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Current results from the Burst and Transient Source Experiment (BATSE) on the Compton Gamma-ray Observatory (CGRO) place rather tight constraints on extended galactic halo models for the spatial distribution of gamma-ray burst (GRB) sources. The observed isotropy and lack of excess bursts from the direction to M31 constrain the BATSE sampling depth in these models to between about 200 - 400 kpc. If such halos do indeed contain the GRB sources, then sensitive searches of the halos of nearby galaxies should reveal them. We show that the proportional counter array (PCA) experiment onboard the X-ray Timing Explorer (XTE), scheduled for launch in August 1995, will have the capability to detect extended halo GRBs, if they exist, around galaxies within about 4 Mpc. Calculations of the expected detection rate, based on the observed BATSE rate, indicate from about 0.08 to 0.24 bursts per day, depending on the details of the model halo distribution and the distance to the host galaxy. This suggests that extended halo models can be seriously confronted with about 15 - 20 days of search data.
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