The Euler-Poisson system of partial differential equations describes the dynamics of a self-gravitating gas. For the energy-critical polytropic pressure law, there is an explicit steady-state solution describing an isolated star. I will discuss recent work which describes the nonlinear phase space around this solution and proves the existence of a new instability mechanism: accelerated shock formation. This talk is based on forthcoming work done in collaboration with Mahir Hadžić, Juhi Jang and Sung-Jin Oh.