A quantitative explanation of the cyclotron-line variation in accreting magnetic neutron stars of super-critical luminosity

Abstract

Magnetic neutron stars (NSs) often exhibit a cyclotron resonant scattering feature (CRSF) in their X-ray spectra. Cyclotron lines are believed to form in the radiative shock in the accretion column. High-luminosity NSs show a smooth anti-correlation between the cyclotron-line centroid ($E_{CRSF}$) and X-ray luminosity ($L_X$). The observed $E_{CRSF}-L_X$ smooth anti-correlation has been in tension with the theoretically predicted one by the radiative shock model. Since there is no other candidate site for the cyclotron-line formation, we re-examine the predicted rate of change of the cyclotron-line energy with luminosity at the radiative shock, taking a closer look at the Physics involved. We developed a purely analytical model describing the overall dependence of the observed cyclotron energy centroid on the shock front's height, including the relativistic boosting effect due to the mildly relativistic motion of the accreting plasma upstream with respect to the shock's reference frame and the gravitational redshift. We find that the CRSF energy varies with a) the shock height due to the dipolar magnetic field, b) the Doppler boosting between the shock and bulk-motion frames, and c) the gravitational redshift. We show that the relativistic effects noticeably weaken the predicted $E_{CRSF}-L_X$ anti-correlation. We use our model to fit the data of the X-ray source V0332+53 and demonstrate that the model fits the data impressively well, alleviating the tension between observations and theory. The reported $E_{CRSF}-L_X$ weak anti-correlation in the supercritical accretion regime may be explained by the combination of the variation of the magnetic-field strength along the accretion column, the effect of Doppler boosting, and the gravitational redshift. Thus, the actual magnetic field on the NS surface may be a factor of $\sim 2$ larger than the naively inferred value from the observed CRSF.