Skip to content

Chemistry of Ti and V compounds

Titanium and vanadium oxides (TiO, VO) are strong optical absorbers thought to cause thermal inversions in highly irradiated atmospheres. VULCAN includes a first kinetic treatment of titanium and vanadium species, used in the WASP-33b case study (Section 2.7 of Tsai et al. 2021 1).

Species and thermodynamic data

The species list is expanded with Ti, TiO, TiO\(_2\), TiH, TiC, TiN, V, and VO. Of these, only Ti, TiO, and TiO\(_2\) have NASA-polynomial thermodynamic data; the remainder are sourced as follows:

Species Thermodynamic source
Ti, TiO, TiO\(_2\) NASA polynomials
TiH Burrows et al. (2005) 2, from ab-initio Gibbs free energy
TiC Woitke et al. (2018) 3
TiN, V, VO Tsuji (1973) 4

Rate-coefficient estimates

High-temperature kinetics data for Ti/V species are essentially nonexistent, so the rates are estimated:

  1. Where an analogous transition-metal (e.g. Fe) reaction is measured at high temperature, the same rate coefficient is adopted.
  2. Otherwise, the temperature dependence is estimated from transition-state theory: for an endothermic reaction the activation energy (the exponential term in the Arrhenius form) is approximated by the enthalpy difference between products and reactants, assuming the transition-state energy increase is small relative to the enthalpy difference for radical reactions.
  3. The pre-exponential factor is then adjusted to match the reference value at low temperature.

The adopted Ti/V reactions and rates are tabulated in Table B1 of the paper 1. For photolysis, TiO, TiO\(_2\), TiH, TiC, and VO are included, with UV cross sections estimated from FeO (Chestakov et al. 2005 5) at 252.39 nm and the threshold scaled by each species' bond dissociation energy.

Implementation in VULCAN

Unlike the transport, photochemistry, and condensation routines, the Ti/V chemistry is not hard-coded in the Python modules. It enters entirely through the data layer that the general machinery already supports:

  • the Ti/V reactions (Table B1) live in the network file and are parsed by op.ReadRate.read_rate like any other reaction, including the modified-Arrhenius and three-body forms;
  • the thermodynamic data sit in thermo/NASA9/ (with the externally sourced fits for TiH, TiC, etc.), and are reversed to enforce equilibrium through the same make_chem_funs.make_Gibbs / op.ReadRate.rev_rate path described in Chemical networks;
  • the Ti/V photodissociation branches are handled by compute_J using their tabulated cross sections, exactly as for the C–H–N–O–S photolysis reactions.

In other words, modeling Ti/V is a matter of selecting a network and thermo dataset that include these species; no dedicated code path is required.

  1. Tsai, S.-M., Malik, M., Kitzmann, D., et al. (2021). A comparative study of atmospheric chemistry with VULCAN. The Astrophysical Journal, 923(2), 264. https://doi.org/10.3847/1538-4357/ac29bc ↩↩

  2. Burrows, A., Dulick, M., Bauschlicher, C. W., et al. (2005). Spectroscopic constants, abundances, and opacities of the TiH molecule. The Astrophysical Journal, 624(2), 988. https://doi.org/10.1086/429366 ↩

  3. Woitke, P., Helling, C., Hunter, G. H., et al. (2018). Equilibrium chemistry down to 100 K. Astronomy & Astrophysics, 614, A1. https://doi.org/10.1051/0004-6361/201732193 ↩

  4. Tsuji, T. (1973). Molecular abundances in stellar atmospheres. II. Astronomy & Astrophysics, 23, 411. ↩

  5. Chestakov, D. A., Parker, D. H., & Baklanov, A. V. (2005). Iron monoxide photodissociation. The Journal of chemical physics, 122(8). https://doi.org/10.1063/1.1844271 ↩