Dynamical stability of infinite homogeneous self-gravitating systems: application of the Nyquist method

TL;DR

This paper analyzes the dynamical stability of infinite homogeneous self-gravitating systems using the Nyquist method, comparing stellar and gaseous systems, and providing analytical expressions for growth and damping rates.

Contribution

It applies the Nyquist method to determine stability of homogeneous stellar systems and gaseous media, extending classical results to symmetric and asymmetric velocity distributions.

Findings

Onset of instability is the same for stellar and gaseous systems with single-humped distributions.

Analytical expressions for growth, damping rates, and pulsation periods are derived for isothermal and polytropic distributions.

Stability diagrams are established for two-stream stellar systems and compared with plasma and gaseous systems.

Abstract

We complete classical investigations concerning the dynamical stability of an infinite homogeneous gaseous medium described by the Euler-Poisson system or an infinite homogeneous stellar system described by the Vlasov-Poisson system (Jeans problem). To determine the stability of an infinite homogeneous stellar system with respect to a perturbation of wavenumber k, we apply the Nyquist method. We first consider the case of single-humped distributions and show that, for infinite homogeneous systems, the onset of instability is the same in a stellar system and in the corresponding barotropic gas, contrary to the case of inhomogeneous systems. We show that this result is true for any symmetric single-humped velocity distribution, not only for the Maxwellian. If we specialize on isothermal and polytropic distributions, analytical expressions for the growth rate, damping rate and pulsation period of the perturbation can be given. Then, we consider the Vlasov stability of symmetric and asymmetric double-humped distributions (two-stream stellar systems) and determine the stability diagrams depending on the degree of asymmetry. We compare these results with the Euler stability of two self-gravitating gaseous streams. Finally, we determine the corresponding stability diagrams in the case of plasmas and compare the results with self-gravitating systems.