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.

\begin{equation}
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 \\
\end{equation}

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

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