Two Cases of Radial Adiabatic Motions of a Polytrope with Gamma=4/3

TL;DR

This paper analyzes radial adiabatic motions of a polytropic star with gamma=4/3, deriving solutions that describe limitless expansion, including a special zero-energy case relevant to red giant formation.

Contribution

It derives exact solutions for radial motions of a polytrope with gamma=4/3, revealing two types of expansion and highlighting a zero-energy solution of astrophysical significance.

Findings

Existence of two solution types: positive energy expansion and zero-energy expansion.

The zero-energy solution models a slowly expanding star similar to a red giant.

Density concentrates more at the center than in a stationary star with the same gamma.

Abstract

A self-gravitating sphere of polytropic gas (polytrope) is considered. The system of equations describing radial motions of this sphere in Lagrangian variables reduces to the only nonlinear PDE of the second order in both variables (Lagrangian coordinate and time). The linearization of this PDE leads to the well-known Eddington's equation of the standard model. The case of no energy exchange between the polytrope and the outer medium is considered, that is, polytrope's motions are adiabatic. If gamma (a ratio of the specific heats of the gas) is 4/3 than PDE obtained allows the separation of variables. There exist two types of solutions of the problem both describing limitless expansion without shock wave formation. The first one is an expansion with positive total energy, and the second one is an expansion with zero total energy. The second solution is of an astrophysical interest. It describes the permanently retarding expansion that, perhaps, is akin to a born of a red giant. The stellar density in this case concentrates to the centre of the star stronger than the density of the stationary star with the same gamma.