Question

I understand that closer orbits around Schwarzschild black holes require faster speeds, culminating in a required speed of c

at the photon sphere (r=3M
in Schwarzschild coordinates). I'd love to be able to work out the orbital speed at the ISCO (6M
) and marginally bound orbit (4M
) and others. Is there a formula for this orbital speed as a function of radius? (I don't mind that these orbits are unstable.)

Answer 1

Yes, there is a formula for the orbital speed as a function of radius around a Schwarzschild black hole. The formula is given by v = sqrt((2GM)/r), where v is the orbital speed, G is the gravitational constant, M is the mass of the black hole, and r is the radius of the orbit.

Step-by-step explanation:

Yes, there is a formula for the orbital speed as a function of radius around a Schwarzschild black hole. The formula is given by:

v = sqrt((2GM)/r)

Where v is the orbital speed, G is the gravitational constant, M is the mass of the black hole, and r is the radius of the orbit.

For example, at the ISCO (Innermost Stable Circular Orbit) with a radius of 6M, the orbital speed would be:

• v = sqrt((2GM)/(6M)) = sqrt(2G/M)

And at the marginally bound orbit with a radius of 4M, the orbital speed would be:

• v = sqrt((2GM)/(4M)) = sqrt(4G/M) = 2(sqrt(G/M))