The physical origin of the maximum stellar size

Gautham N. Sabhahit; Jorick S. Vink

REVIEW

The physical origin of the maximum stellar size

Gautham N. Sabhahit, Jorick S. Vink

Solar and Stellar Astrophysics · Submitted 30 Jul 2026 · 5 references

Pith's verdict: Feeding the PoWRHD Ṁ–Γ kink into MESA recovers the HD radius boundary without imposing it; missing control run is the real soft spot, but the result still deserves referees.

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arXiv ID2607.28013 (version 1)

Pith IDMUSJPCU6, Pith's permanent record of this paper and its review

CitationGautham N. Sabhahit, Jorick S. Vink. "The physical origin of the maximum stellar size." arXiv:2607.28013 (2026). https://pith.science/paper/MUSJPCU6

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Editor's note

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The one thing worth knowing: they take hydrodynamically consistent mass-loss rates that already contain a kink in Ṁ versus Γ, drop them self-consistently into MESA, and the tracks turn back at roughly the observed Humphreys-Davidson boundary—including the slanted segment and the low-luminosity ceiling. That evolutionary demonstration is new. The wind kink itself and the transition-rate idea were already in the literature; earlier evolution papers mostly imposed the HD limit by hand. Here the morphology emerges from the rates.

What works is the causal story and the mass dependence. Higher-mass models hit the high-Γ regime early and stay hot; intermediate-mass models make longer coolward excursions before the kink peels them; stars below ~30 M☉ never hit it and expand to RSGs. The red diamonds in Fig. 3 and Extended Data Fig. 1 line up with that picture. The rates sit near the Arches transition value and the Pauli empirical compilation, so the absolute scale is not free-floating. The implications they flag for BH masses, GW rates, SN types, and JWST ionizing budgets are genuine.

Soft spots in proportion. The stress-test concern lands: there is no control grid with the same MESA physics but a smooth classical Ṁ(Γ). Without it you cannot fully isolate how much of the shape is the kink versus the temperature-fence switches (PoWRHD only between log Teff 3.8–4.9, de Jager below, WR above) or the fixed mixing (αMLT=1.5, fov=0.03, MLT++ off on the MS). Single metallicity is another real limit for a claim this broad. Tracks are on request and the explicit Ṁ formula lives in the companion paper, which slows checks. None of that overturns the central result; it just means the causal attribution is one clean contrast short of airtight.

This is for anyone who needs a physical radius limit inside population synthesis, SN progenitor work, or high-z feedback. It is serious thinking, formally grounded enough, and important enough to send to referees. I would bring it to reading group and cite the evolutionary result once the tracks are public. Engage.

Referee report

Significance

A predictive, physically motivated explanation of the HD limit would be a genuine advance. The limit controls whether the most massive stars remain compact ionizing sources or become cool supergiants, thereby affecting envelope retention, supernova type, binary interaction probability, black-hole mass spectra and the interpretation of JWST spectra of distant populations. The work supplies a concrete, implementable Ṁ(Γ, L, Teff) prescription derived from hydrodynamically consistent atmospheres and demonstrates that MESA tracks using it visually recover the observed HD morphology (Fig. 3). If the causal link holds under scrutiny, the framework can be adopted directly by population-synthesis and spectral-synthesis codes. The explicit anchoring to the Arches transition rate and comparison to the Pauli et al. empirical compilation are welcome strengths.

Major objections (4)

[Fig. 3; Methods; Extended Data Fig. 1] The central causal claim—that the HD kink is the evolutionary manifestation of the PoWRHD Ṁ–Γ kink—is not isolated by a control experiment. Figure 3 and Extended Data Figure 1 show only the successful kink-bearing grid. A parallel MESA grid with identical mixing, rotation and temperature-window switches but a smooth classical Ṁ(Γ) (no upturn) is required to demonstrate that the wind kink is necessary and sufficient for the slanted segment and the ~30 M☉ ceiling. Without that contrast, envelope-inflation treatment (MLT++ off on the MS) or the de Jager switch below log Teff = 3.7 could be co-shaping the morphology.
[Methods (initial metal mass fraction Z = 0.02)] All results are shown at a single metallicity (Z = 0.02). The empirical HD limit is defined in both the Milky Way and the LMC; the Ṁ–Γ kink location and steepness are expected to be metallicity-dependent. At minimum the paper should present one lower-Z grid (or a clear statement of why the Z = 0.02 morphology is representative) before claiming a general first-principles origin.
[Methods (paragraph on temperature regimes)] The mass-loss recipe is applied only inside fixed temperature fences (PoWRHD for 3.8 < log Teff < 4.9, de Jager below 3.7, WR above 5, with smooth interpolation). Because the HD morphology is precisely a statement about how far tracks cool, the location of these fences is load-bearing. Sensitivity of the final HD shape to modest shifts in the fence boundaries (or to continuous use of a single recipe) should be quantified.
[Methods (convection and mixing paragraphs)] Mixing parameters are fixed (αMLT = 1.5, fov = 0.03, αsc = 1, MLT++ off on the MS, 20 % critical rotation). While a full parameter study is beyond scope, the paper should show that the qualitative recovery of the HD kink survives reasonable variations in overshooting and semiconvection, or at least discuss how these choices affect the timing of kink crossing (red diamonds in Fig. 3 and Extended Data Fig. 1).

Minor objections (5)

[Figure 1 caption] Figure 1 caption cites the empirical HD limit as ‘dashed red)1,29,30’; the reference list numbering appears inconsistent with the superscript style used in the main text. Please unify.
[Extended Data Figure 1; main text paragraph on the luminosity ceiling] The main text states that stars below ~30 M☉ ‘do not encounter the mass-loss kink at all’, yet Extended Data Figure 1 still marks red diamonds on the lower-mass tracks. Clarify whether those symbols indicate a formal crossing or are simply omitted for M ≲ 30 M☉.
[Supplementary Information] Supplementary Figure 1 compares the implemented rates to the Pauli et al. compilation, but the figure itself is only described, not shown with axis ranges or a quantitative residual. A brief table of mean offset and scatter would strengthen the anchoring claim.
[Data availability] Data-availability statement says tracks ‘will be made available upon reasonable request’. For a result intended for community population-synthesis use, deposition of the MESA inlists and final tracks in a public repository at acceptance would be preferable.
[Abstract; penultimate main-text paragraph] Typographical: ‘self consistently’ should be hyphenated ‘self-consistently’ in the abstract and opening paragraph; ‘by utilised’ → ‘be utilised’.

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The paper

The paper claims

The kinked Humphreys-Davidson radius limit is the evolutionary face of a kink in how mass-loss rate scales with the Eddington parameter.

The reading

The most massive stars are not the largest; they stop short of red-supergiant sizes along a kinked boundary in the Hertzsprung-Russell diagram called the Humphreys-Davidson limit. For fifty years that shape lacked a first-principles explanation. This paper shows the boundary appears automatically once stellar-evolution calculations use mass-loss rates that themselves contain a sharp upturn: when a star’s luminosity-to-mass ratio pushes it near the Eddington limit, radiation-driven winds switch from classical optically thin outflows to a much stronger, multiple-scattering regime. Very massive stars hit that switch early and stay hot and compact; intermediate-mass stars hit it later after a partial coolward excursion; stars below roughly thirty solar masses never hit it and expand freely. The resulting tracks recover both the slanted segment and the low-luminosity ceiling of the observed limit. Because stellar radius controls binary interaction, black-hole masses, supernova type, and ionising output, a predictive origin for the limit supplies a usable framework for gravitational-wave rates, population synthesis, and high-redshift spectra.

What carries the argument

The Ṁ–Γ kink: the abrupt steepening of mass-loss rate once the electron-scattering Eddington parameter Γe rises high enough for multiple line scattering to dominate. Implemented from hydrodynamically consistent atmosphere models into evolutionary tracks, the delayed or immediate onset of this enhanced-loss regime sets how far each star can expand before strong winds reverse the excursion.

Extended reading notes

Core claim

The characteristic kinked shape of the Humphreys-Davidson limit is simply the evolutionary manifestation of the kink already present in the mass-loss rate versus Eddington-parameter relation. When hydrodynamically consistent rates that contain this kink are fed self-consistently into stellar-structure calculations, the empirical radius boundary—including its slanted portion and its luminosity ceiling—emerges without being imposed by hand.

Load-bearing premise

The mass-loss recipe, including exactly where and how steeply the kink occurs and how the recipe switches with temperature, stays valid all the way from the zero-age main sequence through core-helium burning.

Editorial extensions

If this is right

Binary population synthesis can now decide Roche-lobe overflow, common-envelope evolution and mergers from a physically motivated radius limit rather than an imposed cut.
Black-hole mass distributions and gravitational-wave merger rates inherit a firmer upper bound set by envelope stripping near the Eddington limit.
Ionising-photon budgets of very massive star populations become predictable, improving spectral synthesis for high-redshift galaxies.
The same framework distinguishes which stars keep hydrogen envelopes (Type II supernovae) from those that are stripped (Type Ibc).
Chemical and radiative feedback calculations for the early Universe can include very massive stars that remain compact hot sources instead of cool supergiants.

Reading between the lines

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

Because the kink is tied to Γe, the location of the Humphreys-Davidson boundary should shift with metallicity in a way that can be tested in Magellanic Cloud and metal-poor galaxies.
Single-star tracks that already enforce the radius limit will change the predicted relative numbers of luminous blue variables versus red supergiants at a given age.
Once the same mass-loss switch is inserted into rotating or binary models, the fraction of stars that merge before core collapse may drop, lowering certain gravitational-wave channel rates.

Original abstract

One might expect the most massive stars to also be the largest by size, yet they are not, and this has puzzled astronomers for decades. The Eddington limit sets an upper bound to stellar mass through the balance between radiation pressure and gravity, but a second empirical boundary on stellar radius has equally far-reaching consequences for modern Astrophysics: it affects the black-hole masses and merger rates inferred by LIGO and Virgo, determines whether a star remains a hot, compact ionising source or inflates into a cool supergiant, and thereby shapes how spectra of distant, unresolved stellar populations are interpreted. Half a century ago, this radius limit was shown to trace a characteristic kinked shape in the Hertzsprung-Russell diagram known as the Humphreys-Davidson limit, yet a predictive, first-principle physical explanation has remained elusive. Here we show that this kink is the evolutionary manifestation of a transition from classical stellar outflows to an Eddington-enhanced mass-loss regime implemented self consistently in evolutionary models. With our models accurately reproducing the empirical constraints on the radii of the most massive stars, we now have a framework that can be incorporated into binary population synthesis, black-hole mass predictions, gravitational-wave event rates, and the interpretation of high-redshift James Webb spectra.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,530 input tokens.

What the abstract itself states, as probabilities: claims first 0.18 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.59 · mentions code or data availability 0.04.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

One might expect the most massive stars to also be the largest by size, yet they are not, and this has puzzled astronomers for decades. The Eddington limit sets an upper bound to stellar mass through the balance between radiation pressure and gravity, but a second empirical boundary on stellar radius has equally far-reaching consequences for modern Astrophysics: it affects the black-hole masses and merger rates inferred by LIGO and Virgo, determines whether a star remains a hot, compact ionising source or inflates into a cool supergiant, and thereby shapes how spectra of distant, unresolved stellar populations are interpreted. Half a century ago, this radius limit was shown to trace a characteristic kinked shape in the Hertzsprung-Russell diagram known as the Humphreys-Davidson limit, yet a predictive, first-principle physical explanation has remained elusive. Here we show that this kink is the evolutionary manifestation of a transition from classical stellar outflows to an Eddington-enhanced mass-loss regime implemented self consistently in evolutionary models. With our models accurately reproducing the empirical constraints on the radii of the most massive stars, we now have a framework that can be incorporated into binary population synthesis, black-hole mass predictions, gravitational-wave event rates, and the interpretation of high-redshift James Webb spectra.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	One might expect the most massive stars to also be the largest by size, yet they are not, and this has puzzled astronomers for decades.	background 0.92	0.04	0.02	asserted 0.55
s1	The Eddington limit sets an upper bound to stellar mass through the balance between radiation pressure and gravity, but a second empirical boundary on stellar radius has equally far-reaching consequences for modern Astrophysics: it affects the black-hole masses and merger rates inferred by LIGO and Virgo, determines whether a star remains a hot, compact ionising source or inflates into a cool supergiant, and thereby shapes how spectra of distant, unresolved stellar populations are interpreted.	background 0.94	0.06	0.05	asserted 0.99
s2	Half a century ago, this radius limit was shown to trace a characteristic kinked shape in the Hertzsprung-Russell diagram known as the Humphreys-Davidson limit, yet a predictive, first-principle physical explanation has remained elusive.	gap 0.81	0.03	0.03	asserted 1.00
s3	Here we show that this kink is the evolutionary manifestation of a transition from classical stellar outflows to an Eddington-enhanced mass-loss regime implemented self consistently in evolutionary models.	result 0.63	0.94	0.93	asserted 1.00
s4	With our models accurately reproducing the empirical constraints on the radii of the most massive stars, we now have a framework that can be incorporated into binary population synthesis, black-hole mass predictions, gravitational-wave event rates, and the interpretation of high-redshift James Webb spectra.	interpretation 0.64	0.82	0.15	asserted 0.98

What kind of paper the classifier reports this is: simulation study 0.63 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Hertzsprung-Russell diagram showing the empirical HD-limit kink (dashed red)1,29,30. Constant radius lines (dotted black) are overplotted showcasing the fundamental radius limit imposed by the HD-limit. The location of the Zero Age Main Sequence is shown for comparison (solid black)

**Figure 2.** Predicted mass-loss rates from PoWRHD models as a function of electron scattering Eddington parameter illustrating the kink in the Ṁ-Γ scaling

**Figure 3.** Hertzsprung-Russell diagram tracks of our evolutionary models with dot symbols mark timesteps of every 100,000 years. Red diamond symbols mark the point at which each model crosses the mass-loss kink, above which Ṁ scales steeply with the Eddington parameter

Figures reproduced from arXiv:2607.28013 by the authors.

References

Works cited (5)

[1]

Eddington, A. S. The Internal Constitution of the Stars, Cambridge Univ. Press, (1926). 2. Abbott, B. P. et al. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett. 116, 061102 (2016). 3. Humphreys, R. M. & Davidson, K. Studies of luminous stars in nearby galaxies. III. Comments on the evolution of the most massive stars in...

View on Pith 1926
[12]

Schaerer, D., Guibert, J., Marques-Chaves, R. & Martins, F. Observable and ionizing properties of star-forming galaxies with very massive stars and different initial mass functions. Astron. Astrophys. 693, A271 (2025). 13. Barkana, R. & Loeb, A. In the beginning: the first sources of light and the reionization of the universe. Phys. Rep. 349, 125–238 (200...

View on Pith 2025
[25]

Bestenlehner, J. M. et al. The VLT-FLAMES Tarantula Survey. XVII. Physical and wind properties of massive stars at the top of the main sequence. Astron. Astrophys. 570, A38 (2014). 26. Sabhahit, G. N., Vink, J. S. & Sander, A. A. C. Hydrodynamical mass-loss rates for very massive stars: I. Investigating the wind kink. Astron. Astrophys. 706, A97 (2026). 2...

View on Pith 2014
[37]

Paxton, B. et al. Modules for Experiments in Stellar Astrophysics (MESA). Astrophys. J. Suppl. Ser. 192, 3 (2011). 38. Paxton, B. et al. Modules for Experiments in Stellar Astrophysics (MESA): Planets, Oscillations, Rotation, and Massive Stars. Astrophys. J. Suppl. Ser. 208, 4 (2013). 39. Paxton, B. et al. Modules for Experiments in Stellar Astrophysics (...

View on Pith 2011
[49]

Sander, A. A. C., Hamann, W.-R., Todt, H., Hainich, R. & Shenar, T. Coupling hydrodynamics with comoving frame radiative transfer. I. A unified approach for OB and WR stars. Astron. Astrophys. 603, A86 (2017). 50. de Jager, C., Nieuwenhuijzen, H. & van der Hucht, K. A. Mass loss rates in the Hertzsprung-Russell diagram. Astron. Astrophys. Suppl. Ser. 72, ...

View on Pith arXiv 2017

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A census of massive eclipsing binaries in the Whirlpool Galaxy
astro-ph.SR 2026-09 accept novelty 7.0 of 10

A census of 173 massive eclipsing binary candidates in M51 yields a close binary fraction of 27.4% and a log-flat period distribution, consistent with Milky Way and Magellanic Cloud populations.
Beyond Mergers: the Role of Metallicity in Forming a Reservoir of Massive and Detached Binary Black Hole Systems
astro-ph.SR 2026-10 conditional novelty 6.0 of 10

At moderate metallicities, massive black holes in binaries are predominantly left in wide detached orbits rather than merging, creating a reservoir for dynamical hardening.
Wolf-Rayet formation through L/M-dependent superwinds
astro-ph.SR 2026-10 conditional novelty 5.0 of 10

An L/M-dependent red supergiant wind law drives runaway envelope stripping, allowing ~20 solar-mass stars to form Wolf-Rayet stars as single stars.

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Eddington, A. S. The Internal Constitution of the Stars, Cambridge Univ. Press, (1926). 2. Abbott, B. P. et al. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett. 116, 061102 (2016). 3. Humphreys, R. M. & Davidson, K. Studies of luminous stars in nearby galaxies. III. Comments on the evolution of the most massive stars in...

View on Pith 1926

[2] [12]

Schaerer, D., Guibert, J., Marques-Chaves, R. & Martins, F. Observable and ionizing properties of star-forming galaxies with very massive stars and different initial mass functions. Astron. Astrophys. 693, A271 (2025). 13. Barkana, R. & Loeb, A. In the beginning: the first sources of light and the reionization of the universe. Phys. Rep. 349, 125–238 (200...

View on Pith 2025

[3] [25]

Bestenlehner, J. M. et al. The VLT-FLAMES Tarantula Survey. XVII. Physical and wind properties of massive stars at the top of the main sequence. Astron. Astrophys. 570, A38 (2014). 26. Sabhahit, G. N., Vink, J. S. & Sander, A. A. C. Hydrodynamical mass-loss rates for very massive stars: I. Investigating the wind kink. Astron. Astrophys. 706, A97 (2026). 2...

View on Pith 2014

[4] [37]

Paxton, B. et al. Modules for Experiments in Stellar Astrophysics (MESA). Astrophys. J. Suppl. Ser. 192, 3 (2011). 38. Paxton, B. et al. Modules for Experiments in Stellar Astrophysics (MESA): Planets, Oscillations, Rotation, and Massive Stars. Astrophys. J. Suppl. Ser. 208, 4 (2013). 39. Paxton, B. et al. Modules for Experiments in Stellar Astrophysics (...

View on Pith 2011

[5] [49]

Sander, A. A. C., Hamann, W.-R., Todt, H., Hainich, R. & Shenar, T. Coupling hydrodynamics with comoving frame radiative transfer. I. A unified approach for OB and WR stars. Astron. Astrophys. 603, A86 (2017). 50. de Jager, C., Nieuwenhuijzen, H. & van der Hucht, K. A. Mass loss rates in the Hertzsprung-Russell diagram. Astron. Astrophys. Suppl. Ser. 72, ...

View on Pith arXiv 2017