Fundamentals of Satellite Navigation - Understanding GPS: Principles and Applications - page 39

Once the eccentric anomaly has been computed, the mean anomaly is given by
Kepler's equation

MEe E =− sin (2.9)

As stated previously, the importance of transforming from the true to the mean
anomaly is that time varies linearly with the mean anomaly. That linear relationship
is as follows:

µ

( )

MM −= tt − (2.10)

0 0

a

3

where M 0 is the mean anomaly at epoch t 0 , and M is the mean anomaly at time t .
From Figures 2.9 and 2.10, and (2.8) and (2.9), it can be verified that M = E = = 0

at the time of perigee passage. Therefore, if we let t , (2.10) provides a transfor-

mation between mean anomaly and time of perigee passage:

=

µ

( )

M =− τ − t (2.11)

0 0

a

3

From (2.11), it is possible to characterize the two-body orbit in terms of the
mean anomaly, M 0 , at epoch t 0 , instead of the time of perigee passage τ . GPS makes

use of the mean anomaly at epoch in characterizing orbits.
GPS also makes use of a parameter known as mean motion , which is given the
symbol n and is defined to be the time derivative of the mean anomaly. Since the
mean anomaly was constructed to be linear in time for two-body orbits, mean
motion is a constant. From (2.10), we find the mean motion as follows:

dM
dt

µ 3

n

de =

=

a

t 0 ).
Mean motion can also be used to express the orbital period P of a satellite in
two-body motion. Since mean motion is the (constant) rate of change of the mean
anomaly, the orbital period is the ratio of the angle subtended by the mean anomaly
over one orbital period to the mean motion. It can be verified that the mean anom-
aly passes through an angle of 2

From this definition, (2.10) can be rewritten as M − M 0 = n ( t −

radians during one orbit. Therefore, the orbital

period is calculated as follows:

3

2 π a

P == 2 π (2.12)

n µ

Figure 2.11 illustrates the three additional Keplerian orbital elements that
define the orientation of the orbit. The coordinates in Figure 2.11 could refer either
to an ECI or to an ECEF coordinate system. In the case of GPS, the Keplerian
parameters are defined in relation to the ECEF coordinate system described in Sec-
tion 2.2. In this case, the xy -plane is always the Earth's equatorial plane. The fol-