Analysis and interpretation of the Cramer-Rao lower-bound in astrometry:
  One dimensional case
  In this paper we explore the maximum precision attainable in the location of
a point source imaged by a pixel array detector in the presence of a
background, as a function of the detector properties. For this we use a
well-known result from parametric estimation theory, the so-called Cramer-Rao
lower bound. We develop the expressions in the 1-dimensional case of a linear
array detector in which the only unknown parameter is the source position. If
the object is oversampled by the detector, analytical expressions can be
obtained for the Cramer-Rao limit that can be readily used to estimate the
limiting precision of an imaging system, and which are very useful for
experimental (detector) design, observational planning, or performance
estimation of data analysis software: In particular, we demonstrate that for
background-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 of
the 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 detector
size, and (3) the effect of source de-centering. Putting these results into
context, the theoretical Cramer-Rao lower bound is compared to both ground- as
well as spaced-based astrometric results, indicating that current techniques
approach this limit very closely. Our results indicate that we have found in
the Cramer-Rao lower variance bound a very powerful astrometric "benchmark"
estimator concerning the maximum expected positional precision for a point
source, given a prescription for the source, the background, the detector
characteristics, and the detection process.
