Photochemistry and Spectral Characterization of Temperate and Gas-rich Exoplanets

We use the atmospheric structure and cloud formation model in Hu (2019) to simulate the pressure–temperature profile and potential gas depletion by condensation in temperate and cold exoplanets.

We have updated the model with a routine to compute the condensation of the NH₄SH cloud, in a way similar to that of the equilibrium cloud condensation model of Atreya & Romani (1985). In short, we compare the products of the partial pressure of NH₃ and H₂S with the equilibrium constant of the reaction that produces an NH₄SH solid (Lewis 1969), and partition the NH₃ and H₂S in excess to form the NH₄SH solid cloud in each atmospheric layer. We have verified that the resulting NH₄SH cloud density and pressure level is consistent with those of the previously published models when applied to a Jupiter-like planet (e.g., Atreya et al. 1999).

Another update is that the model now traces the concentration of NH₃ in liquid-water cloud droplets when applicable. The model of Hu (2019) has included the dissolution of NH₃ in the liquid-water droplets. By additionally tracing the concentration of NH₃ in droplets, we have now taken into account the non-ideal effects when the NH₃ solution is non-dilute. When the mass ratio between NH₃ and H₂O in the droplet is >0.05, we replace Henry's law with the vapor pressure of NH₃ in equilibrium with the solution (Perry & Green 2007). The latter merges the solubility in the Henry's law regime to that in the Raoult's law regime smoothly. We also apply the vapor pressure of H₂O in equilibrium with the solution, which can be substantially smaller than that with pure water when the solution is non-dilute (i.e., Raoult's law). While the impact of these processes on the overall atmospheric composition of the planets studied in this paper—planets warmer than Jupiter—is small, these processes may control the mixing ratio of H₂O and NH₃ in the atmospheres of even colder planets (De Pater et al. 1989; Romani et al. 1989).

We use the general-purpose photochemical model in Hu et al. (2012, 2013) to simulate the photochemical products in the middle atmospheres of temperate and cold exoplanets. The photochemical model includes a carbon chemistry network and a nitrogen chemistry network and their interactions (Hu et al. 2012). The photochemical model also includes a sulfur chemistry network and calculates the formation of H₂SO₄ and S₈ aerosols when applicable (Hu et al. 2013).

We have made several updates to the original reaction network (Hu et al. 2012), and they are listed in Table 1. We have checked the main reactions that produce, remove, and exchange C₁ and C₂ hydrocarbons in the Jovian atmosphere (Gladstone et al. 1996; Moses et al. 2005) and updated rate constants when more recent values in the relevant temperature range are available in the NIST Chemical Kinetics Database. We have added low-pressure or high-pressure rate constants for three-body reactions if any of them were missing in the original reaction rate list. Certain reactions important for the hydrocarbon chemistry do not have a directly usable rate constant expression in the NIST database; rather, their rates are fitted on experimental data or estimated by Moses et al. (2005). We have also added several reactions that involve NH because it may be produced by NH₃ photodissociation, and updated the rate constant of an important reaction NH₂ + CH₄ → NH₃ + CH₃ to the latest calculated value. Lastly, we have removed two reactions that were incorrectly included, CH₄ + C₂H₂ → C₂H₃ + CH₃ and C₂H₆ + C₂H₂ → C₂H₃ + C₂H₅, because the reactant should have been CH₂ = C.

Table 1. Reactions and Rate Constants Updated with Respect to Hu et al. (2012)

Reaction	Rate	Reason for Update	Source
H + C2H3 → C2H2 + H2	1.2 × 10−12 T0.5	Revise rate	Moses et al. (2005)
H + C2H4 → C2H3 + H2	$5.0\times {10}^{-12}{\left(T/298\right)}^{1.93}\exp (-6520/T)$	Revise rate	NIST
H + C2H5 → 2 CH3	1.25 × 10−10	Revise rate	NIST
CH + CH4 → C2H4 + H	$9.1\times {10}^{-11}{\left(T/298\right)}^{-0.9}$ (T > 295 K)	Revise rate	NIST
	$1.06\times {10}^{-10}{\left(T/298\right)}^{-1.04}\exp (-36/T)$ (T ≤ 295 K)
CH3 + H2 → CH4 + H	$2.31\times {10}^{-14}{\left(T/298\right)}^{2.24}\exp (-3220.0/T)$	Revise rate	NIST
C2H + CH4 → C2H2 + CH3	$1.2\times {10}^{-11}\exp (-490.7/T)$	Revise rate	NIST
C2H + C2H6 → C2H2 + C2H5	$2.58\times {10}^{-11}{\left(T/298\right)}^{0.54}\exp (180/T)$	Revise rate	NIST
C2H3 + H2 → C2H4 + H	$3.39\times {10}^{-14}{\left(T/298\right)}^{2.56}\exp (-2530.5/T)$	Revise rate	NIST
C2H3 + C2H5 → 2C2H4	8.0 × 10−13	Revise rate	Moses et al. (2005)
C2H3 + C2H5 → C2H2 + C2H6	8.0 × 10−13	Revise rate	Moses et al. (2005)
C2H5 + C2H5 → C2H6 + C2H4	2.4 × 10−12	Revise rate	NIST
NH + NH → N2H2	$8.47\times {10}^{-11}{\left(T/298\right)}^{-0.04}\exp (80.6/T)$	Add reaction	NIST
NH + NH2 → N2H2 + H	$1.5\times {10}^{-10}{\left(T/298\right)}^{-0.27}\exp (38.5/T)$	Add reaction	NIST
NH + CH4 → NH2 + CH3	$1.49\times {10}^{-10}\exp (-10,103/T)$	Add reaction	NIST
NH + C2H6 → NH2 + C2H5	$1.16\times {10}^{-10}\exp (-8420.3/T)$	Add reaction	NIST
NH + OH → NH2 + O	$2.94\times {10}^{-12}{\left(T/298\right)}^{0.1}\exp (-5800/T)$	Revise rate	NIST
NH2 + CH4 → NH3 + CH3	$5.75\times {10}^{-11}\exp (-6952/T)$	Revise rate	NIST
${\rm{H}}+{\rm{H}}\mathop{\to }\limits^{M}{{\rm{H}}}_{2}$	${k}_{0}=\min (8.85\times {10}^{-33}{\left(T/298\right)}^{-0.6},1.0\times {10}^{-32})$	Revise k0;	NIST
	k∞ = 1.0 × 10−11	Add k∞	Moses et al. (2005)
${\rm{H}}+{\mathrm{CH}}_{3}\mathop{\to }\limits^{M}{\mathrm{CH}}_{4}$	${k}_{0}=6.0\times {10}^{-29}\max {\left(T/298,1.0\right)}^{-1.8}$	Revise k0 at T ≤ 298 K;	NIST;
	${k}_{\infty }=1.92\times {10}^{-8}(\max {(T,110)}^{-0.5}\exp (-400/\max (T,110))$	Add k∞	Moses et al. (2005)
	${F}_{c}=0.3\,+\,0.58\exp (-T/800)$
${\rm{H}}+{{\rm{C}}}_{2}{\rm{H}}\mathop{\to }\limits^{M}{{\rm{C}}}_{2}{{\rm{H}}}_{2}$	${k}_{0}=1.26\times {10}^{-18}{T}^{-3.1}\exp (-721/T)$	Add k0	Moses et al. (2005)
	k∞ = 3.0 × 10−10
${\rm{H}}+{{\rm{C}}}_{2}{{\rm{H}}}_{2}\mathop{\to }\limits^{M}{{\rm{C}}}_{2}{{\rm{H}}}_{3}$	${k}_{0}=3.31\times {10}^{-30}\exp (-740/T)$	Add k∞	NIST
	${k}_{\infty }=1.4\times {10}^{-11}\exp (-1300/T)$
	Fc = 0.44
${\rm{H}}+{{\rm{C}}}_{2}{{\rm{H}}}_{3}\mathop{\to }\limits^{M}{{\rm{C}}}_{2}{{\rm{H}}}_{4}$	k0 = 2.3 × 10−24 T−1	Add k0;	Moses et al. (2005)
	k∞ = 1.8 × 10−10	Revise k∞
${\rm{H}}+{{\rm{C}}}_{2}{{\rm{H}}}_{4}\mathop{\to }\limits^{M}{{\rm{C}}}_{2}{{\rm{H}}}_{5}$	${k}_{0}=\max (1.3\times {10}^{-29}\exp (-380/T),3.7\times {10}^{-30})$	Revise k0 at T ≤ 300 K;	NIST;
	${k}_{\infty }=6.6\times {10}^{-15}{T}^{1.28}\exp (-650/T)$	Add k∞	Moses et al. (2005)
	${F}_{c}=0.24\exp (-T/40)+0.76\exp (-T/1025)$
${\rm{H}}+{{\rm{C}}}_{2}{{\rm{H}}}_{5}\mathop{\to }\limits^{M}{{\rm{C}}}_{2}{{\rm{H}}}_{6}$	${k}_{0}=4.0\times {10}^{-19}{T}^{-3}\exp (-600/T)$ (T > 200 K)	Revise k0;	Moses et al. (2005)
	k0 = 2.49 × 10−27 (T ≤ 200 K)
	k∞ = 2.0 × 10−10
${\rm{C}}+{{\rm{H}}}_{2}\mathop{\to }\limits^{M}{\mathrm{CH}}_{2}$	k0 = 6.89 × 10−32	Add k∞	NIST
	${k}_{\infty }=2.06\times {10}^{-11}\exp (-57/T)$
$\mathrm{CH}+{{\rm{H}}}_{2}\mathop{\to }\limits^{M}{\mathrm{CH}}_{3}$	${k}_{0}=9.0\times {10}^{-31}\exp (550/T)$	Revise k0 and k∞	NIST
	${k}_{\infty }=2.01\times {10}^{-10}{\left(T/298\right)}^{0.15}$		Moses et al. (2005)
${\mathrm{CH}}_{3}+{\mathrm{CH}}_{3}\mathop{\to }\limits^{M}{{\rm{C}}}_{2}{{\rm{H}}}_{6}$	${k}_{0}=1.68\times {10}^{-24}{\left(T/298\right)}^{-7}\exp (-1390/T)$ (T > 300 K)	Revise k0 at T ≤ 300 K;	NIST;
	k0 = 6.15 × 10−18 T−3.5 (T ≤ 300 K)	Add k∞	Moses et al. (2005)
	${k}_{\infty }=1.12\times {10}^{-9}{T}^{-0.5}\exp (-25/T)$
	${F}_{c}=0.62\exp (-T/1180)+0.38\exp (-T/73)$

Note. The rate constants of two-body reactions and the high-pressure limiting rate constants of three-body reactions (k_∞) have a unit of cm³ molecule⁻¹ s⁻¹, and the low-pressure limiting rate constants of three-body reactions (k₀) have a unit of cm⁶ molecule⁻² s⁻¹. The rates of three-body reactions are ${k}_{0}{k}_{\infty }[M]/({k}_{\infty }+{k}_{0}[M]){F}_{c}^{\beta }$, where [M] is the number density of the atmosphere, and $\beta ={\left(1+{\left({\mathrm{log}}_{10}({k}_{0}[M/{k}_{\infty })\right)}^{2}\right)}^{-1}$. F_c = 0.6 unless otherwise noted. NIST = NIST Chemical Kinetics Database (http://kinetics.nist.gov).

Download table as:  ASCIITypeset image

The photochemical model is applied to the "stratosphere" of the atmosphere, where the "tropopause" is defined as the pressure level at which the temperature profile becomes adiabatic. We define the lower boundary of the model as the pressure level 10-fold greater than the tropopause pressure, and thus include a section of the "troposphere" in the model. These choices are customary in photochemical studies of giant planets' atmospheres (e.g., Gladstone et al. 1996), and reasonable because the photochemical products in the stratosphere (and above the condensation clouds) are the objective of the study. Including a section of the troposphere makes sure that the results do not strongly depend on the lower boundary conditions assumed.

We apply fixed mixing ratios as the lower boundary conditions for H₂, He, H₂O, CH₄, NH₃, and when applicable, H₂S according to the assumed elemental abundance. When interior sources of CO, CO₂, and N₂ are included in some scenarios (see Section 2.4 for details), fixed mixing ratios are also applied to these gases at the lower boundary. We assume that all other species can move across the lower boundary (i.e., dry deposition when the lower boundary is a surface in terrestrial planet models) at a velocity of K_zz/H, where K_zz is the eddy diffusion coefficient and H is the scale height. This velocity is the upper limit of the true diffusion velocity, which could be damped by the gradient of the mixing ratio (Gladstone et al. 1996); however, the velocity only matters for long-lived species (e.g., C₂H₆ in Jupiter). Our choice of lower boundary conditions thus results in conservative estimates of the abundance of long-lived photochemical gases.

The upper boundary is assumed to be at 10⁻⁴ Pa, i.e., small enough so that the peaks of photodissociation of all species are well within the modeled atmosphere. Following Gladstone et al. (1996), we assume a zero-flux boundary condition for all species except for H, for which we include a downward flux of 4 × 10⁹ cm⁻² s⁻¹ (Waite et al. 1983) to account for ionospheric processes that produce H. This influx of H was calculated for Jupiter and the actual flux can conceivably be different. The impact of this additional H is limited to the upper atmosphere and, in most of our cases, is swamped by the H from the photodissociation of H₂O (see Section 3.4).

Since the modeled domain of the atmosphere includes the stratosphere and a small section of the upper troposphere, the standard mixing-length scaling (Gierasch & Conrath 1985) is not applicable to estimate the eddy diffusion coefficient. We instead anchor our choice of the eddy diffusion coefficient on the value in the upper troposphere of Jupiter (∼1 × 10³ cm² s⁻¹, Conrath & Gierasch 1984) and explore a larger value in the study. Above the tropopause, we assume that mixing is predominantly caused by the breaking of gravity waves and the eddy diffusion coefficient is inversely proportional to the square root of the number density (Lindzen 1981).

Because the pressure range of the photochemical model typically includes the condensation of NH₃ and H₂O, we have added a scheme to account for the condensation of NH₃ into the photochemical model, with that for H₂O already included in the model of Hu et al. (2012). In addition, we have added the schemes of condensation for N₂H₄ and HCN, the two main photochemical gases expected to condense in Jupiter's upper troposphere (e.g., Atreya et al. 1977; Moses et al. 2010). The low-temperature vapor pressures of N₂H₄ and HCN are adopted from Atreya et al. (1977) and Krasnopolsky (2009), respectively. As such, these gases are treated in the photochemical model and their production and removal paths including chemical reactions and condensation are self-consistently computed. This is important because, for example, NH₃ above the clouds in Jupiter is expected to be completely removed by photodissociation and converted to N₂H₄ and N₂, followed by condensation and transport to the deep atmosphere (Strobel 1973; Atreya et al. 1977; Kaye & Strobel 1983a, 1983b; Moses et al. 2010). As we will show in Section 3, the condensation of N₂H₄ and HCN limits their abundance in the middle atmosphere of cold planets like Kepler-167 e. For H₂S, we make a binary choice: if the cloud model indicates NH₄SH formation, we remove sulfur chemistry from the model, because NH₄SH should completely sequester H₂S (Atreya & Romani 1985); and we include the sulfur chemistry if NH₄SH cloud is not formed. This simplifies the calculations of sulfur photochemistry and is broadly valid when N/S > 1 in the bulk atmosphere.

We calculate the cross sections and single scattering albedo of ammonia and water cloud particles using their optical properties (Palmer & Williams 1974; Martonchik et al. 1984) and the radiative properties of the sulfur haze particles in the same way as Hu et al. (2013). NH₄SH and HCN condensates are treated the same way as NH₃ clouds. N₂H₄ condensates have a very low abundance in all models and do not contribute significantly to the opacity. Thus, our model includes the absorption and scattering of cloud and haze particles when calculating the radiation field that drives photochemical reactions in the atmosphere.

As a test case, we have applied the coupled cloud condensation and photochemical model to a Jupiter-like planet and compared the results with the measured gas abundance in Jupiter and previous models of Jupiter's stratospheric composition (Atreya et al. 1977; Kaye & Strobel 1983a, 1983b; Gladstone et al. 1996; Moses et al. 2005, 2010). Figure 1 shows the pressure–temperature profile, eddy diffusion coefficient, and the mixing ratios of CH₄, NH₃, and major photochemical gases of the test case. The atmospheric structure model adequately predicts the tropospheric temperature profile and the pressure level of the tropopause, but it cannot generate a temperature inversion in the middle atmosphere (Figure 1, panel (a)). We have run the photochemical model with the pressure–temperature profile measured in Jupiter and the modeled pressure–temperature profile (i.e., without the temperature inversion) to see how much the photochemical gas mixing ratios change.

Zoom In Zoom Out Reset image size

We find that the photochemical model can predict the mixing ratios of C₂H₆, C₂H₂, and C₂H₄ measured in Jupiter's stratosphere, and the modeled profile of HCN is consistent with the upper limit in Jupiter's upper troposphere when the measured pressure–temperature profile is adopted (Figure 1, panel (c)). The only exception is the C₂H₂ mixing ratio at ∼1 Pa, where the modeled mixing ratio is greater than the measured value by 2σ ∼ 3σ. This less-than-perfect performance may be due to the lack of C₃, C₄, and higher hydrocarbons in our reaction network. For example, Moses et al. (2005) was able to fit the C₂H₂ mixing ratio at ∼1 Pa, together with other mixing ratio constraints, with a more complete hydrocarbon reaction network and specific choices in the eddy diffusion coefficient profiles for Jupiter's stratosphere. In terms of nitrogen photochemistry, our photochemical model finds that NH₃ is depleted by photodissociation to the cloud deck, and the vast majority of the net photochemical removal of NH₃ becomes N₂H₄ and then condenses out. A small fraction becomes N₂ and HCN. The abundance of HCN is low (∼10⁻⁹) in the troposphere due to the photolysis of NH₃ and CH₄ occurring at well separated pressure levels, and is limited by the cold trap near the tropopause (Figure 1). These behaviors are qualitatively similar to the past models of Jupiter's nitrogen photochemistry (Atreya et al. 1977; Kaye & Strobel 1983a, 1983b; Moses et al. 2010).

Figure 1 also indicates that adopting the modeled pressure–temperature profile that does not have a stratosphere, while preserving the overall behavior of the atmospheric photochemistry, would underpredict the mixing ratios of C₂H₆ and C₂H₂ by approximately half an order of magnitude. We use the atmospheric structure model in this study for speedy exploration of the main photochemical behavior, and one should keep this context in mind when interpreting the results shown in Section 3.

Another interesting point to make is that the quantum yield of H in the photodissociation C₂H₂ has been convincingly measured to be 100% by recent experiments (Läuter et al. 2002). When producing the models shown as the solid and dashed lines in Figure 1, panel c, we have applied a quantum yield of 16% so that the top-of-atmosphere rate of C₂H₂ + hν → C₂H + H would match with the models of Gladstone et al. (1996) and Moses et al. (2005). Revising the quantum yield to 100%, as shown by the dotted lines in Figure 1, panel c, slightly reduces the steady-state mixing ratio of C₂H₆ and reduces the mixing ratio of C₂H₂ and C₂H₄ by a factor of ∼5 in the lower stratosphere (∼10³ Pa). The photodissociation of C₂H₂ is the main source of H in the lower stratosphere (e.g., Gladstone et al. 1996) and thus its quantum yield is important for the hydrocarbon chemistry in the lower stratosphere. However, a quantum yield of 100% would result in poor fits to the measured mixing ratios of C₂H₂ and C₂H₄, and this potential discrepancy suggests that additional consideration of the atmospheric photochemistry of Jupiter might be warranted. We adopt the quantum yield of 100% in the subsequent models.

We use the temperate sub-Neptune K2-18 b as a representative case of temperate and gas-rich planets around M dwarf stars, and use the gas giants PH2 b and Kepler-167 e as the representative cases of temperate and cold planets around G and K stars (Table 2). The results for K2-18 b are generally applicable to temperate (mini-)Neptunes of M dwarf stars such as the recently detected TOI-1231 b. For K2-18 b, the interior structures, thermochemical abundances, and atmospheric circulation patterns have been studied (Benneke et al. 2019; Madhusudhan et al. 2020; Piette & Madhusudhan 2020; Blain et al. 2021; Charnay et al. 2021), but the effects of atmospheric photochemistry remain to be studied. Kepler-167 e is considered a "cold" exoplanet because it only receives stellar irradiation 7.5% of Earth's. The equilibrium cloud condensation model would predict NH₃ to condense in its atmosphere and form the uppermost cloud deck, below which NH₄SH solids form and scavenge sulfur from the above-cloud atmosphere. In the atmospheres of K2-18 b and PH2 b, only H₂O is expected to condense and forms the cloud deck—and thus the physical distinction between "temperate" and "cold."

The UV spectrum of K2-18 has not been measured and so we adopt that of GJ 176, a similar M dwarf star with the UV spectrum measured in the MUSCLES survey (France et al. 2016). The reconstructed Lyα flux of GJ 176 is similar to the measured flux of K2-18 (dos Santos et al. 2020). We adopt the UV spectrum of the Sun for the models of PH2 b and Kepler-167 e, even though Kepler-167 is a K star. Figure 2 shows the incident stellar flux at the top of the atmospheres adopted in this study. K2-18 b, while having a similar total irradiation to that of PH2 b, receives considerably less irradiation in the near-ultraviolet.

Zoom In Zoom Out Reset image size

For these planets, we simulate H₂-dominated atmospheres having 1–100×solar metallicities. The higher-than-solar metallicity scenario may be particularly interesting for sub-Neptunes like K2-18 b because of a proposed mass–metallicity relationship that posits a less massive planet should have a higher metallicity (Thorngren et al. 2016). For PH2 b and Kepler-167 e, we assume as fiducial values a surface gravity of 25 m s⁻² and an internal heat flux that corresponds to T_int = 100 K, similar to the parameters of Jupiter. Changing the surface gravity to 100 m s⁻² results in slightly different cloud pressures and above-cloud abundances of gases on these planets, but does not change the qualitative behaviors of the atmospheric chemistry. For K2-18 b we assume an internal heat flux that corresponds to T_int = 60 K, similar to that of Neptune.

In the standard models, we assume that the dominant O, C, N, and S species are H₂O, CH₄, NH₃, and H₂S at the base of the photochemical domain. Gases and aerosols produced in the photochemical domain can be transported through the lower boundary, and thus the standard model setup implicitly assumes that thermochemical recycling in the deep troposphere effectively recycles the photochemical products into H₂O, CH₄, NH₃, and H₂S. Here, we quantitatively assess how realistic this assumption is based on the quench-point theory (e.g., Visscher & Moses 2011; Moses et al. 2013; Hu & Seager 2014; Zahnle & Marley 2014; Tsai et al. 2018). In that theory, the "quench point" is defined as the pressure level where the chemical lifetime of a gas equals the vertical mixing timescale (typically at the pressure of 10⁷ Pa or higher). The gas is close to thermochemical equilibrium at the quench point, and its mixing ratio is carried to the atmosphere above the quench point by vertical mixing.

Figures 3–5 show the pressure–temperature profiles of the three planets calculated by the atmospheric structure model, and the mixing ratios of major C and N molecules at the respective quench points. We adopt the chemical lifetime of the CO${\rm{\longleftrightarrow }}$CH₄ and N₂ ${\rm{\longleftrightarrow }}$NH₃ conversions from Zahnle & Marley (2014) and estimate the eddy diffusion coefficient in the deep troposphere using the mixing-length theory in Visscher et al. (2010). The eddy diffusion coefficient depends on the assumed internal heat flux and has a typical value of ∼10⁴ m² s⁻¹ at the pressure of 10⁶–10⁸ Pa. The quench point of CO₂ follows that of CO, and, similarly, that of HCN occurs at a similar pressure and temperature as N₂ (Zahnle & Marley 2014; Tsai et al. 2018). The mixing ratios of gases at the quench points are calculated using the thermochemical equilibrium model of Hu & Seager (2014).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figures 3–5 show that a solar metallicity atmosphere is likely deep in the CH₄- and NH₃-dominated regime at the quench points on all three planets. Specifically, we find the mixing ratio of CO ≤ 10⁻⁸, that of CO₂ ≤ 10⁻¹¹, and the mixing ratio of NH₃ greater than that of N₂ by >10-fold. With 10 × solar metallicity, the atmosphere remains CH₄ dominated, but the mixing ratio of CO transported from the deep troposphere can be on the order of 10⁻⁶ ∼ 10⁻⁵ and thus non-negligible. With the assumed internal heat flux and the modeled strength of deep-tropospheric mixing, the mixing ratio of N₂ can be comparable to that of NH₃ at the quench point. As N₂ does not have strong spectral features and is not a feedstock molecule for photochemistry, the effect of a hot interior would be mostly seen as a reduction of the mixing ratio of NH₃. The impact of the hot interior is the most significant in the 100 × solar metallicity atmosphere. Both CO and CO₂ have mixing ratios >10⁻⁴ at the quench point, and in the hottest case (PH2 b), the mixing ratio CO is greater than that of CH₄. For nitrogen, the mixing ratio of NH₃ can be reduced by a factor of 10 ∼ 100 at the thermochemical equilibrium in the deep troposphere.

As a general trend, a higher deep-atmosphere temperature favors CO, CO₂, and N₂, and reduces the equilibrium abundance of NH₃. We have thus run variant models for the 10 × and 100 × solar metallicity cases, and used the mixing ratios of CH₄, CO, CO₂, NH₃, and N₂ at the quench points as shown in Figures 3–5 as the lower boundary conditions. Technically the mixing ratio of deep H₂O is also affected, but the photochemical models have lower boundaries that are well above the base of the water cloud, and are thus immune to small changes in the input water abundance. Also, we do not fix the lower boundary mixing ratio of HCN in these models, because the mixing ratio of HCN at the quench point does not exceed the mixing ratio found by the photochemical models at the lower boundary in any case. We emphasize that specific quantities of the input gas abundance depend on the detailed thermal structure of the interior, which is related to the thermal history of the planet and exogenous factors like tidal heating, as well as the strength of vertical mixing in the interior (Fortney et al. 2020). For example, applying an internal heat flux that corresponds to T_int = 30 K (similar to Earth) largely restores the CH₄ and NH₃ dominance for the three planets. While these factors are likely uncertain for many planets to be observed, the standard and variant photochemical models presented in this paper give an account of the range of possible behaviors that manifest in the observable part of the atmosphere.

The abundance profiles of the equilibrium gases (H₂O, CH₄, NH₃, and H₂S) and their main photochemical products are shown in Figures 6–8. In all models, we find that the main photochemical products are C₂H₆, C₂H₂, CO, CO₂, N₂, HCN, N₂O, and elemental sulfur haze, but the abundance of these species and their pressure dependency differ significantly from model to model.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

3.1.1. K2-18 b: a Temperate Planet around an M Star

3.1.2. PH2 b: a Temperate Planet around a G/K Star

3.1.3. Kepler-167 e: a Cold Planet around a G/K Star

From Figures 6–8, we see that NH₃ is depleted to the cloud deck in temperate and cold planets around G/K stars but remains intact in the middle atmosphere of temperate and cold planets around M stars. This finding is significant because it implies that NH₃ should be detectable on temperate planets around M stars but not around G/K stars (see Section 3.5).

The root cause of this different behavior is the M stars (represented by GJ 176 here) emit substantially lower irradiation at the near-ultraviolet wavelengths than do the G/K stars (represented by the Sun here; Figure 2). The radiation that dissociates NH₃ in the H₂-dominated atmosphere is the radiation that is not absorbed by the typically more abundant CH₄ and H₂O. NH₃ has a dissociation limit at ∼230 nm, while for CH₄ it is at ∼150 nm and for H₂O ∼240 nm, but the cross section and the shielding effect of H₂O is small, >200 nm (Hu et al. 2012; Ranjan et al. 2020). C₂H₂ also absorbs photons up to ∼230 nm but it typically does not strongly interfere with the NH₃ photodissociation due to its relatively low abundance. Thus, photons in the range 200–230 nm are the most relevant for the photodissociation of NH₃ in K2-18 b and PH2 b, and photons in the range 150–230 nm are the most relevant for Kepler-167 e. Having similar bolometric irradiation, the photon flux at 200–230 nm received by PH2 b is more than that received by K2-18 b by >2 orders of magnitude (Figure 2). The photon flux received by Kepler-167 e is one order of magnitude more than for K2-18 b, and the removal of NH₃ by condensation further pushes down the pressure of photochemical depletion (see below).

3.2.1. Criterion of Photochemical Depletion

How does the photon flux control the pressure of photochemical depletion? Guided by the numerical results, here we develop a simple theory that estimates the pressure of photochemical depletion. Assuming that photodissociation is the only process that removes NH₃ with no recycling or production, its mixing ratio profile at the steady state should obey the following differential equation:

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dz}}\left({KN}\displaystyle \frac{{df}}{{dz}}\right)={fNJ},\end{eqnarray} \tag{ 1 }$$

where z is altitude, K is the eddy diffusion coefficient, N is the total number density of the atmosphere, f is the mixing ratio, and J is the photodissociation rate (often referred to as the "J-value" in the atmospheric chemistry literature). The number density has a scale height of H, and the equation can be rewritten as

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}f}{{{dz}}^{2}}-\displaystyle \frac{1}{H}\displaystyle \frac{{df}}{{dz}}-\displaystyle \frac{J}{K}f=0.\end{eqnarray} \tag{ 2 }$$

Assuming J, H, and K to be constant with respect to z, the equation above has the analytical solution of

$$\begin{eqnarray}&&f={f}_{0}\exp \left(\displaystyle \frac{z}{2}\left(\displaystyle \frac{1}{H}-\sqrt{\displaystyle \frac{1}{{H}^{2}}+\displaystyle \frac{4J}{K}}\right)\right)\equiv {f}_{0}\exp \left(-\displaystyle \frac{\alpha z}{H}\right),\end{eqnarray} \tag{ 3 }$$

where f₀ is the mixing ratio at the pressure of photochemical depletion (z = 0 for simplicity), and α is

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{1}{2}\left(\sqrt{1+\displaystyle \frac{4{{JH}}^{2}}{K}}-1\right).\end{eqnarray} \tag{ 4 }$$

Therefore, when the product 4J H²/K is small, α → 0 and the mixing ratio profile is close to a constant; and when 4J H²/K is large, α can be ≫1 and thus the mixing ratio drops off very quickly. This explains the vertical profiles of NH₃ seen in Figures 6–8.

Going back to Equation (1), which can be integrated from the pressure of photochemical depletion to the top of the atmosphere, we have

$$\begin{eqnarray}&&{KN}\displaystyle \frac{{df}}{{dz}}{| }_{z=\infty }-{KN}\displaystyle \frac{{df}}{{dz}}{| }_{z=0}={\int }_{0}^{\infty }n(z)J(z){dz},\end{eqnarray} \tag{ 5 }$$

where n ≡ fN is the number density of NH₃. Assuming that the photoabsorption of NH₃ itself is the sole source of opacity, J can be expressed as

$$\begin{eqnarray}&&J(z)={J}_{\infty }\exp (-\sigma {\int }_{z}^{\infty }n(z^{\prime} ){dz}^{\prime} ),\end{eqnarray} \tag{ 6 }$$

where J_∞ is the top-of-atmosphere J-value and σ is the mean cross section of NH₃. The differential of Equation (6) is

$$\begin{eqnarray}&&\displaystyle \frac{{dJ}}{{dz}}=\sigma {nJ}.\end{eqnarray} \tag{ 7 }$$

Combining Equations (5) and (7), and recognizing df/dz vanishes at z = ∞ , we obtain

$$\begin{eqnarray}&&-{KN}\displaystyle \frac{{df}}{{dz}}{| }_{z=0}=\displaystyle \frac{1}{\sigma }{\int }_{0}^{\infty }\displaystyle \frac{{dJ}}{{dz}}{dz}=\displaystyle \frac{{J}_{\infty }-J(z=0)}{\sigma }.\end{eqnarray} \tag{ 8 }$$

With J(z = 0) ∼ 0 (i.e., the J-value immediately below the pressure of photochemical depletion is minimal), and J_∞ = σ I, where I is the photon flux at the top of the atmosphere, we obtain

$$\begin{eqnarray}&&-{KN}\displaystyle \frac{{df}}{{dz}}{| }_{z=0}=I.\end{eqnarray} \tag{ 9 }$$

Note that to derive Equation (9), no specific profiles for J or n(f) need to be assumed.

The physical meaning of Equation (9) is that the number of NH₃ molecules that diffuse through the pressure of photochemical depletion should be equal to the number of photons received at the top of the atmosphere. This physical condition would become evident if one regards the column of NH₃ above the pressure of photochemical depletion as a whole and recognizes that one photon dissociates one molecule. To the extent that the photoabsorption of NH₃ itself is the dominant source of opacity, the criterion expressed by Equation (9) does not depend on the mean cross section. Similarly, the criterion will be applicable to any molecule subject to photodissociation in a wavelength range largely free of interference by other molecules.

It should be noted that Equation (9) cannot be derived by requiring the pressure of photochemical depletion to occur roughly at the optical depth of unity for the photodissociating radiation. This is because the mixing ratio profile in Equation (3) is valid only locally and depends on J, which in turns depends on the vertical profile of the mixing ratio. As such, one cannot integrate Equation (3) directly to find the pressure of photochemical depletion, and the optical-depth-of-unity condition is not as predictive as Equation (9).

The left-hand side of Equation (9) can be evaluated locally using Equation (3), and Equation (9) becomes

$$\begin{eqnarray}&&\displaystyle \frac{\alpha {{KN}}_{0}{f}_{0}}{H}=I,\end{eqnarray} \tag{ 10 }$$

where N₀ and f₀ are the total number density and the mixing ratio at the pressure of photochemical depletion. The pressure is thus P₀ = N₀k_b T where k_b is the Boltzmann constant and T is temperature. The term α can be evaluated with Equation (4) for a J value that corresponds to 5% of the top-of-atmosphere value. Equation (10) thus provides a closed-form criterion that determines the pressure of photochemical depletion, and explains why the pressure of photochemical depletion is sensitive to the top-of-atmosphere flux of photons that drive photodissociation.

Figure 9 shows both sides of Equation (10) for the three planets modeled assuming a solar abundance atmosphere. We can see that the pressure of photochemical depletion implied by Equation (10) for K2-18 b is ∼100 Pa, consistent with the pressure where the photochemical model starts to substantially deviate from the equilibrium cloud condensation model (Figure 6). Figure 6 also shows that the mixing ratio NH₃ decreases very slowly near the pressure of the photochemical depletion, but the decrease becomes faster for lower pressures, where the J value and the 4J H²/K product become greater (see Equations (3) and (4)). The mixing ratio of NH₃ eventually drops below 10⁻⁶ at the pressure lower than the pressure of photochemical depletion by approximately one order of magnitude. The pressures of photochemical depletion implied by Equation (10) for PH2 b and Kepler-167 e are close to or below the cloud deck (i.e., 10⁴–10⁵ Pa), which is consistent with numerical finding that NH₃ is photodissociated to the cloud deck on these planets (Figures 7 and 8). Therefore, although Equation (10) cannot replace the full photochemical calculation due to the underlying assumptions (e.g., no recycling or production, self-shielding only), it provides a guiding estimate of whether a gas is likely depleted by photodissociation in the middle atmosphere.

Zoom In Zoom Out Reset image size

3.2.2. Sensitivity to the Eddy Diffusion Coefficient

From the criterion of photochemical depletion (Equation (10)), we see that when the eddy diffusion coefficient increases, the pressure of photochemical depletion decreases. In other words, stronger mixing would sustain a photodissociated gas (e.g., NH₃) to a lower pressure or higher altitude. We have used the photochemical model to conduct a sensitivity study of the eddy diffusion coefficient, and the results confirm this understanding (Figure 10). The most significant sensitivity happens with PH2 b: the standard model predicts the photodissociation would deplete NH₃ to the cloud deck, while with a 10-fold or 100-fold greater eddy diffusion coefficient, NH₃ would be depleted at 10² ∼ 10³ Pa. For Kepler-167 e, with a 10-fold or 100-fold greater eddy diffusion coefficient, photodissociation can no longer deplete the NH₃ ice cloud, while the mixing ratio of NH₃ in the middle atmosphere remains small due to condensation and photodissociation above the cloud deck.

Zoom In Zoom Out Reset image size

The top of the sulfur haze layer moves up in the atmosphere when the eddy diffusion coefficient increases. For both K2-18 b and PH2 b, the top of the sulfur haze would be at ∼10³ Pa and 10² Pa for 10-fold and 100-fold greater eddy diffusion coefficient (Figure 10). A haze layer that extends to 10² Pa would greatly interfere with transmission spectroscopy and also affect the spectra of the reflected starlight (see Section 3.5). This trend is consistent with the findings of Zahnle et al. (2016) and is produced by two effects acting together. First, the pressure of photochemical depletion of H₂S, the feedstock of sulfur haze, decreases for a greater eddy diffusion coefficient. Second, a stronger eddy diffusion helps increase the lifetime of haze particles against falling (see the formulation in Hu et al. 2012). For PH2 b, the extended sulfur haze layer further keeps NH₃ from photochemical depletion by absorbing the ultraviolet photons that can dissociate NH₃.

The sensitivity of main photochemical gases' abundance to the eddy diffusion coefficient is complex (Figure 10), which indicates several factors at work. For N₂ and HCN (the dominant photochemical gases of nitrogen), their mixing ratios at the lower boundary decrease with the eddy diffusion coefficient. This is because, in our model, gases move across the lower boundary at a velocity that is proportional to the eddy diffusion coefficient, and the loss to the lower boundary is the main loss mechanism for both N₂ and HCN. Their mixing ratios in the middle atmosphere do not necessarily follow the same trend, as they also depend on the photochemical production (see Section 3.3). The abundance of the photochemical gases of carbon does not depend on the eddy diffusion coefficient monotonically, and this is because the formation rates of CO, CO₂, and C₂H₆ largely depend on the abundance of H, OH, and O, which is in turn controlled by the full chemical network involving the photodissociation of CH₄, H₂O, and NH₃ (see Section 3.4). For example, in K2-18 b with the solar metallicity, both CO and CO₂ have very small mixing ratios in the middle atmosphere in the standard case; the two would be substantially more abundant in the middle atmosphere with a 10-fold greater eddy diffusion coefficient, and CO would become more abundant than CO₂ with a 100-fold greater eddy diffusion coefficient. These examples highlight the richness and complexity of atmospheric photochemistry in temperate and cold planets.

A common phenomenon that emerges from the photochemical models is the synthesis of HCN in temperate and H₂-rich exoplanets. The photodissociation of NH₃ in Jupiter leads to N₂ but not significant amounts of HCN, and this is mainly because NH₃ is dissociated at much higher pressures than CH₄ (e.g., Atreya et al. 1977; Kaye & Strobel 1983a, 1983b; Moses et al. 2010). HCN in Titan's N₂-dominated atmosphere mainly comes from the reactions between atomic nitrogen and hydrocarbons and the associated chemical network (Yung et al. 1984; Lavvas et al. 2008; Krasnopolsky 2014; Vuitton et al. 2019). Similar processes, as well as the reactions between CH and NO/N₂O may also lead to formation of HCN on early Earth or rocky exoplanets with N₂-dominated atmospheres irradiated by active stars (Airapetian et al. 2016; Rimmer & Rugheimer 2019). In addition, the formation of HCN has been commonly found in warm and hot H₂-rich exoplanets (e.g., Line et al. 2011; Moses et al. 2011; Venot et al. 2012; Agúndez et al. 2014; Mollière et al. 2015; Moses et al. 2016; Blumenthal et al. 2018; Kawashima & Ikoma 2018; Hobbs et al. 2019; Lavvas et al. 2019; Molaverdikhani et al. 2019), and the mechanisms identified include quench kinetics (Moses et al. 2011; Venot et al. 2012; Agúndez et al. 2014) and photochemistry (Line et al. 2011; Kawashima & Ikoma 2018; Hobbs et al. 2019). Here, we show that HCN can also build up to significant amounts in temperate exoplanets with H₂-dominated atmospheres.

Figure 11 shows the chemical network that starts with the photodissociation of NH₃ and ends with the formation of N₂ and HCN as the main photochemical products. The key condition for the synthesis of HCN is the photodissociation of NH₃ in the presence of CH₄ and at a temperature > ∼200 K. This condition allows CH₃, one of the ingredients for the synthesis of HCN, to be produced locally by the reaction between CH₄ and H, and this H is produced by the photodissociation of NH₃ itself. We describe the details as follows.

Zoom In Zoom Out Reset image size

The photodissociation of NH₃ mainly produces NH₂,

$$\begin{eqnarray}&&{\mathrm{NH}}_{3}\mathop{\to }\limits^{h\nu }{\mathrm{NH}}_{2}+{\rm{H}},\end{eqnarray} \tag{ R1 }$$

and some of the NH₂ produced is returned to NH₃ via

$$\begin{eqnarray}&&{\mathrm{NH}}_{2}+{{\rm{H}}}_{2}\to {\mathrm{NH}}_{3}+{\rm{H}},\end{eqnarray} \tag{ R2 }$$

and

$$\begin{eqnarray}&&{\mathrm{NH}}_{2}+{\rm{H}}\mathop{\to }\limits^{M}{\mathrm{NH}}_{3}.\end{eqnarray} \tag{ R3 }$$

Another channel of the photodissociation of NH₃ is to produce NH

$$\begin{eqnarray}&&{\mathrm{NH}}_{3}\mathop{\to }\limits^{h\nu }\mathrm{NH}+{{\rm{H}}}_{2}.\end{eqnarray} \tag{ R4 }$$

The NH₂ channel requires photons more energetic than 230 nm and the NH channel requires photons more energetic than 165 nm. Therefore, the photons that produce NH are more easily shielded by H₂O and CH₄. For the three planets modeled, the NH channel is important in K2-18 b and Kepler-167 e, but not in PH2 b. This is because the photodissociation of NH₃ occurs at higher pressures in PH2 b and is subject to the shielding effect of both H₂O and CH₄. The NH channel mostly leads to N₂ (Figure 11).

The NH₂ that is not recombined to form NH₃ can undergo

$$\begin{eqnarray}&&{\mathrm{NH}}_{2}+{\mathrm{NH}}_{2}\to {{\rm{N}}}_{2}{{\rm{H}}}_{4},\end{eqnarray} \tag{ R5 }$$

and the N₂H₄ produced (if not condensed out) can then become N₂H₃. N₂H₃ can react with itself to form N₂H₂, whose photodissociation produces N₂, or with H to return to NH₂ (Figure 11). The other loss of NH₂ is to react with CH₃,

$$\begin{eqnarray}&&{\mathrm{NH}}_{2}+{\mathrm{CH}}_{3}\to \mathrm{CH}5{\rm{N}},\end{eqnarray} \tag{ R6 }$$

followed by photodissociation to form HCN,

$$\begin{eqnarray}&&{\mathrm{CH}}_{5}{\rm{N}}\mathop{\to }\limits^{h\nu }\mathrm{HCN}+2{{\rm{H}}}_{2}.\end{eqnarray} \tag{ R7 }$$

Reaction (R6) is the critical step in this HCN formation mechanism, and it requires the CH₃ radical to be available. The CH₃ in Reaction (R6) is mainly produced by

$$\begin{eqnarray}&&{\rm{H}}+{\mathrm{CH}}_{4}\to {{\rm{H}}}_{2}+{\mathrm{CH}}_{3}.\end{eqnarray} \tag{ R8 }$$

Note that the photodissociation of CH₄, which also produces CH₃, does not contribute significantly to the source of CH₃ in Reaction (R6) because the photodissociations of CH₄ and NH₃ typically occur at very different pressures. Another formational path of HCN is through

$$\begin{eqnarray}&&{\mathrm{NH}}_{2}+{{\rm{C}}}_{2}{{\rm{H}}}_{3}\to {{\rm{C}}}_{2}{{\rm{H}}}_{5}{\rm{N}},\end{eqnarray} \tag{ R9 }$$

followed by photodissociation

$$\begin{eqnarray}&&{{\rm{C}}}_{2}{{\rm{H}}}_{5}{\rm{N}}\mathop{\to }\limits^{h\nu }\mathrm{HCN}+{\mathrm{CH}}_{3}+{\rm{H}}.\end{eqnarray} \tag{ R10 }$$

The C₂H₃ in Reaction (R9) is mainly produced by

$$\begin{eqnarray}&&{\rm{H}}+{{\rm{C}}}_{2}{{\rm{H}}}_{2}\mathop{\to }\limits^{M}{{\rm{C}}}_{2}{{\rm{H}}}_{3},\end{eqnarray} \tag{ R11 }$$

and C₂H₂ is ultimately produced by the photodissociation of CH₄ and then transported to the pressure of the photodissociation of NH₃. The HCN produced in Reactions (R7) and (R10) is photodissociated to form CN, but CN quickly reacts with H₂ and C₂H₂ to return to HCN. Thus, HCN does not have significant net chemical loss and is transported together with N₂ through the lower boundary.

The NH₃–CH₄ coupling (Reactions (R6)–(R8)) dominates the formation of HCN over the NH₃−C₂H₂ coupling (Reactions (R9)–(R11)) in temperate H₂-dominated atmospheres by several orders of magnitude. This is because the mixing ratio of C₂H₂ at the pressure of NH₃ photodissociation is typically very small on temperate planets like K2-18 b and PH2 b (Figures 6 and 7). On colder planets like Kepler-167 e, more C₂H₂ is available and the NH₃−C₂H₂ coupling can contribute 1%–10% of the HCN formation, consistent with the results for Jupiter (Moses et al. 2010). We also note that past models of warm and hot H₂-rich exoplanets suggested different reactions to represent the NH₃−CH₄ coupling, including NH + CH₃ (Line et al. 2011) and N + CH₃ (Kawashima & Ikoma 2018; Hobbs et al. 2019); in our models the contribution from N + CH₃ → HCN + H₂ contributes to the formation of HCN less than Reactions (R6)–(R8) by more than three orders of magnitude.

The efficacy of the NH₂ path to produce N₂ and HCN and the branching between N₂ and HCN depend on the abundance of H and the temperature. Reaction (R8) has an activation energy of 33.60 kJ mol⁻¹ (Baulch et al. 1992) and does not occur at very low temperatures. At the pressure of NH₃ photodissociation, the temperature is 220–240 K in K2-18 b and PH2 b, 120–130 K in Kepler-167 e, and ∼110 K in Jupiter. This makes Reaction (R8) faster by six orders of magnitude in K2-18 b and PH2 b than in Kepler-167 e or Jupiter, eventually leading to an efficient HCN production and a high abundance in the middle atmosphere. This is why the HCN production mechanism (Reactions (R6)–(R8)) does not operate efficiently in giant planets in the solar system but can build up HCN in warmer exoplanetary atmospheres.

The abundance of H is another important control. From Figure 11, we can see that a higher abundance of H would enhance the recycling from N₂H₄ to NH₂, produce more NH₃ to react with NH₂, and help the return of NH₂ to NH₃. In other words, a higher abundance of H would reduce the overall efficacy of the NH₂ path but favor the branch that leads to HCN. At the pressure of NH₃ photodissociation, the main source of H is the combination of Reactions (R1) and (R2), whose net result is the dissociation of H₂ but not NH₃. The sink of H is mainly Reaction (R3) and the direct recombination ${\rm{H}}+{\rm{H}}\mathop{\to }\limits^{M}{{\rm{H}}}_{2}$. In high-metallicity atmospheres, another sink of H is Reaction (R5) followed by

$$\begin{eqnarray}&&{{\rm{N}}}_{2}{{\rm{H}}}_{4}+{\rm{H}}\to {{\rm{N}}}_{2}{{\rm{H}}}_{3}+{{\rm{H}}}_{2},\end{eqnarray} \tag{ R12 }$$

and

$$\begin{eqnarray}&&{{\rm{N}}}_{2}{{\rm{H}}}_{3}+{\rm{H}}\to 2{\mathrm{NH}}_{2}.\end{eqnarray} \tag{ R13 }$$

The net result of Reactions (R5), (R12), and (R13) is H + H → H₂. Therefore, the chemical network that starts with the photodissociation of NH₃ is both a source and a sink of H, which feed back to determine the outcome of the network in a non-linear way. For example, the NH₂ channel is a minor pathway to form N₂ in the solar or 10 × solar abundance atmosphere of K2-18 b, but it becomes an important pathway in the 100 × solar atmosphere.

The abundance of H at the pressure of NH₃ photodissociation also explains the different sensitivity of the HCN mixing ratio on the inclusion of a deep-tropospheric source of CO/CO and a partial depletion of NH₃. For K2-18 b, the reduction in the HCN mixing ratio is small or proportional to the reduction in the input NH₃ abundance, but more reduction in the HCN mixing ratio is found for PH2 b (Figures 6 and 7). This is because the photodissociation of NH₃ occurs at higher pressures in PH2 b than in K2-18 b. When abundant CO exists, the reactions $\mathrm{CO}+{\rm{H}}\mathop{\to }\limits^{M}\mathrm{HCO}$ and HCO + H → CO + H₂ efficiently remove H. Note that the first reaction in this cycle is three-body and only significant at sufficiently high pressures. This sink of H results in the reduction of CH₃ production (Reaction R8) and thus disfavors the branch in the NH₂ path that leads to HCN.

To summarize, the numerical models and the HCN formation mechanism presented here indicate that HCN and N₂ are generally the expected outcomes of the photodissociation of NH₃ in gaseous exoplanets that receive stellar irradiance approximately equal to Earth's, regardless of the stellar type.

The formation of CO and CO₂ as the most abundant photochemical gases of carbon on K2-18 b and PH2 b is another significant finding of our numerical models. The photodissociation of CH₄ in colder H₂-dominated atmospheres—such as the giant planets' atmospheres in the solar system—produces hydrocarbons such as C₂H₆ and C₂H₂ but not oxygenated species (e.g., Gladstone et al. 1996; Moses et al. 2005). This is because H₂O condenses out and is almost completely removed from the above-cloud atmosphere (such as in Kepler-167 e, Figure 8). External sources such as comets and interplanetary dust can supply oxygen to the upper atmospheres of Jupiter and the other giant planets (e.g., Moses et al. 2005; Dobrijevic et al. 2020), but we do not include this source in the present study. For warmer planets, however, H₂O is only moderately depleted by condensation. The above-cloud water is photodissociated at approximately the same pressure as is CH₄ (Figures 6 and 7). The photodissociations of CH₄ and H₂O together in H₂-dominated atmospheres produce a chemical network beyond hydrocarbons (Figure 12) and eventually lead to the formation of CO and CO₂. Even warmer atmospheres (e.g., the atmosphere of GJ 1214b with an effective temperature of 500–600 K) may also have CO and CO₂ as the most abundant photochemical gases of carbon (e.g., Kempton et al. 2011; Kawashima & Ikoma 2018).

Zoom In Zoom Out Reset image size

The photodissociation of CH₄ and the subsequent chemical reactions produce a wealth of hydrocarbons and radicals, and many of them (e.g., C, CH, CH₃, C₂H₂, and C₂H₄) lead to chemical pathways that form CO (Figure 12). Between K2-18 b and PH2 b and among the modeled metallicities, we do not see a monotonic trend regarding the relative contribution of these CO forming pathways, probably due to many chemical cycles and feedback in hydrocarbon photochemistry. CO is converted to CO₂ by the reaction with OH:

$$\begin{eqnarray}&&\mathrm{CO}+\mathrm{OH}\to {\mathrm{CO}}_{2}+{\rm{H}}.\end{eqnarray} \tag{ R14 }$$

Reaction (R14) is the dominant source of CO₂ in all models, and the only significant chemical loss of CO₂ is to form CO via either photodissociation or the reaction with elemental sulfur when available (Figure 12). The CO₂ that is not returned to CO is then transported through the lower boundary.

What are the sources of OH, O, and H that power the chemical pathways shown in Figure 12? At the pressure of CH₄ and H₂O photodissociation, the source of OH is the photodissociation of water,

$$\begin{eqnarray}&&{{\rm{H}}}_{2}{\rm{O}}\mathop{\to }\limits^{h\nu }{\rm{H}}+\mathrm{OH},\end{eqnarray} \tag{ R15 }$$

and the main sink is the reaction with H₂,

$$\begin{eqnarray}&&\mathrm{OH}+{{\rm{H}}}_{2}\to {{\rm{H}}}_{2}{\rm{O}}+{\rm{H}}.\end{eqnarray} \tag{ R16 }$$

Reaction (R16) is the main sink of OH in all models, which means that the use of OH in the chemical pathways shown in Figure 12 does not usually become the dominant sink of OH. Reactions (R15) and (R16) together are equivalent to the net dissociation of H₂, which overtakes the photodissociation of CH₄ and subsequent hydrocarbon reactions as the dominant source of H in temperate atmospheres. Lastly, the main source of O is the photodissociation of CO and CO₂, which eventually traces to OH and the photodissociation of water.

At this point we can explain the ratio between CO₂ and CO in the middle atmosphere, which is ∼1 on K2-18 b and ∼10 on PH2 b (Figures 6 and 7). Because Reaction (R14) is the main source of CO₂ and photodissociation is the main sink, the number density of CO₂ is $\sim {k}_{R14}[\mathrm{CO}][\mathrm{OH}]/{J}_{{\mathrm{CO}}_{2}}$, where k is the reaction rate constant and [] means the number density of a molecule. Because Reaction (R15) is the main source of OH, the number density of OH is $\propto {J}_{{{\rm{H}}}_{2}{\rm{O}}}[{{\rm{H}}}_{2}{\rm{O}}]$. Therefore, the ratio between CO₂ and CO is $\propto {J}_{{{\rm{H}}}_{2}{\rm{O}}}[{{\rm{H}}}_{2}{\rm{O}}]/{J}_{{\mathrm{CO}}_{2}}$. For any given metallicity, the abundance of H₂O in the middle atmosphere of PH2 b is 3 ∼ 5-fold greater than that in K2-18 b because PH2 b is slightly warmer (Figures 6 and 7). And, ${J}_{{{\rm{H}}}_{2}{\rm{O}}}$ at the top of the atmosphere on PH2 b is approximately twice that on K2-18 b, while ${J}_{{\mathrm{CO}}_{2}}$ is similar between the two planets (Figure 2). Together, this causes the [CO₂]/[CO] ratio to be greater in the atmosphere of PH2 b than in K2-18 b by ∼10-fold.

This trend to maintain the [CO₂]/[CO] ratio also controls how the atmosphere reacts to a deep-tropospheric source of CO and CO₂ that is applied as input at the lower boundary. The input CO₂ is always less than CO by one or more orders of magnitude (Figures 3–5). On PH2 b, photochemical processes convert CO into CO₂ in the middle atmosphere (∼10² Pa), and cause the steady-state mixing ratio of CO₂ to be greater than that of CO. This conversion even becomes a significant sink of H₂O and causes H₂O to be depleted in the middle atmosphere in the 100 × solar metallicity case (Figure 7). The CO to CO₂ conversion is not so strong in the atmosphere of K2-18 b or Kepler-167 e, and their mixing ratios in the middle atmosphere are largely the input values at the lower boundary (Figure 6).

Finally, let us turn to the impact of H₂O and NH₃ photodissociation on the hydrocarbon chemistry. Compared with Kepler-167 e, the mixing ratio of C₂H₆—the dominant, supposedly long-lived hydrocarbon—in K2-18 b and PH2 b is smaller and sometimes features an additional peak near the cloud deck (Figures 6–8). Particularly, the atmospheres of K2-18 b and PH2 b have a strong sink of C₂H₆ at ∼1–10 Pa, while the atmosphere of Kepler-167 e does not. This sink is ultimately because of the high abundance of H produced by the photodissociation of H₂O (Reactions (R15) and (R16)). The detailed reaction path involves the formation of C₂H₅ from C₂H₆ (by direct reaction with H or photodissociation to form C₂H₄ followed by H addition), and then C₂H₅ + H → 2 CH₃. Because of the abundance of H, CH₃ mostly combines with H to form CH₄, rather than recombines to form C₂H₆. It is well known that the abundance of hydrocarbons is fundamentally controlled by the relative strength between ${\rm{H}}+{\mathrm{CH}}_{3}\mathop{\to }\limits^{M}{\mathrm{CH}}_{4}$ and ${\mathrm{CH}}_{3}+{\mathrm{CH}}_{3}\mathop{\to }\limits^{M}{{\rm{C}}}_{2}{{\rm{H}}}_{6}$ (e.g., Gladstone et al. 1996; Moses et al. 2005). Here, we find that the added H from H₂O photodissociation results in a net sink for C₂H₆ in K2-18 b and PH-2 b at ∼1–10 Pa and limits the abundance of hydrocarbons in their atmospheres. This sink does not exist in the atmosphere of Kepler-167 e, because little H₂O photodissociation occurs in its atmosphere. Additionally, near the cloud deck, the temperature is warmer, and Reaction (R8) that uses H from the photodissociation of NH₃ provides an additional source of CH₃, and some of the CH₃ becomes C₂H₆ and thus it peaks near the cloud deck. The formation of hydrocarbons is thus strongly impacted by the water and nitrogen photochemistry.

3.5.1. Transmission Spectra

Figures 13–15 show the transmission spectra of the temperate and cold planets K2-18 b, PH2 b, and Kepler-167 e, based on the gas and sulfur haze profiles simulated by the photochemical models. These modeled spectra can be regarded as the canonical examples of a temperate (Earth-like insolation) planet irradiated by an M dwarf star (K2-18 b, and also TOI-1231b), a temperate (Earth-like insolation) planet irradiated by a G/K star (PH2 b), and a cold (∼0.1 × Earth insolation) planet irradiated by a G/K star (Kepler-167 e). Here, we focus on the wavelength range 0.5–5.0 μm, where several instruments on JWST will provide spectral capabilities (e.g., Beichman et al. 2014).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For K2-18 b, the equilibrium gases CH₄, H₂O, and NH₃, as well as the photochemical gas HCN have potentially detectable spectral features in the visible to mid-infrared wavelengths (Figure 13). Adding deep-tropospheric source of CO, CO₂, and N₂ and sink of NH₃ does not cause a significant change of the spectrum of a 10 × solar metallicity atmosphere. However, a 100 × solar metallicity atmosphere with deep-tropospheric source and sink would be free of the spectral features of NH₃ or HCN, but instead have potentially detectable features of CO₂ and CO.

Strikingly, the models from 1 × to 100 × solar abundance and with the standard eddy diffusion coefficient provide good fits to the existing transit depth measurements by K2, HST, and Spitzer (Benneke et al. 2019; Tsiaras et al. 2019). The models with a 100-fold greater eddy diffusion coefficient would have the sulfur haze layer extending to 10² Pa and mute the spectral features in 1.1–1.7 μm, at odds with the HST data. Both CH₄ and H₂O contribute to the spectral modulations seen by HST, which may have caused the difficulties in the identification of the gases by spectral retrieval (Benneke et al. 2019; Tsiaras et al. 2019; Blain et al. 2021).

HCN, one of the most abundant photochemical gases in the middle atmosphere, is likely detectable in K2-18 b via its spectral band at ∼3.0 μm. The HCN is produced from the photodissociation of NH₃ in the presence of CH₄. Also at 3.0 μm are the absorption bands of NH₃ and to a lesser extent C₂H₂. It would be possible to disentangle these bands with a reasonably wide wavelength coverage because NH₃ has multiple and more prominent bands in the mid-infrared (Figure 13), and because C₂H₂ should have a minimal abundance in the middle atmosphere (Figure 6) and contribute little to the transmission spectra.

The spectral bands of CO₂ and CO can be seen in the modeled spectra (in 4–5 μm) of K2-18 b only when the atmosphere has supersolar metallicity and the transport from the deep troposphere is taken into account (Figure 13). In other words, the CO and CO₂ that are produced from the photodissociation of CH₄ together with H₂O would have too low mixing ratios to be detected. The photodissociation of CH₄ also produces C₂H₆. While C₂H₆ has strong bands at 3.35 and 12 μm, they would not be detectable due to its relatively low abundance and the strong CH₄ and NH₃ bands at the same wavelength, respectively (Figure 13).

For PH2 b, prominent spectral bands of CH₄, H₂O, and the photochemical gases CO₂ and HCN can be expected (Figure 14). NH₃ is not detectable because it is depleted by photodissociation to the cloud deck (Figure 7). Even though its pressure of photochemical depletion can be reduced to ∼10² Pa for a large eddy diffusion coefficient, the sulfur haze in that case would mute spectral features that are generated from approximately the same pressure levels (Figure 10) and thus cause NH₃ to be undetectable. HCN, CO₂, and CO are the most abundant photochemical gases (Figure 7); but the CO bands are intrinsically weaker and so CO₂ and HCN are the detectable photochemical gases via their spectral bands at 4.2 and 3.0 μm, respectively. Similar to K2-18 b, adding a deep-tropospheric source of CO, CO₂, and N₂ and sink of NH₃ does not cause a significant change of the spectrum of a 10 × solar metallicity atmosphere. However, a 100 × solar metallicity atmosphere with a deep-tropospheric source and sink would not have the spectral features of H₂O or HCN and would have more prominent features of CO₂ and CO, as predicted by the photochemical model (Figure 7).

Lastly for the cold planet Kepler-167 e, the transmission spectra will be dominated by the absorption bands of CH₄ (Figure 15), as H₂O is completely removed by condensation and NH₃ by condensation and photodissociation. For a large eddy diffusion coefficient, the pressure of photochemical depletion of NH₃ can be reduced to ∼10³ Pa (Figure 10) and this can produce a spectral band of NH₃ at ∼3.0 μm. Thus, a search for this absorption band in the transmission spectra may constrain the eddy diffusion coefficient, although to distinguish it with a small peak due to the combined absorption of the photochemical gases HCN and C₂H₂ (Figure 15) may involve quantification through photochemical models. The main photochemical gas in this cold atmosphere C₂H₆ has spectral bands at 3.35 and 12 μm. The 3.35 μm band is buried by a strong CH₄ band, and while not shown in Figure 15, the 12 μm band might be detectable given appropriate instrumentation with the spectral capability in the corresponding wavelength range. Finally, the deep-troposphere-sourced CO₂ and CO in a 100 × solar metallicity atmosphere may produce detectable spectral features at 4–5 μm.

To summarize, transmission spectroscopy from the visible to mid-infrared wavelengths can provide the sensitivity to detect the equilibrium gases CH₄ and H₂O, and the photochemical gases HCN, and in some cases CO₂ in temperate/cold and H₂-rich exoplanets. We do not expect C₂H₆ to be detectable. NH₃ would be detectable on temperate planets around M dwarf stars but not detectable on temperate planets around G/K stars. The deep-tropospheric source and sink can have a major impact only on the transmission spectrum of a 100 × solar metallicity atmosphere, where typically the features of NH₃ and HCN would be reduced and those of CO₂ and CO would be amplified. The detection and nondetection of these gases will thus test the photochemical model and improve our understanding of the photochemical mechanisms as well as tropospheric transport in temperate/cold and H₂-dominated atmospheres.

3.5.2. Spectra of the Reflected Starlight

The temperate and cold planets around G/K stars are widely separated from their host stars and may thus also be characterized in the reflected starlight by direct imaging. Figure 16 shows the geometric albedo spectra of PH2 b and Kepler-167 e in the visible and near-infrared wavelengths that approximately correspond to the Roman Space Telescope's coronagraph instrument (Kasdin et al. 2020) and its potential Starshade Rendezvous (Seager et al. 2019) and the HabEx concept (Gaudi et al. 2020). While PH2 b and Kepler-167 e themselves are not potential targets for these missions, their albedo spectra broadly resemble the targets in the temperate (PH2 b) and cold (Kepler-167 e) regimes.

Zoom In Zoom Out Reset image size

The spectral features of CH₄ and H₂O can be seen in the reflected starlight of PH2 b. This ability to detect H₂O in giant planets warmer than Jupiter is consistent with the finding of MacDonald et al. (2018). In addition to the absorption features of CH₄ and H₂O, the albedo spectra of PH2 b feature the absorption of the sulfur (S₈) haze layer at wavelengths shorter than ∼0.5 μm. This result is consistent with the findings of Gao et al. (2017). For a greater eddy diffusion coefficient, the sulfur haze layer is higher and the spectral features of CH₄ and H₂O become weaker. Interestingly, the absorption features of H₂O are the most prominent in the solar abundance case, and they are somewhat swamped by the adjacent CH₄ features at higher metallicities. This is because, as H₂O condenses out, the above-cloud mixing ratio of H₂O only slightly increases with the metallicity, while that of CH₄ increases proportionally (Figure 7). Only the absorption of CH₄ can be seen in the albedo spectra of Kepler-167 e, as H₂O is depleted by condensation. On both planets, the spectral features of NH₃ are not seen due to its weak absorption (Irwin et al. 2018) and photochemical depletion to the cloud deck (Figures 7 and 8). The deep-tropospheric source and sink has minimal impact on the albedo spectra, unless in the 100 × solar metallicity atmosphere on PH2 b where a reduction of the CH₄ features can be seen.