Cyclotron lines in subcritical X-ray pulsars: Monte Carlo simulations reveal the origin of the observed variability
Observed cyclotron resonant scattering features (CRSFs) in X-ray pulsars (XRPs) exhibit strong variability. In the subcritical luminosity regime, the centroid energy ($E_{CRSF}$) and line width ($σ_{CRSF}$) often show positive correlations with the X-ray luminosity. We investigate the physical origin of the observed variability quantitatively, focusing on the effects of resonant scattering and Doppler shift induced by the plasma flow in the accretion funnel. We developed a relativistic Monte Carlo code to perform detailed radiative transfer calculations in the accretion funnel above the hotspot and derive angle-dependent spectra. Analytical plasma density and velocity profiles were adopted to account for the effects of radiation pressure on the flow. Approximate resonant scattering cross-sections were employed. We varied the accretion luminosity to explore the resulting variability of the CRSF properties. The emergent spectra exhibit a prominent, asymmetric CRSF accompanied by a broad blue wing. The CRSF is systematically redshifted relative to the classical cyclotron energy, with the magnitude of the redshift decreasing at higher luminosities and for larger viewing angles $θ$. Both $E_{CRSF}$ and $σ_{CRSF}$ correlate positively with luminosity for all viewing angles. Their absolute values, however, depend strongly on the viewing angle, indicating substantial variability over the pulse cycle and sensitivity to the system geometry. At fixed luminosity, $E_{CRSF}$ ($σ_{CRSF}$) decreases (increases) with increasing $\cosθ$. Consequently, phase-resolved observations are expected to reveal an anticorrelation between the CRSF centroid energy and width. When applied to the XRP GX 304$-$1, the model reproduces the observed CRSF variability over nearly an order of magnitude in luminosity for geometries in which the accretion funnel is predominantly viewed edge-on.