# Formula of black hole gravitational sphere of influence

Well,

I want to derive the formula

$$r = \frac{GM}{\sigma^{2}}$$

which happens to be the radius of the gravitational sphere of influence of a supermassive black hole inside a galaxy. How can I do that? I'll accept any tips or indications that can help me to do that. If you need to know, 𝜎 is the stellar velocity dispersion, 𝐺 is the gravitational constant, and 𝑀 the mass of the black hole.

Edit 1 - Little context: In this case I have to imagine that I have a black hole at the center of a galaxy. The black hole has a gravitacional influence around it, but it has a finite distance, meaning that at some radius r the biggest gravitational influence changes from the black hole to the one of the galaxy. The r in my formula is exactly this distance. Hope I explained it well.

Edit 2: If you look for Sphere of influence (black hole) at wikipedia you will find a little explanation about this formula.

• What does "sphere of influence" mean? Sep 9 '20 at 20:33
• The actual definition is: Sphere of influence (SOI) is the region around a celestial body where the primary gravitational influence on an orbiting object is that body
– kplt
Sep 9 '20 at 20:54
• to derive this relationship you're going to need a bunch of assumptions about the distribution of stars in the galaxy and their mass density relative to the black hole, right? Sep 9 '20 at 21:03
• The Wikipedia article says that this is one possible definition. Definitions cannot (by definition!) be derived, although they can be motivated. A possible motivation is the virial theorem. Sep 9 '20 at 21:27

The virial theorem tells us that the sum of twice the kinetic energy and gravitational potential energy of an assembly of stars in equilibrium is zero. i.e. $$2K + \Omega =0$$
For the black hole to dominate over the gravitational potential of the rest of the galaxy out to some radius $$r$$, then the potential energy due to the black hole must be numerically less than that of the galaxy (recall the gravitational potential is negative). i.e. $$-\frac{GMM_{\rm BH}}{r} < \Omega,$$ where $$M$$ is the total mass within $$r$$ and $$M_{\rm BH}$$ is the black hole mass. (Assuming spherical symmetry and ignoring numerical factors associated with how the mass is distributed etc).
But $$\Omega = -2K \simeq -M\sigma^2$$ so $$\frac{GMM_{\rm BH}}{r} > M\sigma^2,$$ $$r < \frac{GM}{\sigma^2}.$$