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) (q,P vertical bar beta epsilon(s)) that considers an appropriate deformation of Maxwell-Boltzmann form with inverse temperature beta, in particular, a power-law truncation at the scape energy epsilon(s) with exponent gamma > 0. This deformation is implemented using a generalized gamma-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 gamma < gamma(c) similar or equal to 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.