The interior and atmosphere of the habitable-zone exoplanet K2-18b

Exoplanets orbiting M dwarfs present a valuable opportunity for their detection and atmospheric characterisation. This is evident from recent inferences of H$_2$O in such atmospheres, including that of the habitable-zone exoplanet K2-18b. With a bulk density between Earth and Neptune, K2-18b may be expected to possess a H/He envelope. However, the extent of such an envelope and the thermodynamic conditions of the interior remain unexplored. In the present work, we investigate the atmospheric and interior properties of K2-18b based on its bulk properties and its atmospheric transmission spectrum. We constrain the atmosphere to be H$_2$-rich with a H$_2$O volume mixing ratio of $0.02-14.8$%, consistent with previous studies, and find a depletion of CH$_4$ and NH$_3$, indicating chemical disequilibrium. We do not conclusively detect clouds/hazes in the observable atmosphere. We use the bulk parameters and retrieved atmospheric properties to constrain the internal structure and thermodynamic conditions in the planet. The constraints on the interior allow multiple scenarios between rocky worlds with massive H/He envelopes and water worlds with thin envelopes. We constrain the mass fraction of the H/He envelope to be $\lesssim 6$%; spanning $\lesssim 10^{-5}$ for a predominantly water world to $\sim6$% for a pure iron interior. The thermodynamic conditions at the surface of the H$_2$O layer range from the super-critical to liquid phases, with a range of solutions allowing for habitable conditions on K2-18b. Our results demonstrate that the potential for habitable conditions is not necessarily restricted to Earth-like rocky exoplanets.


INTRODUCTION
Recent exoplanet detection surveys have revealed high occurrence rates of low-mass planets orbiting M Dwarfs (Dressing & Charbonneau 2015;Mulders et al. 2015). The low masses, sizes and temperatures of M Dwarfs also mean that the planet-star contrast is favourable for planetary detection and characterisation. This 'smallstar opportunity' has led to several detections of lowmass planets (< 10M ⊕ ) in the habitable-zones of M Dwarf hosts such as Trappist-1 (Gillon et al. 2017), Proxima Centauri (Anglada-Escudé et al. 2016), K2-18 Foreman-Mackey et al. 2015), and LHS 1140 (Dittmann et al. 2017).
The habitable-zone transiting exoplanet K2-18b is a particularly good example Montet et al. 2015). The close proximity and small size of its host star make precise measurements of the plane- E-mail: nmadhu@ast.cam.ac.uk tary mass, radius, and atmospheric spectra viable (Benneke et al. 2017;Cloutier et al. 2019), as exemplified by the recent detection of H 2 O in its atmosphere (Tsiaras et al. 2019;Benneke et al. 2019). The habitable-zone temperature of K2-18b provides further impetus for detailed characterisation of its interior and atmosphere.
Given its mass (M p = 8.63 ± 1.35 M ⊕ , Cloutier et al. 2019) and radius (R p = 2.610 ± 0.087 R ⊕ , Benneke et al. 2019), K2-18b has a bulk density (2.67 +0.52 −0.47 g/cm 3 , Benneke et al. 2019). This density, between that of Earth and Neptune, may be thought to preclude a purely rocky or icy interior and require a hydrogen-rich outer envelope. However, the extent of such an envelope and the conditions at the interface between the envelope and the underlying interior have not been explored. We note that the mass and radius of the planet have recently been revised (Benneke et al. 2019), which may have impacted inferences made using previous values (Cloutier et al. 2017;Tsiaras et al. 2019).
Previous studies of planets with similar masses and radii, such as GJ 1214b, suggested envelope mass frac-  tions 7% (Rogers & Seager 2010;Nettelmann et al. 2011;Valencia et al. 2013). GJ 1214b is expected to host super-critical H 2 O below the envelope at pressures and temperatures too high to be conducive for life (Rogers & Seager 2010). However, while GJ 1214b has an equilibrium temperature (T eq ) of ∼ 500 K, K2-18b may be more favourable given its lower T eq ∼ 250 − 300 K. In the present work, we conduct a systematic study to constrain both the atmospheric and interior composition of K2-18b based on extant data along with detailed atmospheric retrievals and internal structure models.

ATMOSPHERIC PROPERTIES
We retrieve the atmospheric properties of K2-18b using its broadband transmission spectrum reported by Benneke et al. (2019). The data include observations from the HST WFC3 G141 grism (1.1-1.7 µm), photometry in the Spitzer IRAC 3.6 µm and 4.5 µm bands, and optical photometry in the K2 band (0.4-1.0 µm). We perform the atmospheric retrieval using an adaptation of the AURA retrieval code (Pinhas et al. 2019;. Our model solves line-byline radiative transfer in a plane-parallel atmosphere in transmission geometry. The model assumes hydrostatic equilibrium and considers prominent opacity sources in the observed spectral bands as well as homogeneous/inhomogeneous cloud/haze coverage. Clouds are included through a gray cloud deck with cloud-top pressure (P c ) as a free parameter. Hazes are included as a modification to Rayleigh-scattering through parameters for the scattering slope (γ) and a Rayleigh-enhancement factor (a).  (Barber et al. 2014), and collision-induced absorption due to H 2 -H 2 and H 2 -He (Richard et al. 2012).
The model comprises 16 free parameters: abundances of 5 molecules, 6 parameters for the pressure-

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Note-Four models are considered with different treatments of clouds and hazes. For each model, the volume mixing ratios (log(XH 2 O), log(XCH 4 ), and log(XNH 3 )) are shown along with the Bayesian evidence (ln(Z)) and detection significance (DS). The DS is derived from the Bayesian evidence and a value below 2.0σ is considered weak (Trotta 2008). The preference of the reference model (case 1) over other models is quantified by the DS. For example, the DS for case 2 implies that case 1 is preferred over case 2 at 1.2σ. H2O is detected at 3.25σ and clouds/hazes at only ∼1σ.
temperature (P -T ) profile, 4 cloud/haze parameters, and 1 parameter for the reference pressure P ref at R p (e.g., . The Bayesian parameter estimation is conducted using the Nested Sampling algorithm MultiNest (Feroz et al. 2009) through Py-MultiNest (Buchner et al. 2014). We conduct retrievals for four model configurations: (1) a full model including inhomogeneous clouds and hazes, (2) a clear atmosphere, (3) an atmosphere with an opaque cloud deck but no hazes, and (4) an atmosphere with inhomogeneous clouds but no hazes. The atmospheric constraints are shown in Figure 1 and Table 1. We confirm the high-confidence detection of H 2 O in a H 2 -rich atmosphere as reported by Benneke et al. (2019) and Tsiaras et al. (2019). Our abundance estimates are consistent to within 1σ between all four model configurations and with Benneke et al. (2019). The derived H 2 O volume mixing ratio ranges between 0.02-14.80%, with median values of 0.7-1.6% between the 4 model cases, as shown in Table 1. The case with an opaque cloud deck (a clear atmosphere) retrieves slightly higher (lower) H 2 O abundances as expected . Our derived H 2 O abundance range corresponds to an O/H ratio of 0.2-176.8×solar, assuming all the oxygen is in H 2 O as expected in H 2 -rich atmospheres at such low temperatures (Burrows & Sharp 1999). The median H 2 O abundance is 9.3×solar for the full model, case 1. We cannot compare our results with Tsiaras et al. (2019) as their retrievals were based on only the HST WFC3 data and used older measurements of the planetary mass and radius which could have biased their inferences.
We find a depletion of CH 4 and NH 3 in the atmosphere. For a H 2 -rich atmosphere at ∼300 K, CH 4 and NH 3 are expected to be dominant carriers of carbon and nitrogen, respectively, in chemical equilibrium (Burrows & Sharp 1999), as also seen for the gas and ice giants in the solar system (Atreya et al. 2018). Assuming solar elemental ratios (i.e., C/O = 0.55, N/O = 0.14), the CH 4 /H 2 O (NH 3 /H 2 O) ratio is expected to be ∼0.5 (∼0.1). However, we do not detect CH 4 or NH 3 despite their strong absorption in the HST WFC3 and/or Spitzer 3.6 µm bands. As shown in Figure 1, the retrieved posteriors of the CH 4 and NH 3 abundances are largely sub-solar, with 99% upper limits of 3.47×10 −2 and 5.75×10 −5 , respectively. These sub-solar values are in contrast to the largely super-solar H 2 O, arguing against chemical equilibrium at solar elemental ratios.
We do not find strong evidence for clouds/hazes in the atmosphere. Our model preference for clouds/hazes, relative to the cloud-free case, is marginal (1.2σ) compared to Benneke et al. (2019) (2.6σ). Our retrieved cloud-top pressure (P c ) for the full case is weakly constrained to 0.1 mbar to 2 bar, close to the observable photosphere. Finally, we retrieve P ref for the full case to be 12 − 174 mbar corresponding to R p . The median value of 0.05 bar is used as the surface boundary condition, pressure P 0 , for the internal structure models in section 3.1.

INTERNAL STRUCTURE AND COMPOSITION
In this section we use the observed bulk properties of K2-18b, namely the planetary mass (M p ), radius (R p ), and its atmospheric properties, to constrain its internal structure and thermodynamic conditions.

Internal structure model
We model the interior of the planet with a canonical four-layer structure. The model comprises a twocomponent Fe+rock core consisting of an inner Fe layer and an outer silicate layer, a layer of H 2 O, and an outer H/He envelope. Such a model spans the possible internal structures and compositions of super-Earths and mini-Neptunes (e.g. Valencia et al. 2010Valencia et al. , 2013; Rogers et al. The model solves the standard internal structure equations of hydrostatic equilibrium and mass continuity assuming spherical symmetry. The equation of state (EOS) for each of the two inner layers is adopted from Seager et al. (2007) who use the Birch-Murnaghan EOS (Birch 1952) for Fe (Ahrens 1995) and MgSiO 3 perovskite (Karki et al. 2000). For the H 2 O layer we use the temperature-dependent H 2 O EOS compiled by Thomas & Madhusudhan (2016)  The EOS in the H/He and H 2 O layers can have a significant temperature dependence which we consider in our model. Past studies (Rogers et al. 2011;Valencia et al. 2013) considered analytic P -T profiles for irradiated atmospheres derived using double gray approx-imations (Hansen 2008;Guillot 2010) with the internal and external fluxes and opacities as free parameters. We calculate self-consistent dayside P -T profiles for K2-18b in the H/He envelope using the GENESIS code (Gandhi & Madhusudhan 2017). GENESIS solves lineby-line radiative transfer under assumptions of hydrostatic, radiative-convective and thermochemical equilibrium. We include opacity due to H 2 O (Rothman et al. 2010), as detected in the transmission spectrum (section 2), H 2 Rayleigh scattering, clouds and H 2 -H 2 and H 2 -He collision-induced absorption. We use a H 2 O abundance of 10×solar (see section 2) and also use 10×solar abundances for the cloud species. We include KCl, ZnS and Na 2 S clouds (Morley et al. 2013), for which we obtain opacities from Pinhas & Madhusudhan (2017). We further include water ice clouds using opacities from Budaj et al. (2015).
The P -T profile also depends on the planetary internal flux, which is characterised by the internal temperature T int . We consider values of T int which span the range expected for a planet with the mass and radius of K2-18b and an age of 1 − 10 Gyr, with envelope compositions from solar to water-rich. We choose end-member cases of T int = 25K and 50K, consistent with previous studies on planets of similar mass and radius, e.g., GJ 1214b (e.g., Valencia et al. 2013). The GENESIS models are calculated between pressures of 10 −5 − 10 3 bar, and assume full redistribution of the incident stellar irradiation. We explore a range of P -T profiles and choose two representative cases, with different T int , discussed further in sections 3.2 and 3.3. Where required by the internal structure model, the bottom of the P -T profile of the H/He envelope is continued to deeper pressures using the adiabatic gradient from Chabrier et al. (2019). We also employ an adiabatic temperature profile in the H 2 O layer, following Thomas & Madhusudhan (2016). Figure 2 shows mass-radius relations for models with different interior compositions. We explore the full range of plausible interior compositions in three components: x core = x Fe + x rock , x H2O , and x env , where x i = M i /M p is the mass fraction of each component i. For each atmospheric P -T profile considered, we explore two different core compositions: (1) an Earth-like core made of 33% Fe, 67% rock by mass, and (2) a pure Fe core, the densest possible composition. Here, we discuss results from two end-member cases: (1) a pure Fe core with T int = 25K, and (2) an Earth-like (33% Fe) core with T int = 50K. Solutions for all other cases lie between these two cases.

Constraints on interior composition
As shown in Figure 3, while a wide range of core and H 2 O mass fractions are permitted, we place a stringent upper limit on the mass fraction of the H/He envelope: x env = 6.2%. This maximal x env corresponds to the case of a pure Fe core, with x core ∼ 94%, underlying the H/He envelope with no x H2O ; here it is assumed that the atmospheric H 2 O is not mixed in the envelope. However, if the retrieved atmospheric H 2 O abundance is assumed to be well mixed in the envelope then the maximal x env = 6% with x H2O = 0.4% by mass; low, but still significantly higher than that of the Earth's oceans (∼0.02%). We find that a substantial gaseous H/He envelope is not necessary to explain the density of K2-18b. Figure 3 shows the x env required for different x core . At one extreme, a ∼100% H 2 O interior with no rocky core can explain the data with an x env of just ∼10 −6 , comparable to the mass fraction of the Earth's atmosphere. The presence of a rocky core would necessitate at least a thin H/He envelope. However, even considering a reasonable x core = 10−50% still requires x env of only ∼ 10 −5 −10 −2 , as shown in Figure 3. Model solutions with the hotter P -T profile and/or lower Fe content in the core require smaller x env for a given x core .
We have also considered models with miscible H 2 O and H/He envelopes. We follow the approach of Soubi-ran & Militzer (2015), using an additive volume law for mixtures. Assuming that the median H 2 O mixing ratio in the atmosphere is representative of the mixed (H 2 O-H/He) envelope, we find that the difference in radius between the mixed and non-mixed models is less than half of the measured uncertainty (see Figure 2). The constraint on the envelope mass fraction from this mixed case is x env = 2.5 − 6.4%, consistent with, and a subset of, the constraints discussed above. Note that in this case x env includes both the H/He and H 2 O mass fraction.

Atmosphere-Ocean Boundary
Our constraints on the interior compositions of K2-18b result in a wide range of thermodynamic conditions at the H 2 O-H/He boundary (HHB). The pressure (P HHB ) and Temperature (T HHB ) at the HHB for the model solutions are shown in Figure 4. Each point on the HBB loci denotes the transition from the P -T profile in the H/He envelope to the corresponding H 2 O adiabat. The P HHB and T HHB depend on the H/He envelope mass fraction. For a given P -T profile, larger envelopes result in higher P HHB and T HHB . For example, solutions with x env 1% lead to P HHB and T HHB corresponding to the super-critical phase of H 2 O. As shown in Figure 3, solutions with higher x env correspond to higher x core and lower x H2O . Conversely, solutions with lower x core , and hence lower x env and higher x H2O , lead to lower P HHB and T HHB with H 2 O in vapour or liquid phases at the HHB. For example, an x core 30% leads to a P HHB and T HHB corresponding to the liquid phase of H 2 O, for the cooler P -T profile (with T int = 25K). For x core ∼10% or less, the P HHB and T HHB approach STP conditions for liquid H 2 O. Below the HHB, H 2 O is found in increasingly dense phases spanning liquid, vapour, super-critical, and ice states depending on the location of the HHB and the extent of the H 2 O layer, as shown in Figure 4. In the case of a mixed H 2 O-H/He envelope, the HHB is undefined as it corresponds to an extreme case with no pure H 2 O layer.

DISCUSSION
Our constraints on the interior and atmospheric properties of K2-18b provide insights into its physical conditions, origins, and potential habitability.

Possible Compositions and Origins
Here we discuss three representative classes that span the range of possible compositions, as indicated in Fig-ures 2, 3 and 4. The specific cases chosen here fit the M p and R p exactly, as shown in figure 2. A wider range of solutions exist in each of these classes within the 1σ uncertainties.
Case 1: Rocky World. One possible scenario is a massive rocky interior overlaid by a H/He envelope. For example, a pure Fe core of 94.7% by mass with an almost maximal H/He envelope of 5% explains the data with minimal x H2O = 0.3%, consistent with our retrieved H 2 O abundance in the atmosphere. The HHB in this case is at ∼ 10 6 bar, yielding supercritical H 2 O close to the ice X phase. It is also possible in this case that the H 2 O and H/He are mixed, meaning the HHB is not well-defined. Such a scenario is consistent with either H 2 outgassing from the interior (Elkins-Tanton & Seager 2008;Rogers & Seager 2010) or accretion of an H 2 -rich envelope during formation (Lee & Chiang 2016).
Case 2: Mini-Neptune. There are a range of plausible compositions consisting of a non-negligible H/He envelope in addition to significant H 2 O and core mass fractions, akin to canonical models for Neptune and Uranus (Guillot & Gautier 2015). One such example is a 45% Earth-like core with x env = 0.03% and x H2O = 54.97%. In this case the HHB is at P HHB = 700bar and T HHB = 1500K, with H 2 O in the supercritical phase.
Case 3: Water World. A ∼100% water world with a minimal H 2 -rich atmosphere (x env ∼ 10 −6 ) is permissible by the data. However, such an extreme case is implausible from a planet formation perspective; some amount of rocky core is required to initiate further ice and gas accretion (Léger et al. 2004;Rogers et al. 2011;Lee & Chiang 2016). For example, a planet with x core = 10%, x H2O = 89.994% and a thin H/He envelope (x env = 0.006%) can explain the data. For this case, P HHB = 130bar and T HHB = 560K, corresponding to liquid H 2 O. For the same core fraction, solutions with even smaller H/He envelopes are admissible within the 1σ uncertainties on M p and R p , leading to P HHB and T HHB approaching habitable STP conditions.

Potential Habitability
A notional definition of habitability argues for a planetary surface with temperatures and pressures conducive to liquid H 2 O (e.g., Kasting et al. 1993;Meadows & Barnes 2018). Living organisms are known to thrive in Earth's extreme environments (extremophiles). Their living conditions span the phase space of liquid H 2 O up to ∼1000 bar pressures at the bottom of the Marianas Trench and ∼400 K temperatures near hydrothermal vents (e.g., Merino et al. 2019).
Whether or not habitable conditions prevail on K2-18b depends on the extent of the H/He envelope. The thermodynamic conditions at the surface of the H 2 O layer span a wide range in the H 2 O phase diagram. While most of these solutions lie in the super-critical phase, many others lie in the liquid and vapour phases. Model solutions with core mass fractions <15% and H/He envelopes 10 −3 allow for liquid H 2 O at Earthlike habitable conditions discussed above. One plausible scenario is an ocean world, as discussed in section 4.1, with liquid water approaching STP conditions (e.g., 300 K, ∼1-10 bar) underneath a thin H/He atmosphere (x env 10 −5 ).
A number of studies in the past have argued for potential habitability on planets with H/He-rich atmospheres orbiting M Dwarfs (e.g., Pierrehumbert & Gaidos 2011;Seager et al. 2013;Koll & Cronin 2019). Given our constraints above, we find that K2-18b has a realistic chance of being habitable. Furthermore, our constraints on CH 4 and NH 3 suggest chemical disequilibrium. Among other possibilities for chemical disequilibrium, e.g. photochemistry, the potential influence of biochemical processes may not be entirely ruled out. Future observations, e.g. with the James Webb Space Telescope, will have the potential to refine our findings. We argue that planets such as K2-18b can indeed have the potential to approach habitable conditions and searches for biosignatures should not necessarily be restricted to smaller rocky planets.