Measuring the mass of a planet
 

Let a planet P with orbital semi-major axis a, sidereal period of revolution T and mass m possess a satellite P₁ that moves in an orbit about P with semi-major axis a₁ and sidereal period of revolution T₁. Let the masses of the Sun and satellite be M and m1 respectively.
(1)

Then by equation (1), we have for the planet and the Sun,
(2)
where μ = G(M + m).
For the planet and satellite we have
(3)
where μ1 = G(m + m₁).
Dividing equation (3) by equation (2), we obtain

or

We may write

But we have seen that the ratio m/M is much less than unity and for the satellites in the Solar System, the ratio of their masses to that of their primary is also much less than unity. Hence,
(4)
Only two of the planets, Mercury and Venus, have no known satellites. Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto each have one or more moons and so can have their masses in terms of the Sun’s mass calculated by measuring the satellites’ orbital semi-major axes and sidereal periods of revolution. Other methods have to be adopted in the cases of Mercury and Venus. The masses of Mercury and Venus were first obtained from their perturbing effects on the orbits of other planets; the mass of Venus is now accurately known from observations of artificial satellites of Venus. The triple flypast of Mercury by the spacecraft Mariner 10 has improved our knowledge of that planet’s mass. The mass of Pluto, derived from its moon Charon, is of the order of 0·003 Earth masses.