The Star Formation Factory revisited I. The impact of metallicity on collapsing star-forming clouds

A. Srbljanovi\'c; D. Kom\'anek; J. Palou\v{s}; R. W\"unsch; S. Ehlerov\'a; S. Jim\'enez; S. Mart\'inez-Gonz\'alez

arxiv: 2604.00126 · v1 · submitted 2026-03-31 · 🌌 astro-ph.GA

The Star Formation Factory revisited I. The impact of metallicity on collapsing star-forming clouds

S. Jim\'enez , D. Kom\'anek , R. W\"unsch , J. Palou\v{s} , S. Ehlerov\'a , S. Mart\'inez-Gonz\'alez , A. Srbljanovi\'c This is my paper

Pith reviewed 2026-05-13 22:33 UTC · model grok-4.3

classification 🌌 astro-ph.GA

keywords star formationstellar feedbackmetallicitymolecular cloudsstar clustersradiative coolingstellar windssupernovae

0 comments

The pith

Metallicity regulates stellar feedback in collapsing clouds, raising star formation efficiency at low values through weaker winds and shell stalling.

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

The paper explores how the metal abundance of molecular clouds shapes the interplay between stellar winds, supernovae, and gas collapse. It finds that lower metallicity weakens stellar winds enough to offset poorer radiative cooling, allowing the cloud to form stars over longer periods and at higher overall efficiency. A stalling phase in the expanding shell appears over much of the explored parameter space at low metallicity but vanishes at higher values. This mechanism operates on par with cloud mass and radius in setting outcomes. The result bears on why star clusters show varied properties across different epochs in cosmic history.

Core claim

Metallicity acts as a key regulator of feedback, comparable in importance to cloud mass and radius. In low-metallicity clouds, reduced radiative cooling is offset by weaker stellar winds, leading to prolonged star formation and higher efficiencies. Across a substantial portion of parameter space, the expanding shell undergoes a stalling phase that further enhances the star formation efficiency, an outcome that is not observed at higher metallicities.

What carries the argument

one-dimensional spherical model of feedback-driven bubbles that tracks time-dependent energy and mass injection, self-gravity, radiative cooling with bubble-shell heat transfer, shell instabilities, and triggered star formation

If this is right

Low-metallicity clouds achieve higher star-formation efficiencies than high-metallicity clouds of equal mass and radius.
The expanding shell stalls over a wide range of low-metallicity initial conditions, extending the window for star formation.
Diverse star-cluster properties across cosmic time can arise from the metallicity-dependent balance between feedback and cooling.
Stellar-wind strength and radiative cooling compete directly as regulators of feedback strength.

Where Pith is reading between the lines

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

Early-universe galaxies with lower average metallicity may have converted gas into stars more efficiently than present-day systems.
Galaxy-evolution models that omit metallicity-dependent shell stalling will underpredict star-formation rates in metal-poor environments.
Targeted observations of star-formation efficiency in metal-poor versus metal-rich clouds of matched mass and size could test the predicted offset.
If the one-dimensional stalling phase survives in full three-dimensional geometry, it supplies a concrete target for high-resolution simulations.

Load-bearing premise

The one-dimensional spherical model with its specific treatment of heat transfer across the bubble-shell interface and shell instabilities accurately captures the essential physics of three-dimensional cloud collapse and feedback.

What would settle it

Three-dimensional simulations of the same collapsing clouds at low and high metallicity that show no stalling phase or efficiency increase at low metallicity would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.00126 by A. Srbljanovi\'c, D. Kom\'anek, J. Palou\v{s}, R. W\"unsch, S. Ehlerov\'a, S. Jim\'enez, S. Mart\'inez-Gonz\'alez.

**Figure 1.** Figure 1: Schematic of the model setup of this work. We model the evolution of feedback-driven bubbles, formed by the winds and supernovae from the star cluster forming at the center. The scheme presents a cone of the otherwise spherically symmetric bubble structure, showing the location of the reverse (Rrs) and forward shocks (Rsh), and the contact discontinuity (Rcd) which separates shocked wind from ambient swept… view at source ↗

**Figure 2.** Figure 2: Evolution of the mechanical power (top) and mass input rate (bottom) per unit solar mass of a stellar population with MW (dashed lines), dwarfA (dotted lines) and IZw18 (solid lines) metallicities as obtained using the Bonn Optimized Stellar Tracks (BoOST; Szécsi et al. 2022). Article number, page 3 of 20 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Time evolution of the shell radius (top left), shell mass (top right), stellar mass (bottom left) and thermal energy (bottom right) for the low metallicity (IZw18) models, that were performed for three different values of ϵf f (the first three models listed in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Model evolution for a case with log[Mgas,0(M⊙)] = 6.3, Rcl = 25.7 pc, ϵff = 0.3 and for the IZw18 metallicity. First plot from top to bottom: shell radius (left y-axis), shell velocity (right y-axis). Second: shell mass (Msh), stellar mass formed from core star formation (Msc,c), and total stellar mass (Msc). Third: the outward force produced by the thermal energy (U˙ th, dashed) and the total inward force… view at source ↗

**Figure 5.** Figure 5: Same as Fig. D.1, but for two cases with different cloud gas masses and radii, as shown in the inset (ϵff = 0.3 in both cases). Upper panel: time evolution of Lw (dashed lines) and Qw (solid lines). Bottom panel: Evolution of the shell radius with time for the same models, using the same color as in the top panel. ∼ 5 Myr, with final stellar masses ranging from 103 to 106 M⊙. The most massive clusters form… view at source ↗

**Figure 6.** Figure 6: Standing shell models for the IZw18 metallicity, plotted in the parameter space of the initial cloud radius and mass, with the subplots showing the different values of ϵff adopted for the calculations. Each model is color-coded with the duration of the StSh phase, ∆tStSh (see the color bar). the mechanical power. We verified numerically that, in the simulations, the average cooling rate approaches the mec… view at source ↗

**Figure 7.** Figure 7: The final stellar mass (Msc,final) obtained for IZw18 StSh models, as a function of the original cloud mass, Mgas,0. We plot with different symbols the three values of ϵff for these calculations (as indicated by the inset legend). Each model color shows the initial cloud radius, Rcl,0, as shown by the color bar. Finally, the dashed lines present three different values (as shown in the figure) of the integ… view at source ↗

**Figure 9.** Figure 9: Phase space diagram, showing the emission-weighted density (⟨ρhot⟩) and temperature (⟨Thot⟩) in the hot bubble. The size of the markers represents the feedback efficiency, defined here as ηfeed = Lw/Qw, and the colorbar shows the time. The upper, middle and bottom panel present IZW18, dwarfA and MW metallicity simulations of the same cloud with initial properties: log(Mgas,0[M⊙]) = 6.23, Rcl,0[pc] = 25.7,… view at source ↗

**Figure 10.** Figure 10: presents ϵint as a function of Σgas for all of our MW and the standing shell IZw18 models, as indicated by the inset labels. Note that our models indeed show a strong correlation between these variables, in agreement with the findings by Polak et al. (2024). However, our star formation efficiencies are a bit smaller than theirs, which is perhaps in better agreement with observations, such as in the case … view at source ↗

read the original abstract

Context. Stellar feedback regulates star formation and shapes the interstellar medium, yet its role during the collapse of molecular clouds remains uncertain over a wide range of initial conditions. Aims. We explore how stellar winds and supernovae influence star formation in collapsing gas clouds that span a broad parameter space in mass, size, and metallicity. Methods. Using a one-dimensional numerical model, we follow the evolution of feedback-driven bubbles produced by embedded clusters, incorporating time-dependent energy and mass injection, self-gravity, integrated cloud collapse, radiative cooling, shell instabilities, and triggered star formation. Our treatment of gas cooling in the hot bubble explicitly accounts for heat transfer across the bubble-shell interface. Results. We find that metallicity acts as a key regulator of feedback, comparable in importance to cloud mass and radius. In low-metallicity clouds, reduced radiative cooling is offset by weaker stellar winds, leading to prolonged star formation and higher efficiencies. Across a substantial portion of parameter space, the expanding shell undergoes a stalling phase that further enhances the star formation efficiency, an outcome that is not observed at higher metallicities. Conclusions. Our results suggest that the diverse properties of star clusters across cosmic time may arise from the metallicity-dependent interplay between stellar feedback and gas cooling.

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 1D model finds low metallicity raises star-formation efficiency via weaker winds offsetting poor cooling plus a shell-stalling phase, but the result hinges on spherical symmetry and sub-grid instability handling.

read the letter

The main thing to know is that this paper runs a 1D spherical code over a grid of cloud masses, radii, and metallicities and reports that low-Z clouds reach higher star-formation efficiencies. Weaker stellar winds let the cloud collapse longer even though radiative cooling is less effective, and the shell often stalls for a while, giving extra time for star formation. That stalling phase does not show up at higher metallicities. The survey itself is systematic and the equations are integrated directly from the initial conditions without fitting output parameters to the same runs, which keeps the logic straightforward. They include time-dependent energy and mass injection, self-gravity, integrated collapse, cooling in the hot bubble with explicit heat transfer to the shell, and an effective prescription for shell instabilities. Those choices let them trace how metallicity changes the feedback loop across a useful range of conditions. The clear limitation is the one-dimensional geometry. The stalling and efficiency trends depend on the shell staying coherent long enough for the reported phase to occur. In three dimensions Rayleigh-Taylor and Kelvin-Helmholtz modes can fragment the shell earlier, especially where cooling is inefficient, and the mixing that controls heat flux would be different. The paper's sub-grid treatment of instabilities and the interface heat transfer may or may not reproduce the net momentum coupling once those modes are resolved. Without resolution tests or side-by-side 3D runs shown, it is hard to judge how much the metallicity dependence would shift. This work is aimed at people who need simple, physically motivated prescriptions for star-formation efficiency in galaxy-formation models at varying redshifts. A reader focused on detailed hydrodynamics would want 3D validation before using the numbers. I would send it to peer review. The parameter study is broad enough to be worth referee time, provided the authors address the dimensionality question and show sensitivity checks on the instability and heat-transfer pieces.

Referee Report

2 major / 2 minor

Summary. The paper presents a one-dimensional spherical numerical model that evolves feedback-driven bubbles in self-gravitating, collapsing molecular clouds. It incorporates time-dependent energy and mass injection from stellar winds and supernovae, radiative cooling with explicit heat transfer across the bubble-shell interface, an effective treatment of shell instabilities, and triggered star formation. Across a grid of initial cloud masses, radii, and metallicities, the central claim is that metallicity regulates feedback comparably to mass and radius: low-Z clouds exhibit weaker winds that offset reduced cooling, producing prolonged star formation, higher efficiencies, and a stalling shell phase absent at higher metallicities.

Significance. If the 1D results prove robust, the work supplies a computationally tractable framework for mapping how metallicity modulates the coupling between stellar feedback and cloud collapse, offering a plausible explanation for the diversity of star-cluster properties across cosmic time. The explicit inclusion of heat transfer and instability prescriptions, together with the broad parameter survey, constitutes a useful step beyond purely analytic or fixed-efficiency models.

major comments (2)

[§2] §2 (numerical model) and the description of the bubble-shell interface: the effective sub-grid prescription for shell instabilities and the heat-transfer term across the interface are load-bearing for the reported stalling phase at low Z. No direct calibration or comparison against 3D hydrodynamical simulations that resolve Rayleigh-Taylor and Kelvin-Helmholtz growth is presented, leaving open whether spherical symmetry plus the adopted sub-grid model reproduces the net momentum and energy coupling once 3D fragmentation occurs.
[Results] Results section (discussion of stalling phase): the claim that the expanding shell stalls and thereby boosts SFE over a substantial fraction of parameter space at low metallicity rests entirely on the 1D integration. Without an explicit test showing that the stalling survives when the instability growth rates are varied (e.g., by changing the effective viscosity or mixing coefficient), the metallicity-dependent offset between weaker winds and reduced cooling cannot be regarded as demonstrated.

minor comments (2)

[Abstract] The abstract states that metallicity is 'comparable in importance to cloud mass and radius' but does not quantify the relative effect sizes; a brief statement of the fractional change in SFE per dex in Z versus per dex in M or R would help readers assess the claim.
[Figures] Figure captions and axis labels should explicitly note the adopted solar metallicity value and the range of Z explored so that readers can immediately map the plotted trends onto physical units.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed report. The comments correctly identify the central role of our sub-grid prescriptions and the reliance on the 1D framework. We respond to each major comment below and indicate the changes planned for the revised manuscript.

read point-by-point responses

Referee: [§2] §2 (numerical model) and the description of the bubble-shell interface: the effective sub-grid prescription for shell instabilities and the heat-transfer term across the interface are load-bearing for the reported stalling phase at low Z. No direct calibration or comparison against 3D hydrodynamical simulations that resolve Rayleigh-Taylor and Kelvin-Helmholtz growth is presented, leaving open whether spherical symmetry plus the adopted sub-grid model reproduces the net momentum and energy coupling once 3D fragmentation occurs.

Authors: We agree that direct 3D validation would be the strongest test. Our sub-grid terms for Rayleigh-Taylor and Kelvin-Helmholtz growth follow standard analytic growth-rate expressions (with an effective mixing coefficient) that have been adopted in earlier 1D superbubble models; the heat-transfer term is computed from the local temperature gradient at the interface. The 1D spherical geometry is chosen precisely to enable the wide parameter survey that constitutes the main result. In the revision we will expand the description in §2 with explicit references and derivations for each sub-grid component, and we will add a dedicated limitations paragraph in the discussion that quantifies the possible impact of 3D fragmentation on net momentum coupling and on the duration of the stalling phase. revision: partial
Referee: [Results] Results section (discussion of stalling phase): the claim that the expanding shell stalls and thereby boosts SFE over a substantial fraction of parameter space at low metallicity rests entirely on the 1D integration. Without an explicit test showing that the stalling survives when the instability growth rates are varied (e.g., by changing the effective viscosity or mixing coefficient), the metallicity-dependent offset between weaker winds and reduced cooling cannot be regarded as demonstrated.

Authors: We accept that an explicit sensitivity test is required to substantiate the robustness of the stalling phase. Although the submitted manuscript reports only the fiducial mixing coefficient, we will rerun the low-metallicity grid with the mixing coefficient varied by factors of 0.5 and 2.0. The outcomes of these runs will be presented in a new appendix (or subsection) of the revised manuscript, demonstrating that the stalling phase and the associated elevation in star-formation efficiency persist across this range and thereby confirming that the metallicity-dependent offset is not an artifact of the precise instability parameter choice. revision: yes

standing simulated objections not resolved

Direct calibration or comparison against 3D hydrodynamical simulations that resolve Rayleigh-Taylor and Kelvin-Helmholtz growth

Circularity Check

0 steps flagged

No significant circularity; results follow from forward numerical integration

full rationale

The paper computes results via direct numerical integration of a one-dimensional spherical model whose equations (time-dependent energy/mass injection, self-gravity, radiative cooling, heat transfer across the bubble-shell interface, shell instabilities, and triggered star formation) are evolved forward from specified initial conditions in mass, radius, and metallicity. No output quantity is defined in terms of a parameter fitted to the same run's data, nor is any prediction constructed by renaming or re-using an input. Self-citations to prior work, if present, are not load-bearing for the central claims because the model equations and integration procedure stand independently and are falsifiable against external benchmarks. The reported metallicity-dependent stalling phase and efficiency trends therefore emerge from the physics implementation rather than from any definitional or self-referential loop.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The model rests on standard astrophysical assumptions for cooling functions and wind scaling with metallicity. No new physical entities are introduced. Free parameters are the initial cloud mass, radius, and metallicity values that are scanned across the explored space.

free parameters (3)

initial cloud mass
Varied as a primary input parameter across the studied range.
initial cloud radius
Varied as a primary input parameter across the studied range.
metallicity
Key varied parameter whose effect on cooling and wind strength drives the central claim.

axioms (2)

domain assumption One-dimensional spherical symmetry is sufficient to capture the essential dynamics of bubble expansion and cloud collapse.
Required by the choice of a 1D numerical model.
domain assumption Radiative cooling rates and stellar wind luminosities scale with metallicity according to the implemented prescriptions.
Underlies the claimed offset between cooling and wind strength at low metallicity.

pith-pipeline@v0.9.0 · 5563 in / 1495 out tokens · 62312 ms · 2026-05-13T22:33:07.199242+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Using a one-dimensional numerical model, we follow the evolution of feedback-driven bubbles... incorporating time-dependent energy and mass injection, self-gravity, integrated cloud collapse, radiative cooling, shell instabilities... Our treatment of gas cooling in the hot bubble explicitly accounts for heat transfer across the bubble-shell interface.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

We find that metallicity acts as a key regulator of feedback... reduced radiative cooling is offset by weaker stellar winds, leading to prolonged star formation and higher efficiencies... the expanding shell undergoes a stalling phase

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

4 extracted references · 4 canonical work pages · 1 internal anchor

[1]

The evolution of velocity dispersion in the Sco-Cen OB association

Adamo, A., Bradley, L. D., V anzella, E., et al. 2024, Nature, 632, 513 Airapetian, V ., Carpenter, K. G., & Ofman, L. 2010, ApJ, 723, 1210 Andersson, E. P ., Mac Low, M.-M., Agertz, O., Renaud, F., & Li, H. 2024, A&A, 681, A28 Bisnovatyi-Kogan, G. S. & Silich, S. A. 1995, Reviews of Modern Physics, 67, 661 Cantó, J., Raga, A. C., & Rodríguez, L. F. 2000,...

work page internal anchor Pith review Pith/arXiv arXiv 2024
[2]

is estimated following Weaver et al. (1977). Indeed, for each time step (t), the radial velocity and temperature proﬁles within the hot bubble, speciﬁcally in the region between the reverse shock (Rrs) and the outer shell radius ( Rsh), Rrs ≤ x ≤ Rsh, are the solution to the equations: 1 x2 ∂ ∂x ( x2v ) − ( v − αx t ) 1 T ∂T ∂x = β + δ t , (A.1) 1 Pth x2 ...

work page 1977
[3]

(A.5) Here we describe the method developed to estimate the cool- ing rate in the hot bubble

and: α = d ln Rsh d ln t , (A.3) β = − d ln Pth d ln t , (A.4) and δ = d ln T d ln t . (A.5) Here we describe the method developed to estimate the cool- ing rate in the hot bubble. Equations A.1–A.2 are integrated nu- merically at every time step, to ﬁnd the velocity and temperature proﬁles within the hot wind bubble, i.e., for Rrs ≤ x ≤ Rsh. The thermal ...

work page 1977
[4]

This implies that the remaining poles of Equation C.19 always have negative real parts

(C.26) These inequalities always hold for our models in any case, given that vcl < 0 in our calculations. This implies that the remaining poles of Equation C.19 always have negative real parts. Thus, the homogeneous component of the system (given by Equation C.9) is always marginally stable. Appendix C.3: On the stability of the complete system In the las...

work page 2023

[1] [1]

The evolution of velocity dispersion in the Sco-Cen OB association

Adamo, A., Bradley, L. D., V anzella, E., et al. 2024, Nature, 632, 513 Airapetian, V ., Carpenter, K. G., & Ofman, L. 2010, ApJ, 723, 1210 Andersson, E. P ., Mac Low, M.-M., Agertz, O., Renaud, F., & Li, H. 2024, A&A, 681, A28 Bisnovatyi-Kogan, G. S. & Silich, S. A. 1995, Reviews of Modern Physics, 67, 661 Cantó, J., Raga, A. C., & Rodríguez, L. F. 2000,...

work page internal anchor Pith review Pith/arXiv arXiv 2024

[2] [2]

is estimated following Weaver et al. (1977). Indeed, for each time step (t), the radial velocity and temperature proﬁles within the hot bubble, speciﬁcally in the region between the reverse shock (Rrs) and the outer shell radius ( Rsh), Rrs ≤ x ≤ Rsh, are the solution to the equations: 1 x2 ∂ ∂x ( x2v ) − ( v − αx t ) 1 T ∂T ∂x = β + δ t , (A.1) 1 Pth x2 ...

work page 1977

[3] [3]

(A.5) Here we describe the method developed to estimate the cool- ing rate in the hot bubble

and: α = d ln Rsh d ln t , (A.3) β = − d ln Pth d ln t , (A.4) and δ = d ln T d ln t . (A.5) Here we describe the method developed to estimate the cool- ing rate in the hot bubble. Equations A.1–A.2 are integrated nu- merically at every time step, to ﬁnd the velocity and temperature proﬁles within the hot wind bubble, i.e., for Rrs ≤ x ≤ Rsh. The thermal ...

work page 1977

[4] [4]

This implies that the remaining poles of Equation C.19 always have negative real parts

(C.26) These inequalities always hold for our models in any case, given that vcl < 0 in our calculations. This implies that the remaining poles of Equation C.19 always have negative real parts. Thus, the homogeneous component of the system (given by Equation C.9) is always marginally stable. Appendix C.3: On the stability of the complete system In the las...

work page 2023