TABLE OF CONTENTS Perturbation methods are a study topic in the solution of differential equations and have many applications in the analysis of science and engineering problems. In this application, it is the detailed analysis of the variant characteristics of the Keplerian orbital elements. In Sec. 2.2.3, we showed that the Keplerian orbital elements can be solved using the differential equations of motion for a space vehicle in orbit about a planet with a central gravity field. Equation (2.48) is repeated here for continuity in the discussion and is written as follows:
The Keplerian orbital elements are represented in functional notation at the defining epoch by
Likewise, in Sec. 2.2.4, we showed that the space vehicle position and velocity states can be propagated in its orbit to a new epoch time t by
and
respectively.
When simulating the orbital motion of a space vehicle using Cowell s method, we can include the perturbing accelerations due to the nonspherical gravity harmonics and other accelerations from forces such as aerodynamic drag. We can write the equations of motion for the space vehicle as follows:
where the 
i are the components of all of the other perturbing accelerations combined. Unlike the solution for the Keplerian orbital elements, as described in Sec. 2.2.4 and by Eqs. (4.133) and (4.134), there is no closed-form solution for Eq. (4.135). Numerical methods must be used, which can require large amounts...
