Truncated γ-exponential models for tidal stellar systems

Gravitation is an example of an attractive interaction that is unable to confine particles. It is always possible that some particles acquire sufficient energy (via encounters or any other mechanism) and escape from the gravitational influence of the system. Evaporation is an unavoidable process that drives the dynamical evolution of astrophysical systems¹. As discussed elsewhere [1]–[8], the presence of evaporation is a very important ingredient in explaining the behavior of stellar structures observed in Nature. The effects of evaporation crucially depend on the relaxation mechanisms that are present in the microscopic dynamics, which depend on the concrete conditions of a given particular system. Among such mechanisms, one could mention (i) particle collisions [3]–[5], which are present in stellar systems that are sufficiently dense, such as globular clusters; and (ii) general mechanisms like parametric resonance, which is the origin of the chaotic behavior of microscopic dynamics in nonlinear many-body Hamiltonian systems with bound motions in configuration space [9].

We propose in this work a parametric family of astrophysical models that account for the influence of evaporation under different relaxation regimes: the truncated γ-exponential models. These phenomenological models constitute a suitable generalization of some models of tidal stellar systems available in the literature, such as the Wooley and Dickens truncated isothermal model [8], as well as King's models of globular clusters [4]. This proposal enables us to analyze the influence of the evaporation on the thermodynamic behavior and distribution profiles. Moreover, it provides a unification framework for the known isothermal and polytropic profiles considered in hydrodynamic models of stellar systems. Remarkably, these models are sufficiently tractableto allow an analytical derivation of most of their associated hydrodynamic quantities.

The paper is organized into sections as follows. Section 2 is devoted to introducing the parametric family of quasi-stationary distributions, and the derivation of their associated hydrodynamic quantities, as well as some details about the integration of the associated nonlinear Poisson equation. Afterwards, a third section is devoted to discussing numerical results obtained from this model: information concerning thermodynamical descriptions as well as distribution profiles. Some conclusions are drawn in the fourth section. Finally, mathematical properties concerning the so-called γ-exponential function are discussed in the appendix.

Both observational evidence and theoretical analysis suggest that the distribution profiles of many stellar systems can be explained by considering a quasi-stationary one-particle distribution with a mathematical form close to a Maxwell–Boltzmann distribution [10]:

$$\begin{equation}\label {MB.prof} f_{\rm MB}\left (\mathbf {q},\mathbf {p}\right ) \sim \exp \left [-\beta \varepsilon (\mathbf {q},\mathbf {p})\right ], \end{equation} \tag{ 1 }$$

where ε(q, p) = p²/2m + mϕ q is the mechanical energy of a given particle with momentum p located at the position q with a mean field gravitational potential ϕ q. The presence of evaporation introduces a deformation of the above distribution—specifically, a truncation for energies above a certain escape energy ε_e. Two examples of distributions that introduce this kind of deformation are the truncated isothermal model proposed by Wooley and Dickens [8]:

$$\begin{equation}\label {WD.prof} f_{\rm WD}\left (\mathbf {q},\mathbf {p}\mid \beta ,\varepsilon _{\rm e}\right ) = A\exp \left \{ \beta \left [\varepsilon _{\rm e}- \varepsilon (\mathbf {q},\mathbf {p})\right ] \right \}, \end{equation} \tag{ 2 }$$

and the Michie–King models [3, 4]:

$$\begin{equation}\label {MK.prof} f_{\rm MK}(\mathbf {q},\mathbf {p}\mid \beta ,\varepsilon _{\rm e})= A\left \{ \exp \left \{ \beta \left [\varepsilon _{\rm e}- \varepsilon (\mathbf {q},\mathbf {p})\right ] \right \} -1\right \}, \end{equation} \tag{ 3 }$$

which vanish for energies ε(q, p) > ε_e, where A represents a normalization constant. The first of these corresponds to a Maxwell–Boltzmann distribution (1) that is discontinuously truncated at the escape energy ε_e, while the second one considers a progressive vanishing at this point.

Our interest is in proposing a generalization of the evaporation truncation scheme considered in the above astrophysical models. Firstly, let us express them in a unifying form as follows:

$$\begin{equation}\label {df.model} f_{\gamma }(\mathbf {q},\mathbf {p}\mid \beta ,\varepsilon _{\rm e}) =A E_{\gamma }(x), \end{equation} \tag{ 4 }$$

where x = β ε_e − ε(q, p), while E_γ(x) is a certain function obtained from the truncation of the exponential function exp(x), with γ being a certain parameter. In particular, distributions (2) and (3) are described by the functions

$$\begin{eqnarray}\label {f01} \begin{array}{l} E_{0}(x)=\left \{ \begin{array}{@{}ll@{}} \exp (x), &\quad\qquad \mbox {for }x>0, \\ 0, &\quad\qquad \mbox {otherwise}, \end{array} \right .\quad\qquad \mbox {and}\\ E_{1}(x)=\left \{ \begin{array}{@{}ll@{}} \exp (x)-1, & \quad\qquad \mbox {for }x>0, \\ 0, &\quad\qquad \mbox {otherwise}. \end{array} \right . \end{array} \end{eqnarray} \tag{ 5 }$$

These two truncation schemes can be generalized to any nonnegative integer γ = n by subtracting the first n terms of the power expansion of the exponential function:

$$\begin{equation}\label {n.truncated} E_{n}(x)=\left \{ \begin{array}{@{}ll@{}} \displaystyle \exp (x)-\sum ^{n-1}_{k=0}x^{k}/k!, & \quad\qquad \mbox {for }x>0, \\ 0, & \quad\qquad \mbox {otherwise}. \end{array} \right . \end{equation} \tag{ 6 }$$

It is worthy of note that this ansatz contains the functions (5) as particular cases with γ = 0 and γ = 1, which now guarantees the vanishing of its n − 1 first derivatives at x = 0:

$$\begin{equation}\label {first.derivatives} E_{n}(0)=E_{n}'(0)=\cdots =E_{n}^{(n-1)}(0)=0. \end{equation} \tag{ 7 }$$

For positive values of the argument x, the function (6) can be rewritten into a more convenient integral representation as follows:

$$\begin{equation}E_{n}(x)=\frac {1}{(n-1)!}\int ^{x}_{0}\eta ^{n-1}\exp (x-\eta ) \,{\rm d} \eta . \end{equation} \tag{ 8 }$$

This integral representation enables a direct extension of the above truncation scheme to any nonnegative real number γ, on replacing the factorial (n − 1)! by the known Gamma function Γ(γ):

$$\begin{equation} \Gamma (\gamma )=\int ^{+\infty }_{0}\eta ^{\gamma -1}\exp (-\eta ) \,{\rm d} \eta . \end{equation} \tag{ 9 }$$

The resulting function

$$\begin{equation}\label {integral.gamma} E_{\gamma }(x)\equiv E\left (x;\gamma \right )= \frac {1}{\Gamma (\gamma )}\int ^{x}_{0}\eta ^{\gamma -1} \exp (x-\eta )\,{\rm d} \eta = \sum _{n=0}^{\infty } \frac {x^{\gamma +n}}{\Gamma \left (\gamma +1+n\right )} \end{equation} \tag{ 10 }$$

modifies the usual exponential function using γ as a deformation parameter. Hereinafter, this function will be referred to as a γ-exponential function, and it will be equivalently denoted as E_γ(x) or E x; γ. Definition (10) is simply the Riemann–Liouville fractional integral of the ordinary exponential function [20]. Considering the fractional differentiation rule for the power-function series:

$$\begin{equation} \frac {{\rm d} ^{\alpha }}{{\rm d} x^{\alpha }}\sum _{\mu }a_{\mu }x^{\mu }\equiv \sum _{\mu }a_{\mu }\frac {\Gamma (\mu +1)} {\Gamma (\mu -\alpha +1)}x^{\mu -\alpha }, \end{equation} \tag{ 11 }$$

conditions (7) are now generalized as follows:

$$\begin{equation} \frac {{\rm d} ^{\alpha }}{{\rm d} x^{\alpha }}E\left (0;\gamma \right )=0, \end{equation} \tag{ 12 }$$

where the order of the fractional differentiation α belongs to the interval 0 ≤ α < γ. Readers can find some useful properties of the γ-exponential function E x; γ in the appendix.

The proposed family of quasi-stationary distributions (4) describe the influence of evaporation with two independent parameters: the escape energy ε_e and the deformation parameter γ. This family of distributions is shown schematically in figure 1. They also exhibit a Maxwell–Boltzmann profile (1) for energies ε(q, p) that are sufficiently below the escape energy ε_e. Besides this, they show a power-law truncation at ε_e with exponent γ. The increasing of the deformation parameter γ characterizes a larger deviation from the Maxwell–Boltzmann profile (1). From a phenomenological viewpoint, a larger value of the deformation parameter γ describes a stronger influence of the evaporation, or equivalently, a weaker influence of the relaxation mechanism. As shown below, the ansatz (4) can describe distribution profiles with isothermal cores and polytropic haloes, where the deformation parameter γ determines the polytropic structure of haloes. The escape energy ε_e = mϕ_e can be derived from the so-called tidal potential ϕ_e, which accounts for the gravitational influence of neighboring systems [1]–[3], [5]. According to this energetic truncation, a particle that is trapped under the system gravitational influence will be confined inside a finite region of the space, where the gravitational potential ϕ(q) fulfills the inequality ϕ(q) ≤ ϕ_e. For a system with spherical symmetry, the boundary of this region is the sphere of radius R_t:

$$\begin{equation}\label {tidal radius} R_{\rm t}=-GM/\phi _{\rm e}, \end{equation} \tag{ 13 }$$

which is referred to as the tidal radius [1], with M = Nm being the total mass of the astrophysical system.

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Let us now obtain some hydrodynamic observables associated with the family of distributions (4). Introducing the dimensionless variable τ = βp²/2m as well as the dimensionless potential Φ = Φ q:

$$\begin{equation}\label {dimensionless.pot} \Phi \left (\mathbf {q}\right )=\beta m \left [\phi _{\rm e}-\phi \left (\mathbf {q}\right )\right ], \end{equation} \tag{ 14 }$$

the particle density n(q) can be calculated as follows:

$$\begin{equation}n\left (\mathbf {q}\right ) =\int f_{\gamma }\left (\mathbf {q},\mathbf {p}\right ) \,{\rm d} ^{3}\mathbf {p}=A^{\prime }\int _{0}^{\Phi }E\left (\Phi -\tau ;\gamma \right ) \tau ^{\frac {1}{2}}\,{\rm d} \tau , \end{equation} \tag{ 15 }$$

where A^' = 2πA 2m/β^3/2. The integration can be performed by using the convolution formula (A.11) with parameter ν = 3/2, which leads to an expression for the particle density in terms of the γ-exponential function as follows:

$$\begin{equation}\label {densidad particulas} n\left (\mathbf {q}\right ) =n\left [\Phi (\mathbf {q}); \gamma \right ] =CE\left [\Phi (\mathbf {q});\gamma +\tfrac {3}{2}\right ]. \end{equation} \tag{ 16 }$$

Here, C = A 2mπ/β^3/2 and $\Gamma (3/2)=\sqrt {\pi }/2$, in accordance with the gamma function for half-integer values:

$$\begin{equation}\label {equation17} \Gamma \left (n + \frac {1}{2}\right ) = \frac {(2n)!}{4^{n} n!}\sqrt {\pi }. \end{equation} \tag{ 17 }$$

The dependence of particle density on the dimensionless potential, equation (17), is shown in figure 2. Another important observable is the kinetic energy density υ(q):

$$\begin{equation} \upsilon (\mathbf {q})=\int f_{\gamma }\left (\mathbf {q},\mathbf {p}\right ) \frac {1}{2m} \mathbf {p}^{2}{\rm d} ^{3}\mathbf {p}=\frac {A^{\prime }}{\beta } \int _{0}^{\Phi }E\left (\Phi -\tau ;\gamma \right ) \tau ^{\frac {3}{2}}\,{\rm d} \tau , \end{equation} \tag{ 18 }$$

which is also expressed in terms of the γ-exponential function using the convolution formula (A.11) with parameter ν = 5/2:

$$\begin{equation} \upsilon (\mathbf {q})=\upsilon \left [\Phi (\mathbf {q}); \gamma \right ]=\frac {3}{2\beta }CE\left [\Phi (\mathbf {q}); \gamma +\frac {5}{2}\right ]. \end{equation} \tag{ 19 }$$

Finally, we can introduce the magnitude (q) = υ(q)/n(q):

$$\begin{equation} \epsilon (\mathbf {q}) = \frac {3}{2\beta }\chi \left (\Phi ;\gamma \right ) =\frac {3}{2\beta }\frac {E\left (\Phi ;\gamma +\frac {5}{2}\right )} {E\left (\Phi ;\gamma +\frac {3}{2}\right )}, \end{equation} \tag{ 20 }$$

which represents the kinetic energy per particle.

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According to the kinetic theory for ideal gases, this magnitude (q) can be related to the fluid pressure p(q) as follows:

$$\begin{equation}\label {PresionGamma} p(\mathbf {q}) = \frac {2}{3}n(\mathbf {q})\epsilon (\mathbf {q})=\frac {1}{\beta }CE\left (\Phi ;\gamma +\frac {5}{2}\right ). \end{equation} \tag{ 21 }$$

The hydrostatic equilibrium arises in a fluid when the gravitational force and pressure gradient are balanced as follows:

$$\begin{equation}\label {hidro} \nabla p(\mathbf {q})= \rho (\mathbf {q}) {\bf g}(\mathbf {q}), \end{equation} \tag{ 22 }$$

with p and ρ being its pressure and mass density, respectively. The gravitation intensity vector g(q) can be expressed in terms of the gravitational potential ϕ(q):

$$\begin{equation}\label {gravedad} {\bf g}(\mathbf {q})= - \nabla \phi (\mathbf {q}). \end{equation} \tag{ 23 }$$

Considering the total differentiations for Φ(q) and p(q):

$$\begin{equation} {\rm d} \Phi (\mathbf {q})=\mathbf {\nabla }\Phi (\mathbf {q}) \boldsymbol {\cdot } {\rm d} \mathbf {q}\quad\qquad \mbox {and}\quad\qquad {\rm d} p(\mathbf {q})=\mathbf {\nabla }p(\mathbf {q}) \cdot {\rm d} \mathbf {q}, \end{equation} \tag{ 24 }$$

the hydrodynamic equilibrium condition (22) can also be rewritten as

$$\begin{equation}\label {NuevaEquiHidro} \beta \frac {{\rm d} p\left (\Phi ;\gamma \right )} {{\rm d} \Phi }\equiv n\left (\Phi ;\gamma \right ), \end{equation} \tag{ 25 }$$

where n Φ; γ and p Φ; , γ are the particle density and the pressure expressed in terms of the dimensionless potential Φ. Considering the differentiation identity (A.9), it is straightforwardly verified that the hydrodynamic quantities (16) and (21) derived from the truncated γ-exponential model satisfy the condition for hydrostatic equilibrium (22).

The mass density ρ(q) = mn(q) and gravitational potential ϕ(q) are related via the Poisson equation, Δϕ(q) = 4πGρ(q). Restricting to situations with spherical symmetry, one obtains the following nonlinear Poisson problem in terms of the dimensionless potential (14):

$$\begin{equation}\label {p1} \frac {1}{\xi ^{2}}\frac {{\rm d} }{{\rm d} \xi }\left [\xi ^{2} \frac {{\rm d} }{{\rm d} \xi }\Phi (\xi )\right ]=-4 \pi E\left [\Phi (\xi );\gamma +\frac {3}{2}\right ]. \end{equation} \tag{ 26 }$$

We have considered here the dimensionless radius

$$\begin{equation} \xi =\sqrt {\kappa }r/R_{\rm t}, \end{equation} \tag{ 27 }$$

where r is the physical radius, κ = Gm²βCR² is an auxiliary constant, and R_t is the tidal radius. Problem (26) can be solved numerically by demanding the following conditions at the origin:

$$\begin{equation} \Phi \left (0\right ) =\Phi _{0}, \quad\qquad \Phi ^{\prime }\left (0\right ) =0, \end{equation} \tag{ 28 }$$

and integratinguntil the vanishing of a dimensionless potential at the surface of the system with dimensionless radius $\xi _{\rm c}=\sqrt {\kappa }$ is achieved:

$$\begin{equation}\label {Boundary} \Phi \left (\xi _{\rm c}\right ) =0,\quad\qquad \Phi ^{\prime }\left (\xi _{\rm c}\right ) =-\frac {\eta }{\xi _{\rm c}}. \end{equation} \tag{ 29 }$$

This constant allows us to express the mass density ρ in units of the characteristic density $\rho _{\rm c} = M/R^{3}_{t}$ as follows:

$$\begin{equation} \rho =\frac {\kappa }{\eta }E\left (\Phi ;\gamma +\frac {3}{2}\right ) , \label {den} \end{equation} \tag{ 30 }$$

where η is the dimensionless inverse temperature:

$$\begin{equation} \eta =\beta \frac {GMm}{R_{\rm t}}. \end{equation} \tag{ 31 }$$

The total energy U = K + V can be obtained from the total kinetic K and the total potential energy V :

$$\begin{eqnarray} &&\begin{array}{l} \displaystyle V = -\frac {1}{2}-\frac {1}{2\eta ^{2}\xi _{\rm c}}\int _{0}^{\xi _{\rm c}}E\left [\Phi (\xi );\gamma +\frac {3}{2}\right ] \Phi (\xi )~4\pi \xi ^{2}\,{\rm d} \xi ,\\ \displaystyle K = \frac {3}{2\eta ^{2}\xi _{\rm c}}\int _{0}^{\xi _{\rm c}}E\left [\Phi (\xi );\gamma +\frac {5}{2}\right ] 4\pi \xi ^{2}\,{\rm d} \xi ,\label {K} \end{array} \end{eqnarray} \tag{ 32 }$$

which are expressed in units of the characteristic energy U_c = GM²/R_t.

3.1. Generalities

Many studies concerning astrophysical systems and cosmological problems [1, 5] start from assuming a polytropic dependence between the pressure p and the particle density ρ:

$$\begin{equation}\label {state.eq} p=C \rho ^{\gamma ^{*}}, \end{equation} \tag{ 33 }$$

where C is a certain constant and γ^∗ is the so-called polytropic index. The phenomenological state equation (33) can be combined with the condition for hydrostatic equilibrium (22) to derive a power-law relation between the density and the dimensionless potential Φ:

$$\begin{equation} \rho =K\Phi ^{n}, \end{equation} \tag{ 34 }$$

where K = (γ^∗− 2)/γ^∗mβCⁿ and the exponent n ≡ 1/(γ^∗− 1) ≥ 0. The marginal case γ^∗ = 1 is fully permitted. The condition for hydrostatic equilibrium now predicts an exponential dependence between ρ and Φ:

$$\begin{equation} \rho =K^{*}\exp \left [\Phi \right ], \end{equation} \tag{ 35 }$$

where K^∗ is a certain integration constant. This behavior can be associated with a system under isothermal conditions, on imposing the constraint C ≡ 1/mβ, where m is the mass of the constituent particles and β is the inverse temperature. The ansatz (4) proposed in this study allows us to consider the above dependences in a unifying fashion. According to the asymptotic behavior of the truncated γ-exponential function (A.7), the particle density (30) describes a power-law dependence for Φ small with exponent n = γ + 3/2, while it describes an exponential dependence for Φ large enough:

$$\begin{equation}\label {asymp.density} \rho \left (\Phi ;\gamma \right ) =\left \{ \begin{array}{@{}ll@{}} {\propto }\Phi ^{n}, &\quad\qquad x\ll 1,\\ {\propto }\exp (\Phi ) , &\quad\qquad x\gg 1. \end{array} \right . \end{equation} \tag{ 36 }$$

Accordingly, the deformation parameter γ can be related to the polytropic index γ^∗ of the polytropic state equation (33) as γ^∗ = (2γ + 5)/(2γ + 3). According to the nonlinear Poisson problem (26), the dimensionless potential Φ decreases from the inner region towards the outer region of the system. This means that the truncated γ-exponential models could describe distribution profiles with isothermal cores and polytropic haloes. Since the particle density (30) should vanish at the system surface, where the dimensionless potential Φ(ξ_c) = 0, one should expect the external dimensionless radius ξ_c to be finite: 0 < ξ_c < +∞. However, polytropic profiles with exponent n > 5 are infinitely extended in space and exhibit an infinite mass: ξ_c→ + ∞ and M→ + ∞. Such profiles cannot describe any realistic situation [5]. On the other hand, a finite character of the quasi-stationary distribution function f_qe(q, p) is only possible if the deformation parameter γ ≥ 0. Consequently, the admissible values of the deformation parameter γ must be restricted to the following interval:

$$\begin{equation}0\leq \gamma <\gamma _{\rm m}=7/2. \end{equation} \tag{ 37 }$$

Accordingly, these models can only describe polytropic dependences with exponent n restricted to the interval 3/2 ≤ n < 5. As shown below, this restriction will be manifested in both thermodynamic quantities and distribution profiles.

The nonlinear Poisson problem (26) is integrated using γ and Φ₀ as independent integration parameters. This task was accomplished using the Runge–Kutta fourth-order method, which was implemented using FORTRAN 90 programming. Results from this integration are shown in figure 3—in particular, the dependences of ξ_c and η versus the central value of the dimensionless potential Φ₀ for some values of the deformation parameter γ. According to these results, all dependences of the dimensionless radius ξ_c diverge when the dimensionless potential Φ₀ approaches zero, which means that the admissible values of the parameter Φ₀ are nonnegative. Curiously, the dependences of the dimensionless radius ξ_c corresponding to the deformation parameter γ = 2.5 and γ = 3.0 diverge at a certain value $\Phi ^{\infty }_{0}$ of the dimensionless potential Φ₀ that depends on the deformation parameter γ, while the corresponding dependences of the dimensionless inverse temperature η simultaneously vanish. This means that the values of the dimensionless potential Φ₀ above the point $\Phi ^{\infty }_{0}$ are also nonphysical. A better understanding of the physical meaning of the behaviors observed in the dependences of figure 3 is achieved by analyzing the dependences of the thermodynamic quantities.

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3.2. The thermodynamical behavior

Dependences of the inverse temperature η and central particle densities ρ(0) versus the dimensionless energy U are shown in figure 4 for different values of the deformation parameter γ. All quasi-stationary configurations obtained from these models always have negative values for the energy U. Moreover, one can recognize the existence of three notablepoints:

• The critical point of gravothermal collapse U_A: the quasi-stationary configuration with minimum energy U_A. There are no quasi-stationary configurations for energies below this point. If a system is initially prepared with an energy below this point, it will experience an instability process that leads to a sudden contraction of the system under its own gravitational field, a phenomenon commonly referred to in the literature as gravothermal collapse [21].
• The critical point of isothermal collapse U_B: the quasi-stationary configuration with minimum temperature T_B. There are no stable configurations for temperatures T < T_B. If a system under isothermal conditions (in the presence of a thermostat at constant temperature) is initially put in thermal contact with a heat reservoir with T < T_B, this system will experience an instability process fully analogous to gravitational collapse, which is referred to as isothermal collapse [21]. This type of thermodynamical instability is less relevant than the gravothermal collapse because considering the presence of a thermostat is actually unrealistic in most of astrophysical situations.
• The critical point of evaporation disruption U_C: the quasi-stationary configuration with maximal energy, that is, there are no quasi-stationary configurations for energies U > U_C. The existence of this superior bound is a direct consequence of the presence of evaporation, which imposes a maximum value for the individual mechanical energies of the system constituents, ε < ε_e. If the system is initially prepared with an energy above this, it will experience a sudden evaporation in order to release its excess of energy [7]. Note that the inverse temperature η_C always vanishes at this point regardless of the value of the deformation parameter γ.

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All quasi-stationary configurations are located inside the energy region U_A ≤ U ≤ U_C. Moreover, there is more than oneadmissible value for the dimensionless inverse temperature η for a given energy U near the point of gravothermal collapse U_A. According to results shown in figure 5, stable quasi-stationary configurations belong to the superior branch A–B–C, since these configurations exhibit a higher value of the entropy S for a given total energy. The energy dependence of this thermodynamic potential was evaluated from numerical integration of the expression

$$\begin{equation}\label {entropy.integral} \Delta S=S(U)-S_{\rm C}=\int _{U_{\rm C}}^{U}\eta (U')\, {\rm d} U', \end{equation} \tag{ 38 }$$

which employs as a reference the value S_C corresponding to the critical point of evaporation instability U_C. Stable quasi-stationary configurations inside the energy range U_A ≤ U ≤ U_B exhibit negative heat capacities C < 0. The existence of this thermodynamic anomaly is a remarkable consequence of the long-range character of gravitation—in particular, because of the short-range divergence of its interaction potential energy:

$$\begin{equation} \phi \left (\mathbf {r}_{i},\mathbf {r}_{j}\right )=- Gm_{i}m_{j}/\left |\mathbf {r}_{j}-\mathbf {r}_{i}\right |\rightarrow \infty \end{equation} \tag{ 39 }$$

when the particle separation distance r_j −r_i drops to zero. While such configurations are unstable if the system is put into thermal contact with an environment at constant temperature (canonical ensemble), they are stable if the system is put into energetic isolation (microcanonical ensemble).

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To provide a better understanding of the influence of the deformation parameter γ, we have calculated the dependences of some thermodynamic quantities at notable points. Results are shown in figure 6. It is clearly evident that increase of the deformation parameter γ provokes a systematic decrease of the inverse temperatures (η_A, η_B), and an increase of the absolute values of the energies U_A, U_B, U_C and their associated central particle densities ρ_0A, ρ_0B, ρ_0C. Interestingly, the inverse temperature η_A at the notable point of gravothermal collapse vanishes when γ ≥ γ_c ≃ 2.1. This means that both the total energy U_A and the temperature T_A diverge at this point as a consequence of the divergence of the central density ρ_A. To be precise, the existence of this divergence was also manifested as a divergence of the dimensionless radius ξ_c, which was shown in figure 3 for the particular cases with deformation parameters γ = 2.5 and γ = 3.0.

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The fact that the energy of gravothermal collapse diverges, U_A→−∞, when γ ≥ γ_c significantly reduces the dramatic character of this phenomenon. For admissible values of the deformation parameter γ below the point γ_c, the system develops a gravothermal collapse at the finite energy U_A, which should evolve in a discontinuous way towards a certain collapsed structure that is not describable with the present model. For admissible values of the deformation parameter γ above the point γ_c, the system should release an infinite amount of energy to reach a collapsed structure with a divergent central density associated with the point of gravothermal collapse. However, this collapsed structure is now described within the present models and the transition is developed in a continuous way with the decreasing of the internal energy.

It is noteworthy that an analogous divergence is observed in thermodynamic parameters of other notable points when the deformation parameter γ approaches its maximum admissible value γ_m = 3.5. As expected, this second divergence point is related to the nonphysical character of polytropic dependences when n > 5. A more precise estimation for the critical value γ_c can be obtained from considering an adjustment of the dependence η_A(γ)with a power-law form:

$$\begin{equation}\label {form.power} \eta _{\rm A}\left (\gamma \right )= A|\gamma -\gamma _{\rm c}|^{p}. \end{equation} \tag{ 40 }$$

As shown in the inset panel of figure 6, the proposed form (40) exhibits a great agreement with numerical results for the following set of parameters: A = 0.292 ± 0.004, p = 1.24 ± 0.01 and γ_c = 2.1307 ± 0.003.

3.3. Distribution profiles

We show in figure 7 a distribution profile with deformation parameter γ = 1 at the point of gravothermal collapse, which corresponds to the Michie–King profile with lowest energy. As a consequence of gravitation, the highest concentration of the particles is always located in the inner region of the system, while the particle density gradually decays with increase of the radius r until it vanishes at the tidal radius R_t. As clearly shown in this figure, the inner region—the core—can be well-fitted with an isothermal profile [19], while the outer one—the halo—is well-fitted with a polytropic profile [5].

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Dependences of the distribution profiles on the deformation parameter γ and the internal energy U are illustrated in figure 8—specifically, twelve profiles corresponding to three notable points and four different values of the deformation parameter γ. The particle concentration in the inner regions decreases with increase of the internal energy U. Increase of the deformation parameter γ produces distribution profiles with more dilute haloes, and consequently, denser cores. Not all admissible profiles derived from the present family of models can exhibit isothermal cores. However, all these profiles exhibit polytropic haloes. In fact, distribution profiles near the point of evaporation disruption can be regarded as everywhere polytropic with high accuracy. Finally, distribution profiles corresponding to the point of gravothermal collapse with γ > γ_c are divergent at the origin.

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We have shown in figure 9 the phase diagram of truncated γ-exponential models in the plane of integration parameters γ−Φ₀ of the nonlinear Poisson problem (26). For each value of the deformation parameter γ, the admissible values of the central dimensionless potential Φ₀ are located inside the interval 0 ≤ Φ₀ ≤ Φ_A(γ), where Φ_A(γ) corresponds to the critical point of gravothermal collapse U_A. Central values of the dimensionless potential Φ₀ above the dependence Φ_A(γ) correspond to nonphysical or unstable configurations (white region). Additionally, we have included the dependence Φ_B(γ) associated with the point of isothermal collapse U_B. Configurations between the dependences, Φ_B(γ) ≤ Φ₀ ≤ Φ_A(γ), exhibit negative heat capacities (dark gray region), while those with the central dimensionless potential Φ₀ belonging to the interval 0 < Φ₀ < Φ_B(γ) exhibit positive heat capacities (light gray region). It is remarkable that the dependence Φ_A(γ) is weakly modified by a change in the deformation parameter γ for values below the critical point γ_c. However, this function experiences a sudden change above this critical value. As expected, this behavior accounts for the sudden change in behavior of the distribution profiles: the proposed models can exhibit profiles with isothermal cores for γ < γ_c, while they only exhibit profiles without isothermal cores for γ ≥ γ_c.

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According to equation (36), profiles with isothermal cores are directly related to asymptotic dependence of the particle distribution following an exponential law with regard to the local value of the dimensionless potential Φ(ξ). Such an asymptotic behavior is better described in terms of the deviation function δ(x, γ) with respect to the exponential function, which is introduced in the appendix. For the sake of convenience, we have denoted the dependence Φ_ic(γ) as follows:

$$\begin{equation} \delta \left [\Phi _{\rm ic}(\gamma );\gamma +\tfrac {3}{2}\right ]=\epsilon , \end{equation} \tag{ 41 }$$

where the convergence error was fixed at the value = 1.6 × 10⁻⁴. This small value guarantees the matching of this dependence at the critical value of the deformation parameter γ_c with the central dimensionless potential Φ₀ associated with the point of gravothermal collapse, Φ_ic(γ_c) = Φ_A(γ_c). Quasi-stationary configurations located inside the region with Φ_ic(γ) ≤ Φ₀ ≤ Φ_A(γ) and 0 < γ < γ_c exhibit isothermal cores, that is, the inner regions of the particle distribution can be fitted with an isothermal profile.

We have introduced truncated γ-exponential models to characterize the properties of astrophysical systems in quasi-stationary evolution under the influence of evaporation. Our proposal generalizes models of tidal stellar systems available in the literature, such as Michie–King models. These models exhibit many features observed in other astrophysical models. Due to the truncation of the particle distribution in the configuration space f_qe(q, p), the distribution of particles in the physical space is located inside a finite region limited by the tidal radio R_t. Moreover, the total energy U is also restricted to a finite region U_A ≤ U ≤ U_C. The lower bound U_A represents the energy of gravitational collapse, while the upper one U_C is the energy of evaporation disruption. Caloric curves for these models exhibit an anomalous branch with negative heat capacities for energies U_A ≤ U ≤ U_B, where U_B is the value of the energy corresponding to the point of isothermal collapse, where the system reaches a minimum temperature T_B. Distribution profiles with lower energy could exhibit isothermal cores and polytropic haloes with exponent n = γ + 3/2. Moreover, polytropic profiles are always obtained for energies near the point of evaporation disruption U_C. The admissible values of the deformation parameter γ are restricted to the interval 0 ≤ γ < γ_m = 7/2. This means that this family of models describes polytropic profiles with exponent n inside the interval 3/2 ≤ n < 5. A nontrivial result obtained from these models is that the existence of distribution profiles with isothermal cores is no longer possible if the deformation parameter γ ≥ γ_c ≃ 2.13. The existence of a notable value γ_c for deformation parameter indicates a drastic change in the behavior of the thermodynamic quantities and distribution profiles. Specifically, these cases exhibit a simultaneous divergence of the energy of gravitational collapse U_A, its corresponding temperature T_A and the central density ρ_0A. For larger energies, these values of the deformation parameter γ describe distribution profiles with an almost polytropic form.

Before ending this section, let us briefly refer to some open questions. A realistic improvement of the present models would be the consideration of the mass spectrum for the constituent particles, which would enable us to study the influence of mass segregation on the thermodynamic behavior and distribution profiles [7]. A second improvement would be the consideration of factors that lead to a breakdown of spherical symmetry, such as system rotation or the tidal field of a neighboring astrophysical system [11]–[15]. Finally, we shall perform a comparison of results obtained from these models with experimental data—in particular, particle distributions of tidal stellar systems such as globular clusters and elliptical galaxies. These problems will be analyzed in forthcoming works.

Velazquez thanks for financial support CONICyT/Programa Bicentenario de Ciencia y Tecnología PSD 65. He also wishes to express thanks for partial financial support from the VRIDT-UCN research program.

The γ-exponential function E x; γ is defined with the help of the so-calledgamma distribution as follows:

$$\begin{equation}E\left (x;\gamma \right ) ={\rm e} ^{x}\frac {1}{\Gamma \left (\gamma \right ) } \int _{0}^{x}{\rm e} ^{-\tau }\tau ^{\gamma -1}\,{\rm d} \tau , \label {def} \end{equation} \tag{ A.1 }$$

which vanishes for x < 0. This family of functions is directly related to the lower incomplete gamma function:

$$\begin{equation}\label {gincomplet} \gamma \left (s,x\right ) = \int ^{x}_{0} {\rm e} ^{-t}t^{s-1}\,{\rm d} t, \end{equation} \tag{ A.2 }$$

which only differs from expression (A.1) because of the factor A s, x = e^x/Γ s. Moreover, this can also be obtained applying the fractional integral operator [20]:

$$\begin{equation}\label {dfractorial} \left (J^{\gamma }f\right )=\frac {1}{\Gamma \left (\gamma \right )} \int _{0}^{x}\left (x-t\right )^{\gamma -1}f\left (t\right )\,{\rm d} t \end{equation} \tag{ A.3 }$$

to the exponential function. Changing the integration variable, t→x − t, in this last definition, one obtains

$$\begin{equation} \left (J^{\gamma }f\right )\equiv \frac {1}{\Gamma \left (\gamma \right )} \int _{0}^{x}t^{\gamma -1}f\left (x-t\right )\,{\rm d} t, \end{equation} \tag{ A.4 }$$

which reduces to expression (A.1) when f(t) = exp(t).

Performing the integration by parts, one can obtain from definition (A.1) the recurrent relation

$$\begin{equation}\label {recurrence} E\left (x;\gamma \right ) =\frac {x^{\gamma }}{\Gamma \left (\gamma +1\right ) }+E\left (x;\gamma +1\right ) , \end{equation} \tag{ A.5 }$$

which leads to the power expansion in equation (10). This representation allows us to obtain the following particular expressions:

$$\begin{equation}\label {particular} E\left (x;-n\right ) =E\left (x;0\right ) ={\rm e} ^{x},\quad\qquad E\left (x,\tfrac {1}{2}\right ) ={\rm e} ^{x}{\rm erf}\,(\sqrt {x}) , \end{equation} \tag{ A.6 }$$

where n is any positive integer number. In general, the γ-exponential E x; γ is a continuous and differentiable function for every x > 0. This function vanishes at x = 0 whenever γ > 0, while it exhibits a discontinuity at x = 0 when the deformation parameter γ→0. It is easy to verify that the function E x; γ with γ > 0 exhibits an exponential behavior for sufficiently large x and a power-law dependence for small x:

$$\begin{equation}\label {asymptotic} E\left (x;\gamma \right ) =\left \{ \begin{array}{@{}ll@{}} x^{\gamma }/\Gamma \left (\gamma +1\right ) &\quad\qquad \mbox { if }x\ll 1,\\ {\rm e} ^{x} &\quad\qquad \mbox { if } x\gg 1. \end{array} \right . \end{equation} \tag{ A.7 }$$

According to the recurrence relation (A.5), the γ-exponential function E(x, γ) always diverges at the origin x = 0 if the deformation parameter γ is negative but not an integer. The behavior of this family of functions is shown in panel (a) of figure A.1 for some values of the deformation parameter γ. Moreover, we have also considered the deviation function:

$$\begin{equation}\label {defect} \delta (x,\gamma )=1-\exp (-x)E(x,\gamma ) \end{equation} \tag{ A.8 }$$

that characterizes the relative deviation of the γ-exponential E(x, γ) from the ordinary exponential. The behavior of this latter function is illustrated in panel (b) of figure A.1.

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Using the power-expansion expression (10), one can obtain the integration–differentiation rules:

$$\begin{eqnarray} &&\frac {{\rm d} }{{\rm d} x}E\left (x;\gamma \right ) = E\left (x;\gamma -1\right ),\label {diff} \end{eqnarray} \tag{ A.9 }$$

$$\begin{eqnarray} &&\int E\left (x;\gamma \right ) {\rm d} x = E\left (x;\gamma +1\right ) +C, \end{eqnarray} \tag{ A.10 }$$

and the convolution formula:

$$\begin{equation} \frac {1}{\Gamma \left (\nu \right ) }\int _{0}^{x}E\left (x-\tau ;\gamma \right ) \tau ^{\nu -1}\,{\rm d} \tau =E\left (x;\gamma +\nu \right ). \label {convolution} \end{equation} \tag{ A.11 }$$

This last relation is obtained using the Beta function:

$$\begin{equation} {\rm B}\left (\mu ,\nu \right ) =\int _{0}^{1}\left (1-\tau \right ) ^{\mu -1}\tau ^{\nu -1}\,{\rm d} \tau =\frac {\Gamma \left (\mu \right ) \Gamma \left (\nu \right ) }{\Gamma \left (\mu +\nu \right )}. \end{equation} \tag{ A.12 }$$

Notethat definition (A.1) is a particular case of the convolution formula (A.11) for γ = 0. Moreover, this identity accounts for the fractional integration of the γ-exponential function:

$$\begin{equation} \hat {J}^{\nu }E\left (x;\gamma \right )\equiv \frac {1}{\Gamma \left (\nu \right ) }\int _{0}^{x}E\left (\tau ;\gamma \right ) (x-\tau )^{\nu -1}\,{\rm d} \tau =E\left (x;\gamma +\nu \right ). \end{equation} \tag{ A.13 }$$