Resolving the image of gamma-ray burst afterglows with gravitational microlensing

Microlensing of a gamma-ray burst afterglow by an intervening star can be used to infer the radial structure of the afterglow image. Near the peak of the microlensing event, the outer edge of the image is more highly magnified than its central region, whereas the situation is reversed at later times because of the rapid radial expansion of the image on the sky. Thus, the microlensed afterglow light curve can be inverted to recover the self-similar radial intensity profile of the afterglow image. We calculate the expected errors in the recovered intensity profile as a function of the number of resolution elements, under the assumption that the afterglow and microlensing event parameters are known. For a pointmass lens and uniform source, we derive a simple scaling relation between these parameters and the resultant errors. We find that the afterglow need not be monitored for its entire duration; rather, observations from the peak magnification time t(peak) of the microlensing event until similar to 7t(peak) are sufficient to resolve the majority of the afterglow image. Thus, microlensing events can be alerted by relatively infrequent observations of afterglows and then monitored intensively, without significant loss of information about the afterglow intensity profile. The relative intensity profile of similar to 1% of all afterglows can be measured with 10 resolution elements to an accuracy of O(1%) in the optical and O(10%) in the infrared, using 4 m class telescopes. Weak microlensing events with large impact parameters are more common; we estimate that for similar to 10% of afterglows the image profile may be inverted to a fractional accuracy through frequent optical observations. We also calculate the effects of external shear due to the less than or similar to 20% host galaxy or a binary companion, as well as contamination by background light from the host galaxy.