Short Gamma-Ray Bursts (SGRBs) are expected to form from the coalescence of compact binaries, either of primordial origin or from dynamical interactions in globular clusters. In this paper, we investigate the possibility that the offset and afterglow brightness of a SGRB can help revealing the origin of its progenitor binary. We find that a SGRB is likely to result from the primordial channel if it is observed within 10 kpc from the center of a massive galaxy and shows a detectable afterglow. The same conclusion holds if it is 100 kpc away from a small, isolated galaxy and shows a weak afterglow. On the other hand, a dynamical origin is suggested for those SGRBs with observable afterglow either at a large separation from a massive, isolated galaxy or with an offset of 10-100 kpc from a small, isolated galaxy. We discuss the possibility that SGRBs from the dynamical channel are hosted in intra-cluster globular clusters and find that GRB 061201 may fall within this scenario.
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