Free-fall time of a uniform sphere ($t_{\mathrm{ff}}=\sqrt{3\pi /(32G\rho )}$)

A pressureless uniform sphere initially at rest collapses under Newtonian gravity in a time depending only on its initial mass density. Solving the radial shell equation gives $\sqrt{3\pi /(32G\rho )}$. It is commonly used as a dynamical-time benchmark for cooling and star formation, with the gravitating density rather than gas density alone when other matter supplies gravity.