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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.

-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.

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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...

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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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... 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. ...
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... 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 ...
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... 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. ...
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... 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). ...
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