SATELLITE SYSTEMS - GPS Satellite Surveying - page 65

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coordinate system can be readily computed from X by applying 2.34. We can then
readily compute spherical latitude and longitude ( ) and the trajectories of the

satellites on the sphere. For reasons of convenience, we express the mean motion
of the satellites in revolutions per day,

φ , λ

. The longitude difference between
consecutive equator crossings can then be computed from

n ¯ = n/ ω

1

1
¯

∆λ = π − (3.64)

n

Ta ble 3.1 also lists the change in longitude of the trajectory over a 24-hour period,
de noted by . The number in parentheses on the graphs indicates the number of days

pl otted. In all cases the inclination is i

δλ

= 65°. The maximum and minimum of the

−

tra jectories occur at a latitude of i and i , respectively.

¯ =

2, applies to GPS because the satellite orbits twice per
(si dereal) day. Case 2 has been constructed such that the trajectories intersect the
eq uator at 90°. In case 3 the point at which the trajectory touches, having common
ve rtical tangent, and the point of either maximum or minimum have the same longi-
tu de. The mean motion must be computed from a nonlinear equation, but

Case 1, specified by n

[65

Lin
—
0.8
——
No
PgE

n> 1is
va lid. In case 4 the satellite completes one orbital revolution in exactly one (sidereal)
da y. Case 5 represents a retrograde motion with

¯

n< 1 but with the same properties as
ca se 3. In case 6 the common tangent at the extrema is vertical. The interested reader
m ay verify that

¯

tan − 1 cos i

n sin − 1 sin

sin φ 1

¯

φ
sin i

λ = sin 2 i − (3.65)

sin 2

[65

− φ

and

sin 2 i

sin 2 d φ cos φ − φ

= n ¯ (3.66)

d λ n cos i ¯ − cos 2 φ

is valid for all cases.

3. 1.3 Satellite Visibility and Topocentric Motion

The topocentric motion of a satellite as seen by an observer on the surface of the earth
can be readily computed from existing expressions. Let X S denote the geocentric
position of the satellite in the celestial coordinate system (X). These positions could,
for example, have been obtained from (3.45) in the case of normal motion or from the
integration of perturbed orbits discussed below. The position X S can then be readily
transformed to crust-fixed coordinate system (x), giving x S by applying (2.34). If
we further assume that the position of the observer on the ground in the crust-fixed
coordinate system is x P , then the topocentric coordinate difference

∆ x = x S − x P (3.67)