Truncated γ-exponential models for tidal stellar systems

We introduce a parametric family of models to characterize the properties of astrophysical systems in a quasi-stationary evolution under the incidence evaporation. We start from an one-particle distribution fγ (q, p|β,ϵs) that considers an appropriate deformation of Maxwell-Boltzmann form with inverse temperature β, in particular, a power-law truncation at the scape energy ϵs with exponent γ > 0. This deformation is implemented using a generalized γ-exponential function obtained from the fractional integration of ordinary exponential. As shown in this work, this proposal generalizes models of tidal stellar systems that predict particles distributions with isothermal cores and polytropic haloes, e.g.: Michie-King models. We perform the analysis of thermodynamic features of these models and their associated distribution profiles. A nontrivial consequence of this study is that profiles with isothermal cores and polytropic haloes are only obtained for low energies whenever deformation parameter γ < γc ≃ 2.13. This study is a first approximation to characterize a self- gravitating system, so we consider equal to all the particles that constitute the system.


Introduction
Gravitation is an example of attractive interaction that is unable to confine particles. It is always possible that some particles acquire sufficiently energy and escape from gravitational influence of the system. Evaporation is an unavoidable process that drives dynamical evolution of astrophysical systems in Nature. Needless to say that it is a very important ingredient to explain structure and evolution of the so-called tidal stellar systems, such as globular clusters [1,2]. In general, the effects of evaporation crucially depend on the relaxation mechanisms that are present in the microscopic dynamics, which depend on concrete conditions of a given particular system [3,4,5].
We propose in this work a parametric family of astrophysical models that accounts for the influence of evaporation under different relaxation regimes: the truncated γ-exponential models [6]. These phenomenological models constitutes a suitable generalization of some models of tidal stellar systems available in the literature, such as Wooley & Dickens truncated isothermal model [7], as well as King's models of globular clusters [8,9] and Wilson models [10]. This proposal enable us to analyze the influence of the evaporation on the thermodynamical behavior and distribution profiles. Moreover, it provides a unification framework of the known isothermal and polytropic profiles considered in hydrodynamic models of stellar systems [11]. Remarkably, these models are sufficiently amenable to allow an analytical derivation of most of their associated hydrodynamic quantities.
The paper is organized into sections as follows. Second section is devoted to introduce the parametric family of quasi-stationary distributions, the derivation of their associated hydrodynamic quantities, as well as some details about the integration of the associated nonlinear Poisson equation. Afterwards, third section is devoted to discuss numerical results obtained from this model, both information concerning to thermodynamical description as well as distribution profiles. Some conclusions are drawn in forth section.

A parametric family of quasi-stationary one-body distributions
Observational evidences and theoretical analysis suggest that distribution profiles of many stellar systems can be explained considering a quasi-stationary one-particle distribution with a mathematical form close to Maxwell-Boltzmann distribution [12]: with ε(q, p) = p 2 /2m + mϕ (q) being the mechanical energy of a given particle with momentum p located at the position q with a mean field gravitational potential ϕ (q). The incidence of evaporation introduces a deformation of the above distribution, specifically, a truncation for energies above a certain escape energy ε s . Some examples of distributions that introduce this kind of deformation are truncated isothermal models of Wooley and Dickens [7], Michie-King models [8,9] and Wilson models [10]. Quasistationary one-body distributions of all these models can be expressed using certain truncated exponential function E t (x) as follows: The function E t (x) adopts the following forms: for each ones of these models, respectively, with H(x) being the Heaviside step function. Our interest is to propose a generalization of evaporation truncation scheme considered in the above astrophysical models. Firstly, let us express them into a unifying form as follows: where E γ (x) is a certain function obtained from the truncation of exponential function exp(x), with γ being a deformation parameter. At first glance, the truncation schemes (3) could be generalized for any nonnegative integer γ = n subtracting the first n terms of power-expansion of exponential function: Precisely, definitions (3) correspond to the particular cases γ = 0, γ = 1 and γ = 2. A more general definition for any nonnegative real number γ is the following: which is obtained replacing factorial function (n+k)! by the Gamma function Γ(γ+1+k). Hereinafter, this function will be referred to as γ-exponential function, and the same one will be equivalently denoted as E γ (x) or E (x; γ). Strictly speaking, this function represents fractional integration of ordinary exponential function [13]. The proposed family of quasi-stationary distributions describe the influence of evaporation with two independent parameters: the escape energy ε s and the deformation parameter γ. This family of distributions is schematically shown in figure 1. The same ones exhibit a Maxwell-Boltzmann profile (1) for energies ε(q, p) that are far enough from below the escape energy ε s . Besides, they show a power-law truncation at ε s with exponent γ. The increasing of deformation parameter γ characterizes a larger deviation from Maxwell-Boltzmann profile (1). For a phenomenological viewpoint, a larger value of the deformation parameter γ describes a stronger influence of evaporation, or equivalently, the incidence of a weaker relaxation mechanism.

Structure equation and observables
Truncation due to evaporation leads to spherical-symmetric distributions that are located inside a region of finite radius R t , the so-called tidal radius [1]. Rewriting the escape energy ε s = mϕ s using the surface gravitational potential ϕ s , one can express R t as follows: where M is the system total mass. Introducing the dimensionless potential Φ = Φ (q): one can express the mass density ρ in characteristic units ρ c = M/R 3 t as follows: where η is the dimensionless inverse temperature: and κ a certain integration parameters defined below. Mathematical form of dimensionless potential is obtained from integration of the following nonlinear Poisson problem: 1 We have considered here the dimensionless radius: where r is the physical radius, κ = Gm 2 βCR 2 is an auxiliary constant, R t is the tidal radius. Problem (11) can be solved in a numerical way by demanding the following conditions at the origin: which is integrated till the vanishing of dimensionless potential at the surface of the system with dimensionless radius ξ c = √ κ: The total energy U = K +V can be obtained from the total kinetic K and total potential energy V : which are expressed in characteristic units U c = GM 2 /R t .

Results and discussions
According the asymptotic behavior of truncated γ-exponential function, the particles density (9) describes a power-law dependence for Φ small with exponent n = γ + 3/2, while an exponential dependence for Φ large enough: Consequently, this family of models unifies the known isothermal and polytropic profiles of hydrodynamic astrophysical models [1,11]. Since the deformation parameter γ is nonnegative and the polytropic profiles are unstable for n ≥ 5 [11], the admissible values of the polytropic exponent n are restricted to the interval 3/2 ≤ n < 5, which correspond  Truncated γ-exponential profile with γ = 1 (Michie-King profile) corresponding to the point of gravothermal collapse and its comparison with isothermal and polytropic profiles using log-log scales. Accordingly, the proposed family of models can describe distribution profiles that exhibit isothermal cores and polytropic haloes.
to the interval 0 ≤ γ < 7/2 for the deformation parameter γ. This restriction will be manifested in the thermodynamic quantities and distribution profiles.
Dependencies of the temperature T versus the dimensionless energy U are shown in figure 2 for different values of deformation parameter γ. All quasi-stationary configurations obtained from these models have always negative values for the energy U . Moreover, one can recognize the existence of three notable points: the critical point of gravothermal collapse U A , the critical point of isothermal collapse U B and the critical point of evaporation disruption U C [4,5]. All quasi-stationary configurations are located inside the energy region U A ≤ U ≤ U C . Stable quasi-stationary configurations inside the energy range U A ≤ U ≤ U B exhibit negative heat capacities C < 0, which is a distinguished feature of thermodynamics of astrophysical systems [14,15] sake of a better understanding about the influence of deformation parameter γ, we have calculated dependence of some thermodynamic quantities at the notable points. Such results are shown in figure 3. The increasing of the deformation parameter γ provokes a systematic decreasing of inverse temperatures (η A , η B ), and the increasing of absolute values of energies (U A , U B , U C ) and their associated central particles 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 ≃ 2.13. This means that both the total energy U A and the temperature T A diverge at this point as a consequence of divergence of the central density ρ A . When γ is below the point γ c the system develops a gravothermal collapse at the finite energy U A and when the deformation parameter γ is above the point γ c , the system should release an infinite amount of energy. Additionally, one observes a divergence in the other thermodynamic variables when the deformation parameter γ approaches its maximum admissible value γ m = 7/2, which is related to instability of polytropic profiles for n ≥ 5. We show in figure 4 a distribution profile with deformation parameter γ = 1 at the point of gravothermal collapse, which corresponds to a Michie-King profile with lowest energy. As clearly shown in this figure, the inner region, the core, can be wellfitted with an isothermal profile [14], while the outer one, the halo, is well-fitted with a polytropic profile [11]. Isothermal cores disappears for energies sufficiently large, so that, distribution profiles are practically polytropic. For a better illustration, dependence of distribution profiles on the deformation parameter γ and the internal energy U is shown in figure 5, specifically, twelve profiles corresponding to three notable points and forth different values of deformation parameter γ. Particles concentration in the inner regions decreases with the increasing of the internal energy U . The increasing of the deformation parameter γ produces distribution profiles with more diluted haloes, and consequently, more dense cores.
We have shown in figure 6 what is considered as the phase diagram of truncated γexponential models in the plane γ − Φ 0 . For each value of deformation parameter γ, the admissible values of the central dimensionless potential Φ 0 are located inside the interval 0 ≤ Φ 0 ≤ Φ A (γ), where Φ A (γ) corresponds to the critical point of gravothermal collapse U A . Central values of dimensionless potential Φ 0 above dependence Φ A (γ) correspond to nonphysical on unstable configurations (white region). Additionally, we have included dependence Φ B (γ) associated with the point of isothermal collapse U B . Configuration between dependencies Φ B (γ) ≤ Φ 0 ≤ Φ A (γ) exhibit negative heat capacities (dark gray region), while those one with central dimensionless potential Φ 0 belonging to interval 0 < Φ 0 < Φ B (γ) exhibit positive heat capacities (light gray region). It is remarkable that dependence Φ A (γ) is weakly modified by a change in the deformation parameter γ for values below the notable point γ c . However, this function experiences an sudden change above this notable value. As expected, this behavior accounts for a sudden change in behavior of 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 .