Normalized nodal distance-based J2-perturbed ground-track adjustment with apsidal-altitude constraints

Abstract

This paper presents a novel analytical approach based on normalized nodal distance to obtain the entire set of solutions for the J2-perturbed ground-track adjustment problem under apsidal-altitude constraints. First, by introducing general nodes and considering apsidal-altitude and maximum fuel constraints, a conversion method is proposed to convert these constraints into equivalent linear inequality constraints on normalized nodal distance. A relation curve between post-maneuver apsidal altitudes is also provided. Next, two explicit analytical expressions are derived separately based on Lagrange's expansion for two crucial transcendental equations, one for the time-of-flight concerning the argument of latitude and the other for the impulse concerning the normalized nodal distance. Then, by obtaining a starting node, this constrained ground-track adjustment problem can be transformed into a simple distance assignment problem. Combining with the inequality constraints, the entire solution set concerning the normalized nodal distance can be obtained immediately. Additionally, for the case of an empty solution set, a method for determining the relative longitude coordinates of no-solution intervals is also proposed. The case study demonstrates the effectiveness and accuracy of the proposed method and finds that for a fixed departing point, the most fuel-efficient time-of-flight window is not always the maximum allowable window within the specified time range.