New nonsingular forms of perturbed satellite equations of motion

Abstract

Vector calculus is used to derive two new forms of the equations of motion of perturbed satellites. Both forms are given in terms of nonsingular orbital elements in extended vector space that eliminate the ambiguity in the variation of retrograde orbits. In the first form, the orbital eccentricity, true anomaly, argument of latitude, and orbital inclination are substituted by four fast variables whose unperturbed periods are essentially equal to the orbital period. In the second form, these four fast variables are transformed to four slow variables such that the eccentricity vector variation is independent of the out-of-plane acceleration. The transformation matrix between these new nonsingular elements and Cartesian coordinates is given and perturbing accelerations due to the Earth and a third body are developed. Applications of the perturbation methods of multiple scale and averagings to the developed equations of motion are demonstrated. T HE satellite equations of motion can be expressed in various forms. A simple Cartesian form1'2 is widely used in connection with numerical integration. More sophisticated forms based on the variation of parameters developed by Lagrange3 are particularly useful in connection with perturbation theories. Lagrange's equations can be transformed to canonical forms3'5 or nonsingular noncanonical forms.6'7 The independent variable is usually the time but in some cases, e.g., Earth's zonal harmonic perturbations, it is advantageous to use an independent variable other than time8'10 (e.g., true anomaly or a similar fast angle). The purpose of this paper is to develop two new forms of Lagrange's equations of motion [Eqs. (12) and (19)] in terms of nonsingular orbital elements whose variations are determined by the components (Fr,Fe,FH) of a perturbing acceleration in the Euler-Hill rotating frame depicted in Fig. 1. Section II develops the equations of motion in terms of the fast variables [Eq. (12)]. This form is particularly useful for the application of perturbation method of Lindested-Poincare' and the method of multiple scale.11'13 Section III derives the equations of motion in terms of slow variables [Eq. (19)] in a form suitable for the application of the method of averaging,14 Lie transform,15'16 and Lie series.17 Section IV summarizes the transformation between the Cartesian and the nonsingular orbital elements useful for the evaluation of the initial conditions. Section V develops the form of the perturbing acceleration of the Earth and a third body in terms of the obtained variables. Sections VI and VII demonstrate the application of the method of multiple scale and the methods of averaging to the obtained equations of motion. II. Equations of Motion in Terms of Fast Variables This section starts with the equations of motion in the Cartesian form and applies vector calculus to obtain the equations of motion in terms of a set of fast nonsingular orbital elements. Following Newton's law of gravitation, the equations of motion of a satellite can be written in the vectorial form r = v