CARBON DIOXIDE IN EXOPLANETARY ATMOSPHERES: RARELY DOMINANT COMPARED TO CARBON MONOXIDE AND WATER IN HOT, HYDROGEN-DOMINATED ATMOSPHERES

There is an ongoing debate in the astrophysical community on how to interpret the measured spectra of the atmospheres of exoplanets. The first school of thought uses a series of "forward models": given a set of assumptions about the atmosphere, one computes forward and predicts a synthetic spectrum (e.g., Sudarsky et al. 2000; Burrows et al. 2007; Cahoy et al. 2010; Fortney et al. 2010; Seager & Sasselov 2000; Barman et al. 2011; Marley et al. 2012; Spiegel & Burrows 2012). This approach has the advantage that it is grounded by the laws of physics and chemistry (radiative and chemical equilibria). It draws from a rich heritage of, and has enjoyed success in, interpreting brown dwarf spectra (e.g., Marley et al. 1996; Tsuji et al. 1996; Burrows et al. 1997).

The disadvantage is that Nature may outsmart our preconceived notions of an atmosphere. The second school of thought uses "atmospheric retrieval," which is the attempt to shed these preconceived notions (e.g., Madhusudhan & Seager 2009; Benneke & Seager 2012; Lee et al. 2013; Line et al. 2014). This approach is sound when remote sensing and in situ measurements are available, such as for the Earth and the Solar System bodies, but its robustness is not assured when scrutinizing distant, unresolved point sources of light. For example, workers enforce global energy conservation (e.g., Madhusudhan & Seager 2009), but this does not guarantee local energy conservation (radiative equilibrium³ ; see the Appendix). Generally, the robustness of an interpretation via retrieval depends on the prior knowledge that one is assuming about the atmosphere—one's model is only as good as the assumptions one inserts into it.

In the present study, our goal is to study the gaseous chemistry of hot, hydrogen-dominated atmospheres. It may not apply to atmospheres with rocky surfaces, where outgassing may produce carbon-dioxide-dominated atmospheres. By "hot," we mean that water cannot condense out, at least in the photospheric regions observed by astronomers. Furthermore, if the inert gas of the atmosphere is a mixture of hydrogen and helium, we assume that atmospheric escape does not act to remove hydrogen, leaving behind a helium-dominated atmosphere that may contain more carbon dioxide than water (Hu & Seager 2014; Hu et al. 2015).

To move forward, it would be useful to uncover generic physical and chemical trends that one may insert into these inversion techniques as priors. In the current paper, we focus on the study of carbon dioxide (CO₂), which is typically a minor carrier of carbon in exoplanetary atmospheres. Understanding its abundance relative to other molecules such as carbon monoxide (CO), water (H₂O) and methane (CH₄) is directly relevant to deciphering the carbon-to-oxygen ratio (C/O) of an atmosphere (Line et al. 2013),

$$\begin{eqnarray}&&{\rm{C}}/{\rm{O}}\approx \displaystyle \frac{{\tilde{n}}_{{\mathrm{CH}}_{4}}+{\tilde{n}}_{\mathrm{CO}}+{\tilde{n}}_{{\mathrm{CO}}_{2}}}{{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}+{\tilde{n}}_{\mathrm{CO}}+2{\tilde{n}}_{{\mathrm{CO}}_{2}}},\end{eqnarray} \tag{ 1 }$$

where the mixing ratios are represented by $\tilde{n}$ and labeled with self-explanatory subscripts. Specifically, if CO is the dominant molecule, then we have ${\rm{C}}/{\rm{O}}\approx 1$. By contrast, if CO₂ dominates, then we have ${\rm{C}}/{\rm{O}}\approx 0.5$. For example, this has direct consequences for the debate on whether WASP-12b is carbon-rich (Madhusudhan et al. 2011) or carbon-poor (Line et al. 2014). Madhusudhan et al. (2011) report CO being dominant over CO₂ (see their Figure 2), while Line et al. (2014) report CO₂ being dominant over CO (see their Table 3).

The main conclusion of this study is that carbon dioxide is almost always subdominant in hot, hydrogen-dominated atmospheres, compared to carbon monoxide and water, because

$$\begin{eqnarray}\displaystyle \frac{{\tilde{n}}_{{\mathrm{CO}}_{2}}}{{\tilde{n}}_{\mathrm{CO}}} & = & \displaystyle \frac{{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}}{{K}_{2}^{\prime }}\sim {\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}},\\ \displaystyle \frac{{\tilde{n}}_{{\mathrm{CO}}_{2}}}{{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}} & = & \displaystyle \frac{{\tilde{n}}_{\mathrm{CO}}}{{K}_{2}^{\prime }}\sim {\tilde{n}}_{\mathrm{CO}}.\end{eqnarray} \tag{ 2 }$$

We will elucidate what "almost" means. At this juncture, it is not so important to understand what the dimensionless function ${K}_{2}^{\prime }$ is—this will come later. It is more important to note that it has no dependence on pressure and that $1/{K}_{2}^{\prime }\sim 0.1$–1 for a broad range of temperatures (Figures 1 and 2). Since we typically have ${\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\ll 1$ and ${\tilde{n}}_{\mathrm{CO}}\ll 1$ in a hydrogen- or helium-dominated atmosphere, we may conclude that, in most situations,

$$\begin{eqnarray}&&\displaystyle \frac{{\tilde{n}}_{{\mathrm{CO}}_{2}}}{{\tilde{n}}_{\mathrm{CO}}},\displaystyle \frac{{\tilde{n}}_{{\mathrm{CO}}_{2}}}{{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}}\ll 1.\end{eqnarray} \tag{ 3 }$$

These trends have previously been elucidated by Madhusudhan (2012), Kopparapu et al. (2012), Moses et al. (2013a), Moses et al. (2013b) and Venot et al. (2014), in one form or another, using numerical calculations of chemical kinetics. Our present study takes a complementary approach and corroborates this conclusion via a set of analytical calculations of equilibrium chemistry, spanning a broad range of temperatures, pressures, C/O values and metallicities that are representative of all of the currently observable and characterizable exoplanetary atmospheres. (Whenever it is relevant, we point out the limitations of our analytical approach.)

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Our main technical contribution is to present novel analytical solutions for chemical systems with carbon dioxide, carbon monoxide, water, methane, acetylene (C₂H₂) and molecular hydrogen, which allow us to efficiently explore vast swarths of parameter space. We use tabulated values of the Gibbs free energy to relate the equilibrium constants to temperature and pressure. We then fold these calculations of atmospheric chemistry with analytical models of temperature–pressure profiles to explore its effects in a wide variety of exoplanetary atmospheres.

We consider the conversion of methane to carbon monoxide via the net reaction (Burrows & Sharp 1999; Lodders & Fegley 2002; Moses et al. 2011),

$$\begin{eqnarray}&&{\mathrm{CH}}_{4}+{{\rm{H}}}_{2}{\rm{O}}\leftrightarrows \mathrm{CO}+3{{\rm{H}}}_{2}.\end{eqnarray} \tag{ 4 }$$

This reaction alone is insufficient for modeling carbon-rich atmospheres, because hydrocarbons such as acetylene and hydrogen cyanide (HCN) are expected to appear at high temperatures (Madhusudhan 2012; Moses et al. 2013a; Venot et al. 2015). We supplement it with another net reaction for producing carbon dioxide (Lodders & Fegley 2002; Moses et al. 2011),

$$\begin{eqnarray}&&{\mathrm{CO}}_{2}+{{\rm{H}}}_{2}\leftrightarrows \mathrm{CO}+{{\rm{H}}}_{2}{\rm{O}}.\end{eqnarray} \tag{ 5 }$$

If molecular hydrogen is vastly more abundant than water, then the production of carbon monoxide is favored.

The dimensional equilibrium constant of the first reaction is

$$\begin{eqnarray}&&{K}_{\mathrm{eq}}^{\prime }=\displaystyle \frac{{n}_{\mathrm{CO}}{n}_{{{\rm{H}}}_{2}}^{3}}{{n}_{{\mathrm{CH}}_{4}}{n}_{{{\rm{H}}}_{2}{\rm{O}}}}=\displaystyle \frac{{\tilde{n}}_{\mathrm{CO}}{n}_{{{\rm{H}}}_{2}}^{2}}{{\tilde{n}}_{{\mathrm{CH}}_{4}}{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}},\end{eqnarray} \tag{ 6 }$$

which may be normalized to obtain

$$\begin{eqnarray}&&{K}^{\prime }\equiv \displaystyle \frac{{K}_{\mathrm{eq}}^{\prime }}{{n}_{{{\rm{H}}}_{2}}^{2}}.\end{eqnarray} \tag{ 7 }$$

For the second reaction, the equilibrium constant is already normalized,

$$\begin{eqnarray}&&{K}_{\mathrm{eq},2}^{\prime }\equiv \displaystyle \frac{{n}_{\mathrm{CO}}{n}_{{{\rm{H}}}_{2}{\rm{O}}}}{{n}_{{{\rm{H}}}_{2}}{n}_{{\mathrm{CO}}_{2}}}=\displaystyle \frac{{\tilde{n}}_{\mathrm{CO}}{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}}{{\tilde{n}}_{{\mathrm{CO}}_{2}}}\equiv {K}_{2}^{\prime }.\end{eqnarray} \tag{ 8 }$$

Note that the number densities of the molecules marked by tildes have been normalized by ${n}_{{{\rm{H}}}_{2}}$, while those of the atoms will be normalized by ${n}_{{\rm{H}}}$. The former are the mixing ratios, while the latter are the normalized elemental abundances.

Stoichiometric book-keeping (counting the number of atoms of each species) yields

$$\begin{eqnarray}{n}_{{\mathrm{CH}}_{4}}+{n}_{\mathrm{CO}}+{n}_{{\mathrm{CO}}_{2}} & = & {n}_{{\rm{C}}},\\ {n}_{{{\rm{H}}}_{2}{\rm{O}}}+{n}_{\mathrm{CO}}+2{n}_{{\mathrm{CO}}_{2}} & = & {n}_{{\rm{O}}},\\ 4{n}_{{\mathrm{CH}}_{4}}+2{n}_{{{\rm{H}}}_{2}{\rm{O}}}+2{n}_{{{\rm{H}}}_{2}} & = & {n}_{{\rm{H}}}.\end{eqnarray} \tag{ 9 }$$

Manipulating these equations and using the equilibrium constants yield a cubic equation for the mixing ratio of water,

$$\begin{eqnarray}&&{{ \mathcal C }}_{3}{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}^{3}+{{ \mathcal C }}_{2}{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}^{2}+{{ \mathcal C }}_{1}{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}+{{ \mathcal C }}_{0}=0,\end{eqnarray} \tag{ 10 }$$

where we have defined

$$\begin{eqnarray}{{ \mathcal C }}_{3} & = & \displaystyle \frac{{K}^{\prime }}{{K}_{2}^{\prime }}(2{\tilde{n}}_{{\rm{O}}}-4{\tilde{n}}_{{\rm{C}}}-1)\\ {{ \mathcal C }}_{2} & = & {K}^{\prime }\left[2({\tilde{n}}_{{\rm{O}}}-{\tilde{n}}_{{\rm{C}}})-1+\displaystyle \frac{2}{{K}_{2}^{\prime }}({\tilde{n}}_{{\rm{O}}}-2{\tilde{n}}_{{\rm{C}}})\right],\\ {{ \mathcal C }}_{1} & = & 2{\tilde{n}}_{{\rm{O}}}+4{\tilde{n}}_{{\rm{C}}}-1+2\;{K}^{\prime }({\tilde{n}}_{{\rm{O}}}-{\tilde{n}}_{{\rm{C}}}),\\ {{ \mathcal C }}_{0} & = & 2{\tilde{n}}_{{\rm{O}}}.\end{eqnarray} \tag{ 11 }$$

Unlike when carbon dioxide is excluded, we do not obtain a quadratic equation for the mixing ratio of methane (Heng et al. 2015). We solve this cubic equation using the polynomial.polyroots routine in Python.

The rest of the mixing ratios may be obtained using

$$\begin{eqnarray}{\tilde{n}}_{{\mathrm{CH}}_{4}} & = & \displaystyle \frac{2{\tilde{n}}_{{\rm{C}}}(1+{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}})}{1+{K}^{\prime }{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}+{K}^{\prime }{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}^{2}/{K}_{2}^{\prime }-4{\tilde{n}}_{{\rm{C}}}},\\ {\tilde{n}}_{\mathrm{CO}} & = & {K}^{\prime }{\tilde{n}}_{{\mathrm{CH}}_{4}}{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}},\\ {\tilde{n}}_{{\mathrm{CO}}_{2}} & = & \displaystyle \frac{{\tilde{n}}_{\mathrm{CO}}{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}}{{K}_{2}^{\prime }}.\end{eqnarray} \tag{ 12 }$$

In the low-temperature limit, we obtain ${\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\approx 2{\tilde{n}}_{{\rm{O}}}$ and ${\tilde{n}}_{{\mathrm{CH}}_{4}}\approx 2{\tilde{n}}_{{\rm{C}}}$, such that the ratio of methane to water abundances is the carbon-to-oxygen ratio (Heng et al. 2015). At low temperatures, water and methane each sequester an extra atom of oxygen and carbon, respectively, at the expense of carbon monoxide.

Following Heng et al. (2015), we include acetylene as a proxy for all of the hydrocarbons that may form at high temperatures and in carbon-rich situations, via the net reaction (Lodders & Fegley 2002; Moses et al. 2011),

$$\begin{eqnarray}&&2{\mathrm{CH}}_{4}\leftrightarrows {{\rm{C}}}_{2}{{\rm{H}}}_{2}+3{{\rm{H}}}_{2}.\end{eqnarray} \tag{ 13 }$$

Its equilibrium constant is

$$\begin{eqnarray}&&{K}_{\mathrm{eq},3}^{\prime }=\displaystyle \frac{{n}_{{{\rm{C}}}_{2}{{\rm{H}}}_{2}}{n}_{{{\rm{H}}}_{2}}^{3}}{{n}_{{\mathrm{CH}}_{4}}^{2}}=\displaystyle \frac{{\tilde{n}}_{{{\rm{C}}}_{2}{{\rm{H}}}_{2}}{n}_{{{\rm{H}}}_{2}}^{2}}{{\tilde{n}}_{{\mathrm{CH}}_{4}}^{2}},\end{eqnarray} \tag{ 14 }$$

which we also normalize,

$$\begin{eqnarray}&&{K}_{3}^{\prime }\equiv \displaystyle \frac{{K}_{\mathrm{eq},3}^{\prime }}{{n}_{{{\rm{H}}}_{2}}^{2}}.\end{eqnarray} \tag{ 15 }$$

Stoichiometric book-keeping is generalized to

$$\begin{eqnarray}{n}_{{\mathrm{CH}}_{4}}+{n}_{\mathrm{CO}}+{n}_{{\mathrm{CO}}_{2}}+2{n}_{{{\rm{C}}}_{2}{{\rm{H}}}_{2}} & = & {n}_{{\rm{C}}},\\ {n}_{{{\rm{H}}}_{2}{\rm{O}}}+{n}_{\mathrm{CO}}+2{n}_{{\mathrm{CO}}_{2}} & = & {n}_{{\rm{O}}},\\ 4{n}_{{\mathrm{CH}}_{4}}+2{n}_{{{\rm{H}}}_{2}{\rm{O}}}+2{n}_{{{\rm{H}}}_{2}}+2{n}_{{{\rm{C}}}_{2}{{\rm{H}}}_{2}} & = & {n}_{{\rm{H}}}.\end{eqnarray} \tag{ 16 }$$

Instead of a cubic equation for the mixing ratio of water, we obtain a pair of coupled quadratic equations for the mixing ratios of water and methane,

$$\begin{eqnarray}&&\displaystyle \frac{{K}^{\prime }{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}^{2}{\tilde{n}}_{{\mathrm{CH}}_{4}}}{{K}_{2}^{\prime }}+2\;{K}_{3}^{\prime }{\tilde{n}}_{{\mathrm{CH}}_{4}}^{2}(1-{\tilde{n}}_{{\rm{C}}})-2{\tilde{n}}_{{\rm{C}}}{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\\ &&\quad +{\tilde{n}}_{{\mathrm{CH}}_{4}}+{K}^{\prime }{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}{\tilde{n}}_{{\mathrm{CH}}_{4}}-4{\tilde{n}}_{{\rm{C}}}{\tilde{n}}_{{\mathrm{CH}}_{4}}-2{\tilde{n}}_{{\rm{C}}}=0,\\ &&\displaystyle \frac{2\;{K}^{\prime }{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}^{2}{\tilde{n}}_{{\mathrm{CH}}_{4}}}{{K}_{2}^{\prime }}-2\;{K}_{3}^{\prime }{\tilde{n}}_{{\rm{O}}}{\tilde{n}}_{{\mathrm{CH}}_{4}}^{2}+{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}(1-2{\tilde{n}}_{{\rm{O}}})\\ &&\quad +{K}^{\prime }{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}{\tilde{n}}_{{\mathrm{CH}}_{4}}-4{\tilde{n}}_{{\rm{O}}}{\tilde{n}}_{{\mathrm{CH}}_{4}}-2{\tilde{n}}_{{\rm{O}}}=0.\end{eqnarray} \tag{ 17 }$$

The preceding pair of equations looks deceptively simple, but it is actually difficult to solve numerically as the mixing ratios span more than 20 orders of magnitude in value across the range of temperatures and pressures we are interested in. It is challenging to maintain numerical stability across such an enormous range of values. Taking an analytical approximation is tricky, because it is difficult to judge which terms to drop or neglect. It turns out that we may employ an algebraic trick if we recast the equations in (17) in terms of ${\tilde{n}}_{\mathrm{CO}}$, rather than ${\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}$, which allows us to combine them into a single equation that is essentially a quadratic equation for the mixing ratio of CO,

$$\begin{eqnarray}&&\displaystyle \frac{{\tilde{n}}_{\mathrm{CO}}^{2}}{{K}^{\prime }{K}_{2}^{\prime }}+\displaystyle \frac{{\tilde{n}}_{\mathrm{CO}}}{{K}^{\prime }}(1+2{\tilde{n}}_{{\rm{C}}}-2{\tilde{n}}_{{\rm{O}}})-2\;{K}_{3}^{\prime }{\tilde{n}}_{{\mathrm{CH}}_{4}}^{3}(1-{\tilde{n}}_{{\rm{C}}}+{\tilde{n}}_{{\rm{O}}})\\ &&\quad -{\tilde{n}}_{{\mathrm{CH}}_{4}}^{2}-2{\tilde{n}}_{{\mathrm{CH}}_{4}}({\tilde{n}}_{{\rm{O}}}-{\tilde{n}}_{{\rm{C}}})(2{\tilde{n}}_{{\mathrm{CH}}_{4}}+1)=0.\end{eqnarray} \tag{ 18 }$$

Notice how the mixing ratios of carbon monoxide and methane are no longer coupled to each other within the same equation—there are no "mixed" terms, unlike for each equation in (17) between water and methane. This property has the virtue that we may cleanly take the approximation ${\tilde{n}}_{{\rm{C}}},{\tilde{n}}_{{\rm{O}}}\ll 1$ and end up with a relatively simple expression for the mixing ratio of carbon monoxide,

$$\begin{eqnarray}&&{\tilde{n}}_{\mathrm{CO}}\approx {K}^{\prime }{\tilde{n}}_{{\mathrm{CH}}_{4}}[2\;{K}_{3}^{\prime }{\tilde{n}}_{{\mathrm{CH}}_{4}}^{2}+{\tilde{n}}_{{\mathrm{CH}}_{4}}+2({\tilde{n}}_{{\rm{O}}}-{\tilde{n}}_{{\rm{C}}})].\end{eqnarray} \tag{ 19 }$$

Despite not being exact, the preceding expression retains generality, because the only assumption we have made so far is that the elemental abundances are small compared to hydrogen. It is less obvious how to take the ${\tilde{n}}_{{\rm{C}}},{\tilde{n}}_{{\rm{O}}}\ll 1$ approximation directly using the equations in (17).

The expression for ${\tilde{n}}_{\mathrm{CO}}$ leads to an approximate expression for the mixing ratio of water,

$$\begin{eqnarray}&&{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\approx 2\;{K}_{3}^{\prime }{\tilde{n}}_{{\mathrm{CH}}_{4}}^{2}+{\tilde{n}}_{{\mathrm{CH}}_{4}}+2({\tilde{n}}_{{\rm{O}}}-{\tilde{n}}_{{\rm{C}}}),\end{eqnarray} \tag{ 20 }$$

and a quintic equation for the mixing ratio of methane,

$$\begin{eqnarray}&&\displaystyle \sum _{i=0}^{5}{{ \mathcal A }}_{i}{\tilde{n}}_{{\mathrm{CH}}_{4}}^{i}\approx 0,\end{eqnarray} \tag{ 21 }$$

where the coefficients are

$$\begin{eqnarray}{{ \mathcal A }}_{5} & = & \displaystyle \frac{8\;{K}^{\prime }{K}_{3}^{\prime 2}}{{K}_{2}^{\prime }},\\ {{ \mathcal A }}_{4} & = & \displaystyle \frac{8\;{K}^{\prime }{K}_{3}^{\prime }}{{K}_{2}^{\prime }},\\ {{ \mathcal A }}_{3} & = & \displaystyle \frac{2\;{K}^{\prime }}{{K}_{2}^{\prime }}[1+8\;{K}_{3}^{\prime }({\tilde{n}}_{{\rm{O}}}-{\tilde{n}}_{{\rm{C}}})]+2\;{K}^{\prime }{K}_{3}^{\prime },\\ {{ \mathcal A }}_{2} & = & \displaystyle \frac{8\;{K}^{\prime }}{{K}_{2}^{\prime }}({\tilde{n}}_{{\rm{O}}}-{\tilde{n}}_{{\rm{C}}})+2\;{K}_{3}^{\prime }+{K}^{\prime },\\ {{ \mathcal A }}_{1} & = & \displaystyle \frac{8\;{K}^{\prime }}{{K}_{2}^{\prime }}{({\tilde{n}}_{{\rm{O}}}-{\tilde{n}}_{{\rm{C}}})}^{2}+1+2\;{K}^{\prime }({\tilde{n}}_{{\rm{O}}}-{\tilde{n}}_{{\rm{C}}}),\\ {{ \mathcal A }}_{0} & = & -2{\tilde{n}}_{{\rm{C}}}.\end{eqnarray} \tag{ 22 }$$

The mixing ratios of CO and CO₂ may be obtained using the second and third equations in (12), respectively, while we have ${\tilde{n}}_{{{\rm{C}}}_{2}{{\rm{H}}}_{2}}={K}_{3}^{\prime }{\tilde{n}}_{{\mathrm{CH}}_{4}}^{2}$. An indication that this equation is correct comes from the fact that it automatically yields ${\tilde{n}}_{{\mathrm{CH}}_{4}}\approx 2{\tilde{n}}_{{\rm{C}}}$ when all of the normalized equilibrium constants vanish (i.e., the low-temperature limit). Figure 3 shows the benchmarking of our carbon-rich versus carbon-poor solutions, which match perfectly when ${\rm{C}}/{\rm{O}}\lt 1$.

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This (approximate) quintic equation that governs the mixing ratio of methane is a novel result that generalizes the work of Burrows & Sharp (1999) and Heng et al. (2015). Burrows & Sharp (1999) excluded both acetylene and carbon dioxide in the analytical solution listed in their Appendix, while Heng et al. (2015) did not consider carbon dioxide.

Let the ${{ \mathcal R }}_{\mathrm{univ}}$ be the universal gas constant, μ be the mean molecular weight, $m=\mu {m}_{{\rm{u}}}$ be the mean molecular mass, ${m}_{{\rm{u}}}$ is the atomic mass unit and ${k}_{{\rm{B}}}={{ \mathcal R }}_{\mathrm{univ}}/{N}_{{\rm{A}}}$ be the Boltzmann constant. If we denote the mass density by ρ, the volume by V, the temperature by T, the number of particles by N and the number density by n, then the ideal gas law may be expressed as

$$\begin{eqnarray}&&P={{nk}}_{{\rm{B}}}T=\displaystyle \frac{N{{ \mathcal R }}_{\mathrm{univ}}T}{{{VN}}_{{\rm{A}}}},\end{eqnarray} \tag{ 23 }$$

where ${N}_{{\rm{A}}}=6.02214129\times {10}^{23}$ mol⁻¹ is Avogrado's constant. A mole of matter contains exactly ${N}_{{\rm{A}}}$ particles. By definition, ${m}_{{\rm{u}}}{N}_{{\rm{A}}}\equiv 1$ g mol⁻¹. We write the ideal gas law in this way, because we wish to express the universal gas constant in mks units as ${{ \mathcal R }}_{\mathrm{univ}}=8.3144621$ J K⁻¹ mol⁻¹ in order to utilize the data from the thermodynamic databases. There is some confusion over the definition of the universal gas constant, because one may only switch between its erg K⁻¹ mol⁻¹ and erg K⁻¹ g⁻¹ forms, without paying a unit conversion penalty, when dealing with cgs units.

In a departure from the formalism of Heng et al. (2015), we write G as the Gibbs free energy, rather than the specific Gibbs free energy. It follows that the scaling relation involving it becomes

$$\begin{eqnarray}&&\displaystyle \frac{G}{N/{N}_{{\rm{A}}}}=\displaystyle \frac{{G}_{0}}{N/{N}_{{\rm{A}}}}+{{ \mathcal R }}_{\mathrm{univ}}T\;\mathrm{ln}\left(\displaystyle \frac{P}{{P}_{0}}\right),\end{eqnarray} \tag{ 24 }$$

where G₀ is the Gibbs free energy at the reference pressure (P₀). Typically, it is the molar Gibbs free energy at a reference pressure (${\tilde{G}}_{0}\equiv {G}_{0}{N}_{{\rm{A}}}/N$), and not G₀, that is tabulated in thermodynamic databases. It has units of J mol⁻¹.

We need to distinguish between the different definitions of the equilibrium constant (Visscher & Moses 2011; Heng et al. 2015). The dimensionless equilibrium constant is (e.g., Jacobson 2005)

$$\begin{eqnarray}&&{K}_{\mathrm{eq}}=\mathrm{exp}\left(-\displaystyle \frac{{\rm{\Delta }}{\tilde{G}}_{\mathrm{0,1}}}{{{ \mathcal R }}_{\mathrm{univ}}T}\right),\end{eqnarray} \tag{ 25 }$$

where ${\rm{\Delta }}{\tilde{G}}_{\mathrm{0,1}}\equiv {\rm{\Delta }}{G}_{\mathrm{0,1}}{N}_{{\rm{A}}}/N$ and ${\rm{\Delta }}{G}_{\mathrm{0,1}}$ is the change in Gibbs free energy going from the reactants to the products, at the reference pressure, associated with the first net reaction. ${K}_{\mathrm{eq}}$ should not be confused with the dimensional equilibrium constants we have used so far, which have been rendered dimensionless by dividing by ${n}_{{{\rm{H}}}_{2}}^{2}$.

If we focus on the first net reaction, then the dimensionless (${K}_{\mathrm{eq}}$), dimensional (${K}_{\mathrm{eq}}^{\prime }$) and normalized (${K}^{\prime }$) equilibrium constants are related by (Heng et al. 2015)

$$\begin{eqnarray}&&{K}_{\mathrm{eq}}^{\prime }={n}_{0}^{2}{K}_{\mathrm{eq}}={n}_{{{\rm{H}}}_{2}}^{2}{K}^{\prime },\end{eqnarray} \tag{ 26 }$$

where ${n}_{0}={P}_{0}/{k}_{{\rm{B}}}T$ is the number density corresponding to the reference pressure and $P={n}_{{{\rm{H}}}_{2}}{k}_{{\rm{B}}}T$. Thus, we obtain

$$\begin{eqnarray}&&{K}^{\prime }={\left(\displaystyle \frac{{P}_{0}}{P}\right)}^{2}\mathrm{exp}\left(-\displaystyle \frac{{\rm{\Delta }}{\tilde{G}}_{\mathrm{0,1}}}{{{ \mathcal R }}_{\mathrm{univ}}T}\right).\end{eqnarray} \tag{ 27 }$$

For the reaction involving carbon dioxide, we have

$$\begin{eqnarray}&&{K}_{2}^{\prime }=\mathrm{exp}\left(-\displaystyle \frac{{\rm{\Delta }}{\tilde{G}}_{\mathrm{0,2}}}{{{ \mathcal R }}_{\mathrm{univ}}T}\right).\end{eqnarray} \tag{ 28 }$$

For the reaction involving acetylene, we have

$$\begin{eqnarray}&&{K}_{3}^{\prime }={\left(\displaystyle \frac{{P}_{0}}{P}\right)}^{2}\mathrm{exp}\left(-\displaystyle \frac{{\rm{\Delta }}{\tilde{G}}_{\mathrm{0,3}}}{{{ \mathcal R }}_{\mathrm{univ}}T}\right).\end{eqnarray} \tag{ 29 }$$

Here, the pressure P is interpreted as the total pressure of the atmosphere, which is exerted mostly by H₂ in hydrogen-dominated atmospheres.

Table 1 lists calculations or measurements of the molar Gibbs free energies of the molecules involved in the present study, taken from the NIST-JANAF database.⁴ Table 2 lists the change in molar Gibbs free energy, at the reference pressure of ${P}_{0}=1\;\mathrm{bar}$, for all three net reactions, which allow us to relate ${K}^{\prime }$, ${K}_{2}^{\prime }$ and ${K}_{3}^{\prime }$ to T and P (Figure 2).

Table 2.  Molar Gibbs Free Energies of the Three Net Reactions (${P}_{0}=1$ bar)

T	${\rm{\Delta }}{\tilde{G}}_{\mathrm{0,1}}$	${\rm{\Delta }}{\tilde{G}}_{\mathrm{0,2}}$	${\rm{\Delta }}{\tilde{G}}_{\mathrm{0,3}}$
(K)	(kJ mol−1)	(kJ mol−1)	(kJ mol−1)
500	96.378	20.474	262.934
600	72.408	16.689	237.509
700	47.937	13.068	211.383
800	23.114	9.593	184.764
900	−1.949	6.249	157.809
1000	−27.177	3.021	130.623
1100	−52.514	−0.107	103.282
1200	−77.918	−3.146	75.840
1300	−103.361	−6.106	48.336
1400	−128.821	−8.998	20.797
1500	−154.282	−11.828	−6.758
1600	−179.733	−14.600	−34.315
1700	−205.166	−17.323	−61.865
1800	−230.576	−20.000	−89.403
1900	−255.957	−22.634	−116.921
2000	−281.308	−25.229	−144.422
2100	−306.626	−27.789	−171.898
2200	−331.911	−30.315	−199.353
2300	−357.162	−32.809	−226.786
2400	−382.38	−35.275	−254.196
2500	−407.564	−37.712	−281.586
2600	−432.713	−40.123	−308.953
2700	−457.830	−42.509	−336.302
2800	−482.916	−44.872	−363.633
2900	−507.97	−47.211	−390.945
3000	−532.995	−49.528	−418.243

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To apply our calculations of equilibrium chemistry to atmospheres, we need to know their thermal structures. To this end, we employ the analytical temperature–pressure profiles of Heng et al. (2012, 2014), which generalized the pure-absorption models of Guillot (2010) to include non-isotropic scattering in both the visible and infrared range of wavelengths. These models require a small number of input parameters: the equilibrium temperature (${T}_{\mathrm{eq}}$), the interior/internal temperature (${T}_{\mathrm{int}}$), the visible opacity (${\kappa }_{\mathrm{vis}}$), the infrared opacity (${\kappa }_{\mathrm{IR}}$) and the Bond albedo (${A}_{{\rm{B}}}$). Two of these parameters are directly observable or inferable quantities (${T}_{\mathrm{eq}}$ and ${A}_{{\rm{B}}}$). In principle, the opacities may be the outcome of a retrieval calculation. The internal temperature is a quantity that is unconstrained by the observations for transiting exoplanets, but we will adopt a finite value to include its effect. We do not explore the effects of collision-induced absorption, scattering by large particles in the infrared or Gaussian cloud decks, as these are secondary effects.

It is important to note that these temperature–pressure profiles are formal solutions of the radiative transfer equation and thus enforce local and global energy conservation—with the former being radiative equilibrium—by construction (see Appendix). They are not ad hoc fitting or parametric functions, which are commonly used in retrieval models. Rather, they are first-principle calculations with simplifying approximations taken, the most major of which is that starlight and thermal emission are grouped into separate wavebands.

In Figure 4, we show calculations of the mixing ratios of carbon dioxide, carbon monoxide, water, methane and acetylene. We set ${\tilde{n}}_{{\rm{O}}}=5\times {10}^{-4}$, which is the approximate value at the solar photosphere. For a sub-solar carbon abundance (${\rm{C}}/{\rm{O}}=0.1$), water is the dominant molecule regardless of temperature (Moses et al. 2013b). Methane dominates carbon monoxide at low temperatures (Burrows & Sharp 1999; Lodders & Fegley 2002) and the trend reverses at high temperatures depending on the pressure (Madhusudhan 2012). Carbon dioxide is dominated by methane at low temperatures, but the trend reverses in a pressure-independent manner, because the number of reactants (two) versus products (two) is the same in the chemical reaction involving the production of carbon dioxide. Acetylene always remains subdominant in a carbon-poor environment. As we increase C/O to 1, 10 and 100, the system becomes water-poor (Madhusudhan 2012). The trend between methane and carbon monoxide persists in a qualitative sense, but the transition occurs at different temperatures. At high temperatures, acetylene starts to dominate both methane and carbon dioxide (Madhusudhan 2012; Moses et al. 2013a; Venot et al. 2015).

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A key limitation of our approach is that graphite starts to condense out of the gas when the carbon-to-oxygen ratio and/or metallicity are greatly enhanced (Moses et al. 2013b). For example, Moses et al. (2013b) have shown that, at $P=0.1\;\mathrm{bar}$, graphite formation occurs at ${\rm{C}}/{\rm{O}}\gtrsim 0.7$ and $T\gtrsim 700$ K when the metallicity is 300 times above solar; their Figure 7 also shows that, at solar ${\rm{C}}/{\rm{O}}$, graphite forms when the metallicity is 1000 times solar and $T\lesssim 800$ K. Generally, graphite formation and rain out are expected to maintain the gas-phase ${\rm{C}}/{\rm{O}}$ near unity (see Moses et al. 2013b and references therein). We note that Hu & Seager (2014) also did not include graphite formation in their calculations.

In Figure 5, we divide the mixing ratio of carbon dioxide by that of the other molecules. The striking thing we notice is that carbon dioxide is always subdominant compared to carbon monoxide and water, across the entire range of temperatures considered, and for ${\rm{C}}/{\rm{O}}=0.1$, 1, 10 and 100. Physically, the relative scarcity of carbon dioxide stems from the fact that it both requires the production of carbon monoxide and successfully competing with it and water for oxygen atoms. Furthermore, ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}}$ is insensitive to pressure for the reason previously mentioned, unless acetylene starts to become dominant. By contrast, a higher pressure favors the production of methane over carbon dioxide as the reverse reaction is favored, because the number of products (four) exceeds the number of reactants (two)—this is a manifestation of Le Châtelier's principle.

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Next, we wish to compute the abundance of carbon dioxide relative to carbon monoxide and water in model atmospheres and fully explore the effects of pressure variations. We pick ${\rm{C}}/{\rm{O}}=0.5$ for illustration. In Figure 6, we create temperature–pressure profiles of model atmospheres with three different equilibrium temperatures: ${T}_{\mathrm{eq}}=600$, 1200 and 2500 K. These values are representative of a broad range of currently characterisable exoplanetary atmospheres from GJ 436b to HD 189733b to WASP-12b. For our other parameters, we choose typical, plausible or illustrative values: $g={10}^{3}\;\mathrm{cm}$ s⁻², ${T}_{\mathrm{int}}=300$ K, ${\kappa }_{\mathrm{vis}}=0.01$ cm² g⁻¹ and ${\kappa }_{\mathrm{IR}}=0.02$ cm² g⁻¹. These choices imply that the infrared photosphere resides at a pressure of

$$\begin{eqnarray}&&{P}_{\mathrm{IR}}\sim \displaystyle \frac{g}{{\kappa }_{\mathrm{IR}}}=50\;\mathrm{mbar}.\end{eqnarray} \tag{ 30 }$$

Starlight is mostly deposited at a pressure of $0.63g/{\kappa }_{\mathrm{vis}}$ (Heng et al. 2012, 2014). For each value of ${T}_{\mathrm{eq}}$, we create three variations: pure absorption, a finite Bond albedo (which we set to ${A}_{{\rm{B}}}=0.5$) and a temperature inversion (which we create by doubling the value of the visible opacity to ${\kappa }_{\mathrm{vis}}=0.04$ cm² g⁻¹). This gives us a total of 9 model atmospheres.

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For each model atmosphere, we calculate ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}}$ and ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}$. Across all 9 models (see Figure 6), we find that the mixing ratio of carbon dioxide is always subdominant, compared to carbon monoxide and water, by at least an order of magnitude—often more. Thus, we have demonstrated the conclusion previously stated in Equation (3), at least for elemental oxygen abundances that are solar.

Our chemical-equilibrium calculations do not account for disequilibrium chemistry that may arise from atmospheric mixing. Workers constructing one-dimensional models of the atmosphere have traditionally used a diffusion coefficient, commonly termed an "eddy mixing coefficient," to collectively mimic advection, convection, diffusion and turbulence (e.g., Moses et al. 2011, 2013a, 2013b; Venot et al. 2012, 2014; Agúndez et al. 2014; Hu & Seager 2014; Hu et al. 2015). In three dimensions, atmospheric circulation hardly resembles diffusion with equator-to-pole circulation cells that penetrate down to ∼10 bar across the temperature range we are examining (Showman et al. 2009; Heng et al. 2011a, 2011b; Perna et al. 2012; Kataria et al. 2013; Mayne et al. 2014). In the case of GJ 436b, the atmospheric circulation penetrates down to only ∼1 bar (Lewis et al. 2010). (For a review of the atmospheric dynamics of hot exoplanetary atmospheres, see Heng & Showman 2015.) Furthermore, it is often overlooked that the intense stellar heating associated with close-in, transiting exoplanets flattens their temperature–pressure profiles and suppresses convection (Heng et al. 2011a). Thus, the association between eddy diffusion/mixing and convection becomes even more tenuous.

Nevertheless, for the sake of discussion, we invoke the simplest approach for including atmospheric mixing: the "quenching approximation," where one locates the point in the atmosphere where the chemical and dynamical timescales equate. Above this point (i.e., at lower pressures or higher altitudes), the chemical abundances are frozen to their quench-point values. A caveat is that different chemical species will generally have different quench points. If carbon dioxide is subdominant compared to carbon monoxide and water everywhere in the atmosphere, then locating the quench point becomes irrelevant—quenching will not alter this outcome. Generally, the conclusion reached in Equation (3) is unaffected by atmospheric dynamics and mixing. For atmospheric mixing to alter our conclusion requires substantially enhanced metallicities and low temperatures—in Section 4.3, we will quantify what "substantially enhanced" and "low" means.

Photochemistry is expected to enhance the production of acetylene (Line et al. 2011; Moses et al. 2011),

$$\begin{eqnarray}&&2\mathrm{CO}+{{\rm{H}}}_{2}\to {{\rm{C}}}_{2}{{\rm{H}}}_{2}+2{\rm{O}},\\ &&2{\mathrm{CH}}_{4}\to {{\rm{C}}}_{2}{{\rm{H}}}_{2}+3{{\rm{H}}}_{2},\end{eqnarray} \tag{ 31 }$$

where the first and second net reactions proceed via the photolysis of carbon monoxide (into C and O atoms, which act as radicals) and water (into the O atom and the OH radical), respectively. The first reaction is relevant only for $P\lesssim 1$ μbar, while the second reaction is the dominant one for photospheric pressures (Moses et al. 2011). The second net reaction is essentially the same as the one we consider in the present study for thermochemistry. Thus, if it is enhanced via photochemistry, then it can only lead to enhanced acetylene production at the expense of carbon dioxide. Overall, the presence of photochemistry strengthens, rather than weakens, our conclusion.

Our results, which are analytical except for the use of a numerical solver for the quintic equation, appear to be supported by more sophisticated calculations in the literature that focus on somewhat narrower regimes of parameter space (e.g., Moses et al. 2011; Kopparapu et al. 2012; Madhusudhan 2012; Venot et al. 2012, 2014; Hu & Seager 2014). Exceptions include Line et al. (2011), who focused on the hot Neptune GJ 436b and studied the effects of enhanced metallicities; they reported ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}}\gtrsim 1$ at some locations in their model atmospheres with $50\times$ the solar metallicity. Moses et al. (2013b) demonstrated that carbon dioxide could be the dominant gas by mass, over a temperature range consistent with that found in GJ 436b, if the elemental carbon abundance is enhanced by four orders of magnitude over its solar value. Agúndez et al. (2014) also noted that ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}$ increases dramatically with metallicity.

We interpret an enhanced metallicity to be an increased value of the elemental abundance of oxygen (${\tilde{n}}_{{\rm{O}}}$) relative to its solar value ($5\times {10}^{-4}$). In Figure 7, we repeat our calculations for an enhanced metallicity of $50\times$ solar, but we keep the carbon-to-oxygen ratio at its solar value (${\rm{C}}/{\rm{O}}=0.5$) as metallicity and C/O are independent quantities. We also focused on a lower temperature range that is representative of GJ 436b. We are thus able to reproduce the same qualitative behavior as Line et al. (2011): ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}}\gtrsim 1$ at $T\approx 500$–600 K. The cross-over point appears to occur at about 600 K. Despite the enhanced metallicity, our GJ 436b-like calculation displays ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\ll 1$ throughout. We also repeated the calculations in Figure 6 and find that ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}},{\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\ll 1$ for HD 189733b- and WASP-12b-like atmospheres, despite the enhanced metallicity.

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Figure 8 expresses the effects of metallicity in a more general way. We examine pressures of 1 μbar and 10 bar, because these values bracket the conceivable range of pressures probed by the infrared observations. We examine pressures up to 10 bar, because this is the maximum depth to which atmospheric circulation penetrates for the range of equilibrium temperatures we are interested in (Perna et al. 2012). We see that atmospheres that have T = 600 K always have ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\lt 1$ and only have ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}}\gt 1$ when the metallicity is enhanced by more than an order of magnitude relative to solar. By the time the temperature reaches T = 1000 K, we rarely have ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}}\gt 1$ unless one entertains implausible metallicities reaching $\sim {10}^{3}$ times that of solar.

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A few considerations give pause to contemplating such high metallicities. To begin with, we note that the elemental abundances of carbon and oxygen in Jupiter are enhanced by only about an order of magnitude relative to solar (Wong et al. 2004). Miller & Fortney (2011) have shown, using interior structure calculations applied to mass and radius measurements, that gas-giant exoplanets have bulk enhancements of metallicities ∼1–10 times of their parent stars. Furthermore, carbon-rich stars appear to be rare (Teske et al. 2014).

Our results have important consequences for retrieval studies, which is that chemical abundances cannot be allowed to completely roam free. For atmospheres with temperatures of 1000 K and above, one may wish to set, as a prior in one's retrieval procedure, that the mixing ratio of CO₂ cannot exceed that of CO and H₂O,

$$\begin{eqnarray}&&\displaystyle \frac{{\tilde{n}}_{{\mathrm{CO}}_{2}}}{{\tilde{n}}_{\mathrm{CO}}}\lt 1\;\mathrm{and}\ \displaystyle \frac{{\tilde{{\rm{n}}}}_{{\mathrm{CO}}_{2}}}{{\tilde{{\rm{n}}}}_{{{\rm{H}}}_{2}{\rm{O}}}}\lt 1.\end{eqnarray} \tag{ 32 }$$

Retrieval outcomes that violate these criteria, for hot atmospheres, should be flagged as being unphysical.

We join the debate on the atmosphere of the hot Jupiter WASP-12b. Madhusudhan et al. (2011) has previously reported that ${\rm{C}}/{\rm{O}}\gt 1$. Their Figure 1 shows four best-fit models with ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}},{\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\ll 1$, but an inspection of their Figure 2 reveals that, while ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}}\lt 1$ appears to hold for their ensemble of best-fit models (which are colored purple), the same models also seem to have ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\gt 1$. By contrast, Line et al. (2014) reported that ${\rm{C}}/{\rm{O}}\lt 1$ for WASP-12b, but this appears to be based on retrievals that yield ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}},{\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\gg 1$, as stated in their Table 3 and displayed in their Figure 3. Furthermore, Stevenson et al. (2014) have claimed that Line et al. (2014) do not include acetylene or hydrogen cyanide in their analysis, which drives the retrieval calculations to "unrealistically large" abundances of CO₂. It is thus unsurprising that the two sets of authors reach different conclusions regarding the carbon-to-oxygen ratio of WASP-12b, even without discussing the controversy surrounding the data themselves (Crossfield et al. 2012; Stevenson et al. 2014). In other words, some of these reported retrievals are chemically impossible (but see Moses et al. 2013a). To be fair, Line et al. (2014) did remark: "One might question how realistic solutions with such high abundances of CO₂ may be. Such high CO₂ abundances are generally not thermochemically permissible in highly reducing hot-Jupiter-like atmospheres." Line et al. (2014) then performed a separate retrieval analysis by setting a hard limit on the mixing ratio of CO₂ to be 10⁻⁵ and found that it has no effect on the CH₄ and CO abundances, although the retrieved H₂O abundance is increased by 2 to 3 orders of magnitude.

Stevenson et al. (2014) present an improved analysis of the dayside emission of WASP-12b, which included new data from the Spitzer Space Telescope. They did not enforce the priors in Equation (32), so as to allow for uncertainties associated with the absorption opacities and for the possibility of enhanced helium abundances. This permits both physically plausible and implausible retrievals to be reported and for the rejection of implausible solutions after the fact. Stevenson et al. (2014) concluded that oxygen-rich atmospheres require implausibly high abundances of carbon dioxide. Both Madhusudhan (2012) and Stevenson et al. (2014) concluded that the dayside emission spectrum of WASP-12b favors a carbon-rich interpretation.

More recently, Kreidberg et al. (2015) obtained a near-infrared transmission spectrum, which probes a series of water lines originating from the terminators of the atmosphere of WASP-12b, using the Wide Field Camera 3 on the Hubble Space Telescope. Using a thermochemically consistent retrieval analysis, they ruled out a carbon-rich scenario for the terminator region of WASP-12b at more than $3\sigma$-level confidence.

Within the context of our study, we find that the atmosphere of WASP-12b is too hot for enhanced metallicity to play a role (see Figures 6–8) in boosting the CO₂ abundance over that of CO and H₂O. If we repeat our calculations in Figure 8 for T = 2000–3000 K (not shown), we obtain the result that the metallicity needs to be enhanced by about 4 orders of magnitude for WASP-12b—a statement that is independent of pressure, because of the high temperatures involved. Another consequence is that, since ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{{{\rm{C}}}_{2}{{\rm{H}}}_{2}}$ appears to vary from $\gg 1$ to $\ll 1$ as C/O increases, this pair of molecules may be used as a diagnostic for the carbon-to-oxygen ratio (see also Venot et al. 2015), although this will be challenging as their spectral lines will probably be dominated by those of the other molecules.

Generally, past claims in the literature of ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{\mathrm{CO}}\gg 1$ and/or ${\tilde{n}}_{{\mathrm{CO}}_{2}}/{\tilde{n}}_{{{\rm{H}}}_{2}{\rm{O}}}\gg 1$ should be viewed with skepticism, unless a chemical mechanism (e.g., enhanced metallicity at low temperatures) has been elucidated. If CO₂ is reported to be constrained by a retrieval calculation but CO is not, then it should also be viewed skeptically.