## Vimppq l 2pUs 1 2p qMs rniis ue2335

where p is an index that ranges from 0 to l, q is an index associated with a function that is dependent on the orbit eccentricity, and ue is the rotation rate of the primary body. The orbit element change with time is dependent on eccentricity and inclination functions. The eccentricity-dependent function has a leading term that is dependent on e'9'; thus, for small eccentricity orbits, q is usually considered to range between —2 and +2. The characteristics of the orbit perturbations caused by individual degree and order terms in the gravity field are summarized in Table 2.3.4, where the categories are defiled by the nature of V>lmpq. Note

Perturbation Categories and Sources

Category

Source

Comments

Secular Long period Short period m-daily Resonance

Even-degree zonals Odd-degree zonals All gravity coeffici ents Order m terms Order m terms l-2p= 0; l-2p+q=0; m=0 l-2p=0; l-2p+q=0; m=0

l-2p+q=0 l-2p=0; l-2p+q=0; m=0 (l-2p)w+(l-2p+q)M = -m(il — We )

that the period of the perturbation is . An example of resonance is the geostationary satellite, where M = uie.

A special class of perturbed orbits, known as frozen orbits, have proven useful for some applications. For these orbits, even though the inclination is not critical, J2 and J3 interact to produce cUs = 0 (Cook, 1966; Born et al, 1987) and the mean perigee remains fixed at about 90° or 270° in an average sense. Frozen orbits are useful in applications where it is desired that the mean altitude have small variations for a given latitude. They are required if it is necessary to have an exactly repeating ground track. Otherwise, the time to go between two latitudes will vary as the perigee rotates, and the ground tracks will not repeat.

The nodal period is the time required for a satellite to complete successive equatorial crossings, such as ascending node to ascending node. The nodal period, Pn, is

0 0