Cyclotron lines in subcritical X-ray pulsars: Monte Carlo simulations reveal the origin of the observed variability

¹ University of Crete, Department of Physics & Institute of Theoretical & Computational Physics, 70013 Heraklion, Greece
² Institute of Astrophysics, Foundation for Research and Technology-Hellas, 71110 Heraklion, Crete, Greece
³ Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA
⁴ Max-Planck-Institut für extraterrestrische Physik, Postfach 1312, 85741 Garching, Germany
⁵ University Observatory, Faculty of Physics, Ludwig-Maximilians Universität, Scheinerstr. 1, 81679 Munich, Germany

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Received: 2 February 2026
Accepted: 30 March 2026

Abstract

Context. 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.

Aims. 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.

Methods. We developed a relativistic Monte Carlo code to perform detailed radiative transfer calculations in the accretion funnel above the hotspot and derived angle-dependent spectra. We adopted analytical plasma density and velocity profiles to account for the effects of radiation pressure on the flow, and we employed approximate resonant scattering cross sections. We varied the accretion luminosity to explore the resulting variability of the CRSF properties.

Results. 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 a fixed luminosity, E_CRSF (σ_CRSF) decreases (increases) with increasing cos θ. Consequently, our model predicts that phase-resolved observations will 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.

Conclusions. Resonant scattering of radiation propagating through the accreting plasma in the magnetic funnel provides a natural explanation for the observed CRSF variability in subcritical XRPs.