Newton’s form of Kepler’s third law

Let two planets revolve about the Sun in orbits of semi-major axes a₁ and a₂, with periods of revolution T₁ and T₂. Let the masses of the Sun and the two planets be M, m₁ and m₂ respectively.

(1)

Then by equation (1),

where μ₁ = G(M + m₁). Also,

Hence, 
(2)
Kepler’s third law would have been written as

The only difference between this last equation and (2) is a factor k, where

Dividing top and bottom by M, we obtain

The greatest departure of k from unity arises when we take the two planets to be the most massive and the least massive in the Solar System. The most massive is Jupiter: in this case m1/M = 1/1047·3. Of the planets known to Newton, the least massive was Mercury, giving m₂/M = 1/6 200 000. Hence, to three significant figures, k = 1. Kepler’s third law is, therefore, only an approximation to the truth, though a very good one. Newton’s form of Kepler’s third law, namely equation (2), is much better.