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Appendix A: Fluid truncated isothermal spheres

In this appendix we summarize the properties of spatially truncated self-gravitating fluid isothermal spheres.

The equations for the hydrostatic equilibrium of a spherically symmetric fluid with the equation of state of an ideal gas are We express M(r) by means of the condition of hydrostatic equilibrium (A.1), differentiate it to obtain dM, and equate this to dM = 4πρ₀(r)r². By making the change of variable ρ₀(r) = ρ₀(0)e^− ψ(r), where ρ₀(0) is the central density, and introducing the dimensionless radius ξ = r/λ, where λ = [kT/4πGρ₀(0)m]^1/2, we obtain the differential equation for ψ(ξ), recorded in the main text as Eq. (5). (Because the constant ρ₀(0) is interpreted as the central density, we are considering the boundary condition ψ(0) = 0; the density is taken to be regular at the origin, so that the second boundary condition is ψ′(0) = 0.) The solution of Eq. (5)(called Emden equation) is a monotonic increasing function characterized by logarithmic behavior ψ(ξ) ~ lnξ² and ψ′(ξ) ~ 2/ξ as ξ → ∞.

From Eq. the mass enclosed within the radius ξ is (A.4)From Eq. (A.4)and the asymptotic behavior of ψ, it is clear that a solution with finite total mass is only obtained by truncating the system at a dimensionless radius ξ = Ξ. A truncated isothermal sphere is then identified by two scales T and ρ₀(0) and one dimensionless parameter Ξ. The density profile of a spatially truncated isothermal sphere is given by ρ₀(r) = ρ₀(0)e^− ψ(r) where ψ is the solution to Eq. (5).

Appendix B: Linearization of the hydrodynamic equations

Appendix B.1: Eulerian representation

Here we record the calculations leading to the linearized Eq. (8). The unperturbed density profile is given by Eq. (4). We substitute Eqs. (6)in Eq. (2)and expand to first order in quantities with subscript 1 to obtain Then we look for solutions of the form (7). From Eq. (B.1)we obtain ρ₁: (B.3)and thus eliminate it from Eq. (B.2)to find the radial component of the Navier-Stokes equation: (B.4)where u₁ is the radial component of the velocity. By defining f(r) ≡ ρ₀(r)u₁(r) and introducing the dimensionless radius ξ = r/λ we obtain Eq. (8).

Appendix B.2: Lagrangian representation

Appendix B.2.1: Change of variables

Here we show the change of variables leading from Eqs. (2)(Eulerian representation) to Eqs. (14)(Lagrangian representation). Let us assume spherical symmetry and neglect viscosity. By dropping the nonlinear term (u·∇)u, the Euler equation and the continuity Eq. (2)become, in the Eulerian representation:

(B.5)

Now we perform a change of variables. In the Lagrangian representation, each quantity is described as a function of the new independent variables r₀ and t, where r₀ is the position of the fluid element at t = t₀. The standard rules to transform derivatives of a generic function f are: (B.6)The partial derivatives of r₀ are obtained from the following relation, which expresses the condition that two fluid shells do not cross each other: (B.7)By taking the partial derivative with respect to r of Eq. (B.7)(each side of the equation is considered as a function of r, t) we obtain (B.8)By taking the partial derivative of Eq. (B.7)with respect to t we obtain (B.9)From the continuity equation, Eq. (B.9)then becomes (B.10)From Eqs. (B.10), (B.8), and (B.6)we obtain the equations of hydrodynamics in the Lagrangian representation (14).

Appendix B.2.2: Linearization

Here we approximate Eqs. (14)to first order for small perturbations around the hydrostatic equilibrium states. We substitute Eqs. (15)in Eqs. (14)and expand to first order in quantities with subscript 1. By noting that M(r₀,t) = M(r₀,t = t₀), we obtain (B.11)From the usual rules of derivation, we have: (B.12)By applying Eqs. (B.10)and (B.8), Eq. (B.12)can be written as (B.13)Now we wish to obtain one equation involving only the unknown ρ₁, by combining the three Eqs. (B.11)and (B.13). We assume the modal dependence (16). We eliminate r₁ from (B.13)and the second of (B.11). We then eliminate u₁ from the resulting equation and the first of (B.11), to obtain the following equation {we also used hydrostatic equilibrium to replace }: (B.14)By referring to the dimensionless radius ξ₀ ≡ r₀/λ, we obtain Eq. (17).

Appendix B.3: Temperature expression for the constant {E, V} case

Here we show the steps leading from Eq. (26)to Eq. (27).

From the Poisson and Emden equations, the dimensionless potential ψ is related to the gravitational potential Φ₀ of the unperturbed density distribution in the following way: (B.15)From Eq. (B.15), by recalling that we only consider perturbations that do not change the total mass (), we obtain (B.16)From Eq. (B.3), by setting f = ρ₀u₁ and ξ = r/λ, we obtain Eq. (27).

Appendix C: Detailed angular momentum conservation and collapse

Here, for completeness, we show in detail that the angular momentum barrier prevents a spherically symmetric collisionless system from collapsing. We only consider perturbations that do not break the assumed spherical symmetry of the system.

Consider a collisionless system of particles of individual mass m and total mass M. From the assumption of spherical symmetry, each particle is confined to a plane and we can write the single-particle Lagrangian as (C.1)where V(r,t) is a time-dependent potential, t time, r the distance from the center, and θ the angular coordinate. The Lagrangian (C.1)conserves the angular momentum of the particle, that is, , where C is a constant. Then the equation of the motion is (C.2)where M(r,t) is the total mass contained in the sphere of radius r. Equation (C.2)is the equation of the motion of a particle moving in one dimension and subject to the force F_r = mC²/r³ − GmM(r,t)/r². The following inequality holds: (C.3)By multiplying by ṙ and integrating both sides of Eq. (C.2), we obtain (C.4)Since ṙ²(t)/2 is positive, the righthand side of Eq. (C.5)must be positive. By taking r(t) ≤ r(t₀) (we are not interested in the case in which r(t) is greater than the initial radius) and using Eqs. (C.5)and (C.3), we find (C.5)For given values of r(t₀) and ṙ(t₀), the quantity appearing in the last line of Eq. (C.5)tends to − ∞ as r(t) → 0. Hence, for given initial conditions the particle cannot reach arbitrarily low values of r(t).

Appendix D: Density and velocity profiles of the linear modes

In this appendix we show density and velocity profiles of the normal modes for the linear stability analysis presented in Sect. 4; ρ₁/ρ₀ and u₁ are meant to be on arbitrary scales.

Appendix D.1: Constant {T, V} profiles

Fig. D.1 Relative density perturbation profiles ρ1(ξ)/ρ0(ξ) and velocity profiles of the normal modes for the constant { T,V } case, obtained by solving Eq. (18). The profiles should be truncated at a value ξ = Ξ where the velocity profile vanishes, to satisfy boundary conditions (11). The vertical dotted line indicates where the system should be truncated to obtain the mode of lowest L for fixed Ξ: for L =  −0.02 and L = 0 only this mode is entirely displayed, while for L = 0.02 two modes are displayed, depending on which zero of the velocity profile is chosen. In the case L = 0, the total density ρ(t) = ρ0 + ρ1(t) at the point ξ = 4.07 remains unchanged, that is, unperturbed; this is one of the relevant points listed by Lynden-Bell & Wood (1968).

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Appendix D.2: Constant {E, V} profiles

Fig. D.2 Relative density perturbation profiles ρ1(ξ)/ρ0(ξ) of the normal modes for the constant { E,V } case, obtained by solving Eq. (28). The modes should be truncated at a value ξ = Ξ where the corresponding velocity profile shown in Fig. D.3 vanishes, as marked by the vertical dotted lines, in order to satisfy the boundary conditions (11). Zeros are displayed. Only modes of lowest L at given Ξ are shown. Note that the core-halo structure described in Sect. 4.1.2 disappears between L = 0.021 and L = 0.022.

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Fig. D.3 Velocity profiles of the normal modes for the constant { E,V } case, obtained by solving Eq. (28). The modes should be truncated at a value ξ = Ξ corresponding to the vertical dotted lines, in order to satisfy the boundary conditions (11). Other zeros are displayed. Only modes of lowest L for fixed Ξ are shown. Note that the core-halo structure described in Sect. 4.1.2 disappears when the velocity has no internal zeros, between L = 0.021 and L = 0.022.

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Appendix D.3: constant {T, P} profiles

Fig. D.4 Relative density perturbation profiles ρ1(ξ0)/ρ0(ξ0) of the normal modes for the constant { T,P } case, obtained by solving Eq. (38), calculated and displayed here in the Lagrangian representation. The modes should be truncated at a value ξ0 = Ξ where the density profile vanishes, in order to satisfy the boundary conditions (39). The first zero, which represents the mode of minimum L at given Ξ, is indicated by the vertical dotted line. A mode of higher L for fixed Ξ is shown in the L = 0.03 case; for other cases higher modes can be identified in a similar way.

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Appendix E: Equations for the two-component case

In this appendix we summarize the equations of the linear analysis for the two-component case and show some examples of the density profiles associated with the modes that characterize the onset of the instability. We denote by subscripts A and B the lighter and the heavier component, respectively.

The unperturbed states are the two-component self-gravitating truncated isothermal spheres considered by Taff et al. (1975), Lightman (1977), Yoshizawa et al. (1978), de Vega & Siebert (2002), Sopik et al. (2005). The density profiles can be written as (E.1)\vspace*{1.2mm}where ρ_A0 and ρ_B0 are respectively the density profiles of the lighter and heavier component, ξ ≡ r/λ₂ is the dimensionless radial coordinate, where λ₂ ≡ [kT(1/m_A + 1/m_B)/4πGρ₀(0)]^1/2 and we denote by ρ₀(ξ) ≡ ρ_A0(ξ) + ρ_B0(ξ) the total unperturbed density; Ξ is the value of ξ at the truncation radius, β ≡ m_B/m_A is the ratio of the single-particle masses, ψ is the solution of the following generalization of the Emden Eq. (5): where α = ρ_A0(0)/ρ_B0(0) is the ratio of the unperturbed central densities. The symbol ′ denotes derivative with respect to the argument ξ.

The linearized hydrodynamical equations, governing the evolution of the two-component fluid system for small deviations from the unperturbed states described above, are obtained by generalizing in a straightforward manner the steps leading

from Eqs. (2)to Eq. (18). The result, which generalizes Eq. (18), is the following system of equations that governs the evolution of radial perturbations: (E.4)Here L = ω²/4πGρ₀(0) represents the dimensionless (squared) eigenfrequency, f_A(ξ) ≡ ρ_A0(ξ)u_A1(ξ) and f_B(ξ) ≡ ρ_B0(ξ)u_B1(ξ), where u_A1 and u_B1 are the radial velocity perturbations of the two components. The boundary conditions are: (E.5)Similarly to the one-component case, the two conditions at the center follow from requiring regularity and spherical symmetry, while the two conditions at the truncation radius satisfy the requirement that the radial velocities must vanish at the edge.

The system (E.4)for L = 0 is equivalent to the system that can be obtained by generalizing in a straightforward manner the thermodynamical analysis of Chavanis (2002). The latter analysis can be used to find the points for the onset of instability. This proves that the onset of instability occurs at the same values of Ξ in the dynamical and in the thermodynamical approach.

In Fig. E.1 we show the density profiles for the marginally stable modes (L = 0) in three different situations, that is, with β = 3 and three different values of M_B/M_A. The density perturbation of the heavier component is greater than the density perturbation of the lighter component even for low values of M_B/M_A, indicating that the heavier component is the more important driver of the instability.

Fig. E.1 Relative density perturbation profiles ρ1A(ξ)/ρ0(ξ) (lighter component, dotted line) and ρ1B(ξ)/ρ0(ξ) (heavier component, solid line) of the normal modes for the two-component constant { T,V } case, obtained by solving Eq. (E.4). The plots show marginally stable modes (L = 0) of minimum L at given Ξ. They represent the density profiles that characterize the onset of the instability for fixed value of β = mB/mA = 3 at different values of the total mass ratio MB/MA. Even for low values of MB/MA, the density perturbation of the heavier component dominates, thus suggesting that the heavier component is the more important driver of the instability.

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