Reduction of Stratified Axi-Symmetric Euler-Poisson Equations Under Symmetry

TL;DR

This paper simplifies the complex Euler-Poisson equations governing self-gravitating, rotating, stratified fluids into two manageable equations, enabling analytical solutions that determine the shape of astrophysical objects like stars.

Contribution

It reduces a six-equation nonlinear PDE system to two equations in cylindrical coordinates, providing explicit solutions for the shape of rotating astrophysical bodies.

Findings

Derived analytical solutions for density and gravitational field.

Determined the shape of rotating stars with zero-pressure boundary.

Provided explicit pressure-density relationships.

Abstract

The paper considers Euler-Poisson equations which govern the steady state of a self gravitating, rotating, axi-symmetric fluid under the additional assumption that it is incompressible and stratified. In this setting we show that the original system of six nonlinear partial differential equations can be reduced to two equations, one for the mass density and the other for gravitational field. This reduction is carried out in cylindrical coordinates. As a result we are able to derive also expressions for the pressure as a function of the density. The resulting equations are then solved analytically. These analytic solutions are used then to determine the shape of the rotating star (or interstellar cloud) by applying the boundary condition that the pressure is zero at the boundary.