  The Unruh effect states that a uniformly linearly accelerated observer with
proper acceleration $a$ experiences Minkowski vacuum as a thermal state in the
temperature $T_{\text{lin}} = a/(2\pi)$, operationally measurable via the
detailed balance condition between excitation and de-excitation probabilities.
An observer in uniform circular motion experiences a similar Unruh-type
temperature $T_{\text{circ}}$, operationally measurable via the detailed
balance condition, but $T_{\text{circ}}$ depends not just on the proper
acceleration but also on the orbital radius and on the excitation energy. We
establish analytic results for $T_{\text{circ}}$ for a massless scalar field in
$3+1$ and $2+1$ spacetime dimensions in several asymptotic regions of the
parameter space, and we give numerical results in the interpolating regions. In
the ultrarelativistic limit, we verify that in $3+1$ dimensions
$T_{\text{circ}}$ is of the order of $T_{\text{lin}}$ uniformly in the energy,
as previously found by Unruh, but in $2+1$ dimensions $T_{\text{circ}}$ is
significantly lower at low energies. We translate these results to an analogue
spacetime nonrelativistic field theory in which the circular acceleration
effects may become experimentally testable in the near future. We establish in
particular that the circular motion analogue Unruh temperature grows
arbitrarily large in the near-sonic limit, encouragingly for the experimental
prospects, but the growth is weaker in effective spacetime dimension $2+1$ than
in $3+1$.
