= Free fall in a fixed singular isothermal sphere
{title2=$t_{\rm ff}=(8G\rho_0)^{-1/2}$}
A test particle released from rest at $r_0$ in a fixed spherical <mass density> $\rho(r)=\rho_0(r_0/r)^2$ obeys $\ddot r=-4\pi G\rho_0r_0^2/r$. Integrating once gives $\dot r^2=8\pi G\rho_0r_0^2\ln(r_0/r)$. The substitution $u=\sqrt{\ln(r_0/r)}$ then gives $t(r)=(8G\rho_0)^{-1/2}\operatorname{erf}(u)$, and the arrival time at the centre is $(8G\rho_0)^{-1/2}$. This is <free fall> in a frozen <singular isothermal sphere>, not the self-consistent pressureless collapse of the background. An initially resting material shell with the same initial enclosed <mass> instead arrives after $\sqrt{\pi/(32G\rho_0)}$, because its enclosed <mass> stays fixed while the cloud contracts.
Back to article page
