JWST COMPASS: NIRSpec/G395H Transmission Observations of the Super-Earth TOI-836b

The ExoTiC-JEDI (Exoplanet Timeseries Characterization—JWST Extraction and Diagnostic Investigator) package ¹¹ performs end-to-end extraction, reduction, and analysis of JWST time-series data, from uncal files through to light-curve fitting, to produce planetary spectra. Throughout, NRS1 and NRS2 data are reduced independently, and each visit is treated separately. In all cases, we tried a variety of values for each reduction parameter, and determined the value that resulted in the smallest out-of-transit scatter in the resulting white-light curve.

We begin with a modified version of Stage 1 of the jwst pipeline (v.1.8.6, context map 1078; Bushouse et al. 2022), performing linearity, dark current, and saturation corrections, and using a jump detection threshold of 15. We next perform a custom destriping routine to remove group-level 1/f noise, masking the spectral trace 15σ from the dispersion axis for each integration and subtracting the median pixel value of nonmasked pixels from each detector column in each group. We also perform a custom bias subtraction, building a pseudo-bias image by computing the median of each detector pixel in the first group across every integration in the time series. This median image is then used in place of a bias and subtracted from every group, and was found to improve the out-of-transit scatter for both detector white-light curves for both visits (see also Alderson et al. 2023). We then proceed with the standard ramp-fitting step. ExoTiC-JEDI also utilizes Stage 2 of the jwst pipeline to produce the 2D wavelength map needed to obtain the wavelength solution.

In Stage 3 of ExoTiC-JEDI, we extract our 1D stellar spectra, performing additional cleaning steps and 1/f correction. Using the standard data quality flags produced by the jwst pipeline, we replace any pixels flagged as do not use, saturated, dead, hot, low quantum efficiency, or no gain value with the median of the neighboring 4 pixels in each row. To replace any spurious pixels that have not yet been corrected (such as cosmic rays), we identify any that are outliers from their nearest neighbors on the detector or throughout the time series. We use a 20σ threshold in time and a 6σ threshold spatially, replacing the problem pixel with the median of that pixel in the surrounding 10 integrations or 20 pixels in the row, respectively. Any remaining 1/f noise and background are removed by subtracting the median unilluminated pixel value from each column in each integration. To extract the 1D stellar spectra, we fit a Gaussian to each column to obtain the center and width of the spectral trace across the detector, fitting a fourth-order polynomial to each. The spectral trace centers and widths are then smoothed with a median filter of window size 11 to determine the aperture region. For both visits and both detectors, we used an aperture five times the FWHM of the trace, approximately 8 pixels wide from edge to edge. An intrapixel extraction is used to obtain the 1D stellar spectra, where "intrapixel" is defined as the fraction of the FWHM that falls on each pixel such that at the edge of the aperture, the flux included from any intersected pixel is equal to the fraction of the pixel within the aperture, multiplied by the total flux value of that pixel. The 1D stellar spectra are then cross-correlated to obtain the x- and y-positional shifts throughout the observation for use in systematic light-curve detrending.

Finally, we fit white-light curves for both NRS1 and NRS2, as well as spectroscopic light curves across the full NIRSpec/G395H wavelength range at a variety of resolutions for both visits. For the white-light curves (spanning 2.814–3.717 μm for NRS1 and 3.824–5.111 μm for NRS2), we fit for the system inclination, i, ratio of semimajor axis to stellar radius, a/R_*, center of transit time, T₀, and ratio of planet to stellar radii, R_p /R_*, holding the period and eccentricity, e, and argument of periastron, ω, fixed to values presented in Hawthorn et al. (2023). The stellar limb-darkening coefficients are held fixed to values calculated using the ExoTiC-LD package (Grant & Wakeford 2022), based on the stellar T_*, log(g), and [Fe/H]_* presented in Hawthorn et al. (2023; see Table 1), with Set One of the MPS-ATLAS stellar models (Kostogryz et al. 2022, 2023) and the nonlinear limb-darkening law (Claret 2000). We used a least-squares optimizer to fit for a batman (Kreidberg 2015) transit model simultaneously with our systematic model S(λ), which took the form

$$\begin{eqnarray*}&&S(\lambda )={s}_{0}+({s}_{1}\times t)+({s}_{2}\times {x}_{s}| {y}_{s}| ),\end{eqnarray*}$$

where x_s is the x-positional shift of the spectral trace, ∣y_s ∣ is the absolute magnitude of the y-positional shift of the spectral trace, t is the time, and s₀, s₁, s₂ are coefficient terms, as previously used for the ExoTiC-JEDI analysis in May & MacDonald et al. (2023). For the spectroscopic light curves, we fit for R_p /R_*, holding T₀, i, and a/R_* fixed to the respective white-light-curve fit value, as shown in Table 2.

Table 1. System Properties for TOI-836b

Property	Value
K (mag)	6.804 ± 0.018
R* (R⊙)	0.665 ± 0.010
T* (K)	4552 ± 154
log(g)	4.743 ± 0.105
[Fe/H]*	−0.284 ± −0.067
Period (days)	3.81673 ± 0.00001
MP (M⊕)	${4.5}_{-0.86}^{+0.92}$
RP (R⊕)	1.704 ± 0.067
Teq (K)	871 ± 36
e	0.053 ± 0.042
ω (°)	9 ± 92
Semimajor axis (au)	0.04220 ± 0.00093

Note. Values used in the light-curve fitting are shown in Table 2. All values from Hawthorn et al. (2023).

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For both the white- and spectroscopic light curves, we removed any data points that were greater than 4σ outliers in the residuals and refit the light curves until no such points remained. We also rescaled the flux time-series errors, using the beta value (Pont et al. 2006) as measured from the white- and red-noise values calculated using the extra_functions.noise_calculator() in ExoTiC-JEDI, to account for any remaining red noise in the data. We removed the first 370 integrations (∼22 minutes) of Visit 1 and the first 440 integrations (∼26 minutes) of Visit 2 to remove settling ramps at the start of the observations. We additionally removed 259 integrations from the end of Visit 1 (∼15 minutes) and 508 integrations from the end of Visit 2 (∼30 minutes), which removed a slight linear slope in the residuals of the NRS1 fits and removed a ∼10 ppm offset between the transit depths of NRS1 and NRS2. The ExoTiC-JEDI-fitted white-light curves and residuals for each visit are shown in Figure 1.

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For our second independent reduction, we utilize Eureka!, an end-to-end pipeline for analyzing JWST transiting planet data. We used the default procedures for Stage 1 and Stage 2 from the Eureka! wrapper of the jwst pipeline, following the same procedures as for ExoTiC-JEDI, but using the standard superbias subtraction. We also used the aforementioned ExoTiC-JEDI group-level background subtraction to account for 1/f noise. Following this, we use Eureka! Stage 3 to extract the stellar spectra and produce the broadband and spectroscopic light curves. Note that while ExoTiC-JEDI maintains the slightly curved shape of the NIRSpec/G395H spectral trace throughout spectral extraction, the Eureka! pipeline flattens this trace by bringing the center of mass of each column to the center of the subarray, allowing for a straight box extraction to be used. Eureka! allows for the customization of a variety of reduction parameters in Stage 3, including the trace extraction width, the region and method for the background extraction, and trace extraction parameters. To find the best combination of values, we tested extraction apertures consisting of combinations of 4–8 pixels from the center of the flattened trace, background apertures of 8–11 pixels, thresholds for optimal extraction outlier rejection of 10σ and 60σ (which approximate the standard extraction), and two different methods of background subtraction (an additional column-by-column mean subtraction and a full-frame median subtraction). We select the final reduction parameters as the combination that minimizes the scatter in the resulting white-light curves, doing this separately for each detector and each visit. We find that for both visits, both NRS1 and NRS2 favored an additional column-by-column background subtraction using a threshold of 60σ (which approximates a standard box extraction). The optimal aperture half-widths for the trace extraction for NRS1 and NRS2 were 6 pixels for Visit 1 and 4 pixels for Visit 2. The background-subtraction region spans from the upper and lower edge of the detector subarray to an inner bound a number of pixels away from the flattened trace. This inner bound was found to be 8 pixels in both detectors for Visit 1 and 9 pixels for NRS1 and 8 pixels for NRS2 for Visit 2. We then extract white-light and 30 pixel binned (∼0.02 μm, R ∼200) spectroscopic light curves for both visits for both detectors.

During the light-curve fitting stage, we move away from the Eureka! pipeline and utilize a custom light-curve fitting code, but refer to this reduction as the "Eureka!" reduction for simplicity. We fit each white- and spectroscopic light curve separately using emcee (Foreman-Mackey et al. 2013), fitting for i, a/R_*, T₀, and R_p /R_* and fixing the other orbital parameters to those from Table 1, using the batman package (Kreidberg 2015). We utilize quadratic limb-darkening coefficients calculated using ExoTiC-LD and the stellar parameters from Table 1. We fit our transit model and a systematic model simultaneously, which took the form

$$\begin{eqnarray*}&&S(\lambda )={s}_{0}+{s}_{1}\times T+{s}_{2}\times X+{s}_{3}\times Y,\end{eqnarray*}$$

where X and Y are the normalized positions of the trace on the detector and s_i are the free parameters in our systematic noise model. We use an iterative rolling median outlier rejection with a 50 data-point-wide window three times on both the white- and spectroscopic light curves, removing outliers more than 3σ from the median. We initialize our Markov Chain Monte Carlo (MCMC) walkers using the best-fit results from a Levenberg–Marquardt least-squares minimization. We utilize three times the number of free parameters as the number of walkers (resulting in 27 walkers) and run a burn-in of 50,000 steps, which is discarded, followed by a production run of 50,000 steps, with uninformed priors on all of the parameters. We trim the initial 444 points (∼20 minutes) from all the light curves to remove any initial ramp that may be present, and we removed the last 259 points of Visit 1 and 508 points of Visit 2 (see Section 3.1). The fitted white-light curves and residuals resulting from the Eureka! reduction for each visit are shown in Figure 1, while the fitted white-light-curve parameters are shown in Table 2.

During our spectroscopic light-curve fits, we once again fit for i, a/R_*, T₀, and R_p /R_*, resulting in parameters that are consistent with the fitted white-light-curve values to within 2σ. We utilize a prior when fitting the i, a/R_*, T₀ for the spectroscopic light curves. To obtain priors for these fits, we utilize the posterior distribution for each free parameter that resulted from the MCMC chains of the white-light-curve fits for each visit. We use the median 3σ values from combining the chains from NRS1 and NRS2 as Gaussian priors for each astrophysical parameter, meaning that the prior represents the combined constraints from NRS1 and NRS2. For R_p /R_*, we use a flat uninformed prior to not bias our transmission spectrum.

3.2.1. Joint Fit of the Eureka! Light Curves

For the synthetic model fits, we use the MLFriends statistic sampler (Buchner 2016, 2019) implemented in the open-source code UltraNest (Buchner 2021). For each visit and each data reduction, we fit: (1) a one-parameter, zero-slope line; (2) a two-parameter step function composed of two zero-sloped lines, one each for NRS1 and NRS2; and (3) a two-parameter-sloped line. The best-fit results are shown in Table 3 and Figure 4.

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The zero-slope fits for the ExoTiC-JEDI Visit 1 and 2 data are consistent to within 1σ, resulting in a transit depth baseline of ${({R}_{p}/{R}_{* })}^{2}$ = 608 ± 3 ppm and 609 ± 3 ppm for Visits 1 and 2, respectively. The Eureka! Visit 1 data are also consistent within the 1σ ExoTiC-JEDI range at ${({R}_{p}/{R}_{* })}^{2}$ = 605 ± 3, but the Eureka! Visit 2 data have a somewhat higher (∼2σ) baseline of ${({R}_{p}/{R}_{* })}^{2}=620\pm 4$.

The step function fits for both ExoTiC-JEDI and Eureka! Visits 1 and 2 are not consistent—Visit 1 has a positive step function, while Visit 2 has a negative step function relative to NRS2. However, comparing across pipelines, the ExoTiC-JEDI and Eureka! reductions give consistent steps within 1σ for each visit. For Visit 1, ExoTiC-JEDI and Eureka! produce an offset of +13 ± 7 ppm and +11 ± 7 ppm, respectively, while for Visit 2, ExoTiC-JEDI and Eureka! produce an offset of −18 ± 7 ppm and −31 ± 7 ppm, respectively. This leads to a similar discrepancy when fitting the sloped line, where there is agreement across the reduction methods, but not from visit to visit. Within 1σ, both Visit 1 reductions produce a positive slope, while Visit 2 produces a negative slope. This largely suggests that the slope and step function are not astrophysical in nature, unlike those that have been seen in other observations of small exoplanet atmospheres with NIRSpec/G395H (e.g., Moran & Stevenson et al. 2023). Regardless, the sizes of the offsets obtained for both ExoTiC-JEDI visits and for Eureka! Visit 1 are smaller than the corresponding median transit depth uncertainty.

To confirm that the apparent offsets between NRS1 and NRS2 need not be a major consideration in our final interpretation of TOI-836b's atmosphere, we can also assess which of the zero-slope, step, and slope models are statistically preferred by the data. Table 3 lists the likelihoods for each of these fits. For both ExoTiC-JEDI and Eureka! Visit 1 reductions, and the ExoTiC-JEDI Visit 2 reduction, the step function and slope models are not preferred or are only weakly preferred over the zero-slope model—i.e., the data are well described by a flat line given their comparative Bayes factors (${\mathrm{lnB}}_{12}=\mathrm{ln}$Z₁ [Model 1]−ln Z₂ [Model 2]). For the Eureka! reduction of the Visit 2 data, comparisons of the Bayes factors suggest that both the step function and slope models are at least moderately preferred over the zero slope with lnB₁₂ = 6.5 and 2.7, ¹² respectively. Though Bayes factors do not directly map to σ-significance for non-nested models (Trotta 2008), for the step function and slope models these roughly translate to strong and moderate preferences, respectively, over the zero-slope model in this context (see Table 1 of Trotta 2008). This is an understandable result, given that the Eureka! Visit 2 offset is larger than the transit depth uncertainty near the detector gap.

As both ExoTiC-JEDI reductions produce consistent results for our synthetic modeling, we proceed with a weighted average transmission spectrum of the two visits for the ExoTiC-JEDI reduction. As there is "moderate" preference for a step function offset in the Eureka! reduction of Visit 2, we choose to leverage the joint fit transmission spectrum from the Eureka! reduction for the rest of our analysis.

We also ran our synthetic modeling on the weighted average ExoTiC-JEDI and joint fit Eureka! transmission spectra to confirm that they are in agreement. In the case of the joint fit, we find that the Eureka! data now obtain an offset smaller than the median transit depth uncertainty of 25 ppm, resulting in the step function and slope models no longer being preferred over the zero-slope model. In the case of the weighted average, the ExoTiC-JEDI data continue not to prefer the step function or slope models, with the calculated offset between NRS1 and NRS2 now consistent with 0 ppm. The combined-visit transmission spectra of TOI-836b are therefore well described by a flat line regardless of the reduction pipeline used, and we can proceed with our physically motivated modeling.

The spectral feature sizes of the transmission spectra are largely driven by the scale height (=kT/μg) and potential muting by aerosols (e.g., Sing et al. 2016). In order to understand what region of parameter space we can rule out for this system, we create a grid of spectral models as a function of metallicity and "opaque pressure level." For the former parameter, metallicity, it is unlikely that this system (R = 1.7 R_⊕) has a large hydrogen–helium envelope. However, similar to the analysis of the TRAPPIST-1 system (Moran et al. 2018) and of other small planets (e.g., Lustig-Yaeger & Fu et al. 2023; Moran & Stevenson et al. 2023), metallicity offers a suitable proxy for the mean molecular weight. For example, for our given assumption in the temperature–pressure profile, 100 times solar corresponds to a mean molecular weight of 4.3 g mol⁻¹, while 1000 times solar corresponds to a mean molecular weight of 15.7 g mol⁻¹. The other parameter, "opaque pressure level," is a term adapted from Lustig-Yaeger & Fu et al. (2023) and represents a pressure below which the atmosphere is opaque (e.g., Seager & Sasselov 2000; Charbonneau et al. 2002; Berta et al. 2012; Knutson et al. 2014; Kreidberg et al. 2014b). ¹³ Other manuscripts have referred to this as a "cloud-top pressure" (e.g., Kreidberg et al. 2014b; Moran et al. 2018). Here we use a more general term, as we cannot differentiate a cloud-top pressure from a surface pressure.

Using a range of metallicities (1–1000 times solar, log-spaced with 26 grid points) and opaque pressure levels (100–10⁻⁴ bar, log-spaced with 5 grid points), we compute a grid of transmission spectra using the open-source PICASO package (Batalha et al. 2019). For the pressure–temperature profile, we use a 1D five-parameter double-gray analytic formula (Guillot 2010). We also test whether or not the results are sensitive to our choice of pressure–temperature profile by computing a similar grid with simple isothermal pressure–temperature profiles. Our conclusions do not change depending on this choice. Given the pressure–temperature profile, we fix the elemental C/O ratio to solar (=0.55; Asplund et al. 2009) and obtain the chemistry by interpolating on a precomputed chemical equilibrium grid. The chemistry grid was computed by Line et al. (2013) with NASA's CEA (Chemical Equilibrium with Applications) code (Gordon & McBride 1994). This grid is publicly available on GitHub as part of CHIMERA's open-source code. ¹⁴ Of note to this analysis, the molecules that absorb from 3–5 μm are H₂O, CH₄, CO₂, and CO, which are all included in the grid.

Figure 5 shows the results of the grid analysis, where we show how our confidence level, expressed as a σ-level, changes as a function of metallicity for an intermediate opaque pressure level of 0.1 bar. The general behavior of the significance curves in Figure 5(a), which peak toward 10–50 times solar and decrease toward 1000 times solar, is well documented (e.g., Moran et al. 2018). Toward 10 times solar, the magnitude of spectral features increases, because the added molecular opacity is better able to surpass the contribution from the H₂/He continuum without affecting the mean molecular weight, increasing the significance to which the spectral features can be ruled out. Beyond ∼50 times solar, the contribution from the heavier metals starts to increase the mean molecular weight of the atmosphere and results in overall smaller spectral features, which are harder to more confidently rule out.

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We choose to show 0.1 bar for reference, as it is synonymous with the tropopause of all solar system objects (Robinson & Catling 2014). Here, σ is computed by converting χ²/N to a p-value and then to a σ-significance. For each individual visit, we are able to rule out metallicities lower than 100–160 times solar, depending on the visit and the reduction. For example, in the case of the first visit, the individual ExoTiC-JEDI and Eureka! spectra enable metallicities to be ruled out at the <130 times solar and <160 times solar levels, respectively. With the combined spectra, we are further able to rule out metallicities <250 time solar and <380 times solar, for ExoTiC-JEDI and Eureka!, respectively, corresponding to mean molecular weights of ∼6–9 g mol⁻¹. Combining both visits enables us to rule out nearly double the parameter space in metallicity, demonstrating the potential of multivisit observing strategies. As shown in Figure 5(a), the reported 3σ lower limit on metallicity is dependent on the reduction method. However, the overall scientific conclusions are agnostic to the data reduction, as in all cases we are able to rule out H₂-dominated atmospheres with mean molecular weights less than ∼6 g mol⁻¹.

Figure 5 shows the results for an intermediate cloud case, though we ran a full grid of both cloud-free and highly cloudy cases. For an effective cloud-free atmosphere, in which the atmosphere only becomes opaque below 100 bar, the combined ExoTiC-JEDI spectrum is able to rule out the 300 times solar case. For cases where the opaque pressure level is 10⁻⁴ bar, comparable to a highly lofted cloud, we can rule out cases ≤100 times solar. Both the 100 bar and 10⁻⁴ bar cases represent unlikely physical scenarios, as clouds are expected to form in super-Earth atmospheres (Mbarek & Kempton 2016; i.e., atmospheres are highly unlikely to be effectively cloud free), and clouds lofted to high altitudes are unlikely to be completely opaque (Robinson & Catling 2014). As end-member cases, however, they demonstrate that solar-like composition atmospheres are not plausible for TOI-836b, regardless of the height of any opaque pressure level. Additionally, we tested whether or not our conclusions are affected by the choice of binning scheme and found that the conclusions are unchanged.

Figure 5(b) shows four of the spectra used to compute the σ-significance curves in 5(a), for reference, along with the weighted average spectra from ExoTiC-JEDI. The main features shown are those of CH₄ and CO₂, in the 1 times solar case, and primarily H₂O and CO₂ in the other cases. Figure 5(b) also lists the χ²/N and sigma rejection thresholds for each of the metallicity cases, demonstrating, for example, that we cannot confidently distinguish between atmospheres with 250 times solar and 1000 times solar metallicities, given the constraints we achieve with two combined transit observations.

TOI-836b's mass and radius place it at a very intriguing position in the mass–radius diagram, toward the lower edge of the radius valley. Furthermore, despite its low density, its parameters are compatible with a pure rock composition (no iron core) at the 1σ level. To infer the bulk properties of TOI-836b, we use the SMINT (Structure Model INTerpolator) package from Piaulet et al. (2021), which performs an MCMC retrieval of planetary bulk compositions from precomputed grids of theoretical interior structure models. We consider two possible compositions for the interior: (1) an Earth-like core with an H₂–He envelope of solar metallicity (Lopez & Fortney 2014); and (2) a refractory core with a variable core mass fraction and a pure H₂O envelope and atmosphere on top (Aguichine et al. 2021). These compositions represent end-member cases between an envelope that would form by accreting nebular gas with a Sun-like composition and a high-mean-molecular-weight envelope, where water is used as a proxy for all volatiles. Based on its bulk properties and these scenarios, we find that TOI-836b could have an envelope mass fraction of at most 0.1% in the case of gas of solar composition, or a water mass fraction of 9% ± 5% in the pure H₂O case, as shown in Figure 6.

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The two planets of the TOI-836 system are located on opposite sides of the radius valley, which translates into very different bulk compositions when inferred from interior structure models. The interior of TOI-836b is compatible with a pure rock (no iron core) composition at 1σ, meaning that the possibility of TOI-836b being a terrestrial planet cannot be excluded. From our interior modeling, we also find that the planet could be made of at most 0.1% solar-metallicity gas or 9% ± 5% pure water. In contrast, the possible bulk compositions of TOI-836 c are ${1.74}_{-0.48}^{+0.55} \%$ in the case of solar-metallicity gas or ${52}_{-14}^{+15} \%$ in the pure water case (Wallack & COMPASS et al. 2024). Figure 6 shows these possible interior compositions for both planets, which represent end-member cases for hydrogen-dominated and pure water atmospheres such that intermediate compositions are also possible.

Given TOI-836b's high equilibrium temperature and the stellar insolation flux received, photoevaporation could likely be responsible for the observed lack of a hydrogen-dominated atmosphere. Applying the photoevaporation model of Rogers et al. (2021) to a planet with the properties of TOI-836b, and including a core mass of 4.5 M_⊕, we find that any initial envelope mass fraction in the range 2%–30% is blown away in <400 Myr. Applying the model to a planet with the properties of TOI-836 c, and including a core mass of 9.4 M_⊕, we find that hydrogen envelopes of up to 10% can be retained. Given the reported age of TOI-836 of ${5.4}_{-5.0}^{+6.3}$ Gyr (Hawthorn et al. 2023), this analysis strongly suggests the absence of a hydrogen-dominated atmosphere for TOI-836b, which is in line with the apparent >6 g mol⁻¹ mean molecular weight derived in Section 5.2. The bulk composition of TOI-836 c, however, is still degenerate, with low-mean-molecular-weight atmospheres still being plausible, particularly in the presence of clouds and hazes (Wallack & COMPASS et al. 2024). These models provide the first clues as to this system's possible evolution, although we caution that transmission spectra alone are unable to determine whether the TOI-836 planets formed with their current masses, or if the present-day difference in bulk compositions is the consequence of photoevaporation.

Here we examine how our results can inform the planning of future observations of small planets with high atmospheric metallicities that orbit bright stars, comparing our measured data to PandExo JWST simulations, which are used by the community for planning observations. Using the grid of models described in Section 5.2, we determine how many additional transits would be needed to rule out a certain metallicity model. This is an identical exercise to that performed in Figure 5, except here our "data" comprise a PandExo simulation of a featureless spectrum. We compute noise simulations with PandExo (Batalha et al. 2017) using an identical observational setup to our program here, assuming that each additional visit provides a gain in precision of $\sqrt{{\rm{n}}\_\mathrm{transit}}$. For the "real data" case, we use the noise budget measured in this program from Visit 1, assuming that the increase in precision is equivalent to what we have measured moving from individual visits to the combined joint Eureka! reduction. Doing so results in a measured precision gain of $\sim 98 \% \sqrt{2}$ for TOI-836b. Using each of the estimates for noise, we compute simulated observations of each modeled spectrum and then compute the number of transits needed to rule out a zero-sloped line with 3σ confidence. Additionally, we include random noise and repeat the test for 1000 different random noise instances. Figure 7 shows the median result for the case of an opaque pressure level of 0.1 bar. Overall, ruling out cases up to 1000 times solar metallicity for TOI-836b would require up to eight additional transits, assuming that the data continued to result in a precision gain of $\sim 98 \% \sqrt{n\_\mathrm{transit}}$. For lower-metallicity cases (<100 times solar), the PandExo predictions are in line with those based on the real data. However, for higher metallicities, the PandExo data appear to be somewhat optimistic, resulting in estimates requiring one to two fewer transits than the real data. This result should be noted for future observation planning for planets that are expected to be heavily enriched in metals, particularly for those around bright stars with more complex noise properties.

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