Gravitational potential energy and escape velocity

Gravitational potential energy

Near Earth's surface, gravitational potential energy is often written $U=mgh$. In astronomy, distances change too much for that approximation. For two masses $M$ and $m$ separated by distance $r$, the gravitational potential energy is

U(r)=-Grac{Mm}{r}

The zero of energy is chosen at infinite separation. The negative sign means gravity is attractive: a bound object has less energy than it would at infinity.

Force from potential energy

The radial force is related to potential energy by

F_r=-rac{dU}{dr}

Using $U=-GMm/r$ gives

F_r=-Grac{Mm}{r^2}

where the negative sign indicates attraction toward smaller $r$.

Total mechanical energy

The total mechanical energy of an orbiting body is

$E=K+U$

For speed $v$,

K=rac{1}{2}mv^2

so

E=rac{1}{2}mv^2-Grac{Mm}{r}

Bound orbits have $E<0$. Escape trajectories have $Egeq0$.

Circular orbit energy

For a circular orbit, gravity supplies centripetal force:

v_c=sqrt{rac{GM}{r}}

The kinetic energy is then

K=rac{1}{2}mrac{GM}{r}=Grac{Mm}{2r}

Since

U=-Grac{Mm}{r}

we get total energy

E=-Grac{Mm}{2r}

A circular orbit is bound with energy equal to half the potential energy.

Elliptical orbit energy

For an elliptical orbit with semi-major axis $a$, the total energy is

E=-Grac{Mm}{2a}

This is similar to the circular result with $r$ replaced by $a$. The energy depends on orbit size, not eccentricity. Different ellipses with the same semi-major axis have the same total energy.

Escape velocity

Escape velocity is the minimum speed needed to reach infinity with zero speed remaining. Set total energy equal to zero:

0=rac{1}{2}mv_{esc}^2-Grac{Mm}{r}

Solving gives

v_{esc}=sqrt{rac{2GM}{r}}

Notice that

$v_{esc}=sqrt{2}v_c$

at the same radius. Escape speed does not depend on the mass of the escaping object, assuming air resistance and propulsion after launch are ignored.

Potential wells

A massive body creates a gravitational potential well. To escape, an object must climb out of this well. More massive and more compact bodies have higher escape velocities. For black holes, the escape speed at the Schwarzschild radius equals $c$ in a Newtonian analogy, though the full explanation requires general relativity.

The big idea

Gravitational energy gives a powerful view of orbits. Bound systems have negative total energy, circular orbits satisfy $E=-GMm/(2r)$, and escape occurs when total energy reaches zero. Escape velocity follows directly from energy conservation and scales as $sqrt{GM/r}$.