12.16 - Kepler’s third law
The law of periods: The square of the period of
an orbit is proportional to the cube of the
semimajor axis of the orbit.
Kepler’s third law, proposed in 1619, states that the period of an orbit around a central
body is a function of the semimajor axis of the orbit and the mass of the central body.
The semimajor axis a is one half the width of the orbit at its widest. In a circular orbit,
the semimajor axis is the same as the radius r of the orbit.
We illustrate this in Concept 1 using the Earth and the Sun. Given the scale of
illustrations in this section, the Earth’s nearly circular orbit appears as a circle.
The length of the Earth’s period í a year, the time required to complete a revolution
about the Sun í is solely a function of the mass of the Sun and the distance a shown in
Concept 2.
Kepler’s third law states that the square of the period is proportional to the cube of the
semimajor axis, and inversely proportional to the mass of the central body. The law is
shown in Equation 1. For the equation to hold true, the mass of the central body must
be much greater than that of the satellite.
This law has an interesting implication: The square of the period divided by the cube of
the semimajor axis has the same value for all the bodies orbiting the Sun. In our solar
system, that ratio equals about 3×10í^34 years^2 /meters^3 (where “years” are Earth years)
or 3×10í^19 s^2 /m^3. This is demonstrated in the graph in Concept 3. The horizontal and
vertical scales of the coordinate system are logarithmic, with semimajor axis measured
in AU and period measured in Earth years.
Orbital period
Time of one revolution
Kepler’s third law
Square of orbital period proportional to
cube of semimajor axisGraph of Kepler’s third law
Orbital size versus period for planets
orbiting Sun