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

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.