Elliptical Orbits – Deriving Kepler’s Equation

Here is that drawing again, showing the eccentric anomaly E and the true anomaly f. What we’ve done so far in this post and in that post is just use Kepler’s equation

M = E – e Sin E

to move between position and time on an elliptical orbit.

Let’s derive the equation using geometry and trigonometry. (Both Conway & Prussing, and Bate, Mueller & White – see my bibliography page – have this derivation.)


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Elliptical orbits: given time, find position

For a change, I decided to split two problems into two posts.

We have found time from position; now let’s reverse that and find position from time. The following problem is Example 2.1 on p. 31 of Prussing & Conway, “Orbital Mechanics” (see my bibliography page).

“An Earth satellite orbit has a semimajor axis a = 4R, and a perigee radius 1.5R, where R is Earth radius. Find the true anomaly at t=4 hours after perigee passage.”
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Elliptical orbits: given position, find time

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

n = \sqrt{\mu/a^3}\ .

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

T = 2\pi \sqrt{a^3/\mu}\ ,

so

n = 2 \pi / T\ .

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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Orbits: the elliptical orbit

I want to show you the geometry of an elliptical orbit.

Most of this post is reference, but note that it includes a calculation of the Hohmann transfer orbit between two circular orbits.

Let me start with a simple drawing:

We measure the angle \nu\ from the horizontal, counterclockwise. All our training in trigonometry says we should label the horizontal line as the x-axis — and I will subsequently do so, but it’s not essential.
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Two-body orbits: example from initial scalars

Introduction

I want to work out a problem similar to the previous one — but different.

Instead of being given the position and velocity vectors, we will be given their magnitudes, i.e. the distance (from the center of mass of the primary) and the speed — and the flight path angle.

These are enough to let us determine the scalar orbit; and that is what I want to show you how to do.
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Two-body orbits: example from initial vectors

Edit: 2011 Apr 29, mu = GM not Gm!

Introduction & Review

I am going to work a numerical example of the equations I worked out in the previous post. To be specific, I am going to assume we are given the position and velocity vectors for an object in orbit, at one instant of time.

Let me begin by doing what I should have done at the end of the previous post: here’s a summary of the previous post.

We took the vector differential equation

\ddot{\vec{r}} + \frac{\mu}{r^3}\ \vec{r} = 0
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Orbits: Vector derivations for 2-body orbits

Introduction

First, let’s take it for granted that the differential equation for the two-body problem (e.g. the earth orbiting the sun in an otherwise empty universe) is

\ddot{\vec{r}} + \frac{\mu}{r^3}\ \vec{r} = 0

where \vec{r}\ is the vector from the primary (e.g. the sun with mass M) to the secondary (e.g. the earth, with mass m) and \mu = G(M+m)\ ; and a simple r\ is the magnitude of the vector \vec{r}\ .

I want to derive

  • conservation of energy
  • conservation of angular momentum
  • a neat equation for r and v
  • the scalar equation for the orbit.

Furthermore, I want to use vector operations wherever possible. (I like vectors!)
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