Certainly! Let's go through the step-by-step solution to determine the volume inside the event horizon of a black hole with a given mass [tex]\( M \)[/tex].
### Step 1: Understand the Inputs and Constants
1. Gravitational Constant ([tex]\( G \)[/tex]): [tex]\( G = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex]
2. Speed of Light ([tex]\( c \)[/tex]): [tex]\( c = 299792458 \, \text{m/s} \)[/tex]
3. Mass of the Sun ([tex]\( M_{\odot} \)[/tex]): [tex]\( M_{\odot} = 1.989 \times 10^{30} \, \text{kg} \)[/tex]
4. Mass of the Black Hole ([tex]\( M \)[/tex]): Let's assume it is 10 times the mass of the Sun. Therefore, [tex]\( M = 10 \, M_{\odot} = 10 \times 1.989 \times 10^{30} \, \text{kg} = 1.989 \times 10^{31} \, \text{kg} \)[/tex].
### Step 2: Calculate the Schwarzschild Radius
The Schwarzschild radius [tex]\( R_s \)[/tex] is given by the formula:
[tex]\[ R_s = \frac{2 G M}{c^2} \][/tex]
Plug in the known values:
[tex]\[ R_s = \frac{2 \times 6.67430 \times 10^{-11} \times 1.989 \times 10^{31}}{(299792458)^2} \][/tex]
After performing the calculation:
[tex]\[ R_s = 29541.27 \, \text{meters} \][/tex]
### Step 3: Calculate the Volume Inside the Event Horizon
The volume [tex]\( V \)[/tex] inside the event horizon is given by the formula for the volume of a sphere:
[tex]\[ V = \frac{4}{3} \pi R_s^3 \][/tex]
Plug in the Schwarzschild radius:
[tex]\[ V = \frac{4}{3} \pi (29541.27)^3 \][/tex]
### Step 4: Perform the Final Calculation
[tex]\[ V = \frac{4}{3} \pi (29541.27)^3 \][/tex]
After performing the calculation:
[tex]\[ V = 1.07988 \times 10^{14} \, \text{m}^3 \][/tex]
### Summary of the Solution
- Gravitational Constant (G): [tex]\(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex]
- Speed of Light (c): [tex]\(299792458 \, \text{m/s} \)[/tex]
- Mass of the Sun ([tex]\( M_{\odot} \)[/tex]): [tex]\(1.989 \times 10^{30} \, \text{kg} \)[/tex]
- Mass of the Black Hole (M): [tex]\(1.989 \times 10^{31} \, \text{kg} \)[/tex]
- Schwarzschild Radius ([tex]\( R_s \)[/tex]): [tex]\(29541.27 \, \text{m} \)[/tex]
- Volume inside the Event Horizon: [tex]\(1.07988 \times 10^{14} \, \text{m}^3 \)[/tex]
This comprehensive calculation determines the volume inside the event horizon of a black hole with a mass 10 times that of the Sun.
