How to quantify the similarity of 2D distributions: Comparison of spatial distribution of Dark Matter and Intracluster light

In studying the dynamical evolution of galaxy clusters, one intriguing approach is to compare the spatial distributions of various components, such as the dark matter, the member galaxies, the gas, and the intracluster light (ICL; the diffuse light from stars, which are not bound any individual cluster galaxy). If we find a visible component whose spatial distribution coincides with the dark matter distribution, then we could draw a dark matter map without requiring laborious weak lensing analysis. Furthermore, if the component traces the dark matter distribution better for more relaxed galaxy cluster, we could use the similarity as a dynamical stage estimator of the galaxy cluster. We present a novel new methodology to quantify the similarity of two or more 2-dimensional spatial distributions. We apply the method to a sample of galaxy clusters at different dynamical stages simulated within N-cluster Run, which is an N-body simulation using the galaxy replacement technique. Among the various components (stellar particles, galaxies, ICL), the velocity defined ICL+ brightest cluster galaxy (BCG) component traces the dark matter best. Between the sample galaxy clusters, the relaxed clusters show stronger similarity of the spatial distribution between the dark matter and ICL+BCG than the dynamically young clusters.