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.
