Space methods - Physical Geodesy - page 259

rotates around the earth, but much slower than the satellite itself; a typical
period of ω is two months. Therefore, terms containing cos ω, sin ω ,orsin2 ω
are called long-periodic.
The first equation of (7-23) shows that the semimajor axis of the orbit
does not change secularly or long-periodically. The eccentricity and the in-
clination undergo long-period, but not secular, variations, whereas Ω and ω
change both secularly and long-periodically.
Equations (7-23) are linear in J 2 ,J 3 ,J 4 , ... . For practical applications,
nonlinear terms containing J 2 ,J 2 J 3 ,J 2 J 4 , etc., must also be taken into ac-
count, since J 2 is of the order of J 4 . The derivation of these nonlinear terms
is much more dicult, and their expressions are different in the various or-
bital theories that have been proposed. For these reasons, such expressions
will not be given here.
Equations (7-23), supplemented by certain nonlinear terms, can be used
to determine coecients J 2 ,J 3 ,J 4 , etc. Since the secular or long-periodic
variations ∆Ω , ∆ ω, ∆ e, ∆ i are known from observation for a sucient num-
ber of satellites, we obtain equations of the form

+ a 22 J 2 + a 23 J 2 J 3 + a 2 J 2 + a 3 J 3 + a 4 J 4 + ··· ··· = A,

+ b 22 J 2 + b 23 J 2 J 3 +

(7-24)

b 2 J 2 + b 3 J 3 + b 4 J 4 + ··· ··· = B,

. . . . . .

which can be solved for J 2 ,J 3 ,J 4 , ... . Since there can be only a finite
number of these equations, we must neglect all J n with n greater than a
certain number n 0 , which depends on the number of equations available, on
their degree of mutual independence, etc. This used to be a di culty with
this method, but it has been overcome long ago by least-squares collocation
(Moritz 1980 a: Sect. 21). For details see Schwarz (1976).
From (7-23) it is seen that the coecients of the J n depend essentially
on the inclination i . It is, therefore, important to use satellites with a wide
variety of inclinations, in order to get equations with a high mutual inde-
pendence.
Now the question arises which orbital elements are to be used for deter-
mining the coecients J n . The semimajor axis a clearly cannot be used at
all. As for the other elements, we must distinguish between coecients of
even and of odd degree n . The even coecients J 2 ,J 4 , ... can be determined
well from the regression of the node, ∆Ω, and the rotation of perigee, ∆ ω .
To see this, inspect (7-23). The even harmonics cause secular disturbances
of Ω and ω , which are much larger than the long-periodic effects of the odd
coecients, since J 3 ,J 5 , ... are multiplied by the small eccentricity e .