# How does Kepler’s improved III law allow determining the masses of planets with satellites and the mass of the Sun?

According to Kepler’s III law, it is possible to determine the ratio between the mass of a planet and the mass of the Sun if the planet has at least one satellite and its distance from the planet and the period of revolution around it are known.

T ^ 2 (M + m) / t ^ 2c (Mc + mc) – a ^ 3 / ac ^ 3

where М, m, mс are the masses of the Sun, the planet and its satellite, T and tc are the periods of the planet’s revolutions around the Sun and the satellite around the planet, and and ac are the distances of the planet from the Sun and the satellite from the planet, respectively. From the equation follows

(M / m +1) / (1 + mc / m) = tc ^ 2 * a ^ 3 / T ^ 2 * ac ^ 3

The M / m ratio for all planets is very large; the ratio m / mc is very small (except for the Earth and the Moon, Pluto and Charon), and it can be neglected. The M / m ratio is easy to find from the equation.