Orbit -- from Eric Weisstein's World of Physics

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Orbit

For an elliptical orbit in the two-body problem with mass of the central body much greater than mass of the orbiting body , the orbital period is

where a is the semimajor axis Eric Weisstein's World of Math and G is the gravitational constant. The angular frequency is

This result is Kepler's third law, and is often written in the simple form

where n is the mean motion.

The total energy of an orbiting body is the sum of the kinetic energy K and the gravitational potential energy U,

where m is the mass of the body, v its speed, G is the gravitational constant, M is the mass of the central body, and r is the orbital distance.

For a circular Eric Weisstein's World of Math orbit, the speed of orbit is

so

For the general case of an elliptical Eric Weisstein's World of Math orbit,

E may be evaluated at the pericenter of the particle's orbit, where . Therefore, from (0)
where a is the semimajor axis Eric Weisstein's World of Math of the orbit. The energy per unit mass is then

Central Orbit, Clarke Orbit, Geostationary Orbit, Geostationary Transfer Orbit, Geosynchronous Orbit, High Earth Orbit, Kepler's Laws, Kepler's Third Law, Low Earth Orbit, Molniya Orbit, n-Body Problem, Planar Orbit, Polar Orbit, Restricted Three-Body Problem, Retrograde Orbit, Sun-Synchronous Orbit, Three-Body Problem, Two-Body Problem