Fermionic King model: critical masses resulting from the competition of quantum and evaporation effects

Abstract

We study the influence of evaporation (the escape of constituents) on the thermodynamics of a self-gravitating non-relativistic gas of fermions in the framework of Newtonian gravitation. For this purpose, it is reconsidered as the so-called fermionic King model introduced by Ruffini and Stella in the context of dark matter halos problems (1983 Astron. Astrophys. 119 3). As the usual King model, the present model predicts density profiles that drop to zero at a certain finite radius R, which is referred to as the tidal radius. Our interest is focused on the thermodynamics of this model at constant total mass M and constant tidal radius R. A special interest is devoted to study the incidence of mass on the collective phenomena of gravitational collapse and evaporation disruption. The underlying physics is driven by the Fermi mass M F ∼ ℏ 6/G 3 m 8 R 3, which corresponds to the total mass of a self-gravitating degenerate Fermi gas of non-relativistic point particles with mass m, whose linear size is equal to the tidal radius R of evaporation. Numerical calculations predict three critical values for the total mass, M 1 > M 2 > M 3, which hallmark the thermodynamic stability. The lower bound is a critical value of the total mass below the which the system gravitation is unable to confine its constituents, so that the system with total mass M < M 3 undergoes an evaporation disruption. The other bounds, M 2 ≃ 90.9M F and M 1 ≃ 8.9 × 106 M F, are critical values that hallmark the continuous (or discontinuous) character of the microcanonical phase transition associated with gravothermal collapse and the presence of states with negative heat capacities.