  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
$\gamma<\gamma_{c}\simeq 2.13$.
