  The equation of state for a degenerate gas of fermions at zero temperature in
the non relativistic case is a polytrope, i.e. $p=\gamma \rho^{5/3}/m_F^{8/3}$.
If dark matter is modelled by such non interacting fermion, this dependence in
the mass of the fermion $m_F$ explains why if dark matter is very heavy the
effective pressure of dark matter is negligible. Nevertheless, if the mass of
the dark matter is very small, the effective pressure can be very large, and
thus, a system of self-gravitating fermions can be formed. In this work we
model the dark matter halo of the Milky-Way by solving the
Tolman-Oppenheimer-Volkoff equations, with the equation of state for a
partially degenerate ultralight non interacting fermion. It is found that in
order to fit its rotational velocity curve of the Milky Way, the mass of the
fermion should be in the range $29 ~\mbox{eV} < m_F < 33~$eV. Moreover, the
central density is constrained to be in the range of $46 < \rho_0<61$
GeV/cm$^3$. The fermionic dark matter halo has a very different profile as
compared with the standard Navarro-Frenk-White profile, thus, the possible
indirect signals for annihilating dark matter may change by orders of
magnitude. We found bounds for the annihilation cross section in this case by
using the Saggitarius A* spectral energy distribution. Those limits are very
strong confirming the idea that the lighter the dark matter particle is, the
darkest it becomes.
