Kepler's laws

Kepler's planetary laws of motion tell us many important things. Among them: the outer planets move more slowly.

Kepler discovered these laws after many months of painstaking calculations, starting from the data compiled through astronomical observations. He may or may not have used logarithms, which were back then the latest and greatest mathematical novelty.

One important discovery: The planets usually move on ellipses (only in rare cases do they move on almost perfect circles).

Another important discovery: The planets move faster while they are near the sun. Watch the animation to the left to observe this!

This animation is actually a real simulation: It solves Newton's equations of motion for the planet, inside your browser! Newton discovered those about 70 years after Kepler had found the laws mentioned here. Kepler's laws follow from Newton's equations.

From the modern perspective, Newton's equations are not very complicated. Actually, in your browser, it is just four simple lines of program code that make the planet move in the right way. Still, they are very powerful: the same equations can be used to describe the whole solar system with all its planets and moons and asteroids and comets. When you wait long enough, you see that the motion of the solar system is chaotic (small changes can have big effects later on!). So there is complexity from simplicity.

Notes for the close observer: You may notice that the ellipse on which the planet moves will tilt slowly over time. This is not a consequence of Newton's equations and not a part of Kepler's laws. It is simply a result of the numerical errors in our very simple scheme of solving the equations. So, not the fault of Kepler or Newton (rather the 'fault' of Euler, though he knew very well that his way of solving the equations was only approximate). There are much better ways by now, and they only need a few more lines of code.

Florian Marquardt 2020