative stress at the center is 3,000,000 times that at the surface, there is a series of persistently recurring stress-differences, greatest at the center and declining outwards, superposed on enormous static stresses, also intensest at the center and declining outwards. Now, if the earth material were once made up of a mixture of minerals of different fusibility, some of which became more mobile (whether fluid or viscous) than others under the rising temperature of the interior, it seems that the more mobile portion must have tended to move from the regions of greater stress-differences to those of lesser stress-differences. The persistence and the rhythmical nature of the tidal stress-differences seem well suited to aid the mobile parts in gradually working their way outwards. At the same time the more solid and resistant portions should remain behind, and thus come to constitute the dominant material of the central regions where stress-differences were greatest, and so, as it were, concentrate rigidity there. The process may still be in action.
If it be assumed that the rhythmical stresses have thus developed a resistance to deformation proportional to their intensity, we may combine this with density to form the basis of another hypothetical distribution of rigidity, as follows:
| Distances from center in terms of radius. | Densities under Laplace’s law. | Density ratios. | Ratios adjusted to stress-differences. 1:8) | Deduced rigidities. |
|---|---|---|---|---|
| 1.00 | 2.8 | 1 | 1 | 0.16 Steel |
| .75 | 5.7 | 2 | 3.5 | 0.58 ” |
| .50 | 8.39 | 3 | 5.4 | 0.90 ” |
| .25 | 10.27 | 3.7 | 7 | 1.16 ” |
| .00 | 10.95 | 3.9 | 8 | 1.33 ” |
The average rigidity is here also much less than that of steel, but its distribution is such as to render it ideally fitted to resist tidal distortion.
These hypothetical distributions of rigidity have no claims to special value in themselves, for the grounds on which they are based are quite inadequate, but they are not without importance in giving tangible form to considerations that bear vitally not only on tidal problems, but on many others connected with the internal constitution and dynamics of the earth.