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.

  • 1
    $\begingroup$ What does "sphere of influence" mean? $\endgroup$ Sep 9 '20 at 20:33
  • $\begingroup$ 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 $\endgroup$
    – kplt
    Sep 9 '20 at 20:54
  • $\begingroup$ 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? $\endgroup$ Sep 9 '20 at 21:03
  • $\begingroup$ 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. $\endgroup$
    – G. Smith
    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 $$\frac{GMM_{\rm BH}}{r} > M\sigma^2,$$ $$ r < \frac{GM}{\sigma^2}.$$


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