Question

According to Kepler's third law of planetary motion, the period of revolution of a planet is related to the planet's what?

Answer 1

Kepler's third law relates the orbital period of a planet, or moon to its semi major axis distance.

Explanation:

Kepler's first law states that orbits are ellipses. The third law relates the orbital period `T` to the semi-major axis distance `a`. The semi-major axis being half the distance of greatest width of the ellipse.

Kepler wrote the law as a proportionality.

`T^2 prop a^3`

Newton used his theory of gravity to find the constant of proportionality.

`T^2 = (4 pi^2)/(G(M+m))a^3`

Where `G` is the gravitational constant, `M` is the mass of the body being orbiting and `m` is the mass of the orbiting body.

In the case of a solar system the mass of the sun `M` is significantly greater than the mass of a planet `m`. The Sun contains over 99% of the mass of the solar system. So, the value of `m` can be ignored to give.

`T^2 = (4 pi^2)/(GM)a^3`

For our solar system if the value of the period `T` is measured in years and the value of the semi-major axis `a` is measured in astronomical units AU then.

`(4 pi^2)/(GM)=1`

Hence for everything orbiting the Sun.

`T^2 = a^3`