The $K_s$-band luminosity function of the rich cluster VC04 in the Vela supercluster

To learn more about the properties of the Vela Supercluster (VSCL) located behind the Milky Way at $cz\sim 18000$~km~s$^{-1}$, we determined the $Ks$-band Luminosity Function (LF) of VC04, the richest known galaxy cluster in the VSCL, and two other VSCL clusters (VC02 and VC08). The galaxy sample is based on NIR observations which are complete to an extinction-corrected absolute magnitude of $M{Ks}^o<-21.5$ mag ($\sim 2.5$ mag below $MK^*$), within the clustercentric radius of $rc<1.5~$Mpc ($\sim 70 \%$ of the Abell radius). For VC04 we obtained 90 new spectroscopic redshifts of galaxies observed with the 11-m Southern African Large Telescope. We found the Schechter parameters of the VC04 LF to be $M^*=-24.41\pm 0.44$ mag, $\alpha=-1.10\pm 0.20$ and $\phi^*=8.84\pm 0.20$. Both the redshift data and the LF confirm VC04 to be a rich, not yet fully relaxed cluster. We independently determined the LF of VC04 on membership defined by the Red-Sequence method and demonstrated that this method can be used in the absence of high spectroscopic coverage over a cluster. This allowed us to determine the LFs of VC02 and VC08. We also derived the LFs of the Coma, Norma and Virgo clusters to similar depth and extent as the VSCL clusters. We found that the Schechter parameters of VC04 are within $1\sigma$ uncertainties of these local clusters, as well as VC02 and VC08. We do not find significant differences between the LFs in the different cluster environments probed in this work down to $M_K^*+2.5$ mag.