3.4 Perburbed Satellite Motion

In the two body problem (or Keplerian motion) we got the basic equation:

which integration formally gives the equation

But the assumed ideal conditions to get the satellite motion in the two body problem is only a first approximation. In general we have to take into account an additional acceleration produced from the resulting perturbing forces:

The perturbing forces and the corresponding accelerations are:

1. Due to the non-spherically and inhomogeneous mass distribution within the earth (central body), . The equatorial bulge of the Earth has two main effects: a) the non-radial component of the force, producing a torgue which results in the rotation of the line of nodes (1.2 deg/month for GPS satellites). And b), twice-per-orbit harmonic perturbation, producing as well the rotation of the major axis in the orbital plane.
   
2. Due to other celestial bodies, mainly the sun ( ) and the moon ( ).
3. Earth and oceanic tides , .
4. Atmospheric drag .
5. Direct and earth-reflected (albedo) solar radiation pressure , .

The effect of the perturbations on the orbit can be computed numerically, directly on the satellite coordinates or we can assume that the effect is that now the orbital elements are time dependent: analytical integration, named variation of constants.

In this last approach (which is taken in the navigation message to transmit the perturbation corrections to the orbit with the osculating elements):

3-hours orbit 3-days orbit

2 km 14 km

Further harmonics 50-80 m 100-1500 m

Solar & Lunar grav. 5-150 m 1000-3000 m

Body tides - 0.5-1.0 m

Ocean tides - 0.0-2.0 m

Solar rad. press. 5-10 m 100-800 m

Albedo - 1.0-1.5 m

Effects of perturbations on GPS satellites after 4 hours:

geopotential Moon pressure

e 1600 m 5 m 140 m 5 m

i 800 m 5 m 80 m 2 m

4800 m 3 m 80 m 5 m

1200 m 4 m 500 m 10 m

Such parameters are provided in the GPS navigation message, which are described in the GPS Inter Control Document (ICD).

Typically there is an upload per satellite once per day, containing ephemerides predicted for 14 days, but they degrade if they are not updated (new satellites incorporate cross-links measurements making feasible extending the quality of the autonomous orbits.

NIMA and International GPS Service (IGS) networks compute postprocessed orbits at decimeter-level of accuracy. IGS computes as well ultra-rapid (accurate predicted) orbits with typical errors of few decimeters.

In particular each satellite broadcast its own ephemeris, and a coarse set (almanac) for the overall set of satellites for rapid acquisition purposes, which are updated each several days. The 7-parameter set broadcasted (the 6 keplerian elements plus the line of nodes right ascension drift) provides a typical error of 1-2 km, enough for constellation visibility computation purposes.