You probably already know that the force of gravity is in inverse square to the distance between the two objects involved:

introduction to gravity

Analysing a circular orbit, where this force is constant, is easy. In the general case it is not however, and the orbit is often elliptical. This was one of the great questions of science in the 17th century. Kepler had shown that planets orbits are ellipses with the sun at one focus, but did not know why this should be the case.

He was of the opinion (as was everyone else of his time) that for an object to move, it must have a force pushing it from behind. It was suspected later that an inverse square force toward the focus (the sun) could be causing the elliptical orbits, but no-one could prove it. Isaac Newton proved it in 1687 in his book Philosophae Naturalis Principia Mathematica, and named the force gravity.

This article sets out the proof (probably not the same as Newton’s!) that if an object moves under the influence of a central inverse square force, its path is a conic section (don’t worry, conic sections will be explained later), also how in real life this relates to gravity.

First of all you’ll need to understand motion described in polar co-ordinates and conics. I know this site is called astronomy for beginners but you don’t mind if I put one complicated bit in do you?

Good. This one’s for you mathematicians and/or physicists out there, knowledge of calculus and a grounding in mechanics is needed to understand the following page (in particular you’ll need to know about conics, the polar equation of a conic and using radial unit vectors) that is, proof that an inverse square attractive force produces conic motion. If you don’t feel up to that you might be able to skip the proof and just have a look at the last page, the actual maths behind orbits.


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