Larson–Penston Self-similar Gravitational Collapse

Abstract

Using numerical integration, in 1969 Penston (Mon Not R Astr Soc 144:425–448, 1969) and Larson (Mon Not R Astr Soc 145:271–295, 1969) independently discovered a self-similar solution describing the collapse of a self-gravitating asymptotically flat fluid with the isothermal equation of state $$p=k\varrho $$ p = k ϱ , $$k>0$$ k > 0 , and subject to Newtonian gravity. We rigorously prove the existence of such a Larson–Penston solution.

Gravitational Collapse for Polytropic Gaseous Stars: Self-Similar Solutions

“…Recently, the first three authors were able to construct LP solutions in the case γ = 1 in [12]. The main result of this paper is to show that Yahil solutions exist for the full physical range γ ∈ (1, 4 3 ), including the finite energy range (γ > 6 5 ).…”

“…In the context of the isothermal problem (γ = 1), the demand that the solution be regular produces two possible algebraic "branches" for the Taylor expansion coefficients at the sonic point. The LP-solution constructed in [12] belongs to one of them, all the Hunter solutions to the other, and the branches intersect at exactly one point. When γ > 1, we will show that there are two analogous branches.…”

“…1, parametrised by ω 0 (equivalently, by y * ). In this case, the LP solution constructed in [12] lies in the region of the LP branch for which ω 0 < 1 2 (equivalently y * > 2) while the numerically constructed Hunter solutions all lie in the region ω 0 > 1 2 (equivalently y * < 2), compare also [16,Fig. 2].…”

“…Once we have correctly identified the branch of solutions on which the LPHtype solution should lie, we seek the globally defined Yahil-type solution whose Taylor expansion at the sonic point is of LPH-type (see Definition 2.12 for the precise meaning). We then proceed in four key steps, as in the earlier work of the first three authors, [12].…”

“…When clear from the context, we shall occasionally drop the dependence on y * in the notation above. In comparison to [12], the convergence of the Taylor series is significantly complicated by the presence of the term ρ γ −1 with its non-integer power. Various technical tricks are employed, using the Faà di Bruno formula, to control the size of the coefficients arising in the expansion, while interval arithmetic is employed to control rigorously the sign of three key quantities (see (2.64)-(2.66) and Appendix C.2 below).…”