Beyond Equilibrium Temperature: How the Atmosphere/Interior Connection Affects the Onset of Methane, Ammonia, and Clouds in Warm Transiting Giant Planets

Even 25 yr after the discovery of gas giant exoplanets (Mayor & Queloz 1995), we are still in our infancy in characterizing the atmospheres of these worlds. Over the past two decades, astronomers have made fantastic strides to obtain spectra of exoplanets, but we still have much to do. In the realm of transiting planets, observers have often been hindered by instruments aboard space- and ground-based telescopes that were never designed for precision time series spectrophotometry. Even as dozens of planets have been seen in transmission spectroscopy (e.g., Sing et al. 2016) and occultation spectroscopy or photometry (e.g., Kreidberg et al. 2014; Garhart et al. 2020), our ability to understand the physics and chemistry of hydrogen-dominated atmospheres has been limited, principally by low signal-to-noise ratio observations and limited wavelength coverage. On the side of the directly imaged planets, telescopes like Keck, the Very Large Telescope, and Gemini have allowed more robust atmospheric spectroscopy, but with a sample size that is so far limited in number (e.g., Konopacky et al. 2013; Macintosh et al. 2015; Gravity Collaboration et al. 2019).

It is with brown dwarfs, now numbering over 1000, with temperatures down to 250 K (Luhman 2014; Skemer et al. 2016), where robust atmospheric characterization has taken place over the past 25 yr. The major transitions in atmospheric chemistry and cloud opacity have now been unveiled (Burrows et al. 2001; Kirkpatrick 2005; Helling & Casewell 2014; Marley & Robinson 2015), although major open questions still exist on the role of clouds in shaping the spectra across a range of T_eff and surface gravity. However, it should be clear that relying solely on the classic "stellar" fundamental quantities of T_eff, log g, and metallicity has already shown its faults for these objects. For instance, time variability can reach tens of percent, and effects due to rotation rate (Artigau 2018) and viewing angle have now been seen as important to take into account for atmospheric characterization (Vos et al. 2017).

To understand the atmospheres of giant planets, we will certainly need a larger sample size than the brown dwarfs, for a similar level of understanding, as planets have many additional complicating factors (Marley et al. 2007). For instance, substantial recent work has gone into assessing the Spitzer IRAC 3.6/4.5 colors of cooler transiting planets, in order to better assess atmospheric metallicity and the role of CH₄ and CO absorption (Kammer et al. 2015; Triaud et al. 2015; Wallack et al. 2019; Dransfield & Triaud 2020). The wide diversity of colors at a given T_eq, much wider than is seen in brown dwarfs at a given T_eff (Beatty et al. 2014; Dransfield & Triaud 2020), has been interpreted as needing a large dispersion in atmospheric metallicity and potentially C/O ratio.

Planets present additional complicating physics, such as heating from above, across a range of incident stellar spectral types (Mollière et al. 2015), in addition to a range of UV fluxes. The planets will have diverse day–night contrasts and circulation regimes, likely with a very wide range of atmospheric metallicities (Fortney et al. 2013; Kreidberg et al. 2014) and nonsolar abundance ratios (Öberg et al. 2011; Madhusudhan et al. 2014; Espinoza et al. 2017). The cooling of the interiors of giant planets—even the cooler giant planets not affected by the hot Jupiter radius anomaly—is also still not fully understood (e.g., Vazan et al. 2015; Berardo & Cumming 2017).

Key science goals of the James Webb Space Telescope (JWST) and ARIEL are to obtain spectra of a wide range of planetary atmospheres (Beichman et al. 2014; Greene et al. 2016; Tinetti et al. 2018). In the realm of transiting giant planets, which have predominantly accreted their atmospheres from the protostellar nebula, one aspect of this science will be characterizing planets over a wide range of temperatures, to sample a wide range of transitions in atmospheric chemistry and cloud formation. A significant amount of previous theoretical and modeling work has gone into trying to predict and understand trends in the atmospheres of these planets, going back to important early works such as Marley et al. (1999) and Sudarsky et al. (2000), supplemented by later works like Fortney et al. (2008), Madhusudhan et al. (2011b), and Mollière et al. (2015). Most of these papers have pointed to planetary equilibrium temperature, T_eq, as the dominant physical parameter that determines atmospheric physics and chemistry, somewhat akin to T_eff in stars. While there are good reasons to think that this is indeed true, there are equally good reasons to think that T_eq is only a starting point and that other physical parameters can have a crucial effect on determining the atmospheric spectra that we will see.

Of course, T_eq is only part of the energy budget, and it is well understood that ${T}_{\mathrm{eff}}^{4}={T}_{\mathrm{eq}}^{4}+{T}_{\mathrm{int}}^{4}$, with T_int parameterizing the intrinsic flux from the planetary interior, and T_eq from thermal balance with the parent star. In Jupiter, for instance, T_eq and T_int are similar, with neither dominating the energy budget (Pearl & Conrath 1991; Li et al. 2018). Recently, Thorngren et al. (2019, 2020) pointed out that the radii of "hot" and "warm" Jupiter population can be used to assess the intrinsic flux coming from planetary interiors. Often Jupiter-like values of T_int (100 K) had been chosen for convenience, but the inflated radius of a typical hot Jupiter goes hand in hand with a hotter interior and much higher T_int values (assuming convective interiors).

This work gives us the ability to better assess the depth of the radiative–convective boundary (RCB) in these strongly irradiated planets. A key finding of Thorngren et al. (2019) was that the T_int values are typically larger (sometimes much larger) than previous expectations, which moves the RCB to lower pressures. A higher T_int can remove or weaken cold traps in these atmospheres, which can alter atmospheric abundances and the depth at which clouds form. Much additional work needs to be considered for these hot planets, perhaps much of it in the 3D context, given the large day–night temperature contrasts (Parmentier & Crossfield 2018).

The role of the current paper is to serve as a complement of sorts, and extension to, the work of Thorngren et al. (2019), but mostly for cooler planets. For planets below T_eq ∼ 1000 K, a wide range of chemical and cloud transitions should occur (Marley et al. 1999; Sudarsky et al. 2000; Morley et al. 2012). What is not as appreciated, however, is that temperatures in the deeper atmosphere, which are typically not visible, can play as large a role, or even a larger role, in determining atmospheric abundances as the visible atmosphere, which is dominated by absorbed starlight.

The temperatures of the deep atmosphere, while typically not measurable, can be constrained in a variety of ways. Observationally, flux from the deep interior can potentially be seen at wavelengths where the opacity is low ("windows"). This has been constrained for GJ 436b emission photometry (Morley et al. 2017a) and could potentially be done for a small number of other planets (Fortney et al. 2017). Another is cold-trapping gases into condensates via crossing a condensation curve in the deep atmosphere (Burrows et al. 2007; Fortney et al. 2008; Beatty et al. 2019; Sing et al. 2019; Thorngren et al. 2019).

As was done in Thorngren et al. (2019), the planetary radius can be used as a constraint, with assumptions about interior energy transport. Planetary thermal evolution/contraction models aim to understand the cooling of the planetary interior with time (e.g., Fortney et al. 2007; Baraffe et al. 2008). Furthermore, there are planets for which thermal evolution models can be made more uncertain—those that are undergoing tidal eccentricity damping. If this energy is dissipated in the planet's interior, the temperature of the deep atmosphere can be significantly enhanced compared to simple predictions. Lastly, one can assess the role of disequilibrium chemistry tracers. Recently, Miles et al. (2020) have used observations of disequilibrium CO in cold brown dwarfs to understand atmospheric dynamics and temperature structures. They constrain the rate of atmospheric vertical mixing as a function of T_eff, providing strong evidence for a detached radiative zone, below the visible atmospheres, long predicted in these atmospheres (Marley et al. 1996; Burrows et al. 1997). It is these disequilibrium tracers that we turn to next, in more detail.

Due to nonequilibrium chemistry via vertical mixing, deep atmosphere temperatures can matter as much as temperatures in the visible atmosphere in determining observable abundances. This well-understood process affects abundances when the mixing timescale for a parcel of gas, t_mix, is shorter than the chemical conversion timescale, t_chem, for a given chemical reaction. Well-studied reactions are CO to CH₄ and N₂ to NH₃. These timescales can be so long that the gas in the visible atmosphere (at, say, 1 mbar) will be representative of pressure–temperature (P–T) conditions at ∼1–1000 bars, as we will readily show. The effects of nonequilibrium chemistry on the atmospheric abundances and resulting spectra in giant planet (both solar system and extrasolar) and brown dwarf atmospheres have previously been extensively studied (Fegley & Lodders 1996; Saumon et al. 2003, 2006; Visscher et al. 2010; Madhusudhan et al. 2011b; Moses et al. 2011; Visscher & Moses 2011; Venot et al. 2012; Moses et al. 2013; Miguel & Kaltenegger 2014; Zahnle & Marley 2014; Macdonald & Madhusudhan 2017; Molaverdikhani et al. 2019, 2020; Miles et al. 2020; Venot et al. 2020), and here we will not break new ground on the chemistry. Rather, following the carbon and nitrogen chemistry work of Zahnle & Marley (2014), we will point out several novel complexities that arise when applying nonequilibrium chemistry to the quite inhomogeneous exoplanet population. Given the very large uncertainties in vertical mixing speeds, in particular for these irradiated atmospheres that are mostly radiative rather than convective (where mixing-length theory could plausibly be used), in addition to uncertainties in thermal evolution models, as well as the currently unknown atmospheric metal enrichments, we will show that a very wide range of behavior should be expected. For instance, one should not expect a single transition temperature in T_eq from CO-dominated to CH₄-dominated atmospheres, an area of active study already with Hubble and Spitzer (Stevenson et al. 2010; Morley et al. 2017a; Kreidberg et al. 2018; Benneke et al. 2019).

We can first look at an illustrative example of why vertical mixing from different atmospheric depths can strongly affect observed abundances and spectra, by exploring the behavior of CO, CH₄, and H₂O. Figure 1 shows the atmospheric P–T profile for a planet at 0.15 au from the Sun, with T_eq = 710 K. Five models are shown, with decreasing T_int, leading to cooler interior adiabats. Underplotted in light gray are curves of constant volume mixing ratio (mole fraction) for CO, to the lower left, following the chemical equilibrium calculations of Visscher et al. (2010) and Visscher (2012). Underplotted in dark gray is the same for CH₄, to the upper right. The thick dashed black curve shows the equal-abundance boundary, where the mixing ratio of CO = CH₄:

$$\begin{eqnarray}&&{\mathrm{log}}_{10}P\approx 5.05-5807.5/T+0.5[\mathrm{Fe}/{\rm{H}}],\end{eqnarray} \tag{ 1 }$$

for P in bars, T in K, and [Fe/H] as the metallicity (Visscher 2012). When we turn to nitrogen chemistry in Section 4.2, we will use the analogous N₂ = NH₃ equal-abundance curve:

$$\begin{eqnarray}&&{\mathrm{log}}_{10}P\approx 3.97-2721.2/T+0.5[\mathrm{Fe}/{\rm{H}}].\end{eqnarray} \tag{ 2 }$$

Zoom In Zoom Out Reset image size

Numbered black filled circles in Figure 1 have been placed along the profiles. Point 1 is at 1 mbar, a pressure that would be readily probed in transmission spectroscopy. Point 2 is at 700 K, where the local temperature is equal to T_eff, a good representation of the mean thermal photosphere in emission. Points 1 and 2 are in the CH₄-dominated region, with point 2 having ∼10× more CO. Moving down to point 3, all profiles are now in the CO-dominated regime, where the CH₄ abundance falls off dramatically with temperature. Point 4 is deeper in the atmosphere along the hottest adiabat, in the CO-rich region, with a decrease in CH₄ compared to point 3. Points 5 and 6 are along cooler adiabats, with point 5 having abundances quite similar to point 3. Point 6 is quite interesting, in that, while it is in the deep part of the atmosphere, it is clearly within the CH₄-dominant region and has the same CH₄ and CO abundances as point 2. This complexity should be contrasted with the profile of a T_eff  = 1000 K, log g = 5 brown dwarf, plotted by the thick orange curve. For the brown dwarf, as a parcel of gas moves along from high pressure to low, there is a monotonic increase in CH₄ and decrease in CO.

As one would expect, the spectra that use the quenched abundances, brought up to the visible atmosphere from the black points of Figure 1, vary considerably as the abundances of CO and CH₄ vary by orders of magnitude. In addition, the abundance of H₂O changes depending on whether CO is present as well. We demonstrate this for five different models shown in Figure 2. For points to the "right" of the CO/CH₄ equal-abundance curve, like points 3, 5, and especially 4, the CO band is much stronger and CH₄ is weaker. The spectra from points 1 and 6 are substantially similar, given their relative positions in CO/CH₄ phase space. The lack of monotonic behavior in the mixing ratio (and observability) of CH₄ as a function of the quench pressure was also pointed out by Molaverdikhani et al. (2019, see their Figure 2), although they did not explore variations in the lower boundary condition, which is our focus here.

Zoom In Zoom Out Reset image size

Such a wide range of internal adiabats, for a given upper atmosphere, is quite possible owing to the differences in cooling histories in giant planets. It is by now widely appreciated that giant planets cool over time, most dramatically at young ages, and that more massive planets take longer to cool (Marley et al. 1996; Burrows et al. 1997; Chabrier & Baraffe 2000). For reference, in Figure 3 we plot cooling tracks for planets from 10 to 0.1 M_J (32 M_⊕) for ages from 10⁷ to 10¹⁰ yr, using the models of Fortney et al. (2007) and Thorngren et al. (2016). At an age of 3 Gyr, for instance, T_int values of 50–350 K span the population. Such model planets would in reality all have different surface gravities, which would then yield different P–T profile shapes, even at the same orbital separation, as shown in Figure 4. This plot is for the expected surface gravity for the five planet masses (at an age of 3 Gyr) shown in Figure 3.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Taken as a whole, these simple examples serve as motivation to explore a wider range of parameter space for H/He-dominated atmospheres. The aim then is to show that a range of factors other than equilibrium temperature can have significant impacts, even dominant impacts, on atmospheric abundances and spectra. We also explore how nonequilibrium chemistry can serve as a tracer for understanding the deep temperature structure for these atmospheres, at pressures far below where one can probe directly. After describing our methods in a bit more detail, we investigate these factors, first for well-known transiting Neptune-class planets GJ 436b, GJ 3470b, and WASP-107. After that, we will explore carbon chemistry more generally, followed by nitrogen chemistry more generally, before our discussion (with caveats) and conclusions.

The model atmosphere methods used here have previously been extensively described in the literature. We compute planet-wide average ("4πぱい re-radiation of absorbed stellar flux") 1D radiative–convective equilibrium models using the model atmosphere code described in the papers of Marley & McKay (1999), Marley et al. (1996), and Fortney et al. (2005, 2008) and the general review of Marley & Robinson (2015). The radiative transfer methods are described in McKay et al. (1989). The model uses 90 layers, typically evenly spaced in log pressure from 1 μみゅーbar to 1300 bars. The equilibrium chemical abundances follow the work of Lodders & Fegley (2002), Visscher et al. (2006, 2010), and Visscher (2012). The opacity database is described in Lupu et al. (2014) and Freedman et al. (2014). Transmission spectra are calculated using the 1D code described in Morley et al. (2017b).

As already mentioned, the giant planet thermal evolution models use the methods of Fortney et al. (2007) and Thorngren et al. (2016). These thermal evolution calculations use an extensive grid of 1D nongray solar-composition radiative–convective atmosphere models, which serve as the upper boundary condition. The interior H/He equation of state is that of Saumon et al. (1995). We make the standard, typical assumption of a fully convective H/He envelope, and these evolution models also have a 10 M_⊕ ice/rock core.

Tidal heating, to be investigated in a Section 3, uses the extensive tidal evolution equations derived in Leconte et al. (2010). We determine the tidal heating rate (in energy per second) with Equation (13) in this work. We will show that for some planets this tidal heating flux from the interior can be orders of magnitude higher than that calculated from normal secular cooling of the interior.

When treating nonequilibrium chemistry, an important topic in this paper, we make extensive use of the findings of Zahnle & Marley (2014). These authors provide quenching relations that are derived by fitting to the complete chemistry of a full ensemble of 1D kinetic chemistry models. We use the standard "quench pressure" formalism, where we assume chemical equilibrium where the chemical conversion time, t_chem, is shorter than the vertical mixing time, t_mix. The local values of t_mix along a P–T profile use the standard assumption that t_mix = L²/K_zz, where L is the length scale of interest, here assumed to be the local pressure scale height, H, and K_zz is the vertical diffusion coefficient. Other, potentially smaller values of L could be used (Smith 1998; Visscher & Moses 2011); however, as we discuss below, uncertainties in K_zz dwarf any uncertainty in L, so, following Zahnle & Marley (2014), we make the simplest choice.

For these strongly irradiated planets, atmospheres can be radiative until depths of tens of bars, even beyond ∼1 kbar, depending on the the value of T_int. The lower the value of T_int, the deeper the radiative zone, as shown in Figure 1. While in convective zones mixing-length theory can be used as a guide to values of K_zz (Gierasch & Conrath 1985), in radiative regions no such readily usable theory exists, although it is generally expected that radiative regions will have orders of magnitude lower K_zz values.

Some 3D circulation model simulations of hot Jupiters have attempted to gauge reasonable K_zz values. Parmentier et al. (2013) suggested a fit to models of planet HD 209458b that yielded ${K}_{{zz}}=5\times {10}^{8}/\sqrt{{P}_{\mathrm{bar}}}$ cm² s⁻¹. They suggest that cooler planets, like the ones treated here, should have slower vertical wind speeds and smaller values of K_zz. More recent work has tried to estimate K_zz from first principles (Zhang & Showman 2018a, 2018b; Menou 2019).

The chemical kinetics literature for irradiated planets shows a range of K_zz choices. These include basing values tightly on 3D simulations but, more commonly, choosing a wide range of constant-with-altitude K_zz values, to bracket a reasonable parameter space. It is this bracketing choice that we make here, as we aim to make the point that nonequilibrium chemistry must be important for a wide range of objects. For calculations for particular planets of interest it may be worthwhile to generate K_zz predictions from GCM simulations. We return to this point in Section 5. Follow-up work that couples planetary temperature structures with detailed predictions of K_zz profiles (Zhang & Showman 2018a, 2018b; Menou 2019), to predict atmospheric abundances, would be important and fruitful work.

Before exploring a wide range of planets, we first investigate how our models can be used to understand the atmospheric abundances of three (relatively) well-studied Neptune-class transiting planets, which have been the targets of many observations with Spitzer and Hubble.

In Section 3 we examined the CO–CH₄ boundary for specific tidally heated Neptune-class planets. Objects with tidal heating are special cases, but they certainly will be common enough that they cannot simply be ignored, when looking at general trends. But here we can examine the general trends in the absence of tidal heating, for a range of planet masses and ages. As we will see, the range of cooling histories, as well as the lack of clarity with how vertical mixing will change with planet mass, can lead to important complexities.

4.1.1. Effects of T_eq and Vertical Mixing

We first examine the general case of a Saturn-like exoplanet as a function of distance from a Sun-like star. Here we have chosen a 10× solar atmosphere, surface gravity of 10 m s⁻², and T_int  = 75 K, representative of a several-gigayear-old Saturn-mass exoplanet. We choose this as our "base planet" since these kinds of giant planets would be excellent targets for atmospheric characterization via transmission. Atmospheric P–T profiles are shown in Figure 7, for planets from 0.06 to 2 au. The three sets of black filled circles show quench pressures corresponding to log K_zz values of 4, 8, and 11. Most importantly, at lower pressures, the atmospheres diverge quiet widely, owing to the factor of ∼1100 difference in incident flux across these models.

Zoom In Zoom Out Reset image size

As one looks deeper, it is apparent that profiles modestly converge as the pressure increases, followed by a dramatic "squeezing together" as the planets fall on nearly identical adiabats. This is a generic behavior for g/T_int pairs, and one could make a plot like this for any Jupiter-like planet, super-Jupiter, or sub-Saturn. Why this behavior occurs requires some discussion. To our knowledge this effect was first noted in Figure 3 of Fortney et al. (2007), who described the effects of these "bunched-up" deep profiles on the mass–radius relation for warm transiting giant planets, but they did not identify a cause for the similarity of the deep temperatures.

A study of the gray analytic temperature profiles of Guillot (2010) suggests, via their Equation (29), a relation between the temperature (T) and optical depth τたう that is a function of only three quantities: the irradiation temperature (which is directly related to T_eq), T_int, and γがんま, the ratio of the visible to thermal opacities. If γがんま is relatively constant, and at a given T_int value, decreasing T_eq cools the entire atmosphere at every τたう, including the deep region that here transitions to an adiabat. However, if γがんま were to dramatically decrease with decreasing T_eq, the deep T − τたう profile (analogous to our deep T–P profile) could remain nearly constant at depth with an upper atmosphere that was colder with decreasing T_eq. Indeed, Figure 5 of Freedman et al. (2014) shows a factor of ∼60 falloff in γがんま from ∼1400 to 700 K, due to the loss of alkali metals Na and K from the vapor phase, with γがんま relatively constant at hotter and colder temperatures. This 700–1400 K temperature range corresponds reasonably well to what is seen in our Figure 7 and the "middle region" of Figure 3 of Fortney et al. (2007). Therefore, we suggest that this change in visible opacity is the dominant physical effect that keeps the deep atmosphere temperatures relatively constant across this T_eq range. However, additional work on this point is surely needed.

Of particular interest is that the coldest profiles are mostly in the CH₄-dominant region at lower pressures, but along the atmospheric adiabat, as one reaches hotter layers, one finds gradually more CO. This is the "typical" case for brown dwarfs (Saumon et al. 2003; Phillips et al. 2020) and for Jupiter as well (Prinn & Barshay 1977; Lodders & Fegley 2002). However, for the hottest models, this typical trend is reversed, and when one probes quite deeply, one reaches more CH₄-rich gas, in particular at P > 1 bar, where the isothermal regions are reached.

We can examine how atmospheric abundances are affected by making plots of volume mixing ratio as a function of planetary T_eq. Such a plot is shown in Figure 8 and includes all the profiles shown in Figure 7. The mixing ratios at 1 mbar for H₂O, CO, and CH₄ are plotted, for equilibrium chemistry and for log K_zz of 4 and 8. In the equilibrium chemistry case (dashed curves), the changeover from CO dominant to CH₄ dominant is at about T_eq  = 850 K. As one goes cooler, this also leads to an increase in the H₂O abundance, as oxygen is liberated from CO (and CO₂).

Zoom In Zoom Out Reset image size

If we include quite sluggish vertical mixing, with log K_zz  = 4 (thin solid line), this boundary shifts dramatically left, to a much lower T_eq value of only 475 K. The slopes of the CH₄ and CO curves, versus T_eq, are both quite shallow compared to the equilibrium chemistry case, and one might readily expect both molecules to be seen from ∼800 to 200 K. Of course, how "detectable" a molecule is depends strongly on the wavelength being investigated, the spectral resolution, and the impact on other opacity sources, like clouds. Given the nondetections of CH₄ with HST at mixing ratios of ∼10⁻⁶ in the Neptune-class planets (See Section 3), here we suggest ∼10^−5.5. However, the 3.3 and 7.8 μみゅーm bands of CH₄ and the 4.5 μみゅーm band of CO are strong and could likely yield detections at lower mixing ratios, in particular at high spectral resolution.

Interestingly, a look back to Figure 7 might suggest that the log K_zz = 8 case might be a bit less extreme in altering abundances, even though we are mixing up from even hotter layers. The modest pinching together of the P–T profiles yields a behavior in Figure 8 (solid line) that is intermediate between the two previous behaviors, with a crossover T_eq of 680 K. Both CO and CH₄ may be seen from T_eq  ∼ 900 to 400 K. The upshot here is that the value of K_zz in these atmospheres and its depth dependence, which is currently unknown, will have a significant effect on the atmospheric abundances as a function of T_eq, and a wide range of behavior is expected. As discussed later, given that K_zz is unlikely to be constant with altitude, more realistic mixing further complicates this picture.

4.1.2. Effects of Planet Mass at a Given Age

In the previous section we examined one particular planet, a Saturn-like object at different distances from the Sun. However, we have already discussed in some detail in the Introduction that planets of different masses are expected to have quite different cooling histories (Figure 3).

We can begin to address the question of planet mass with three disparate planet examples, with planets of 10 M_J (a super-Jupiter), 1 M_J, and 0.1 M_J (32 M_⊕, a super-Neptune). For now we limit ourselves to the same 10× atmospheric metallicity, so as not to change too many parameters at once. Similar to Figure 7 above, we have computed a range of atmospheric P–T profiles for these three planets, at different distances from the Sun, assuming an age of 3 Gyr and the T_int values from Figure 3. These profiles are shown in Figure 9. For clarity, profiles are only shown at three distances, 0.1, 0.5, and 2 au. Along each profile, colored filled circles, from lower to higher pressure, show log K_zz of 4, 8, and 11, respectively. The more massive the planet, the higher the surface gravity, and the higher the pressure at a given temperature, in the outer atmosphere. This, however, is reversed in the deep atmosphere and interior, as the higher-mass planets take longer to cool, so they have a higher T_int (333, 117, and 52 K, respectively, for the 10, 1, and 0.1 M_J models) and "hotter" (higher specific entropy) interior adiabat. The much larger scale heights for the low-gravity models mean greater physical distances for mixing, thus longer mixing times for a fixed K_zz, and hence lower quench pressures.

Zoom In Zoom Out Reset image size

What we are particularly interested in here is how the roles of surface gravity and cooling history work to dramatically change the ratio of CO/CH₄ in these atmospheres. We address this scenario in Figure 10. This abundance ratio is plotted versus planetary T_eq, and we will first examine the abundances for equilibrium chemistry at 1 mbar. The "transition" T_eq value is 950 K at 10 M_J and 850 K at 1 and 0.1 M_J. With sluggish vertical mixing (log K_zz = 4), the story becomes more complex, however. The 10 M_J planet has a relatively hot interior adiabat, which is essentially the same for all values of T_eq, as seen in orange in Figure 9. For such a large value of T_int, the smaller values of T_eq become essentially irrelevant. For the lower-mass planets, the transition T_eq is much lower than in the equilibrium case, reaching 500 K. For more vigorous mixing (log K_zz = 8), more CH₄-rich gas is brought up, leading to a hotter transition temperature, at 700 K.

Zoom In Zoom Out Reset image size

4.1.3. Effects of Planet Age at a Given Mass

Up until this point, we have examined "old" planetary systems that to date make up the vast majority of the transiting population. However, studying younger transiting planets to better understand evolutionary histories is extremely important. First, this would yield connections to the directly imaged self-luminous planets, which are predominantly young (Bowler 2016). Second, understanding atmospheric abundances as a function of planet age would give us new insight into planetary thermal evolution. Third, since parent stars are much more active when they are young, high X-ray and ultraviolet fluxes for young systems could drive quite interesting photochemistry.

In the absence of tidal heating giant planet interiors inexorably cool as they age, meaning cooler interior adiabats and lower T_int values. In the face of vertical mixing, we should expect atmospheric abundances to change then as well. We also note here that Table 1 may prove a useful guide to the range of atmosphere models and figures in the paper. We examine the effect on a range of P–T profiles for a Jupiter-like example (1 M_J, 3× solar) at 0.15 au in Figure 11. The values of T_int are taken from every half dex in planetary thermal evolution from an age of 10 Myr to 10 Gyr, yielding seven models from T_int of 501 to 84 K. For moderately irradiated planets like these, the cooling of the interior has little effect on the upper atmosphere (Sudarsky et al. 2003), but we should expect quite different atmospheric abundances when including vertical mixing. The three sets of black filled circles in Figure 11 show log K_zz of 4, 8, and 11.

Table 1.  Guide to Model Parameters

Figure	Teq (K)	Tint (K)	MJ	g (m s−2)	m	Age (Gyr)
1	710	60, 100, 200, 300, 400	1	25	10×
4, 23	710	52, 77, 117, 182, 333	0.1, 0.3, 1, 3, 10	5.8, 9.8, 24, 65, 225	10×	3
7, 13	1120 to 180	75	0.3	10	10×	3
9, 15	870, 380, 180	52, 117, 333	0.1, 1, 10	5.8, 24, 225	10×	3
11, 17	710	501, 383, 283, 212, 156, 117, 84	1	13, 16, 19, 21, 23, 24, 26	3×	0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0
19	870, 380	52, 117, 333	0.1, 1, 10	5.8, 24, 225	1, 3, 50×	3

Note. In each figure, a range of planetary models is considered explored across different planetary parameters. The metallicity factor m is defined as m = 10^[Fe/H].

Download table as:  ASCIITypeset image

Zoom In Zoom Out Reset image size

In Figure 12 we examine the corresponding chemical abundances for equilibrium and the three values of vertical mixing strength, as a function of planetary age. In equilibrium at 1 mbar, the atmosphere is CH₄ dominated, and the CO mixing ratio is nearly off the bottom of the plot. However, even very modest vertical mixing (log K_zz  = 4, thin lines) changes the picture. The atmosphere becomes modestly CO dominated, and we lose essentially all sensitivity to the deeper atmosphere of the planet—the abundances depend very little on T_int. However, with more vigorous vertical mixing, we see a picture emerge that has much in common with our understanding of nonequilibrium chemistry in brown dwarfs. Higher T_int values and hotter interiors lead to more CO and less CH₄. The plot shows a changeover from CO dominated to CH₄ dominated at ∼200 Myr, at a T_int value of ∼250 K. Again, this is generic behavior, as more massive objects would transition later in life (but at higher T_int values given their higher-pressure photospheres and the positions of the CO and CH₄ iso-composition curves), and less massive objects earlier (but at higher T_int values, given their lower-pressure photospheres). While we expect that building up a large sample of atmospheric spectra size as a function of planetary age will be a challenge, it will be rewarding to have a statistical sample to compare to the typical several-gigayear-old systems. This could yield important insights into planetary cooling history and the vigor of vertical mixing with age.

Zoom In Zoom Out Reset image size

Nitrogen chemistry is predominantly a balance between N₂ and NH₃ and has been explored and validated in the brown dwarf context (e.g., Saumon et al. 2000, 2003; Cushing et al. 2006; Hubeny & Burrows 2007; Zahnle & Marley 2014). N₂ is favored at high temperatures (and low pressures), while NH₃ is favored at low temperatures (and high pressures). The transition from N₂ to NH₃ at cooler temperatures has a similar character to that of CO converting to CH₄, but it occurs at lower temperatures. Understanding nonequilibrium nitrogen chemistry in brown dwarfs has typically been hampered by two constraints. The first is that N₂, with no permanent dipole, has no infrared absorption features, unlike CO. The second is that NH₃ iso-composition curves have slopes that lie nearly along interior H/He adiabats, meaning that one typically cannot assess a given atmosphere's quench pressure, as all pressures along the adiabat correspond to nearly the same NH₃ mixing ratio.

However, in some sense irradiated planets have the advantage of having relatively more isothermal P–T profiles, which can remain nonadiabatic to pressure of ∼1 kbar. And, if these predominantly radiative atmospheres have K_zz values less than their mostly convective brown dwarf cousins, then it may be these more isothermal radiative parts of the atmosphere where one may quench the chemistry. We can examine this with the same Saturn-like P–T profiles we first examined in Figure 7. These profiles, but now with quench pressures for N₂–NH₃ chemistry (Zahnle & Marley 2014), are shown in Figure 13.

Zoom In Zoom Out Reset image size

Underplotted in black are curves of constant NH₃ abundance, falling off at higher temperature and lower pressure. Underplotted in gray are curves of constant N₂ abundance, falling off at lower temperature and higher pressure. A detailed look at Figure 13, compared to Figure 7, shows that the NH₃ iso-composition curves are more "spread out" than similar curves for CH₄, suggesting a more gradual change in nitrogen chemistry, with temperature, than for carbon. As the chemical conversion times for N₂ → NH₃ are longer than for CO → CH₄, the corresponding quench pressures for log K_zz = 4, 8, and 11 cm² s⁻¹ are at somewhat higher pressures. While for vigorous mixing (log K_zz = 11) all profiles converge to the same quench pressure (and hence changes in T_eq across this range would yield no change in the NH₃ abundance), there are broad ranges of N₂ and NH₃ mixing ratios for the log K_zz = 4 and K_zz = 8 cases.

Figure 14 shows the mixing ratios of N₂ and NH₃ as a function of planetary T_eq. Equilibrium chemistry (at 1 mbar) shows a crossover from N₂ dominant to NH₃ dominant at around 475 K. However, even sluggish vertical mixing keeps all of these atmospheres N₂ dominant, while also increasing the NH₃ mixing ratio for all T_eq values >600 K. More vigorous mixing (log K_zz = 8) further flattens the slope of the NH₃ curve, leading to relatively abundant NH₃ at essentially all T_eq values, as expected from the grouping of most of the log K_zz = 8 black filled circles in Figure 13. Across the entire phase space, the NH₃ mixing ratios are similar to those of CH₄ (see Figure 8) and are actually even higher for NH₃ than for CH₄ for the higher T_eq values. This suggests that onset of detectable CH₄ in these planets should be accompanied by NH₃ as well—one will not need to wait for particularly cold temperatures, compared to the brown dwarfs. For those interested in determining the relative abundances of C, N, and O, to compare to Jupiter's values (Wong et al. 2004), we note that in these models NH₃ never becomes the dominant nitrogen carrier compared to N₂, such that the nitrogen abundance determined from NH₃ would only be a lower limit.

Zoom In Zoom Out Reset image size

4.2.1. Effects of Planet Mass at a Given Age

Previously, in Section 4.1.2 and Figures 9 and 10, we investigated the role that surface gravity and cooling history have for the planets. Here we examine the same profiles, but for nitrogen chemistry. Figure 15 shows these sample P–T profiles for the 0.1, 1.0, and 10 M_J planets, with log K_zz = 4, 8, and 11. Compared to the carbon example from Figure 9, the quench pressures are higher. For the high-gravity (10 M_J) planet in particular, the quench pressure is within the deep atmosphere adiabat for log K_zz = 8 and 11, and near it for log K_zz = 4. We might expect that the NH₃ abundance will change little with K_zz, similar to a brown dwarf case (Zahnle & Marley 2014). The deeper one probes, the closer one comes to these adiabats, which lie nearly parallel to curves of constant NH₃ abundance. Instead, the NH₃ mixing ratio is in some sense a probe of the current specific entropy of the adiabat, which could prove useful in constraining thermal evolution models.

Zoom In Zoom Out Reset image size

We can examine the N₂/NH₃ ratio as a function of T_eq for these three planets in Figure 16. The crossover T_eq for nitrogen chemistry, in equilibrium, would be ∼550 K at 10 M_J, 500 K at 1 M_J, and 475 K at 0.1 M_J. However, even modest vertical mixing dramatically changes this picture. As the T_eq decreases, the quench pressure falls near or into the deep atmosphere adiabat, even at low gravity. On Figure 15 this manifests as the N₂/NH₃ ratio asymptoting to values that depend solely on the specific entropy of the adiabat, as one might have expected for the specific cases investigated for the Saturn-like planet in Figure 14. Much like the brown dwarfs, at cool temperatures (and especially at high surface gravity) planets here are insensitive to K_zz.

Zoom In Zoom Out Reset image size

4.2.2. Effects of Planet Age at a Given Mass

Previously in Section 4.1.3 and Figures 11 and 12 we found that planet age, and hence the cooling history and specific entropy of the interior adiabat, can have dramatic effects on the carbon chemistry. Young planets would have quite different abundances (richer in CO) than older planets at the same T_eq, all things being equal. We can investigate the role of cooling history on the nitrogen chemistry with these same profiles. In Figure 17 we plot the 1 M_J profiles from 10 Myr to 10 Gyr, this time with the nitrogen quench pressures labeled. The figure is quite similar to Figure 11, but with higher quench pressures, at hotter temperatures. At log K_zz  = 4, the levels are in the radiative part of the atmosphere but are relatively pinched together. At log K_zz  = 8 and 11, we find all quench pressure in or very near the deep atmosphere adiabats.

Zoom In Zoom Out Reset image size

The effects on the atmospheric mixing ratios of N₂ and NH₃, shown in Figure 18, are quite straightforward but different from that found for the carbon chemistry in Figure 12. In equilibrium at 1 mbar, as the atmosphere changes negligibly in temperature, the NH₃ mixing ratio (dashed line) changes little with age. The same is true at log K_zz  = 4, albeit at a higher NH₃ abundance. Since both the log K_zz  = 8 and 11 quench pressures sample the deep adiabat, which are nearly parallel NH₃ abundance curves, we find essentially the same behavior of mixing ratio as a function of age, independent of (high) K_zz. This is essentially the same as the well-understood brown dwarf behavior.

Zoom In Zoom Out Reset image size

So far we have aimed, as much as possible, to investigate the physical and chemical effects of only altering one or two quantities at a time, including distance from the Sun, surface gravity, and T_int. Atmospheric metallicity will also play an important role in altering these boundaries. This chemistry has certainly be explored before, or a very wide range of compositions (e.g., Moses et al. 2013). In this section we attempt to explore a composition phase space, but in a narrower sense.

It is strongly suggested from the bulk densities of transiting giant planets that there is a bulk "mass–metallicity relation" for the planets (Thorngren et al. 2016), with the lower-mass giant planets being more metal-rich. The effect of such a relation at atmospheric abundances is not yet clear (Kreidberg et al. 2014; Wakeford et al. 2017; Welbanks et al. 2019), but there is such a relation in the solar system for carbon (e.g., Atreya et al. 2016) and from standard models of core accretion planet formation theory, albeit with a large spread (Fortney et al. 2013).

For both the carbon and nitrogen chemistry discussed in Sections 4.1.2 and 4.2.1, for the three planet masses at 10× solar, we can examine how an increasing metallicity with lower planet masses may alter the previously examined trends. Figure 19 shows P–T profiles for planets at 0.5 and 2 au from the Sun, with the top panel showing carbon quench pressures and the bottom panel showing nitrogen quench pressures. The profiles themselves differ somewhat from those shown in Figures 9 and 15, as the models here use 50× solar (0.1 M_J), 3× solar (1 M_J), and 1× (10 M_J). Since the plots use three different metallicities, we also show three different CO/CH₄ equal-abundance curves (dashed).

Zoom In Zoom Out Reset image size

Compared to our previous investigations into chemistry at 10× solar metallicity (Figures 10 and 16), the two panels in Figure 20 show a much wider range of behavior. At higher metallicity, the cooler models "hang on" to CO and N₂ to much cooler T_eq values. In equilibrium the carbon transitions would occur between 1100 and 700 K in these models. Even sluggish vertical mixing shows a large impact. For instance, with more vigorous mixing (log K_zz = 8), these three transition T_eq values are ∼1100, 800, and 450 K.

Zoom In Zoom Out Reset image size

We can examine the N₂/NH₃ ratio as a function of T_eq for these three planets in Figure 19. The crossover T_eq for nitrogen chemistry, in equilibrium, would be ∼600 K at 10 M_J, 530 K at 1 M_J, and 420 K at 0.1 M_J. However, even modest vertical mixing dramatically changes this picture. As the T_eq decreases, the quench pressure falls near or into the deep atmosphere adiabat, even at low gravity. On Figure 15 this manifests as the N₂/NH₃ ratio asymptoting to values that depend solely on the metallicity and the specific entropy of the adiabat, as one might have expected for the specific cases investigated for the Saturn-like planet in Figure 14.

We can summarize, at least for the "old" 3 Gyr planets that have been the baseline for many of calculations, the expected rise of detectable CH₄ and NH₃ abundances. It is by now well understood that for the atmospheres of brown dwarfs the onsets of CH₄ and NH₃ are well separated in T_eff-space. Indeed, the rise of near-infrared CH₄ and NH₃ defines the T and Y spectral classes, at ∼1300 and ∼600 K, respectively (Kirkpatrick 2005; Stephens et al. 2009; Line et al. 2017), although the much stronger mid-IR bands can appear at 1700 K (CH₄ at 3.3  μみゅーm) and 1200 K (NH₃ at 10.5 μみゅーm).

However, significantly different P–T profiles of irradiated giant planets lead to much different behavior. This is shown in Figure 21, both for planets at a fixed 10× solar metallicity (top panel) and for planets that use the notional mass–metallicity relation (bottom panel), with both panels using log K_zz of 8. For the higher-gravity planets with a large thermal reservoir in their interior, the giant planet behavior is at least similar to that of brown dwarfs, with CH₄ coming on for T_eq a few hundred kelvin hotter for the 1× solar case at 10 M_J (bottom panel). However, beyond that example, a different and richer behavior, driven mostly by the altered temperature structure of irradiated planets, is seen. For all other example planets in both panels, CH₄ and NH₃ onset is at a similar T_eq, and at the higher metallicities (bottom panel) NH₃ can arise at warmer T_eq values than CH₄.

Zoom In Zoom Out Reset image size

Figure 21 is in some ways the central prediction of the paper, albeit for a relatively constrained example, as we describe at some length in the Discussion section. The oddly shaped and radiative P–T profiles lead to an expectation of significantly different behavior from that already known for brown dwarfs.

A lesson well learned from observations of transiting planet atmospheres to date is that clouds and hazes can readily obscure molecular absorption features. This has typically been thought of as a hindrance. However, early work in this field suggested that the atmospheres of giant planets could potentially be classified based on the presence or absence of clouds (Marley et al. 1999; Sudarsky et al. 2000, 2003). In the end, it seems likely that some mixture will be true—in some ways clouds will help us understand temperature structures and transport in these atmospheres, but they will also obscure features due to atoms and molecules.

However, it seems clear that the role of clouds will not be a simple function of T_eq, as cloud condensation curves can be crossed at a variety of pressures. At a low pressure, perhaps little condensable material will exist. At a high pressure, perhaps all cloud material in an optically thick cloud will be below the visible atmosphere. These effects will depend on the shape of the atmospheric P–T profile, and hence on the specific entropy of the adiabat (which depends on planet mass and age), in addition to the role of atmospheric metallicity (more metals means more cloud-forming material), and even the spectral type of the parent star, which can also alter profile shapes, as discussed below.

In some ways this topic is beyond the scope of the paper, which is focused on 1D models, but we can motivate that there will be a diversity in behavior at a given planetary T_eq with plots that focus on P–T profiles and condensation curves. First, we will examine our trio of warm Neptunes, GJ 436b, GJ 3470b, and WASP-107b. In Figure 22 we replot the same P–T profiles from Figure 5, with chemical information removed, but now including RCB depths with squares, and condensation curves for potential cloud-forming materials. These "cooler" clouds, for planets cooler than the hot Jupiters, have been studied in Morley et al. (2012, 2013). Note, however, that Gao et al. (2020) have suggested that most of these cloud species (save KCl) may not nucleate and form. Lee et al. (2018) suggest that Cr, KCl, and NaCl (instead of Na₂S) will form across this temperature range. These predictions can be corroborated by future detailed spectroscopic observations of brown dwarfs and planets.

Zoom In Zoom Out Reset image size

The KCl and ZnS cloud bases move little with or without tidal heating, as the upper atmospheres change little. The Na₂S cloud base, however, can move dramatically. Without tidal heating, the cloud base would be ∼300 bars in all three planets. However, for tidal heating with Q = 10⁴, the Na₂S cloud base moves to ∼0.1 bars, in the visible atmosphere. A similar effect is seen for MnS and Cr.

We have previously investigated generic Saturn-like-planet P–T profiles at 0.15 au from the Sun. Figure 23 shows the same profiles that were explored in Figure 4, but now with a focus on RCBs and cloud condensation, rather than chemical abundances. The interface between these profiles and condensation depends strongly on surface gravity. For instance, the denser, higher-pressure photosphere of the highest-gravity models yields a detached convective zone near 0.2 bars, coincidentally at the region of ZnS and KCl clouds, which is not seen in the lower-gravity models. Potentially more vigorous mixing here could lead to thicker clouds and larger particle sizes. If these profiles were calculated at greater orbital distances, yielding cooler atmospheres, all would develop this detached convective zone (Fortney et al. 2007). The Na₂S case is also interesting for these profiles. The cloud base is found in the deep atmosphere for the two higher-gravity models, but at a few tenths of a bar in the three lower-gravity models. This clearly shows that at a given T_eq the depth of cloud formation can be significantly impacted by the temperature of the deep atmosphere, which is mitigated by the interior cooling. One could readily imagine other examples where the cloud formation depth is affected by planetary age, at a given mass, as is seen in brown dwarfs and self-luminous imaged planets.

Zoom In Zoom Out Reset image size