Dynamics equations of relative motion around an oblate earth with air drag

Under the perturbations of the J ₂ zonal harmonics term and air drag, to obtain a set of dynamics equations that have a simple and concise form and can be used to exactly propagate the relative motion of a chaser spacecraft with respect to a reference spacecraft in an arbitrary elliptical reference orbit, Newtonian Hamiltonian mechanics are adopted in the derivation process. On the basis of the principle of Newtonian mechanics, six first-order differential equations are derived to accurately describe the inertial motion of a reference spacecraft under the affect of J ₂ term and air drag. By using the principle of Hamiltonian mechanics, another six first-order differential equations are obtained to precisely propagate the relative motion together with inertial motion dynamics equations of the reference spacecraft, without the equation of the right ascension of ascending node Ω. Because the spherical gravitational potential, the J ₂ zonal harmonics gravitational potential, and air drag are axisymmetric about the Earth rotation axis and independent of the motion of Ω, the relative motion is independent of Ω. So Ω is unnecessary in the relative motion propagation. The affect of J ₂ zonal harmonics term and air drag on precession of the reference orbital frame also is analyzed. On the basis of derivation and analysis, the dynamics equation of relative motion around an oblate Earth with air drag are promoted to four application areas: A1, long-time relative motion prediction; A2, field of view analysis for relative measurement sensor; A3, the analysis of affect of coordinate frame precession on reference spacecraft attitude control; A4, the analysis of affect of of coordinate frame precession on attitude synchronization mission. Simulation results in application (1) confirm the validity and exactness of the dynamics equations derived in this paper. All applications indicate that the dynamics equations have a good prospect of application in formation flying mission.