# Radiative equilibrium in orbit of a black hole

According to Life under a black sun, Miller's planet from Interstellar, with a time dilation factor of 60,000, should be heated to around 890C by blue-shifted cosmic background radiation.

How they arrive at that number, however, seems to me a little opaque.

As the article describes, there are two major effects to consider: gravitational blueshifting, and blue- and redshifts due to the planet's orbital motion.

Calculating the purely gravitational effects seems straightforward (although I admit I may still be missing something); given that radiative power is proportional to $T^4$, and power should scale linearly with the time dilation factor, the apparent CMB temperature should be $2.7K * 60,000^{1/4} = 42.26K$. Considering that a cold black hole occupies part of the sky, the equilibrium temperature of the planet should be slightly lower. That's clearly a long way from 890C!

It appears, then, that the majority of the heating must be a result of the circular motion of the planet in orbit. Now, it seems fairly obvious that getting precise answers will require numerical simulation, but it should be possible to at least get a close order-of-magnitude estimate based on a model of a planet moving at constant velocity through a background of the temperature calculated from gravitational effects alone. Unfortunately, though the article doesn't quote speeds, and I haven't been able to figure out how to calculate the relevant velocities for a planet is a low orbit around a rotating black hole.

So, can anybody help me fill in the blanks? If I start with a black hole of a given mass and angular momentum, and a planet in a stable circular orbit at some given radius, how do I get to an estimate of equilibrium temperature?

• See Section 3 of this paper for calculating the velocity of circular orbits in Kerr spacetime. – Michael Feb 2 '17 at 16:15