Question

What is the Schwarzschild radius of a black hole that has a mass eight times that of our Sun? Note that stars must be more massive than the Sun to form black holes as a result of a supernova.

Short answer

The Schwarzschild radius of a black hole is obtained as \({\rm{24 km}}\).

Step-by-step solution

Schwarzschild radius

Schwarzschild radius is given by,

\({R_s} = \frac{{2GM}}{{{c^2}}}\)

Here\({R_s}\)is Schwarzschild radius,\(G\)is the universal gravitational constant,\(M\)is the mass and\(c\)is the speed of the light.

Evaluating the Schwarzschild radius

The Schwarzschild radius is given by,

\({R_s} = \frac{{2GM}}{{{c^2}}} \)

Putting the value \({\rm{M = 8}}{{\rm{M}}_{{\rm{sun}}}}\), mass of the sun is \(2 \times {10^{30}}\,{\rm{kg}}\)

\(\begin{align}{R_s} &= {\rm{ }}\frac{{{\rm{2 \times 6}}{\rm{.67 \times 1}}{{\rm{0}}^{{\rm{ - 11}}}}\frac{{{\rm{N}}{{\rm{m}}^{\rm{2}}}}}{{{\rm{k}}{{\rm{g}}^{\rm{2}}}}}{\rm{ \times 8 \times 2 \times 1}}{{\rm{0}}^{{\rm{30}}}}{\rm{kg}}}}{{{{{\rm{(3 \times 1}}{{\rm{0}}^{\rm{8}}}\frac{{\rm{m}}}{{\rm{s}}}{\rm{)}}}^{\rm{2}}}}}{\rm{ }}\\ &= {\rm{ 24 km}}\end{align}\)

Therefore, the Schwarzschild radius is \({\rm{24 km}}\).