## Kepler’s Laws Of Planetary Motion

Johanes Kepler

Johannes Kepler analysed the astronomical observations of Tycho Brahe. Tycho is remembered as being the most skilled observational astronomers of his time, but never

managed to describe his results mathematically. Kepler was a highly skilled mathematician, and it is said that he easily recognised the elliptical nature of the planetary orbits. This is not a trivial matter, given that the earth is also a moving object.

Kepler proposed three ‘laws’ of planetary motion. Modern astronomical observation shows that the paths of the planets are slightly disturbed by the gravitation of other nearby planets and by the effects of General Relativity. Otherwise, Kepler’s laws are still found to hold true.

# Kepler’s Three Laws

## Law 1

The orbital paths of the planets are shaped as ellipses. The Sun is located at one focus of the ellipse.

An ellipse is a very precisely defined geometric object. Mathematically, the shape is defined by:

$r=\frac{a(1-\epsilon^2)}{1+\epsilon\cos\theta}$

The terms in this equation are:

$r$ = the distance from the Sun.
$a$ = the semi-major axis (half the length of the long side of the ellipse).
$\epsilon$ = the eccentricity of the ellipse (how “squashed” the shape is).
$\theta$ = the angular distance traveled around the Sun.

This formula allows us to describe and predict the path of a moving planet. Some planetary orbits are more elliptic than others; some may be very (but not quite) circular. An ellipse has two foci (singular focus – these are a bit like the centre of a circle).

## Law 2

The imaginary line between the Sun and the planet sweeps out equal areas in equal amounts of time.

This simply means that when the planet is close to the sun, it moves much faster than when it is further away. This may seem obvious, but this is in fact a very precise statement which allows us to calculate how quickly a planet will be moving at another point in its orbital path.

## Law 3

The square of the orbital period is directly proportional to the semi-major axis cubed.

This is written mathematically:

$p^2=a^3$

Where:

$p$ = the orbital period (the time in Earth years to complete one orbit.
$a$ = in this context, the average distance from the Sun in AU.  The Earth is 1 AU from the Sun.

Notice that what seems to be a complicated statement is expressed very simply in the technical language of mathematics. When you read a science book, the equations are there to make things more simple. Don’t be scared of them.

This formula allows us to calculate how long a year is on another planet after we have measured how far away it is. Alternatively, we can time how long it takes a planet to return to its starting position, and calculate how far away it it from the sun. This is an example of how physics is amazing – you can measure the orbital time of a planet using a watch, and calculate how hot it is for the people that live there!

Happy Calculating!