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We consider the possibility that dark matter halos are made of quantumparticles such as fermions or bosons in the form of Bose-Einstein condensates.In that case, they generically have a "core-halo" structure with a quantum corethat depends on the type of particle considered and a halo that is relativelyindependent of the dark matter particle and that is similar to the NFW profileof cold dark matter. We model the halo by an isothermal gas with an effectivetemperature $T$. We then derive the core mass -- halo mass relation $M_c(M_v)$of dark matter halos from an effective thermodynamical model by extremizing thefree energy $F(M_c)$ with respect to the core mass $M_c$. We obtain a generalrelation that is equivalent to the "velocity dispersion tracing" relationaccording to which the velocity dispersion in the core $v_c^2\sim GM_c/R_c$ isof the same order as the velocity dispersion in the halo $v_v^2\sim GM_v/r_v$.We provide therefore a justification of this relation from thermodynamicalarguments. In the case of fermions, we obtain a relation $M_c\propto M_v^{1/2}$that agrees with the relation found numerically by Ruffini {\it et al.}. In thecase of noninteracting bosons, we obtain a relation $M_c\propto M_v^{1/3}$ thatagrees with the relation found numerically by Schive {\it et al.}. In the caseof bosons with a repulsive self-interaction in the Thomas-Fermi limit, wepredict a relation $M_c\propto M_v^{2/3}$ that still has to be confirmednumerically. We also obtain a general approximate core mass -- halo massrelation that is valid for bosons with arbitrary repulsive or attractiveself-interaction. For an attractive self-interaction, we determine the maximumhalo mass that can harbor a stable quantum core (dilute axion "star").

Derivation of the core mass -- halo mass relation of fermionic and bosonic dark matter halos from an effective thermodynamical model