  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.
