Total Energy of an Orbiting Satellite

A satellite in a circular orbit of radius $r$ possesses both kinetic energy (from its orbital motion) and gravitational potential energy (from its position in the Earth's gravitational field); adding the two together gives its total mechanical energy in orbit.

Kinetic energy. Using the orbital speed found in Section 7.11, $v_o = \sqrt{GM/r}$,

Potential energy. From Section 7.8, directly,

Total energy. Adding these two together:

Two features of this result deserve comment. First, the total energy is exactly HALF the potential energy in magnitude, and always negative -- and this negative sign is the single most important feature of the result: it is the signature of a gravitationally bound system. A satellite with $E < 0$ simply does not have enough total energy to reach infinity (where, by definition, $U=0$) with any kinetic energy left over at all; it is permanently trapped in some closed orbit around the Earth unless energy is deliberately added to it (for instance, by firing a rocket engine). A total energy of exactly zero would correspond to the limiting escape trajectory of Section 7.10 (just barely reaching infinity with zero speed left over); a positive total energy would correspond to an unbound, hyperbolic trajectory that escapes to infinity with speed to spare -- neither of which is a closed orbit at all.

Second, the magnitude of this total energy, $GMm/2r$, is exactly the binding energy of the satellite: the minimum extra energy that would have to be supplied to it, from outside, to raise its total energy from its current negative value up to exactly zero and thereby let it escape to infinity. …