Potential energy for gravitational forces between two bodies

The gravitational potential function, also known as gravitational potential energy, is:

${\displaystyle U=-<!--\; -\; -->\frac{GMm}{r},}$

The negative sign follows the convention that work is gained from a loss of potential energy.

Derivation

The gravitational force between two bodies of mass M and m separated by a distance r is given by Newton's law

${\displaystyle \mathbf{F}=-<!--\; -\; -->\frac{GMm}{r^2}\stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{r}},}$

where ${\displaystyle \stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{r}}}$ is a vector of length 1 pointing from M to m and G is the gravitational constant.

Let the mass m move at the velocity v then the work of gravity on this mass as it moves from position r(t₁) to r(t₂) is given by

${\displaystyle W=-<!--\; -\; -->{\int <!--\; \int \; -->}_{\mathbf{r}(t_1)}^{\mathbf{r}(t_2)}\frac{GMm}{r^3}\mathbf{r}\cdot <!--\; \cdot \; -->d\mathbf{r}=-<!--\; -\; -->{\int <!--\; \int \; -->}_{t_1}^{t_2}\frac{GMm}{r^3}\mathbf{r}\cdot <!--\; \cdot \; -->\mathbf{v}\mathrm{d}t.}$

The position and velocity of the mass m are given by

${\displaystyle \mathbf{r}=r{\mathbf{e}}_r,\mathbf{v}=\stackrel{\dot{}<!--\; \dot{}\; -->}{r}{\mathbf{e}}_r+r\stackrel{\dot{}<!--\; \dot{}\; -->}{\mathrm{\theta <!--\; \theta \; -->}}{\mathbf{e}}_t,}$

where e_r and e_t are the radial and tangential unit vectors directed relative to the vector from M to m. Use this to simplify the formula for work of gravity to,

${\displaystyle W=-<!--\; -\; -->{\int <!--\; \int \; -->}_{t_1}^{t_2}\frac{GmM}{r^3}(r{\mathbf{e}}_r)\cdot <!--\; \cdot \; -->(\stackrel{\dot{}<!--\; \dot{}\; -->}{r}{\mathbf{e}}_r+r\stackrel{\dot{}<!--\; \dot{}\; -->}{\mathrm{\theta <!--\; \theta \; -->}}{\mathbf{e}}_t)\mathrm{d}t=-<!--\; -\; -->{\int <!--\; \int \; -->}_{t_1}^{t_2}\frac{GmM}{r^3}r\stackrel{\dot{}<!--\; \dot{}\; -->}{r}\mathrm{d}t=\frac{GMm}{r(t_2)}-<!--\; -\; -->\frac{GMm}{r(t_1)}.}$

This calculation uses the fact that

${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}{r}^{-<!--\; -\; -->1}=-<!--\; -\; -->r^{-<!--\; -\; -->2}\stackrel{\dot{}<!--\; \dot{}\; -->}{r}=-<!--\; -\; -->\frac{\stackrel{\dot{}<!--\; \dot{}\; -->}{r}}{r^2}.}$