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,∊s) that considers an appropriate deformation of Maxwell-Boltzmann form with inverse temperature β, in particular, a power-law truncation at the scape energy ∊s 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.