Natural units of star clusters angular momentum and energy
Using unit analysis, I've found that a natural unit of angular momentum should be
\begin{equation}\tag{1}
L_0 = \frac{G M_{\odot}}{c} \approx 8.816 \times 10^{41} \, \text{kg} \, \text{m}^2/\text{s}.
\end{equation}

In the case of a bounded system of two equal masses, circularly moving around their center of mass, the total angular momentum (relative to the center of mass) is
\begin{equation}\tag{2}
L_{\text{bound}} = 2 M \, v \, r = \frac{2M \, v^2 \, r}{v}.
\end{equation}
Now, using Newton's equation and the gravitationnal force, we have
\begin{equation}
M \, v^2 = \frac{G M^2}{4 \, r}.
\end{equation}
Thus, equation (2) becomes :
\begin{equation}\tag{3}
L_{\text{bound}} = \frac{G M^2}{2 \, v}.
\end{equation}
The classical theory cannot be trusted when $v \sim c$, and since $v &lt; c$ (forgetting the 2 factor) :
\begin{equation}\tag{4}
L_{\text{bound}} \approx \frac{G M^2}{v} &gt; \frac{G M^2}{c} \equiv L_0.
\end{equation}
So $L_0$ appears to be a good unit.  When $L \sim L_0$, we then need relativity.  We're in a classical regime when $L \gg L_0$.

The system isn't bounded when $L \gg L_{\text{bound}}$, which implies $L \ggg L_0$.

I don't know if this is a good way to introduce the $L_0$ unit defined by (1).  Any opinion/suggestions/comments on this ?

Do astronomers use $G M^2 / c$ as natural unit of angular momentum, when they study star clusters and globulars ?