How the spin-phase variability of cyclotron lines shapes the pulsed fraction spectra: Insights from 4U 1538–52

Aims. We aim to study the energy-dependent pulse profile of the X-ray accreting pulsar 4U 1538–52 and its phase-dependent spectral variability, with a particular emphasis on the behavior around the cyclotron resonant scattering feature at Ecyc ∼ 21 keV. Methods. We analyzed all available NuSTAR observations of 4U 1538–52. We decomposed the energy-resolved pulse profiles into Fourier harmonics to study their energy dependence. Specifically, we computed the pulsed fraction spectra, cross-correlation, and lag spectra, identifying discontinuities and linking them to features in the phase-averaged spectra. We performed both phase-averaged and phase-resolved spectral analyses to probe spectral variability and its relation to pulse profile changes. Finally, we interpreted our findings based on a physical modeling of the energy- and angle-dependent pulse profile emission, performing radiative transfer in a homogeneous slab-like atmosphere under conditions relevant to 4U 1538–52. The emission is projected onto the observer’s sky plane to derive the expected observables. Results. In contrast to the dips in pulsed fraction spectra observed in other sources (e.g., Her X-1), we find a broad bump near the cyclotron resonance energy in 4U 1538–52. This increase is driven primarily by phase-dependent spectral variability, especially by strong variations in cyclotron line depth across different phase intervals. We interpreted the observed contrast between dips and bumps in various sources as arising from phase-dependent variations of cyclotron line depth relative to the phase-modulated flux. We modeled the X-ray emission from an accreting neutron star and found that our simulations indicate high values of both the observer’s inclination and the magnetic obliquity, along with a ∼10 − 15° asymmetry between the locations of the magnetic poles. Assuming this geometry, we were able to adequately reproduce the observed pulse profiles and introduce general trends in the observables resulting from the system’s geometry.