The orbital period of an object is, quite simply, the time taken for that object to orbit another. N.B. The mathematics below assume point masses and circular orbits. As the planets in the solar system are neither (although they do have centres of mass, these calculations would not work for them. There are, however, some objects in the solar system which do have circular orbits, for example Triton, one of the moons of Neptune.
For the gravitational force:
F = G M1M2 / r2
Where G is the gravitational constant, M1 and M2 are the masses of objects 1 and 2, and r is the separation of these objects.
For centripetal forces:
F = M1ω2r
Where M is the mass of the orbiting object, ω is the angular velocity and r is once more, the separation.
As we know the Gravitational force is keeping the object in orbit, we can say it is equal to the centripetal force (from here, only circular orbits!), therefore:
G M1M2 / r2 = M1ω2r
or
G M2 / r3 = ω2
Which is equal to:
ω = √( G M2 / r3 )
Gives ω in terms of radians of orbit per second. There are 2π radians in an orbit, so 2π/ω is equal to seconds per orbit, which can easilly be converted into longer time periods.
When dealing with two orbiting bodies A and B, you can find the ratio of orbital periods by finding the ratio ωA:ωB, so that if one orbital period is known, the other can be found. To do this calculation, only the distances need to be known:
ωA = √( G M2 / rA3 )
ωB = √( G M2 / rB3 )
ωA/ωB = √( rA3/rB3 )
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