Characterisation of the hydrospheres of TRAPPIST-1 planets
Lorena Acuña, Magali Deleuil, Olivier Mousis, Emmanuel Marcq, Maëva Levesque, Artyom Aguichine
AAstronomy & Astrophysics manuscript no. arxiv © ESO 2021January 21, 2021
Characterisation of the hydrospheres of TRAPPIST-1 planets
Lorena Acuña , Magali Deleuil , Olivier Mousis , Emmanuel Marcq , Maëva Levesque , and Artyom Aguichine Aix Marseille Univ, CNRS, CNES, LAM, Marseille, Francee-mail: [email protected] LATMOS / IPSL, UVSQ Université Paris-Saclay, Sorbonne Université, CNRS, Guyancourt, FranceReceived 12 November 2020; accepted -
ABSTRACT
Context.
Planetary mass and radius data are showing a wide variety in densities of low-mass exoplanets. This includes sub-Neptunes,whose low densities can be explained with the presence of a volatile-rich layer. Water is one of the most abundant volatiles, which canbe in the form of di ff erent phases depending on the planetary surface conditions. To constrain their composition and interior structure,it is required to develop models that calculate accurately the properties of water at its di ff erent phases. Aims.
We present an interior structure model that includes a multiphase water layer with steam, supercritical and condensed phases.We derive the constraints for planetary compositional parameters and their uncertainties, focusing on the multiplanetary systemTRAPPIST-1, which presents both warm and temperate planets.
Methods.
We use a 1D steam atmosphere in radiative-convective equilibrium with an interior whose water layer is in supercriticalphase self-consistently. For temperate surface conditions, we implement liquid and ice Ih to ice VII phases in the hydrosphere. Weadopt a MCMC inversion scheme to derive the probability distributions of core and water compositional parameters
Results.
We refine the composition of all planets and derive atmospheric parameters for planets b and c. The latter would be in apost-runaway greenhouse state and could be extended enough to be probed by space mission such as JWST. Planets d to h presentcondensed ice phases, with maximum water mass fractions below 20%.
Conclusions.
The derived amounts of water for TRAPPIST-1 planets show a general increase with semi-major axis, with the exceptionof planet d. This deviation from the trend could be due to formation mechanisms, such as migration and an enrichment of water in theregion where planet d formed, or an extended CO -rich atmosphere. Key words. planets and satellites: interiors – planets and satellites: composition – planets and satellites: atmospheres – planets andsatellites: individual: TRAPPIST-1 – methods: statistical – methods: numerical
1. Introduction
Ongoing space missions such as CHEOPS (Benz 2017) andTESS (Ricker et al. 2015), and their follow-up with ground-based radial velocity telescopes, are confirming the existence oflow-mass exoplanets with a wide range of densities. These den-sities range from the values typically inferred for the Earth orMercury to those measured in Uranus and Neptune. The exo-planets in the former class are mainly composed of a Fe-richcore and a silicate mantle, while the latter class has a layer thatis rich in volatiles. Water is the most abundant and least densevolatile after H and He (Forget & Leconte 2014), which makes ita likely species to constitute the volatile reservoir in these plan-ets. Several studies have investigated the interior structure andcomposition of water-rich planets (Sotin et al. 2007; Seager et al.2007; Dorn et al. 2015; Zeng et al. 2019), but focused mainlyon its condensed phases. Nonetheless, many sub-Neptunes areclose to their host star and receive enough irradiation to triggera runaway greenhouse state in which water is present as steam.In some cases, the high pressure and temperature conditions canrender the hydrosphere in supercritical and plasma, or even su-perionic phases (Mazevet et al. 2019; French et al. 2016). There-fore, it is crucial to include the modeling of all possible phasesof water to provide an accurate description of its presence onthe planetary surface. Moreover, the surface conditions are de-termined by the greenhouse e ff ect caused by atmospheric gases,making the modelling of radiative-convective equilibrium in at-mospheres a key parameter to determine in which phase water could be present on the surface. Most of interior structure mod-els represent the planetary atmosphere as a gas layer with a sim-plified isothermal temperature profile (Dorn et al. 2018, 2017b),which is very di ff erent from the temperature profile in the con-vective deep layers of thick atmospheres (Marcq 2012).Multiplanetary systems are unique environments that presentboth planets that can hold condensed phases as well as highly-irradiated planets with steam atmospheres. In this study, we de-velop a planet interior model suitable for the di ff erent conditionsat which water can be found in low-mass planets. Our implemen-tation includes a supercritical water layer, introduced in Mousiset al. (2020), coupled with a 1D radiative-convective atmospheremodel (Marcq 2012; Marcq et al. 2017; Pluriel et al. 2019) to cal-culate the total radius of the highly-irradiated planets with waterself-consistently. Furthermore, for temperate planets, we haveupdated the interior model presented in Brugger et al. (2016,2017) to include ice phases Ih, II, III, V and VI. We introducethese models in a MCMC Bayesian analysis scheme adaptedfrom Dorn et al. (2015). This allows us to derive the water massfraction (WMF) and core mass fraction (CMF) with their asso-ciated confidence intervals that reproduce the observed radius,mass and stellar composition measurements.We use this model to explore the possible water content ofthe TRAPPIST-1 system, an ultra-cool M dwarf that hosts sevenlow-mass planets in close-in orbits. Three of these planets arelocated in the habitable zone (Grimm et al. 2018), meaning thatthey can hold liquid water or ice Ih on their surfaces. Although Article number, page 1 of 11 a r X i v : . [ a s t r o - ph . E P ] J a n & A proofs: manuscript no. arxiv all planets in TRAPPIST-1 system have masses and radii thatare characteristic of rocky planets, their di ff erences in densityindicate that each planet has a di ff erent volatile content. Thismakes this planetary system ideal for testing planet interior, at-mospheric structure and formation scenarios.In Sect. 2 we describe the complete interior structure model,including the new updates for the supercritical and ice phases,the coupling between the interior and the atmosphere for steamand supercritical planets, and the MCMC Bayesian algorithm.The parameters for the TRAPPIST-1 planets used in this studyare summarized in Sect. 3, including mass, radius and Fe / Simolar ratio. The results of our analysis of the hydrospheres ofTRAPPIST-1 planets are described in Sect. 4. We compare ourresults with previous works and discuss the implications of ourwater estimates for planet formation in Sect. 5. We finally exposeour conclusions in Sect. 6.
2. Planetary structure model
For consistency, we recall the main principles of the interiorstructure model. The basis of our model is explained in Brug-ger et al. (2016, 2017). The 1D interior structure model takes asinput the mass and the composition of the planet, which is pa-rameterized by the CMF and WMF. The structure of the planetis stratified in 3 layers: a core, a mantle and a hydrosphere. Thepressure, temperature, gravity acceleration and density are com-puted at each point of the one-dimensional spatial grid along theradius of the planet. The pressure, P ( r ), is obtained by integrat-ing the hydrostatic equilibrium (Eq. 1); the gravitational acceler-ation, g ( r ), by solving Gauss’s theorem (Eq. 2); the temperature, T ( r ), with the adiabatic gradient (Eq. 3); and the density, ρ ( r ),with the Equation of State (EOS). m is the mass at radius r , G is the gravitational constant, and γ and φ are the Gruneisen andthe seismic parameters, respectively. Their formal macroscopicdefinitions are shown in equation 4, where E is the internal en-ergy and V is the volume. The Gruneisen parameter is a thermo-dynamic parameter that describes the dependence of the vibra-tional properties of a crystal with the size of its lattice. It relatesthe temperature in a crystalline structure to the density, which iscalculated by the EOS. The seismic parameter defines how seis-mic waves propagate inside a material. It is related to the slopeof the EOS at constant temperature (Brugger et al. 2017; Sotinet al. 2007). dPdr = − ρ g (1) dgdr = π G ρ − Gmr (2) dTdr = − g γ T φ (3) φ = dPd ργ = V (cid:32) dPdE (cid:33) V (4)The boundary conditions are the temperature and pressure atthe surface, and the gravitational acceleration at the center of theplanet. The value of the latter is zero. The total mass of the planet is calculated with Eq. 5, which is derived from the conservationof mass (Brugger et al. 2017; Sotin et al. 2007). Once the totalinput mass of the planet is reached and the boundary conditionsare fulfilled, the model has converged. dmdr = π r ρ (5)Depending on the surface conditions, the hydrosphere canbe present in supercritical, liquid or ice states. For each of thesephases of water, we use a di ff erent EOS and Gruneisen parameterto compute their P-T profiles and density accurately. In Sect. 2.1we describe the updates to the supercritical water layer with re-spect to the model depicted in Mousis et al. (2020), while inSect. 2.2 we present the implementation of the hydrosphere inice phases. Finally, the coupling between the atmosphere andthe interior model with planets whose hydrosphere is in steamor supercritical phases is explained in Sect. 2.3, followed by thedescription of the MCMC algorithm in Sect. 2.4. If the planet is close enough to its host star, the upper layerof the hydrosphere corresponds to a hot steam atmosphere,whose temperature at the base is determined by the radiative-convective balance calculated by the atmosphere model (Marcq2012; Marcq et al. 2017). When the pressure and temperatureat the surface, which is defined as the base of the hydrospherelayer, are above the critical point of water, we include a super-critical water layer extending from the base of the hydrosphereto a height corresponding to the phase change to steam (Mousiset al. 2020). We updated the EOS for this layer to the EOS intro-duced by Mazevet et al. (2019), which is a fit to the experimentaldata provided by the International Association for the Propertiesof Water and Steam (IAPWS) (Wagner & Pruß 2002) for thesupercritical regime, and quantum molecular dynamics (QMD)simulations data for plasma and superionic water (French et al.2009). The IAPWS experimental data span a temperature rangeof 251.2 to 1273 K and of 611.7 to 10 Pa in pressure, whiletheir EOS can be extrapolated up to 5000 K in temperature and10 Pa in pressure (Wagner & Pruß 2002). The validity range ofthe EOS presented in Mazevet et al. (2019) includes that of theIAPWS plus the region in which the QMD simulations are appli-cable, which corresponds to a temperature from 1000 K to 10 K and densities in the 1–10 g / cm range. These densities arereached at high pressures, i.e., in the 10 –10 Pa range. Follow-ing Eq. 3, the adiabatic gradient of the temperature is specifiedby the Gruneisen and the seismic parameters. These are depen-dent on the derivatives of the pressure with respect to the densityand the internal energy (Eq. 4). We make use of the specific in-ternal energy and density provided by Mazevet et al. (2019) tocalculate them.
We extended the hydrosphere in Brugger et al. (2016, 2017) withliquid and high pressure ice VII by adding 5 more condensedphases: ice Ih, II, III, V and VI. EOS for ice Ih has been devel-oped by Feistel & Wagner (2006) with minimization of the Gibbspotential function from the fit of experimental data. It covers allthe pressure and temperature range in which water forms ice Ih.Fei et al. (1993) proposed a formalism to derive the EOS ofices II, III and V. These EOS have the form of V = V ( P , T ), Article number, page 2 of 11orena Acuña et al.: Characterisation of the hydrospheres of TRAPPIST-1 planets
Table 1: EOS and reference thermal parameters for ices Ih, II, III, V and VI. This includes the reference values for the density ρ ,the temperature T , the bulk modulus K T and its derivative K (cid:48) T , the heat capacity C p ( T ), and the thermal expansion coe ffi cient α .Phase ρ [kg m − ] T [K] K T [GPa] K (cid:48) T C p ( T ) [J kg − K − ] α [10 − K − ] ReferencesIh 921.0 248.15 9.50 5.3 1913.00 147 1, 8II 1169.8 237.65 14.39 6.0 2200.00 350 1, 2, 7III 1139.0 237.65 8.50 5.7 2485.55 405 3, 4, 5, 7V 1235.0 237.65 13.30 5.2 2496.63 233 1, 4, 5, 7VI 1270.0 300.00 14.05 4.0 2590.00 146 4, 6, 7 References. (1) Gagnon et al. (1990); (2) Báez & Clancy (1995); (3) Tulk et al. (1997); (4) Tchijov et al. (2004); (5) Shaw (1986); (6) Bezacieret al. (2014); (7) Choukroun & Grasset (2010); (8) Feistel & Wagner (2006) which can be found by integrating the following di ff erentialequation (Tchijov et al. 2004): dVV = α dT − β dP (6)where α is the thermal expansion coe ffi cient and β the isothermalcompressibility coe ffi cient. If the relationship between the spe-cific volume, V , and the pressure, P , at a constant temperature T = T is determined, Eq. 6 can be integrated as: V ( P , T ) = V ( P , T ) exp (cid:32)(cid:90) TT α ( P , T (cid:48) ) dT (cid:48) (cid:33) (7)Fei et al. (1993) proposed the following expression for thethermal expansion coe ffi cient α : α ( P , T ) = α ( P , T ) (cid:32) + K (cid:48) T K T P (cid:33) − η (cid:32) + K (cid:48) T K T P (cid:33) − η = − ρ d ρ ( T ) dT (cid:32) + K (cid:48) T K T P (cid:33) − η (cid:32) + K (cid:48) T K T P (cid:33) − η (8)where η is an adjustable parameter estimated from the fitting ofexperimental data. Its value is 1.0 for ice II and ice III (Leonet al. 2002) and 7.86 for ice V (Shaw 1986). ρ is the density, α ( P , T ) is the coe ffi cient of thermal expansion at a referencepressure P , K T is the isothermal bulk modulus at the referencetemperature T , and K (cid:48) T is the first derivative of the isothermalbulk modulus at the reference temperature. Hence, by substitut-ing Eq. 8 in Eq. 7 and integrating, we obtain the following EOSfor high-pressure ice: V ( P , T ) = V ( P , T ) exp ln (cid:32) ρ ( T ) ρ ( T ) (cid:33) (cid:32) + K (cid:48) T K T P (cid:33) − η (cid:32) + K (cid:48) T K T P (cid:33) − η (9)The final expression (Eq. 9) requires the knowledge of the vari-ation of the specific volume, V ( P , T ), with pressure at the refer-ence temperature T . Moreover, the variation of the density withtemperature, ρ ( T ), and the bulk modulus with its derivative atthe reference temperature, K T and K (cid:48) T , must also be provided.In Table 1 we specify the data and references to obtain theseparameters for each ice phase.In the case of ice VI, we adopt the second-order Birch-Murnaghan (BM2) formulation, which is: P = K T (cid:32) ρρ (cid:33) − (cid:32) ρρ (cid:33) , (10)where ρ is the reference density for ice VI. We also introduce athermal correction to the density since the pressure also dependson the temperature: ρ ( T ) = ρ exp ( α ( T − T )) (11)where α is the reference coe ffi cient of thermal expansion. In-terfaces between liquid and ice layers are established by phasetransition functions from Dunaeva et al. (2010). We use a one-dimensional atmosphere model designed to com-pute radiative transfer and pressure-temperature ( P , T ) profilesfor water and CO atmospheres (Marcq 2012; Marcq et al.2017). The formation of water clouds is considered in the com-putation of the albedo. The atmosphere is in radiative equilib-rium, and presents a composition of 99% water and 1% CO .The density of steam is obtained using a non-ideal EOS (Haaret al. 1984).If the surface pressure is below 300 bar, the atmosphere andthe interior are coupled at the atmosphere-mantle boundary andwater does not reach the supercritical regime. However, if thesurface pressure is greater than 300 bar, the atmosphere and theinterior are coupled at this pressure level and a layer of water insupercritical phase forms between the atmosphere and the man-tle. The pressure level at 300 bar is close enough to the criticalpoint of water at 220 bar to avoid the atmosphere model takeover pressures and temperatures where the temperature profile isadiabatic.The top-of-atmosphere pressure is set to 20 mbar, which cor-responds to the observable transiting radius (Mousis et al. 2020;Grimm et al. 2018). We denote the radius and mass from thecenter of the planet to this pressure level the total radius andmass, R total and M total , respectively. We also define the radius andthe mass that comprise the core, mantle and supercritical layersas the bulk radius and mass, R bulk and M bulk , respectively. Theatmosphere model provides the Outgoing Longwave Radiation(OLR), albedo, thickness and mass of the atmosphere as a func-tion of the bulk mass and radius, and the surface temperature. Ifthe atmosphere of the planet is in radiative equilibrium, the OLRis equal to the radiation the planet absorbs from its host star, F abs . The OLR depends on the e ff ective temperature since OLR Article number, page 3 of 11 & A proofs: manuscript no. arxiv
Fig. 1: Structural diagram of the coupling between the interiorstructure model and the atmosphere model. T base is the tem-perature at the bottom of the steam atmosphere in radiative-convective equilibrium. z and M atm denote the atmosphericthickness and mass, respectively. R bulk and M bulk correspond tothe planet bulk radius and mass, respectively. R guess refers to theinitial guess of the bulk radius, while R interior is the output bulkradius of the interior structure model in each iteration. = σ T ff , where σ corresponds to the Stefan-Boltzmann constant.To calculate the absorbed radiation F abs , we first compute theequilibrium temperature, which is T eq = (1 − A B ) . (cid:32) R (cid:63) a d (cid:33) . T (cid:63) , (12)where A B is the planetary albedo, R (cid:63) and T (cid:63) are the radius ande ff ective temperature of the host star, respectively. a d is the semi-major axis of the planet. The absorbed radiation is then calcu-lated as F abs = σ T . (13)Figure 1 shows the algorithm we implemented to couplethe planetary interior and the atmosphere. The interior struc-ture model calculates the radius from the center of the planetto the base of the steam atmosphere. For a fixed set of bulk massand radius, the OLR depends on the surface temperature. Con-sequently, the surface temperature at which the OLR equals theabsorbed radiation corresponds to the surface temperature thatyields radiative equilibrium in the atmosphere. This is estimatedwith a root-finding method. Since the bulk radius is an outputof the interior model ( R interior ) and an input of the atmospheremodel, we first need to calculate the surface temperature for acertain mass and composition with an initial guess bulk radius.Then this surface temperature is the input for the interior model,which provides the bulk radius. With this bulk radius, we cangenerate a new value of the surface temperature. This scheme isrepeated until the bulk radius converges to a constant value, towhich we add the thickness of the atmosphere, z , to get the totalradius of the planet R total . The total mass M total is obtained asthe sum of the bulk mass M bulk plus the atmospheric mass M atm .The tolerance used to determine if the bulk radius has achievedconvergence is 2% of the bulk radius in the previous iteration.This is approximately 0.02 R ⊕ for an Earth-sized planet. We adapted the MCMC Bayesian analysis algorithm describedin Dorn et al. (2015) to our coupled interior and atmo-sphere model. The input model parameters are the bulk plan-etary mass M bulk , the CMF and the WMF. Therefore, m = { M bulk , CMF , W MF } , following the notation in Dorn et al.(2015). Depending on the planetary system and their availabledata, we can have observational measurements of the total plan-etary mass and radius and the stellar composition, or only thetotal planetary mass and radius. The available data in the formercase is denoted as d = { M obs , R obs , Fe / S i obs } , while the data inthe latter case is represented as d = { M obs , R obs } . The uncertain-ties on the measurements are σ ( M obs ) , σ ( R obs ), and σ ( Fe / S i obs ).The CMF and WMF prior distributions are uniform distri-butions between 0 and a maximum limit. This maximum limit is75% for the CMF, which is derived from the maximum estimatedFe / Si ratio of the proto-Sun (Lodders et al. 2009). With this limiton the CMF, we are assuming that the exoplanets have not beenexposed to events during or after their formation that could havestripped away all of their mantle, such as mantle evaporation orgiant impacts. In addition, the maximum WMF is set to 80%,which is the average water proportion found in comets in the so-lar system (McKay et al. 2019). The prior distribution for themass is a Gaussian distribution whose mean and standard devi-ations correspond to the central value and uncertainties of theobservations.
Article number, page 4 of 11orena Acuña et al.: Characterisation of the hydrospheres of TRAPPIST-1 planets
The MCMC scheme first starts by randomly drawing avalue for each of the model parameters from its respectiveprior distributions. This set of values is designated as m = (cid:8) M bulk , , CMF , W MF (cid:9) . The index i = n = / Si mole ratio, which are the set of output parametersg( m ) = { R , M , Fe / S i } . The likelihood of a set of model pa-rameters is then calculated via the following relationship (Dornet al. 2015): L ( m i | d ) = C exp (cid:32) − (cid:34)(cid:32) ( R i − R obs ) σ ( R obs ) (cid:33) + (cid:32) ( M i − M obs ) σ ( M obs ) (cid:33) + (cid:32) ( Fe / S i i − Fe / S i obs ) σ ( Fe / S i obs ) (cid:33) (cid:35)(cid:33) (14)where the normalization constant of the likelihood function C isdefined as: C = π ) / (cid:2) σ ( M obs ) · σ ( R obs ) · σ ( Fe / S i obs ) (cid:3) / . (15)When the Fe / Si mole ratio is not available as data, the squareresidual term of the Fe / Si mole ratio is removed from Eq. 14, aswell as its squared uncertainty in Eq. 15.Subsequently we draw a new set of input parameters, m = (cid:8) M bulk , , CMF , W MF (cid:9) from the prior distributions within thesame chain, n . We assure that the absolute di ff erence betweenthe values for i = i = m is uniformly bounded and centered around theold state, m . The maximum perturbation size is selected so thatthe acceptance rate of the MCMC, which is defined as the ra-tio between the number of models that are accepted and thenumber of proposed models, is above 20%. After m is chosen,the forward model calculates its corresponding output parame-ters and obtains their likelihood L ( m | d ), as shown in Eq. 14.The acceptance probability is estimated with the log-likelihoods l ( m | d ) = log ( L ( m | d )) as: P accept = min (cid:110) , e ( l ( m | d ) − l ( m | d )) (cid:111) (16)If P accept is greater than a number drawn from a uniform dis-tribution between 0 and 1, m is accepted and the chain movesto the state characterised by m , starting the next chain n + m and a di ff erentset of model parameters is proposed as m . To make sure thatthe posterior distributions converge and that all parameter spaceis explored, we run 10 chains. In other words, with acceptancerates between 0.2 and 0.6, the MCMC proposes between 1.6 and5 × sets of model inputs.
3. System parameters of TRAPPIST-1
Agol et al. (2020) have performed an analysis of TTVs that in-cludes all transit data from
Spitzer since the discovery of thesystem. We adopt these data for the mass, radius and semi-majoraxis in our interior structure analysis (Table 2).TRAPPIST-1 does not have available data regarding itschemical composition. However, the Fe / Si abundance ratio canbe estimated assuming that TRAPPIST-1 presents a similar chemical composition to that of other stars of the same metallic-ity, age and stellar population. As proposed by Unterborn et al.(2018), we select a sample of stars from the Hypatia Catalog(Hinkel et al. 2014, 2016, 2017). We choose the set of stars byconstraining the C / O mole ratio to be less than 0.8, and the stel-lar metallicity between -0.04 and 0.12, as this is the metallicityrange calculated for TRAPPIST-1 by Gillon et al. (2017). Wediscard thick disk stars since TRAPPIST-1 is likely a thin diskstar. Our best-fit Gaussian to the distribution of the Fe / Si moleratio shows a mean of 0.76 and a standard deviation of 0.12.Since this Fe / Si value is an estimate based on the chemical com-position of a sample of stars that belong to the same stellar popu-lation of TRAPPIST-1, we present two scenarios for each planet.In scenario 1, the only available data are the planetary mass andradius, while scenario 2 includes the estimated stellar Fe / Si moleratio to constrain the bulk composition.For temperate planets that cannot have a steam atmosphere,we set the surface temperature in our interior model to theirequilibrium temperatures assuming an albedo zero (Table 2). Al-though surface temperatures for thin atmospheres are lower thanthat obtained with this assumption, the dependence of the bulkradius on surface temperature for planets with condensed wateris low. For example, if we assume a pure water planet of 1 M ⊕ with a surface pressure of 1 bar, the increase in radius due toa change of surface temperature from 100 K to 360 K is 0.002 R ⊕ , which is less than 0.2% of the total radius, i.e., 10 times lessthan our convergence criterion. Additionally, the atmospheres ofTRAPPIST-1 planets in the habitable zone and farther are signif-icantly thinner than those of the highly-irradiated planets. Lin-cowski et al. (2018) estimate thicknesses of approximately 80km for temperate planets in TRAPPIST-1, which is negligiblecompared to their total radius. Therefore, we only calculate theatmospheric parameters (OLR, surface temperature, albedo andthickness of the atmosphere) for planets that present their hydro-spheres in steam phase.
4. Characterisation of hydrospheres
Tables 3 and 4 show the retrieved parameters, including the totalplanetary mass and radius, and the Fe / Si mole ratio. In both sce-narios, we retrieve the mass and radius within the 1 σ –confidenceinterval of the measurements for all planets. In scenario 1, whereonly the mass and radius data are considered, we retrieve Fe / Simole ratios without any assumptions on the chemical composi-tion of the host star. Although the uncertainties of these estimatesare more than 50% in some cases, we can estimate a commonFe / Si mole ratio for the planetary system. This common Fe / Sirange is determined by the overlap of the 1 σ confidence inter-vals of all planets, which corresponds to Fe / Si = / Si mole ratio of 0.76 ± σ –confidence regions derivedfrom the 2D marginalized posterior distributions of the CMF andWMF. The minimum value of the common CMF is determinedby the lower limit of the confidence region of planet g, which isapproximately 0.23, whereas the common maximum CMF valuecorresponds to the upper limit of planets b and c, which is 0.4.This is partially in agreement with the CMF obtained in scenario2, when we assume the Fe / Si mole ratio proposed by Unterbornet al. (2018), which are found between 0.2 and 0.3 (Table 4).Thus, the CMF of the TRAPPIST-1 planets could be compatiblewith an Earth-like CMF (CMF ⊕ = Article number, page 5 of 11 & A proofs: manuscript no. arxiv
Planet M [ M ⊕ ] R [ R ⊕ ] a d [10 − AU] T eq [ K ]b 1.374 ± + . − . ± + . − . ± + . − . ± + . − . ± + . − . ± + . − . ± ± ff ective temperature, stellar radius and semi-major axis provided by Agol et al. (2020).Planet M ret [ M ⊕ ] R ret [ R ⊕ ] CMF WMF Fe / Si ret b 1.375 ± ± ± + . − . ) 10 − ± ± ± ± + . − . ) 10 − ± ± ± ± ± + . − . e 0.699 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± + . − . + . − . Table 3: Output parameters retrieved by the MCMC method for all TRAPPIST-1 planets: total mass ( M ret ) and radius ( R ret ), CMF,WMF and Fe / Si molar ratio. In this case the mass and radius are considered as input data (scenario 1).Planet M ret [ M ⊕ ] R ret [ R ⊕ ] CMF WMF Fe / Si ret b 1.359 ± ± ± + . − . ) 10 − ± ± ± ± + . − . ) 10 − ± ± ± ± ± ± ± ± ± + . − . ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± M ret ) and radius ( R ret ), CMF,WMF and Fe / Si molar ratio. In this case the Fe / Si mole ratio estimated by following Unterborn et al. (2018) is also included as data(scenario 2).In scenario 1, the retrieved WMF for all planets in the sys-tem are below 20% within their uncertainties. This maximumWMF limit reduces to 10% for scenario 2. This indicates thatthe TRAPPIST-1 system is poor in water and other volatiles, es-pecially the inner planets b and c. Both planets are compatiblewith a dry composition in both scenarios, although the presenceof an atmosphere cannot be ruled out given the possible CMFrange estimated in scenario 1.
Figure 3 shows the OLR calculated by the atmosphere modeland the absorbed radiation (Eq. 12 and 13) for planets b, c and d.For temperatures lower than ∼ T sur f = ff ect (Ingersoll 1969).We obtain a constant OLR or an OLR limit (Nakajima et al.1992) of 274.3, 273.7 and 254.0 W / m for planets b, c and d, re-spectively. These are close to the OLR limit obtained by Katyalet al. (2019) of 279.6 W / m for an Earth-like planet. The smalldi ff erence is due to their di ff erent surface gravities. As explainedin Sect. 2.3, if the atmosphere can find a surface temperature at which the OLR and the absorbed radiation are equal, their atmo-spheres are in global radiative balance. This is the case for plan-ets b and c, whose surface temperatures are approximately 2450K and 2250 K, respectively. These are above the temperatureswhere the blanketting e ff ect is e ff ective, named T ε in Marcq et al.(2017) implying that the atmospheres of planets b and c are in apost-runaway state. However, planet d is not in global radiativebalance since its absorbed radiation never exceeds its OLR. Thismeans that planet d would be cooling down, and an internal fluxof approximately 33 W / m would be required to supply the extraheat to balance its radiative budget. TRAPPIST-1 inner planetsare likely to present an internal heat source due to tidal heating(Barr et al. 2018; Dobos et al. 2019; Turbet et al. 2018). The tidalheat flux estimated for planet d is F tidal = . W / m (Barr et al.2018), which is one order of magnitude lower than needed forradiative-convective balance of a steam atmosphere. Due to theblanketting e ff ect of radiation over the surface of planet d, theOLR limit is larger than the absorbed radiation and hence theplanet can cool enough to present its hydrosphere in condensedphases.Figure 4 shows the ( P , T ) profiles and the di ff erent phasesof water we can find in the hydrospheres of the TRAPPIST-1planets. The maximum WMF of planets b and c are 8 . × − Article number, page 6 of 11orena Acuña et al.: Characterisation of the hydrospheres of TRAPPIST-1 planets
M M F C M F WMF defg hbc1.00.90.80.70.60.50.40.30.20.10.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.01.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
M M F C M F WMF
Fig. 2: Top panel:1 σ –confidence regions derived from the two-dimensional posterior distributions for the first scenario, whereonly the masses and radii are available as data. Bottom panel: 1 σ confidence regions derived from the two-dimensional posteriordistributions for the second scenario, where the Fe / Si abundanceratio from Unterborn et al. (2018) is considered together with themass and radius for each planet. The axis of the ternary diagramindicate the CMF, the WMF and the mantle mass fraction MMF = . × − , which correspond to a surface pressure of 128.9bar and 4.85 bar, respectively.The thermal structure of their steam atmospheres are domi-nated by a lower, unsaturated troposphere where water conden-sation does not occur. Then the atmosphere consists of a mid-dle, saturated troposphere where cloud formation would be pos-sible, extending up to 10 mbar, and finally an isothermal meso-sphere above. Since we consider a clear transit radius of 20 mbar(Grimm et al. 2018; Mousis et al. 2020) the presence of cloudsabove this pressure level would flatten the water features in theplanetary spectrum (Turbet et al. 2019; Katyal et al. 2020). Onthe other hand, planets d and e could present water in liquidphase, which could be partially or completely covered in ice Ih.While the hydrosphere of planet h is not massive enough to at-tain the high pressures required for ice VII at its base, planets d,to g can reach pressures up to a 100 GPa. Figure 5 shows the output atmospheric parameters (surface tem-perature, atmospheric thickness, albedo and atmospheric mass)
Surface temperature [K] F [ W / m ] OLR (b)Absorbed (b)OLR (c)Absorbed (c)OLR (d)Absorbed (d)
Fig. 3: OLR and absorbed radiation as a function of surface tem-perature for the steam atmospheres of TRAPPIST-1 b, c and d.Vertical dotted lines indicate the surface temperature at whichthe absorbed flux is equal to the OLR for planets b and c.
200 400 1000 2000 4000
Temperature [K] P r e ss u r e [ P a ] SupercriticalIce VIILiquid VaporIce IhIce II-VI bcdefgh
Fig. 4: ( P , T ) profiles of the hydrospheres of TRAPPIST-1 plan-ets. The dashed-dotted grey horizontal line indicates the 20 mbarpressure level (see text). Thicker lines indicate the profile for theminimum WMF estimated for each planet in scenario 1, whilethinner lines mark the profile for the maximum WMF under thesame scenario. The minimum WMF of planets b, c and h is zero.of TRAPPIST-1 b and c for a water-dominated atmosphere inscenario 1. The total thickness of an atmosphere is related to itsscale height, which is defined as H = RT /µ g , where R = / Kmol is the gas constant, T is the mean atmospheric temperature, µ the mean molecular mass and g the surface gravity accelera-tion. For planets b and c, the mean atmospheric temperatures are940.4 and 499.4 K, and their surface gravities are 10.8 and 10.7 m / s , respectively. The mean molecular mass for a 99% waterand 1% CO atmosphere is 18.3 g / mol. The mean temperatureincreases with surface temperature, while the mean molecularmass is determined by the composition of the atmosphere. Article number, page 7 of 11 & A proofs: manuscript no. arxiv
Fig. 5: 2D and 1D marginal posterior distributions for the atmospheric parameters (surface temperature T sur f , atmospheric thickness z atm , albedo and atmospheric mass M atm ), and bulk mass and radius, M bulk and R bulk , of TRAPPIST-1 b (left panel) and c (rightpanel). These have been derived under scenario 1, where we do not consider Fe / Si data.For the same composition and surface gravity, the scaleheight and therefore the thickness of the atmosphere is directlycorrelated to the surface temperature. As shown in Fig. 5, the at-mospheric thickness, z atm increases with the surface temperature T sur f . This is known as the runaway greenhouse radius inflatione ff ect (Goldblatt 2015; Turbet et al. 2019), where a highly irra-diated atmosphere is more extended than a colder one despitehaving similar compositions. For planet b, its atmosphere canextend up to 450 km, while planet c presents a maximum ex-tension of 300 km. The minimum limit for the thicknesses iszero, which corresponds to the case of a dry composition. Or-tenzi et al. (2020) estimated that for a planet of 1-1.5 M ⊕ themaximum atmospheric thickness due to the outgassing of an ox-idised mantle is 200 km, which is compatible with the ranges wehave obtained for the atmospheric thicknesses. Scenario 2 showsthe same trends for the atmospheric parameters but with loweratmospheric mass and surface pressure. With their WMF pos-terior distributions centered in zero and low standard deviation,the surface pressure is below 1 bar and atmospheric thicknessesbelow 100 km in most of the accepted models, which means thatin scenario 2 planets b and c are most likely dry rocky planets.
5. Discussion
Agol et al. (2020) use the interior and atmosphere models pre-sented in Dorn et al. (2018) and Turbet et al. (2020b) to obtainthe WMF estimated of the TRAPPIST-1 planets with updatedand more precise radii and masses data from
Spitzer
TTVs (Agolet al. 2020). We thus limited the comparison to the sole resultsof Agol et al. (2020) with the same input values. By doing so,we can be certain that the variations in WMF estimates are dueto our di ff erent modelling approach. Figure 6 shows that plan-ets b and c are most likely dry in scenario 2, where the result-ing CMF are between 0.2 and 0.3 for the whole system. We a d [10 AU] W M F [ % ] b c d e f g h This work - Scenario 2Agol et al. 2020 - CMF = 0.25 a d [10 AU] W M F [ % ] b c d e f g h This work - Scenario 1Barr et al. 2018
Fig. 6: Water mass fraction as a function of the distance to thestar for the TRAPPIST-1 system. Upper panel shows our esti-mates for scenario 1 and those of Barr et al. (2018), where onlymass and radius data were taken into account. The lower panelcorresponds to scenario 2, whose CMF is constrained in a nar-row range between 0.2 and 0.3, while for Agol et al. (2020) weshow the WMF for a CMF of 0.25.obtain maximum estimates of 3.4 × − and 2.7 × − for band c, respectively. For the same density, the estimated value ofthe WMF depends on the CMF that is considered. Therefore wecompare WMF estimates for similar CMF between this work and Article number, page 8 of 11orena Acuña et al.: Characterisation of the hydrospheres of TRAPPIST-1 planets
Agol et al. (2020). We show our WMF in scenario 2, since theCMF of all planets spans a narrow range between 0.2 and 0.3,which are the most similar values to one of the CMF assumedby Agol et al. (2020), CMF = − for aconstant CMF of 0.25. We are able to reduce the maximum limitof the water content of the highly irradiated planets comparedto previous studies and establish the most likely WMF with ourcoupled atmosphere-interior model. The calculation of the totalradius requires a precise determination of the atmospheric thick-ness. This depends strongly on the surface temperature and thesurface gravity, which are obtained with radiative transfer in theatmosphere, and the calculation of the gravity profile for a bulkmass and composition in the interior self-consistently.In the case of planet d, we estimate a WMF of 0.036 ± − .The latter estimate considers that water is in vapor form, whichis less dense than condensed phases, while our model shows thatthe surface conditions allow liquid or ice phases, resulting in ahigher WMF. This discrepancy in the possible water phases onthe surface of planet d is due to di ff erent atmospheric composi-tions. We consider a water-dominated atmosphere with 1% CO ,while Agol et al. (2020) and Turbet et al. (2020b) assume a N and H O mixture. This di ff erence in composition changes radia-tive balance since CO is a strong absorber in the IR comparedto N , which is a neutral gas. Nonetheless, N is subject to stel-lar wind-driven escape and it is unlikely to be stable for the in-ner planets of TRAPPIST-1, while CO is more likely to survivethermal and ion escape processes (Turbet et al. 2020a).Our WMF for planets e to h are in agreement within uncer-tainties with Agol et al. (2020), although their central values aresignificantly lower. The EOS employed to compute the densityof the water layers in Agol et al. (2020) is also used in Dornet al. (2018) and Vazan et al. (2013), which agrees well with thewidely-used SESAME and ANEOS EOSs (Bara ff e et al. 2008).These EOS are not consistent with experimental and theoreticaldata since they overestimate the density at pressures higher than70 GPa (Mazevet et al. 2019). This yields an underestimation ofthe WMF for the same total planetary density and CMF.For the specific case of scenario 1, with no assumptions onthe stellar composition and the Fe / Si mole ratio, we comparedour CMF and WMF with those obtained in Barr et al. (2018)(Figure 6 and Table 5). These authors use masses and radii datagiven by Wang et al. (2017). They obtain lower masses comparedto Agol et al. (2020) while their radii are approximately similar,which would explain why Barr et al. (2018) tend to overesti-mate the water content of the TRAPPIST-1 planets. Moreover,most of the mass uncertainties in Wang et al. (2017) are 30-50%,while the mass uncertainties obtained by Agol et al. (2020) are3-5%. This causes Barr et al. (2018) to calculate wider CMFand WMF 1 σ confidence intervals. In addition, there are di ff er-ences between our interior modelling approach and that of Barret al. (2018). For example, according to their results, planet bcan have up to 50% of its mass as water. This high WMF valueis due to the assumption that the hydrosphere is in liquid andice I phases, and high-pressure ice polymorphs (HPPs), whichare more dense than the steam atmosphere we consider. In con-trast, the CMF seems to be closer to our estimates, especially forplanet b, d and e, where their maximum CMF is approximately0.40, in agreement with our CMF 1 σ intervals.We can also discuss the possible habitability of the hydro-spheres of the TRAPPIST-1 planets by comparing our WMF es-timates with the layer structure as a function of planetary mass Planet CMFBarr et al. (2018) This study (2020)b 0.00-0.43 0.12-0.41c 0.00-0.98 0.16-0.32d 0.00-0.39 0.24-0.58e 0.00-0.40 0.32-0.57f 0.00 0.24-0.58g 0.00 0.26-0.54h 0.00 0.15-0.53Table 5: Comparison between our one-dimensional 1 σ confi-dence regions for the CMF and those of Barr et al. (2018). Weshow only estimates for scenario 1, since Barr et al. (2018) didnot consider any constraints on the Fe / Si ratio based on stellarcomposition. M ( M )0.650.700.750.800.850.90 R ( R ) TRAPPIST-1 d
10% Core+CO atm.20% Core+CO atm.30% Core+CO atm.100% Mantle10% Core20% Core30% Core Fig. 7: Mass-radius relationships for planets with CO -dominated atmospheres assuming di ff erent CMF. The surfacepressure is 300 bar. The black dot and its error bars indicate thelocation and uncertainties of planet d in the mass-radius diagram.and water content obtained by Noack et al. (2016) . Accordingto Noack et al. (2016), a habitable hydrosphere must be struc-tured in a single liquid water ocean or in several ice layers thatenable the formation of a lower ocean layer. This lower oceanwould be formed by the heat supplied by the mantle that meltsthe high pressure ice in the ice-mantle boundary (Noack et al.2016). For planet d, a surface liquid ocean would form for allits possible WMF if the atmosphere allows for the presence ofcondensed phases. For planets e, f and g, the hydrosphere couldbe stratified in a surface layer of ice Ih and a liquid or an ice II-VI layer. In the case we had low-pressure ices II-VI, their basecould be melted by the heat provided by the mantle, and form alower ocean layer as suggested by Noack et al. (2016). At WMF ≥ ≥ In the case of scenario 1, where no Fe / Si data is assumed, theWMF increases with the distance to the star with the exceptionof planet h, whose WMF is similar to that of planet d. In the case
Article number, page 9 of 11 & A proofs: manuscript no. arxiv of scenario 2, where a common Fe / Si of 0.76 ± ffi cient in the case of planet d, comparedto planet e (Coleman et al. 2019). On the other hand, the gasat the distance at which planet d formed could have been moreenriched in volatiles than the outer planets, accreting more wa-ter ice than planet e in a ’cold finger’ (Stevenson & Lunine 1988;Cyr et al. 1998). Pebble formation in the vicinity of the water ice-line can induce important enhancements of the water ice fractionin those pebbles due to the backward di ff usion of vapor throughthe snowline and the inward drift of ice particles. Therefore, ifa planet forms from this material, it should be more water-richthan those formed further (Mousis et al. 2019). These formationscenarios could explain the high WMF of planet d when we as-sume that its water layer is in condensed phases. Post-formationprocesses could also have shaped the trend of the WMF withaxis, such as atmospheric escape due to XUV and X-ray emis-sion from their host star. Bolmont et al. (2017) estimated a max-imum water loss of 15 Earth Oceans (EO) for TRAPPIST-1 band c and 1 EO for planet d. If we assume that the current WMFare the central values of the posterior distributions we derived inscenario 1, planets b, c and d would have had an initial WMFof 2.37 × − , 2.50 × − and 0.085, respectively. Therefore,atmospheric escape would have decreased the individual WMFof each planet, but the increase of WMF with distance from thestar would have been preserved.In addition to the WMF-axis trend, we can di ff erentiate thevery water-poor, close-in planets, b and c, from the outer, water-rich planets, d to h. This has been reported as a consequenceof pebble accretion in the formation of other systems, such asthe Galilean moons. While Io is dry, Callisto and Ganymedeare water-rich, with Europa showing an intermediate WMF of8% (Ronnet et al. 2017). Pebble-driven formation can produceplanets with WMF ≥
15% if these are formed at the water iceline (Coleman et al. 2019; Schoonenberg et al. 2019). In con-trast, planets formed within the ice line would present WMF lessthan 5% (Liu et al. 2020; Coleman et al. 2019), which is closeto the mean value we calculated for planet h, 5.5%. The max-imum WMF limit in the first scenario is approximately 20%.This maximum limit is significantly lower than the typical WMFgenerated by planetesimal accretion scenario, which is 50-40%(Miguel et al. 2020). Therefore, our results are consistent withthe pebble-driven formation scenario.
However, the atmosphere of planet d could be dominated byother atmospheric gases di ff erent from H O-based mixtures,which could produce an extended atmosphere and increase thetotal planetary radius. Hydrogen-dominated atmospheres havebeen deemed unlikely as one of the possible atmospheric com-positions for all planets in the TRAPPIST-1 system, both cloud-free (de Wit et al. 2016, 2018) and with high-altitude clouds andhazes (Grimm et al. 2018; Ducrot et al. 2020). Similarly, CH -dominated atmospheres are not probable given the photometrydata of the Spitzer
Space Telescope (Ducrot et al. 2020). There-fore, our best candidate to explain the low density of planet d ina water-poor scenario is CO . We find that a CO -dominated at- mosphere with 1% water vapor in planet d would be in radiative-convective equilibrium by computing the OLR and absorbed ra-diation, as we did for water-dominated atmospheres. The re-sulting surface temperature is approximately 950 K, which isslightly higher than the surface temperature of Venus (700 K)with a higher water vapor mixing ratio. Figure 7 introduces themass-radius relationships for di ff erent CMF, assuming a CO -dominated atmosphere with a surface pressure of 300 bar. Itshows that planet d is compatible with a planet with a CO -dominated atmosphere and CMF between 0.2 and 0.3, which isa very likely CMF range for TRAPPIST-1 planets based on ouranalysis. Surface pressures lower than 300 bar would yield loweratmospheric thicknesses, so it would be necessary to consider alower CMF to explain the observed density of planet d. CO inthe case of planet d can be provided by volcanic outgassing (Or-tenzi et al. 2020), since its internal heat flux produced by tidalheating is in the range 0.04-2 W / m , which favours plate tecton-ics (Papaloizou et al. 2018). Secondary CO -dominated atmo-spheres could have traces of O , N and water vapor.
6. Conclusions
We presented an interior structure model for low-mass planetsat di ff erent irradiations that is valid for a wide range of waterphases, derived from the approaches of Brugger et al. (2017) andMousis et al. (2020). For highly-irradiated planets, we couple a1D water steam atmosphere in radiative-convective equilibriumwith a high-pressure convective layer in supercritical phase. Thedensity in this layer is computed by using an accurate EOS forhigh-pressure and high-temperature water phases. For temper-ate planets whose surface conditions allow the formation of con-densed phases, we implemented a hydrosphere with liquid waterand ice phases Ih, II, III, V, VI and VII. In addition, we adaptedthe MCMC Bayesian algorithm described in Dorn et al. (2015)to our interior model to derive the posterior distributions of thecompositional parameters, WMF and CMF, given mass, radiusand stellar composition data. We then applied our interior modelto the particular case of TRAPPIST-1 planets using the latestmass and radius data from Spitzer (Agol et al. 2020).The hydrospheres of TRAPPIST-1 planets have been char-acterised by calculating their
P-T profiles and thermodynamicphases. Planets b and c are warm enough to present steam atmo-spheres. They could hold post-runaway greenhouse atmosphereswith thicknesses up to 450 km and surface temperatures up to2500 K, which means that they are extended enough to be suit-able targets for atmospheric characterisation by future space-based facilities such as JWST. Moreover, planets d to g presenttheir hydrospheres in condensed phases. These hydrospheres cancontain high-pressure ices that start to form at 10 − Pa.We have obtained CMF and WMF probability distributionsfor all planets in the system. We found that the Fe / Si mole ratioof the system is in the 0.45-0.97 range without considering anyassumption on the chemical composition of the stellar host. ThisFe / Si range corresponds to a CMF value in the 0.23-0.40 range,making the CMF of TRAPPIST-1 planets compatible with anEarth-like value (0.32). In addition, our WMF estimates agreewithin uncertainties with those derived by Agol et al. (2020), al-though their most likely values are considerably lower for plan-ets with condensed phases. In the case of planets with steam hy-drospheres, their densities are compatible with dry rocky plan-ets with no atmospheres. Nevertheless, we cannot rule out thepresence of an atmosphere with the Fe / Si range we have derivedwithout any assumption on the chemical composition of the hoststar. When considering a possible estimate of the Fe / Si ratio of
Article number, page 10 of 11orena Acuña et al.: Characterisation of the hydrospheres of TRAPPIST-1 planets the host star (scenario 2), we obtained lower maximum limits ofthe WMF for planets b and c compared to previously calculatedlimits by Agol et al. (2020) for a similar CMF of 0.25. Our esti-mated WMF in steam and condensed phases are consistent withan increase of WMF with progressing distance from the host star.This trend, as well as the maximum WMF we calculated, favourpebble-driven accretion as a plausible formation mechanism forthe TRAPPIST-1 system. However, planet d presents a slightlyhigher WMF than planet e. This could be due to processes thattook place during planet formation, such as migration, a low-e ffi cient ablation of pebbles and gas recycling or an enhancementof the water ice fraction in pebbles at the distance of the discwhere planet d formed. An extended atmosphere dominated bygreenhouse gases di ff erent from a water-dominated atmosphere,such as CO , could also explain the low-density of planet d com-pared to planet e.Future work should include more atmospheric processesand species that determine the mass-radius relations of planetswith secondary atmospheres in the Super-Earth and sub-Neptuneregime. These can vary the atmospheric thickness and increasethe total planetary radius with varying atmospheric masses whileother compositional parameters change the bulk radius. Theseshould be integrated in one single interior-atmosphere model,combined in a MCMC Bayesian framework such as the one weused in this study. This statistical approach has been employedwith interior models for planets with H / He-dominated atmo-spheres (Dorn et al. 2017b,a, 2018), or dry planets (Plotnykov& Valencia 2020), but not for planets with secondary, CO andsteam-dominated atmospheres. The integrated model should alsoinclude a description of escape processes, such as hydrodynamicor Jeans escapes, which is particularly interesting to explore thelifetime of secondary atmospheres. Close-in, low-mass planetsare likely to outgas atmospheric species, such as CO , and formO via photodissociation of outgased H O, during their magmaocean stage or due to plate tectonics (Chao et al. 2020). Thus, amixture of these gases should be considered to study the thermalstructure of planets with secondary atmospheres. Planets b and cin the TRAPPIST-1 system could present magma oceans due totheir high surface temperatures (T ≥ Acknowledgements.
MD and OM acknowledge support from CNES. We ac-knowledge the anonymous referee whose comments helped improve and clarifythis manuscript.
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