Fundamentals of Satellite Navigation - Understanding GPS: Principles and Applications - page 38

Although the Keplerian integrals of two-body motion use time of perigee pas-
sage as one of the constants of motion, an equivalent parameter used by the GPS sys-
tem is known as the mean anomaly at epoch. Mean anomaly is an angle that is
related to the true anomaly at epoch, which is illustrated in Figure 2.9 as the angle ν.

After defining true anomaly precisely, the transformation to mean anomaly and the
demonstration of equivalence to time of perigee passage will be shown.
True anomaly is the angle in the orbital plane measured counterclockwise from
the direction of perigee to the satellite. In Figure 2.9, the true anomaly at epoch is =

∠ PFA . From Kepler's laws of two-body motion, it is known that true anomaly does

not vary linearly in time for noncircular orbits. Because it is desirable to define a
parameter that does vary linearly in time, two definitions are made that transform
the true anomaly to the mean anomaly, which is linear in time. The first transforma-
tion produces the eccentric anomaly, which is illustrated in Figure 2.10 with the true
anomaly. Geometrically, the eccentric anomaly is constructed from the true anom-
aly first by circumscribing a circle around the elliptical orbit. Next, a perpendicular
is dropped from the point A on the orbit to the major axis of the orbit, and this per-
pendicular is extended upward until it intersects the circumscribed circle at point B .
The eccentric anomaly is the angle measured at the center of the circle, O , counter-
clockwise from the direction of perigee to the line segment OB . In other words, E =

∠ POB . A useful analytical relationship between eccentric anomaly and true anom-

aly is as follows [14]:

1
1

−
+

e
e

1
2

E = 2 arctan tan ν (2.8)

B

A

r

E

ν

O P

F

Figure 2.10 Relationship between eccentric anomaly and true anomaly.