On the derivations of the Equation of Hydrostatic Equilibrium

TL;DR

This paper provides both geometrical and analytical derivations of the hydrostatic equilibrium equation for stars, highlighting errors in existing textbook derivations and offering a more accurate formulation.

Contribution

It introduces a new derivation based on the Navier-Cauchy equation and critically analyzes textbook methods, correcting common errors.

Findings

Identifies errors in textbook derivations of hydrostatic equilibrium.

Provides a geometrical derivation for spherically symmetric stars.

Offers an analytical derivation using the Navier-Cauchy equation.

Abstract

We develop and present a geometrical and an analytical derivation of the equation of hydrostatic equilibrium in spherically symmetric stars with a generalized stress tensor. The analytical derivation is based on the Navier-Cauchy equation. We also critically examine the derivation of this equation found in textbooks on stellar astrophysics and show that there are errors in many of the derivations presented in textbooks.