Perturbation Analysis of Gravitational Collapse of Gaseous Stars in the Pressureless Euler-Poisson System

The compressible Euler-Poisson system describes the full evolutionary processes of inviscid compressible gases, including compression, expansion, fluid motion, equilibrium states and gravitational collapse, under the combined effects of self-gravitation, pressure and inertial forces. It constitutes the fundamental macroscopic mathematical model for describing stellar structures, nebular evolution and gravitational collapse of gaseous celestial bodies. The system comprises mass conservation, momentum conservation, energy conservation laws coupled with the Poisson equation. Following the research in \cite{huang2022}, this paper investigates the quantitative relations among the relative perturbation amplitude of initial velocity, critical time and collapse time for gaseous stars undergoing gravitational collapse within the framework of the pressureless Euler-Poisson system. The closed-form formulas are verified against direct numerical integration, the accuracy of the asymptotic power law is quantified, and the relation between the two time scales is examined.