  The magnetosphere of strongly magnetized neutron stars, such as magnetars,
can sustain large electric currents. The charged particles return to the
surface with large Lorentz factors, producing a particle bombardment. We
investigate the transport of radiation in the atmosphere of strongly magnetized
neutron stars, in the presence of particle bombardment heating. We solve the
radiative transfer equations for a gray atmosphere in the Eddington
approximation, accounting for the polarization induced by a strong magnetic
field. The solutions show the formation of a hot external layer and a low
(uniform) temperature atmospheric interior. This suggests that the emergent
spectrum may be described by a single blackbody with the likely formation of a
optical/infrared excess (below ~ 1 eV). We also found that the polarization is
strongly dependent on both the luminosity and penetration length of the
particle bombardment. Therefore, the thermal emission from active sources, such
as transient magnetars, in which the luminosity decreases by orders of
magnitude, may show a substantial variation in the polarization pattern during
the outburst decline. Our results may be relevant in view of future X-ray
polarimetric missions such as IXPE and eXTP.
