The position and resolvability of blended point sources

TL;DR

This work analyzes blending of close point sources under elongated Gaussian PSFs, as relevant for Gaia-like instruments. It reduces the 2D brightness profile to a 1D profile along the line between sources using an orientation-dependent effective width $\gamma(\phi)$, enabling a unified treatment of circular and elongated PSFs via the dimensionless separation $\rho=r/\gamma$ and light ratio $l$. The authors derive analytic conditions for resolvability, including a critical light ratio $l_c(\rho)$ and a numerical recipe to locate extrema; they also quantify how fixed pairs produce photocenter offsets $x_m$ and an excess noise $D$ across scan angles, with clear regimes where $k$ and $D$ are significant. The results illuminate how blending biases astrometric measurements, potentially inflating RUWE and producing apparent motion or parallax signals, and offer practical guidance for diagnosing and mitigating blends in large surveys. While based on Gaussian PSFs, the framework captures key scaling with $\gamma$ and provides a tractable pathway to incorporate more realistic PSFs in future work.

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.

Figures (8)

• Figure 1: A grid of example images (as given by $b(x,y)$, equation \ref{['eq:b_xy']}) of two sources with light-ratio $l=0.5$ observed at different separations $r$ (rows) and angles $\phi$ (columns) by an instrument with an elongated PSF with relative width $\beta=3\alpha$. We work in coordinates $x,y$, on-sky distances are in units of $\alpha$, with the primary at the origin and the secondary on the line $y=0$. The two white circles show the positions of the two point sources. Red $+$ symbols denote maxima of the brightness profile, and where present orange $\times$ symbols denote a saddle-point (in which cases there are two maxima). The red-orange-yellow contour lines show 10 contours of equal brightness evenly spaced between the maximum value and 0. Straight lines show the along- and across- scan positions of the primary (black), primary maximum (red), and where present the saddle-point (dashed orange) and secondary maximum (dashed red). Unresolved systems (no orange $\times$ or line) occur at low separation, but also at larger separations if the PSF is oriented such that the long axis is close to parallel with the line separating the pair.
• Figure 2: One-dimensional brightness profiles of two sources along their connecting line ($y=0$ and thus $b(x,y) \Rightarrow b(x,0) = b_x(x)$, equation \ref{['eq:bx']}). We work in units of the effective width, $\gamma(\phi)$ (see equation \ref{['eq:gamma']}), $\rho=r/\gamma$ and $\chi=x/\gamma$, and thus are able to remove any $\phi$ dependence from the functional form. The top panel shows sources of fixed $\rho=3$ and varying $l$ and the bottom panel shows sources of fixed $l=0.5$ and varying $\rho$. The black dashed line shows the profile of the primary (always fixed), coloured dotted lines show the profile of the secondary, and solid coloured lines the total brightness. Maxima are denoted by full circles and minimum (where present) by an open circle. Black vertical lines show the position of the primary, and in the first panel also the (fixed) position of the secondary.
• Figure 3: Here we show, as a function of $\rho$, the relative location of the primary maxima (solid line) and where present the secondary maximum (dashed line) and the minimum (dotted line), for various $l$. The dashed vertical lines show the critical $\rho_c(l)$ above which a minimum exists. The wide translucent horizontal lines show $x_m = x_0$, the asymptotic solution for the position of the primary maximum when $\rho \ll 1$.
• Figure 4: Numerically derived positions of the primary maximum, $x_1$, and where they exist ($l>l_c(\rho)$) the minimum $x_{min}$ and secondary maximum $x_2$. Here we show all quantities in units of $r$. The solid black line corresponds to the critical point beyond which three solutions (two peaks and a minima) exist, as described by equations \ref{['eq:l_c']} and \ref{['eq:chi_c']}). The dashed black line shows the $\rho$ that maximises $x_1$ for a given $l$, as detailed in appendix \ref{['ap:max_offset']}.
• Figure 5: The variation of the position of the observed maxima, as a function of scan angle $\phi$, of a system with $l=\frac{1}{2}$ and a range of $r$. As in figure \ref{['fig:col_grid']} we use $\beta /\alpha=3$. The two largest separation systems (darkest blues) are resolved at some angles (small $|\mathrm{s}_{\phi}|$) as shown with a thicker line, and with the position of the minima and secondary maxima also shown as dotted and solid lines respectively. As all expressions involved are even functions of $\phi$ we need only show the behaviour for $0\leq \phi \leq \pi$.