Issue 
A&A
Volume 638, June 2020



Article Number  A41  
Number of page(s)  10  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201937151  
Published online  09 June 2020 
Revised massradius relationships for waterrich rocky planets more irradiated than the runaway greenhouse limit
^{1}
Observatoire astronomique de l’Université de Genève,
51 chemin des Maillettes,
1290
Sauverny,
Switzerland
email: martin.turbet@unige.ch
^{2}
Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geoffroy SaintHilaire,
33615
Pessac,
France
Received:
20
November
2019
Accepted:
1
April
2020
Massradius relationships for waterrich rocky planets are usually calculated assuming most water is present in condensed (either liquid or solid) form. Planet density estimates are then compared to these massradius relationships, even when these planets are more irradiated than the runaway greenhouse irradiation limit (around 1.1 times the insolation at Earth for planets orbiting a Sunlike star), for which water has been shown to be unstable in condensed form and would instead form a thick H_{2}Odominated atmosphere. Here we use a 1D radiativeconvective inverse version of the LMD generic numerical climate model to derive new theoretical massradius relationships appropriate for waterrich rocky planets that are more irradiated than the runaway greenhouse irradiation limit, meaning planets endowed with a steam, waterdominated atmosphere. As a result of the runaway greenhouse radius inflation effect introduced in previous work, these new massradius relationships significantly differ from those traditionally used in the literature. For a given watertorock mass ratio, these new massradius relationships lead to planet bulk densities much lower than calculated when water is assumed to be in condensed form. In other words, using traditional massradius relationships for planets that are more irradiated than the runaway greenhouse irradiation limit tends to dramatically overestimate possibly by several orders of magnitude their bulk water content. In particular, this result applies to TRAPPIST1 b, c, and d, which can accommodate a water mass fraction of at most 2, 0.3 and 0.08%, respectively, assuming planetary core with a terrestrial composition. In addition, we show that significant changes of massradius relationships (between planets less and more irradiated than the runaway greenhouse limit) can be used to remove bulk composition degeneracies in multiplanetary systems such as TRAPPIST1. Broadly speaking, our results demonstrate that nonH_{2}/Hedominated atmospheres can have a firstorder effect on the massradius relationships, even for rocky planets receiving moderate irradiation. Finally, we provide an empirical formula for the H_{2}O steam atmosphere thickness as a function of planet core gravity and radius, water content, and irradiation. This formula can easily be used to construct massradius relationships for any waterrich, rocky planet (i.e., with any kind of interior composition ranging from pure iron to pure silicate) more irradiated than the runaway greenhouse irradiation threshold.
Key words: planets and satellites: terrestrial planets / planets and satellites: composition / planets and satellites: atmospheres / planets and satellites: individual: TRAPPIST1 / planets and satellites: interiors / methods: numerical
© ESO 2020
1 Introduction
With the discovery of the nearby TRAPPIST1 system (Gillon et al. 2016, 2017; Luger et al. 2017), we now have seven rocky planets in temperate orbits for which both radii (Gillon et al. 2017; Luger et al. 2017; Delrez et al. 2018) and masses (Grimm et al. 2018) have been measured with unprecedented accuracy for planets of this nature. Grimm et al. (2018) and Dorn et al. (2018) compared TRAPPIST1 planets’ bulk density^{1} estimates with massradius relationships of rocky planets endowed with thick condensed water layers and inferred from the comparison that most of the seven planets are likely enriched in volatiles (e.g., water) up to several tens of percent of planetary mass.
We were motivated by these studies to recalculate massradius relationships for waterrich rocky planets in cases where all water is vaporized, forming a thick H_{2}Odominated steam atmosphere. This situation has been shown to occur for planets receiving more irradiation from their host star than the theoretical runaway greenhouse irradiation limit (Kasting et al. 1993; Goldblatt & Watson 2012; Kopparapu et al. 2013). In the TRAPPIST1 system, the three innermost planets (TRAPPIST1 b, c, and d) are thought to receive more irradiation than the theoretical runaway greenhouse irradiation limit for ultracool stars (Kopparapu et al. 2013; Wolf 2017; Turbet et al. 2018), even when considering the possible negative feedback of substellar water clouds (Yang et al. 2013; Kopparapu et al. 2016) expected on tidally locked planets.
Traditionally, massradius relationships (Seager et al. 2007; Sotin et al. 2007; Grasset et al. 2009; Mordasini et al. 2012; Swift et al. 2012; Zeng & Sasselov 2013; Zeng et al. 2016) for waterrich rocky planets are calculated assuming water is either in solid or liquid form, depending on planet equilibrium temperatures. Some studies (Dorn et al. 2018; Zeng et al. 2019) included the effect of a H_{2}Orich atmosphere on the planetary radius estimate by assuming an isothermal steam atmosphere at the equilibrium planet temperature. Thomas & Madhusudhan (2016) explored the effect that a thick H_{2}O atmosphere may have on the massradius relationships of Earth to superEarthmass planets. To do this, they used a structural model forced at various surface temperatures and in various pressureboundary conditions. Their model takes convection processes into account, but lacks a radiative transfer. As a result, the surface temperatures assumed in Thomas & Madhusudhan (2016) are significantly lower than those calculated selfconsistently in the standard atmospheric numerical simulations taking into account the radiative exchanges both in shortwave and longwave ranges (Kopparapu et al. 2013; Goldblatt et al. 2013; Turbet et al. 2019). Radiative transfer is a necessary component to ensure that atmospheric states have reached topofatmosphere radiative balance, and thus describe physically realistic planets.
All the aforementioned approaches most likely underestimate the physical size of a H_{2}Odominated steam atmosphere for planets receiving more irradiation than the runaway greenhouse limit (Turbet et al. 2019). Using a 1D numerical radiativeconvective climate model, Turbet et al. (2019) in fact recently showed that waterrich planets receiving more irradiation than the runaway greenhouse irradiation threshold should suffer from a strong atmospheric expansion compared to planets receiving less irradiation than this threshold. The effect, which they named the runaway greenhouse radius inflation effect, originates from the cumulative effect of four distinct causes: (i) a significant increase in the total atmospheric mass; (ii) a significant increase in the atmospheric temperatures; (iii) an increase in optical thickness at low atmospheric pressure; and (iv) a decrease in the mean molecular mass.
To the best of our knowledge, the study of Valencia et al. (2013) is the only work ever to have selfconsistently considered the effect of a steam H_{2}O atmosphere on massradius relationships in the Earth to superEarth mass regime planets. However, this work focused on highly irradiated planets only (around 20 times the insolation at Earth), with the aim of improving our understanding of the nature of the exoplanet GJ 1214b. Although they did not directly calculate massradius relationships, Nettelmann et al. 2011 (based on previous results from MillerRicci & Fortney 2010) also carried out interioratmosphere calculations selfconsistently taking into account the effect of a H_{2}Odominated steam atmosphere to evaluate the possible nature of GJ 1214b. The results of Nettelmann et al. (2011) and Valencia et al. (2013) are qualitatively in agreement (and quantitatively in agreement in the case of GJ 1214b), that planets endowed with a steam H_{2}O atmosphere have a significantly larger radius than icy or liquid ocean planets, for a given watertorock ratio. Thomas & Madhusudhan (2016) also recovered qualitatively similar results, that planets endowed with a steam H_{2}O atmosphere have a significantly larger radius than icy or liquid ocean planets, for a given watertorock ratio.
Here we make use of the 1D inverse radiativeconvective model previously introduced in Turbet et al. (2019), coupled to massradius relationships of rocky interiors from Zeng et al. (2016), to produce revised massradius relationships for rocky planets in temperate orbits endowed with thick H_{2}O steam envelopes, as predicted for waterrich planets receiving more irradiation than the runaway greenhouse limit (Turbet et al. 2019). While Turbet et al. (2019) focused on the theoretical and numerical ground of the runaway greenhouse radius inflation, as well as observational tests to detect it in the exoplanet population, here we derive and make available to the community massradius relationships aimed at better interpreting the nature of terrestrialsize planets, for which we are beginning to have increasingly accurate measurements of masses and radii.
In Sect. 2, we describe the method we used to calculate massradius relationships for planets endowed with steam, waterdominated atmospheres. These new massradius relationships are then presented and discussed in Sect. 3. Lastly, we present the conclusions of this work and discuss future perspectives in Sect. 4.
2 Methods
In this section, we describe first the method we used to calculate massradius relationships for planets endowed with steam, waterdominated atmospheres. We then provide the empirical massradius relationship fitted to these calculations.
2.1 Procedure to derive revised massradius relationships
We calculated the massradius relationships for waterrich rocky planets that are more irradiated than the runaway greenhouse irradiation limit in four main steps: firstly, we retrieved massradius relationships of dry, rocky planets. In this paper, we chose to use the massradius relationships of Zeng et al. (2016)^{2} for (i) pure silicate (MgSiO_{3}) planets; (ii) terrestrial core composition planets; and (iii) pureiron (Fe) planets. However, any type of rocky interior composition (from pure iron to pure silicate) could be used.
Secondly, for each set of rocky interior mass M_{core} and radius R_{core}, we calculated the transit thickness z_{atmosphere} and the mass M_{atmosphere} that a pure H_{2}O atmosphere would have for a wide range of possible water atmospheric pressures, using a 1D inverse radiativeconvective version of the LMD Generic model. The model was adapted in Turbet et al. (2019) to simulate the vertical structure of steam atmospheres, taking into account the condensation of water vapor using a nondilute moist lapse rate formulation as in Marcq et al. (2017) and a radiative transfer using the waterdominated absorption coefficients of Leconte et al. (2013). For the calculation of the atmospheric profile, the change in gravity with altitude is also taken into account. For more details on the model, we refer the reader to Appendix A of Turbet et al. (2019). This second step is discussed in more detail below.
Thirdly, for each set of rocky interior mass and radius, and for each possible water atmospheric pressure, we calculated the resulting mass M_{planet} and transit radius R_{planet} by using M_{planet} = M_{core} + M_{atmosphere,} and assuming that R_{planet} = R_{core} + z_{atmosphere}. In Appendix A, we discuss in detail how the relationship R_{planet} = R_{core} + z_{atmosphere} – with R_{core} calculated neglecting the effect of the atmosphere – remains valid as soon as the mass of the H_{2}Odominated atmosphere is significantly lower than the total mass of the planet.
Lastly, we drew massradius relationships for rocky planets with various watertorock mass ratios. This last step was performed by carrying out a logarithmic interpolation of the watertorock mass ratio for each possible rocky core mass and radius using the array of transit radius calculated for a wide range of possible total H_{2}O atmospheric pressures^{3}.
The second step of our procedure (i.e., the calculation of z_{atmosphere}) was achieved through a number of substeps listed below: Firstly, we estimated the surface temperature T_{surf} of a H_{2}Odominated steam atmosphere as a function of H_{2}O atmospheric pressure , surface gravity g and irradiation received by the planet S_{eff}, following the same approach as in Turbet et al. 2019 (Fig. 3). To do this, we first performed 1D inverse radiativeconvective calculations for a wide range of surface temperatures (from 300 to 4300 K), irradiations (roughly from 1 to 40× the irradiation received on Earth), surface gravities (from 2 to 50 m s^{−2}), and water vapor pressures (from 2.7 × 10^{5} to 2.7 × 10^{9} Pa). We then fit a polynomial (see Methods in Appendix B) on all these parameters to derive the following empirical equation for the surface temperature T_{surf} (in Kelvins): (1)
with x = (k_{1})/k_{2} with the H_{2}O partial pressure expressed in bar units, y = (log_{10}(g) − k_{3})/k_{4} with g the surface gravity (at the interioratmosphere boundary) in m s^{−2}, and z = (log_{10}(S_{eff}) − k_{5})/k_{6} with S_{eff} the irradiation received by the planet (S_{eff} is in Earth insolation units; i.e., S_{eff} = 1 when the planet receives the same insolation as Earth of 1366 W m^{2}). The empirical coefficients are shown in Table 1.
This empirical relationship provides an estimate of the surface temperature (see Fig. 1) of a H_{2}Odominated steam atmosphere as a function of surface gravity, water vapor surface pressure and irradiation. It is valid within a few percent for most of the parameter space (maximum error ~ 10%; see Fig. B.2, left panel), for irradiation from ~ 1 to 30 S_{⊕} (assuming the irradiation received by the planet is above the runaway greenhouse irradiation threshold), surface gravity from 0.2 to 6 g_{⊕}, and water vapor pressure from 2.7 bar to 27 kbar, and as far as surface temperature remains between 300 and 4300 K.
Secondly, for each possible rocky core mass and radius pair taken from Zeng et al. (2016), and for a wide range of H_{2}O atmospheric pressures (from 2.7 × 10^{1} to 2.7 × 10^{5} bars), we built the atmospheric structure following the approach presented in Turbet et al. (2019) (Appendix A), and originating from Marcq (2012), Marcq et al. (2017) and Pluriel et al. (2019).
Lastly, we evaluated the transit radius of each possible planet (made of each possible combination of rocky interior and water atmospheric pressure) by integrating these atmospheric profiles in the hydrostatic approximation, using nonideal thermodynamic properties of H_{2}O (Haar et al. 1984)as in Turbet et al. (2019), and assuming the transit radius is controlled by the altitude of the upper water cloud layer. For this, we used the altitude of the top of the moist convective layer as a proxy. The total atmospheric transit thickness of a thick H_{2}Odominated atmosphere has been shown to be roughly unchanged whether a cloudy or cloudfree atmosphere is considered (Turbet et al. 2019).
Fig. 1 Surface temperature as a function of the effective flux received on a planet (xaxis) and the surface pressure of its steam H_{2}O atmosphere (yaxis), for three different surface gravities (0.3, 1 and 3× the gravity on Earth). The surface temperature was estimated using Eq. (1). The small black dots indicate the parameter space for which atmospheric numerical simulations have been carried out, and on which the fit of the surface temperature is based. 
2.2 An empirical massradius relationship formula
Motivated to make our revised massradius relationships accessible to the community, we constructed an empirical massradius relationship formula (provided below) for waterrich rocky planets receiving more irradiation than the runaway greenhouse irradiation limit. This formula was constructed in two steps:
Firstly, we derived an analytic expression of the massradius relationships, assuming (i) the perfect gas law approximation, and (ii) an isothermal temperature profile: (2)
with R_{core} and g_{core} the core (or surface) radius and gravity of the planet, respectively, R the gas constant (=8.314 J K^{−1} mol^{−1}), the molar mass of water (=1.8 × 10^{−2} kg mol^{−1}), G the gravitational constant (=6.67 × 10^{−11} m^{3} kg^{−1} s^{−2}), and the water mass fraction (between 0 and 1) of the planet. P_{transit} is the pressure at the transit radius. T_{eff} is the temperature of the isothermal atmosphere. The procedure to derive this equation is detailed in Appendix C. This equation well describes (see hereafter) the family of possible behaviors of the massradius relationships for H_{2}O steam atmosphere planets.
Secondly, we fit the free parameters (T_{eff} and P_{transit}) using the range of simulations described in the previous subsection. Our simulations show that P_{transit} varies little across the range of parameters we explored and is roughly equal to 10^{−1} Pa. We thus set it to this value. T_{eff} is an effective atmospheric temperature that we empirically fit (see Methods in Appendix B) as follows: (3)
with x = ()/α_{2}, with the mass water fraction of the planet (between 0 and 1), y = (log_{10}(g) − α_{3})/α_{4}, with g the surface gravity (at the interioratmosphere boundary) in m s^{−2}, and z = (log_{10}(S_{eff}) − α_{5})/α_{6}, with S_{eff} the irradiation received by the planet (S_{eff} is in Earth insolation units). The empirical coefficients are shown in Table 2.
These relationships (Eqs. (2) and (3)) are valid within a few percent for most of the parameter space (again, maximum error of ~ 10%; see Fig. B.2, right panel), for irradiation from 1 to 30 S_{⊕} (assuming the irradiation received by the planet is above the runaway greenhouse irradiation threshold), surface gravity from 0.2 to 6 g_{⊕}, and water vapor pressure from 2.7 bar to 27 kbar, and as far as surface temperature remains between 300 and 4300 K.
We propose in Appendix D a tutorial on how to use these massradius relationships.
3 Results
3.1 Revised massradius relationships
The main result of this work is summarized in Fig. 2, which shows how massradius relationships can vary depending on if water is treated as a condensed layer (Zeng et al. 2016) or as an atmosphere (this work). As a direct consequenceof the runaway greenhouse radius inflation introduced in Turbet et al. (2019), massradius relationships in the steam atmosphere configuration give for a given planet mass a significantly larger radius than in the condensed water configuration. This translates in two main consequences: firstly, traditional massradius relationships (Seager et al. 2007; Sotin et al. 2007; Grasset et al. 2009; Mordasini et al. 2012; Swift et al. 2012; Zeng & Sasselov 2013; Zeng et al. 2016) for waterrich rocky planets (i.e. where most water is considered to be in the solid or liquid form) tend to significantly overestimate their bulk density if the planets are more irradiated than the runaway greenhouse irradiation limit. Secondly, comparing these traditional massradius relationships for waterrich rocky planets with real planet measured densities tend to overestimate the evaluation of their watertorock mass fraction, possibly by several orders of magnitude.
In Eqs. (2) and (3), we provide an empirical formula for the H_{2}O steam atmosphere thickness as a function of planet core gravity and radius, water content and irradiation. This formula can easily be used (seethe procedure in Appendix D) to construct massradius relationships for waterrich, rocky planets that are more irradiated than the runaway greenhouse irradiation threshold, for any type of planet interior.
Lastly, our revised massradius relationships for steam planets indicate that small rocky planets (M_{planet} ≾ 0.5 M_{⊕}) that are more irradiated than the runaway greenhouse irradiation threshold should be unable to retain more than a few percent water by mass. This is because for these small planets the runawaygreenhouseinduced radius inflation is so extreme that the upper atmosphere becomes gravitationally unbounded for steam atmospheres only a few percent by mass and efficient atmospheric escape mechanisms should take place. For instance, for a 0.3 M_{⊕} pure silicatecore planet (located just above the runaway greenhouse irradiation threshold) with a 5% watertorock ratio, Fig. 2 (right panel) indicates that the transit radius lies around 1.2 R_{⊕}. The gravity at the transit radius is thus as low as 20% of that at the surface of the Earth, so ~2 m s^{−2}, meaning atmospheric escape can be very strong. In fact, the Ushape of the massradius relationships (in the upperleft part of the massradius relationships for steam planets in Fig. 2) is symptomatic of the fact that the atmosphere becomes gravitationally unbounded. This Ushape has already been predicted for H_{2} /Herich planets (Fortney et al. 2007; Baraffe et al. 2008; Lopez & Fortney 2014; Zeng et al. 2019), but we show here that it is also expected for planets endowed with H_{2}Orich atmospheres. This Ushape can be described well at first order by Eq. (2).
3.2 Application to the TRAPPIST1 system
The fact that the use of traditional massradius relationships for waterrich rocky planets tend to overestimate the evaluation of a planet watertorock mass fraction is particularly relevant for our understanding of the nature of the TRAPPIST1 planets, the only system known to date (as of November 2019) of temperateorbit Earthsize planets (Gillon et al. 2017) for which both radii and masses have been measured (Grimm et al. 2018). Based on comparisons of TRAPPIST1 planet bulk densities with traditional massradius relationships, it has been speculated that some planets in the system may be enriched with water, possibly up to tens of percent for some of them (Grimm et al. 2018; Dorn et al. 2018).
Our results suggest that the three innermost planets of the TRAPPIST1 system and more particularly, TRAPPIST1 b and d, for which TTVs measurements point toward particularly low bulk densities (Grimm et al. 2018) do not necessarily need to be highly enriched with water to reach their measured density. In fact, Table 3 provides quantitative estimates for the maximum water content of the three TRAPPIST1 innermost planets, for several core compositions. For a core composition similar to that of the solar system terrestrial planets, TRAPPIST1 b, c, and d cannot accommodate more than 2, 0.3, and 0.08%, respectively,of water. Specifically, TRAPPIST1 d cannot be composed of more than 2% water whatever the core composition assumed. For comparison, Bourrier et al. (2017) evaluated that the current rate of water loss can be as high as 0.19, 0.06, and 0.18% per gigayear by mass for TRAPPIST1 b, c, and d, respectively. Putting these pieces of information together, it is likely that the three inner TRAPPIST1 planets may all be completely dry today.
A direct consequence of this result is that if the planets of the TRAPPIST1 system are all rich in water, as supported by planet formation and migration models for which TRAPPIST1 planets formed far from their host star, beyond water and other volatile ice lines, and subsequently migrated forming a resonant chain (Ormel et al. 2017; Unterborn et al. 2018; Coleman et al. 2019), then our revised massradius relationships – leading to much lower water content for TRAPPIST1 inner planets than previous calculations (Grimm et al. 2018; Dorn et al. 2018) showed – can be reconciled with the fact that outer planets are expected to be more volatilerich and waterrich than inner planets (Unterborn et al. 2018), due both to planet formation and migration (Ormel et al. 2017; Unterborn et al. 2018; Coleman et al. 2019), and atmospheric escape processes (Bolmont et al. 2017; Bourrier et al. 2017). This would avoid the need for exotic planet formation and water delivery processes (Dorn et al. 2018; Schoonenberg et al. 2019) to explain apparent density variation with irradiation among TRAPPIST1 planets.
As of November 2019, the uncertainties on the masses of TRAPPIST1 planets are still large (see the 2σ uncertainty ellipses on Fig. 2, from Grimm et al. 2018). However, it is expected that these uncertainties will significantly decrease in the near future, either through a followup on the transit timing variations (Spitzer Proposal ID 14223, PI: Eric Agol) or using radial velocity measurements with nearinfrared groundbased spectrographs (Klein & Donati 2019) such as SPIRou (Artigau et al. 2014) or NIRPS (Wildi et al. 2017).
Below, and with the support of Fig. 3, we discuss, as a proof of concept, one example of a possible scenario for the masses and radii of each of the seven TRAPPIST1 planets, that of the case where all the TRAPPIST1 planets closely follow an isocomposition interior massradius relationship. The baseline interior composition (10% Fe, 90% MgSiO_{3}, i.e., the solid gray line in Fig. 3) was chosen to ensure that this scenario remains compatible with the Grimm et al. (2018) 95% confidence ellipses^{4}.
This assumption, however, does not guarantee in principle that the planets do have an interior composition of 10% Fe and 90% MgSiO_{3}. This stems from the fact that the composition of the planets is, in principle, highly degenerate, because their positions in the massradius diagram (Fig. 3) can be explained either by (i) dry planets with a 10% Fe + 90% MgSiO_{3} core (solid gray line in Fig. 3), or (ii) wet planets with a denser core (e.g., 6% water with a terrestrial core, as illustrated by the solid blue line in Fig. 3). This is illustrated in Fig. 3 where the solid blue (6% water with a terrestrial core) and gray (10% Fe + 90% MgSiO_{3} core) lines are almost superimposed.
However, this degeneracy is removed here bearing in mind that the three innermost planets of the TRAPPIST1 system are more irradiated than the runaway greenhouse limit and should therefore follow a different isocomposition massradius relationship (e.g., the upper dashed purple line in Fig. 3, for planets with 6% water and a terrestrial core). The black arrows in Fig. 3 indicate the new positions of TRAPPIST1 b, c, and d in the massradius diagram taking into account the revised massradius relationship. In other words, the runaway greenhouse transition allow planets to jump from one massradius relationship to another, which makes it possible to break the composition degeneracy.
This demonstrates, to a certain extent, that in our scenario all the TRAPPIST1 planets should all be very dry, because (i) if all planets were to be waterrich, then they would have to follow a different massradius relationship (purple dashed lines for the three innermost planets, versus solid blue line for the four outermost planets in Fig. 3); (ii) if only some of the planets were to be waterrich and others were not, then the planets should not follow an isocomposition massradius relationship anyway. In our scenario, we evaluate that, assuming that all TRAPPIST1 planets have the same mass composition (for the rocky interior and water content), the planets cannot accommodate more than 10^{−3}% of water by mass in order to fit all the small circles in Fig. 3. This argument – that all planets are very dry – should hold unless we are dealing with a finetuned scenario where, for each of the planets, all processes (water delivery, runaway greenhouse radius inflation effect, water loss, different core composition) compensate each other exactly.
A direct consequence is that any significant deviation of planetary densities from an interior isocomposition massradius relationship would be a strong indication that (i) there is either today large reservoirs of water or volatiles on at least some planets of the system, or that (ii) there are significant differences in TRAPPIST1 planets’ core composition. In some cases (e.g., a significant trend in planets density with irradiation), the first interpretation would be favored.
Fig. 2 Massradius relationships for various interior compositions and water content, assuming water is in the condensed form (left panel) and water forms an atmosphere (right panel). The silicate composition massradius relationship assumes a pure MgSiO_{3} interior and was taken from Zeng et al. (2016). The waterrich massradius relationships for water in condensed form (left panel) were derived using the data from Zeng et al. (2016). The waterrich massradius relationships for water in gaseous form (right panel) are the result of the present work. All massradius relationships with water were built assuming a pure MgSiO_{3} interior. For comparison, we added the measured positions of the seven TRAPPIST1 planets measured from Grimm et al. (2018), with their associated 95% confidence ellipses. Based on the irradiation they receive compared to the theoretical runaway greenhouse limit (Kopparapu et al. 2013; Wolf 2017; Turbet et al. 2018), TRAPPIST1 e, f, g, and h should be compared with massradius relationships on the left, while TRAPPIST1b, c, and d should be compared with those on the right. To emphasize this, we indicated, on each panel and in black (and solid line ellipses), the planets (and their associated 95% confidenceellipses) for which massradius relationships (with water) are appropriate. In contrast, we indicated on each panel in gray (and dashed line ellipses) the planets (and their associated 95% confidence ellipses) for which massradius relationships (with water) are not appropriate. For reference, we also added a terrestrial composition that resembles that of the Earth, but also that of Mars and Venus. We note that massradiusrelationships for steam planets (right panel) can be easily built following the procedure described in Appendix D. 
Maximum water content of TRAPPIST1 b, c, and d, depending on the assumed core composition.
Fig. 3 Example of a scenario for TRAPPIST1 planets where masses and radii follow an interior isocomposition line (gray line; 10% Fe, 90% MgSiO_{3} composition) chosen to be consistent with 2σ uncertainty ellipses of Grimm et al. (2018). While the seven large ellipses indicate the known current estimates (95% confidence) for the masses and radii of the seven TRAPPIST1 planets, the seven small circles indicate the positions (in the mass, radius diagram) of the seven planets as speculated in our scenario. Each planet (associated with a current uncertainty ellipse, and a speculated position) is identified by a distinct color. This figure also shows massradius relationships for terrestrial core planets, in some cases endowed with either a condensed layer of water (solid blue line) or a steam H_{2}O atmosphere of various masses (dashed purple lines). The massradius relationships for steam planets can be built following the procedure described in Appendix D. We note that the massradius relationships for steam H_{2}Orich atmosphere planets can slightly change depending on the level of stellar irradiation they receive. However, because these changes are low (for the levels of irradiation on TRAPPIST1 b, c, and d) compared to the runawaygreenhousetransitioninduced massradius relationship change, we decided no to show them for clarity. 
4 Conclusions
In this work, we calculated revised massradius relationships for waterrich, rocky planets, which are more irradiated than the runaway greenhouse irradiation limit. This was performed by coupling the massradius relationships for rocky interior of Zeng et al. (2016) with our estimates of the atmospheric thickness of H_{2}Odominated atmospheres with a 1D radiativeconvective model.
For a given watertorock mass ratio, our revised massradius relationships lead to planet bulk densities much lower than calculated when most water is assumed to be in condensed form, which is the common standard in the literature (Seager et al. 2007; Sotin et al. 2007; Grasset et al. 2009; Mordasini et al. 2012; Swift et al. 2012; Zeng & Sasselov 2013; Zeng et al. 2016). This means that using traditional massradius relationships for planets that are more irradiated than the runaway greenhouse irradiation limit tends to dramatically overestimate possibly by several orders of magnitude their bulk water content.
More specifically, this result has important consequences for our understanding of the nature of the TRAPPIST1 planets. Our work shows that the measured density (yet to be confirmed) of the three innermost planets of the TRAPPIST1 system indicates their bulk water content should be significantly lower than what was previously speculated in Grimm et al. (2018) and Dorn et al. (2018). More generally, these results demonstrate that nonH_{2}/Hedominated atmospherescan have a firstorder effect on the massradius relationships even for Earthmass planets receiving moderate irradiation.
Future work should focus on more carefully taking into account possible interactions and feedback between the planet interior and the steam atmosphere, and should aim to extend our work to more irradiated, more massive planets (socalled superEarth planets), for which mass and radius measurements have been performed for a much larger number of planets. For this, an interior model could be coupled to a steam atmosphere model to account for (1) the greenhouse effect feedback of the atmosphere on the interior structure; (2) the planetary core cooling; and (3) the possible outgassing or accumulation through photodissociation of various gases such as O_{2}, N_{2}, CO_{2}, etc. Future work should also reexamine our results with 3D global climate models, consistently taking into account the effect of clouds and shortwave absorption in the upper atmosphere. This is in order to improve the estimate of the thermal structure and thus the true radius of the planet. Meanwhile, in Sect. 2 we provide empirical formulae for the surface temperature and the thickness of a H_{2}O steam atmosphere, as well as a tutorial on how to correctly use them in Appendix D. These formulae can be used in interior models to better capture the boundary effect of a thick H_{2}Odominated atmosphere.
Acknowledgements
This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie SklodowskaCurie Grant Agreement No. 832738/ESCAPE. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 724427/FOUR ACES and No. 679030/WHIPLASH). This work has been carried out within the framework of the National Centre of Competence in Research PlanetS supported by the Swiss National Science Foundation. The authors acknowledge the financial support of the SNSF. M.T. is grateful for the computing resources on OCCIGEN (CINES, French National HPC). This research has made use of NASA’s Astrophysics Data System. M.T. thanks the Gruber Foundation for its generous support to this research. M.T. thanks BriceOlivier Demory for providing useful feedbacks on the manuscript. Last but not least, we thank the reviewer for his/her insightful remarks and comments on our manuscript.
Appendix A Why and when the R_{planet} = R_{core}+z_{atmosphere} approximation is valid
In order to calculate massradius relationships for waterrich rocky planets, we assumed that the transit radius of a planet can be approximated by the sum of the core radius (directly taken from the Zeng et al. 2016 dry massradius relationships) and the thickness of the water layer, calculated independently. This approach remains valid only if the feedback of the water layer on the rocky interior physical size is negligible. The presence of a water layer (in solid, liquid, or gaseous form) can have two distinct impacts:
 1.
The water layer can exert a pressure force that compresses the rocky interior. Figure A.1 shows the equation of state (EOS) for silicate (MgSiO_{3}, denoted by a solid brown line). The density of MgSiO_{3} is roughly constant until pressure reaches ~10^{10} Pascals (denotedby the vertical, dashed line). In other words, it means that if the water layer exerts a basal pressure that is significantly lower than this ~10^{10} Pascals limit, then the presence of the water layer should have a negligible effect on the silicate interior density profile and thus its physical size. To check that this effect does not significantly affect the massradius relationships presented in Fig. 2, we calculated massradius relationships of waterrich planets with a pure silicate interior, with water assumed to be present in a solid layer (using the water EOS shown in Fig. A.1, denoted by a solid blue line), and compared these calculations with those of Zeng et al. (2016) that selfconsistently take into account the pressure feedback on the interior. The result of this comparison is shown in Fig. A.2. For a 5% watertorock mass ratio, the approximation made in our work leads to a 1% error maximum (for a 2 M_{⊕} core planet) for the range of planets discussed in Figs. 2 and A.2. Finally, this demonstrates that the approximation discussed here is largely acceptable to establish the massradius relationships presented in Fig. 2.
 2.
The water layer can change the thermal structure and possibly even the physical state of the interior. This is particularly relevant in the H_{2}O steam atmosphere case where the surface temperature can reach thousands of Kelvin (Kopparapu et al. 2013; Goldblatt et al. 2013; Turbet et al. 2019), which imposes an extreme surface boundary condition on the interior. While the direct interior temperature profile change should have a limited impact on the radius of the rocky core (Seager et al. 2007; Zeng & Sasselov 2014), the temperature change could lead to a phase change of the interior (e.g., melting) that could significantly increase its physical radius (Bower et al. 2019). Taking this effect into account in a selfconsistent way requires the use of an interior atmosphere coupled model. We leave this for future work.
Fig. A.1 Equations of state (EOS) for iron, silicate (MgSiO_{3}; perovskite phase and its highpressure derivatives), and H_{2}O (Ice Ih, Ice III, Ice V, Ice VI, Ice VII, Ice X, and superionic phase along its melting curve, i.e., solidliquid phase boundary). These EOS were taken from Zeng & Sasselov (2013) (Fig. 1). The vertical, dashed line denotes the typical pressure at which the density of iron and MgSiO_{3} starts to deviate from a constant value. 
Fig. A.2 Comparisons of massradius relationships for waterrich planets (with water in condensed phase) of Zeng et al. (2016) (solid blue lines) with massradius relationships calculated in the present work (dashed blue lines), and assuming a layer of condensed water is added on top of massradius relationships for pure silicate (MgSiO_{3}) planets of Zeng et al. (2016). 
Appendix B Procedure for the polynomial fits
The polynomial fit of our data (surface temperature and effective temperature) was performed in four distinct steps and makes use of the scikitlearn python library (Pedregosa et al. 2011).
As a first step, we recentered and normalized the distribution of values for each parameter (water pressure or water content, surface gravity, stellar flux) of the fit. For this, we used the StandardScaler python tool^{5}. As a second step, we built a matrix of all possible terms of polynomials of degree n or lower. This matrix was constructed using the PolynomialFeatures python tool^{6}. As a reminder, for a polynomial of degree n constructed on k parameters, there is a total number of N = = polynomial terms. In practice, we constructed a matrix of all possible terms of polynomials of degree n = 8 and lower (on our k = 3 parameters; i.e., for water pressure or water content, surface gravity, and stellar flux), reaching a total of N = = 165 polynomial terms for n = 8.
Fig. B.1 Maps of the root mean square error (RMSE) for the fits on the surface temperature T_{surf} (left panel) and the effective temperature T_{eff} (right panel), as a function of the number of polynomial components and the initial degree of the polynomial. 
Fig. B.2 Probability distributions (blue histogram) of residuals for the fits on the surface temperature T_{surf} (left panel) and the effective temperature T_{eff} (right panel). A total of 16 488 1D numerical atmospheric simulations were used. The orange curves indicate the normal distribution laws that best fit the distributions. 
As a third step, we used the recursive feature elimination (RFE) iterative method (Pedregosa et al. 2011) to derive the optimal polynomial fit of our data, using the RFE python tool^{7}. The RFE method was implemented following the recursive steps described below, for each n (from n = 8 to 1):
 1.
We performed a linear fit of our modeled data (surface temperature and effective temperature) with the polynomial of N terms (initially, and for reference, N = ).
 2.
We calculated the RMS of the fit.
 3.
We evaluated the absolute contribution of all N polynomial terms to the fit.
 4.
We removed the polynomial term with the smallest absolute contribution.
 5.
We restarted the procedure iteratively with N1 terms, and until N = 1.
As the final step, we compared the RMS of the fit for each polynomial in order to derive the best compromise between the value of the RMS and the number N of polynomial terms. Based on Fig. B.1, we decided that the fit that gives the best compromise is found for N = 10 for both the surface temperature and the effective temperature. Figure B.1 shows the distribution of the residuals of the fit for the surface temperature and the effective temperature, thus making it possible to evaluate the goodness of the fit (mean error ~ 2.5%; max error ~ 10%).
Appendix C Procedure to derive the empirical formula of the thickness of a steam H_{2}O atmosphere
To construct Eq. (2), we first assumed the hydrostatic equilibrium: (C.1)
with P the atmospheric pressure, r the radial coordinate, g the gravity, and r the radial coordinate. g can be written as (C.2)
We then assumed the atmosphere follows the perfect gas law: (C.3)
with R the gas constant and the molecular weight of H_{2}O (here the dominant gas). T_{eff} is the effective atmospheric temperature and assumed to be constant, for simplicity.
Combining the three previous equations, we derived: (C.4)
We then integrated this equation (assuming P = P_{surf} at r = R_{core}): (C.5)
At the transit radius, R = R_{p} and P = P_{transit}, which gives (C.6)
which can be rewritten as (C.7)
With R_{p} = R_{core} + z_{atmosphere}, we have (C.8)
with M_{atmosphere} the mass of the steam H_{2}Odominated atmosphere (in kg). This relationship does not hold for inflated atmospheres, but for simplicity, we assumed it is valid anyway. Moreover, we have (C.10)
Combining the three previous equations leads to Eq. (2).
Appendix D Quick guide on how to build massradius relationships for waterrich rocky planets more irradiated than the runaway greenhouse limit
In this appendix, we provide a procedure that can be followed to build massradius relationships for waterrich rocky planets more irradiated than the runaway greenhouse limit:
 1.
Choose a core composition.
 2.
Retrieve(or calculate) the massradius relationship corresponding to this core composition. For instance, Zeng et al. (2016)^{8} provides ascii tables of massradius relationships for a wide range of interior composition.
 3.
Choose the water mass fraction () of your planets, as well as the irradiation (S_{eff}) they receive. We note that the irradiation must be larger than the runaway greenhouse irradiation limit, which depends on the type of host star (Kopparapu et al. 2013) and on the mass and radius of the planetary core (Kopparapu et al. 2014). Moreover, the water mass fraction must be “reasonable” (see discussions in Appendix A).
 4.
For each datapoint of the selected core massradius relationship (i.e., for each set of core mass and radius), calculate the corresponding surface gravity (g_{core}).
 5.
For each datapoint of the selected core massradius relationship, compute the thickness z_{atmosphere} of the H_{2}O atmospheric layer using Eq. (2). Equation (2) makes use of Eq. (3) and the empirical coefficients provided in Table 2.
 6.
For each datapoint of the selected core massradius relationship (M_{core}, R_{core}), compute the new massradius relationship (M_{planet}, R_{planet}) by assuming that R_{planet} = R_{core} + z_{atmosphere} and M_{planet} = M_{core}/(1).
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The densities of TRAPPIST1 planets were measured with the transit timing variations (TTVs) technique. They are therefore absolute densities, and are thus not affected by inaccuracy on the stellar mass and radius measurements (Grimm et al. 2018).
Userfriendly data is provided on the personal website of Li Zeng (https://www.cfa.harvard.edu/~lzeng/planetmodels.html).
Userfriendly data is provided on the personal website of Li Zeng (https://www.cfa.harvard.edu/~lzeng/planetmodels.html)
All Tables
Maximum water content of TRAPPIST1 b, c, and d, depending on the assumed core composition.
All Figures
Fig. 1 Surface temperature as a function of the effective flux received on a planet (xaxis) and the surface pressure of its steam H_{2}O atmosphere (yaxis), for three different surface gravities (0.3, 1 and 3× the gravity on Earth). The surface temperature was estimated using Eq. (1). The small black dots indicate the parameter space for which atmospheric numerical simulations have been carried out, and on which the fit of the surface temperature is based. 

In the text 
Fig. 2 Massradius relationships for various interior compositions and water content, assuming water is in the condensed form (left panel) and water forms an atmosphere (right panel). The silicate composition massradius relationship assumes a pure MgSiO_{3} interior and was taken from Zeng et al. (2016). The waterrich massradius relationships for water in condensed form (left panel) were derived using the data from Zeng et al. (2016). The waterrich massradius relationships for water in gaseous form (right panel) are the result of the present work. All massradius relationships with water were built assuming a pure MgSiO_{3} interior. For comparison, we added the measured positions of the seven TRAPPIST1 planets measured from Grimm et al. (2018), with their associated 95% confidence ellipses. Based on the irradiation they receive compared to the theoretical runaway greenhouse limit (Kopparapu et al. 2013; Wolf 2017; Turbet et al. 2018), TRAPPIST1 e, f, g, and h should be compared with massradius relationships on the left, while TRAPPIST1b, c, and d should be compared with those on the right. To emphasize this, we indicated, on each panel and in black (and solid line ellipses), the planets (and their associated 95% confidenceellipses) for which massradius relationships (with water) are appropriate. In contrast, we indicated on each panel in gray (and dashed line ellipses) the planets (and their associated 95% confidence ellipses) for which massradius relationships (with water) are not appropriate. For reference, we also added a terrestrial composition that resembles that of the Earth, but also that of Mars and Venus. We note that massradiusrelationships for steam planets (right panel) can be easily built following the procedure described in Appendix D. 

In the text 
Fig. 3 Example of a scenario for TRAPPIST1 planets where masses and radii follow an interior isocomposition line (gray line; 10% Fe, 90% MgSiO_{3} composition) chosen to be consistent with 2σ uncertainty ellipses of Grimm et al. (2018). While the seven large ellipses indicate the known current estimates (95% confidence) for the masses and radii of the seven TRAPPIST1 planets, the seven small circles indicate the positions (in the mass, radius diagram) of the seven planets as speculated in our scenario. Each planet (associated with a current uncertainty ellipse, and a speculated position) is identified by a distinct color. This figure also shows massradius relationships for terrestrial core planets, in some cases endowed with either a condensed layer of water (solid blue line) or a steam H_{2}O atmosphere of various masses (dashed purple lines). The massradius relationships for steam planets can be built following the procedure described in Appendix D. We note that the massradius relationships for steam H_{2}Orich atmosphere planets can slightly change depending on the level of stellar irradiation they receive. However, because these changes are low (for the levels of irradiation on TRAPPIST1 b, c, and d) compared to the runawaygreenhousetransitioninduced massradius relationship change, we decided no to show them for clarity. 

In the text 
Fig. A.1 Equations of state (EOS) for iron, silicate (MgSiO_{3}; perovskite phase and its highpressure derivatives), and H_{2}O (Ice Ih, Ice III, Ice V, Ice VI, Ice VII, Ice X, and superionic phase along its melting curve, i.e., solidliquid phase boundary). These EOS were taken from Zeng & Sasselov (2013) (Fig. 1). The vertical, dashed line denotes the typical pressure at which the density of iron and MgSiO_{3} starts to deviate from a constant value. 

In the text 
Fig. A.2 Comparisons of massradius relationships for waterrich planets (with water in condensed phase) of Zeng et al. (2016) (solid blue lines) with massradius relationships calculated in the present work (dashed blue lines), and assuming a layer of condensed water is added on top of massradius relationships for pure silicate (MgSiO_{3}) planets of Zeng et al. (2016). 

In the text 
Fig. B.1 Maps of the root mean square error (RMSE) for the fits on the surface temperature T_{surf} (left panel) and the effective temperature T_{eff} (right panel), as a function of the number of polynomial components and the initial degree of the polynomial. 

In the text 
Fig. B.2 Probability distributions (blue histogram) of residuals for the fits on the surface temperature T_{surf} (left panel) and the effective temperature T_{eff} (right panel). A total of 16 488 1D numerical atmospheric simulations were used. The orange curves indicate the normal distribution laws that best fit the distributions. 

In the text 
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