Time-explicit representation of relative motion between elliptical orbits

The classic treatment of rendezvous mechanics and other problems involving the relative motion of two spacecraft assumes a circular reference orbit, allowing a simple closed-form description of the motion. A solution is developed using an elliptical reference orbit, expanding the state transition matrix in powers of eccentricity, while retaining the explicit time dependence of the three-dimensional motion. The solution includes separate matrix elements for first- and second-order terms in eccentricity and for both Cartesian and cylindrical coordinates. Assessment of the maximum errors in position and velocity components over one complete revolution of the reference satellite shows that the solution is accurate for practical purposes with eccentricities in the range 0-0.3. An example application is given for the proposed laser interferometer space antenna gravity wave experiment.