  The most promising indirect search for the existence of axion dark matter
uses radio telescopes to look for narrow spectral lines generated from the
resonant conversion of axions in the magnetospheres of neutron stars.
Unfortunately, a large list of theoretical uncertainties has prevented this
search strategy from being fully accepted as robust. In this work we attempt to
address major outstanding questions related to the role and impact of the
plasma, including: $(i)$ does refraction and reflection of radio photons in the
magnetosphere induce strong inhomogeneities in the flux, $(ii)$ can refraction
induce premature axion-photon de-phasing, $(iii)$ to what extent do
photon-plasma interactions induce a broadening of the spectral line, $(iv)$
does the flux have a strong time dependence, and $(v)$ can radio photons
sourced by axions be absorbed by the plasma. We present an end-to-end analysis
pipeline based on ray-tracing that exploits a state-of-the-art
auto-differentiation algorithm to propagate photons from the conversion surface
to asymptotically large distances. Adopting a charge symmetric Goldreich-Julian
model for the magnetosphere, we show that for reasonable parameters one should
expect a strong anisotropy of the signal, refraction induced axion-photon
de-phasing, significant line-broadening, a variable time-dependence of the
flux, and, for large enough magnetic fields, anisotropic absorption. Our
simulation code is flexible enough to serve as the basis for follow-up studies
with a large range of magnetosphere models.
