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REVIEW 2 major objections 5 minor 19 references

Cold planets top out at Jupiter size by a hydrogen-atom argument

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A hydrogen-atom variational calculation shows cold planets reach a maximum radius near Jupiter's radius at a mass of about 3.6 Jupiter masses.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A clean toy-model derivation that explains Jupiter's radius cap with a hydrogen-atom analogy, but the advertised 'no free parameters' coefficient actually depends on an unstated lattice-geometry assumption. the 2 major comments →

arxiv 2509.07048 v1 pith:4N25AK6D submitted 2025-09-08 astro-ph.IM astro-ph.EP

What is the maximum radius of cold planets?

David Garfinkle , Alberto G. Rojo This is my paper
classification astro-ph.IM astro-ph.EP
keywords cold planetsmaximum radiusJupiter radiusvariational principlehydrogen atomgravitational compressionmass-radius relationwhite dwarfs
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 tries to show that the observed fact that planets have radii no larger than Jupiter's is not a coincidence of planetary formation but a consequence of the same variational principle that fixes the size of a hydrogen atom. Treating a cold pure-hydrogen planet as N non-interacting hydrogen atoms squeezed by gravity, the authors write the total energy as a function of a single scale a, balance the electrostatic, gravitational, and quantum kinetic terms, and derive a closed-form maximum radius Rmax = sqrt(5/(2π)) N*^(1/3) a0 ≈ 0.75 RJ and corresponding mass Mmax ≈ 3.6 MJ. The gravity-to-electric ratio enters only through the dimensionless number N* = (e²/(4πϵ0 G mH²))^{3/2}, which is fixed by fundamental constants. A sympathetic reader would care because the result explains a Jupiter-scale upper limit without free parameters and is within tens of percent of more detailed cold-sphere models. The paper is explicitly a pedagogical order-of-magnitude estimate, not a full equation-of-state calculation.

Core claim

On the paper's own terms, the discovery is that the maximum radius and mass of a cold planet follow from a variational calculation nearly identical to the hydrogen atom. The trial wavefunction e^(−r/a) gives a single-atom energy E(a); packing N such atoms in a cubic lattice of spacing 2a and adding the uniform-density gravitational energy −3GM²/(5R) changes the effective charge e² in the Coulomb term to e-tilde² = e²[1 + (3/5)(πN²/(6N*²))^{1/3}]. Minimizing the total energy over a yields a modified Bohr radius, and maximizing R(N) = a0(6N/π)^{1/3}/[1 + (3/5)(πN²/(6N*²))^{1/3}] gives Eqs. (14)–(15). Numerically, Rmax ≈ 0.75 RJ and Mmax ≈ 3.6 MJ, which the authors compare with more sophisticat

What carries the argument

The central object is the dimensionless ratio N* = (e²/(4πϵ0GmH²))^{3/2}, whose 2/3 power is the ratio of electric repulsion to gravitational attraction between two protons. It carries the argument by replacing the single charge e in the hydrogen-atom energy with an effective charge e-tilde that grows with N; the Bohr-radius minimization then becomes a radius maximization, and the maximum radius and mass are algebraic functions of N*, a0, and mH. The companion object is the trial wavefunction e^(−r/a): it lets compression be represented by rescaling one length scale a.

Load-bearing premise

The argument rests on assuming a cold pure-hydrogen planet can be described as N non-interacting hydrogen atoms whose only response to compression is a rescaling of the single Bohr radius a, with uniform density for gravity; if dense hydrogen's real equation of state makes interactions or degeneracy dominate, the derived maximum radius and mass lose their grounding.

What would settle it

Compute the zero-temperature mass-radius relation for pure hydrogen using a realistic dense-matter equation of state. If the maximum radius is not within tens of percent of 0.75 RJ, or occurs at a mass far from 3.6 MJ, then the single-rescaled-Bohr-radius toy model is not the mechanism behind Jupiter's maximum radius.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Below N*, radius grows as N^{1/3} and volume scales linearly with particle number; above N*, radius shrinks as N^{−1/3}, so adding mass compresses the planet.
  • The maximum mass is independent of ħ: only the radius scale depends on quantum mechanics through the Bohr radius a0.
  • The same energy balance applies to cold degenerate objects without fusion, such as cool white dwarfs and iron cores, giving a qualitative upper radius for those objects too.
  • The derivation produces a parameter-free estimate of the Jupiter scale—0.75 RJ and 3.6 MJ for pure hydrogen—with more detailed models giving 1.165 RJ and 3.31 MJ.
  • The toy model is designed to be usable in undergraduate courses as an accessible derivation of why gas giants share a common radius scale.

Where Pith is reading between the lines

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

  • Including helium, the actual gas-giant composition, would change the effective particle mass and mean charge, shifting the numeric prefactor but leaving the N* structure intact; one test is whether a helium-fraction-corrected version lands closer to the observed ~1.2 RJ.
  • The framework suggests that any cold object made of atoms with Bohr radius a0 and particle mass mH has a maximum radius proportional to a0 times a fundamental-constant ratio; comparing objects made of heavier atoms would probe this scaling.
  • The maximum-radius mass may show up as a characteristic turnover in the mass–radius diagram of cool exoplanets, so a survey of transiting planets could test whether the turnover sits near a few Jupiter masses.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper presents a toy variational model for the maximum radius of cold hydrogen planets. A planet is approximated as N non-interacting hydrogen atoms on a cubic lattice of spacing 2a, with total energy N times the single-atom hydrogen expectation value evaluated at a rescaled length a, plus the uniform-density gravitational energy. Minimizing this energy with respect to a gives a 'modified Bohr radius' and hence R(N); maximizing R(N) over N yields Rmax = sqrt(5/(2pi)) N_star^(1/3) a0 ~ 0.75 RJ and Mmax = (5/3) sqrt(10/pi) N_star mH ~ 3.6 MJ. These are compared with detailed cold-sphere models (Rmax ~ 1.165 RJ, Mmax ~ 3.31 MJ) and the agreement is described as surprisingly good.

Significance. The qualitative insight that the Jupiter-scale maximum radius follows from a balance between electrostatic, gravitational, and quantum kinetic energies is appealing and pedagogically valuable. The variational algebra is internally consistent, and the scaling Rmax ~ N_star^(1/3) a0 is robust; the comparison with detailed models is made after the derivation, so the logic is not circular. The paper has no fitted parameters, and the presentation is transparent. However, the numerical coefficient of the scaling law is fixed by an unstated geometric assumption, and the many-body energy model is asserted rather than derived, so the 'quantitative' claim is weaker than the abstract implies.

major comments (2)
  1. [Section II, Eq. (10)] The numerical coefficients in Eqs. (14) and (15) depend on the arbitrary relation a = (1/2)(V/N)^(1/3). Replacing it with a general relation a = beta (V/N)^(1/3) changes the results to Rmax = (1/2) sqrt(5/(4pi)) beta^(-3/2) N_star^(1/3) a0 and Mmax = ((5 gamma0)/(3 beta))^(3/2) N_star mH with gamma0 = (3/(4pi))^(1/3). For beta = 0.25, 0.5, 0.62, 1.0, Rmax is approximately 2.1, 0.75, 0.53, and 0.26 RJ, respectively. Thus the 'no free parameters' or quantitative precision claim is not justified unless the geometric factor is independently derived. The paper's toy-model framing mitigates this, but the text should explicitly state that the coefficient carries an O(1) geometric uncertainty and that the comparison with 1.165 RJ is order-of-magnitude, not a precise prediction.
  2. [Section II, Eq. (10)] The total energy E = N [hbar^2/(2ma^2) - tilde(e)^2/(4pi epsilon_0 a)] is assumed, not derived. The paper does not specify a many-body Hamiltonian for the collection of atoms, so the variational calculation is not an upper bound on the true ground-state energy of the planet; it is a heuristic energy model. This is a legitimate simplification for an order-of-magnitude paper, but the wording 'variational principle very similar to hydrogen atom' can overstate the rigor. Please add an explicit sentence clarifying that Eq. (10) is a model energy, not the expectation value of an actual many-body Hamiltonian.
minor comments (5)
  1. [Section II, Eq. (1)] The integral notation in Eq. (1) is cramped; please define dv as the volume element and add spaces for readability.
  2. [Section II, text after Eq. (2)] The line 'H psi_0(r) = E_i psi_0(r)' should likely be 'H psi_0(r) = E_0 psi_0(r)' or the subscript i should be defined.
  3. [Section II, Eq. (11)] The text calls tilde(e) a 'modified charge,' but Eq. (11) defines tilde(e)^2. Please say 'modified squared charge' to avoid confusion.
  4. [Conclusions] The concluding paragraph says the approach applies to white dwarfs and iron cores, but the derivation uses hydrogen atoms with a specific mH and N_star. The scaling may generalize, but the numerical maximum mass is a Jupiter mass, not a white-dwarf mass; clarify the intended scope.
  5. [Introduction] The phrase 'the size starts to decrease instead of increase' would be clearer as 'the size starts to decrease instead of continuing to increase.'

Circularity Check

0 steps flagged

No circularity: the variational derivation is self-contained; the lattice-geometry ansatz is an assumption, not a fitted input.

full rationale

The paper's central claim is derived by minimizing an energy functional built from the hydrogen-atom expectation value (Eq. 3), a geometric relation between lattice spacing and volume (Eq. 5), and a uniform-density gravitational energy (Eq. 7). The variational parameter a is minimized, not tuned to any target radius or mass. The maximum radius and mass, Eqs. (14)-(15), follow from calculus on R(N) and N* is defined from fundamental constants in Eq. (8). Jupiter's radius and mass appear only in the final numerical comparison, after the derivation, and the more detailed models are cited as external benchmarks rather than as inputs. The only discretionary element is the cubic-lattice assumption a = (1/2)(V/N)^{1/3}, which affects the numerical coefficient in Eqs. (14)-(15); a different geometric factor would change the coefficient without making the argument circular. This is a modeling assumption or robustness concern, not a fitted parameter or a self-citation. There are no load-bearing self-citations and no importation of the target result. The derivation is therefore self-contained against external benchmarks and receives a circularity score of 0.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The derivation is self-contained and uses no fitted constants. Its main input is an ad hoc energy model with a single variational scale, which is the key cost of the paper. The effective charge e~ is a bookkeeping device, not a new physical entity.

axioms (5)
  • standard math The quantum variational principle: for any trial wavefunction, the expectation value of the Hamiltonian is an upper bound on the ground state energy (Eq. 1).
    Used to justify minimizing E(a) and equating the optimum to the physical scale.
  • standard math The hydrogen atom energy with trial wavefunction exp(-r/a) is E(a) = hbar^2/(2 m a^2) - e^2/(4 pi epsilon0 a) (Eq. 3).
    Textbook standard calculation, invoked in Section II.
  • ad hoc to paper A cold hydrogen planet can be modeled as N non-interacting hydrogen atoms in a cubic lattice of spacing 2a, with total energy N times the single-atom energy of Eq. (3).
    This is the load-bearing toy-model assumption; it is stated in Section II immediately after Eq. (4) and acknowledged not to be rigorous.
  • domain assumption The planet has uniform density, so gravitational potential energy is E_grav = -3 G M^2/(5 R) (Eq. 7).
    Stated 'For simplicity' in Section II; planetary interiors are not uniform, but this is a standard crude estimate.
  • domain assumption A cold planet's radius is the value of a that minimizes total energy, i.e., hydrostatic equilibrium as energy minimization.
    The paper differentiates E(a) and sets dE/da = 0 after Eq. (10); valid for zero-temperature equilibrium but not discussed.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of What is the maximum radius of cold planets?." pith.science (2026). https://pith.science/paper/4N25AK6D

@misc{pith2026250907048,
  author       = {Pith},
  title        = {Pith review of: What is the maximum radius of cold planets?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4N25AK6D}},
  note         = {Machine review of arXiv:2509.07048}
}
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read the original abstract

Planets have maximum radii close to that of Jupiter. Qualitatively, the reason for this maximum size is that, as one adds mass, the force of gravity becomes sufficiently strong to cause the radius to decrease. We show that this effect can be understood quantitatively using a simple variational principle very similar to that used to compute the size of the hydrogen atom.

discussion (0)

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Reference graph

Works this paper leans on

19 extracted references · 17 canonical work pages

  1. [1]

    H. S. Zapolsky and E. E. Salpeter, `` The Mass-Radius Relation for Cold Spheres of Low Mass " , Astrophysical Journal, 58 , pp. 809-813, (1969)

    work page 1969

  2. [2]

    J. J. Fortney, M. S. Marley, and J. W. Barnes, ``Planetary Radii across Five Orders of Magnitude in Mass and Stellar Insolation: Application to Transits", The Astrophysical Journal, 659 , pp. 1661-1672, (2007). Also in https://arxiv.org/pdf/astro-ph/0612671 https://arxiv.org/pdf/astro-ph/0612671

    Pith/arXiv arXiv 2007

  3. [3]

    See F. J. Dyson, ``Ground‐State Energy of a Finite System of Charged Particles ", J.Math.Phys. 8 , 1538–1545, (1967)

    work page 1967

  4. [4]

    F. J. Dyson and A. Lenard. ``Stability of Matter, Parts I and II ", J. Math. Phys., 8 , 423–434 (1967); J. Math. Phys., 9 , 698–711 (1968) )

    work page 1967

  5. [5]

    E. H. Lieb and W. E. Thirring, ``Bound for the kinetic energy of fermions which proves the stability of matter", Phys. Rev. Lett. 35 (1975), 687-689. Errata ibid. 35 (1975), 1116

    work page 1975

  6. [6]

    Landau, ``On the theory of Stars”, Physikalische Zeitschrift der Sowjetunion, 1 , pp

    L. Landau, ``On the theory of Stars”, Physikalische Zeitschrift der Sowjetunion, 1 , pp. 285-288, (1932)

    work page 1932

  7. [7]

    See for example Marc L

    This calculation is in many elementary textbooks. See for example Marc L. Kutner , ``Astronomy: A Physical Perspective", Cambridge University Press, Second Edition, p. 158 (2003)

    work page 2003

  8. [8]

    In rigor this approximation is valid for large Z

    For this expression we evaluate the energy of Z electrons approximating the sum over states as an integral in k space. In rigor this approximation is valid for large Z . See Neil Ashcroft and David N. Mermin, Solid State Physics. New York: Holt, Rinehart and Winston, pp. 32--38 (1976)

    work page 1976

  9. [9]

    115 (2018)

    James Patterson and Bernard Bailey , ``Solid State Physics, Introduction to the Theory", Springer, p. 115 (2018)

    work page 2018

  10. [10]

    A. M. Low, ``The Chandrasekhar limit: a simplified approach'' Phys. Educ. 58 , 045008 (2023)

    work page 2023

  11. [11]

    For data on exoplanets see https://exoplanet.eu/home/ https://exoplanet.eu/home/ )

    For example, the recently discovered exoplanet Kepler-91b has a radius of 1.37 R_ Jup . For data on exoplanets see https://exoplanet.eu/home/ https://exoplanet.eu/home/ )

    work page

  12. [12]

    Fortney, Rebekah I

    See for example Jonathan J. Fortney, Rebekah I. Dawson, Thaddeus D. Komacek, ``Hot Jupiters: Origins, Structure, Atmospheres", Journal of Geophysical Research: Planets: 126 , Issue 3, pp. 1-28 (2001)>

    work page 2001

  13. [13]

    Chandrasekhar, ``The Maximum Mass of Ideal White Dwarfs", Astrophysical Journal., 74 , pp

    S. Chandrasekhar, ``The Maximum Mass of Ideal White Dwarfs", Astrophysical Journal., 74 , pp. 81-82, (1931)

    work page 1931

  14. [14]

    E. G. Thomas, ``Edmund Stoner and white dwarf stars". Philosophical Magazine, 91 (26), pp. 3416–3422, (2011). https://doi.org/10.1080/14786435.2011.586377 https://doi.org/10.1080/14786435.2011.586377

    arXiv 2011

  15. [15]

    Nauenberg, ``Edmund C

    M. Nauenberg, ``Edmund C. Stoner and the Discovery of the Maximum Mass of White Dwarfs". Journal for the History of Astronomy, 39 pp. 297-312. (2008) https://doi.org/10.1177/002182860803900302 https://doi.org/10.1177/002182860803900302

    work page doi:10.1177/002182860803900302 2008

  16. [16]

    Weisskopff , ``Search for Simplicity: Mountains, water waves and leaky ceilings, Am

    V. Weisskopff , ``Search for Simplicity: Mountains, water waves and leaky ceilings, Am. J. Phys. 54 , p.2, (1986)

    work page 1986

  17. [17]

    D. J. Mullan, ``Why is the Sun so large?" Am. J. Phys 74 , pp. 10-13, (2006)

    work page 2006

  18. [18]

    David Garfinkle, ``The Planck mass and the Chandrasekhar limit" Am. J. Phys. 77 , pp. 683–687. (2009)

    work page 2009

  19. [19]

    Lieb, ``The stability of matter: from atoms to stars" , Bulletin of the American Mathematical Society

    See Elliot H. Lieb, ``The stability of matter: from atoms to stars" , Bulletin of the American Mathematical Society. 22 , pp. 1-49, (1990)

    work page 1990

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.