Modeling Synthetic Spectra for Transiting Extrasolar Giant Planets: Detectability of H₂S and PH₃ with the James Webb Space Telescope

We use a one-dimensional diffusion–kinetic model developed in Wang et al. (2015, 2016) to compute the vertical profiles of molecular abundances. The code solves the equation

$$\begin{eqnarray}&&\displaystyle \frac{\partial {Y}_{i}}{\partial t}=\displaystyle \frac{1}{\rho }\displaystyle \frac{\partial }{\partial z}\left(\rho {K}_{\mathrm{eddy}}\displaystyle \frac{\partial {Y}_{i}}{\partial z}\right)+{P}_{i}-{L}_{i},\end{eqnarray} \tag{ 1 }$$

where Y_i is the mass fraction of species i, ρ is the density of the atmosphere, z is the vertical coordinate, ${K}_{\mathrm{eddy}}$ is the vertical eddy diffusion coefficient, P_i is the chemical production rate of species i, and L_i is the chemical loss rate of species i. Two physical processes are modeled by the equation. One is the chemical production or loss of species i, and the other is its vertical transport. The mixing ratio of species in the atmospheres is determined by the dynamic balance between these two physical processes.

We neglect the effect of photochemistry. The effect on the chemical abundances is the photodissociation of hydrogen-bearing species (e.g., H₂O, CH₄, NH₃) and the production of photochemical products (e.g., C₂H₆, C₂H₂, HCN) (Moses et al. 2011, 2013; Kopparapu et al. 2012; Venot et al. 2012; Agúndez et al. 2014; Miguel & Kaltenegger 2014). Photochemistry changes the abundances only in the upper atmosphere, which is at millibar levels. Therefore, we expect our computed abundance profiles to be valid below ∼10 mbar.

The diffusion–kinetic model requires three kinds of input. First is the temperature–pressure (T–P) profile; second is a list of thermodynamic properties and a list of reactions between these species; third is the elemental compositions and the eddy diffusion coefficient. We detail how we choose the inputs below.

• 1.  
  T–P profile: we compute this using the model developed in Parmentier & Guillot (2014) and Parmentier et al. (2015), which is a non-gray analytical model.
• 2.  
  Thermodynamic properties and reaction rates: the thermodynamic properties are used to compute the equilibrium abundances as well as the backward reaction rates. The thermodynamic properties are compiled from Burcat & Ruscic (2005), McBride et al. (1993), Dean & Bozzelli (2000), and Venot (2012). The kinetic network used for modeling the C/N/O/H chemistry consists of 108 species and 1000 reactions, originally from Venot et al. (2012). The H/P/O reaction network consists of 24 species and 175 reactions, originally from Twarowski (1995). A more detailed description of the C/N/O/H and H/P/O reaction networks used in this paper can be found in Wang et al. (2016), and both reaction networks can be downloaded at the KIDA database (http://kida.obs.u-bordeaux1.fr/networks.html).
• 3.  
  Elemental abundances: we assume that the elemental composition of the atmosphere is solar. The solar elemental abundances are from Asplund et al. (2009).
• 4.  
  Eddy diffusion coefficient: ${K}_{\mathrm{eddy}}$ is used in the one-dimensional chemical models for parameterizing the vertical transport. There is no observational constraint on the eddy diffusion coefficient on exoplanets. However, its values can be approximated by multiplying the vertical convective velocity derived from 3D general circulation models with the pressure scale height (e.g., Moses et al. 2011; Venot et al. 2012; Parmentier et al. 2013). This mixing length theory approximation has an uncertainty on the order of 10 in the estimated eddy diffusion coefficient (Smith 1998). In this paper, we choose to use a constant profile for the ${K}_{\mathrm{eddy}}$, with values equal to 1 × 10⁹ cm² s⁻¹ throughout the atmospheres. We explore the dependence on ${K}_{\mathrm{eddy}}$ in Section 6.

In each simulation, we provide the elemental abundances, the ${K}_{\mathrm{eddy}}$, and the T–P profile to set up the code, then we initialize the Y_i of species with chemical equilibrium mass fractions. Y_i are evolved toward a steady state where the time derivative term in Equation (1) is zero. In the code, we terminate the simulation once the relative changes of mole fractions in successive Δt are smaller than 1 × 10⁻³, where Δt is the overturning timescale, defined as ${H}^{2}/{K}_{\mathrm{eddy}}$, where H is the pressure scale height at 1 bar level. The output is the vertical profiles of Y_i for each species in the model.

To simulate the synthetic spectra of transiting exoplanets, we modified the Smithsonian Astrophysical Observatory 1998 (SAO98) radiative transfer code (see Traub & Stier 1976; Traub & Jucks 2002; Kaltenegger & Traub 2009 and references therein for details). The line-by-line radiative transfer code calculates the atmospheric emergent spectra and also transmission of stellar radiation through the atmosphere with disk-averaged quantities at high spectral resolution. The atmosphere is divided in different layers, where the transmission is calculated using Beer's law. Updates include a new database with molecules relevant for giant planets that include H₂O, CH₄, CO, CO₂, NH₃, N₂, HCN, PH₃, and H₂S taken from the HITRAN (Rothman et al. 2013) and HITEMP (Rothman et al. 2010) databases.

The overall high-resolution spectrum is calculated with 0.1 cm⁻¹ wavenumber steps. We smear them out to a resolving power of 100 to simulate the resolution that we will obtain with the MIRI instrument. For NIRISS and NIRCam, we assume the observed spectra are binned to a resolution of 100. The smoothing was done using a triangular smoothing kernel.

We neglect the effect of cloud and hazes. Their influence on our results is discussed in the Section 6.

2.3.1. Greene et al. (2016) Noise Model

The noise of primary and secondary transit spectra is simulated following the recipes in Greene et al. (2016). Here we provide a compact summary of the noise modeling methodology, along with parameters in the model, summarized in Table 1. The selected JWST observing modes are from Table 4 of Greene et al. (2016). We did not cover the NIRISS SOSS observing mode. The features of H₂S and PH₃ are not as strong as those of H₂O and hence we expect their effects to be masked by H₂O. Therefore, we do not expect this part of the spectrum to be relevant for detection of PH₃ and H₂S. The NIRCAM instrument with LW grism mode covers the wavelength 2.5 ∼ 5.0 μm with a native resolution of ∼1700; the MIRI instrument with slitless mode covers the wavelength 5.0 ∼ 11 μm with a resolution of ∼100. We adopted a cutoff at 11 μm for MIRI slitless mode because the transmission becomes low at longer wavelength (Kendrew et al. 2015). The selected JWST modes provide a wavelength coverage between 2.5 and 11 μm. We choose a binned resolution of R = 100 for all modes to ensure each bin contains enough photons in our simulation.

Table 1.  Parameters for Computing Noise; Extension of Table 4 in Greene et al. (2016)

Instrument	Mode	λ (μm)	native R	${A}_{\mathrm{pix}}$ (arcsec2)a	${n}_{\mathrm{pix}}$ b	b (${{\rm{e}}}^{-}$s−1 arcsec−2)c	Nd (CDS)	${n}_{\mathrm{groups}}$	Noise Floor (ppm)d
NIRCam	LW grism F322W2	2.5−3.9	1200@2.5 μm	0064 × 0064	4	203.62	18	12	30
NIRCam	LW grism F444W	3.9−5.0	1550@4.5 μm	0064 × 0064	4	308.74	18	25	30
MIRI	Slitless LRS prism	5.0−11.0	40@5 μm, 160@10 μm	0110 × 0110	4	5156.0	46	8	50

Notes.

^a http://www.stsci.edu/jwst/instruments/. ^bTwo times the spatial extent of point source spectrum. ^cThe background photon rate is computed at the JWST exposure time calculator https://demo-jwst.etc.stsci.edu/. ^dThe adopted noise floor values are from Greene et al. (2016).

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There are four noise components: the signal photon shot noise, the background photon shot noise, the detector noise, and the systematic noise. The equations for computing each component are from Greene et al. (2016). For completeness, we present these equations below, and describe how we choose the parameter values in these equations.

• 1.  
  The number of signal photons in each spectral bin is computed following the equation

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

  where ${S}_{\lambda }$ is the number of signal photons in each spectral bin, ${F}_{\lambda }$ is the flux of the signal as received at the telescope, ${A}_{\mathrm{tel}}$ is area of the aperture of the JWST, t is the integration time, R is the binned spectral resolution, and τ is the total system transmission. The integration time t is adopted as the full transit duration T₁₄ (assuming equal integration time for both in and out transit). The transmission τ is sensitive to wavelength and is obtained from the JWST documentation (https://jwst-docs.stsci.edu/display/JTI/NIRCam+Filters).The signal flux ${F}_{\lambda }$ is measured at three configurations, namely, in-transit, out-transit, and in-eclipse. We assume the in-transit and out-transit integration time to be unity. The signal shot noise is equal to the square root of ${S}_{\lambda }$.
• 2.  
  The background signal is computed following the equation

  $$\begin{eqnarray}&&{B}_{\lambda }={b}_{\lambda }{{tA}}_{\mathrm{pix}}{n}_{\mathrm{pix}}{R}_{\mathrm{native}}/R,\end{eqnarray} \tag{ 3 }$$

  where ${B}_{\lambda }$ is the background photon numbers in each spectral bin, ${b}_{\lambda }$ is the background electron flux, ${A}_{\mathrm{pix}}$ is the area subtended by each pixel, ${n}_{\mathrm{pix}}$ is two times the number of spacial pixels covered by the spectrum, and ${R}_{\mathrm{native}}$ is the native resolution of the spectrum before binning. The values of the above parameters are summarized in Table 1. The background shot noise is equal to the square root of ${B}_{\lambda }$.
• 3.  
  The total detector noise in single transit observation is calculated as

  $$\begin{eqnarray}&&{N}_{{\rm{d}},\mathrm{tot}}={N}_{d}\sqrt{{n}_{\mathrm{pix}}{n}_{\mathrm{ints}}{R}_{\mathrm{native}}/R},\end{eqnarray} \tag{ 4 }$$

  where N_d is the total detector noise in one integration, and ${n}_{\mathrm{ints}}$ is the number of integrations in one transit observation. The parameter ${n}_{\mathrm{ints}}$ depends on the total transit duration, the brightness limit of each instrument mode, and the brightness of the star. The parameter values are summarized in Table 1.
• 4.  
  The systematic noise cannot be reduced by summing over more observations. We adopt the systematic noise floor as suggested by Greene et al. (2016), as presented in Table 1.

The four noise components are combined quadratically to compute the total noise in each spectral bin for a single transit observation.

2.3.2. PandExo Noise Model

Assuming solar elemental abundances for phosphorus, hydrogen, and oxygen, we compute the abundance profiles of H/P/O-bearing species for different levels of insolation. We present our results for the phosphorus chemistry in Figure 2. The most abundant H/P/O-bearing species are PH₃, PH₂, PH, HOPO, H₃PO₄, and P₂. For solar composition atmospheres with ${T}_{\mathrm{eq}}=500$ K and ${T}_{\mathrm{eq}}=1000$ K, the abundances are out of chemical equilibrium due to the effect of vertical mixing. For solar composition atmospheres with ${T}_{\mathrm{eq}}=1500$ K and ${T}_{\mathrm{eq}}=2000$ K, the abundances are in chemical equilibrium. The vertical mixing still exists; however, the mixing timescale is longer than the chemical timescale, and the abundances quickly re-equilibrate after mixing. The major phosphorus species are different for different levels of insolation. For a solar composition atmosphere with ${T}_{\mathrm{eq}}=500$ K, the dominant phosphorus-containing species is PH₃. This is similar to Jupiter, which has an equilibrium temperature at approximately 160 K. For a solar composition atmosphere with ${T}_{\mathrm{eq}}=1000$ K, PH₃ and P₂ are the most abundant phosphorus-containing species above 1 bar. Below this level, PH₃ is still the dominant species. At this temperature, part of the PH₃ is thermally decomposed into PH₂. The reactions between radicals can produce molecules with two or more phosphorus atoms such as P₂. At ${T}_{\mathrm{eq}}=1500$ K and 2000 K, the temperature is high enough that most PH₃ is thermally decomposed into PH₂ and PH. The photodissociation of PH₃ is potentially important at low pressure levels. For Jupiter and Saturn, the PH₃ abundance decreases near 0.1 bar level (Fletcher et al. 2009). There is no PH₃ photochemical modeling for hot Jupiters. In our case of solar composition Jupiter-size planets, we assume our abundance profile is valid below 0.1 bar. Above this, the profile is subject to change due to photodissociation.

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Assuming solar composition atmospheres, we compute the equilibrium abundances of sulfur species along the T–P profile in order to identify the most abundant sulfur-bearing species. We consider species H₂S, HS, H₂S₂, CH₃SH, S, S₂, SO, SO₂, CS, CS₂, COS, and SN. The results are shown in Figure 3. For solar composition atmospheres with ${T}_{\mathrm{eq}}=500$ K, 1000 K, and 1500 K, H₂S is the dominant species at pressure levels between 1 × 10⁻⁴ bar and 1 × 10⁴ bar. For solar composition atmospheres with ${T}_{\mathrm{eq}}=2000$ K, H₂S is the dominant species below 0.01 bar. Above this level, atomic S is the dominant species. Since vertical mixing has the effect of homogenizing the abundances, the addition of vertical mixing into the model is not expected to change the result for H₂S. However, we ignore the effect of photochemistry, which may affect the vertical profile at low pressure levels. Zahnle et al. (2009, 2016) have done photochemical modeling of sulfur species in the atmospheres of hot Jupiters and warm Jupiters. From their calculations, H₂S is largely photodissociated at $P\gtrsim 0.01\,\mathrm{bar}$, but remains the dominant sulfur carrier at $P\lesssim 0.01\,\mathrm{bar}$. Zahnle et al. (2009) showed that the abundances are not sensitive to temperature and insolation over the parameter ranges (T = 1200 ∼ 2000 K and I = 1 ∼ 1000). Following those results, we assume that H₂S is the dominant sulfur-bearing species below 0.01 bar in our calculations.

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It is necessary to carefully model the contribution of C/N/O/H-bearing species to the transit spectra if we want to identify spectral features of H₂S and PH₃. Molecules such as H₂O or CO are more abundant than H₂S and PH₃, and thus contribute the most to the transit spectra. In order to find molecules that are more abundant than H₂S and PH₃, we performed independent calculations for C/N/O/H chemistry. Our results are in general consistency with those reported in the literature (e.g., Moses et al. 2011; Venot et al. 2012; Hu & Seager 2014; Miguel & Kaltenegger 2014). We find the major C/N/O/H-bearing molecules are H₂O, CO, CH₄, CO₂, N₂, NH₃, and HCN. These molecules must be included in the spectra calculation in order to cover all important opacity sources. We present our computed vertical profile of these molecules along with PH₃ and H₂S in Figure 4. The results are shown for four different levels of stellar insolation.

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For solar composition atmospheres with ${T}_{\mathrm{eq}}=500$ K, the most abundant species are H₂O, CH₄, NH₃, H₂S, N₂, and PH₃. The abundances are nearly homogeneous in the vertical direction down to ∼10 bar. The atmosphere is strongly homogenized by vertical mixing, and species are in a disequilibrium state. CH₄ is the primary carbon-bearing species, H₂O is the primary oxygen-bearing species, NH₃ is the primary nitrogen-bearing species, H₂S is the primary sulfur-bearing species, and PH₃ is the primary phosphorus-bearing species.

For solar composition atmospheres with ${T}_{\mathrm{eq}}=1000$ K, the most abundant species are H₂O, CO, CH₄, N₂, NH₃, H₂S, HCN, and PH₃. Abundances are nearly homogeneous down to 1–10 bar due to the effect of vertical mixing. CO carries about 2/3 of the total carbon abundance, and CH₄ carries the other 1/3. H₂O is the dominant oxygen-bearing species. N₂ and NH₃ each carry about 1/2 of the total nitrogen abundance. This temperature marks the transition temperature for CO/CH₄ conversion and N₂/NH₃ conversion. For ${T}_{\mathrm{eq}}$ ≲ 1000 K, CH₄, and NH₃ are the major carbon and nitrogen carriers; for ${T}_{\mathrm{eq}}$ ≳1000 K, CO, and N₂ are the major carbon and nitrogen carriers.

For solar composition atmospheres with ${T}_{\mathrm{eq}}=1500$ K, the most abundant species are CO, H₂O, N₂, and H₂S. CO is the primary carrier of both oxygen and carbon. The rest of the oxygen is in the form of H₂O. N₂ is the primary carrier of nitrogen. CH₄ and NH₃ are much less abundant in the atmospheres.

For solar composition atmospheres with ${T}_{\mathrm{eq}}=2000$ K, the most abundant species are CO, H₂O, N₂, and H₂S. The abundances are nearly in chemical equilibrium due to the high temperature. CH₄, NH₃, and PH₃ are much less abundant.

There are three regimes for the abundance profiles depending on the level of insolation. For highly irradiated atmospheres (e.g., ${T}_{\mathrm{eq}}\gt 1500$ K), the chemical abundances are in local chemical equilibrium. Therefore, when doing retrieval on atmosphere composition and T–P profiles, assumptions of chemical equilibrium should be valid. For moderately irradiated atmospheres (e.g., ${T}_{\mathrm{eq}}\lt 1000$ K), the vertical mixing tends to produce homogeneous abundances in the atmospheres. It should be valid to assume a constant mixing ratio profile when doing atmospheric retrieval. In between is the transition regime where both chemical conversions and vertical mixing are important in the atmospheres. In this regime, the abundance profiles will depend on the vertical eddy diffusion coefficient as well as the T–P profile. In Figure 5 we show the computed abundances at 1 bar as a function of ${T}_{\mathrm{eq}}$. From the figure, CO, CO₂, and N₂ abundances increase as ${T}_{\mathrm{eq}}$ increases, while CH₄, NH₃, and PH₃ abundances decrease. H₂O and H₂S abundances remain approximately unchanged relative to the change of ${T}_{\mathrm{eq}}$. The HCN abundance increases and then decreases as ${T}_{\mathrm{eq}}$ increases.

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Figure 6 we present the primary and secondary transit spectra for the planetary systems in Table 2, with ${T}_{\mathrm{eq}}=500$, 1000, and 1500 K. The planets are assumed to have solar composition atmospheres. The vertical abundance profiles are taken from Figure 4. We compare the spectra simulated with all species, and the spectra simulated with all species except PH₃. The difference between these two spectra indicates the absorption from PH₃. For a solar composition atmosphere with ${T}_{\mathrm{eq}}=500$ K, the absorption from PH₃ occurs between 4 and 5 μm. The absorption depth is about 40 ppm in the primary transit spectra, and about 20 ppm in the secondary transit spectra. For the atmosphere with ${T}_{\mathrm{eq}}=1000$ K, the absorption is about 5 ppm in the primary transit spectra, while in the secondary transit spectra it is too small to be seen in the figure. For the atmosphere with ${T}_{\mathrm{eq}}=1500$ K, there is no apparent PH₃ absorption feature in the spectra.

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The lack of a PH₃ spectral feature for the solar composition atmospheres with ${T}_{\mathrm{eq}}=1000$ K and 1500 K is due to the thermal decomposition of PH₃ under high temperatures. From Figure 2, for the atmospheres with ${T}_{\mathrm{eq}}=500$ K, almost all of the phosphorus is in the form of PH_3, while for the atmospheres with ${T}_{\mathrm{eq}}=1000$ K and 1500 K, most of the phosphorus is in the form of P₂ and PH₂. Therefore, the spectral features of PH₃ are only expected in moderately irradiated atmospheres.

In Figure 7, we present the synthetic primary and secondary transit spectra for the solar composition atmospheres with ${T}_{\mathrm{eq}}=500$, 1000, and 1500 K in Table 2. We compare the spectra simulated with all species and the spectra simulated with all species except H₂S. For the atmospheres with ${T}_{\mathrm{eq}}=500$ K, the absorption depth is very small. In the primary transit spectra, there is a 5 ppm absorption at 2.6 ∼ 2.8 μm and a 10 ppm absorption at 3.9 ∼ 4.3 μm. In the secondary transit spectra, there is a 10 ppm absorption at 3.9 ∼ 4.3 μm. For the atmospheres with ${T}_{\mathrm{eq}}=1000$ K, the absorption depths are also very small for both primary transit and secondary transit spectra. For the atmospheres with ${T}_{\mathrm{eq}}=1500$ K, the absorption depths are much bigger. In the primary transit spectra, the absorption depth is about 15 ppm at 2.6 ∼ 2.8 μm, and 100 ppm at 3.5 ∼ 4.1 μm. In the secondary spectra, the absorption depth is about 10 ppm at 2.6 ∼ 2.8 ppm and 100 ppm at 3.5 ∼ 4.1 μm.

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The spectral feature of H₂S is more prominent in highly irradiated atmospheres. What determines the relevance of H₂S in the spectra is other species. In cold atmospheres, H₂S has to compete with the more abundant NH₃ and CH₄ to absorb photons while in the hottest case those two molecules are less abundant, leaving more space to H₂S to absorb photons and be seen in the spectra. Another factor that may contribute is the larger pressure scale height in hotter atmospheres.

In Figure 8 we present the synthetic primary and secondary transit spectra for solar composition atmospheres with ${T}_{\mathrm{eq}}=500$, 1000, and 1500 K listed in Table 2. For the atmospheres with ${T}_{\mathrm{eq}}=500$ and 1500 K, there is little absorption from HCN, mainly because the mixing ratio of HCN is very low, as shown in Figure 4. For the atmospheres with ${T}_{\mathrm{eq}}=1000$ K, there are small absorption features between 12 and 16 μm. The absorption depth in the primary transit spectra is about 15 ppm, and in the secondary transit spectra it is about 80 ppm.

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We used both PandExo and the Greene et al. (2016) noise model to compute the error on the spectrum. In Figure 9, we show the noise level as a function of wavelength for a range of instrument modes covering wavelengths ranging from 2.5 to 11 μm. The parameters of the planetary system being modeled are listed in Table 2. We assume two transits, each with equal in-transit and out-transit integration time. The transit duration is 7.2 hr, and the total (transit + baseline) time is 14.4 hr. The number of groups and number of integrations are optimized by PandExo. Besides the results calculated using PandExo, we also compute the error using the Greene et al. (2016) noise model for the NIRCam F322W2 mode and F444W mode. The Greene et al. (2016) model gives slightly lower noise levels, but similar to the results from PandExo.

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In Figure 10, we show the synthetic primary and secondary transit spectra with simulated JWST noise and compare the spectra with and without PH₃. The planetary system being modeled is listed in Table 2 with ${T}_{\mathrm{eq}}=500$ K. The observation parameters are summarized in Table 3. The spectral absorption feature of PH₃ is between 4.0 and 4.7 μm. In the primary transit spectra, the absorption depth for PH₃ is approximately one standard deviation of the noise. Since there are ∼10 measurements within this feature, the significance level for detecting PH₃ is more than 3σ. In order to cover the 4.0–4.7 μm wavelength range, the NIRCam LW grism mode with F444W filter can be used in the observation. To achieve this level of error, two transits are needed. The integration time for each transit is about 7.2 hr. We assume equal integration time for in-transit and out-transit, and that leads to a total observing time of 7.2 hr × 2 in- and out-transit integration × 2 transits =28.8 hr. The absorption feature in the secondary transit spectra is harder to detect since the absorption depth from PH₃ is only about half the standard deviation of the noise. Therefore, it is most effective for detecting PH₃ to use the primary transit spectra with the NIRCam LW grism mode and F444W filter, and get the spectra between 3.9 μm and 5.0 μm. However, the feature is very weak and requires long integration time (7.2 hr) to achieve the error level shown in Figure 10. Since CH₄ absorption is strong between 4 and 5 μm, additional observation using the NIRCam LR F444W may be needed to break the degeneracy between the PH₃ absorption and CH₄ absorp-tion. For higher equilibrium temperatures (${T}_{\mathrm{eq}}\gt 1000$K), the spectral feature of PH₃ is below 5 ppm, thus the JWST is unlikely to detect it.

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In Figure 11, we show the synthetic primary and secondary transit spectra with simulated JWST noise and compare the spectra with and without H₂S. The spectral absorption feature of H₂S is between 3.5 and 4 μm. We show the spectra with simulated noise for the planetary system in Table 2 with ${T}_{\mathrm{eq}}=1500$ K. The star in the system is a Sun-like star with K-band magnitude of 6.8. The observational parameters are summarized in Table 4. In the primary transit spectra, the absorption depth is about two times the standard deviation of the noise. The shape of the absorption feature can be resolved with the binned spectral resolution of R ∼ 100. To cover the spectral feature of H₂S, one can use the NIRCam LW grism mode with the F322W2 filter, obtaining spectra between 2.4 μm and 4.0 μm. In the secondary transit spectra, the absorption depth is also about two times the standard deviation of the noise. Therefore, we are also likely to detect H₂S in these spectra. The same mode of NIRCam can be used to obtain the secondary transit spectra. To achieve this level of error, five transits are needed. The integration time for each transit is about 2.4 hr. We calculate the total observing time in the same way as in Section 5.2, to obtain 24.0 hr. Since H₂O absorption is also important in the 2.4–4.0 μm region, NIRISS SOSS mode observations may be needed to determine the base and peak of water features, in order to verify the spectra features of H₂S. We only show the case for ${T}_{\mathrm{eq}}=1500$ K in Figure 11, while for atmospheres with lower equilibrium equilibrium temperature (500 K, 750 K, 1000 K), the absorption from H₂S is below 10 ppm and the JWST is unlikely to detect it.

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