Analysis of the maximum hyperbolic residual velocity for small thrust capture orbits

In response to the problem of small thrust capture in capture orbit design, considering the limitations of thrust magnitude, this analysis studies the maximum hyperbolic remaining velocity for achieving capture, namely the maximum speed of the spacecraft when entering the sphere of influence of the target body, enabling the spacecraft to be captured in an orbit around the target with a given semi-major axis under small thrust. First, a trajectory optimization model to solve for the maximum remaining velocity was established, and the optimal control problem was transformed into a two-point boundary value problem using the indirect method for solution. The characteristics of co-state variables were analyzed, and the two-point boundary value problem was modeled as a nonlinear equation with the eccentricity of the target orbit and true anomaly as variables. Taking Mars capture as an example, the trends of maximum remaining velocity with variations in thrust acceleration magnitude, capture time, and long-axis of the orbit were revealed, analyzing the upper and lower limits of capture time and their corresponding optimal control, presenting the multi-pass optimal trajectory for the spacecraft to enter the sphere of influence and being captured into the target orbit. Finally, the application of maximum hyperbolic remaining velocity analysis in mission design and future research prospects were summarized.