Fundamentals of Satellite Navigation - Understanding GPS: Principles and Applications - page 40

lowing three Keplerian orbital elements define the orientation of the orbit in the
ECEF coordinate system:

i = inclination of orbit

= longitude of the ascending node

= argument of perigee

Inclination is the dihedral angle between the Earth's equatorial plane and the
satellite's orbital plane. The other two Keplerian orbital elements in Figure 2.11 are
defined in relation to the ascending node , which is the point in the satellite's orbit
where it crosses the equatorial plane with a z component of velocity (i.e., going

from the southern to the northern hemisphere). The orbital element that defines the
angle between the

+

x -axis and the direction of the ascending node is called the right
ascension of the ascending node (RAAN). Because the

+

x -axis is fixed in the direc-
tion of the prime meridian (0° longitude) in the ECEF coordinate system, the right
ascension of the ascending node is actually the longitude of the ascending node, .
The final orbital element, known as the argument of perigee, , measures the angle
from the ascending node to the direction of perigee in the orbit. Notice that

+

is

measured in the equatorial plane, whereas is measured in the orbital plane.
In the case of GPS satellites, the orbits are nearly (but not quite) circular, with
eccentricities of no larger than 0.02 and semimajor axes of approximately 26,560
km. From (2.12), we compute the orbital period to be approximately 43,080 sec-
onds or 11 hours, 58 minutes. The orbital inclinations are approximately 55° for the

Normal to the
orbital plane

z

Equatorial
plane

Direction of
perigee

i

y

Orbital
plane

Ω

Ascending
node

x

Figure 2.11 The three Keplerian orbital elements defining the orientation of the orbit.