A Roller Coaster Ride through Relativity/Appendix H

The Gravitational Potential at the surface of a star

Gravitational Potential

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

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

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

$W=\underset{r}{\overset{\mathrm{\infty }}{\int }}\frac{GMm}{r^2}dr$

This works out to be

$W=\frac{GMm}{r}$

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

$\mathrm{\Delta }\varphi =\frac{w}{m}=\frac{GM}{R}$

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

${\varphi }_s=-\frac{GM}{R}$

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