Periodic Orbits in the Elliptical Relative Motion Problem with Space Surveillance Applications

Abstract

Periodic orbits with respect to an object in an eccentric Keplerian reference orbit can be found in a variety of ways, including the use of Tschauner-Hempel equations and orbital element differences, which both admit linearized solutions, as well as through direct analyses of two orbits. An alternative parameterization of the last approach is proposed in the current study, using an inertial frame and simple geometrical constructs inherent to the relative motion problem. In this manner, an intuitive and straightforward characterization of periodic orbits is established that retains the nonlinear dynamics. The resulting multidimensional space that defines the periodic orbits is surveyed, archiving 336 orbits and their characteristics, and direct comparisons are made with the Tschauner-Hempel equations to assess the linear region of validity. Applications focus on resident space object (RSO) surveillance and circumnavigation orbits. An orbit's effectiveness is analyzed in terms of object coverage using coverage figures of merit, including the concept of "RSO tracks", the analogy of ground tracks on the bodyfixed surface of an RSO. Nomenclature a = semimajor axis, LU C = chief body-fixed frame e = chief eccentricity vector e = chief eccentricity e D = deputy eccentricity H = Hill frame or local-vertical-local-horizontal frame M = number of grid points distributed over the chief's surface N = number of points evaluated along a trajectory n = chief mean motion, rad · TU −1 P = unit vector directed along the chief eccentricity vector P = percentage of chief surface covered by the deputy P = perifocal framê Q =Ŵ ×P B Q A = rotation matrix from the A frame to the B frame B q A = quaternion describing the orientation of the A frame with respect to the B framê R = unit vector directed along the chief position vector R = Earth-centered rotating frame r = inertial position vector, LU r = magnitude of inertial position vector, LÛ S =Ŵ ×R T = orbital period, TU v = inertial velocity vector, LU · TU −1 v = magnitude of inertial velocity vector, LU · TU −1 W = unit vector normal to the plane of motion of the chief orbit, along the angular momentum vector x, y, z = deputy coordinates in the Hill frame, LU α = deputy position in-plane colatitudelike angle relative to the chief, rad β = deputy position out-of-plane longitudelike angle relative to the chief, rad δ = precedes a deputy quantity defined relative to the chief λ = longitude, rad μ = gravitational parameter of primary body, LU 2 · TU −3 ν = chief true anomaly, rad σ = standard deviation, LU ϕ = latitude, rad B ω A = angular velocity of the A frame relative to the B frame, rad · TU −1 B ω A = magnitude of B ω A , rad · TU −1 Subscripts C = quantity associated with the chief D = quantity associated with the deputy ds = inertially dual-axis spin attitude mode NL = quantity computed using the full, nonlinear equations of motion for Keplerian orbits np = inertially nadir-pointing attitude mode ns = inertially nonspinning attitude mode s = angular velocity spin component about t...