Table of Contents

Orbital Mechanics

Student Activity

Kepler's 3rd Law

Complete the following chart which uses Kepler's 3rd Law to compute the semi-major axes of the orbital ellipses of the planets in the solar system.

Note: Kepler's 3rd law is derived from the expression

Force of gravity = centripetal force

Using Newtonian mechanics this can be written as

GMm/r² = mv²/r (eq.1)

The orbital velocity v can be calculated using the circumference of the orbit divided by the orbital period p.

For circular orbits (or nearly circular orbits) of radius r = a

v = 2a/p (eq.2)

Substituting the expression for v into the Newton's expression above and simplifying we get

M = 4²a³/Gp² (eq.3)

In the case of Kepler's discovery he used each planet's orbital radius in AUs and orbital period in years.

If the mass of the Sun is (arbitrarily) set equal to unity (1) then one can substitute into equation (eq.3) and get

4²/G = 1

then from expression (eq.3)

a³= p² (Kepler's 3rd Law)

The expression M = a³/p² is also a very useful form of Kepler's 3rd Law.

It can be used to determine the mass M (in solar mass) of a massive astronomical object which is orbited by a much less massive object at an orbital radius a (in AUs) and having an orbital period p (in years).