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Hydrodynamical mass-loss rates for very massive stars II. New theoretical mass-loss predictions at solar metallicity (Z = 0.02)

T0 review · 3 major / 7 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read A single continuous mass-loss recipe captures the kink from optically thin O-star winds to optically thick WNh winds at solar metallicity.

desk verdict Solid, usable solar-Z mass-loss recipe from a large PoWR-HD grid; kink and bistability confirmed across L/X, anchored at Arches and Pauli, with the main systematic being prescribed clumping/turbulence. read the letter →

arxiv 2607.28012 v1 pith:Y2LMQKJ7 submitted 2026-07-30 astro-ph.SR astro-ph.GAastro-ph.HE

Gautham N. Sabhahit , Jorick S. Vink , Andreas A. C. Sander This is my paper · ORCID
classification astro-ph.SRastro-ph.GAastro-ph.HE
keywords verymassivestarsmasslossstellarwindsEddingtonparameterWolf-RayethydrodynamicalatmospheremodelsbistabilityO
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

The fates of the most massive stars are set largely by how much mass their radiatively driven winds remove. This paper builds a grid of 178 hydrodynamically consistent non-LTE wind models and turns the results into one continuous fitting formula for mass-loss rate that depends on the Eddington parameter, luminosity, temperature, and surface hydrogen abundance. The formula encodes a shallow-to-steep kink when winds become optically thick, plus two iron bistability features, and it matches the model-independent transition mass-loss rate in the Arches Cluster. Applied on the zero-age main sequence it also tracks recent empirical mass-loss versus Eddington trends. The result is a drop-in, empirically anchored recipe for stellar evolution calculations from roughly 20 to 500 solar masses at solar metallicity.

What carries the argument

A log-sum-exponential bridge between low- and high-Eddington power laws (the paper’s Eq. 5), fed by hydrodynamically consistent PoWR-HD solutions that predict mass-loss rate and velocity structure instead of assuming a prescribed beta-law.

What would settle it

Obtain independent mass-loss rates and Eddington parameters for Of/WNh transition stars in other young massive clusters; if the rates at the spectral thin-to-thick transition systematically miss log(Mdot) ≈ −5.15 at the matching luminosity, the absolute calibration fails.

Watch

Extended reading notes

Core claim

Across 178 PoWR-HD models at Z=0.02, mass loss versus classical Eddington parameter shows a kink: a shallow scaling near 2.8 for optically thin O-star winds becomes a steep scaling near 10 once winds are optically thick. The kink occurs where the flux-weighted optical depth at the sonic point is order unity and the wind efficiency is about 0.4. One continuous fit captures that kink, two iron ionisation bistability dips, and explicit luminosity and hydrogen-abundance scalings, and it recovers the Arches transition rate of about log(Mdot) = −5.2.

Load-bearing premise

Clumping and turbulent velocity are fixed by hand in every model rather than predicted, so the entire fitted recipe inherits those prescribed choices.

Editorial extensions

If this is right

  • Stellar evolution codes can replace separate O-star and VMS wind recipes with one continuous function over ~20–500 solar masses at Z=0.02.
  • Above the kink, steeper high-Eddington mass loss will change retained mass, black-hole masses, and pair-instability outcomes relative to older shallow recipes.
  • Below the transition the new rates lie below classic Vink et al. (2001) rates, reducing main-sequence mass loss for ordinary O stars.
  • Auxiliary relations convert surface Teff and high-T Eddington estimates into the inner-boundary quantities the fit needs, enabling direct use in structure codes.
  • The same grid supplies terminal velocities and ionising photon rates as secondary outputs across the parameter space.

Reading between the lines

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

  • Because the kink tracks sonic-point optical depth of order unity, lower-metallicity hydro grids should shift the kink in Eddington space and change the initial-mass threshold for WNh-type winds.
  • Once depth-dependent turbulence from multi-D envelope simulations is solved inside the same hydro code, absolute rates may move by tenths of a dex, especially at cool temperatures.
  • Agreement on the ZAMS with empirical Mdot–Γe slopes implies that, at solar metallicity, the dominant remaining uncertainty for very massive star endpoints is less the main-sequence wind law than binary mass transfer and post-main-sequence mass loss.
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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

3 major / 7 minor

Summary. The authors present a continuous theoretical mass-loss recipe for massive and very massive stars at Z=0.02, derived from a grid of 178 hydrodynamically consistent PoWR-HD non-LTE wind models spanning M⋆≈16–500 M⊙, log(L⋆/L⊙)=5.5–6.8, T⋆=12–50 kK, and X=0.01–0.9. They confirm a kink in the Ṁ–Γe relation (shallow slope ∼2.8 below the kink; steep ∼10 above), linked to τF,sonic∼1 and η∼0.4, and two Fe-driven bistability features near T⋆≈25 and 17 kK. These behaviours, plus explicit L⋆ and X scalings, are captured in a single fitting formula (Eq. 5) with auxiliary T⋆–Teff(τR=2/3) and incomplete-ionisation Γe corrections for evolutionary implementation. Absolute scale is tested against the model-independent Arches O-to-WNh transition mass-loss rate (predicted log Ṁ≈−5.15 vs ≈−5.2) and against the Pauli et al. (2025) empirical Ṁ–Γe trend on the ZAMS.

Significance. Mass loss is the dominant uncertainty for VMS evolution, pair-instability boundaries, and heavy black-hole formation. A hydrodynamically self-consistent, continuous recipe that joins optically thin O-star and optically thick WNh regimes—and is anchored to the Arches transition point rather than only to internal model normalisations—is of clear practical value for stellar evolution. Strengths include the large documented grid, explicit coefficients with uncertainties and quoted RMSE (0.12 dex on Ṁ), auxiliary implementation relations, and two external anchors (Arches transition; Pauli et al. 2025) not used to set the hydro solutions. Terminal-velocity and ionising-flux predictions add further utility. The work is a natural and useful extension of Paper I and of the η-framework of Sabhahit et al. (2022).

major comments (3)
  1. [§2, Appendix A, Eq. (5)] §2 and Appendix A: Every hydrodynamic solution (hence every coefficient in Eq. 5) is computed with prescribed micro-clumping (Dcl=1→10, onset τcl=0.1) and a fixed radially constant vturb=70.71 km s−1. Appendix A already shows that vturb variations can shift Ṁ at the ∼0.2 dex level (especially at cool T⋆). The Arches τF,sonic=1 anchor (Fig. 8) remains persuasive and is partly robust to vturb, but the manuscript does not quantify how changing the clumping law (factor or onset) would move the absolute Ṁ(Γe) curve or the high-Γe slope. For a recipe intended for evolution codes, please add a short, explicit systematic-uncertainty floor (or a limited clumping sensitivity test) so users know the precision claimed beyond the 0.12 dex fit RMSE.
  2. [§3.1, §3.4, Fig. 6] §3.3–3.4 and Fig. 6: The models systematically under-predict terminal velocities relative to Arches transition objects (η≈0.4 at the kink vs the η≈0.6 used in the Vink & Gräfener 2012 argument). The paper correctly notes that the recipe is parametrized in Γe rather than η, so Ṁ at the kink is still well matched. However, evolutionary applications often need both Ṁ and v∞ (wind momentum, mechanical feedback). Please state more clearly whether v∞ from the grid (or a recommended scaling) should be used with Eq. 5, and flag the v∞ under-prediction as a separate limitation rather than only as an explanation for low η.
  3. [§3.3, Table 2, §4.3] §3.3 and §4.3: Implementation requires iterating Eqs. (5), (6), and (9) from structure-code Teff(τR=2/3) and Γe,high-T. The text describes the procedure but does not demonstrate numerical stability or provide a minimal worked example (e.g., one ZAMS row from Table 2 with intermediate T⋆, Γe, and Ṁ iterates). Given the steep high-Γe branch, a 10% Γe error is stated to cost ∼0.5 dex in Ṁ; a short convergence note or pseudocode would make the “suitable for stellar evolution” claim load-bearing rather than aspirational.
minor comments (7)
  1. [Abstract, §5] Abstract and Conclusions quote suitability for 20–500 M⊙ (abstract) and 30–500 M⊙ (conclusions). Align the stated mass range.
  2. [Fig. 1] Fig. 1 caption refers to “best-fit relations described in Sect. 3.3” before that section; a forward reference is fine but ensure all solid-line sequences are uniquely identifiable against Table 1 sequence indices (as in Fig. 2).
  3. [§3.3, Eq. (5)] Eq. (5): T2 is written as depending on Γe/Γe,ref while T1 is fixed; a one-sentence physical motivation for allowing only the cooler dip centre to shift would help readers.
  4. [Fig. 9, §4.2] Fig. 9 vs Fig. 1: the apparent high-Γe slope difference at fixed Teff versus fixed T⋆ is important for users; consider a brief callout in the Fig. 9 caption pointing to the T⋆–Teff correction in §3.3.
  5. [Appendix C] Table C.1 is very long; if journal length is an issue, consider moving the bulk to CDS/Zenodo and retaining a representative subset plus a machine-readable full table statement in the text.
  6. [Throughout] Minor typography: “3 turb”, “3∞”, “PoWR hd/PoWRhd” appear in several spellings; unify notation for vturb, v∞, and PoWR-HD across text and tables.
  7. [§3.2, §4.2] References to Vink et al. (2026, submitted) and related in-press items should be updated or marked consistently at proof stage.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the recipe is an explicit fit to independent hydro models, externally checked against Arches and Pauli et al.

full rationale

The central product is a multi-parameter fit (Eq. 5) to 178 PoWR-HD hydrodynamic solutions in which Mdot and the velocity law are outputs of the equation of motion, not prescribed inputs. That fit describing its own training grid is ordinary recipe construction, not a circular derivation. Absolute scale is not forced by the Arches transition criterion: the models independently place τF,sonic≈1 and η≈0.4 at the kink and yield log Mdot(τF,sonic=1)≈−5.15 in the Arches L–T window (Fig. 8), which is then compared to the externally estimated transition rate ≈−5.2; the paper even reports a mild η mismatch (∼0.4 vs ∼0.6) rather than tuning to the Vink & Gräfener factor. ZAMS application uses MESA structure L and Teff and is compared to the independent Pauli et al. (2025) empirical Mdot–Γe relation without refitting. Clumping and vturb are fixed priors that set a systematic floor (App. A), which is a modeling assumption, not a definitional loop. Overlap with Vink & Gräfener (2012) and Paper I is normal sequential self-citation and does not make the hydro solutions or the external numerical anchors reduce to inputs by construction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The load-bearing content is a large grid of stationary non-LTE hydro wind models plus a phenomenological fit. Almost all physics (radiative driving, Fe line lists, micro-clumping, fixed vturb, solar-scaled Z=0.02, τR,cont=20 inner boundary) is inherited domain machinery; the free parameters are the numerous fit coefficients and the prescribed clumping/turbulence numbers. No new physical entity is postulated.

free parameters (4)
  • log Mdot_0 and kink/bistability coefficients in Eq. 5 = log Mdot_0=-5.448±0.021; fhigh slope=9.243±0.277; Γe,ref base 0.43; etc.
    Simultaneous fit of ~15 coefficients (log Mdot_0, flow/fhigh slopes, fL, fX, fT, A1,k1, A2,k2, T2 shift, Γe,ref terms, etc.) to the 178-model grid; RMSE 0.12 dex.
  • Clumping factor Dcl and onset τcl = Dcl,max=10, τcl=0.1
    Prescribed micro-clumping stratification Dcl=1→10 with onset at τcl=0.1, motivated by prior spectral fits but not varied in the main grid.
  • Turbulent velocity vturb = 70.71 km/s (main); α_min=0.062, α_diff=0.713, T0=47.8 kK
    Main grid fixes vturb=70.71 km/s; Appendix A fits a correction α(T*,Γe) from 44 extra models.
  • T*–Teff and Γe ionization correction coefficients = e.g. 0.542±0.012 in Γe correction; a,b coefficients in Eq. 6
    Auxiliary fits (Eqs. 6–9) mapping structure-code Teff and high-T Γe to the atmosphere quantities used in Eq. 5.
assumptions (6)
  • domain assumption Stationary, spherically symmetric, non-LTE CMF radiative transfer with hydro equation of motion solved self-consistently (PoWR-HD framework).
    §2; entire grid depends on this modelling paradigm versus time-dependent multi-D winds.
  • domain assumption Micro-clumping formalism adequately represents wind inhomogeneity for mass-loss prediction.
    §2; clumping fixed from prior R136/R144 fits rather than predicted.
  • domain assumption Iron-group line driving at Z=0.02 with Grevesse & Sauval (1998) scaled abundances dominates the wind force; recent solar O revision is negligible.
    §2; authors note a test with oxygen set to zero changes Mdot by only 0.03 dex.
  • domain assumption Classical Eddington parameter Γe (electron-scattering) is the primary organizing variable for Mdot, with secondary explicit L and X terms.
    §3.1; motivates the log-sum-exponential kink form.
  • domain assumption Transition mass-loss diagnostic η≈0.6 τF,sonic with τF,sonic~1 gives a nearly model-independent Mdot at the O/WNh boundary (Vink & Gräfener 2012).
    §1, §4.1; used as external absolute-scale benchmark.
  • ad hoc to paper Log-sum-exponential connection plus two Gaussian-like bistability dips in T* is a sufficient phenomenological form for the grid.
    Eq. 5; chosen to fit the models, not derived from a closed-form wind theory.

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Pith. "Pith review of Hydrodynamical mass-loss rates for very massive stars II. New theoretical mass-loss predictions at solar metallicity (Z = 0.02)." pith.science (2026). https://pith.science/paper/Y2LMQKJ7

@misc{pith2026260728012,
  author       = {Pith},
  title        = {Pith review of: Hydrodynamical mass-loss rates for very massive stars II. New theoretical mass-loss predictions at solar metallicity (Z = 0.02)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2LMQKJ7}},
  note         = {Machine review of arXiv:2607.28012}
}
abstract

The evolutionary pathways and ultimate fates of very massive stars are governed primarily by mass loss through radiatively-driven winds. We present a new theoretical mass-loss prescription for (very) massive stars, capturing the complex dependence on the Eddington parameter $\Gamma_e$, luminosity, temperature, and hydrogen abundance. We calculated an extensive grid of 178 hydrodynamically consistent wind-atmosphere models in non-local thermodynamic equilibrium using the PoWR-HD code, predicting wind properties such as the mass-loss rate and terminal velocity self-consistently. The grid spans masses $M_*$ = 16-500 Msun, luminosities $\log(L_*/L_\odot) = 5.5-6.8$, inner boundary temperatures $T_* = 12-50$ kK, and hydrogen mass fractions X = 0.01-0.9, at a fixed metallicity of Z=0.02. We confirm the presence of a mass-loss kink in the $\dot{M}-\Gamma_e$ relation across the explored parameter space. The kink marks the transition from a shallow scaling ($\sim 2.8$) at low $\Gamma_\mathrm{e}$ for optically thin O-star winds to a steeper scaling ($\sim 10$) for optically thick winds at high $\Gamma_e$. We derive comprehensive fitting relations capturing both the kink behaviour and two bistability jumps arising from iron ionisation changes, and provide auxiliary relations for implementation into stellar evolutionary calculations. Our prescription correctly reproduces the model-independent transition mass-loss rate in the Arches Cluster, confirming the accuracy of our predicted rates at the O-to-WNh transition. Application of our recipe to the Zero Age Main Sequence provides excellent agreement with recent empirical $\dot{M}-\Gamma_e$ relation obtained for a wide range of temperatures and Eddington parameters. We provide a physically motivated, continuous, and empirically anchored mass-loss recipe for (very) massive stars, suitable for stellar evolution calculations in the 20-500 Msun range.

Figures

Figures reproduced from arXiv: 2607.28012 by the authors.

Figure 1
Figure 1. Predicted mass-loss rates from PoWRhd models as a function of the classical Eddington parameter, Γe . We examine the effects of varying L⋆ and X while keeping the inner boundary temperature fixed at T⋆ = 35 kK for all the model sequences shown. (Top:) Model sequences in which the stellar mass, M⋆, is varied at fixed values of L⋆ (left panel) and X (right panel). (Bottom:) Model sequences in which L⋆ (left panel) and… view at source ↗
Figure 2
Figure 2. Wind efficiency parameter, η, and wind optical depth, τF,sonic, as a function of the classical Eddington parameter, Γe . All sequences from [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Predicted mass-loss rates from PoWRhd models as a function of the inner boundary temperature T⋆ for select-few model sequences from our grid. All stellar parameters are held fixed while T⋆ is varied. The stellar mass of each sequence is labelled in the plot, while the luminosity is given in the legend. The H mass fraction is fixed at X = 0.7. The dashed black lines indicate a reference M˙ ∝ T −6 ⋆ scaling. X have fu… view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: Percentage difference between the Γe predicted in the inner wind of our PoWRhd models (accounting for ionisation effects) and the high￾temperature limit Γe,high-T obtained by assuming complete H and He ion￾isation (Eq. 8). The solid red line is our best fit relation ac…
Figure 6
Figure 6. Figure 6: (Top:) Terminal velocity and (bottom:) terminal to escape ve￾locity ratio as a function of temperature T⋆ for all models in our grid. Individual symbols are colour-coded according to their mass-loss rate. The dashed red line marks where the flux-mean optical depth at t…
Figure 7
Figure 7. Figure 7: (Top:) H i and (bottom:) He ii ionising flux counts (in log of num￾ber of photons s−1 ) as a function of temperature T⋆ of all models in our grid. Individual symbols are colour coded according to their mass-loss rate. Article number, page 8 of 18 [PITH_FULL_IMAGE:figu…
Figure 8
Figure 8. Figure 8: Predicted mass-loss rate as a function of the flux-weighted op￾tical depth at the sonic point. All models with 5.7 < log(L⋆/L⊙) < 6.3 and 25 kK < Teff(τR = 2/3) < 35 kK are plotted regardless of surface H abundance, mass (and therefore Γe), or turbulent velocity. The r…
Figure 10
Figure 10. Figure 10: Comparison of our mass-loss predictions with oft-used massive star recipes as a function of Teff(τR. The stellar mass and luminosity are fixed to M⋆ = 50 M⊙ and log(L⋆/L⊙) = 5.6 respectively. 4.2. Comparison to other recipes An accurate reproduction of the transition …
Figure 11
Figure 11. Figure 11: Mass-loss prediction on the ZAMS as a function of Edding￾ton parameter Γe (black solid line with grey squares). The regression relation from Pauli et al. (2025) relation based on empirical masses is shown as a red dashed line. ing us to capture the complex Γ and tempe…

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Reviewed July 31, 2026 · model on record in the stance chip above.