4.6: Perturbation Methods

4.6 Perturbation Methods

We have on several occasions mentioned that truly Keplerian orbits are essentially nonexistent and have given methods for analyzing some of the perturbations to Keplerian orbits that are important in spacecraft and mission design. Perturbation theory forms an elaborate structure in astrodynamics and classical celestial mechanics and, indeed, comprises much of the current literature in the latter subject. Such topics are completely beyond the scope of this text. However, many of the results cited earlier are due to perturbation theory, and a brief outline of this topic is in order.

Perturbation methods are broadly divided into special and general theories. Special perturbation theory is ultimately characterized by the direct numerical integration of the equations of motion due to a dominant acceleration and one or more small perturbing accelerations. As with all numerical analysis, results are unique to the given case, and it is often unclear how to extrapolate the results of one situation to another case of interest.

General perturbation analysis, historically the first approach to be developed, proceeds as given earlier, except that the perturbing accelerations are integrated analytically, at least to some given order of accuracy. Because closed-form integration of given perturbing accelerations will rarely be possible, series expansion to a desired order of accuracy is used to represent the perturbation, and the series integrated term by term. Analytical results are thus available, and broader applications and more general conclusions are possible. Nearly all important results have been obtained through general perturbation methods; on the...