The White Dwarf Opportunity: Robust Detections of Molecules in Earth-like Exoplanet Atmospheres with the James Webb Space Telescope

We model the atmosphere of our Earth-like exoplanet, which evolved around a WD, using EXO-Prime (Kaltenegger & Sasselov 2010)—a 1D radiative-convective terrestrial atmosphere code. Given an incident stellar/host spectrum and planetary outgassing rates, EXO-Prime couples 1D climate and photochemistry models to compute the vertical temperature structure and atmospheric mixing ratio profiles. The application of EXO-Prime to Earth-like planets evolving around WD hosts from T_eff = 6000–4000 K is extensively described in Kozakis et al. (2018, 2020). Here, we define "Earth-like" to refer to an Earth-radius and Earth-mass planet with similar outgassing rates to the modern Earth.

The irradiation environment around a WD changes the atmospheric composition of an Earth-like planet compared to the modern Earth (Kozakis et al. 2018). Given the temperature of WD  1856+534 (${T}_{\mathrm{eff}}=4710\,$K, Vanderburg et al. 2020), we use the 5000 K WD HZ terrestrial planet model from Kozakis et al. (2020) for our analysis. We note that WD hosts display largely featureless spectra⁷ (Saumon et al. 2014), similar to blackbodies, below 5000 K (Kepler et al. 2016). The atmosphere of a rocky planet receiving S = S_⊕ around a 5000 K WD host, with Earth-like outgassing rates, shows higher levels of CH₄, lower O₃, and similar H₂O concentrations compared to Earth, along with a temperature inversion. We note that the surface UV environment of a WD planet is time dependent, impacting the atmospheric composition (see Kozakis et al. 2018). We tested our model sensitivity to UV levels by artificially increasing the flux in our Lyα bin (1200–1300 Å) by 1000×. Such an unphysical increase only slightly lowers the CH₄, H₂O, O₃, and N₂O abundances in the upper atmosphere (<a factor of 3), so our results are robust to higher UV environments expected around younger WDs. A detailed account of our climate-photochemistry model is provided in Kozakis et al. (2018).

Our model transmission spectrum is computed with the POSEIDON radiative transfer code (MacDonald & Madhusudhan 2017). POSEIDON has been widely applied to the modeling and interpretation of giant planet atmospheres, ranging from hot Jupiters to exo-Neptunes (e.g., Sedaghati et al. 2017; Kilpatrick et al. 2018; MacDonald & Madhusudhan 2019). For the present study, we extended the opacity database in POSEIDON to include all molecular species with prominent absorption features expected in the atmospheres of Earth-like planets orbiting WDs (Kozakis et al. 2020). We pre-computed cross sections for O₂, O₃, H₂O, CH₄, N₂O, CO₂, CO, and CH₃Cl from HITRAN2016 line lists (Gordon et al. 2017) using the HITRAN Application Programming Interface (Kochanov et al. 2016). Our cross-section grid covers temperatures from 100 to 400 K (20 K spacing), pressures from 10⁻⁶ to 10 bar (1 dex spacing), and wavelengths from 0.4 to 50 μm (0.01 cm⁻¹ resolution). All spectral lines are modeled as air-broadened Voigt profiles, with the line wings calculated to either 500 half-width at half-maximum or 30 cm⁻¹ from the line core (whichever deviation is smaller). Given the importance of optical wavelength O₃ opacity in terrestrial planet transmission spectra (e.g., Kaltenegger 2017; Meadows et al. 2018; Kozakis et al. 2020), not currently included in HITRAN, we employ the temperature-dependent O₃ cross sections from Serdyuchenko et al. (2014). The latest HITRAN collision-induced absorption (CIA) data, covering N₂–N₂, O₂–O₂, O₂–N₂, N₂–H₂O, O₂–CO₂, CO₂–CO₂, and CO₂-CH₄, are also included (Karman et al. 2019).

A high-resolution transmission spectrum is derived from radiative transfer through our model Earth-like atmosphere. The equation of radiative transfer is solved by integrating the wavelength-dependent extinction coefficient along the line of sight for successive atmospheric annuli. The mixing ratio profiles and temperature structure described in Section 2.1 were interpolated onto an altitude grid with 10 layers per pressure decade, uniformly spaced in log-pressure, from the surface to 10⁻⁷ bar. For layers above 0.1 mbar, the atmosphere was assumed to possess the same temperature and composition as present at 0.1 mbar.⁸ The resultant high-resolution spectrum matches the spectra modeled by Kozakis et al. (2020)—with the addition of CIA, which has been added to the transmission spectra presented here.

We modify our radiative transfer prescription to correct for the effect of refraction. Light rays probing sufficiently dense, deep regions of an atmosphere do not contribute to transmission spectra due to refraction preventing rays reaching a distant observer (e.g., Bétrémieux & Kaltenegger 2014; Robinson et al. 2017). For a given planet and atmosphere, this effect is most prominent where the angular size of a star from the perspective of its planet is small. For example, a remote observation of the Earth–Sun system would yield a transmission spectrum that can only be probed down to altitudes of about 12.7 km (e.g., Bétrémieux & Kaltenegger 2014). In the case of our WD planet (R_*/a < 10⁻²), we employ Equation (14) from Robinson et al. (2017) to estimate a refractive surface that occurs at 0.523 bar (5.2 km altitude). We simulate the effect of refraction by neglecting the contribution of impact parameters below the refractive surface (e.g., Bétrémieux & Kaltenegger 2014; Robinson et al. 2017; Macdonald & Cowan 2019). Following this procedure, we computed a high-resolution (R = 50,000) transmission spectrum from 0.4 to 5.4 μm, shown in Figure 1 (right panel).

Imagine now observing an Earth-like planet transiting WD 1856+534 with JWST. With a 2.2 minute transit duration, and a twice-transit out-of-transit baseline, a minimum exposure time of 6.6 minutes per transit results. However, JWST time constrained observations⁹ incur a 1 hr penalty per visit to ensure a sufficient baseline to characterize instrument systematics. We therefore conservatively allocate 1.5 hr observing time per transit. Further allowing a 40% overhead¹⁰ above the science time, we estimate 12 transits can fit in a small program (<25 hr), 35 in a medium program (<75 hr), and 35+ in a large program (>75 hr). In what follows, we consider the atmospheric characterization potential of representative small, medium, and large JWST programs.

We simulate JWST observations with the NIRSpec Prism using PandExo (Batalha et al. 2017). The NIRSpec Prism covers a wide spectral range (0.6–5.3 μm) in a single transit, yielding high information content for terrestrial exoplanet atmospheres (Batalha et al. 2018). We focus on the Prism due to its coverage of a wide array of spectral features, including O₂, O₃, H₂O, CO₂, CH₄, and N₂O (see Figure 2). Other instrument modes, such as NIRISS SOSS, may provide complementary information in narrower spectral regions. We generated synthetic JWST observations for 1, 2, 3, 5, 10, 25, 50, and 100 transits using the NIRSpec Prism's 512 subarray for the model transmission spectrum from Section 2.2. Example noise instances for the 10- and 50-transit cases are shown in Figure 1 (right panel). For the host WD spectrum, we assume a 4780 K blackbody, normalized to J = 15.677 (Cutri et al. 2003). For a total integration time (inside and outside transit) of 1.5 hr, PandExo's optimizer determined a ∼38 s exposure time and 166 groups per integration. We generated simulated observations at their native resolution (R ∼ 30–300) to avoid information loss from binning (see Benneke & Seager 2013; Tremblay et al. 2020).

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The spectral uncertainties generated by PandExo can be used in two ways. One approach produces simulated observations for a specific (Gaussian) noise instance (e.g., Figure 1). However, as noted by Feng et al. (2018), results derived whenceforth are specific to this noise instance. They showed that a data set with the same uncertainties, but centered on the model, results in derived probability distributions representative of the average over many noise instances (via the central limit theorem). We follow this second approach in reporting our predicted detection significances and abundance constraints, ensuring our results are unbiased by individual noise realizations. Nevertheless, we verified that consistent results occur using data with Gaussian scatter. We now subject our simulated JWST data sets to a series of atmospheric retrievals, assessing the ability to recover atmospheric information from such observations.

Atmospheric retrieval refers to the inversion of atmospheric properties from a planetary spectrum. Retrieval codes couple a parametric atmospheric model and radiative transfer solver with a statistical sampling algorithm. Retrieval techniques are commonly applied in solar system remote sounding (e.g., Irwin et al. 2018) and to spectroscopy of giant exoplanets (see Madhusudhan 2018 for a review). Here, we employ atmospheric retrievals to rigorously conduct Bayesian parameter estimation and nested model comparisons.

Our retrievals are computed using the retrieval module of POSEIDON (MacDonald & Madhusudhan 2017). This utilizes a parametric version of the transmission spectrum model described in Section 2.2, coupled with the nested sampling algorithm MultiNest (Feroz & Hobson 2008; Feroz et al. 2009, 2019). The atmosphere is divided into 81 layers uniformly spaced in log-pressure from 10⁻⁷ to 10 bar. The mixing ratios of O₂, O₃, H₂O, CO₂, CH₄, N₂O, CO, and CH₃Cl, assumed uniform in altitude, are ascribed as free parameters, with the remaining gas composed of N₂. Each mixing ratio has a log-uniform prior from 10⁻¹² to 1, with mixing ratio sums exceeding unity rejected. The temperature structure is parameterized by the six-parameter function from Madhusudhan & Seager (2009), with priors as in MacDonald & Madhusudhan (2019) but for a surface (1 bar) temperature parameter (uniform prior from 100 to 400 K). Two parameters are also assigned for the planetary radius at 1 bar (uniform prior from 0.8 to 1.2 R_⊕) and the pressure of the refractive surface (log-uniform prior from 10⁻⁷ to 10 bar). This yields a total of 16 free parameters. Radiative transfer is evaluated at R = 2000 from 0.5 to 5.4 μm, with each model spectrum convolved to the resolving power of the NIRSpec Prism and integrated over its sensitivity function for comparison with each data point. The parameter space exploration is conducted via the PyMultiNest package (Buchner et al. 2014) with 2000 live points.

We conducted eight atmospheric retrievals for each synthetic JWST data set, for a total of 64 retrievals. For each number of transits, one "full" retrieval was computed for parameter estimation, along with seven retrievals each excluding one molecule. These nested retrievals yield prediction detection significances for N₂, O₂, O₃, H₂O, CO₂, CH₄, and N₂O (CO and CH₃Cl are unconstrained in all cases) via Bayesian model comparisons (see Trotta 2017). A typical retrieval involved the computation of 1–5 million transmission spectra. We show an example of the best-fitting spectrum from the 25-transit case in Figure 2, demonstrating the ability to correctly infer the principal molecular species present in our model atmosphere. We now proceed to present the results from our series of atmospheric retrievals.

Many atmospheric molecules, including prospective biosignatures, can be readily detected in Earth-like planets orbiting WDs with JWST. Our predicted detection significances are shown in Figure 3, categorized according to an adaptation of the Jeffreys scale for Bayesian model comparisons (e.g., Trotta 2017). The easiest molecules to detect are H₂O and CO₂, owing to their multiple strong absorption features throughout the infrared (see Figure 2). A single transit is sufficient to provide evidence for H₂O and CO₂ at >2σ. A five-transit reconnaissance program can yield >5σ detections of H₂O and CO₂, providing guidance for subsequent larger programs. Detecting other species becomes possible with only a few additional transits. CH₄ and O₃ are detectable at >3σ in 10 transits, accessible within a small JWST program. The strong detectability of O₃ is enabled by the prominent Chappuis and Wulf bands at visible wavelengths (see Serdyuchenko et al. 2014), with a lesser contribution from the 4.8 μm feature. N₂O becomes detectable at >4σ in 25 transits, owing to its features around 2.9, 3.9, and 4.5 μm. Detecting N₂ and O₂ is more challenging, with >5σ detections requiring 50 transits for N₂ and 100 transits for O₂, respectively.

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It is therefore possible to detect multiple combinations of biosignatures, namely, CH₄ in combination with O₃ and CH₄ in combination with N₂O (see Kaltenegger 2017) within the remit of a medium JWST program.

The absorption features due to CIA are crucial to correctly identify the dominant atmospheric constituents. The detectability of N₂ stems from the relative strengths of O₂ and N₂ collisional pairs, as an O₂-dominated atmosphere would produce strong O₂–O₂ features at 1.06 and 1.25 μm, inconsistent with the simulated data. Without these CIA features, distinguishing between O₂ and N₂ is challenging due to their similar molecular weights (Benneke & Seager 2013). A secondary contribution to N₂ detectability is the broad N₂–N₂ feature around 4.3 μm (e.g., Schwieterman et al. 2015). The detectability of O₂, even at the low resolution of the NIRSpec Prism, arises from the combination of the A-band (0.76 μm), O₂–O₂ (1.06, 1.25 μm), and O₂–N₂ (1.25, 4.3 μm) features.

Beyond detecting molecular species, transiting WD planets would provide the opportunity to precisely measure molecular abundances in Earth-like atmospheres. Our predicted abundance constraints¹¹ for representative small (10-transit), medium (25-transit), and large (50-transit) JWST programs are shown in Figure 4.

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The abundances of spectrally prominent trace gases can be constrained with only a handful of transits. A single transit provides relatively loose constraints on H₂O and CO₂ abundances, spanning 2 orders of magnitude to 1σ. Constraints improve considerably with five transits, with H₂O, CO₂, CH₄, and O₃ measurable to within an order of magnitude. Within the remit of a small JWST program (10 transits), N₂O can also be constrained within an order of magnitude. A medium program (25 transits) results in abundance constraints approaching a factor of 2 in precision (0.3 dex). Finally, a large 100-transit program can measure most abundances to ≲0.15 dex (40% precision). CO and CH₃Cl are unconstrained in all cases, due to their low abundances and negligible contributions to the true model spectrum. The predicted values for precise trace gas abundance constraints (≤1 dex), as a function of the number of transits, are as follows:

• 1.  
  H₂O: N_trans =  5 (0.6 dex), 10 (0.4 dex), 25 (0.24 dex), 50 (0.22 dex), 100 (0.12 dex).
• 2.  
  CO₂: N_trans = 5 (0.8 dex), 10 (0.5 dex), 25 (0.34 dex), 50 (0.30 dex), 100 (0.16 dex).
• 3.  
  CH₄: N_trans = 5 (0.6 dex), 10 (0.3 dex), 25 (0.19 dex), 50 (0.16 dex), 100 (0.09 dex).
• 4.  
  O₃: N_trans = 5 (1.0 dex), 10 (0.4 dex), 25 (0.20 dex), 50 (0.17 dex), 100 (0.11 dex).
• 5.  
  N₂O: N_trans = 10 (0.8 dex), 25 (0.36 dex), 50 (0.29 dex), 100 (0.19 dex).

Constraining the composition of the background gases, N₂ and O₂, is comparatively challenging. An upper bound on the O₂ abundance is possible even with 10 transits, due to the opacity contributions of O₂–O₂ CIA. However, the lack of a lower bound on the O₂ abundance for ≲25 transits results in the preferred solution being an N₂-dominated atmosphere. Once sufficiently precise observations are obtained, the abundances of N₂ and O₂ can be correctly inferred and precisely constrained. With a 50-transit program, the fractions of O₂ and N₂ in the atmosphere, ${X}_{{{\rm{O}}}_{2}}$ and ${X}_{{{\rm{N}}}_{2}}$, can be measured with uncertainties of ∼7%. This improves to ∼4% for a 100-transit program. It would therefore be eminently possible to precisely measure the molecular composition of Earth-like planets orbiting WDs.

Within 25 pc of the Sun, there are more than 220 WDs, of which ∼24% are multiple systems (Holberg et al. 2016). The closest, Sirius B (25,000 K), resides 2.6 pc away, with the closest single WD, van Maanen's Star (6000 K), at 4.3 pc. Table 1 shows the closest five single WDs with similar surface temperatures to WD 1856+534, including WD 1856+534 for comparison. Due to their higher brightnesses (smaller magnitudes), planets orbiting closer WDs would present more promising characterization opportunities. For the closest WD in Table 1 (WD 0552-041), we estimate NIRSpec Prism error bars can be ∼5× smaller than for WD 1856+534. Therefore, our results for 25 transits of WD 1856+534 (i.e., strong detections of H₂O, CO₂, CH₄, and O₃) could be achieved with a single transit of WD 0552-041.

Table 1.  Closest Five White Dwarfs with Comparable Surface Temperatures to WD 1856+534

WD Identifier	${T}_{{\rm{eff}}}$	Age	Distance	log(L/L⊙)	Radius	J	Spectral	Alias
	(K)	(Gyr)	(pc)		${R}_{\odot }$	(mag)	Type
WD 0552-041	5182 ± 81	7.31	6.4411	−4.22	0.0097	13.05	DZ10.0	LHS 32
WD 1334+039	4971 ± 83	5.38	8.2372	−4.03	0.0130	13.06	DA11	LHS 46
WD 1132-325	5000 ± 500	5.69	9.5471	−4.05	0.0126	13.56	DC10	vB 4/LHS 309
WD 0245+541	5139 ± 82	6.51	10.8661	−4.14	0.0107	13.87	DAZ9.5	LHS 1446
WD 0821-669	5088 ± 137	6.00	10.6751	−4.10	0.0115	13.79	DA9.8	SCR J0821-6703
WD 1856+534	4710 ± 60	5.85	24.7541	−4.13	0.0131	15.68	DC4	LP 141-14

Note. WD 1856+534 is shown for comparison (bold entry). Ages and spectral types are from Holberg et al. (2016).

References. (1) Gaia Collaboration et al. (2018), (2) Holberg et al. (2008), (3) Holberg et al. (2016), (4) McCook & Sion (2016).

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Earth-like planets around cool small M dwarfs, such as TRAPPIST-1, are promising targets for characterization with the upcoming Extremely Large Telescopes (ELTs) and JWST (e.g., Barstow & Irwin 2016; Lustig-Yaeger et al. 2019; Lin & Kaltenegger 2020). However, there remain outstanding challenges in interpreting transmission spectra of M-dwarf terrestrial planets, notably contamination from unocculted starspots (e.g., Rackham et al. 2018). Planets around WDs, on the other hand, are less susceptible to unocculted spots due to the greater WD area occulted by the planet during transit. WD planets also present increased planet to host radius ratios, consequently lowering the required time to achieve a signal-to-noise ratio sufficient to remotely detect biosignatures, such as O₃ + CH₄ and CH₄ + N₂O (e.g., Figure 3).

The relative potential of the M-dwarf and WD opportunities are ultimately modulated by the number of characterizable targets. While the occurrence rate of rocky planets in the HZ of WDs is unknown, one can estimate the relative number of characterizable planets:

• 1.  
  Number of stars/hosts. Winters et al. (2019) report 48 M-dwarfs within 5 pc, while Hollands et al. (2018) report 139 WDs within 20 pc. Comparing their respective local space-densities, WDs are ∼20× less common than M-dwarfs.
• 2.  
  Planetary occurrence rate. M-dwarf HZ terrestrial planet occurrence rate estimates range from ≈20%–50% (e.g Dressing & Charbonneau 2015; Hsu et al. 2020). The equivalent rate for WDs is currently unknown, with K2 suggesting a limit of <28% (van Sluijs & Van Eylen 2018).
• 3.  
  Transit probability. Earth-sized planets in the M-dwarf HZ have a ∼1%–2% transit probability (e.g., Dittmann et al. 2017; Gillon et al. 2017). The smaller R_* of WDs and smaller HZ orbital separation yields similar transit probabilities of ∼1%.
• 4.  
  Characterization horizon. Detecting O₃, for example, would require ≳100 JWST transits for a modern Earth atmosphere around the M-dwarf TRAPPIST-1 (12.4 pc) (Lustig-Yaeger et al. 2019). Our results for WD 1856+534 (24.8 pc) show this can be achieved in 25 transits. Using PandExo, we estimated the limiting WD magnitude at which an O₃ detection would also require 100 transits. We found a J  ≈  17.7 (≈62 pc) WD is roughly equivalent to TRAPPIST-1e in atmospheric characterization potential. Earth-sized planets transiting WDs therefore offer a characterization horizon ∼5× further than for M-dwarfs.

The greater characterization horizon for WD HZ planets overcomes the lower space-density of WD hosts. Our characterization horizon conservatively assumes a similar JWST observing time per transit for WD and M-dwarf planets. Higher time efficiency per transit would expand this characterization volume. Assuming a comparable occurrence rate for temperate rocky planets in the WD HZ to the M-dwarf HZ, the above factors suggest ∼6× more characterizable transiting planets may orbit in the WD HZ than the equivalent M-dwarf HZ. Gaia DR2 has identified ∼4700 WDs within 62 pc (Gaia Collaboration et al. 2018), with some cool WDs likely missing from this sample. Assuming a 10% occurrence rate, and 1% transit probability, this would result in five WD systems with transiting Earth-sized planets within JWST's characterization horizon.

The transit duration of planets in the HZ of temperate WDs is remarkably short, lasting only several minutes for WD with surface temperatures below 6000 K (Kozakis et al. 2018). However, time constrained observations with JWST (<1 hr start window) require at least a 1 hr temporal baseline to correctly characterize instrument systematics. While this additional out-of-transit baseline can be used to access about one-tenth of the full orbital lightcurve, it results in marginal improvements to the transmission spectrum precision. Present or future instruments capable of observing WD planet transits in short ≲10 minute exposures would require ∼10× less observing time per transit compared to JWST. Ground-based facilities observing in atmospheric windows would provide one such avenue, capable of observing 100 transits with ≲20 hr exposure time. Short exposure spectroscopy of planets orbiting in the WD HZ provides a powerful avenue to rapidly conduct atmospheric reconnaissance of terrestrial exoplanets.

The short transit duration of rocky planets in the HZ of cool (≲6000 K) WDs requires high-cadence observations to enable their detection. Several ground-based and space-based surveys have undertaken preliminary searches (Fulton et al. 2014; Xu et al. 2015; van Sluijs & Van Eylen 2018). TESS's new 20 s cadence mode will be instrumental to identify such planets in the near future. The large transit depths of these planets (≳50%) also render them eminently detectable by upcoming high-cadence ground-based surveys. For instance, the Vera Rubin Observatory (formerly LSST), even with relatively sparse sampling, could detect a subset of the transiting rocky planets in the WD HZ (Cortés & Kipping 2019).

Our results demonstrate that terrestrial planets transiting WDs offer an exceptional opportunity to characterize and remotely detect the presence of life on exoplanets. WD planets may be second-generation planets, forming after the stellar main sequence. Revealing their atmospheric composition will offer a treasure trove of insights into planet formation, atmospheric chemistry, the lifespans of any biota, and the possibility of a second genesis after a star's demise.

While small planets around WDs have yet to be found, the detection of the first planetary-mass object around a WD (Vanderburg et al. 2020) motivates the search for smaller planets around WDs with missions like the Transiting Exoplanet Survey Satellite (TESS). With around 4700 WDs within 62 pc (Gaia Collaboration et al. 2018), there are abundant opportunities to realize the WD opportunity. Our search for life in the universe is bound to offer surprises. Somewhere, in the vast expanse of the cosmos, life may yet flourish; illuminated by the remnant core of a long-forgotten star.