Kepler’s Laws

Johannes Kepler was one of the last pre-telescope astronomers, plotting the movements of the stars and planets across the heavens, taking advantage of the clear, dark skies of the late 1500s/early 1600s before cities became clouded with light and smog in the industrial revolution. Under the mentorship of (and money from) Tycho Brahe, Kepler analysed the movements of the planets, attempting to prove, and improve upon, the Copernican heliocentric model.

I made a youtube video explaining each of his Laws, which I’ve linked here, but I will also explain it briefly in this post for easier reference!

So Kepler described the motions of the planets in 3 laws:

1. The Orbit of a Planet is an ellipse with the sun at one focus

To understand this, and all of the other laws, we first have to understand some features of an ellipse- a stretched out circle. The two foci (F1 & F2) are equidistant from the centre of the ellipse (C), and the sun is at one of them (F1). The other is just a random point in space- it doesn’t have any physical property, as they are just a way to mathematically describe an ellipse.

Other features are the semi major axis (longest radius) and the semi minor axis (shortest radius) which is less commonly used. Along the plane of the semi major axis also lies perihelion (the point on the orbit closest to the sun) and aphelion (the point on the orbit furthest from the sun)

2. A line joining a Planet and the Sun sweeps out equal areas in equal intervals of time

The further from the sun, the weaker the force of gravity pulling the planet in, so it travels more slowly. This means closer to aphelion, it covers a shorter distance in the same amount of time than when closer to perihelion.

Mars is one of the most eccentric planets, so when modelled as a circle as Copernicus did, it didn’t match up with its movements across the sky, because there is a larger difference in speed between perihelion and aphelion, and it is closer to earth than the other planets, so the effect was more noticeable. However the new model with the orbit as an ellipse does work!

3. The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis of its orbit

This is what I spend most of the video explaining. Kepler discovered the relationship by analysing the data for each planets year in earth days and relative distance from the sun.

He found an equation a^3= k T^2

Where a is the semi major axis, T is the time it takes to complete one orbit (the year) and k is the constant that you multiply by.

Although he could find the value of k (I talk about this more in the video), he didn’t know where it came from, fundamentally- it was up to Newton to explain that, using his laws of motion and gravitation

You can then check this with any orbiting body! I’ve chosen Jupiter

It is also true for moons, but you replace the mass of and distance from the sun with whatever the central body it’s orbiting. So if you want to check for Europa’s orbit around Jupiter, you need to measure in relation to Jupiter. Remember if you want to try with a system like Pluto and Charon, the distance you measure is to the centre of mass, not the distance from Pluto, as the barycentre/ centre of mass is above Pluto’s surface.

Let me know in the comment which planet/moon you tried it with and if it worked!