Fully time-dependent cloud formation from a non-equilibrium gas-phase in exoplanetary atmospheres (Corrigendum)

S. Kiefer; H. Lecoq-Molinos; Ch. Helling; N. Bangera; L. Decin

doi:10.1051/0004-6361/202554447e

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Erratum

This article is an erratum for:
[https://doi.org/10.1051/0004-6361/202347441]

Issue		A&A Volume 696, April 2025


Article Number		C1
Number of page(s)		5
Section		Planets, planetary systems, and small bodies
DOI		https://doi.org/10.1051/0004-6361/202554447e
Published online		28 March 2025

A&A, 696, C1 (2025)

Fully time-dependent cloud formation from a non-equilibrium gas-phase in exoplanetary atmospheres (Corrigendum)

S. Kiefer¹^,2^,3^★, H. Lecoq-Molinos²^,3^,1, Ch. Helling²^,3, N. Bangera²^,3 and L. Decin¹

¹ Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
² Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria
³ Institute for Theoretical Physics and Computational Physics, Graz University of Technology, Petersgasse 16 8010 Graz, Austria

^★ Corresponding author; sven.kiefer@kuleuven.be

Key words: astrochemistry / methods: analytical / planets and satellites: atmospheres / errata, addenda

Erratum for: A&A, 682, A150 (2024), https://doi.org/10.1051/0004-6361/202347441

A follow-up study on TiO₂ nucleation revealed a unit conversion error in the calculation of the original TiO₂ nucleation rates. This error concerns all two-body association and dissociation reaction rates of TiO₂ clusters, which were overestimated in the numerical evaluations.

A re-analysis with the correct nucleation rates was performed and confirmed the qualitative conclusions of the original manuscript. However, several results changed quantitatively. The corrected rates for TiO₂ cluster sizes 1 to 4 can be seen in Fig. 1, where $NU (i, j) : {({TiO}_{2})}_{i} + {({TiO}_{2})}_{i} \leftrightarrow {({TiO}_{2})}_{(i + j)} .$ ${\rm{NU}}(i,j):{\left( {{\rm{Ti}}{{\rm{O}}_2}} \right)_i} + {\left( {{\rm{Ti}}{{\rm{O}}_2}} \right)_i} \leftrightarrow {\left( {{\rm{Ti}}{{\rm{O}}_2}} \right)_{(i + j)}}.$ (1)

Evaluating the kinetic chemistry with the correct reaction rates leads to a slower nucleation of TiO₂ (see Fig. 2). The depletion of oxygen-bearing gas-phase species is not affected by this change. The change in the CH₄ number density due to the SiO-SiO₂ cycle is slightly less significant (see Fig. 3). Our analysis of the kinetic nucleation of TiO₂ reflects the longer nucleation timescales and the nucleation rate peaks between 0.1 to 1 second, which is two orders of magnitudes higher than before (see Fig. 4). However, the temperature range for which the formation of larger TiO₂ clusters is favoured changes by less than 50 K compared to the original manuscript (see Fig. 5). Lastly, our simulations of HD 209458 b show a lower nucleation rate (see Figs. 6, 7, and 8). The lower nucleation rate also leads to a temporary increase in the average cloud particle size. This is relevant if the cloud formation timescales are to be compared to other chemical or dynamical timescales in exoplanet atmospheres.

Still, the conclusion of the original work stands, that cloud particle surface reactions allow a catalytic SiO-SiO₂ cycle, which affects the CH₄ gas-phase number density. Nucleation is now found to reach constant cloud particle number densities within 100 seconds.

Fig. 1

Reaction rate coefficients for the formation of TiO₂ clusters up to (TiO₂)₄. All solid lines assume p_gas = 0.002 bar. Top: association reactions. Bottom: dissociation reactions.

Table 1

Offset values between number densities of given molecular species for different chemical networks.

Fig. 2

Concentrations of selected gas-phase species for p_gas = 0.002 bar at T_gas = 1378 K using different chemical kinetics networks. The diamond marker shows GGchem results that including equilibrium condensation.

Fig. 3

Differences in CH₄ abundance between chemical equilibrium and the full network for various temperatures and pressures.

Fig. 4

Cloud particle number densities (top) and nucleation rates (bottom) for TiO₂ nucleation with different N_max at p_gas = 0.02 bar and T_gas = 1379 K.

Fig. 5

Cloud particle concentrations for TiO₂ nucleation for a range of temperatures and pressures.

Fig. 6

Concentrations of selected gas-phase species (top), cloud particle number density (upper middle), mean cloud particle size (lower middle), and selected volume fractions (bottom) at the sub-stellar point, evening terminator, anti-stellar point, and morning terminator at p_gas = 0.002 bar. The sub-stellar point does not form clouds.

Fig. 7

Concentrations of selected gas-phase species (top), cloud particle number density (upper middle), mean cloud particle size (lower middle), and selected volume fractions (bottom) for logarithmically spaced pressures along the evening terminator.

Fig. 8

Volume fractions for T_gas - p_gas points of HD 209458 b. The sub-stellar point (p_gas = 0.002 bar, T_gas = 2026 K) is not shown since no cloud formation occurs.

Table 2

Bulk growth reactions considered for this study.

Acknowledgements

The authors thank Laura X. Worutowicz for her help with finding and correcting the unit conversion error.

References

Helling, Ch., Gourbin, P., Woitke, P., & Parmentier, V. 2019, A&A, 626, A133 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]

Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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All Tables

Table 1

Offset values between number densities of given molecular species for different chemical networks.

In the text

Table 2

Bulk growth reactions considered for this study.

In the text

All Figures

	Fig. 1 Reaction rate coefficients for the formation of TiO₂ clusters up to (TiO₂)₄. All solid lines assume p_gas = 0.002 bar. Top: association reactions. Bottom: dissociation reactions.
In the text

	Fig. 2 Concentrations of selected gas-phase species for p_gas = 0.002 bar at T_gas = 1378 K using different chemical kinetics networks. The diamond marker shows `GGchem` results that including equilibrium condensation.
In the text

	Fig. 3 Differences in CH₄ abundance between chemical equilibrium and the full network for various temperatures and pressures.
In the text

	Fig. 4 Cloud particle number densities (top) and nucleation rates (bottom) for TiO₂ nucleation with different N_max at p_gas = 0.02 bar and T_gas = 1379 K.
In the text

	Fig. 5 Cloud particle concentrations for TiO₂ nucleation for a range of temperatures and pressures.
In the text

	Fig. 6 Concentrations of selected gas-phase species (top), cloud particle number density (upper middle), mean cloud particle size (lower middle), and selected volume fractions (bottom) at the sub-stellar point, evening terminator, anti-stellar point, and morning terminator at p_gas = 0.002 bar. The sub-stellar point does not form clouds.
In the text

	Fig. 7 Concentrations of selected gas-phase species (top), cloud particle number density (upper middle), mean cloud particle size (lower middle), and selected volume fractions (bottom) for logarithmically spaced pressures along the evening terminator.
In the text

	Fig. 8 Volume fractions for T_gas - p_gas points of HD 209458 b. The sub-stellar point (p_gas = 0.002 bar, T_gas = 2026 K) is not shown since no cloud formation occurs.
In the text

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[1] Helling, Ch., Gourbin, P., Woitke, P., & Parmentier, V. 2019, A&A, 626, A133 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]