Formula of black hole gravitational sphere of influence

Question

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, ? is the stellar velocity dispersion, ? is the gravitational constant, and ? 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.

ProfRob · Accepted Answer

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 $$Omega = -2K simeq -Msigma^2$$
so
$$frac{GMM_{rm BH}}{r} > Msigma^2,$$
$$ r < frac{GM}{sigma^2}.$$

Formula of black hole gravitational sphere of influence

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