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The joint detection of GW 170817 and GRB 170817A indicated that at least afraction 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 possiblecentral engine for GRBs. For a rapidly rotating magnetar central engine, thedeposition of the rotation energy into the ejecta launched from the mergercould lead to bright radio emission. The brightness of radio emission yearsafter an SGRB would provide an estimation of the kinetic energy of ejecta andhence, a possible constraint on the BNS merger product. We perform a moredetailed calculation on the brightness of radio emission from the interactionbetween 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 toconstrain the allowed parameter space for 15 SGRBs that have late radioobservations. Our results show that an injected energy $E_{\rm inj} \sim10^{52}$ erg is allowed for all the cases, which suggests that the possibilityof a supra-massive or hyper-massive neutron star remnant is not disfavored bythe available radio data.

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