# H2-dominated Atmosphere as an Indicator of Second-generation Rocky White
Dwarf Exoplanets

Zifan Lin Department of Earth, Atmospheric, and Planetary Sciences,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA
02139, USA  Sara Seager  Department of Earth, Atmospheric, and Planetary
Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue,
Cambridge, MA 02139, USA  Department of Physics, Massachusetts Institute of
Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA  Department of
Aeronautics and Astronautics, Massachusetts Institute of Technology, 77
Massachusetts Avenue, Cambridge, MA 02139, USA  Sukrit Ranjan Northwestern
University, Center for Interdisciplinary Exploration and Research in
Astrophysics, Evanston, 60201, USA  Northwestern University, Department of
Astronomy & Astrophysics, Evanston, 60201, USA  Blue Marble Space Institute of
Science, Seattle, 98154, USA  Thea Kozakis DTU Space, National Space
Institute, Technical University of Denmark, Elektrovej 328, DK-2800 Kgs.
Lyngby, Denmark  Lisa Kaltenegger Carl Sagan Institute and Department of
Astronomy, Cornell University, Ithaca, NY 14853, USA

###### Abstract

Following the discovery of the first exoplanet candidate transiting a white
dwarf (WD), a “white dwarf opportunity” for characterizing the atmospheres of
terrestrial exoplanets around WDs is emerging. Large planet-to-star size
ratios and hence large transit depths make transiting WD exoplanets favorable
targets for transmission spectroscopy – conclusive detection of spectral
features on an Earth-like planet transiting a close-by WD can be achieved
within a medium James Webb Space Telescope (JWST) program. Despite the
apparently promising opportunity, however, the post-main sequence (MS)
evolutionary history of a first-generation WD exoplanet has never been
incorporated in atmospheric modeling. Furthermore, second-generation planets
formed in WD debris disks have never been studied from a photochemical
perspective. We demonstrate that transmission spectroscopy can identify a
second-generation rocky WD exoplanet with a thick ($\sim 1$ bar) H2-dominated
atmosphere. In addition, we can infer outgassing activities of a WD exoplanet
based on its transmission spectra and test photochemical runaway by studying
CH4 buildup.

##  1 Introduction

An exciting opportunity for characterizing the atmospheres of terrestrial
exoplanets transiting WDs is emerging. The first planet candidate transiting a
WD was discovered in the WD 1856+534 system (Vanderburg et al., 2020),
followed by a recent microlensing detection of a gas giant in a Jupiter-like
orbit around a WD (Blackman et al., 2021). kaltenegger_white_2020Kaltenegger,
MacDonald et al. (2020) explored the possibility of observing transiting
Earth-like WD planets with JWST and described a “white dwarf opportunity” of
detecting biosignature gases on such planets. Due to large planet-to-star
radius ratio, WD exoplanets have much larger transit depths compared to
planets around MS hosts, making them favorable targets for transmission
spectroscopy (e.g., Agol, 2011; Loeb & Maoz, 2013). For a hypothetical Earth-
sized planet with Earth-like atmosphere transiting WD 1856+534, JWST can
detect H2O and CO2 with just a few transits and detect biosignature gases such
as the O3 \+ CH4 pair in 25 transits (kaltenegger_white_2020Kaltenegger,
MacDonald et al. 2020). For comparison, JWST would struggle to detect the O3
\+ CH4 biosignature pair on a terrestrial planet orbiting a M dwarf such as
TRAPPIST-1e even with 100 transits (e.g., Lin et al., 2021).

There have been some pioneering works on photochemical modeling of WD
exoplanets, but those works are naturally limited in the parameter space
explored. Kozakis et al. (2018, 2020) modeled Earth-mass planets orbiting WDs
with N2-dominated atmospheres, modern Earth-like O2 and CO2 concentrations,
and modern-Earth like outgassing rates for key spectral species including CH4
and N2O. kaltenegger_white_2020Kaltenegger, MacDonald et al. (2020) analyzed
the detectability of spectral signatures by JWST for Earth-like rocky WD
exoplanets, based on the models developed by Kozakis et al. (2018, 2020).

Figure 1: Irradiation received at the top of atmosphere for planets orbiting
WDs at 1 au equivalent distance. In our models, 1 au equivalent distance means
the WD planets receive the same integrated flux as modern Earth around the
Sun. The stellar spectra are shown for (left) UV wavelengths and (right)
150-1600 nm. The solar spectrum is shown as black solid lines. The spectra for
WDs with effective temperatures of 6000 K, 5000 K, and 4000 K are shown as
blue, orange, and red solid lines, respectively. The UV spectra (left) show
that 6000 K WD has similar EUV ($<124$ nm) intensity as the Sun, while 5000 K
WD and 4000 K WD have $\sim 10^{3}$ and $>10^{10}$ times lower EUV intensity
than the Sun, respectively (Saumon et al., 2014). The overall spectra (right)
shows that WD spectra are almost perfect blackbodies while the solar spectrum
shows abundant absorption lines, which explains the difficulty of constraining
masses of WD exoplanets using the radial velocity method (Vanderburg et al.,
2020).

While Earth-like composition is an important possibility to include, the
atmosphere of an Earth-mass rocky planet may span different oxidation states,
from reducing to oxidizing. Therefore, here we attempt to expand our knowledge
of putative Earth-mass exoplanets in WD systems by exploring the vast
uncharted parameter space from an atmospheric modeling perspective. We follow
the “exoplanet benchmark cases” outlined by Hu et al. (2012) and consider
three types of atmospheric compositions, including reducing (H2-dominated),
weakly oxidizing (N2-dominated), and oxidizing (CO2-dominated) atmospheres.
Furthermore, the evolutionary history of a first-generation WD exoplanet is
distinct from a MS planet like Earth, and the formation origin of a second-
generation WD exoplanet is also different from Earth’s. Affected by its origin
and evolution, WD exoplanets can have atmospheric composition and surface
emissions distinct from Earth. Therefore, it is necessary to incorporate the
unique evolutionary history of WD exoplanets when modeling their atmospheres.

Due to the large scale height of H2-dominated atmospheres, H2 atmospheres on
Earth-mass planets are much easier for JWST to detect and characterize
compared to high mean molecular weight (MMW) atmospheres, so we place special
emphasis on H2-dominated atmospheres. In addition, for H2 atmospheres, low
ultraviolet (UV) radiation environment can facilitate biosignature gases
accumulation (e.g., Seager et al., 2013), and we discuss the photochemical
implications of H2 atmospheres around cool WDs, which have extremely low UV.

In what follows, we introduce our input WD stellar spectra and models in
Section 2. We present our main results for H2-dominated atmospheres in Section
3 and main results for N2\- and CO2-dominated atmospheres in Section 4.
Section 5 contains the discussion, and we summarize our conclusions in Section
6.

##  2 Methods

###  2.1 Stellar Model

Following Kozakis et al. (2018, 2020) we use cool WD spectral models
calculated by Saumon et al. (2014) for WD temperatures of 6000, 5000, and 4000
K to represent WD cooling throughout time. These models assume the average WD
mass of 0.6 M⊙ (Kepler et al., 2016) with pure hydrogen atmospheres and
surface gravities of $\log\,g=8.0$. Due to the high surface gravity these WD
atmospheres are highly differentiated and only display Balmer absorption lines
for temperatures $\gtrsim$ 5000 K, with hydrogen becoming neutral at lower
temperatures (as seen in Figure 1). For further discussions on spectral
modeling methods for cool WD atmospheres see e.g., Saumon et al. (2014),
Kozakis et al. (2018), and references therein.

Figure 2: Chemical mixing ratios for our model planets. We compare high
outgassing rates (solid lines) with low outgassing rates (dotted lines). H2-,
N2-, and CO2-dominated atmospheres are shown in the top, middle, and bottom
rows, respectively. The left, middle, and right columns show WDs with
effective temperatures of 6000 K, 5000 K, and 4000 K, respectively. Mixing
ratios of volcanically produced gases, such as CH4, CO2, and SO2, are
generally higher in the high outgassing models. Mixing ratios of
photochemically destroyed gases such as CH4 are generally higher in the cooler
WD models, due to less UV irradiation.

###  2.2 Photochemistry Model

We use a one-dimensional photochemistry model that was validated by simulating
the atmospheric compositions of modern Earth and Mars (Hu et al., 2012), with
CO2 and H2O cross-sections updated by Ranjan et al. (2020). The photochemistry
model was manually coupled with an analytical atmospheric temperature model
formulated by Guillot (2010) implemented as a part of the petitRADTRANS
radiative transfer package (Mollière et al., 2019). Converged atmospheric
chemical profiles are shown in Figure 2.

We assume three types of anoxic atmospheres following the exoplanet benchmark
scenarios presented by Hu et al. (2012). The benchmark scenarios include a
reducing (90% H2, 10% N2), a weakly oxidizing ($>99$% N2), and an oxidizing
(90% CO2, 10% N2) atmosphere. For each atmospheric scenario, we assume two
sets of outgassing rates for CO2, H2, SO2, CH4, and H2S, one corresponding to
modern Earth-like volcanic emission rates, and the other corresponding to a
less geologically active planet outgassing at 1000 times lower rates.

We assume Earth-like parameters – 1 $M_{\oplus}$ mass, 1 $R_{\oplus}$ radius,
and 1 bar surface pressure – for all planets modeled. For the N2-dominated
atmosphere models, we assume the planet is located at the 1 au equivalent
distance, which means that the planet receives the same integrated flux as
Earth’s irradiation on top of its atmosphere. Due to enhanced greenhouse
effect, planets with CO2\- and H2-dominated atmospheres are placed at 1.3 au
and 1.6 au equivalent distance to maintain similar surface temperature
conditions. For more details of the photochemistry model, refer to Appendix
A.1.

###  2.3 Transmission Spectra Model

We use a publicly available Python package petitRADTRANS (Mollière et al.,
2019) to calculate the transmission spectra for all our planet models, using
our photochemistry code results as inputs. We apply the low-resolution
correlated-k method to generate spectra with $\lambda/\Delta\lambda=1000$. We
generate 100 atmospheric layers with pressures distributed equidistantly in
log space, starting from the surface (1 bar) to the top of atmosphere ($\sim
10^{-7}$ bar).

Cloud layers mute spectral signatures by blocking molecules below from the
view of an observer (see e.g., Komacek et al. 2020; Suissa et al. 2020 for
recent 3D explorations). In addition, atmospheric refraction may prevent
transmission spectroscopy from accessing the lower atmosphere by bending light
rays away from a distant observer (e.g., Bétrémieux & Kaltenegger, 2014;
Robinson et al., 2017). Effects of clouds and refraction are both accounted
for in our transmission spectra model. For more details of the transmission
spectra model, refer to Appendix A.2.

Figure 3: Flowchart showing evolution of H2 atmospheres on first- and second-
generation rocky WD exoplanets. Plausible evolutionary pathways are shown as
solid black arrows, while improbable pathways are shown as dashed grey arrows.
The main takeaway from this flowchart is that retention of H2 atmospheres on
first-generation Earth-mass rocky planets requires a sequence of unlikely
coincidences, while second-generation rocky WD planets have a viable pathway
to produce and maintain a H2-dominated atmosphere.

###  2.4 Simulated JWST Observations

We simulate JWST observations of our model rocky WD exoplanets with the
NIRSpec Prism instrument using PandExo (Batalha et al., 2017). For all
simulations, we limit saturation level to 80% and do not bin the output
spectra. For stellar input parameters, we use the physical properties of WD
1856+534 reported by Vanderburg et al. (2020): J band magnitude = 15.677 and
stellar radius = 0.0131 $R_{\odot}$. For planet parameters, we use the
transmission spectra generated by petitRADTRANS as the input spectra, assuming
a 1 $R_{\oplus}$ planet radius and a 2-minute transit duration, which is the
transit duration of a planet in the HZ of WD 1856+534 (e.g.,
kaltenegger_white_2020Kaltenegger, MacDonald et al. 2020). All other
observational parameters are optimized by PandExo under the default settings.
We simulated a range of JWST campaign sizes: 1 transit, 5 transits, 10
transits, and 25 transits, which corresponds to 2, 10, 20, and 50 minutes of
total in-transit integration time, respectively.

##  3 Results

Here we demonstrate in Section 3.1 that a thick ($\sim 1$ bar) H2-dominated
atmosphere on a rocky WD exoplanet is an indicator of a second-generation
Earth-mass planet. We show that such H2 atmospheres are easily detectable by
JWST and discuss the implications of our simulated transmission spectra in
Section 3.2–3.4. Detectability of key spectral features by JWST is summarized
in Appendix C.

###  3.1 H2-dominated Atmospheres as Indicators of Second-generation Planets

Here we we qualitatively discuss the evolution of a H2-dominated atmosphere.
Quantitative results supporting the qualitatively analysis are shown in
Appendix B.

####  3.1.1 Evolution of H2-dominated Atmospheres on WD Exoplanets

We begin our evolutionary analysis by considering an Earth-mass planet formed
around the young MS progenitor (top row, Figure 3). Primary H2 envelopes of an
Earth-mass rocky planet may experience photoevaporation due to the excessive
EUV radiation produced by young stars (e.g., Owen & Wu, 2013; Lopez & Fortney,
2013). A terrestrial planet either loses its hydrogen atmosphere entirely and
remains as a barren core with a radius distribution that peaks at $\sim
1.3\,R_{\oplus}$, or retains a very thick envelope that doubles the core’s
radius, creating an “evaporation valley” in the radius distribution of known
exoplanets (e.g., Owen & Wu, 2017). A more recent work argues that $\sim
2\,M_{\oplus}$ rocky planets with H2 envelopes may have already been
discovered, and even lower mass planets with voluminous H2 atmospheres
possibly exist (Owen et al., 2020). Hence, first-generation H2-rich rocky
planets cannot be ruled out on the basis of photoevaporation.

Even if the H2 envelope on an Earth-mass exoplanet survives the MS phase, it
is likely to be lost in the post-MS evolution. The red giant phase and the hot
WD phase can both threaten H2 atmospheres. As a red giant loses its gaseous
envelope, stellar winds due to mass loss can erode atmospheres (e.g., Ramirez
& Kaltenegger, 2016; Kozakis & Kaltenegger, 2019). Outward migration is
therefore necessary for atmospheric retention. Such orbital expansion can be
triggered by red giant mass loss (e.g., Schröder & Smith, 2008).

A hydrogen-dominated atmosphere on an Earth-mass planet is very unlikely to
survive the hot WD phase, even if it fortuitously survived both the MS phase
and the red giant phase. Young WDs initially have very high effective
temperatures ($T_{\rm eff}$) and intense EUV radiation. When a solar mass star
turns into a WD, it will start with $T_{\rm eff}\gtrsim 100,000$ K and then
experience quasi-exponential radiative cooling to $\sim 30,000$ K in a
timespan of approximately 10 Myr (Fontaine et al., 2001). High $T_{\rm eff}$
results in extreme EUV intensity. An exoplanet orbiting a hot WD will be
bombarded by excessive EUV radiation up to a million times higher than modern
solar levels, resulting in rapid atmospheric mass loss even if the planet
migrates to an orbital distance of 50–100 au (Schreiber et al., 2019).
Volcanic H2 emission cannot compensate such loss (Figure 4). We therefore
conclude that the EUV radiation from hot WDs will cause total erosion of
hydrogen atmospheres. In Appendix B we quantitatively prove this conclusion.

Given that the survival of a primary hydrogen atmosphere on an Earth-mass
first-generation WD exoplanet is highly unlikely, replenishing a H2-dominated
atmosphere via outgassing remains as the only possibility. We will show in
Section 3.1.2 that replenishing hydrogen by outgassing is unlikely for a
first-generation planet.

Figure 4: Volcanic emission rates and escape rates of hydrogen in atmospheres
of $1\,M_{\oplus}$ rocky WD exoplanets. The energy-limited escape rates
(purple and black solid lines) are functions of EUV flux received by the
planet, while all other lines are independent of flux or assume a fixed flux.
The red solid line and the red dotted line represent high and low volcanic
emission rates of H2, respectively. The high emission rate is 20 times higher
than modern Earth H2 emission rate and 1000 times higher than the low emission
rate. The blue solid line, blue dashed line, and blue dotted line demarcate
the Jeans escape rates assuming exobase temperatures of 1500 K (roughly the
maximum exospheric temperature on Earth), 700 K (roughly the minimum
exospheric temperature on Earth), and 350 K (exobase temperature on Mars),
respectively. The cyan solid line is the energy-limited escape rate of a
planet orbiting a 40,000 K hot WD at a distance of $\sim 50$–$80$ au
(Schreiber et al., 2019). The black solid line is the energy-limited escape
rate calculated based on the Watson et al. (1981) and Sengupta (2016)
formulas, which are only solvable for a certain range. The purple solid line
shows a range of energy-limited escape rates assuming different
$\epsilon_{EUV}$ and $R_{EUV}$ for a sensitivity test (Luger & Barnes, 2015).
The approximate ranges of EUV radiation received by planets at 1 au equivalent
distance orbiting 4000 K, 5000 K, and 6000 K WDs are shown as red, orange, and
blue rectangles, respectively. Implications of this figure are: (i) energy-
limited escape driven by excessive EUV of hot WDs will erode H2 atmospheres
even if the planets have high H2 emission rate, and (ii) maintaining
H2-dominated atmospheres is possible for planets with high outgassing rates
orbiting all three types of WDs, and is possible for planets with low
outgassing rates orbiting cool ($\lesssim 5000$ K) WDs, assuming suitable
exobase temperatures. For numerical details, see Appendix B.

We now shift the focus to second-generation Earth-mass planets and show that
planets formed in the debris disks of WDs can possibly maintain a detectable
H2-dominated atmosphere (bottom row, Figure 3). Protoplanets can form in tight
orbits around WDs via coagulation of viscously spreading disk materials, and
then the protoplanets can further accrete disk material to form major planets
just outside the Roche limit of the host WD. If a super-Earth is tidally
destroyed, this formation mechanism can potentially recycle materials,
including volatile materials such as water, from the disrupted planet to form
an Earth-mass planet (Bear & Soker, 2015; van Lieshout et al., 2018).

Second-generation planets can form at any time during a WD’s lifetime.
Formation of second-generation WD planets is possible whenever planetary
materials are delivered into a WD’s Roche limit and be tidally disrupted (Bear
& Soker, 2015; van Lieshout et al., 2018). Such delivery can occur at any time
during a WD’s lifetime, because WD pollutants have been detected on WDs across
temperature range. The coolest polluted WD has an effective temperature of
$\sim 4000$ K (Coutu et al., 2019), evidencing that some of the oldest known
WDs are actively accrete materials, so second-generation planet formation can
occur around cool WDs.

Figure 5: Transmission spectra models and simulated JWST transit observations
for H2-dominated atmospheres, assuming a $1\,R_{\oplus}$ planet transiting WD
1856+534. We compare high outgassing rate models (red solid lines) with low
outgassing rate models (blue solid lines) and show simulated JWST data points
with their $1\,\sigma$ error bars. We assume 1 NIRSpec Prism transit for the
H2-dominated atmosphere models. In the H2-dominated atmospheres, spectral
signatures have high detectability, and differentiating between high and low
outgassing scenarios is achievable with 1 transit.

Second-generation planets formed around $>6000$ K WDs are also vulnerable to
photoevaporation because of their modern Sun-like UV radiation (Figure 1).

A hydrogen atmosphere can survive, however, if the planet forms around a cool
($\lesssim 5000$ K) WD. EUV intensity of a 5000 K WD is $\sim 100$–$1000$
times less than modern Sun, while a 4000 K WD emits $\gtrsim 10^{10}$ times
less EUV radiation than modern Sun (Figure 1). Both types of dominant neutral
atmospheric escape mechanisms – Jeans escape (e.g., Hunten, 1973) and
hydrodynamic escape (Watson et al., 1981; Luger & Barnes, 2015; Sengupta,
2016) – are powered by EUV radiation. We will show quantitatively in Appendix
B that given the low EUV intensity of $\lesssim 5000$ K WDs, high to moderate
volcanic hydrogen emission rates are sufficient to sustain a H2-dominate
atmosphere (Figure 4).

In summary, the most viable path to a roughly 1 bar H2-dominated atmosphere on
an Earth-mass WD exoplanet is provided by second-generation planets formed
around cool WDs (Figure 3). Survival of H2 atmospheres on first-generation
rocky WD exoplanets requires a sequence of improbable coincidence.

####  3.1.2 First-generation Planets Cannot Replenish H2 Atmosphere via
Outgassing

Once the H2-dominated atmosphere on a first-generation rocky planet is lost,
replenishing a secondary H2 atmosphere via outgassing is unlikely.
Replenishing a reduced hydrogen atmosphere requires reducing volcanic
emissions, neglecting other sources such as biogenic emission. Reduced
volcanic emissions require reduced mantles. The dominant form of hydrogen
outgassing on a planet with an oxidized mantle, such as present-day Earth, is
H2O (Holland, 1984). Molecular hydrogen is the dominant form of hydrogen
emission only when the planet’s mantle is highly reduced (Ramirez et al.,
2014; Ortenzi et al., 2020).

Table 1: $\sigma$ significance of ruling out a flat line based on transmission
spectra High outgassing
---
| N2 atmospheres | CO2 atmospheres | H2 atmospheres
| 4000 K | 5000 K | 6000 K | 4000 K | 5000 K | 6000 K | 4000 K | 5000 K | 6000 K
1 NIRSpec Prism transit | 2.7 | 5.2 | 6.6 | 5.1 | 4.7 | 2.3 | $>10$ | $>10$ | $>10$
5 NIRSpec Prism transits | $>10$ | 6.8 | 7.5 | 7.5 | 7.8 | 7.4 | $>10$ | $>10$ | $>10$
Low outgassing
| N2 atmospheres | CO2 atmospheres | H2 atmospheres
| 4000 K | 5000 K | 6000 K | 4000 K | 5000 K | 6000 K | 4000 K | 5000 K | 6000 K
1 NIRSpec Prism transit | 3.4 | 3.7 | 3.6 | 4.3 | 3.7 | 4.4 | $>10$ | $>10$ | $>10$
5 NIRSpec Prism transits | 6.5 | 2.8 | 6.6 | 4.2 | 7.1 | 6.6 | $>10$ | $>10$ | $>10$

However, first-generation Earth-mass WD exoplanets are generally expected to
have oxidized mantles because of mantle self-oxidation. The mantle of a $\sim
1\,M_{\oplus}$ rocky planet starts reduced, but becomes more oxidized overtime
due to gradual or stepwise self-oxidation (Scaillet & Gaillard, 2011). Such
self-oxidation is thought to be a natural consequence of the size of Earth-
mass rocky planets (Wood et al., 2006). Similarity in bulk composition between
WD exoplanets and the Earth is inferred by WD pollution (e.g., Farihi et al.,
2016; Doyle et al., 2019). Due to compositional resemblance, we expect Earth-
mass first-generation WD planets would undergo the same self-oxidation
process.

Mantle self-oxidation of Earth-mass rocky planets occurs within $\sim 100$ Myr
(Trail et al., 2011), which is much shorter compared to the age of first-
generation WD exoplanets. The age of a first-generation WD exoplanet is the
sum of MS progenitor lifetime and WD cooling time, which are both on the order
of Gyr. Therefore, first-generation rocky WD planets do not have the right
conditions to outgas hydrogen in the form of H2.

###  3.2 Transmission Spectra Can Differentiate Between First- and Second-
generation Planets

We have demonstrated that theoretically, the presence of H2-dominated
atmospheres on Earth-mass WD exoplanets indicates second-generation planets.
Observationally, JWST can potentially differentiate between first- and second-
generation Earth-mass planets using H2 atmospheres using only 1 NIRSpec Prism
transit, assuming a $1\,R_{\oplus}$ planet transiting WD 1856+534.

An atmosphere is detectable if a null assumption (flat spectrum) can be ruled
out conclusively. For each simulated NIRSpec Prism transmission observation,
we find the best-fit horizontal line representing a flat spectrum and
determine the $\sigma$ significance of ruling out this flat line fit using a
$\chi^{2}$ analysis. Results are summarized in Table 1 and model transmission
spectra overplotted with simulated observation data are shown in Figure 5. Due
to the large scale height of hydrogen atmospheres, 1 transit is sufficient to
rule out a flat spectrum for all H2-dominated atmosphere models at
$>10\,\sigma$.

High-altitude clouds or hazes layers can hide the atmosphere below, hence
decreasing the detectability of the atmosphere. Our models assume an opaque
global cloud layer at 0.47 bar ($\approx 35$ km in H2-dominated atmospheres).
Organic haze formation in atmospheres on rocky Earth-like exoplanets is
studied (e.g., Arney et al., 2016, 2017, 2018) but not included in our
atmospheric models, and will be discussed in Section 5.4.

###  3.3 Transmission Spectra Can Infer Outgassing Activities

Rocky WD exoplanets can have drastically different outgassing rates depending
on their tectonic states. An Earth-mass rocky planet can possibly have three
tectonic states: (i) active lid, where lithosphere strength is overwhelmed by
convective stresses and surface materials can be recycled into the mantle,
(ii) stagnant lid, where the lithosphere is too rigid to be deformed and
recycled into the mantle, and (iii) an episodic regime characterized by
occasional lithosphere overturn (see e.g., Lenardic et al. 2016 and references
therein).

Conceptually, planets with Earth-like active plate tectonics can recycle
atmospheric and oceanic volatiles back into the mantle to sustain continuous
outgassing over long timescales (e.g., Kasting & Catling, 2003), while
stagnant lid planets cannot. Indeed, some argued that Earth-mass stagnant lid
planets would deplete mantle volatiles rapidly and the outgassing rates would
drop to negligible levels within $\sim 1$–$2$ Gyr (e.g., Foley & Smye, 2018;
Dorn et al., 2018). Therefore, differentiating between active lid and stagnant
lid regimes on rocky WD exoplanets is ostensibly possible, if we can
differentiate between the different outgassing scenarios with transmission
spectroscopy.

However, the topic of rocky planet tectonics remains hotly debated, and the
possibility of inferring tectonic activities from outgassing rates is
challenged by several uncertainties. One major uncertainty is that rocky
planets can alternate between multiple tectonic states over Gyr timescales
(Weller et al., 2015), so we are agnostic about a planet’s tectonic history
given only a snapshot of its present-day outgassing rates. In addition, even
for the best-studied rocky planet – Earth – the onset and end of plate
tectonics are highly uncertain (see e.g., Rey et al. 2014; Weller et al. 2015
and references therein).

Due to the uncertainties of rocky planet tectonic states, we do not intend to
infer the tectonic activities or evolutionary history of rocky WD exoplanets
from their atmospheres. Instead, we focus on the capability of transmission
spectra to differentiate between high and low outgassing scenarios, and leave
the linkage between outgassing rates and tectonic states for future
investigation. Here we consider planets with a thick ($\sim 1$ bar) atmosphere
and two outgassing rates: a high outgassing rate corresponding to modern
Earth-like volcanic emission rates (following Hu et al. 2012), and a low
outgassing rate that is reduced by a factor of 1000.

Transmission spectra can differentiate between our high and low outgassing
scenarios assuming H2-dominated atmospheres. The two outgassing scenarios can
be differentiated because strong spectral absorbers such as CH4 and CO2 reach
higher equilibrium mixing ratio when surface emission fluxes are higher. Trace
gases in H2 atmospheres with low UV irradiation can easily accumulate (e.g.,
Seager et al., 2013), which further amplifies the difference between the two
scenarios.

We demonstrate the difference between our high and low outgassing scenarios
and the ability of JWST to differentiate them in Figure 5, assuming 1 NIRSpec
Prism transit. CH4 accumulates to very high levels in the cool (4000 and 5000
K) WD models for reasons we will discuss in Section 3.4. As a result, the high
and low outgassing scenarios can be differentiated at $\sim 3\,\sigma$ and
$>5\,\sigma$ in the 4000 K and 5000 K models, respectively, based on several
CH4 features in 1.0–2.5 $\micron$. For the 6000 K WD model, CO2 features at
2.8 and 4.2 $\micron$ and the CO feature at 4.7 $\micron$ are the keys
differences. Conclusive differentiation is not achievable for the 6000 K WD
model with 1 NIRSpec Prism transit due to low singal-to-noise ratio (SNR), but
is achievable with 5 transits.

Figure 6: Molecular contributions to the overall model transmission spectra
for H2-dominated atmospheres. Only the 6000 K WD high outgassing rate scenario
is shown for simplicity. The left panel shows features in the bandpass of
NIRSpec Prism (0.6-5 $\micron$) to allow comparison with Figure 5, and the
right panel shows infrared features in 5-20 $\micron$.

###  3.4 H2-Dominated Atmospheres Around WDs Can Test Photochemical Runaway

Table 2: Column-integrated CH4 Mixing Ratios in Photochemistry Models | H2 atmospheres | CO2 atmospheres | N2 atmospheres
---|---|---|---
| high outgassing | low outgassing | high outgassing | low outgassing | high outgassing | low outgassing
4000 K | 0.73 | $9.5\times 10^{-7}$ | 0.1 | $1.0\times 10^{-7}$ | $7.1\times 10^{-4}$ | $1.2\times 10^{-7}$
5000 K | $4.2\times 10^{-3}$ | $2.4\times 10^{-5}$ | $7.2\times 10^{-6}$ | $2.8\times 10^{-9}$ | $1.2\times 10^{-4}$ | $1.7\times 10^{-13}$
6000 K | $1.0\times 10^{-5}$ | $3.0\times 10^{-5}$ | $3.8\times 10^{-8}$ | $5.7\times 10^{-10}$ | $8.5\times 10^{-5}$ | $7.4\times 10^{-13}$

Gases whose major sinks are photochemical reaction powered by UV photons can
potentially undergo so-called “photochemical runaway” in H2-dominated
atmospheres around cool WDs. Photochemical runaway occurs when the emission of
a gas saturates its photochemical sink. Under such conditions, trace species,
such as biosignature gases, can accumulate to detectable levels (e.g., Segura
et al., 2005; Sousa-Silva et al., 2020; Zhan et al., 2021; Ranjan et al.,
2022). It is debated whether photochemical runaway is a physically realistic
process or an artifact of photochemistry models. For a detailed quantitative
exploration in support of the physicality of photochemical runaway, see Ranjan
et al. (2022).

Photochemical runaway can occur on exoplanets around cool WDs because the UV-
powered photochemical sinks can be easily saturated. UV photons generally play
a key role in the photochemical removal of atmospheric gases for planets and
moons in the Solar System, either by dissociating molecules directly or by
producing reactive radicals (e.g., Catling & Kasting, 2017; Ranjan et al.,
2022). As the effective temperature of a WD drops from 6000 K to 4000 K, UV
radiation plummets by a factor of $\sim 10^{10}$ (Figure 1).

Extremely low UV radiation from cool WDs implies that a physically plausible
surface emission flux can trigger photochemical runaway. Indeed, runaway
buildup is observed for CH4, CO2, and CO in our H2-dominated atmosphere models
(Figure 2). Here we use CH4 as an example to study photochemical runaway in H2
atmospheres around cool WDs, because CH4 buildup has the strongest impact on
transmission spectra. Rapid CH4 buildup as $T_{\rm eff}$ of the host WD
decreases agrees with previous study assuming Earth-like atmospheres (Kozakis
et al., 2018). Column integrated mixing ratios of CH4 are summarized in Table
2.

The nonlinear nature of photochemical runaway is observed in our models. Once
past the runaway threshold, CH4 increase is thought to increase drastically
given a small increase in surface emission (Ranjan et al., 2022). Indeed, a
1000-fold increase in outgassing rate results in a $\sim 10^{6}$ increase in
CH4 mixing ratio in our 4000 K WD H2 atmosphere model. For comparison, the
same increase only results in a $\sim 100$ times increase in mixing ratio in
the 5000 K WD model, which can be explained by higher runaway threshold
because 5000 K WD produces higher UV radiation. In the 6000 K WD model, an
increase in CH4 outgassing rate results in a slight decrease in CH4 mixing
ratio. This is likely because outgassing rates of oxidizing gases such as CO2
are increased by the same factor, and the supply of UV photons is sufficient
to catalyze the reaction of CH4 with oxidants. In other words, for $>6000$ K,
pCH4 is limited by oxidant flux, not UV flux, as pO2 is limited by reductant
flux on modern Earth. Given the modern Sun-like UV radiation of a 6000 K WD,
an increase in oxidizing gas emission has a stronger impact on the
photochemical equilibrium than an increase in CH4.

Runaway buildup of CH4 on transiting cool WD exoplanets also implies that
spectral signatures of CH4 are easily detectable. For H2-dominated models, 1
transit with NIRSpec Prism can detect CH4 conclusively (Figure 5) for a
hypothetical rocky planet orbiting WD 1856+534 in the WD HZ. For reference,
Figure 6 shows the contribution by each important absorbing species to the
overall transmission spectra. For the H2 6000 K scenario, models fit simulated
observations very well at the 1.2 and 1.4 $\micron$ CH4 features, with the
highest data points $\sim 5\,\sigma$ from the baseline. The CH4 feature at 3.4
$\micron$ is also accessible despite larger error bars. The CH4 feature at 2.3
$\micron$, however, overlaps with a H2-H2 CIA feature and hence is challenging
to detect. For the H2 5000 K scenario, the high CH4 mixing ratio ($4.2\times
10^{-3}$) combined with the extended scale height of a H2-dominated atmosphere
produce very strong CH4 feature for the high outgassing case. The strength of
the features make very high $\sigma$ significance detection possible with 1
NIRSpec Prism transit, especially in the shorter wavelengths due to high SNR.
For the H2 4000 K model, the runaway buildup of CH4 that reaches a mixing
ratio of 0.73 significantly changes the atmosphere’s mean molecular weight and
shrinks its scale height. CH4 features in the high outgassing model are weaker
than in the 5000 K scenario, but still strong enough for $>5\,\sigma$
detection. In the low outgassing model, CH4 features are considerably weaker,
but highest data points at the 1.4 $\micron$ feature are still $\sim
3\,\sigma$ from baseline.

Our examination on CH4 is only a case study – photochemical runaway can occur
for any gas with high enough production rate to saturate its photochemical
sink, which is usually powered by UV photons. The extremely low UV radiation
of cool WDs means that many gases, including CO, PH3, NH3, and isoprene, can
easily enter photochemical runaway (e.g., Ranjan et al., 2022). Among these
gases, PH3, NH3, and isoprene have been suggested as potential biosignatures
(Sousa-Silva et al., 2020; Huang et al., 2021; Zhan et al., 2021), while CH4
in combination with O2 or O3 is considered as a strong biosignature pair
(e.g., Lederberg, 1965). While we only consider volcanic emission, on
habitable WD planets, additional biogenic emission may trigger photochemical
runaway more easily. Therefore, transiting terrestrial WD exoplanets may be
the most accessible targets for biosignature detection.

##  4 Results for N2\- and CO2-dominated Atmospheres

To explore more possible oxidation states of exoplanet atmospheres, we also
run our photochemical model assuming N2\- and CO2-dominated atmospheres. The
key difference between these two high MMW atmospheric compositions and the
H2-dominated scenario is that a WD exoplanet with a high MMW atmosphere can
either be first- or second-generation. Here we discuss why this degeneracy is
unlikely to be resolved. Additional results for high MMW atmospheres,
including outgassing activities, photochemical runaway, and location and
detectability of transmission spectra features, are summarized in Appendix D.

Ambiguity in the formation origin comes from the high survivability of high
MMW atmospheres. Both Jeans escape and energy-limited hydrodynamic escape have
sensitive dependence on molecular weight. The Jeans escape flux
$\Phi_{J}\propto e^{-\lambda_{J}}$, where $\lambda_{J}$ is the Jeans escape
parameter that is linearly dependent on molecular mass (e.g., Catling &
Kasting, 2017). Therefore, an 14-fold increase of MMW from mass of H2 (2 amu)
to N2 (28 amu) would result in a $e^{-14}$ ($\sim 10^{-7}$) decrease in Jeans
escape flux. The decrease would be more drastic for CO2 (44 amu). Energy-
limited escape of hydrogen can drag heavy molecules along with it. H2 mixing
ratios in our N2\- and CO2-dominated atmosphere models do not exceed $\sim
10^{-3}$, so even in a physically unlikely scenario that all hydrogen atoms
are lost, and each H atom drags along a N or C atom, the impact of escape on
the whole atmosphere is limited. The exact escape rate of heavy molecules,
however, would depend on the mass of the heavy molecule, upward H escape flux,
thermospheric temperature, and hydrogen diffusion through a high MMW
atmosphere (Luger & Barnes, 2015). Quantitative constraints on hydrodynamic
escape rates of heavy species is therefore beyond the scope of this work.

Even if a high MMW atmosphere is lost during the post-MS evolution, a rocky WD
planet can potentially replenish the atmosphere via volcanic outgassing,
adding another layer of uncertainty to the planet’s origin. Emission of
oxidizing heavy molecules can continue at high rates for geological timescale.
CO2, for example, can be recycled from the atmosphere to sustain long-term
high CO2 emission rates on an active lid planet (e.g., Kasting & Catling,
2003). N2 is also produced from volcano arcs and mid-ocean ridges on Earth at
a rate of $\sim 10^{8}$ cm-2 s-1 (Fischer, 2008), and the geological nitrogen
cycle appears to be well balanced by the interplay between volcanic emission
and sedimentary burial (Catling & Kasting, 2017). We therefore conclude that
high MMW atmospheres dominated by N2 and CO2 can potentially be replenished
after erosion.

Even though N2\- and CO2-dominated atmospheres cannot uniquely constrain the
evolutionary history of a rocky WD exoplanet, these high MMW atmospheres can
be conclusively detected by JWST with a small number of transits. For most
scenarios, even 1 transit with NIRSpec Prism can rule out a flat spectrum and
hence imply the presence of an atmosphere with high significance (Table 1).
Assuming high outgassing rates, flat spectra can be ruled out conclusively for
N2 5000 K WD, N2 6000 K WD, CO2 4000 K WD, and CO2 5000 K WD scenarios at
$\gtrsim 5\,\sigma$. For the low outgassing models, a barren rock planet can
be ruled out at $>3\,\sigma$ for all cases, despite weaker spectral features
due to reduced emissions. We note two exceptions – the high outgassing N2 4000
K WD and CO2 6000 K WD models have low significance of 2.7 and $2.3\,\sigma$,
respectively. This deviation from the general trend can be explained by
stochastic noises added to each PandExo simulation. Our simulated JWST
observations represent a set of random samples, rather than a set of typical
samples. Fluctuations in calculated significance are therefore normal.

With 5 NIRSpec Prism transits, conclusive ($>5\,\sigma$) detection is
achievable for almost all scenarios for a hypothetical rocky planet orbiting
WD 1856+534 in the WD HZ. There are two exceptions, N2 5000 K WD and CO2 4000
K WD scenarios, where the significance of ruling out a flat spectrum with 5
transits is lower than the significance with 1 transit. Because other models
with similar transit depths have very high significance of ruling out a flat
spectrum ranging from 6.5 to $7.8\,\sigma$, we conclude that these two cases
represent the pessimistic end of stochastic distribution of PandExo noises.
For a typical observation, JWST would be able to conclusively detect a N2 or
CO2 atmosphere with 5 or less transits.

##  5 Discussion

###  5.1 H2 Escape Mechanisms

Atmospheric loss driven by red giant stellar winds has been quantitatively
studied for $\sim 0.5\,M_{\oplus}$ exomoons, Earth-mass planets, and
$>5\,M_{\oplus}$ super-Earths (Ramirez & Kaltenegger, 2016). The authors
considered Earth-like high MMW atmospheres and concluded that $\sim 100\%$
atmospheric loss will occur for planets receiving Mars-like stellar radiation
orbiting F5 or later type stars, when the host stars evolve into red giants.
For K5 or later type MS progenitors, even a Saturn equivalent separation
cannot prevent total atmospheric loss. Due to the complex nature of turbulent
mixing, efficiency of atmospheric loss due to interactions with stellar winds
is not well constrained, so we leave quantitative study of H2 atmosphere
survivability around red giants for future investigation.

Note that present-day WDs cannot have originated from M dwarfs, because MS
lifespans of M dwarfs are considerably longer than the age of the universe.
Therefore, escape mechanisms specific to M dwarfs such as ion escape (e.g.,
Airapetian et al., 2017; Dong et al., 2017) do not apply to WD exoplanets.

###  5.2 H2 Production and Retention Mechanisms

Because rates of volcanic H2 emissions have been quantitatively discussed in
Section 3.1, here we focus on outgassing during accretion (e.g., Elkins‐Tanton
& Seager, 2008). A modeling study on various types of common meteorites have
shown that a substantial H2-dominated atmosphere can be outgassed via the
reaction between water and metallic iron, if sufficient water is added to
reduced meteoritic materials (Elkins‐Tanton & Seager, 2008). Further studies
on mixtures of meteoritic materials showed that H2 would be the dominant form
of hydrogen outgassing when oxygen fugacity (a measurement for rock oxidation
state) of the mixture is roughly lower than or equal to the oxidation state of
O chondrites (e.g., Schaefer & Fegley, 2010).

Compositional study of WD pollutants, which represent the building blocks for
second-generation WD exoplanets, implies that second-generation rocky WD
planets are capable of outgassing substantial H2. WD pollutants are similar to
solar system meteorites in bulk elemental composition and oxygen fugacity
(Doyle et al., 2019). Measured oxygen fugacity of planetary materials accreted
by six polluted WDs are consistent with C chondrites, O chondrites, bulk
Earth, and bulk Mars. This compositional similarity implies that at least some
second-generation planets formed from WD debris disks have Earth- and Mars-
like geophysical and geochemical properties (Doyle et al., 2019). Both Earth
and Mars are thought to have had an early H2-rich phase (e.g., Ramirez et al.,
2014), so it is not surprising if a geochemically similar second-generation
rocky WD planet is born with a hydrogen envelope.

Addition of water to a forming second-generation rocky WD planet will
facilitate H2 emission (Elkins‐Tanton & Seager, 2008), and water is indeed
available in large quantities in WD debris disks. Observations of polluted WDs
have shown that accreted materials can consist of 20% or more water by mass
(e.g., Farihi et al., 2016), in agreement with model predictions that water in
minor planets or exomoons can survive post-MS evolution in large quantities
(e.g., Malamud & Perets, 2017). Accretion of such water-rich minor planets or
comets onto a second-generation WD planet may yield a water reservoir
equivalent to $10^{-5}$ to $10^{-2}$ times the mass of Earth’s ocean (Veras et
al., 2014; van Lieshout et al., 2018). The initial water content of a second-
generation WD planet may be enhanced if it accreted a tidally disrupted water-
rich super-Earth. How much water can a second-generation planet recycle from
the water-rich debris disk, however, remains an open question.

Kite et al. (2019, 2020) proposed that the abundance of $2$–$3\,R_{\oplus}$
sub-Neptunes can be explained by H2 dissolving into long-lived magma ocean
under excessive pressure. When H2 atmospheres suffer from atmospheric loss on
such planets, hydrogen exsolution from the magma ocean can increase the
survivability of H2 envelopes. Therefore, our results cannot be directly
adopted to indicate the second-generation origin of H2-rich sub-Neptunes.
Nevertheless, planets considered by Kite et al. (2019, 2020) are in a
completely different physical regime ($\gtrsim 4\,M_{\oplus}$ planets with
$\gtrsim 400$ K equilibrium temperature) compared to the temperate
$1\,M_{\oplus}$ rocky planets we consider, so magma-atmosphere interaction
does not affect our results.

###  5.3 Origin of Close-in Orbits of WD Exoplanets

We consider planets that receive similar irradiation as the Earth. Due to the
low luminosity of WDs, these planets need to be in or near the WD HZ ($\sim
0.005$–$0.01$ au) to receive sufficient heating (Agol, 2011). The origin of
such close-in orbits needs to be accounted for.

Second-generation WD planets naturally have close-in orbits because they form
at the 2:1 mean motion resonance with the WD Roche limit. Coincidentally, the
2:1 resonance fits in the HZ of $T_{\rm eff}\lesssim 6000$ K WDs (van Lieshout
et al., 2018).

First-generation planets, however, must be delivered into a close-in orbit by
some migration mechanism, because any first-generation planets born close to
the star would be engulfed in the red giant phase. There are two mechanisms
that can deliver a remote planet to close-in orbits around WDs, namely binary
star system interactions and multiplanet scattering. In a wide binary stellar
system, the distant stellar companion can perturb the orbit of a major planet
orbiting the evolving post-MS star and potentially deliver it into the Roche
limit of its host. Under this mechanism, Neptune-like planets or Kuiper Belt
analog objects would spend the first $\sim 200$ Myr of the WD lifetime of its
host on a $\sim 100$ au orbit. Subsequently, the planet can be delivered to a
close-in orbit with $\sim 10^{-2}$ au periapsis by relatively rapid inward
migration that takes $\sim 5$ Myr (Stephan et al., 2017). Alternatively, in a
closely packed multiplanet system, due to the chaotic nature of scattering, a
planet can remain on a $\sim 10$ au orbit for $>10$ Gyr and then be scattered
onto a highly eccentric orbit with periapsis distance only a few percent of an
au (Veras & Gänsicke, 2015). Tidal interactions with the WD can then
circularize its orbit in under $10^{3}$ years. Efficient tidal interaction
also means that planets in WD HZ are tidally locked (e.g., Agol, 2011).

###  5.4 Clouds and Hazes in WD Exoplanet Atmosphere

Pervasive high cloud decks pose a major challenge to the detection of spectral
features, because molecules below cloud decks are not accessible. Recent
general circulation model (GCM) simulations concluded that transit
observations of tidally locked M dwarf terrestrial exoplanets, especially
water-rich planets, would be strongly affected by clouds (Komacek et al.,
2020; Suissa et al., 2020). In the moist greenhouse regime, however, highly
saturated stratosphere improves the detectability of H2O despite thick
tropospheric clouds (Chen et al., 2019). Clouds also affect our photochemistry
models indirectly by controlling the atmospheric energy budget and by
influencing the amount of UV reaching the lower atmosphere. Planets in WD HZ
are rapid rotators with $\approx 4$–32 hr period (Agol, 2011), in contrast to
the slowly rotating ($\gtrsim 12$ days) M dwarf planets on which cloud effects
are most pronounced (Komacek et al., 2020).

Our photochemistry models imply that strong haze formation may be prevalent in
the atmospheres of WD exoplanets. In an Archean Earth-like N2-dominated anoxic
atmosphere, a CH4/CO2 ratio of above 0.1 would lead to strong hydrocarbon haze
production driven by CH4 photolysis (e.g., Arney et al., 2016). Indeed, our
photochemistry results show that in N2-dominated models with high outgassing
rates, CH4/CO2 equals 0.72, 1.0, and 5.9 for 6000 K, 5000 K, and 4000 K WDs,
respectively. In N2-dominated models with low outgassing rates, CH4/CO2 ratios
are $6.2\times 10^{-6}$, $1.4\times 10^{-6}$, and 1.0 for 6000 K, 5000 K, and
4000 K WDs, respectively. Given the high CH4/CO2 ratios for all high
outgassing scenarios and the 4000 K low outgassing scenario, extensive organic
haze production in the atmospheres of WD exoplanets may be common.

Even though the production of organic hazes in CO2\- and H2-dominated
atmospheres is not well studied, CH4/CO2 ratios in our photochemistry models
for these types of atmospheres often exceed 0.1. In a CO2-dominated atmosphere
with high outgassing rates, CH4/CO2 equals $4.3\times 10^{-8}$, $8.1\times
10^{-6}$, and 0.13 for 6000 K, 5000 K, and 4000 K WDs, respectively. In a
H2-dominated atmosphere with high outgassing rates, CH4/CO2 equals 0.13, 41,
and $2.3\times 10^{4}$ for 6000 K, 5000 K, and 4000 K WDs, respectively. If
haze production is also positively correlated with CH4/CO2 ratio in highly
reducing or highly oxidizing atmospheres, the high CH4/CO2 ratios for
H2-dominated models and for the 4000 K WD CO2-dominated model imply that
organic hazes on WD exoplanets are common regardless of atmospheric oxidation
state.

Scattering from hazes can complicate detectability of spectral features, but
the presence of hazes also provides a new approach to characterize the
atmospheres of WD exoplanets and can even indicate biological activity. At
visible wavelengths, scattering from hazes produce a slope at $\sim
0.5\,\micron$ and shorter wavelengths (e.g., Marley et al., 2013), which has a
similar shape compared to the Rayleigh scattering slope. In an atmosphere with
Earth-like concentration of oxygenic species, such haze slopes may also
obscure the O3 Chappuis band at 0.6 $\micron$, complicating the detection of
the CH4 \+ O3 biosignature pair (e.g., Lin et al., 2021). In infrared
wavelengths, organic haze has a feature at approximately 6 $\micron$, just
outside of the bandpass of NIRSpec Prism (Arney et al., 2016, 2017). If
organic haze features are detected on a WD exoplanet with low CH4/CO2 ratio,
it is possible that the planet has large emissions of biogenic sulfur gases
(Arney et al., 2018).

###  5.5 Habitability of WD Exoplanets

Here we discuss how the changing luminosity, UV radiation, volatile reservoir,
and orbital dynamics would affect the habitability of WD exoplanets.

The slow cooling process of WDs provides stable environments for planets
around them for billions of years. Agol (2011) estimated that a planet can
stay in the continuously HZ of WDs for $>3$ Gyr. Kozakis et al. (2018) showed
that the maximum time a planet spends within the HZ of a 0.6 M⊙ WD is $\sim 6$
Gyr, assuming a conservative HZ defined by the runaway greenhouse and maximum
greenhouse effects of H2O and CO2 (e.g., Kasting et al., 1993). Furthermore,
cool single WDs are photometrically stable (Fontaine & Brassard, 2008), which
rules out high energy flares that may cause atmospheric erosion. Flares, which
impact habitability of M dwarf planets, should not affect WD planets we
consider.

WDs remain cool and quiescent during most of their lifetime, providing a
stable low UV environment for any planet orbiting it. High energy UV photons
can potentially erode atmospheres and place the water reservoir at risk, due
to the photodissociation of water and subsequent hydrogen escape (e.g.,
Airapetian et al., 2017; Dong et al., 2017). Some studies have suggested that
high UV flux may be necessary for triggering complicated prebiotic chemistry
reactions, which are essential for the emergence of life (e.g., Fossati et
al., 2012; Ranjan et al., 2018). WDs with $T_{\rm eff}=6000$ K have UV
radiation comparable to the modern Sun (Figure 1), implying that younger,
hotter WDs can output UV photons on the same levels as the young Sun, which is
presumed to trigger abiogenesis on Earth.

A rocky WD exoplanet can have an abundant volatile reservoir, which is another
requirement for habitability. First-generation rocky WD planets may require
either some migration mechanism that delivers the planet from an outer orbit
into WD HZ (e.g., Veras & Gänsicke, 2015), or a large initial water fraction.
In the latter case, steam-dominated atmospheres on water-rich “super-Venuses”
can survive for geologic timescale even when exposed to intense EUV fluxes
(Harman et al., 2021). Analogous to planets with high MMW atmospheres
considered here, transiting WD planets with steam atmospheres can be readily
characterized by JWST, offering an opportunity to study the volatile evolution
on terrestrial planets. For a second-generation WD planet, whether it has
sufficient volatiles depends on whether the WD debris disk from which the
planet formed is volatile-rich. As discussed in Section 5.2, volatiles such as
H2O are indeed abundant in WD debris disks, according to WD pollution
observations.

Short tidal circularization and tidal locking timescales imply that planets in
the WD HZ are expected to be tidally locked, with orbital periods of $\approx
4$–32 hr (Agol, 2011). Rapid rotation allows for efficient heat
redistribution, preventing nightside atmospheric collapse and hence increasing
habitability. Rapid rotation also leads to the formation of narrow global
cloud bands, in contrast to thick substellar cloud decks on slowly rotating
planets, which have secondary effects on habitability that are explored by
GCMs (e.g., Yang et al., 2014).

###  5.6 Future Opportunities

To date, no Earth-mass transiting WD planet has been discovered. Earth-sized
rocky exoplanets transiting a cool WD in the HZ have transit durations of only
$\sim 2$ minutes, so detection of such planets require high-cadence
observations. Some have searched for such planets using both ground- and
space-based facilities (e.g., van Sluijs & Van Eylen, 2018). In addition, the
feasibility of discovering WD exoplanets has been studied for astrometric
detection by Gaia (Perryman et al., 2014) and for large-scale survey detection
by the Large Synoptic Survey Telescope (LSST, Cortés & Kipping 2019). The new
20 s cadence mode available during TESS extended mission will provide another
avenue of detecting transiting WD exoplanets with short transit durations.

The search for Earth-mass transiting WD exoplanets may be a fruitful one.
About 5 Earth-mass transiting WD exoplanets are expected to be detected within
the characterization horizon of JWST, assuming Earth-like atmospheres
(kaltenegger_white_2020Kaltenegger, MacDonald et al. 2020). For H2
atmospheres, the characterization horizon may be extended. The half-sky survey
by LSST may produce a more optimistic number of $\sim 100$ transiting WD
exoplanets (Cortés & Kipping, 2019), although some of those targets may be too
dim for JWST to characterize.

##  6 Conclusion

In this work, we present a photochemical modeling exploration of Earth-mass
exoplanets transiting WDs under the context of WD system evolution. We present
1D photochemistry models coupled with an analytical climate model, simulated
transmission spectra, and JWST observation models for three types of anoxic
atmospheres with different oxidation states. We show that detection of a
H2-dominated thick ($\sim 1$ bar) atmosphere indicates a second-generation WD
rocky planet, while the detection of a high MMW (N2\- or CO2-dominated)
atmosphere is degenerate. Detecting H2O, H2, CO2, CO, and CH4 features in H2
atmospheres with JWST requires only 1 transit with NIRSpec Prism. For N2\- and
CO2-dominated atmospheres, $\sim 25$ NIRSpec Prism transits are required to
conclusively detect the above molecules. Buildup of CH4 via photochemical
runaway is observed in most models, an effect that can be used to
differentiate between high and low outgassing scenarios with 1 JWST transit
for H2 atmosphere models and with 25 JWST transits for most N2 and CO2
atmosphere models. For more details on molecule detectability in H2-dominated
atmospheres, see Appendix C. For additional results and figures for N2\- and
CO2-dominated atmospheres, see Appendix D.

Earth-mass transiting WD exoplanets are among the most favorable targets for
atmospheric characterization of terrestrial planets via transmission
spectroscopy. Besides an opportunity to search for biosignature gases on
inhabited WD exoplanets, here we show that there is also a “white dwarf
opportunity” for constraining the evolutionary history of abiotic and
prebiotic rocky WD exoplanets. Rocky exoplanets transiting WDs are yet to be
found. We intend for the results here to motivate the search for these unique
worlds circling dead stars and follow-up atmospheric reconnaissance by JWST.

We thank the anonymous reviewer for constructive comments that improved the
clarity of this paper. We thank Sujan Sengupta for co-developing a hydrogen
escape code that contributed to this study. Z.L. acknowledges support from the
MIT Presidential Fellowship. S. R. thanks Northwestern University for support
via the CIERA Postdoctoral Fellowship.

##  Appendix A Model Details

###  A.1 Photochemistry Model

Our photochemistry model includes $>800$ chemical reactions, photochemical
processes, and emission and thermal escape mechanisms. The model also solves
chemical-transport equations for 111 O, H, C, N, and S species, as well as S8
and H2SO4 aerosols, linked by 645 bimolecular reactions, 85 ter-molecular
reactions, and 93 thermal dissociation reactions (see Hu et al. 2012 for a
full list of species and reactions). The reaction network has recently been
updated to include nitrogenous chemistry, and the rate laws of several
reactions have been updated as well (see Ranjan et al. 2022 for details). Note
that because photochemistry of higher hydrocarbons and organic haze production
involve many uncertainties, Hu et al. (2012) excluded reactions involving
molecules containing more than two carbon atoms in an ad hoc fashion by
assuming a high ($10^{-5}$ cm s-1) deposition velocity for C2H6. Implications
of hazy atmospheres are discussed in Section 5.4.

For each atmospheric oxidation state, we use a subset of the full reaction
network that is relevant. We assume the planets are covered with a substantial
surface liquid water ocean and water vapor is transported upwards at a
constant flux of $10^{-2}$ cm-2 s-1 due to evaporation. We consider a zero
rainout rate to simulate an ocean that is saturated with H2, CO, CH4, C2H6,
and O2 on an abiotic planet (following Hu et al., 2012). Dry deposition
velocities assumed in our model follow the exoplanet benchmark cases
parameters (Table 5, Hu et al. 2012). The photochemical model is considered to
be converged when the variation timescale of each species at each altitude
exceeds the diffusion timescale of the entire atmosphere. Key model parameters
are summarized in Table 3.

Our high and low outgassing rates are differ by a factor of 1000. This factor
is not arbitrary. Volcanic production rate of modern Earth is $\sim$ 100–1000
times higher than Venus (Gaillard & Scaillet, 2014; Gillmann & Tackley, 2014).
To explore a wider parameter space, we choose a factor of 1000 instead of 100.

The required inputs of the temperature model include the planet’s equilibrium
temperature, interior temperature, mean thermal opacity of the atmosphere, and
mean optical opacity. Equilibrium temperatures of our models are summarized in
Table 3. For all models, we assume an Earth-like interior temperature of 35.7
K, which is calculated based on an estimated Earth’s total surface heat flow
of $47\pm 2\times 10^{12}$ W (Davies & Davies, 2010). Mean thermal and optical
opacities were calculated based on mixing ratios of key absorbing species,
such as CH4, CO2, H2O, and H2, produced by the photochemistry model, in
combination with cross section data from the Exoclimes Simulation Platform
(Grimm & Heng, 2015) and the MPI-Mainz Spectral Atlas (Keller-Rudek et al.,
2013). Opacities calculated from photochemistry model outputs are inputted to
the temperature model as initial conditions, while temperature-pressure
profile generated by the temperature model is in turn inputted to the
photochemistry model. The two models were run iteratively until surface
temperature predicted by both models differ by less than 1 K. We run several
extra iterations after convergence to ensure stable solution. Surface
temperatures for all three 6000 K WD models are fixed at 288 K, and the
thermal and optical opacities of the 6000 K WD models were used as benchmark
for the 5000 K and 4000 K WD models. The cooler WD models have higher
concentration of gases with high thermal opacities due to lower UV radiation
from host stars and hence lower photodissociation rate. This would increase
the thermal opacities of cooler WD models relative to the 6000 K WD model,
causing temperatures to increase. The final converged surface temperatures for
all the models are summarized in Table 3.

Table 3: Model Parameters for the Three Anoxic Atmospheric Compositions Parameters | Reducing | Weakly oxidizing | Highly oxidizing
---|---|---|---
(90% H2, 10% N2) | ($>99$% N2) | (90% CO2, 10% N2)
Outgassing scenario | high | low | high | low | high | low
Equivalent semi-major axis (au) | 1.6 | 1.6 | 1.0 | 1.0 | 1.3 | 1.3
Mass ($M_{\oplus}$) | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
Radius ($R_{\oplus}$) | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
Surface pressure (bar) | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
Surface temperature (K) | 4000 K WD | 347 | 288 | 299 | 284 | 290 | 288
5000 K WD | 290 | 288 | 289 | 284 | 288 | 288
6000 K WD | 288 | 288 | 288 | 284 | 288 | 288
MMW (amu) | 4000 K WD | 13 | 4.6 | 28 | 28 | 39 | 42
5000 K WD | 4.7 | 4.6 | 28 | 28 | 42 | 42
6000 K WD | 4.6 | 4.6 | 28 | 28 | 42 | 42
Gas emission (cm-2 s-1) | CO2 | $3\times 10^{11}$ | $3\times 10^{8}$ | $3\times 10^{11}$ | $3\times 10^{8}$ | N/A | N/A
H2 | N/A | N/A | $3\times 10^{10}$ | $3\times 10^{7}$ | $3\times 10^{10}$ | $3\times 10^{7}$
SO2 | $3\times 10^{9}$ | $3\times 10^{6}$ | $3\times 10^{9}$ | $3\times 10^{6}$ | $3\times 10^{9}$ | $3\times 10^{6}$
CH4 | $3\times 10^{8}$ | $3\times 10^{5}$ | $3\times 10^{8}$ | $3\times 10^{5}$ | $3\times 10^{8}$ | $3\times 10^{5}$
H2S | $3\times 10^{8}$ | $3\times 10^{5}$ | $3\times 10^{8}$ | $3\times 10^{5}$ | $3\times 10^{8}$ | $3\times 10^{5}$
Water and rainout | f(H2O) surface emission | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01
rainout rate | 0 | 0 | 0 | 0 | 0 | 0

###  A.2 Transmission Spectra Model

Our transmission spectra model includes the most spectroscopically relevant
species: C2, C2H2, C2H4, CH, CH3, CH4, CN, CO, CO2, CS, H2, H2O, H2O2, H2S,
HCN, HNO3, NH, NH3, O3, OH, SO2, and SO3. We include the following Rayleigh
scattering species: H2O, CO, CH4, CO2, H2, O2, and N2. Collision-induced
absorption (CIA) plays an important role in transmission spectroscopy,
especially when one or both species in a CIA pair exist in high concentration
in the modeled atmosphere. We therefore include the following CIA pairs:
N2-N2, H2-H2, O2-O2, N2-O2, CO2-CO2, and H2-He.

Detailed vertical and horizontal distribution of realistic 3D clouds on
exoplanets is highly uncertain. For generality, we assume a single homogeneous
cloud layer located at 6 km in the N2-dominated atmosphere (following Lin et
al., 2021). For consistency across atmospheres with different scale heights,
we place the cloud layer at 0.47 bar in the CO2\- and H2-dominated models,
which corresponds to 6 km in the N2-dominated atmospheres. We account for the
most pessimistic scenario by assuming the cloud layer is fully opaque and
covers the entire day-night terminator.

Effect of atmospheric refraction is dependent on $R_{*}/a$, the inverse of the
scaled semimajor axis, and is strongest when this value is small (Bétrémieux &
Kaltenegger, 2014; Robinson et al., 2017). Applying Equation (14) from
Robinson et al. (2017), we find that the maximum pressures accessible to
transmission spectroscopy are $\approx 0.7$, $\approx 4$, and $\approx 0.3$
bar, for N2, H2, and CO2 atmospheres, respectively, with uncertainties coming
from the exact choice of orbital distance, MMW, and atmospheric refractivity.
The refractive cutoffs in N2\- and H2-dominated atmospheres are below the
cloud layer at 0.47 bar and therefore do not affect the transmission spectra.
The refractive cutoff in CO2-dominated atmosphere, however, is above the cloud
deck, so we introduce a completely opaque layer at 0.3 bar.

##  Appendix B Quantitative Analysis of H2-dominated Atmosphere as Indicator
of Second-generation Planet

###  B.1 Survivability of Hydrogen Atmospheres on First-generation WD
Exoplanets

In this section, we quantitatively discuss hydrogen escape on a first-
generation rocky planet during the hot WD phase. We show that even if the
planet migrates to a 50–100 au orbit, excessive EUV radiation from the hot WD
can lead to complete erosion of a $\sim 1$ bar hydrogen atmosphere.

Two types of neutral atmospheric escapes are relevant for rocky planets,
namely Jeans escape and hydrodynamic escape. Hydrodynamic escape dominates at
high incident fluxes and can produce extremely high mass loss rates that can
account for the evaporation of entire atmospheres (Owen & Wu, 2013). We
therefore focus on the hydrodynamic escape mechanism and calculate the
“energy-limited” mass loss rate given the EUV irradiation from a hot WD.

Intense EUV radiation from hot WDs results in extreme mass loss rates. EUV
flux from a hot ($\gtrsim 30,000$ K) WD exceeds EUV flux of the young Sun by a
factor of $10^{2}$–$10^{3}$, while the young Sun is thought to be $10^{3}$
times more active than modern Sun (Schreiber et al., 2019). Even at a distance
of $\sim 50$–$100$ au, EUV flux from a 40,000–80,000 K WD can be as strong as
$\sim 0.1$–$10$ W m-2. In this EUV flux range, atmospheric escape is dominated
by the so-called “energy-limited” escape (e.g., Luger & Barnes, 2015;
Sengupta, 2016). Here we adopt equation (2) in Luger & Barnes (2015) to
calculate the mass loss rate on rocky exoplanets orbiting hot WDs:

$\dot{M}_{\rm EL}=\frac{\epsilon_{\rm EUV}\pi\mathcal{F}_{\rm EUV}R_{\rm
p}R_{\rm EUV}^{2}}{GM_{\rm p}K_{\rm tide}}$ (B1)

where $\mathcal{F}_{\rm EUV}$ is the EUV flux, $R_{\rm p}$ is the planet
radius, $R_{\rm EUV}$ is the radius at which the bulk of EUV energy is
deposited, $\epsilon_{\rm EUV}$ is the EUV absorption efficiency, and $K_{\rm
tide}$ is the tidal correction term. We make several simplifications here:
$R_{\rm EUV}$ is assumed to be equal to $R_{p}$ because scale height
$H<<R_{p}$ on a rocky planet, $\epsilon_{\rm EUV}$ is assumed to be 0.3, and
$K_{\rm tide}$ is assumed to be 1 (following Luger & Barnes, 2015). We assume
$1\,R_{\oplus}$ radius and $1\,M_{\oplus}$ mass. We consider a scenario that
the planet migrates to a $\sim 50$–$100$ au orbit around a $\sim 40,000$ K WD,
which is optimistic for hydrogen survival. EUV flux received by the planet in
this case is assumed to be 0.1 W m-2 (following Schreiber et al., 2019).
Hydrogen escape flux in this scenario is $4.1\times 10^{7}$ kg s-1, which is
equivalent to $2.4\times 10^{31}$ H atoms per s. In addition, we perform a
sensitivity test assuming different values of $\epsilon_{\rm EUV}$ and $R_{\rm
EUV}$. Luger & Barnes (2015) considered $0.15\leq\epsilon_{EUV}\leq 0.3$ as
typical for H2-rich atmospheres. Our model H2 atmospheres have maximum
altitudes of $\sim 1200$ km, and we choose half of the maximum altitude (600
km) as the maximum $R_{\rm EUV}$. Pressures at 600 km in our models are $\sim
10^{6}$ times lower than Earth’s exobase pressure. The sensitivity test
results in a factor of $\approx 2.4$ change in the energy-limited escape rate
(Figure 4).

A first-generation WD planet can maintain its H2-dominated atmosphere if
sources of hydrogen overwhelm sinks. We make a first-order assumption that the
sole sink for hydrogen is escape at the top of atmosphere and the sole source
is volcanic emission. We consider an optimistic case that H2 outgassing is 20
times higher than modern Earth volcanic emission. The volcanic H2 flux on
modern Earth is estimated to be $1.5\times 10^{9}$ cm-2 s-1 (James & Hu,
2018). On young planets with more reduced mantles or planets with additional
internal heating due to tidal dissipation, $\sim 20$ times higher production
is plausible. We therefore assume $3.0\times 10^{10}$ cm-2 s-1 flux as a “high
outgassing” scenario (following James & Hu, 2018). We also consider a
pessimistic “low outgassing” scenario where H2 outgassing is 1000 times lower
than the high outgassing scenario. The high and low outgassing rates
translates to global H atom production rates of $3.1\times 10^{29}$ s-1 and
$3.1\times 10^{26}$ s-1 (Figure 4).

An immediate observation from Figure 4 is that even in the optimistic high
outgassing scenario, hydrogen production is significantly less than hot WD
EUV-driven hydrogen loss by a factor of $\sim 100$. Therefore, net H2 loss
will occur even in a the most optimistic scenario for hydrogen retention,
where the planet migrates to a separation of $\sim 50$–$100$ au, orbits a
relatively cool ($\sim 40,000$ K) young WD, and emits 20 times more H2 than
modern Earth. The mass loss rate in this scenario is $\sim 4\times 10^{4}$ kg
s-1, or $\sim 10^{12}$ kg yr-1. Total mass of Earth’s atmosphere is on the
order of $\sim 10^{18}$ kg, so a H2 atmosphere on an Earth-mass first-
generation WD exoplanet will be evaporated entirely in a timescale of $\sim 1$
Myr. This timescale is shorter than the $\sim 10$ Myr cooling time for the
effective temperature of a 100,000 K WD to drop to 30,000 K (Fontaine et al.,
2001). Cooling below 30,000 K is much slower. It takes $\sim 2$ Gyr to cool a
WD to 6000 K (Bergeron et al., 2001; Fontaine et al., 2001), and at this
temperature the WD still emits EUV radiation comparable to modern Sun (Figure
1). The extended exposure to high levels of EUV photon flux guarantees total
evaporation of hydrogen envelopes on first-generation rocky WD planets.

###  B.2 Escape Mechanisms on Rocky WD Exoplanet

Here we quantitatively study the escape mechanisms on a rocky planet orbiting
cool ($T_{\rm eff}\lesssim 5000$ K) WDs. We conclude that maintaining a
H2-dominated atmosphere around a cool WD is possible for planets with high
outgassing rates and Earth-like exobase temperature ($T_{\rm exo}$), or
planets with low outgassing rates and Venus- or Mars-like $T_{\rm exo}$. We
break down our quantitative analysis in the order of Jeans escape, energy-
limited escape, and diffusion. H2 emission and escape fluxes are visualized in
Figure 4.

On a planet with low EUV irradiation, Jeans escape is the dominant escape
mechanism. Jeans escape is controlled by the Jeans escape parameter, which is
inversely proportional to $T_{\rm exo}$ (Hunten, 1973; Tian et al., 2008). At
low exobase temperatures ($T_{\rm exo}\lesssim 1000$ K), Jeans escape flux is
a very sensitive function of exobase temperature, where a $\sim$ 300 K
decrease in $T_{\rm exo}$ can lead to two orders of magnitude decrease in
Jeans escape flux.

It should be reasonable to assume that $T_{\rm exo}$ on cool WD exoplanets
with anoxic atmospheres are lower than Earth’s $T_{\rm exo}$, despite the lack
of model constraints. Earth’s exobase is heated by several mechanisms,
including UV-induced photoionization and photodissociation of CO2, N2, O, O2
and O3, where the most efficient heat source among all mechanisms is
photoionization and photodissociation of O2 (see e.g., Kulikov et al. 2007 and
references therein). In anoxic atmospheres, such as CO2-dominated Venusian and
Martian atmospheres, $T_{\rm exo}$ can be as low as 275 K and 350 K,
respectively (de Pater & Lissauer, 2001). For comparison, Earth’s exobase is
heated to a temperature of $\sim 1000$ K (e.g., Tian et al., 2008). Because
exospheric heating is powered by UV, $T_{\rm exo}$ varies with stellar
activity. Earth’s exobase temperature ranges from $\approx 700$ K to $\approx
1500$ K depending on solar activity, with some uncertainties from the assumed
eddy diffusion coefficient (Roble et al., 1987). To explore a wide parameter
space, we consider Jeans escape for three $T_{\rm exo}$: 1500 K, 700 K, and
350 K. Note that all three types of atmospheres we consider are anoxic, and
the EUV levels of cool WDs are significantly lower than modern Sun, so even
the 700 K “solar min” exobase temperature is likely to be an overestimate for
cool WDs. But we stick with these temperature choices to account for $\gtrsim
6000$ K WDs with roughly solar level UV radiation.

We calculate Jeans escape rates on WD exoplanets based on formalism derived by
Hunten (1973) and Watson et al. (1981). The Jeans escape rates are shown in
Figure 4 alongside with H2 emission rates for comparison. The high outgassing
rate is $\sim 3$ times higher than the “solar max” Jeans escape rate and $\sim
100$ times higher than the “solar min” Jeans escape rate. The low H2
outgassing rate is lower than the “solar min” Jeans escape rate but is $\sim
1000$ times higher than the Jeans escape rate assuming $T_{\rm exo}=350$ K.
The low H2 outgassing rate also is comparable to the energy-limited escape
rate around a 5000 K WD, but is many orders of magnitudes higher than the
energy-limited escape rate around a 4000 K WD. Given that even the “solar min”
escape rate is likely an overestimate for cool WDs, we conclude that a rocky
planet around cool WDs can maintain a hydrogen atmosphere if it has modern
Earth-like or higher H2 emissions. Even a planet with 1000 times reduced H2
emission can maintain a H2-dominated atmosphere around a 4000 K WD. Survival
of H2 atmosphere on a low outgassing planet orbiting a 5000 K WD would depend
on the EUV levels of the particular WD and heating mechanisms in the planet’s
exosphere.

Energy-limited hydrodynamic escape is important when EUV flux is high. In
Figure 4, we show the energy-limited escape rate as a function of WD EUV
radiation, following two analytical models (Luger & Barnes, 2015; Sengupta,
2016). Hydrogen escape flux due to energy-limited escape only becomes
comparable to Jeans escape for $\gtrsim 6000$ K WDs. Energy-limited escape
flux drops logarithmically as EUV flux decreases and is orders of magnitudes
lower than Jeans escape flux for $\lesssim 5000$ K WDs and is therefore
neglected.

On Earth, hydrogen escape is not limited by the rate at which H atoms diffuse
into space, but rather by the rate at which hydrogen is delivered to the
exobase from the collisional lower atmosphere (e.g., Catling & Kasting, 2017).
In an atmosphere where H2 is the dominant constituent, however, supply of
hydrogen is practically infinite. Thus, diffusion-limited escape is neglected.

##  Appendix C Detectability of Spectral Features in H2-Dominated Atmospheres

Here we discuss the locations and detectability of key spectral features in
H2-dominated atmospheres based on Figure 5 and 6. The most prominent feature
in our model spectra is the H2-H2 CIA extending from $\sim 6$ to 20 $\micron$.
Rayleigh scattering slope at $\lesssim 1\,\micron$ is also dominated by H2
opacity, except in the 4000 K high outgassing case, where CH4 becomes
pervasive in the atmosphere with a mixing ratio of 0.73. CH4 features at 1.7,
2.3, 3.4, and 7.5 $\micron$ are strong in most scenarios. CO2 features are
strong in the high outgassing models, with the 4.2 $\micron$ feature being the
most prominent because the 15 $\micron$ feature is partially obscured by the
broad H2-H2 CIA wing. H2O features at 2.6 and 6.4 $\micron$ are present in
some models, especially in the 6000 K WD models where the CH4 feature at 7.5
$\micron$ is relatively weak and does not completely overlap with the 6.4
$\micron$ water feature. CO shows a strong feature at 4.7 $\micron$, which has
comparable strength as the 4.2 $\micron$ CO2 feature and is not obscured by
any overlapping stronger features. The potential of detecting CO in large
amount (the mixing ratios of CO range from $8.0\times 10^{-5}$ to $3.7\times
10^{-4}$ in the high outgassing models) in a H2-dominated atmosphere is
noteworthy, because CO is the best-studied species under the context of
photochemical runaway (Schwieterman et al., 2019; Ranjan et al., 2022).

Figure 7: Transmission spectra models and simulated JWST transit observations
for N2-dominated atmospheres, assuming a $1\,R_{\oplus}$ planet transiting WD
1856+534. We compare high outgassing rate models (red solid lines) with low
outgassing rate models (blue solid lines) and show simulated JWST data points
with their $1\,\sigma$ error bars. We assume 25 NIRSpec Prism transits for the
N2-dominated atmosphere models. Major spectral features are detectable in the
N2-dominated atmosphere models. High and low outgassing scenarios are
potentially distinguishable from different strengths of CH4 features in all
three models. In the 4000 K WD model, high and low outgassing scenarios are
most easily distinguishable due to runaway buildup of CH4.  Figure 8:
Molecular contributions to the overall model transmission spectra for
N2-dominated atmospheres.  Figure 9: Transmission spectra models and simulated
JWST transit observations for CO2-dominated atmospheres, assuming a
$1\,R_{\oplus}$ planet transiting WD 1856+534. We compare high outgassing rate
models (red solid lines) with low outgassing rate models (blue solid lines)
and show simulated JWST data points with their $1\,\sigma$ error bars. We
assume 25 NIRSpec Prism transits for the CO2-dominated atmosphere models. CO2
features are detectable with 25 transits for all models, but distinguishing
between high and low outgassing scenarios is only achievable for the 4000 K WD
model, because CH4 only builds up to detectable level in the 4000 K WD high
outgassing scenario.  Figure 10: Molecular contributions to the overall model
transmission spectra for CO2-dominated atmospheres.

##  Appendix D Additional Results for N2\- and CO2-dominated Atmospheres

###  D.1 Inferring Outgassing Activities of Planets with High MMW Atmospheres

As demonstrated in Section 3.3, high and low outgassing rates can be
differentiated on rocky WD planets with H2 atmospheres, although inferring
tectonic activities of rocky exoplanets from outgassing rates is limited by
many uncertainties. Here, we focus on weakly and highly oxidizing atmospheres
dominated by N2 or CO2 and quantitatively discuss the ability of JWST to
distinguish between high and low outgassing scenarios for the N2\- and
CO2-dominated models.

Transmission spectra can differentiate between our high and low outgassing
scenarios assuming N2 or CO2 atmospheres, albeit more telescope time is
required due to the smaller extent of high MMW atmospheres. We compare
simulated transmission spectra for our high and low outgassing scenarios in
Figure 7 and shows molecular contributions to the overall spectra in Figure 8
for N2-dominated atmospheres. We show the same information for CO2-dominated
atmospheres in Figure 9 and Figure 10. We model an array of campaign sizes
with NIRSpec Prism, including 1, 5, 10, and 25 transits, and choose to show
the 25-transit program. This is because programs with 10 or less transits have
low SNR due to small scale height of N2 and CO2 atmospheres.

For the N2-dominated models, CH4 features at 1.7, 2.3, and 3.4 $\micron$ are
the keys to distinguish between the high and low outgassing scenarios (Figure
7). CO2 feature at 4.2 $\micron$ also differs significantly between the two
scenarios but is a less effective indicator due to lower SNR. For the 6000 K
WD models, CH4 features differ by $\sim 2.5\,\sigma$ between high and low
outgassing scenarios. CH4 mixing ratio rapidly increases when outgassing rates
are high as effective temperature of the host WD decreases, due to the
photochemical runaway mechanism discussed in Section 3.4. Therefore, high and
low outgassing scenarios can be differentiated with larger significance for
the 5000 K and 4000 K WD models.

For the CO2-dominated models, high and low outgassing scenarios are not
distinguishable with 25 NIRSpec Prism transit for the 6000 K and 5000 K WD
models (Figure 9). In an oxidizing environment dominated by CO2, CH4 cannot
efficiently accumulate even if the emission flux increases by a factor of
1000, explaining the lack of difference between the two scenarios. For the
4000 K WD models, however, CH4 provides opportunity for distinguishing between
high and low outgassing rates, likely because emission of CH4 passes the low
runaway threshold on a planet with extremely low UV radiation. Using the CH4
features between 0.9 and 2.5 $\micron$ as indicators, we may differentiate
between the two outgassing scenarios at a $\sim 10\,\sigma$ significance with
25 transits, and at $\sim 5\,\sigma$ with 10 transits.

###  D.2 Testing Photochemical Runaway in Oxidizing Atmospheres

Redox state impacts photochemical runaway (e.g., Ranjan et al., 2022). The
main sink of CH4 is hydroxyl (OH) radicals, which is typically produced by
O(1D) reacting with water. More oxidizing environments naturally lead to
higher production of OH radicals, making the photochemical sink of CH4 harder
to saturate. Indeed, the mixing ratio of CH4 is generally much lower in N2\-
and CO2-dominated models, compared to H2-dominated models with the same
outgassing rate and WD host (Table 2).

Nevertheless, CH4 runaway can occur in weakly oxidizing and oxidizing
atmospheres, as evidenced by strong CH4 features in the 4000 K WD models for
both atmospheric redox states (Figure 7 and 9). As a result of photochemical
runaway, CH4 is easily detectable by JWST on rocky exoplanets transiting cool
WDs. Here we quantitatively discuss CH4 detectability by NIRSpec Prism in our
various models.

Detecting CH4 is more challenging in N2-dominated atmospheres compared to
H2-dominated atmospheres but is still achievable within a small JWST program
for the high outgassing scenarios. For the 4000 K WD model, which has the
highest CH4 mixing ratios due to low UV radiation, 5 NIRSpec Prism transits
can detect CH4 features at $\sim 3$–$4\,\sigma$. For the N2 5000 K WD and N2
6000 K WD high outgassing models, detecting CH4 is more complicated – CH4
features at $<2\,\micron$ partially or totally overlap with H2O features with
similar strength, while CH4 features at $>2\,\micron$ suffer from higher noise
levels. Therefore, confidently detecting CH4 in N2-dominated atmospheres with
high outgassing rates based on the 1.7 and 2.3 $\micron$ features requires 25
transits. For all the low outgassing scenarios, CH4 is not detectable.

For the CO2-dominated models, CH4 only reach detectable levels via
photochemical runaway in the 4000 K high outgassing scenario. In this case, 5
NIRSpec Prism transits can detect CH4 at $\sim 3$–$4\,\sigma$ and 10 transits
can achieve a conclusive $\sim 5\,\sigma$ detection.

###  D.3 Detectability of Spectral Features in N2\- and CO2-dominated
Atmospheres

Here we summarize the locations and detectability of key spectral features in
N2\- and CO2-dominated atmospheres based on simulated transmission spectra and
JWST observations (Figure 7 for N2 atmospheres and Figure 9 for CO2
atmospheres). We also show the molecular contributions to the overall spectra
(Figure 8 for N2 atmospheres and Figure 10 for CO2 atmospheres). Note that in
Figure 5 we show JWST observations with 1 NIRSpec Prism transit for the H2
models, but for N2\- and CO2-dominated models we show 25 transits, because
high MMW N2 and CO2 atmospheres have lower atmospheric scale heights. Also
note that we assume the stellar parameters of WD 1856+534 (Vanderburg et al.,
2020). A brighter or more close-by host can increase SNR and hence
detectability of spectral features.

In N2-dominated atmosphere models, H2O have strong features in all scenarios.
The strongest water features are located at 2.6 and 6.4 $\micron$, with a few
weaker features located at approximately 1.0, 1.2, 1.3, and 1.9 $\micron$, as
well as the broad continuum from about 10 to 20 $\micron$. Water features at
1.0-2.6 $\micron$ coincide with a bandpass in which noise levels are low and
are likely detectable by NIRSpec Prism. CO2 features at 4.2 and 15 $\micron$
are present but weaker than water features in the reduced outgassing models,
but are predominant in the high outgassing models, where CO2 outgassing rates
are 1000 times higher than in the reduced outgassing models. In the high
outgassing models, CH4 also shows strong features at 3.4 and 7.5 $\micron$, as
well as several weaker features in the visible wavelength range. As discussed
in Section D.2, these CH4 features are detectable by NIRSpec Prism at high
significance and can potentially differentiate between the high and low
outgassing scenarios. In addition, N2-N2 has a CIA feature at 4.3 $\micron$,
which overlaps with the 4.2 $\micron$ CO2 feature. In the reduced outgassing
models, the N2-N2 CIA and the CO2 feature have similar strength and may lead
to detection degeneracy.

In a CO2-dominated atmosphere (90% CO2), CO2 features unsurprisingly dominate
the transmission spectra. The strongest CO2 feature is at 15 $\micron$, with
broad wings extending to about 11 and 20 $\micron$ on each side. The second
strongest CO2 feature is located at 4.2 $\micron$ and is likely the most
easily detectable feature for NIRSpec Prism. In addition, CO2 has a few weaker
features at 1.4, 1.6, 1.9, 2.1, 2.8, 3.0, 4.8, 5.2, and 10.3 $\micron$. Other
than single-molecule absorption features, a CO2-CO2 CIA feature is also
present at about 7.5 $\micron$. CO has a feature at about 4.7 $\micron$ but is
obscured by the stronger CO2 feature at 4.8 $\micron$. H2O has a broad feature
at 6.4 $\micron$, which is the strongest feature of water and does not overlap
with any other features. H2O also has a continuum from about 10 to 20
$\micron$, several weak features in visible wavelengths and a strong feature
at 2.6 $\micron$, but all these features are obscured by nearby CO2 features.
CH4 has two broad features at 3.4 and 7.5 $\micron$, as well as several
features in the visible and near-IR wavelengths spanning from 0.9 to 2.5
$\micron$. CH4 features are only strong enough to be detectable in the 5000 K
and 4000 K high outgassing cases. Especially in the 4000 K high outgassing
scenario, CH4 features have comparable strength as the strongest CO2 features
and dominate the visible to near-IR wavelength range probed by NIRSpec Prism.

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