The quenching rate is known to depend on galaxy stellar mass and environment, however, possible dependences on the hosting halo properties, such as mass, richness, and dynamical status, are still debated. The determination of these dependences is hampered by systematics, induced by noisy estimates of cluster mass or by the lack of control on galaxy stellar mass, which may mask existing trends or introduce fake trends. We studied a sample of local clusters (20 with 0.02<z<0.1 and log(M200/Msun)>14), selected independent of the galaxy properties under study, having homogeneous optical photometry and X-ray estimated properties. Using those top quality measurements of cluster mass, hence of cluster scale, richness, iron abundance, and cooling time/presence of a cool-core, we study the simultaneous dependence of quenching on these cluster properties on galaxy stellar mass M and normalised cluster-centric distance r/r200. We found that the quenching rate can be completely described by two variables only, galaxy stellar mass and normalised cluster-centric distance, and is independent of halo properties (mass, richness, iron abundance, presence of a cool-core, and central cooling time). These halo properties change, in most cases, by less than 3% the probability that a galaxy is quenched, once the mass-size (M200-r200) scaling relation is accounted for through cluster-centric distance normalisation.
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