# Interior Structure Models Of Venus.

Oliver Shah  Center for Theoretical Astrophysics and Cosmology, University of
Zurich, Switzerland  Center for Space and Habitability, University of Bern,
Switzerland  Ravit Helled  Center for Theoretical Astrophysics and Cosmology,
University of Zurich, Switzerland  Yann Alibert  Center for Space and
Habitability, University of Bern, Switzerland  Klaus Mezger  Center for Space
and Habitability, University of Bern, Switzerland  Institut für Geologie,
University of Bern, Switzerland

###### Abstract

Venus’ mass and radius are similar to those of Earth. However, dissimilarities
in atmospheric properties, geophysical activity and magnetic field generation
could hint towards significant differences in the chemical composition and
interior evolution of the two planets. Although various explanations for the
differences between Venus and Earth have been proposed, the currently
available data are insufficient to discriminate among the different solutions.
Here we investigate the possible range of Venus structure models. We assume
that core segregation happened as a single-stage event. The mantle composition
is inferred from the core composition using a prescription for metal-silicate
partitioning. We consider three different cases for the composition of Venus
defined via the bulk Si and Mg content, and the core’s S content. Permissible
ranges for the core size, mantle and core composition as well as the
normalized moment of inertia (MoI) are presented for these compositions. A
solid inner core could exist for all compositions. We estimate that Venus’ MoI
is 0.317-0.351 and its core size 2930-4350 km for all assumed compositions.
Higher MoI values correspond to more oxidizing conditions during core
segregation. A determination of the abundance of FeO in Venus’ mantle by
future missions could further constrain its composition and internal
structure. This can reveal important information on Venus’ formation and
evolution, and possibly, the reasons for the differences between Venus and our
home planet.

Solar system terrestrial planets (797) — Planetary interior (1248) — Planetary
structure (1256)

##  1 Introduction

Venus is visible in the night sky by the naked eye and belongs to one of the
longest studied objects by astronomers. Due to its similarity to Earth in
size, mass, density, and orbital distance it is often referred to as Earth’s
twin. However, its surface conditions are strongly dissimilar to those of
Earth. Its atmosphere is about one hundred times denser than Earth’s
atmosphere and consists predominantly of $\rm CO_{2}$ and CO while being
almost completely devoid of $\rm H_{2}O$. These discrepancies in atmospheric
composition hint to significantly different internal and atmospheric evolution
histories of the two planets (e.g. Gaillard & Scaillet [2014]). Furthermore,
unlike Earth, Venus currently lacks a significant magnetic dynamo in the core
and is tectonically inactive (e.g. Davaille et al. [2017], Dumoulin et al.
[2017], Stevenson [1983]). In addition, Venus has no moon and performs
retrograde rotation (e.g. Raymond et al. [2013], Gillmann et al. [2016]).
Moreover, its density seemingly falls out of line with the other terrestrial
planets. Although Venus is closer to the Sun, its density is $\sim$2% lower
than predicted if the same bulk composition as Earth is assumed (Lewis [1972],
Ringwood & Anderson [1977], Anderson [1980], Goettel et al. [1981], Kaula
[1990], Kaula [1994], Basilevsky & Head [2003], Aitta [2012], Dumoulin et al.
[2017]).

Physical segregation and equilibrium condensation in the solar nebula could
lead to a decrease of the intrinsic density of planetary bodies with
heliocentric distance (Kovach & Anderson [1965], Lewis [1972], Ringwood &
Anderson [1977]). Such a strict trend, however, is not observed in the solar
system. Various mechanisms as possible explanation for the density deficit of
Venus have been presented. In particular, it was proposed that Venus’ density
deficit could be attributed to a lower S content relative to Earth (Lewis
[1972]). This idea was challenged since the composition of Venus’ atmosphere
could hint to an interior that is enriched in S (Ringwood & Anderson [1977]).
It was further pointed out by these authors that Venus’ S content as estimated
by Lewis [1972] is insufficient to explain its density. The density deficit of
Venus was attributed to differences in the oxidation sate of the mantle. This
conclusion was based on the assumption that Venus’ lower mass lead to overall
lower accretion temperatures than for the Earth. It was concluded that the
[FeO]/[FeO + MgO] ratio of Venus’ mantle is about a factor of two higher than
that of Earth’s mantle (Ringwood & Anderson [1977]). Consequently, Venus would
possess a smaller core than Earth.

Morgan & Anders [1980] predicted the composition of the planets in the solar
system using a more general framework. It was argued that elements with
similar cosmochemical properties exhibit similar fractionation behaviour in
the solar nebula. These elements can be divided into five groups, dominantly
based on their respective condensation temperature. The bulk composition of a
planet can then be determined by knowing the abundances of one element in each
of these groups. It was found that Venus could contain less Fe and S than the
Earth. This would results in a density that is $\sim$1.5% lower than the
density of the Earth.

Goettel et al. [1981] suggested that the thermal state of the interior played
a key role in constraining the planetary composition. Venus’ internal
temperature is expected to be higher than Earth’s (Ringwood & Anderson [1977],
Goettel et al. [1981], Stevenson [1983]). This is in agreement with more
recent estimates based on the composition of basaltic surface rocks on Venus
obtained from the Venera 13 and 14 missions in 1981-1982 (Lee et al. [2009]).
Higher temperatures and lower pressures inside Venus could be sufficient to
have prevented core crystallization which might explain the absence of a
magnetic dynamo (Stevenson [1983], Stevenson et al. [1983], Stevenson [2003],
Aitta [2012], Jacobson et al. [2017]). Goettel et al. [1981] proposed that in
the absence of subduction the internal temperature might be high enough to
explain Venus’ density. However, there is experimental and observational
evidence that some form of subduction may occur on Venus (e.g. Davaille et al.
[2017]).

None of these models can be excluded from currently available observational
constraints. It was pointed out early on that definite resolution of the
ambiguity would require seismic data of Venus’ interior (Goettel et al.
[1981]). This conclusion was based on the notion that non-hydrostatic effects
in the slowly rotating Venus render reliable measurements of the normalized
moment of inertia factor $C/MR^{2}$ (hereafter MoI) very difficult (Kaula
[1979]). However, recent progresses in radar observations have opened new
possibilities. Margot et al. [2021] used radar data, obtained by Earth-based
observations in 2006-2020, and estimated Venus’ MoI to be $0.337\pm 0.024$.
This is an important step towards understanding the differences between Earth
and Venus. However, the current uncertainty of about 7% on the MoI is still
too large to place firm enough constraints on interior models. Future missions
to Venus could drastically improve upon this estimate. EnVision, ESA’s
candidate mission for the cosmic vision program, is expected to constrain the
MoI within $\sim$1.4% uncertainty (Rosenblatt et al. [2021]).

In this study we investigate possible interior models for Venus. In
particular, we focus on differences in the chemical composition and the
thermal state of the mantle and the core. A single-stage core segregation
model is employed to self-consistently compute the mantle composition from the
core composition using recent metal-silicate partitioning data for O and Si in
multi-component fluids. This approach constrains the light element content of
the core, the oxidation state of the mantle, the size of the core, and the
presence of a solid inner core.

##  2 Model

###  2.1 The structure equations

The planets internal structure is inferred by integrating the one-dimensional
structure equations for spherical objects in hydrostatic equilibrium:

$\frac{dP}{dr}=-\frac{Gm\rho}{r^{2}},$ (1)

$\frac{dm}{dr}=4\pi\rho r^{2},$ (2)

$\frac{dT}{dr}=\frac{dP}{dr}\left(\frac{dT}{dP}\right)_{ad},$ (3)

where $P(r)$, $T(r)$, $\rho(r)$, and $m(r)$ are the pressure, temperature,
density, and enclosed mass at radial distance $r$ from the center. $G$ is the
gravitational constant and $(dT/dP)_{ad}$ the adiabatic gradient. To close the
system of differential equations eq. 1-3 an equation of state (EoS) of the
form $\rho(r)=\rho(T(r),P(r))$ for all the constituent materials is required.
These EoS are described in Appendix C. A 4th order Runge-Kutta scheme was then
employed from the center outward to obtain the pressure, temperature and
density profiles within a planet. The (non-normalized) moment of inertia $C$
is computed according to:

$\frac{dC}{dr}=\frac{8}{3}\pi\rho(r)r^{4}.$ (4)

###  2.2 Bulk composition and boundary conditions

Figure 1: Sketches of Venus’ interior structure. Left: The planetary models
are divided into four distinct layers: Upper mantle, lower mantle, liquid
outer core, and solid inner core. Right: Composition of each layer and layer
transitions. The parameters used to define each layer transition are indicated
in the right and are further explained in the main text.

The planetary composition is similar to that presented in Shah et al. [2021].
The interior is divided into four distinct layers (see Fig. 1): An inner solid
and an outer liquid iron core, a lower mantle and an upper mantle. The lower
mantle consists of the minerals $\rm(Mg,Fe)SiO_{3}$ (perovskite) and (Mg,Fe)O
(wüstite) while the minerals considered in the upper mantle are
$\rm(Mg,Fe)_{2}SiO_{4}$ (olivine) and $\rm(Mg,Fe)_{2}Si_{2}O_{6}$ (pyroxene)
with variable Fe content. This allows a variable [Si]/[Mg] ratio in the
mantle. In the outer core S, O and Si are considered as possible impurities.
The Earths core is thought to contain other elements such as Ni, C, and H
(e.g. Hirose et al. [2021]). These elements were not considered here. This is
justified as it was shown that accounting for Si and O is sufficient to
reproduce the Earth’s mantle composition from the core segregation model (see
section 2.3). Furthermore, if applied to the Earth our model is capable of
reproducing the content of impurities in the core other than Si, S and O
reported in Hirose et al. [2021] if some of the S is replaced by these
elements based on mass balance (see Appendix D for details). The transition
between the upper and lower mantle in the mantle transition zone (MTZ) is
determined by the phase transition from olivine to perovskite and wüstite
taken at $P_{\rm diss}^{\rm olivine}=$ 25 GPa. The value for the
Grüneisenparameter in the core is set to $\gamma=1.36$ (Sotin et al. [2007]).
Details on the EoS of the different materials considered and the numerical
procedures employed to solve for the interior structure are given in Appendix
C and Shah et al. [2021]. The model accounts for the separation of the core
into an inner solid part and an outer liquid part using the melting curve of
the Fe-Si-O-S chemical system (see Section 2.3). The core mantle boundary
(CMB) is defined via the core mass. The mole fractions of $\rm SiO_{2}$ and
FeO in the mantle were obtained from chemical equilibrium between mantle and
core using experimental data for metal-silicate partitioning of O and Si (see
Section 2.4).

The boundary conditions are defined via the surface pressure $P_{\rm S}$, the
temperature at the thermal boundary layer (TBL) at the top of the mantle
$T_{\rm TBL}$, the Mg-number $\rm Mg\\#\equiv$ [Mg]/[Fe + Mg] and the total
planetary mass $M$. Eq. 1-3 for a given set of boundary conditions are solved
is the same way as in Shah et al. [2021]. The main degrees of freedom in the
present model are: the core segregation pressure $P_{\rm CS}$ (see section
2.4), the temperature at the thermal boundary layer $T_{\rm TBL}$, the
temperature discontinuity across the mantle transition zone $\Delta T_{\rm
MTZ}$, the temperature discontinuity across the core-mantle-boundary (CMB)
$\Delta T_{\rm CMB}$, and the amount of FeO, FeS and FeSi in the core,
quantified by the mole fractions $X_{\rm FeO}^{\rm Core}$, $X_{\rm FeS}^{\rm
Core}$, $X_{\rm FeSi}^{\rm Core}$.

The corresponding value ranges that were considered in this study are
summarized in Table 3 (see also Table 4 for an overview of all relevant
parameters). For Earth, the core segregation pressure was inferred to be 30-70
GPa (Fischer et al. [2015], Siebert et al. [2011], Wade & Wood [2005],Wood et
al. [2006], Li & Agee [1996]). Given the similar masses of Earth and Venus,
and in order to allow for a direct comparison between the two planets, we use
the same range for Venus. The sensitivity of the results on the chosen range
of the core segregation pressure is presented in Appendix B. The pressure and
temperature conditions of basaltic magma generation on terrestrial planets can
be constrained from the composition of surface basalts which allows inferring
the temperature at the TBL (Lee et al. [2009]). These authors have used
basalts found on Earth and data obtained from the Venera landers to estimate a
range of $\approx 1400-1800$ K for the Earth and $\approx 1600-1800$ K for
Venus. Due to the sparse availability of compositional data for Venus the
range for the TBL temperature of Venus could be larger. As a result, we
consider a range of $T_{\rm TBL}$ between 1400 and 1800 $K$ . The temperature
change across the mantle transition zone of Earth, $\Delta T_{\rm MTZ}$, was
estimated to be $\sim$ 300 K (e.g. Sotin et al. [2007] and references
therein). This result corresponds to the assumption that the MTZ arises due to
a transition between two different convective regimes in the mantle. However,
strong evidence points towards whole-mantle convection within the Earth (e.g.
Phipps Morgan & Shearer [1993], Arnould et al. [2018], Loper [1985]).
Therefore we do not consider a temperature discontinuity across the MTZ in
this study. The temperature drop across the CMB, $\Delta T_{\rm CMB}$, for
Earth was estimated to be 500–1800 K (Lay et al. [2008]). Another study
proposed a simple scaling law to compute the temperature drop as $\Delta
T_{\rm CMB}\sim 1400\left(M/M_{\oplus}\right)$ (Stixrude [2014]). In order to
introduce the maximum variability into our model, we consider the wide range
of 500–1800 K. No models with $X_{\rm FeO}^{\rm Core}>0.2$ that matched the
boundary conditions for Venus were found. Therefore, for the results presented
below, a range for $X_{\rm FeO}^{\rm Core}$ between 0 and 0.2 was probed. The
Si content of the metal in the silicate-metal partitioning experiments of
Fischer et al. [2015] was $\leq$9 wt%. This corresponds to a mole fraction of
FeSi in the core of $\sim$0.25. Here we have set a larger range of 0-0.3 for
the maximum FeSi content in the core. The range for the S content in the core
is $X_{\rm FeS}^{\rm Core}=0-0.5$ and was constrained from the solar ratio
[S]/[Fe] assuming most of the S and Fe to be present in the core (Lodders
[2019]).

Due to the very low abundance or even absence of water on Venus it is expected
to have only developed a basaltic crust (density $\sim$2.9-3.1 $\rm g\
cm^{-3}$) while the Earth with its higher water content developed also a
granitic continental crust (density of $\sim$ 2.7-2.9 $\rm g\ cm^{-3}$)
(Rudnick [2018], Stolper & Walker [1980], Smrekar et al. [2018], Moore
[2001]). Therefore, the density contrast between the mantle and the crust is
likely smaller on Venus than on Earth. As a result, the crust is not
explicitly modelled in this study but implicitly incorporated into the upper
mantle. Whether granitic regions in the crust are present on Venus (indicating
the presence of water in the past) is an open question and could be resolved
by future space missions.

The initial values for these parameters are chosen by uniform sampling within
the selected ranges for each interior model. Depending on the specific
combination of $P_{\rm CS}$, $X_{\rm FeO}^{\rm Core}$, $X_{\rm FeS}^{\rm
Core}$, and $X_{\rm FeSi}^{\rm Core}$ it is possible to obtain unrealistic
values for the FeO and $\rm SiO_{2}$ content in the mantle. Here the FeO
content in the mantle is limited such that $\rm[FeO]/[FeO+MgO]<0.5$. The Si-
number of the mantle is defined as $\rm
Si\\#_{Mantle}\equiv[SiO_{2}]/[MgO+SiO_{2}]$. It is computed from the $\rm
SiO_{2}$ content according to ${\rm Si\\#_{Mantle}}=X_{\rm SiO_{2}}^{\rm
Mantle}/(1-X_{\rm FeO}^{\rm Mantle})$. For the silicate composition in the
upper and lower mantle considered in this study ${\rm Si\\#}_{\rm Manlte}$
must be between $1/(3-2{\rm Fe}\\#_{\rm Mantle})$ and $1/(2-{\rm Fe}\\#_{\rm
Mantle})$ where $\rm Fe\\#_{Mantle}\equiv[FeO]/[FeO+MgO]$. In each numerical
run the sampling of the initial parameters is re-iterated until these
conditions are met.

The surface pressure is fixed to 1 bar for Earth and 100 bar for Venus.
Variations in the surface pressure of a few tens of bars have no impact on the
results presented in this study. The bulk composition was defined as
$0.47\leq\rm Mg\\#\leq 0.53$ and $0.47\leq\rm Si\\#\leq 0.56$ (Sotin et al.
[2007]). Three different cases for the core composition defined via the S
content were considered. A nominal composition was defined as $X^{\rm
Core}_{\rm FeS}$ = 0.08-0.15 (see Appendix D for details). In order to assess
the effect of different S contents on the possible models for Venus S-rich
models are defined as $X^{\rm Core}_{\rm FeS}$ = 0.2-0.5 and S-free models as
$X^{\rm Core}_{\rm FeS}$ = 0. Convergence of the structure integration is
secured via the total mass $M$, $\rm Mg\\#$, and $T_{\rm TBL}$: only models
for which these parameters match within 0.1%, 0.1% and 1% are considered.

###  2.3 Single-stage core segregation and inner core boundary

We adopt the simplifying assumption that chemical equilibrium between the core
and the mantle during core segregation is a single-stage process (e.g. Fischer
et al. [2015], Suer et al. [2017], Rubie et al. [2011], Schaefer et al.
[2017]). This allows omitting explicit modelling of the temporal evolution of
the cooling magma ocean. Such a single-stage core formation model was found to
yield the correct concentrations of $\rm SiO_{2}$ and FeO of the Earth’s
mantle (Rubie et al. [2011], Fischer et al. [2015]). However, in these studies
some trace elements in the mantle such as Ni, Co, W, and Cr could not be
reproduced. Our study focuses on the abundances of the major elements Si, Mg,
O and Fe using the partitioning data from Fischer et al. [2015] and does not
consider the trace elements mentioned above in the mantle. We apply the model
to Earth as a reference and confirm that the $\rm SiO_{2}$ and FeO content of
the mantle can be reproduced with this model (see Appendix D).

The temperature at which equilibrium is reached in the model is given by the
pyrolite-liquidus. It is described in its general form using the Simon-Glatzel
equation for the melting temperature of solids (Simon & Glatzel [1929]):

$T_{\rm melt}(P)=T_{0}\left(\frac{P-P_{0}}{a}+1\right)^{1/b}.$ (5)

Table 1: Parameters to compute the melting temperature of pyrolite and pure iron from eq. 5. Material | $a$ | $b$ | $T_{0}$ | $P_{0}$
---|---|---|---|---
| [GPa] |  | [K] | [bar]
Iron(a) | 23 | 2.26 | 1811 | 1
Pyrolite(b) | 29 | 1.9 | 1940 | 0
References. (a)Li et al. [2020], (b)Andrault et al. [2011]. |  |  |  |

In the single-stage core segregation model it is assumed that the chemical
equilibration of sinking iron droplets building up the core occurred at an
average pressure of $P_{\rm CS}$ (see Fig. 2 for illustration). For a given
value of $P_{\rm CS}$ the core segregation temperature $T_{\rm CS}$ is
computed via eq. 5 using the parameters from Table 1. The inner core boundary
(ICB) is given by the melting curve of the Fe-Si-S-O system described by eq. 5
and the parameters for pure-iron given in Table 1. The effect of impurities on
the melting curve in the system are taken from Andrault et al. [2016]. In
particular, the melting temperature of the system is reduced by 30 K, 50 K,
and 100 K for every weight percent of Si, O, and S, respectively, dissolved in
the metal melt.

Figure 2: A schematic illustration of the core segregation process. The iron
core is formed from sinking iron droplets with dissolved Si, O, and S. The Si,
O, and S content of the droplets is dictated by the solubility of these
elements in liquid iron at the corresponding temperature and pressure
conditions. As the mantle solidifies, the lowermost pressure level at which
equilibration takes place moves outward. This pressure corresponds to the
liquidus of the silicates. In the single stage core segregation model, the
equilibration is assumed to have taken place at an average pressure of $P_{\rm
CS}$ which is treated as a free parameter.

Figure 3: FeO and $\rm SiO_{2}$ content in the mantle as function of the core
segregation pressure $P_{\rm CS}$ computed from the metal-silicate
partitioning model given by eqs. 6 and 7 for different core compositions. The
colors denote different O contents, the line styles different Si contents, and
the three rows correspond to 0, 5 and 10 wt% of S in the core, respectively.

Figure 4: Internal temperature (left), pressure (center), and density (right)
profiles of the S-free Venus models ($X_{\rm FeS}^{\rm Core}=0$) as a function
of the radial distance from the center $R$ in units of earth radii
$R_{\oplus}$. The top row shows all possible profiles for the allowed range of
the MoI. The middle and bottom row show only the profiles within $\pm 1\%$ of
the lowest and highest possible value of MoI. The MoI value is indicated by
different colors. The grey shaded regions indicate the possible ranges for the
size of the core along the x-axes and the corresponding temperature, pressure
and density ranges along the y-axes at the CMB. The dashed lines denote the
density profile for the Earth from the Preliminary Earth Reference Model
(PREM).

Figure 5: Same as Fig. 4 but for the nominal Venus models ($X_{\rm FeS}^{\rm
Core}=0.08-0.15$).

Figure 6: Same as Fig. 5 but for the S-rich Venus models ($X_{\rm FeS}^{\rm
Core}=0.2-0.5$).

Figure 7: Overview of the permissible value ranges of the most relevant
structure and composition parameters for the S-free (left), nominal (center),
and S-rich (right) models. The pressures and temperatures given for each layer
transition correspond to the values in the lower of the two layers at the
boundary. The top row shows all possibilities within the permissible MoI
range. The middle and bottom row only show the models within $\pm 1\%$ of the
lowest and highest possible MoI value. The pressure and temperature values at
each layer transition correspond to the values in the layer below the
boundary.

Figure 8: Core radius as a function of the redox state of the mantle (left)
and the FeO content in the mantle (right) for the different compositions. The
value for the MoI is indicat ed by different colors. The permissible ranges of
$\log{f_{\rm O_{2}}}$ relative to the IW buffer and $X_{\rm FeO}^{\rm Mantle}$
for the Earth estimated from our model are shown as grey shaded regions. Even
without further constraining the MoI, the core size of Venus could be
constrained from the mantle composition alone.

Figure 9: A schematic comparison between Earth and the different interior
models of Venus. The S-free models correspond to $X_{\rm FeS}^{\rm Core}=0$
while the nominal and S-rich compositions are defined as $X_{\rm FeS}^{\rm
Core}=0.08-0.15$ and $X_{\rm FeS}^{\rm Core}=0.2-0.5$. The distances from the
center for each layer and the core and mantle compositions are indicated for
each case. The brown, red, and black dashed segments indicate the lower bound
for the mantle transition zone, the core-mantle boundary and the inner-to-
outer core boundary. The relative sizes of the different layers are on scale.
The resulting permissible ranges for the MoI and some bulk elemental ratios
are given next to the each sketch of the interior. Figure 10: Estimated ranges
of Venus’ MoI from Zhang & Zhang [1995], Aitta [2012], and Dumoulin et al.
[2017] and the three different bulk compositions we consider. Measured values
for the MoI for Earth, Venus, Mercury, and Mars are shown for comparison. The
shaded region denotes the current uncertainty of Venus’ MoI from observations
(Margot et al. [2021]).

###  2.4 Mantle and core composition

Table 2: Parameters for eq. 8 and 9 from Fischer et al. [2015]. Element $i$ | $e_{\rm O}^{i}$ | $e_{\rm Si}^{i}$ | $a_{i}$ |  $b_{i}$ [K] |  $c_{i}$ [K/GPa]
---|---|---|---|---|---
Si | -0.06 | … | 0.6 | -11,700 | …
O | -0.12 | -0.11 | 0.1 | -2200 | 5

The results for metal-silicate partitioning of O and Si from Fischer et al.
[2015] are used to compute the mantle composition from the core composition.
Chemical equilibrium between the core and the mantle is assumed to have been
achieved in a single-stage process. The mantle composition was defined via the
mole fractions $X_{\rm FeO}^{\rm Mantle}$ and $X_{\rm SiO_{2}}^{\rm Mantle}$:

$X_{\rm SiO_{2}}^{\rm Mantle}=\frac{X_{\rm Si}^{\rm Core}}{K_{\rm D}^{\rm
Si}}\frac{X_{\rm FeO}^{\rm Mantle}}{X_{\rm Fe}^{\rm Core}}$ (6)

$X_{\rm FeO}^{\rm Mantle}=\frac{X_{\rm Fe}^{\rm Core}X_{\rm O}^{\rm
Core}}{K_{\rm D}^{\rm O}},$ (7)

where $K_{\rm D}^{\rm Si}$ and $K_{\rm D}^{\rm O}$ are the partition
coefficients of O and Si. In a mixture of $N$ components the partition
coefficient of component $i$ is given by (Fischer et al. [2015], Ma [2001]):

$\begin{split}\log K_{\rm
D}^{i}(T,P)&=a_{i}+\frac{b_{i}}{T}+\frac{c_{i}P}{T}+\frac{\epsilon_{i}^{i}\ln{\left(1-X_{i}\right)}}{2.303}\\\
&+\frac{1}{2.303}\sum_{k=2,k\neq
i}^{N}\epsilon_{k}^{i}X_{k}\left(1+\frac{\ln{\left(1-X_{k}\right)}}{X_{k}}-\frac{1}{1-X_{i}}\right)\\\
&-\frac{1}{2.303}\sum_{k=2,k\neq
i}^{N}\epsilon_{k}^{i}X_{k}^{2}X_{i}\left(\frac{1}{1-X_{i}}+\frac{1}{1-X_{k}}+\frac{X_{i}}{2\left(1-X_{i}\right)^{2}}-1\right).\end{split}$
(8)

The parameter $\epsilon_{k}^{i}$, is given by:

$\epsilon_{k}^{i}(T)=1+\frac{e_{k}^{i}m_{i}}{0.242}\frac{T_{\rm
ref}}{T}-\frac{m_{i}}{55.85}$ (9)

Table 3: Probed ranges for the input parameters for the Venus models. Mg# | $P_{\rm CS}$ | $T_{\rm TBL}$ | $\Delta T_{\rm CMB}$ | $X_{\rm FeO}^{\rm Core}$ | $X_{\rm FeS}^{\rm Core}$ | $X_{\rm FeSi}^{\rm Core}$
---|---|---|---|---|---|---
| [GPa] | [K] | [K] |  |  |
0.47-0.53 | 30–70 | 1400-1800 | 500–1800 | 0-0.2 | 0-0.5 | 0-0.3

where $m_{i}$ is the molar mass of species $i$ in grams per mole, $T_{\rm
ref}=1873$ K, and $e_{k}^{i}$ is an interaction parameter between species $i$
and $k$. The corresponding parameters are given in Table 2. With this the mole
fractions of FeO and $\rm SiO_{2}$ in the mantle are uniquely constrained from
the core composition and the conditions at the bottom of the magma ocean
during core segregation. The presence of S in the core could impact the
partitioning behaviour of Si and O (Fischer et al. [2015]). However, to our
knowledge, no self-consistent prescription for these effects are currently
available. In order to study the influence of S in the Fe-Si-O-S system a
series of challenging experiments would be required (pers. comm. D. Frost).
The importance of such experiments was already noted by Fischer et al. [2015].
Incorporating the effect of S into the partitioning model could reduce the
redundancy of possible interior models for Venus and should be investigated in
the future.

Fig. 3 shows the mantle composition resulting from eq. 6 and 7. The different
colors correspond to different O contents in the core while the different
linestyles denote different values for the Si content. The three rows
correspond to 0, 5 and 10 wt% S. The oxygen fugacity of the manlte is
estimated as a function of temperature and pressure using a mineral redox
buffer. Here the iron-wüstite buffer (IW) was used. It is based on the
equlibrium reaction $2{\rm Fe}+{\rm O}_{2}=2{\rm Fe}{\rm O}$. The oxygen
fugacity is typically given in log units relative to a chosen redox buffer.
Negative values correspond to oxygen fugacities that are lower than the
equilibrium of the buffer at a given temperature and pressure. The oxygen
fugacity relative to the iron-wüstite buffer is calculated as (Rose-weston et
al. [2009]):

$\log f_{\rm O_{2}}=2\log\left(\frac{a_{\rm FeO}^{\rm sil}}{a_{\rm Fe}^{\rm
met}}\right),$ (10)

where:

$a_{\rm Fe}^{\rm met}=\gamma_{\rm Fe}^{\rm met}X_{\rm Fe}^{\rm met}$ (11)

$a_{\rm FeO}^{\rm sil}=\gamma_{\rm FeO}^{\rm sil}X_{\rm FeO}^{\rm sil}.$ (12)

$a_{\rm FeO}^{\rm sil}$ and $a_{\rm Fe}^{\rm met}$ are the activities of FeO
and Fe in the silicate melt (sil) and the metal (met), respectively. The
$\gamma$’s are the activity coefficients. For non-ideal mixtures they are
obtained from the asymmetric Margules equations using the fit parameters
presented in Frost et al. [2010]. To account for ideal mixing the Margules
parameters are smoothed above 50 GPa to drive the activity coefficients to
unity at higher pressures (Schaefer et al. [2017]). Including the dependence
on temperature, pressure and composition of the activity coefficients has no
significant impact on the resulting estimates for the oxygen fugacity (e.g.,
Fischer et al. [2015], Siebert et al. [2012]).

Table 4: Summary of the parameters considered in this study to characterize the interior structure and composition of Venus. Parameter | Unit | Description
---|---|---
$\rm Mg\\#$ |  | Total magnesium number [Mg]/[Fe + Mg]
$\rm Si\\#$ |  | Total silicon number [Si]/[Si + Mg]
$P_{\rm CS}$ | GPa | Core segregation pressure
$T_{\rm TBL}$ | K | Temperature at the TBL
$P_{\rm S}$ | GPa | Surface pressure
$\Delta T_{\rm MTZ}$ | K | Temperature difference across the MTZ (fixed at 0 K)
$\Delta T_{\rm CMB}$ | K | Temperature difference across the CMB
$X_{\rm FeS}^{\rm Core}$ |  | Mole fraction of FeS in core [FeS]/[Fe + FeO + FeS + FeSi]
$X_{\rm FeSi}^{\rm Core}$ |  | Mole fraction of FeSi in core [FeSi]/[Fe + FeO + FeS + FeSi]
$X_{\rm FeO}^{\rm Core}$ |  | Mole fraction of FeO in core [FeO]/[Fe + FeO + FeS + FeSi]
$X_{\rm Fe}^{\rm Core}$ |  | Mole fraction of Fe in core [Fe]/[Fe + S + Si + O]
$X_{\rm S}^{\rm Core}$ |  | Mole fraction of S in core [S]/[Fe + S + Si + O]
$w_{\rm S}$ |  | Weight fraction of S in core
$w_{\rm Si}$ |  | Weight fraction of Si in core
$w_{\rm O}$ |  | Weight fraction of O in core
$X_{\rm FeO}^{\rm Mantle}$ |  | Mole fraction of FeO in mantle [FeO]/[FeO + MgO + $\rm SiO_{2}$]
$X_{\rm SiO_{2}}^{\rm Mantle}$ |  | Mole fraction of $\rm SiO_{2}$ in mantle [$\rm SiO_{2}$]/[FeO + MgO + $\rm SiO_{2}$]
$T_{\rm CS}$ | K | Core segregation temperature
$R_{\rm Core}$ | km | Core radius
$R_{\rm ICB}$ | km | Inner-to-outer core boundary
$C/MR^{2}$ |  | Normalized moment of inertia (referred to as MoI)
$P_{\rm C}$ | GPa | Central pressure
$T_{\rm C}$ | K | Central temperature
$f_{\rm O_{2}}$ |  | Oxygen fugacity of the mantle
Table 5: Summary of the different cases for the composition referred to in this study. Details on the Earth reference models can be found in Appendix D. The bulk composition is defined via $\rm Mg\\#$ and $\rm Si\\#$. Composition | Description
---|---
Nominal | Bulk composition and S content in the core from the Earth reference models.
S-free | Bulk composition from the Earth reference models but no S in the core.
S-rich | Bulk composition from the Earth reference models with 20-50 mol% FeS in the core.

##  3 Results and discussion

###  3.1 Venus models

Venus’ MoI is estimated to be 0.337 $\pm$ 0.024 (Margot et al. [2021]). Our
analysis considers all models for which Venus’ radius matches within
$\delta_{R}=$ 1%. Three different cases for the core composition (S-free,
nominal, S-rich) defined via the S content were considered (see section 2.2
for details). For these compositions three different cases are considered for
the MoI: Models within the entire allowed range of 0.337 $\pm$ 0.024 (All
MoI), models within 1% of the lowest possible value (Low MoI), and models
within 1% of the highest possible value (High MoI). We find that not all
values within 0.337 $\pm$ 0.024 are possible. As a result, the highest and
lowest MoI values are different for the various assumed compositions. The
effects of turning off the core segregation model from section 2.3 on the main
results are discussed in Appendix A.

####  3.1.1 Internal profiles

Figures 4 to 6 show the possible temperature, pressure, and density profiles
of the different ranges of MoI values and for the three compositions as
defined above. The first row depicts all possible profiles for which the MoI
lies in the allowed range. The second and third rows only show the profiles
that lie within 1% of the lowest and largest possible value for the MoI. The
MoI is indicated in colors and the density profile for the Earth from the
Preliminary Earth Reference Model (PREM) is shown for comparison. Furthermore,
the resulting ranges for the core size and the conditions at the CMB are
indicated by the shaded regions.

The most prevalent trend that emerges for the MoI in all cases are an overall
increase with the mantle density. This behaviour is expected as higher mantle
densities lead to a smaller density contrast between the mantle and the core.
Increasing the size of the core can lead to both an increase and decrease of
the MoI. The density of the mantle minerals is mainly influenced by the Fe
content of the silicates. It is quantified by the amount of FeO, and hence
related to the oxidation state of the mantle. The core size strongly depends
on its composition. For a given Mg#, larger amounts of lighter elements in the
core lead to larger cores. Therefore, the MoI indirectly correlates with the
core composition, and hence, the pressure gradient within the core. This
typically leads to lower (higher) core pressures for high (low) MoI values.

The presence of a solid inner core and even a fully crystallized core cannot
be excluded for any of the considered compositions. The highest possible
pressures in the core are achieved for low MoI values while no obvious trends
are observed for the central temperature $T_{\rm C}$ in most cases (Figures
4-6). The ICB is dictated by the melting curve of Fe-alloys which shifts to
lower temperatures with increasing S content (see Section 2.3). A completely
molten core without intrinsic heat sources cannot sustain thermal convection
over long periods of time (Stevenson et al. [1983]). In this case, the
temperature profile within Venus’ core could have become subadiabatic
$\sim$1.5 Gyr ago.

####  3.1.2 Internal structure and chemical composition

Figures 7 provide a schematic overview of the main composition and structure
parameters. The left, middle, and right column correspond to the S-free,
nominal, and S-rich models. The top, middle, and bottom rows correspond to all
values, the lowest values, and the highest values of the MoI. The pressures
and temperatures given for each layer transition correspond to the values in
the lower of the two layers at the boundary.

For a S-free composition, the core contains up to $\sim$0.73 wt% O and 13 wt%
Si. The amount of FeO in the mantle is poorly constrained between 0.05 and 7
mol%. This leads to a large range of possible redox states of the mantle with
$-\log{f_{\rm O_{2}}}=1.5-5.5$. If Venus is depleted in S, the radius of its
core is predicted to be 2930-3610 km. The size of a solid inner core in this
case would be 0-3330 km. The absence of S in the core leads to higher melting
temperatures of the Fe-alloys. This allows for larger inner cores in
comparison to the other compositions (see below). Finally, in the S-free case,
the central pressure could be as high as 351 GPa, which is similar to the
lower bound inferred for reference models of the Earth (see Appendix D).

If the true value of Venus’ MoI is low, the possible ranges for the core size,
the FeO content in the mantle, and the core composition would be much
narrower. In particular, the core size would have to be between 3120 and 3610
km and contains no more than 0.39 wt% O. Furthermore, the mantle could not
contain more than 3.6 mol% FeO. As expected, the FeO content in the mantle for
the highest possible MoI can be larger and between 0.06 and 7 mol%. In this
case the density contrast between the core and the mantle is minimal. The core
size is then predicted to be 2930-3510 km. An inner solid core of up to 3180
km could exist.

The middle column of Fig. 7 shows the results for the nominal composition. For
these models a core composition with 0-1.3 wt% O, 0.12–13 wt% Si, and 3.9-7.9
wt% S was obtained. This leads to a FeO content in the mantle of 0.03-9.8
mol%. The size of the core in this case is between 2960 and 3890 km. If an
inner core is present, it could not be larger than 3300 km. The possible core-
and mantle compositions strongly depend on the actual value of the MoI.

The second and third panels in the middle column in Fig. 7 show the bracketing
cases of the MoI with 0.317-0.323 and 0.331-0.338 for the nominal composition.
For the lowest possible MoI the O content in the core is restricted to 0-0.42
wt%, while the Si and S contents remain similar to the case that considers the
entire range for the MoI. This results in 0.04-3.3 mol% FeO in the mantle and
hence a larger density contrast between the core and the mantle. The size of
the core is 3120-3740 km. In this case the mantle of Venus is likely more
reduced than the mantle of the Earth.

If the maximum MoI value for a nominal composition is assumed, the density
contrast between the mantle and the core is smaller. This leads to much higher
FeO contents in the mantle (4.3-9.8 mol%) and an O content of 0.52-1.3 wt% in
the core. Furthermore, larger amounts of S in the core (4.3–7.8 wt%) further
decrease the density contrast by decreasing the core density. In this case the
core size is 2980-3380 km and an inner core, if present, is limited to 2380
km.

In the third column in Fig. 7 the results for the S-rich models are shown. In
this case the core could contain 0-2.9 wt% O, 0.04-12 wt% Si, and 9.1-22 wt%
S. This results in similar ranges for the MgO and $\rm SiO_{2}$ content of the
mantle than for the S-free and nominal composition but a higher upper bound
for the FeO content. This is due to the fact the O content of the cores of the
S-rich models can be considerably larger than for the other compositions. It
is possible that the presence of S influences the distribution of O and Si
between the metal and the silicates (e.g. Fischer et al. [2015]). This could
affect the correlations between the amount of S in the core and the
composition of the mantle.

The core size of the S-rich models is 3010-4350 km. A core of Venus that is
smaller than $\sim$3600 km would be consistent with a core depleted in S. A
core size larger than about 3900 km could hint towards an enrichment in S with
respect to the Earth reference models. The possible range for the MoI inferred
for the S-rich models is 0.32-0.351. The upper bound for the MoI is
considerably higher than for the other compositions while the lower bound is
similar. This is expected as the main difference between these models is a
larger amount of S in the core which decreases the density contrast between
the mantle and the core.

The bracketing cases of the MoI for the S-rich models yield similar trends as
for the other compositions. Lower MoI values generally require a larger
density contrast between the core and the mantle. With this the FeO content in
the mantles of the S-rich models with a low MoI cannot be higher than 3.9 mol%
and the maximum O content in the core is 0.56 wt%. The possible range for the
core size in this case lies between 3090 and 3920 km and a fully molten core
is not possible. If the largest possible value for MoI is assumed, this range
becomes 3060-3810 km. In our sample 677 models match the boundary conditions
for Venus for an S-rich composition and the high MoI. Of these models only two
were found for which an inner core could exist. An O content in the core of
1-2.9 wt% and correspondingly a larger FeO content in the mantle, i.e. 5.5-15
mol%, is required. Furthermore, the Si content of the core becomes very small
for high MoI values.

####  3.1.3 Mantle composition as a proxy for the core size

In this section the relation between the mantle composition and core size is
investigated. Fig. 8 presents the core radius as a function of the oxygen
fugacity $\log f_{\rm O_{2}}$ (left) and the FeO content in the mantle $X_{\rm
FeO}^{\rm Mantle}$ (right) for the different compositions. The MoI value is
indicated by different colors. The grey shaded region denotes the allowed
values for the oxygen fugacity and FeO content of the Earth reference models
(see Appendix D for details).

Different amounts of S primarily lead to lower core densities and larger cores
at otherwise equal properties. Since potential effects of S on the metal-
silicate partitioning of O and Si are not included, the mantle composition is
not very sensitive to the S content. Larger amounts of S in the core lead to
larger upper bounds of the O content. It is possible that additional trends
for the different compositions would arise if partitioning effects are
considered.

The major trends evident in Fig. 8 are the negative correlation between the
FeO content in the mantle and the core size and a decrease of the MoI value
for smaller FeO contents. Both effects are readily explained by the fact that
higher amounts of FeO in the mantle increase the mantle density and reduce the
fraction of Fe in the core at a fixed bulk composition. This correlation
between the mantle composition and the core size holds despite the large
variability of the core composition. If the FeO content in Venus’ mantle is
similar to that of the Earth reference models and the nominal composition is
assumed, its core radius is expected to be $\sim$2900-3600 km. Including the
S-rich and S-free models this range becomes $\sim$2900-4000 km. This estimate
does not depend on any knowledge of the S content in the core or further
constraints on the MoI value. The mantle composition could, in principle, be
further constrained from remote surface surveys and surface samples.

####  3.1.4 Venus versus Earth

Fig. 9 compares the main parameters for the S-free, nominal, and S-rich
composition to the Earth reference models. The composition of the core and
mantle of Earth are taken from our model while the layer boundaries are taken
from the PREM data (rounded to three significant digits). The MoI value is the
measured value for the Earth. The relative sizes of the individual layers are
to-scale and the possible ranges for the Venus models are indicated by dashed
segments. The red segments denote lower bounds for the core size while the
dark grey segments correspond to the lower bounds of the inner core.

Here, we focus on possible structural and compositional differences between
Earth and Venus that could explain their measured properties. All the models
presented in this study match the mean density of Venus by construction and
hence incorporate the 2% density deficit with respect to Earth. None of the
inferred core or mantle compositions of the Earth can be excluded for Venus.
Therefore, Venus’ mean density could be explained with a very similar bulk and
core composition to that of Earth.

Given the sparse data available for Venus it is also possible that the
interior composition and element distribution significantly differ from those
of Earth. For example, Venus’ mantle could be more reduced than the mantle of
the Earth. This scenario could possibly be excluded if a high MoI value is
measured (see Fig. 7). In this case our model predicts that the redox state of
the mantle could be similar to that of Earth, and possibly more oxidized. For
the S-free models a high MoI would also be consistent with a more reduced
mantle. The O content in Earth’s core derived from our model is below 1 wt%
while for Venus up to 2.9 wt% is predicted. The size of Venus’ core cannot be
constrained well from the models. This is due to the large variability in the
possible core compositions and the element distribution between the core and
mantle.

If more stringent constraints on the abundances of the major oxides FeO, MgO
and $\rm SiO_{2}$ were imposed it could be possible to narrow down the
permissible range for the core size, even without more accurate MoI
measurements or seismic data. In particular, for the nominal models a similar
FeO content in the mantle than for the Earth would indicate that Venus’ core
could be similar in size to that of Earth (see Fig. 8). The Si content in the
cores of the nominal models could be much larger than is inferred for the
Earth. If Venus has a more oxidized mantle than the Earth, its core must be
considerably smaller for the nominal composition, but could be larger for
S-rich compositions. For the S-free models Venus’ core would be smaller than
Earth’s core within the entire range of the FeO content in the mantle of Earth
reference models.

###  3.2 Comparison to previous studies

The models for the lowest possible MoI value for a S-free or nominal
composition resemble the scenario from Lewis [1972]. In this scenario a low
oxidation state of the mantle and a core that was depleted in S relative to
the Earth was proposed to explain Venus’ density. The models with S and the
highest possible value of the MoI are akin to that of Ringwood & Anderson
[1977]. In this case the oxidation state of the mantle would be higher. It was
inferred that Venus would currently contain about 13.4 wt% FeO in the mantle
or [FeO]/[FeO + MgO] = 0.24 and $\sim$16 mol% S in the core (Ringwood &
Anderson [1977]). This would lead to a core mass fraction of $\sim$0.23 in
comparison to 0.32 for Earth. These predictions are within or close to the
inferred ranges from our models for the highest MoI of the S-rich models. In
particular, for these models we predict a core mass fraction of 0.22-0.38, a S
content in the core of 21-32 mol%, and [FeO]/[FeO + MgO] = 0.11-0.28. The
oxygen fugacity is $-\log{f_{\rm O_{2}}}=0.7-2.1$ which is more oxidized than
the Earth reference models which have $-\log{f_{\rm O_{2}}}=1.8-2.8$ (see
Appendix D).

Models with lower oxygen fugacities tend to result in a lower MoI due to the
larger mass concentration in the core. It seems therefore likely that the
different scenarios proposed by Lewis [1972] and Ringwood & Anderson [1977]
are quantitatively represented by the bracketing cases for the MoI found in
this study. The idea that reliable measurements of the MoI could allow to
distinguish between these two (and other) scenarios has been entertained early
on (e.g. Ringwood & Anderson [1977], Goettel et al. [1981]). If Venus’ MoI
value is measured to be at the high end of the possible values, this could be
strongly in favor of the scenario proposed by Ringwood & Anderson [1977].

Previous studies used the density profile of the Earth to construct a scaled
model for the density as a function of depth for Venus (e.g. Aitta [2012],
Steinberger et al. [2010]). Dumoulin et al. [2017] presented a series of
different interior models for Venus and computed the core radius and MoI for
the different cases. They employed four different composition models which
were based on assumptions regarding the composition of the solar nebula and
the accretion history. Furthermore, in this study only two end-member
temperature profiles (referred to by the authors as ”hot” and ”cold”) were
considered. These profiles were inferred from assumptions about the similarity
of Earth and Venus in combination with recent data on topography and gravity
anomalies (Steinberger et al. [2010], Armann & Tackley [2012]). Amongst their
models the largest variation of the moment of inertia was found to be no more
than about 5%, which is too low to be resolved within the current
observational uncertainty.

The central temperature and pressure of Venus were estimated to be 5200 K and
275 GPa (Aitta [2012]). Steinberger et al. [2010] obtained a central pressure
of about 285 GPa. A more recent estimate predicted a range of $271-299$ GPa
(Dumoulin et al. [2017]). The latter was obtained from previously published
temperature profiles (Steinberger et al. [2010], Armann & Tackley [2012]).
Furthermore, the core radius of Venus was estimated to be $\sim$3186 km by
Steinberger et al. [2010] and 3228 km by Aitta [2012]. Dumoulin et al. [2017]
obtained values between 2941 km and 3425 km. These estimates are well within
the possible ranges obtained here. Our lower limit for the core size differs
from that obtained by Dumoulin et al. [2017] only by 10-70 km for the three
compositions considered here. However, our model predicts a much larger upper
limit for the core size. This is because large amounts of lighter elements
were allowed to be present in the core and more variable mantle compositions
were considered here. For the nominal composition and an FeO content in the
mantle similar than in Earth’s mantle, the core size is estimated to be
$\sim$2900-3600 km. This range is only slightly larger than that inferred by
Dumoulin et al. [2017].

The MoI is estimated to be 0.317-0.33 for a S-free, 0.317-0.338 for a nominal
and 0.32-0.351 for a S-rich composition. These ranges are somewhat larger, but
compatible with previous estimates of 0.327-0.342 and 0.329-0.341 (Dumoulin et
al. [2017], Zhang & Zhang [1995]). Fig. 10 shows the predicted ranges for MoI
from various authors including the results obtained here for the different
assumed compositions. The shaded region depicts the currently allowed range
from observations (Margot et al. [2021]) while the black bars show the
inferred ranges from different models. The MoI of Earth, Mars, Mercury, and
the Moon are shown for reference. The models from Zhang & Zhang [1995], Aitta
[2012] and Dumoulin et al. [2017] assumed a composition similar to the Earth
and obtained the density profile in the mantle from scaling the density of the
Earth. This leads to a narrower range for the MoI. Although the model
approaches in these studies are different the inferred ranges are similar.
This illustrates the redundancy in Venus structure models and confirms that an
accurate MoI measurement alone, although highly desirable, would not strongly
constrain Venus’ internal structure.

##  4 Summary

This study presents interior structure models for Venus for different bulk
compositions. The mantle composition was obtained from chemical equilibrium
calculations using published metal-silicate partitioning data. Core-mantle
equilibration was assumed to have occurred as a single-stage process. We
further assume that the pressure range over which core segregation occurred in
Venus was similar to that of Earth. This model was employed to infer possible
interior models for Venus and assesses potential structural and compositional
differences between Venus and Earth.

Three different compositions, S-free, nominal, and S-rich, were considered.
While we can impose constraints on some of the composition and internal
structure parameters from available data, considerable redundancies in
possible structure models for Venus remain. A better understanding of the
influence of S and other elements on the metal-silicate partitioning of O and
Si could help to mitigate these redundancies. Furthermore, constraining the
composition and chemical state of Venus’ mantle could provide tighter
constraints on possible interior models and in particular on the core size.
Our key results can be summarized as follows:

  1. 1.

In the framework of our model, higher concentrations of S in the core
correspond to higher concentrations of O in the core while the range for the
Si concentration remains unchanged. This result may need modification if
partitioning of Si and O into the core is affected by the presence of S in the
metal.

  2. 2.

Venus could be much more oxidized than Earth. As a consequence the FeO content
in Venus’ mantle could be significantly higher than that of Earth. High values
of the MoI would be in favor of models with high oxygen fugacities. Low values
of the MoI could indicate a more reduced state.

  3. 3.

Currently available constraints cannot rule out the absence of a solid inner
core.

  4. 4.

Modelling the influence of S and other elements on the metal-silicate
partitioning of O and Si could considerably mitigate the redundancy of
possible composition and differentiation models for Venus. Experiments to
quantify these effects are highly desirable.

  5. 5.

The core size of the Venus models from all considered compositions is
estimated to be 2930-4350 km. If the S content in the core and the FeO content
of the mantle are similar to those inferred from the Earth reference models,
the permissible range for the core size is approximately 2900-3600 km.

  6. 6.

The pressure and temperature conditions at the center of Venus could be
similar to those inferred for Earth.

  7. 7.

The inferred ranges for the MoI depend on the assumed bulk compositions and
are found to be 0.317-0.33, 0.317-0.338, and 0.32-0.351 for a S-free, a
nominal, and a S-rich composition, respectively. These ranges are in good
agreement with previous studies.

Currently, there are two future space missions dedicated for the exploration
of Venus. NASA has recently announced the selection of DAVINCI (Deep
Atmosphere Venus Investigation of Noble gases, Chemistry, and Imaging) and
VERITAS (Venus Emissivity, Radio Science, InSAR, Topography, and Spectroscopy)
for further studies, with a launching date by the end of this decade. EnVision
is a planned low altitude orbiter and currently a candidate for ESA’s cosmic
vision program. Numerical simulations suggest that it could reach a precision
for the tidal Love number and the moment of inertia of Venus of 0.3% and 1.4%,
respectively (Rosenblatt et al. [2021]). Although, EnVision is expected to be
launched after the mid 2030’s, it will constrain the composition and structure
of our twin planet. This could promote our understanding of planet formation
mechanisms and shed further light into why Venus and Earth have evolved so
differently despite their similarities in size, mass, and heliocentric
distance.

In preparation for such missions theoretical predictions of possible
compositions and internal structures are crucial for interpreting the acquired
data. The model presented here could be used to guide such interpretations in
the near future. Accurate measurements of Venus’ gravity field and further
constraints on its mantle composition would narrow down the possibilities for
its core size and composition as well as its thermal state. In combination
with a more complex chemical and thermal model for core segregation, including
the effects on metal-silicate partitioning of S and other elements, it could
be possible to constrain the chemical composition and state of the core of
Venus even in the absence of seismic data.

## Acknowledgement

We thank Dave Stevenson and Morris Podolak for discussions. This work has been
carried out within the framework of the National Centre of Competence in
Research (NCCR) PlanetS supported by the Swiss National Science Foundation
(SNSF).

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##  Appendix A The effect of the core segregation model

Here the sensitivity of our results to the model for chemical equilibration
during core segregation is assessed. To this end an additional set of Venus
models probing the same parameter ranges as given in Table 3 was created
without invoking the core segregation model from section 2.3. For these models
the mantle composition is not constrained from the core composition. To
illustrate the effects on the main results a comparison of the two cases is
given for the nominal composition ($X_{\rm FeS}^{\rm Core}=0.08-0.15$). Fig.
11 shows the interior profiles for the nominal models using the core
segregation model (top panels) and without the core segregation model (lower
panels). Relaxing the constraints on the mantle composition has little effect
on the overall range of possible interior structures. Fig. 12 shows a
comparison of the correlation between the mantle composition and the core
size. The spread of permissible core sizes at a given mantle composition is
enhanced if the core segregation model is turned off. A comparison of the
permissible ranges for a selection of parameters is given in Table 7. The
first column shows the inferred ranges for the nominal bulk composition as
presented in the main text. The second column corresponds to the ranges
obtained when the core segregation model is turned off and the third column
considers a wider range for the core segregation pressure (see Appendix B).
The most prominent effect of the CS model is observed for the O content in the
core and the oxygen fugacity of the mantle. This is not surprising as the O
and Si content in the core are controlled by the core segregation process.

Figure 11: Effect of the core segregation model on the interior profiles for
the nominal composition. The top panels are equivalent to the top panels in
Fig. 5 and the bottom panels correspond to the results obtained without
invoking the core segregation model. Figure 12: Effect of the core segregation
model on the correlation between the mantle composition and the inferred core
size of Venus for the nominal composition. The top panels are equivalent to
the top panels Fig. 8 and the bottom panels correspond to the results obtained
without invoking the core segregation model. Figure 13: Effect of the range of
the core segregation pressure on the interior profiles for the nominal
composition. The top panels are equivalent to the top panels Fig. 5 and the
bottom panels correspond to the results obtained with $P_{\rm CS}=10-100$ GPa.
Figure 14: Effect of the range of the core segregation pressure on the
correlation between the mantle composition and the inferred core size of Venus
for the nominal composition. The top panels are equivalent to the top panels
Fig. 8 and the bottom panels correspond to the results obtained with $P_{\rm
CS}=10-100$ GPa.

##  Appendix B The effect of the chosen range of the core segregation pressure

The Venus models presented in this study were constructed assuming the core
segregation process to have been similar to that in the Earth. In this section
the effects of relaxing this assumption on the main results are discussed. To
this end a set of Venus models with the nominal bulk composition was created
for a wider range of the core segregation pressure of $P_{\rm CS}=10-100$ GPa.
Pressures above 100 GPa were not probed by the experiments of Fischer et al.
[2015]. A comparison of the resulting pressure, temperature, and density
profiles with the models presented in the main text are shown in Fig. 13. Fig.
14 shows the same comparison for the correlation between the mantle
composition and the core size. The third column in Table 7 shows the resulting
ranges for some of the major parameters investigated in this study.
Interestingly, allowing a much wider range for the core segregation pressure
has only minor effects on most of the studied parameters. The only significant
effect was observed for the O content in the core which can become larger if a
wider pressure range is considered. The effect of turning off the core
segregation model on the core composition and oxygen fugacity is larger than
the effect of considerably increasing the probed range of the core segregation
pressure. This means that the main results of this study are less sensitive to
the chosen range of the core segregation pressure. It is therefore likely that
using a more complex multi-stage core segregation model would only have minor
effects on the main results. This should be investigated further.

##  Appendix C Thermal equations of state

The EoS used in this study are identical to the ones used in a previous study
(see Appendix B in Shah et al. [2021]). The EoS parameters for the materials
that have been added to the model or for which changes were adopted are
summarized in Table 6. The parameters are given for the Mie-Grüneisen-Debye
EoS (MGD) and the 3rd order Birch-Murnaghan EoS (BM3). The mean densities of
mixtures were computed using a linear mixing law.

Table 6:  EoS parameters for the materials that have not been described in
Shah et al. [2021] or for which modifications have been adopted.

material | EoS | $\rho_{0}$ | $K_{T,0}$ | ${K^{\prime}}_{T,0}$ | $\gamma_{0}$ | $\theta_{D,0}$ | $q$ | $a_{T}$ | $b_{T}$ | $c_{T}$ | $a_{P}$
---|---|---|---|---|---|---|---|---|---|---|---
|  | $\rm[kg\ m^{-3}]$ | [GPa] | [K] |  | [K] |  | $\rm[K^{-1}]$ | $\rm[K^{-2}]$ | [K] | $\rm[GPa\ K^{-1}$
FeS | MGD | 4900(a) | 135(a) | 6(a) | 1.36(a) | 998(a) | 0.91(a) | … | … | … | …
FeSi | MGD | 6543(b) | 230(b) | 4.17(b) | 1.3(b) | 417(b) | 1.7(b) | … | … | … | …
FeO | MGD | 5862(c) | 137.8(c) | 4(d) | 1.45(a) | 430(a) | 3(a) | … | … | … | …
MgO | MGD | 3583(d) | 160.2(d) | 3.99(d) | 1.524(d) | 773(d) | 1.65(d) | … | … | … | …
$\rm Mg_{2}Si_{2}O_{6}^{(a)}$ | BM3 | 3215 | 111 | 7 | … | … | … | 2.86e-5 | 7.2e-9 | 0 | 0
$\rm Fe_{2}Si_{2}O_{6}^{(a)}$ | BM3 | 4014 | 111 | 7 | … | … | … | 2.86e-5 | 7.2e-9 | 0 | 0

References. (a) Alibert [2014], (b) Fischer et al. [2015], (c) Fischer et al.
[2011], (d) Dorogokupets [2010]

##  Appendix D Earth reference models

In order to validate our model and to define the nominal bulk composition used
in this study the model was applied to the interior of modern-day Earth. For
this additional constraints provided by the PREM data were imposed on the
interior structure. In particular, we require the ICB, the CMB, and the MTZ to
be matched with no more than 200 km deviation, respectively. Furthermore, the
densities at these locations were required to agree with the PREM data within
5%. The model uncertainty on the radius and MoI of Earth are set to be 1%. It
is commonly acknowledged that pure iron overestimates the density of the inner
core of the Earth by $\sim$3%. This hints towards the presence of some lighter
elements in the inner core. Here, the inner core was modelled as pure iron for
simplicity. This results in a slight overestimation of the inner core density
and a slight underestimation of the outer core density in our models. Our
models furthermore somewhat overestimate the density of the lowermost mantle
and underestimate the density in the MTZ. This could be explained by the
finding that the MTZ of Earth could be enriched in Fe with respect to the rest
of the mantle (Liu et al. [2020], Zhu et al. [2019]). This possibility was not
considered here. The ranges for the probed parameters are listed in Table 8. A
total of 82,821 models were created within this ranges, 28 of which satisfied
the above conditions. The resulting permissible ranges for all relevant input
and output parameters are summarized in Tables 9 and 10. The temperature,
pressure, and density profiles as a function of the normalized radius are
shown in Fig. 16.

Ranges for the central pressure and central temperature of 352-374 GPa and
5330-5750 K are obtained. The FeO and $\rm SiO_{2}$ contents in the mantle are
$\sim 3.4-7.2\%$ and $\sim 44-50\%$. These ranges are somewhat wider but in
good agreement with estimates of the composition of basaltic rocks on Earth.
The bulk Mg and Si contents are $\rm Mg\\#=0.48-53$ and $\rm Si\\#=0.47-0.52$.
The outer core is estimated to contain 4.6-7.6 wt% S, 0.21-1.3 wt% Si, and
0.37-0.95 wt% O. The estimates for O and Si are consistent with recently
published values (Hirose et al. [2021]) of 0-4 wt% Si and 0.8-5.3 wt% O. The S
content in our model is larger than the 1.7 wt% obtained by these authors.
This is due to the fact that the presence of Ni, C, and H in the core was not
accounted for in this study. Hirose et al. [2021] estimated a total of $\sim
5.5$ wt% contained in these elements. Replacing some of the S with Ni, C, and
H in our model (neglecting the effect on the outer core density) in order to
match the Ni, C, and H content from Hirose et al. [2021] would lead to
$\lesssim 2$ wt% S in the outer core. This is consistent with their estimate
for the S content. These results show that our model is capable of reproducing
the composition of the Earth but with the constraints employed in this study
the inferred allowed ranges for some parameters are larger than those obtained
for the Earth from measurements.

In addition to the presence of Ni, C, and H, the partitioning of lighter
elements between the inner and the outer core was not considered in this study
and should be investigated further. The total S content corresponds to a molar
FeS content of roughly 8.6-15%. The inferred composition of the Earth
reference models were used to constrain the compositions of the Venus models
presented in section 3. The allowed ranges of $\rm Si\\#=0.47-0.56$ and $\rm
Mg\\#0.47-0.53$ were taken from the bracketing cases of Sotin et al. [2007]
for all Venus models (see Tables 9 and 10). The FeS content of the core for
the nominal composition was constrained from the Earth reference models to
$X_{\rm FeS}^{\rm Core}=0.08-0.15$.

It is interesting to note that the possible range for the core segregation
pressure spans the total range of 30 - 70 GPa but the chemical composition of
both the core and the mantle are rather restricted. This is a desirable
outcome for potential applications of our model to exoplanets. If the results
were very sensitive to the core segregation pressure the model would need to
be constrained to a narrower range. This would be challenging for exoplanets.
The fact that the Earth can be reproduced within a rather wide pressure range
indicate that only a very crude estimation of $P_{\rm CS}$ is required to
characterize the composition of a planet (at least in terms of the major
oxides FeO, MgO, and $\rm SiO_{2}$). On this level of precision a simple
scaling law $P/P_{\oplus}\sim\left(M/M_{\oplus}\right)^{\eta}$ is likely
sufficient to adopt adequate pressure ranges for planets over a wider mass
range. With this the difference between the Earth and Venus in terms of the
core segregation pressure is likely of minor importance because of their
similar masses. The effect is expected to become relevant only if planets over
a wider mass range, say 0.1 - 10 $M_{\oplus}$ for terrestrial exoplanets, is
considered. It is expected that this would give rise to systematic trends in
the composition and chemical states of exoplanets as a function of their total
mass (see also Schaefer et al. [2017]). These trends would be based on the
differences in chemical equilibrium between core and mantle due to the
differences in the core segregation pressure. Such trends could assist in
characterizing exoplanets and should be investigated further.

##  Appendix E Testing convergence

Our data set consists of a total of 98,046 individual models distributed over
the parameter space summarized in Table 3. From this we estimate the maximum
allowed ranges for each parameter. However, due the finite number of models it
has to be ensured that these ranges are representative and that they would not
get significantly enlarged if the total number of models was increased.

We therefore employ a simple test to confirm the convergence of the results.
Random subsets of the data of variable sizes have been selected to reassess
the resulting parameter ranges. The chosen sample sizes were:

$\displaystyle N_{j}=N_{\rm tot}/2^{(1+j)},$ (E1)

where $N_{\rm tot}$ denotes the total number of models and $j=0,1,2,3,4,5$.
For each chosen sample size we iterate this procedure $k=10,000$ times and
compute the root mean square deviation of the lower and upper bounds for each
parameter $Q$:

$\displaystyle{\rm{RMSD}}_{j}(Q)=\sqrt{\frac{\sum_{i=1}^{k}\left(x_{ij}-\bar{x}_{j}\right)^{2}}{k-1}},$
(E2)

where $x_{ij}$ is the inferred upper or lower bound for the sample $i$ of size
$N_{j}$, $\bar{x}_{j}$ is the mean upper- or lower bound inferred for the
sample size $N_{j}$ and $k$ is the number of samples for each sample size. The
relative variability is then defined as:

$\displaystyle\sigma_{j}(Q)\equiv\frac{{\rm{RMSD}}_{j}(Q)}{{\bar{x}}_{j}}.$
(E3)

The resulting $\sigma_{j}(Q)$ for some of the tracked parameters are shown in
Fig. 15. The solid and dashed curves correspond to the results for the lower
and upper bounds on each parameter, respectively. We find that the variability
monotonically decreases for all parameters. The $\sigma_{0}(Q)$ can therefore
be viewed as a measure for the variability of the parameter ranges estimated
in this study. For the cases where the only high or low MoI were considered
the number of models that match the boundary conditions for Venus is smaller.
The corresponding variability of the parameter ranges from this approach would
be larger in these cases.

Table 7: Comparison of the permissible ranges for some of the major parameters of the results presented in the main text (Main), cases where no core segregation model was used (No CS) and cases where a core segregation pressure of 10–100 GPa was used (Wide $P_{\rm CS}$). Parameter | Unit | Main | No CS | Wide $P_{\rm CS}$
---|---|---|---|---
$C/MR^{2}$ |  | 0.317-338 | 0.317–0.34 | 0.317–0.339
$X_{\rm FeO}^{\rm Mantle}$ | mol% | 0.03-9.8 | 0–10 | 0–10
$X_{\rm SiO_{2}}^{\rm Mantle}$ | mol% | 38-50 | 38–50 | 38–50
$R_{\rm Core}$ | km | 2960-3890 | 2960–3870 | 2930–3920
$R_{\rm ICB}$ | km | 0-3300 | 0–3330 | 0–3300
$-\log(f_{\rm O_{2}})$ |  | 1.4-5.6 | 0.94-11 | 1–5.7
$w_{\rm S}$ | wt% | 3.9-7.9 | 3.8-7.8 | 3.9–7.9
$w_{\rm Si}$ | wt% | 0.12-13 | 0-12 | 0.06–13
$w_{\rm O}$ | wt% | 0-1.3 | 0–2.6 | 0–1.7
$P_{C}$ | GPa | 267-350 | 262-351 | 268-348
$T_{C}$ | K | 3290-5710 | 3320-5690 | 3370–5700
Table 8: Probed ranges for input parameters for the Earth models. Mg# | $P_{\rm CS}$ | $T_{\rm TBL}$ | $\Delta T_{\rm CMB}$ | $X_{\rm FeO}^{\rm Core}$ | $X_{\rm FeS}^{\rm Core}$ | $X_{\rm FeSi}^{\rm Core}$ |
---|---|---|---|---|---|---|---
| [GPa] | [K] | [K] | [K] |  |  |
0.47-0.53 | 30-70 | 1400-1800 | 500–1800 | 0-0.1 | 0-0.2 | 0-0.1 |
Table 9: Permissible ranges for the input parameters for the Earth reference
models.

Parameter | Unit | Model range | Literature
---|---|---|---
$\rm Mg\\#$ |  | $0.48-0.53$ |  $0.47-0.53$ (a)⋆
$P_{\rm CS}$ | GPa | 30-70 | 30-70 (b,c,d,e,f,g)
$T_{\rm TBL}$ | K | 1620–1800 | 1600-1800 (h)
$\Delta T_{\rm CMB}$ | K | 1510-1800 | 500-1800 (i,s)
$X_{\rm FeS}^{\rm Core}$ | mol% | 8.6-15 | 3.6-36(l,m,d,e,r,p,q,n,o,r)
$X_{\rm FeSi}^{\rm Core}$ | mol% | 0.46-2.7 | 2.1-27(l,m,d,e,r,p,q,n,o,r)
$X_{\rm FeO}^{\rm Core}$ | mol% | 1.4-3.6 | 1.8-12(l,m,d,e,r,p,q,n,o,r)
$w_{\rm S}$ | wt% | 4.6-7.6 | 1.7-15 (l,m,n,o,r,t)
$w_{\rm Si}$ | wt% | 0.21-1.3 | 0-10 (l,m,d,e,r,t)
$w_{\rm O}$ | wt% | 0.37-0.95 | 0.5-5.3 (l,m,d,p,q,r,t)

References. (a) Sotin et al. [2007]; (b) Siebert et al. [2011]; (c) Siebert et
al. [2012]; (d) Fischer et al. [2015]; (e) Wade & Wood [2005]; (f) Li & Agee
[1996]; (g) Wood et al. [2006]; (h) Lee et al. [2009]; (i) Stixrude [2014];
(j) Palme et al. [2010]; (k) Lodders [2019]; (l) Zhang et al. [2016]; (m) Wood
et al. [2006]; (n) Stewart & Schmidt [2007]; (o) Seagle et al. [2006]; (p)
Tsuno et al. [2013]; (q) Ricolleau et al. [2011]; (r) Rubie et al. [2011]; (s)
Lay et al. [2008]; (t) Hirose et al. [2021]

$\star$ Al, Ca, and Ni replaced by Si, Mg, and Fe (see Sotin et al. [2007])

s

Table 10: Permissible ranges for the output parameters for the Earth reference
model.

Parameter | Unit | Model range | Literature
---|---|---|---
$\rm Si\\#$ |  | 0.47–0.52 | 0.47-0.56 (d)⋆
$X_{\rm FeO}^{\rm Mantle}$ | mol% | 3.4-7.2 | 5.4-6 (e,l)⋆
$X_{\rm SiO_{2}}^{\rm Mantle}$ | mol% | 44-50 | 41-43 (e,l)⋆
$T_{\rm CS}$ | K | 3030-3690 | 2400-4200 (a,b,h,i,j,k)
$R_{\rm Core}$ | km | 3390-3620 | 3485(n)
$R_{\rm ICB}$ | km | 1040-1430 | 1215(c,n)
$-\log(f_{\rm O_{2}})$ |  | 1.8–2.8 | 0.6-2.3 (a,b,m)

References. (a) Siebert et al. [2011]; (b) Siebert et al. [2012]; (c)
Monnereau et al. [2010]; (d) Sotin et al. [2007]; (e) Allègre et al. [1995];
(f) Marty [2012]; (g) Ohtani [2019]; (h) Fischer et al. [2015]; (i) Wade &
Wood [2005]; (j) Li & Agee [1996]; (k) Wood et al. [2006]; (l) Workman & Hart
[2005]; (m) Doyle et al. [2019]; (n) Jordan & Anderson [1974]

$\star$ Al, Ca, and Ni replaced by Si, Mg, and Fe (see Sotin et al. [2007])

Figure 15: Variability in the ranges for the a selection of important
parameters as a function of the size of the sample $N/2^{(1+n)}$ considered.
The shown data corresponds to the models with a nominal composition. Figure
16: Temperature (left), pressure (center), and density (right) profiles of the
Earth reference models. The PREM data (grey dashed curve) is shown as a
reference. The density in the inner core is somewhat overestimated as it was
assumed to consist of pure iron. In order to match the mean density and the
MoI of the Earth, this results in a underestimation of the density in the
outer core.