In this paper, we derive analytical expressions and numerical recipes for finding the effective observed position of sources close enough on-sky that their point spread functions (PSFs), modelled as Gaussian profiles, overlap. In particular, we derive these for an elongated PSF, with a long and short axis, such as we would see from an instrument with a rectangular or elliptical mirror (relevant, for example, for the Gaia mission). We show that in this case the problem can be reduced to a one-dimensional brightness profile with extrema along the line connecting the two sources, with an effective PSF width that depends on the relative orientation of the PSF and its degree of elongation. The problem can then be expressed in units of this effective width to be a function of the relative separation and light ratio alone (thus reducing to a rescaling of the un-elongated case). We derive the minimum light ratio, for a given separation and effective width, above which two sources will be resolved. We map out numerical procedures for finding the positions of these extrema across all possible cases. Finally, we derive the positional offset and deviance associated with observing a fixed pair of blended sources from a variety of orientations, showing that this can be a significant source of excess noise.
Zephyr Penoyre (Thu,) studied this question.