# Numerous chondritic impactors and oxidized magma ocean set Earth’s volatile
depletion

Haruka Sakuraba  Department of Earth and Planetary Sciences, Tokyo Institute
of Technology, Ookayama, Meguro-ku, Tokyo, 152-8551, Japan
<EMAIL_ADDRESS>Hiroyuki Kurokawa  Earth-Life Science
Institute, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo,
152-8550, Japan  Hidenori Genda  Earth-Life Science Institute, Tokyo Institute
of Technology, Ookayama, Meguro-ku, Tokyo, 152-8550, Japan  Kenji Ohta
Department of Earth and Planetary Sciences, Tokyo Institute of Technology,
Ookayama, Meguro-ku, Tokyo, 152-8551, Japan

###### Abstract

Earth’s surface environment is largely influenced by its budget of major
volatile elements: carbon (C), nitrogen (N), and hydrogen (H). Although the
volatiles on Earth are thought to have been delivered by chondritic materials,
the elemental composition of the bulk silicate Earth (BSE) shows depletion in
the order of N, C, and H. Previous studies have concluded that non-chondritic
materials are needed for this depletion pattern. Here, we model the evolution
of the volatile abundances in the atmosphere, oceans, crust, mantle, and core
through the accretion history by considering elemental partitioning and impact
erosion. We show that the BSE depletion pattern can be reproduced from
continuous accretion of chondritic bodies by the partitioning of C into the
core and H storage in the magma ocean in the main accretion stage and
atmospheric erosion of N in the late accretion stage. This scenario requires a
relatively oxidized magma ocean ($\log_{10}f_{\rm O_{2}}$ $\gtrsim$ $\rm
IW$$-2$, where $f_{\rm O_{2}}$ is the oxygen fugacity, ${\rm IW}$ is
$\log_{10}f_{\rm O_{2}}^{\rm IW}$, and $f_{\rm O_{2}}^{\rm IW}$ is $f_{\rm
O_{2}}$ at the iron-wüstite buffer), the dominance of small impactors in the
late accretion, and the storage of H and C in oceanic water and carbonate
rocks in the late accretion stage, all of which are naturally expected from
the formation of an Earth-sized planet in the habitable zone.

## Introduction

Earth’s major volatile elements—carbon (C), nitrogen (N), and hydrogen (H)—are
the main components of the atmosphere and oceans and the key elements for
life. The budget of these major volatiles in the bulk silicate Earth
(including the atmosphere, oceans, crust, and mantle: hereafter BSE)
influences the volume of the oceans and the atmospheric inventory of C (CO2)
and N (N2), and consequently, Earth’s habitable environment [1, 2, 3]. Similar
isotopic compositions of volatiles in Earth and chondrites suggests that
delivery was made chiefly by chondritic materials [4]. In contrast, although
their absolute abundances are largely uncertain, the abundances of C-N-H in
BSE relative to chondrites are known to have a V-shaped depletion pattern [5,
6]. Owing to this discrepancy, the origin of the major volatile elements on
Earth remains unclear [5].

Figure 1: Cartoon of element partitioning processes during Earth’s accretion
according to our model. Accreting planetesimals and giant impactors deliver
volatiles and simultaneously form a vapour plume eroding the atmosphere. a:
Model for the main accretion stage (10% to 99.5% of the Earth’s mass).
Equilibration among the magma ocean (silicate melt), liquid metal droplets
transiting to the core, and the overlying atmosphere are achieved according to
each metal-silicate partitioning coefficient and solubility. b: Model for the
late accretion stage after the solidification of the magma ocean (the last
0.5%). We consider the liquid water oceans and the carbonate-silicate cycle to
be driven by plate tectonics on the surface. In this stage, most H and C on
Earth are stored in the oceans and carbonate rocks, respectively. Numerous
impactors can selectively erode N.

The composition of major volatiles in the BSE should have been modified by
element partitioning processes, including removal by core-forming metals and
by atmospheric escape (Fig. 1). Although the origins and timing of volatile
delivery are still debated, the delivery during the main accretion stage is
expected based on the planet formation theory, which involves large-scale
dynamic evolution, such as the Grand Tack model [7]. Recent isotopic analyses
point to the late-accreting bodies being composed of enstatite chondrites [8]
or carbonaceous chondrites[9], suggesting that volatile delivery continued
into the late stage. On the growing proto-Earth with planetesimal accretion
and several giant impacts, the formation of magma oceans allowed volatiles to
be stored within the magma ocean [10]. Core-forming metal could have removed
some of the iron-loving elements (siderophiles) from the magma ocean during
the main accretion stage [11]. Volatiles partitioned into the atmosphere
(atmophiles) were continuously removed via atmospheric erosion caused both by
small planetesimal accretion[12] and giant impacts[13]. The successive late
accretion after the solidification of the magma ocean further removed and
replenished volatile elements[14].

Alhough several previous studies have attempted to explain the depletion
patterns of major volatile elements in BSE [15, 5, 16], the evolution of the
volatile composition through the full accretion history has not been
simulated. The previous studies employed ad hoc models where a single-stage
metal-silicate equilibration event and complete/negligible atmospheric loss
were assumed. Hirschmann[5] showed that the combination of core segregation
and atmospheric blow off would leave BSE with low C/H and C/N ratios compared
with accreted material, and he concluded that the BSE’s high C/N ratio
requires late accreting bodies with elevated C/N ratios compared with
chondrites. Other works have attributed the discrepancy largely to the
accretion of thermally processed or differentiated, non-chondritic bodies[15,
16], which are hypothetical and may not satisfy the isotopic constraints[8].
Here, we consider another mechanism that can fractionate C/N: the preferential
loss of N relative to C and H by impacts during the late accretion stage,
where N is partitioned into the atmosphere, while C and H are partitioned into
the oceans and carbonate rocks[14]. This work builds on previous studies[15,
5, 16] in terms of volatile element partitioning, but makes improvements to
simulate core formation and atmospheric loss as continuous processes rather
than single stage events.

In this study, we aimed to reproduce the V-shaped C-N-H pattern by considering
realistic processes to the extent of today’s observational uncertainties. We
modelled the evolution of the volatile abundances in the atmosphere, oceans,
crust, mantle, and core through the full accretion by taking elemental
partitioning and impact erosion into account. Fig. 1 shows a schematic image
of our model setting. The main and late accretion stages were modelled
separately, and the masses of C, N, and H in each reservoir were computed
using a multiple-boxes model (Methods). We assumed the existence of the oceans
and the active carbonate-silicate cycle in the late accretion stage; the
validity of this assumption is discussed. We explored the plausible accretion
scenarios that reproduce the current BSE’s C-N-H composition pattern from the
accretion of chondritic bodies. The major parameters were the size
distribution of planetesimals in each stage; the number of giant impacts; the
redox state of the magma ocean, which controls the solubility and metal-
silicate partitioning coefficient of volatiles; the composition of impactors;
and the total mass of the late accretion. Our nominal model, which is a
successful case, assumes a volatile supply from CI chondrite-like building
blocks, an oxidized magma ocean ($\log_{10}f_{\rm O_{2}}$ $\sim$IW+1, where
$f_{\rm O_{2}}$ is the oxygen fugacity, ${\rm IW}$ is defined hereafter as
$\log_{10}f_{\rm O_{2}}^{\rm IW}$, and $f_{\rm O_{2}}^{\rm IW}$ is $f_{\rm
O_{2}}$ at the iron-wüstite buffer), a single giant impact, and a change in
planetesimal size distribution with time, and 0.5 wt.% late accretion. Fig. 2
and Fig. 3 show the evolution of major volatile abundances for this successful
case. As the composition of building blocks, a mixture of CI chondrite-like
impactors (12 wt.%) and dry objects (88 wt.%) was fixed by exploring the best
fit homogeneous accretion (see Supplementary Information). In order to
understand the physical behaviours of the volatile element partitioning, we
also calculated the evolution for other cases with different impactor size
distributions, accretion scenarios, amounts of late accretion, and redox
states of the magma ocean (Fig. 4). In the Supplementary Information, we show
the results for the cases where we assume different source for volatile
elements (enstatite chondrites) and the range of partitioning coefficients and
solubilities. We confirmed that other parameters such as the magma ocean
depth, metal/silicate ratio, surface temperature during magma ocean stage, and
efficiency of impact erosion by a giant impact had only minor effects (Fig.
S1). The uncertainties in the final volatile abundances in BSE caused by these
minor parameters, except for the magma ocean depth, are smaller than 10%. As
to the magma ocean depth, the uncertainties differ by species and the redox
state of the magma ocean: a factor of $\sim$2 for H in the oxidized model,
$\sim$40% for N and $\sim$15% for H in the reduced model, and smaller than 10%
for the others (Fig. S1 for the oxidized model and the figure not shown for
the reduced model) .

##  Origin of the V-shaped C-N-H depletion pattern

Figure 2: Evolution of major volatile abundances in the bulk silicate Earth
(BSE) scaled by those of CI chondrites in the nominal model. The abundances
are normalized by each planetary mass at each time for a, the main accretion
stage, from 10% to 99.5% of Earth’s accretion, and b, the late accretion stage
defined as the last 0.5% of accretion after the magma ocean solidification.
The time sequence is shown by lines from top to bottom with snapshots. The
thick orange and red lines correspond to the end of main and late accretion
stages, respectively. The range in the current BSE composition estimate[5, 17,
4] is shown for comparison (green area). The mean value of Hirschmann [5] is
shown as a reference for the relative depletion pattern (green line). See
Table. 1 for the composition of BSE and chondrites.

The V-shaped C-N-H depletion pattern compared with chondrites in the current
BSE (Fig. 2) can be successfully reproduced from chondritic building blocks by
considering both the element partitioning between reservoirs and the impact-
induced atmospheric erosion simultaneously over all accretion stages. The
successful case (the nominal model) assumed three conditions: an oxidized
magma ocean (IW+1), a change in impactor size distribution with time, and the
storage of H and C in oceanic water and carbonate rocks after magma ocean
solidification. The nominal model assumed CI chondrite-like building blocks,
but enstatite chondrite-like impactors’ case, where a smaller amount of H is
present in the building blocks, also successfully reproduced the BSE
composition under more limited conditions (see Supplementary Fig. S2, Fig. S3,
Fig. S4, and Fig. S5). Element partitioning among the overlying atmosphere,
magma ocean, and suspended metal droplets during the main accretion phase led
to a subtle C deficit, N excess, and an adequate amount of H at the end of the
main accretion stage (Fig. 2a). In the late accretion stage after
solidification, the interplay of H and C storage in oceanic water and
carbonate rocks and the preferential loss of N due to atmospheric erosion
finally solved the remaining issue: N excess (Fig. 2b).

Figure 3: Evolution of the abundances of C, N, and H in the surface and
interior reservoirs over the full accretion obtained from the nominal model.
Dashed lines correspond to the amounts in the atmosphere (light-blue), magma
ocean (orange), and metallic core (grey) for the main accretion phase and in
the surface reservoirs (the atmosphere, oceans, and carbonate rocks: red solid
line) for the late accretion phase. Solid lines mean the net cumulated into
the bulk silicate Earth (BSE; red) and delivered by impactors (brown). The
green areas denote the amounts in the current BSE. Plotted abundances are
scaled by the planetary mass at a given time.

Earth’s C and H abundances were set chiefly during the main accretion stage
(Fig. 3). Although the first kink of volatile abundances in each reservoir is
set by the initial conditions (see Methods), the system soon evolves towards a
quasi-steady state between the gain and loss of volatile elements. The highly
siderophile property of C[18] and high solubility of H[19] in silicate melt
under the oxidized condition in the nominal model caused those elements to be
removed by core segregation. As the remaining part of C was partitioned into
the atmosphere owing to its low solubility[20], the atmospheric erosion led to
C being more depleted than H in BSE. The low solubility of N[21] in magma led
to almost all N in BSE being partitioned into the atmosphere soon after the
magma ocean solidification. For N, the impact-induced erosion governs the
abundance evolution[14], while the transport to the core governs for C and H.
N also has a siderophile property, which has been proposed to cause N
depletion in the BSE[22, 16, 23]. However, we found that partitioning into the
core is less important compared with the atmospheric escape, even considering
the uncertainty in the partitioning coefficient (see Supplementary
Information). A Mars-sized, moon-forming giant impact[24] was assumed in the
nominal model, which corresponds to the kink at 90% Earth mass, but it did not
modify the BSE volatile abundances significantly. We considered a completely
molten mantle[25] in the element partitioning after the giant impact, while a
smaller molten fraction of 30 wt.% was assumed for the planetesimal accretion.
Thus, the larger mass of the magma ocean after the giant impact allowed
increases in the abundances of all volatile elements in the magma ocean.

The abundance of N was decreased by approximately one order of magnitude
during the late accretion phase owing to impact-induced atmospheric escape.
The formation of oceans and the initiation of the carbonate-silicate cycle
right after the solidification of the magma ocean trapped H and C into the
surface reservoirs, and subsequently facilitated preferential N erosion from
the atmosphere. Since N neither condense nor become incorporated into any
solid or liquid reservoirs in our model, the final N abundance is determined
by the balance between the supply by impactors and the loss by atmospheric
erosion. From this result, we argue that the presence of oceans and carbonate
formation in the late accretion stage are requirements to explain the current
high C/N and H/N ratios of the BSE.

The exact timing of the ocean formation and the initiation of the carbonate-
silicate cycle are unknown, but these assumptions are supported by
theoretical, geological, and geochemical studies. As the timescale for the
magma ocean solidification is short (0.2–7 Myrs after the last giant
impact[26, 27, 28]), the oceans would have persisted in the late accretion
stage. Archean sediments imply that the oceans[29] and plate tectonics[30]
already existed at least 3.8 Gyr ago. Geochemical studies of Hadean zircons
suggest the existence of oceans and the active plate boundary interactions in
the Hadean[31, 32].  Although plate tectonics might not have started on the
Hadean Earth[33], the storage of C by carbonate precipitation is possible even
without plate tectonics as long as liquid water is present[34]. The
continental crust, which releases water-soluble cations such as Ca2+ and Mg2+
on modern Earth, might not be present in the Hadean, but efficient carbonate
formation is possible owing to seafloor weathering [35]. However, the presence
of liquid water which allows silicate weathering to occur is required to drive
the carbonate-silicate cycle[3]. Furthermore, if marine pH was neutral to
alkaline ($>\sim$7), most of the total atmosphere plus ocean C inventory
($>\sim$90%) would have dissolved into the oceans as bicarbonate and carbonate
ions[36], as proposed for the preferential N loss by a giant impact[37]. We
note, however, that a giant impact will vaporize the oceans, which could end
up losing some C.

Another key assumption of our model is the slow (negligible) N fixation
compared to C on early Earth during late accretion, which led to the
preferential erosion of atmospheric N. A combined model of atmospheric and
oceanic chemistry determined the lifetime of molecular N in anoxic atmospheres
to be $>10^{9}$ years[38]. After Earth’s accretion ceased, N cycling between
the atmosphere and mantle over a billion-year timescale[39] would lead to
lower N partial pressures in the later period (e.g., those recorded in the
Archean[40]). In contrast, the timescale of carbonate precipitation even in
the cation supply-limited regime ($\sim 10^{6}$ years[3]) is shorter than the
duration of late accretion ($\sim 10^{7}$–$10^{8}$ years[41]).

## Size distribution of impactors and accretion models

Figure 4: Dependence of final volatile composition of the bulk silicate Earth
(BSE) on the accreting conditions. a) Effects of the impactor’s size
distribution. P2-G1-L3 (the nominal model, red line): planetesimal accretion
($q=2$ and $q=3$ for in the main and late accretion stages, respectively) and
one giant impact. G10-L3 (orange line): ten giant impacts and the late
accretion of planetesimals . P2-G1-L2 (blue line): shallower planetesimal size
distribution ($q=2$ throughout the full accretion) and one giant impact. Here
we assumed $\mathrm{d}N/\mathrm{d}D\propto D^{-q}$, where $N(D)$ is the number
of objects of diameter smaller than $D$, and $q$ is the power law index. b)
Dependence on volatile accretion scenarios. Late volatile accretion (dark-
purple): volatiles are delivered only by late accretion with CI chondrite-like
bodies. Heterogeneous accretion model (purple): volatiles are supplied in the
last 30 wt.% accretion with CI chondrite-like bodies. c) Dependence on the
late accretion mass. The mass of late accretion was varied from 0.5 wt.%
(brown) to 2.5 wt.% (orange). d) Dependence on the redox state of the magma
ocean. Oxidized (the nominal model, red line), intermediate (light blue line),
and reduced (blue line) conditions are compared. The solubilities and
partitioning coefficients are summarised in Table. 1.

Our results suggest the dominance of small (km-sized) impactors during the
late accretion. The nominal model assumed a change in the size distribution of
impactors from shallower ($q=2$ in $\mathrm{d}N/\mathrm{d}D\propto D^{-q}$,
where $N(D)$ is the number of objects of diameter smaller than $D$, and $q$ is
the power law index) for the main accretion phase to steeper ($q=3$, main
asteroid belt-like) for the late accretion phase (see Methods, Fig. 4a). The
impact erosion by the late accretion which has a shallow size distribution is
not sufficient to reproduce the BSE’s N-depletion because km-sized small
bodies are the most efficient in ejecting the atmosphere per unit mass of the
impactor[12, 42].

The case where Earth formed by multiple giant impacts followed by late
accretion also reproduced the current BSE volatile composition (Fig. 4a);
however, atmospheric erosion by small bodies during late accretion is needed
to obtain the V-shaped C-N-H pattern anyway. Since atmospheric loss per unit
impactor mass is less efficient in giant impacts than in planetesimal
accretion, larger amounts of volatiles remained in the BSE. In addition,
incomplete mixing between the impactor’s core and magma ocean lowered the
storage in the core.

The combination of a moon-forming giant impact with continuous planetesimal
accretion in our nominal model is a plausible accretion scenario for Earth.
The dominance of planetesimal accretion and the change in the size
distribution with time are naturally expected from the planet formation
models. The giant impact stage was set in the time when the total mass in
planetesimals remained comparable to the mass in protoplanets[43, 44, 42].
About half of giant impacts are not accretionary, but hit-and-run collisions
which produce many fragments[45]. The change in size distribution from
shallower to steeper is consistent with the inference that asteroids are born
big and evolved through collisional cascades[46, 47]. A size distribution of
late accretion impactors shallower than that of our nominal model ($q\sim 2$
vs. $q=3$) has been proposed as the origin for the lunar depletion of highly
siderophile elements (HSE) relative to Earth[48], but the long-lived magma
ocean on the Moon also serves alternative explanation[41]. Our scenario, in
which Earth chiefly formed from a swarm of planetesimals, is in contrast to
the pebble accretion model[49] and the idea that attributes the Late Veneer to
a single big impact[50], although impact fragments may act as small
"planetesimals" to eject atmospheric N efficiently in those cases.

The timing of the volatile delivery onto the accreting terrestrial planets
remains an open question[51]. We assumed homogeneous accretion of CI
chondritic building blocks combined with dry objects in the nominal model and
investigated the dependence on the variety of Earth’s accretion models for
both the volatile-rich late accretion[52, 9] and the heterogeneous accretion
model[53, 54, 55] (Fig. 4b). The latter two models, which assumed later
addition of 100% CI chondrites accretion, result in much larger amounts of
volatiles, especially for N, than the current BSE inventory. This means that
the volatile content fraction of the late accretion impactors appears more
significant factor than the total accumulated amount for explaining Earth’s N
depletion. The uncertainty in the late accretion mass (0.5 wt.% to 2.5
wt.%[56]) is considered in Fig. 4c. Greater late accretion can erode more N
and accumulate more C, but V-shaped patterns were obtained over the range of
mass estimates.

##  Magma ocean redox state

We explored how the redox state of the magma ocean affects the final volatile
composition by considering oxidized (IW+1), intermediate (IW-2), and reduced
(IW-3.5) conditions[5] (see Methods). The current C-N-H depletion pattern can
be obtained under the oxidized or intermediate magma ocean, while we ruled out
the reduced condition (Fig. 4d). The redox state of the magma ocean, which
successfully reproduces the BSE’s abundance of major volatile elements, has a
relatively oxidized condition ($\log_{10}f_{\rm
O_{2}}{\color[rgb]{0,0,0}\gtrsim}{\color[rgb]{0,0,0}\rm IW-2}$). In the
reduced model, the final amount of H corresponding to 1.15 ocean mass (0.15
ocean mass in the mantle) was obtained; this is even smaller than the minimum
estimate for present-day Earth’s mantle water content of > 0.56 ocean masses
[57, 58, 5, 17, 4]. The behaviour of element partitioning during the main
accretion stage is determined by solubilities in the magma ocean and the
partitioning coefficients between silicate melt and metal liquid, both of
which depend on the pressure-temperature-$f_{\rm O_{2}}$ conditions of the
magma ocean. According to what is currently understood, C becomes soluble and
less siderophile, N becomes insoluble, and H becomes far more soluble in
silicate liquids under oxidized conditions compared with reduced conditions[5,
59, 22, 21, 60]. We note that assuming enstatite chondrites as volatile
sources fits only with the oxidized model (Fig. S5).

Since the flux from the atmosphere to the core is proportional to the product
of solubility and partitioning coefficient (see Methods), their influences on
the resulting C cancelled each other out. The change in molecular masses
(e.g., CO2 = 44 amu in the oxidized model to CH4 = 16 amu in the reduced
model) also influences partial pressure and, consequently, effective
solubility slightly, but the influence is not significant. For H, the storage
in the magma ocean is important to reproduce the current BSE abundance and so
high solubility under oxidizing conditions is required. We emphasize that even
when we employed smaller values for the partitioning coefficient of H as
argued by a few previous high-pressure metal-silicate partitioning
experiments[61, 62], the final H amount does not increase under the reduced
magma ocean (see Supplementary Fig. S6, case ’m’) because the low solubility
in silicate melt limits the influence of partitioning into metal on the BSE’s
budget, supporting the idea of a relatively oxidized magma ocean during
Earth’s accretion.

The oxidizing magma ocean has been proposed by several studies and mechanisms
to oxidize the magma ocean as Earth grew have been suggested[63, 64, 65].
Crystallization of perovskite at the depth of the lower mantle induces
disproportion of ferrous to ferric iron plus iron metal, the latter of which
would then sink to the core, with the remnant in the mantle being oxidized
(named self-oxidation[66]). The pressure effect on a fixed Fe3+/Fetotal ratio
leads to a redox gradient where the surface becomes more oxidized than the
depth[67, 65]. Both are natural outcomes of the formation of an Earth-sized
planet. An increase in the mantle oxygen fugacity on growing Earth has been
proposed by many heterogeneous accretion models to explain Earth’s bulk
composition of refractory and moderately siderophile elements[54, 66, 68, 69].
As shown in Fig. 3, the volatile distribution between the atmosphere, magma
ocean, and metal soon converged towards quasi-steady state balancing of in/out
fluxes of volatile elements within the first $\sim$20 wt.% accretion. Given 20
wt.% accretion being the mass required to reach the steady state for a give
condition, the requirement for the redox state of the magma ocean
($\log_{10}f_{\rm O_{2}}\gtrsim{\color[rgb]{0,0,0}\rm IW-2}$) should be
considered as that for the final $\sim$20 wt.% planetesimal accretion before
the giant impact. This result does not contradict the initial reduced
condition followed by a more oxidized state.

##  Volatile predictions for Earth’s core, bulk Venus, and Mars

Our scenario for the origin of Earth’s volatile depletion pattern is testable
with the further constraints of the composition of light elements in the core.
The final mass fractions in the metallic core in our nominal model, assuming
an oxidized magma ocean (IW+1), were 0.9 wt.%, 0.004 wt.%, and 0.2 wt.%for C,
N, and, H, respectively. These predicted contents of light elements are within
the range of each element’s content allowance and account for approximately
30% of the Earth’s core density deficit[70]. Therefore, other light elements
such as oxygen, silicon, and sulphur should also contribute to the core
density deficit. The relatively oxidized magma ocean required from our results
may support the large contribution of oxygen[64]. With upcoming data of
solubilities and partitioning coefficients, this model will provide a more
accurate estimate of Earth’s core composition (see also Supplementary
Information).

Our model also predicts the different depletion patterns of major volatile
elements in bulk Venus and Mars. Venus might never experience the condensation
of liquid water and, consequently, carbonate precipitation on the surface[28].
The lack of H and C storage leads to atmospheric loss of these elements as
well as N. If atmospheric CO2 is the dominant C reservoir of bulk Venus, the
total amount of C for Venus is $\sim$0.4 times the value for Earth[71],
supporting the model prediction. In contrast, the formation of H2O and CO2 ice
on early Mars[72] allows the preferential loss of atmospheric N by impact
erosion in the same manner as Earth[14]. From the analysis of SNC
meteorites[73, 74], the Martian mantle is thought to be very reduced with
(IW-1), and previous models have predicted a more reduced magma ocean for Mars
than for Earth[75, 76]. The reduced mantle would lead to a smaller storage of
H in the magma ocean and a subsequent deficit of bulk H.

## Methods

We calculated the abundances of C, N, and H in Earth growing from a Mars-sized
embryo via accretion of planetesimals and protoplanets with a box model
considering the partitioning between reservoirs and atmospheric erosion caused
by impacts. We defined two stages: main accretion and late accretion stages
(e.g., the growth from 10% to 99.5% and the final 0.5% of current Earth mass,
respectively) in our nominal model. The former is also called the magma ocean
stage. For each finite mass step of planetesimal accretion, we solved the
deterministic differential equations formally given by,

$\frac{{\rm d}M_{\rm i}^{\rm BSE}}{{\rm d}M_{\rm p}}=x_{\rm i}-\sum_{\rm
sinks}F^{\rm i}_{\rm k},$ (1)

where $M_{\rm i}^{\rm BSE}$, $M_{\rm p}$, $x_{\rm i}$, and $F^{\rm i}_{\rm k}$
are the total mass of element i in BSE, planetary mass, the abundance of
volatiles in impactors, and outflux per unit mass accretion by the process k,
respectively. As the volatile loss processes (sinks), we considered the
atmospheric escape $F_{\rm esc}$ through the full accretion and the core
segregation $F_{\rm core}$ only for the magma ocean stage. For each accretion
step, the element partitioning between surface and interior reservoirs is
calculated by the mass balance modelling,

$\displaystyle M_{\rm i}^{\rm BSE}=\sum_{\rm j}M_{\rm i}^{\rm j},$ (2)

where the atmosphere, silicate melt, and suspended metal in the magma ocean in
the main accretion stage, the atmosphere, ocean, and sedimentary carbonate
rocks in the late accretion stage are considered as reservoirs j (see below
for details). Each giant impact is treated separately from the statistically
averaged planetesimal accretion. We note that we confirmed numerical
convergence by changing the step size of the cumulative accreted mass in our
simulations.

Accretion model. In our model, Earth grows by accreting planetesimals and
protoplanets during the main accretion phase, followed by late accretion
composed only of planetesimals. For planet growth, we considered a change in
the bulk density caused by pressure by using Eq. 3, which expresses the mass-
radius relationship for planets with an Earth-like composition. Seager et
al.[77] showed a power-law relation between the masses and radii of solid
planets by modelling their interior structures. They provided fitted formulas
for rocky planets with 67.5 wt.% MgSiO3 \+ 32.5 wt.% Fe as,

$\displaystyle{\color[rgb]{0,0,0}\log\tilde{R}=k_{1}+\frac{1}{3}\log\tilde{M}+k_{2}\tilde{M}^{k_{3}},}$
(3)

where $\tilde{R}=R/R_{\rm s}$ and $\tilde{M}=M/M_{\rm s}$ are the normalized
radius and mass of terrestrial planets, $k_{\rm i}$ is the fitting constants
($k_{1}$ = 0.20945, $k_{2}=$ 0.0804, $k_{3}=$ 0.394), and $r$ and $M_{\rm s}$
are the conversion factors obtained as $R_{\rm s}=3.19\ R_{\rm Earth}$ and
$M_{\rm s}=6.41\ M_{\rm Earth}$; we used modified $R_{\rm s}=3.29\ R_{\rm
Earth}$ in our study to match Earth’s mass and radius without changing the
power-law index. The growth by planetesimal accretion was investigated by a
statistical method[14] where the contribution of each impact was averaged over
their size and velocity distributions. The size distribution is given by a
single power-law $\mathrm{d}N/\mathrm{d}D\propto D^{-q}$, where $N(D)$ is the
number of objects of diameter smaller than $D$ and the index $q$ is a
parameter. We assumed a shallow size distribution with $q=2$ for the main
accretion phase and a steep size distribution with $q=3$, which corresponds to
that of the present-day main belt asteroids[46], for the late accretion phase
in the nominal model, and investigated the dependence of the results by
changing the power-law index (see main text). The minimum and maximum sizes
are assumed to be $10^{-1.5}$ km and $10^{3}$ km, respectively. We assumed a
Rayleigh distribution for the random velocity of impactors as modelled by
Sakuraba et al.[14], which corresponds to the Gaussian eccentricity
distribution for impactors from the terrestrial planet feeding zone excited by
a protoplanet[78]. Each giant impact is included in a discrete way separately
from continuous planetesimal accretion.

Volatiles have been delivered by the accreted bodies that have formed Earth.
The building blocks of Earth are not fully understood and so we treat the
abundance of volatiles in an impactor as an unknown parameter. We considered
CI chondrites containing volatile elements which accreted with dry
planetesimals[11]. CI chondrites contain 2–5 wt.% C[79, 80, 81], 500–2000 ppm
N[79, 80, 81], and 0.47–1.01 wt.% ppm H[79, 80, 82], and their atomic ratio of
C/N is 17.0$\pm$3.0 [15]. Enstatite chondrites contain 0.2–0.7 wt.% C[80, 81],
100–500 ppm N[81], and 90–600 ppm H[83, 84], and their C/N is 13.7$\pm$
12.1[15]. In our model, we assumed the reference abundances as listed in
Table. 1. The fraction of CI chondrites was used as a parameter and set to be
12 wt.% in the nominal model (Supplementary Fig. S2). The results for cases
where enstatite chondritic impactors are considered as volatile sources are
also shown in the Supplementary Information (Supplementary Fig. S3 and Fig.
S4).

Giant impactors would have experienced core-mantle differentiation and
volatile loss by atmospheric erosion. We calculated the abundances of C, N,
and H in protoplanets by running our model for the growth by planetesimal
accretion in advance from 0.05 to 0.1 Earth masses and then adapted the result
to the compositions of giant impactors.

We assumed that 32.5 wt.% of the impactor mass is added to Earth as metallic
iron regardless of the impactor type to reproduce the mass fraction of Earth’s
core. The metal mass fraction is not necessarily equal to that of impactors
because the former would be controlled by redox reactions (namely, the oxygen
fugacity of the magma ocean), which are not explicitly modelled in our study.

Atmospheric erosion and loss of impactors.

Atmospheric erosion and loss of the impactors themselves through planetesimal
accretion are computed as compiled by Sakuraba et al.[14]. The net impact-
induced escaping flux per unit impactor mass is given by

$\displaystyle F_{\rm esc}=\zeta x_{\rm i}+\eta,$ (4)

where $x_{\rm i}$ is the abundance of volatiles in the impactor, $\eta$ is the
erosion efficiency of the atmosphere, and $\zeta$ is the escaping efficiency
of the impactor vapor, and calculated for each element i. The erosion and
escaping efficiencies for a single impact can be estimated from input
impactor’s size and impact velocity. We calculated the efficiencies
statistically-averaged over both impactor’s size and velocity distributions
assumed in the above accretion model. We adopted the scaling laws obtained by
numerical simulations for the atmospheric escape and for loss of an impactor’s
vapour. Realistic numerical simulations of atmospheric erosion were given in
3D geometry by Shuvalov[85], and in 2D cylindrical geometry by Svetsov[86]. In
Sakuraba et al.[14], the Svetsov model[87, 86] and Shuvalov model[85] are
adopted for the atmospheric erosion efficiency $\eta$ (Equation 5 in Sakuraba
et al.[14]) and the impactor’s escaping efficiency $\zeta$ (Equation 8 in
Sakuraba et al.[14]), respectively. The effect of oblique impacts that enhance
the impact erosion is considered by including the angle-averaged factor
estimated by Shuvalov model[85]. The volatiles in the fraction of impactors
that avoided the loss are released to the atmosphere and then partitioned into
other reservoirs. Since the atmospheric loss mass is proportional to the
abundance of each species in the atmosphere, element partitioning between the
atmosphere and the other solid or liquid reservoirs is important for
atmospheric erosion[14]. We used the surface temperatures, 1,500 K for the
main accretion stage and 288 K for the late accretion stage, given below to
compute the atmospheric density and scale height, but we confirmed that the
results are insensitive to the atmospheric temperature (Supplementary Fig.
S1a).

For each giant impact, we calculated the atmospheric loss from the mixture of
the proto-atmosphere and the impactor’s atmosphere caused by the global ground
motion by using the model of Schlichting et al.[42]. We assumed Mars-sized
(0.1 Earth masses) impactors whose impact velocity is 1.1 times the mutual
escape velocity as commonly considered for the Moon-forming impact (see, for
example, Hosono et al.[88]). We note that the estimates for the giant impact
velocity has uncertainty from 1.0 to 1.2 times the escape velocity[89]. We
also note that recent 3D smoothed particle hydrodynamics simulations[90, 91]
suggest a lower erosion efficiency than that of Schlichting model by a factor
of 3. However, we found that the dependence on the atmospheric erosion
efficiency by a giant impact was negligible even if the atmospheric erosion
was more or less efficient by a factor of three (Supplementary Fig. S1b) owing
to the range in uncertainty arising from the velocity of a giant impact and/or
the model dependency.

Core segregation. We considered the core segregation under the solidified
mantle in the magma ocean stage by excluding the segregated metal from the
equilibrium partitioning calculation. Core-forming liquid metal droplets carry
volatile elements from the magma ocean to the metal pond, and consequently, to
the core. We assumed the suspended metal fraction in the magma ocean to be
constant; then, the net segregation flux per unit impactor mass is given by

$\displaystyle F_{\rm core}=C_{\rm i}^{\rm met}\left(x_{\rm met}-X_{\rm
MO}X{\rm{}_{met}^{MO}}\right),$ (5)

where $C_{\rm i}^{\rm met}$, $x_{\rm met}$, $X_{\rm MO}$, and
$X{\rm{}_{met}^{MO}}$ are the volatile concentration in the metal, the metal
fraction of the newly accreted mass (assumed to be 32.5 wt.%), the melting
fraction of the planet, and the metal mass fraction in the magma ocean,
respectively. Our model is based upon previous studies[15, 5] that assumed
that core-mantle separation is a single stage event as a simplification, but
we improved the model to track time evolution through accretion, where core
formation is a contentious process.

During growth by planetesimal accretion, the mass fraction of the molten magma
ocean $X_{\rm MO}$ was fixed to 30 wt.% of the planetary mass. This was
derived from the estimated depth of the magma ocean (30%-40% of the mantle)
constrained from the abundance of refractory siderophile elements (Ni and Co)
in the present-day mantle[66]. A deeper magma ocean is also suggested[68], but
we confirmed that the results do not change significantly even if we consider
a deeper magma ocean of up to 60 wt.% (see Supplementary Fig. S1c).

As the Earth grows, the metal droplets descend through the deep magma ocean,
continuously equilibrating with the silicate liquid[92]. Metal droplets that
have reached the base of the magma ocean forms metal ponds and subsequently
descend as large diapirs to the growing core without further equilibration
with the surrounding silicate[66]. The metal in the magma ocean is assumed to
be in equilibrium with the entire magma ocean and the mass fraction of the
suspended metal $X{\rm{}_{met}^{MO}}$ is fixed to $10^{-6}$. This reference
value is estimated from the typical timescales of metal droplets settling
($\tau_{\rm rain-out}\sim 10^{1}$ yrs[93]) and accretion of the Earth
($\tau_{\rm accretion}\sim 10^{7}$ yrs[94]) by,

$\displaystyle
X{\rm{}_{met}^{MO}}\sim(1-\zeta)x{\rm{}_{met}}M_{\oplus}\cdot\frac{\tau_{\rm
rain-out}}{\tau_{\rm accretion}}\sim 10^{-6}.$ (6)

The settling timescale of the metal droplets was estimated from the magma
ocean depth $\sim$102–103 km divided by the settling velocity $\sim$0.5
cm/s[92]. In fact it evolves with time, but we confirmed that the results do
not depend on the suspended metal fraction unless it exceeds 10-2 because
metal droplets carry almost the same mass of volatiles in total anyway
(Supplementary Fig. S1d). The sum of remaining fraction of metal is assumed to
have segregated from the magma ocean into the metal pond and, subsequently,
into the core and are treated as one isolated reservoir.

In contrast to planetesimal accretion, giant impacts lead to a deeper magma
ocean and incomplete mixing between the magma ocean and the core of the
impactor. Since the Mars-sized moon-forming giant impact is thought to have
delivered enough energy to completely melt the whole mantle[95], we assumed a
fully molten mantle after the giant impacts (see, for example, Canup[25]).
Moreover, we confirmed that assuming partial melting for a giant impact does
not change our results significantly (Supplementary Fig. S1c). For
metal–silicate mixing in re-equilibration between the impactor’s core and
proto-magma ocean, we considered turbulent entrainment by applying the formula
of the metal fraction in the metal-silicate mixture to the completely molten
mantle (Equation 6 in Degen et al.[96]).

The solidified mantle is assumed to be volatile free and is not considered as
a reservoir because of its low capacity compared with the magma ocean (see,
for example, Elkins-Tanton[10]). In terms of H and C, it is suggested that
considerable amounts could be retained in the residual solid mantle as melt
inclusions upon magma ocean solidification[97], but this would not affect the
final BSE inventory estimates. While we did not consider the volatile trapping
by the solid silicate reservoir during the crystallization of the magma ocean,
we did consider efficient trapping of outgassed H and C into the oceans and
the crustal carbonate reservoirs as well as the mantle, respectively, which
are included as BSE abundances. Whether H and C are trapped in the mantle or
in the surface reservoirs does not affect the evolution of the BSE volatile
contents. In the case of N, since the N partitioning coefficient between
mantle minerals and silicate melt is smaller than unity by orders of magnitude
even under high temperature[98], almost all N would have been enriched in the
melt during the crystallization and subsequently outgassed to form the early
atmosphere. Hier-Majumder and Hirschmann[97] showed that owing to extremely
low solubility of N2, N retention into the residual mantle is inefficient. For
these reasons, the incorporation of volatiles into the solidified mantle does
not change our conclusions.

Equilibrium partitioning in magma ocean. We calculated the partitioning of
elements between the magma ocean, core-forming alloy, and overlying atmosphere
(Fig. 1a), assuming equilibrium partitioning. For the element partitioning
between these three reservoirs, the mass balance equation (Eq. 2) can be
written as,

$\displaystyle M_{\rm i}^{\rm BSE}=M_{\rm i}^{\rm atm}+M_{\rm i}^{\rm
sil}+M_{\rm i}^{\rm met}=\frac{4\pi R^{2}m_{\rm i}^{\rm atm}}{g\bar{m}r_{\rm
i}}P_{\rm i}+C_{\rm i}^{\rm sil}M_{\rm sil}+C_{\rm i}^{\rm met}M_{\rm met},$
(7)

where $P_{i}$ is the partial pressures, $g$ is the gravitational acceleration
given by $g=\frac{GM}{R^{2}}$, where $G$ is the gravitational constant,
$\bar{m}$ is the mean molecular mass of the atmosphere, $m_{\rm i}^{\rm atm}$
is the molecular mass of the volatile species, $r_{\rm i}=m_{\rm i}^{\rm
atm}/m_{\rm i}$ is the mass ratio of the volatile species to the element of
interest (e.g., $r_{\rm C}={m_{\rm CO_{2}}}/{m_{\rm C}}=44/12$ for C, $r_{\rm
N}={m_{\rm N_{2}}}/2{m_{\rm N}}=1$ for N, $r_{\rm H}={m_{\rm H_{2}O}}/2{m_{\rm
H}}=18/2$ for H in the oxidized condition), $C_{\rm i}^{\rm j}=M_{\rm i}^{\rm
j}/M_{\rm j}$ is the mass concentration in the silicate and metal, and $M_{\rm
sil}$ and $M_{\rm met}$ are the masses of the magma ocean and the metal liquid
in the magma ocean. The partition coefficient which characterizes elemental
partitioning between silicate and metal can be written as,

$\displaystyle D_{\rm i}^{\rm met/sil}=\frac{C_{\rm i}^{\rm met}}{C_{\rm
i}^{\rm sil}}.$ (8)

Equilibrium between the atmosphere and the underlying silicate melt is given
by a solubility law, that in many cases can be approximated by a Henrian
constant,

$\displaystyle S_{i}\ (x=1,2)={\color[rgb]{0,0,0}\frac{C_{\rm i}^{\rm
sil}}{P_{i}^{\frac{1}{x}}}},$ (9)

where $x$ is the ratio of the number of atoms between gas and solute phases
for the element of interest (e.g., $x=1$ for C and N, $x=2$ for H, see below).

We defined three models for the redox state of the magma ocean: the oxidized
($\log_{10}f_{\rm O_{2}}\sim{\color[rgb]{0,0,0}\rm IW+1}$), intermediate
(IW-2), and reduced (IW-3.5) conditions[5], where $f_{\rm O_{2}}$ is the
oxygen fugacity, ${\rm IW}$ is defined hereafter as $\log_{10}f_{\rm
O_{2}}^{\rm IW}$, and $f_{\rm O_{2}}^{\rm IW}$ is $f_{\rm O_{2}}$ at the iron-
wüstite buffer. We call the redox state of $\log_{10}f_{\rm
O_{2}}\sim\log_{10}f_{\rm O_{2}}^{\rm IW}-2$ an ’intermediate’ condition
because it corresponds to that estimated for a single stage core formation
scenario[99]. As proposed by several studies[65, 64], we assumed an oxidized
magma ocean in the nominal model, and investigated the dependence of the
results by changing the redox state. The conditions for the metal-silicate
equilibrium at the bottom of Earth’s magma ocean were estimated to be at
$\sim$40 GPa from Ni and Co partitioning, and at $\sim$3750 K from V and Cr
partitioning[54]. We assumed the partitioning coefficients thought to be
applicable to these high $P$-$T$ conditions mentioned above for each redox
state model (ref.[100, 101, 102, 103]) as tabulated by Hirschmann [5] (see
Table. 1). Since the dependence of the partitioning coefficients on
$P$-$T$-$f_{\rm O_{2}}$ conditions has not been fully understood, we
additionally investigated the sensitivity of our model by varying partitioning
coefficients for a wide range suggested from the literature:
$D{\rm{}_{C}}^{\rm met/sil}$ = 0.5–5670[18, 100, 101, 104, 22, 59, 62, 105],
$D_{\rm N}^{\rm met/sil}$ = 0.003–150[103, 106, 22, 107, 108, 16, 23] , and
$D_{\rm H}^{\rm met/sil}$ = 0.2–100[102, 109, 61, 62, 110, 111](Supplementary
Text, Supplementary Tab. S1 and Supplementary Fig. S6).

We considered different atmospheric components and used the constant
solubilities and partitioning coefficients for each redox state model: CO2,
N2, and H2O for the nominal oxidized model; CO, N2, and H2 for the
intermediate model; and CH4, NH3, and H2 for the reduced model. All these
species of molecules in the atmosphere are assumed by following Hirschmann[5].
As summarised in Table. 1, we fixed solubilities for C and N and used the
Moore model[19] for H by following Hirschman[5]. Considering the range of
solubilities reported by experimental studies, we also tested the model
sensitivity for $S_{\rm C}$ = (0.002–8) $\left(\frac{P_{\rm CO_{2}}}{\rm
MPa}\right)$ ppm[20, 60, 112, 113, 101], $S_{\rm N}$ = (0.2–46)
$\left(\frac{P_{\rm N_{2}}}{\rm MPa}\right)$ ppm[21, 114, 115]. For $S_{\rm
H}$, we tested for more and less soluble cases than the Moore model by a
factor of two (Supplementary Text, Supplementary Tab. S1, and Fig. S6). By
substituting Eq. (8) and (9) to Eq. (5), we can get the relationship of
$F_{\rm core}\propto C_{\rm i}^{\rm met}\propto S_{\rm i}D_{\rm i}^{\rm
met/sil}$ which means that the volatile flux of core segregation is
proportional to the product of solubility and the partitioning coefficient of
each element i[5].

Late accretion stage surface environment. As we defined the late accretion as
the last 0.5 wt.% accretion, we examined whether the surface could melt again
after the magma ocean solidification by roughly estimating the energy balance
between the gravitational accretion and the planetary radiation from the
oceanic surface during late accretion. The input accretional energy is
estimated by $E_{\rm in}=(GM_{\rm Earth}M_{\rm LA}/R_{\rm Earth})\tau_{\rm
LA}^{-1}$, where $M_{\rm LA}$ is 0.5 wt.% of the Earth mass and $\tau_{\rm
LA}\sim 10^{8}$ yrs[48] is the timescale of the late accretion. The maximum
outgoing planetary radiation flux is estimated by assuming the radiation limit
from the saturated atmosphere ($\sim$300 W/m2[116, 117, 118]). As a result,
the latter exceeded the former by $\sim$2 orders of magnitude and confirmed
that the planetary surface would not melt again during the late accretion.

We assumed the formation of the oceans and onset of the carbonate-silicate
cycle (Fig. 1b) as modelled by Sakuraba et al.[14]. For the surface
conditions, it is suggested that the oceans have formed within the several
Myrs after the Moon-forming giant impact by calculating the timescale of the
magma ocean solidification [26, 27, 28]. Furthermore, voluminous oceans
underlain by a dry mantle, like that right after the magma ocean
solidification, create an ideal situation to drive plate tectonics[33, 119].
The carbonate-silicate cycle is believed to have stabilized the Earth’s
climate over a long geological timescale against the increase in solar
luminosity over the past 4 billion years[120]. As long as its negative
feedbacks of temperature-dependent continental and seafloor weathering has
been driven throughout Earth history as assumed here, the early climate was
likely temperate, with temperatures on Earth is estimated to be 273 K to 370
K[121, 35]. Since we confirmed that dependence on the temperature within this
range is negligible (the figure not shown), we assumed the present-day global
mean temperature of $T$ = 288 K as the reference value. We calculated the
partitioning of elements between the atmosphere, oceans, and crust plus
mantle. The vapour-liquid equilibrium sets an upper limit to the partial
pressure of H2O in the atmosphere, $P^{\rm crit}_{\rm H_{2}O}=1.7\times
10^{-2}\ {\rm bar}$[122], calculated for the assumed surface temperature of
288 K. We impose the negative feedback of the carbonate-silicate cycle by
simply setting an upper limit to the partial pressure of $P^{\rm crit}_{\rm
CO_{2}}=10$ bar, as expected for the steady state [121]. Neglecting the time
lag to reach the steady state is justified by considering the short timescale
of carbonate precipitation compared with the duration of late accretion (see
main text). Atmospheric H and C in excess from the upper limits are
partitioned into oceans and crust plus mantle reservoirs, respectively.

Initial condition. We prepared the initial condition for the elemental
abundances by assuming a chondritic bulk composition the same as planetesimal
impactors and equilibrium partitioning between the atmosphere, the fully
molten magma ocean, and the core. While this is a crude assumption that
neglects the complexity of how planetary embryos formed, we have confirmed
that the results are insensitive to the initial condition because the system
soon evolves towards the quasi-steady state between the gain and loss of
volatile elements.

Table 1: List of key parameters for the model calculations. Observational data and model reference values for the abundances of C, N, and H in the bulk silicate Earth (BSE) and chondrites (assumed as building blocks, CI chondrites and enstatite chondrites). We set CI chondritic model, solubilities $S_{\rm i}$ (the units are $({P_{\rm i}}/{\rm 1MPa})^{1/2}\ {\rm ppm}$ for reduced $S_{\rm H}$ and $\left({P_{\rm i}}/{\rm 1MPa}\right)\ {\rm ppm}$ for the others, where $P_{\rm i}$ is the partial pressure of each molecule ${\rm i}$), and the partitioning coefficients $D_{\rm i}$ under each redox state by following Hirschmann (2016)[5]. a: Hirschmann (2016)[5], b: Marty (2012)[4], c: Hirschmann (2018)[17], d: Bergin et al. (2015)[15], e: Hirschmann & Dasgupta (2009), f: Grady & Wright, (2003)[81], g: Vacher et al. (2020)[82], h: Kerridge (1985) [79], i: Piani et al. (2020)[84], j: Stolper & Holloway (1988)[123], k:Pan et al. (1991)[20], l:Libourel et al. (2003)[114] m:Moore et al. (1998) [19], n:Chi et al. (2014)[100], o:Armstrong et al. (2015)[101], p:Roskosz et al. (2013)[103], q:Okuchi, (1997)[102], r: Wetzel et al. (2013)[112] s:Hirschmann et al. (2012)[67]. | C | N | H | C/N | C/H
---|---|---|---|---|---
BSE [$\mu$g/g-Earth] |  42–730a,b,c |  0.83–2.5a,b,c |  44–450a,b,c |  40 $\pm$ 8a,d |  1.3 $\pm$ 0.3 a,e
CI chondrites [ppm] |  20,000–50,000f |  500–2,000 f |  4,700–10,500g |  14.5 $\pm$ 2.5d |  2–8h
CI model [ppm] | 35,000 | 1,500 | 6,900 | 23 | 5
Enstatite chondrites [ppm] |  2,000–7,000f |  100–500 f |  90–600 i |  13.7$\pm$ 12.1d |
EC model [ppm] | 4,000 | 250 | 400 | 16 | 10
Oxidised model |  |  |  |  |
$S_{\rm i}$ |  1.6j,k |  1.0l |  M98 modelm |  |
$D_{\rm i}^{\rm met/sil}$ |  500n,o |  20p |  6.5q |  |
Intermediate model |  |  |  |  |
$S_{\rm i}$ | 0.55o | 5.0p | M98 modell |  |
$D_{\rm i}^{\rm met/sil}$ | 1000n,o | 20p | 6.5q |  |
Reduced model |  |  |  |  |
$S_{\rm i}$ |  0.22r |  50l |  5.0s |  |
$D_{\rm i}^{\rm met/sil}$ |  3,000n,o |  20p |  6.5q |  |

## References

  * [1] Abbot, D. S., Cowan, N. B. & Ciesla, F. J.  Indication of insensitivity of planetary weathering behavior and habitable zone to surface land fraction.  _The Astrophysical Journal_ 756, 178 (2012).
  * [2] Wordsworth, R. & Pierrehumbert, R.  Hydrogen-nitrogen greenhouse warming in Earth’s early atmosphere.  _science_ 339, 64–67 (2013).
  * [3] Foley, B. J.  The role of plate tectonic–climate coupling and exposed land area in the development of habitable climates on rocky planets.  _The Astrophysical Journal_ 812, 36 (2015).
  * [4] Marty, B.  The origins and concentrations of water, carbon, nitrogen and noble gases on Earth.  _Earth and Planetary Science Letters_ 313, 56–66 (2012).
  * [5] Hirschmann, M. M.  Constraints on the early delivery and fractionation of Earth’s major volatiles from C/H, C/N, and C/S ratios.  _American Mineralogist_ 101, 540–553 (2016).
  * [6] Halliday, A. N.  The origins of volatiles in the terrestrial planets.  _Geochimica et Cosmochimica Acta_ 105, 146–171 (2013).
  * [7] Walsh, K. J., Morbidelli, A., Raymond, S. N., O’brien, D. P. & Mandell, A. M.  A low mass for Mars from Jupiter’s early gas-driven migration.  _Nature_ 475, 206–209 (2011).
  * [8] Dauphas, N.  The isotopic nature of the Earth’s accreting material through time.  _Nature_ 541, 521–524 (2017).
  * [9] Fischer-Gödde, M. _et al._ Ruthenium isotope vestige of Earth’s pre-late-veneer mantle preserved in archaean rocks.  _Nature_ 579, 240–244 (2020).
  * [10] Elkins-Tanton, L. T.  Linked magma ocean solidification and atmospheric growth for Earth and Mars.  _Earth and Planetary Science Letters_ 271, 181–191 (2008).
  * [11] Dasgupta, R. & Grewal, D. S.  Origin and Early Differentiation of Carbon and Associated Life-Essential Volatile Elements on Earth.  In _Deep Carbon_ , 4–39 (Cambridge University Press, 2019).
  * [12] de Niem, D., Kührt, E., Morbidelli, A. & Motschmann, U.  Atmospheric erosion and replenishment induced by impacts upon the Earth and Mars during a heavy bombardment.  _Icarus_ 221, 495–507 (2012).
  * [13] Genda, H. & Abe, Y.  Enhanced atmospheric loss on protoplanets at the giant impact phase in the presence of oceans.  _Nature_ 433, 842–844 (2005).
  * [14] Sakuraba, H., Kurokawa, H. & Genda, H.  Impact degassing and atmospheric erosion on Venus, Earth, and Mars during the late accretion.  _Icarus_ 317, 48–58 (2019).
  * [15] Bergin, E. A., Blake, G. A., Ciesla, F., Hirschmann, M. M. & Li, J.  Tracing the ingredients for a habitable earth from interstellar space through planet formation.  _Proceedings of the National Academy of Sciences_ 112, 8965–8970 (2015).
  * [16] Grewal, D. S., Dasgupta, R., Sun, C., Tsuno, K. & Costin, G.  Delivery of carbon, nitrogen, and sulfur to the silicate Earth by a giant impact.  _Science advances_ 5, eaau3669 (2019).
  * [17] Hirschmann, M. M.  Comparative deep Earth volatile cycles: The case for C recycling from exosphere/mantle fractionation of major (H2O, C, N) volatiles and from H2O/Ce, CO2/Ba, and CO2/Nb exosphere ratios.  _Earth and Planetary Science Letters_ 502, 262–273 (2018).
  * [18] Dasgupta, R., Chi, H., Shimizu, N., Buono, A. S. & Walker, D.  Carbon solution and partitioning between metallic and silicate melts in a shallow magma ocean: Implications for the origin and distribution of terrestrial carbon.  _Geochimica et Cosmochimica Acta_ 102, 191–212 (2013).
  * [19] Moore, G., Vennemann, T. & Carmichael, I.  An empirical model for the solubility of H2O in magmas to 3 kilobars.  _american Mineralogist_ 83, 36–42 (1998).
  * [20] Pan, V., Holloway, J. R. & Hervig, R. L.  The pressure and temperature dependence of carbon dioxide solubility in tholeiitic basalt melts.  _Geochimica et Cosmochimica Acta_ 55, 1587–1595 (1991).
  * [21] Miyazaki, A., Hiyagon, H., Sugiura, N., Hirose, K. & Takahashi, E.  Solubilities of nitrogen and noble gases in silicate melts under various oxygen fugacities: implications for the origin and degassing history of nitrogen and noble gases in the Earth.  _Geochimica et Cosmochimica Acta_ 68, 387–401 (2004).
  * [22] Dalou, C., Hirschmann, M. M., von der Handt, A., Mosenfelder, J. & Armstrong, L. S.  Nitrogen and carbon fractionation during core–mantle differentiation at shallow depth.  _Earth and Planetary Science Letters_ 458, 141–151 (2017).
  * [23] Grewal, D. S., Dasgupta, R., Hough, T. & Farnell, A.  Rates of protoplanetary accretion and differentiation set nitrogen budget of rocky planets.  _Nature Geoscience_ 14, 369–376 (2021).
  * [24] Canup, R. M. & Asphaug, E.  Origin of the Moon in a giant impact near the end of the Earth’s formation.  _Nature_ 412, 708–712 (2001).
  * [25] Canup, R. M.  Simulations of a late lunar-forming impact.  _Icarus_ 168, 433–456 (2004).
  * [26] Salvador, A. _et al._ The relative influence of H2O and CO2 on the primitive surface conditions and evolution of rocky planets.  _Journal of Geophysical Research: Planets_ (2017).
  * [27] Nikolaou, A. _et al._ What factors affect the duration and outgassing of the terrestrial magma ocean?  _The Astrophysical Journal_ 875, 11 (2019).
  * [28] Hamano, K., Abe, Y. & Genda, H.  Emergence of two types of terrestrial planet on solidification of magma ocean.  _Nature_ 497, 607–610 (2013).
  * [29] Appel, P. W., Fedo, C. M., Moorbath, S. & Myers, J. S.  Early Archaean Isua supracrustal belt, West Greenland: pilot study of the Isua multidisciplinary research project.  _Geology of Greenland Survey Bulletin_ 180, 94–99 (1998).
  * [30] Komiya, T. _et al._ Plate tectonics at 3.8–3.7 Ga: Field evidence from the Isua accretionary complex, southern West Greenland.  _The Journal of Geology_ 107, 515–554 (1999).
  * [31] Wilde, S. A., Valley, J. W., Peck, W. H. & Graham, C. M.  Evidence from detrital zircons for the existence of continental crust and oceans on the Earth 4.4 Gyr ago.  _Nature_ 409, 175–178 (2001).
  * [32] Hopkins, M., Harrison, T. M. & Manning, C. E.  Low heat flow inferred from> 4 Gyr zircons suggests Hadean plate boundary interactions.  _Nature_ 456, 493–496 (2008).
  * [33] Korenaga, J.  Initiation and evolution of plate tectonics on Earth: theories and observations.  _Annual review of earth and planetary sciences_ 41, 117–151 (2013).
  * [34] Foley, B. J.  Habitability of Earth-like stagnant lid planets: Climate evolution and recovery from snowball states.  _The Astrophysical Journal_ 875, 72 (2019).
  * [35] Krissansen-Totton, J., Arney, G. N. & Catling, D. C.  Constraining the climate and ocean ph of the early Earth with a geological carbon cycle model.  _Proceedings of the National Academy of Sciences_ 201721296 (2018).
  * [36] Pierrehumbert, R. T.  _Principles of planetary climate_ (Cambridge University Press, 2010).
  * [37] Tucker, J. M. & Mukhopadhyay, S.  Evidence for multiple magma ocean outgassing and atmospheric loss episodes from mantle noble gases.  _Earth and Planetary Science Letters_ 393, 254–265 (2014).
  * [38] Hu, R. & Diaz, H. D.  Stability of nitrogen in planetary atmospheres in contact with liquid water.  _The Astrophysical Journal_ 886, 126 (2019).
  * [39] Stüeken, E. E., Kipp, M. A., Koehler, M. C. & Buick, R.  The evolution of Earth’s biogeochemical nitrogen cycle.  _Earth-Science Reviews_ 160, 220–239 (2016).
  * [40] Catling, D. C. & Zahnle, K. J.  The Archean atmosphere.  _Science Advances_ 6, eaax1420 (2020).
  * [41] Morbidelli, A. _et al._ The timeline of the lunar bombardment: Revisited.  _Icarus_ 305, 262–276 (2018).
  * [42] Schlichting, H. E., Sari, R. & Yalinewich, A.  Atmospheric mass loss during planet formation: The importance of planetesimal impacts.  _Icarus_ 247, 81–94 (2015).
  * [43] Goldreich, P., Lithwick, Y. & Sari, R.  Final stages of planet formation.  _The Astrophysical Journal_ 614, 497 (2004).
  * [44] Kenyon, S. J. & Bromley, B. C.  Terrestrial planet formation. I. The transition from oligarchic growth to chaotic growth.  _The Astronomical Journal_ 131, 1837 (2006).
  * [45] Kokubo, E. & Genda, H.  Formation of terrestrial planets from protoplanets under a realistic accretion condition.  _The Astrophysical Journal Letters_ 714, L21 (2010).
  * [46] Bottke, W. F. _et al._ The fossilized size distribution of the main asteroid belt.  _Icarus_ 175, 111–140 (2005).
  * [47] Bottke, W. _et al._ Dating the Moon-forming impact event with asteroidal meteorites.  _Science_ 348, 321–323 (2015).
  * [48] Bottke, W. F., Walker, R. J., Day, J. M., Nesvorny, D. & Elkins-Tanton, L.  Stochastic late accretion to Earth, the Moon, and Mars.  _Science_ 330, 1527–1530 (2010).
  * [49] Levison, H. F., Kretke, K. A., Walsh, K. J. & Bottke, W. F.  Growing the terrestrial planets from the gradual accumulation of submeter-sized objects.  _Proceedings of the National Academy of Sciences_ 112, 14180–14185 (2015).
  * [50] Brasser, R., Mojzsis, S., Werner, S., Matsumura, S. & Ida, S.  Late veneer and late accretion to the terrestrial planets.  _Earth and Planetary Science Letters_ 455, 85–93 (2016).
  * [51] Albarede, F.  Volatile accretion history of the terrestrial planets and dynamic implications.  _Nature_ 461, 1227–1233 (2009).
  * [52] Chou, C. L.  Fractionation of siderophile elements in the Earth’s upper mantle.  _Lunar and Planetary Sciences_ IX, 219–230 (1978).
  * [53] Wänke, H., Dreibus, G. & Jagoutz, E.  Mantle chemistry and accretion history of the Earth.  In _Archaean geochemistry_ , 1–24 (Springer, 1984).
  * [54] Wade, J. & Wood, B.  Core formation and the oxidation state of the Earth.  _Earth and Planetary Science Letters_ 236, 78–95 (2005).
  * [55] Schönbächler, M., Carlson, R., Horan, M., Mock, T. & Hauri, E.  Heterogeneous accretion and the moderately volatile element budget of Earth.  _Science_ 328, 884–887 (2010).
  * [56] Marchi, S., Canup, R. & Walker, R.  Heterogeneous delivery of silicate and metal to the Earth by large planetesimals.  _Nature Geoscience_ 11, 77 (2018).
  * [57] Korenaga, J., Planavsky, N. J. & Evans, D. A.  Global water cycle and the coevolution of the Earth’s interior and surface environment.  _Phil. Trans. R. Soc. A_ 375, 20150393 (2017).
  * [58] Peslier, A. H., Schönbächler, M., Busemann, H. & Karato, S.-I.  Water in the Earth’s interior: distribution and origin.  _Space Science Reviews_ 212, 743–810 (2017).
  * [59] Tsuno, K., Grewal, D. S. & Dasgupta, R.  Core-mantle fractionation of carbon in Earth and Mars: The effects of sulfur.  _Geochimica et Cosmochimica Acta_ 238, 477–495 (2018).
  * [60] Bureau, H., Pineau, F., Métrich, N., Semet, M. & Javoy, M.  A melt and fluid inclusion study of the gas phase at Piton de la Fournaise volcano (Réunion Island).  _Chemical geology_ 147, 115–130 (1998).
  * [61] Clesi, V. _et al._ Low hydrogen contents in the cores of terrestrial planets.  _Science advances_ 4, e1701876 (2018).
  * [62] Malavergne, V. _et al._ Experimental constraints on the fate of H and C during planetary core-mantle differentiation. Implications for the Earth.  _Icarus_ 321, 473–485 (2019).
  * [63] Siebert, J., Badro, J., Antonangeli, D. & Ryerson, F. J.  Terrestrial accretion under oxidizing conditions.  _Science_ 339, 1194–1197 (2013).
  * [64] Badro, J., Brodholt, J. P., Piet, H., Siebert, J. & Ryerson, F. J.  Core formation and core composition from coupled geochemical and geophysical constraints.  _Proceedings of the National Academy of Sciences_ 112, 12310–12314 (2015).
  * [65] Armstrong, K., Frost, D. J., McCammon, C. A., Rubie, D. C. & Ballaran, T. B.  Deep magma ocean formation set the oxidation state of Earth’s mantle.  _Science_ 365, 903–906 (2019).
  * [66] Wood, B. J., Walter, M. J. & Wade, J.  Accretion of the Earth and segregation of its core.  _Nature_ 441, 825 (2006).
  * [67] Hirschmann, M., Withers, A., Ardia, P. & Foley, N.  Solubility of molecular hydrogen in silicate melts and consequences for volatile evolution of terrestrial planets.  _Earth and Planetary Science Letters_ 345, 38–48 (2012).
  * [68] Fischer, R. A. _et al._ High pressure metal–silicate partitioning of Ni, Co, V, Cr, Si, and O.  _Geochimica et Cosmochimica Acta_ 167, 177–194 (2015).
  * [69] Rubie, D. C. _et al._ Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water.  _Icarus_ 248, 89–108 (2015).
  * [70] Hirose, K., Labrosse, S. & Hernlund, J.  Composition and state of the core.  _Annual Review of Earth and Planetary Sciences_ 41, 657–691 (2013).
  * [71] Dauphas, N. & Morbidelli, A.  6.1 - Geochemical and Planetary Dynamical Views on the origin of Earth’s Atmosphere and Oceans.  In Holland, H. D. & Turekian, K. K. (eds.) _Treatise on Geochemistry (Second Edition)_ , 1–35 (Elsevier, Oxford, 2014), second edition edn.
  * [72] Forget, F. _et al._ 3d modelling of the early Martian climate under a denser CO2 atmosphere: Temperatures and CO2 ice clouds.  _Icarus_ 222, 81–99 (2013).
  * [73] Wadhwa, M.  Redox state of Mars’ upper mantle and crust from Eu anomalies in shergottite pyroxenes.  _Science_ 291, 1527–1530 (2001).
  * [74] Herd, C. D., Borg, L. E., Jones, J. H. & Papike, J. J.  Oxygen fugacity and geochemical variations in the martian basalts: Implications for martian basalt petrogenesis and the oxidation state of the upper mantle of Mars.  _Geochimica et Cosmochimica Acta_ 66, 2025–2036 (2002).
  * [75] Hirschmann, M. M. & Withers, A. C.  Ventilation of CO2 from a reduced mantle and consequences for the early Martian greenhouse.  _Earth and Planetary Science Letters_ 270, 147–155 (2008).
  * [76] Zhang, H., Hirschmann, M., Cottrell, E. & Withers, A.  Effect of pressure on Fe3+/$\Sigma$Fe ratio in a mafic magma and consequences for magma ocean redox gradients.  _Geochimica et Cosmochimica Acta_ 204, 83–103 (2017).
  * [77] Seager, S., Kuchner, M., Hier-Majumder, C. & Militzer, B.  Mass-radius relationships for solid exoplanets.  _The Astrophysical Journal_ 669, 1279 (2007).
  * [78] Ida, S. & Makino, J.  N-body simulation of gravitational interaction between planetesimals and a protoplanet: I. velocity distribution of planetesimals.  _Icarus_ 96, 107–120 (1992).
  * [79] Kerridge, J. F.  Carbon, hydrogen and nitrogen in carbonaceous chondrites: Abundances and isotopic compositions in bulk samples.  _Geochimica et Cosmochimica Acta_ 49, 1707–1714 (1985).
  * [80] Pepin, R. O.  On the origin and early evolution of terrestrial planet atmospheres and meteoritic volatiles.  _Icarus_ 92, 2–79 (1991).
  * [81] Grady, M. M. & Wright, I. P.  Elemental and isotopic abundances of carbon and nitrogen in meteorites.  _Space Science Reviews_ 106, 231–248 (2003).
  * [82] Vacher, L. G. _et al._ Hydrogen in chondrites: Influence of parent body alteration and atmospheric contamination on primordial components.  _Geochimica et Cosmochimica Acta_ 281, 53–66 (2020).
  * [83] Robert, F.  The D/H ratio in chondrites.  _Space Science Reviews_ 106, 87–101 (2003).
  * [84] Piani, L. _et al._ Earth’s water may have been inherited from material similar to enstatite chondrite meteorites.  _Science_ 369, 1110–1113 (2020).
  * [85] Shuvalov, V.  Atmospheric erosion induced by oblique impacts.  _Meteoritics & Planetary Science_ 44, 1095–1105 (2009).
  * [86] Svetsov, V.  Atmospheric erosion and replenishment induced by impacts of cosmic bodies upon the Earth and Mars.  _Solar System Research_ 41, 28–41 (2007).
  * [87] Svetsov, V.  On the efficiency of the impact mechanism of atmospheric erosion.  _Solar Syst. Res._ 34, 398–410 (2000).
  * [88] Hosono, N., Karato, S.-i., Makino, J. & Saitoh, T. R.  Terrestrial magma ocean origin of the Moon.  _Nature Geoscience_ 1 (2019).
  * [89] Kaib, N. A. & Cowan, N. B.  The feeding zones of terrestrial planets and insights into Moon formation.  _Icarus_ 252, 161–174 (2015).
  * [90] Kegerreis, J. _et al._ Planetary giant impacts: convergence of high-resolution simulations using efficient spherical initial conditions and swift.  _Monthly Notices of the Royal Astronomical Society_ 487, 5029–5040 (2019).
  * [91] Kegerreis, J. A. _et al._ Atmospheric erosion by giant impacts onto terrestrial planets: A scaling law for any speed, angle, mass, and density.  _The Astrophysical Journal Letters_ 901, L31 (2020).
  * [92] Rubie, D., Melosh, H., Reid, J., Liebske, C. & Righter, K.  Mechanisms of metal–silicate equilibration in the terrestrial magma ocean.  _Earth and Planetary Science Letters_ 205, 239–255 (2003).
  * [93] Rubie, D., Nimmo, F. & Melosh, H.  _Formation of Earth’s Core. Treatise on Geophysics. V. 1/Ed. in Chief G. Schubert_ (Elsevier B.V., 2015).
  * [94] Kleine, T., Münker, C., Mezger, K. & Palme, H.  Rapid accretion and early core formation on asteroids and the terrestrial planets from hf–w chronometry.  _Nature_ 418, 952 (2002).
  * [95] Pritchard, M. & Stevenson, D.  Thermal aspects of a lunar origin by giant impact.  _Origin of the Earth and Moon_ 1, 179–196 (2000).
  * [96] Deguen, R., Olson, P. & Cardin, P.  Experiments on turbulent metal-silicate mixing in a magma ocean.  _Earth and Planetary Science Letters_ 310, 303–313 (2011).
  * [97] Hier-Majumder, S. & Hirschmann, M. M.  The origin of volatiles in the Earth’s mantle.  _Geochemistry, Geophysics, Geosystems_ 18, 3078–3092 (2017).
  * [98] Li, Y., Wiedenbeck, M., Shcheka, S. & Keppler, H.  Nitrogen solubility in upper mantle minerals.  _Earth and Planetary Science Letters_ 377, 311–323 (2013).
  * [99] Palme, H. & O’Neill, H.  Cosmochemical Estimates of Mantle Composition. Planets, Asteriods, Comets and The Solar System, Volume 2 of Treatise on Geochemistry. Edited by Andrew M. Davis (2014).
  * [100] Chi, H., Dasgupta, R., Duncan, M. S. & Shimizu, N.  Partitioning of carbon between Fe-rich alloy melt and silicate melt in a magma ocean–implications for the abundance and origin of volatiles in Earth, Mars, and the Moon.  _Geochimica et Cosmochimica Acta_ 139, 447–471 (2014).
  * [101] Armstrong, L. S., Hirschmann, M. M., Stanley, B. D., Falksen, E. G. & Jacobsen, S. D.  Speciation and solubility of reduced C–O–H–N volatiles in mafic melt: Implications for volcanism, atmospheric evolution, and deep volatile cycles in the terrestrial planets.  _Geochimica et Cosmochimica Acta_ 171, 283–302 (2015).
  * [102] Okuchi, T.  Hydrogen partitioning into molten iron at high pressure: implications for Earth’s core.  _Science_ 278, 1781–1784 (1997).
  * [103] Roskosz, M., Bouhifd, M. A., Jephcoat, A., Marty, B. & Mysen, B.  Nitrogen solubility in molten metal and silicate at high pressure and temperature.  _Geochimica et Cosmochimica Acta_ 121, 15–28 (2013).
  * [104] Li, Y., Dasgupta, R., Tsuno, K., Monteleone, B. & Shimizu, N.  Carbon and sulfur budget of the silicate Earth explained by accretion of differentiated planetary embryos.  _Nature Geoscience_ 9, 781–785 (2016).
  * [105] Fischer, R. A., Cottrell, E., Hauri, E., Lee, K. K. & Le Voyer, M.  The carbon content of Earth and its core.  _Proceedings of the National Academy of Sciences_ 117, 8743–8749 (2020).
  * [106] Li, Y.-F., Marty, B., Shcheka, S., Zimmermann, L. & Keppler, H.  Nitrogen isotope fractionation during terrestrial core-mantle separation.  _Geochemical Perspectives Letters_ 2 (2016).
  * [107] Speelmanns, I. M., Schmidt, M. W. & Liebske, C.  The almost lithophile character of nitrogen during core formation.  _Earth and Planetary Science Letters_ 510, 186–197 (2019).
  * [108] Grewal, D. S. _et al._ The fate of nitrogen during core-mantle separation on Earth.  _Geochimica et Cosmochimica Acta_ 251, 87–115 (2019).
  * [109] Shibazaki, Y., Ohtani, E., Terasaki, H., Suzuki, A. & Funakoshi, K.  Hydrogen partitioning between iron and ringwoodite: Implications for water transport into the Martian core.  _Earth and Planetary Science Letters_ 287, 463–470 (2009).
  * [110] Li, Y., Vočadlo, L., Sun, T. & Brodholt, J. P.  The Earth’s core as a reservoir of water.  _Nature Geoscience_ 13, 453–458 (2020).
  * [111] Tagawa, S. _et al._ Experimental evidence for hydrogen incorporation into Earth’s core.  _Nature communications_ 12, 1–8 (2021).
  * [112] Wetzel, D. T., Rutherford, M. J., Jacobsen, S. D., Hauri, E. H. & Saal, A. E.  Degassing of reduced carbon from planetary basalts.  _Proceedings of the National Academy of Sciences_ 110, 8010–8013 (2013).
  * [113] Stanley, B. D., Hirschmann, M. M. & Withers, A. C.  Solubility of C–O–H volatiles in graphite-saturated martian basalts.  _Geochimica et Cosmochimica Acta_ 129, 54–76 (2014).
  * [114] Libourel, G., Marty, B. & Humbert, F.  Nitrogen solubility in basaltic melt. Part I. Effect of oxygen fugacity.  _Geochimica et Cosmochimica Acta_ 67, 4123–4135 (2003).
  * [115] Boulliung, J. _et al._ Oxygen fugacity and melt composition controls on nitrogen solubility in silicate melts.  _Geochimica et Cosmochimica Acta_ 284, 120–133 (2020).
  * [116] Abe, Y. & Matsui, T.  Evolution of an impact-generated H2O–CO2 atmosphere and formation of a hot proto-ocean on Earth.  _Journal of the Atmospheric Sciences_ 45, 3081–3101 (1988).
  * [117] Kasting, J. F.  Runaway and moist greenhouse atmospheres and the evolution of Earth and Venus.  _Icarus_ 74, 472–494 (1988).
  * [118] Nakajima, S., Hayashi, Y.-Y. & Abe, Y.  A study on the “runaway greenhouse effect” with a one-dimensional radiative–convective equilibrium model.  _Journal of Atmospheric Sciences_ 49, 2256–2266 (1992).
  * [119] Kurokawa, H., Foriel, J., Laneuville, M., Houser, C. & Usui, T.  Subduction and atmospheric escape of Earth’s seawater constrained by hydrogen isotopes.  _Earth and Planetary Science Letters_ 497, 149–160 (2018).
  * [120] Walker, J. C., Hays, P. & Kasting, J. F.  A negative feedback mechanism for the long-term stabilization of Earth’s surface temperature.  _Journal of Geophysical Research: Oceans_ 86, 9776–9782 (1981).
  * [121] Kasting, J. F.  Earth’s early atmosphere.  _Science_ 920–926 (1993).
  * [122] Murphy, D. M. & Koop, T.  Review of the vapour pressures of ice and supercooled water for atmospheric applications.  _Quarterly Journal of the Royal Meteorological Society_ 131, 1539–1565 (2005).
  * [123] Stolper, E. & Holloway, J. R.  Experimental determination of the solubility of carbon dioxide in molten basalt at low pressure.  _Earth and Planetary Science Letters_ 87, 397–408 (1988).

## Acknowledgements (not compulsory)

This work was supported by JSPS KAKENHI Grant No. 19J21445 and 18K13602, and
MEXT KAKENHI Grant No. JP17H06457.

## Author contributions statement

H.S., H.K., H.G., and K.O. designed the project; H.S. performed the numerical
simulations and analysed the results; H.S. and H.K. interpreted the results
and wrote the manuscript.

## Additional information

Competing interests: The authors declare no competing interests.

## Code availability

The codes used to generate these results and the data that support the
findings of this study are available at
http://www.geo.titech.ac.jp/lab/okuzumi/sakuraba/Contents.html.


* [22] T. Kugo, I. Ojima: Local covariant operator formalism of non-abelian gauge theories and quark confinement problem. Suppl. Prog. Theor. Phys. 66 (1979) 1–130.
  * [23] J. Mund, K.-H. Rehren, B. Schroer: Helicity decoupling in the massless limit of massive tensor fields. Nucl. Phys. B924 (2017) 699–727.
  * [24] J. Mund, K.-H. Rehren, B. Schroer: Gauss’ Law and string-localized quantum field theory. JHEP 01 (2020) 001.
  * [25] J. Mund, K.-H. Rehren, B. Schroer: Infraparticle fields and the formation of photon clouds. JHEP 04 (2022) 083.
  * [26] J. Mund, B. Schroer, J. Yngvason: String-localized quantum fields and modular localization. Commun. Math. Phys. 268 (2006) 621–672.
  * [27] M.E. Peskin, D.V. Schroeder: An Introduction to Quantum Field Theory, Pegasus Books, Reading (MA) 1995.
  * [28] H. Ruegg, M. Ruiz-Altaba: The Stueckelberg field. Int. J. Mod. Phys. A19 (2004) 3265–3347.
  * [29] G. Scharf: Quantum Gauge Theories: A True Ghost Story, Wiley (2001).
  * [30] B. Schroer: An alternative to the gauge theoretic setting. Found. of Phys. 41 (2011) 1543–1568.
  * [31] B. Schroer: The role of positivity and causality in interactions involving higher spin. Nucl. Phys. B941 (2019) 91–144.
  * [32] M.D. Schwartz: Quantum Field Theory and the Standard Model, Cambridge University Press 2014.
  * [33] F. Tippner, “Scalar QED with String-Localised Potentials”, Bachelor’s Thesis, Göttingen University, 2019.
  * [34] S. Weinberg: The Quantum Theory of Fields, Cambridge University Press 1995.
  * [35] E.P. Wigner: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. (2nd Ser.) 40 (1939) 149–204.
  * [36] J. Yngvason: Zero-mass infinite spin representations of the Poincaré group and quantum field theory, Commun. Math. Phys. 18 (1970) 195
  * [37] T.T. Wu, S.L Wu: Comparing the $R_{\xi}$ gauge and the unitary gauge for the standard model: An example. Nucl. Phys. B914 (2017) 421–445.