When a planets orbit a star, theoretically, their orbit may be circular. This case is dealt with under circular motion.

In reality, planets orbit in ellipses. An ellipse is a shape which has two foci (singular 'focus'). The total of the distances from any point on an ellipse to its foci is constant. All orbits take an elliptical shape, with the sun as one of the foci. As the planet approaches its star, its speed increases. This is because gravitational potential energy is being converted into kinetic energy.

A circle is an ellipse, in the special case when both foci are at the same point.

## Kepler's Third Law

Kepler's third law states that:

 “ The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. ”

Mathematically, for orbital period T and semi-major axis R:

$T^{2}\propto R^{3}$

This result was derived for the special case of a circular orbit as the fourth circular motion problem. The semi-major axis is the distance from the centre of the ellipse (the midpoint of the foci) to either of the points on the edge of the ellipse closest to one of the foci.

## Questions

Planet Mercury Venus Earth Mars
Picture
Mean distance from Sun (km) 57,909,175 108,208,930 149,597,890 227,936,640
Orbital period (years) 0.2408467 0.61519726 1.0000174 1.8808476

1. The semi-major axis of an elliptical orbit can be approximated reasonably accurately by the mean distance of the planet for the Sun. How would you test, using the data in the table above, that the inner planets of the Solar System obey Kepler's Third Law?

2. Perform this test. Does Kepler's Third Law hold?

3. If T2 α R3, express a constant C in terms of T and R.

4. Io, one of Jupiter's moons, has a mean orbital radius of 421600 km, and a year of 1.77 Earth days. What is the value of C for Jupiter's moons?

5. Ganymede, another of Jupiter's moons, has a mean orbital radius of 1070400 km. How long is its year?