Warm Dark Matter Galaxies with Central Supermassive Black-Holes
  We generalize the Thomas-Fermi approach to galaxy structure to include
self-consistently and non-linearly central supermassive black holes. This
approach naturally incorporates the quantum pressure of the warm dark matter
(WDM) particles and shows its full powerful and clearness in the presence of
supermassive black holes (SPMHs). We find the main galaxy and central black
hole magnitudes: halo radius r_h , halo mass M_h, black hole mass M_BH,
velocity dispersion, phase space density, with their realistic astrophysical
values, masses and sizes over a wide galaxy range. The SMBH masses arise
naturally in this framework. Our extensive numerical calculations and detailed
analytic resolution show that with SMBH's, both WDM regimes: classical
(Boltzmann dilute) and quantum (compact) do necessarily co-exist in any galaxy:
from the smaller and compact galaxies to the largest ones. The transition from
the quantum to the classical region occurs precisely at the same point r_A
where the chemical potential vanishes. A novel halo structure with three
regions shows up: A small quantum compact core of radius r_A around the SMBH,
followed by a less compact region till the BH influence radius r_i, and then
for r> r_i the known halo galaxy shows up with its astrophysical size. Three
representative families of galaxy plus central SMBH solutions are found and
analyzed:small, medium and large galaxies having SMBH masses of 10^5, 10^7 and
10^9 M_sun respectively. A minimum galaxy size and mass ~ 10^7 M_sun larger
than the one without SMBH is found. Small galaxies in the range 10^4 M_sun <
M_h < 10^7 M_sun cannot harbor central SMBHs. We find novel scaling M_BH - r_h
- M_h relations. The galaxy equation of state is derived: The pressure P(r)
takes huge values in the SMBH vecinity and then sharply decreases entering the
classical region following a local perfect gas behaviour.(Abridged)
