Page 71 - Satellite Communications, Fourth Edition

Orbits and Launching Methods 51

only the magnitude of the position vector is required. From the geome-
try of the ellipse (see App. B), this is found to be

2
a(1 e )
r (2.23)
1 e cos
The true anomaly is a function of time, and determining it is one of
the more difficult steps in the calculations.
The usual approach to determining proceeds in two stages. First, the
mean anomaly M at time t is found. This is a simple calculation:

) (2.24)
M n(t T p
Here, n is the mean motion, as previously defined in Eq. (2.8), and T p
is the time of perigee passage. The time of perigee passage T can be
p
eliminated from Eq. (2.24) if one is working from the elements specified
by NASA. For the NASA elements,

M n(t T )
0
p
0
Therefore,
M 0
t (2.25)
T p 0 n

Substituting this in Eq. (2.24) gives

M M n(t t ) (2.26)
0
0
Consistent units must be used throughout. For example, with n in
degrees/day, time (t t ) must be in days and M in degrees, and M will
0
0
then be in degrees.
Example 2.12 Calculate the time of perigee passage for the NASA elements given
in Table 2.1.
Solution The specified values at epoch are mean motion n 14.23304826 rev/day,
mean anomaly M 0 246.6853°, and t 0 223.79688452 days. In this instance it
is only necessary to convert the mean motion to degrees/day, which is 360n.
Applying Eq. (2.25) gives
246.6853
T 223.7968452
14.23304826 360
223.74874044 days
Once the mean anomaly M is known, the next step is to solve an
equation known as Kepler’s equation. Kepler’s equation is formulated