  Axion dark matter can resonantly convert to photons in the magnetosphere of
neutron stars, possibly giving rise to radio signals observable on Earth. This
method for the indirect detection of axion dark matter has recently received
significant attention in the literature. The calculation of the radio signal is
complicated by a number of effects; most importantly, the gravitational infall
of the axions onto the neutron star accelerates them to semi-relativistic
speed, and the neutron star magnetosphere is highly anisotropic. Both of these
factors complicate the calculation of the conversion of axions to photons. In
this work, we present the first fully three-dimensional calculation of the
axion-photon conversion in highly magnetised anisotropic media. Depending on
the axion trajectory, this calculation leads to orders-of-magnitude differences
in the conversion compared to the simplified one-dimensional calculation used
so far in the literature, altering the directionality of the produced photons.
Our results will have important implications for the radio signal one would
observe in a telescope.
