Two of the previous posts about orbits are particularly relevant to this one. Canonical units are described at the end of this post; the geometry of elliptical orbits is described in this post.

I propose to show you how to use “Kepler’s Equation” before I derive it. As we use it, we will see some of the things that enter into the derivation; just as importantly for some of us, we will see why the equation is useful. The equation actually has two parts… it is usually written

M = E – e Sin[E],

where M is called the **mean anomaly**, but that’s not much use unless we know M:

M = n t,

where t is the time since periapse passage (i.e. since the last time the object was at periapse), and

.

okay… but what is that? Well, the period of an elliptical orbit is

,

so

.

In other words, n is the average angular speed in radians per second, averaged over one orbit (or any number of complete orbits).

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