# Detailed chemical compositions of planet-hosting stars: II. Exploration of
the interiors of terrestrial-type exoplanets

H. S. Wang1,2, S. P. Quanz1,2, D. Yong3, F. Liu4, F. Seidler1, L. Acuña5, and
S. J. Mojzsis6,7

1Institute for Particle Physics and Astrophysics, ETH Zürich, Wolfgang-Pauli-
Strasse 27, 8093 Zürich, Switzerland
2National Center of Competence in Research PlanetS (www.nccr-planets.ch)
3Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia
4Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn, Victoria 3122, Australia
5Aix-Marseille Univ., CNRS, CNES, LAM, Marseille, France
6Origins Research Institute, Research Centre for Astronomy and Earth Sciences,
H-1112 Budapest, Hungary
7Department of Geological Sciences, University of Colorado, Boulder, CO
80309-0399, USA
E-mail<EMAIL_ADDRESS>
(Accepted 2022 April 19. Received 2022 April 15; in original form 2021 October
6)

###### Abstract

A major goal in the discovery and characterisation of exoplanets is to
identify terrestrial-type worlds that are similar to (or otherwise distinct
from) our Earth. Recent results have highlighted the importance of applying
devolatilisation – i.e. depletion of volatiles – to the chemical composition
of planet-hosting stars to constrain bulk composition and interiors of
terrestrial-type exoplanets. In this work, we apply such an approach to a
selected sample of 13 planet-hosting Sun-like stars, for which high-precision
photospheric abundances have been determined in the first paper of the series.
With the resultant devolatilised stellar composition (i.e. the model planetary
bulk composition) as well as other constraints including mass and radius, we
model the detailed mineralogy and interior structure of hypothetical,
habitable-zone terrestrial planets ("exo-Earths") around these stars. Model
output shows that most of these exo-Earths are expected to have broadly Earth-
like composition and interior structure, consistent with conclusions derived
independently from analysis of polluted white dwarfs. The exceptions are the
Kepler-10 and Kepler-37 exo-Earths, which we predict are strongly oxidised and
thus would develop metallic cores much smaller than Earth. Investigating our
devolatilisation model at its extremes as well as varying planetary mass and
radius (within the terrestrial regime) reveals potential diversities in the
interiors of terrestrial planets. By considering (i) high-precision stellar
abundances, (ii) devolatilisation, and (iii) planetary mass and radius
holistically, this work represents essential steps to explore the detailed
mineralogy and interior structure of terrestrial-type exoplanets, which in
turn are fundamental for our understanding of planetary dynamics and long-term
evolution.

###### keywords:

planets and satellites: composition – planets and satellites: interiors –
planets and satellites: terrestrial planets – stars: abundances

††pubyear: 2022††pagerange: Detailed chemical compositions of planet-hosting
stars: II. Exploration of the interiors of terrestrial-type exoplanets–14

##  1 INTRODUCTION

An important threshold has been crossed for detailed studies of interior
structure and composition of terrestrial exoplanets following the ever more
precise measurements of both planetary mass and radius (Weiss et al., 2016;
Stassun et al., 2017, 2018; Otegi et al., 2020) and of host stellar
photospheres that reveal the primordial elemental compositions of the systems
(Nissen, 2015; Brewer & Fischer, 2016; Liu et al., 2016, 2020; Delgado Mena et
al., 2017; Bedell et al., 2018; Clark et al., 2021; Adibekyan et al., 2021).
Indeed, the past decade has witnessed a number of interior models that follow
this lead (e.g. Dorn et al., 2015; Santos et al., 2015; Unterborn et al.,
2016; Dorn et al., 2017; Brugger et al., 2017; Unterborn et al., 2018; Wang et
al., 2019b; Acuna et al., 2021; Wang et al., 2022). We can anticipate that
future observations of planetary atmospheres with JWST (Morley et al., 2017;
Gialluca et al., 2021) and other innovative ground- and space-based missions
and mission concepts such as ELT/METIS (Quanz et al., 2015; Bowens et al.,
2021), ARIEL (Tinetti et al., 2018; Turrini et al., 2021), PLATO (Rauer et
al., 2014; Nascimbeni et al., 2022), and LIFE (Quanz et al., 2021, 2022) will
reveal new details of the surface and interior characteristics of terrestrial
exoplanets, given the evolutionary outcomes of the dynamic interactions
between the interior, surface, atmosphere, and possible hydrosphere or even
biosphere (Shahar et al., 2019; Bower et al., 2019; Ortenzi et al., 2020; Dyck
et al., 2021; Hakim et al., 2021; Acuna et al., 2021; Kacar et al., 2021).

In the present work, we build upon the procedures outlined in Wang et al.
(2019a); Wang et al. (2019b) (W19a and b, thereafter) that introduced the idea
of using the devolatilised host stellar abundances, rather than the unaltered
host stellar abundances, to constrain the bulk composition and interior
modelling of hypothetical habitable-zone terrestrial exoplanets ("exo-
Earths"). This idea was established based on the observations of bulk
composition differences and similarities of the Solar System’s rocky bodies
relative to the Sun (Grossman & Larimer, 1974; Bland et al., 2005; Davis,
2006; Carlson et al., 2014; Wang et al., 2018; Sossi & Fegley, 2018).
Importantly, among the 10 major rock-forming elements (Mg, Si, Fe, Ni, Al, Ca,
Na, O, S, and C), only Ca and Al (the two most refractory ones) are not
observed to be depleted in rocky bodies relative to the Sun. All other
elements have been depleted to some degree: e.g. for Mg, Si, Fe and Ni the
depletion is by $\sim$ 10-20%; for volatiles like O, S and C the depletion is
over 80% (W19a). It is reasonable to argue that the devolatilisation process
is not unique to the Solar System and may be a universal process in the
formation of rocky (exo)planets. A recent study of the major rock-forming
elements (including oxygen) for a sample of six white dwarfs (Doyle et al.,
2019) shows that the bulk composition of the planetary debris polluting these
white dwarfs resembles those of rocky planets in the Solar System. This
suggests that the parent rocky bodies of these debris must also be the
devolatilised pieces of their host stars (also see Harrison et al., 2021;
Bonsor et al., 2021), although the exact devolatilisation factors for such
planetary systems are beyond what can be constrained with the existing data.
Hence, starting with the best-known calibration of the devolatilisation based
on our Solar System (in particular, Sun and Earth; W19a), we can apply it, to
first order, to other Sun-like star systems and estimate the potential rocky
planetary bulk composition from the measurable host stellar photospheric
abundances. Subsequently, planetary interior composition and structure can be
modelled, as shown for a sample of 4 planet-hosting stars in W19b. Even
further, interior dynamics and thermo-chemical evolution (Spaargaren et al.,
2020, 2021; Wang et al., 2020b), carbon cycle modelling (Hakim et al., 2021),
and habitability arguments (Kacar et al., 2021) ensue from such an analysis.

Here, we extend the analysis to a further sample of 13 planet-hosting Sun-like
stars for which detailed and precise chemical compositions (for up to 18
elements including all major rock-forming elements at a typical precision of
$\sim$ 0.025 dex) have been determined with high-quality spectra in the first
paper of the series (Liu et al., 2020) (hereafter Paper I). We are interested
in how diverse the interiors of the model exo-Earths around these stars would
be.

The paper is organised by presenting our methodology and analysis in Section 2
and results in Section 3, followed by a discussion of the effect of varying
devolatilisation scaling factors (within plausible bounds) and planetary size
(within the terrestrial regime) on the interiors and of the model restrictions
in Section 4. We summarise and conclude in Section 5.

##  2 METHODOLOGY AND ANALYSIS

###  2.1 Methodology

To carry out the analysis, two sets of software are employed: ExoInt (W19b)
for devolatilising stellar abundances and modelling stoichiometric mantle and
core compositions as well as core mass fractions; Perple_X (Connolly, 2009)
for modelling of a detailed mantle mineralogy and interior structure (e.g.
self-consistent density, pressure and temperature profiles as well as core
radius fraction). It is important to note that the devolatilisation model of
W19a that we adopted is empirically quantified by the bulk elemental abundance
ratio ($f$) between Earth and proto-Sun as a function of 50% condensation
temperature ($T_{c}$; Lodders, 2003):

$\log(f)=\alpha\log(T_{c})+\beta$ (1)

where, the best-fit coefficients $\alpha=3.676\pm 0.142$ and $\beta=-11.556\pm
0.436$.

An application of such a model to other planetary systems is by all means a
simplification of devolatilisation processes and outcomes that may vary across
different systems and, in principle, even at different orbital distances
within one system. We therefore limit such an application to only habitable-
zone, terrestrial-type exoplanets around Sun-like stars – i.e. exo-Earths by
our definition. It has been shown in the literature (e.g. Wang, 2018; Sossi &
Fegley, 2018; Yoshizaki & McDonough, 2020) that the bulk compositional
differences of Venus and Mars from the Sun are within the uncertainty of such
a difference between Earth and Sun. We emphasise that the empirical
devolatilisation model of W19a is the first-order quantitative model of such
an important process. A sophisticated model of devolatilisation that involves
disc evolution, accretion and hydrodynamic escape processes is still awaiting
formalisation (e.g. Wang, 2022). To account for potential variation in
devolatilisation scales for the exo-Earths considered here, we vary the
uncertainty range of the adopted W19a model arbitrarily by a factor of 3 to
assess how this may affect the interior modelling results (to be discussed in
Sect. 4.1).

For the detailed procedure of ExoInt, we refer the reader to Fig. 3 and
Appendix A of W19b. It is briefly summarised here that we adopt the chemical
networks of Na2O-CaO-MgO-Al2O3-SiO2-FeO-NiO-SO3-CO2-C(graphite/diamond)-metals
for the mantle composition (in terms of first-order oxides and reduced phases)
and of Fe-Ni-Si-S alloy for the core composition of a terrestrial-type
exoplanet. The core mass fraction is determined by mass balance after
distributing the bulk planetary composition into the stoichiometric mantle and
core compositions. The difference to W19b is that Si is not firstly oxidised
in the oxidation sequence of major elements, but follows Na, Ca, Mg, and Al
and precedes Fe, Ni, and S. This approach allows for the possibility that Si
can be partially oxidised in the case of a reduced mantle (i.e. at a low
oxidation state) and complies with the fact that it can be an important light
element constituent of metallic cores (McDonough, 2003; Hirose et al., 2013;
Li & Fei, 2014; Wang et al., 2018). This update has been applied in Wang et
al. (2022) and the code is publicly accessible111https://github.com/astro-
seanwhy/ExoInt/tree/master/v1.2 (IDL version); https://github.com/astro-
seanwhy/ExoInt/tree/master/pyExoInt (Python version)..

Having determined stoichiometric mantle and core compositions as well as core
mass fractions, planetary mineralogy (i.e. complex mineral assemblages) and
structure are then modeled with Perple_X by assuming that these exo-Earths are
all Earth-like in mass and radius. Later in the discussion (Sec. 4.2), we
explore the effect of varied planetary mass and radius (within the terrestrial
density regime) on the predictions of detailed interiors. The underlying
method for computing mantle mineralogy given the mantle composition of major
oxides and the pressure and temperature profiles for a terrestrial-type planet
is Gibbs free energy minimisation (Connolly, 2009). The mineral equations of
state and thermodynamic parameters – essential for the Gibbs free energy
minimisation – are adopted from Stixrude & Lithgow-Bertelloni (2011). For a
given Fe-Ni-Si-S-alloy core, we adopt the equation of state from Kuwayama et
al. (2020). We adopt an adiabatic thermal gradient, as similarly practiced in
Dorn et al. (2015), Unterborn et al. (2018), and Lorenzo (2018), and integrate
it with a mantle potential temperature of 1700 K at 1 bar, arbitrarily set to
be approximate to that of the modern Earth (Anderson, 2000). Hinkel &
Unterborn (2018) found that different setups of a mantle potential temperature
(at a typical range of 1500 K and 1900 K) for a terrestrial-type planet only
introduce accountable effect towards the mineralogies in the transition zone
between the upper and lower mantle – we have verified this finding with Earth
as an example (Fig. 10). We also introduce a temperature jump at the core
mantle boundary (CMB) by following Noack & Lasbleis (2020) (for all of these
exo-Earths) and Stixrude (2014) (for the tested cases with masses out of
$[0.8,2]M_{\oplus}$ – the range in which the Noack & Lasbleis (2020) model is
parameterised). It has been found that a variable thermal profile plays a
negligible role in changing the interior structure (Dorn et al., 2015) – we
have further verified this finding with varied temperature jumps at the CMB in
the case of Earth (Fig. 11).

###  2.2 Sample selection and analysis

Figure 1: C/O vs. Mg/Si – abundance ratios by number – for the sample of
stars. The vertical and horizontal dashed lines respectively indicate the
critical value of Mg/Si (= 1.0) and C/O (= 0.8) (Bond et al., 2010; Suárez-
Andrés et al., 2018). HD179079 and HD190360 (colored in red), for which C/O >
0.8, are excluded from our detailed analysis. The Sun (‘$\sun$’; Asplund et
al., 2021) is shown for reference.

Our sample for this study is based on the planet-hosting stars studied in
Paper I: Kepler-21, Kepler-37, Kepler-68, Kepler-93, Kepler-96, Kepler-K100,
Kepler-131, K2-222 (EPIC220709978), K2-277 (EPIC212357477), Kepler-408,
HD1461, and HD219828. The addition to this list is Kepler-10, for which the
detailed elemental abundances are obtained in Liu et al. (2016) in the same
fashion as in Paper I. We have excluded Kepler-409 because its oxygen
abundance – essential for our analysis – is undetermined. We have also
excluded HD179079 and HD190360, for which the values of C/O are larger than
0.8 (Fig. 1) and thus around which carbide planets may be developed (Bond et
al., 2010; Teske et al., 2014; Brewer & Fischer, 2016). Our model is based on
silicate chemistry and thus does not apply to plausible carbide planets. For
the reader interested in the interiors of carbide planets we refer to Hakim et
al. (2018); Hakim et al. (2019) for more details. Therefore, we have a sample
of 13 Sun-like stars (with $5400<T_{\textrm{eff}}<6400$ K, $4.0<\log g<4.5$
$\mathrm{c}\mathrm{m}\,\mathrm{s}^{-2}$, and $-0.3<\textrm{[Fe/H]}<+0.3$),
each of which has been confirmed to host at least one planet with a mass
$<10M_{\oplus}$ (except for HD 219828 that hosts a planet with a minimum mass
of $\sim 21M_{\oplus}$). Since these detected planets are all in the proximity
(with an orbital period $<50$ days) of their host stars, they are not included
in this study that focuses on hypothetical, (habitable-zone) exo-Earths.

As shown in Fig. 1, the values of Mg/Si (abundance ratio by number) in our
sample are distributed within a narrow range from $\sim$ 0.9 to $\sim$ 1.4.
Mg/Si modulates the dominant mineral phases in the mantle of a silicate
planet: pyroxene (MgSiO3) and various feldspars for Mg/Si < 1, a mixture of
olivine (Mg2SiO4) and pyroxene assemblages for 1 < Mg/Si <2, and olivine with
other Mg-rich species for Mg/Si > 2 (Bond et al., 2010; Suárez-Andrés et al.,
2018). At first glance, therefore, the mantles of exo-Earths around our sample
of stars would most likely be made of a mixture of olivine and pyroxene
assemblages, while some (with Mg/Si < 1) may be slightly enriched in
pyroxenes.

Following the study of W19b, we apply the Sun-to-Earth devolatilisation model
(W19a) to the sample of stars for 10 major rock-forming elements: Mg, Si, Fe,
Ni, Al, Ca, Na, O, S, and C. The differential abundances determined in Paper I
(their Table 2) are firstly converted to the absolute abundances by referring
to the latest solar abundances of Asplund et al. (2021). We ignore the
diffusion effect since this effect is equivalent for those major rock-forming
elements for the Sun (Asplund et al., 2009) and presumably for Sun-like stars
as well. We also do not consider the effect of Galactic chemical evolution
(GCE) on the host stellar abundances, since planets are fundamentally
correlated with the properties (and formation environment) of individual host
stars. In other words, whatever a GCE effect may have with the host stellar
abundances should have also been an inherent part of the formation histories
of planets around these stars. Thus, the GCE effect has validly shaped the
chemical compositions of these planets. The resultant, "devolatilised" stellar
abundances – i.e., the model planetary bulk composition – are listed in Table
1 and used as a principal set of constraints for a detailed modelling of the
interiors of (hypothetical) exo-Earths around these stars.

##  3 RESULTS

###  3.1 Key planetary geochemical ratios

The abundance ratio of carbon to oxygen (hereafter C/O) in a planet host star
is useful, as illustrated in Fig. 1, to indicate to first order if a potential
rocky world around it would be dominated by silicates or by carbides. However,
upon the application of the devolatilisation, both oxygen and carbon become so
severely depleted that the remaining atoms are principally locked in planetary
mineral assemblages. C/O in a rocky planet is therefore no longer a valid
indicator of the mantle oxidation state, which, however, is essential to
understand the planetary interiors.

For a silicate planet including our own, MgO and SiO2 are the foremost mineral
oxides in the mantle, with Fe being distributed between its oxidised form
(e.g. FeO) in the mantle and its reduced, metallic/liquid iron phase in the
core, depending on the oxygen fugacity ($f\textrm{O}_{2}$) of the planet
(McDonough & Sun, 1995; Palme & O’Neill, 2014; Dorn et al., 2015; Wang et al.,
2019b). The calculation of $f\textrm{O}_{2}$, often relative to either the
quartz-faylite-magnetite buffer (O’Neill, 1987; Cottrell & Kelley, 2011) or
the iron-wüstite buffer (e.g. Doyle et al., 2019), requires the prior
knowledge of the relative fractions between different phases of iron, which we
do not have in the first place for exoplanets. Based on the estimated bulk
elemental composition (Table 1) and considering that O, Mg, Si, and Fe are the
foremost abundant, rock-forming elements (Palme & O’Neill, 2014; Wang et al.,
2018), we propose the bulk (O-Mg-2Si)/Fe as a simple alternative of oxygen
fugacity to indicate the oxidation state of a silicate terrestrial exoplanet.

Planetary Mg/Si (having no significant difference from its host stellar Mg/Si)
is still critical to modulate the dominant mineral assemblages (olivine vs.
pyroxene) in the mantle of a silicate planet. Further, Fe/Mg is preferred over
Fe/Si as an indicator of the degree of core-mantle fractionation (determining
the core size), owing to the fact that Si may be present as a major light
element in the core of a rocky planet (McDonough, 2003; Hirose et al., 2013;
Li & Fei, 2014; Wang et al., 2018) and is indeed considered in the core
compositional model in this work.

Figure 2: (O - Mg - 2Si)/Fe vs. Mg/Si and (O - Mg - 2Si)/Fe vs. Fe/Mg diagrams
for the sample of model exo-Earths. The dashed, vertical and horizontal lines
classify these planets into different categories, in which different planetary
interior properties may be expected (see text as well as the brief notations
on the plot). Earth ("E", the blue dot; Wang et al., 2018) and Mars ("M", the
brown dot; Yoshizaki & McDonough, 2020) are shown for reference.

Fig. 2 shows the distributions of these model exo-Earths on the diagram of (O
- Mg - 2Si)/Fe vs. Mg/Si and (O - Mg - 2Si)/Fe vs. Fe/Mg. First, it shows that
all of the sample planets are significantly below the unit line of (O - Mg -
2Si)/Fe, except Kepler-10 exo-Earth ("K10-exoE") and Kepler-37 exo-Earth
("K37-exoE"). The direct implication is that most of these planets would have
a large iron core and potentially an Earth-like structure while iron in
K10-exoE and K37-exoE may be much more oxidised and thus these two planets
would develop comparably smaller cores. The core size of K10-exoE would be the
smallest due to its significantly high (O - Mg - 2Si)/Fe that would cause most
of the iron to be oxidised and locked in the mantle. Considering that the core
size is also modulated by Fe/Mg (right panel of Fig. 2), K37-exoE – which has
the equivalent oxidation state as Mars, but lower Fe/Mg – would potentially
develop a core that is smaller, in relative terms, than the core:mantle ratio
of Mars. Frank et al. (2014) referred to this class of terrestrial-type
exoplanets as "Super-Lunas". Among those planets clumped around the Earth’s
loci on both panels of the diagram, their mantle mineralogies would be more or
less the same, with K93-exoE being likely the most olivine-rich due to its
relatively high Mg/Si. These qualitative analyses are further verified by the
following, detailed interior modelling.

###  3.2 Mantle and core compositions and core mass fraction

Our estimates of the mantle composition for all studied exo-Earths are
presented in Table 2, with the normalised composition of the foremost major
oxides (i.e. normalising the sum of SiO2, MgO, and FeO to be 100 wt%) shown in
the ternary diagram Fig. 3. Overall, the mantle compositions of these exo-
Earths are very similar, except for K10-exoE and K37-exoE. The latter two are
particularly enriched in FeO, consistent with the analysis above based on key
geochemical ratios.

Figure 3: Ternary diagrams showing the estimated mantle compositions of major
mineral oxides (normalized by SiO2 + MgO + FeO = 100 wt%) for the sample of
model exo-Earths (for clarity, the sample is divided arbitrarily to two
groups). The range of the squares indicates the modelled 1$\sigma$
uncertainties in the normalised mantle compositions of the individual planets.
The normalised compositions of SiO2, MgO, and FeO of Earth mantle (McDonough &
Sun, 1995) and of Martian mantle (Yoshizaki & McDonough, 2020) are shown for
reference.

The estimates of core composition and core mass fraction (CMF; i.e. the core-
to-planet mass ratio) of these exo-Earths are also presented in Table 2. We
find that the core compositions of these exo-Earths are more or less the same,
with the concentration of Fe ranging from $\sim$ 80 to $\sim$90 wt%. In
contrast, the values of CMF are diverse, ranging from $\sim$ 0 wt% to $\sim$
40 wt% (Fig. 4). K10-exoE and K37-exoE, for which the concentrations of FeO in
the mantle are the highest (as mentioned above), are the ones with the lowest
CMFs – $0.0^{+9.4}_{-0.0}$ and $11.9^{+15.0}_{-11.9}$ \- broadly mimicking
Moon-like and Mars-like structures, respectively. The CMFs of eight exo-Earths
(K21-exoE, K93-exoE, K96-exoE, K100-exoE, K408-exoE, K2-222-exoE, K2-227-exoE,
and HD219828) are consistent with the CMF ($32.5\pm 0.3$ wt%; Wang et al.
2018) of the Earth within uncertainties. The remaining three exo-Earths
(K68-exoE, K131-exoE, and HD 1461-exoE) have CMFs statistically higher than
that of the Earth, with K131-exoE – appearing at the bottom right (i.e. with
the highest Fe/Mg and lowest (O-Mg-2Si)/Fe) in the right panel of Fig. 2 –
being the highest ($39.7^{+4.4}_{-4.7}$ wt%) .

Figure 4: Estimates of the core mass fraction (CMFs) of the sample of model
exo-Earths under three different scenarios: (i) applying the standard Sun-to-
Earth devolatilisation pattern (i.e., with the 1$\sigma$-uncertainty range) of
W19a (filled circles with error bars in black); (ii) applying the 3$\sigma$
upper limit of the standard devolatilisation model (open diamonds with error
bars in grey); (iii) applying the 3$\sigma$ lower limit of the standard
devolatilisation model (open circles with error bars in grey). Earth’s core
mass fraction (32.5 $\pm$ 0.3 wt%; Wang et al., 2018) and Mars’ core mass
fraction ($18^{+8}_{-3}$ wt%; Yoshizaki & McDonough, 2020) are shown for
reference.

The minor oxides – Na2O, CaO, and Al2O3 – will be combined with the
aforementioned three major oxides to model the detailed mineralogy of these
exo-Earths. Other minor/trace end-members including NiO, SO3, CO2, C and
metals are however not involved in the subsequently detailed interior
modelling but their realisation is essential to correctly distribute oxygen
into those more abundant oxides.

###  3.3 Mineralogy and internal structure

Using Kepler-21 exo-Earth (K21-exoE) as an example, we present its best-fit
mineralogy and structure (in terms of self-consistent pressure, temperature
and density profiles) in Fig. 5. It shows that in the upper mantle, pyroxenes
(orthopyroxene – "opx", clinopyroxene – "cpx" and high-pressure clinopyroxene
– "hp-cpx") are relatively enriched over olivine ("ol"), being consistent with
the planet’s relatively low Mg/Si ratio (the left panel of Fig. 2). The lower
mantle (starting from the density jump at $\sim$ 0.9 $R_{\oplus}$) is
dominated by magnesium (post-)perovskite ("mg-pv" and "mg-postpv") and is
similar to the Earth’s lower mantle composition (Palme & O’Neill, 2014). By
comparing its density profile with the Earth’s (in both cases a (plausible)
inner solid core has been ignored; Fig. 5), K21-exoE has a best-fit core
slightly smaller than that of Earth. Please note that the best-fit result is
obtained at the mean values of the first-order major oxides and core mass
fraction (Table 2) and at a radius of 1 $R_{\oplus}$ (with its self-consistent
mass, returned together with the mineralogy, equivalent to 1 $M_{\oplus}$ as
well). The uncertainties of the best-fit results for each mineral in the
mantle, the radius fraction of the core as well as other structural profiles
(Fig. 12) are obtained at the 16% and 84% quartiles of a population of the
mineralogy and structure analyses repeated from a random draw of the estimates
of the mantle and core compositions as well as core mass fractions (Table 2).

Figure 5: The best-fit interior mineralogy and structure of Kepler-21 exo-
Earth (as an example). The solid black, blue and red curves are respectively
the density, pressure and temperature profiles of the planet. For reference,
the corresponding profiles of the Earth are shown as dashed curves. The grey
regime on the left panel represents the core, which is presumably composed of
Fe, Ni, Si, and S (the sum of which is normalised to 1). The temperature jumps
at the CMBs are introduced by following Noack & Lasbleis (2020), resulting in
1513 K for K21-exoE and 1537 K for Earth. The abbreviations for mantle
mineralogy stand for: ol - olivine, plg - plagioclase, opx - orthopyroxene,
cpx - clinopyroxene, hp-cpx - high-pressure clinopyroxene, wad - wadsleyite,
ring - ringwoodite, gt - garnet, stv - stishovite, wus - magnesiowüstite
(ferropericlase), mg-pv - magnesium perovskite (bridgmanite), mg-postpv -
magnesium postperovskite, cf - calcium-ferrite structured phase, and ca-pv -
calcium perovskite. For other details of the model see text.

The best-fit results for all planets, as shown in Figs. 13 and 14, reveal that
most of these exo-Earths are Earth-like in both the mineralogy and structure.
The notable exceptions are K10-exoE and K37-exoE. The former has a negligible
core – as shown in Fig. 4 as well – and a gigantic mantle, whereas the latter
has a core that is comparably smaller than all cases other than K10-exoE.
Further, both have a deep mantle dominated by the high-pressure, "mg-postpv"
phase – compared to the "mg-pv" phase (bridgemanite) for other cases. For the
uppermost mantle (above $\sim$ 0.95 $R_{\oplus}$), it is dominated by olivine
over pyroxene assemblages (also similar to that of Earth; Palme & O’Neill
2014) for all exo-Earth cases, except K2-277-exoE (and to a lesser degree,
K21-exoE as well). Particularly, K2-277-exoE is also the only planet with a
mean value of Mg/Si < 1 (Fig. 2). This highlights the significance of Mg/Si in
determining the mantle mineralogy, as investigated in Hinkel & Unterborn
(2018) and Spaargaren et al. (2020) as well. The intermediate range between
upper and lower mantle ($\sim$ 0.9-0.95 $R_{\oplus}$; i.e. the mantle
transition zone) is dominated by wadsleyite ("wad") and ringwoodite ("ring")
(also similar to Earth’s scenario), except for K10-exoE, K37-exoE, K68-exoE,
and K2-277-exoE. Considering that both wadsleyite and ringwoodite can store
water in their crystal structures by about one order of magnitude higher than
any other minerals including olivine, pyroxene and perovskite (Bercovici &
Karato, 2003; Pearson et al., 2014; Fei et al., 2017), these exceptional
planets may also be the ones among the sample with the least water-storage
capacity in its interior, although this assessment needs to be exercised with
caution considering the yet-large uncertainty in the modelled mantle
mineralogy.

Broadly speaking and by considering the uncertainty level of such an analysis
(as assessed for K21-exoE as an example), all of these cases (except K10-exoE
and K37-exoE) share both the internal structure and mineralogy of a broadly
Earth-like planet.

##  4 DISCUSSION

###  4.1 The effect of varying devolatilisation scales on the interiors

Planet formation is a complex process. A variety of outcomes for the bulk
composition of a rocky planet may result from composition-, location-, and
time-scale-dependent differences in various devolatilisation processes (W19b;
Dorn et al., 2019; Harrison et al., 2021). Our empirical understanding of the
devolatilisation from the protosolar to terrestrial abundances cannot be a
true reflection of the devolatilisation that occurred in other planetary
system, even for "exo-Earths" by our definition. However, we suppose that the
discrepancy is not dramatic concerning the similar Earth-like composition for
rocky exoplanets as revealed by the abundance measurements on polluted white
dwarfs (Doyle et al., 2019).

To explore the effect of varying the devolatilisation scales on the interiors
of exo-Earths, we apply the upper and lower limits of the 3$\sigma$ range of
the adopted devolatilisation model to the abundances of these host stars. As a
result, we obtain two alternative sets of planetary bulk compositions,
corresponding to the "less-depleted" (Fig. 6) and "more-depleted" (Fig. 7)
scenarios, respectively. In the "less-depleted" scenario (Fig. 6), the mantle
compositions (normalised by MgO + SiO2 \+ FeO = 100 wt%) of these exo-Earths
shift towards the direction where SiO2 and FeO are more enriched (relative to
the scenario under the standard devolatilisation model – Fig. 3). In such a
case, oxygen – acting as a critical element to the mantle composition estimate
– is much less depleted, resulting in a much higher mantle oxidation state and
thus more Si and Fe in the planet being oxidised. Likewise, in the "more-
depleted" scenario (Fig. 7), the normalised mantle compositions shift towards
the direction where SiO2 and FeO are more depleted (relative to the scenario
under the standard devolatilisation model – Fig. 3). In this case oxygen is
much more depleted, thus resulting in a reduced mantle and an increased amount
of Si as well as Fe partitioned into the core.

The effect of such varied devolatilisation scales is also reflected onto the
modelled core mass fractions (Fig. 4). In the "less-depleted" scenario (shown
as open diamonds withe error bars in Fig. 4), the core mass fractions are
systematically smaller than those under the "standard" scenario (i.e. filled
circles with error bars in Fig. 4), because the mantle is more oxidised with a
reduced fractionation of metallic Fe and its alloy elements (Ni, Si, and S)
into the core. Similarly, in the "more-depleted" scenario (shown as open
circles with error bars in Fig. 4), the core mass fractions are systematically
larger than those under the "standard" scenario. In such a case, the mantle is
more reduced and a larger fraction of Fe and its alloy elements is partitioned
into the core. It is also noteworthy that the limits of our calculated CMFs
(0–46%) under the different scenarios are statistically broader than those
constrained by the unaltered stellar compositions in the literature – e.g.
20–46% (Plotnykov & Valencia, 2020) and 21–41% (Schulze et al., 2021) – and
narrower than those constrained purely by mass and radius measurements – e.g.
1–92% (Plotnykov & Valencia, 2020) and 0–73% (Schulze et al., 2021) for
(potentially) rocky planets. These discrepancies are fundamentally attributed
to i) the difference in the available oxygen budget relative to other rock-
forming elements in a planet (W19b) and ii) the inherent degeneracy in
constraining planetary interiors with only mass and radius measurements (Dorn
et al., 2015). For a detailed discussion of model and observational
uncertainties on the determination of planetary interiors, we refer to Otegi
et al. (2020).

Such a variance in the devolatilisation scales also induces a change to the
modelled internal structure and mineralogy (Fig. 8; K21-exoE is taken for
example, in comparison with Fig. 5). In the "less-depleted" scenario (left
panel), the modelled core-mantle boundary (in line with the horizontal part of
the solid black curve) is relatively deeper than that modelled under the
"standard" scenario (dashed black curve), while it is opposite in the "more-
depleted" scenario. For the mantle mineralogies as modelled under various
scenarios, there are no significant differences, except for the minerals (e.g.
ringwoodite – "ring", wadsleyite – "wad", and stishovite – "stv") in the
mantle transition zone and the high-pressure "mg-postpv" phase just above the
core-mantle boundary. Overall, the variability of the devolatilisation scaling
factors seems to influence more the internal structure than the mantle
mineralogy. As explained in Sect. 3.1, Mg/Si plays a critical role in
modulating the mantle mineralogy, whereas this ratio is only negligibly
altered by devolatilisation (W19a). On the other hand, the fractionation of Fe
between core and mantle is directly related to the mantle oxidation state
(Righter, 2003) and oxygen is a volatile (sensitive to devolatilisation);
consequently, the internal structure may be more affected when the
devolatilisation scale varies.

It is noteworthy that our exploration of the effect of varied devolatilisation
scales may not be applicable to peculiar planets such as super-Mercuries (e.g.
Adibekyan et al., 2021), nor to any type of planets around M stars such as the
TRAPPIST-1 system (Gillon et al., 2017) and Proxima b (Anglada-Escudé et al.,
2016). This is a consequence from potentially dramatic differences in planet
formation histories and/or in stellar properties (e.g. XUV fluxes). To take
into account these factors, a comprehensive investigation of nebular
condensation (e.g. Wang et al., 2020a), disc evolution (e.g. Bergner et al.,
2020), hydrodynamic escape (e.g. Benedikt et al., 2020), accretionary dynamics
(e.g. Emsenhuber et al., 2021), and impacts (Helffrich et al., 2019) is
warranted. Before such a comprehensive investigation sheds more light on
devolatilisation (in a quantitative manner), however, the inclusion of such an
empirical model is a first-order but integral part of the efforts in reducing
the modelling degeneracies of interiors of terrestrial-type exoplanets.

Figure 6: Similar to Fig. 3, but constrained with the 3$\sigma$ upper limit of
the Sun-to-Earth devolatilisation pattern (W19a). The open arrow indicates the
overall direction that the mantle composition shifts towards in the parameter
space of the ternary diagram. For details see Sect. 4.1. Figure 7: Similar to
Fig. 3, but constrained with the 3$\sigma$ lower limit of the Sun-to-Earth
devolatilisation pattern (W19a). The open arrow indicates the overall
direction that the mantle composition shifts towards in the parameter space of
the ternary diagram. For details see Sect. 4.1. Figure 8: The modelled
mineralogies and structures of Kepler-21 exo-Earth under two different
devolatilisation scenarios: the 3$\sigma$ upper limit (left) and lower limit
(right) of the Sun-to-Earth devolatilisation pattern (W19a). For comparison,
the density profile for the planet under the "standard" scenario (Fig. 5) is
replotted as a dashed curve in each panel.

###  4.2 The effect of varying mass and radius (within the terrestrial regime)
on the interiors

As analysed in Sect. 3.3, these hypothetical exo-Earths are assumed to have a
radius of 1 $R_{\oplus}$ and a mass of 1 $M_{\oplus}$. In reality, habitable-
zone terrestrial planets are unlikely to be an ideal Earth 2.0 and their mass
and radius may vary. To test the effect of varying mass and radius (within the
terrestrial regime) on the interiors, we assume two extreme cases for the size
of a model terrestrial planet by referring to the definition of a rocky planet
orbiting within the (empirical) habitable zone: 0.5 $R_{\oplus}$ and 1.5
$R_{\oplus}$ for the LIFE targets (Quanz et al., 2021). The mass is not
predefined but computed together with the interiors by keeping the planet to
be in the terrestrial regime. The modelling results are shown in Fig. 9.

For a model terrestrial planet of 0.5 $R_{\oplus}$ orbiting in the habitable
zone of Kepler-21, its mantle mineralogy and self-consistent mass ($\sim$ 0.1
$M_{\oplus}$) approximately resemble those of Mars (Yoshizaki & McDonough,
2020). For a model terrestrial planet of 1.5 $R_{\oplus}$ orbiting in the
habitable zone of Kepler-21, however, there is no exact analogue in our Solar
System, since its self-consistent mass of $\sim$ 4.3 $M_{\oplus}$ falls in the
regime of a super-Earth and it has a lower mantle mineralogy distinctly
dominated by the high-pressure "mg-postpv" phase, while its upper mantle
mineralogy resembles that of Earth (McDonough & Sun, 1995; Palme & O’Neill,
2014).

Please note that, both cases are conducted under the standard devolatilisation
model. Namely, the same set of the first-order mantle and core compositions as
well as core mass fraction of Kepler-21 exo-Earth (Table 2) are input for
producing Figs. 5 and 9. In other words, the significant differences in mantle
mineralogy between these scenarios shown in Figs. 5 and 9 are dictated only by
the size (and implicitly the mass) of the model planet. The trivial
differences in core radius fraction are related to the difference in density
compression between the smaller and larger (model) planets.

Figure 9: The modelled mineralogies and structures of Kepler-21 exo-Earth
under the standard devolatilisation scenario but with different assumptions of
planetary size: 0.5$R_{\oplus}$ (left) and 1.5$R_{\oplus}$ (right). The
returned, self-consistent masses are 0.1$M_{\oplus}$ and 4.3$M_{\oplus}$,
respectively, enabling the planet to be within the terrestrial regime. Please
note that the CMB temperature jumps are introduced by following Stixrude
(2014), instead of Noack & Lasbleis (2020) (which is parameterised for rocky
planets of $[0.8,2]M_{\oplus}$), for these two cases. Please also note the
differences in the scales of radius and density between the two panels.

###  4.3 On the precision requirement of using (devolatilised) host stellar
abundances to constrain exoplanet interiors

In W19b, a precision level of $\lesssim$ 0.04 dex (differentially) or
$\lesssim$ 10% is proposed for the host stellar abundances to be used, upon
devolatilisation, to constrain and distinguish the interior compositions and
structures of hypothetical terrestrial exoplanets. However, it is worth
clarifying that this precision requirement has not taken into account the
uncertainty in the reference solar abundance, which is typically at the level
of $\sim$ 0.03 dex (Asplund et al., 2009, 2021) for the aforementioned major
rock-forming elements. If the latter were taken into account, the precision
requirement for the host stellar abundances should be as small as $\sim$ 0.025
dex (differentially) or $\sim$ 6%, which has been achieved with high-precision
spectroscopic analysis for bright Sun-like stars (e.g. Paper I; Spina et al.,
2021) and binary stars (Morel, 2018; Liu et al., 2018, 2021). Apart from the
concern with the precision, there are also nontrivial systematic differences
between different reference solar abundances (e.g. Asplund et al., 2021;
Adibekyan et al., 2021; Lodders, 2020; Wang et al., 2019a). Therefore, one
must clarify which solar abundances are applied for converting the stellar
differential abundances to the absolute abundances, which are then used for
further modelling of the properties of individual planets. For a detailed
discussion of the systematic differences between stellar abundances determined
with different techniques and their impacts on many aspects including
modelling planetary properties, please refer to Hinkel et al. (2016).

###  4.4 Remaining limitations

For our interior structure model, we must emphasize that it is a rigid two-
layer model - i.e. mantle + core, without the layers such as crust and water,
nor that we have divided the core to an inner component and an outer one.
Considering the currently typical uncertainties of 5-10% and 10-20%,
respectively, for radius and mass measurements (Otegi et al., 2020), such a
simplification is practical for interior modelling and has a trivial effect on
our estimates of mantle composition and internal structure (in terms of core
radius/mass fraction). The modelling of crust formation and composition is an
active field and usually involves the modelling of partial melts, tectonics,
and even weathering (Rozel et al., 2017; Brugman et al., 2021; Hakim et al.,
2021), which are however beyond the scope of this paper. Nonetheless, our
estimates of mantle mineralogy and core size provide essential information for
future modelling of the crust owing to the fact that the crust is
fundamentally an extraction product of the upper mantle through magmatic
processes (Rozel et al., 2017; Noack et al., 2012) and is also (ultimately)
influenced by core formation (Dyck et al., 2021).

A water layer (while without a crust layer) has been considered in other
interior models (e.g. Tian & Stanley, 2013; Noack et al., 2016; Dorn et al.,
2017; Brugger et al., 2017; Unterborn et al., 2018; Acuna et al., 2021). Based
on numerical modelling, Shah et al. (2020) found that for a rocky planet with
a mass of 0.1 – 3 $M_{\oplus}$, the effect of an isolated surface water (i.e.
oceans) on the radius is $\leq$ 5%, while the effect of a hydration case (i.e.
water being chemically mixed with minerals) is $\leq$ 2.5% – see also Vazan et
al. (2022). Such effects are still within the currently typical uncertainty
for radius but may become more profound with the continuous improvement of the
precisions of mass and radius measurements (e.g. up to 5% and 3%,
respectively; Stassun et al. 2017). The effect of water inclusion on the mass
of a terrestrial-type planet should be nonetheless negligible. However, water
(regardless its amount) should be critically taken into account while
extending a study on mineralogy and structure to that on internal dynamics
(Evans et al., 2014; Spaargaren et al., 2020) and/or crust formation (Collins
et al., 2020).

Finally, we treat the core to be completely molten and homogeneously composed
of Fe, Ni, Si and S, without differentiating a plausible inner solid portion
made of pure Fe and Ni from an outer liquid portion that contains the light
elements (McDonough, 2003; Hirose et al., 2013; Wang et al., 2018). This may
inevitably underestimate the core mass and thus overestimate core radius
should the planet’s core be differentiated. However, we envisage that the
under-/over-estimation should not be significant, concerning that Earth’s
inner solid core just accounts for 5% of the mass of the core (Yoder, 1995;
McDonough, 2017) and that even a planet as small as Mars has been suggested to
be in a molten state based on the most recent seismic data from the InSight
mission (Stähler et al., 2021). Conflicting views on liquid/solid cores for
super-Earths have been presented (e.g. Valencia et al., 2006; Morard et al.,
2011), so do the models underpinning the two scenarios – c.f., ’solid’ (Dorn
et al., 2017; Brugger et al., 2017) vs. ’liquid’ (Unterborn et al., 2018;
Lorenzo, 2018). The phase of a planet’s core should be not only related to the
planet’s mass and size but also its formation history and age (Stevenson,
2008; Stixrude, 2014). Before we know better such information collectively,
this issue will remain open for further discussion.

##  5 Summary and Conclusions

Based on the detailed chemical compositions of planet-hosting stars obtained
in Paper I and Liu et al. (2016), this work extends the analysis of W19a,b and
goes beyond the estimates of the first-order mantle oxide composition, core
composition and core mass fraction to the detailed mineralogy (i.e. complex
mineral phases that are often seen in a rocky planet like Earth) and interior
structure (in terms of not only core mass fraction, but also core radius
fraction and self-consistent density, pressure and temperature profiles). We
have also assessed the uncertainties of the detailed interior estimates, which
are contributed from the uncertainties associated with the host stellar
abundances, the devolatilisation pattern, as well as the interior modelling
degeneracy. Further, by respectively varying the devolatilisation scales and
the planetary size, we test how such variabilities will affect the modelled
interiors of terrestrial-type planets.

We find that among the 13 model exo-Earths, 11 are broadly Earth-like in both
interior composition and structure, whereas Kepler-10 and Kepler-37 exo-Earths
(both with high (O-Mg-2Si)/Fe, implying a high planetary oxidation state) are
predicted to have substantially smaller cores. On the effect of varying
devolatilisation scaling factors on planetary interiors, we find that interior
structure is more affected than the mantle mineralogy, probably because the
fractionation of Fe between mantle and core is more sensitive to the planetary
oxidation state – thus determining the core size, whereas the mantle
mineralogy is crucially modulated by Mg/Si (Hinkel & Unterborn, 2018;
Spaargaren et al., 2020) while this ratio is negligibly altered by
devolatilisation (W19a). The further test of varying the planetary size (and
implicitly mass, by keeping within the terrestrial regime) reveals the
potential diversity of the mantle mineralogy of terrestrial planets even if
they might have experienced an equivalent devolatilisation. We also recommend
a precision level of $\lesssim$ 0.025 dex for the stellar differential
abundances and a clarification of the reference solar abundances (and their
uncertainties) in modelling planetary bulk composition and interiors.

Our model is nonetheless limited by its rigid assumption of a two-layer
structure – i.e., mantle and core, with no crust (lithosphere) or water
(potentially biosphere) considered yet, nor have we differentiated the core to
be an inner component and an outer one based on their extent of
solidification/crystallisation. However, such a simplification should have
little impact on our modelling results of the static, mantle mineralogy and
structure, while the model may be sophisticated further when the mass and
radius measurements of real terrestrial planets become much more precise (e.g.
up to 5% and 3%; Stassun et al. 2017; Rauer et al. 2014).

Exercised with caution, such an analysis (with a yet-large uncertainty)
nevertheless offers an insight, in terms what we can learn already with the
available data (essentially host stellar composition as well as planetary mass
and radius) and with the bulk/interior models (with sensibly simplifications)
that we can build, into the detailed properties of habitable-zone,
terrestrial-type exoplanets, thus providing guidance for the target selections
for future missions, such as PLATO (Rauer et al., 2014; Nascimbeni et al.,
2022), Ariel (Turrini et al., 2021), and LIFE (Quanz et al., 2021, 2022).

## Acknowledgements

We thank the reviewers, particularly Lena Noack, for their helpful comments,
which have greatly improved the quality of the manuscript. This work has been
carried out within the framework of the National Centre of Competence in
Research PlanetS supported by the Swiss National Science Foundation (SNSF).
H.S.W and S.P.Q acknowledge the financial support of the SNSF. FL acknowledges
the support of the Australian Research Council through Future Fellowship grant
FT180100194. S.J.M. thanks the Research Centre for Astronomy and Earth
Sciences (Budapest, Hungary) for support.

## Data availability

The spectral data underlying this article are available in Keck Observatory
Archive at https://koa.ipac.caltech.edu/cgi-bin/KOA/nph-KOAlogin. They can be
accessed with Keck Program ID: Z148 (Semester: 2016B, PI: Yong) and Z279
(Semester: 2018A, PI: Yong). Other data underlying this article are available
in the article or in the specified references.

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##  Appendix A Model planetary bulk compositions

Table 1: Model bulk compositions (normalised to Al=100) of hypothetical
habitable-zone terrestrial exoplanets ("exoE"), as devolatiliseda from their
host stellar abundancesb.
The reported uncertainties are 1$\sigma$c. | C | O | S | Na | Si | Mg | Fe | Ni | Ca | Al
---|---|---|---|---|---|---|---|---|---|---
K10-exoE | 41${}^{+10}_{-8}$ | 3871${}^{+445}_{-403}$ | 31${}^{+4}_{-4}$ | 12${}^{+1}_{-1}$ | 819${}^{+67}_{-62}$ | 1048${}^{+92}_{-84}$ | 679${}^{+72}_{-65}$ | 38${}^{+4}_{-4}$ | 66${}^{+5}_{-5}$ | 100${}^{+7}_{-7}$
K21-exoE | 57${}^{+15}_{-12}$ | 4862${}^{+642}_{-574}$ | 46${}^{+6}_{-5}$ | 23${}^{+3}_{-3}$ | 1490${}^{+128}_{-117}$ | 1515${}^{+135}_{-124}$ | 1225${}^{+138}_{-124}$ | 66${}^{+8}_{-7}$ | 110${}^{+9}_{-8}$ | 100${}^{+7}_{-7}$
K37-exoE | 30${}^{+7}_{-6}$ | 2936${}^{+339}_{-307}$ | - | 10${}^{+1}_{-1}$ | 670${}^{+56}_{-51}$ | 1013${}^{+85}_{-78}$ | 617${}^{+65}_{-59}$ | 34${}^{+4}_{-3}$ | 59${}^{+5}_{-4}$ | 100${}^{+28}_{-22}$
K68-exoE | 22${}^{+5}_{-4}$ | 1857${}^{+213}_{-193}$ | 20${}^{+2}_{-2}$ | 10${}^{+1}_{-1}$ | 649${}^{+53}_{-49}$ | 744${}^{+61}_{-56}$ | 597${}^{+63}_{-57}$ | 35${}^{+4}_{-3}$ | 51${}^{+4}_{-3}$ | 100${}^{+35}_{-26}$
K93-exoE | 30${}^{+8}_{-7}$ | 2889${}^{+332}_{-301}$ | 28${}^{+3}_{-3}$ | 12${}^{+1}_{-1}$ | 776${}^{+64}_{-59}$ | 996${}^{+83}_{-76}$ | 716${}^{+75}_{-68}$ | 41${}^{+4}_{-4}$ | 65${}^{+6}_{-5}$ | 100${}^{+12}_{-11}$
K96-exoE | 36${}^{+9}_{-7}$ | 3176${}^{+361}_{-327}$ | 34${}^{+4}_{-4}$ | 14${}^{+1}_{-1}$ | 1000${}^{+81}_{-75}$ | 1152${}^{+95}_{-87}$ | 1026${}^{+107}_{-97}$ | 54${}^{+6}_{-5}$ | 86${}^{+6}_{-6}$ | 100${}^{+10}_{-9}$
K100-exoE | 42${}^{+10}_{-9}$ | 2785${}^{+361}_{-323}$ | 35${}^{+4}_{-4}$ | 16${}^{+2}_{-1}$ | 906${}^{+80}_{-73}$ | 985${}^{+86}_{-79}$ | 759${}^{+84}_{-76}$ | 46${}^{+5}_{-5}$ | 64${}^{+5}_{-5}$ | 100${}^{+10}_{-9}$
K131-exoE | 32${}^{+9}_{-8}$ | 2459${}^{+291}_{-263}$ | 28${}^{+3}_{-3}$ | 13${}^{+1}_{-1}$ | 861${}^{+71}_{-65}$ | 963${}^{+82}_{-75}$ | 847${}^{+91}_{-82}$ | 49${}^{+5}_{-5}$ | 71${}^{+6}_{-5}$ | 100${}^{+8}_{-7}$
K2-222-exoE | 54${}^{+22}_{-17}$ | 3809${}^{+471}_{-424}$ | 40${}^{+5}_{-5}$ | 18${}^{+2}_{-2}$ | 1125${}^{+96}_{-88}$ | 1293${}^{+114}_{-104}$ | 960${}^{+106}_{-96}$ | 51${}^{+6}_{-5}$ | 84${}^{+7}_{-6}$ | 100${}^{+12}_{-11}$
K2-277-exoE | 44${}^{+13}_{-11}$ | 3012${}^{+394}_{-353}$ | 37${}^{+5}_{-4}$ | 14${}^{+2}_{-1}$ | 966${}^{+86}_{-78}$ | 958${}^{+87}_{-79}$ | 851${}^{+96}_{-86}$ | 51${}^{+6}_{-5}$ | 70${}^{+7}_{-6}$ | 100${}^{+8}_{-7}$
K408-exoE | 44${}^{+11}_{-9}$ | 3418${}^{+434}_{-390}$ | 38${}^{+5}_{-4}$ | 15${}^{+2}_{-2}$ | 1125${}^{+101}_{-92}$ | 1332${}^{+116}_{-107}$ | 957${}^{+105}_{-95}$ | 50${}^{+6}_{-5}$ | 93${}^{+8}_{-7}$ | 100${}^{+9}_{-8}$
HD1461-exoE | 36${}^{+9}_{-8}$ | 2426${}^{+293}_{-264}$ | 30${}^{+4}_{-3}$ | 17${}^{+2}_{-2}$ | 867${}^{+72}_{-66}$ | 928${}^{+78}_{-72}$ | 762${}^{+82}_{-74}$ | 49${}^{+5}_{-5}$ | 59${}^{+5}_{-4}$ | 100${}^{+10}_{-9}$
HD219828-exoE | 33${}^{+9}_{-7}$ | 2482${}^{+318}_{-286}$ | 28${}^{+3}_{-3}$ | 16${}^{+2}_{-1}$ | 840${}^{+71}_{-65}$ | 868${}^{+77}_{-70}$ | 730${}^{+81}_{-73}$ | 44${}^{+5}_{-4}$ | 60${}^{+5}_{-5}$ | 100${}^{+9}_{-9}$
a The model of devolatilisation: W19a – Wang et al. (2019a); the specific
devolatilisation factors for these listed elements can also be found in Table
1 of W19b – Wang et al. (2019b).
b Sources of host stellar abundances: Paper I – Liu et al. (2020) (Table 2),
except for Kepler 10 (Liu et al., 2016) (Table 2, derived with HET data)
c The uncertainties are propagated from the 1$\sigma$ uncertainties in both
host stellar abundances and in the devolatilisation model.

##  Appendix B Modelling details of planetary interiors

Table 2: Interior compositions and core mass fractions (with 1$\sigma$ uncertainties) of hypothetical exo-Earths (exoE) around the sample of stars, in comparison with Earth. | Mantle composition (wt%)
---|---
| Na2O | CaO | MgO | Al2O3 | SiO2 | FeO | NiO | SO3 | CO2 | C | Metals
K10-exoE | 0.24${}^{+0.04}_{-0.03}$ | 2.41${}^{+0.36}_{-0.24}$ | 27.4${}^{+4.3}_{-2.6}$ | 3.32${}^{+0.47}_{-0.30}$ | 31.8${}^{+5.1}_{-2.9}$ | 29.02${}^{+2.99}_{-5.32}$ | 1.66${}^{+0.23}_{-0.38}$ | 1.39${}^{+0.26}_{-0.42}$ | 0.68${}^{+0.18}_{-0.17}$ | 0.15${}^{+0.05}_{-0.05}$ | -
K21-exoE | 0.41${}^{+0.09}_{-0.09}$ | 3.52${}^{+0.62}_{-0.73}$ | 35.2${}^{+6.2}_{-7.7}$ | 2.92${}^{+0.51}_{-0.60}$ | 47.8${}^{+5.3}_{-8.3}$ | 8.62${}^{+6.89}_{-6.58}$ | 0.42${}^{+0.47}_{-0.27}$ | 0.50${}^{+0.10}_{-0.10}$ | 0.03${}^{+0.01}_{-0.01}$ | 0.38${}^{+0.12}_{-0.10}$ | 0.15${}^{+0.28}_{-0.13}$
K37-exoE | 0.27${}^{+0.06}_{-0.04}$ | 2.87${}^{+0.67}_{-0.45}$ | 35.0${}^{+9.3}_{-5.5}$ | 4.45${}^{+1.58}_{-1.15}$ | 34.4${}^{+7.3}_{-5.5}$ | 21.01${}^{+6.14}_{-11.56}$ | 1.19${}^{+0.42}_{-0.75}$ | - | 0.20${}^{+0.06}_{-0.07}$ | 0.27${}^{+0.08}_{-0.07}$ | 0.02${}^{+0.04}_{-0.02}$
K68-exoE | 0.49${}^{+0.07}_{-0.06}$ | 4.32${}^{+0.55}_{-0.47}$ | 45.6${}^{+5.6}_{-5.0}$ | 7.82${}^{+2.73}_{-2.13}$ | 41.0${}^{+5.7}_{-6.9}$ | 0.22${}^{+0.43}_{-0.18}$ | 0.03${}^{+0.02}_{-0.01}$ | 0.04${}^{+0.01}_{-0.03}$ | - | 0.41${}^{+0.10}_{-0.09}$ | 0.03${}^{+0.15}_{-0.02}$
K93-exoE | 0.35${}^{+0.06}_{-0.07}$ | 3.46${}^{+0.61}_{-0.66}$ | 38.4${}^{+6.5}_{-7.7}$ | 4.82${}^{+0.94}_{-0.96}$ | 42.2${}^{+5.0}_{-7.2}$ | 9.29${}^{+7.87}_{-7.10}$ | 0.53${}^{+0.51}_{-0.33}$ | 0.61${}^{+0.11}_{-0.12}$ | 0.02${}^{+0.01}_{-0.01}$ | 0.34${}^{+0.11}_{-0.09}$ | 0.02${}^{+0.13}_{-0.02}$
K96-exoE | 0.38${}^{+0.05}_{-0.05}$ | 4.31${}^{+0.56}_{-0.59}$ | 41.5${}^{+5.3}_{-5.8}$ | 4.53${}^{+0.68}_{-0.66}$ | 45.7${}^{+4.2}_{-5.5}$ | 2.47${}^{+2.90}_{-1.94}$ | 0.12${}^{+0.16}_{-0.07}$ | 0.25${}^{+0.05}_{-0.11}$ | - | 0.38${}^{+0.10}_{-0.09}$ | 0.37${}^{+0.52}_{-0.30}$
K100-exoE | 0.49${}^{+0.08}_{-0.08}$ | 3.64${}^{+0.54}_{-0.57}$ | 40.8${}^{+5.6}_{-6.2}$ | 5.20${}^{+0.83}_{-0.83}$ | 46.5${}^{+4.4}_{-6.1}$ | 2.37${}^{+3.20}_{-1.89}$ | 0.19${}^{+0.20}_{-0.10}$ | 0.30${}^{+0.06}_{-0.13}$ | - | 0.51${}^{+0.14}_{-0.12}$ | 0.03${}^{+0.05}_{-0.02}$
K131-exoE | 0.47${}^{+0.07}_{-0.06}$ | 4.56${}^{+0.59}_{-0.54}$ | 44.7${}^{+5.5}_{-5.2}$ | 5.85${}^{+0.77}_{-0.69}$ | 43.5${}^{+5.2}_{-6.2}$ | 0.39${}^{+0.61}_{-0.32}$ | 0.04${}^{+0.04}_{-0.02}$ | 0.07${}^{+0.02}_{-0.05}$ | - | 0.43${}^{+0.14}_{-0.11}$ | 0.05${}^{+0.37}_{-0.04}$
K2-222-exoE | 0.40${}^{+0.08}_{-0.08}$ | 3.45${}^{+0.57}_{-0.65}$ | 38.5${}^{+5.9}_{-7.7}$ | 3.72${}^{+0.71}_{-0.73}$ | 45.5${}^{+4.8}_{-7.1}$ | 6.92${}^{+6.38}_{-5.37}$ | 0.31${}^{+0.43}_{-0.19}$ | 0.51${}^{+0.09}_{-0.13}$ | 0.02${}^{+0.01}_{-0.01}$ | 0.46${}^{+0.20}_{-0.16}$ | 0.19${}^{+0.31}_{-0.17}$
K2-277-exoE | 0.42${}^{+0.08}_{-0.08}$ | 3.72${}^{+0.62}_{-0.67}$ | 36.8${}^{+5.6}_{-6.9}$ | 4.83${}^{+0.74}_{-0.85}$ | 48.7${}^{+4.5}_{-6.7}$ | 4.25${}^{+5.32}_{-3.37}$ | 0.31${}^{+0.33}_{-0.17}$ | 0.45${}^{+0.09}_{-0.15}$ | - | 0.49${}^{+0.16}_{-0.14}$ | 0.02${}^{+0.14}_{-0.02}$
K408-exoE | 0.39${}^{+0.07}_{-0.06}$ | 4.30${}^{+0.63}_{-0.58}$ | 44.2${}^{+6.1}_{-5.8}$ | 4.19${}^{+0.59}_{-0.59}$ | 44.2${}^{+4.8}_{-6.2}$ | 1.55${}^{+1.89}_{-1.19}$ | 0.07${}^{+0.10}_{-0.04}$ | 0.17${}^{+0.03}_{-0.08}$ | - | 0.44${}^{+0.12}_{-0.10}$ | 0.45${}^{+0.64}_{-0.36}$
HD1461-exoE | 0.61${}^{+0.10}_{-0.08}$ | 3.88${}^{+0.52}_{-0.46}$ | 43.8${}^{+5.8}_{-5.1}$ | 5.97${}^{+0.89}_{-0.80}$ | 44.7${}^{+5.2}_{-6.5}$ | 0.33${}^{+0.69}_{-0.27}$ | 0.05${}^{+0.04}_{-0.02}$ | 0.07${}^{+0.02}_{-0.05}$ | - | 0.51${}^{+0.14}_{-0.12}$ | 0.06${}^{+0.08}_{-0.04}$
HD219828-exoE | 0.57${}^{+0.09}_{-0.09}$ | 3.85${}^{+0.57}_{-0.55}$ | 40.4${}^{+5.6}_{-5.6}$ | 5.86${}^{+0.91}_{-0.85}$ | 46.9${}^{+4.5}_{-6.2}$ | 1.64${}^{+2.26}_{-1.37}$ | 0.13${}^{+0.14}_{-0.07}$ | 0.20${}^{+0.04}_{-0.11}$ | - | 0.45${}^{+0.13}_{-0.11}$ | 0.03${}^{+0.15}_{-0.02}$
Eartha | 0.36 | 3.58 | 38.1 | 4.49 | 45.4 | 8.11 | - | - | - | - | -
| Core composition (wt%) and core mass fraction (CMF, wt%) |  |  |
|  | Fe | Ni | Si | S |  | CMF |  |  |  |
K10-exoE |  | 90.4${}^{+2.1}_{-3.1}$ | 5.5${}^{+0.1}_{-0.2}$ | 0.2${}^{+0.2}_{-0.1}$ | 3.9${}^{+3.5}_{-2.0}$ |  | 0.0${}^{+9.4}_{-0.0}$ |  |  |  |
K21-exoE |  | 89.7${}^{+1.7}_{-7.8}$ | 5.2${}^{+0.3}_{-0.8}$ | 3.3${}^{+3.1}_{-2.3}$ | 1.7${}^{+0.5}_{-0.4}$ |  | 28.8${}^{+6.5}_{-15.8}$ |  |  |  |
K37-exoE |  | 93.4${}^{+0.0}_{-1.1}$ | 5.7${}^{+0.0}_{-0.5}$ | 0.9${}^{+1.1}_{-0.7}$ | - |  | 11.9${}^{+15.0}_{-11.9}$ |  |  |  |
K68-exoE |  | 80.3${}^{+7.1}_{-6.3}$ | 5.0${}^{+0.7}_{-0.7}$ | 13.2${}^{+6.1}_{-6.5}$ | 1.5${}^{+0.3}_{-0.2}$ |  | 38.7${}^{+4.2}_{-4.4}$ |  |  |  |
K93-exoE |  | 90.6${}^{+1.1}_{-5.8}$ | 5.5${}^{+0.2}_{-0.7}$ | 2.1${}^{+2.3}_{-1.5}$ | 1.8${}^{+0.6}_{-0.5}$ |  | 26.9${}^{+7.1}_{-14.2}$ |  |  |  |
K96-exoE |  | 88.0${}^{+4.9}_{-6.7}$ | 4.9${}^{+0.7}_{-0.6}$ | 5.5${}^{+4.2}_{-3.5}$ | 1.6${}^{+0.3}_{-0.3}$ |  | 36.4${}^{+4.4}_{-6.7}$ |  |  |  |
K100-exoE |  | 85.2${}^{+7.0}_{-8.2}$ | 5.4${}^{+0.4}_{-0.8}$ | 7.3${}^{+5.2}_{-4.6}$ | 2.1${}^{+0.4}_{-0.4}$ |  | 33.7${}^{+5.0}_{-7.7}$ |  |  |  |
K131-exoE |  | 82.5${}^{+7.1}_{-6.1}$ | 5.0${}^{+0.7}_{-0.7}$ | 10.9${}^{+5.5}_{-5.8}$ | 1.5${}^{+0.3}_{-0.2}$ |  | 39.7${}^{+4.4}_{-4.7}$ |  |  |  |
K2-222-exoE |  | 89.3${}^{+2.0}_{-8.1}$ | 5.1${}^{+0.4}_{-0.8}$ | 3.6${}^{+3.2}_{-2.5}$ | 1.9${}^{+0.5}_{-0.4}$ |  | 29.4${}^{+5.7}_{-13.6}$ |  |  |  |
K2-277-exoE |  | 87.8${}^{+3.9}_{-8.3}$ | 5.4${}^{+0.3}_{-0.7}$ | 4.8${}^{+4.1}_{-3.1}$ | 2.0${}^{+0.5}_{-0.5}$ |  | 33.6${}^{+5.4}_{-10.8}$ |  |  |  |
K408-exoE |  | 84.1${}^{+8.5}_{-8.1}$ | 4.6${}^{+0.8}_{-0.6}$ | 9.4${}^{+6.1}_{-5.6}$ | 1.9${}^{+0.4}_{-0.3}$ |  | 34.1${}^{+4.7}_{-5.9}$ |  |  |  |
HD1461-exoE |  | 81.0${}^{+7.7}_{-6.7}$ | 5.4${}^{+0.6}_{-0.7}$ | 11.9${}^{+6.2}_{-6.4}$ | 1.8${}^{+0.3}_{-0.3}$ |  | 38.3${}^{+4.5}_{-4.8}$ |  |  |  |
HD219828-exoE |  | 84.5${}^{+8.1}_{-7.6}$ | 5.3${}^{+0.5}_{-0.7}$ | 8.5${}^{+5.4}_{-5.1}$ | 1.8${}^{+0.3}_{-0.3}$ |  | 35.9${}^{+4.9}_{-6.5}$ |  |  |  |
Earthb |  | 87.5 | 5.3 | 5.2 | 1.9 |  | $32.5\pm 0.3$ |  |  |  |
a Refer to McDonough & Sun (1995) and normalise the contents of the adopted six major oxides to 100 wt%. |  |  |  |
b Refer to Wang et al. (2018) and normalise the contents of the adopted four elements in the core to 100 wt%. |  |  |  |

Figure 10: The comparison of the modelled mantle mineralogies in the case of
Earth given different mantle potential temperatures ($T_{0}$): 1500 K, 1700 K,
and 1900 K, which respectively correspond to the dashed, solid, and dotted
curves in the rightmost panel for the temperature profiles. The core-mantle
temperature jump is fixed at 1537 K for all cases. The core in each case is
only partially shown to highlight the mantle mineralogy. Figure 11: The
comparison of the modelled interior profiles for density (black curves),
pressure (blue curves), and temperature (red curves) in the case of Earth
given different core-mantle temperature jumps: 0 K, 1000 K, 1500 K, and 2000
K, which respectively correspond to the dotted, dashed, solid, and dash-dotted
curves of these profiles. The mantle potential temperature is fixed at 1700 K
for all cases. Figure 12: The uncertainty assessment of the modelled
mineralogy and structure (taking K21-exoE as an example). The best-fit
mineralogy/structure profiles (corresponding to Fig. 5) are shown in dotted
lines in each panel, with the 50th percentile and the range of 16th and 84th
percentiles of the estimates shown in a solid line and a shadow area. The
best-fit profiles of pressure and density are exactly coincident with their
50th percentiles. The vertical dashed line in each panel indicates the 50th
percentile of the core size ($R_{\mathrm{core}}$), with its uncertainty shown
in the light grey area. The illustration of the 50th percentile and the
embraced range of [16th, 84th] percentiles of the lower mantle mineralogy has
been cut off beyond the 1$\sigma$ lower limit of the radius. Figure 13: The
best-fit mineralogies and structures of half of the sample of exo-Earths. The
dashed curves indicate the Earth density profile. For the explanation of other
details of these individual diagrams, please refer to Fig. 5. The additional
mineral name abbreviations – sp and neph – stand for spinel and nepheline,
respectively. Figure 14: Similar to Fig. 13 but for the other half of the
sample.