To what fraction of its current radius would the sun have to be compressed to create a black hole?
You want to determine the radius of the sphere that the present volume of the sun would have to be compressed in, to achieve a black hole.
We can determine this using a formula developed by the mathematician named Karl Schwarzschild.
According to the formula, to create a black hole starting with a body of mass M, the radius of the sphere it has to be compressed to is given by Rs = 2*G*M/c^2, where G is the gravitational constant, M is the mass of the body we want to compress and c is the speed of light.
The mass of the Sun is 1.99*10^30 kg, G= 6.674*10*10^-11 N*(m/kg) ^2 and the speed of light is 299792458 m/s.
Now using these values in the formula, we get Rs = 2*1.99*10^30*6.674*10^-11/ (299792458) ^2
= 2955 m
The present radius of the Sun is 7*10^5 km = 7*10^8 m.
Therefore we need to compress this into a sphere with a radius that is 2955/7*10^8 = 4.2*10^-6 times the present radius.
The ratio of the radius of the sphere required to achieve a black hole to the present radius of the Sun is equal to 4.2*10^-6.