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

Theoretical studying of the very inner structure of faint satellite galaxy requires very high-resolution hydro-dynamical simulations with realistic models for star formation, which are beginning to emerge recently. In this work we present an analytical description to model the inner kinematic of satellites in the Milky Way (MW). We use a Monte-Carlo method to produce merger trees for MW mass halo and analytical models to produce stellar mass in the satellite galaxies. We consider two important processes which can significantly modify the inner mass distribution in satellite galaxy. The first is baryonic feedback which can induce a flat inner profile depending 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 MW satellite galaxies in both CDM and thermal relic WDM models. It is found that tidal heating must be effective to produce a relatively flat distribution of the satellite circular velocities, to agree with the data. The constraint on WDM mass depends on the host halo mass. For a MW halo with dark matter mass lower than $2\times 10^{12}M_{\odot}$, a 2 keV WDM model can be safely excluded as the predicted satellite circular velocities are systematically lower than the data. For WDM with mass of 3.5 keV, it requires the MW halo mass to be larger than $1.5\times 10^{12}M_{\odot}$, otherwise the 3.5 Kev model can also be excluded. Our current model can not exclude the WDM model with mass larger than 10 Kev.