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# -Ternary diagram including the mass and radius uncertainties for a planet of a fixed mass and radius. This example is for a planet with M p ¼ 10 AE 0:5 M ⊕ and R p ¼ 2 AE 0:1 M ⊕ , showing the 1, 2, and 3 σ uncertainty curves. This Figure shows the continuous distribution of possible combinations of iron, silicate, and water with increasing uncertainties according to the color bar. Notice that considering the 3 σ uncertainties almost the entire ternary diagram is covered-in other words there is no constraint on the planet internal composition. See text for a discussion of the direction and spacing of the curves.

Source publication

The prospects for finding transiting exoplanets in the range of a few to 20
Earth masses is growing rapidly with both ground-based and spaced-based
efforts. We describe a publicly available computer code to compute and quantify
the compositional ambiguities for differentiated solid exoplanets with a
measured mass and radius, including the mass and...

## Contexts in source publication

**Context 1**

... third subroutine plots the ternary dia- grams; this subroutine calls a ternary diagram plotting routine 4 that plots a single line for each of the 1, 2, and 3 σ contour lines (see § 4.1). An example from this code is shown in Figure 4. ...

**Context 2**

... we have assumed that the uncertainties in mass and radius are independent from each other and have assumed the linearity of the superposition of small uncertainties. Figure 4 shows a planet with M p ¼ 10 AE 0:5 M ⊕ and R p ¼ 2 AE 0:1 M ⊕ . We see that taking the 3 σ limit, almost the entire ternary diagram is filled. ...

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## Citations

... In general, in the literature to determine the internal structure of the planets, models based on EOS of minerals and metals obtained either theoretically or experimentally in the laboratory are used. Several previous models of solid planets have been built using pure constituents, like solid Fe-in the core and pv and ppv in the mantle, without contemplating the presence of impurities or other elements, and considering a complete differentiation of layers that do not occur on real planets (Zeng & Seager 2008;Zeng & Sasselov 2013). These theoretical models do not represent the Earth characteristics properly. ...

For a planet to be considered habitable on its surface, it is an important advantage for it to have a magnetic field that protects its atmosphere from stellar winds as well as cosmic rays. Magnetic protection of potentially habitable planets plays a key role in determining the chances of detecting atmospheric biosignatures. This paper proposes to use the Preliminary Reference Earth Model (PREM) internal structure as the base of a numerical model. With this model, we estimate the magnetic properties of dry and water-rich Earth-like and Super-Earth-like planets. We apply it to those of this kind at the first 176 planets confirmed by TESS. Using PREM as a reference, we estimate the internal structure of dry and water-rich rocky planets. This model provides an estimation of the average density and core size of the planet and, with them, its magnetic moment depending on whether it is tidally locked or not. Our model estimates the thermodynamic variables as a function of pressure and includes saltwater as a component of water-rich exoplanets. We have not used the perfect layer differentiation approximation. We have validated our model with those planets and satellites in the Solar system with similar characteristics. The differences with the observed values in the internal structure characteristics, mass, average density, moment of inertia factor, and local Rossby number are remarkably low or even negligible. The estimated magnetic moments are also very similar to the observed ones. We have applied the model to the first 176 planets confirmed by the TESS, finding that, from an astrobiological perspective TOI-700 d and TOI-2257 b are the most interesting ones as being located in the habitable zone (HZ), although their magnetic moments are only about 0.01 of the Earth's magnetic moment.

... Though the predicted helium signal is too large, the predicted mass loss rates in Table 3 are all reasonable. We assume a 2% hydrogen/helium envelope, which is consistent with the planet's radius, mass, and equilibrium temperature assuming a rocky core (Zeng & Seager 2008). The mass loss timescales predicted by Microthena would then range from 1.6-2.4 ...

We report helium absorption from the escaping atmosphere of TOI 560.01 (HD 73583b), an R = 2.8 R ⊕ , P = 6.4 day mini-Neptune orbiting a young (∼600 Myr) K dwarf. Using Keck/NIRSPEC, we detect a signal with an average depth of 0.68% ± 0.08% in the line core. The absorption signal repeats during a partial transit obtained a month later, but is marginally stronger and bluer, perhaps reflecting changes in the stellar wind environment. Ingress occurs on time, and egress occurs within 12 minutes of the white light egress, although absorption rises more gradually than it declines. This suggests that the outflow is slightly asymmetric and confined to regions close to the planet. The absorption signal also exhibits a slight 4 km s ⁻¹ redshift rather than the expected blueshift; this might be explained if the planet has a modest orbital eccentricity, although the radial velocity data disfavors such an explanation. We use XMM-Newton observations to reconstruct the high-energy stellar spectrum and model the planet’s outflow with 1D and 3D hydrodynamic simulations. We find that our models generally overpredict the measured magnitude of the absorption during transit, the size of the blueshift, or both. Increasing the metallicity to 100× solar suppresses the signal, but the dependence of the predicted signal strength on metallicity is non-monotonic. Decreasing the assumed stellar EUV flux by a factor of three likewise suppresses the signal substantially.

... We assume a core made of Fe and FeS. Many previous studies assume pure iron (Zeng & Seager 2008;Dorn et al. 2015;Rogers & Seager 2010). However, the addition of light alloys in the Earth's core (Dziewonski & Anderson 1981) can reduce the core density between 5 to 10%. ...

We demonstrate that the deep volatile storage capacity of magma oceans has significant implications for the bulk composition, interior, and climate state inferred from exoplanet mass and radius data. Experimental petrology provides the fundamental properties of the ability of water and melt to mix. So far, these data have been largely neglected for exoplanet mass–radius modeling. Here we present an advanced interior model for water-rich rocky exoplanets. The new model allows us to test the effects of rock melting and the redistribution of water between magma ocean and atmosphere on calculated planet radii. Models with and without rock melting and water partitioning lead to deviations in planet radius of up to 16% for a fixed bulk composition and planet mass. This is within the current accuracy limits for individual systems and statistically testable on a population level. Unrecognized mantle melting and volatile redistribution in retrievals may thus underestimate the inferred planetary bulk water content by up to 1 order of magnitude.

... The plotted planets have orbital periods smaller than 100 days, radius between 1.5 R ⊕ and 5 R ⊕ , and <25% mass uncertainty. The solid lines are planet composition lines fromZeng & Seager (2008). The red dashed line corresponds to the mass-radius relation given byWeiss & Marcy (2014). ...

We present the discovery of Kepler-129 d (P_d = 7.2_(-0.3)^(+0.4) yr, m sin i_d = 8.3_(-0.7)^(+1.1), M_(Jup), e_d = 0.15_(-0.05)^(+0.07)) based on six years of radial-velocity observations from Keck/HIRES. Kepler-129 also hosts two transiting sub-Neptunes: Kepler-129 b (P_b = 15.79 days, r_b = 2.40 ± 0.04 R_⊕) and Kepler-129 c (P_c = 82.20 days, r_c = 2.52 ± 0.07 R_⊕) for which we measure masses of m_b < 20 M_⊕ and m_c = 43₋₁₂⁺¹³ M_⊕. Kepler-129 is a hierarchical system consisting of two tightly packed inner planets and a massive external companion. In such a system, two inner planets precess around the orbital normal of the outer companion, causing their inclinations to oscillate with time. Based on an asteroseismic analysis of Kepler data, we find tentative evidence that Kepler-129 b and c are misaligned with stellar spin axis by ≳38°, which could be torqued by Kepler-129 d if it is inclined by ≳19° relative to inner planets. Using N-body simulations, we provide additional constraints on the mutual inclination between Kepler-129 d and inner planets by estimating the fraction of time during which two inner planets both transit. The probability that two planets both transit decreases as their misalignment with Kepler-129 d increases. We also find a more massive Kepler-129 c enables the two inner planets to become strongly coupled and more resistant to perturbations from Kepler-129 d. The unusually high mass of Kepler-129 c provides a valuable benchmark for both planetary dynamics and interior structure, since the best-fit mass is consistent with this 2.5 R_⊕ planet having a rocky surface.

... To explore the implications of these outgassing calculations for the Trappist-1 system, we estimated the implied ocean depth and seafloor pressures for the Trappist-1 planets using mass and radius constraints (Agol et al. 2021) and a modified version of the publicly available interior model described in Zeng & Seager (2008) and Seager et al. (2007). A complete description of our approach is provided in Appendix A. To briefly summarize, mass and radius constraints are used to solve for iron core, silicate mantle, and water/ice mass fraction distributions, along with self-consistent pressure-depth profiles. ...

... Separate probability distributions are shown assuming iron core fractions of 10%-20%, 20%-30%, 30%-40%, and 40%-50%. The top panel shows the results using the original EOSs for iron and silicates from Zeng & Seager (2008), while the bottom panel shows results using EOSs that are empirically adjusted to fit Earth's interior composition (see Appendix A). These two scenarios are considered endmembers for plausible interior structures. ...

... Outgassing fluxes from magmatic sources are determined by the integrated loss of volatiles over Figure 3. Pressure at the silicate-water interface for Trappist-1f and Trappist-1g. The top row shows results using the unmodified equations of state from Zeng & Seager (2008). The bottom row shows results using density profiles empirically modified to match Earth's pressure-density profile (see the Appendices). ...

Terrestrial planets with large water inventories are likely ubiquitous and will be among the first Earth-sized planets to be characterized with upcoming telescopes. It has previously been argued that waterworlds-particularly those possessing more than 1% H$_2$O-experience limited melt production and outgassing due to the immense pressure overburden of their overlying oceans, unless subject to high internal heating. But an additional, underappreciated obstacle to outgassing on waterworlds is the high solubility of volatiles in high-pressure melts. Here, we investigate this phenomenon and show that volatile solubilities in melts probably prevent almost all magmatic outgassing from waterworlds. Specifically, for Earth-like gravity and oceanic crust composition, oceans or water ice exceeding 10-100 km in depth (0.1-1 GPa) preclude the exsolution of volatiles from partial melt of silicates. This solubility limit compounds the pressure overburden effect as large surface oceans limit both melt production and degassing from any partial melt that is produced. We apply these calculations to Trappist-1 planets to show that, given current mass and radius constraints and implied surface water inventories, Trappist-1f and -1g are unlikely to experience volcanic degassing. While other mechanisms for interior-surface volatile exchange are not completely excluded, the suppression of magmatic outgassing simplifies the range of possible atmospheric evolution trajectories and has implications for interpretation of ostensible biosignature gases, which we illustrate with a coupled model of planetary interior-climate-atmosphere evolution.

... To explore the implications of these outgassing calculations for the Trappist-1 system, we estimated the implied ocean depth and seafloor pressures for the Trappist-1 planets using mass and radius constraints (Agol et al. 2021) and a modified version of the publicly available interior model described in Zeng & Seager (2008) and Seager et al. (2007). A complete description of our approach is provided in Appendix A. To briefly summarize, mass and radius constraints are used to solve for iron core, silicate mantle, and water/ice mass fraction distributions, along with self-consistent pressure-depth profiles. ...

... Separate probability distributions are shown assuming iron core fractions of 10%-20%, 20%-30%, 30%-40%, and 40%-50%. The top panel shows the results using the original EOSs for iron and silicates from Zeng & Seager (2008), while the bottom panel shows results using EOSs that are empirically adjusted to fit Earth's interior composition (see Appendix A). These two scenarios are considered endmembers for plausible interior structures. ...

... Outgassing fluxes from magmatic sources are determined by the integrated loss of volatiles over Figure 3. Pressure at the silicate-water interface for Trappist-1f and Trappist-1g. The top row shows results using the unmodified equations of state from Zeng & Seager (2008). The bottom row shows results using density profiles empirically modified to match Earth's pressure-density profile (see the Appendices). ...

Terrestrial planets with large water inventories are likely ubiquitous and will be among the first Earth-sized planets to be characterized with upcoming telescopes. It has previously been argued that waterworlds-particularly those possessing more than 1% H2O-experience limited melt production and outgassing due to the immense pressure overburden of their overlying oceans, unless subject to high internal heating. But an additional, underappreciated obstacle to outgassing on waterworlds is the high solubility of volatiles in high-pressure melts. Here, we investigate this phenomenon and show that volatile solubilities in melts probably prevent almost all magmatic outgassing from waterworlds. Specifically, for Earth-like gravity and oceanic crust composition, oceans or water ice exceeding 10-100 km in depth (0.1-1GPa) preclude the exsolution of volatiles from partial melt of silicates. This solubility limit compounds the pressure overburden effect as large surface oceans limit both melt production and degassing from any partial melt that is produced. We apply these calculations to Trappist-1 planets to show that, given current mass and radius constraints and implied surface water inventories, Trappist-1f and -1g are unlikely to experience volcanic degassing. While other mechanisms for interior-surface volatile exchange are not completely excluded, the suppression of magmatic outgassing simplifies the range of possible atmospheric evolution trajectories and has implications for interpretation of ostensible biosignature gases, which we illustrate with a coupled model of planetary interior-climate-atmosphere evolution.

... The plotted planets have orbital periods smaller than 100 days, radius between 1.5 R⊕ and 5 R⊕, and < 25% mass uncertainty. The solid lines are planet composition lines fromZeng & Seager (2008). The red dashed line corresponds to the mass radius relation given byWeiss & Marcy (2014). ...

We present the discovery of Kepler-129 d ($P_{d}=7.2^{+0.4}_{-0.3}$ yr, $m\sin i_{d}=8.3^{+1.1}_{-0.7}\ \rm M_{Jup}$, $ e_{d}=0.15^{+0.07}_{-0.05} $) based on six years of radial velocity (RV) observations from Keck/HIRES. Kepler-129 also hosts two transiting sub-Neptunes: Kepler-129 b ($P_{b}=15.79$ days, $r_{b}=2.40\pm{0.04}\ \rm{R_{\oplus}}$) and Kepler-129 c ($P_{c}=82.20$ days, $r_{c}=2.52\pm{0.07}\ \rm{R_{\oplus}}$) for which we measure masses of $m_{b}<20\ \rm{M_{\oplus}}$ and $m_{c}=43^{+13}_{-12}\ \rm{M_{\oplus}}$. Kepler-129 is an hierarchical system consisting of two tightly-packed inner planets and an external companion whose mass is close to the deuterium burning limit. In such a system, two inner planets precess around the orbital normal of the outer companion, causing their inclinations to oscillate with time. Based on an asteroseismic analysis of Kepler data, we find tentative evidence that Kepler-129 b and c are misaligned with stellar spin axis by $\gtrsim 38$ deg, which could be torqued by Kepler-129 d if it is inclined by $\gtrsim 19$ deg relative to inner planets. Using N-body simulations, we provide additional constraints on the mutual inclination between Kepler-129 d and inner planets by estimating the fraction of time during which two inner planets both transit. The probability that two planets both transit decreases as their misalignment with Kepler-129 d increases. We also find a more massive Kepler-129 c enables the two inner planets to become strongly coupled and more resistant to perturbations from Kepler-129 d. The unusually high mass of Kepler-129 c provides a valuable benchmark for both planetary dynamics and interior structure, since the best-fit mass is consistent with this $\rm{2.5\ R_{\oplus}}$ planet having a rocky surface.

... A possible restriction in our adopted methodology is that our core composition analysis has implemented the Fortney et al. (2007) mass-radius relationships, which only allow two species to be present in a core at one time: Either an ice-silicate mixture or a silicate-iron mixture. In reality, all three species are likely to coexist in cores and hence we use an exoplanet interior model from Zeng & Seager (2008) to calculate the allowed composition fractions for a planet with our inferred bulk core density ∼5 g cm −3 . Fig. 13 shows the iron and silicate mass fractions as a function of ice mass fraction for a 4M ⊕ planet with a ∼1.63R ⊕ core radius -not only does this combination ensure matching bulk core densities with our inferred distributions, but also is a typical mass and radius for a super-Earth. ...

... Furthermore, if we wish to match ice mass fractions predicted by formation models beyond the Figure 13. Mass fractions of iron and silicate as a function of ice mass fraction for a 4M ⊕ planet according to core interior models from Zeng & Seager (2008). We fix the core radius to be ∼1.63R ...

The radius distribution of small, close-in exoplanets has recently been shown to be bimodal. The photoevaporation model predicted this bimodality. In the photoevaporation scenario, some planets are completely stripped of their primordial H/He atmospheres, whereas others retain them. Comparisons between the photoevaporation model and observed planetary populations have the power to unveil details of the planet population inaccessible by standard observations, such as the core mass distribution and core composition. In this work, we present a hierarchical inference analysis on the distribution of close-in exoplanets using forward-models of photoevaporation evolution. We use this model to constrain the planetary distributions for core composition, core mass and initial atmospheric mass fraction. We find that the core-mass distribution is peaked, with a peak-mass of ∼4 M⊕. The bulk core-composition is consistent with a rock/iron mixture that is ice-poor and “Earth-like”; the spread in core-composition is found to be narrow ($\lesssim 16\%$ variation in iron-mass fraction at the 2σ level) and consistent with zero. This result favours core formation in a water/ice poor environment. We find the majority of planets accreted a H/He envelope with a typical mass fraction of $\sim 4\%$; only a small fraction did not accrete large amounts of H/He and were “born-rocky”. We find four-times as many super-Earths were formed through photoevaporation, as formed without a large H/He atmosphere. Finally, we find core-accretion theory over-predicts the amount of H/He cores would have accreted by a factor of ∼5, pointing to additional mass-loss mechanisms (e.g. “boil-off”) or modifications to core-accretion theory.

... Iron Silicate Figure 13. Mass fractions of iron and silicate as a function of ice mass fraction for a 6M ⊕ planet according to core interior models from Zeng & Seager (2008). We fix the core radius to be ∼ 1.63R ⊕ in this calculation as this requires that an ice-free core has a silicate-to-iron composition ratio of ∼ 3:1 and hence aligns with our constrained distributions. ...

... A possible restriction in our adopted methodology is that our core composition analysis has implemented the Fortney et al. (2007) mass-radius relationships, which only allow two species to be present in a core at one time: either an ice-silicate mixture or a silicate-iron mixture. In reality, all three species are likely to coexist in cores and hence we use an exoplanet interior model from Zeng & Seager (2008) to calculate the allowed composition fractions for a planet with our inferred bulk core density ∼ 5g cm −3 . Figure 13 shows the iron and silicate mass fractions as a function of ice mass fraction for a 6M ⊕ planet with a ∼ 1.63R ⊕ core radius -not only does this combination ensure matching bulk core densities with our inferred distributions, but also is a typical mass and radius for a super-Earth. ...

The radius distribution of small, close-in exoplanets has recently been shown to be bimodal. The photoevaporation model predicted this bimodality. In the photoevaporation scenario, some planets are completely stripped of their primordial H/He atmospheres, whereas others retain them. Comparisons between the photoevaporation model and observed planetary populations have the power to unveil details of the planet population inaccessible by standard observations, such as the core mass distribution and core composition. In this work, we present a hierarchical inference analysis on the distribution of close-in exoplanets using forward-models of photoevaporation evolution. We use this model to constrain the planetary distributions for core composition, core mass and initial atmospheric mass fraction. We find that the core-mass distribution is peaked, with a mean-mass of $\sim 6$ M$_\oplus$. The bulk core-composition is consistent with a rock/iron mixture that is ice-poor and "Earth-like"; the spread in core-composition is found to be narrow ($\lesssim 16\%$ variation in iron-mass fraction at the 2$\sigma$ level) and consistent zero. This result favours core formation in a water/ice poor environment. We find the majority of planets accreted a H/He envelope with a typical mass fraction of $\sim 4\%$; only a small fraction did not accrete large amounts of H/He and were "born-rocky". We find four-times as many super-Earths were formed through photoevaporation, as formed without a large H/He atmosphere. Finally, we find core-accretion theory over-predicts the amount of H/He cores would have accreted by a factor of $\sim 5$, pointing to additional mass-loss mechanisms (e.g. "boil-off") or modifications to core-accretion theory.

... Mass-radius curves. The mass-radius curves (Figs. 1 & 2) allow calculations of average densities and further constraining the planet bulk compositions and internal structures following the methods that we published earlier (106)(107)(108)(109) and similar efforts from other researchers in our field (110)(111)(112)(113)(114)(115)(116)(117)(118)(119)(120). The H2-He is assumed to be a cosmic mixture of 75% H2 and 25% He by mass (121)(122)(123). ...

The radii and orbital periods of 4000+ confirmed/candidate exoplanets have been precisely measured by the Kepler mission. The radii show a bimodal distribution, with two peaks corresponding to smaller planets (likely rocky) and larger intermediate-size planets, respectively. While only the masses of the planets orbiting the brightest stars can be determined by ground-based spectroscopic observations, these observations allow calculation of their average densities placing constraints on the bulk compositions and internal structures. Yet an important question about the composition of planets ranging from 2 to 4 Earth radii still remains. They may either have a rocky core enveloped in a H2-He gaseous envelope (gas dwarfs) or contain a significant amount of multi-component, H2O-dominated ices/fluids (water worlds). Planets in the mass range of 10-15 Earth masses, if half-ice and half-rock by mass, have radii of 2.5 Earth radii, which exactly match the second peak of the exoplanet radius bimodal distribution. Any planet in the 2-4 Earth radii range requires a gas envelope of at most a few mass percentage points, regardless of the core composition. To resolve the ambiguity of internal compositions, we use a growth model and conduct Monte Carlo simulations to demonstrate that many intermediate-size planets are water worlds.