Swept-Volume Geometry and Overlap Corrections in Protostellar Binary Accretion: A Kinematic Upper Bound and Its Limitations

This paper examines how orbital motion alone affects the ability of protostellar binary systems to encounter surrounding material. The analysis deliberately isolates geometry and kinematics, treating accretion as a swept-volume problem and ignoring all physical capture mechanisms such as gravity, gas dynamics, and thermodynamics. The central result is that orbital motion increases the effective path length through the medium, providing a purely kinematic upper bound on encounter capacity relative to a single object moving through the same environment. This enhancement is modest but systematic and exists independently of any detailed physical model of accretion. The study further demonstrates that for sufficiently close binaries, overlap between the regions swept by the two components can completely eliminate this kinematic advantage. This explains why numerical simulations of binary accretion do not show a clear geometric signature associated with orbital motion: observed enhancements are dominated by dynamical effects rather than encounter geometry. The framework is intended as a diagnostic tool. It establishes what kinematics alone can contribute to material encounter rates and clarifies where purely geometric reasoning breaks down, requiring physical processes to take over. The results place firm bounds on the role of orbital motion and help separate kinematic effects from genuinely dynamical mechanisms in studies of star formation.