$g/e=5\times10^{-17}\,\text{keV}^{-1}$. Moreover, for our benchmark point relevant to SGR 1806-20, the analytic solution with $|\Delta_ar_{a\to\gamma}|\gtrsim0.45$ is valid in the soft X-ray regime (*i.e.* from its lower boundary at $\omega\sim0.1\,\text{keV}$ to $\omega\lesssim3.89\,\text{keV}$) while the analytic solution with $|\Delta_ar_{a\to\gamma}|\lesssim0.45$ is valid in the hard X-ray regime (*i.e.* from $\omega\gtrsim3.89\,\text{keV}$ to its upper boundary at $\omega\sim200\,\text{keV}$) and beyond. Both analytic solutions overestimate $R$ in the region around $\omega=3.89\,\text{keV}$.

Observational Outlook {#SObs}
=====================

We now turn to some comments on the observational outlook of our methods.

The field of X-ray polarimetry is currently very active, with several new instrument designs being proposed and detectors that have either been launched or are in the planning stage [@fabiani]. The theoretical targets include understanding the emission from white dwarfs and neutron stars in binary systems, and the coupling of the plasma to the magnetic field in accreting X-ray pulsars [@gnedin; @Marin:2017vdv].

The targets most germane to our work are neutron stars and magnetars. For these stellar objects, there are two sources of polarization that have been explored theoretically: the polarization due to non-linear QED effects [@gnedin2] and the polarization due to the anisotropic opacities of the surface plasma [@Lai:2003nd; @Taverna:2015vpa]. While polarization due to ALP-photon mixing has 