Satellite kinematics – II. The halo mass–luminosity relation of central galaxies in SDSS

The kinematics of satellite galaxies reflect the masses of the extended dark matter haloes in which they orbit, and thus shed light on the mass–luminosity relation (MLR) of their corresponding central galaxies. In this paper, we select a large sample of centrals and satellites from the Sloan Digital Sky Survey and measure the kinematics (velocity dispersions) of the satellite galaxies as a function of the r -band luminosity of the central galaxies. Using the analytical framework presented in More, van den Bosch & Cacciato, we use these data to infer both the mean and the scatter of the MLR of central galaxies, carefully taking account of selection effects and biases introduced by the stacking procedure. As expected, brighter centrals on average reside in more massive haloes. In addition, we find that the scatter in halo masses for centrals of a given luminosity,  σ_{log  M}   , also increases with increasing luminosity. As we demonstrate, this is consistent with  σ_{log  L}   , which reflects the scatter in the conditional probability function   P ( L _c| M )  , being independent of halo mass. Our analysis of the satellite kinematics yields  σ_{log  L} = 0.16  ±  0.04  , in excellent agreement with constraints from clustering and group catalogues, and with predictions from a semi-analytical model of galaxy formation. We thus conclude that the amount of stochasticity in galaxy formation, which is characterized by  σ_{log  L}   , is well constrained, independent of halo mass and in a good agreement with current models of galaxy formation.