ELASTODYNAMICS OF SELF-GRAVITATING MATTER - NONRADIAL VIBRATIONS OF ASTAR MODELED BY A HEAVY SPHERICAL MASS OF AN ELASTIC SOLID

The continuum dynamics of self-gravitating elastic substance is modele d by the closed system of elastodynamic equations and Poisson's equati on of the Newtonian gravity. Instead of the Lame's equation, which des cribes small-amplitude vibrations of an isotropic elastic solid, the e quations of the elastodynamics are introduced as a natural extension o f the hydrodynamic equations: the continuity equation for the bulk den sity and Euler's equation for the velocity held are supplemented by th e equation for the tensor of elastic stresses. The emphasis is placed on the study of nonradial spheroidal and torsional `gravitation-elasti c vibrations of a star modeled by a heavy spherical mass of a perfectl y elastic substance. It is found that eigenfrequencies of spheroidal v ibrations 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 w here G denotes the gravitational constant. To reveal similarities and differences between the seismology of stars with elastodynamic and flu id-dynamic properties of medium, the vibrational dynamics of a self-gr avitating elastic globe is considered in juxtaposition with Kelvin's t heory for the small-amplitude oscillations of a heavy spherical drop o f an incompressible inviscid liquid.