Virial theorem for a cloud of stars obtained from Jeans equations with the second correlation moments

TL;DR

This paper develops a hydrodynamic model for star clouds incorporating gravitational correlations, deriving a virial theorem consistent with stability conditions and identifying acoustic oscillation modes.

Contribution

It introduces a novel model including second correlation moments of gravitational fields and derives a virial theorem from Einstein equations in a star cloud context.

Findings

Derived a non-relativistic equation for gravitational field correlations.

Identified three acoustic oscillation modes in star clouds.

Established a critical radius for cloud stability.

Abstract

A hydrodynamic model for small acoustic oscillations in a cloud of stars is built, taking into account the self-consistent gravitational field in equilibrium with a non-zero second correlation moment. It is assumed that the momentum flux density tensor should include the analog of the anisotropic pressure tensor and the second correlation moment of both longitudinal and transverse gravitational field strength. The non-relativistic temporal equation for the second correlation moment of the gravitational field strength is derived from the Einstein equations using the first-order post-Newtonian approximation. One longitudinal and two transverse branches of acoustic oscillations are found in a homogeneous and isotropic star cloud. The requirement for the velocity of transverse oscillations to be zero provides the boundary condition for the stability of the cloud. The critical radius of the spherical cloud of stars is obtained, which is precisely consistent with the virial theorem.