To estimate the equilibrium temperature of a planet orbiting a black hole, we need to account for both gravitational blueshifting and Doppler shifting due to the planet's orbital motion. Here's a step-by-step breakdown:

1. Gravitational Time Dilation (γ_grav)

The gravitational time dilation factor for a static observer at radius ( r ) (distance from the black hole's center) is given by: [ ...

Short answer up front:

• The 42 K you got is far too low because you used the fourth root of the total time‑dilation factor. In this problem the energy density that sets the equilibrium temperature does not scale that way.
• Near a black hole the CMB is seen only through a very small solid-angle “escape cone,” but each ray that does get through is hugely blueshifted. Those two effects almost cance...

This is a fascinating question about the extreme physics near a black hole! Let me help break down the calculation step by step.

Key Physics Concepts

The main effects to consider are:

1. Gravitational redshift/blueshift from the gravitational field
2. Doppler shift from orbital motion
3. Beaming effects from relativistic motion

Setting Up the Problem

For a rotating (Kerr) bla...

There are two quite different things that have to be kept apart

1. the frequency shift suffered by every single photon that climbs out of the gravitational potential and is then seen by a rapidly moving planet, and

2. the solid angle under which the planet can actually see the rest of the universe, i.e. the part of the sky that is not hidden behind the black-hole shadow.

Both effects enter ...

Excellent question! You've done a great job breaking down the problem and your initial calculation is a perfectly logical first step. You are correct that the orbital motion is the dominant factor, and your intuition that this requires a more detailed model is spot on.

Let's walk through the physics to see how we get from 42 K to the paper's 890°C (1163 K).

The Flaw in the Initial Calculatio...

Radiative Equilibrium Near a Black Hole: Understanding the Temperature Calculation

This is a fascinating question that touches on general relativity, thermodynamics, and relativistic effects. Let me walk through the calculation step by step.

Breaking Down the Problem

You're right that there are two major effects to consider:

1. Gravitational blueshifting of the CMB
2. Doppler shifting due ...

The user has calculated the gravitational effect alone: T_g = T_CMB * (time_dilation_factor)^{1/4} = 2.7 K * (60,000)^{1/4} ≈ 42.26 K

But this is much lower than 89...

Here is a “cook‐book’’ for getting the equilibrium temperature that a little planet in circular orbit around a Kerr (or Schwarzschild) black hole will settle to, when the only “heater’’ is the cosmic microwave background at 2.7 K as seen from infinity. In outline you do precisely the same thing you would do in special relativity (blue‐shift plus Doppler beaming of an otherwise isotropic bath) exc...

Below is (i) the physical reason why the simple Τ ∝ (60000)¼ estimate is wrong,
(ii) the correct way to combine gravitational blue–shift and orbital Doppler effects, and
(iii) a worked example that reproduces the 8-–9 × 10² °C quoted in “Life under a Black Sun”.

Symbols
T0 = 2.725 K  (current CMB temperature at infinity)
σ = Stefan–Boltzmann constant,   a = 4σ/c
M, a = mass and (speci...