Stability analysis of the Lane--Emden equation of index n=5 describing polytropic gas spheres
Abstract
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 θλ(ξ)=λ1/2θ(λξ) generates a neutral mode ϕ0=1/2θ+ξθ' about the Schuster profile. Schuster's transformation and a logarithmic variable reduce the equation to the autonomous Hamiltonian system ν̈=1/4(ν−ν5) with a first integral. Besides the Schuster (separatrix) and Srivastava solutions, every orbit crossing ν=0 is given in closed form by a Weierstrass ℘-function, and the closed forms agree with direct numerical integration to about 10−12. Up to scaling and sign, the Schuster solution is the only solution regular at the center, and it separates positive solutions from solutions with infinitely many zeros. In homology-invariant variables the singular solution θ=(2ξ)−1/2 is a center exactly at n=5, where its stability changes. For a distance-varying Emden--Fowler system with a shifted equilibrium we prove that at n=5 the unique equilibrium is linearly and nonlinearly unstable. Finally, the linearization about the Schuster profile is the Schrödinger operator L=−Δr−5θS4 on L2((0,∞),ξ2dξ). Its Friedrichs extension has essential spectrum [0,∞), a threshold resonance at $0$ carried by the non-normalizable scaling mode, and exactly one negative eigenvalue, λ0≈−1.21, so the Schuster profile is a saddle point of the associated energy functional.
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BibTeX
@misc{mancas2026stability,
title = {{Stability analysis of the Lane--Emden equation of index $n=5$ describing polytropic gas spheres}},
author = {S. C. Mancas and Ronald Adams},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35932},
eprint = {2610.35932},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35932},
}
RIS
TY - UNPB
TI - Stability analysis of the Lane--Emden equation of index n=5 describing polytropic gas spheres
AU - S. C. Mancas
AU - Ronald Adams
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.35932
UR - https://arxiv.org/abs/2610.35932
AB - 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^1/2θ(λξ) generates a neutral mode ϕ_0=1/2θ+ξθ' about the Schuster profile. Schuster's transformation and a logarithmic variable reduce the equation to the autonomous Hamiltonian system ν̈=1/4(ν−ν^5) with a first integral. Besides the Schuster (separatrix) and Srivastava solutions, every orbit crossing ν=0 is given in closed form by a Weierstrass ℘-function, and the closed forms agree with direct numerical integration to about 10^−12. Up to scaling and sign, the Schuster solution is the only solution regular at the center, and it separates positive solutions from solutions with infinitely many zeros. In homology-invariant variables the singular solution θ=(2ξ)^−1/2 is a center exactly at n=5, where its stability changes. For a distance-varying Emden--Fowler system with a shifted equilibrium we prove that at n=5 the unique equilibrium is linearly and nonlinearly unstable. Finally, the linearization about the Schuster profile is the Schrödinger operator L=−Δ_r−5θ_S^4 on L^2((0,∞),ξ^2dξ). Its Friedrichs extension has essential spectrum [0,∞), a threshold resonance at $0$ carried by the non-normalizable scaling mode, and exactly one negative eigenvalue, λ_0≈−1.21, so the Schuster profile is a saddle point of the associated energy functional.
ER -