Minimum Orbital Radius Around Black Holes

[GoBluePhysics]

Hi there,

I was reading one of my textbooks and I had a thought. For a black hole, there is minimum orbiting radius of $R_{min}=3R_s$ where $R_s$ is the Schwarzschild Radius. This minimum orbit is created by the fact that in order to obtain an orbit of that radius around a black hole, you would need to be traveling at a velocity exceeding $c$. That's all good and dandy. However, wouldn't it be true that there exists a minimum orbit for any massive body due to the the velocity limit of c? In other words, if I am in an orbit around the Earth such that my velocity is equal to $c$, what is my radius? Just something I thought up.

 

[Chronos]

The short answer is It's the same as the photon sphere around a black hole [hint, less than the radius of earth].

 

[GoBluePhysics]

The short answer is It's the same as the photon sphere around a black hole [hint, less than the radius of earth].

When you say "photon sphere" do you mean the sphere created by a radius such that $r=1.5R_s$ (for a black hole)?

Wait. I think I understand. If I think of it in terms of Birkhoff's theorem, if the density of a black hole were expanded to have the density of another object (say, the earth) then the photon sphere and $R_s$ would be unmoved but they would also be inside of the object. Is that kind of the hand-wavy gist of it?

 

[Chronos]

Yes.