The Fundamental Equilibrium Equation For Gaseous Stars And The Tolman-Oppenheimer-Volkoff Equation -- Derivations And Applications With Emphasis On Optimisational-Variational Methods

TL;DR

This paper reviews the fundamental differential equations governing gaseous star equilibrium, including Newtonian and relativistic forms, emphasizing various derivation methods especially variational approaches, and discusses their astrophysical applications.

Contribution

It provides a comprehensive overview of derivation techniques for stellar equilibrium equations, highlighting variational methods and their applications in astrophysics.

Findings

Derivation of the Newtonian equilibrium equation via multiple methods

Derivation of the Tolman-Oppenheimer-Volkoff equation from various principles

Discussion of applications and astrophysical implications of these equations

Abstract

Stars are essentially gravitationally stabilised thermonuclear reactors in hydrostatic equilibrium. The fundamental differential equation for all Newtonian gaseous stars in equilibrium is \begin{align} \frac{dp(r)}{dr}=-\frac{\mathscr{G}\mathcal{M}(r)\rho(r)}{r^{2}}\nonumber \end{align} where $p(r),\rho(r)$ are the pressure, density at radius $r$ and $\mathcal{M}(r)$ is the mass contained within a shell of radius $r$ given by $\mathcal{M}(r)=\int_{0}^{r}4\pi \overline{r}^{2} \rho(\overline{r})d\overline{r}$, and $\mathscr{G}$ is Newton's constant. This simple but crucial differential equation for the pressure gradient within any star, underpins much of astrophysical theory and it can derived by various methods:via a simple heuristic argument; via the Euler-Poisson equations for a self-gravitating fluid/gas; via a variational method by taking the 1st variation of the sum of the thermal and gravitational energies of the star; via the 2nd variation of the Massiue thermodynamic functional for a self-gravitating isothermal perfect-gas sphere; from conservation of the virial tensor; as the non-relativistic limit of the Tolman-Oppenheimer-Volkoff equation (TOVE). The TOVE for equilibrium of relativistic stars in general relativity can in turn be derived by various methods: from the energy-momentum conservation constraint on the Einstein equations applied to a spherically symmetric perfect fluid/gas; via a constrained optimization method on the mass and nucleon number; via a maximum entropy variational method for a sphere of self-gravitating perfect fluid/gas or radiation. An overview is given of all derivations with emphasis on variational methods. Many important applications and astrophysical consequences of the Newtonian equilibrium equation are also reviewed.