Smoluchowski-Poisson system with a perfectly absorbing central body: steady-state analytical solution in one dimension

Abstract

Self-gravitating systems have been extensively studied over the past century due to their unique thermodynamic properties. While these systems were primarily used to model astrophysical structures such as star clusters and galaxies or systems with negligible friction, recent decades have seen a renewed interest in the high-friction limit. Notably, Chavanis demonstrated that overdamped self-gravitating Brownian systems, governed by the Smoluchowski-Poisson equations, can describe the dynamics of chemotactic bacterial populations. Furthermore, he argued that these equations could model dust particles in protoplanetary nebulae, where friction and turbulence are dominant. In this study, we solve for the steady state of a Smoluchowski-Poisson system in the presence of a perfectly absorbing central body and a particle reservoir at the outer radius. This configuration could model a chemotactic bacterial population attracted to a central source coated with a lethal toxin. We find that, in the steady state, the governing equation reduces to the Airy equation; consequently, the density and mass profiles are described by Airy functions. We also demonstrate that the flux through the system is constant and uniquely determined by the central body mass μ and the gravity-temperature ratio η. Finally, we verify our analytical results by solving the governing equations numerically.