  The present effort addresses the question about the existence of a
well-defined thermodynamic limit for the astrophysical systems with the
following power law form: to tend the number of particles, N, the total energy,
E, and the characteristic linear dimension of the system, L, to infinity,
keeping constant E/N^{\Lambda_{E}} and L/N^{\Lambda_{L}}, being \Lambda_{E} and
\Lambda_{L} certain scaling exponent constant. This study is carried out for a
system constituted by a non-rotating fluid under the influence of its own
Newtonian gravitational interaction. The analysis yields that a thermodynamic
limit of the above form will only appear when the local pressure depends on the
energy density of fluid as $p=\gamma\epsilon$, being $\gamma$ certain constant.
Therefore, a thermodynamic limit with a power law form can be only satisfied by
a reduced set of models, such as the selfgravitating gas of fermions and the
Antonov isothermal model.
