Probability P(<v>) that objects have capture-permitting velocities

Determine the probability P(<v>) that an interstellar object (e.g., a rogue planet) crossing the capture cross section near the L1/L2 channels has a velocity magnitude within the ranges (as set by Jacobi constants C≲C2) required to yield permanent weak capture about the Sun under the planar circular restricted three-body model with PMW and S.

Background

The paper provides an overall probability estimate for permanent weak capture as a product of geometric and environmental factors, but it isolates a key velocity-dependent factor P(<v>) as unknown.

The authors note that P(<v>) is an observable that could be constrained by surveys for rogue planets (e.g., via JWST), directly impacting quantitative capture-rate predictions.