The analysis of relative motion of two spacecraft in Earth-bound orbits is usuallycarried out on the basis of simplifying assumptions. In particular, the referencespacecraft is assumed to follow a circular orbit, in which case the equationsof relative motion are governed by the well-known Hill-Clohessy-Wiltshire (HCW) equations. Circular motion is not, however, a solution whenthe Earth's flattening is accounted for, except for equatorial orbits, where inany case the acceleration term is not Newtonian. Several attempts have beenmade to account for the J_2 effects, either by ingeniously taking advantage oftheir differential effects, or by cleverly introducing ad-hoc terms in the equationsof motion on the basis of geometrical analysis of the J_2 perturbing effects.Analysis of relative motion about an unperturbed elliptical orbit is thenext step in complexity. Relative motion about a J_2-perturbed elliptic referencetrajectory is clearly a challenging problem, which has received little attention.All these problems are based on either the HCW equations for circular referencemotion, or the de Vries/Tschauner-Hempel equations for elliptical referencemotion, which are both approximate versions of the exact equations ofrelative motion. The main difference between the exact and approximate formsof these equations consists in the expression for the angular velocity and theangular acceleration of the rotating reference frame with respect to an inertialreference frame. The rotating reference frame is invariably taken as the localorbital frame, i.e., the RTN frame generated by the radial, the transverse, andthe normal directions along the primary spacecraft orbit. Some authors havetried to account for the non-constant nature of the angular velocity vector, buthave limited their correction to a mean motion value consistent with the J_2 perturbationterms. However, the angular velocity vector is also affected in direction,which causes precession of the node and the argument of perigee, i.e., ofthe entire orbital plane. Here we provide a derivation of the exact equations ofrelative motion by expressing the angular velocity of the RTN frame in termsof the state vector of the reference spacecraft. As such, these equations arecompletely general, in the sense that the orbit of the reference spacecraft needonly be known through itsephemeris, and therefore subject to any force field whatever. It is also shown thatthese equations reduce to either the Clohessy-Wiltshire, or the HCW equations,depending on the level of approximation. The explicit form of the equations ofrelative motion with respect to a J_2-perturbed reference orbit is also introduced.
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