Two Bodies Orbiting Each Other
In the more general case where the bodies need not be a large one and a small one (the two-body problem), we define:
- the vector r is the position of one body relative to the other
- r, v, and in the case of an elliptic orbit, the semi-major axis a, are defined accordingly (hence r is the distance)
- μ = Gm1 + Gm2 = μ1 + μ2, where m1 and m2 are the masses of the two bodies.
Then:
- for circular orbits, rv2 = r3ω2 = 4π2r3/T2 = μ
- for elliptic orbits, 4π2a3/T2 = μ (with a expressed in AU; T in Earth years and M the total mass relative to that of the Sun, we get a3/T2 = M)
- for parabolic trajectories, rv2 is constant and equal to 2μ
- for elliptic and hyperbolic orbits, μ is twice the semi-major axis times the absolute value of the specific orbital energy, where the latter is defined as the total energy of the system divided by the reduced mass.
Read more about this topic: Standard Gravitational Parameter
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