Can thermodynamic equilibrium be established in planet-forming disks?

Christiane Helling; Inga Kamp; Jayatee Kanwar; Paul Rimmer; Peter Woitke

REVIEW 4 major objections 8 minor 1 cited by

Can thermodynamic equilibrium be established in planet-forming disks?

T0 review · 4 major / 8 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read No region of a typical planet-forming disk reaches thermodynamic equilibrium; only a tiny, warm, dense pocket shielded from cosmic rays behind the inner rim comes close.

desk verdict Useful new network and analysis, but the 'no TE anywhere' headline leans on the slowest N2 formation routes being complete, which the authors themselves doubt. read the letter →

arxiv 2505.13705 v1 pith:5XXEUDMU submitted 2025-05-19 astro-ph.EP

Jayatee Kanwar , Peter Woitke , Inga Kamp , Paul Rimmer , Christiane Helling This is my paper · ORCID

classification astro-ph.EP

keywords thermodynamicequilibriumprotoplanetarydiskschemicalnetworksChaiTeainnerdiskchemistrydetailedbalancecosmicraysJWST

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether thermodynamic equilibrium can ever be established in planet-forming disks, and its answer is no: in a typical T Tauri disk model no region attains equilibrium, with one near-exception in a tiny, warm, high-density pocket just behind the inner rim where cosmic rays are shielded. To make this test the authors build a new chemical network, ChaiTea, that merges disk chemistry with planetary-atmosphere chemistry and enforces detailed balance by pairing every gas-phase reaction with a reverse rate derived from Gibbs free energies. They verify that the network relaxes to equilibrium when radiation is removed, then show how photochemistry and cosmic rays push abundances away from it. This matters because the same inner-disk molecules probed by JWST, such as C2H2, HCN, CH4, CO2 and N2H+, change by orders of magnitude when the new network replaces the standard disk network.

What carries the argument

The load-bearing object is the ChaiTea chemical network, a merged 239-species network that combines interstellar and disk reaction databases with the STAND planetary network. Its defining mechanism is reaction reversal: every invertible gas-phase reaction is paired with its reverse, and the reverse rate coefficient is computed from the forward rate and the reaction's Gibbs free energy, enforcing detailed balance at equilibrium. This is supplemented by a Lindemann-Hinshelwood treatment of termolecular reactions that captures pressure-dependent stabilisation in the dense inner disk. The network is embedded in a thermochemical disk model, benchmarked against an independent equilibrium chemistry code, and diagnosed with two tools: a deviation measure $\sigma$ that averages logarithmic abundance differences between kinetic and equilibrium solutions, and eigenvalue-based chemical relaxation timescales obtained from the Jacobian of the rate equations.

What would settle it

Measure or compute the rate of the slowest N2-forming neutral-neutral reactions near 1200 K; a fast channel that shortens the $\sim 10^8$-year relaxation timescale would weaken the central 'no equilibrium' claim. Observationally, JWST mid-infrared spectra of a disk whose inner rim is directly viewed should show the ChaiTea prediction of reduced C2H2, reduced HCN, and increased CO2 inside 1 au; a spectrum matching the old standard network would refute the predicted kinetic structure.

Watch

Extended reading notes

Core claim

The central claim is that thermodynamic equilibrium cannot be established anywhere in a planet-forming disk. In the full 2D disk model, kinetic chemistry and equilibrium abundances never fall below about one order of magnitude apart everywhere ($\sigma \gtrsim 1$), and they differ by tens of orders of magnitude in the observable warm molecular layer. Only a small, warm, dense region in the midplane directly behind the inner rim ($r \lesssim 0.1$ au), shielded from cosmic rays, approaches equilibrium. At a density of $7.6\times 10^{13}$ cm$^{-3}$, the chemical relaxation timescale rises from about a month at 2000 K to roughly $10^8$ years at 1200 K, and exceeds the age of the universe below about 1100 K, with the slowest mode being the formation of N2 from other nitrogen carriers. Because kinetic equilibrium distributes carbon into CO and CO2, nitrogen into N2, and oxygen into H2O and CO, whereas thermodynamic equilibrium favours CH4, NH3 and H2O, switching to the new network changes the inner-disk abundances of JWST-visible molecules.

Load-bearing premise

The conclusion that thermodynamic equilibrium is nowhere established rests on the completeness of the ChaiTea network's slowest relaxation pathways, especially the neutral-neutral reactions that form N2 from other nitrogen-bearing molecules; if faster routes exist, more of the warm inner disk could relax toward equilibrium.

Editorial extensions

If this is right

Thermodynamic equilibrium cannot be assumed when modelling or interpreting inner-disk chemistry; kinetic chemistry with pressure-dependent and reverse reactions is required.
JWST-visible abundances change by orders of magnitude with the new network: C2H2 drops in the inner midplane, CO2 increases within 1 au, and N2H+ and CH+ become more abundant in the warm molecular layer and outer disk.
Spurious endothermic reactions with zero or tiny activation energies are automatically corrected by Gibbs-free-energy pairing; the paper identifies 45 such reactions in the large DIANA-standard network and 68 once STAND reactions are added, eliminating artefacts such as excess atomic hydrogen in the outer midplane.
Below roughly 1100 K, chemical relaxation toward equilibrium takes longer than the age of the universe, so cold disk regions cannot be interpreted as equilibrium mixtures even if cosmic rays and photochemistry were absent.

Reading between the lines

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

If faster neutral-neutral routes to N2 exist than the network currently contains, the $10^8$-year relaxation at 1200 K would shorten, and the warm inner-disk region approaching equilibrium could be larger than the paper's fiducial answer allows.
The same Gibbs-free-energy reversal and $\sigma$ diagnostic could be carried back to exoplanet atmosphere chemistry to map, in a common metric, where equilibrium assumptions for hot Jupiter spectra break down.
Because the auto-cleaning step found dozens of spurious endothermic reactions in an established disk network, applying the same audit to other astrochemical networks could change predicted outer-disk abundances of H, N2H+, and related tracers.
Running the same 2D disk model across a range of carbon-to-oxygen ratios would test whether the size and location of the tiny equilibrium pocket track the thermochemical shifts, giving observers a concrete target to confirm or refute the prediction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Jayatee et al. have built something the field can actually use: ChaiTea, a 239-species network that folds STAND's pressure-dependent termolecular reactions and Gibbs-derived reverse rates into the UMIST 2022 and DIANA disk chemistry, implemented in ProDiMo. The benchmark against GGchem is clean — <1% deviations down to 600 K when photorates and cosmic rays are off — and the auto-detection of 45–68 spurious endothermic reactions in standard disk networks is a tangible service to the community. The 2D sigma maps are a nice way to see where TE fails, and the changes they find for C2H2, CO2, and HCN in the warm inner disk plausibly matter for JWST interpretation. This deserves serious referee time.

The soft spots are just as real. First, the grid_0 agreement is partly built in: both ProDiMo and GGchem use the same Burcat thermodynamic polynomials, so the check demonstrates detailed-balance self-consistency, not independent validation. Second, the headline that "thermodynamic equilibrium cannot be established anywhere in the disk" rests on the slowest chemical mode being gas-phase N≡N bond formation. At 1200 K the relaxation time is ~10^8 yr by their own visual estimate, but they concede in Sect. 4.3 that missing neutral-neutral N2 formation routes could shorten this substantially — they tried N2H, NCN, HNCO and got only a factor 2–3, but that does not rule out faster pathways. Because the benchmark is time-independent, it cannot catch this; a missing fast pathway changes the steady-state composition not at all but can crash the relaxation timescale down to disk ages. If such routes exist, the warm inner rim region could relax toward TE, directly softening the abstract's claim. The paper itself is honest about this, which I credit, but the claim in the abstract goes beyond what the network can currently establish.

Third, the full network and data are not released with the paper, which is a real problem for a paper whose main deliverable is a network. The species tables and reaction lists are in the text, but not machine-readable.

My advice: send it to review, but tell the authors to soften the "anywhere" claim to "in the regions sampled by this network," release the network and input files, and add a dedicated test of N2 relaxation with an independent route, even a toy one. The paper is useful and the thinking is clear; it just overstates the certainty of its central negative result.

Referee Report

4 major / 8 minor

Summary. The paper develops a new chemical network, ChaiTea, for planet-forming disk models by merging the STAND exoplanet network, UMIST 2022, and the DIANA disk network, and by adding reverse reactions derived from Gibbs free energies and termolecular reactions via a Lindemann-Hinshelwood treatment. The authors verify that, in the absence of photorates, cosmic rays, and X-rays, the network relaxes to thermodynamic equilibrium by comparing single-point ProDiMo results with the equilibrium code GGchem. They then quantify deviations from equilibrium induced by a Planck radiation field and by cosmic-ray ionization, and compute chemical relaxation timescales. Finally, they compare 2D disk abundance maps from the standard DIANA network, the DIANA network with reverse reactions, and the full ChaiTea network. The headline result is that thermodynamic equilibrium cannot be established anywhere in a typical T Tauri disk, although a small, warm, high-density region behind the inner rim, shielded from cosmic rays, approaches equilibrium; a secondary result is that the slowest relaxation mode is N2 formation, with timescales as long as ~1e8 yr at 1200 K.

Significance. If the results hold, this is a timely and useful contribution. It brings pressure-dependent and reverse reactions, standard in exoplanet chemistry, into disk modeling, and it directly addresses a question of broad interest: when can equilibrium chemistry be used for inner disk observations with JWST and ALMA. The paper is strong on transparency: it defines a quantitative deviation metric σ, computes detailed balance tests (Table 6), reports multiple relaxation-timescale definitions (Table 7), and explicitly identifies spurious endothermic reactions in existing networks. The authors also acknowledge the main weakness of their own analysis, namely the possible incompleteness of the network for N≡N bond formation. For these reasons the work is a valuable reference point even if its strongest conclusion needs qualification.

major comments (4)

[§4.3, Table 7, Fig. 2] The central claim that thermodynamic equilibrium is never established anywhere in the disk rests on the identification of N2 formation as the slowest chemical mode, with relaxation timescales of ~1e8 yr at 1200 K and >1e10 yr below ~1100 K. However, the authors themselves state in §4.3 that 'we may miss some other neutral-neutral reaction pathways to form or destroy N≡N bonds.' The grid_0 benchmark in §4.1 cannot detect such a missing pathway because it solves the time-independent chemical equilibrium problem: a missing fast reaction leaves the converged equilibrium composition unchanged but can shorten the relaxation timescale by orders of magnitude. If an unlisted route (e.g., N+NH2, NH+NO, or grain-surface N2 formation) is fast at 1200 K, the warm inner-disk region behind the inner rim could relax to thermodynamic equilibrium on disk-evolutionary timescales, weakening the abstract's 'cannot be established anywhere' statement. The authors should either conduct a targeted literature search for N-N bond forming/destroying reactions and demonstrate that they do not change the conclusions, or consistently qualify all headline statements as 'within the current ChaiTea network.'
[Abstract vs §4.7 and §6] There is a direct numerical inconsistency in the reported deviation in the warm intermittent molecular layer. The Abstract states 'In the warm intermittent molecular layer, which is observable, σ≥10,' whereas §4.7 states 'there is ... the warm molecular intermittent surface layer where σ again reaches values as low as 5,' and the bullet list in §6 states 'There is a warm intermittent molecular layer where σ∼5.' Since this quantity is a headline result about the observability of disequilibrium, the authors must decide on the correct value and use it consistently.
[§4.1, §2.3] The verification against GGchem is not fully independent: §4.1 states that GGchem was run 'with a new option to use the same Burcat thermochemical data as used in this work.' Because both codes share the same Gibbs free energy data, the <1% agreement in grid_0 is primarily a detailed-balance self-consistency check of the kinetic network, not a validation of the thermochemical data themselves. The authors do compare formation enthalpies with NIST and ATcT for some species, which partially mitigates this, but the manuscript should explicitly frame the grid_0 benchmark as an internal consistency check and should justify that the shared thermochemical data are not the source of the reported disk conclusions.
[§4.2, Table 7] The three relaxation timescale definitions in Table 7 disagree by three to four orders of magnitude at 1200 K (τchem,1 = 1.68×10^4 yr, τchem,2 = 4.03×10^7 yr, visual τchem,3 ~1e8 yr). The paper explains that τchem,1 is misleading because it tracks individual species rather than collective modes, but it then admits that even the Jacobian-based τchem,2 is 'not fully satisfactory' due to numerical ill-conditioning of the Jacobian. Given that these timescales are load-bearing for the conclusion that TE cannot be established, the manuscript should report the condition numbers or another convergence diagnostic for the eigen-decomposition and should present the resulting uncertainty on the derived temperature below which relaxation exceeds the disk age.

minor comments (8)

[§4.1] The word 'studed' is a typo and should read 'studied.'
[Fig. 2 caption] The caption gives the density as 7.65×10^13 cm^-3, while the text and Table 7 use 7.56×10^13 cm^-3; the values should be harmonized.
[§4.4, Fig. 3] The text says the grid_0 deviations are '<1%' but the color scale in Fig. 3 spans 10^-4 to 10^2; a zoomed panel or an explicit color-bar range for the converged models would make the claim easier to verify.
[§4.6] The sentence 'the CRIs scale with n⟨H⟩' is imprecise; the ionization rate per unit volume scales linearly with n⟨H⟩, while the underlying rate coefficient is independent of density.
[§3.2] The reaction count summary '5539 (2462 +375+1155+1547)' is mathematically correct but could be clearer if the subtotals (2837+1155+1547 = 5539) were stated explicitly in the text.
[§3.3, Eq. (9)] The sentence after Eq. (9) says the auto-cleaning multiplies both Kf and Kr by a common factor when ΔRG°/R > γf; it would help to state explicitly that this preserves the ratio Kf/Kr and therefore detailed balance at equilibrium, since that property is central to the paper's verification.
[§5.1, Eq. (22)-(23)] The erroneous reaction is said to have a reaction enthalpy of '+6.5 eV' while the correct reaction is '+0.43 eV'; the second value lacks explicit units and should read '+0.43 eV'.
[§4.5, Eq. (20)] The radiative association H + e− → H− + hν is listed as a source of deviation from detailed balance, but the sentence would be clearer if it stated that this reaction is treated as non-invertible, which is why it breaks detailed balance.

Circularity Check

2 steps flagged · score 4.0 of 10

The grid_0 'verification' is a built-in consistency check because the reverse rates and the GGchem benchmark share the same Gibbs free-energy data; the paper's central no-TE claims are independent of that check.

self definitional [Sect. 4.1 (verification), using Eq. (8) of Sect. 2.2]
"For the latter, we used GGchem (Woitke et al. 2018) with a new option to use the same Burcat thermochemical data as used in this work."

The reverse rate coefficients are constructed from the reaction Gibbs free energy via Eq. (8), and those Gibbs energies are evaluated from the Burcat polynomials. GGchem's equilibrium abundances are computed from the same Burcat Gibbs data. Therefore any converged kinetic steady state in which every forward/reverse pair reaches detailed balance automatically satisfies the same mass-action equations that GGchem solves; the grid_0 agreement (sigma < 1%) is a self-consistency test of the implementation rather than an independent confirmation that the network 'achieves thermodynamic equilibrium'. The paper does not use this agreement to derive the central no-TE results, which instead come from the CR/photo grids and the 2D disk model.
fitted input called prediction [Sect. 2.3 (thermochemical data), Table B.1; benchmark in Sect. 4.1]
"In these cases we (i) fitted the thermochemical data found in Gleich-Gewichts-Chemie (GGchem) and exported them in Burcat format"

For roughly 30 species, the Gibbs free energies that enter Eq. (8) were taken from GGchem, and GGchem is then used as the 'independent' equilibrium reference in grid_0. For those species the agreement is enforced by construction. The impact is limited because the paper excludes Mg-, Na-, and Fe-bearing species from sigma and the dominant contributors are common molecules with standard Burcat data, so this fitted-input circularity does not drive the headline claims about cosmic rays and photo-processes.

full rationale

The central claims—deviations from thermodynamic equilibrium due to photoreactions and cosmic rays, the chemical relaxation timescales, and the 2D sigma structure—are outputs of the ChaiTea network with radiation processes that break detailed balance; they are not equivalent to the Gibbs-energy inputs. The grid_0 comparison is a built-in consistency check because both the reverse rates and the GGchem benchmark use the same Burcat data, and for about 30 species the data were fitted from GGchem itself. This is a real but bounded circularity in the verification step, so the score is moderate rather than zero. The N2-network incompleteness noted in Sect. 4.3 is a correctness risk (missing fast pathways could shorten the slowest relaxation mode), not a circularity, and should be weighed separately. No load-bearing self-citation chain or uniqueness import was found; citations to STAND and GGchem are normal prior-work references, not circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on thermodynamic consistency of the network and on the completeness of the reaction list; neither is externally validated, and the N2 problem shows the latter is not fully satisfied.

free parameters (1)

sigma abundance weight exponent = 0.2
Ad hoc exponent in Eq. (19) that sets how strongly trace species contribute to the deviation metric; chosen by hand, not from data.

assumptions (5)

standard math The law of mass action and detailed balance relate forward and reverse rate coefficients via Gibbs free energies (Eq. 8; Appendix A).
Standard thermodynamics for ideal gas mixtures, assumed throughout.
domain assumption All gas-phase reactions in the network are reversible, and reverse rates can be computed from thermodynamic data even when no measured reverse rate exists.
Central to ChaiTea; this is the exoplanet-chemistry assumption imported from STAND.
ad hoc to paper The kinetic network contains all chemically important pathways, especially for N2 formation.
The authors themselves note the N2 formation timescale might shrink with additional neutral-neutral reactions (Section 4.3); the 1200 K relaxation time and the no-TE conclusion depend on this.
domain assumption The single T Tauri disk model (Table C.1) is representative of planet-forming disks.
The statement that TE is nowhere established is drawn from one set of stellar and disk parameters.
domain assumption The adopted rate coefficients from UMIST 2022, STAND, and additional literature are correct.
Rate data are taken from databases without independent validation for disk conditions.

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Cite this review

Pith. "Pith review of Can thermodynamic equilibrium be established in planet-forming disks?." pith.science (2026). https://pith.science/paper/5XXEUDMU

@misc{pith2026250513705,
  author       = {Pith},
  title        = {Pith review of: Can thermodynamic equilibrium be established in planet-forming disks?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5XXEUDMU}},
  note         = {Machine review of arXiv:2505.13705}
}

read the original abstract

The inner regions of planet-forming disks are warm and dense. The chemical networks used for disk modelling so far were developed for a cold and dilute medium and do not include a complete set of pressure-dependent reactions. The chemical networks developed for planetary atmospheres include such reactions along with the inverse reactions related to the Gibb's free energies of the molecules. The chemical networks used for disk modelling are thus incomplete in this respect. We want to study whether thermodynamic equilibrium can be established in a planet-forming disk. We identify the regions in the disk most likely to reach thermodynamic equilibrium and determine the timescale over which this occurs. We employ the theoretical concepts used in exoplanet atmosphere chemistry for the disk modelling with PROtoplanetary DIsk MOdel ({\sc ProDiMo}). We develop a chemical network called CHemistry Assembled from exoplanets and dIsks for Thermodynamic EquilibriA ({\sc ChaiTea}) that is based on the UMIST 2022, STAND, and large DIscANAlysis (DIANA) chemical networks. It consists of 239 species. From the STAND network, we adopt the concept of reversing all gas-phase reactions based on thermodynamic data. We use single-point models for a range of gas densities and gas temperatures to verify that the implemented concepts work and thermodynamic equilibrium is achieved in the absence of cosmic rays and photoreactions including radiative associations and direct recombinations. We then study the impact of photoreactions and cosmic rays that lead to deviations from thermodynamic equilibrium. We explore the chemical relaxation timescales towards thermodynamic equilibrium. Lastly, we study the predicted 2D chemical structure of a typical T\,Tauri disk when using the new {\sc ChaiTea} network instead of the large DIANA standard network, including photorates, cosmic rays, X-rays, and ice....

Figures

Figures reproduced from arXiv: 2505.13705 by the authors.

**Figure 1.** Abundances of various species, ϵ(X) = nX/n⟨H⟩ , obtained with the time-independent ProDiMo models from grid_0 for density n⟨H⟩ = 7.56 × 1013 cm−3 (dots), compared to the corresponding GGchem models (dashed lines). The deviations at the lowest temperatures are due to numerical problems in converging the ProDiMo models. tained for C2H2 solving the rate network in the time-independent mode at T = 1200 K and n⟨H⟩ = 7.5… view at source ↗

**Figure 2.** Abundances, ϵ(X) = nX/n⟨H⟩ , of various species (X) as a function of time obtained with PRODIMO at 1200 K and 2000 K at a density of n⟨H⟩ =7.56 × 1013 cm−3 . The large coloured dots represent the abundances in thermodynamic equilibrium obtained with GGchem. war et al. (2024b) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Deviation σ (see Eq. 19), between particle densities obtained with time-independent ProDiMo (ni) and GGchem ( ◦ ni) at various temperatures and densities from grid_0. The blank rectangles at the bottom mark the models which do not converge. The numbers on the x-axis correspond to the ascending list of densities in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Same as [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Impact of cosmic-ray ionisation rate on deviations from thermodynamic equilibrium at different densities and temperatures from grid_cr. The adopted values of CRIs are listed at the top of each panel. The x- and y-axes are the same as in [PITH_FULL_IMAGE:figures/full_f…

**Figure 6.** Figure 6: Abundances of selected species in the 2D disk model calculated using chemical kinetics (left) and thermodynamic equilibrium (right). The white contours correspond to the tick values on the colorbars. have a binary character: either a molecule is very abundant and is th…

**Figure 8.** Figure 8: Deviation (σ) between kinetic chemical equilibrium and thermodynamic equilibrium in the 2D disk model. σ = 1 denotes a deviation of one order of magnitude between the ProDiMo and GGchem abundances. low as 5. Here, the gas-phase chemical reactions can compete with the…

**Figure 9.** Figure 9: The abundances of various species in the three models calculated using different chemical networks and rate databases outlined in [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: Continued from [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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, " * 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 ...

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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....

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