Question

Bonus questions can add +2 to your overall score if your answer is correct. Calculate the Schwarzschild radius for fictional planet Druidia (mass = 3.654×1025 kg ) to become a black hole.

Answer 1

Approximately
`5.427 * 10^(-2)\; {\rm m}`.

Step-by-step explanation:

For a planet of mass
`M`, the Schwarzschild Radius would be:

`\displaystyle r_{\text{s}} = (2\, G\, M)/(c^(2))`,

Where:

• 
  `G \approx 6.674 * 10^(-11)\; {\rm m^(3)\cdot kg^(-1)\cdot s^(-2)}` is the gravitational constant, and
• 
  `c \approx 2.998 * 10^(8)\; {\rm m\cdot s^(-1)}` is the speed of light in vacuum.

It is given that the mass of this planet is
`M = 3.654 * 10^(25)\; {\rm kg}`. Substitute this value into the expression for
`r_{\text{s}}` and evaluate to obtain:

`\begin{aligned} r_{\text{s}} &= (2\, G\, M)/(c^(2)) \\ &\approx \frac{2\, (6.674 * 10^(-11)\; {\rm m^(3)\cdot kg^(-1)\cdot s^(-2)})\, (3.654 * 10^(25)\; {\rm kg})}{\left(2.998 * 10^(8)\; {\rm m\cdot s^(-1)}\right)^(2)}\\ &\approx 5.427 * 10^(-2)\; {\rm m}\end{aligned}`.

In other words, if the radius of this planet goes below approximately
`5.427 * 10^(-2)\; {\rm m}`, the planet would become a black hole.