7 -15 -10 -5 0 5 ln(1/ 0 100 200 300 400 n=3 n=2 n=1 d=1 0 10 20 30 40 50 1 2 3 4 e d=1  > 1  < 1 FIG. 3: Inverse temperature η as a function of the central density 1 /λ for the first three branches in d = 1. FIG. 1: The function e $_{−}$ψ for d = 1 and λ = 0 . 5 < 1 (bottom) or λ = 2 > 1 (top). In d = 1, the oscillations are undamped. FIG. 2: The function e $_{−}$ψ for d = 2 and λ = 0 . 5 < 1 (bottom) or λ = 2 > 1 (top). In d ≥ 2, the oscillations are damped. The case d = 3 (not represented) is similar. 0 25 50 75 100 0.5 1 1.5 2 2.5 3 e - d=2  > 1  < 1 concentrations, the Newtonian solution provides a good approximation of the modified Newtonian solution in the core (see Appendix E). We must now relate the normalized central density 1 /λ to the temperature T . Recalling that ρ = M/V with V = 1 $_{d}$S$_{d}$ R $^{d}$, we obtain Introducing the normalized temperature we find the relation is determined, the density profile is given by Eq. (46). The density profile is extremum at the center and at the boundary. On the n -th branch, the density profile shows n “clusters” corresponding to the oscillations of e − ψ ( ξ $^{)}$. Close to the origin, the density increases for λ > 1 while it decreases for λ < 1. The homogeneous state ψ = 0 cor- responds to λ = 1. This solution is degenerate because the boundary condition (49) is satisfied for any α . Recalling that α = α$_{n}$ ( λ ) for the n -th branch, this equa- tion gives the relation between the inverse temperature η and the central density 1 /λ for the n -th branch. In Figs. 3, 4 and 5, we plot the inverse temperature η as a func- tion of the central density 1 /λ for the first three branches n = 1 , 2 , 3 in different dimensions of space d = 1 , 2 , 3. Let us discuss the asymptotic behaviors of the tem- perature (we only describe the first branch n = 1) and compare with the Newtonian model (see, e.g., [29]): C. The temperature λ = ρ ρ$_{0}$ = dM S$_{d}$R d 1 ρ$_{0}$ = d GMmβ R d − 2 1 α $_{2}$. (50) η ≡ βGMm R d − 2 , (51) η = 1 d λα $^{2}$. (52) • In d = 1: for the ordinary Newtonian model, the series of equilibria is parameterized by α , which is a measure of the central density. When α → + ∞ , the distribution tends to a Dirac peak ρ = Mδ ( x ) and the inverse temperature η → + ∞ . When α → 0, the dis- tribution is homogeneous and the inverse temperature η → 0. For the modified Newtonian model, the se- ries of equilibria is parameterized by the central den- sity 1 /λ . When 1 /λ → + ∞ , the distribution tends to Remark: When λ → 0, corresponding to large values of the central density, we expect to obtain results similar to those obtained for the usual Newtonian model since the differential equation (47) reduces to the ordinary Emden equation. However, the results are different because the boundary conditions are not the same. In the Newto- nian model, the force at the boundary is non zero (for a spherically symmetric system, according to the Gauss theorem, we have Φ $^{′}$( R ) = GM/R d − $^{1}$) while in the mod- ified Newtonian model the force at the boundary is zero (Φ $^{′}$( R ) = 0). Therefore, strictly speaking, the Newtonian and the modified Newtonian models behave differently even when ρ$_{0}$ → + ∞ . Nevertheless, for large central