New Methods of Celestial Mechanics

Our first objective is to reformulate the age-old problem of a satellite orbiting a primary (of a substantially bigger mass) by means of quaternion algebra.1 In subsequent chapters, this will facilitate a relatively simple construction of the problem’s iterative solution. Here we also review some details of constructing the unperturbed solution.

Starting with the unperturbed solution, we now build a general, arbitrarily accurate analytic solution to the perturbed Kepler problem, assuming that the perturbation is proportional to a small parameter denoted ε.1 This is done in an iterative manner, where the crucial step is constructing an ε-accurate solution; subsequent iterations then follow the same pattern. The chapter involves a lot of tedious but inevitable algebra, and needs to be read in detail only by those who want to understand the derivation of all key formulas. Anyone interested only in applications of the technique should proceed directly to the chapter’s last section (the technique’s summary).

In terms of actually solving (1.29), it is convenient to treat separately two distinct cases:

1. The autonomous case, in which the perturbing force εf has no explicit time dependence.

2. The case of εf being an explicit (we will assume periodic) function of time.

In either case, we find it convenient to work in the orbit’s Kepler frame, introduced in (1.16); we have yet to elaborate on some of the details.

In this chapter we derive expressions for common perturbing forces. Note that these are defined in terms of (1.27), i.e. per unit of satellite’s mass.

We compute the effect of perturbations of the type discussed in Section 3.1. Since the masses of all planets are small (compared to Sun’s mass), only first-order effects are considered.1 We also utilize yet another approximation, facilitated by the fact that the resulting changes (circulation, libration and nutation) of the orbital elements are substantially (several orders of magnitude) slower than the orbital periods of the perturbing planets. As a result, the time-dependent coefficients of the Q andWdo not affect the perturbed planet’s long-run behavior and can be dropped out of the corresponding differential equations. This is equivalent to replacing each perturbing force by its long-run average (computed by averaging over one period of each of the perturbing planets). The perturbing forces thus become effectively time-independent, implying that only the autonomous-case formulas will be required in this chapter. In general, one must be cautious when applying this approximation, and refrain from using it when any one of the perturbing forces is commensurable with the motion of the perturbed planet (meaning that the two periods are in a simple ratio, such as 2:5).

The fact that Earth is not a perfect sphere (it is slightly flattened and otherwise distorted, as discussed in a previous chapter), introduces extra terms into the expansion of its gravitational potential, which in turn affect the motion of near-Earth (i.e. artificial) satellites. The first one of these terms (due to Earth’s oblateness) is by far the largest, and therefore quite often the only one considered (finding the corresponding solution is referred to as the main problem of satellite motion)1. At the same time, it is the quintessential example of time-independent perturbations, and the main topic of this chapter. The remaining terms are of two basic types: those corresponding to axially symmetric (e.g. pear shaped) distortions are called zonal harmonics, and remain time-independent; those which vary with longitude (and thus rotate with Earth) are called tesseral harmonics. For these, we only mention a few basic results.

In this chapter we investigate complexities of Moon’s motion [11], [49], [53]. The perturbing body is Sun itself; the magnitude of the perturbing force is thus relatively large. Here, we can no longer use the averaging principle (its error would skew some of our results by as much as a factor of two), thus, the time-dependent version of our formulas will be occasionally required. Traditionally, this problem has always been considered a benchmark for any new technique.

Resonances occur when period of the perturbing force is commensurable with the perturbed body’s orbital motion (2/1 resonance corresponds to a perturbing force whose one cycle is completed in two orbits of the perturbed body; two cycles are completed during one orbit in the case of 1/2 resonance, etc.). They constitute a rather special (and difficult) category of time-dependent perturbations, as we demonstrate using several examples of an asteroid perturbed by Jupiter’s gravitational pull. We start with the 1/1 resonance, which has some unique features, not shared by other resonances.

In this chapter we briefly discuss most of the remaining perturbing forces. None of them have an explicit time dependence, which simplifies the resulting differential equations for the orbital elements (making them autonomous). Furthermore, the results are developed only to the first-order-in-ε accuracy. With the exception of the distant-source radiation, all perturbing forces discussed here act within the orbital plane; we can thus treat all quantities as complex (no quaternions needed) — this also eliminates θ and φ from our orbital-element list. Finally (with the same single exception), the perturbing forces involve no special, fixed direction, which eliminates ψ from the right hand side of the resulting differential equations, making them functions of a and β only.

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