V. The gravitational stability of the Earth

Abstract

If in a gravitating body there occurs a displacement which involves alteration of density, there must be a tendency for the material to move towards the places where the density is increased, and away from the places where the density is diminished. The effect of this tendency, if it were not held in check, would be to accentuate local alterations of density. In any body the tendency is partially held in check by the elasticity of the body, and, in particular, by the elastic resistance which the body offers to compression. If this resistance is sufficiently great, the body is stable, in spite of the tendency to instability which arises from gravitation. It is important to determine the conditions of stability for bodies of various forms and constitutions, with various distributions of density. The problem of the stability of spherically symmetrical configurations of a quantity of gravitating gas has been investigated by J. H. Jeans, and he has drawn from his investigations some interesting conclusions in regard to the course of evolution of stellar and planetary systems. In a subsequent memoir he proceeded to investigate a similar problem in regard to gravitating bodies of a more coherent character. A gravitating solid body, such as a planet may be conceived to he, might exist in a spherical shape with a spherically symmetrical distribution of density. In the absence of gravitation there could he no question of instability. The effect of any local condensation would be to set up vibrations, and the frequency of the vibration of any spherical harmonic type would depend upon the elasticity of the material. If the resistance of the material to compression is sufficiently high the stability persists in spite of gravitation. There are thus two competing agencies: gravitation, tending to instability, and the elasticity of the material, tending to stability. In a general way it is clear that, as the elasticity diminishes, the frequency of vibration of any type also diminishes; and, if the frequency can vanish for sufficiently small elasticity, the planetary body possessing such elasticity cannot continue to exist in the spherically symmetrical configuration. The problem is to determine the conditions as regards elasticity in which the instability occurs. A grave difficulty presents itself at the outset. In the equilibrium configuration the gravitating planet is in a state of stress; and, in a body of such dimensions as the Earth, this stress is so great that the total stress existing in the body when it vibrates cannot be calculated by the ordinary methods of the theory of elasticity. In that theory it is ordinarily assumed that the body under investigation is in a state so little removed from one of zero stress that the strain, measured from this state as a zero of reckoning, is proportional to the stress existing at any instant. In order that this assumption may he valid, it is necessary that the strain which is calculated by means of it should be so small that its square may be neglected. Now if we apply the equations of the ordinary theory to the problem of a solid sphere strained by its own gravitation, and if we take the sphere to he of the same size and mass as the Earth, and the material of which it is composed to possess moduluses of elasticity as great as those of ordinary steel, we find that the strains may be as great as and thus the strains are much too great for the assumption to he valid. The initial stress existing in the gravitating planet, the stress by which the self-attraction of the body is equilibrated, is much too great to perm it of the application of the ordinary theory. The same difficulty presents itself in every problem concerning the elasticity of a gravitating planet, for example, in the problem of tidal deformation or of the stress produced in the interior by the w eight of continents. In these problems the difficulty was turned by Lord Kelvin and Sir G. H. Darwin by taking the modulus of compression to be much greater than that of any known material, in other words, by taking the material to be incompressible. Their object was to determine the degree of rigidity which must be assigned to the Earth , and for that object it is permissible to turn the difficulty in this way. When the problem is that of gravitational instability this artifice cannot be adopted, because the whole question is that of the degree of compressibility which is admissible if the gravitating planet is to be stable in a spherically symmetrical configuration. The artifice adopted by Jeans (1903) consisted in annulling the initial stress by introducing an imagined external field of force to equilibrate the self-attraction of the planet.