Analytic solution to the dynamical friction acting on circularly moving perturbers

TL;DR

This paper derives an exact analytic solution for the dynamical friction experienced by a circularly moving perturber in a gaseous medium, revealing force behaviors across different Mach regimes and implications for numerical simulations.

Contribution

It provides the first analytic solution for circular-orbit dynamical friction, including force components and divergence analysis, extending understanding beyond linear motion models.

Findings

Steady-state is reached after one sound-crossing time.

Radial force dominates at high Mach numbers, inward and opposing motion.

Logarithmic divergence occurs in the supersonic regime, affecting short-distance force calculations.

Abstract

We present an analytic approach to the dynamical friction (DF) acting on a circularly moving point mass perturber in a gaseous medium. We demonstrate that, when the perturber is turned on at $t=0$, steady-state (infinite time perturbation) is achieved after exactly one sound-crossing time. At low Mach number $\mathcal{M}~\ll~1$, the circular-motion steady-state DF converges to the linear-motion, finite time perturbation expression. The analytic results describe both the radial and tangential forces on the perturbers caused by the backreaction of the wake propagating in the medium. The radial force is directed inward, toward the motion centre, and is dominant at large Mach numbers. For subsonic motion, this component is negligible. For moderate and low Mach numbers, the tangential force is stronger and opposes the motion of the perturber. The analytic solution to the circular-orbit DF suffers from a logarithmic divergence in the supersonic regime. This divergence appears at short distances from the perturber solely (unlike the linear motion result which is also divergent at large distances) and can be encoded in a maximum multipole. This is helpful to assess the resolution dependence of numerical simulations implementing DF at the level of Li\'enard-Wiechert potentials. We also show how our approach can be generalised to calculate the DF acting on a compact circular binary.