A Tisserand-Based Reachability Criterion for Gravity-Assist Selection: Why a Single Venus Flyby Is Ineffective for Rendezvous with (16) Psyche

This study investigates why single planetary gravity assists fail to provide competitive impulsive architectures for rendezvous with the metallic main-belt asteroid (16) Psyche. A decade-long Lambert-based trajectory survey (2028–2037) is performed using a patched-conic model with powered gravity assists to compare direct Earth–Psyche, Earth–Mars–Psyche, and Earth–Venus–Psyche transfer architectures. The optimal direct transfer requires 9.65 km s−1, whereas Venus-assisted trajectories demand 15.3–15.7 km s−1. For Mars, the survey shows that no feasible near-ballistic solution exists within the entire decade: the low-energy Earth–Mars legs arrive at Mars with the hyperbolic excess velocity directed against the planet’s motion, so that redirecting it towards the asteroid belt requires turn angles of approximately 115–120°, far exceeding the ≈65° that Mars can deliver at its minimum allowable flyby periapsis, defined as the planetary mean radius plus a 250 km altitude margin. Two complementary analytical results explain this behaviour. A Tisserand-based reachability criterion shows that reaching Psyche requires a minimum hyperbolic excess speed of approximately 9.8 km s−1 at Venus but only 4.1 km s−1 at Mars, an energetic penalty that is fixed before the encounter because an unpowered flyby preserves the magnitude of the excess velocity. A directional feasibility condition then shows that satisfying the scalar criterion is necessary but not sufficient: the excess-velocity vector delivered by the inbound leg must also lie within the attainable deflection cone of the assisting planet, a condition that Venus fails energetically and Mars fails geometrically. Together, the two conditions form a predictive analytical framework for screening gravity-assist planets prior to computationally intensive optimisation, and they explain why low-thrust propulsion—rather than any single impulsive flyby—underpins the trajectory of NASA’s Psyche mission.