From Cavitation to Astrophysics: Explicit Solution of the Spherical Collapse Equation

Abstract

Differential equations of the form $\ddot R=-kR^\gamma$, with a positive constant $k$ and real parameter $\gamma$, are fundamental in describing phenomena such as the spherical gravitational collapse ($\gamma=-2$), the implosion of cavitation bubbles ($\gamma=-4$) and the orbital decay in binary black holes ($\gamma=-7$). While explicit elemental solutions exist for select integer values of $\gamma$, more comprehensive solutions encompassing larger subsets of $\gamma$ have been independently developed in hydrostatics (see Lane-Emden equation) and hydrodynamics (see Rayleigh-Plesset equation). I here present a universal explicit solution for all real $\gamma$, invoking the beta distribution. Although standard numerical ODE solvers can readily evaluate more general second order differential equations, this explicit solution reveals a hidden connection between collapse motions and probability theory that enables further analytical manipulations, it conceptually unifies distinct fields, and it offers insights into symmetry properties, thereby enhancing our understanding of these pervasive differential equations.