The Detectability and Constraints of Biosignature Gases in the Near- and Mid-infrared from Transit Transmission Spectroscopy

We simulate the transmission spectrum of a terrestrial planet, with mass and radius equivalent to TRAPPIST-1e, residing within the habitable zone of TRAPPIST-1 with an approximate modern Earth-like atmosphere (observational system setup/signal-to-noise discussed below). We acknowledge, up front, that this may not be a physically plausible scenario due to the vastly different incident spectral energy distribution. However, given the overwhelming number of unknowns involved in self-consistent planetary atmosphere modeling (e.g., star–planet interactions, surface fluxes, formation conditions, bulk elemental composition, 3D atmospheric dynamical effects, cloud microphysics, dynamical evolution/history) we simply choose to treat our atmosphere as Earth-like (as has also been done in previous works; Morley et al. 2017; Krissansen-Totton et al. 2018).

To produce model transmission spectra, we leverage a variant⁷ of the CHIMERA transmission spectrum routine (Line et al. 2013; Greene et al. 2016; Line & Parmentier 2016; Batalha & Line 2017; Line et al. 2013; Greene et al. 2016; Line & Parmentier 2016; Batalha & Line 2017). Specifically, we re-parameterize the code (Table 1) to make it more amenable for temperate worlds by including as free parameters the constant-with-altitude mixing ratios of H₂O, CO₂, CO, CH₄, O₃, N₂O and an unknown background gas with a free-parameter molecular weight (taken to be Earth's N₂+O₂ value), an isothermal scale-height temperature, planetary radius at the surface (or scaling thereof), and an opaque gray cloud-top pressure (nominal truth values are given in Table 1). We mimic refraction with this cloud-top pressure (here, 0.56 bars) using the prescription from Robinson et al. (2017), but otherwise assume a cloud-free atmosphere (the refractive layer is in effect, mimicking a cloud, as well as its inclusion as a free parameter). CHIMERA uses correlated-K opacities (computed at the given constant resolving powers below), here derived from a grid of pre-computed line-by-line (≤0.01 cm⁻¹, 70–410 K, 10⁻⁷–30 bar) cross-sections generated with the high-resolution transmission molecular absorption database (HITRAN) Application Programming Interface (HAPI) routine (Kochanov et al. 2016) and the HITRAN2016 line database (Gordon et al. 2017). We do not include collision-induced opacities, though they may spectrally present themselves throughout the mid-infrared (Schwieterman et al. 2015).

We benchmarked our transmission-forward model against those available from the Virtual Planetary Laboratory⁸ (Figure 1).

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We perform Bayesian parameter estimation and model selection, on the simulated data sets described below, using the PyMultiNest routine (Buchner et al. 2014) following the methods described in Benneke & Seager (2013). We initialized our retrievals with 3000 live points. An advantage of nested sampling algorithms is the ease of evidence computation, which can be used to assess model complexity. We use Bayesian nested model comparison (by removing one gas at a time) to determine the detection significance of each molecular species (Trotta 2008; Benneke & Seager 2013), and it is what we utilize as a metric for assessing instrument performance. An advantage of utilizing the detection significance via the Bayesian evidence over more traditional methods (e.g., line or band height above a continuum relative to the noise) is that it fully utilizes all of the spectral information included in all of the relevant bands as well as encompasses the model degeneracies.

We use the JWST radiometric noise model described in Greene et al. (2016) to produce spectrophotometric uncertainties. The collecting area of 25 m² was retained to reflect the identical collecting area of JWST in order to provide a baseline with which to easily scale the results of this study. Modifications were made to parameters of the code to reflect the expected performance of a next-generation mid-infrared telescope. These parameters included a 2.85–30.0 micron wavelength range (HgCdTe detectors below 10.5 microns and SiAs detectors above 10.5 microns), a zodiacal background estimated by Glass & Blair (2015) and Glass et al. (2015), and an intrinsic resolving power of 300. These modifications reflect the capabilities of new detector technologies discussed by Matsuo et al. (2018).

Our input stellar spectrum derives from the PHOENIX stellar models using the PySynPhot software package (STScI Development Team 2013). We adopted the values for TRAPPIST-1 (T = 2550K,M/H = 0.40, log(g) = 4.0) and a stellar radius of 81373.5 km from Gillon et al. (2016). Additionally, we have scaled the stellar spectrum to a K-band magnitude of 8.0 rather than mag_K = 10.3, TRAPPIST-1's native magnitude. While this may be representative of an optimistic scenario, mag_K = 8.0 is the mean magnitude of the brightest 10 M-dwarf stars in the Barclay catalog of the predicted TESS yield (Barclay et al. 2018).

We produce synthetic transmission spectra of our Earth-like atmosphere model over a grid of spectral resolution and bandpass choices. The nominal wavelength coverage was chosen to cover the near- and mid-infrared regimes spanning 1–30 μm, shown in Figure 2. We divide this wavelength range into eight separate regimes to explore the influence of additional bands of a given molecule and their overlap with features of other molecules. Additionally, we test low and moderate spectral resolutions ranging from R = 30 to R = 300. The spectral resolution (λ/δλ) dictates the degree to which individual molecular bands can be resolved and overlap degeneracies broken. Figure 3 compares the spectra at the resolutions explored in this work. We simulate different resolutions within each spectral band in order to explore the resolution trade along with each wavelength regime. This produces a grid of 23 test cases (shown in Table 2) to evaluate through our retrieval algorithm described in 2.2.

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In addition to resolution and wavelength coverage, we also explored the signal-to-noise trade, parameterized here as the number of transits (1–100), which can be considered as a proxy for the mirror diameter or source magnitude, where the noise on a single transit is defined by the nominal noise model setup described in Section 2.3. Utilizing this trade, we explored the minimum number of transits necessary to achieve a threshold 3.6σ confidence level detection of each molecule. We provide Equations (1) and (2) as a means of converting our retrieval results to stars of differing magnitudes or for different sized collecting mirrors.

$$\begin{eqnarray}&&{M}_{\mathrm{eff}}={M}_{\mathrm{nom}}-2.5{\mathrm{log}}_{10}\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{nom}}}\right)\end{eqnarray} \tag{ 1 }$$

where T_eff/T_nom is the ratio determining how many more or less transits are equivalent to a change from the native magnitude (M_nom = 8.0) to a new magnitude (M_eff).

$$\begin{eqnarray}&&{A}_{\mathrm{eff}}={A}_{\mathrm{nom}}\sqrt{\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{nom}}}}\end{eqnarray} \tag{ 2 }$$

where T_eff/T_nom is the ratio of transits equivalent to a change from the native collecting area (A_nom = 25 m²) to a new mirror size (A_eff).

Water has spectral features spanning nearly the entire wavelength range in this study, with the most prominent features centered at 2.65 and 6.3 μm. Additionally, there are two smaller features extending into the near-infrared at 1.4 and 1.85 μm as well as a series of features upwards of 20 μm. The most relevant comparison is between the near-infrared (which we define as the 1–5 μm bandpass) and the mid-infrared (5–11 μm). In isolation from other water features in the 5–11 μm region, the broad 6.3 μm feature produces comparatively underwhelming results (for both detection significance and abundance constraints) likely due to the limited contribution from the few other prominent features in this bandpass (Benneke & Seager 2012). In contrast, the 1–5 μm region encompasses the prominent 2.65 μm feature as well as the two smaller features. The precision on the abundance constraints improves by approximately 60% both when increasing the resolution from R = 50 to R = 100 and again from R = 100 to R = 300. However, it is worth noting that these near-infrared features are very narrow and are thus significantly more sensitive to changes in the choice of spectral resolution. At a resolution of R = 100, a single H₂O feature in the near-infrared consists of between 10 and 20 resolution elements compared to 44 for the 6.3 μm feature.

While the near-infrared (1–5 μm bandpass) produces the tightest abundance constraints, it does not provide the greatest detection significance values given the same spectral resolution and number of transits (refer to Table 3 for a comparison of the detection significance results). By comparison, the 3–11 μm range (which does not include additional H₂O features) requires 33.9% less observation time at R = 100 and 50.0% less at R = 50 than the 1–5 μm bandpass. Alternatively, by looking further into the mid-infrared, the 5–30 μm bandpass offers a 3.4% decrease at R = 100 and a 24.5% decrease at R = 50 compared to the 1–5 μm bandpass.

Due to its prominent spectral features across most of the near- and mid-infrared, CO₂ is the easiest molecule to detect within each of our bandpass choices, with the exception of 5–11 μm. CO₂ possess four prominent spectral features centered at 2.0, 2.7, 4.3, and 15um. The 5–11 μm region is the only bandpass choice that does not include a spectral feature for CO₂. Any other choice of wavelength coverage and spectral resolution evaluated in our study proved to be sufficient to detect CO₂ in less than seven transits. The inclusion of the 4.3 μm feature ensures a 3.6σ detection within the observation time of approximately a single transit.

Despite the strong detection of CO₂, the abundance constraints are, ironically, not as precise as would be expected. This is largely due to the fact that deriving narrow abundance constraint relies on a change in the strength of the spectral features. Line & Parmentier (2016) derive the function for spectral modulation with respect to wavelength for multiple absorbing molecular species, reproduced here for convenience (Equation (3)).

$$\begin{eqnarray}&&\displaystyle \frac{d{\alpha }_{\lambda }}{d\lambda }=\displaystyle \frac{2{R}_{p}}{{R}_{\star }^{2}}H\displaystyle \frac{1}{1+\tfrac{{\xi }_{2}{\sigma }_{\lambda ,2}}{{\xi }_{1}{\sigma }_{\lambda ,1}}}\left(\displaystyle \frac{d\mathrm{ln}({\sigma }_{\lambda ,1})}{d\lambda }+\displaystyle \frac{{\xi }_{2}{\sigma }_{\lambda ,2}}{{\xi }_{1}{\sigma }_{\lambda ,1}}\displaystyle \frac{d\mathrm{ln}({\sigma }_{\lambda ,2})}{d\lambda }\right)\end{eqnarray} \tag{ 3 }$$

where dα_λ/dλ is the wavelength-dependent slope of the transmission spectra, H is the scale height given by $\tfrac{{k}_{b}T}{\mu g}$, and ${\sigma }_{\lambda ,i}$ and ξ_i are the absorption cross-section and abundance, respectively, of a given molecular species.

In the case of CO₂, the ξ₁σ_λ,1 term is so much greater than the other absorbers that the spectral modulation due to CO₂ becomes insensitive to even large changes in the abundance. The retrieval thus determines that a wide range of abundance values can reproduce the shape of the observed features within the error bars. Therefore, the only means to narrow the constraint is to increase the number of observed transits.

At 50 transits, the 1–5 μm bandpass produces approximately a single order-of-magnitude (1.01 dex) constraint at R = 100 or 0.81 dex at R = 300. The abundance constraints produced by the 1–5 μm bandpass cases at both R = 50 and R = 100 are comparable to those of both the 3–11 μm and 3–30 μm bandpasses.

Due to its low abundance in the modern Earth atmosphere, CH₄ is difficult to detect and ultimately determines the lower limit on the observing time required to detect all five bio-indicator molecules. Despite the larger number of transits required to detect CH₄, we can place the best constraints on its abundance. Indeed, it is notable that our results indicate that for the 3–11 μm at R = 100 case, a 3.6σ detection is only possible after 61.7 observed transits while at only 50 transits we can place an order-of-magnitude constraint on its abundance. This appears initially paradoxical, until we examine the underlying dependencies of these two different results. Effectively, the detectability (via the Bayes factor) depends on the amplitude of the spectral features with respect to the continuum or adjacent features of other molecules (e.g., the wavelength-dependent derivative of transit depth shown in Equation (3)). The mixing ratio constraints on the other hand are dictated by the derivative of the transit depth with respect to a given species abundance (or rather, the inverse, e.g., Line et al. 2012), with each species having varying sensitivity. Therefore, one could observe small features resulting in a low detectability (as in the case of CH₄) while having a large abundance derivative, resulting in a tight constraint. Figure 10 illustrates the molecule-specific relationship between the abundance constraints (log) and detection significance under the 1–30 μm, R = 100 scenario for varying numbers of transits. Figure 10 illuminates the key differences between CH₄ and CO₂ in that while CO₂ has a high detection significance, it does not necessarily have a precise constraint. Alternatively, even low confidence detections of CH₄ can yield precise abundance constraints.

At Earth-like abundances, CH₄ has only three discernible spectral features in the near- and mid-infrared: 2.3, 3.3, and 7.6 μm. The most prominent of these is centered at 7.6 μm and directly overlaps with a larger N₂O feature centered at 7.7 μm. Bandpass choices that exclude additional methane features will lack the ability to overcome this degeneracy. Fortunately, the 3.3 μm (ν₃) feature provides an unambiguous marker for the presence of CH₄. We find that the inclusion of this feature is crucial to detecting CH₄ as neither the 5–11 μm nor the 5–30 μm bandpasses are capable of detecting methane within 100 transits, despite the presence of the 7.6 μm (ν₄) feature. There is an additional feature at 2.3 μm which, when combined with the 3.3 μm band, greatly improves the detection significance.

When observing only the 3–5 μm bandpass, at R = 300, CH₄ is detectable with 54.9 transits and provides an abundance constraint of 1.04 dex at 50 transits. Maintaining this narrow wavelength coverage and reducing to R = 100, increases the necessary observation time to beyond 100 transits. However, expanding our bandpass either to the near- or mid-infrared regimes proves to provide significant benefits. The near-infrared bandpass (1–5 μm) requires 38.5 transits and provides a 0.63 dex (at 50 transits) abundance constraint or 54.4 transits and 0.81 dex for R = 300 and R = 100, respectively. Alternatively, observing the 3–11 μm bandpass requires 34.3 transits and provides a 0.79 dex (at 50 transits) abundance constraint or 61.7 transits and 0.99 dex for R = 300 and R = 100, respectively.

Based on these results, we find that including the ν₃ feature, within the chosen bandpass, is essential to detecting and constraining CH₄. In the scenario that spectral resolution is limited to R = 100, the addition of the 2.3 μm feature is incredibly beneficial, providing a 0.5 dex improvement in the abundance constraint and requiring only a third of the observational time. However, while it is certainly ideal to include both features if possible, it is quite noteworthy that the benefit of a higher resolution (R = 300) choice, in the absence of the 2.3 μm feature, outweighs the benefit of including this feature at a lower resolution (R = 100).

Ozone is different from the other molecular species we have addressed thus far, in that its most prominent features extend further into the mid-infrared. The three primary features for O₃ are at 4.75, 9.6, and 14.2 μm with the 9.6 μm feature being the strongest. Conveniently, due to the location of these spectral features and our bandpass choices, we can evaluate the benefit of each additional feature, individually, on the retrieval results.

Isolating the 9.6 μm feature, in the 5–11 μm bandpass, requires a relatively low number of transits at only 14.9 and 18.2 transits for resolutions of R = 100 and R = 50, respectively. However, the abundance constraints for this bandpass exceed an order of magnitude at 1.41 dex (R = 100) and 1.58 dex (R = 50). If we choose to observe further into the mid-infrared (5–30 μm), including the features at 9.6 and 14.7 μm, we notice only slight reductions in the required observation time and maintain larger than an order-of-magnitude constraint on both abundance values. The one noticeable benefit of including these longer wavelengths is that less than 20 transits are required to detect O₃ even at a resolution of R = 30. Alternatively, extending the bandpass to 3–11 μm to include the 4.75 μm feature results in a reduction of the necessary observation time to 10.5 transits for a resolution of R = 100 (12.9 for R = 50) and improves our abundance constraints by nearly a factor of four (0.82 and 0.99 dex) for both R = 100 and R = 50. It is worth noting that the 4.75 μm feature has significant overlap with a minor CO₂ feature, thereby causing a degeneracy if observed without additional O₃ features. We conclude that the optimal wavelength coverage for detecting and constraining O₃ is the 3–11 μm bandpass choice. Additionally, this bandpass would only require a resolution of R = 50 to place an order-of-magnitude constraint on the volume mixing ratio for O₃.

Similar to O₃, the most prominent features for N₂O extend out into the longer wavelengths. N₂O's strongest spectral features are located at 4.5 and 7.7 μm, each with an adjacent smaller feature at 3.9 μm and 8.6 μm, respectively. Although there is a broad feature at 17 μm, it is overlapped by a much larger CO₂ feature.

Due to the large degree of degeneracy with the 7.6 μm CH₄ feature, isolating the 7.7 and 8.6 μm features in the 5–11 μm bandpass proves to be inadequate. At a spectral resolution of R = 100, the 5–11 μm bandpass requires nearly 12 times as many transits (∼94 transits) as the 3–11 μm bandpass would require to obtain a 3.6σ detection. Extending to 30 μm incorporates the 17 μm feature and only reduces the observational time to 72.8 transits while the constraints still exceed 1.50 dex.

We find that the 3–5 μm bandpass is sufficient to detect N₂O at 14.4 transits (R = 100) or 23.8 transits (R = 50), making it one of the only two molecular species that can be constrained in that bandpass within relatively few transits. However, that region is too narrow to provide adequate abundance constraints, resulting in a 1.27 dex constraint at 50 transits. If we expand to either the 1–5 μm or the 3–11 μm bandpass, our required number of transits drops to 9.2 and 7.9, respectively. The corresponding constraints at 50 transits are 0.97 dex (1–5 μm) and 1.02 dex (3–11 μm), roughly an order of magnitude for both. The 3–11 μm region would thus allow for a 3.6σ detection in fewer transits than would be required for either O₃ or H₂O within the same regime. Likewise, the 1–5 μm bandpass requires only 39.5% of the observational time, to detect N₂O, as needed to detect H₂O in the same regime. With comparable abundance constraints, the 3–11 μm bandpass proves to be the optimal bandpass, requiring only 74%–86% (R = 50 to R = 100) of the observation time as the 1–5 μm bandpass choice.