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A&A
Volume 674, June 2023
Article Number A137
Number of page(s) 12
Section Planets and planetary systems
DOI https://doi.org/10.1051/0004-6361/202245387
Published online 16 June 2023

© The Authors 2023

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1 Introduction

In recent decades, thousands of exoplanets have been discovered by transit observations such as Kepler (Borucki et al. 2010), TESS (Ricker et al. 2014), and other instruments. Mass and radius are currently the two primary direct observables for exoplanets. Mass measurements are usually obtained from ground-base telescope radial velocity(RV) measurements, while the radius is mainly measured by transit methods. From measurements of radius and mass, exoplanets are generally classified into the most massive planets (H/He-rich gas giants), intermediate-mass planets (Neptunes), super-Earths and mini-Neptunes, as well as low-mass planets that are also knowns as terrestrial planets (Spiegel et al. 2014; Zeng et al. 2019). The Kepler survey revealed that super-Earths and mini-Neptunes are among the most common planet types occurring in the galaxy (Petigura et al. 2013; Silburt et al. 2015; Fulton et al. 2017). The essential difference among them is that super-Earths are rocky planets with small radial fractions of volatiles while mini-Neptunes are planets with thick volatile layers (Dorn et al. 2018). In this work, we focus our attention on super-Earths that are larger than Earth but smaller than two Earth radii (R⊕).

The chemical composition of rocky exoplanets can be constrained from the observed planet mass and radius using interior models (Valencia et al. 2006; Seager et al. 2007; Sotin et al. 2007). However, the interior models of rocky exoplanets have an inherent degeneracy issue because two interior models with different interior compositions could yield the same mass and radius. For example, a super-Earth can have the same radius and mass of another despite the latter having larger iron-rich cores and more volatiles. To break this inherent degeneracy, more observables of exoplanets are needed to constrain their interior models. One feasible way is to use the elemental abundances of host stars to constrain the bulk elemental abundance ratios of their hosted planets (Rogers & Seager 2010; Dorn et al. 2015, 2017; Santos et al. 2015; Brugger et al. 2017). This is based on the planetary formation scenario positing that planets are formed in a protoplanetary disk that has a similar composition to that of the star. Furthermore, Earth and Mars have the bulk molar Fe/Mg ratios to within ~ 10–15% of the solar abundance (Wanke & Dreibus 1994; Wang et al. 2018; Asplund et al. 2009). However, Mercury exhibits a quite different Fe/Mg of roughly 3.9–5.8 (Unterborn & Panero 2019), which suggests that the interior composition of planets could also deviate from that of their host stars. Therefore, it is of great importance to explore the compositional correlation between rocky exoplanets and their host stars in order to characterize planetary interiors and identify habitable planets.

Recently, Plotnykov & Valencia (2020) and Schulze et al. (2021) have compared the composition of individual rocky exo-planets to their host stars or to a population of planet host stars. Adibekyan et al. (2021) have found a planet-star composition relationship for rocky exoplanets around FGK stars. Owing to the prominence and number of M dwarf stars in the galaxy and the apparent excess of terrestrial planets formed around them, the number of rocky exoplanets detected around M stars has been rising rapidly (Amado et al. 2021; Childs et al. 2022). The planetary disk around M star have lower disk mass and smaller scale height because of its low star-mass. This could lead to a different disk structure. However, the disk frequency of M stars is found to be similar to Sun-like stars to the point that raw materials for planet formation appear to be available (Luhman et al. 2006). It has been suggested that M dwarfs host more inner planets than FGK host stars (Childs et al. 2022). Some studies have been devoted to investigating the M dwarf exoplanet population via their planetary system architectures, where the architectures derived for M dwarf hosts can be compared to those of FGK stars (Mulders et al. 2018; He et al. 2021). It is therefore a matter of interest to take into account more rocky exoplanets and discern the compositional correlation between rocky exoplanets and their host stars.

In this work, we compute the bulk refractory elemental abundance ratios of rocky exoplanets and discuss the statistical relationship between the compositions of rocky exoplanets and their host stars. It is not only helpful to reduce the degeneracy of interior models, but this also provides valuable information on the process of planet formation.

2 Sample selection and methodology

As of March 2022, there are over 5000 confirmed planets in the NASA Exoplanet Archive1 and that number continues to grow. In this work, we restrict our attention to the planets with masses less than ten Earth masses (M⊕) and radii R < 2R⊕, which are most likely composed of rocky materials (Zeng et al. 2019). We found 112 planets satisfying the criterion. We further limited our sample by requiring the uncertainties in mass and radius are both less than 30%, ultimately yielding 74 planets. We can notice that some planets have more than one measurements in the archive. For one planet, if the difference between these measurements was larger than 50%, we removed it from our sample to ensure the accuracy of our results (as much as possible). For example, the difference between the mass measurements is greater than 50% for CoRoT-7 b, Kepler-80 d, TOI-561 b, and TRAPPIST-1 f, hence, they are outside of our sample. For the remaining, we chose the measurements of mass and radius with the smallest relative error, most of which are the latest observables. In addition, 55 Cnc e, Kepler-60 c, and TOI-1634 b were removed from our sample because some of their measurements suggest they are probably rocky, while the others suggest they are probably mini-Neptunes. The host star HD 80653 has no spectrum data available to date, and hence HD 80653 b is outside of our sample. Figure 1 shows the properties of 66 planets in our sample. The largest radius of rocky exoplanets is also displayed by blue dashed lines for given planet masses, corresponding to the silicate planets without iron-rich cores. We can see that nine planets denoted by stars (K2-111 b, KOI-1599.01, Kepler-138 d, Kepler-60 b, Kepler-60 d, Kepler-114 c, L 98-59 d, TOI-1685 b, and TOI-776 b) are larger than the largest size of rocky exoplanets. They are generally identified as ocean planets or mini-Neptunes. We do not consider them and restrict our attention to rocky exoplanets. Finally, 57 planets form the sample for our analyses.

These 57 planets orbit around 48 host stars. More specifically, 22 of them are located nearby 15 M stars and 35 of them are around nearby 33 FGK stars. For the stellar parameters of these 48 host stars, we obtained their relative iron abundance [Fe/H] from the NASA Exoplanet Archive as well and chose the measurements with the smallest uncertainties. The relative iron abundance [Fe/H] is defined as (1)

where NFe and NH are the number of iron and hydrogen atoms, respectively. The solar elemental abundances are taken from Asplund et al. (2009) for reference. The relative abundances of other elements have the similar definition. We also searched for the relative abundances of rock-building elements, [Si/H] and [Mg/H]. There are only data available for some FGK stars in the sample, because high-resolution spectrum are needed to determine their abundances. Adibekyan et al. (2021) collected high-resolution optical spectra from seven spectrographs and derived the abundances of Fe, Mg, and Si for 20 host stars. In view of this, we then replaced the [Fe/H] data by the results from Adibekyan et al. (2021). The parameters of these 48 host stars are summarized in Table A.1. The iron-mass fraction Festar% of host stars is inferred using the absolute stellar abundances (Adibekyan et al. 2021): (2)

where NX denotes the number of atoms for each species X (Fe, Si, and Mg), and mY denotes the molar mass of corresponding materials Y (Fe, SiO2, and MgO).

thumbnail Fig. 1

Sample of selected exoplanets with radius measurements plotted as a function of mass and equilibrium temperature. The color of the points represents the equilibrium temperature, Teq, of a planet (zero albedo assumed), and most of the sample planets have Teq large than the Earth. Blue dashed lines denote the M – R curve of pure-silicate planets calculated by the interior model. The planets larger than the largest size of rocky planets are marked by stars.

2.1 Interior model of rocky exoplanets

We used a open-source interior model named ExoPlex (Lorenzo 2018; Unterborn et al. 2023) to model the interior structure of rocky exoplanets. The interior of rocky exoplanets is assumed as an iron-rich core overlaid with a silicate mantle. The internal structure of rocky exoplanets is governed by the following hydrostatic and mass-continuity equations: (3) (4)

where P(r) is the pressure profile as a function of radius r, ρ(r) is the density profile as a function of radius r, m(r) is the mass enclosed within the radius r, and g(r) = Gm(r)/r2 is the gravity at the radius r. The temperature profile is calculated using an adiabatic temperature gradient, (5)

where α(P, T) and CP(P, T) are the thermal expansivity and coefficient of specific heat of the constituent minerals at a given pressure, P, and temperature, T. The surface boundary conditions are expressed as P(R) = 1 bar and T(R) = TPot, where R is the final radius of the planet and TPot is the potential temperature. The potential temperature have little effect on the radius of a planet (Unterborn & Panero 2019) and so it is set to the potential temperature of the Earth’s mantle, TPot = 1600 K.

To solve this set of equations, the equation of state (EoS) for the constituent minerals, ρ(P, T), is needed. For the core, a minor proportion of light elements, such as sulfur, oxygen, and silicon, are ignored since little is known about exoplanetary cores. The core EoS adopted in the ExoPlex is based on the fourth order Birch-Murnaghan EoS of liquid iron (Anderson & Ahrens 1994). The mantle, separated into upper and lower mantles, is regarded to comprise major rock-building elements (Fe, Mg, Si, Al, and Ca) in addition to oxygen. The stable phase of the oxides SiO2, FeO, MgO, CaO, and Al2O3 is calculated by the minimization of Gibbs free energy (Connolly 2009). The EoS and other physical parameters are then calculated by the method of Stixrude & Lithgow-Bertelloni (2005, 2011).

2.2 Bayesian inference of interior composition

The interior model ExoPlex calculates the radius of an exoplanet from its mass, core mass fraction (CMF) and mantle elemental ratios. The input parameters into ExoPlex contain mass, CMF, and mantle elemental ratios Fe/Mg, Si/Mg, Al/Mg, and Ca/Mg. In view of the degeneracy in the interior composition of rocky planets, we employed a Bayesian statistical approach to explore possible interior compositions that would fit the observed mass and radius.

Here, we use the affine invariant Markov chain Monte Carlo (MCMC) sampler emcee (Foreman-Mackey et al. 2013) to sample the space of parameters. The MCMC method draws samples from the given prior distributions and maximizes the likelihood function, yielding the posterior probability distributions of the parameters for each planet. As for the prior distribution, all input parameters are assumed to be independent and their prior distributions are based on the current knowledge of them. For example, the input mass is set to a Gaussian distribution with mean Mp and standard deviation σM, where Mp is the observed mass of the planet and σM is the mass uncertainty. The CMF is uniformly distributed between 0.01 and 0.99. For the element compositions in the mantle, the Fe/Mg ratio has primary influence on the radius of the planet (Lorenzo 2018). It is uniformly distributed in the range 0–1.6, corresponding to the Mg# number from 0.38 to 1.0. Such a wide range covers all possible Mg# numbers of Sotin et al. (2007) and Adibekyan et al. (2021). The Si/Mg ratio is set to a Gaussian distribution with the mean of the solar value 0.81 (Asplund et al. 2009), for which the standard deviation is taken as 0.081. Considering CaO and Al2O3 are generally minor components in the mantle, we ignore them in the calculations, yielding the largest Mg content.

The likelihood function P is assumed to be of a normal form, determined by the observed mass and radius: (6)

where Rp and σR are the observed value of radius and its uncertainty. For some observed values with asymmetric uncertainties, σR is set as the larger uncertainty of Rp. According to the Bayesian theorem, the output posterior distribution of model parameters is completely specified by the log-likelihood and the prior distributions, serving as the solution of the inverse problem. In general, the MCMC model runs for about 1.5 × 105 iterations for each planet to reach a good convergence.

3 Results

In this work, we mainly focus attention on the bulk elemental ratios of rocky exoplanets and, hence, the model parameters of interest contain the bulk molar ratios Fe/Mg, Si/Mg, and Fe/(Mg+Si), as well as the iron-mass fraction of a planet in addition to the planet mass and radius. The iron-mass fraction, Feplanet%, of a planet is given by (7)

Taking Kepler-20 b as an example, Appendix B shows the results from the MCMC simulation. For the bulk compositional parameters, we adopt the median values of their posterior distributions for our analysis. The detailed results for the 57 exoplanets in our sample are summarized in Table A.2.

Adibekyan et al. (2021) used the normalized density (ρ/ρEarth-like) to distinguish super-Mercuries from super-Earths because it can account for the effect of different masses on compression. Figure 2 illustrates the normalized densities (ρ/ρEarth-like) of the selected 57 exoplanets as a function their iron-mass fractions calculated from the MCMC. Then, ρEarth-like denotes the density of the planet with Earth-like compositions (an iron core with mass fractions of 32.3% and mantle elemental ratios of Fe/Mg= 0.12 and Si/Mg=0.81). For each rocky planet, it is calculated using the interior model ExoPlex: given the known mass of one planet and specifying the Earth-like composition, the ExoPlex model yields the radius of the planet and then ρEarth–like is simply calculated from the resulting radius and known mass. As additional information, the distributions of Feplanet% and ρ/ρEarth–like are shown in the top and right panels of Fig. 2. It can be seen that the distributions are not uniform but have two peaks. One peak at Feplanet% ≈ 0.35 and ρ/ρEarth–like ≈ 0.9 corresponds to the planets with Earth-like compositions. The other peak at Feplanet% ≈ 0.70 and ρ/ρEarth–like ≈ 1.5 corresponds to the planets with a greater iron content than even Mercury. In view of the different formation process of Earth and Mercury, we conservatively classified the exoplanets with both Feplanet% and ρ/ρEarth–like larger than Mercury as super-Mercuries and the others as super-Earths. In this way, the super-Mercuries contains GJ 367 b, Kepler-105 c, Kepler-107 c, Kepler-406 b, and HD 137496 b, as labeled in Fig. 2.

thumbnail Fig. 2

Normalized densities ρ/ρEarth–like of the 57 exoplanets as a function of their iron-mass fractions Feplanet%. The normalized density, ρ/ρEarth–like, of a planet is directly calculated from its mass and radius and ρEarth–like is the density of an Earth-like composition (core mass fractions of 32.3% and mantle molar ratios of Fe/Mg=0.12 and Si/Mg=0.81). The data of Mercury is marked as black star for reference. The exoplanets with both Feplanet% > FeMercury% and ρ/ρEarth–like> (ρ/ρEarth–like)Mercury are circled as super-Mercuries while the others are super-Earths. The distributions of Feplanet% and ρ/ρEarth–like are showed in the top and right panels, respectively.
thumbnail Fig. 3

Iron-mass fractions, Feplanet%, of the 57 rocky exoplanets as a function of their equilibrium temperature, Teq. The data points are divided by the spectral type of their host stars into two groups: planets around M stars (red squares) and around FGK stars (blue triangles). The Feplanet% of the two groups show different correlations with equilibrium temperature. Green stars denote the results for the four terrestrial planets in our Solar System.

3.1 Radiation effect on the interior composition of rocky exoplanets

Figure 3 shows the iron-mass fractions Feplanet% of the 57 planets as a function of the equilibrium temperature, Teq. Since the bulk albedo of rocky exoplanets is affected by the spectrum type of host star radiation, we divided the sample into two groups: planets around M dwarf stars and planets around FGK host stars. The planets in the FGK star group all have the equilibrium temperature exceeding 500 K. There is no correlation between Teq and Feplanet% for the FGK star group. This is quite consistent with the finding of Adibekyan et al. (2021). The planets around M stars generally have lower Teq than those around FGK stars, since M stars generally exhibit smaller radii and lower effective temperatures. Their iron fractions show a rising tendency with increasing equilibrium temperature. To gain more insight into this tendency, we apply an ordinary least square linear fit to quantify the relation y = αx+β and then use Pearson test to assess its significance. We then minimized the reduced residual: (8)

where n is the number of data points and . is the error of yi. A strong correlation between them has been found, namely, Feplanet% = (3.57 ± 0.65) × 10−4Teq + (0.19 ± 0.03). This fitting has a reduced residual of 0.56 and a Pearson correlation coefficient of 0.77. It means Feplanet% have a strong positive correlation with Teq. The null hypothesis that the coefficient is equal to zero has a p-value of 2.6 × 10−5. That is to say that we can reject the null hypothesis at a probability of 1 – p. For the terrestrial planets in our Solar System, the iron-mass fractions of Mercury, Venus, Earth, and Mars are, respectively, evaluated as 0.64, 0.31, 0.32, and 0.28 (Morgan & Anders 1980; Wanke & Dreibus 1988), and they also show the similar correlation with the equilibrium temperature as the planets around M stars.

The positive slope of the Feplanet% – Teq relationship for the planets around M stars probably reflects a combined impact – or photoevaporation – driven density enhancement in the planet’s history. This has been found by Swain et al. (2019) from an increasing trend of the bulk density of small bodies and terrestrial planets with levels of insolation. The planetesimal impacts bring in volatile plumes that are then stripped by insolation. Loss of volatiles through impact is likely more efficient for higher levels of insolation, and furthermore this process is more considerable for small planets and small bodies (Swain et al. 2019). The trend of planets around M stars with lower Teq might exhibit substantial volatiles in their interiors or gas-rich envelopes, and the planets with increasing Teq are likely to exhibit less volatiles and even become purely rocky. Indeed, Acuña et al. (2021) used interior-atmosphere models to constrain the water mass fraction of TRAPPIST-1 planets and found the water mass fraction is generally increased with decreasing Teq. The similar trend for the terrestrial planets in our Solar System suggests that planets with lower Teq formed around FGK stars could also experience the similar process. Along with more exoplanets with Teq ≤ 500 K detected around FGK stars, it is of great interest to further address the Feplanet% – Teq relationship. The volatiles or gas-rich envelope are beyond the scope of the interior model presented here and the two groups begin to intersect around Teq ≈ 500 K. With these in mind, we do not consider the planets with Teq < 500 K when comparing the respective iron content among rocky exoplanets and their host stars, leaving 48 purely rocky planets for the following analysis. The planets removed here, all orbiting M stars, may exhibit a volatile-rich layer. Among the 48 purely rocky planets, 13 of them are around M stars and 35 of them around FGK stars.

3.2 Correlation between the iron-mass fractions of rocky exoplanets and the metallicities of their host stars

The relative iron abundance [Fe/H] of host stars is often regarded as the stellar metallicity. The stellar metallicity determines the total mass of iron ingredient in the protoplanetary disk. Observational and theoretical studies show that planet properties are correlated with the metallicity of host stars, such as planet occurrence rates – especially for giant planets and planet orbital distributions (Cabral et al. 2019). Thus, the stellar populations showing different metallicities could produce planets with different properties. For solar-type stars, Wang & Fischer (2015) found that the occurrence rates of various sized planets, including gas-giant planets, gas-dwarf planets, and terrestrial planets, are higher for metal-rich stars than for metal-poor stars and this property is more considerable for hotter stars. Lu et al. (2020) found that the period-average occurrence rate of rocky planets is higher for metal-rich stars than for metal-poor stars by a factor of , although this enhancement is not statistically significant enough to confirm the correlation between planet occurrence rates and host-star metallicities. Besides, some studies have been devoted to characterizing the link between the composition of planet building blocks and the metallicity of host stars (Santos et al. 2017; Cabral et al. 2019; Bitsch & Battistini 2020). Cabral et al. (2019) found a clear dependence of iron-to-silicate mass fractions on the initial a abundances of host stars. Combined with the observation that iron-poor stars hosting planets preferentially present enhanced alpha-element compositions (Adibekyan et al. 2012), iron-poor stars might produce planets with lower iron contents. The work by Bitsch & Battistini (2020) explored the chemical composition of solid planetary building blocks around stars with different metallicities, where solids formed completely exterior to the water ice line show an overall reduced water ice content and a rising elemental mass fraction of Fe–Mg–Si with increasing host star metallicity.

Figure 4 illustrates the iron-mass fractions Feplanet of the 48 rocky planets as a function of the metallicities [Fe/H] of their host stars. We can see that the metallicities of host stars are generally correlated with the iron-mass fractions of their hosted rocky planets. The exoplanet WASP-47 e in the right bottom corner is an outlier. We will exclude this planet in the subsequent analysis and make specific discussions in Sect. 4.2 later. Following Adibekyan et al. (2021), we apply Orthogonal Distance Regression (ODR, Boggs & Rogers 1990) in SciPy ODRPACK package to do linear regression to this sample. Comparing to the ordinary least square fitting, the ODR fitting is more reasonable for data with comparable errors in explanatory and response variables. It determines the parameters of linear functions y = αx + β by minimizing the reduced residual (9)

where n is the number of data points, and are the error of each data point, and δi is the fitting error of xi. Here, the fitting process is converted to the mathematical problem of minimizing the function χ(α,β δi) with n + 2 independent variables. The ODRPACK solves this problem numerically and then returns the best fit of the model parameters (α,β δi). A strong correlation is found for Feplanet% = (0.49 ± 0.12) × [Fe/H] + (0.46 ± 0.02) with a reduced residual of χ2 = 1.02. A fitting with a reduced residual close to 1 is generally regarded as a good fit. We also calculated the t-statistic from the slope and obtained a p-value of 6.4 × 10−5. This means that we can reject the null hypothesis that the slope is equal to zero at a probability of 1 – p. Therefore, the relationship derived from our fitting is statistically significant. In view of the fact that the super-Mercuries could have different experiences during their formation, we also fit the sample without the super-Mercuries. This yields the relation Feplanet% = (0.30 ± 0.11) × [Fe/H] + (0.42 ± 0.02) with a reduced residual of χ2 = 0.67. The normalized residual of each data point can be defined as (10)

where sgn(δi) stands for the sign function that returns the sign of δi. The distribution of χi is shown in the lower panel of Fig. 4. As we can see, the χi distribution is uniform and stochastic, showing a good fitting. And the t-statistic of this slope have a p-value of 0.0048. Under the premise that the modelling of a planet’s iron content is accurate, we conclude that the exoplanets orbiting a star with higher metallicities are more likely to have higher iron-mass fractions. The stellar metallicity has been measured for most of stars in our neighborhood, which makes it a useful tool for quickly gaining knowledge of the iron content of hosted planets.

Hinkel et al. (2014) built a catalog of elemental abundances of FGKM and planet host stars, known as the Hypatia Catalog2. This catalog contained about 8000 stars when it was accessed. Among them, there are 1326 stars hosting planet-forming planetary systems. For each star with multiple observations, we used the median value of their elemental abundances. Although the sample in this catalog is not directly link to our exoplanet sample, we can still compare the general trends of elemental abundance ratios with the stellar metallicity between planet-hosting stars and rocky exoplanets. Here we used the logarithm relative abundance [Fe/Mg] = [Fe/H] – [Mg/H] as the elemental abundance ratio. Figure 5 illustrates the variations of the abundance ratios [Fe/Mg] with the stellar metallicity [Fe/H] for the 48 rocky exoplanets of our sample (navy triangles and squares) and for the 1326 planet-hosting stars (orange and red circles). The metallicity of 1326 planet-hosting stars ranges from −1.0 to 0.6. As shown in Fig. 5, the ratio [Fe/Mg] shows a increasing tendency with increasing the stellar metallicity for both the planet-hosting stars and rocky exoplanets. The increasing tendency of the planet-hosting stars can be easily understood as iron-rich stars hosting planets prefer to present reduced alpha-element compositions (Adibekyan et al. 2012; Bashi & Zucker 2019; Cabral et al. 2019). Stars with different chemical compositions are expected to form solid planetary building blocks with different compositions. Thus, it is not surprising that the [Fe/Mg] of the rocky exoplanets shows a rising trend with the increase in the stellar metallicity, such as the stellar [Fe/Mg]. Plotnykov & Valencia (2020) found the probability density distribution of Fe/Mg shows higher Fe/Mg values for purely rocky super-Earths than for their host stars. Our results are consistent with this finding and also suggest that the spread of Fe/Mg is influenced by the stellar metallicity. The [Fe/Mg] enhancement for rocky exoplanets reflects iron has been concentrated into the interiors of rocky exoplanets during the formation and evolution of planetary systems, such as iron-rich projectiles and collisional stripping. In the following, we focus our discussion on a direct planet-star comparison.

thumbnail Fig. 4

Iron-mass fraction, Feplanet%, of rocky planets versus the metal-licity [Fe/H] of their host stars. Among the sample of the 48 rocky exoplanets, five super-Mercuries are separated from the others using the different backgrounds (brown and blue), which are drawn by eye. The ODR fitting is performed for the whole sample (dashed lines) and for the sample without the super-Mercuries (solid lines), respectively. Normalized residuals from the fit without super-Mercuries are plotted in the bottom panel.
thumbnail Fig. 5

Logarithm relative abundances [Fe/Mg] as a function of the stellar metallicity [Fe/H] for the 48 rocky exoplanets (triangles for FGK planets and squares for M planets) and for the 1326 planet-hosting stars (orange circles for FGK stars and red circles for M stars). The elemental abundance of planet-hosting stars are taken from the Hypatia Catalog (Hinkel et al. 2014) and the logarithm relative abundances [Fe/Mg] of the 48 rocky exoplanets are calculated from the interior model using the MCMC method.

3.3 Correlation between the iron content of rocky exoplanets and their host stars

For the 48 planets presented in Sect. 3.2, there are only 25 host stars with a complete set of elemental abundances for Fe, Si, and Mg. The iron-mass fraction Festar% of hypothetical planets with the same composition of host stars can be calculated by using Eq. (2). Figure 6 shows the direct comparison of the iron-mass fractions between the 26 rocky exoplanets and their host stars. As in the previous analysis, we achieve a strong correlation between them, Feplanet% = (10.80 ± 3.56) × Festar% + (−3.09 ± 1.16) with a reduced residual of χ2 = 0.84. The t-statistic of this slope yields a significant p-value of 0.0029. For the 26 rocky exoplanets, there are three exoplanets (Kepler-105 c, Kepler-107 c, and Kepler-406 b) classified as super-Mercuries. For the sample without the super-Mercuries, we obtain the relation: Feplanet% = (5.52 ± 1.43) × Festar% + (−1.41 ± 0.47) with a reduced residual of χ2 = 0.54. The χi distribution for all the data points (illustrated in the lower panel of Fig. 6) is uniform and stochastic. And the í-statistic of the slope yields a significant p-value of 4.7 × 10−4, suggesting the strong correlation between the iron-mass fractions from the MCMC and from the stellar elemental abundances. The slopes of these two relations are both larger than one as demonstrated in Adibekyan et al. (2021). In Fig. 7, we also compare the bulk molar ratios Fe/(Mg+Si) between the rocky exoplanets and their host stars. The Fe/(Mg+Si) ratio of the host stars is confined in a narrow range between 0.3 and 0.5, while the Fe/(Mg+Si) ratio of the exoplanets spans a wider range. The super-Mercuries may even have a Fe/(Mg+Si) ratio as large as 3.0 and there exits a clear gap between the super-Earths and super-Mercuries. For the 23 Super-Earths, there occurs a strong correlation Fe/(Mg+Si)planet = (5.91 ± 1.88) × Fe/(Mg + Si)star – 1.89 ± 0.78 with a reduced residual of χ2 = 0.41 and a t-test p-value of 0.0025 for the slope. The χi distribution shown in Fig. 7 is also uniform and stochastic. The slope greater than one implies there is a mechanism concentrating iron during the formation and evolution of rocky exoplanets (see more in Sect. 4.3).

thumbnail Fig. 6

Direct comparison of the iron-mass fractions between the 26 rocky exoplanets and their host stars. The iron-mass fractions of rocky exoplanets are computed from the interior model using the MCMC method, and the iron fractions of host stars are calculated from stellar elemental abundance ratios using Eq. (2). Different colors denote the equilibrium temperature Teq of rocky exoplanets. The ODR fitting is performed for the whole sample (dashed lines) and for the sample without the super-Mercuries (solid lines), respectively. Normalized residuals from the fit without super-Mercuries are plotted in the bottom panel.
thumbnail Fig. 7

Direct comparison of the bulk molar ratio Fe/(Mg+Si) between the 26 rocky exoplanets and their host stars. Different colors denote the equilibrium temperature Teq of rocky exoplanets. The solid line denotes the ODR fitting of the sample without the super-Mercuries, and normalized residuals from the fit are plotted in the bottom panel.
thumbnail Fig. 8

Total iron masses of the 48 rocky exoplanets as a function of the stellar metallicity [Fe/H] of their host stars. The red dashed line, which is draw by eye, denotes the potential upper limit of the iron masses in the rocky exoplanets.

4 Discussion

4.1 Influence of stellar iron abundance to total iron mass

The stars with higher metallicities have a protoplanetary disk with more iron-rich materials. The exoplanets formed in such a metal-rich environment are likely to have higher iron masses. Combining the entire mass and iron-mass fraction of planets, we can achieve the total iron mass of each individual planet by MFe = Mp × Feplanet%. It is not surprising that the iron mass is dependent upon the stellar metallicity to some extent. In Fig. 8, the planets with small MFe appear around both low- and high-metallicity stars, while the planets with large MFe values only appear around high-metallicity stars. The absence of the planets with large MFe around metal-poor stars indicates that the stellar metallicity plays a dominate role for the maximum iron content of rocky exoplanets formed around it.

4.2 WASP-47 e

We may note an exception from the general trend, which is marked in green in Figs. 4 and 5. The host star WASP-47 has a high metallicity of about 0.4, while the iron-mass fraction of WASP-47 e is as low as 0.14, as shown in Fig. 4. The low iron-mass fraction of WASP-47 e cannot be explained by an extra water-ice or gas layer because of its high Teq. The [Fe/Mg] ratio of the star WASP-47 is lower than that of most of the other metal-rich stars and the [Fe/Mg] ratio of WASP-47 e is significantly lower than its host star unlike the other planets, as shown in Fig. 5. Dorn et al. (2019) suggested that planets similar to WASP-47 e, which formed close to their host stars, could be enriched in Ca and Al, because Ca-Al-rich minerals have higher condensation temperatures. It should be noted that our two-layer interior models do not consider any of the Al and Ca bearing minerals, which brings in the largest Mg content and, hence, the rather low [Fe/Mg] value for WASP-47 e.

4.3 Discrepancy between stellar and planetary composition

In our Solar System, the relative elemental abundances of CI carbonaceous chondrites are more similar to the solar values with respect to the terrestrial planets (Asplund et al. 2009). Hence, CI carbonaceous chondrites are regarded as the remnant of plan-etesimals. Devolatilization has been quantified as a function of elemental condensation temperatures in going from the solar nebula to the formation of the Earth (Wang et al. 2019a). And the devolatilization mechanism has been applied to infer the bulk composition of terrestrial planets from the elemental abundances of host stars (Wang et al. 2019b, 2022). Devolatilization plays a critical role in determining the bulk composition of terrestrial exoplanets. For example, the devolatilization of oxygen has direct influence on the mantle and core compositions for purely rocky planets due to the trade-off between oxygen and refractory elements. It determines the fractionation of Fe between mantle and core and, thereby, the core mass fractions in modeling exo-planet interiors. The difference of the refractory elemental ratios between rocky exoplanets and their host stars presented here can be attributed to the composition differentiation both during the gas-dust fractionation process in the protoplanetary disk and in the subsequent formation process of rocky planets.

At the early stage of planet formation, the formation of plan-etesimals in different region of the protoplanetary disk can lead to composition differentiation. Solid materials in protoplanetary disk are migrating inward due to gas drag force. During the inward migration, solid materials crossed rocklines of different minerals (the regions where refractory materials condensate or sublimate). The vaporized minerals were no longer affected by drag force and stopped migration. The different condensation temperature of minerals causes the differentiation of the Mg-Fe-Si composition in different positions, separating the Fe-rich from Mg-rich materials. The disk transport model in Aguichine et al. (2020) shows that inward migration of solid materials in the protoplanetary disk concentrated different materials in the vicinity of different rocklines and formed planetesimals. The composition of the planetesimals formed by the separate materials therefore deviates from the composition of their host stars. The separated materials can also explain the diversity of the bulk composition of cosmic spherules, chondrules, and chondrites in our Solar System. However, this formation mechanism is not sufficient to explain the existence of extreme iron-rich planets such as super-Mercuries (Aguichine et al. 2020). Our work suggests that the differentiation of Mg-Fe-Si might happen in most of the exoplanet systems and only plays a partial role in the iron enrichment in rocky exoplanets.

The composition differentiation could also happen during the late stage of planet formation, where planets are formed by planetary embryo collisions. The results of collisions are affected by encounter velocity and impact factor. Most of the collisions are imperfect so that not all of the materials merged together. The debris ejected by the collision are mostly composited by silicates from the mantle. The CMF of the final planet after losing silicates can spread from 0.8 to 1.7 of the initial CMF (0.333, Scora et al. 2020), which enhanced the fraction of iron-rich cores and enriched the iron content of the planet. It has been shown by Chau et al. (2018) that iron-rich Mercuries can also be formed for a small range of encounter velocity and impact factor. In view of different protoplanetary disc conditions in the formation and evolution of planetary systems, planets around M dwarfs and around FGK-type stars could undergo various volatile losses and bombardment histories. For M-dwarf planets, both planetesimals and larger protoplanets are subject to extensive devolatilization processes owing to the longer super-lunimous pre-main-sequence phase of their host stars. The heat produced by impacts can make protoplanets further devolatilized, leading to the compositional deviation of protoplanets. The late-stage impact bombardment simulations of Lichtenberg & Clement (2022) found a decreasing rate of late-accreting reducing impacts with decreasing stellar mass, suggesting that late-stage impact bombardments are favored around FGK stars with respect to M dwarfs. Due to the trade-off between these two factors, rocky planets around M dwarfs and FGK stars may result in the similar iron-enrichment, as shown in Fig. 6. Moreover, it should be noted that the ratio of Feplanet% to Festar% spans over a wider range than the CMF ratio presented in Scora et al. (2020). Therefore, the iron content enhancement in rocky exoplanets could be the combined result of these two mechanisms.

5 Conclusions

In this work, we focus our attention on the larger sample of rocky exoplanets with masses M < 10 M⊕ and quantify the compositional correlation between them and their host stars. The interior of these planets is modeled as an iron-rich core overlaid with a silicate mantle. And the refractory element abundance ratios and iron-mass fraction of these planets are constrained from the measured mass and radius using the MCMC method. Then the rock-building element abundances of their host stars are collected and the hypothetical iron-mass fractions are calculated from the stellar elemental ratios. We made statistical analyses of the bulk compositions between these rocky planets and their host stars. It is found that:

  1. For the planets around M stars, their iron-mass fractions, Feplanet, are found to correlate with equilibrium temperature Teq. They are likely to maintain volatiles. However, there is no correlation between Feplanet and Teq for the planets around FGK stars with Teq ≥ 500 K;

  2. Some rocky exoplanets are shown to have large iron-mass fractions and are identified as an iron-enriched superMercury;

  3. The iron content of rocky exoplanets has a strong positive correlation with the metallicity [Fe/H] of their host stars. The planets formed around a higher metallicity star generally span a wider range of iron masses and exhibit higher iron-mass fractions;

  4. The iron-mass fraction of rocky exoplanets is also related to the relative abundance ratios Fe/Mg/Si of their host stars, showing the iron enrichment in rocky exoplanets.

The planet–star compositional correlation of the refractory elements confirms that it is feasible to use stellar abundance ratios to break the degeneracy in exoplanetary interiors. The compositional difference between rocky exoplanets and their host stars suggests that iron content enhancement is common during planet formation. The inward migration of solid materials in the pro-toplanetary disk concentrated different minerals to their own rocklines, contributing to the iron concentration in rocky exo-planets. The existence of super-Mercuries still requires the effect of late-stage impacts.

The compositional correlation between rocky exoplanets and their host stars ought to be further explored. For example, improved detections of planet-hosting stars and rocky exoplanets could provide us a larger sample to quantify this correlation. New observables of rocky exoplanets, such as tidal Love numbers, k2, could further break the degeneracy in interior compositions. In addition, the effects of significant populations of planet host stars in the galaxy are worthy of investigation.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 12022517), the Science and Technology Development Fund, Macau SAR (File No. 0048/2020/A1), and the Pre-Research Projects on Civil Aerospace Technologies of China National Space Administration (Grant No. D020308 and D020303). This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. The research shown here acknowledges use of the Hypatia Catalog Database, an online compilation of stellar abundance data as described in Hinkel et al. (2014), which was supported by NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network and the Vanderbilt Initiative in Data-Intensive Astrophysics (VIDA).

Appendix A Sample and data

The stellar parameters of the sample stars are presented in Table A.1. The MCMC results for the sample planets are summarized in Table A.2.

Table A.1

Parameters of the 48 host stars.

Table A.2

Summary of the MCMC results for the 57 rocky exoplanets.

Appendix B MCMC results for Kepler20 b

In this section, we take Kepler20 b as an example and show the results from the MCMC simulation. Figure B.1 illustrates the marginalized posterior distributions of the bulk parameters for Kepler20 b. As expected, the posterior distributions of mass and radius are both converged to the Gaussian form concentrated around the observed values.

thumbnail Fig. B.1

MCMC corner plot showing marginalized posterior distributions for the variables obtained from the interior model for Kepler20 b, including mass, radius, bulk molar ratios: Fe/Mg, Si/Mg, and Fe/(Mg+Si), and the iron-mass fractions. The median values are indicated in red and the ±1σ uncertainties are represented by dashed lines.

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All Tables

Table A.1

Parameters of the 48 host stars.

In the text
Table A.2

Summary of the MCMC results for the 57 rocky exoplanets.

In the text

All Figures

thumbnail Fig. 1

Sample of selected exoplanets with radius measurements plotted as a function of mass and equilibrium temperature. The color of the points represents the equilibrium temperature, Teq, of a planet (zero albedo assumed), and most of the sample planets have Teq large than the Earth. Blue dashed lines denote the M – R curve of pure-silicate planets calculated by the interior model. The planets larger than the largest size of rocky planets are marked by stars.
In the text
thumbnail Fig. 2

Normalized densities ρ/ρEarth–like of the 57 exoplanets as a function of their iron-mass fractions Feplanet%. The normalized density, ρ/ρEarth–like, of a planet is directly calculated from its mass and radius and ρEarth–like is the density of an Earth-like composition (core mass fractions of 32.3% and mantle molar ratios of Fe/Mg=0.12 and Si/Mg=0.81). The data of Mercury is marked as black star for reference. The exoplanets with both Feplanet% > FeMercury% and ρ/ρEarth–like> (ρ/ρEarth–like)Mercury are circled as super-Mercuries while the others are super-Earths. The distributions of Feplanet% and ρ/ρEarth–like are showed in the top and right panels, respectively.
In the text
thumbnail Fig. 3

Iron-mass fractions, Feplanet%, of the 57 rocky exoplanets as a function of their equilibrium temperature, Teq. The data points are divided by the spectral type of their host stars into two groups: planets around M stars (red squares) and around FGK stars (blue triangles). The Feplanet% of the two groups show different correlations with equilibrium temperature. Green stars denote the results for the four terrestrial planets in our Solar System.
In the text
thumbnail Fig. 4

Iron-mass fraction, Feplanet%, of rocky planets versus the metal-licity [Fe/H] of their host stars. Among the sample of the 48 rocky exoplanets, five super-Mercuries are separated from the others using the different backgrounds (brown and blue), which are drawn by eye. The ODR fitting is performed for the whole sample (dashed lines) and for the sample without the super-Mercuries (solid lines), respectively. Normalized residuals from the fit without super-Mercuries are plotted in the bottom panel.
In the text
thumbnail Fig. 5

Logarithm relative abundances [Fe/Mg] as a function of the stellar metallicity [Fe/H] for the 48 rocky exoplanets (triangles for FGK planets and squares for M planets) and for the 1326 planet-hosting stars (orange circles for FGK stars and red circles for M stars). The elemental abundance of planet-hosting stars are taken from the Hypatia Catalog (Hinkel et al. 2014) and the logarithm relative abundances [Fe/Mg] of the 48 rocky exoplanets are calculated from the interior model using the MCMC method.
In the text
thumbnail Fig. 6

Direct comparison of the iron-mass fractions between the 26 rocky exoplanets and their host stars. The iron-mass fractions of rocky exoplanets are computed from the interior model using the MCMC method, and the iron fractions of host stars are calculated from stellar elemental abundance ratios using Eq. (2). Different colors denote the equilibrium temperature Teq of rocky exoplanets. The ODR fitting is performed for the whole sample (dashed lines) and for the sample without the super-Mercuries (solid lines), respectively. Normalized residuals from the fit without super-Mercuries are plotted in the bottom panel.
In the text
thumbnail Fig. 7

Direct comparison of the bulk molar ratio Fe/(Mg+Si) between the 26 rocky exoplanets and their host stars. Different colors denote the equilibrium temperature Teq of rocky exoplanets. The solid line denotes the ODR fitting of the sample without the super-Mercuries, and normalized residuals from the fit are plotted in the bottom panel.
In the text
thumbnail Fig. 8

Total iron masses of the 48 rocky exoplanets as a function of the stellar metallicity [Fe/H] of their host stars. The red dashed line, which is draw by eye, denotes the potential upper limit of the iron masses in the rocky exoplanets.
In the text
thumbnail Fig. B.1

MCMC corner plot showing marginalized posterior distributions for the variables obtained from the interior model for Kepler20 b, including mass, radius, bulk molar ratios: Fe/Mg, Si/Mg, and Fe/(Mg+Si), and the iron-mass fractions. The median values are indicated in red and the ±1σ uncertainties are represented by dashed lines.
In the text

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