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In the last section we showed that the orbits are ellipses
following the equation
 |
(21) |
In this equation
is the angle of the semi-major axes
with the ordinate,
is called the true
anomaly. The angle
is also commonly denoted by
.
One focus sits at the origin of the coordinate system.
The earth is presumed to be at this focus. The perigee is the
point on the ellipse closest to this focus, the apogee, the point
which is farthest. The true anomaly is the angle between a point
on the ellipse and the perigee. Often
is replaced by
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.
Figure 2:
The relation between the true anomaly
and the eccentric anomaly
![\includegraphics[scale=0.5]{kepler.eps}](img143.png) |
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) |
A relation between the true and the eccentric anomaly.
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.
Next: Kepler's Equation
Up: Kepler's Equation
Previous: Kepler's Equation
Werner Horn
2006-06-06