Non-extensive statistics and the stellar polytrope index

Polytropic distribution functions provide the simplest models for self-gravitating stellar systems [1]. These models are characterized by an equation of the state of the form p = Kρ , where K and γ are constants. The polytropic index n is defined by the following expression: γ = 1 + 1 n which in the limit n → ∞ or, equivalently, γ = 1 results in an equation of state identical to the one of an isothermal body of gases, i.e., p = Kρ. The distribution function associated to these systems is given by [1]