Analytical Optimality Criteria for Low-Energy Capture via the Weak Stability Boundary

Low-energy capture via the weak stability boundary (WSB) is a critical technique for reducing the terminal orbit-insertion cost in planetary exploration missions. However, under the influence of multibody chaotic dynamics, the relationship between achievable propellant savings and high-dimensional design parameters remains inadequately characterized. Thus, identifying globally optimal capture opportunities is challenging, as large-scale numerical searches are computationally expensive and lack convergence guarantees. This paper presents a layered-bounded search (LBS) approach that incorporates analytical convergence conditions to efficiently identify the discrete globally optimal feasible solution under the prescribed discretization. The main contributions are threefold: First, a closed-form expression for the propellant savings of WSB capture relative to direct capture is derived, establishing an explicit correspondence between design parameters and propellant savings. Second, leveraging this expression, analytical optimality criteria for low-energy capture are developed, providing the theoretically optimal combination of design parameters. Finally, analytical convergence conditions are formulated, and the proposed LBS strategy initiates numerical refinement in the neighborhood of the theoretical optimum, rapidly locating the globally feasible optimum under real dynamics and substantially reducing computational cost. The proposed criteria and method are validated in the Sun–Jupiter and Sun–Venus systems, confirming their ability to provide a convergent, interpretable guide for designing low-energy capture trajectories.