A satellite orbiting a much more massive central body — a moon around a planet, a planet around a star — is really a two-object system interacting only through gravity. When the satellite's mass is negligible compared to the central object's mass, the central object's own motion is negligible too: for all practical purposes, it just sits there while the satellite moves around it.
What governs that motion are the same conservation laws you've used all unit — energy and angular momentum — now applied to a system bound by gravity instead of a spring, a collision, or a rigid connection.
Toggle between a circular orbit and an elliptical orbit to see which quantities stay constant throughout the motion.
| Quantity | Constant? |
|---|---|
| Total mechanical energy (E) | ✓ constant |
| Angular momentum (L) | ✓ constant |
| Gravitational PE (U) | ✓ constant |
| Kinetic energy (K) | ✓ constant |
Recall from Unit 2 that a satellite in a circular orbit has its centripetal acceleration provided entirely by gravity, which links the orbital period and radius to the central body's mass: T² = 4π²r³/GM. That relationship still holds here — it's the circular-orbit special case of the same gravitational interaction driving everything in this lesson.
Before we can talk about a satellite's energy, we need a zero point. By convention, the gravitational potential energy of a satellite–central-object system is defined to be zero when the satellite is an infinite distance from the central object:
For a satellite in a circular orbit specifically, there's a fixed relationship between its kinetic energy and the system's potential energy — you can derive it by setting the gravitational force equal to the centripetal force requirement. The result:
That lets you write the system's total energy — kinetic plus potential — in terms of just one of them:
A 1000 kg satellite circles an Earth-mass central body (M ≈ 5.97 × 10²⁴ kg). Slide the orbital radius from low Earth orbit toward geostationary altitude and watch U, K, and Etotal all move together.
A 1000 kg satellite orbits Earth (M ≈ 5.97 × 10²⁴ kg) in a circular geostationary orbit at r = 4.22 × 10⁷ m. Find the system's total mechanical energy.
What if a satellite has just enough speed to break free of the central body entirely? The escape velocity is defined as the speed at which the mechanical energy of the satellite–central-object system is exactly zero:
Solving that equation for vesc gives the escape velocity from a distance r away from a central body of mass M:
Adjust the central body's mass and the launch distance from its center to see how escape velocity compares to the speed needed for a circular orbit at that same distance.
vesc = √2 · vorbit — escaping takes about 41% more speed than staying in a circular orbit at that same distance.
Find the escape velocity for an object launched from Earth's surface (M ≈ 5.97 × 10²⁴ kg, R ≈ 6.37 × 10⁶ m).