On the Probability That a Rocky Planet’s Composition Reflects Its Host Star

The bulk density of a planet, as measured by mass and radius, is a result of planet structure and composition. Relative proportions of iron core, rocky mantle, and gaseous envelopes are degenerate for a given density. This degeneracy is reduced for rocky planets without significant gaseous envelopes when the structure is assumed to be a differentiated iron core and rocky mantle, in which the core mass fraction (CMF) is a first-order description of a planet’s bulk composition. A rocky planet’s CMF may be derived both from bulk density and by assuming the planet reflects the host star’s major rock-building elemental abundances (Fe, Mg, and Si). Contrasting CMF measures, therefore, shed light on the outcome diversity of planet formation from processes including mantle stripping, out-gassing, and/or late-stage volatile delivery. We present a statistically rigorous analysis of the consistency of these two CMF measures accounting for observational uncertainties of planet mass and radius and host-star chemical abundances. We find that these two measures are unlikely to be resolvable as statistically different unless the bulk density CMF is at least 40% greater than or 50% less than the CMF as inferred from the host star. Applied to 11 probable rocky exoplanets, Kepler-107 c has a CMF as inferred from bulk density that is significantly greater than the inferred CMF from its host star (2σ) and is therefore likely an iron-enriched super-Mercury. K2-229b, previously described as a super-Mercury, however, does not meet the threshold for a super-Mercury at a 1σ or 2σ level.


Major Rock-building Elements in Stars and Planets
Rocky planet composition is degenerate with respect to mass and radius, the primary direct observables of small exoplanets. To break this degeneracy, rocky planets are often assumed to be made of predominantly Fe and MgSiO 3 with proportions determined by the relative abundances of the major rockbuilding elements, Fe, Mg, and Si, observed in the host star (e.g., Dorn et al. 2015;Unterborn et al. 2016;Brugger et al. 2017). The primary foundations for these assumptions come from the relationship between the compositions of the solar system rocky planets and the relative solar Fe, Mg, and Si abundances, and the fact that, together with oxygen, these elements make up 95 mol % of the Earth (McDonough 2003).
The Earth's relative bulk composition of major rock-building elements reflects that of the Sun . Upon condensation, the major hosts for Mg and Si are forsterite and enstatite (Mg 2 SiO 4 and MgSiO 3 ) while Fe initially condenses as a metal, each with 50% condensation temperatures between 1300 and 1350 K (Lodders 2003). Further, the most chemically primitive remnants from solar system formation, CI-chondrites, have abundances of the refractory and major rock-building elements that are within ∼10% the relative abundances found in the Sun (Lodders 2003). Their Fe/Mg and Si/Mg ratios reflect the solar photospheric ratios to within 2% and 4%, respectively (Putirka & Rarick 2019).
Mars, like Earth, has a molar Fe/Mg ratio to within ∼10%-15% of the Sun's abundance (Lodders 2003;McDonough 2003;Wanke & Dreibus 1994;Bertka & Fei 1998;Zharkov & Gudkova 2005;Yoshizaki & McDonough 2020). While the Fe/Mg ratio for Venus is poorly constrained, it is consistent with the Earth (Zharkov 1983). Thus, the bulk chemical compositions of Venus, Earth, and Mars appear to be consistent with the hypothesis that these planets initially formed from chondrites, and are thus reflective of the initial relative abundances of the major rock-building elements of the solar photosphere.
In contrast, Mercury has an Fe concentration ∼200%-400% greater than expected relative to silicates (e.g., Morgan & Anders 1980). Therefore, not all rocky planets in the solar system reflect the relative solar abundances of the major rockbuilding elements. The case for Mercury's chemical anomaly suggests an opportunity to study the diversity of the outcomes of planet formation: searching for the chemical anomaly in a large sample of exoplanets.
Starting from a hypothesis that rocky planet compositions mirror their host star's major rock-building element abundances, model-dependent planet masses and radii can be inferred. For example, iron enrichment relative to magnesium and silicon is invoked to explain higher than expected density (e.g., Santerne et al. 2018). Given the hypothesis of compositional mirroring of its host star, a star with a relatively high or low Fe/Mg ratio will form relatively Fe-rich (denser) or Fe-poor (less dense) planets, respectively. Therefore, planets whose apparent relative iron content is statistically greater than predicted by the host require alternative formation and/or evolutionary mechanisms to explain their compositions. Where the model-based masses and radii result in lower densities than predicted from host-star abundances, these planets are suggested to have thick surface ice/water layers (Unterborn et al. 2018), to be planets enriched in those minerals condensing at the highest temperatures with depleted iron , to be core-free planets (Elkins-Tanton & Seager 2008), or to be magma ocean planets (Bower et al. 2019).
In this work, we present a rigorous statistical method to test the null hypothesis,  0 : the measured mass and radius of a given exoplanet is statistically consistent with a model of a barren planet consisting of only an iron core and iron-free silicate mantle in proportions identical to the host star's measured photospheric Fe/Mg and Si/Mg abundance ratios. This approach determines the likelihood that a given planet with well-measured mass and radius satisfies or refutes the oftinvoked assumption that small, dense planets reflect the relative abundances of the major rock-building elements of its host star. In cases where  0 fails, the planet may either have significant atmospheric layers or nonstellar relative abundances of Fe, Mg, and Si, but our approach makes no attempt to infer the cause.
In the case of planets that do not satisfy the null hypothesis, we discuss the range of possible interpretations, including whether such a planet requires a super-stellar iron abundance relative to its host or is instead consistent with either a smaller than expected core or an outer volatile layer.

Sample Selection
To test the hypothesis, we identify planets with wellconstrained mass and radius measurements that are most likely to have rocky surfaces without a significant gas layer. There are over 4000 confirmed planets in the NASA Exoplanet Archive, 4 in which 761 planets have both mass and radius measurements. From this sample, we choose planets that are unlikely to retain significant H/He envelopes because of their low surface gravity and the radiation received from their host stars (e.g., Jin & Mordasini 2018). We use the period-dependent radius gap (Van Eylen et al. 2018) as an upper bound on planet radius. This period-dependent radius gap corresponds to R p ; 2.3, 1.9, and 1.5 R ⊕ at orbital periods of 1, 10, and 100 days, respectively. We find 74 planets with measured masses that meet this radius criterion.
We further limit our sample to 28 planets with uncertainties of 20% and 10% in planetary mass and radius, respectively. Among them, only 11 planets have host stars with reported Fe, Mg, and Si abundance measurements for their host stars. These 11 planets form the sample for our subsequent analyses, and their properties are summarized in Table 1.
Of this sample, the orbital periods range from 0.58 to 6.76 days, radii vary from 1.197 to 1.897 R ⊕ , and the observational uncertainties in mass and radius range from 4% to 19% and 1.5% to 6.4%, respectively. The associated uncertainty in bulk density of these planets ranges from 8% (55 Cnc e) to 21% (K2-229b). All of the identified planets are in orbit around FGK stars.

Calculating CMF
To first-order, the composition of a rocky planet can be described by the relative amount of iron to silicates (Plotnykov & Valencia 2020). Assuming all Fe is in the core and all silicates reside in the mantle, the composition of a rocky planet can be quantified by its core mass fraction (CMF) given present-day precision in mass and radius (Unterborn et al. 2016;Dorn et al. 2015).
Therefore, we test  0 through comparison of two independent calculations of the CMF: (1) the fraction core required to explain the average density of the planet, CMF ρ , and (2) the mass fraction of core as predicted by the Mg, Si, and Fe relative abundances of the star, CMF star . We determine that the hypothesis  0 is refuted when these two measures for CMF differ given the limits of the observational data.
We use the thermodynamically self-consistent ExoPlex 5 mass-radius software (Unterborn et al. 2018) to solve for CMF ρ . ExoPlex solves the five coupled differential equations: the mass within a sphere, hydrostatic equilibrium, adiabatic temperature profile, Gauss's law of gravity in one dimension, and the thermally dependent equation of state. We fix the planetary mass and set a radius convergence criterion of 0.0001R ⊕ , more than two orders of magnitude more precise than the planetary radius uncertainties in our sample. CMF ρ calculations assume a pure solid Fe core and an oxidized Fe-free silicate mantle (MgSiO 3 ) with a solid surface (i.e., it is not a magma ocean planet). For this calculation, we make a simplifying assumption that the mantle has a fixed molar ratio of Si/Mg = 1. Si/Mg ratios between 0.5 and 2 affect the calculation in planet mass by no more than 2% (Dorn et al. 2015;Unterborn et al. 2016), less than the observational uncertainties. We also do not include minor mantle elements (i.e., Ca and Al) in our models as these also do not significantly affect inferred masses. We adopt the iron Vinet equation of state from Smith et al. (2018) for the core and the equation of state developed in Stixrude & Lithgow-Bertelloni (2005) for the mantle. In the simplified, two-layer model of a rocky planet, CMF ρ is the mass of iron, M Fe , divided by the mass of the planet, M p , The expected proportion of Fe in a rocky planet's core and Mg and Si making up the mantle as a mixture of oxides, CMF star , can be expressed as where compositions (X/Y) are the stellar molar ratio for elements X and Y, m i is the molar mass of species i. This approach makes a parallel assumption to the calculation of CMF ρ , assuming the core is pure iron and the mantle reflects fully oxidized Mg and Si. A mantle composed of fully oxidized Mg and Si with a metallic Fe core implies an oxygen content controlled by the Si and Mg content of the planet (Unterborn & Panero 2017). Atomic diffusion in main-sequence stars can result in surface abundances that are different than the bulk composition, as some elements experience preferential gravitational settling. However, Mg, Si, and Fe are all expected to be affected by about the same amount in FGK stars (e.g., Liu et al. 2019) and the ratios of these elements reflect the bulk stellar composition.

The Impact of Observational Uncertainties on CMF Calculations
The comparison between a planet's composition as inferred from our simple model and its host-star's abundances is limited by the observational uncertainties of planetary mass and radius, as well as the uncertainties in host-star abundances. We, therefore, quantify the relationship between these observational uncertainties and the proportional impact on planetary structure as described by the relative proportions of rocky mantle and metallic core (Tables A1 and A2).
For each mass and radius of the planets in our sample ( Table 1), we calculate the 1σ uncertainties in CMF ρ from the errors in their mass and radius measurements using the joint 1σ mass-radius elliptical distribution. We sample 1000 massradius pairs along the 1σ M-R ellipse, from which we derive the 1σ mean density uncertainty and its corresponding CMF ρ uncertainty.
The comparable uncertainty in CMF star with respect to Fe/Mg and Si/Mg is found through a propagation of uncertainties in Equation (2), These uncertainties are independent of planet size, and a weak function of composition (Table A2).

Hypothesis Testing
We quantify the probability that a planet's composition reflects the major rock-building element composition of its host star by calculating the amount of overlap between CMF ρ and CMF star normalized by the null hypothesis that both distributions have the same mean values that we fix here to 0.5, star CMF star are the probability distributions of CMF ρ and CMF star , respectively. For calculations, we assume that all distributions are Gaussian.
This approach incorporates the mutual uncertainties of mass, radius, and host-star abundances, in which the probability is proportional to the similarity between modeled CMF ρ and predicted CMF star . Large uncertainties increase the likelihood that a planet will be indistinguishable from  0 . A graphical interpretation of Equation (4) is illustrated in Figure A1.
The calculation of CMF ρ , CMF star , their uncertainties, and ( )  P 0 are calculated with the publicly available ExoLens 6 code developed for this work. The code uses inputs of planet mass, radius, stellar abundances, and the uncertainty in each of these observables.
If P(CMF ρ = CMF star ) 32% then we assert a planet deviates from what is expected at the 1σ significance level. Similarly, if P(CMF ρ = CMF star ) 5% then a planet deviates from its host star at the 2σ significance level. We consider planets that deviate from their host stars at the 2σ level to be statistically inconsistent with the null hypothesis.
Our strict use of [0,1] bounds in Equation (4) without renormalization of f(CMF) outside this range accounts for the fact that negative CMF values require planets with compositions that are inconsistent with any CMF star value (as defined by Equation (2)), and thus are inconsistent with our null hypothesis. If we were to renormalize the probability distribution for CMF ρ to be equal to unity over [0,1], we would be enforcing the planet to be rocky by our definition, and thus would not be testing  0 as defined. Rather, we would be testing the conditional probability that, if the planet is rocky (CMF ρ 0), that its inferred composition is consistent with that of the host star, i.e., that the probability distribution of CMF ρ normalized to unity over [0,1] is consistent with the probability distribution of CMF star at the 2σ level. While testing this alternative hypothesis is a reasonable approach, it is more restrictive than our approach, as we do not require that the planet is rocky in the sense that its mass and radius can be fit by a model consisting purely of an iron core and iron-free silicate mantle.

Results
The average molar Fe/Mg for FGK-type stars is 0.7 ± 0.18 (Hinkel et al. 2014 Plotnykov & Valencia (2020). For this sample set, we find a relatively narrow range of CMF star from 0.26 to 0.33, with uncertainties between 0.06 and 0.10. We find a significantly wider range of CMF ρ from 0.004-0.70 and uncertainties from 0.10-0.24.
Despite the large range of CMF ρ , the mutual uncertainties of pairs of CMF values are such that all planets but Kepler-107 c are consistent with the null hypothesis at the 2σ level (Table 2; Figure 1). Therefore, within the limits of the observational measurements, 91% of planets meeting the selection criteria for this study are not distinguishable from their stellar host in major rock-building element composition. The one exception, Kepler-107 c, has a 1% likelihood of satisfying  0 , with a CMF star < CMF ρ (Figure 1) implying greater than expected density relative to that predicted by its host star's major rockbuilding element composition. We, therefore, classify this planet as a super-Mercury (SM). The rest of the planets are indistinguishable from the host star (IHS).
At the 1σ significance level, the set of those planets inconsistent with the null hypothesis (P(H 0 ) 32%) increases by one ( Table 2). 55 Cnc e has a 9% probability of being consistent with the null hypothesis, in which CMF star > CMF ρ . This suggests 55 Cnc e has a lower than expected density relative to its host star's major rockbuilding element abundances given the a priori assumption that the planet is rocky. This result is consistent with previous results that the planet has a potential lower than expected density (e.g., Demory et al. 2016aDemory et al. , 2016bAngelo & Hu 2017a;Bourrier et al. 2018;Dai et al. 2019;Dorn et al. 2019). The failure of the null hypothesis does not assess the cause of the inferred low density as being a result of a primary or secondary gaseous envelope or of a chemical deviation from the major rock-building element abundances of the host star.

Cases for a Rejected Null Hypothesis
For planets that deviate from  0 in Table 2, there are two cases, (1) in which CMF star < CMF ρ , suggesting a planet with a larger core than expected, a super-Mercury, and (2) in which CMF star > CMF ρ , a low-density small planet, which then suggests a region of nonunique solutions. We define and use the term low-density small planet (LDSP) for planets with an apparent density deficit, but distinguished from super-Puffs, a class of super-Earth mass planets with gas-giant transit radii leading to bulk densities of 0.1 g cc −1 (e.g., Wang & Dai 2019, and references therein). In contrast, LDSPs have bulk densities that can only be explained with a rockdominated composition but sufficiently low that they are inconsistent with  0 . For example, the candidate LDSP 55 Cnc e has a mean bulk density of ∼6.25 g cc −1 which, for its mass, is consistent with a rocky (pure MgSiO 3 ) composition, but is ∼1.76 g cc −1 lower than its expected bulk density per  0 .
Kepler-107 c and 55 Cnc e, the two planets inconsistent with  0 at the >1σ level, differ in that Kepler 107 c has a mass excess relative to what would be predicted by its host star, while 55 Cnc e has a relative mass deficit. This demands multiple explanations for these planets and suggests a diversity of planetary outcomes for planets that orbit close to their stars.

Super-Mercuries
Kepler-107 c has a greater core mass fraction than predicted by its host star's Fe/Mg abundances. There are few explanations for increasing the density of a planet beyond excess iron relative to MgO and SiO 2 . The simplifying assumption of a solid, pure iron core means that ( P 0 ) is an upper bound. Therefore, this planet is likely a super-Mercury.
Considering the actual core properties of the rocky solar system planets, it is likely that the cores of exoplanets contain some amount of alloying light elements and are at least partially liquid. The cores of the terrestrial solar system planets contain ∼10% light elements and are volumetrically dominated by liquid (e.g., Lehmann 1936;Birch 1952;Aitta 2012;Smith et al. 2012;Helffrich 2017). Including these factors only strengthens the case for a large core for Kepler-107c.
Both light element incorporation and liquid iron reduce the density of the core. For models that conserve both mass and radius, our simplifying assumption of overestimating the core density  Table A3. Planets are listed in order of increasing radius as in Table 1. Figure 1. Probability density functions (f) as a function of core mass fraction given 1σ distributions for CMF ρ (red) and CMF star (purple). All CMF calculations assume that the cores are pure iron and silicate mantles are iron-free, such that CMFs with these assumptions are equivalent to iron mass fraction. Figures are as listed in Table 1, in order of increasing radius. underestimates core volume due to systematically bounding the density as a likely maximum. CMF ρ values assuming a pure and solid Fe core are ∼0.02-0.04 lower than the liquid and lightelement-enriched core cases for a given planetary mass and radius. As an example, assuming a liquid core for K2-229b, the second most probable super-Mercury, increases CMF ρ by ∼0.02 relative to a solid core, corresponding to a 5% decrease in ( )  P 0 . An additional assumption is that all iron is in the core and there is none in the mantle. In the calculation of CMF star , Fe/Mg is constant, but the oxidation of iron removes Fe from the core as well as adds oxygen to the planet, decreasing the effective CMF star by 0.02-0.03 compared to the FeO-free case. The comparable CMF ρ calculation, however, oxidizes Fe from the core incorporating it into the mantle as FeO. This increases the average density of the mantle while decreasing the density of the planet through added oxygen, decreasing compressibility of the oxide compared to the metal, and decreasing core mass. For example, an Earth-like, whole-rock assemblage with 4 mol % FeO, accounting for oxidation of 13 mol% of Earth's iron, has a bulk modulus of 250 GPa (Lee et al. 2004), while the bulk modulus of solid, hcp iron is 178 GPa (Smith et al. 2018). As a result, following the simplifying assumption that all iron is in the core, the resulting model for CMF ρ at a given mass and radius is an underestimate for planet iron fraction by ∼0.01-0.04. For example, the assumption of an Fe-free mantle in K2-229b predicts CMF ρ = 0.565. 10 mol% oxidation of the available iron, removing it from the core and incorporating it into the mantle as an iron oxide, requires a 0.032 increase in planet Fe/Mg for K2-229b at its given mass to reproduce this planet's radius. Given constant stellar Fe/Mg, this increase in iron corresponds to an 8% decrease in ( )  P 0 relative to the Fe-free mantle case.
For both assumptions of a solid, pure iron core and iron-free mantle, the probability assigned to  0 is an upper bound in the case of CMF ρ > CMF star , and, therefore, the 5% probability criterion for super-Mercuries in this situation is a conservative measure.

Low-density Small Planets
Four compositional variations can potentially explain LDSPs, (1) significant oxidization of iron such that it is removed from the core and incorporated into the mantle (e.g., Rogers & Seager (2010)), (2) a calcium-aluminum oxide dominated planet that formed from only these highest-temperature refractory materials with significantly depleted Fe As with the impact of oxidation on K2-229b, we explore the degree to which core oxidation affects our analysis of the null hypothesis in the case of LDSPs. In the case that 55 Cnc e is core-free due to oxidation of all iron, ( )  P 0 increases from 9% to 10% ( Figure 2). As in Section 6.1, this is a consequence of the iron being incorporated in a lower density, lower compressibility oxide as compared to the metal. At the same time, this results in an associated decrease in CMF star due to an added oxygen atom per iron atom.
A major objection to the possibility of a planet made of ultrahigh temperature condensates as proposed by Dorn et al. (2019) is that there is an insufficient mass of Al and Ca present within the protoplanetary disk available to produce the observed masses of these planets. For instance, reproducing the mass of 55 Cnc e assuming it formed from a minimum mass solar nebula (Kuchner 2004), which likely overestimates the disk mass available to planets forming around its K-dwarf host, requires a factor of ∼3.5 (0.55 dex) and ∼2 (0.3 dex) increase in the already super-solar Ca and Al abundances of 55 Cnc, respectively. That being said, the mean density of 55 Cnc e is consistent with a virtually iron-free planet of MgSiO 3 , similar to the compositional prediction in this hypothesis.
Where the lower than expected density cannot be explained by oxidation or iron deficit, the most likely explanation is a combination of H/He or higher-mass atmospheric compositions For 55 Cnc e, the radius deficit between what is observed and what is expected per  0 can be explained by an ∼1000 km thick atmosphere. Given the proximity of 55 Cnc e to its host, an H/Heor H 2 O-dominated atmosphere would result in escaping hydrogen, which has not been observed (e.g., Bourrier et al. 2018, and references therein). A water-rich atmosphere has recently been ruled out at the 3σ level (Jindal et al. 2020): if 55 Cnc e does have an atmosphere, it must be dominated by heavier species including CO, CO 2 , or N 2 .
We note that earlier, larger, measurements of the radius of 55 Cnc e and WASP-47e place these planets as inconsistent with  0 at greater than 2σ and 1σ level, respectively (Table A3), which would predict both planets as LDSPs. Both planet radii have been revised downward using updated stellar parameters from Gaia parallaxes ).

Necessary Observational Improvements to Reject the Null Hypothesis
Current efforts are underway to improve mass and radius measurements for rocky exoplanets. We investigate here the needed improvements in the observational uncertainties of mass, radius, and host-star abundances to help quantify the range of rocky planet compositions. We find that increasing precision in planetary radius measurements is the most critical.
Across the radius and mass range of the analysis (2.5-9.7 M ⊕ and 1.1-1.9 R ⊕ ), we calculate the relative effects of uncertainties in measured mass, radius, and stellar abundances in CMF ρ and CMF star (Figure 3). Uncertainties arising in mass and/or radius resulting from the uncertainties in the underlying equations of state for each layer are minimal (Unterborn & Panero 2019). The uncertainty in CMF ρ is weakly dependent upon the planetary mean density in the considered range (Table A1).
For a typical planet with CMF ρ = 0.35, we determine that a 20% uncertainty in mass for a planet leads to a CMF ρ uncertainty of 0.15 (Figure 3(a), Table A1). The observational uncertainties in planet radius have the greatest impact on inferred CMF ρ . A 10% radius uncertainty, again for a typical CMF ρ of 0.35, leads to an uncertainty that is as large as the central value, i.e., CMF = 0.35 ± 0.35.
The observational uncertainties in host-star Fe, Mg, and Si abundances have a proportionally smaller effect on CMF star (Table A2). A 40% uncertainty in molar Fe/Mg leads to a CMF star uncertainty <0.10 (Figure 3(b)). We find that the uncertainty in molar Si/Mg has minimal impact.
Improving mass and radius uncertainties for small planets is a resource-intensive process, which will benefit from increased signal-to-noise ratios of TESS (Ricker et al. 2015) and CHEOPS (Benz et al. 2018) for bright hosts, along with parallax measurements from Gaia (Gaia Collaboration et al. 2016), which may improve mean planet density observational uncertainty to as little as 4% (Stevens et al. 2018 Assuming a near best case precision of 4% in mass, 1% in radius (5% ρuncertainty), and 8% in Fe/Mg and Si/Mg stellar abundances (corresponding to ∼solar Fe/Mg and Si/Mg uncertainties), for planets in the range of 2.5-9.7 M ⊕ and 1.1-1.9 R ⊕ , the null hypothesis cannot be refuted when 0.5 < CMF ρ /CMF star < 1.4. For the planets in our sample set, assuming accurate central values for each measurement, Kepler-10b and 55 Cnc e may be conclusively determined to have lower than expected densities, while HD 219134c and K2-229b may be sufficiently resolved as planets with greater than expected densities, or super-Mercuries (Figure 4, Table A4). The remaining planets, HD 219134 b, K2-265b, WASP-47e, HD 15337 b, HD 213885b, and Kepler 20b cannot be distinguished from the null hypothesis at the 2σ level using mass, radius, and stellar abundances.
Next, we consider Mercury, Earth, and Mars. While these planets are smaller than those in our sample, their geophysical constraints on CMF provide test cases for the validity of our methods and assumptions. MESSENGER constrained Mercury's CMF to ∼0.69-0.77, corresponding to CMF ρ /CMF star = 2.17-2.43 (Nittler et al. 2019). Were Mercury to be viewed as an exoplanet with near best case observation precisions, it should, therefore, be resolvable as denser than expected. Applying the methods and assumptions outlined in this work, we indeed find Mercury would be resolvable as denser than expected with CMF ρ /CMF star = 1.92 and ( )  P 0 indistinguishable from zero. Earth has a CMF ρ /CMF star of 1.02 (McDonough 2017, and references therein). Thus, even with best case observational precisions, the Earth as an exoplanet would be indistinguishable from the Sun. Using our approach on Earth, we find CMF ρ /CMF star = 1.03 and ( ) =  P 99% 0 . Last, while Mars has a molar Fe/Mg ratio to within ∼10%-15% of the Sun's, much of its Fe is oxidized, leading to a smaller CMF than the Earth. Geophysical constraints give CMF ρ /CMF star ∼ 0.57-0.67 for Mars. Thus, even with significant oxidation of iron, Mars as an exoplanet should still be indistinguishable from the Sun at the 2σ level. Our approach suggests values on the lower end without iron oxidation, CMF ρ /CMF star = 0.57 and ( ) =  P 11% 0 (Wanke & Dreibus 1994;Bertka & Fei 1998;Zharkov & Gudkova 2005;Yoshizaki & McDonough 2020).

Discussion and Conclusions
We assess the statistical consistency of a planet's composition and structure as inferred from its mass and radius with what is expected from its host's major rock-building elemental abundance ratios. We test the hypothesis directly and demonstrate that for just one planet, Kepler 107 c, the mass and radius cannot be described as a terrestrial planet with the same relative major rock-building element abundances as its host, demanding a two-fold excess of iron. This approach is complementary to the Bayesian approaches in Dorn et al. (2015) and Otegi et al. (2020), which leverage stellar abundances to reduce planetary structure degeneracy, in which the null hypothesis tested here is assumed a priori. Once the conditions under which a terrestrial planet is describable by its host's abundances are understood, a more complete Bayesian analysis of planetary composition and structure will be warranted.
We demonstrate that super-Mercuries with an iron overabundance that is at least 40% greater than CMF star may be resolvable as having different from host-star major rock-building element compositions. For example, we show that HD 219134c will be a resolvable super-Mercury while HD 213885b will remain unresolvable despite these planets having the same mean CMF ρ . The difference arises from HD 213885 being more Fe-rich relative to silicates than HD 219134 leading to 135% and 150% iron overabundances for HD 213885b and HD 219134b, respectively.
Several planetary formation and evolution mechanisms may explain super-Mercury planets. Each model identifies mechanisms by which planets are enriched in iron relative to their host star. These models include giant impacts (Marcus et al. 2010;Leinhardt & Stewart 2011), a series of smaller impacts (Chau et al. 2018;Swain et al. 2019), mantle evaporation of hot planets (e.g., Santerne et al. 2018, and references therein), iron enrichment in the inner regions of planet-forming disks due to iron condensing at a higher temperature than silicate material (Lewis 1972) or via photophoresis (e.g., Ebel & Stewart 2019, and references therein), and mantle stripping via planet-star tidal interactions (Jia & Spruit 2016) or planet-planet tidal interactions (Deng 2020). Each model to explain the formation of Mercury or super-Mercuries involves compositional sculpting of such planets relative to their host star. The frequency and orbital properties of exo-super-Mercuries are a crucial test of these theories. Figure 4. Planet density as a function of radius for the 11 planets in our sample (circles) and the solar system terrestrial planets (stars). The observational radius and density uncertainties are plotted for each planet. Open circles indicate planets that may be distinguishable from  0 at the 2σ significance level should s = 4% Mp , s = 1% Rp , and σ Fe/Mg = 8% precisions be attained. The dark-gray shaded region covers the density range of our sample for which CMF ρ /CMF star = 1.4, while the light gray shading reflects the middle 95% range of CMF star from the stellar abundances in the Hypatia catalog. Similarly, the cyan shading reflects the range of this sample and of the equivalent Hypatia range where CMF ρ /CMF star = 0.5. Gray crosses are the 17 planets that meet our mass and radius criteria but lack Fe, Mg, and Si abundance measurements of their host stars. These planets are listed in Table A5. HD 219134c is potentially resolvable as a super-Mercury only when analyzed relative to its host's abundances. HD219134 is a star with proportionally little Fe with Fe/Mg = 0.69 ± 0.25, yet slightly iron-enriched relative to solar, with [Fe/H] = 0.09 (Hinkel et al. 2014). Had the planetary structure analysis focused solely on the CMF ρ ( -+ 0.42 0.14 0.13 ), this planet may have been missed as a planet that underwent significant chemical sculpting, as it is less than 1σ greater than an Earth-like CMF of 0.32. Notably, this planet orbits outside that of HD 219134 b, which is indistinguishable from its host star. Similarly, K2-229b may have been prematurely misidentified as a super-Mercury, even with the updated mass and radius values used in this study, had its host star's abundances not been considered. K2-229b and HD 219134c show that stellar major rock-building element composition must be carefully considered when investigating the outcome diversity of rocky planet formation to avoid failing to or misidentifying a planet as a super-Mercury or LDSP.
We also show that we can identify low-density, small planets whose CMF ρ is 50% less than predicted by CMF star . The source of the low density is degenerate, but the 1σ mass deficit for 55 Cnc e cannot be explained through oxidation of all iron. The relative influence of atmospheric layers, a dramatic iron deficit, or global magma oceans remain degenerate. Such planets offer important clues as to planetary system evolution, demonstrating that small planets below the radius gap cannot be exclusively rocky planets.
This approach of comparing expected density based on host-star composition to that of the planet only addresses the most extreme cases of compositional sculpting. The remaining planets in the sample set, more than half, are not distinguishable from the null hypothesis, nor will they be distinguishable from their host star with respect to composition based on mass and radius measurements alone. For potential LDSPs, further constraint on the differences between host-star composition and rocky planet composition must come from alternate methods. First, the measurement of day-to-night side temperature contrast could reveal a thick atmosphere and thus an LDSP (Koll et al. 2019). Second, the measurement of planet atmospheric composition and relative abundance ratio (Morley et al. 2017) can be compared to that of the host star. Plotnykov & Valencia (2020) and Scora et al. (2020) have recently presented complementary approaches to detecting rocky planets with nonstellar compositions. Plotnykov & Valencia (2020) compare the mean planetary CMF distribution (〈CMF ρ 〉) for likely rocky planets with s M p and s < 25% R p to the mean stellar CMF distribution (〈CMF star 〉) of the stars in the Hypatia Catalog (Hinkel et al. 2014). The authors find that the CMF ρ distribution peaks at a lower value and is more broad than the CMF star distribution, requiring an explanation in planet formation theories. Scora et al. (2020) investigate whether the diversity of 〈CMF ρ 〉 relative to what is expected from 〈CMF star 〉 can be explained through the cumulative effects of collisions during formation, finding that collisions alone cannot explain the diversity of rocky planet compositions.
While comparing statistical average values for CMF ρ and CMF star is useful for assessing planets in which no stellar abundance measurements are made, it fails to consider the composition deviation in Fe/Mg from star to planet on an individual basis. Assuming a best case CMF ρ uncertainty of 0.03 (Kepler-107 c  (Plotnykov & Valencia 2020), only planets with CMF ρ < 0.02 or >0.67 (ratios of <0.06 and >2.1, respectively) may be resolvable as statistically inconsistent with 〈CMF star 〉. As observational precisions increase and Fe/Mg and Si/Mg values become more widely reported, our approach will be able to identify these same compositional extremes and more modest cases of compositional sculpting.
To assess the formation processes that result in measurable compositional deviations for planets smaller than the radius gap will require a larger, more precise sample with both extreme and more moderate cases of compositional sculpting. In our sample selection process, we identified 17 small planets with highprecision mass and radius measurements, but without host-star abundance measurements beyond [Fe/H] ( Table A5). The sample set, therefore, may be doubled rapidly with targeted, high-precision stellar abundance analysis. With a larger data set, deviations from the expected can be addressed statistically much like the radius populations are able to do now. It is imperative that host-star Fe, Mg, and Si abundance measurements are included along with mass and radius measurements of likely rocky planets as part of the discussion and inference on their structure and composition. Without mass-radius and host-star abundance measurements, it is impossible to determine if a planet's composition reflects that of its host and, in turn, determine its most likely formation pathways.
C.T.U. acknowledges the support of Arizona State University through the SESE Exploration fellowship. The results reported herein benefited from collaborations and/or information exchange within NASA's Nexus for Exoplanet System Science (NExSS) We provide the reader a schematic representation of Equation (4), shown as Figure A1. Figure A1. We illustrate the meaning of Equation (4) with an example of a planet whose CMF ρ is similar to its CMF star value (top) and one whose CMF ρ is greater than its CMF star value (bottom). The numerator of Equation (4) computes the integral of the product of the distribution of each CMF value. In the case where they are similar, the overlaps are significant, yet where the CMF values are different, there is nearly no overlap. The denominator for each case accounts for the consequences of variable distribution widths so as to normalize the numerator to 1 in the case that CMF ρ = CMF star .

A.3. Previous Studies
We calculate CMF ρ estimates for planets in our sample with multiple reported mass and radius measurements on the NASA Exoplanet Archive. We only consider studies where both mass and radius, and thus bulk density, are measured. All CMF ρ values presented here are calculated using ExoLens. The results of these calculations are listed in Table A3.

A.4. Minimum Uncertainties
Here we calculate the minimum uncertainties needed to classify the planets in our sample at the 2σ significance level (Table A4).

A.5. Figure 4 Planet Sample
We list the planets plotted in Figure 4 ( Table A5). This sample includes the 11 analyzed in this work and 17 additional planets without reported host-star abundance measurements beyond [Fe/H], that otherwise meet our selection criteria. . We vary σ Fe/Mg = σ Si/Mg until ( )   P 5% 0 stopping at the optimistic value of σ Fe/Mg = σ Si/Mg = 8% if this ( )  P 0 criteria is not met. For these planets, the ( )  P 0 corresponding to s = 4% Mp , s = 1% Rp , and σ Fe/Mg = σ Si/Mg = 8% is recorded. This stellar abundance criteria is derived from the recommended abundances of Lodders (2003). While such precisions will be difficult to obtain for extrasolar stars from spectroscopic abundance measurements alone, σ Fe/Mg = σ Si/Mg = 8%, nonetheless, provides a good stopping point for our minimum precision calculations. Planets with no value recorded in the abundance ratio uncertainty column do not require increased abundance precisions to classify at better than the 2σ level if their mass and radius uncertainties reach 4% and 1%, respectively.