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

¹ Dr. Karl-Remeis-Observatory and Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Sternwartstr. 7, 96049 Bamberg, Germany
² CRESST and Center for Space Sciences and Technology, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA
³ NASA Goddard Space Flight Center, Astrophysics Science Division, Greenbelt, MD 20771, USA
⁴ University of Maryland, Department of Astronomy, College Park, MD 20742, USA
⁵ INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 19 December 2025
Accepted: 23 March 2026

Abstract

Context. The X-ray spectra of neutron stars with moderate magnetic fields (B ∼ 10¹² G) in high-mass X-ray binaries (HMXBs) at low X-ray luminosities (L_X ≲ 10³⁵ erg s⁻¹) 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.

Aims. 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.

Methods. 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.

Results. 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.

Conclusions. 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 via our proposed 2-B-12 rule. Additional observations of XRBs and dedicated modeling efforts will be necessary to determine whether this approach is valid beyond the B-field range of a few 10¹²–10¹³ G that was tested in this work.