RMS asymmetry: a robust metric of galaxy shapes in images with varied depth and resolution

Structural disturbances, such as galaxy mergers or instabilities, are key candidates for driving galaxy evolution, so it is important to detect and quantify galaxies hosting these disturbances spanning a range of masses, environments, and cosmic times. Traditionally, this is done by quantifying the asymmetry of a galaxy as part of the concentration-asymmetry-smoothness system, $A_{\rm{CAS}}$, and selecting galaxies above a certain threshold as merger candidates. However, in this work, we show that $A_{\rm{CAS}}$, is extremely dependent on imaging properties -- both resolution and depth -- and thus defining a single $A_{\rm{CAS}}$ threshold is impossible. We analyze an alternative root-mean-squared asymmetry, $A_{\rm{RMS}}$, and show that it is independent of noise down to the average SNR per pixel of 1. However, both metrics depend on the resolution. We argue that asymmetry is, by design, always a scale-dependent measurement, and it is essential to define an asymmetry at a given physical resolution, where the limit should be defined by the size of the smallest features one wishes to detect. We measure asymmetry of a set of $z\approx0.1$ galaxies observed with HST, HSC, and SDSS, and show that after matching the resolution of all images to 200 pc, we are able to obtain consistent $A_{\rm{RMS, 200pc}}$ measurements with all three instruments despite the vast differences in the original resolution or depth. We recommend that future studies use $A_{\rm{RMS, x pc}}$ measurement when evaluating asymmetry, where $x$ is defined by the physical size of the features of interest, and is kept consistent across the dataset, especially when the redshift or image properties of galaxies in the dataset vary.