Gravitational collapse

Gravitational collapse is contraction driven by a system's own gravitational attraction or an enclosing potential. Pressure, angular momentum and other stresses can oppose it. A free-fall timescale measures the dynamical contraction rate when pressure support is removed; gas cooling determines whether thermal support can be removed fast enough.

A test particle released from rest at $r_0$ in a fixed spherical mass density $\rho (r)={\rho }_0(r_0/r{)}^2$ obeys $\ddot{r}=-4\pi G{\rho }_0{r}_0^2/r$. Integrating once gives ${\dot{r}}^2=8\pi G{\rho }_0{r}_0^2\ln (r_0/r)$. The substitution $u=\sqrt{\ln (r_0/r)}$ then gives $t(r)=(8G{\rho }_0{)}^{-1/2}\mathrm{erf}(u)$, 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.

A collapse front propagates out through an initially singular isothermal envelope at $r_f=a_{T}t$. The only steady accretion scale is $\dot{M}\propto a_T^3/G\propto T^{3/2}$ for fixed mean particle mass. The front sweeps original mass at $2a_T^3/G$, but the exact central accretion coefficient requires the dynamical similarity solution and cannot be identified with front sweeping alone.

For a singular isothermal sphere with $\rho =a_s^2/(2\pi G{r}^2)$, the initial mass inside the outward front $r_{\mathrm{ew}}=a_{s}t$ is $M_{\mathrm{ew}}=2a_s^{3}t/G$. This is not all immediately accreted by the central object. The inside-out collapse of a singular isothermal sphere instead has ${\dot{M}}_{\ast }=m_0{a}_s^3/G$, with $m_0$ fixed by its dynamical similarity solution; the remainder can stay in the infalling envelope. Equating front sweeping with central accretion loses that mass budget.

Where a central mass dominates and radial infall is locally steady, $|v_r|\simeq \sqrt{2G{M}_{\star }/r}$. A conserved mass flux $\dot{M}=4\pi r^2\rho |v_r|$ then gives $\rho \simeq \dot{M}[4\pi \sqrt{2G{M}_{\star }}{]}^{-1}{r}^{-3/2}$. Constant accretion gives $M_{\star }\propto t$ and an inner normalization proportional to $t^{-1/2}$.

In a spherical collapse model a freely falling surface can cross its event horizon after finite proper time, while outgoing signals received far away have increasing delay and redshift. Schwarzschild-coordinate freezing does not mean collapse physically stops or leaves a permanently bright surface at the horizon.

A uniform spherical cloud with negligible pressure and velocity $\mathbf{u}=(\dot{R}/R)\mathbf{r}$ remains uniform: the continuity equation gives $\rho R^3={\rho }_{00}{R}_0^3$. The shell theorem gives interior gravitational acceleration $-(4\pi G\rho /3)\mathbf{r}$, so its outer radius satisfies the displayed equation, with fixed mass $M=4\pi {\rho }_{00}{R}_0^3/3$. Collapse from rest has ${\dot{R}}^2=2GM(1/R-1/R_0)$. Writing $C^2=2GM/R_0$, its cycloid parametrization is $R=R_0(1+\cos \varphi )/2$ and $Ct=R_0(\varphi +\sin \varphi )/2$, for $0\le \varphi \le \pi$. Complete collapse occurs after $\sqrt{3\pi /(32G{\rho }_{00})}$. This Newtonian dust solution ceases to be a regular fluid configuration at zero radius.

A Fourier mode advected with a homologous spherical flow has phase $\mathbf{q}\cdot \mathbf{r}/R(t)$. Its material derivative in the background vanishes because $\dot{\mathbf{k}}=-(\dot{R}/R)\mathbf{k}$. Thus contraction shortens its physical wavelength while preserving its phase on each fluid element. Linearizing the continuity equation and pressureless momentum equation gives $\dot{\delta }=-i\mathbf{k}\cdot {\mathbf{u}}_1$ and ${\dot{\mathbf{u}}}_1+(\dot{R}/R){\mathbf{u}}_1=-i\mathbf{k}{\Phi }_1$. Combining these with the Poisson equation for Newtonian gravity yields $\ddot{\delta }+2(\dot{R}/R)\dot{\delta }-4\pi G\rho \delta =0$ for nonzero compressive wavenumber. It is a local bulk calculation, without the moving cloud-boundary conditions.

For pressureless homologous spherical collapse, let $\varphi$ be its cycloid parameter. The density contrast equation becomes ${\delta }_{\varphi \varphi }-\tan (\varphi /2){\delta }_{\varphi }-3\delta /(1+\cos \varphi )=0$. One solution is ${\delta }_g=\sin \varphi /(1+\cos \varphi {)}^2$. Reduction of order gives an independent solution that decays at collapse,For $ϵ=\pi -\varphi$, the integral is ${ϵ}^5/40+O({ϵ}^7)$, while ${\delta }_g\sim 4{ϵ}^{-3}$. Therefore ${\delta }_d\sim {ϵ}^2/10$. Since $R\sim R_0{ϵ}^2/4$, the displayed powers follow. The growing branch is unique up to scale and addition of the decaying branch; many mixed solutions grow, so it is not literally the only function with growth. Finite perturbations eventually leave the linear regime.

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.