# Natural units of star clusters angular momentum and energy

I'm calculating the total angular momentum of a bunch of theoretical stars of a few solar masses, separated by a few light years, moving at velocities of about a few dozens of km / sec.

Mechanical angular momentum relative to an arbitrary origin (the cluster's center of mass is a good choice) is this : $$\vec{L} = \sum_i \vec{r}_i \times m_i \, \vec{v}_i,$$ so the SI units are $\mathrm{kg} \cdot \mathrm{m}^2 / \mathrm{sec}$. But in real astronomy, what should be the most simple/natural units for star clusters ? $M_{\odot} \cdot (\mathrm{ly})^2 / \mathrm{year} \equiv M_{\odot} \, c^2 \cdot \mathrm{year}$ ?

What are the angular momentum units used by astronomers ?

And what about energy units for a cluster of stars ? I'm interested in kinematical energy and gravitationnal potential energy of the whole cluster. Certainly not joules !? And $M_{\odot} \, c^2$ would certainly be too much, I suppose !

• I think I've found a simple answer. I'll add an answer to my own query. – Cham Jun 3 '17 at 12:08

Using unit analysis, I've found that a natural unit of angular momentum should be $$\tag{1} L_0 = \frac{G M_{\odot}}{c} \approx 8.816 \times 10^{41} \, \text{kg} \, \text{m}^2/\text{s}.$$
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 $$\tag{2} L_{\text{bound}} = 2 M \, v \, r = \frac{2M \, v^2 \, r}{v}.$$ Now, using Newton's equation and the gravitationnal force, we have $$M \, v^2 = \frac{G M^2}{4 \, r}.$$ Thus, equation (2) becomes : $$\tag{3} L_{\text{bound}} = \frac{G M^2}{2 \, v}.$$ The classical theory cannot be trusted when $v \sim c$, and since $v < c$ (forgetting the 2 factor) : $$\tag{4} L_{\text{bound}} \approx \frac{G M^2}{v} > \frac{G M^2}{c} \equiv L_0.$$ 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 ?