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arxiv: 2606.30157 · v1 · pith:7FM47YFYnew · submitted 2026-06-29 · 🌌 astro-ph.EP

CO snow lines are stabilised by the vertical transport of volatiles

Alfie Robinson , James E. Owen , Richard A. Booth This is my paper

Pith reviewed 2026-06-30 03:55 UTC · model grok-4.3

classification 🌌 astro-ph.EP
keywords protoplanetary discsCO snow linesnow surfacevolatile transportthermal instabilitydisc chemistryvertical structure
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The pith

Vertical transport of volatiles in protoplanetary discs creates two stable CO snow surfaces but prevents limit-cycle oscillations.

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

Protoplanetary discs have snow lines where CO and other volatiles freeze onto dust grains below certain temperatures. Earlier one-dimensional models predicted that thermal instability could cause these lines to oscillate in position over time. This work adds an ice-vapour chemistry solver to a two-dimensional disc code and finds that the full vertical structure produces two steady-state equilibrium positions for the snow surface. In time-dependent runs, however, the disc avoids the predicted limit cycles because the curved two-dimensional snow surface and the vertical movement of volatiles between layers damp the instability. Snow line locations are therefore expected to shift mainly through gradual changes in the disc's dust distribution rather than through regular periodic jumps.

Core claim

The CO snow line instability produces two stable equilibrium solutions for the snow surface once vertical disc structure is included, yet dynamically evolving simulations do not enter limit cycles because the geometry of the two-dimensional snow surface and the vertical transport of volatiles suppress the oscillatory behaviour.

What carries the argument

The two-dimensional snow surface together with vertical volatile transport in the cuDisc evolution code that includes an ice-vapour chemistry solver.

If this is right

  • Dynamically evolving snow lines due to instability are restricted to transient, stochastic events instead of regular oscillations.
  • The snow surface still changes substantially over the disc lifetime solely through evolution of the dust spatial distribution and grain sizes.
  • Two distinct steady-state stable positions exist for the snow surface when vertical structure is accounted for.
  • No limit-cycle behaviour appears in full dynamical calculations of the disc.

Where Pith is reading between the lines

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

  • Compositional changes in forming planets may be steadier than one-dimensional instability models suggested.
  • Similar damping by vertical transport could affect snow lines of other volatiles such as water.
  • Adding turbulence or magnetic fields to the model might change whether the two stable equilibria remain.
  • Comparing snow line radii across discs of different ages could reveal whether shifts are mostly transient or driven by dust evolution.

Load-bearing premise

The model's treatment of vertical transport and ice-vapour chemistry accurately represents the main physical processes at work.

What would settle it

A long-term simulation or observation that records regular, periodic shifts in the radial location of the CO snow line over the disc lifetime would show the claimed stabilisation does not hold.

Figures

Figures reproduced from arXiv: 2606.30157 by Alfie Robinson, James E. Owen, Richard A. Booth.

Figure 1
Figure 1. Figure 1: The left-hand panel shows grain-size-independent mass densities of ice and dust for vapour condensing onto a static distribution of dust grains. The dashed blue line shows the initial non-icy dust distribution, whilst the dashed green line is proportional to the total grain surface in a bin, i.e. ∝ 𝑛𝑘𝑎 2 𝑘 . The right-hand panel shows grain-size-independent mass densities of ice and dust evolving through c… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic showing the method used for updating the vapour column during hydrostatic equilibrium calculations. The left two cartoons show how the column is affected by a temperature decrease that causes the gas density to increase at lower altitudes. The dashed lines show the new altitudes of the column values that were initially at the heights 𝑍0 up to 𝑍3. The black line on the plot on the right shows the … view at source ↗
Figure 3
Figure 3. Figure 3: The optical constants used for CO when calculating opacities. The data (red dashed line) is taken from Gavdush et al. (2022) and the black line shows the extended optical constants used for calculations. The optical constants for H2O ice from Warren & Brandt (2008) are plotted for comparison. 10 5 10 4 10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 O p a city [c m 2 g 1 ] Wavelength [cm] Absorption MRN up to 1 cm… view at source ↗
Figure 4
Figure 4. Figure 4: Grain-size-distribution-averaged absorption and scattering opacities for grains with and without CO as a part of their composition. The grain-size distributions are set as MRN (Mathis et al. 1977) distributions with maximum grain-sizes of 1 𝜇m (orange lines) and 1 cm (blue lines). The base grain composition (without CO, shown by dashed lines) is the DSHARP composition (Birnstiel et al. 2018) and the mixed … view at source ↗
Figure 5
Figure 5. Figure 5: 2D ice density and temperature structure of stable equilibrium snow line solutions in the single-grain size model. Temperature contours (in K) are over-plotted on the density maps. 10 1 10 0 Solid: R<Rsnow Dashed: R>Rsnow s i 10 2 10 1 10 0 Height [au] 10 2 10 1 10 0 C u m ulativ e o ptic al d e pth, Dotted: hot soln. Dot-dashed: cold soln [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Cumulative optical depths to mid-plane and surface radiation, 𝜏𝑖 and 𝜏𝑠, as functions of height. For the optical depth to mid-plane radiation, the column is summed from the mid-plane up to the surface, and vice versa for the optical depth to the surface. The top panel shows how the optical depth changes on either side of the snow line, which lies at radius 𝑅snow, due to ice condensation affecting the solid… view at source ↗
Figure 7
Figure 7. Figure 7: Mid-plane snow line radii for the hot and cold solutions as a function of total volatile surface density. We pick a reference radius of 30 au for the total volatile surface density as this lies between the snow line solutions. The black circles indicate the two snow line solutions for the initial surface density. 4 DYNAMICAL EVOLUTION We now construct simulations to study whether the CO snow line can be un… view at source ↗
Figure 8
Figure 8. Figure 8: Snapshots of the vertically-isothermal simulation in the pre-jump cold state, during the hot state, and in the post-jump cold state. The top row shows the surface densities of dust (ice included), ice and vapour. The middle row shows the mid-plane temperature, 𝑇mid, whilst the bottom row shows the ice mass-fraction of the dust, 𝑓ice. timescales greater than those associated with the limit-cycles (kyr), mea… view at source ↗
Figure 10
Figure 10. Figure 10: The ice mass-fraction, vertical optical depth to surface and mid￾plane radiation fields (𝑇surf & 𝑇mid respectively), and mid-plane temperature as functions of time, at a radius of 39.6 au. After each jump back to the cold state from the hot state (at around 42 kyr and 155 kyr), the optical depth to surface radiation takes a longer amount of time to reach a maximum than that to the mid-plane radiation. Thi… view at source ↗
Figure 11
Figure 11. Figure 11: S-curves of stable equilibrium solutions for one cycle in mid-plane temperature (left panel), CO mass-flux, (right panel circles) and ice-density￾averaged Stokes number (right panel crosses) vs. CO surface density space for the vertically-isothermal simulation, at a radius of 39.6 au. There are no values for the average Stokes number whilst the disc is in the hot state. The points marked with black crosse… view at source ↗
Figure 12
Figure 12. Figure 12: Evolution of the fiducial full 2D simulation. The simulation has run for ∼ 4 Myr; line opacity increases with time in intervals of 0.4 Myr. The top row shows the surface densities of dust (ice included), ice and vapour, whilst the second row shows the mid-plane temperature. The small non￾continuous deviations in the surface density appear due to integration over the tightly-constrained, discontinuous vert… view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the fiducial full 2D simulation after perturbation. The left column shows the evolution after a perturbation to the ice density, whilst the right column shows the evolution after a perturbation to the total volatile density. Line opacity increases with time, with snapshots at 𝑡 = 0, 4, 27, 176 and 1150 yr. The top row shows the surface densities of dust (ice included), ice and vapour, whilst … view at source ↗
Figure 16
Figure 16. Figure 16: Spatial distributions of CO ice and vapour density in the full 2D simulation with lower turbulence (𝛼 = 10−4 ). Temperature contours (in K) are over-plotted for reference. 10 4 10 3 10 2 10 1 S urfa c e d e n sity [g c m 2 ] Dust Ice Vapour 40 60 80 Radius [au] 20 25 Mid-plane temperature [K] [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The steady-state reached for the full 2D simulation with a lower fragmentation velocity (𝑣frag = 1 m s−1 ). The top row shows the surface densities of dust (ice included), ice and vapour, whilst the second row shows the mid-plane temperature. 5 DISCUSSION The perturbation tests show that the disc can be unstable and would be able to enter a limit-cycle if the vapour reservoir was unable to supply the mid-… view at source ↗
Figure 18
Figure 18. Figure 18: The left panel shows the radial CO flux at a range of heights (height increases with line darkness); 0, 0.13, 0.3, 0.57, 0.85, 1.2 and 1.6 au. The flux is defined as positive in the radially-inward direction, i.e. towards the star. The right panel shows the the vertical CO flux at a radius of 60 au. The flux is defined as positive in the vertically-downward direction, i.e. towards the mid-plane. The verti… view at source ↗
read the original abstract

Volatile evolution in protoplanetary discs determines the compositional evolution of forming planets. Below their sublimation temperatures, volatiles freeze out from the vapour phase onto dust grains in the disc and transition to being dynamically-coupled to the dust component as opposed to the gas. The boundary between the ice and vapour phases is referred to as the snow line, when thought of as the mid-plane radius at which the phase transition occurs, or the snow surface, when viewed as a 2D (radial and vertical) structure in the disc. We investigate whether the CO snow line (and therefore snow surface) is thermally unstable and therefore liable to changes in its location during disc evolution using the disc evolution code cuDisc, to which we have added an ice-vapour chemistry solver. We find that the instability does lead to there being two steady-state stable equilibrium solutions for the snow surface when including the vertical structure. However, in dynamically-evolving simulations, the disc does not enter a limit-cycle - as seen in previous 1D models - due to the shape of the 2D snow surface and the vertical transport of volatiles. We therefore expect that dynamically evolution of snow lines due to instability is limited to transient, stochastic events rather than oscillatory behaviour with a regular period. However, we also expect the snow surface to evolve substantially during the disc lifetime solely due to changes in the thermal structure driven by evolution of the dust spatial structure and grain-size distribution - this we will explore in future models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper augments the cuDisc disc-evolution code with an ice-vapour chemistry solver and performs 2D simulations of CO snow surfaces in protoplanetary discs. It reports that vertical structure yields two stable equilibrium snow-surface solutions, yet dynamically evolving runs do not enter the limit-cycle oscillations previously seen in 1D models; the stabilization is attributed to the geometry of the 2D snow surface combined with vertical volatile transport. The authors conclude that any instability-driven evolution is limited to transient stochastic events and that longer-term snow-surface changes will be dominated by dust evolution.

Significance. If the numerical result is robust, the work shows that thermal instabilities do not produce periodic snow-line migration once vertical transport is included, thereby constraining the role of such instabilities in volatile delivery and planet-composition models. The finding redirects attention to dust spatial and grain-size evolution as the dominant driver of snow-surface migration over disc lifetimes.

major comments (2)
  1. [model setup and numerical implementation] Simulation description (model setup and numerical implementation): no resolution study, convergence test, or recovery of 1D analytic limits is presented for the vertical transport operator or the ice-vapour chemistry solver. Because the headline claim that vertical transport suppresses the limit cycle rests entirely on the fidelity of this operator, the absence of such tests is load-bearing for the central result.
  2. [dynamically-evolving simulations] Dynamically-evolving simulations section: the assertion that the 2D snow-surface shape prevents limit-cycle entry is not accompanied by a quantitative comparison of vertical mixing timescale versus thermal adjustment timescale, leaving the mechanistic link between vertical transport and stabilization incompletely demonstrated.
minor comments (1)
  1. [abstract] Abstract: the statement that 'the disc does not enter a limit-cycle' would benefit from a brief parenthetical note on the simulation duration or number of orbits examined to allow readers to gauge the strength of the negative result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive comments. We address each of the major comments below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: Simulation description (model setup and numerical implementation): no resolution study, convergence test, or recovery of 1D analytic limits is presented for the vertical transport operator or the ice-vapour chemistry solver. Because the headline claim that vertical transport suppresses the limit cycle rests entirely on the fidelity of this operator, the absence of such tests is load-bearing for the central result.

    Authors: We agree with the referee that additional numerical validation is necessary to support the reliability of our results. In the revised manuscript, we will include a resolution study for the vertical transport operator, convergence tests, and a demonstration of recovery of the 1D analytic limits for both the transport and chemistry components. These will be presented in an expanded model setup section. revision: yes

  2. Referee: Dynamically-evolving simulations section: the assertion that the 2D snow-surface shape prevents limit-cycle entry is not accompanied by a quantitative comparison of vertical mixing timescale versus thermal adjustment timescale, leaving the mechanistic link between vertical transport and stabilization incompletely demonstrated.

    Authors: We acknowledge that providing a quantitative comparison between the vertical mixing timescale and the thermal adjustment timescale would better illustrate the mechanism by which vertical transport stabilizes the snow surface. We will add this analysis, including timescale estimates derived from the simulation parameters, to the dynamically-evolving simulations section of the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from forward simulation of transport/chemistry equations

full rationale

The paper reports outcomes from numerical integration of ice-vapour chemistry and vertical transport operators inside cuDisc. The abstract and described results (two stable 2D equilibria, absence of limit cycles) follow directly from evolving the stated physical equations rather than any self-definitional mapping, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equation or claim reduces to its own inputs by construction. This is the expected non-finding for a simulation-based study whose central claim is externally falsifiable via the implemented operators.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the correctness of the cuDisc thermal and dynamical solver plus the newly added chemistry module; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The cuDisc code with the added ice-vapour chemistry solver accurately represents the thermal structure, dynamics, and volatile transport in the disc.
    All reported equilibria and the absence of limit cycles rest on this numerical model being faithful to the underlying physics.

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