Clohessy-Wiltshire Equations Modified to Include Quadratic Drag

Abstract

The relative-motion equations of a spacecraft in the vicinity of a satellite in an orbit that is not highly eccentric but decays as a result of drag are investigated. Because the initial orbit is not highly eccentric, theequations of motion of both the satellite and the spacecraft can be approximated by simpler equations. Some transformations are then applied to the equations of relative motion. If the drag is quadratic in the magnitude of the velocity and varies inversely with the distance from the center of attraction, the equations simplify further. There are some interesting consequences if the two objects in orbit have the same drag constant. In these cases, the transformed equations of relative motion generalize the Tschauner-Hempel equations and asymptotically approach an extension of the Clohessy-Wiltshire equations, modified to include the quadratic drag model. If the initial orbit is circular, the relative-motion equations become the new modified Clohessy-Wiltshire equations. The closed-form solution of these equations has similar structure to that of the Clohessy-Wiltshire equations. Among the potential applications of these new equations are fast preliminary studies for terminal rendezvous, station keeping, formation flying, and constellations of satellites.