Fifty Years of CAK

Jorick S. Vink (Armagh Observatory; Planetarium)

REVIEW 4 major objections 7 minor 24 references

Fifty Years of CAK

T0 review · 4 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that L/M-dependent wind physics, replacing empirical mass-loss recipes, yields vertical HRD tracks for the most massive stars and caps the maximum Galactic black hole mass at about 30 solar masses, from 35-45 solar mass…

desk verdict A readable proceedings synthesis of the author's own prior work; the new RSG-kink conjecture is interesting but under-supported, and the whole framework rests on the unexamined Vink+11 high-Gamma slope. read the letter →

arxiv 2506.06421 v1 pith:GBPQ7TYN submitted 2025-06-06 astro-ph.SR astro-ph.GAastro-ph.HE

Jorick S. Vink (Armagh Observatory , Planetarium) This is my paper

classification astro-ph.SRastro-ph.GAastro-ph.HE

keywords massivestarsstellarmasslossline-drivenwindsCAKtheoryEddingtonparameterblackholeredsupergiantsHertzsprung-Russelldiagram

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 argues that massive star evolution should be driven by the physics of radiation-driven winds—specifically by the Eddington parameter $\Gamma$, the ratio of radiation pressure to gravity, which is proportional to $L/M$—rather than by empirical mass-loss recipes. When the steep $\Gamma$-dependent mass-loss law of Eq. (2) is implemented in stellar evolution calculations, the most massive stars evolve vertically in the Hertzsprung-Russell diagram at nearly constant effective temperature, matching observations of very massive stars in young clusters. The same mass loss makes the heaviest Galactic black holes, around $30\,M_\odot$, come from modest $35$–$45\,M_\odot$ progenitors rather than from the most massive stars, because the most massive stars shed so much mass that their cores erode. The paper closes by conjecturing that the recently observed red supergiant mass-loss kink has the same underlying $L/M$ cause as the hot-star kink.

What carries the argument

The load-bearing object is the steep high-$\Gamma$ mass-loss law of Eq. (2), $\log \dot M \propto 4.77 \log(L/L_\odot) - 3.99 \log(M/M_\odot)$, paired with the criterion that the optically thin-to-thick wind transition occurs where the wind efficiency and optical depth both equal unity, $\eta=\tau=1$, with a calibration factor $f=0.6$ at the transition. The Eddington parameter $\Gamma = \kappa L/(4\pi G c M)$ measures how close a star is to being blown apart by its own radiation, so mass loss raises $\Gamma$ by reducing $M$, creating a positive feedback loop. Inserting this law into stellar evolution calculations is what turns CAK wind physics into vertical HRD evolution, truncates the black hole masses of the most massive stars, and supplies the template for the proposed RSG kink.

What would settle it

A decisive test is to measure wind momentum rates for stars spanning the transition region, roughly $\log(L/L_\odot)$ from 5 to 6.5, at Milky Way and LMC metallicities. If the observed steepening of $\dot M$ with $\Gamma$ is shallower than Eq. (2), or if the transition point lies outside the calibrated factor $0.6\pm0.2$, then the vertical HRD tracks and the $30\,M_\odot$ black-hole peak would not follow.

Watch

Extended reading notes

Core claim

The paper's central claim is that the dividing line between ordinary O-star winds and the much stronger winds of very massive stars is a transition at wind efficiency $\eta = \dot M v_\infty / (L/c)$ equal to the wind optical depth $\tau$, both crossing unity at the spectral Of/WN boundary. Above that point the mass-loss rate follows the steep relation $\log \dot M \propto 4.77 \log(L/L_\odot) - 3.99 \log(M/M_\odot)$, and this is the physics that should replace empirical recipes in stellar evolution models. Stars on the steep branch lose so much mass that they move vertically rather than redwards in the HR diagram; stars below it evolve in the traditional horizontal way. As a result the final black hole mass peaks near $30\,M_\odot$ for zero-age main sequence masses of $35$–$45\,M_\odot$ at Galactic metallicity, with the most massive stars ending as stripped stars that produce only $10$–$15\,M_\odot$ black holes. The same $\Gamma$-dependent logic is then applied to red supergiants, where a new prescription reproduces the Humphreys–Davidson limit and resolves the red supergiant supernova problem.

Load-bearing premise

Everything downstream rests on the steep mass-loss law $\log \dot M \propto 4.77 \log(L/L_\odot) - 3.99 \log(M/M_\odot)$, together with the claim that the switch to this steeper law happens at the $\eta=\tau=1$ transition with a correction factor of $f=0.6$; if that steep slope or transition point is an artifact of the wind models or their calibration, the vertical evolution, the $30\,M_\odot$ black-hole peak, and the red-supergiant kink interpretation all lose their quantitative foundation.

Editorial extensions

If this is right

Stars with initial masses above roughly $80$–$100\,M_\odot$ lose so much mass that even their cores are eroded; their Wolf–Rayet remnants form black holes of only about $10$–$15\,M_\odot$.
The maximum Galactic black hole mass is about $30\,M_\odot$, produced by $35$–$45\,M_\odot$ zero-age main-sequence stars, while the most massive progenitors produce lighter black holes.
Vertical HRD evolution removes the need to finely balance envelope inflation against wind stripping to keep very massive stars at their observed effective temperatures.
A $\Gamma$-dependent red supergiant mass-loss prescription reproduces the Humphreys–Davidson limit and resolves the red supergiant supernova problem.
At lower metallicity, weaker winds allow heavier black holes, up to about $93\,M_\odot$ just below the pair-instability gap.

Reading between the lines

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

The paper leaves implicit that the $\eta=\tau=1$ transition should show up in other wind diagnostics, such as X-ray or radio indicators, and would predict a corresponding break in those observables at the same $L/M$ values; a multi-wavelength survey of stars across the kink could test this directly.
If red supergiant mass loss is governed by current $L/M$ rather than luminosity alone or initial mass, then two red supergiants of equal luminosity but different current masses should have different wind strengths; this is testable with eclipsing or astrometric binaries where current masses are measured.
An explicit counterfactual prediction is that stellar models with a flat, weak mass-loss law above the transition would fail to keep very massive stars hot and blue, producing envelope inflation and redward evolution instead; reproducing the observed near-constant effective temperatures in young clusters would then require ad hoc tuning.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a readable StanFest proceedings piece that repackages the author's earlier mass-loss and evolution results into one narrative. The genuinely new element is the RSG-kink conjecture, and it is also the softest part. It is not a replacement for reading Vink et al. 2011, Vink & Gräfener 2012, Sabhahit et al. 2022/23, and Vink et al. 2024, but as a synthesis it is honest about what it is.

The paper does some things well. It states the physical case clearly: once winds become optically thick, mass loss depends on L/M (or Gamma), and putting that into MESA produces vertical HRD tracks and a non-monotonic initial-final mass relation. Figure 3 is a nice summary of why the most massive ZAMS stars do not make the heaviest BHs. The suggestion that a similar L/M dependence drives the RSG kink is testable, and the acknowledgment that current mass is not directly observable but still the physically right variable is fair.

Where it gets soft: nearly all the quantitative content is 'as shown in my cited papers.' Eq. (2) appears without normalization or derivation; the exact MESA switch is not given; no error bars or sensitivity analysis are included. That is okay for a proceedings chapter, but it means this manuscript alone cannot support the headline numbers. The RSG section is the weakest: the new prescription is asserted to reproduce the Humphreys-Davidson limit and solve the RSG supernova problem with no citation and no model description. And the two blue lines in Fig. 4 are based on current masses of 8 and 30 Msun, but we aren't told why those masses are relevant to the Yang et al. sample. The whole enterprise rests on the steep Vink+11 high-Gamma slope and the eta=tau=1 transition with f=0.6. If that slope or transition is off, the vertical evolution, the 30 Msun peak, and the RSG analogy all shift. The reader's 'circularity burden' is a bit strong—using your own earlier calibrated relations to build evolution tracks is normal—but the absence of independent validation is real.

Audience: people working on massive star evolution or black hole formation who want a quick map of the Armagh group's framework. It is a useful entry point, not a definitive paper. As a proceedings piece it is fine to include, but a journal referee would want the missing MESA details and an error analysis. I would not desk-reject it outright; I would send it to review as a perspective or invited review, with a clear request to either cut the unsupported RSG claims or add the underlying model.

Referee Report

4 major / 7 minor

Summary. Vink proposes a new framework for massive-star evolution in which mass loss is set by the Eddington parameter Γ (equivalently L/M) rather than by empirical 'Dutch' recipes. Section 2 introduces the hot-star mass-loss kink at the η=τ=1 transition and the steep high-Γ relation log Ṁ ∝ 4.77 log(L/L⊙) − 3.99 log(M/M⊙) in Eq. (2). Section 3 reports that MESA models with this implementation produce vertical HRD evolution at the highest masses and an initial-final mass relation whose black hole masses peak near 30 M⊙ for ZAMS masses of 35–45 M⊙. Section 4 proposes that a recently identified red supergiant (RSG) kink at log(L/L⊙)≈4.6 shares the same underlying L/M physics and claims that a new RSG prescription reproduces the Humphreys-Davidson limit and resolves the RSG supernova problem.

Significance. If the central claims hold, the paper offers a physically motivated alternative to recipe-based mass loss and makes concrete, falsifiable predictions: vertical HRD tracks for very massive stars, a maximum Galactic BH mass near 30 M⊙, and a specific interpretation of the RSG kink. The framework is clearly stated, and the dependence on published MESA models (Sabhahit et al. 2022, 2023; Vink et al. 2024) is transparent. However, the quantitative headline results are not derived or tested within this manuscript: the vertical tracks and the BH peak ride entirely on the calibration of Eq. (2) and the transition criterion in Section 2, while the RSG claims in Section 4 are presented without equations, model tracks, or quantitative comparisons. The paper therefore reads as a programmatic summary rather than a self-contained derivation; its value depends on whether the cited companion papers contain the missing details and on whether the additional tests suggested below are performed.

major comments (4)

[Section 2, Eq. (2)] The steep high-Γ mass-loss relation log Ṁ ∝ 4.77 log(L/L⊙) − 3.99 log(M/M⊙) is the single load-bearing input for the vertical evolution and BH peak, but the manuscript does not give the normalization constant, the exact functional form, the range of Γ over which it is applied, or how the switch between this relation and the Vink et al. (2000) relation is implemented in MESA. Without these details, a reader cannot reproduce or verify the central claims, and small changes in the slope or in the transition location will alter which stars evolve vertically and what BH masses result.
[Section 3, Fig. 3] The claim that the maximum Galactic BH mass is about 30 M⊙ for ZAMS masses of 35–45 M⊙ is presented as a result of "our mass-loss implementation," but the figure and text rely on previous model runs (Sabhahit et al. 2022, 2023; Vink et al. 2024) without specifying the input physics (mixing, rotation, mass-loss normalization, metallicity Z=0.02) or providing an error budget. No comparison to observed BH mass distributions from gravitational waves or X-ray binaries is made, so the predicted peak is not falsifiably tested within this manuscript.
[Section 4, Fig. 4] The RSG kink is interpreted by overlaying "current mass" lines of 8 and 30 M⊙ on the Yang et al. (2023) data, but this is an L/M scaling argument, not a derivation from cool-star opacity physics. The statement that the newly proposed RSG prescription "correctly reproduces the Humphreys-Davidson limit" and "resolves the RSG supernova problem" is not supported by equations, model tracks, or quantitative comparisons; these claims need to be either substantiated in the text or explicitly deferred to a companion paper.
[Sections 2 and 5] The transition criterion η=τ=1 with correction factor f=0.6 is taken from Vink & Gräfener (2012), and the same empirical fits are later used to interpret both the hot-star kink and the RSG kink as evidence for the same physics. This creates a risk of circularity: data are used to calibrate the model and then cited as confirmation of the model. The circularity can be broken by an out-of-sample test, such as predicting the BH mass distribution and comparing it with GWTC-3, or predicting RSG mass-loss rates in a galaxy whose data were not used in the calibration.

minor comments (7)

[Section 2, Fig. 1] The axis label "log(Edd)" should be "log Γ_Edd" or "log Γe", and the text "At transition, s 1" appears to be a typo for "τ≈1".
[Section 3, first sentence] "displayed in Fig.,3" contains a typographical error; it should read "displayed in Fig. 3".
[Section 4] The term "RSG supernova problem" is used without definition; a one-sentence explanation would help readers who are not specialists.
[Section 4] The "Humphreys-Davidson limit" is mentioned but not defined; please provide a brief definition or a reference at first use.
[Section 5] The suggestion that yellow and red supergiants should be subjected to Γ-dependent mass loss is speculative and should be framed explicitly as a conjecture rather than a demonstrated result.
[Figure 2] The label "de Jager" in Figure 2 does not appear in the reference list; either add the source (e.g., de Jager et al. 1988) or remove the label.
[General] The manuscript appears to be a StanFest proceedings contribution; if it is intended for a regular archival journal, the informal style (e.g., "Stan-the-Man" in the acknowledgments) and the lack of a methods section should be revised.

Circularity Check

0 steps flagged · score 0.0 of 10

No exhibited circular reduction: the hot-star kink, transition criterion, and steep high-Γ slope are prior Monte-Carlo results, and the MESA tracks/BH peak/RSG kink are forward applications, not definitions of the inputs.

full rationale

The paper's derivation chain is not circular by the strict standard of exhibiting a reduction of a predicted quantity to a fitted input or a self-citation that is load-bearing by construction. Equation (2), log Mdot ∝ 4.77 log L − 3.99 log M, is quoted from Vink et al. (2011), and the η=τ=1 transition with correction factor f=0.6 from Vink & Gräfener (2012); these are independent Monte-Carlo and atmospheric-model results, not quantities defined in terms of the targets (vertical HRD evolution, 30 Msun BH peak, or the RSG kink location). The MESA tracks in Figs. 2 and 3 are forward stellar-evolution simulations using those prescriptions, with no parameter fitted to the claimed outcomes. The RSG-kink discussion in Section 4 explicitly imports the hot-star steep relation as a model to be compared with Yang et al. (2023) empirical data; the two blue 'current mass' lines are an illustrative overlay, not a derivation that presupposes the kink position. The self-citations to Sabhahit et al. (2022, 2023) and Vink et al. (2024) are references to prior numerical implementations and results, which is normal for a proceedings summary, and no uniqueness theorem or ansatz is smuggled in via those citations. The dependence of the quantitative conclusions on the steep slope or transition point is a sensitivity/robustness concern, not a circularity. Because no specific reduction of a claim to its own input is exhibited in the text, the appropriate circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on previously fitted mass-loss relations and on a conceptual extrapolation to red supergiants. The paper introduces no new particles, forces, or conserved quantities, and it adds no new parameter-free derivation of its own.

free parameters (3)

Vink et al. (2011) mass-loss slope coefficients = 4.77 (log L) and -3.99 (log M) in Eq. (2)
The central steep mass-loss relation adopted by the framework is an empirical fit to radiation-driven wind models from prior work, and all MESA results inherit these coefficients.
Mass-loss transition correction factor f = 0.6 +/- 0.2
Vink and Graefener (2012) correction factor used to define the eta equals tau equals 1 transition and calibrate the switch between optically thin and thick wind mass-loss relations, cited in footnote 1.
RSG kink luminosity threshold = log(L/Lsun) about 4.6
The RSG mass-loss prescription uses a shallow relation below and a steep relation above the observed kink in Yang et al. (2023) data, with the threshold taken from that data as shown in Figure 4.

assumptions (4)

domain assumption Time-averaged CAK theory describes hot-star mass-loss rates and terminal velocities.
Invoked at the start of Section 1 to justify the framework, relying on prior work by Owocki et al. (1988).
domain assumption Final stellar mass at the end of core helium burning corresponds to the black hole mass.
Stated at the start of Section 3, citing Fryer et al. (2012); the claimed maximum black hole mass of about 30 solar masses depends on this mapping.
ad hoc to paper The steep Gamma-dependent mass-loss relation above the transition (Eq. 2) applies to very massive stars and, by L/M scaling, to red supergiants.
This extrapolation is the core of the proposed universal trend; the paper offers a conceptual argument rather than a derivation, and the RSG application is new in this paper.
domain assumption Current mass Mcurrent, not initial mass, is the relevant variable for RSG mass loss.
Stated in Section 4; the paper acknowledges that current masses cannot be readily observed, making the prescription difficult to test directly.

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Cite this review

Pith. "Pith review of Fifty Years of CAK." pith.science (2026). https://pith.science/paper/GBPQ7TYN

@misc{pith2026250606421,
  author       = {Pith},
  title        = {Pith review of: Fifty Years of CAK},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GBPQ7TYN}},
  note         = {Machine review of arXiv:2506.06421}
}

read the original abstract

We present a new framework for massive star evolution that is no longer driven by Dutch or other mass-loss rate Recipes, but which take the physics of Gamma or L/M dependent mass loss consistently into account. We first discuss the hot-star mass-loss kink and the transition mass loss rate between optically thin and thick winds, before discussing vertical stellar evolution, mass evaporation, and the maximum black hole (BH) mass. We end with a suggestion that a recently uncovered red supergiant (RSG) kink might be related to similar underlying L/M physics as the hot-star kink.

Figures

Figures reproduced from arXiv: 2506.06421 by the authors.

**Figure 2.** The HR diagram for MESA models incorporates the mass-loss recipes shown [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** The initial-final mass relation applies to initial ZAMS masses of 20, 25, 30, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Observational mass-loss rates from (Yang et al., 2023) show a kink at log(L/L⊙) = 4.6. We utilize a Γ-dependent mass-loss relation informed by our understanding of radiation-driven winds, employing a shallow mass-loss relation below the kink and a steep one above it, a…

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Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

C., Lucy, L

Abbott, D. C., Lucy, L. B., ApJ 288, 679 (1985)

work page 1985
[2]

R., et al., MNRAS 492, 4, 5994 (2020)

Beasor, E. R., et al., MNRAS 492, 4, 5994 (2020)

work page 2020
[3]

M., et al., A&A 570, A38 (2014)

Bestenlehner, J. M., et al., A&A 570, A38 (2014)

work page 2014
[4]

I., Abbott, D

Castor, J. I., Abbott, D. C., Klein, R. I., ApJ 195, 157 (1975)

work page 1975
[5]

A., et al., MNRAS 408, 2, 731 (2010)

Crowther, P. A., et al., MNRAS 408, 2, 731 (2010)

work page 2010
[6]

L., et al., ApJ 749, 1, 91 (2012)

Fryer, C. L., et al., ApJ 749, 1, 91 (2012)

work page 2012
[7]

G., Owocki, S

Gayley, K. G., Owocki, S. P., Cranmer, S. R., ApJ 442, 296 (1995) Gr¨afener, G., Owocki, S. P., Vink, J. S., A&A 538, A40 (2012)

work page 1995
[8]

R., Sander, A

Higgins, E. R., Sander, A. A. C., Vink, J. S., Hirschi, R., MNRAS 505, 4, 4874 (2021)

work page 2021

Show all 24 references

[9]

B., Solomon, P

Lucy, L. B., Solomon, P. M., ApJ 159, 879 (1970)

1970
[10]

P., Castor, J

Owocki, S. P., Castor, J. I., Rybicki, G. B., ApJ 335, 914 (1988)

1988
[11]

Paxton, B., et al., ApJS 208, 1, 4 (2013)

2013
[12]

S., Najarro, F., A&A Rev

Puls, J., Vink, J. S., Najarro, F., A&A Rev. 16, 3-4, 209 (2008)

2008
[13]

N., Vink, J

Sabhahit, G. N., Vink, J. S., Higgins, E. R., Sander, A. A. C., MNRAS 514, 3, 3736 (2022)

2022
[14]

N., Vink, J

Sabhahit, G. N., Vink, J. S., Sander, A. A. C., Higgins, E. R., MNRAS 524, 1, 1529 (2023)

2023
[15]

Sander, A. A. C., Vink, J. S., Hamann, W. R., MNRAS 491, 3, 4406 (2020)

2020
[16]

O., Bj ¨orklund, R., Puls, J., Najarro, F., A&A 632, A126 (2019)

Sundqvist, J. O., Bj ¨orklund, R., Puls, J., Najarro, F., A&A 632, A126 (2019)

2019
[17]

S., A&A 615, A119 (2018)

Vink, J. S., A&A 615, A119 (2018)

2018
[18]

S., ARA&A 60, 203 (2022)

Vink, J. S., ARA&A 60, 203 (2022)

2022
[19]

S., de Koter, A., Lamers, H

Vink, J. S., de Koter, A., Lamers, H. J. G. L. M., A&A 362, 295 (2000)

2000
[20]

S., Gr ¨afener, G., ApJ 751, 2, L34 (2012)

Vink, J. S., Gr ¨afener, G., ApJ 751, 2, L34 (2012)

2012
[21]

S., Sabhahit, G

Vink, J. S., Sabhahit, G. N., Higgins, E. R., A&A 688, L10 (2024)

2024
[22]

S., et al., A&A 531, A132 (2011)

Vink, J. S., et al., A&A 531, A132 (2011)

2011
[23]

Winch, E. R. J., Vink, J. S., Higgins, E. R., Sabhahitf, G. N., MNRAS 529, 3, 2980 (2024)

2024
[24]

Yang, M., et al., A&A 676, A84 (2023) fys.kuleuven.be/ster/events/conferences/2024/stanfest/ StanFest Proceedings ⋆ 7

2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

C., Lucy, L

Abbott, D. C., Lucy, L. B., ApJ 288, 679 (1985)

work page 1985

[2] [2]

R., et al., MNRAS 492, 4, 5994 (2020)

Beasor, E. R., et al., MNRAS 492, 4, 5994 (2020)

work page 2020

[3] [3]

M., et al., A&A 570, A38 (2014)

Bestenlehner, J. M., et al., A&A 570, A38 (2014)

work page 2014

[4] [4]

I., Abbott, D

Castor, J. I., Abbott, D. C., Klein, R. I., ApJ 195, 157 (1975)

work page 1975

[5] [5]

A., et al., MNRAS 408, 2, 731 (2010)

Crowther, P. A., et al., MNRAS 408, 2, 731 (2010)

work page 2010

[6] [6]

L., et al., ApJ 749, 1, 91 (2012)

Fryer, C. L., et al., ApJ 749, 1, 91 (2012)

work page 2012

[7] [7]

G., Owocki, S

Gayley, K. G., Owocki, S. P., Cranmer, S. R., ApJ 442, 296 (1995) Gr¨afener, G., Owocki, S. P., Vink, J. S., A&A 538, A40 (2012)

work page 1995

[8] [8]

R., Sander, A

Higgins, E. R., Sander, A. A. C., Vink, J. S., Hirschi, R., MNRAS 505, 4, 4874 (2021)

work page 2021