High-fidelity linearized J2 model for satellite formation flight

With the recent flurry of research on satellite formation flight, a need has become apparent for a set of linearized equations that describe the relative motion of satellites under the effect of the J(2) geopotential disturbance. In the past, Hill's linearized equations of relative motion have been used to analyze relative motion between satellites, but they were not designed to capture the effect of the J(2) potential. A new set of constant-coefficient, linearized, differential equations of motion is derived. Although surprisingly similar in form to Hill's equations, they are able to capture the effects of the J(2) potential. A numerical. simulator is employed to check the fidelity of the equations. It is shown that with the appropriate initial conditions, the new linearized equations of motion have periodic errors of only 0.4% over all inclinations, radii, and cluster configurations. The new linearized equations of motion also allow for insights into the effects of the J(2) disturbance on a satellite cluster including an effect called tumbling, where the cluster as a whole rotates about the vector normal to the orbital plane of the reference orbit. The differential J(2) effects are also analyzed, and the cluster configurations that minimize this effect can be determined from the new equations. Overall, a new high-fidelity set of linearized differential equations is produced that is well suited to model satellite relative motion in the presence of the J(2) potential.