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

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.