In the last section we showed that the orbits are ellipses
following the equation

where is the semi-major and the semi-minor axis.

It is often more useful to consider an ellipse in a coordinate
system whose origin is at the center of the ellipse. The **
eccentric anomaly** of a point on the ellipse is the angle
shown in the picture below.

We will next derive the connections between **true anomaly** and
**eccentric anomaly**. Either angle can be used to describe the
properties of an ellipse.
In the above illustration, the point has coordinates and , where is the true anomaly. The distance
between the center and the focus is given by

Therefore, the -coordinate of the center is . The -coordinate of the point can be computed using or to get:

For the -coordinate we observe:

i.e.

To continue we see that

Hence,

To continue, we apply the double angle formula in the equation for to get

Combining this with the equation for in terms of we get

In the same way we get

Combining these we have

(22) |

The final angular measure to compute is the **mean anomaly**
. To do this we note that the satellite does not move with a
constant angular velocity. If it were moving at a constant
velocity then

Assuming that is the time at which the satellite moves through the perigee, the mean anomaly at time is the angle

The mean anomaly is specifically nice since it is a geometric quantity which is directly proportional to the time. To get the position of the satellite at a given time we need to compute either the eccentric or the true anomaly from the time. This relationship is called Kepler's equation.