«Wo+1 JJfiRla "7/2(l - e2)"2^-1 sin^)(/ -/„) + © (3-43c)

= «„+ 2.06474X 10l4a"7/2(l - e2)'2^- | surt)(/-/0) + 0(/|) (3-43d)

where the units and variables are the same as those in Eq. (3-41) and «„ is the value of to at epoch /„. For an equatorial orbit, the line of apsides will rotate in the direction of motion and for a polar orbit it will rotate opposite the direction of motion. At ¿=63.435 deg, 2- fsin^O, and the line of apsides does not rotate.

Third-Body Interactions. Perturbations due to the oblateness of the Earth become less important with increasing distance from the Earth. However, as the distance from the Earth increases, perturbations from the gravitational force of the Moon and the Sun become more important Such third-body interactions are the source of the major orbital perturbations for interplanetary flight and, as is clear from the discussion of Lagrange point orbits in Section 3.3, dominate the motion entirely in some circumstances. Unfortunately, the problem of three interacting gravitational objects is intractable; even series expansions for small perturbations from Keplerian orbits generally have very small radii of convergence. A wide variety of special cases and approximation methods have been studied and are discussed in the references for this chapter. In practice, most work involving significant third-body interactions is done by numerical integration of the equations of motion.

To determine when simple two-body solutions are appropriate, it is convenient to divide space into approximate regions, called spheres of influence, in which various orbital solutions are nearly valid. Specifically, consider the case shown in

Fig. 3-20 of a spacecraft of negligible mass moving in the vicinity of two masses, m and M, where There exists an approximately spherical Region I of radius

Rx about m such that within this region the perturbing force due to M is less than t times the force due to m, where t< I is a parameter chosen to reflect the desired accuracy. Within Region I, the motion will be approximately that of a satellite in a Keplerian orbit about m. Similarly, there exists a Region III outside a sphere of radius R2 centered on m, such that outside this sphere the perturbing force due to m is less than c times the force due to M. Thus, in Region III the motion will be approximately that of a satellite in a Keplerian orbit about M. Within Region II, between Rt and R2, the gravitational force from both objects is significant.

Fig. 3-20. Spheres of Influence About the Moon for <=0.01. In Region 1, orbits are approximately Keplerian about the Moon; in Region III, they are approximately Keplerian about the Earth.

Simplifications such as the sphere of influence are not precise because the boundaries of the regions are rather arbitrarily defined and are not exactly spherical, and the magnitude of the perturbations within any of the regions is difficult to estimate. Nonetheless, it is a convenient concept for estimating where Keplerian orbits are valid. Approximate formulas for the radii Rt and R2 are given by (see, for example, Roy [1965]): 1. Region I/II Boundary:

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