### Kepler’s Laws of Planetary Motion in Motion

I’m not exactly the staunchest believer in UFO’s – although I won’t swear they don’t exist just because I’ve never seen one and have a hard time believing they do – but I respect and value greatly what is actually within our grasp due to mathematics and physics. I have to admit, and let’s face it folks, space is damn intriguing!

Johannes Kepler was one bloke – smart bloke, mind you – who had a respect for this kind of thing as well. He came up with what’s known as the Laws of planetary motion, and together with Newton’s pertinent theories and laws, are part of the basis of modern astronomy and physics. Here they are:

1. The Law of Orbits. The orbit of every planet is an ellipse with the Sun at one of the two foci.

2. The Law of Areas. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

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

So let’s break that down a bit. First of all, up until Copernicus, Kepler and a century later, Newton, proved that the orbits of planets were elliptical, everyone thought they were circular. Not only that, but the ancient Greeks and Chinese believed in a “geocentric model” with the Earth as the center of the universe, and everything else orbiting around it.

What’s an Ellipse?

Well, think of an ellipse as a flattened circle of sorts. It’s defined as “a plane curve such that the sums of the distances of each point in its periphery from two fixed points, the foci, are equal.” This means that if you were to draw 2 straight lines from any point on the orbit periphery to the 2 foci, generally forming a “V”, the consequent sum of both line segments would always be equal.

As the 2 foci in an ellipse move further apart, the ellipse is said to become more eccentric, and as the 2 foci move closer together, it begins resembling the shape of a circle. This explains why an orbit may look like a circle – because the 2 foci are so close together. This does not however, mean that it is a circle, as all planetary orbits are ellipses – not circles.

Ok, we know what an ellipse is, and due to Kepler’s first law, we also know that the Sun is one of the 2 foci. The second law is a bit easier to understand. This law is saying that as a planet orbits the Sun, an imaginary line between it and the Sun will sweep out the same exact area regardless of where on the orbit periphery it happens to be, in the same amount of time.

Huh? How’s that possible – what happens when the planet is on the opposite end of the orbit? Well, as you might have guessed, they speed up as they near the Sun, and slow down as they get further from it – thus preserving Kepler’s second law. Now the third law is yet another law rife with technical jargon and may need some deciphering.

What’s a Semi-major Axis?

A semi-major axis is simply half – or the radius – of the longest diameter of an ellipse – also known as the “major axis“. Obviously, you can’t just say radius, as depending on where in the ellipse you measure the diameter, the length will change. So the third law can be stated mathematically as R3/T2=K.

Where R is the semi-major axis and T is the period of time needed for one revolution, K is the constant, and has the same value for any planet or object orbiting the Sun, and equals roughly 2.5m3/day2. So as with any equation, you can use this to calculate related problems – in this case you can find out the average orbit radius of a planet if you knew the orbit period.