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arxiv: 2606.21878 · v1 · pith:ITCIRKKVnew · submitted 2026-06-20 · 🌌 astro-ph.SR

The efficiency per free-fall time as a ratio of the Star Formation Rate to the gas-infall rate in collapsing cores: dependence on the core definition, accretion, and radial structure

Fabi\'an Quesada-Z\'u\~niga , Manuel Zamora-Avil\'es , Enrique V\'azquez-Semadeni , Gilberto C. G\'omez , Aina Palau , Javier Ballesteros-Paredes This is my paper

Pith reviewed 2026-06-26 11:50 UTC · model grok-4.3

classification 🌌 astro-ph.SR
keywords efficiency per free-fall timestar formation ratemolecular cloud coresgravitational collapsenumerical simulationsgas accretioncore definition
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The pith

The efficiency per free-fall time in collapsing cores depends sensitively on core definition and external gas supply.

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

The paper examines the quantity called efficiency per free-fall time, which is formally the ratio of the star formation rate to the gas-infall rate into a core. It performs simulations of isolated core collapse with different initial densities and with closed or open boundaries that control whether fresh gas can enter. The measured value changes with the density threshold chosen to mark the core edge and with whether mass can accrete from outside. This matters because the quantity is used to characterize star formation activity in molecular clouds, so its dependence on these choices affects how simulation results and observations are compared.

Core claim

In simulations that begin with a central Gaussian overdensity and evolve toward a density profile n proportional to r to the minus two, a sink forms at the center. After sink formation the efficiency per free-fall time rises and then stays relatively stable while accretion continues to replenish core mass, but increases once the reservoir is exhausted. The value is higher in low-mass cores because the larger infall rates onto high-mass cores offset their higher star formation rates. Both the boundary conditions and the density threshold used to define the core alter the inferred efficiency.

What carries the argument

The ratio defined as average star formation rate divided by (core mass divided by free-fall time), where core mass is the gas mass above a chosen density threshold.

If this is right

  • After a sink particle forms, the efficiency per free-fall time becomes stable during ongoing accretion but rises sharply once the gas reservoir is depleted.
  • Low-density cores are more sensitive to boundary conditions than high-density cores because they have smaller mass reservoirs.
  • The efficiency per free-fall time ends up higher in low-mass cores than in high-mass cores because higher infall rates onto the latter compensate for their higher star formation rates.
  • Changing the density threshold that defines the core directly changes the measured core mass and therefore the inferred efficiency.

Where Pith is reading between the lines

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

  • Observed differences in efficiency per free-fall time between cores could partly reflect differences in how the cores are bounded or supplied rather than differences in intrinsic star formation physics.
  • The power-law radial structure that develops during collapse directly sets the infall rate entering the efficiency calculation, so any process that alters that structure would change the measured value.

Load-bearing premise

That defining the core as all gas above a density threshold in simplified isolated simulations with Gaussian initial overdensity captures the physics needed to measure the efficiency in real molecular cloud cores.

What would settle it

Measuring the efficiency in a set of simulations that vary only the density threshold while keeping all other parameters fixed and checking whether the predicted change in the ratio matches the change seen when the same threshold variation is applied to observed core samples with independent accretion-rate estimates.

Figures

Figures reproduced from arXiv: 2606.21878 by Aina Palau, Enrique V\'azquez-Semadeni, Fabi\'an Quesada-Z\'u\~niga, Gilberto C. G\'omez, Javier Ballesteros-Paredes, Manuel Zamora-Avil\'es.

Figure 1
Figure 1. Figure 1: The density profile (measured as the mean density in thin annuli at each radius) for each simulation is presented, highlighting five different moments in time: one immediately after sink formation (green dot-dashed lines), along with two time points preceding and two following this time. Each panel shows an 𝑟 −2 profile (solid black line), with a dashed horizontal line indicating the threshold number densi… view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of the star formation rate (𝑀¤ ★) for each simulation, which is represented with different colors: red for the C100 simulation, green for the O100 simulation, blue for the C1000 simulation and black for the O1000 simulation. stable even for the lowest density threshold, yielding SFE values of SFE ≲ 0.4. For the highest threshold (𝑛thr = 105 cm−3 ), the core became more compact and its mass became… view at source ↗
Figure 3
Figure 3. Figure 3: Core mass and SFE evolution (left and right vertical labels, respectively) for each simulation. Here, we show three different core definitions as follows: the core with 𝑛thr = 103 cm−3 with dotted lines, the core with 𝑛thr = 104 cm−3 with dashed lines, and the core with 𝑛thr = 105 cm−3 with solid lines . The top left panel corresponds to the simulation labeled as C100, the top right panel to O100, the bott… view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the ratio of the SFR to the gas-infall rate (𝜖ff) for each simulation. As in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Profiles of the ratio of the SFR to the gas-infall rate (𝜖ff) for each simulation. The top left panel corresponds to the simulation labeled as C100, the top right panel to O100, the bottom left panel to the C1000, and the bottom right panel to the O1000. The dotted colored lines depict the expected slopes for the indicated values of the density-profile slope, 𝑝. ous times in each simulation, as resulting f… view at source ↗
read the original abstract

A parameter used to characterise star formation activity in MCs is the efficiency per free-fall time, $\epsilon_{\rm ff}$, although commonly referred to as an efficiency, it is formally the ratio between the star formation rate (SFR) and the gas-infall rate. Here we numerically study the collapse of cores and define $\epsilon_{\rm ff}\equiv\langle\dot{M}_\star\rangle/(M_{\rm core}/\tau_{\rm ff})$, where $\langle\dot{M}_\star\rangle$ is the average SFR, $M_{\rm core}$ is the gas mass within the core (as the gas cells above a density threshold), and $\tau_{\rm ff}$ is the free-fall time of the core gas. We perform simplified numerical experiments of the gravitational collapse of an isolated core, varying the initial mean number density ($n_0=100$ and $1000~\rm cm^{-3}$) and adopting closed/open BCs to (dis)allow fresh gas accretion into the domain. The simulations start with a slight central Gaussian overdensity that evolved into a power-law profile, $n\propto r^{-p}$ with $p\to2$. As the collapse proceeds, a sink particle forms in the center of the core. We find that both the BCs and the adopted core definition modify the measured core properties and, consequently, the inferred $\epsilon_{\rm ff}$. Low-density models have less mass available, and their accretion histories are therefore much more sensitive to the choice of BCs, while high-density runs, with their larger mass reservoirs, maintain similar accretion histories regardless of the BCs. In all models, after sink formation, $\epsilon_{\rm ff}$ rises and then remains relatively stable while accretion continues to replenish the core's mass, but increases once the gas reservoir is exhausted. Somewhat counterintuitively, $\epsilon_{\rm ff}$ is higher in the low-mass cores, since the larger gas infall rates onto the high-mass cores compensate for their higher SFR. We conclude that the inferred $\epsilon_{\rm ff}$ depends sensitively on both the adopted core definition and external mass supply

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.

Referee Report

2 major / 2 minor

Summary. The manuscript reports controlled numerical experiments on the gravitational collapse of isolated cores initialized with a central Gaussian overdensity that evolves to a power-law density profile. It defines ε_ff ≡ ⟨Ṁ⋆⟩ / (M_core / τ_ff), where M_core is the mass of gas cells above a chosen density threshold, and examines how the measured ε_ff responds to initial mean density (n0 = 100 or 1000 cm^{-3}), closed versus open boundary conditions, and the density threshold adopted for the core. The central result is that ε_ff depends sensitively on both the core definition and the presence of external mass supply, with higher values found in lower-mass cores because their lower infall rates are not offset by proportionally lower SFRs.

Significance. If the reported sensitivities hold, the work supplies a concrete, internally consistent demonstration that ε_ff inferred from simulations is not robust to standard modeling choices. This is useful for the field because it supplies a controlled test of how core definition and mass reservoir affect the ratio of SFR to gas-infall rate, directly addressing a quantity widely used to compare simulations and observations. The use of clearly defined, reproducible numerical setups with no free parameters fitted to data is a methodological strength.

major comments (2)
  1. [Abstract / Results] Abstract and results section: the claim that ε_ff is higher in low-mass cores because 'larger gas infall rates onto the high-mass cores compensate for their higher SFR' is the key counter-intuitive result; however, the manuscript does not report the actual numerical values of ⟨Ṁ⋆⟩, M_core, and τ_ff (or their ratios) for the different n0 and BC cases, making it impossible to verify the magnitude of the compensation or to judge whether the difference is statistically significant given the sink-particle accretion implementation.
  2. [Methods] Methods: the core is defined as all gas cells above a chosen density threshold, yet the specific threshold values, the resulting M_core(threshold) curves, and the time evolution of the enclosed mass are not tabulated or plotted; without these data the statement that 'both the BCs and the adopted core definition modify the measured core properties' cannot be assessed quantitatively.
minor comments (2)
  1. [Results] The evolution of the density profile to n ∝ r^{-p} with p → 2 is stated but not illustrated; a figure showing radial profiles at several times before and after sink formation would clarify how the power-law index is measured and whether it is the same across the n0 and BC runs.
  2. [Abstract / Methods] Notation: the symbol ⟨Ṁ⋆⟩ is introduced as an average SFR, but the time window over which the average is taken (e.g., from sink formation to a fixed multiple of τ_ff) is not stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and recommendation of minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract / Results] Abstract and results section: the claim that ε_ff is higher in low-mass cores because 'larger gas infall rates onto the high-mass cores compensate for their higher SFR' is the key counter-intuitive result; however, the manuscript does not report the actual numerical values of ⟨Ṁ⋆⟩, M_core, and τ_ff (or their ratios) for the different n0 and BC cases, making it impossible to verify the magnitude of the compensation or to judge whether the difference is statistically significant given the sink-particle accretion implementation.

    Authors: We agree that explicit numerical values are needed to substantiate the compensation claim. The revised manuscript will add a table reporting the time-averaged ⟨Ṁ⋆⟩, M_core, τ_ff and resulting ε_ff (with standard deviations) for every n0–BC combination. This will permit direct verification of the magnitude of the effect and assessment of its robustness given the sink implementation. revision: yes

  2. Referee: [Methods] Methods: the core is defined as all gas cells above a chosen density threshold, yet the specific threshold values, the resulting M_core(threshold) curves, and the time evolution of the enclosed mass are not tabulated or plotted; without these data the statement that 'both the BCs and the adopted core definition modify the measured core properties' cannot be assessed quantitatively.

    Authors: We acknowledge that quantitative support for the sensitivity to core definition is currently insufficient. The revision will include (i) explicit statement of the density thresholds used in each model and (ii) a new figure displaying M_core versus threshold at representative times together with the time evolution of enclosed mass for the adopted thresholds under both boundary conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper performs numerical hydrodynamical experiments of isolated core collapse under varying initial densities and boundary conditions, then directly computes ε_ff from the simulation outputs using the explicit definition ε_ff ≡ ⟨Ṁ_star⟩ / (M_core / τ_ff), where all quantities are measured quantities extracted from the runs. The reported sensitivity of ε_ff to core density-threshold definition and to open/closed boundaries follows immediately from those measurements; no parameter is fitted to data and then re-labeled as a prediction, no uniqueness theorem or ansatz is imported via self-citation, and the central claim is scoped to the internal behavior of the chosen numerical setups rather than asserted as an external physical law. The derivation chain is therefore self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of self-gravitating hydrodynamics and the free-fall time definition; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption Gravitational collapse of an isolated core can be modeled with standard hydrodynamics and a sink particle for the central star.
    Invoked throughout the numerical experiments described in the abstract.
  • domain assumption Free-fall time τ_ff is computed from the mean density of gas cells above the chosen threshold.
    Used in the explicit definition of ε_ff.

pith-pipeline@v0.9.1-grok · 5986 in / 1291 out tokens · 15156 ms · 2026-06-26T11:50:18.380079+00:00 · methodology

discussion (0)

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