# Pebble Cloud Angular Momentum

Gravitational collapse of a pebble cloud occurs when the Roche density is reached inside the cloud. At that point, the cloud has mass $$m$$, radius $$\sim R_{\textrm{H}}$$ and density $$\rho_{\textrm{R}}$$.

The Hill radius, $$R_{\textrm{H}}$$, is proportional to the heliocentric distance and the cube root of cloud mass; the Roche density, $$\rho_{\textrm{R}}$$, is proportional to the inverse heliocentric distance cubed.

$$R_{\textrm{H}}\propto a\,m^{1/3},\quad\rho_{\textrm{R}}=\frac{9\Omega^{2}}{4\pi G}\propto\left(\frac{a}{\textrm{AU}}\right)^{-3}\nonumber \\$$

The critical spin frequency above which the pebble cloud will disperse scales as the square root of its density.

$$\omega_{\textrm{crit}}\propto\rho^{1/2}=\rho_{\textrm{R}}^{1/2}\propto a^{-3/2}\nonumber \\$$

At that point, the cloud angular momentum is

\begin{align} L_{\textrm{crit}} & \propto m\,r^{2}\,\omega\notag \\ & =m\,R_{\textrm{H}}^{2}\,\omega_{\textrm{crit}}\notag \\ & =m\,a^{2}\,m^{2/3}\,a^{-3/2}\notag \\ & =m^{5/3}\,a^{1/2}\notag \\ \end{align}