Context. Accretion onto magnetic neutron stars results in X-ray spectra that often exhibit a cyclotron resonance scattering feature (CRSF) and, sometimes, higher harmonics of it. Two places are suspect for the formation of a CRSF: the surface of the neutron star and the radiative shock in the accretion column. Aims. Here we explore the first possibility: reflection at the neutron-star surface of the continuum produced at the radiative shock. It has been proposed that for high-luminosity sources, as the luminosity increases, the height of the radiative shock increases, thus a larger polar area is illuminated, and as a consequence the energy of the CRSF decreases because the dipole magnetic field decreases by a factor of two from the pole to the equator. This model has been specifically proposed to explain the observed anticorrelation of the cyclotron line energy and luminosity of the high-luminosity source V 0332+53. Methods. We used a Monte Carlo code to compute the reflected spectrum from the atmosphere of a magnetic neutron star, when the incident spectrum is a power-law one. We restricted ourselves to cyclotron energies ≪mec2 and used polarization-dependent scattering cross sections, allowing for polarization mode change. Results. As expected, a prominent CRSF is produced in the reflected spectra if the incident photons are in a pencil beam, which hits the neutron-star surface at a point with a well-defined magnetic field strength. However, the incident beam from the radiative shock has a finite width and thus various magnetic field strengths are sampled. As a result of overlap, the reflected spectra have a CRSF, which is close to that produced at the magnetic pole, independent of the height of the radiative shock. Conclusions. Reflection at the surface of a magnetic neutron star cannot explain the observed decrease in the CRSF energy with luminosity in the high-luminosity X-ray pulsar V 0332+53. In addition, it produces absorption lines much shallower than the observed ones.

Context. Radiative transfer calculations in strong (few ×1012 G) magnetic fields, which are observed in X-ray pulsars, require accurate differential cross-sections of resonant scattering. While such cross-sections exist, their application is cumbersome. Aims. Here, we compare the classical (non-relativistic) with the quantum-mechanical (relativistic) resonant differential scattering cross-sections and offer a prescription for the use of the much simpler classical expressions with impressively accurate results. Methods. We expanded the quantum-mechanical differential cross-sections and kept the terms up to the first order in ϵ ≡ E/mec2 and B ≡ ℬ/ℬcr, where E is the photon energy and ℬcr is the critical magnetic field. We recovered the classical differential cross-sections along with the terms that are due to spin flip, which is a pure quantum-mechanical phenomenon. Results. When adding the spin-flip terms to the polarization-dependent classical differential cross-sections by hand, we find that they are in excellent agreement with the quantum mechanical ones for all energies near resonance and all angles. We plotted both of them and the agreement is impressive. Conclusions. We give a prescription for the use of the classical differential cross-sections for radiative transfer calculations that guarantees accurate results.

Context. The nature of dark matter (DM) is still under intense debate. Subgalactic scales are particularly critical, as different, currently viable DM models make diverse predictions on the expected abundance and density profile of DM haloes on these scales. Aims. We investigate the ability of subgalactic DM haloes to act as strong lenses on background compact sources, producing gravitational lensing events on milli-arcsecond scales (milli-lenses), for different DM models. For each DM scenario, we explore whether a sample of ∼ 5000 distant sources is sufficient to detect at least one milli-lens. Methods. We developed a semi-analytical model to estimate the milli-lensing optical depth as a function of the source's redshift for various DM models. We employed the Press-Schechter formalism, as well as results from recent N-body simulations to compute the halo mass function, taking into account the appropriate spherically averaged density profile of haloes for each DM model. We treated the lensing system as a point-mass lens and invoked the effective surface mass density threshold to calculate the fraction of a halo that acts as a gravitational lens. We studied three classes of dark matter models: cold DM, warm DM, and self-interacting DM. Results. We find that haloes consisting of warm DM turn out to be optically thin for strong gravitational milli-lensing (zero expected lensing events). Cold DM haloes may produce lensing events depending on the steepness of the concentration-mass relation. Selfinteracting DM haloes can efficiently act as gravitational milli-lenses only if haloes experience gravothermal collapse, resulting in highly dense central cores.