Warm dark matter model with a few keV mass is bad for the too-big-to-fail problem

Studying the very inner structure of faint satellite galaxy requires very high-resolution hydro-dynamical simulations with realistic model for star formation, which are beginning to emerge recently. In this work we develop an analytical description to model the inner kinematic of galaxy and apply it to the MW satellites. We use a Monte-Carlo method to produce merger trees of MW mass halo and analytical models to produce stellar mass in the satellite galaxies. We consider two important processes which can significantly affect the inner mass distribution in satellite galaxy. The first is the baryonic feedback to induce a flat inner profile with dependence on the star formation efficiency in the galaxy. The second is the tidal stripping to reduce and re-distribute the mass inside satellite. We apply this model to satellite galaxies in MW halo in both CDM and WDM models. It is found that tidal heating is the dominant process to get a relatively flat distribution of the satellite circular velocities, to agree with the data. Due to lower halo concentration and baryonic induced core, a WDM model with sterile neutrino mass of $m_{\nu} < 10$ keV can be excluded as it produces a too low circular velocity than observed data, in contrast to the too-big-to-fail problem. We conclude that more tight constraints require accurate information on the star formation history and orbit evolution of the observed satellite galaxies.