Galaxy Luminosity Function of the Coma Cluster from Deep $u^{\mathrm{\prime }}-g^{\mathrm{\prime }}-r^{\mathrm{\prime }}$ Wendelstein Imaging Data

We derive the $g^{\mathrm{\prime }}$ band galaxy luminosity function (GLF) of quenched red sequence in the Coma cluster from a deep-imaging survey with $\approx 1.5\mathrm{d}\mathrm{e}{\mathrm{g}}^2$ around the cluster center. The dataset comprises deep $u^{\mathrm{\prime }}$-, $g^{\mathrm{\prime }}$-, and $r^{\mathrm{\prime }}$-band data obtained with the Wendelstein Wide Field Imager on the 2.1,m Fraunhofer Wendelstein Telescope reaching median $3\sigma$ surface brightness limits in $10\mathrm{"}\times 10\mathrm{"}$ boxes of $(30.0{\mathrm{u}}^{\mathrm{\prime }},29.6{\mathrm{g}}^{\mathrm{\prime }},28.7{\mathrm{r}}^{\mathrm{\prime }})\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{e}{\mathrm{c}}^{-2}$. We measure structural parameters across a large dynamic range in galaxy brightness ($-24.5g^{\mathrm{\prime }}\mathrm{m}\mathrm{a}\mathrm{g}⪅M⪅-11.3g^{\mathrm{\prime }}\mathrm{m}\mathrm{a}\mathrm{g}$), from the brightest cluster galaxy to low-luminosity dwarfs, including compact dwarf galaxies and ultra-diffuse galaxies. We automatically identify 5161 cluster member candidates based on their membership on the quiescent sequence in the $u^{\mathrm{\prime }}-g^{\mathrm{\prime }}$ versus $g^{\mathrm{\prime }}-r^{\mathrm{\prime }}$ color–color diagram and their red sequence membership. The structural parameters of bright galaxies are obtained via isophotal modeling, and fully automated parametric image fitting for faint ones. Injection-recovery tests and two identically analyzed reference fields provide statistical corrections for completeness and contamination, yielding a representative GLF that reliably probes the faint end and may serve as a benchmark for future studies. We report a best-fit double Schechter $g^{\mathrm{\prime }}$ band GLF with a comparatively steep faint-end slope ${\alpha }_2=-1.539_{-0.024}^{+0.024}$. We perform an apples-to-apples comparison with the Coma counterpart in the SLOW simulations, finding overall agreement but a deficit of bright and an excess of dwarf galaxies in the simulation, despite its shallower faint-end slope, highlighting the need for true apples-to-apples comparisons that consider both normalization and slope for a consistent interpretation of the GLF.