The position and resolvability of blended point sources

TL;DR

This paper analytically and numerically investigates the effective observed position of overlapping Gaussian PSF sources, especially elongated ones, providing formulas and procedures to determine resolvability and positional offsets.

Contribution

It introduces new analytic expressions and numerical methods for analyzing the position and resolvability of blended elongated Gaussian sources, applicable to instruments like Gaia.

Findings

Derived formulas for effective source position with elongated PSF

Mapped minimum light ratio for source resolution

Quantified positional offsets and noise contributions

Abstract

In this work we derive analytic expressions and numerical recipes for finding the effective observed position of sources close enough on sky that their Point Spread Functions (PSF), modelled as Gaussian profiles, overlap. In particularly 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.