# Unveiling shrouded oceans on temperate sub-Neptunes via transit signatures
of solubility equilibria vs. gas thermochemistry

Renyu Hu Jet Propulsion Laboratory, California Institute of Technology,
Pasadena, CA 91109, USA  Division of Geological and Planetary Sciences,
California Institute of Technology, Pasadena, CA 91125, USA  Mario Damiano Jet
Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109,
USA  Markus Scheucher Jet Propulsion Laboratory, California Institute of
Technology, Pasadena, CA 91109, USA  Institut für Planetenforschung, Deutsches
Zentrum für Luft- und Raumfahrt, D-12489 Berlin, Germany  Edwin Kite
Department of the Geophysical Sciences, University of Chicago, Chicago, IL
60637, USA  Sara Seager Department of Earth, Atmospheric, and Planetary
Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Department of Physics and Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Department of
Aeronautics and Astronautics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Heike Rauer  Institut für Planetenforschung,
Deutsches Zentrum für Luft- und Raumfahrt, D-12489 Berlin, Germany  Institut
für Geologische Wissenschaften, Freie Universität Berlin, D-12249 Berlin,
Germany

###### Abstract

The recent discovery and initial characterization of sub-Neptune-sized
exoplanets that receive stellar irradiance of approximately Earth’s raised the
prospect of finding habitable planets in the coming decade, because some of
these temperate planets may support liquid water oceans if they do not have
massive H2/He envelopes and are thus not too hot at the bottom of the
envelopes. For planets larger than Earth, and especially planets in the
$1.7-3.5\ R_{\oplus}$ population, the mass of the H2/He envelope is typically
not sufficiently constrained to assess the potential habitability. Here we
show that the solubility equilibria vs. thermochemistry of carbon and nitrogen
gases results in observable discriminators between small H2 atmospheres vs.
massive ones, because the condition to form a liquid-water ocean and that to
achieve the thermochemical equilibrium are mutually exclusive. The dominant
carbon and nitrogen gases are typically CH4 and NH3 due to thermochemical
recycling in a massive atmosphere of a temperate planet, and those in a small
atmosphere overlying a liquid-water ocean are most likely CO2 and N2, followed
by CO and CH4 produced photochemically. NH3 is depleted in the small
atmosphere by dissolution into the liquid-water ocean. These gases lead to
distinctive features in the planet’s transmission spectrum, and a moderate
number of repeated transit observations with the James Webb Space Telescope
should tell apart a small atmosphere vs. a massive one on planets like K2-18
b. This method thus provides a way to use near-term facilities to constrain
the atmospheric mass and habitability of temperate sub-Neptune exoplanets.

Exoplanet atmospheres — Extrasolar rocky planets — Extrasolar ice giants —
Habitable Planets — Ocean Planets — Transmission spectroscopy

##  1 Introduction

The exoplanet community already has ways to detect an H2 atmosphere by
transmission spectroscopy via its pressure scale height one order of magnitude
larger than that of an N2 or CO2 atmosphere (Miller-Ricci et al., 2008).
However, the mass of the H2 atmosphere – the parameter that controls the
temperature at the bottom of the atmosphere and thus the possibility for
liquid water (Pierrehumbert & Gaidos, 2011; Ramirez & Kaltenegger, 2017; Koll
& Cronin, 2019) – is not directly measurable from the transmission spectrum.
Also, a planet’s mass and radius typically allow multiple models of the
interior structure (e.g., Rogers & Seager, 2010; Valencia et al., 2013). It is
unclear whether the planets in the $1.7-3.5\ R_{\oplus}$ population (Fulton &
Petigura, 2018) are mostly rocky planets with massive H2/He gas envelopes
(Owen & Wu, 2017; Jin & Mordasini, 2018) or planets with a massive water layer
($\sim 50$ wt. %) that do not require a large H2 envelope to explain their
radius (e.g., referred to as “ocean planets” thereafter; Zeng et al., 2019;
Mousis et al., 2020; Venturini et al., 2020). Direct-imaging observations in
the future may provide means to detect a surface underneath a thin atmosphere
on temperate planets, via the ocean glint (Robinson et al., 2010) or surface
heterogeneity (Cowan et al., 2009; Fan et al., 2019). However, these methods
are not applicable to the near-term capabilities such as the JWST and may pose
challenges on precision even for ambitious direct-imaging mission concepts
(Gaudi et al., 2020).

The temperate sub-Neptune K2-18 b is a harbinger of the class of planets that
might be habitable and exemplifies the need for a near-term method to measure
the size of an H2 atmosphere. The planet of $8.6\ M_{\oplus}$ and $2.6\
R_{\oplus}$ is in the habitable zone of an M dwarf star, and has a
transmission spectrum (obtained by Hubble at $1.1-1.7\ \mu m$) with confirmed
spectral features, which indicates that the planet should host an atmosphere
dominated by H2 (Tsiaras et al., 2019; Benneke et al., 2019). Interior
structure models showed that the planet can have a massive ($>\sim 1000$ bar)
H2 atmosphere overlaying a rocky/Fe core and a possibly supercritical water
layer, or a smaller ($<100$ bar) H2 atmosphere with a water-dominated interior
(Madhusudhan et al., 2020; Mousis et al., 2020; Nixon & Madhusudhan, 2021).
For K2-18 b, specifically, a $\sim 10-100$ bar H2 atmosphere overlaying a
water layer would cause $>200$ bar of water to evaporate into the atmosphere,
resulting in a hot steam atmosphere inconsistent with the observed
transmission spectrum (Scheucher et al., 2020). An even smaller, $\sim 1$ bar
H2 atmosphere would prevent this steam atmosphere and produce a liquid-water
ocean (see Section 3), but this requires a very small rocky/Fe core and may be
disfavored from the planet formation standpoint (e.g., Lee & Chiang, 2016).
However, a planet slightly more massive or smaller than K2-18 b – such as
those at the center of the $1.7-3.5\ R_{\oplus}$ planet population – does not
have this small-core difficulty to have a small atmosphere (Zeng et al., 2019;
Nixon & Madhusudhan, 2021), and many such planets and planet candidates have
been detected and will soon be available for transmission spectroscopy (Figure
1, panel a).

Here we propose that transit observations of temperate sub-Neptunes in the
near- and mid-infrared wavelengths, which will soon commence with JWST, can
detect small H2 atmospheres that support liquid-water oceans and distinguish
them from massive atmospheres (Figure 1, panel b). A companion paper has
studied the atmospheric chemistry and spectral features of temperate planets
with massive H2 atmospheres (Hu, 2021), and now we turn to temperate planets
with small H2 atmospheres. A recent paper might have similar intent as our
work: Yu et al. (2021) studied the chemistry of temperate H2 atmospheres with
varied surface pressures, with assumed zero flux for all species at the lower
boundary. The theories of Yu et al. (2021) may thus be more applicable to arid
rocky planets without substantial volcanic outgassing, and here we instead
focus on ocean planets, and address how to identify them observationally. As
we will show later, a small atmosphere on a temperate sub-Neptune will have a
distinctive composition because of its interaction with the ocean underneath.

Figure 1:  Temperate exoplanets amenable for atmospheric characterization via
transmission spectroscopy. (a) Purple dots are confirmed planets with measured
masses, and blue dots are planets with unknown masses or planet candidates.
Data are taken from the NASA Exoplanet Archive and and the TESS Objects of
Interest Catalog. The marker sizes are scaled with the expected S/N of the
spectral features of an H2 atmosphere observed by JWST at $2\ \mu m$. Most of
the temperate planets and planet candidates suitable for atmospheric
characterization are larger than Earth and thus more likely to have H2
atmospheres. (b) A roadmap to characterize the mass of the atmospheres and the
habitability of temperate sub-Neptunes by detecting signature gases. See text
for details.

##  2 Mutual exclusivity of habitability and thermochemical equilibrium

Figure 2:  Interior structures of temperate H-rich exoplanets and the
associated ranges of atmospheric composition. If the planet has a massive H2
atmosphere, the deep atmosphere would be hot – enabling thermochemical
recycling – but a liquid-water surface would not be possible. If the planet
has a small H2 atmosphere, a liquid-water surface may be possible. On these
planets, the equilibrium abundance of atmospheric CO2 is set by the oceanic
chemistry and that of N2 by atmospheric evolution.

On temperate sub-Neptunes, the condition to form a liquid-water ocean and that
to achieve the thermochemical equilibrium of carbon and nitrogen molecules are
mutually exclusive. The CO2-CO-CH4 and N2-NH3 conversion rates are primarily a
function of the temperature and to a lesser extent the pressure (Zahnle &
Marley, 2014; Tsai et al., 2018), and in a temperate sub-Neptune like K2-18 b,
the thermochemical equilibrium of carbon and nitrogen molecules are typically
achieved at the pressure of $10^{7}\sim 10^{8}$ Pa, where the temperature is
$>1000$ K (i.e., substantially higher than the critical point of water;
Fortney et al., 2020; Yu et al., 2021; Hu, 2021). Therefore, the gas-phase
thermochemical equilibrium would be achieved in the deep and hot part of a
massive atmosphere, and in contrast, it would not be achieved in a small
atmosphere overlying a liquid-water ocean. Instead, NH3 and sulfur species
would be sequestered by the ocean (Loftus et al., 2019, and also see Section
3) and the abundance of CO2 would be set by the ocean chemistry (Figure 2,
with the cosmochemical and geological constraints detailed in Appendix A).
This fundamental difference, coupled with atmospheric photochemistry, leads to
distinctive gas abundances in the observable part ($<\sim 0.1$ bar) of the
atmosphere.

If the planet has a massive H2 atmosphere, thermochemical reactions in the
deep atmosphere recycle O, C, N, S species into H2O, CH4, NH3, and H2S
(Burrows & Sharp, 1999; Heng & Tsai, 2016; Woitke et al., 2021; Blain et al.,
2021). H2O can form a cloud and the above-cloud H2O may be partially depleted
as a result (Morley et al., 2014; Charnay et al., 2021; Hu, 2021). Recent
calculations have shown that the photodissociation of NH3 in the presence of
CH4 leads to the formation of HCN and N2, and that CO and CO2 are produced by
the photodissociation of CH4 together with H2O (Hu, 2021). The
photodissociation of H2S leads to the formation of elemental sulfur haze (Hu
et al., 2013; Zahnle et al., 2016), but the haze would likely be close to the
cloud deck and would not mute transmission spectral features (Hu, 2021). These
photochemical products are transported to the deep atmosphere and recycled
back to CH4, NH3, and H2S. An exception is that planets with super-solar
atmospheric metallicity and appreciable internal heat may have additional CO,
CO2, and N2 transported from the deep troposphere and incomplete recycling to
NH3 (Fortney et al., 2020; Yu et al., 2021; Hu, 2021).

If the planet instead has a small atmosphere and a liquid-water ocean, the
thermochemical recycling cannot occur. Instead, CO2 is the preferred form of
carbon in equilibrium with a massive amount of H2O (Hu & Seager, 2014; Woitke
et al., 2021), and NH3 is dissolved in the ocean and largely depleted from the
atmosphere (see Section 3). The abundance of atmospheric CO2 is controlled by
the oceanic pH (Kitzmann et al., 2015; Krissansen-Totton & Catling, 2017; Kite
& Ford, 2018; Isson & Planavsky, 2018) and that of N2 is probably a combined
result of the initial endowment and atmospheric escape. A reasonable lower
bound of the total mass of CO2 in the H2 and H2O layers can be derived from
the cosmochemical constraints of planetary building blocks and the
partitioning between the iron core, the silicate mantle, and the water layer
(Appendix A). Also, the “seafloor” of this thin-atmosphere, H2O-rich sub-
Neptune will not be not a sharp interface in density and composition, but
instead have a finite thickness (Vazan et al., 2020). The interface will be
compositionally stratified with denser material underlying less dense
material, and material transport across this “fuzzy layer” is inhibited due to
the stratification. Thus, any carbon or nitrogen added to the H2 and H2O
envelope by planetesimal accretion late in planet growth will remain in the
envelope, and will not be stirred down into the silicate layer. Meanwhile,
transit observations can straightforwardly identify H2-dominated atmospheres
and rule out CO2 or N2-dominated ones only from the size of spectral features
(Miller-Ricci et al., 2008).

One might also consider the intermediate situation between massive atmospheres
with thermochemical equilibrium and small atmospheres with liquid-water
oceans, e.g., the atmospheres with a surface pressure from a few to $\sim 100$
bars on K2-18 b. For many sub-Neptunes, this intermediate-atmosphere scenario
would still require a massive water layer underneath to explain their mass and
radius. If water was in the liquid form at the interface with the atmosphere,
the evaporation of this ocean would make the atmosphere H2O-dominated
(Scheucher et al., 2020). If water is supercritical, any H2 layer of
intermediate mass should be well mixed with the water layer. Therefore, such
an intermediate endowment of H2 would most likely result in a non-H2-dominated
atmosphere, which is, again, distinguishable with transmission spectroscopy
(Miller-Ricci et al., 2008).

##  3 Ocean Planet Models

Table 1: Summary of the photochemical model parameters and results.

Model | Name | CO2 | CO flux | H2O | CO | CH4 | C2H6
---|---|---|---|---|---|---|---
1 | Low-CO2 | $4\times 10^{-4}$ | $0$ | $2.3\times 10^{-3}$ | $1.4\times 10^{-5}$ | $1.5\times 10^{-2}$ | $3.0\times 10^{-6}$
1a | Low-CO2 Variant | $4\times 10^{-4}$ | $1.0\times 10^{9}$ | $3.3\times 10^{-3}$ | $2.9\times 10^{-4}$ | $2.9\times 10^{-2}$ | $5.1\times 10^{-6}$
2 | High-CO2 | $0.1$ | $0$ | $1.1\times 10^{-4}$ | $9.5\times 10^{-3}$ | $5.3\times 10^{-2}$ | $4.0\times 10^{-7}$

Note. — The volume mixing ratio of CO2 (as inputs) is at the lower boundary,
and those of H2O, CO, CH4, and C2H6 (as results) are column-averaged in
$10-10^{3}$ Pa. The CO flux has a unit of cm-2 s-1.

Figure 3:  Modeled pressure-temperature profiles (a) and abundance profiles of
main gases and photochemical products (b) in a temperate sub-Neptune like
K2-18 b that has a small H2 atmosphere. Solid, dashed, and dash-dot lines show
the results for the low-CO2 case (Model 1 in Table 1), the low-CO2 case with
additional CO sources (Model 1a), and the high-CO2 case (Model 2). For the
stellar irradiance, we use $S=S_{\rm Earth}*(1-A_{\rm B})$, with a Bond albedo
of $A_{\rm B}=0.3$ (similar to Earth), to account for the radiative effects
clouds would have in the otherwise cloud-free climate model. The surface
albedo reflects a dark ocean (0.06). The surface temperatures in these models
are consistent with a liquid-water ocean. The photochemical models use the UV
spectrum of the M dwarf star GJ 176 (France et al., 2016) (similar to K2-18;
dos Santos et al., 2020). The steady-state mixing ratio CH4 is high and those
of nitrogen molecules such as NH3 and HCN is $<10^{-12}$.

We have used an atmospheric photochemical model (Hu et al., 2012) coupled with
a radiative-convective model (Scheucher et al., 2020) to determine the steady-
state abundances of photochemical gases in small and temperate H2 atmospheres,
for a cosmochemically and geologically plausible range of CO2 abundance, and
compared the compositions and transmission spectra with the massive H2
atmosphere models published in Hu (2021). The massive atmosphere models
explored the atmospheric metallicity of $1-100\times$solar and included
possible deep-tropospheric source CO, CO2, and N2 and incomplete reclycing of
NH3 in super-solar atmospheres.

The photochemical model includes a comprehensive reaction network for O, H, C,
N, and S species (including sulfur aerosols, hydrocarbons, and the reactions
important in H2 atmospheres), and it has been used to study the lifetime and
equilibrium abundance of potential biosignature gases in H2 atmospheres
(Seager et al., 2013). We have updated the reaction network and tested the
model with the measured photochemical gas abundance in the atmosphere of
Jupiter (i.e., a low-temperature H2 atmosphere; Hu, 2021).

The pressure-temperature profiles (Figure 3) used as the basis for the
photochemical model are calculated with the climate module of 1D-TERRA
(Scheucher et al., 2020). The module uses a correlated-k approach with the
random overlap method to include molecular absorption, collision-induced
opacities, and the continuum of water vapor to calculate the radiative
equilibrium, and the appropriate (moist or dry) adiabatic lapse rate to apply
the convection adjustment. The module has been tested against the cases of
Earth, Venus, and Mars, as well as with other radiative-convective and 3D
climate models for modeling steam atmospheres (Scheucher et al., 2020).

As examples, we study H2 atmospheres of 1 bar on a sub-Neptune planet that has
a stellar irradiance similar to Earth and orbits around an early M star
similar to K2-18. A 1-bar H2 atmosphere on such a planet would likely have a
surface temperature consistent with a liquid-water ocean (Figure 3). We adopt
the “ocean-planet” interpretation of the $1.7-3.5\ R_{\oplus}$ planet
population that centers at $10\ M_{\oplus}$, and $2.5\ R_{\oplus}$ (Zeng et
al., 2019; Venturini et al., 2020), and assume 50% of water by mass in this
study. In this interpretation, sub-Neptunes may be ocean planets with deep
oceans that do not require a massive H2 envelope to explain their radius, and
can conceivably have moderate-size H2 atmospheres. This may not be directly
applicable for K2-18 b, which resides on the low-density side of the $1.7-3.5\
R_{\oplus}$ population. The specific choices of these parameters are however
unimportant, because atmospheric chemistry is not sensitive to moderate
changes in the surface gravity.

CO2 is the main form of carbon in thermochemical equilibrium with H2O (Hu &
Seager, 2014; Woitke et al., 2021). If a liquid-water ocean exists, the
partial pressure of CO2 is set by atmosphere-ocean partitioning, which in turn
is mainly controlled by the oceanic pH (Kitzmann et al., 2015; Krissansen-
Totton & Catling, 2017; Kite & Ford, 2018; Isson & Planavsky, 2018). The pH is
affected by the abundance of cations in the ocean, which come from complex
water-rock reactions and dissolution of the seafloor. The rates of the
processes involved are uncertain; therefore, we explore the mixing ratio of
CO2 from 400 ppm to 10%, corresponding to the pCO2 range from the present-day
Earth to early Earth (Catling & Kasting, 2017) and including the predicted
range for ocean planets (Kite & Ford, 2018) that is still consistent with an
H2-dominated atmosphere. The $4\times 10^{-4}$ bar partial pressure of CO2 in
the low-CO2 case, while not the absolute lower limit, is a cosmochemically and
geologically reasonable lower bound of the CO2 partial pressure on an ocean
planet (Appendix A).

The mixing ratio of N2 on the modeled planet is probably set by atmospheric
evolution (as opposed to the solubility equilibrium or geological recycling)
and is assumed here to be 1%. As N2 only minimally participates in the
chemical cycles and does not have strong spectral features in the infrared,
its exact abundance is not our main concern. The photochemical model indicates
that the NH3 produced by photodissociation of N2 in H2 atmospheres has
negligible mixing ratios ($<10^{-12}$).

The pressure at the water-rock boundary of a $10-M_{\oplus}$ and
$2.5-R_{\oplus}$ planet is $\sim 500$ GPa (Sotin et al., 2007; Levi et al.,
2014), and this overloading pressure should suppress volcanism completely
(Kite et al., 2009; Noack et al., 2017; Kite & Ford, 2018). Therefore we do
not include any volcanic outgassing in the standard models. As variant models,
we consider the possibility of minor and intermittent sources of CO into the
atmosphere. Evaporation of meteorites may provide a source of CO and CO2
(Schaefer & Fegley, 2017), and water-rock reactions at the temperature
relevant to the “fuzzy layer” may produce CO (and not CH4 as it is
thermochemically disfavored at high temperatures). The rates of these
processes are unknown, but numerical experiments with the photochemical model
indicate that an additional CO source of $10^{10}$ molecule cm-2 s-1 would
lead to a steady-state abundance of CO greater than that of H2, effectively
resulting in a CO-dominated atmosphere. A CO source of $10^{9}$ molecule cm-2
s-1 would produce the CO-dominated atmosphere in the 10%-CO2 case but not in
the 400ppm-CO2 case. We therefore include a low-CO2 case with the CO source of
$10^{9}$ molecule cm-2 s-1 as a variant model.

Table 1 summarizes the input parameters and results of the photochemical
models, and Figure 3 shows the profiles of temperature and mixing ratios of
main gases and photochemical products. CO is produced from the
photodissociation of CO2 and can build up to $10^{-5}$ and $10^{-2}$ mixing
ratio level for the low-CO2 and the high-CO2 cases. OH from the
photodissociation of H2O destroys CO and maintains its steady-state mixing
ratio. CH4 is also produced photochemically and can build up to a substantial
mixing ratio ($10^{-3}\sim 10^{-2}$). This effectiveness in producing CH4 from
CO in temperate H2 atmospheres has also been noted in Yu et al. (2021).
Together with the high CH4 mixing ratio, C2H6 is produced and can accumulate
to a mixing ratio of $\sim 10^{-6}$. C2H2, as expected, is short-lived and
only has significant mixing ratios in the upper atmosphere. Here we have
applied a deposition velocity of $10^{-5}$ cm s-1 for C2H6 to account for the
loss of carbon due to organic haze formation and deposition (Hu et al., 2012);
removing this sink does not substantially change the results shown in Figure
3. The additional source of CO would result in moderately more CO, CH4, and
C2H6 in the atmosphere (Model 1a in Table 1 and Figure 3). The photochemical
CO and CH4 can build up to the mixing ratio levels that cause significant
features in the planet’s transmission spectrum (Section 4).

Before closing this section, we address whether NH3 can be produced
substantially by water-rock reactions and then emitted into the atmosphere.
Hydrothermal systems on early Earth may produce NH3 from the reduction of
nitrite and nitrate (Summers & Chang, 1993; Summers, 2005). On a planet with
an H2-dominated atmosphere, however, atmospheric production of the oxidized
nitrogen including nitrite and nitrate should be very limited. Moreover, the
storage capability of NH3 by the ocean is vast and limits the emission into
the atmosphere. At the pH value of 8 (a lower pH would further favor the
partitioning of NH3 in the ocean), $10^{-6}$ bar of atmospheric NH3 requires a
dissolved ammonium concentration of $10^{-3}$ mol/L in equilibrium (Seinfeld &
Pandis, 2016). The mass of NH3 in the atmosphere and ocean is then $\sim
10^{-5}$ of the planetary mass. This would only be possible if much of the
planet’s rocky core begins with a volatile composition similar to carbonaceous
chondrites, and most of this nitrogen is partitioned into the atmosphere and
ocean as NH3 (Marty et al., 2016), which is highly unlikely as N2 is
thermochemically favored. Therefore, the concentration of dissolved NH3 should
be small and so is the atmospheric NH3 on a planet with a massive ocean.

##  4 Spectral Characterization

Figure 4:  Modeled transmission spectrum of temperate sub-Neptune planets of M
dwarf stars, using K2-18 b as an example and comparing with the planet’s
transit depth observed by Hubble (Benneke et al., 2019). The
massive-H2-atmosphere models (black lines) and the small-H2-atmosphere models
(colored lines) differ in three spectral regions: in (a) and (b), the massive-
atmosphere models have absorption features of NH3 and HCN, while the small-
atmosphere models do not; in (c), the small-atmosphere models with a low
mixing ratio of CO2 (400 ppm) have prominent features of CO2 and CO, while the
massive-atmosphere models only have small features of NH3 and HCN. The
$100\times$solar massive atmosphere with deep-tropospheric source and sink may
have subdued NH3 and HCN features and prominent CO2 and CO features. The
small-atmosphere models with a high mixing ratio of CO2 (10%) has a high mean
molecular weight ($\sim 6$) and a high cloud top (Figure 3) and thus muted
spectral features.

Figure 4 compares the expected spectra for the massive-atmosphere scenarios
and the small-atmosphere scenarios. For K2-18 b, the massive-atmosphere models
with $1-100\times$solar metallicity and the small-atmosphere models with a low
mixing ratio of CO2 (400 ppm) provide good fits to the transmission spectrum
measured by Hubble.

Measuring the transmission spectra in an expanded wavelength range of $1-5\
\mu$m will distinguish the small atmospheres from massive ones. Using K2-18 b
as an example for temperate sub-Neptunes, we see that the massive-atmosphere
models and the small-atmosphere models, while having differences within each
group, can be distinguished using the spectral regions of $1.9-2.1$,
$2.7-3.1$, and $4.1-5.0\ \mu$m (the shaded areas a, b, and c in Figure 4).
Both the massive-atmosphere and small-atmosphere models show spectral features
of H2O and CH4, and so observing these two gases alone is unlikely to separate
the massive versus small scenarios.

At $1.9-2.1$ and $2.7-3.1\ \mu$m, the transmission spectra show NH3 and HCN
absorption in massive atmospheres but not in small atmospheres. If the
$100\times$solar massive atmosphere has incomplete NH3 recycling in the deep
troposphere, it will have much weaker NH3 and HCN features in these spectral
regions. The transmission spectra of small atmospheres show small CO2 features
at $\sim 2.0$ and $\sim 2.75\ \mu$m, but the feature at $\sim 2.75\ \mu$m is
combined with a part of the H2O feature with similar strength. The
transmission spectra of small atmospheres also show a small C2H2 feature at
$\sim 3.05\ \mu$m, and given enough precision, it might be distinguishable
with the HCN feature at $\sim 3.0\ \mu$m.

At $4.1-5.0\ \mu$m, the transmission spectra of small atmospheres (the low-CO2
cases) have prominent features of CO2 and CO, while the spectra of massive
atmospheres have weak features of NH3 and HCN. If the $100\times$solar massive
atmosphere has CO and CO2 transported from the deep troposphere, it can have
prominent spectral features of CO2 and CO in this region as well.

From the above, we see that the $100\times$solar massive atmosphere with deep-
tropospheric effects may resemble a small atmosphere in their transmission
spectra (Figure 4), i.e., the lack of NH3 or HCN and the prominence of CO2 and
CO. Would this potential “false positive” be avoidable? The answer may be yes
given enough precision and spectral resolution. First, the spectrum of the
massive atmosphere with deep-tropospheric effects still has weak spectral
features of HCN, while none of the small atmospheres does. Second, the massive
atmosphere has CO2/CO$<\sim 0.1$, because CO always dominates over CO2 in the
deep H2 troposphere of a temperate planet, and photochemical processes driven
by an M dwarf star do not significantly raise the CO2 mixing ratio in the
observable part of the atmosphere (Hu, 2021). In contrast, the small
atmospheres typically have CO2/CO$\geq 1$ (Table 1). In the more likely
scenario without any volcanic outgassing, CO2/CO$\sim 10$, because CO is
produced photochemically from CO2. Therefore, by measuring the abundance of CO
and CO2 independently, one could tell whether they are sourced from the deep
troposphere.

Furthermore, a massive atmosphere with $\gg 100\times$ solar metallicity will
have a mean molecular weight much higher than that of an H2 atmosphere and is
thus also distinguishable by transmission spectroscopy.

With moderate time investment (i.e., $<100$ hours), JWST will provide the
sensitivity to detect the signature gases aforementioned and distinguish
massive versus small atmospheres on planets like K2-18 b. As an example, we
have used PandExo (Batalha et al., 2017) to simulate the expected photometric
precision using JWST’s NIRSpec instrument. If combining two transit
observations with NIRSpec’s G235H grating and four transits with the G395H
grating, the overall photometric precision would be $\sim 20$ ppm per spectral
element at a resolution of $R=100$ in both channels that cover a wavelength
range of $1.7-5.2\ \mu$m. These observations would distinguish the small-
atmosphere scenarios versus the massive-atmosphere scenarios in Figure 4 with
high confidence.

Additionally, we have performed spectral retrievals based on simulated
observations using Tau-REx (Waldmann et al., 2015). We find that the mixing
ratio of NH3 and HCN and the lack of CO2 or CO in the solar-abundance massive
atmosphere would be usefully constrained (Figure 5). For the $100\times$solar
atmosphere, the CO2 and CO transported from the deep troposphere would be
identified, and the posteriors suggest that CO is likely more abundant than
CO2. The reduction in the mixing ratios of NH3 and HCN due to incomplete
recycling could also be seen in the retrieval, although the constraints on the
mixing ratio of HCN is not accurate. For the small atmosphere, the retrieval
yields degenerate solutions and thus double peaks in some posterior
distributions. Despite this, it is clear from the posteriors that the
atmosphere likely has high mixing ratios of both CO2 and CH4, has more CO2
than CO, and has very little NH3 or HCN (Figure 5). In addition to JWST, the
dedicated exoplanet atmosphere characterization mission ARIEL could also
provide the sensitivity to detect these gases with more repeated transit
observations (Changeat et al., 2020). This example shows that transit
observations in the coming years can tell apart temperate sub-Neptunes with
small H2 atmospheres versus the planets with massive atmospheres and reveal
their distinct atmospheric composition.

Figure 5:  Retrieved posterior distributions of the abundances of the main
chemical compounds and the cloud pressure in example massive-atmosphere and
small-atmosphere scenarios. The input transmission spectra are calculated by
Tau-REx using the atmospheric composition in Figure 3 and the expected
uncertainties are calculated using PandExo (Batalha et al., 2017), assuming to
combine two transits of K2-18 b with NIRSpec/G235H and four transits with
NIRSpec/G395H. The vertical red lines show the input value of the parameter,
and the quantities on the top of each panel show the median and $1\sigma$
values summarized from the posterior. The cloud pressure ($P_{\rm clouds}$)
has a unit of Pa. A detailed characterization of the atmosphere of K2-18 b,
including distinguishing a small atmosphere versus a massive one and measuring
the abundances of H2O, CH4, NH3, HCN, CO2, and CO, will be achievable with
moderate time investment of JWST.

##  5 Discussion and Conclusion

Taken together, the results presented above identify a near-term path to
detect small H2 atmospheres that can be consistent with liquid-water oceans on
temperate exoplanets. H2 atmospheres are probably the only type of temperate
atmospheres readily within the reach of JWST and ARIEL for detailed studies,
since to characterize a heavier H2O, N2, or CO2 atmosphere will require co-
adding a few tens transits – something not impossible but probably very hard
(Belu et al., 2011; Krissansen-Totton et al., 2018; Wunderlich et al., 2019;
Pidhorodetska et al., 2020; Gialluca et al., 2021). The mass of the H2
atmospheres – a parameter that is not directly measured by transits but
critical for habitability if the planet is moderately irradiated – can be
inferred from transmission spectra via the signature gases that indicate
solubility equilibria versus gas-phase thermochemical recycling. The biggest
uncertainty is probably the temperature at the $100\sim 1000$-bar pressure
level in the massive-atmosphere scenarios, which may be affected by ad hot
heating mechanisms such as tidal heating. Detailed models of the interior
temperature and mixing may further constrain this uncertainty (Fortney et al.,
2020; Yu et al., 2021). Based on the range of the parameter space explored, we
suggest that the sensitivity of multiple gases provided by future
observatories’ expanded wavelength coverage over Hubble would enable broad
categorization of small versus massive atmospheres, summarized as a roadmap in
Figure 1, panel b.

How many sub-Neptunes could we expect to be ocean planets in the first place?
The current population statistics of planets provide indirect evidence that
most sub-Neptunes are not ocean planets (Fulton & Petigura, 2018; Owen & Wu,
2017; Jin & Mordasini, 2018), but most known planets are hotter than planets
that can be habitable. Even if the current statistics apply to temperate
planets, there is plenty of room to have 10-20% of sub-Neptunes be ocean
planets, which will still be a lot of planets. Also, some planets in or just
below the “radius valley” may be sub-Neptunes that have evolved into ocean
planets (Kite & Schaefer, 2021) and retained some residual H2 (Misener &
Schlichting, 2021). For these reasons, this possibility of an ocean planet
shrouded by a small H2 atmosphere should motivate detailed observations of
temperate planets with radius from near the “radius valley” ($\sim 1.7\
R_{\oplus}$) to the main sub-Neptune population ($\sim 2.5\ R_{\oplus}$). If
some of the temperate planets in the aforementioned group have small H2
atmospheres, their relative ease for transit observations would significantly
enhance the prospect of detecting and characterizing potentially habitable
exoplanets within the next decade.

## Acknowledgments

The authors thank helpful discussions with Fabrice Gaillard and Sukrit Ranjan.
RH conceived and designed the study, simulated the photochemical models,
interpreted the results, and wrote the manuscript. MD performed the JWST
observation simulations and atmospheric retrievals. MS computed the pressure-
temperature profiles. EK derived the cosmochemical and geological lower bounds
for the carbon content. SS contributed interior structure models and insights.
HR oversaw the development of the radiative-convective model used in the
study. All authors commented on the overall narrative of the paper. The raw
data that are used to generate the figures in this paper are available from
the corresponding author upon reasonable request. This work was supported in
part by NASA Exoplanets Research Program grant #80NM0018F0612. The research
was carried out at the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with the National Aeronautics and Space
Administration.

## References

  * Batalha et al. (2017) Batalha, N. E., Mandell, A., Pontoppidan, K., et al. 2017, Publications of the Astronomical Society of the Pacific, 129, 064501
  * Belu et al. (2011) Belu, A., Selsis, F., Morales, J.-C., et al. 2011, Astronomy & Astrophysics, 525, A83
  * Benneke et al. (2019) Benneke, B., Wong, I., Piaulet, C., et al. 2019, The Astrophysical Journal Letters, 887, L14
  * Bergin et al. (2014) Bergin, E., Cleeves, L. I., Crockett, N., & Blake, G. 2014, Faraday discussions, 168, 61
  * Bergin et al. (2015) Bergin, E. A., Blake, G. A., Ciesla, F., Hirschmann, M. M., & Li, J. 2015, Proceedings of the National Academy of Sciences, 112, 8965
  * Blain et al. (2021) Blain, D., Charnay, B., & Bézard, B. 2021, Astronomy & Astrophysics, 646, A15
  * Burrows & Sharp (1999) Burrows, A., & Sharp, C. 1999, The Astrophysical Journal, 512, 843
  * Catling & Kasting (2017) Catling, D. C., & Kasting, J. F. 2017, Atmospheric evolution on inhabited and lifeless worlds (Cambridge University Press)
  * Changeat et al. (2020) Changeat, Q., Edwards, B., Al-Refaie, A. F., et al. 2020, arXiv preprint arXiv:2003.01486
  * Charnay et al. (2021) Charnay, B., Blain, D., Bézard, B., et al. 2021, Astronomy & Astrophysics, 646, A171
  * Cowan et al. (2009) Cowan, N. B., Agol, E., Meadows, V. S., et al. 2009, The Astrophysical Journal, 700, 915
  * Dasgupta (2013) Dasgupta, R. 2013, Reviews in Mineralogy and Geochemistry, 75, 183
  * Dasgupta & Grewal (2019) Dasgupta, R., & Grewal, D. S. 2019, Deep Carbon, 4
  * dos Santos et al. (2020) dos Santos, L. A., Ehrenreich, D., Bourrier, V., et al. 2020, Astronomy & Astrophysics, 634, L4
  * Fan et al. (2019) Fan, S., Li, C., Li, J.-Z., et al. 2019, The Astrophysical Journal Letters, 882, L1
  * Fischer et al. (2020) Fischer, R. A., Cottrell, E., Hauri, E., Lee, K. K., & Le Voyer, M. 2020, Proceedings of the National Academy of Sciences, 117, 8743
  * Fortney et al. (2020) Fortney, J. J., Visscher, C., Marley, M. S., et al. 2020, The Astronomical Journal, 160, 288
  * France et al. (2016) France, K., Loyd, R. P., Youngblood, A., et al. 2016, The Astrophysical Journal, 820, 89
  * Fulton & Petigura (2018) Fulton, B. J., & Petigura, E. A. 2018, The Astronomical Journal, 156, 264
  * Gaudi et al. (2020) Gaudi, B. S., Seager, S., Mennesson, B., et al. 2020, arXiv e-prints, arXiv:2001.06683.  https://arxiv.org/abs/2001.06683
  * Gialluca et al. (2021) Gialluca, M. T., Robinson, T. D., Rugheimer, S., & Wunderlich, F. 2021, Publications of the Astronomical Society of the Pacific, 133, 054401
  * Heng & Tsai (2016) Heng, K., & Tsai, S.-M. 2016, The Astrophysical Journal, 829, 104
  * Hirschmann (2016) Hirschmann, M. M. 2016, American Mineralogist, 101, 540
  * Hu (2021) Hu, R. 2021, The Astrophysical Journal, in press
  * Hu & Seager (2014) Hu, R., & Seager, S. 2014, ApJ, 784, 63, doi: 10.1088/0004-637X/784/1/63
  * Hu et al. (2012) Hu, R., Seager, S., & Bains, W. 2012, The Astrophysical Journal, 761, 166
  * Hu et al. (2013) —. 2013, The Astrophysical Journal, 769, 6
  * Isson & Planavsky (2018) Isson, T. T., & Planavsky, N. J. 2018, Nature, 560, 471
  * Jin & Mordasini (2018) Jin, S., & Mordasini, C. 2018, The Astrophysical Journal, 853, 163
  * Kempe & Degens (1985) Kempe, S., & Degens, E. T. 1985, Chemical Geology, 53, 95
  * Keppler & Golabek (2019) Keppler, H., & Golabek, G. 2019, Geochem. Perspect. Lett., 11, 12
  * Kite & Ford (2018) Kite, E. S., & Ford, E. B. 2018, The Astrophysical Journal, 864, 75
  * Kite et al. (2009) Kite, E. S., Manga, M., & Gaidos, E. 2009, The Astrophysical Journal, 700, 1732
  * Kite & Schaefer (2021) Kite, E. S., & Schaefer, L. 2021, The Astrophysical Journal Letters, 909, L22
  * Kitzmann et al. (2015) Kitzmann, D., Alibert, Y., Godolt, M., et al. 2015, Monthly Notices of the Royal Astronomical Society, 452, 3752
  * Koll & Cronin (2019) Koll, D. D., & Cronin, T. W. 2019, The Astrophysical Journal, 881, 120
  * Krissansen-Totton & Catling (2017) Krissansen-Totton, J., & Catling, D. C. 2017, Nature communications, 8, 1
  * Krissansen-Totton et al. (2018) Krissansen-Totton, J., Garland, R., Irwin, P., & Catling, D. C. 2018, The Astronomical Journal, 156, 114
  * Lee & Chiang (2016) Lee, E. J., & Chiang, E. 2016, The Astrophysical Journal, 817, 90
  * Levi et al. (2014) Levi, A., Sasselov, D., & Podolak, M. 2014, The Astrophysical Journal, 792, 125
  * Loftus et al. (2019) Loftus, K., Wordsworth, R. D., & Morley, C. V. 2019, The Astrophysical Journal, 887, 231
  * Madhusudhan et al. (2020) Madhusudhan, N., Nixon, M. C., Welbanks, L., Piette, A. A., & Booth, R. A. 2020, The Astrophysical Journal Letters, 891, L7
  * Marchi et al. (2019) Marchi, S., Raponi, A., Prettyman, T., et al. 2019, Nature Astronomy, 3, 140
  * Marty et al. (2016) Marty, B., Avice, G., Sano, Y., et al. 2016, Earth and Planetary Science Letters, 441, 91
  * Miller-Ricci et al. (2008) Miller-Ricci, E., Seager, S., & Sasselov, D. 2008, The Astrophysical Journal, 690, 1056
  * Misener & Schlichting (2021) Misener, W., & Schlichting, H. E. 2021, Monthly Notices of the Royal Astronomical Society, 503, 5658
  * Morley et al. (2014) Morley, C. V., Marley, M. S., Fortney, J. J., et al. 2014, The Astrophysical Journal, 787, 78
  * Mousis et al. (2020) Mousis, O., Deleuil, M., Aguichine, A., et al. 2020, The Astrophysical journal letters, 896, L22
  * Nixon & Madhusudhan (2021) Nixon, M. C., & Madhusudhan, N. 2021, Monthly Notices of the Royal Astronomical Society, 505, 3414
  * Noack et al. (2017) Noack, L., Rivoldini, A., & Van Hoolst, T. 2017, Physics of the Earth and Planetary Interiors, 269, 40
  * Owen & Wu (2017) Owen, J. E., & Wu, Y. 2017, The Astrophysical Journal, 847, 29
  * Pearson et al. (2006) Pearson, V., Sephton, M., Franchi, I., Gibson, J., & Gilmour, I. 2006, Meteoritics & Planetary Science, 41, 1899
  * Pidhorodetska et al. (2020) Pidhorodetska, D., Fauchez, T. J., Villanueva, G. L., Domagal-Goldman, S. D., & Kopparapu, R. K. 2020, The Astrophysical Journal Letters, 898, L33
  * Pierrehumbert & Gaidos (2011) Pierrehumbert, R., & Gaidos, E. 2011, The Astrophysical Journal Letters, 734, L13
  * Ramirez & Kaltenegger (2017) Ramirez, R. M., & Kaltenegger, L. 2017, The Astrophysical Journal Letters, 837, L4
  * Robinson et al. (2010) Robinson, T. D., Meadows, V. S., & Crisp, D. 2010, The Astrophysical Journal Letters, 721, L67
  * Rogers & Seager (2010) Rogers, L., & Seager, S. 2010, The Astrophysical Journal, 716, 1208
  * Schaefer & Fegley (2017) Schaefer, L., & Fegley, B. 2017, The Astrophysical Journal, 843, 120
  * Scheucher et al. (2020) Scheucher, M., Wunderlich, F., Grenfell, J. L., et al. 2020, The Astrophysical Journal, 898, 44
  * Seager et al. (2013) Seager, S., Bains, W., & Hu, R. 2013, The Astrophysical Journal, 777, 95
  * Seinfeld & Pandis (2016) Seinfeld, J. H., & Pandis, S. N. 2016, Atmospheric chemistry and physics: from air pollution to climate change (John Wiley & Sons)
  * Sotin et al. (2007) Sotin, C., Grasset, O., & Mocquet, A. 2007, Icarus, 191, 337
  * Summers (2005) Summers, D. P. 2005, Origins of Life and Evolution of Biospheres, 35, 299
  * Summers & Chang (1993) Summers, D. P., & Chang, S. 1993, Nature, 365, 630
  * Tsai et al. (2018) Tsai, S.-M., Kitzmann, D., Lyons, J. R., et al. 2018, The Astrophysical Journal, 862, 31
  * Tsiaras et al. (2019) Tsiaras, A., Waldmann, I. P., Tinetti, G., Tennyson, J., & Yurchenko, S. N. 2019, Nature Astronomy, 3, 1086
  * Valencia et al. (2013) Valencia, D., Guillot, T., Parmentier, V., & Freedman, R. S. 2013, The Astrophysical Journal, 775, 10
  * Vazan et al. (2020) Vazan, A., Sari, R., & Kessel, R. 2020, arXiv preprint arXiv:2011.00602
  * Venturini et al. (2020) Venturini, J. E., Guilera, O. M., Haldemann, J., Ronco, M. P., & Mordasini, C. 2020, Astronomy and astrophysics, 643, L1
  * Waldmann et al. (2015) Waldmann, I. P., Tinetti, G., Rocchetto, M., et al. 2015, The Astrophysical Journal, 802, 107
  * Woitke et al. (2021) Woitke, P., Herbort, O., Helling, C., et al. 2021, Astronomy & Astrophysics, 646, A43
  * Wunderlich et al. (2019) Wunderlich, F., Godolt, M., Grenfell, J. L., et al. 2019, Astronomy & Astrophysics, 624, A49
  * Yu et al. (2021) Yu, X., Moses, J. I., Fortney, J. J., & Zhang, X. 2021, The Astrophysical Journal, 914, 38
  * Zahnle et al. (2016) Zahnle, K., Marley, M. S., Morley, C. V., & Moses, J. I. 2016, The Astrophysical Journal, 824, 137
  * Zahnle & Marley (2014) Zahnle, K. J., & Marley, M. S. 2014, The Astrophysical Journal, 797, 41
  * Zeng et al. (2019) Zeng, L., Jacobsen, S. B., Sasselov, D. D., et al. 2019, Proceedings of the National Academy of Sciences, 116, 9723

##  Appendix A Reasonable lower bound of CO2

Is the 400-ppm CO2, or $4\times 10^{-4}$ bar partial pressure in a 1-bar
atmosphere, a reasonable lower bound of the CO2 partial pressure on an ocean
planet? We consider this question from a cosmochemical and geochemical
perspective. Assuming equilibrium (during planet formation) between a Fe-core,
a silicate mantle, and a well-mixed supercritical volatile envelope, the
partitioning of C mass between reservoirs is described by

${\rm C_{total}=C_{core}+C_{silicate}+C_{envelope},}$ (A1)

where all reservoir masses are in kg, and

${\rm C_{core}/M_{core}=D_{C}(C_{silicate}/M_{silicate}),}$ (A2)

where ${\rm D_{C}}$ is a dimensionless partition coefficient, ${\rm M_{core}}$
(kg) is the mass of the Fe-dominated core, and ${\rm M_{silicate}}$ (kg) is
the mass of the silicate mantle (molten during planet formation). For the
partitioning between the envelope and the silicate mantle,

${\rm
C_{envelope}k(g_{esi}/A_{esi})(\mu_{avg}/\mu_{C})s_{C}=C_{silicate}/M_{silicate},}$
(A3)

where ${\rm k}$ is a stochiometric correction from C mass to the mass of the
C-bearing species in the envelope (i.e., $44/12\sim 3.7$ for CO2), ${\rm
g_{esi}}$ is gravitational acceleration at the envelope-silicate interface in
m s-2, ${\rm A_{esi}}$ is the area of the envelope-silicate interface in m2,
${\rm\mu_{avg}}$ is the average molecular weight of the envelope (in Da),
${\rm\mu_{C}}$ is the molecular weight of the C-bearing species (in Da), and
${\rm s_{C}}$ is the solubility of the C-bearing species (in Pa-1). Here we
have assumed that the molten silicate layer is well-stirred.

Supposing ${\rm M_{core}/M_{silicate}\sim 0.5}$ (like Earth) and ${\rm
D_{C}\sim 10^{3}}$ (Dasgupta, 2013), then ${\rm C_{core}/C_{silicate}\sim
500}$. If ${\rm C_{silicate}/M_{silicate}\sim 50}$ ppm then ${\rm
C_{core}/M_{core}\sim 2.5}$ wt%, or ${\rm
C_{total}/(M_{core}+M_{silicate})\sim 1}$ wt%, which is a reasonable lower
bound for the primordial carbon endowment (see below). For ${\rm s_{C}=0.55}$
ppm/Mpa (Dasgupta & Grewal, 2019), the envelope partial pressure of the C
species ${\rm(=C_{envelope}k(g_{esi}/A_{esi})(\mu_{avg}/\mu_{C}))}$ is $\sim
10^{3}$ bars. For a ${\rm 5-M_{\oplus}}$ and ${\rm 1.5-R_{\oplus}}$
core+mantle (Zeng et al., 2019) that defines the envelope-silicate boundary,
and ${\rm\mu_{avg}/\mu_{C}}=0.4$ (appropriate for CO2 in a H2O-dominated
supercritical layer during planet formation), the CO2 mass in the envelope is
0.2% of an Earth mass. This estimate shows that even though most C is in the
core, still-significant reservoirs of C exist both in the silicate and in the
envelope (Dasgupta & Grewal, 2019; Bergin et al., 2015; Keppler & Golabek,
2019; Hirschmann, 2016). Recent indications that the partition coefficient
${\rm D_{C}}$ is $\ll 10^{3}$ at the pressures and temperatures that are
relevant for assembly of sub-Neptunes (Fischer et al., 2020) would imply even
more envelope C enrichment.

Following the formation of the liquid-water ocean, almost all of the CO2 will
be dissolved in the ocean. For a ${\rm 5-M_{\oplus}}$ water layer, the CO2
mass in the envelope estimated above corresponds to a concentration of $\sim
0.01$ mol/L of dissolved CO2. Here we have also assumed that the ocean is
well-stirred. A higher oceanic pH leads to more effective dissolution and less
CO2 in the atmosphere. As an extreme, if cations are leached from the silicate
and not charge-balanced by chloride ions, then an ocean composition with a pH
of 9 – 10 (“a soda lake”) will result (Kempe & Degens, 1985). Using the
equilibrium constant of carbonate and bicarbonate dissociation (Seinfeld &
Pandis, 2016), the CO2 partial pressure in equilibrium with this ocean would
be $5\times 10^{-5}\sim 7\times 10^{-4}$ bar, which is consistent with the
assumed lower bound.

The partition coefficient gives the ratios of concentration of a species in
the Fe-dominated core to the concentration of the same species in the silicate
mantle. Therefore doubling the total amount of C in the core+mantle will
double the concentration in the magma. What is the whole-planet C content? In
principle, a planet can form without accreting volatiles. However, a thin-
atmosphere sub-Neptune must have a thick volatile (H2O) layer in order to
match density data. It is very likely that a world that forms with 10s of wt%
H2O will also accrete abundant C. We develop this point in more detail in the
following paragraph.

At $T_{\rm eff}\sim 300$ K, the minimum liquid water content to explain most
sub-Neptune masses and radii is $>\sim 50$ wt% even if there is no Fe-metal
core (Mousis et al., 2020). This is more H2O than can possibly be produced by
hydrogen-magma reactions (Kite & Schaefer, 2021), and instead implies a
contribution of planet building blocks from the temperature range beyond the
water ice snowline. This is a zone where (in the Solar System), abundant
refractory carbon is found. Specifically, the carbon content of primitive
chondrite meteorites (CI and CM type) is 2-6 wt% (Pearson et al., 2006).
Although we do not fully understand the origin of this refractory carbon,
proposed mechanisms for forming this refractory carbon would also apply to
exoplanetary systems (Bergin et al., 2014). Therefore we assume a planet bulk
composition of ${\rm(2-6\ wt\%)\times(1-x)}$ carbon, where ${\rm x}$ is the
H2O mass fraction, and the remainder of the planet’s building blocks are
assumed to have a C content similar to that of primitive chondrites. This is a
conservative lower limit on bulk C content for a thin-H2-atmosphere sub-
Neptune, for the following two reasons. (i) It considers only refractory C,
not C ices (e.g., CO2 ice) which could be important in the case of whole-
planet migration. (ii) Some primitive bodies in the Solar System appear to be
more C-rich than the most primitive chondrite meteorites; for example, the
surface of the dwarf planet Ceres may contain 20 wt% C (Marchi et al., 2019).
These large bulk C contents map to substantial envelope C contents (Equations
A1-A3). As such, the $4\times 10^{-4}$ bar partial pressure of CO2, while not
the absolute lower limit, is a cosmochemically and geologically reasonable
lower bound of the CO2 partial pressure on an ocean planet.