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The intrinsic beauty of polytropic spheres in reduced variables

The concept of reduced variables is revisited with regard to van der Waals' theory and an application is made to polytropic spheres, where the reduced radial coordinate is ${\rm red}(r)=r/R=ξ/Ξ$, $R$ radius, and the reduced density is ${\rm red}(ρ)=ρ/λ=θ^n$, $λ$ central density. Reduced density profiles are plotted for several polytropic indexes within the range, $0\le n\le5$, disclosing two noticeable features. First, any point of coordinates, $({\rm red}(r),{\rm red}(ρ))$, $0\le{\rm red}(r)\le1$, $0\le{\rm red}(ρ)\le1$, belongs to a reduced density profile of the kind considered. Second, sufficiently steep i.e. large $n$ reduced density profiles exhibit an oblique inflection point, where the threshold is found to be located at $n=n_{\rm th}=0.888715$. Reduced pressure profiles, ${\rm red}(P)=P/\varpi=θ^{n+1}$, $\varpi$ central pressure, Lane-Emden fucntions, $θ=(ρ/λ)^{1/n}$, and polytropic curves, ${\rm red}(P)={\rm red}(P)({\rm red}(ρ))$, are also plotted. The method can be extended to nonspherical polytropes with regard to a selected direction, ${\rm red}(r)(μ)=r(μ)/R(μ)=ξ(μ)/Ξ(μ)$. The results can be extended to polytropic spheres made of collisionless particles, for polytropic index within a more restricted range, $1/2\le n\le5$.