The Schwarzschild radius and how to calculate it

When you look up at the sky at night, you can see thousands of stars. Each one of these stars has the potential to end its life in several different ways.  The scope of this particular article does not include the life cycle of a star, which is exciting enough to deserve it’s own dedicated future article, but this article is going to talk about those stars that end their long lives with a bang and become a black hole, or singularity to give it its proper name.

What is a black hole?

Although not wishing to understate the complexity of black holes, the simplest definition of one would be a region of space from which nothing can escape. The name ‘black hole’ comes from the fact that not even light, the fastest thing we know of, can escape from this region. A black hole contains a very large mass in an extremely compact space.

The Schwarzchild radius is the radius an object would have if you compacted its mass down such that the escape velocity this new object would be equal to the speed of light – just like a black hole.

Calculating the Schwarzchild radius

From the definition that we mentioned above, we first equate the kinetic energy of a particle somewhere outside of the newly squashed object with the gravitational pull it feels from the object.

where is the mass of our squashed object, is the mass of our imagined particle, is the gravitational constant, is the velocity of our imagined particle and is the distance between our particle and our squashed object.

Our definition of the Schwarzchild means that in our previous equation we can replace our velocity with the speed of light, given by . We can also cancel out the mass of our small particle and so we are left with:

As we want the radius of our squashed object, we rearrange the above equation for and finally arrive at the object of our desires, the Schwarzchild radius.

Putting this into perspective

We can now use this equation to put things we know into perspective. Firstly, let us take our Sun. It has a mass of 1.99×10³⁰ kg and plugging this into our equation, with the usual values of 6.67×10^-11 m³ kg^-1 s^-2 for and 3×10⁸ m s^-1 for , we find that our Sun, if it were to collapse into a black hole at the end of its lifetime would have a radius of 3km – the point of no return.

This is incredibly small on the kind of scales we are talking about! 3km is 0.000004 times its original radius! If we plug in the mass of the Earth, we obtain a Schwarzchild radius of 9mm – 0.000000001 times the original radius! Imagine all of the Earth squashed down to the size of a marble!

Further your knowledge

Black holes and light are interesting things. Read up about some of the following things to learn more!

• Collapse of a star
• Supernova
• Chandrasekhar mass