Chapter 33, Problem 19P

To determine

To Calculate: The Schwarzschild radius for a typical galaxy.

Answer to Problem 19P

  $R\approx 5.90\times {10}^{15}\text{km}$

Explanation of Solution

Formula used:

Schwarzschild radius can be calculated as:

  $R=\frac{2GM}{c^2}$

Where,

  $G$ = gravitational constant

  $M$ = mass of galaxy

  $c$ = speed of light

Calculation:

Mass of Milky way galaxy = $2\times {10}^{12}$ mass of the sun

Mass of the sun, $m_s=1.99\times {10}^{30}\text{kg}$

Total mass of the galaxy is:

  $\begin{array}{l}=2\times {10}^{12}\times 1.99\times {10}^{30}\\ =3.98\times {10}^{42}\text{kg}\end{array}$

Speed of light, $c=3\times {10}^8\text{m/s}$

Schwarzschild radius can be calculated as:

  $\begin{array}{l}R=\frac{2\times 6.67\times {10}^{-11}\times 3.98\times {10}^{42}}{{\left(3\times {10}^8\right)}^2}\\ R\approx 5.90\times {10}^{15}\text{km}\end{array}$

Conclusion:

Schwarzschild radius for galaxy, $R=5.90\times {10}^{15}\text{km}$