{
  "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 $θ_λ(ξ)=λ^{1/2}θ(λξ)$ generates a neutral mode $ϕ_0=\\tfrac12θ+ξθ'$ about the Schuster profile. Schuster's transformation and a logarithmic variable reduce the equation to the autonomous Hamiltonian system $\\ddotν=\\tfrac14(ν-ν^5)$ with a first integral. Besides the Schuster (separatrix) and Srivastava solutions, every orbit crossing $ν=0$ is given in closed form by a Weierstrass $\\wp$-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 $L^2((0,\\infty),ξ^2dξ)$. Its Friedrichs extension has essential spectrum $[0,\\infty)$, a threshold resonance at $0$ carried by the non-normalizable scaling mode, and exactly one negative eigenvalue, $λ_0\\approx-1.21$, so the Schuster profile is a saddle point of the associated energy functional.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35932",
  "author_total": 2,
  "authors": [
    {
      "name": "S. C. Mancas"
    },
    {
      "name": "Ronald Adams"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/097przds66y2f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35932",
  "fulltext_url": null,
  "id": "097przds66y2f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35932",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "astro-ph.SR"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7224125620",
  "pdf_url": "https://arxiv.org/pdf/2610.35932",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/097przds66y2f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:50Z",
      "native_id": "2610.35932",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35932"
    }
  ],
  "subjects": [
    {
      "field": "astronomy",
      "origin": "arxiv:astro-ph.SR",
      "subfield": "stellar"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Stability analysis of the Lane--Emden equation of index $n=5$ describing polytropic gas spheres",
  "title_text": "Stability analysis of the Lane--Emden equation of index n=5 describing polytropic gas spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/097przds66y2f",
  "venue": "arXiv",
  "versions": []
}