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In this paper we explore the maximum precision attainable in the location ofa point source imaged by a pixel array detector in the presence of abackground, as a function of the detector properties. For this we use awell-known result from parametric estimation theory, the so-called Cramer-Raolower bound. We develop the expressions in the 1-dimensional case of a lineararray detector in which the only unknown parameter is the source position. Ifthe object is oversampled by the detector, analytical expressions can beobtained for the Cramer-Rao limit that can be readily used to estimate thelimiting precision of an imaging system, and which are very useful forexperimental (detector) design, observational planning, or performanceestimation of data analysis software: In particular, we demonstrate that forbackground-dominated sources, the maximum astrometric precision goes as$B/F^2$, where $B$ is the background in one pixel, and $F$ is the total flux ofthe source, while when the background is negligible, this precision goes as$F^{-1}$. We also explore the dependency of the astrometric precision on: (1)the size of the source (as imaged by the detector), (2) the pixel detectorsize, and (3) the effect of source de-centering. Putting these results intocontext, the theoretical Cramer-Rao lower bound is compared to both ground- aswell as spaced-based astrometric results, indicating that current techniquesapproach this limit very closely. Our results indicate that we have found inthe Cramer-Rao lower variance bound a very powerful astrometric "benchmark"estimator concerning the maximum expected positional precision for a pointsource, given a prescription for the source, the background, the detectorcharacteristics, and the detection process.

Analysis and interpretation of the Cramer-Rao lower-bound in astrometry: One dimensional case