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The detection of GW170817 in gravitational waves and gamma rays revealed thatshort gamma-ray bursts are associated with the merger of neutron-stars. Gammarays are thought to result from the formation of collimated jets, but thedetails of this process continue to elude us. One fundamental observable is theemission profile of the jet as a function of viewing angle. We present twomethods to measure the effective angular width, $\theta_B$, of short gamma-rayburst (sGRB) jets using gravitational wave and gamma-ray data, assuming allsGRBs have the same angular dependence for their luminosities. The first is acounting experiment, where we combine the known detection thresholds of theLIGO/Virgo and Fermi Gamma Ray Burst Monitor detectors to infer parameters ofsystems that are detected in gravitational waves. This method requires minimalknowledge about each event, beyond whether or not they were detected ingamma-rays. The second method uses additional information from thegravitational-wave and electromagnetic data to estimate parameters of thesource, and thereby improve constraints on jet properties. Applying our methodsto GW170817, we find only weak constraints on the sGRB luminosity profile, withstatistical uncertainty dominating differences between models. We also analyzesimulated events from future observing runs, and find that with 5 and 100 BNSdetections, the counting method constrains the relative uncertainty in$\theta_B$ to within 51% and 12%, respectively. Incorporatinggravitational-wave parameter estimation would further tighten these constraintsto 43% and 9.6%. In the limit of many detections, incorporating parameterestimation achieves only marginal improvements; we conclude that the majorityof the information about jet structure comes from the relative sensitivities ofgravitational-wave and gamma-ray detectors as encoded in simple countingexperiments.

Counting on Short Gamma-Ray Bursts: Gravitational-Wave Constraints of Jet Geometry