Elastodynamics of self-gravitating matter: Nonradial vibrations of a star modeled by a heavy spherical mass of an elastic solid

The continuum dynamics of self-gravitating elastic substance is modeled by the closed system of elastodynamic equations and Poisson's equation of the Newtonian gravity. Instead of the Lame's equation, which describes small-amplitude vibrations of an isotropic elastic solid, the equations of the elastodynamics are introduced as a natural extension of the hydrodynamic equations: the continuity equation for the bulk density and Euler's equation for the velocity held are supplemented by the equation for the tensor of elastic stresses. The emphasis is placed on the study of nonradial spheroidal and torsional `gravitation-elastic vibrations of a star modeled by a heavy spherical mass of a perfectly elastic substance. It is found that eigenfrequencies of spheroidal vibrations are given by omega(s)(2) = omega(G)(2)[2(3L + 1)(L - 1)/(2L + 1)]; the torsional gravitation-elastic modes are found to be omega(t)(2) = omega(G)(2)(L - 1), where omega(G)(2) = 4 pi G rho(0)/3 is the basic frequency for the star with uniform equilibrium density po and where G denotes the gravitational constant. To reveal similarities and differences between the seismology of stars with elastodynamic and fluid-dynamic properties of medium, the vibrational dynamics of a self-gravitating elastic globe is considered in juxtaposition with Kelvin's theory for the small-amplitude oscillations of a heavy spherical drop of an incompressible inviscid liquid.