  The cores of dense stars are a powerful laboratory for studying feebly
coupled particles such as axions. Some of the strongest constraints on
axionlike particles and their couplings to ordinary matter derive from
considerations of stellar axion emission. In this work we study the radiation
of axionlike particles from degenerate neutron star matter via a
lepton-flavor-violating coupling that leads to muon-electron conversion when an
axion is emitted. We calculate the axion emission rate per unit volume
(emissivity) and by comparing with the rate of neutrino emission, we infer
upper limits on the lepton-flavor-violating coupling that are at the level of
$|g_{ae\mu}| \lesssim 10^{-6}$. For the hotter environment of a supernova, such
as SN 1987A, the axion emission rate is enhanced and the limit is stronger, at
the level of $|g_{ae\mu}| \lesssim 10^{-11}$, competitive with laboratory
limits. Interestingly, our derivation of the axion emissivity reveals that
axion emission via the lepton-flavor-violating coupling is suppressed relative
to the familiar lepton-flavor-preserving channels by the square of the plasma
temperature to muon mass ratio, which is responsible for the relatively weaker
limits.
