A Roller Coaster Ride through Relativity/Appendix H

The Gravitational Potential at the surface of a star

The force of gravity on a mass m at a distance r away from a star of mass M is equal to:

$F={GMm \over r^{2}}$

The work done in pulling the mass from the surface of the star out to infinity is therefore:

$W=\int _{r}^{\infty }{GMm \over r^{2}}dr$

This works out to be

$W={GMm \over r}$

and therefore the gravitational potential difference Δφ between the surface and ∞ is

$\Delta \phi ={w \over m}={GM \over R}$

Now by definition, the gravitational potential at ∞ is zero so the potential at the surface is negative, hence:

$\phi _{s}=-{GM \over R}$