In recent years, gravitational lensing has been used as a means to detect substructure in galaxy-sized halos, via anomalous flux ratios in quadruply-imaged lenses. In addition to causing anomalous flux ratios, substructure may also perturb the positions of lensed images at observable levels. In this paper, we numerically investigate the scale of such astrometric perturbations using realistic models of substructure distributions. Substructure distributions that project clumps near the Einstein radius of the lens result in perturbations that are the least degenerate with the best-fit smooth macromodel, with residuals at the milliarcsecond scale. Degeneracies between the center of the lens potential and astrometric perturbations suggest that milliarcsecond constraints on the center of the lensing potential boost the observed astrometric perturbations by an order of magnitude compared to leaving the center of the lens as a free parameter. In addition, we discuss methods of substructure detection via astrometric perturbations that avoid full lens modeling in favor of local image observables and also discuss modeling of systems with luminous satellites to constrain the masses of those satellites.
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