Interstellar Objects from Planetary Scattering: An Analytical Solution

The gravitational scattering of planetesimals by a planet—arguably the primary channel for producing interstellar objects—remains a fundamental challenge within the circular restricted three-body problem (CR3BP). Although the underlying dynamics are deterministic, the extreme sensitivity of orbital-element changes to each encounter’s exact distance and orientation renders the process highly chaotic. Consequently, most investigations have relied exclusively on numerical integrations, especially once particles cross the planet’s orbit. Here, we introduce a novel analytical framework employing a patched-conic approximation to model the collective scattering of an ensemble of test particles. By averaging over all possible flyby parameters, we derive explicit expressions for the drift and diffusion coefficients of the normalized orbital energy. We then solve the resulting Fokker–Planck equation to obtain a closed-form solution for the time evolution of the particle distribution. The characteristic scattering timescale emerges naturally, scaling as ${P_{p}}{M_{p}^{2}}$, where $P_{p}$ and $M_{p}$ are the planet’s orbital period and mass ratio. Our analytical solution constitutes a universal law that can be applied to any exoplanetary system to estimate ejection rates and the velocity distribution of interstellar objects. This work provides a fast alternative to expensive simulations, opening a new avenue for studying how interstellar objects are distributed across the galaxy.