Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

An analytic formalism to describe the $N_{\rm eff}(\rm H)$-$n_{\rm H}$ relationship in molecular clouds

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper derives a closed-form expression for the effective hydrogen column density that attenuates FUV radiation inside molecular clouds, as a function of local gas density, and shows the formula reproduces simulation-based fits.

desk verdict A useful analytic prescription for Neff-nH, but the turbulent column is a hand-made ansatz that needs testing before the central claim is fully supported. read the letter →

arxiv 2507.16931 v2 pith:WXZ3WZCD submitted 2025-07-22 astro-ph.GA

Brandt A. L. Gaches This is my paper · ORCID
classification astro-ph.GA
keywords ISM:generalstructurecloudsastrochemistrymoleculareffectivecolumndensityFUVshieldingturbulencelengthscales
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

Astrochemical models need to know how much gas lies between a point inside a molecular cloud and the external ultraviolet radiation; ray tracing through 3D simulations is expensive, so 0D models often set this effective column density by hand. This Letter proposes a two-component analytic formula, $N_{\rm eff}(n_{\rm H}) = N_{\rm turb} + S_n N_J$, in which a turbulence-dominated envelope column and a gravitational (Jeans) column are added with a smooth transition. With fiducial Milky Way-like cloud parameters the formula reproduces the previously fitted simulation relation within a factor of two and gives high-density power-law slopes $\gamma \approx 0.4$–$0.5$, consistent with ray-traced simulations. The intended payoff is a physically motivated replacement for a free parameter in chemical models and a sub-grid prescription for simulations that skip radiative transfer.

What carries the argument

The load-bearing object is a density-dependent effective length scale $\ell_{\rm turb}$ built from the virial parameter and the linewidth–size relation, limited by the cloud radius, so that $N_{\rm turb}\approx(\tfrac12 L_{\rm cl} - \tfrac12 \bar\ell)\, n_0$; the companion object is the Jeans half-column $N_J = \tfrac12\lambda_J n_{\rm H}$. A transition density $n_T$, derived by equating the Jeans length with the turbulence sonic scale, feeds a soft threshold $S_n = (n_{\rm H}/n_T)/(1+n_{\rm H}/n_T)$ that blends the two regimes. The model's shape is controlled by six cloud parameters: size, temperature, mean density, virial parameter, velocity-dispersion normalization, and linewidth–size index.

What would settle it

Take a three-dimensional molecular cloud simulation with known size, mean density, temperature, virial parameter, and linewidth–size slope; ray-trace the effective column for an isotropic field, bin by local density, and compare with $N_{\rm turb} + S_n N_J$ using the same parameters. The model is falsified if the low-density portion deviates by more than the factor-of-two band quoted against the fiducial fit, or if the measured high-density slope in $10^3$–$10^5\ \mathrm{cm^{-3}}$ lies outside roughly $\gamma = 0.3$–$0.6$.

Watch

Extended reading notes

Core claim

The central claim is that the volume-averaged relation between effective shielding column and local density is not a purely numerical input but is set by two competing length scales: the depth of the turbulent envelope a parcel sits in, controlled by the virial parameter and the linewidth–size relation, and the Jeans length of the gravitationally bound gas around it. The paper writes this as $N_{\rm eff}(n_{\rm H}) = N_{\rm turb}(n_{\rm H}) + S_n(n_{\rm H}/n_T)\, N_J(n_{\rm H})$, with $N_J = \tfrac12 \lambda_J n_{\rm H} \propto c_s n_{\rm H}^{1/2}$ in isothermal gas, a smoothed step $S_n$ that turns gravity on above a transition density $n_T$, and an optional resolution floor $N_{\Delta x}$. Against the fiducial simulation fit the model lands within a factor of two at intermediate densities, and its logarithmic slope in the dense regime is consistent with the $\gamma \approx 0.4$–$0.5$ power laws extracted from ray-traced simulations; the asymptotic Jeans-only slope is exactly $1/2$, with the fitted slope flattened by the surviving turbulent contribution.

Load-bearing premise

The load-bearing premise is that the effective shielding column is controlled by the shortest path through a turbulent envelope whose depth is fixed by the local density through the virial parameter and the linewidth–size relation; if other directions contribute substantially, or if local density does not set the envelope depth, the low- and intermediate-density part of the relation fails.

Editorial extensions

If this is right

  • Zero-dimensional chemical models can replace a free shielding parameter with a value computed from the local density and a few cloud-scale parameters.
  • Hydrodynamic simulations that cannot afford ray tracing can use the formula as a sub-grid estimate of FUV attenuation; the error from ignoring the full directional average shrinks at high density, where the spread in $A_V$ at fixed density is small.
  • The slope of the $N_{\rm eff}$–$n_{\rm H}$ relation asymptotes to the isothermal Jeans value $1/2$ at high density, yet a log-log power-law fit biased by the turbulent component lands near $0.4$, explaining the simulation-derived indices.
  • Cloud size and mean ambient density shift the whole column, temperature controls the Jeans contribution, and the virial parameter has little net effect because it changes the turbulent length and the transition density in opposite directions.

Reading between the lines

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

  • Because the transition density is tied to the sonic scale, the model predicts that the density where the $N_{\rm eff}$–$n_{\rm H}$ slope bends should shift with Mach number; this is a testable trend in existing ray-traced simulations.
  • Given an observed column-density PDF and cloud depth, the relation could be inverted to estimate the effective turbulent envelope column, offering an observational route to the same quantity.
  • The model's 'minimum column out of the cloud dominates' assumption implies that anisotropic or clumpy boundary illumination would break the predicted relation; a targeted comparison against directional ray tracing in non-spherical clouds would bound the error.
  • Extensions to non-isothermal equations of state would change the Jeans column's density exponent away from $1/2$; the paper's own variable-sound-speed runs indicate the main corrections appear at low and intermediate densities.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a closed-form analytic model for the mean effective hydrogen-column density N_eff(H) that attenuates an external isotropic FUV field, expressed as a function of local hydrogen-nuclei volume density n_H. The model combines a turbulence-dominated contribution N_turb = (R_cl - (1/2) \bar{\ell}) n_0, a gravitationally dominated Jeans contribution N_J,T = S_n(n_H/n_T) (1/2) \lambda_J n_H, and an optional numerical-resolution floor N_\Delta x = C_\Delta x n_H \Delta x. The Jeans component follows from an isothermal Jeans-length estimate, while the turbulent component uses the virial parameter and a linewidth-size relation to define a density-dependent length scale. The fiducial model is compared to the analytic fit of Bisbas et al. (2023, B23), with a 0.05 mag floor imposed for the low-density comparison, and to published high-density power-law slopes \gamma \approx 0.4-0.5 from ray-traced simulations. The central claim is that the model reproduces the simulation-based N_eff(H)-n_H relation and can serve as a subgrid prescription for astrochemical models.

Significance. If the central claim holds, the model would give a fast, physically motivated replacement for ray-tracing in 0D and subgrid astrochemical applications. A genuine strength is the Jeans component: Equation (9) is a clean, essentially parameter-free derivation in the isothermal high-density limit and correctly produces the asymptotic slope 0.5. The parameter study in Section 3 and Appendix B also provides a useful map of how cloud-scale inputs such as L, T, n_0, \alpha_V, \sigma_0, and \beta affect the relation. The comparison to the B23 fit and to published slopes is a good first step, but the central claim currently rests on an unvalidated geometric ansatz in Equation (8), on a post-hoc numerical floor in Section 2.4, and on comparison to fits rather than to direct simulation data. With targeted validation, the paper could be a useful contribution; in its present form the load-bearing turbulent term needs additional support.

major comments (4)
  1. [§2.1, Eq. (8)] The geometric ansatz \ell_turb = R_cl - (1/2)\bar{\ell} is asserted rather than derived from the ray-averaging definition in Eq. (1). It assumes that a parcel at local density n_H sits at a characteristic depth R_cl - (1/2)\bar{\ell} below the cloud surface, so that the lowest column-density direction out of the cloud is determined by the turbulence length scale. This mapping drives the low- and intermediate-density behavior of the model, and the high-density slope flattening from 0.5 to \sim0.4 inherits the constant turbulent contribution n_0 R_cl. The paper provides no test of this mapping against ray-traced simulations, and the statement that the effective column is most sensitive to the lowest column density out of the cloud does not by itself justify the specific form of Eq. (8). Please add either a derivation from the ray-averaging integral or a direct numerical test using a simulation with ray-traced N_eff values.
  2. [§2.3, §2.4, and Fig. 2] The claimed agreement with B23 through the curve labeled "Total w/ biases" depends on a resolution limit N_\Delta x and on a 0.05 mag floor, but the adopted values of \Delta x and C_\Delta x are not stated anywhere in the text, Table 1, or figure caption. Moreover, the floor is introduced specifically "for comparison to B23", and the B23 fit in Eq. (2) has a low-density asymptote of exactly 0.05 mag. Thus part of the low-density agreement is built into the model rather than predicted from its physical inputs. Please report the fiducial \Delta x and C_\Delta x values, and discuss whether the 0.05 mag floor is a physical dust-extinction floor, a numerical floor inherited from the simulations, or an imposed matching parameter.
  3. [§2.2 and Appendix B] The softening threshold S_n(z_n) = z_n/(1+z_n) and the damping function S_\ell(z_\ell) = erf((\sqrt{\pi}/2) z_\ell) are chosen by hand and are not derived from turbulence or gravitational instability theory. The transition region, where the slope changes from superlinear to \sim0.4-0.5, is controlled by S_n and by the pairing of the two components, so the high-density slope comparison in Fig. B.2 is partly sensitive to these ad hoc choices. The manuscript does not test the robustness of the predicted slope to alternative functional forms. Please either justify these forms from a physical argument or show that the main conclusions are insensitive to the specific choice of smoothing and threshold functions.
  4. [§3 and Appendix B] The comparison to previous simulations is made through the B23 analytic fit and through published power-law indices, rather than through direct N_eff(H)-n_H data from ray-traced simulations. Because the fitted slope of the fiducial model varies between roughly 0.3 and 0.6 across the dense-gas range (Fig. B.2), the statement that the model is "consistent with" \gamma \approx 0.4-0.5 would be substantially strengthened by overlaying the model on the actual simulation data of, e.g., Hu et al. (2021) or Safranek-Shrader et al. (2017). This would also provide a test of the turbulent term at intermediate densities, where the current validation is weakest.
minor comments (5)
  1. [Abstract and Figs. 2, 3, B.1, B.2] The axis labels and abstract use "cm 3" where "cm^{-3}" is meant; several panels in Figs. 3 and B.1 show the same typographical issue.
  2. [Eq. (1)] The typesetting of Eq. (1) is malformed: the summation index appears as "NraysX" and the fraction inside the logarithm is unclear. Please set the expression as (1/N_rays) \sum_{i=1}^{N_rays} e^{-\gamma A_{V,i}} explicitly.
  3. [§2.4] The sentence "For comparison to B23, we also used a floor of 0.05 mag" is ambiguous because Eq. (14) already contains a resolution-dependent floor; state clearly whether the 0.05 mag is an additional extinction floor and at which densities it activates.
  4. [§1, last paragraph] The text reads "Finally, in 4, we discuss..." and should read "in Section 4".
  5. [Fig. 1 and Table 1] Figure 1 mentions an "Imposed column density floor N_min" but this quantity is not listed in Table 1 and is not given a fiducial value in Section 2; please either define it in Table 1 or remove it from the chart.

Circularity Check

2 steps flagged · score 6.0 of 10

The low-density agreement with the B23 fit is produced by importing B23's own 0.05 mag floor into the model; the high-density Jeans slope remains independent.

  1. self definitional [Section 2.4 (Combined model), Eq. (13)-(14); Section 3, first paragraph]
    "For comparison to B23, we also used a floor of 0.05 mag. ... At moderate densities, the model matches B23 within a factor of two, although towards low density, a floor is needed to remain consistent with B23. ... The inclusion of the relevant resolution limit and floor brings the model into excellent agreement with B23."

    B23's fit, Eq. (2), is A_V,eff = 0.05 exp[1.6 (n_H/cm^-3)^0.12], whose low-density limit is exactly the 0.05 mag floor inserted into the model. The model's own turbulent component vanishes at low density (N_turb -> 0 as bar-ell -> L_cl), and its gravitational component is well below 0.05 mag for n_H less than about 10 cm^-3. Therefore, in the low-density regime the floor is the sole contributor to the claimed A_V,eff, so the model reproduces B23 in that regime by construction rather than by independent prediction. The headline claim that the model well-reproduces B23 is thus partly true only because the comparison target's own constant was loaded into the model as an input.

  2. other [Appendix A]
    "We used the 1D AV− nH density distribution from B23 (Eq. 2) with the public photodissociation code 3d-pdr ... The result of both is that the Neff(H)− nH is brought slightly closer to the B23 fit."

    The variable sound-speed and mean-molecular-weight corrections that improve agreement with B23 are computed from a PDR model whose input is B23's A_V-nH relation. Those corrections are then inserted into the analytic model and used to claim closer agreement with B23, which is circular in direction: the target relation has generated the correction that produces the improved match. The paper explicitly labels this 'not a self-consistent treatment,' and the appendix is qualitative, so this step is secondary rather than load-bearing for the main high-density result.

full rationale

The high-density part of the model is genuinely independent: the Jeans component, N_J = (1/2) lambda_J n_H proportional to c_s n_H^{1/2}, is parameter-free, and the flattening of a power-law fit from 0.5 to about 0.4 over n_H > 10^3 cm^-3 follows arithmetically from adding the constant turbulent column n_0 R_cl to the Jeans term. The transition density n_T follows Burkhart & Mocz (2019), and the slopes from Safranek-Shrader et al. (2017) and Hu et al. (2021) are external checks. There is no load-bearing self-citation chain. The circularity is confined to the low-density regime and the qualitative appendix: the 0.05 mag floor imported from B23 makes the low-density match to B23 true by definition, and the Appendix A correction is generated using B23 itself. Because the headline claim of reproducing B23 depends on that floor, the score reflects partial circularity rather than a wholly independent derivation.

Assumptions & free parameters 8 free parameters · 9 assumptions · 0 invented entities

The model rests on standard physical inputs (cloud size, temperature, density, turbulence parameters) plus several unproved structural choices: the turbulent column formula, the smooth threshold, the damping function, and the post-hoc floor. No new physical entity is introduced. The high-density Jeans term is the only component derived directly from first principles without adjustable parameters.

free parameters (8)
  • L (cloud size) = 15 pc (fiducial, range 1-20 pc)
    Sets the maximum turbulent envelope depth through R_cl = L/2; not fitted to the target relation, but chosen to represent Milky Way-like clouds.
  • T (gas temperature) = 10 K (fiducial, range 10-100 K)
    Sets the sound speed and hence Jeans length; chosen as a standard cold cloud temperature.
  • n0 (mean cloud density) = 100 cm^-3 (fiducial, range 10-1000 cm^-3)
    Normalizes the turbulent column and enters the transition density n_T; the paper notes it globally shifts the attenuating column.
  • alpha_v (virial parameter) = 1 (fiducial, range 0.5-5)
    Relates local density to turbulent length scale in Eq. (6) and enters n_T; the parameter study shows weak net impact.
  • sigma0, L0 (turbulence normalization) = 0.381 km/s at L0 = 1 pc (fiducial)
    Normalizes the linewidth-size relation used to derive the turbulent length scale.
  • beta (linewidth-size exponent) = 0.51 (fiducial, range 0.3-0.7)
    Sets the scaling of sigma with length; taken from Rice et al. (2016) rather than fitted to the target relation.
  • A_V floor = 0.05 mag
    Added in Section 2.4 specifically 'for comparison to B23'; post-hoc adjustment to match the target fit at low density.
  • C_Delta_x (resolution floor prefactor) = unspecified
    Introduced in Eq. (12) for the resolution limit; the prefactor depends on how self-contribution is counted but is never quantified, and is included when comparing to simulations.
assumptions (9)
  • domain assumption Linewidth-size relation sigma(ell) = sigma0 (ell/L0)^beta holds across the densities sampled.
    Used in Eq. (5) to convert a density-dependent length scale into a velocity dispersion; recent surveys support beta 0.3-0.7, but the relation is empirical.
  • domain assumption Virial balance alpha_v = 5 sigma(ell)^2 / (G mu_H m_H n_H ell^2) with M approximately rho ell^3 connects local density to a characteristic length.
    Eq. (4) assumes mass and density are connected through M approximately rho ell^3, i.e., a hierarchical density structure; this is stated in the text as an assumption.
  • ad hoc to paper Turbulent shielding column equals N_turb = ell_turb n0 with ell_turb = R_cl - (1/2) bar-ell.
    Eq. (8) is asserted rather than derived; it sets the low/intermediate density part of N_eff and is the main unproved component.
  • domain assumption For an isotropic radiation field, the effective column is dominated by the lowest column direction, so ell_turb is capped at R_cl.
    Stated in Section 2.1 after Eq. (8); no calculation from the ray average in Eq. (1) is given.
  • ad hoc to paper Smooth damping S_ell(z) = erf(sqrt(pi) z / 2) approximates a step filter for the length scale.
    Adopted in Eq. (7) with the rationale that it is smooth, not because it follows from cloud structure.
  • ad hoc to paper Softened threshold S_n(z) = z/(1+z) controls the onset of the gravitational component.
    Adopted after Eq. (11); the paper states the choice impacts how rapidly the transition occurs without further justification.
  • domain assumption Transition density n_T/n0 = pi^2/(15 alpha_v M_s^2) from Burkhart and Mocz applies to these clouds.
    Eq. (10) is imported from the cited turbulence literature; it is a theory result, not tested against the simulations discussed.
  • ad hoc to paper A numerical column floor of 0.05 mag is relevant for matching B23.
    Introduced in Section 2.4 with the explicit purpose of comparing to B23; it is a modeling choice rather than a measured quantity.
  • domain assumption Resolution limit N_Delta_x = C n_H Delta x with an unspecified prefactor C bounds the column.
    Eq. (12) is physically reasonable but C is left undefined, so the comparison to simulations with resolution limits cannot be reproduced exactly.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An analytic formalism to describe the $N_{\rm eff}(\rm H)$-$n_{\rm H}$ relationship in molecular clouds." pith.science (2026). https://pith.science/paper/WXZ3WZCD

@misc{pith2026250716931,
  author       = {Pith},
  title        = {Pith review of: An analytic formalism to describe the $N_\rm eff(\rm H)$-$n_\rm H$ relationship in molecular clouds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WXZ3WZCD}},
  note         = {Machine review of arXiv:2507.16931}
}
abstract

Context. Astrochemical modeling requires, as input, the effective column density of gas (or extinction) that attenuates an external, isotropic, far-ultraviolet radiation field. In three-dimensional simulations, this can be calculated through ray-tracing schemes, while in 0D chemical models it is often treated as a free parameter. Aims. We aim to produce an analytic, physically motivated formalism to predict the average relationship between the effective hydrogen-nuclei column density, $N_{\rm eff}({\rm H})$, and the local hydrogen-nuclei number density, $n_{\rm H}$. Methods. We construct an analytic model utilizing characteristic length scales that connects the turbulence-dominated regime and the gravitational-dominated regime at high-density. Results. The model well-reproduces a previous analytic fit to simulation results and is consistent with the high-density power-law indices, e.g., $N_{\rm eff}(H) \propto n^{\gamma}$, of $\gamma \approx 0.4 - 0.5$ found in previous numerical simulations utilizing ray-tracing. Conclusions. We present an analytic model relating the average effective column density, $N_{\rm eff}$, to the local number density, $n_{\rm H}$, which reproduces the behaviors found in three-dimensional simulations. The analytic model can be utilized as a sub-grid prescription for shielded molecular gas or in astrochemical models for a physically motivated estimation of the attenuating column density.

Figures

Figures reproduced from arXiv: 2507.16931 by the authors.

Figure 1
Figure 1. Chart showing the different components and where parameters enter into the model. turbulence and dynamics, an effective column density can be es￾timated that is in general agreement with the results from three￾dimensional simulations. In Section 2 we present the underly￾ing model components. In Section 3, we show the results of the model in comparison to the previous three-dimensional calcula￾tions. Finally, in 4, w… view at source ↗
Figure 2
Figure 2. shows the fiducial model in comparison to the B23 fit. At moderate densities, the model matches B23 within a factor of two, although towards low density, a floor is needed to remain consistent with B23. The slope flattens around nH ≈ 500 cm−3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Parameter space results for the Neff(H)−nH relationship. Each panel varies the annotated variable while keeping the rest fixed to the fiducial values shown in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Emergence of high-mass stars in complex fiber networks (EMERGE) VI. Turbulence dissipation and the formation of dense fibers

    astro-ph.GA 2026-07 accept novelty 6.0 of 10

    In Orion, turbulence dissipates in high-shear regions near dense fibers, so the transition to coherence occurs at the fiber level before cores form.

Reference graph

Works this paper leans on

31 extracted references · 22 canonical work pages · cited by 1 Pith paper

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

  3. [3]

    G., Bell , T

    Bisbas , T. G., Bell , T. A., Viti , S., Yates , J., & Barlow , M. J. 2012, , 427, 2100

    work page 2012

  4. [4]

    G., Tan , J

    Bisbas , T. G., Tan , J. C., & Tanaka , K. E. I. 2021, , 502, 2701

    work page 2021

  5. [5]

    G., van Dishoeck , E

    Bisbas , T. G., van Dishoeck , E. F., Hu , C.-Y., & Schruba , A. 2023, , 519, 729

    work page 2023

  6. [6]

    & Grassi , T

    Bovino , S. & Grassi , T. 2024, Astrochemical Modeling: Practical Aspects of Microphysics in Numerical Simulations

    work page 2024

  7. [7]

    & Mocz , P

    Burkhart , B. & Mocz , P. 2019, , 879, 129

    work page 2019

  8. [8]

    Draine , B. T. 1978, , 36, 595

    1978

Show all 31 references
  1. [9]

    Draine , B. T. 2011, Physics of the Interstellar and Intergalactic Medium

    2011

  2. [10]

    2024, , 685, A82

    Edenhofer , G., Zucker , C., Frank , P., et al. 2024, , 685, A82

    2024

  3. [11]

    Gaches , B. A. L., Bisbas , T. G., & Bialy , S. 2022, , 658, A151

    2022

  4. [12]

    Glover , S. C. O., Federrath , C., Mac Low , M. M., & Klessen , R. S. 2010, , 404, 2

    2010

  5. [13]

    Glover , S. C. O. & Mac Low , M.-M. 2007, , 169, 239

    2007

  6. [14]

    & Dame , T

    Heyer , M. & Dame , T. M. 2015, , 53, 583

    2015

  7. [15]

    Hu , C.-Y., Sternberg , A., & van Dishoeck , E. F. 2021, , 920, 44

    2021

  8. [16]

    2017, , 603, A89

    Kauffmann , J., Pillai , T., Zhang , Q., et al. 2017, , 603, A89

    2017

  9. [17]

    Larson , R. B. 1981, , 194, 809

    1981

  10. [18]

    Nelson , R. P. & Langer , W. D. 1997, , 482, 796

    1997

  11. [19]

    & Nordlund , A

    Padoan , P. & Nordlund , A . 2011, , 730, 40

    2011

  12. [20]

    Padovani , M., Galli , D., & Glassgold , A. E. 2009, , 501, 619

    2009

  13. [21]

    , S., Beuther , H., Benjamin , R

    Rezaei Kh. , S., Beuther , H., Benjamin , R. A., et al. 2024, arXiv e-prints, arXiv:2405.09634

    2024 arXiv

  14. [22]

    S., Goodman , A

    Rice , T. S., Goodman , A. A., Bergin , E. A., Beaumont , C., & Dame , T. M. 2016, , 822, 52

    2016

  15. [23]

    P., Bell , T., et al

    R \"o llig , M., Abel , N. P., Bell , T., et al. 2007, , 467, 187

    2007

  16. [24]

    R., Kim , C.-G., et al

    Safranek-Shrader , C., Krumholz , M. R., Kim , C.-G., et al. 2017, , 465, 885

    2017

  17. [25]

    2017, , 472, 4797

    Seifried , D., Walch , S., Girichidis , P., et al. 2017, , 472, 4797

    2017

  18. [26]

    2022, , 667, A110

    Spilker , A., Kainulainen , J., & Orkisz , J. 2022, , 667, A110

    2022

  19. [27]

    Tielens , A. G. G. M. 2013, Reviews of Modern Physics, 85, 1021

    2013

  20. [28]

    J., & Tan , J

    Van Loo , S., Butler , M. J., & Tan , J. C. 2013, , 764, 36

    2013

  21. [29]

    M., & Herbst , E

    Wakelam , V., Cuppen , H. M., & Herbst , E. 2013, arXiv e-prints, arXiv:1309.7792

    2013 arXiv

  22. [30]

    2024, Frontiers in Astronomy and Space Sciences, 11, 1346812

    W \"u nsch , R. 2024, Frontiers in Astronomy and Space Sciences, 11, 1346812

    2024

  23. [31]

    2021, , 919, 35

    Zucker , C., Goodman , A., Alves , J., et al. 2021, , 919, 35

    2021

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.