Radio emission from colliding outflows in high-mass X-ray binaries with strongly magnetized neutron stars

We present a toy model for radio emission in HMXBs with strongly magnetized neutron stars (NS) where a wind-collision region is formed by the NS outflow and the stellar wind of the massive companion. Radio emission is expected from the synchrotron radiation of shock-accelerated electrons and the free-free emission of the stellar wind. We found that the predicted relation between the GHz luminosity ($L_R$) and the accretion X-ray luminosity ($L_X$) can be written as $L_R \propto L_X^b$ for most parameters. No correlation with X-rays is expected ($b=0$) when the thermal emission of the stellar wind dominates in radio. We typically find a steep correlation ($b=12/7$) for sub-Eddington X-ray luminosities and a more shallow one ($b=2(p-1)/7$) for super-Eddington X-ray luminosities, where $p$ is the power-law index of accelerated electrons. The maximum predicted radio luminosity is independent of the NS properties, while it depends on the stellar wind momentum, binary separation distance, and the minimum electron Lorentz factor. Using a Bayesian approach we modelled the radio observations of \sj that cover a wide range of mass accretion rates. Our results support a shock origin for the radio detections at sub-Eddington X-ray luminosities. However, no physically meaningful parameters could be found for the super-Eddington phase of the outburst, suggesting a different origin. Future observations with more sensitive instruments might reveal a large number of HMXBs with strongly magnetized NSs in radio, allowing determination of the slope in the $L_R-L_X$ relation, and putting the wind-collision scenario into test.