Constraining the long-lived magnetar remnants in short gamma-ray bursts from late-time radio observations

The joint detection of GW 170817 and GRB 170817A indicated that at least a fraction of short gamma ray bursts (SGRBs) originate from binary neutron star (BNS) mergers. One possible remnant of a BNS merger is a rapidly rotating, strongly magnetized neutron star, which has been discussed as one possible central engine for GRBs. For a rapidly rotating magnetar central engine, the deposition of the rotation energy into the ejecta launched from the merger could lead to bright radio emission. The brightness of radio emission years after an SGRB would provide an estimation of the kinetic energy of ejecta and hence, a possible constraint on the BNS merger product. We perform a more detailed calculation on the brightness of radio emission from the interaction between the merger ejecta and circumburst medium in the magnetar scenario, invoking several important physical processes such as generic hydrodynamics, relativistic effects, and the deep Newtonian phase. We use the model to constrain the allowed parameter space for 15 SGRBs that have late radio observations. Our results show that an injected energy $E_{\rm inj} \sim 10^{52}$ erg is allowed for all the cases, which suggests that the possibility of a supra-massive or hyper-massive neutron star remnant is not disfavored by the available radio data.