  We consider the motion of a circularly-moving perturber in a
self-gravitating, collisional system with 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 steady-state and for subsonic velocities
$v_p<c_s$, the dynamical friction torque $F_\varphi\propto v_p^3$ is suppressed
for perturbers orbiting in an isothermal sphere relative to the infinite,
homogeneous medium expectation $F_\varphi\propto v_p$. 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 $10^6 M_\odot$ located at
the center of a (truncated) isothermal sphere, this GW signal could be
comparable to the vacuum GW emission of the black hole binary at sub-nanohertz
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.
