We present a detailed analysis of the influence of the environment and of the environmental history on quenching star formation in central and satellite galaxies in the local Universe. We take advantage of publicly available galaxy catalogues obtained from applying a galaxy formation model to the Millennium simulation. In addition to halo mass, we consider the local density of galaxies within various fixed scales. Comparing our model predictions to observational data (SDSS), we demonstrate that the models are failing to reproduce the observed density dependence of the quiescent galaxy fraction in several aspects: for most of the stellar mass ranges and densities explored, models cannot reproduce the observed similar behaviour of centrals and satellites, they slightly under-estimate the quiescent fraction of centrals and significantly over-estimate that of satellites. We show that in the models, the density dependence of the quiescent central galaxies is caused by a fraction of backsplash centrals which have been satellites in the past (and were thus suffering from environmental processes). Turning to satellite galaxies, the density dependence of their quiescent fractions reflects a dependence on the time spent orbiting within a parent halo of a particular mass, correlating strongly with halo mass and distance from the halo centre. Comparisons with observational estimates suggest relatively long gas consumption time scales of roughly 5 Gyr in low mass satellite galaxies. The quenching time scales decrease with increasing satellite stellar mass. Overall, a change in modelling both internal processes (star formation and feedback) and environmental processes (e.g. making them dependent on dynamical friction time-scales and preventing the re-accretion of gas onto backsplash galaxies) is required for improving currently used galaxy formation models.
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