The Stability of Relativistic, Spherically Symmetric Star Clusters

Citation

Ipser, James Reid (1969) The Stability of Relativistic, Spherically Symmetric Star Clusters. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/GF6C-JN05. https://resolver.caltech.edu/CaltechETD:etd-10152002-160828

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. It has been suggested that very dense star clusters might play important roles in quasi-stellar sources and in the nuclei of certain galaxies, where violent events occur. Such star clusters should become unstable against relativistic gravitational collapse when, in the course of evolution, they contract down to a certain critical density. In this thesis the study of the relativistic instability which triggers such collapse is initiated: The theory of the stability of a spherically symmetric star cluster against small radial perturbations is developed within the framework of general relativity. Collisions between stars in the cluster are neglected, since in realistic situations the time scale for collisions should be much greater than the time scale for the growth of the relativistic instability. The equation of motion governing the small radial perturbations of a spherical cluster is derived and is shown to be self-conjugate. Associated with the equation of motion is a dynamically conserved quantity, and a multidimensional variational principle for the normal modes of radial pulsation. The variational principle provides a necessary and sufficient criterion for the stability of the cluster. Also derived are much simpler, one-dimensional, sufficient (but not necessary) criteria for stability. The most important sufficient criterion is this: A relativistic, spherical cluster is stable against radial perturbations if the gas sphere with the same distributions of density and pressure is stable against radial perturbations with adiabatic index [Gamma][subscript 1] = ([rho] + p)p[subscript -1](dp/dr) (d[rho]/dr)[superscript -1]. The stability criteria are used to diagnose numerically the stability of (i) clusters of identical stars with heavily-truncated Maxwell-Boltzmann velocity distributions, and (ii) clusters whose densities and isotropic pressures obey polytropic laws of index 2 or 3. The calculations show that a cluster of either type is unstable against collapse if the redshift of a photon emitted from its center and received at infinity is z[subscript c][...] 0.5. The cluster is stable if z[subscript c][...]0.5. For purposes of motivation, two new theorems on the theory of the stability of highly relativistic stars (not star clusters!) are also presented in this thesis. The first theorem states that a highly relativistic, spherical star is stable if and only if its adiabatic index (assumed to be constant in the interior regions) is greater than a certain critical value, [gamma][subscript crit], which depends in a specified way on the high-density equation of state. Because of relativistic effects this critical value is somewhat larger than the Newtonian value [gamma][subscript crit] = 4/3. The second theorem shows that, at high central densities, the curves of - (binding energy) versus radius for certain hot, isentropic sequences of stellar models must exhibit damped clockwise spirals. This spiraling reflects the onset of instability in one radial mode of pulsation after another as the central density increases along the sequence.

Thesis Files