Truncated $\gamma$-exponential models for tidal stellar systems

Abstract

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 $f_{\gamma}\left(\mathbf{q},\mathbf{p}|\beta,\varepsilon_{s}\right)$ that considers an appropriate deformation of Maxwell-Boltzmann form with inverse temperature $\beta$, in particular, a power-law truncation at the scape energy $\varepsilon_{s}$ with exponent $\gamma>0$. This deformation is implemented using a generalized $\gamma$-exponential function obtained from the \emph{fractional integration} of ordinary exponential. As shown in this work, this proposal generalizes models of tidal stellar systems that predict particles distributions with \emph{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 $\gamma<\gamma_{c}\simeq 2.13$.