## Kepler's Third Law

8.6 - Be able to use Kepler’s third law in the form: a constant__T__= a constant

^{2}r

^{3}

where

*T*is the orbital period of an orbiting body and

*r*is the mean radius of its orbit

8.7 - Understand that the constant in Kepler’s third law depends inversely on the mass of the central body

**Law #3. The orbital period of a planet squared is proportional to its mean distance from the Sun cubed, when different planets are compared.**

There is a relationship between the distance of a planet and the time it takes to orbit the Sun.

**Terms you need to know first...**

**SQUARE** = a number multiplied by itself e.g. 2 x 2 = 4. 4 is the square of 2

**CUBE** = a number multiplied by itself and then multiplied by itself again. e.g. 3 x 3 x 3 = 27. 27 is the cube of 3.

**T** = Period/ Time it takes to orbit the Sun

**r** = mean radius from Sun

Although we use the Sun as our example, this equally applies to any primary body e.g. the Earth and calculating the orbit of the Moon around it.

Kepler's Third Law is this: The square of the Period is approximately equal to the cube of the Radius.

**T**^{2} = r^{3}

^{2}= r

^{3}

##### The role of mass

The constant above depends on the influence of mass. Gravitation attraction depends on mass. So we must take into account the mass of the primary body (e.g. the Sun) and add to the mass of the secondary body (e.g. the planet). This amount is not often substantial because the Sun has many times the mass of the planets it is not often used but we can apply it to a Moon orbiting a Planet. However even then we would have to account for the Sun's mass also. The inverse square law applies to working out the gravitational influence.

###### Summary

**T**^{2} = r^{3}

^{2}= r

^{3}

Note: Incidentally some astronomers, textbooks and websites uses "P" for Period rather than "T" for Time. Don't worry if you see this being used elsewhere

###### Animation

###### Mix & Match

###### Links

University of Rochester Newtonian Gravitation and the Laws of Kepler

Vik Dhillon, University of Sheffield Newton's derivation of Kepler's laws

Skywise Unlimited Kepler's Three Laws of Planetary Motion

UBC's Interactive Mathematics Interactive Simulator