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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
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(1) |
where a is the semimajor axis  and G is the gravitational constant. The angular frequency is
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(2) |
This result is Kepler's third law, and is often written in the simple form
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(3) |
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,
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(4) |
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 orbit, the speed of orbit is
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(5) |
so
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(6) |
For the general case of an elliptical orbit,
E may be evaluated at the pericenter of the particle's orbit, where
 . Therefore, from (0)
where a is the semimajor axis of the orbit. The energy per unit mass is then
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(9) |
 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
© 1996-2007 Eric W. Weisstein
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