8 FIG. 4: Inverse temperature η as a function of the central density 1 /λ for the first three branches in d = 2. FIG. 5: Inverse temperature η as a function of the central density 1 /λ for the first three branches in d = 3. a Dirac peak ρ = M δ ( x ) and η → + ∞ with the same asymptotic behavior as in the Newtonian model (see Ap- pendix E). When λ = 1, the distribution is homoge- neous and η = η ∗ c = π 2 ≃ 9 . 8696044 (see Appendix F). When 1 /λ → 0, the distribution tends to a Dirac peak ρ = M 2 ( δ ( x − R ) + δ ( x + R )) concentrated at the box and η → + ∞ . • In d = 2: for the ordinary Newtonian model, the series of equilibria is parameterized by α . When α → + ∞ , the distribution tends to a Dirac peak ρ = M δ ( r ) and the inverse temperature tends to η$_{c}$ = 4. When α → 0, the distribution is homogeneous and the inverse temperature η → 0. For the modified Newtonian model, the series of equilibria is parameterized by 1 /λ . When 1 /λ → + ∞ , the distribution tends to a Dirac peak ρ = M δ ( r ) and η → η$_{c}$ = 4 (since the density is very much concentrated, the boundary conditions do not matter and we recover the same results as in the Newtonian case). When λ = 1, the distribution is homogeneous and η = η ∗ c = 1 $_{2}$j 2 11 ≃ 7 . 3410008 (see Appendix F). When 1 /λ → 0, the distribution is concentrated at the boundary and η → + ∞ . Recalling that α = α$_{n}$ ( λ ) and η = η$_{n}$ ( λ ) for the n -th branch, this equation gives the relation between the en- ergy Λ and the central density 1 /λ for the n -th branch. we obtain Introducing the dimensionless variables defined previ- ously, recalling that r = ξR/α , and introducing the nor- malized energy The total energy E = K + W is therefore given by Using the modified Poisson equation (43) and an integra- tion by parts, the potential energy can be written Using the Maxwell-Boltzmann distribution (10), the ki- netic energy is simply We must also relate the normalized central density 1 /λ to the energy E . The total energy is given by (see Ap- pendix C): • In d = 3: for the ordinary Newtonian model, the se- ries of equilibria is parameterized by α . When α → + ∞ , the distribution tends to the singular isothermal sphere ρ$_{s}$ ( r ) = 1 / (2 πGβmr $^{2}$) and the inverse temperature η → η$_{s}$ = 2. The curve η ( α ) displays damped oscillations around this value. When α → 0, the distribution is ho- mogeneous and the inverse temperature η → 0. For the modified Newtonian model, the series of equilibria is pa- rameterized by 1 /λ . When 1 /λ → + ∞ , the distribution is concentrated at the center and we numerically find that η → 3 . 05 ... (the value is different from the Newto- nian result η$_{s}$ = 2 due to different boundary conditions). The curve η ( λ ) displays damped oscillations around this value. When λ = 1, the distribution is homogeneous and η = η ∗ c = 1 $_{3}$x 2 1 ≃ 6 . 7302445 (see Appendix F). When 1 /λ → 0, the distribution is concentrated at the bound- ary and we numerically find that η → + ∞ . -15 -10 -5 0 5 10 ln(1/ 0 25 50 75 100 n=1 n=2 n=3 d=2 $_{c}$=4 -10 -5 0 5 10 15 20 ln(1/ 0 10 20 30 40 50 n=3 n=2 n=1 d=3 Λ = − d 2 η + 1 2 η 2 1 α d − 2 ∫ α 0 ( dψ dξ ) 2 ξ d − $^{1}$dξ. (58) Λ ≡ − ER d − 2 GM 2 , (57) E = d 2 Nk$_{B}$T − 1 2 S$_{d}$G ∫ ( ∇ Φ) 2 d r . (56) W = − 1 2 S$_{d}$G ∫ ( ∇ Φ) 2 d r . (55) K = d 2 Nk$_{B}$T. (54) E = ∫ f v 2 2 d r d v + 1 2 ∫ ( ρ − ρ ) Φ d r . (53) D. The energy