  Active Brownian engines rectify energy from reservoirs composed of
self-propelling non-equilibrium molecules into work. We consider a class of
such engines based on an underdamped Brownian particle trapped in a power-law
potential. The energy they transform has thermodynamic properties of heat only
if the non-equilibrium reservoir can be assigned a suitable effective
temperature consistent with the second law and thus yielding an upper bound on
the engine efficiency. The effective temperature exists if the total force
exerted on the particle by the bath is not correlated with the particle
position. In general, this occurs if the noise autocorrelation function and the
friction kernel are proportional as in the fluctuation-dissipation theorem. But
even if the proportionality is broken, the effective temperature can be defined
in restricted, fine-tuned, parameter regimes, as we demonstrate on a specific
example with harmonic potential.
