CHARACTERIZING TRANSITING EXOPLANET ATMOSPHERES WITH JWST

Signals and noise of transmission and eclipse spectra were estimated for each instrument mode (Table 4) for each planetary system (Table 1). We combined these synthetic spectra to mimic the observational sequence of observing a single transit or eclipse event with each instrument mode (a total of four events for each transit or eclipse), producing a complete simulated λ = 1–11 μm spectrum and error bars estimating the uncertainty in each spectral channel.

We computed signals for the separate stages of a transit or eclipse event: star only or star + planet. Model stellar flux spectra were constructed by interpolating Nextgen (Hauschildt et al. 1999) models with surface gravities and effective temperatures spanning the observed values for each system. The in-transit star + planet flux was defined as $F{(\ast +p)}_{\mathrm{Tr},\lambda }={F}_{* ,\lambda }\times (1-{({R}_{p,\lambda }/{R}_{* })}^{2})$. The star + planet emission flux is simply the sum of the model star flux and the model planet flux (see Section 3) at each wavelength, $F{(\ast +p)}_{\mathrm{Em},\lambda }={F}_{* ,\lambda }+{F}_{p,\lambda }$. Estimated detected signals of each of these astrophysical events were computed for each system using the chosen system parameters (Table 4), with

$$\begin{eqnarray}&&{S}_{\lambda }={F}_{\lambda }{A}_{\mathrm{tel}}t\displaystyle \frac{{\lambda }^{2}}{{hcR}}\tau \end{eqnarray} \tag{ 2 }$$

where S_λ is the signal per spectral element in electrons, F_λ is the spectral flux density of the star or star + planet, t is the total exposure time during the event, λ is the wavelength of the spectral element, R is the resolution of the spectral bin, A_tel is the area of the JWST aperture (25 m²), and τ is the total system transmission (photon conversion efficiency) at that wavelength. Total exposure time t was calculated as the sum of the photon collecting time of all integrations that are executed during 0.9× the event duration (T₁₄ in Table 2). Individual integration times were computed from the host star brightnesses (Table 2) and the correlated double-sample minimum integration times and bright limits of each instrument mode (see Beichman et al. 2014), including the resulting readout efficiency.

We assumed that the star would also be observed for a time t before and/or after the transit or eclipse event, yielding total integration time 2t for the visit. The system transmission τ was computed to be the product of that of the three-element JWST telescope (set to 0.9) and that of the selected instrument. The transmission τ of NIRISS single-object slitless spectroscopy (SOSS) was estimated to be 0.35 at λ = 1.25 μm, falling off gradually at longer and shorter wavelengths in an approximation of the first-order blaze function of its new GR700XD grism. Similarly, the NIRCam was estimated to have τ = 0.30 at the 3.7 μm blaze peak of its grisms, falling off at longer and shorter wavelengths in an approximation of the first-order grism blaze function (Greene et al. 2007). The slitless photon conversion efficiency curve (τ ≃ 0.3) in Figure 9 of Kendrew et al. (2015) was adopted for the MIRI LRS τ values.

The JWST observatory is expected to have minimal natural (zodiacal) and telescope background emission at near-IR wavelengths, but the telescope background will increase quickly at wavelengths λ ≳ 10 μm. Background emission can become an important noise term when observing faint stars with the MIRI LRS because its slitless mode detects the full background of its λ = 5–12 μm bandpass. We computed the background signal for each observation with the equation

$$\begin{eqnarray}&&\mathrm{Bkg}={{BtA}}_{\mathrm{pix}}{n}_{\mathrm{pix}}{R}_{\mathrm{native}}/R\end{eqnarray} \tag{ 3 }$$

where Bkg is the background signal detected in each spectral bin, B is the background of the instrument mode in electrons arcsec⁻² s⁻¹, t is the total exposure time during the event, A_pix is the area subtended by each pixel in arcsec², n_pix is the number of spatial × 2 spectral pixels summed in each R_native native two-pixel resolution element of the selected observing mode, and R is the final binned spectral resolution of the observation. The B background values were computed from the average zodiacal and telescope backgrounds of each instrument mode provided by the STScI JWST prototype exposure time calculator.¹³

Noise values were computed at each wavelength as the sums of photoelectron Poisson noise, background Poisson noise, and total (read and dark) detector noise N_d,tot:

$$\begin{eqnarray}&&{N}_{\lambda }=\sqrt{{S}_{\lambda }+\mathrm{Bkg}+{N}_{d,\mathrm{tot}}^{2}}\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}&&{N}_{d,\mathrm{tot}}={N}_{d}\sqrt{{n}_{\mathrm{pix}}{n}_{\mathrm{ints}}{R}_{\mathrm{native}}/R}.\end{eqnarray} \tag{ 5 }$$

N_d is the total (read and dark current) detector noise of a single integration, and n_ints is the number of integrations during exposure time t. We set N_d = 18 electrons for the HgCdTe detectors (NIRISS and NIRCam modes) and N_d = 28 electrons for MIRI; these correlated double-sample noise values are expected maxima for nearly all observations. The numbers of integrations n_ints were set using the bright limits, efficiencies, and frame times of each instrument mode given in Beichman et al. (2014) such that the total real time fit within 0.9 times the transit duration (T₁₄ in Table 2), summing to total exposure time t for the event (plus equal time on the host star alone).

The signals and backgrounds of the astrophysical events were combined as follows to estimate the final transmission (Tr_λ) and emission (Em_λ) spectra:

$$\begin{eqnarray}&&{\mathrm{Tr}}_{\lambda }=\displaystyle \frac{{S}_{\lambda }({F}_{* })+\mathrm{Bkg}-({S}_{\lambda }(F{(\ast +p)}_{\mathrm{Tr}})+\mathrm{Bkg})}{{S}_{\lambda }({F}_{* })+\mathrm{Bkg}-\langle \mathrm{Bkg}\rangle }\end{eqnarray} \tag{ 6 }$$

and

$$\begin{eqnarray}&&{\mathrm{Em}}_{\lambda }=\displaystyle \frac{{S}_{\lambda }(F{(\ast +p)}_{\mathrm{Em}})+\mathrm{Bkg}-({S}_{\lambda }({F}_{* })+\mathrm{Bkg})}{{S}_{\lambda }({F}_{* })+\mathrm{Bkg}-\langle \mathrm{Bkg}\rangle }.\end{eqnarray} \tag{ 7 }$$

Note that Tr_λ ≡ (R_p,λ/R_*)² and ${\mathrm{Em}}_{\lambda }\equiv {F}_{p,\lambda }/{F}_{* }$ in noiseless cases. These Tr_λ and Em_λ signals were computed for the chosen system, instrument, and observation parameters for each planetary model in Table 1. We binned spectral channels to achieve a final resolution of R ≤ 100 per wavelength bin; instrument modes with higher intrinsic R were binned to R = 100, while ones with lower R (i.e., MIRI LRS at λ < 8 μm; see Kendrew et al. 2015) were not binned. We chose R = 100 as a compromise value low enough to maximize signal-to-noise while also being high enough to resolve molecular band spectral features.

We did perform a preliminary test of how binning impacts the retrieved uncertainties of H₂O, CO, and CO₂ in the cloudy transmission spectrum of a hot Jupiter of solar composition. We found that binning λ = 1–5 μm (NIRISS + NIRCam) spectra to R = 350 (two NIRISS resolution elements) did not produce any mixing ratio uncertainties that were less than ones retrieved for data binned to R = 100 in two trials using different instances of N_λ (no systematic noise was included). However, the ideal binning of each observing mode will likely be sensitive to actual in-flight noise performance, and this will not be known until after JWST begins operations.

The large aperture of JWST will ensure that the observatory will collect a large number of photons from bright stars, potentially resulting in the detection of ∼10¹⁰ or more photoelectrons per spectral bin when summing a significant number of observations of transit or eclipse events. In such cases, stellar photon noise will dominate N_λ, resulting in signal-to-noise ratios S/N ∼ 10⁵ or N_λ values being only 10 parts per million (10 ppm) of signal values. Existing observations of transiting planets have not yielded such low noise values. Astrophysical noise (e.g., Barstow et al. 2015) and/or instrumental noise (e.g., decorrelation residuals) produce systematic noise floors that are not lowered when summing more data. The best HST WFC3 G141 observations of transiting systems to date have noise of the order of 30 ppm (Kreidberg et al. 2014a), while observations of transiting planets with the Spitzer space telescope Si:As detectors showed noise as low as ∼65 ppm (Knutson et al. 2009).

We adopt reasonably optimistic systematic noise floor values of 20, 30, and 50 ppm for NIRISS SOSS (λ = 1–2.5 μm), NIRCam grism (λ = 2.5–5.0 μm), and MIRI LRS (λ = 5.0–11 μm) observations, respectively. These are less than or equal to the values estimated by Deming et al. (2009) for the JWST NIRSpec and MIRI instruments. The excellent spatial sampling of the NIRISS GR700XD SOSS grism approaches that of the HST WFC3 G141 spatial scanning mode, and both instruments have reasonably similar HgCdTe detectors. We anticipate that decorrelation techniques will continue to improve, so we assign a 20 ppm noise floor value to NIRISS even though HST has not yet done quite this well. NIRCam will have similar detectors but will not be sampled as well (see Table 4), so we assign it a systematic noise floor equivalent to HST's best performance to date. The MIRI imager/LRS detector is similar (in materials, architecture, and sampling) to the Spitzer IRAC Si:As detector used by Knutson et al. (2009), and we use this as the basis for assigning a 50 ppm noise floor to MIRI. We will not know the actual performance of these instruments until after JWST commissioning, but Beichman et al. (2014) have demonstrated decorrelated noise precision similar to the adopted NIRISS and NIRCam values in laboratory tests devised to simulate their observations of transiting planets.

A single instance of noise was computed for each transmission or emission observation, using the expression for N_λ and propagating through the relation for Tr_λ or Em_λ. We then added the appropriate systematic noise floor in quadrature, using the values given above. The single noise instance was drawn from a Poisson distribution with this amplitude, and the instance was added to the Tr_λ or Em_λ signal to produce the final simulated spectrum for each planet model in Table 1. Uncertainties were set to the total noise amplitude determined for each bin; Figures 4 and 5 show the simulated spectra for each case with these uncertainties plotted as error bars for one example. We computed different noise instances for each instrument mode (NIRISS SOSS, NIRCam LW grism, and MIRI slitless LRS). The resultant spectra and uncertainties were then used to retrieve the parameters of each model planet's atmosphere as discussed in Section 3.

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We now investigate whether the chosen atmospheric parameterization and priors may impact the retrieved results. Kreidberg et al. (2015) used both the "free" and "chemically consistent" retrieval approaches to determine the C/O in the transmission spectrum of WASP-12b. They found consistent results between the two approaches, suggesting a robust solution. We perform a similar analysis here. In addition to retrieving the mixing ratios of the individual molecules independent of the chemistry that links them (the "free" approach performed thus far), we also retrieve log(C/O) and [Fe/H] directly as a test case (with prior ranges from −2 to 2, and −4 to 4 respectively). These quantities, together with the temperature and pressure at each level in the atmosphere, permit the determination of the thermochemical equilibrium gas mixing ratios. These solutions are advantageous as they are chemically plausible and thermochemically self-consistent. However, such an approach does not account for the nearly limitless possible combinations of physical and chemical processes occurring in planetary atmospheres (e.g., vertical mixing, photochemistry, ion chemistry, 3D transport, cloud-gas microphysics interactions, interior-atmosphere chemistry coupling, escape processes, non-LTE, etc.) so we have not adopted it as our primary technique.

While not comprehensive, we perform one example using this chemically consistent approach and examine the impact it has on our ability to infer the atmospheric metallicity and C/O ratio. We retrieve on the full 1–11 μm wavelength emission spectrum of the warm sub-Neptune for the clear solar-composition atmosphere. Figure 10 illustrates the resulting marginalized log(C/O) and [Fe/H] metallicity posteriors compared with those derived from the molecular abundances in our baseline free retrievals (i.e., Figure 8, top row). The C/O histograms are qualitatively similar; both suggest a weak constraint of the C-to-O ratio. Perhaps more interesting is the comparison of the metallicity histograms. The chemically consistent approach provides a metallically constraint that is several orders of magnitude better than that derived from retrieving the molecular mixing ratios freely. This is because the chemically consistent approach rules out combinations of molecular abundances that do not abide by thermochemical equilibrium. Effectively, more prior information is being added to the chemically consistent retrieval system in the form of a more sophisticated parameterization with more assumptions but with fewer free parameters. More generally, it would be possible to apply chemically consistent models on all posterior "free" retrieval histograms (Figures 6–9) to rule out non-physical parameter spaces within the equilibrium framework. This would be an intermediate step between the classic retrieval and forced self-consistency, but we do not implement it here.

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Given a high enough signal-to-noise ratio and sufficient spectral resolution along with correct physics and chemistry constraints, one would expect the two approaches to produce the same distributions of the retrieved quantities. That would suggest true independence from any prior assumptions, and this would be the ideal regime for learning more about these atmospheres.

Figure 11 shows the range of T–P profiles retrieved from the simulated emission spectra over the three different wavelength ranges. Figure 12 shows normalized thermal emission contribution functions for the solar-composition hot Jupiter and warm sub-Neptune planets to illustrate where their thermal emissions originate. The contribution function for the solar-composition warm Neptune is very similar to that for the warm sub-Neptune.

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Table 5 lists the 68% confidence intervals of the retrieved parameters of the transmission and emission scenarios (see Section 3, Table 1). For cases in which a molecule is not particularly abundant (e.g., CH₄ in the hot Jupiter and CO or CO₂ in the cooler objects), only upper limits could be obtained. We quote the 3σ upper limits instead of confidence widths in these cases. We caution, however, that many of these upper limits are relatively soft and that one should really look at the histograms to get a sense of the distribution.

These numbers are meant to be a guide to illustrate how the constraints change from one object to the next. We also note that there is typically a ∼10% uncertainty on these uncertainty values due to the particular instances of random noise applied. In the remainder of this section, we examine how well the directly retrieved parameters constrain the different planetary atmospheres.

Figures 6–9 and Table 5 show that the mixing ratios of the dominant molecules for a particular planet type are well constrained (bounded posterior rather than an upper limit) in the transmission spectra of all clear solar atmospheres in most cases. Furthermore, observations with NIRISS alone (λ ≤ 2.5 μm) can constrain H₂O in these atmospheres nearly as well as observations out to 11 μm wavelengths in some cases. For example, Table 5 shows that the retrieved H₂O mixing ratio of the hot Jupiter with a clear solar atmosphere has a 68% uncertainty of 0.16 dex (45%) for a λ = 1–11 μm spectrum, and this degrades to only 0.20 dex (58%) when only using the 1–2.5 μm wavelengths in a NIRISS observation. However, CO is well constrained (uncertainty ∼0.3 dex for retrievals done with the full 1–11 μm or 1–5 μm spectra), but the uncertainty is much worse (∼1.8 dex) when only the NIRISS data are used.

The H₂O and CH₄ uncertainties of the warm Neptune with clear solar atmosphere increase by ∼0.3 dex (factor of ∼2), providing uncertainties of ∼1 dex (factor of ∼10) when using only the NIRISS wavelengths. These species have similar NIRISS-only uncertainties in the warm sub-Neptune with clear solar atmosphere, but their full λ = 1–11 μm spectra provide better uncertainties (∼0.2 versus ∼0.5 dex for the warm Neptune). These values are about ∼1.5 dex for the full 1–11 μm wavelengths for the cool super-Earth with clear solar atmosphere, and the CH₄ uncertainty does not change much when considering only the NIRISS data range. However, the retrieved H₂O mixing ratio value becomes an upper limit. Overall it is encouraging that these atmospheres can be constrained so well in these modest observations (one transit at each wavelength), and the wavelength range of NIRISS alone is sufficient to detect dominant species with good confidence or even measure mixing ratios to good precision in some cases (i.e., the chosen hot Jupiter). Even for gases that are not abundant in the atmospheres (CH₄ and NH₃ in the hot Jupiter, CO and CO₂ in the cooler objects), relatively low upper limits can be obtained. For instance, most scenarios within the hot Jupiter planet type can detect, or rather rule out, NH₃ and CH₄ at the ∼1 ppm level.

Planets with cloudy atmospheres will be more challenging to constrain with transmission spectra; this is already apparent from numerous HST observations of GJ 1214b (e.g., Croll et al. 2011; Kreidberg et al. 2014a, and references therein) and other planets that exhibit weak or no spectral features. Figures 6–9 and Table 5 show that the mixing ratios of most molecules are not constrained well (uncertain by more than 1.0 dex) with λ ≤ 2.5 μm (NIRISS only) transmission spectra for exoplanet atmospheres of solar composition with clouds. Mixing ratio uncertainties improve somewhat when λ ≥ 5 μm data (NIRCam or NIRCam+MIRI) are added in many cases. However, the resulting constraints are considerably worse than those of the clear solar atmospheres (as good as 0.2–0.5 dex as discussed above). Mixing ratio uncertainties improve to ∼1 dex for some molecules in the cloudy atmospheres of the hot Jupiter and warm sub-Neptune when these longer wavelengths are also included in the retrievals. The mixing ratio uncertainties of the molecules in the warm Neptune system also improve with these longer wavelength retrievals, but none gets close to 1.0 dex. The cool super-Earth system retrievals only produce molecular mixing ratio limits for the cloudy atmosphere case regardless of wavelength range used.

Cloud-top pressure P_cloud is also retrieved for the transmission cases, and Table 5 shows that this is constrained to ∼1 dex for the hot Jupiter and warm sub-Neptune systems when using complete λ = 1–11 μm spectra of cloudy solar atmospheres. This allows the position of clouds to be located to an order of magnitude in pressure in their atmospheres. Reducing the wavelength range to 1–5 and 1–2.5 μm increases the uncertainty to ∼1.4 and ∼1.7 dex, respectively. The warm Neptune system is not constrained quite as well (mostly due to its less favorable transmission parameters), having an uncertainty in P_cloud of 1.4–1.9 dex for the three different wavelength ranges of the cloudy solar atmosphere case. The retrievals provide lower limits for P_cloud (highest cloud altitudes) for all clear atmospheres (solar or HMMW) of the hot Jupiter, warm Neptune, and warm sub-Neptune systems. Clouds in the clear solar atmospheres of all three systems are constrained to P ≳ 1 mbar, a useful limit given that transmission spectra probe only high-altitude/low-pressure atmospheric regions. P_cloud is not constrained well for any atmospheres of the cool super-Earth system. The P_cloud histograms in Figures 6–9 illustrate these constraints graphically.

Emission spectra retrievals constrain the mixing ratios of the most dominant species of the solar composition atmospheres of the hot Jupiter (better than 0.3 dex for CO, CO₂, and H₂O) and warm Neptune (to ∼1 dex for CH₄, H₂O, and NH₃). Figures 6–9 and Table 5 show that λ > 5 μm data (e.g., MIRI) are required to obtain these good constraints. This can be understood by examining Figure 5; emission spectra have low S/N at shorter wavelengths, and there are numerous strong molecular absorption features at λ > 2.5 μm. Note that the NIRISS-only (λ = 1–2.5 μm) emission spectrum of the hot Jupiter gives a false peak in its CH₄ mixing ratio, and this disappears when longer wavelength data are added (see Figure 6). Molecular mixing ratio uncertainties are worse than ∼1 dex for the emission retrievals of the warm sub-Neptune planet, and the S/N of the emission spectrum of the cool super-Earth was too low to perform useful retrievals at any wavelengths (see also Figure 5).

These results show that JWST λ > 2.5 μm emission spectra with moderate-to-high S/N will be very useful for atmospheric characterization. Emission spectra are also required to retrieve T–P profiles (Figure 11), which are particularly important for understanding energy absorption and transport in strongly insolated atmospheres. Fortney (2005) showed that the direct, face-on geometry relevant to emission spectra is much less impacted by clouds or hazes than the slant geometry of transmission spectroscopy observations. Therefore we assume that observations and retrievals of emission spectra from a cloudy solar atmosphere will be very similar to those from a clear solar-composition atmosphere studied here. Given that, retrievals of JWST λ = 1–11 μm emission spectra provide significantly better constraints than the transmission spectra on the mixing ratios of most molecules for cloudy hot-Jupiter and warm-Neptune atmospheres (see Table 5). Each star+planet system should be evaluated to determine whether emission or transmission observations are more favorable for detecting spectral features and constraining parameters of interest via atmospheric retrievals. It would be ideal to acquire and combine both transmission and emission spectra to probe planetary atmospheres over the broadest possible pressure range and to use all data to constrain compositions and chemistries (Griffith 2014; Kreidberg et al. 2014b).

Spectral absorption features are clearly suppressed over the entire λ = 1–11 μm wavelength range of the hot-Jupiter emission spectrum with a temperature inversion (Figure 5), and the inversion is detected at modest to high S/N in the retrievals. As shown in Figure 11, the temperature of the hot-Jupiter atmosphere decreases to T = 1260 K at P = 10⁻² bar, and then increases to T = 2100 K at P = 10⁻⁴ bar for the inversion model. This difference of ΔT = 840 K is detectable in the retrieved T–P profiles of all three wavelength ranges. At these pressures, the 1σ temperature uncertainties of the 1–2.5, 1–5, and 1–11 μm wavelength ranges are approximately 215 K, 40 K, and 30 K respectively. Therefore the temperature inversion is detected at better than 4σ, 21σ, and 28σ respectively when using multiple retrieved T–P points in this pressure range. It is clear that the λ > 2.5 μm data constrain the hot-Jupiter T–P profile much better than the λ = 1–2.5 μm data alone, and Figure 11 shows that λ = 1–11 μm data constrain T–P profiles somewhat better than λ = 1–5 μm data for all three planets studied.

It will also be possible to constrain the compositions of HMMW (including 100% H₂O) atmospheres with JWST spectra. Figures 7–9 show that the transmission retrievals of the HMMW atmospheres of the warm Neptune, warm sub-Neptune, and cool super-Earth have significantly different molecular mixing ratio distributions than the solar ones (with or without clouds; as expected from Table 3). Likewise, their [Fe/H] distributions are also significantly different, reflective of the high metallicity ([Fe/H] ≃ 3) required to produce HMMW atmospheres. These differences are readily apparent in the λ = 1–2.5 μm results alone in most cases, but they become clearer when longer wavelength data are included. Figures 7–8 and Table 5 clearly show that the emission retrievals constrain the clear HMMW atmospheres of the warm Neptune and warm sub-Neptune less well than transmission ones. These figures also show that the derived [Fe/H] values have uncertainties of ∼0.5 dex (a factor of 3) or better for the clear (HMMW and solar composition) atmospheres of the hot and warm planets studied with λ = 1–5+ μm transmission spectra.

Real planets may have atmospheric species not included in the forward models or retrievals we have performed here. If this were the case, performing these retrievals on real JWST data would produce wavelength-dependent residual differences between observed and modeled spectra. Additional species could then be introduced into the model to minimize these residuals. Computing the Bayes factor between the models with simpler and more complex compositions would reveal whether adding the additional species is statistically justified. Uncertain molecular opacities may also skew the retrieved compositions for observed planets. However, this error is likely to be of the order of 10% but can be as high as a factor of a few in some cases (Grimm & Heng 2015, Section 3.4). This is mostly less than or equal to our best 68% mixing ratio confidence widths (see Table 5).

The C-to-O ratio (C/O) of a planetary atmosphere could be a tracer of its formation and migration history within a protoplanetary disk (Öberg et al. 2011). Determining the C/Os of a diverse set of exoplanets over a wide range of conditions is important for applying this theory to understand planet formation scenarios. There has yet to be unambiguous evidence for high C/O (C/O > 1) within the current ensemble of observed exoplanets (Madhusudhan et al. 2011; Konopacky et al. 2013; Line et al. 2014b; Stevenson et al. 2014; Barman et al. 2015; Benneke 2015; Kreidberg et al. 2015). Only upper limits (Benneke 2015; Kreidberg et al. 2015) have been derived from transmission spectra. Emission spectra retrievals have also provided upper limits for some planets that are otherwise unconstrained, and a specific C/O value has only been determined for HD189733b, which has broad near-continuous wavelength coverage from ∼1–20 μm (Lee et al. 2012; Line et al. 2014b; Waldmann et al. 2015).

There are several ways of determining or diagnosing the C/O in exoplanet atmospheres. The most straightforward, direct approach is to simply determine the abundance of all of the carbon- and oxygen-bearing species present in the spectra and compute it directly by summing the carbon atoms in all of the carbon-bearing species and dividing by the oxygen atoms in all the oxygen-bearing species. Certainly it is possible for some carbon or oxygen to be locked away in condensates or other species that may not necessarily present themselves spectroscopically, resulting in a potential bias. In any case, this is the approach we take. Figures 6–9 show the C/O histograms derived from the retrieved molecular abundances. In some cases, when the abundances are not well constrained, a two-peak distribution exists. This is simply due to the propagation of uniform-in-log priors through the ratio used to compute the C/O (see Line et al. 2013b, for an in-depth discussion). Another approach is to retrieve the C/O directly as done by Kreidberg et al. (2015), Benneke (2015), and in Section 5.1 for a test case. However, doing so requires the assumption of the chemical processes (e.g., thermochemical equilibrium, vertical mixing, photochemistry) via a chemical model that relates the elemental abundances to the molecular abundances (see Section 5.1).

We find that for all of the hot-Jupiter transmission and emission scenarios observed with wavelengths longer than 2.5 μm (e.g., NIRISS+NIRCam and NIRISS+NIRCam+MIRI) the C/O can be constrained to better than 0.2 dex (a factor of 1.6) and is largely unbiased by the two-peak problem. Wavelengths below 2.5 μm do not capture the strong CH₄, CO, or CO₂ vibrational bands that occur at longer wavelengths. The low abundance of CH₄ requires observation of the strong 3.3 and/or 7.7 μm band to provide a meaningful upper limit as CH₄ is not thermochemically dominant in hot atmospheres. While CO and CO₂ are dominant in hot atmospheres, they have a relatively narrow influence across wavelength. Furthermore, there is little improvement in extending coverage beyond 5 μm as the strongest CO or CO₂ bands over the full 1–11 μm wavelength range occur at ∼4.5 μm (see Figure 1). Therefore, for hot Jupiters with comparable S/Ns, observations over λ = 1–5 μm are more than sufficient to provide a meaningful constraint on their C/O.

The transmission spectra of clear atmospheres of low mean molecular weight (solar composition) on the warm Neptune and sub-Neptune also unbiasedly constrain the C/O to ∼0.2 dex or better in all three wavelength ranges. This is because the CH₄ is largely constrained to abundances that are greater than the CO or CO₂ abundances and thus the C/O ratio is mainly set by the methane-to-water ratio (CH₄/H₂O). Because of the large abundance of CH₄ in the cooler planets, it presents strong spectral features at wavelengths below 2.5 μm (see Figure 1). Even in the cloudy scenarios the log(C/O) constraints are only a factor of 1.2 worse.

Emission spectra constrain C/O less well in these small warm planets. 1–11 μm emission spectra constrain C/O to ∼0.5 dex in the solar-composition atmosphere of the warm Neptune, and 1–5 μm emission spectra also provide some constraint (to only ∼1 dex). The λ ≥ 5 μm emission spectra provide only loose constraints on C/O in the clear solar atmosphere of the warm sub-Neptune. Even complete 1–11 μm emission spectra produce C/O distributions very similar to the priors for the HMMW atmospheres of these planets. These relatively cool planets have low flux and therefore low S/N near the λ ≤ 4 μm CH₄ bands, so these results are not particularly surprising.

In general, the C/O constraints are poor for the cool super-Earth atmospheres. None of the transmission scenarios constrain C/O to significantly better than an order of magnitude.

Finally, Figure 13 shows how the retrieved H₂O mixing ratio uncertainties compare to the predicted thermochemical H₂O values for two different C/O cases (see Kreidberg et al. 2015). The uncertainties in H₂O mixing ratios are shown for the different planets' transmission cases (their positions are located for clarity and are not indicative of true temperatures or compositions). The H₂O mixing ratio depends on metallicity and contains no direct information on carbon abundance, so it cannot provide an unambiguous C/O constraint by itself. However, it is instructive to show how one could use H₂O to rule out a high C/O scenario in some cases. For hot Jupiters, where the difference between solar and high C/O chemical models is largest, a measurement in a clear atmosphere could distinguish the difference between a solar and high C/O atmosphere by greater than 50σ for all three wavelength ranges. Using H₂O alone becomes difficult for planets with scale height temperatures below T ∼ 1000 K as the predicted thermochemical H₂O abundances converge for both solar value and high C/O. Other proxies such as the ratio of H₂O to CH₄ (under the assumption that oxidized carbon will be in low abundance; see Figure 2 of Madhusudhan 2012) or a direct determination of the C/O (as above) will be required.

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Regardless of the methodology used, it appears that JWST spectra will be very useful in determining the C-to-O ratios in a wide array of planetary atmospheres and will allow us to begin to address quantitatively the formation location relative to ice lines and any subsequent migration.

Understanding planet formation requires determining and interpreting the mass–metallicity relationship for a varied sample of planets. Establishing this relationship over the exoplanet population provides insight into the formation mechanism of core accretion (e.g., Ida & Lin 2005). There has been some progress in establishing the role that planetary mass and parent star metallicity have in determining the bulk metallicity of transiting gas giants (Miller & Fortney 2011) through the use of planetary evolution models. A corresponding constraint on planetary atmospheric metallicity, from spectroscopy, would help us to understand whether most of these metals are found in a core or are mixed within the H/He-dominated envelope.

Using population synthesis models (Mordasini et al. 2012), Fortney et al. (2013) suggest that as the mass of a planet decreases, the atmospheric metallicity increases. Lower mass planets are unable to accrete substantial envelopes within the core-accretion theory and thus are more susceptible to pollution by infalling planetesimals. We do indeed find tantalizing evidence for such a trend within our own solar system as seen in Figure 14. However, in our own solar system we are limited to using CH₄ as the proxy for metallicity as water is largely sequestered in deep clouds. Exoplanet atmospheres, due to their high temperatures, permit us to access a wide array of molecules, allowing for more precise constraints on the envelope metallicity. Kreidberg et al. (2014b) provided additional leverage on this relationship via a relatively good constraint on the water abundance in a 2 M_J hot Jupiter. Under the assumption of solar C/O, the retrieved water abundance was used as a proxy for metallicity. This proxy-metallicity was found to be consistent with the observed solar system trend. Furthermore, constraints on Neptune-mass objects (GJ 436b by Stevenson et al. 2010 and HAT-P-11b by Fraine et al. 2014) are also suggestive of this trend.

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Recently, Benneke (2015) demonstrated that the water abundance alone cannot constrain the atmospheric metallicity because of the degeneracy of metallicity with the carbon-to-oxygen ratio. The broad wavelength coverage and high S/N of JWST data enable us to constrain not only the water abundance, but carbon species as well. This, in essence, breaks the C/O–metallicity degeneracy. We directly compute the metallicity ([Fe/H]) from the retrieved molecular mixing ratios, and the metallicity histograms are shown in Figures 6–9 (see Section 5). Figure 14 compares a typical metallicity constraint for the hot Jupiter of solar composition and the warm Neptune of HMMW. Observing just five planets spaced logarithmically between a few Jupiter masses and a Neptune mass with such constraints (0.4 dex) would allow us to determine the mass–metallicity slope (in log space) with a 1σ uncertainty of 0.13.

Disequilibrium processes are likely to play a role in sculpting the molecular abundances in exoplanet atmospheres (Liang et al. 2003; Zahnle et al. 2009; Line et al. 2010, 2011; Moses et al. 2011; Venot et al. 2012; Hu et al. 2013; Agúndez et al. 2014). These processes come in a variety of flavors including, but not limited to, vertical and horizontal mixing (Prinn & Barshay 1977; Cooper & Showman 2006), photochemistry (Yung & Demore 1999), ion chemistry (Lavvas et al. 2008), and biology (e.g., Holland 1994). The predicted dominant disequilibrium process in Jovian-type planets is due to vertical mixing and thus we focus on the ability of JWST to infer disequilibrium due to vertical mixing. Vertical mixing tends to set (or quench) the upper atmospheric abundance of certain molecules to a particular deep atmospheric value, which can result in an orders-of-magnitude enhancement (or depletion) relative to the expected equilibrium abundance at a higher altitude in the atmosphere. Line & Yung (2013) devised a scheme to determine the degree to which an atmosphere is out of equilibrium given the retrieved abundances of H₂O, CH₄, CO, and H₂. If these values form a ratio, α,

$$\begin{eqnarray}&&\alpha ({f}_{i},P)=\displaystyle \frac{{f}_{{\mathrm{CH}}_{4}}{f}_{{{\rm{H}}}_{2}{\rm{O}}}}{{f}_{\mathrm{CO}}{f}_{{{\rm{H}}}_{2}}^{3}{P}^{2}}={K}_{\mathrm{eq}}(T)\end{eqnarray} \tag{ 8 }$$

where f_i are the molecular mixing ratios of species i, P is the pressure in bar, and T the temperature in K, which is equivalent to the thermochemical equilibrium constant K_eq(T), then those species are considered to be in chemical equilibrium and thus disequilibrium mechanisms are weak or non-existent.

Figure 15 summarizes the constraints on α as provided by the different wavelength regions for emission spectra. We also note, as in Line & Yung (2013), that there could be up to a factor of ∼100 more uncertainty due to the spread in the probed pressure levels. Furthermore, in some cases only an upper limit on one of the three retrieved gases can be obtained. In these cases α would also only have an upper or lower limit. We find that the hot-Jupiter scenarios will have the best constraint on α. Unfortunately hot Jupiters are generally predicted to be in equilibrium (Line & Yung 2013), so the small error bar is not very useful for identifying disequilibrium in those planets. The objects cooler than ∼1000 K will likely show signs of disequilibrium due to the dredging up of CO (and also N₂ in the NH₃–N₂ chemical system). This will result in an α that falls below the equilibrium line. The maximum expected deviation due to strong vertical mixing is only ∼2σ larger than the smallest warm-Neptune error bar, making the definitive detection of disequilibrium due to vertical mixing difficult. Perhaps the best way of approaching this problem is to determine α over a wide range of planetary effective temperatures to identify where deviations from equilibrium begin to occur. This transition is likely to occur between 1000 and 1200 K (e.g., Moses et al. 2011). This temperature region is also a good place to observe because the CH₄ and CO abundances begin to thermochemically trade places. Their nearly equal abundances and the higher temperatures (relative to the warm Neptunes) will likely allow bounded constraints on both the CO and CH₄ abundances, providing an actual bounded constraint on α.

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We have broadly explored the detectability of disequilibrium due to vertical mixing. Certainly, photochemistry can produce species not included in this investigation (e.g., C_mH_n, HCN), especially in cooler planets in which CH₄ is readily available for photolysis (Line et al. 2011; Miller-Ricci Kempton et al. 2012; Moses et al. 2013; Hu & Seager 2014; Venot et al. 2014). Shabram et al. (2011) determined that some of these photoproducts are potentially detectable in JWST observations of planets similar to the warm Neptune (GJ 436b) we have considered here.

The precision of the retrieved parameters is limited by the particular star–planet systems chosen as well as by the planets themselves. For example, the H₂O mixing ratio uncertainty using all wavelengths is considerably better for the clear solar atmosphere of the warm sub-Neptune (0.25 dex) versus the warm Neptune (0.59 dex) in transmission. These two planets have similar equilibrium temperatures and scale heights (see Table 2), so this difference is likely due to the higher S/N of the simulations for the warm sub-Neptune caused by the relatively high brightness and small size of the adopted host star, GJ 1214. Each star+planet system should be evaluated to determine whether emission or transmission observations are more favorable for detecting spectral features and constraining parameters of interest. The planetary system parameters given in Table 2 should be useful for scaling these results to other systems.

We do not know how well co-adding observations of multiple transits or secondary eclipses will improve our results at this time. The simulated single-transit and single-eclipse observations of our selected systems typically have total noise values only 10%–50% larger than our adopted noise floors (Section 4), so systematic noise assumptions have already significantly influenced the precision of our simulated data and retrieved information. Given this, co-adding more data would not substantially improve the results for these very observationally favored systems with bright host stars. We believe that our systematic noise assumptions are reasonable, but we will not know their exact values in different conditions (e.g., co-adding multiple spectra versus binning to lower resolution) until after JWST becomes operational.

Other instrumental and astrophysical noise sources may also impact JWST data. Barstow et al. (2015) considered the impact of systematic wavelength-invariant offsets between the spectra acquired in different JWST instrument modes. They found that the impact on retrieved parameters was minimal if even small regions of overlap between the modes were used to correct such errors to below the noise of their simulations. This is also likely to apply in our study if residual offsets between the different spectral regions are (corrected to) less than their respective adopted systematic noise floors. We note that the different instrument modes have more spectral overlap than the regions we selected in Table 4.

Barstow et al. (2015) also note that star spots can impact transmission (but not emission) spectra considerably (see also Pont et al. 2013) and they assess their impact on retrievals of planet parameters for both solar- and M-type (T_eff = 3000 K) host stars. They find that the quality of retrieved information is not seriously degraded for a hot Jupiter with a Sun-like star that has a relatively high spot fraction (3% of its area). However, Barstow et al. (2015) find that the retrieved H₂O mixing ratio can be up to an order of magnitude too high for the transmission spectrum of a hot Neptune planet with an M-type host star with a spot coverage of 10% if the planet does not occult any spots during transit. This error is considerably larger than the H₂O mixing ratio uncertainty we expect for the warm Neptune with a clear solar-composition atmosphere and full 1–11 μm wavelength coverage (Table 5), and we share the concern with Barstow et al. (2015) that stellar activity could be the limiting factor for accurate retrievals of JWST observations of active stars. Simultaneous photometric observations at relatively short wavelengths (e.g., see Fraine et al. 2014) could be used to correct this effect, and this may be possible with NIRCam λ ≤ 2.4 μm imaging during spectral observations with the NIRCam λ = 2.5–5.0 μm grism.