Si. Bastrukov, ELASTODYNAMICS OF SELF-GRAVITATING MATTER - NONRADIAL VIBRATIONS OF ASTAR MODELED BY A HEAVY SPHERICAL MASS OF AN ELASTIC SOLID, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 53(2), 1996, pp. 1917-1922
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.