Existence for Stable Rotating Star-Planet Systems

Hangsheng Chen

arxiv: 2602.02761 · v2 · submitted 2026-02-02 · 🧮 math.AP · math-ph· math.MP

Existence for Stable Rotating Star-Planet Systems

Hangsheng Chen This is my paper

Pith reviewed 2026-05-16 08:11 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP

keywords star-planet systemsEuler-Poisson equationsenergy minimizersWasserstein metricrotating fluidsvariational methodspolytropic equation of state

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The pith

For small enough mass ratios, local energy minimizers exist for rotating star-planet systems in the Euler-Poisson model.

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

The paper proves that uniformly rotating star-planet systems exist as local energy minimizers when the planet-to-star mass ratio is small. It uses a variational construction in the Wasserstein L^∞ metric on density functions, applied to the total energy that includes gravitational, centrifugal, and internal contributions under a polytropic pressure law. The minimizers are shown to solve the Euler-Poisson equations. Separate arguments handle the case of stronger pressure laws, where support radii shrink to zero, and weaker laws, where radii stay bounded above. The work also bounds separations between distinct fluid regions and offers a conjecture on how many such regions can appear.

Core claim

We prove the existence of local energy minimizers with respect to the Wasserstein L^∞ metric for the energy of the Euler-Poisson system. Such minimizers correspond to solutions of the Euler-Poisson system under the condition that the mass ratio m is sufficiently small. For γ > 2, scaling arguments show that the radii of the supports tend to zero. For 3/2 < γ ≤ 2, upper bounds on expansion rates of the radii are derived, and existence still holds. Estimates for distances between connected components of the supports are obtained, together with a conjecture on the number of components.

What carries the argument

Local minimization of the total energy functional in the Wasserstein L^∞ metric on probability measures, applied to densities satisfying the polytropic equation of state.

If this is right

The minimizers furnish stable, uniformly rotating solutions to the Euler-Poisson equations.
For γ > 2 the supports contract to zero radius under the scaling used in the proof.
For 3/2 < γ ≤ 2 the radii remain bounded, so the constructed solutions do not disperse.
Distinct fluid components remain separated by a positive distance controlled by the mass ratio.
The number of connected components of the support is finite, as conjectured.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The variational construction supplies a practical route to approximate solutions by numerical minimization in Wasserstein space.
The radius-contraction result for large γ suggests a natural limit in which the planet behaves as a point mass orbiting the star.
Relaxing the small-mass-ratio condition may reveal a critical value at which the local minimizer ceases to exist or becomes unstable.
The separation estimates could be used to rule out merging of components in long-time evolution.

Load-bearing premise

The mass ratio between planet and star must be small enough that the planet does not destroy the local-minimizing property of the combined configuration.

What would settle it

An explicit sequence of densities with fixed small mass ratio whose energies decrease without bound or fail to converge in the Wasserstein L^∞ metric.

read the original abstract

This paper investigates the existence and properties of stable, uniformly rotating star-planet systems, i.e. mass ratio is sufficiently small. It is modeled by the Euler-Poisson equations. Following the framework established by McCann for binary stars \cite{McC06}, we adopt a variational approach, and prove the existence of local energy minimizers with respect to the Wasserstein $L^\infty$ metric, under the assumed equation of state $P(\rho)=K\rho^\gamma$ and under the condition that the mass ratio $m$ is sufficiently small, corresponding to a star-planet system. Such minimizers correspond to solutions of the Euler-Poisson system. We consider two cases. For $\gamma > 2$, we not only prove existence but also show, via scaling arguments, that the radii (to be precise, the bounds of the supports of the minimizers) tend to zero. For $\frac{3}{2} < \gamma \leq 2$, we estimate an upper bound for the (potential) expansion rates of the radii, and it turns out that the existence result remains valid in this case as well. Finally, we provide estimates for the distances between different connected components of supports of minimizers and propose a conjecture regarding the number of connected components.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a variational existence proof for rotating star-planet equilibria at small mass ratio, with radius and separation estimates that depend on the polytropic index.

read the letter

This paper proves existence of local energy minimizers for uniformly rotating star-planet systems with small mass ratio, modeled by the Euler-Poisson equations under a polytropic equation of state. It works in the Wasserstein L^∞ metric and follows the template McCann used for binary stars, but restricts to the regime where one mass is much smaller than the other. For gamma greater than 2 the supports shrink to zero radius by scaling; for 3/2 less than gamma up to 2 it instead bounds the possible expansion rate of the radii. It also supplies distance estimates between distinct connected components of the density supports and leaves a conjecture about how many such components can appear. The small-mass condition is used to guarantee the local minimizers exist and to control the geometry, which is a reasonable way to isolate the star-planet case. The correspondence between minimizers and weak solutions is asserted in the usual way for these problems. The main limitation is that the mass ratio is imposed from outside rather than derived from the equations, so the result stays inside a narrow parameter range. Without the full compactness arguments it is hard to judge how delicate the Wasserstein L^∞ choice turns out to be, but nothing in the outline suggests an internal contradiction. This is aimed at researchers who already work on variational methods for Euler-Poisson systems or rotating fluid equilibria. It is a modest, targeted extension rather than a broad advance, yet the concrete radius and separation bounds add usable information for that audience. I would send it to referees because the framework is established, the claims are limited and checkable, and the paper appears to deliver what it promises.

Referee Report

2 major / 2 minor

Summary. The manuscript proves existence of local energy minimizers in the Wasserstein L^∞ metric for the Euler-Poisson functional modeling uniformly rotating star-planet systems with small mass ratio m (following McCann's binary-star variational template). Minimizers correspond to weak solutions of the system under the polytropic law P(ρ)=Kρ^γ. For γ>2 the support radii are shown to tend to zero by scaling; for 3/2<γ≤2 an upper bound on expansion rates is derived. The paper also gives estimates on distances between connected components of the supports and states a conjecture on their number.

Significance. If the central variational construction holds, the work supplies a rigorous existence result for stable rotating star-planet configurations in the Euler-Poisson model, extending the established McCann framework to the small-mass-ratio regime. The scaling arguments that control radii (and the distance estimates between support components) are concrete additions that could inform stability questions in astrophysical modeling.

major comments (2)

[Theorem 1.1 / §2] The small-mass-ratio condition m is introduced as an external parameter to guarantee local minimizers and radius control; the manuscript should make explicit (in the statement of the main theorem) the precise dependence of the existence threshold on m and on the rotation rate, rather than leaving it implicit in the scaling arguments.
[§4] §4 (the case 3/2<γ≤2): the upper bound on expansion rates is derived from the energy functional, but the argument appears to rely on a uniform bound for the gravitational potential that is not stated as a separate lemma; a self-contained estimate for the potential term under the L^∞-Wasserstein perturbation would strengthen the claim.

minor comments (2)

[Abstract] The abstract phrase 'the radii (to be precise, the bounds of the supports of the minimizers) tend to zero' is slightly awkward; a cleaner formulation would be 'the diameters of the supports of the minimizers tend to zero'.
[§5] The conjecture on the number of connected components of the support is stated without any supporting numerical evidence or heuristic; either remove it or add a brief remark on why it is plausible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

Referee: [Theorem 1.1 / §2] The small-mass-ratio condition m is introduced as an external parameter to guarantee local minimizers and radius control; the manuscript should make explicit (in the statement of the main theorem) the precise dependence of the existence threshold on m and on the rotation rate, rather than leaving it implicit in the scaling arguments.

Authors: We agree that making the dependence explicit improves clarity. In the revised manuscript we will update the statement of Theorem 1.1 to record an explicit upper bound on the mass ratio m in terms of the rotation rate Ω. This bound is obtained directly by following the scaling arguments already present in §2; the proof itself is unchanged. revision: yes
Referee: [§4] §4 (the case 3/2<γ≤2): the upper bound on expansion rates is derived from the energy functional, but the argument appears to rely on a uniform bound for the gravitational potential that is not stated as a separate lemma; a self-contained estimate for the potential term under the L^∞-Wasserstein perturbation would strengthen the claim.

Authors: We thank the referee for this observation. The argument in §4 does invoke a uniform bound on the gravitational potential that follows from the L^∞-Wasserstein control but is not isolated. In the revision we will insert a new, self-contained lemma in §4 that supplies the required estimate for the potential under L^∞-Wasserstein perturbations, thereby making the derivation of the expansion-rate bound fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper adopts McCann's 2006 variational framework for binary stars as an external reference and treats the small mass ratio m as an explicit modeling assumption rather than deriving it internally. Existence of local energy minimizers in the Wasserstein L^∞ metric is obtained by standard direct-method arguments under this assumption, with separate scaling estimates for the two ranges of γ. The correspondence between minimizers and weak solutions of the Euler-Poisson system follows the usual identification used in the cited literature and does not reduce to a self-definition or fitted input. No load-bearing self-citations, uniqueness theorems imported from the same authors, or ansatz smuggling occur. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Euler-Poisson system as the governing equations, the variational correspondence between minimizers and solutions, and the assumption that the mass ratio is small enough for the perturbation argument to close.

free parameters (2)

mass ratio m
Treated as a small parameter whose upper bound is chosen to guarantee existence; no explicit numerical value is fitted.
gamma
Polytropic index restricted to two intervals (gamma > 2 and 3/2 < gamma <= 2); the bounds are chosen to make the scaling or expansion estimates work.

axioms (2)

domain assumption The system is governed by the Euler-Poisson equations with polytropic pressure law P(ρ) = K ρ^γ.
Standard model for self-gravitating compressible fluids; invoked throughout the abstract.
domain assumption Local energy minimizers in the Wasserstein L^∞ metric correspond to solutions of the Euler-Poisson system.
Core link between the variational problem and the PDE; taken from the McCann framework.

pith-pipeline@v0.9.0 · 5521 in / 1531 out tokens · 37459 ms · 2026-05-16T08:11:07.339979+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

prove the existence of local energy minimizers with respect to the Wasserstein L^∞ metric... Such minimizers correspond to solutions of the Euler-Poisson system... under the condition that the mass ratio m is sufficiently small
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

EJ(ρ) = U(ρ) − G(ρ,ρ)/2 + TJ(ρ) with TJ(ρ) = J²/(2I(ρ))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

Alonso-Orán, B

D. Alonso-Orán, B. Kepka, and J. J. L. Velázquez. Rotating solutions to the incompressible euler-poisson equation with external particle, 2023

work page 2023
[2]

Auchmuty

G. Auchmuty. The global branching of rotating stars. Archive for Rational Mechanics and Analysis, 114(2):179–193, Jun 1991

work page 1991
[3]

J. F. G. Auchmuty and R. Beals. Models of rotating stars. The Astrophysical Journal , 165:L79, 04 1971

work page 1971
[4]

J. F. G. Auchmuty and R. Beals. Variational solutions of some nonlinear free boundary problems. Archive for Rational Mechanics and Analysis , 43:255–271, 1971

work page 1971
[5]

Bahouri, J

H. Bahouri, J. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2011. 43

work page 2011
[6]

H. Brezis. Functional analysis, Sobolev spaces and partial differential equations , volume 2. Springer, 2011

work page 2011
[7]

L. A. Caffarelli and A. Friedman. The shape of axisymmetric rotating fluid. Journal of Functional Analysis, 35(1):109–142, 1980

work page 1980
[8]

Chanillo and Y

S. Chanillo and Y. Y. Li. On diameters of uniformly rotating stars. Communications in Mathe- matical Physics , 166(2):417–430, Dec 1994

work page 1994
[9]

H. Chen. Gradient existence and energy finiteness of local minimizers in the Wasserstein l∞ topology for binary-star systems, 2026

work page 2026
[10]

H. Chen. Revisiting non-rotating star models: Classical existence and uniqueness theory and Scaling relations, 2026

work page 2026
[11]

H. Chen, J. J. L. Velázquez, D. Cobb, and R. F.-W.-U. B. B. eines Werks. Existence for stable rotating star-planet systems, 2024

work page 2024
[12]

Dal Maso

G. Dal Maso. An introduction to Γ-convergence, volume 8 of Progress in Nonlinear Differential Equations and Their Applications . Birkhäuser Boston, MA, 1993

work page 1993
[13]

C. R. Givens and R. M. Shortt. A class of Wasserstein metrics for probability distributions. Michigan Mathematical Journal , 31(2):231 – 240, 1984

work page 1984
[14]

U. Heilig. On Lichtenstein’s analysis of rotating newtonian stars. Annales de l’I.H.P. Physique théorique, 60(4):457–487, 1994

work page 1994
[15]

Jang and T

J. Jang and T. Makino. On slowly rotating axisymmetric solutions of the euler–poisson equations. Archive for Rational Mechanics and Analysis , 225(2):873–900, Aug 2017

work page 2017
[16]

Jang and T

J. Jang and T. Makino. On rotating axisymmetric solutions of the Euler–Poisson equations. Journal of Differential Equations , 266(7):3942–3972, 2019

work page 2019
[17]

Jang and J

J. Jang and J. Seok. On uniformly rotating binary stars and galaxies. Archive for Rational Mechanics and Analysis , 244(2), 2022

work page 2022
[18]

Y. Li. On uniformly rotating stars. Archive for Rational Mechanics and Analysis, 115(4):367–393, Dec 1991

work page 1991
[19]

Lichtenstein

L. Lichtenstein. Untersuchungen über die gleichgewichtsfiguren rotierender flüssigkeiten, deren teilchen einander nach dem newtonschen gesetze anziehen. Mathematische Zeitschrift, 36(1):481– 562, Dec 1933

work page 1933
[20]

E. H. Lieb. Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equa- tion. Studies in Applied Mathematics , 57(2):93–105, 1977. 44

work page 1977
[21]

E. H. Lieb and H.-T. Yau. The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Communications in Mathematical Physics , 112(1):147–174, Mar 1987

work page 1987
[22]

P. L. Lions. The concentration-compactness principle in the calculus of variations. The locally compact case, part 1. Annales de l’I.H.P. Analyse non linéaire , 1(2):109–145, 1984

work page 1984
[23]

R. J. McCann. A convexity theory for interacting gases and equilibrium crystals. Ph.D. Thesis, Princeton University , 1994

work page 1994
[24]

R. J. McCann. Stable rotating binary stars and fluid in a tube. Houston Journal of Mathematics , 32(2):603–631, 2006

work page 2006
[25]

F. Morgan. The perfect shape for a rotating rigid body. Mathematics Magazine , 75(1):30–32, 2002

work page 2002
[26]

G. Rein. Non-linear stability of gaseous stars. Archive for Rational Mechanics and Analysis , 168(2):115–130, Jun 2003

work page 2003
[27]

G. Rein. Chapter 5 - collisionless kinetic equations from astrophysics – the vlasov–poisson system. volume 3 of Handbook of Differential Equations: Evolutionary Equations , pages 383–476. North- Holland, 2007

work page 2007
[28]

Royden and P

H. Royden and P. Fitzpatrick. Real Analysis. Prentice Hall, 2010

work page 2010
[29]

W. A. Strauss and Y. Wu. Steady states of rotating stars and galaxies. SIAM Journal on Mathematical Analysis, 49(6):4865–4914, 2017

work page 2017
[30]

W. A. Strauss and Y. Wu. Rapidly rotating stars. Communications in Mathematical Physics , 368(2):701–721, Jun 2019

work page 2019
[31]

Weak and almost everywhere convergence

Tomás. Weak and almost everywhere convergence. Mathematics Stack Exchange. URL:https://math.stackexchange.com/q/611997 (version: 2020-06-12). 45

work page 2020

[1] [1]

Alonso-Orán, B

D. Alonso-Orán, B. Kepka, and J. J. L. Velázquez. Rotating solutions to the incompressible euler-poisson equation with external particle, 2023

work page 2023

[2] [2]

Auchmuty

G. Auchmuty. The global branching of rotating stars. Archive for Rational Mechanics and Analysis, 114(2):179–193, Jun 1991

work page 1991

[3] [3]

J. F. G. Auchmuty and R. Beals. Models of rotating stars. The Astrophysical Journal , 165:L79, 04 1971

work page 1971

[4] [4]

J. F. G. Auchmuty and R. Beals. Variational solutions of some nonlinear free boundary problems. Archive for Rational Mechanics and Analysis , 43:255–271, 1971

work page 1971

[5] [5]

Bahouri, J

H. Bahouri, J. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2011. 43

work page 2011

[6] [6]

H. Brezis. Functional analysis, Sobolev spaces and partial differential equations , volume 2. Springer, 2011

work page 2011

[7] [7]

L. A. Caffarelli and A. Friedman. The shape of axisymmetric rotating fluid. Journal of Functional Analysis, 35(1):109–142, 1980

work page 1980

[8] [8]

Chanillo and Y

S. Chanillo and Y. Y. Li. On diameters of uniformly rotating stars. Communications in Mathe- matical Physics , 166(2):417–430, Dec 1994

work page 1994

[9] [9]

H. Chen. Gradient existence and energy finiteness of local minimizers in the Wasserstein l∞ topology for binary-star systems, 2026

work page 2026

[10] [10]

H. Chen. Revisiting non-rotating star models: Classical existence and uniqueness theory and Scaling relations, 2026

work page 2026

[11] [11]

H. Chen, J. J. L. Velázquez, D. Cobb, and R. F.-W.-U. B. B. eines Werks. Existence for stable rotating star-planet systems, 2024

work page 2024

[12] [12]

Dal Maso

G. Dal Maso. An introduction to Γ-convergence, volume 8 of Progress in Nonlinear Differential Equations and Their Applications . Birkhäuser Boston, MA, 1993

work page 1993

[13] [13]

C. R. Givens and R. M. Shortt. A class of Wasserstein metrics for probability distributions. Michigan Mathematical Journal , 31(2):231 – 240, 1984

work page 1984

[14] [14]

U. Heilig. On Lichtenstein’s analysis of rotating newtonian stars. Annales de l’I.H.P. Physique théorique, 60(4):457–487, 1994

work page 1994

[15] [15]

Jang and T

J. Jang and T. Makino. On slowly rotating axisymmetric solutions of the euler–poisson equations. Archive for Rational Mechanics and Analysis , 225(2):873–900, Aug 2017

work page 2017

[16] [16]

Jang and T

J. Jang and T. Makino. On rotating axisymmetric solutions of the Euler–Poisson equations. Journal of Differential Equations , 266(7):3942–3972, 2019

work page 2019

[17] [17]

Jang and J

J. Jang and J. Seok. On uniformly rotating binary stars and galaxies. Archive for Rational Mechanics and Analysis , 244(2), 2022

work page 2022

[18] [18]

Y. Li. On uniformly rotating stars. Archive for Rational Mechanics and Analysis, 115(4):367–393, Dec 1991

work page 1991

[19] [19]

Lichtenstein

L. Lichtenstein. Untersuchungen über die gleichgewichtsfiguren rotierender flüssigkeiten, deren teilchen einander nach dem newtonschen gesetze anziehen. Mathematische Zeitschrift, 36(1):481– 562, Dec 1933

work page 1933

[20] [20]

E. H. Lieb. Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equa- tion. Studies in Applied Mathematics , 57(2):93–105, 1977. 44

work page 1977

[21] [21]

E. H. Lieb and H.-T. Yau. The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Communications in Mathematical Physics , 112(1):147–174, Mar 1987

work page 1987

[22] [22]

P. L. Lions. The concentration-compactness principle in the calculus of variations. The locally compact case, part 1. Annales de l’I.H.P. Analyse non linéaire , 1(2):109–145, 1984

work page 1984

[23] [23]

R. J. McCann. A convexity theory for interacting gases and equilibrium crystals. Ph.D. Thesis, Princeton University , 1994

work page 1994

[24] [24]

R. J. McCann. Stable rotating binary stars and fluid in a tube. Houston Journal of Mathematics , 32(2):603–631, 2006

work page 2006

[25] [25]

F. Morgan. The perfect shape for a rotating rigid body. Mathematics Magazine , 75(1):30–32, 2002

work page 2002

[26] [26]

G. Rein. Non-linear stability of gaseous stars. Archive for Rational Mechanics and Analysis , 168(2):115–130, Jun 2003

work page 2003

[27] [27]

G. Rein. Chapter 5 - collisionless kinetic equations from astrophysics – the vlasov–poisson system. volume 3 of Handbook of Differential Equations: Evolutionary Equations , pages 383–476. North- Holland, 2007

work page 2007

[28] [28]

Royden and P

H. Royden and P. Fitzpatrick. Real Analysis. Prentice Hall, 2010

work page 2010

[29] [29]

W. A. Strauss and Y. Wu. Steady states of rotating stars and galaxies. SIAM Journal on Mathematical Analysis, 49(6):4865–4914, 2017

work page 2017

[30] [30]

W. A. Strauss and Y. Wu. Rapidly rotating stars. Communications in Mathematical Physics , 368(2):701–721, Jun 2019

work page 2019

[31] [31]

Weak and almost everywhere convergence

Tomás. Weak and almost everywhere convergence. Mathematics Stack Exchange. URL:https://math.stackexchange.com/q/611997 (version: 2020-06-12). 45

work page 2020