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A turning point principle for liquid Lane-Emden stars

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Pith's one-line read For liquid Lane–Emden stars, the number of radial growing modes equals the negative Morse index and jumps only at mass extrema of the mass–radius curve.

desk verdict Solid turning-point theorem for liquid Lane–Emden stars; local jump law is self-contained, global spiral claims ride on the author’s prior planar-tail analysis. read the letter →

arxiv 2607.24530 v1 pith:3OPQYJRQ submitted 2026-07-27 math-ph math.APmath.MP

King Ming Lam This is my paper · ORCID
classification math-phmath.APmath.MP MSC 35Q3585A1576E2035B35
keywords Lane-EmdenstarsEuler-Poissonsystemturningpointprinciplemass-radiuscurveliquidfreeboundaryradialstabilitygrowingmodesMorseindex
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a turning point principle for spherically symmetric liquid stars modeled by the free-boundary Euler–Poisson system with stiffened polytropic pressure. Fixing the adiabatic index, the steady states form a one-parameter family labeled by central density, and the liquid surface breaks the self-similarity that makes gaseous stars scale-invariant. The result is that the count of radially unstable modes is locally constant along the family and can change only where the mass has an extremum; the orientation of the bend on the mass–radius curve decides whether a growing mode is gained or lost. At large central density a planar dynamical system for the gaseous tail decides whether that curve spirals—driving the mode count to infinity—or settles with finitely many turns. The law recovers and sharpens known radial stability thresholds and, with existing non-radial work, yields a mass–radius criterion for full linear stability.

What carries the argument

The reduced radial operator L_κ in comoving coordinates, whose energy form includes a liquid-surface boundary term; the growing-mode count is its negative Morse index, and the marginal mode along the family detects mass turning points via a boundary identity linking L_κ ν_κ to ∂_κ M_κ.

What would settle it

For a fixed γ below the mass-critical index and dimension three, numerically continue the mass–radius curve through the first mass maximum and compute the lowest eigenvalue of L_κ on either side: the eigenvalue must cross from positive to negative exactly there, and further crossings must track later mass extrema (or accumulate if the curve spirals).

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Extended reading notes

Core claim

Along the liquid Lane–Emden family, the number of radial growing modes equals the negative Morse index of the linearised radial operator, stays constant between mass critical points, and changes by exactly the jump of a turning index at each nondegenerate mass extremum, according to the sign change of the product of the mass and radius derivatives. In the infinite-support spiral regime the mode count tends to infinity; otherwise it stabilises to a finite value, odd and at least one when the curve returns to the origin.

Load-bearing premise

The large-density spiral-versus-node picture and the onset of the first mass maximum rest on the phase portrait of a planar system for the gaseous tail already established in earlier work on the same family.

Editorial extensions

If this is right

  • When γ is at least the mass-critical index, every liquid Lane–Emden star is radially linearly stable, with no mass turning points.
  • When γ is below that index (and the discriminant condition holds, automatic in dimensions below ten), stars are stable up to the first mass maximum and unstable immediately beyond it and at large central density.
  • In the spiral regime the number of radial growing modes tends to infinity with central density.
  • In three dimensions every exponentially growing mode is radial, so the same mass–radius turning-point rule governs full linear stability.
  • The precise mode count at any non-critical central density equals the net number of counter-clockwise horizontal crossings of the mass–radius tangent up to that point.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The open node regime in dimensions ten and higher is the natural next target: a single sign computation along the attracting eigendirection would close the large-density stability question there.
  • The same comoving mass-preserving reduction that eliminates the winding index may simplify turning-point arguments for other free-boundary stellar models with a density jump.
  • Because the liquid surface term is what creates the turning points, any equation of state that retains a fixed surface density and an asymptotically polytropic tail should inherit an analogous mass–radius counting law.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper studies the one-parameter family (parametrised by central density κ∈(1,∞)) of spherically symmetric steady states of the free-boundary Euler–Poisson system with liquid equation of state p=ρ^γ−1. Theorem 2.7 identifies the number of radial growing modes with the negative Morse index of the linearised radial operator Lκ — the comoving formulation builds in mass conservation, so no constraint or winding-index subtraction is needed — and shows ker Lκ is non-trivial exactly at critical points of κ↦Mκ, spanned by the marginal mode νκ. Theorem 2.8 is a turning point principle: n_u(κ) is locally constant off mass critical points and jumps by ±1 across nondegenerate ones according to the sign change of (∂κM)(∂κR); the paper further proves stability for all κ when γ≥γ*=2(d−1)/d, onset of instability at the first mass maximum for γ<γ* (under γ≥γ♯ or D(γ,d)<0), and a large-κ spiral/node dichotomy governed by the explicit discriminant (1.8), with n_u→∞ along the spiral. Corollaries recover and sharpen the radial (in)stability results of [36] and, via [37], give a non-radial stability criterion.

Significance. This is, to my knowledge, the first complete and rigorous turning point principle for a Newtonian free-boundary stellar family, and a faithful Newtonian analogue of the Hadžić–Lin–Rein relativistic theory, including the mass–radius spiral and n_u→∞. Particular strengths: the local theory (Lemmas 3.1–3.5) is fully self-contained and variational, with the marginal-mode identity (2.19) carrying the explicit nonzero constant cκ=−Mκ/(4πR^{d−2}); the comoving reduction yields a plain Morse-index count with no winding subtraction, a genuine bookkeeping simplification; the spiral/node dichotomy is decided by an explicit parameter-free discriminant with a clean dimension-ten threshold; the borderline case γ=γ♯ is exactly solvable (κ1 given in closed form, stabilised count 1), supplying falsifiable predictions; and the open node case d≥10 is honestly delineated (Remark 2.8.3) rather than glossed over. The result also sharpens prior work (large-κ instability for d≥10, γ∈[γ♯,γ*) is new relative to [36]).

minor comments (6)
  1. [Lemma 3.4] Step 5, displayed limit for µ(κj)/M′(κj): the factor ℓκ0[˜νκ0,˜νκ0] appears in the numerator, but dividing (3.22) by M′ and passing to the limit gives µ/M′ → cκ0 R′(κ0)/(R^{d+3}_{κ0} ℓκ0[˜νκ0,˜νκ0]) — the ℓ-factor should be in the denominator. The conclusion is unaffected (ℓ>0, so the limit is still a nonzero number of sign opposite to R′(κ0)), but the displayed formula should be corrected.
  2. [Proposition 3.7] Step 1 imports the phase portrait of (3.27) from [36] (rest point v*, exponential asymptotic stability for γ<γ♯, orbit convergence, support dichotomy; also the explicit γ=γ♯ profile used in Lemma 3.8(ii)). Since Theorem 2.8(ii)(b)–(iii) hinge entirely on these inputs, please give theorem-level references within [36] rather than a global citation, so the dependence is directly checkable.
  3. [Corollary 2.10] The proof of Corollary 2.10 relies on the spherical-harmonic block decomposition and non-negativity of the l≥1 blocks from [37], which at present is a preprint by the same author. Please state precisely which statements of [37] are used and note its publication status, since this corollary is one of the headline applications.
  4. [Abstract] The phrase 'creating extrema of the mass when γ<2(d−1)/d' (also in §1.3) is unconditional, but per Remark 2.8.3 the existence of a mass maximum is not guaranteed in the node case d≥10, D(γ,d)≥0. A brief qualification would avoid overstating the scope.
  5. [Remark 2.8.4] The extension to general liquid equations of state is asserted ('can be used to prove', tails 'would match') rather than shown. Consider softening the language or labelling it as a programme, since the tail analysis for asymptotically polytropic P is not carried out here.
  6. [Lemma 3.2] Real-analyticity at the centre is justified via a 'classical majorant argument' with a citation to [28], which gives the series expansions; a reference to a convergence result (e.g., Hukuhara-type theory for the regular singular point) would make this step fully self-supporting. Also, notation 'f0' vs 'f0,κ' is used somewhat loosely in §2.1.

Circularity Check

1 steps flagged · score 1.0 of 10

Local turning-point jump law is self-contained; global spiral/node claims import the planar tail portrait from the author's prior work [36] as an independent analytic input, not a definitional loop.

  1. self citation load bearing [Proposition 3.7, Step 1; Theorem 2.8(ii)(b)–(iii); cf. also Lemma 3.6]
    "The rest points, the eigenvalue formula, the exponential asymptotic stability of v∗ for γ<γ♯, and the convergence of the gaseous orbit v(τ)→v∗ with an exponential rate |v(τ)−v∗|≲e−cτ, c>0, are all established in [36] (proven there by a Poincaré–Bendixson and Bendixson–Dulac argument)."

    Not circular by construction: the paper does not define the phase portrait from nu or fit it to the mass–radius curve. It imports the tail dynamics of (3.27) from the author's prior work [36] and transcribes them via the diffeomorphism Φ of Lemma 3.6 into the geometry of C. This is load-bearing for the spiral (iii)(a), node (iii)(b), and existence of a first mass maximum in (ii)(b), but the local jump law (Lemma 3.4) and Morse count (Theorem 2.7) are independent of that citation. Flagged only as the single self-citation on which the global dichotomy rests.

full rationale

The core identities of the paper do not reduce to their inputs by construction. Theorem 2.7 equates nu(κ) with the negative Morse index n−(Lκ) via the energy form (2.13), compact resolvent of (Lκ+α)−1, and Courant–Fischer; that is standard spectral theory, not a restatement of M(κ) or R(κ). The marginal-mode identity (2.19)–(2.20) and kernel dichotomy (2.21) are derived by differentiating the hydrostatic ODE and the mass constraint, then integrating by parts against the energy form. The jump law (Lemma 3.4) and winding formula (3.25) follow from a simple-eigenvalue crossing argument (pairing χκ with νκ, sign µ = −sign(M′R′)) plus bookkeeping; both are internal. The only self-citation dependence is for the global geometry of C: rest point v∗, exponential attraction for γ<γ♯, support dichotomy γ≷γ♯, and the focus/node dichotomy are explicitly attributed to the Poincaré–Bendixson/Bendixson–Dulac analysis of the planar system (3.27) in the author's prior paper [36], and non-radial block positivity in Corollary 2.10 comes from [37]. Those are independent analytic inputs used as black boxes; the present paper does not redefine them in terms of nu, nor fit parameters to recover the spiral. This is ordinary cumulative mathematics, not circularity. Score 1 only to mark that the spiral/nu→∞ and onset-of-first-maximum statements are load-bearing on [36] and would fail if that tail analysis were incomplete, while the local jump principle would survive.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

Load-bearing content is standard PDE/spectral theory plus the liquid Euler–Poisson model with polytropic stiffened EOS, spherical symmetry, and profile existence from the hydrostatic ODE. No fitted parameters. Invented objects are definitional (marginal mode, turning index, reduced operator Lκ), not new physical entities. The main external analytic inputs are the author’s prior linear theory and the classical planar Lane–Emden dynamical system.

assumptions (5)
  • domain assumption Existence, uniqueness, and joint smoothness of liquid Lane–Emden profiles for κ∈(1,∞), including finite radius where density hits 1 (Definition 2.1, Lemma 3.2; existence cited from [36]).
    The one-parameter family and mass–radius curve are undefined without this; smoothness is used for analyticity of Mκ,Rκ and isolation of critical points.
  • domain assumption Liquid polytropic EOS p=ργ−1 with surface density normalised to 1, γ∈[1,2), d≥3, and gravitational Poisson equation with the paper’s 4π normalisation.
    Model choice throughout; Remark 2.8.4 sketches extension to more general P(ρ) but theorems are proved for this EOS.
  • standard math Compact resolvent / Rellich–Kondrachov embedding of the weighted H1 space into L2 on the ball, and Courant–Fischer for self-adjoint operators bounded below with compact resolvent.
    Lemma 3.1 uses these to get discrete real spectrum and nu=n−(Lκ).
  • domain assumption Phase portrait of the planar gaseous-tail system (3.27): rest point v∗, exponential attraction for γ<γ♯, and focus/node dichotomy by sign of D(γ,d), as established in [36].
    Proposition 3.7 imports this portrait; spiral vs node and lim nu=∞ rest on it.
  • domain assumption Non-radial angular blocks of degree ℓ≥1 are non-negative for the full linearised operator in d=3 ([37]).
    Used only for Corollary 2.10 equating full and radial growing-mode counts.
invented entities (2)
  • Marginal mode νκ independent evidence
    purpose: Infinitesimal mass-preserving deformation along the steady family; detects mass critical points as the kernel of Lκ.
    Definition 2.5; constructed from ∂κρ̄κ via the continuity constraint. Standard variational object, not a new physical field.
  • Turning index iκ independent evidence
    purpose: Bookkeeping of bending orientation of the mass–radius curve at regular points.
    Definition 2.6; pure combinatorial device for the jump rule, analogous to relativistic winding bookkeeping.

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Pith. "Pith review of A turning point principle for liquid Lane-Emden stars." pith.science (2026). https://pith.science/paper/3OPQYJRQ

@misc{pith2026260724530,
  author       = {Pith},
  title        = {Pith review of: A turning point principle for liquid Lane-Emden stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3OPQYJRQ}},
  note         = {Machine review of arXiv:2607.24530}
}
abstract

Upon fixing the adiabatic index, the spherically symmetric liquid Lane-Emden stars governed by the Euler-Poisson system with a "stiffened gas" equation of state $p=\rho^\gamma-1$ form a one-parameter family, naturally parametrised by the central density $\kappa=\bar\rho_\kappa(0)\in(1,\infty)$. In contrast to the gaseous case, the liquid free boundary breaks the self-similarity of the family and bends the mass-radius curve, creating extrema of the mass when $\gamma<2(d-1)/d$. We formulate a turning point principle in the spirit of Zel'dovich, Wheeler and Thorne, and of its rigorous relativistic counterpart from the works of Had\v{z}i\'c, Lin and Rein: the number of radial growing modes equals the negative Morse index of the linearised operator, is locally constant along the family, and can change only at the turning points of the mass-radius curve, the bending orientation there dictating whether a growing mode is gained or lost. We describe the large central density limit in which the gaseous tail (read off from a planar dynamical system) determines whether the mass-radius curve spirals, and hence whether the number of growing modes remains bounded or tends to infinity. As corollaries we recover and sharpen the known radial (in)stability results for liquid Lane-Emden stars and, in combination with existing non-radial analysis, obtain a turning point criterion for non-radial stability.

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Works this paper leans on

63 extracted references · 2 linked inside Pith

  1. [36]

    K. M. Lam. Linear stability of liquid Lane–Emden stars.Quart. Appl. Math., 82(4):639–672, 2024

    2024

  2. [37]

    K. M. Lam. Nonradial linear stability of liquid Lane–Emden stars, 2026. Preprint, arXiv:2603.03548

    arXiv 2026

  3. [1]

    J. M. Bardeen, K. S. Thorne, and D. W. Meltzer. A catalogue of methods for studying the normal modes of radial pulsation of general-relativistic stellar models.Astrophys. J., 145:505–513, 1966

    1966

  4. [2]

    Chandrasekhar

    S. Chandrasekhar. The maximum mass of ideal white dwarfs.Astrophys. J., 74:81–82, 1931

    1931

  5. [3]

    Chandrasekhar.An Introduction to the Study of Stellar Structure

    S. Chandrasekhar.An Introduction to the Study of Stellar Structure. University of Chicago Press, Chicago, IL, 1939

    1939

  6. [4]

    Chandrasekhar

    S. Chandrasekhar. The dynamical instability of gaseous masses approaching the Schwarzschild limit in general relativity.Astrophys. J., 140:417–433, 1964

    1964

  7. [5]

    Cheng, X

    M. Cheng, X. Cheng, and Z. Lin. Expanding solutions near unstable Lane–Emden stars.Commun. Math. Phys., 406:105, 06 2025. 29

    2025

  8. [6]

    Christodoulou

    D. Christodoulou. Self-gravitating relativistic fluids: a two-phase model.Arch. Rational Mech. Anal., 130(4):343–400, 1995

    1995

Show all 63 references
  1. [7]

    Coutand, J

    D. Coutand, J. Hole, and S. Shkoller. Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit.SIAM J. Math. Anal., 45(6):3690– 3767, 2013

    2013

  2. [8]

    Coutand and S

    D. Coutand and S. Shkoller. Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum.Arch. Ration. Mech. Anal., 206(2):515–616, 2012

    2012

  3. [9]

    A. S. Eddington. Liquid stars.Nature, 121:278, 1928

    1928

  4. [10]

    Emden.Gaskugeln: Anwendungen der mechanischen W ¨armetheorie auf kosmologische und meteorologische Probleme

    R. Emden.Gaskugeln: Anwendungen der mechanischen W ¨armetheorie auf kosmologische und meteorologische Probleme. B. G. Teubner, Leipzig and Berlin, 1907

    1907

  5. [11]

    J. L. Friedman, J. R. Ipser, and R. D. Sorkin. Turning-point method for axisymmetric stability of rotating relativistic stars.Astrophys. J., 325:722–724, 1988

    1988

  6. [12]

    Ginsberg, H

    D. Ginsberg, H. Lindblad, and C. Luo. Local well-posedness for the motion of a compressible, self-gravitating liquid with free surface boundary.Arch. Ration. Mech. Anal., 236(2):603–733, 2020

    2020

  7. [13]

    Goldreich and S

    P. Goldreich and S. Weber. Homologously collapsing stellar cores.Astrophys. J., 238:991–997, 1980

    1980

  8. [14]

    Gu and Z

    X. Gu and Z. Lei. Local well-posedness of the three dimensional compressible Euler–Poisson equations with physical vacuum.J. Math. Pures Appl. (9), 105(5):662–723, 2016

    2016

  9. [15]

    G ¨unther, C

    S. G ¨unther, C. Straub, and G. Rein. Collisionless equilibria in general relativity: Stable configura- tions beyond the first binding energy maximum.Astrophys. J., 918(2):48, 2021

    2021

  10. [16]

    Guo and G

    Y . Guo and G. Rein. Isotropic steady states in galactic dynamics.Comm. Math. Phys., 219(3):607– 629, 2001

    2001

  11. [17]

    Had ˇzi´c and J

    M. Had ˇzi´c and J. Jang. Nonlinear stability of expanding star solutions in the radially-symmetric mass-critical Euler–Poisson system.Comm. Pure Appl. Math., 71(5):827–891, 2018

    2018

  12. [18]

    Had ˇzi´c and J

    M. Had ˇzi´c and J. Jang. A class of global solutions to the Euler–Poisson system.Comm. Math. Phys., 370(2):475–505, 2019

    2019

  13. [19]

    Had ˇzi´c, J

    M. Had ˇzi´c, J. Jang, and K. M. Lam. Nonradial stability of self-similarly expanding Goldreich– Weber stars, 2022. Preprint, arXiv:2212.11420

    2022 arXiv

  14. [20]

    Had ˇzi´c and Z

    M. Had ˇzi´c and Z. Lin. Turning point principle for relativistic stars.Comm. Math. Phys., 387(2):729–759, 2021

    2021

  15. [21]

    Had ˇzi´c, Z

    M. Had ˇzi´c, Z. Lin, and G. Rein. Stability and instability of self-gravitating relativistic matter distributions.Arch. Ration. Mech. Anal., 241(1):1–89, 2021

    2021

  16. [22]

    Hao and S

    Z. Hao and S. Miao. On nonlinear instability of liquid Lane–Emden stars.Calc. Var. Partial Differential Equations, 63(6):Paper No. 157, 54, 2024

    2024

  17. [23]

    B. K. Harrison, K. S. Thorne, M. Wakano, and J. A. Wheeler.Gravitation Theory and Gravitational Collapse. University of Chicago Press, Chicago, 1965

    1965

  18. [24]

    Hartman.Ordinary Differential Equations, volume 38 ofClassics in Applied Mathematics

    P. Hartman.Ordinary Differential Equations, volume 38 ofClassics in Applied Mathematics. SIAM, Philadelphia, 2002

    2002

  19. [25]

    J. M. Heinzle. (in)finiteness of spherically symmetric static perfect fluids.Classical Quantum Gravity, 19(11):2835–2851, 2002

    2002

  20. [26]

    J. M. Heinzle, N. R ¨ohr, and C. Uggla. Spherically symmetric relativistic stellar structures.Classi- cal Quantum Gravity, 20(21):4567–4586, 2003

    2003

  21. [27]

    J. M. Heinzle and C. Uggla. Newtonian stellar models.Ann. Physics, 308(1):18–61, 2003. 30

    2003

  22. [28]

    C. Hunter. Series solutions for polytropes and the isothermal sphere.Mon. Not. R. Astron. Soc., 328(3):839–847, 2001

    2001

  23. [29]

    Ifrim and D

    M. Ifrim and D. Tataru. The compressible Euler equations in a physical vacuum: a comprehensive Eulerian approach.Ann. Inst. H. Poincar ´e C Anal. Non Lin´eaire, 41(2):405–495, 2024

    2024

  24. [30]

    J. Jang. Nonlinear instability in gravitational Euler–Poisson system forγ= 6/5.Arch. Ration. Mech. Anal., 188(2):265–307, 2008

    2008

  25. [31]

    J. Jang. Nonlinear instability theory of Lane–Emden stars.Comm. Pure Appl. Math., 67(9):1418– 1465, 2014

    2014

  26. [32]

    Jang and T

    J. Jang and T. Makino. Linearized analysis of barotropic perturbations around spherically sym- metric gaseous stars governed by the Euler–Poisson equations.J. Math. Phys., 61(5):051508, 25, 2020

    2020

  27. [33]

    Jang and N

    J. Jang and N. Masmoudi. Well-posedness of compressible Euler equations in a physical vacuum. Comm. Pure Appl. Math., 68(1):61–111, 2015

    2015

  28. [34]

    J. H. Jeans. On liquid stars and the liberation of stellar energy.Mon. Not. R. Astron. Soc., 87(5):400–414, 1927

    1927

  29. [35]

    J. H. Jeans. Liquid stars.Nature, 121:173, 1928

    1928

  30. [38]

    J. H. Lane. On the theoretical temperature of the sun under the hypothesis of a gaseous mass main- taining its volume by its internal heat and depending on the laws of gases as known to terrestrial experiment.Am. J. Sci. Arts, 50(148):57–74, 1870

  31. [39]

    Lemou, F

    M. Lemou, F. M ´ehats, and P. Rapha¨el. Orbital stability of spherical galactic models.Invent. Math., 187(1):145–194, 2012

    2012

  32. [40]

    S.-S. Lin. Stability of gaseous stars in spherically symmetric motions.SIAM J. Math. Anal., 28(3):539–569, 1997

    1997

  33. [41]

    Z. Lin, Y . Wang, and H. Zhu. Nonlinear stability of non-rotating gaseous stars.Math. Ann., 391:843–880, 07 2024

    2024

  34. [42]

    Lin and C

    Z. Lin and C. Zeng. Instability, index theorem, and exponential trichotomy for linear Hamiltonian PDEs.Mem. Amer. Math. Soc., 275(1347), 2022

    2022

  35. [43]

    Lin and C

    Z. Lin and C. Zeng. Separable hamiltonian pdes and turning point principle for stability of gaseous stars.Comm. Pure Appl. Math., 75(2):306–376, 2022

    2022

  36. [44]

    Lindblad

    H. Lindblad. Well posedness for the motion of a compressible liquid with free surface boundary. Comm. Math. Phys., 260(2):319–392, 2005

    2005

  37. [45]

    Luo and J

    T. Luo and J. Smoller. Existence and nonlinear stability of rotating star solutions of the compress- ible Euler–Poisson equations.Arch. Ration. Mech. Anal., 191(3):447–496, 2009

    2009

  38. [46]

    T. Makino. Blowing up solutions of the Euler–Poisson equation for the evolution of gaseous stars. Transport Theory Statist. Phys., 21(4-6):615–624, 1992

    1992

  39. [47]

    T. Makino. On the spiral structure of the(R, M)-diagram for a stellar model of the Tolman– Oppenheimer–Volkoff equation.Funkcial. Ekvac., 43(3):471–489, 2000

    2000

  40. [48]

    S. Miao, S. Shahshahani, and S. Wu. Well-posedness for free boundary hard phase fluids with Minkowski background and their Newtonian limit.Camb. J. Math., 9(2):269–350, 2021

    2021

  41. [49]

    U. S. Nilsson and C. Uggla. General relativistic stars: polytropic equations of state.Ann. Physics, 286(2):292–319, 2000

    2000

  42. [50]

    T. A. Oliynyk. Dynamical relativistic liquid bodies, 2019. Preprint, arXiv:1907.08192. 31

    2019 arXiv

  43. [51]

    Ramming and G

    T. Ramming and G. Rein. Mass-radius spirals for steady state families of the Vlasov–Poisson system.Arch. Ration. Mech. Anal., 224(3):1127–1159, 2017

    2017

  44. [52]

    G. Rein. Non-linear stability of gaseous stars.Arch. Ration. Mech. Anal., 168(2):115–130, 2003

    2003

  45. [53]

    G. Rein. Stability and instability results for equilibria of a (relativistic) self-gravitating collisionless gas—a review.Classical Quantum Gravity, 40:193001, 2023

    2023

  46. [54]

    J. S. Schiffrin and R. M. Wald. Turning point instabilities for relativistic stars and black holes. Classical Quantum Gravity, 31:035024, 2014

    2014

  47. [55]

    S. L. Shapiro and S. A. Teukolsky.Black Holes, White Dwarfs and Neutron Stars. The Physics of Compact Objects. John Wiley & Sons, Inc., New York, 1983

    1983

  48. [56]

    R. Sorkin. A criterion for the onset of instability at a turning point.Astrophys. J., 249:254–257, 1981

    1981

  49. [57]

    R. Sorkin. A stability criterion for many parameter equilibrium families.Astrophys. J., 257:847, 1982

    1982

  50. [58]

    Enrico Fermi

    K. S. Thorne. The general-relativistic theory of stellar structure and dynamics.Proc. Internat. School of Physics “Enrico Fermi”, Course XXXV, pages 166–280, 1966

    1966

  51. [59]

    Trakhinin

    Y . Trakhinin. Local existence for the free boundary problem for the non-relativistic and relativis- tic compressible Euler equations with a vacuum boundary condition.Comm. Pure Appl. Math., 62(11):1551–1594, 2009

    2009

  52. [60]

    Ya. B. Zel’dovich. The equation of state at ultrahigh densities and its relativistic limitations.Soviet Phys. JETP, 14:1143–1147, 1962

    1962

  53. [61]

    Ya. B. Zel’dovich. Hydrodynamical stability of star.Voprosy Kosmogonii, 9:157–170, 1963

    1963

  54. [62]

    Ya. B. Zel’dovich and I. D. Novikov.Relativistic Astrophysics. Vol. 1: Stars and Relativity. Uni- versity of Chicago Press, Chicago, 1971

    1971

  55. [63]

    Ya. B. Zel’dovich and M. A. Podurets. The evolution of a system of gravitationally interacting point masses.Soviet Astronomy—AJ, 9:742–749, 1965. Translated from Astronomicheskii Zhurnal, V ol. 42. 32

    1965

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