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Let us imagine a population of stars around a massive black hole. If one of those orbiting stars does not feel the effect of its peers, it will continue to move in a Keplerian orbit. However, when two stars chance upon one another, they exert reciprocal force that may push one star closer to the black hole, and the other further away from it. We will refer to this interaction as a collision, even though the two star do not come into contact. These collisions enable accretion of stars onto the black hole. We will show that in steady state those collisions determine the the density of stars as a function of the distance from the black hole.

The collision rate is

$ \frac{1}{\tau} \approx n \left( r \right) v \left( r \right) \Sigma \left( r \right) $

where $ n \left( r \right) $ is the density, $ v $ is the velocity and $ \Sigma $ is the cross section. The density is assumed to be power law $ n \propto r^{-s} $, the velocity is assumed to be Keplerian $ v \propto \frac{1}{r^{1/2}} $ and the cross section is assumed to be the Bondi cross section $ \Sigma \propto \frac{1}{v^4} $. All in all, the collision rate is proportional to $ \frac{1}{\tau} \propto r^{3/2-s} $

Since the star feed the black hole, in steady state the luminosity should be constant and equal to the energy dissipation rate

$ L \propto N\left(r\right) E \left(r\right) / \tau $

where $ N \propto n r^3 $ is the number stars and $ E \propto 1/r $ is their energy. Combining everything and requiring that the luminosity be independent of the radius yields $ n \propto r^{-7/4} $