What is a black hole?

Every object has an attractive force called gravity. In order to escape from an object’s gravitational pull one must achieve an initial minimum speed called escape velocity. A black hole is basically any object whose escape velocity is greater than the speed of light; thus nothing can escape it. How small would the earth have to be in order to become a black hole? Let’s find out.

When an object is outside of a planet’s gravitational pull, the sum of its kinetic and potential energy would have to be zero (or equal). Note the left hand side is kinetic energy and the right hand side is potential.

\frac{1}{2} m v^{2} = \frac{GMm}{r}

  • m is the mass of the object attempting to escape
  • v is the object’s speed
  • G is the gravitational constant
  • M is the mass of the larger object
  • r is the radius of the larger object

Now we need to move stuff around to solve for the radius and set the velocity at the speed of light. Notice that the mass of the object escaping doesn’t matter.

r = \frac{2GM}{v^2}  or  r = \frac{2GM}{c^2}

Now we plug in the necessary values. The radius on which a mass becomes a black hole is called the Schwarzschild radius. Let’s calculate the Schwarzschild radius for the earth. The earth’s mass is about 5.97 \times 10 ^ {24} kg.

r = \frac{2GM}{c^2} = \frac{2 * 6.67 \times 10^{-11} * 5.97 \times 10^{24}}{300,000,000^2} = 0.0087

This would mean the whole earth would have to be squished into an 8 millimeter radius sphere, that’s about the size of a small marble. If we did this calculation with the sun we would get a sphere of 3 kilometers in radius.

Fun Fact: Did you know at the center of the milky way galaxy there is actually a super massive black hole? Don’t worry it’s not close enough to harm us.