We generalize the Thomas-Fermi approach to galaxy structure to include central supermassive black holes and find selfconsistently and non-linearly the gravitational potential of the galaxy plus central black hole ( BH ) system .
This approach naturally incorporates the quantum pressure of the fermionic warm dark matter ( WDM ) particles and shows its full powerful and clearness in the presence of supermassive black holes .
We find the main galaxy and central black hole magnitudes as the halo radius r _ { h } , halo mass M _ { h } , black hole mass M _ { BH } , velocity dispersion \sigma , phase space density , with their realistic astrophysical values , masses and sizes over a wide galaxy range .
The supermassive black hole masses arise naturally in this framework .
Our extensive numerical calculations and detailed analytic resolution of the Thomas-Fermi equations show that in the presence of the central BH both DM regimes : classical ( Boltzmann dilute ) and quantum ( compact ) do necessarily co-exist generically in any galaxy : from the smaller and compact galaxies to the largest ones .
The ratio { \cal R } ( r ) of the particle wavelength to the average interparticle distance shows consistently that the transition , { \cal R } \simeq 1 , 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 : In the vicinity of the BH , WDM is always quantum in a small compact core of radius r _ { A } and nearly constant density .
In the region r _ { A } < r < r _ { i } till the BH influence radius r _ { i } , WDM is less compact and exhibits a clear classical-Boltzmann like behaviour .
For r > r _ { i } , the WDM gravity potential dominates and the known halo galaxy shows up with its astrophysical size , DM is a dilute classical gas in this region .
As an illustration , three representative families of galaxy plus central BH solutions are found and analyzed : small , medium and large galaxies with realistic supermassive BH masses of 10 ^ { 5 } M _ { \odot } , 10 ^ { 7 } M _ { \odot } and 10 ^ { 9 } M _ { \odot } respectively .
In the presence of the central BH , we find a minimum galaxy size and mass M _ { h } ^ { min } \simeq 10 ^ { 7 } M _ { \odot } , larger ( 2.2233 10 ^ { 3 } times ) than the one without BH , and reached at a minimal non-zero temperature T _ { min } .
The supermassive BH heats-up the DM and prevents it to become an exactly degenerate gas at zero temperature .
Colder galaxies are smaller , warmer galaxies are larger .
Galaxies with a central black-hole have large masses M _ { h } > 10 ^ { 7 } M _ { \odot } > M _ { h } ^ { min } ; compact or ultracompact dwarf galaxies in the range 10 ^ { 4 } M _ { \odot } < M _ { h } < 10 ^ { 7 } M _ { \odot } can not harbor central BHs .
We find novel scaling relations M _ { BH } = DM _ { h } ^ { \frac { 3 } { 8 } } and r _ { h } = CM _ { BH } ^ { \frac { 4 } { 3 } } , and show that the DM galaxy scaling relations : M _ { h } = b \Sigma _ { 0 } r _ { h } ^ { 2 } , M _ { h } = a { \sigma _ { h } } ^ { 4 } / \Sigma _ { 0 } hold too in the presence of the central BH , \Sigma _ { 0 } being the constant surface density scale over a wide galaxy range .
The galaxy equation of state is derived : The pressure P ( r ) takes huge values in the BH vecinity region and then sharply decreases entering the classical region following in consistently a self-gravitating perfect gas P ( r ) = { \sigma } ^ { 2 } \rho ( r ) behaviour .