Existence and continuous dependence on parameters of radially symmetric solutions to astrophysical model of self‐gravitating particles

Abstract

In this paper, we obtain the existence of a radial solution for some elliptic nonlocal problem with constraints. The problem is described as stationary state of some evolutionary models provided the pressure function conveys some form arising from statistical mechanics. To be more specific, we consider the following forms of the statistics: Michie-King, Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein, and polytrope. The most recent models were suggested by H. J.de Vega at al. and P. H. Chavanis et al.. We prove the continuous dependence of stationary solutions on parameters for the given statistics. KEYWORDS Fermi-Dirac, Michie-King statistics, nonlocal elliptic equation MSC CLASSIFICATION 35Q85; 47H10; 85A05 ASTROPHYSICAL MOTIVATIONAs it is well established, in the universe, there is vast preponderance of the dark matter and the dark energy. To describe it at the galactic level, numerous models were proposed.The classical Michie-King model was introduced by Michie 1 and King 2 as the natural extension of the Maxwell-Boltzmann case. The classical Michie-King model is a modification of the Maxwell-Boltzmann distribution that takes into account the escape of high-energy stars in a stellar system.Following a long list of works on the subject, de Vega et al 3-5 modeled dark matter halos as self-gravitating gas of fermions at finite temperature and confronted it with observations that correspond to warm dark matter, eg, composed of sterile neutrinos. In the approach by de Vega et al, 3-5 the system does not confine within a box, and halos extend to infinity and have infinite mass. They do not need to compute the total mass in their study, 3-5 so they do not have to deal with this problem.The study that considers self-gravitating fermions in the box is Chavanis, 6 which is concerned with phase transition in the self-gravitating Fermi gas. If we want to study phase transitions between different equilibrium states, it is necessary to have configurations with a finite mass, and this is why Chavanis 6 had to close the system in the artificial box. In more recent papers, Chavanis et al 7,8 studied phase transitions of self-gravitating fermions without the artifice of the material box by using the fermionic King distribution. This approach alleviates the problem by extension of the model with the pressure governed by the Fermi-Dirac statistics to the fermionic Michie-King model. The ad hoc considerations of Ruffini and Stella 9 can be also justified by the statistical mechanics and the kinetic theory once the evaporation of the high energy particles is taken into account as was done by Chavanis et al. 7,8 Although the fermionic King model was introduced heuristically by Ruffini and Stella, 9 a derivation from kinetic theory was first made by Chavanis. 10 To alleviate drawbacks Math Meth Appl Sci. 2019;42:7381-7394.wileyonlinelibrary.com/journal/mma

Mathematical model for Sagittarius A* and related Tolman–Oppenheimer–Volkoff equations

“…To introduce the abovementioned formulas, recall that numerous pressure formulas coming from statistical mechanics of self‐interacting particles have been collected in Biler [12] including Maxwell–Boltzmann, Bose–Einstein, Fermi–Dirac (cf., previous works [13–18]), Michie–King classical (cf., previous works [3, 19, 20]) and fermionic (cf., previous works [21–24]), relativistic fermionic (Fermi–Dirac) model [25–27], polytropic distributions (cf., Chavanis [28]). In the model considered above, we start from Maxwell–Boltzmann distribution and may introduce some modifications beginning with three modifications: •one following from Pauli principle introducing bound in the phase space for the number of particles at given place, with fixed velocity at defined time, leading to the Fermi–Dirac distribution function, •another one making the particles with too high velocities evaporate from the system leading to the classical Michie–King distribution. •adding the relativistic effect leads to the relativistic Maxwell–Boltzmann distribution [29], …”

“…To introduce the abovementioned formulas, recall that numerous pressure formulas coming from statistical mechanics of self-interacting particles have been collected in Biler [12] including Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac (cf., previous works [13][14][15][16][17][18]), Michie-King classical (cf., previous works [3,19,20]) and fermionic (cf., previous works [21][22][23][24]), relativistic fermionic (Fermi-Dirac) model [25][26][27], polytropic distributions (cf., Chavanis [28]). In the model considered above, we start from Maxwell-Boltzmann distribution and may introduce some modifications beginning with three modifications:…”