| Abstract
| - We use a complete set of deep narrow-band imaging data for 384 galaxies gathered during the Virgo Environmental Survey Tracing Ionised Gas Emission (VESTIGE) to derive the first H α luminosity function of the Virgo cluster within its virial radius. The data, which are sensitive to the emission of a single O-early B ionising star, allow us to cover the whole dynamic range of the H α luminosity function (10 36 ≤ L(H α)≤10 42 erg s −1). After they are corrected for [NII] contamination and dust attenuation, the data are used to derive the star formation rate function in the range 10 −4 ≲ SFR ≲ 10 M⊙ yr −1. These luminosity functions are derived for gas-rich and gas-poor systems and for objects belonging to the different substructures of the Virgo cluster. They are then compared to those derived at other frequencies or using different tracers of star formation in Virgo, in other nearby and high- z clusters, in the field, and finally to those predicted by the IllustrisTNG cosmological hydrodynamical simulations (TNG50 and TNG100). The H α luminosity function of the Virgo cluster is fairly flat ( α = −1.07 when fitted with a Schechter function) in the range 10 38.5 ≲ L(H α)≲10 40.5 erg s −1, and it abruptly decreases at lower luminosities. When compared to those derived for other nearby clusters and for the field, the slope and the characteristic luminosity of the Schechter function change as a function of the dynamical mass of the system, of the temperature of the X-rays gas, and of the dynamical pressure exerted on the interstellar medium of galaxies moving at high velocity within the intracluster medium. All these trends can be explained in a scenario in which the activity of star formation of galaxies is reduced in massive clusters due to their hydrodynamical interaction with the surrounding medium, suggesting once again that ram-pressure stripping is the dominant mechanism affecting galaxy evolution in local clusters of dynamical mass Mcluster ≳ 10 14 M⊙. The comparison with the IllustrisTNG cosmological hydrodynamical simulations shows a more pronounced decrease at the faint end of the distribution. If the Virgo cluster is representative of typical nearby clusters of similar mass, this difference suggests that the stripping process in simulated galaxies in these environments is more efficient than observed.
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