Acoustic Wake in a Singular Isothermal Profile: Dynamical Friction and Gravitational-wave Emission

Abstract We consider the motion of a circularly moving perturber in a self-gravitating, collisional system with a spherically symmetric density profile. We concentrate on the singular isothermal sphere, which, despite its pathological features, admits a simple polarization function in linear response theory. This allows us to solve for the acoustic wake trailing the perturber and the resulting dynamical friction, in the limit where the self-gravity of the response can be ignored. In a steady state and for subsonic velocities v p &lt; c s , the dynamical friction torque <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>φ</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∝</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> is suppressed for perturbers orbiting in an isothermal sphere relative to the infinite, homogeneous medium expectation F φ ∝ v p . For highly supersonic motions, both expectations agree and are consistent with a local approximation to the gravitational torque. At fixed resolution (a given Coulomb logarithm), the response of the system is maximal for Mach numbers near the constant circular velocity of the singular isothermal profile. This resonance maximizes the gravitational-wave (GW) emission produced by the trailing acoustic wake. For an inspiral around a massive black hole of mass 10 6 M ⊙ located at the center of a (truncated) isothermal sphere, this GW signal could be comparable to the vacuum GW emission of a black hole binary at subnanohertz frequencies when the small black hole enters the Bondi sphere of the massive one. The exact magnitude of this effect depends on departures from hydrostatic equilibrium and on the viscosity present in any realistic astrophysical fluid, which are not included in our simplified description.