LAUSR

Hyperbolic encounters play a crucial role in diverse dynamical systems, from asteroid flybys ($\mu \lesssim \times {{10}^{ - 6}}$) to galaxy encounters ($0.3 \lesssim \mu \le 0.5$). This paper proposes an analytical framework for material exchange in the hyperbolic restricted three-body problem, showing that the opening of the neck region near L1​ is governed by a time-dependent energy boundary. By introducing the novel libration point energy curve (LPEC) as an analytical generalisation of the zero-velocity curve, a global energy criterion is established for material exchange between primary bodies without requiring extensive numerical integration. It is found that the derived critical velocity threshold provides a clear analytical condition for the transition between confined and exchangeable motion, while the novel concept of conditional Hill stability introduces a combined time and energy constraint determining when exchange remains dynamically feasible. Analysis reveals that increasing mass ratio reduces the conditional Hill stable domain, whereas a larger eccentricity expands it in both time and energy dimensions. In asteroid flybys, it is found that feasible spacecraft trajectories occur only when the velocity remains near the critical velocity threshold​, with periapsis being the optimal epoch for asteroid exploration. Analytical results in galaxy encounters demonstrate that higher mass ratios and periapsis passages enhance material exchange, consistent with the development of tidal bridges between galaxies. These findings provide an analytical framework that links the energy topology of hyperbolic encounters with observable transfer phenomena, offering new insight for both asteroid mission design and tidal structure formation in galaxy encounters.