Gravitational Collapse for Polytropic Gaseous Stars: Self-Similar Solutions

Abstract

AbstractIn the supercritical range of the polytropic indices $$\gamma \in (1,\frac{4}{3})$$ γ ∈ ( 1 , 4 3 ) we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler–Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson–Penston collapsing solutions in the isothermal case $$\gamma =1$$ γ = 1 . They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof.