Determining the viewing angle of neutron star merger jets with VLBI radio images

Very long base interferometry (VLBI) radio images recently proved to be essential in breaking the degeneracy in the ejecta model for the neutron star merger event GW170817. We discuss the properties of synthetic radio images of merger jet afterglow by using semi-analytic models of laterally spreading or non-spreading jets. The image centroid initially moves away from the explosion point in the sky with an apparent superlumianal velocity. After reaching a maximum displacement its motion is reversed. This behavior is in line with that found in full hydrodynamics simulations. Since the evolution of the centroid shift and jet image size are significantly different in the two jet models, observations of these characteristics for very bright events might be able to confirm or constrain the lateral expansion law of merger jets. We explicitly demonstrate how $\theta_{\rm obs}$ is obtained by the centroid shift of radio images or its apparent velocity provided the ratio of the jet core size $\theta_{c}$ and the viewing angle $\theta_{\rm obs}$ is determined by afterglow light curves. We show that a simple method based on a point-source approximation provides reasonable angular estimates ($10-20\%$ errors at most). By taking a sample of structured Gaussian jet results, we find that the model with $\theta_{\rm obs} \sim 0.32$ rad can explain the main features of the GW170817 afterglow light curves and the radio images.