Nonlinear Dynamical Friction in a Gaseous Medium

Abstract

Using high-resolution, two-dimensional hydrodynamic simulations, we investigate nonlinear gravitational responses of gas to, and the resulting drag force on, a very massive perturber M p moving at velocity V p through a uniform gaseous medium of adiabatic sound speed a ∞ . We model the perturber as a Plummer potential with softening radius r s , and run various models with differing A = GM p /(a 2 ∞ r s ) and M = V p /a ∞ by imposing cylindrical symmetry with respect to the line of perturber motion. For supersonic cases, a massive perturber quickly develops nonlinear flows that produce a detached bow shock and a vortex ring, which is unlike in the linear cases where Mach cones are bounded by low-amplitude Mach waves. The flows behind the shock are initially non-steady, displaying quasi-periodic, overstable oscillations of the vortex ring and the shock. The vortex ring is eventually shed downstream and the flows evolve toward a quasi-steady state where the density wake near the perturber is in near hydrostatic equilibrium. We find that the detached shock distance δ and the nonlinear drag force F depend solely on η = A/(M 2 − 1) such that δ/r s = η and F/F lin = (η/2) −0.45 for η > 2, where F lin is the linear drag force of Ostriker (1999). The reduction of F compared with F lin is caused by front-back symmetry in the nonlinear density wakes. In subsonic cases, the flows without involving a shock do not readily reach a steady state. Nevertheless, the subsonic density wake near a perturber is close to being hydrostatic, resulting in the drag force similar to the linear case. Our results suggest that dynamical friction of a very massive object as in a merger of black holes near a galaxy center will take considerably longer than the linear prediction.

Dynamical friction in a magnetized gas

“…Numerical simulations have confirmed the results of the linear analysis for the drag force (e.g. Sánchez‐Salcedo & Brandenburg 1999; Kim & Kim 2009). The standard approach has been extended to cases with a circular‐orbit perturber (Kim 2010), with double perturbers (Kim, Kim & Sánchez‐Salcedo 2008) and with accelerated motion in a straight line (Namouni 2010).…”

Planet-disc interaction in highly inclined systems

“…The linear equations are valid in the limit that α, |β|, |∇ρ 0 /ρ 0 |≪ 1. As shown by Kim & Kim (2009), this is justified as long as the dimensionless non‐linearity parameter η is less than unity. For an Earth‐like planet on a 90° orbit, we have The background steady state is a stratified disc in the z direction with , where H is the thickness of the disc.…”

Gravitational drag on a point mass in hypersonic motion through a gaseous medium

“…Tanaka & Haiman (2009) combined the prescriptions of Ostriker (1999) and Escala et al (2004) into a formula that is used as a prescription of the gaseous drag on black holes in numerical simulations. In order to isolate the physical reason of the failure of Ostriker’s formula, Kim & Kim (2009) and Kim (2010) carried out axisymmetrical simulations of a massive body in rectilinear orbit with different values of the strength of the gravitational perturbation due to the body as measured by where r s is the softening radius of the Plummer perturber. They find that the functional form of the gravitational drag is not so peaked as the linear theory predicts and conclude that the discrepancy between the numerical and Ostriker results are most likely due to the non‐linear effect.…”

“…They find that the functional form of the gravitational drag is not so peaked as the linear theory predicts and conclude that the discrepancy between the numerical and Ostriker results are most likely due to the non‐linear effect. It is important to note that in the simulations of Escala et al (2004), Kim & Kim (2009) and Kim (2010), the perturber simply provides a smooth gravitational potential and does not hold any absorbing surface. Without any absorbing inner boundary condition, a hydrostatic envelope with front‐back symmetry is formed near the perturber.…”