  A new, second-order solution in curvilinear coordinates is introduced for the
relative motion of two spacecraft on eccentric orbits. The second-order
equations for unperturbed orbits are derived in spherical coordinates with true
anomaly as the independent variable, and solved by the method of successive
approximations. A comparison of error trends against eccentricity and
inter-spacecraft separation is presented between the new solution and prominent
Cartesian, curvilinear, and orbital element based solutions from the
literature. The second-order curvilinear solution offers a thousand-fold
improvement in accuracy over the first-order curvilinear solution, and still
greater improvement over first- and second-order rectilinear solutions when
large along-track separations are present.
