An Introduction to Modern Astrophysics · Problem 17.22

Close Binary Star Systems — Carroll & Ostlie

Physics — Astrophysics · black-holes-hawking-radiation

Problem

By combining gravitation (

G

), thermodynamics (

k

), and quantum mechanics (

\hbar

), Stephen Hawking calculated the temperature,

T

, of a nonrotating black hole to be

kT = \frac{\hbar c^3}{8\pi GM} = \frac{\hbar c}{4\pi R_S}, \tag{17.34}

where

R_S

is the Schwarzschild radius. (a) Verify that the expression has the right units. (b) It was mentioned in the text that if a primordial black hole formed at the beginning of the universe 13.7 Gyr ago with a mass of

1.7 \times 10^{11}

kg, it would be reaching the end of its life now. Compute the temperature of a primordial black hole having a mass of

1.7 \times 10^{11}

kg. (c) Approximately what portion of the electromagnetic spectrum would this blackbody temperature correspond to? (d) What would the radius of a sphere having the density of water be if it had a mass of

1.7 \times 10^{11}

kg? (e) Compute the temperature of a

10\,M_\odot

black hole.

Related theory

Primordial Black Holes

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