JuanCastillo29/space-mission-intelligence-data huggingface dataset:
|  | H2O: 4.48, CO2: 760.19, CO: 182.62, H2: 0.13, CH4: 0.00 H2O: 5.08, CO2: 851.88, CO: 119.81, H2: 0.11, CH4: 0.00 H2O: 18.75, CO2: 1155.71, CO: 63.72, H2: 0.25, CH4: 0.00 H2O: 63.67, CO2: 1300.49, CO: 61.41, H2: 0.79, CH4: 0.00 ATMOSPHERE |  |
| --- | --- | --- |
| Time [Myr] t=0 t=0.013 t=0.7 t=30 Solidus (Reference) Liquidus (Reference) Final Psurf = 1429 bar |  | MANTLE |

|  | H2O: 4.54, CO2: 762.05, CO: 183.07, H2: 0.13, CH4: 0.00 H2O: 5.21, CO2: 847.48, CO: 128.39, H2: 0.12, CH4: 0.00 H2O: 6.84, CO2: 919.42, CO: 110.98, H2: 0.14, CH4: 0.00 H2O: 593.74, CO2: 1128.82, CO: 121.41, H2: 11.18, CH4: 0.00 ATMOSPHERE |  |
| --- | --- | --- |
| Time [Myr] t=0 t=0.0096 t=0.032 t=0.83 Solidus (IW 2) Liquidus (IW 2) Final Psurf = 1862 bar |  | MANTLE |

Pressure [bar]
Pressure [bar]
Pressure [GPa]
Pressure [GPa]

[Figure: Fig. 3. T–P profiles for two simulations of a 3 M⊕planet showing the coupled interior–atmosphere evolution from an initially global magma ocean to radiative equilibrium (partial pressures in bar). Left: Baseline melting curves from Andrault et al. (2011); Hamano et al. (2013). The mantle remains partially molten for an extended period and solidifies through fractional crystallisation, reaching steady state after ∼30 Myr with a surface pressure of ≈1429 bar. Right: Redox-dependent melting curves shifted to higher temperatures (IW-2.0). Earlier deep-mantle crystallisation confines melt to a shallow magma ocean sustained by greenhouse warming from outgassed water. Steady state is reached after 0.83 Myr with a surface pressure of ≈1862 bar.]
viscosity convecting melt from a high-viscosity, partially crystal-
lized layer. This transition arises from the viscosity parametriza-
tion as a function of melt fraction, with a sharp increase occur-
ring around a critical melt fraction of ∼40%, where the solid
phase starts to control the dynamic timescale (Bower et al. 2022).
In the IW-2.0 case, water initially dissolved in the deep magma is
progressively exsolved as crystallisation proceeds, increasing the
atmospheric opacity and enhancing greenhouse warming. This
feedback allows the thin, near-surface magma layer to persist
even after most of the deeper mantle has solidified. The compar-
-dependent shift of the melting
of magma ocean evolution occur within the first tens to hundreds
of of megayears (Nicholls et al. 2024). The converged simula-
tions all reach radiative equilibrium well within this timescale,
suggesting that extending the cutoff further would not signif-
icantly affect our results. Fig
4 shows the distribution of in-
terior properties as a function of the main physical and com-
positional drivers, with rows representing the independent pa-
/IW), orbital
First, the rheological front depth (Fig.4, third column; panels
A3–F3) exhibits a pronounced bi modality, with peaks near the
core–mantle boundary and close to the surface. This bi modal-
ity reflects the existence of two dominant evolutionary pathways
(first column; panels A1–F1): planets that reach radiative equi-
librium at very early stages while remaining fully molten, char-
acterized by near-unity melt fractions and rheological fronts an-
chored at the core-mantle boundary (located at ∼ 0.55 in normal-
ized planetary radius, corresponding to the adopted core radius
fraction), and planets that undergo partial crystallisation near the
surface, leading to reduced melt fractions and shallow viscos-
ity contrasts. To complement the density distributions (Fig 4),
we additionally showed cumulative distributions (Fig A.1 in Ap-
pendix) to quantify the fraction of the simulation ensemble that
As a next step, we characterised the diversity of final states and
magma ocean lifetimes by examining the global structure of the
parameter grid. We analysed a suite of 505 out of 600 simula-
tions that successfully ran to completion, where the 600 mod-
els result from the combination of all parameter values listed in
Table 1. The remaining 95 simulations did not converge, pre-
dominantly at the most reducing melting curve (IW−4.0), where
the elevated solidus temperatures led to numerical instabilities
at high pressures. All converged simulations reached radiative
equilibrium before fully solidifying, indicating the presence of
long-lived magma oceans of varying sizes. Regarding the choice
Article number, page 8 of 19

CMB
Separation [AU]
C3
0.07
0.5
0.5
0.5
0.5
0.0153
0.0
103 104 105 106 107 108
0 20 40 60 80 100 0.0
0.0 0.1 0.2 0.3 0.4 0.0
1 2 3 4 0.0
CMB
E3
Planet mass [M ]
3.0
0.5
0.5
0.5
0.5
1.0
0.0
103 104 105 106 107 108
0 20 40 60 80 100 0.0
0.0 0.1 0.2 0.3 0.4 0.0
1 2 3 4 0.0
CMB
0.5
0.5
0.5
0.5
-2
-4
0.0
0.0
0.0
0.0
103 104 105 106 107 108
0 20 40 60 80 100 Melt fraction [%]
0.0 0.1 0.2 0.3 0.4 Rheological front depth [R-d]
1 2 3 4 Magma temperature [kK]
Time to radiative equilibrium [yr]

[Figure: Fig. 4. Smoothed distributions of simulation outcomes across the parameter grid. Rows show variations in the driving parameters (C/H ratio, surface oxidation state fOsurf]
Similarly, Fig 5 and A.2 shows the distribution of atmospheric
outcomes across the full parameter grid as a function of the main
physical and compositional drivers in the PROTEUS simulations.
Each column represents a key atmospheric quantity, including
The overlapping probability and cumulative distributions pro-
vide a sensitivity map of how different input parameters regulate
volatile outgassing and atmospheric composition.
Initially, the bulk C/H ratio distributions primarily reflect
the melt state established by volatile-rich cases that reach ra-
diative equilibrium early in their evolution. Within this frame-
work, variations in C/H modulate how carbon is partitioned be-
tween the interior and the atmosphere. Because carbon is only
weakly soluble in silicate melt, simulations that remain largely
molten preferentially retain carbon-bearing species in the atmo-
sphere, leading to enhanced surface pressures in carbon-rich sys-
tems. These effects are compounded by carbon’s high molecular
weight, which further enhances surface pressure in C-dominated
atmospheres, consistent with the results shown in Bower et al.
(2022).
exerts a primary control
bital separation shows only a weak influence on atmospheric
outcomes (row C). While more distant planets tend to occupy
slightly cooler regimes, the surface and partial pressures remain
broadly similar across the explored range. Therefore, during the
magma ocean stage, volatile in/outgassing are primarily con-
trolled by internal thermochemical processes rather than exter-
nal irradiation, for these two modestly-irradiated scenarios. The
initial hydrogen inventory strongly regulates atmospheric mass
and thermal state through greenhouse feedbacks. Large volatile
tions toward high surface pressures, near-unity melt fractions,
and elevated surface temperatures.
Planetary mass (row E) introduces only minor variations in
atmospheric outcomes across the explored parameter space. In
our model, mass enters the interior structure through hydrostatic
equilibrium, where it determines the radial pressure profile and
thus the mantle grid, and through the gravitational acceleration
(g), which controls the pressure–depth relation and influences
volatile solubility and degassing. In the atmosphere, g further
affects the pressure structure and radiative properties. Despite
these dependencies, the distributions of total atmospheric mass,
surface pressure, and volatile partial pressures remain broadly
and structural effects play a limited role in regulating outgassing
in these scenarios.
(row F) af-
oceans. While the impactograms in Fig 5 summarize global sen-
sitivities across the parameter space, the following figures re-
solve individual volatile contributions and quantify how differ-
ent species partition into the atmosphere under contrasting re-
dox and inventory regimes. This allows us to directly link deep-
mantle oxidation state and volatile budgets to observable atmo-
spheric compositions.
and atmospheric composition for simulations with 3, 10, and
to the total pressure.
The results reveal a systematic decrease in total surface pres-
, consis-
strongly affect the
rate and extent of crystallization, and therefore directly control
the mass of volatiles that can be retained in the interior versus
released to the atmosphere. This behaviour is consistent with the
trends identified in the impactogram A.1, and supports a sim-
plified interpretation in which deep-mantle redox state governs
the mass balance of volatile partitioning through its control on
bulk melt fraction, while surface redox conditions regulate the
chemical speciation of the atmosphere.
Overall, these results demonstrate that the deep-mantle oxi-
and the initial volatile inventory jointly reg-
ulate the timing and the amount of volatiles released by con-
trolling how the bulk melt fraction evolves over time. As the
interior progressively solidifies, volatiles are released at differ-
ent rates, while the surface temperature further influences at-
mospheric composition and, in turn, the cooling rate. By con-
trolling the mantle melting curves and therefore the melt frac-
governs how long and how extensively the man-
fOmelt
0.5
0.5
0.5
-2
-4
0.0
0.0
0.0
100 101 102 103 104 105
100 101 102 103 104 105
100 101 102 103 104 105
Total surface pressure [bar]
H2O partial pressure [bar]
CO2 partial pressure [bar]

[Figure: Fig. 5. Smoothed density distributions of atmospheric outcomes across the parameter grid. Rows show variations in the driving parameters (C/H ratio, surface oxidation state fOsurf]

relative to the iron–wüstite buffer, orbital separation, volatile inventory in H-ocean mass, planetary mass, and deep-mantle oxidation state fOmelt
), while columns show interior outcomes (time to radiative equilibrium, bulk melt fraction, rheologicalfront depth, and surface temperature). Coloured curves represent kernel density estimates of simulations sharing a given parameter value. This smoothing is used only for visualisation to highlight trends and clustering across the discrete simulation outcomes.
negative values indicate preferential retention within the man-
tle. This representation provides a direct, quantitative measure
of degassing efficiency and reveals how mantle redox conditions
and volatile inventory jointly regulate volatile partitioning dur-
ing magma ocean outgassing.
transferred to the atmosphere. This behaviour reflects the high
solubility of water in silicate melt and its limited degassing ef-
ficiency under high melt fractions (Bower et al. 2022, 2025). In
contrast, carbon-bearing species exhibit a pronounced behaviour
atmosphere, with typical ratios of log M /M ≃ +1.5 to +0.8

1.0
0.5
Hoceans = 3
0.0
1.0
tention at high oxidation states due to high melt fractions.
Hydrogen-bearing species show the strongest partitioning
contrast. Molecular hydrogen is highly atmosphere-dominated
soluble in silicate melt and therefore absent from the interior
also preferentially partitions into the atmosphere, but exhibits
-dependent behaviour similar to that of carbon-bearing
species, with higher log M /M ratios under reducing condi-

Hoceans = 100
0.985
100 101 102 103 104 105 106 107 108
Median Time for Equlibrium [yr]

[Figure: Fig. 7. Evolution of the mantle melt fraction with time for three initial water inventories (Hoceans = 3, 10, 100; top to bottom) and different redox-dependent melting scenarios (fOmelt]
+1.9
+1.9
+1.9
+1.8

CH4
-4 -2 0 +2 +4 fOmelt
-4 -2 0 +2 +4 fOmelt
-4 -2 0 +2 +4 fOmelt

[Figure: Fig. 8. Atmosphere-interior partitioning of major volatile species as a function of deep-mantle oxidation state fOmelt]
associated with reduced conditions (IW-2.0) shift the melting re-
gion to higher temperatures, leading to earlier progression to-
wards solidification at depth and reorganizing the distribution
of melt within the planet. Instead of sustaining a deep magma
ocean as in the reference case, reduced melting curves pro-
mote more extensive interior solidification, reducing the depth of
the molten region and confining melt to a shallow near-surface
layer. At early times, however, the migration of the rheologi-
cal front is controlled by the atmospheric cooling rate, as the
interior efficiently resupplies heat to the surface. In this configu-
ration, water released during interior solidification increases at-
mospheric opacity and enhances greenhouse warming, thereby
slowing cooling and allowing the shallow surface magma layer
to persist. In contrast, cooler melting curves associated with ox-
idized conditions (IW+2.0 to IW+4.0) allow extensive melting
to persist in the deep interior for longer, maintaining a deep and
vertically extended magma ocean.
influences not
melting temperature differences can meaningfully influence
planetary evolution. Our results show that the answer depends
critically on the volatile inventory. Redox state undoubtedly af-
fects the melting curve: reducing conditions shift it to higher
temperatures, whereas oxidizing conditions lower it. This off-
set determines when the temperature profile first intersects the
solidus and, consequently, the conditions under which solidifica-
tion initiates. However, whether this difference propagates into
large-scale evolutionary consequences depends on whether the
thermal profile intersects the melting curves before the system
approaches thermal equilibrium.
profile remains far above the solidus for most of the evolution.
These simulations never reach the melting region at all, because
the surface cools into radiative equilibrium long before the in-
terior thermal profile approaches the melting point. As a result,
even a several-hundred-Kelvin redox shift produces little effect
on melt fraction, cooling timescales, or the depth of the con-
does not strongly influence
volatile speciation: hydrogen and carbon remain volatile under
reducing conditions, but oxidize into less volatile species that
dissolve more readily in the melt (Sossi et al. 2020; Nicholls
et al. 2024; Bower et al. 2025; Lichtenberg et al. 2025).
These compositional trends directly affect the atmospheric
mean molecular weight and total surface pressure. Reduced
cases generate the most massive atmospheres, often exceeding
one kilobar for the volatile-rich cases, while oxidized cases gen-
In reality, substantial hydrogen loss
may occur on relatively short timescales, potentially reducing
fer et al. 2016). In particular, predictions concerning the abso-
mospheres would benefit most from a self-consistent treatment
of escape processes.
In addition, we do not explicitly resolve a depth-dependent
oxygen fugacity structure. In the present formulation, the redox
dependence of melting is implemented as a temperature shift
produce a uniform offset while preserving the underlying pres-
function of depth, this would lead to local modifications of the
solidus temperature at each pressure level, effectively producing
a depth-dependent perturbation to the melting relations and po-
tentially shifting the depth at which the interior adiabat intersects
the solidus. Near-surface reservoirs are generally expected to be
more oxidized than the deep interior. While our parameter grid
explores all combinations as a broad sensitivity study, scenarios
are the more geophysically re-
plausible interior-atmosphere redox configurations.
Beyond the effects of oxygen fugacity on the melting curves,
dissolved water in the melt would further modify our results
through two main mechanisms. First, as referenced in Section 1
and Section 2.6 of this study, water depresses the solidus through
hydroxylation of the silicate network (Katz et al. 2003), whereby
bonds and stabilizing the liquid phase, with volatile-saturated sil-
icates exhibiting solidus depressions of up to 600−800 K (Das-
gupta & Hirschmann 2007; Myhill et al. 2017; Xie et al. 2024).
This would tend to maintain higher melt fractions for longer, ex-
tending magma ocean lifetimes and affecting outgassing rates,
particularly in the volatile-poor regime where the thermal pro-
file approaches the solidus. Second, and not accounted for in our
model, water significantly lowers melt viscosity, which would
enhance convective heat transport through the mantle. In this
way, a less viscous melt would increase the efficiency of con-
vection, therefore accelerating cooling.
Despite these assumptions, the physical trends identified
in this work remain robust. Reduced planets may present
favourable observational signatures. For low to intermediate
produce strong molecular absorption features in the 3–5 µm
range accessible to JWST and future ELT instruments (Katyal
et al. 2020; Nicholls et al. 2024). More extremely reduced and
ences suggest that future observations may help constrain deep-
interior redox states and volatile inventories through atmospheric
spectra. Therefore, the initial volatile inventory emerges as the
on the volatile spe-
able volatile budget. As recent JWST observations highlight, the
atmospheric properties of close-in rocky exoplanets are not yet
fully explained. Integrating interior redox evolution, volatile par-
titioning, water-dependent melting feedbacks and atmospheric
escape will therefore be essential for interpreting upcoming de-
tections of secondary atmospheres and for linking observable
spectra to the geochemical state of rocky exoplanet interiors like
in the case of GJ 1132 b, TOI-561 b, and TOI-6255 b.
ventory jointly shape magma ocean evolution and atmospheric
composition in rocky exoplanets, using GJ 1132 b as a reference
tle melting curves, allowing us to assess how variations in re-
dox state propagate into the thermal structure, melt fraction, and
atmospheric composition. We explored a wide range of redox
states (IW-4.0 to IW+4.0) and volatile inventories to determine
under which conditions redox-dependent melting curves signifi-
cantly influence interior evolution and potentially observable at-
mospheric properties.
Our results reveal a hierarchy of controls on magma ocean
evolution, where volatile inventory sets a primary control on
whether the redox-dependent melting relations modulate or not
bulk melt fraction, and consequently the size of the melt reser-
voir available for volatile dissolution and outgassing. Our main
conclusions are as follows:
portant for volatile-poor planets where the mantle ther-
mal profile approaches the solidus. In this regime, hot-
ter melting curves associated with reducing mantle con-
ditions trigger earlier crystallisation, promote a shallow-
surface magma ocean and enhance interior–atmosphere par-
titioning, whereas cooler melting curves associated with oxi-
dizing conditions prolong the fully molten state and promote
a more radially distributed melt reservoir. In contrast, for
volatile-rich planets, the mantle thermal profile remains far
above the melting region and reaches radiative equilibrium
before crystallisation occurs, making the thermal evolution
.
dependent redox evolution, solid-phase volatile trapping, and at-
mospheric escape to enable fully self-consistent predictions of
thermal evolution and spectroscopic observables for rocky exo-
planets.

Acknowledgements. The authors thank the anonymous referee for their constructive comments, which helped improve the manuscript. MS thanks Laura Article number, page 15 of 19

Kreidberg, Mara Attia, Rob Spaargaren and Christopher Boettner for useful comments and suggestions. This work was partially supported by the Branco Weiss Foundation, the Netherlands eScience Center (PROTEUS project, NLESC.OEC.2023.017), the Alfred P. Sloan Foundation (AEThER project, G202114194), NASA’s Nexus for Exoplanet System Science research coordination network (Alien Earths project, 80NSSC21K0593), and the NWO NWAORC PRELIFE Consortium (PRELIFE project, NWA.1630.23.013). HN acknowledges support from STFC grant UKRI1184. We also thank the Center for Information Technology of the University of Groningen for their support and for providing access to the Habrok high performance computing cluster.
stood, as we now overview.
| ˙ωSRF| ∝ 1 (1 −e2)n , (1) where n = 1 for GR, n ≃4.5 for equilibrium tides, and n = 1.5 for rotational bulges.

One can include SRFs in the inner orbit dynamics in ZLK calculations.

In the DA approximation, the dynamics are again integrable.

The two constants of motion are now jz, as in the baseline case described above, and D, which differs from the baseline CK only at extremely high eccentricity (see §2.1).

The upshot is that SRFs always lower the maximum eccentricity emax relative to its no-SRF DA value (Fabrycky & Tremaine 2007) (answering question (i) above) and they leave the secular period PK unchanged (answering question (ii)).

These ‘DA+SRF’ dynamics were distilled into a simple recipe by Liu et al. (2015).

Their paper has become a standard reference in population-syntheses studies of secularly driven hot Jupiters, compact-object mergers, and tidal-disruption channels (e.g., Anderson et al. 2016; Petrovich 2015; Stephan et al. 2016; Mu˜noz et al. 2016; Owen & Lai 2018; Teyssandier et al. 2019; Vick et al. 2019; Stephan et al. 2021; O’Connor et al. 2021; Melchor et al. 2024).
1.2.

Single-averaging (SA) Parallel to this, many authors have also recognized that the DA approximation itself can fail (we will discuss the criterion for this in §2.2).

In these circumstances, the natural next step is to use the single-averaging (SA) approximation, which averages only over the inner orbit.

In an SA calculation, the torque due to the tertiary is equal to the DA expectation plus ‘SA fluctuations’ which occur on the timescale of the outer orbital period Pout.

These in turn produce (typically very small) fluctuations in CK and jz — which are now near-adiabatic invariants, rather than true integrals of motion — and hence in the binary orbital elements.

As a result, the answer to question (i) changes: the SA-corrected eccentricity maximum is approximately emax ≃eDA max + δe, where eDA max is the DA prediction and δe is the SA fluctuation amplitude (Ivanov et al. 2005; Katz & Dong 2012; Antonini et al. 2014; Haim & Katz 2018; Grishin et al.
1.1.

Short range forces (SRFs)

SRF Catalysis of Extreme Orbital Evolution 3

[Figure: Figure 1. Four dynamical regimes of the test-particle quadrupole ZLK problem (assuming |jz| ≪1). The columns separate the averaging level used (DA on the left, SA on the right) and the rows separate the physics included (no SRFs on top, SRFs on the bottom). Each panel sketches the inner binary eccentricity e(t) over several secular cycles. Quadrants 1, 2, and 3 represent integrable or near-integrable problems that are well-understood (semi-)analytically. The focus of this paper is the fourth quadrant — SA + SRFs — wherein adiabatic invariance is completely broken.]