A simple relation: Neutron star magnetic field strength and spectral shape at low mass accretion rates

The X-ray spectra of neutron stars with moderate magnetic fields ($B\sim 10^{12}$ G) in high-mass X-ray binaries (HMXBs) at low X-ray luminosities ($L_\mathrm{X}\lesssim 10^{35}$ erg/s) are characterized by a double humped shape. This shape has been explained either as the radiation from a two-temperature magnetized atmosphere, where thermal radiation dominates at soft X-rays below about 10 keV, and cyclotron radiation with an imprinted cyclotron line dominates at high energies, or by the complex redistribution of primary X-rays in a structured atmosphere. The theoretical explanations of the double humped structure predict the spectra to depend on the magnetic field. We aim to connect the model predictions with observations. We analyzed archival NuSTAR observations of four HMXBs consisting of a neutron star and a Be star (BeXRBs), with known magnetic fields at luminosities low enough to show the characteristic double-hump spectrum. We modeled these spectra empirically and derived a relation between the energy of the intersection of the two humps and the magnetic field strength. In a second step, we tested whether this correlation is supported by fitting synthetic spectra simulated with the physically self-consistent polcap model. We find a linear correlation between the magnetic field strength and the intersection energy for the real BeXRB NuSTAR spectra and polcap-based simulated NuSTAR spectra alike. The effect of the magnetic field on spectral formation results in an observable correlation between the field strength and spectral shape. This derived positive correlation between intersection energy and magnetic field strength also allowed us to roughly estimate the magnetic field strength. Additional observations of XRBs and dedicated modeling efforts will be necessary to determine whether this approach is valid beyond the B-field range that was tested in this work.