  The most efficient axion production mechanism in a supernova (SN) core is the
nucleon-nucleon bremsstrahlung. This process has been often modeled at the
level of the vacuum one-pion exchange (OPE) approximation. Starting from this
naive recipe, we revise the calculation including systematically different
effects, namely a non-vanishing mass for the exchanged pion, the contribution
from the two-pions exchange, effective in-medium nucleon masses and multiple
nucleon scatterings. Moreover, we allow for an arbitrary degree of nucleon
degeneracy. A self consistent treatment of the axion emission rate including
all these effects is currently missing. The aim of this work is to provide such
an analysis. Furthermore, we demonstrate that the OPE potential with all the
previous corrections gives rise to similar results as the on-shell T-matrix,
and is therefore well justified for our and similar studies. We find that the
axion emissivity is reduced by over an order of magnitude with respect to the
basic OPE calculation, after all these effects are accounted for. The
implications for the axion mass bound and the impact for the next generation
experimental axion searches is also discussed.
