Analytic Theory for High-Inclination Orbitsin the Restricted Three-Body Problem

Abstract

DOI: 10.2514/1.41628 This paper explores the analytical solution properties surrounding a hypothetical orbit in an invariant plane perpendicular to the line joining the two primaries in the circular restricted three-body problem. Assuming motion can be maintained in the plane, Jacobi’s integral equation can be analytically integrated, yielding a closed-form expression for the period and path of the third body expressed with elliptic integral and elliptic function theory. In this case, the third body traverses a circular path with nonuniform speed. In a strict sense, the in-plane assumption cannot be maintained naturally. However, the hypothetical orbit is shown to satisfy Jacobi’s integral equation and thetangentialmotionequation exactlyandtheothertwomotionequationsapproximatelyin bounded-averagedand banded sense. More important, the hypothetical solution can be used as the basis for an iterative analytical solution procedure for the three-dimensional trajectory where corrections are computable in closed form. In addition, the in-plane assumption can be strictly enforced with the application of a modulated thrust acceleration which is expressible in closed form. Presented methodology is primarily concentrated on halo-class orbits. Nomenclature a = radius of circular orbit C = Jacobi’s constant G = universal gravitational constant J = Jacobi’s function m1 = mass of first primary m2 = mass of second primary,m2 � m1 r = position vector of third mass relative to center of mass r1 = position vector of first primary relative to center of mass r2 = position vector of second primary relative to center of mass r12 = distance between two primaries � 1 = position vector of third mass relative to first primary � 2 = position vector of third mass relative to second primary ! = angular velocity vector of rotating fame