  Spiral galaxies M31 and M33 are Fermi/LAT-detected gamma-ray sources. We
model the broadband non-thermal (NT) emission of the central region of M31 (R <
5.5 kpc) and of the disk of M33 (R ~ 9 kpc). For either galaxy, we
self-consistently model the broadband SED of the diffuse NT emission based on
published radio and gamma-ray data. All relevant radiative processes involving
relativistic and thermal electrons (synchrotron, Compton scattering,
bremsstrahlung, and free-free emission and absorption), along with relativistic
protons (neutral-pion decay following interaction with thermal protons), are
considered, using exact emissivity formulae. We also use the Fermi/LAT
validated gamma-ray emissivities for pulsars. We find that, in both sources,
the radio emission is composed of primary and secondary electron synchrotron
and thermal bremsstrahlung. The M33 gamma-ray emission appears to be mainly
hadronic, similar to the Magellanic Clouds (Persic & Rephaeli 2022). In
contrast, we find suggestions of a more complex situation in the central region
of M31, whose emission could be a mix of pulsar emission and hadronic emission,
with the latter possibly originating from both the disk and the vicinity of the
nuclear black hole. The alternative modelling of the spectra of M31 and M33 is
motivated by the different hydrogen distribution in the two galaxies: the
hydrogen deficiency in the central region of M31 partially unveils emissions
from the nuclear BH and the pulsar population in the bulge and inner disk. If
this were to be the case in M33 as well, these emissions would be outshined by
diffuse pionic emission originating within the flat central-peak gas
distribution in M33.
