Pith. sign in

REVIEW 2 major objections 4 minor 2 cited by

Lyman-alpha resonant-line radiative transfer in expanding media

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read Homologous expansion alters Lyman-alpha escape scalings from $(a\tau_0)^{1/3}$ to powers of $\beta$, with closed-form solutions for spherical and cosmological flows.

desk verdict Solid moving-media Lyα analytic solutions with a genuine numerical validation method, but the claimed validity boundary is asserted rather than independently mapped and the abstract overstates one comparison; referee it. read the letter →

arxiv 2501.01928 v2 pith:765SYDUY submitted 2025-01-03 astro-ph.GA astro-ph.CO

Aaron Smith (1) , Kevin Lorinc (1) , Olof Nebrin (2) , Bing-Xin Lao (3) ((1) UT Dallas , (2) Stockholm , (3) Princeton) This is my paper · ORCID
classification astro-ph.GAastro-ph.CO
keywords Lyman-alpharadiativetransferresonantscatteringexpandingmediahomologousexpansiondiffusionapproximationMonteCarlocosmologicalLy-alphaescape
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 tries to show that bulk expansion of an optically thick hydrogen cloud can be folded into the standard diffusion picture of Lyman-$\alpha$ radiative transfer, producing closed-form solutions and simple power-law scalings where only static solutions existed. It derives series solutions for homologous expansion in a sphere, for both central point and uniform sources, and a Bessel-function solution for finite-temperature cosmological flows, then validates them with a gridless Monte Carlo code. If correct, the scaling laws $x_{\rm esc}\sim(a\tau_0\beta\sqrt{\pi})^{1/3}$, $M_F\sim\beta^{-1/3}$, and $t_{\rm trap}\sim\beta^{-2/3}$ give analytic benchmarks and correction factors for interpreting outflowing galaxies.

What carries the argument

The central object is the diffusion-approximation transfer equation in the comoving frame, where expansion enters only through the dimensionless velocity divergence $w$ and acts as a constant frequency drift. The solution is built from separable eigenfunctions $\sin(\lambda_n\tilde r)/\tilde r$ and a new special function, the Sigma function $\varsigma(w,s)$, that sums the mode weights and controls the $\beta$-scaling of every physical observable. For the cosmological case the machinery is a four-dimensional Fourier transform whose inversion yields a modified Bessel function $K_1$, giving the finite-temperature solution in Eq. (75).

What would settle it

Run a gridless Monte Carlo calculation without core skipping at $a\tau_0/\beta^2$ just above $10^3$ and compare the force multiplier to Eq. (55); if $M_F$ deviates systematically and the deviation grows with increasing $\beta$ at fixed $a\tau_0/\beta^2$, the delta-function replacement of the line profile is the failing step.

Watch

Extended reading notes

Core claim

In the diffusion limit, a homologous velocity gradient enters the transfer equation as a constant drift term in frequency space, so the central equation becomes $\tilde\nabla^2\tilde J+\partial^2\tilde J/\partial\tilde x^2+2w\,\partial\tilde J/\partial\tilde x=-\eta\,\delta(\tilde x)/k$. The paper solves this equation by eigenfunction expansion for spherical clouds, giving the full radiation field, emergent spectra, red-flux fraction, energy density, force multiplier, trapping time, and number of scatterings. For expansion velocities above thermal, the characteristic escape frequency and trapping time scale as $\beta^{1/3}$ and $\beta^{-2/3}$ rather than the static $(a\tau_0)^{1/3}$, and the force multiplier scales as $\beta^{-1/3}$. The paper also generalizes the zero-temperature cosmological solution to finite temperature, producing an expression with a modified Bessel kernel that regularizes the line-center singularity and adds trapping at small radii.

Load-bearing premise

Everything rests on the radiation field being nearly isotropic and diffusive in both space and frequency, so Fick's law, the Eddington tensor, and the delta-function replacement of the line profile hold; the paper itself notes this breaks down when expansion is so fast that photons free-stream out before many scatterings, i.e. when $a\tau_0/\beta^2$ falls below roughly $10^3$.

Editorial extensions

If this is right

  • For homologous outflows, red-peak dominance increases with $w$, and the paper gives explicit formulas for the red flux fraction $f_{\rm red}$, including the low-velocity limits $f_{\rm red}\approx 1/2+(\ln 2/\pi)w$ for a point source and $f_{\rm red}\approx 1/2+3\zeta(3)w/\pi^3$ for a uniform source.
  • Trapping time and force multiplier in rapidly expanding clouds depend on velocity much more weakly than earlier moving-slab scalings suggested: $t_{\rm trap}\propto\beta^{-2/3}$ and $M_F\propto\beta^{-1/3}$, implying expansion does not reduce Lyman-alpha trapping as drastically as previously claimed.
  • Finite temperature removes the unphysical singular behavior of the zero-temperature cosmological solution near line center, so the new solution is usable at small radii and small frequencies where the old one diverged.
  • The gridless Monte Carlo validation delimits the domain of the analytic solutions: agreement is excellent up to $w\approx 50$--$60$ for trapping time and $w\approx 1800$ for the point-source force multiplier, with discrepancies at high velocity or low optical depth tied to the breakdown of spatial diffusion.
  • The derived scaling relations provide a direct rule of thumb for observers: in the dynamically optically thick regime $a\tau_0/\beta^2\gtrsim 10^3$, peak shifts and red-to-blue ratios can be predicted without running a full Monte Carlo simulation.

Reading between the lines

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

  • Because the diffusion closure depends on the combination $a\tau_0/\beta^2$, observers could translate a measured peak separation into an effective expansion parameter without running radiative-transfer simulations, provided the cloud is in the diffusive regime.
  • The same eigenfunction machinery could be applied to accelerating or conically expanding outflows by letting $w$ vary with radius; the constant-$w$ solutions would then serve as local benchmarks for subgrid models.
  • If the $\beta^{-1/3}$ force-multiplier scaling holds in real outflows, radiation-pressure feedback in low-metallicity galaxy formation remains effective even at high expansion speeds, strengthening the case for Lyman-alpha feedback in early galaxies.
  • A testable extension would be to measure the emergent lab-frame spectrum from a simulated expanding cloud and compare the power-law exponent of the angular distribution against the paper's Fig. B3 fits, which predict a steepening exponential slope for point sources and a mild flattening for uniform sources.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper derives closed-form series solutions to the diffusion-approximation radiative transfer equation for Lyman-alpha photons in homologously expanding, optically thick spheres, and a finite-temperature generalization of the Loeb-Rybicki cosmological solution. For point and uniform sources it presents explicit expressions for the emergent spectrum, red flux fraction, energy density, trapping time, characteristic radius, force multiplier, and scattering number, together with scaling laws x_esc ~ beta^{1/3}, M_F ~ beta^{-1/3}, and t_trap ~ beta^{-2/3} relative to the static (a tau0)^{1/3} behavior. It also introduces a gridless Monte Carlo radiative transfer (GMCRT) method that integrates optical depth exactly along continuously Doppler-shifted paths and compares GMCRT results with the analytic solutions. The central claims are the new analytic benchmarks, the GMCRT method, and the identification of the regime where diffusion-based approximations hold.

Significance. If the claimed results hold, this paper provides valuable analytic benchmarks for Lyman-alpha transport in moving media and correction factors useful for interpreting observations and for subgrid modeling. The derivations reduce correctly to known limits (Lao & Smith 2020 at w=0; Loeb & Rybicki 1999 at T=0), and the GMCRT method with exact optical depth integration is a useful numerical contribution in its own right. The paper also makes explicit falsifiable scaling predictions. However, the numerical validation of the claimed domain of validity and of the source-dependent force-multiplier scaling is not yet conclusive, so the advertised results are not fully supported by the evidence presented.

major comments (2)
  1. [§2.1, §5.3, §6(i)] The asserted domain of validity a tau0 / beta_bar^2 >= 10^3 is not independently mapped. The GMCRT comparisons are presented along Russian-doll tracks in which tau0(r) is proportional to r and beta(r) is proportional to r, so each track is a ray in the (tau0, beta) plane and tau0/beta^2 varies as 1/r along a track. Although different V_max values sample different directions, the paper never reports the onset of disagreement as a function of a tau0/beta^2, nor does it present runs with beta held fixed while tau0 is varied. This matters because the threshold is derived from the same constant-opacity random-walk picture (Eqs. 4-7) whose breakdown is flagged in Eq. (6); without an independent scan, the threshold is self-confirmed. As a concrete inconsistency, the Fig. 10 caption reports excellent agreement up to w ~ 1800 for T = 100 K and tau0,max = 5e8; at the outer edge this corresponds to a tau0/beta^2 ~ 1, well below the stated 10^3 threshold. Please add a dedicated validity scan (fixed beta with varying tau0, and vice versa) and plot the agreement metric as a function of a tau0/beta^2 to establish the claimed boundary.
  2. [§5.3.2, Eq. (67), Fig. 11] The uniform-source force multiplier, which underlies the source-dependent scaling M_F ~ beta^{-2/3} in Eq. (67) and its discussion in Section 6, is not numerically validated by the presented runs. The text explicitly states that core skipping with x_crit = 1 compromised the numerical results for this quantity, and Fig. 11 shows disagreement that the authors attribute to this convergence issue. Because this is the only GMCRT comparison for the uniform-source force multiplier, the beta^{-2/3} scaling is left without an independent check. Please either rerun this quantity with core skipping disabled or with a controlled x_crit study, or restrict the scaling claim to the point-source case until such a check exists.
minor comments (4)
  1. [Appendix D1] The contour shift is justified by saying that 'the factor xi_tilde is small' and that the shift does not add or remove poles. Since the poles of the integrand in the theta' variable lie at +/- i sqrt(|eta|^2 + xi_tilde^2/4), the contour can be shifted from Im theta' = -xi_tilde/2 to the real axis without crossing any poles for any xi_tilde > 0. Please replace the heuristic statement with this explicit argument.
  2. [Abstract] The abstract states the scaling relations x_esc ~ beta^{1/3}, M_F ~ beta^{-1/3}, and t_trap ~ beta^{-2/3} without the domain condition a tau0 / beta_bar^2 >= 10^3 stated in Section 6(i). To avoid misuse by readers applying the scalings outside the intended regime, please include the condition or a pointer to it in the abstract.
  3. [§5.3.3 and Appendix B] The lab-frame comparison in Fig. 13 depends on an assumed angular distribution P(mu) for the emergent photons (Eq. B3). The text should state explicitly which P(mu) is used for the analytic curves in Fig. 13 and clarify that the comoving-frame comparison in Fig. 12 is the primary test of the diffusion solution.
  4. [Fig. 14] The Wasserstein distance is described as a percentage error after scaling by 100/(a tau0)^{1/3}. Please clarify the normalization and the sense in which this is a percentage, and consider using a logarithmic w-axis because the displayed range spans more than two decades.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the analytic solutions are derived from stated diffusion closures and checked against an independent gridless MCRT method; self-citations are frequent but not load-bearing.

full rationale

The derivation chain is self-contained. The governing equation (Eq. 16) is obtained from the full comoving-frame transfer equation (Eq. 1) via explicit closures -- Eddington approximation, Fick's law (Eq. 10), Fokker-Planck frequency diffusion (Eq. 11), and the delta-function replacement of 3 tau0 H^2(x) -- rather than by assuming the final scaling results. The spherical-cloud solutions (Eqs. 23, 34-68) follow from a standard eigenfunction expansion with stated boundary conditions, and the cosmological solution (Eq. 75) follows from an explicit Fourier inversion of Eq. (72). The headline scaling exponents x_esc ~ beta^{1/3}, M_F ~ beta^{-1/3}, and t_trap ~ beta^{-2/3} come from the asymptotics of the independently defined Sigma function (Appendix A), not from fits to the GMCRT histograms. GMCRT solves Eq. (1) directly without the spatial/frequency diffusion approximations, so the agreement reported in Section 5 is not enforced by construction. The stated validity condition a tau0 / beta^2 >= 10^3 is a self-consistency estimate from the random-walk picture and the GMCRT runs do not exhaustively map the full boundary of that regime; this is a validation limitation rather than a circular reduction. Self-citations to Lao & Smith (2020) and Nebrin et al. (2025) appear frequently, but they are used for bookkeeping, for static-limit checks, and for one auxiliary convergence claim (uniform-source force multiplier in Section 5.3.2); the central derivations do not reduce to those citations. No equation is defined in terms of its own output, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central results rest on the standard diffusion-approximation toolkit for resonant line transfer (Eddington and Fokker-Planck closures, delta-function line profile), the choice of homologous expansion in isothermal uniform-density spheres, and several numerical or hand choices (x_crit=1, Sigma prefactor 1.52). These are physical modeling assumptions rather than fits to the target observables. The regime of validity a tau0 / beta^2 >= 10^3 is asserted from the same diffusion theory being validated, which is the main self-referential element. No new physical entities are introduced: the Sigma function sigma(w,s) (Eq. 32) is a defined mathematical series and GMCRT is a numerical algorithm.

free parameters (3)
  • Sigma-function prefactor for point-source force multiplier = 1.52 (numerical integral gives 1.594)
    Appendix A, Eq. (A2): the large-w prefactor of sigma(w,0) is hand-adjusted for 30 <= w <= few 100; affects M_F amplitude at high velocity, not the exponents.
  • Boundary condition constant f = sqrt(3) or 3/2 (unspecified)
    Eq. (17): two-stream or Eddington choice; the paper argues results are insensitive to f, so it is a modeling convention rather than a fitted value.
  • Core-skipping threshold x_crit in GMCRT = 1
    Section 5.3.2: chosen by hand to balance cost and accuracy; the paper says this choice compromised the uniform-source force-multiplier comparison, so it affects the validation figures.
assumptions (7)
  • standard math Eigenfunction expansion of the diffusion operator on the sphere; 4D Fourier transform with modified Bessel function identity for the integral
    Used in Section 3 (Eqs. 18-23) and Section 4 and Appendix D (Eqs. 73-75, D1-D3); these are textbook results.
  • domain assumption Eddington closure K approximately J I/3 and Fick's-law flux H_x approximately -nabla J/(3 k_x)
    Section 2.2, Eqs. (9)-(10); assumes near-isotropy of the radiation field, which fails in the free-streaming regime documented in Section 5.3.
  • domain assumption Fokker-Planck approximation for the redistribution integral, with atomic recoil neglected
    Eq. (11) and footnote 3; stated to be good at moderate to high gas temperatures.
  • domain assumption Replacement of the sharply peaked 3 tau0 H^2(x) profile by a delta function
    Section 2.2, after Eq. (14), following Harrington (1973); standard in the static case and carried over here.
  • domain assumption Homologous velocity field with constant dimensionless divergence w, with the velocity entering only through the scalar divergence of the flow
    Section 3 (w = 3 V_max / sqrt(6) v_th); the trace-only coupling follows from the Eddington tensor assumption in Eq. (9).
  • domain assumption Isothermal, uniform-density spherical cloud with constant temperature so that a and k(x) are spatially constant
    Section 2.2 and Section 3.1; the paper notes temperature gradients are deferred to future work.
  • ad hoc to paper The inverse Fourier contour in Appendix D1 may be shifted to the real axis without adding or removing poles
    Appendix D1, Eq. (D1): the poles lie off-axis by i xi/2; the real-axis treatment is plausible but not proven, and it is specific to this derivation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Lyman-alpha resonant-line radiative transfer in expanding media." pith.science (2026). https://pith.science/paper/765SYDUY

@misc{pith2026250101928,
  author       = {Pith},
  title        = {Pith review of: Lyman-alpha resonant-line radiative transfer in expanding media},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/765SYDUY}},
  note         = {Machine review of arXiv:2501.01928}
}
read the original abstract

The Lyman-alpha (LyA) line of atomic hydrogen encodes crucial information about both the intrinsic sources and surrounding environments of star-forming regions throughout the Universe. Due to the complexity of resonant scattering, analytic solutions remain scarce, with most studies focusing on idealized, static configurations. However, observations of LyA emitting galaxies consistently reveal signatures of outflows, imprinted through red-peak dominance in spectral line profiles. In this paper, we derive novel analytic solutions for resonant-line radiative transfer in moving media, specifically homologous-like cloud expansion and unbounded cosmological flows, which capture the main physics of velocity gradients. To validate these analytic solutions and identify regimes where diffusion-based assumptions hold, we introduce a robust Gridless Monte Carlo Radiative Transfer (GMCRT) method. By integrating optical depths exactly in the comoving frame, GMCRT updates photon frequencies continuously to account for Doppler shifts induced by velocity gradients. We demonstrate excellent consistency between GMCRT and our analytic solutions in regimes where diffusion approximations apply. At higher velocities or lower optical depths, discrepancies highlight the limitations of simplified formalisms. We also provide scaling relations for a point source in a cloud with a maximum-to-thermal velocity ratio beta = V_max / v_th, as modifying the standard dependence on line centre optical depth of (atau_0)^1/3 by additional factors, e.g. characteristic escape frequency scale as x_esc ~ beta^1/3, force multipliers as M_F ~ beta^-1/3, and trapping time as t_trap ~ beta^-2/3. Our work complements numerical simulations by improving physical intuition about nonstatic environments when interpreting LyA observations and guiding future subgrid prescriptions in galaxy formation models.

Figures

Figures reproduced from arXiv: 2501.01928 by the authors.

Figure 1
Figure 1. Top panel: The Sigma function 𝜍 (𝑤, 𝑠) for different values of 𝑠, a series that frequently appears throughout this work. The nearly horizontal dashed lines represent the Riemann Zeta function values derived from Taylor expanding the terms inside the series. The negatively sloped lines show the asymptotic power-law scaling as 𝑤 → ∞, showing that for 𝑠 = 0, 𝜍 ∝ 𝑤−1/3 , while for 𝑠 > 1/3, 𝜍 ∝ 𝑤−2/3 . Bottom panel: The … view at source ↗
Figure 2
Figure 2. The solution of Eq. (16), giving us the internal spectrum 𝐽 (𝑟, 𝑥) as a function of frequency 𝑥 / (𝑎𝜏0 ) 1/3 and radius 𝑟/𝑅 for a homogeneous sphere undergoing homologous expansion with 𝑤 = 3𝑉max/ √ 6𝑣th = 10. For a point source (top panel, using Eq. 44), most of the intensity is clustered around the line centre at 𝑥 / (𝑎𝜏0 ) 1/3 = 0 and near the centre of the sphere. The expansion causes an excess of radiation shif… view at source ↗
Figure 4
Figure 4. The analytical energy density for a central point source (top panel, using Eq. 51) and a uniform source (bottom panel, using Eq. 62) as a function of 𝑟/𝑅 for various values of the expansion parameter, 𝑤 ≡ 3𝑉max/ √ 6𝑣th. The point source energy density has a singularity at the origin due to the emission profile, and the series diverges to +∞. The uniform source has an even distribution of energy out to large relative… view at source ↗
Figures from the paper (10 more)
Figure 5
Figure 5. Figure 5: The average internal spectrum for a central point source (top panel, using Eq. 52) and a uniform source (bottom panel, using Eq. 64) for various values of the expansion parameter, 𝑤 ≡ 3𝑉max/ √ 6𝑣th. For both types of sources, the spectrum shifts from symmetry about the…
Figure 6
Figure 6. Figure 6: Comparison of the radiative transfer solution 𝐽¯(𝑟,¯ 𝜈¯) in an infinite expanding Universe, using the parameter-space conventions of Loeb & Rybicki (1999). Top Panel: Illustration of how finite temperature affects the solution, especially closer to line centre, shown f…
Figure 7
Figure 7. Figure 7: Schematic of Gridless Monte Carlo Radiative Transfer (GMCRT). During transport, the optical depth is integrated exactly accounting for con￾tinuous Doppler shifting as opposed to the piecewise-constant static approx￾imation that is commonly employed by MCRT codes but on…
Figure 8
Figure 8. Figure 8: Trapping time (𝑡trap), normalized by the light-crossing time (𝑡light), as a function of 𝑎𝜏0 for a point source with 𝑇 = 100 K and 𝜏0,max = 5 × 108 . The thin lines correspond to the analytical predictions (Eq. 53) plotted for different values of 𝑎𝜏0 and 𝑤 ≡ 3𝑉max/ √ 6𝑣…
Figure 9
Figure 9. Figure 9: Trapping time (𝑡trap) for a uniform source normalized by the average light-crossing time ⟨𝑡light⟩ = (3/4) 𝑅/𝑐, as a function of 𝑎𝜏0 with 𝑇 = 104 K and 𝜏0,max = 5 × 108 . The thin lines are the analytical predictions from Eq. (65) plotted for different values of 𝑎𝜏0 and…
Figure 11
Figure 11. Figure 11: Force multiplier (𝑀F) for a uniform source as a function of 𝑎𝜏0 with the same setup as described in [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: Emergent spectra presented in the internal comoving frame for several expansion values of 𝑤 ≡ 3𝑉max/ √ 6𝑣th = {1, 20, 80, 320}. The thin lines represent the analytical solutions for a central point source (top panel, Eq. 46) and a uniform source (bottom panel, Eq. 60)…
Figure 14
Figure 14. Figure 14: One-dimensional Wasserstein distance (Eq. 87) between the nor￾malized emergent spectra from GMCRT simulations (𝐽n, i.e. the histogram of escaping frequencies) and analytic solutions (𝐽a, from Eq. 46 for the point source results at Eq. 60 for the uniform results), for …
Figure 13
Figure 13. Figure 13: Emergent spectra in the external lab frame after Doppler-shifting (see Appendix B for a full discussion). The same setup is used as in [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 15
Figure 15. Figure 15: Fraction of the total flux that is redward of line centre in the co￾moving frame, 𝑓red ≡ ∫ 0 −∞ 𝐽 (𝑥) d𝑥/ ∫ ∞ −∞ 𝐽 (𝑥) d𝑥. The solid lines represent the analytic predictions from Eq. (47) (point source red fraction) and Eq. (37) (uniform source red fraction). The dott…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Force convergence in Monte Carlo Lyman-alpha radiative transfer

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

    A moment-based hierarchy (zeroth, first, second order) diagnoses convergence of Lyman-alpha MCRT momentum-transfer estimators, showing that core-skipping biases internal forces and that statistical precision, cost, an...

  2. THOR: a GPU-accelerated and MPI-parallel radiative transfer code

    astro-ph.GA 2025-07 conditional novelty 6.0 of 10

    A new GPU-accelerated, MPI-parallel Monte Carlo radiative transfer code (THOR) for resonant emission lines is validated against analytic solutions and benchmarked across idealized and cosmological setups, with measure...

Reference graph

Works this paper leans on

65 extracted references · 5 canonical work pages · cited by 2 Pith papers

  1. [1]

    F., 1972, @doi [ ] 10.1086/151503 , https://ui.adsabs.harvard.edu/abs/1972ApJ...174..439A 174, 439

    Adams T. F., 1972, @doi [ ] 10.1086/151503 , https://ui.adsabs.harvard.edu/abs/1972ApJ...174..439A 174, 439

    doi:10.1086/151503 1972

  2. [2]

    F., 1975, @doi [ApJ] 10.1086/153891 , 201, 350

    Adams T. F., 1975, @doi [ApJ] 10.1086/153891 , 201, 350

    doi:10.1086/153891 1975

  3. [3]

    M., 2002, @doi [ApJ] 10.1086/338497 , 567, 922

    Ahn S.-H., Lee H.-W., Lee H. M., 2002, @doi [ApJ] 10.1086/338497 , 567, 922

    doi:10.1086/338497 2002

  4. [4]

    M., 1981, @doi [Astrophysics] 10.1007/BF01014298 , https://ui.adsabs.harvard.edu/abs/1981Ap.....17...69B 17, 69

    Basko M. M., 1981, @doi [Astrophysics] 10.1007/BF01014298 , https://ui.adsabs.harvard.edu/abs/1981Ap.....17...69B 17, 69

    doi:10.1007/bf01014298 1981

  5. [5]

    C., 2014, @doi [ ] 10.1051/0004-6361/201322949 , https://ui.adsabs.harvard.edu/abs/2014A&A...563A..77B 563, A77

    Behrens C., Dijkstra M., Niemeyer J. C., 2014, @doi [ ] 10.1051/0004-6361/201322949 , https://ui.adsabs.harvard.edu/abs/2014A&A...563A..77B 563, A77

    doi:10.1051/0004-6361/201322949 2014

  6. [6]

    Bithell M., 1990, , https://ui.adsabs.harvard.edu/abs/1990MNRAS.244..738B 244, 738

    work page 1990

  7. [7]

    Blaizot J., et al., 2023, @doi [ ] 10.1093/mnras/stad1523 , https://ui.adsabs.harvard.edu/abs/2023MNRAS.523.3749B 523, 3749

    doi:10.1093/mnras/stad1523 2023

  8. [8]

    Bonilha J. R. M., Ferch R., Salpeter E. E., Slater G., Noerdlinger P. D., 1979, @doi [ ] 10.1086/157426 , https://ui.adsabs.harvard.edu/abs/1979ApJ...233..649B 233, 649

    doi:10.1086/157426 1979

Show all 65 references
  1. [9]

    I., 2004, Radiation Hydrodynamics

    Castor J. I., 2004, Radiation Hydrodynamics

    2004

  2. [10]

    Chandrasekhar S., 1945, @doi [ ] 10.1086/144771 , https://ui.adsabs.harvard.edu/abs/1945ApJ...102..402C 102, 402

    1945 doi

  3. [11]

    S., Dijkstra M., Ciardi B., Gronke M., 2016, @doi [ ] 10.1093/mnras/stv2340 , https://ui.adsabs.harvard.edu/abs/2016MNRAS.455..884C 455, 884

    Chung A. S., Dijkstra M., Ciardi B., Gronke M., 2016, @doi [ ] 10.1093/mnras/stv2340 , https://ui.adsabs.harvard.edu/abs/2016MNRAS.455..884C 455, 884

    2016 doi

  4. [12]

    Dijkstra M., 2014, @doi [ ] 10.1017/pasa.2014.33 , https://ui.adsabs.harvard.edu/abs/2014PASA...31...40D 31, e040

    2014 doi

  5. [13]

    Dijkstra M., Haiman Z., Spaans M., 2006, @doi [ ] 10.1086/506243 , https://ui.adsabs.harvard.edu/abs/2006ApJ...649...14D 649, 14

    2006 doi

  6. [14]

    B., 1959, @doi [ ] 10.1086/146654 , https://ui.adsabs.harvard.edu/abs/1959ApJ...129..551F 129, 551

    Field G. B., 1959, @doi [ ] 10.1086/146654 , https://ui.adsabs.harvard.edu/abs/1959ApJ...129..551F 129, 551

    1959 doi

  7. [15]

    N., Forero-Romero J

    Garavito-Camargo J. N., Forero-Romero J. E., Dijkstra M., 2014, @doi [ ] 10.1088/0004-637X/795/2/120 , https://ui.adsabs.harvard.edu/abs/2014ApJ...795..120G 795, 120

    2014 doi

  8. [16]

    Garel T., Michel-Dansac L., Verhamme A., Mauerhofer V., Katz H., Blaizot J., Leclercq F., Salvignol G., 2024, @doi [ ] 10.1051/0004-6361/202450654 , https://ui.adsabs.harvard.edu/abs/2024A&A...691A.213G 691, A213

    2024 doi

  9. [17]

    H., 2017, @doi [ ] 10.1093/mnras/stx2074 , https://ui.adsabs.harvard.edu/abs/2017MNRAS.472.2773G 472, 2773

    Ge Q., Wise J. H., 2017, @doi [ ] 10.1093/mnras/stx2074 , https://ui.adsabs.harvard.edu/abs/2017MNRAS.472.2773G 472, 2773

    2017 doi

  10. [18]

    F., 1973, @doi [ ] 10.1086/152344 , https://ui.adsabs.harvard.edu/abs/1973ApJ...184..461G 184, 461

    Gray D. F., 1973, @doi [ ] 10.1086/152344 , https://ui.adsabs.harvard.edu/abs/1973ApJ...184..461G 184, 461

    1973 doi

  11. [19]

    Gronke M., Dijkstra M., 2016, @doi [ ] 10.3847/0004-637X/826/1/14 , https://ui.adsabs.harvard.edu/abs/2016ApJ...826...14G 826, 14

    2016 doi

  12. [20]

    Gronke M., Bull P., Dijkstra M., 2015, @doi [ ] 10.1088/0004-637X/812/2/123 , https://ui.adsabs.harvard.edu/abs/2015ApJ...812..123G 812, 123

    2015 doi

  13. [21]

    P., 2006, @doi [ ] 10.1111/j.1365-2966.2005.09870.x , https://ui.adsabs.harvard.edu/abs/2006MNRAS.367..979H 367, 979

    Hansen M., Oh S. P., 2006, @doi [ ] 10.1111/j.1365-2966.2005.09870.x , https://ui.adsabs.harvard.edu/abs/2006MNRAS.367..979H 367, 979

    2006

  14. [22]

    P., 1973, @doi [ ] 10.1093/mnras/162.1.43 , https://ui.adsabs.harvard.edu/abs/1973MNRAS.162...43H 162, 43

    Harrington J. P., 1973, @doi [ ] 10.1093/mnras/162.1.43 , https://ui.adsabs.harvard.edu/abs/1973MNRAS.162...43H 162, 43

    1973 doi

  15. [24]

    G., 1962, @doi [MNRAS] 10.1093/mnras/125.1.21 , 125, 21

    Hummer D. G., 1962, @doi [MNRAS] 10.1093/mnras/125.1.21 , 125, 21

    1962 doi

  16. [25]

    Kakiichi K., Gronke M., 2021, @doi [ ] 10.3847/1538-4357/abc2d9 , https://ui.adsabs.harvard.edu/abs/2021ApJ...908...30K 908, 30

    2021 doi

  17. [26]

    Kimm T., Haehnelt M., Blaizot J., Katz H., Michel-Dansac L., Garel T., Rosdahl J., Teyssier R., 2018, @doi [ ] 10.1093/mnras/sty126 , https://ui.adsabs.harvard.edu/abs/2018MNRAS.475.4617K 475, 4617

    2018 doi

  18. [27]

    Kimm T., Blaizot J., Garel T., Michel-Dansac L., Katz H., Rosdahl J., Verhamme A., Haehnelt M., 2019, @doi [ ] 10.1093/mnras/stz989 , https://ui.adsabs.harvard.edu/abs/2019MNRAS.486.2215K 486, 2215

    2019 doi

  19. [28]

    Kimm T., Bieri R., Geen S., Rosdahl J., Blaizot J., Michel-Dansac L., Garel T., 2022, @doi [ ] 10.3847/1538-4365/ac426d , https://ui.adsabs.harvard.edu/abs/2022ApJS..259...21K 259, 21

    2022 doi

  20. [29]

    Lao B.-X., Smith A., 2020, @doi [ ] 10.1093/mnras/staa2198 , https://ui.adsabs.harvard.edu/abs/2020MNRAS.497.3925L 497, 3925

    2020 doi

  21. [30]

    O., Sommer-Larsen J., 2009, @doi [ApJ] 10.1088/0004-637X/696/1/853 , 696, 853

    Laursen P., Razoumov A. O., Sommer-Larsen J., 2009, @doi [ApJ] 10.1088/0004-637X/696/1/853 , 696, 853

    2009 doi

  22. [31]

    Laursen P., Duval F., \"O stlin G., 2013, @doi [ ] 10.1088/0004-637X/766/2/124 , https://ui.adsabs.harvard.edu/abs/2013ApJ...766..124L 766, 124

    2013 doi

  23. [32]

    B., 1999, @doi [ ] 10.1086/307844 , https://ui.adsabs.harvard.edu/abs/1999ApJ...524..527L 524, 527

    Loeb A., Rybicki G. B., 1999, @doi [ ] 10.1086/307844 , https://ui.adsabs.harvard.edu/abs/1999ApJ...524..527L 524, 527

    1999 doi

  24. [33]

    C., Davis S

    McClellan B. C., Davis S. W., Arras P., 2022, @doi [ ] 10.3847/1538-4357/ac7724 , https://ui.adsabs.harvard.edu/abs/2022ApJ...934...37M 934, 37

    2022 doi

  25. [34]

    Mihalas D., Auer L., 2001, Journal of Quantitative Spectroscopy and Radiative Transfer, 71, 61

    2001

  26. [35]

    K., 2025, @doi [ ] 10.1093/mnras/staf038 , https://ui.adsabs.harvard.edu/abs/2025MNRAS.537.1646N 537, 1646

    Nebrin O., Smith A., Lorinc K., H \"o rnquist J., Larson A ., Mellema G., Giri S. K., 2025, @doi [ ] 10.1093/mnras/staf038 , https://ui.adsabs.harvard.edu/abs/2025MNRAS.537.1646N 537, 1646

    2025 doi

  27. [36]

    A., 1990, @doi [ ] 10.1086/168375 , https://ui.adsabs.harvard.edu/abs/1990ApJ...350..216N 350, 216

    Neufeld D. A., 1990, @doi [ ] 10.1086/168375 , https://ui.adsabs.harvard.edu/abs/1990ApJ...350..216N 350, 216

    1990 doi

  28. [37]

    A., McKee C

    Neufeld D. A., McKee C. F., 1988, @doi [ ] 10.1086/185241 , https://ui.adsabs.harvard.edu/abs/1988ApJ...331L..87N 331, L87

    1988 doi

  29. [38]

    P., Haiman Z., 2002, @doi [ ] 10.1086/339393 , https://ui.adsabs.harvard.edu/abs/2002ApJ...569..558O 569, 558

    Oh S. P., Haiman Z., 2002, @doi [ ] 10.1086/339393 , https://ui.adsabs.harvard.edu/abs/2002ApJ...569..558O 569, 558

    2002 doi

  30. [39]

    Omukai K., 2001, @doi [ ] 10.1086/318296 , https://ui.adsabs.harvard.edu/abs/2001ApJ...546..635O 546, 635

    2001 doi

  31. [40]

    E., 1962, @doi [ApJ] 10.1086/147258 , 135, 195

    Osterbrock D. E., 1962, @doi [ApJ] 10.1086/147258 , 135, 195

    1962 doi

  32. [41]

    Padmanabhan H., Loeb A., 2024, @doi [ ] 10.1088/1475-7516/2024/10/059 , https://ui.adsabs.harvard.edu/abs/2024JCAP...10..059P 2024, 059

    2024 doi

  33. [42]

    B., Peebles P

    Partridge R. B., Peebles P. J. E., 1967, @doi [ ] 10.1086/149079 , https://ui.adsabs.harvard.edu/abs/1967ApJ...147..868P 147, 868

    1967 doi

  34. [43]

    C., Meszaros P., 1986, @doi [ ] 10.1086/164682 , https://ui.adsabs.harvard.edu/abs/1986ApJ...310..284P 310, 284

    Phillips K. C., Meszaros P., 1986, @doi [ ] 10.1086/164682 , https://ui.adsabs.harvard.edu/abs/1986ApJ...310..284P 310, 284

    1986 doi

  35. [44]

    Planck Collaboration et al., 2020, @doi [ ] 10.1051/0004-6361/201833910 , https://ui.adsabs.harvard.edu/abs/2020A&A...641A...6P 641, A6

    2020 doi

  36. [45]

    arXiv:1509.02237

    Ramdas A., Garcia N., Cuturi M., 2015, @doi [arXiv e-prints] 10.48550/arXiv.1509.02237 , https://ui.adsabs.harvard.edu/abs/2015arXiv150902237R p. arXiv:1509.02237

    2015 doi arXiv

  37. [46]

    B., 2006, @doi [ ] 10.1086/505327 , https://ui.adsabs.harvard.edu/abs/2006ApJ...647..709R 647, 709

    Rybicki G. B., 2006, @doi [ ] 10.1086/505327 , https://ui.adsabs.harvard.edu/abs/2006ApJ...647..709R 647, 709

    2006 doi

  38. [47]

    B., Dell'Antonio I

    Rybicki G. B., Dell'Antonio I. P., 1994, @doi [ApJ] 10.1086/174170 , 427, 603

    1994 doi

  39. [48]

    B., Lightman A

    Rybicki G. B., Lightman A. P., 1979, Radiative processes in astrophysics . John Wiley & Sons, Ltd

    1979

  40. [49]

    Semelin B., Combes F., Baek S., 2007, @doi [ ] 10.1051/0004-6361:20077965 , https://ui.adsabs.harvard.edu/abs/2007A&A...474..365S 474, 365

    2007 doi

  41. [50]

    Seon K.-i., Kim C.-G., 2020, @doi [ ] 10.3847/1538-4365/aba2d6 , https://ui.adsabs.harvard.edu/abs/2020ApJS..250....9S 250, 9

    2020 doi

  42. [51]

    Smith A., Safranek-Shrader C., Bromm V., Milosavljevi \'c M., 2015, @doi [ ] 10.1093/mnras/stv565 , https://ui.adsabs.harvard.edu/abs/2015MNRAS.449.4336S 449, 4336

    2015 doi

  43. [52]

    Smith A., Bromm V., Loeb A., 2017, @doi [ ] 10.1093/mnras/stw2591 , https://ui.adsabs.harvard.edu/abs/2017MNRAS.464.2963S 464, 2963

    2017 doi

  44. [53]

    Smith A., Tsang B. T. H., Bromm V., Milosavljevi \'c M., 2018, @doi [ ] 10.1093/mnras/sty1509 , https://ui.adsabs.harvard.edu/abs/2018MNRAS.479.2065S 479, 2065

    2018 doi

  45. [54]

    L., Hopkins P

    Smith A., Ma X., Bromm V., Finkelstein S. L., Hopkins P. F., Faucher-Gigu \`e re C.-A., Kere s D., 2019, @doi [ ] 10.1093/mnras/sty3483 , https://ui.adsabs.harvard.edu/abs/2019MNRAS.484...39S 484, 39

    2019 doi

  46. [55]

    Smith A., Kannan R., Garaldi E., Vogelsberger M., Pakmor R., Springel V., Hernquist L., 2022a, @doi [ ] 10.1093/mnras/stac713 , https://ui.adsabs.harvard.edu/abs/2022MNRAS.512.3243S 512, 3243

    doi

  47. [56]

    Smith A., et al., 2022b, @doi [ ] 10.1093/mnras/stac2641 , https://ui.adsabs.harvard.edu/abs/2022MNRAS.517....1S 517, 1

    doi

  48. [57]

    S., 2020, @doi [ ] 10.3847/1538-4357/abac02 , https://ui.adsabs.harvard.edu/abs/2020ApJ...901...41S 901, 41

    Song H., Seon K.-I., Hwang H. S., 2020, @doi [ ] 10.3847/1538-4357/abac02 , https://ui.adsabs.harvard.edu/abs/2020ApJ...901...41S 901, 41

    2020 doi

  49. [58]

    Tasitsiomi A., 2006a, @doi [ApJ] 10.1086/504460 , 645, 792

    doi

  50. [59]

    Tasitsiomi A., 2006b, @doi [ ] 10.1086/505682 , https://ui.adsabs.harvard.edu/abs/2006ApJ...648..762T 648, 762

    doi

  51. [60]

    M., Ferrara A., 2021, @doi [ ] 10.1093/mnras/stab876 , https://ui.adsabs.harvard.edu/abs/2021MNRAS.504...89T 504, 89

    Tomaselli G. M., Ferrara A., 2021, @doi [ ] 10.1093/mnras/stab876 , https://ui.adsabs.harvard.edu/abs/2021MNRAS.504...89T 504, 89

    2021 doi

  52. [61]

    Unno W., 1952, PASJ, 4, 100

    1952

  53. [62]

    Verhamme A., Schaerer D., Maselli A., 2006, @doi [ ] 10.1051/0004-6361:20065554 , https://ui.adsabs.harvard.edu/abs/2006A&A...460..397V 460, 397

    2006 doi

  54. [63]

    Yajima H., Li Y., Zhu Q., Abel T., 2012, @doi [ ] 10.1111/j.1365-2966.2012.21228.x , https://ui.adsabs.harvard.edu/abs/2012MNRAS.424..884Y 424, 884

    2012

  55. [64]

    Zheng Z., Miralda-Escud\' e J., 2002, @doi [ApJ] 10.1086/342400 , 578, 33

    2002 doi

  56. [65]

    Zheng Z., Wallace J., 2014, @doi [ ] 10.1088/0004-637X/794/2/116 , https://ui.adsabs.harvard.edu/abs/2014ApJ...794..116Z 794, 116

    2014 doi

  57. [66]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.