Question

The Schwarzschild radius is the distance from an object at which the escape velocity is equal to the speed of light. A black hole is an object that is smaller than its Schwarzschild radius, so not even light itself can escape a black hole. The Schwarzschild radius ???? depends on the mass ???? of the black hole according to the equation ????=2????????????2 where ???? is the gravitational constant and ???? is the speed of light. Consider a black hole with a mass of 5.7×107M⊙. Use the given equation to find the Schwarzschild radius for this black hole. Schwarzschild radius: m What is this radius in units of the solar radius

Answer 1

1. $1.69\cdot 10^{11} m$

The Schwarzschild radius of an object of mass M is given by:

$r_s = \frac{2GM}{c^2}$ (1)

where

G is the gravitational constant

M is the mass of the object

c is the speed of light

The black hole in the problem has a mass of

$M=5.7\cdot 10^7 M_s$

where

$M_s = 2.0\cdot 10^{30} kg$ is the solar mass. Substituting,

$M=(5.7\cdot 10^7)(2\cdot 10^{30}kg)=1.14\cdot 10^{38} kg$

and substituting into eq.(1), we find the Schwarzschild radius of this black hole:

$r_s = \frac{2(6.67\cdot 10^{-11})(1.14\cdot 10^{38} kg)}{(3\cdot 10^8 m/s)^2}=1.69\cdot 10^{11} m$

2) 242.8 solar radii

We are asked to find the radius of the black hole in units of the solar radius.

The solar radius is

$r_S = 6.96\cdot 10^5 km = 6.96\cdot 10^8 m$

Therefore, the Schwarzschild radius  of the black hole in solar radius units is

$r=\frac{1.69\cdot 10^{11} m}{6.96\cdot 10^8 m}=242.8$