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Stability analysis of the Lane--Emden equation of index $n=5$ describing polytropic gas spheres
We study the Lane--Emden equation of index $n=5$, the critical polytropic case separating finite-radius from infinite-radius self-gravitating equilibria. The scaling family $θ_λ(ξ)=λ^{1/2}θ(λξ)$ generates a neutral mode $ϕ_0=\tfrac12θ+ξθ'$ about the Schuster profile. Schuster's transformation and a
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