computed as a linear function of entropy between these bounds (Eq. 6).

This representation allows SPIDER to flexibly incorporate melting relations from experimental petrology (e.g., peridotite and chondritic solidus/liquidus data) and to smoothly transition material properties across phase boundaries using numerical smoothing routines (Bower et al. 2018).

Therefore, the lookup tables encode the full thermodynamic structure of the Article number, page 5 of 19 showed a strong dependence of the solidus on fOmelt under oxidizing conditions (IW+2.0)

at pressures of 16-26 GPa.

Comparing these results with previous experiments under more reducing conditions (Trønnes 2000; Trønnes & Frost 2002; Ishii et al. 2018), they find a shift of several log units in fOmelt
melting region and directly determine where the mantle is solid, partially molten, or fully liquid such that

1, if S > S l S −S s S l −S s , if S l ≥S ≥S s
between the two regimes.

This behaviour can be expressed as:
ϕ =
(6)
0, if S s > S
T fO2
solidus (±383K) = T IW+2.0 solidus + 340 Conduction is described by Fourier’s law, with thermal diffusivity κt, specific heat capacity c and the adiabatic temperature gradient (∂T/∂r)S , and is expressed as � −ρcκt
�∂T ∂r
�
�∂S ∂r
Fcond = −ρTκt
. (7)
S
The total heat flux is expressed as the sum of the contributions of each heat flux associated to heat transport processes during the transition of a molten mantle:
Ftotal = Fconv + Fmix + Fcond + Fgrav. (8) At each grid cell Ftotal can be positive or negative, defining the cooling or heating trajectory of individual layers within the planet’s interior.

Therefore, each of the fluxes will have a fundamental contribution to the total heat budget of the planet.

We do not directly model the geophysical evolution of the metallic core, and we assumed the core-mantle boundary (CMB) to be thermally coupled to the mantle based on energy balance such that CMBFCMB mcoreccore ˆTcore (9)
dTCMB dt = −4πr2 where mcore, ccore, ˆTcore, and FCMB are the mass, heat capacity, thermal structure correction factor of the core, and heat flux at the CMB, respectively (Bower et al. 2018).

The core heat capacity ccore is prescribed as a constant value representative of liquid iron (880 J kg−1 K−1), assuming an isentropic, well-mixed core.

Material properties are computed using separate equations of state for the melt and solid phases.

The MgSiO3 melt is described using the RTpress equation of state (Wolf & Bower 2018), which spans a wide pressure–entropy (P–S ) range and is calibrated to experimental data using a limited set of physical parameters.

For the solid phase, we adopted the MgSiO3 equation of state from Mosenfelder et al. (2009).

For the mixed-phase region, the properties are computed according to the local melt fraction Eq. (6), and assuming volume additivity when computing the two-phase adiabat (Solomatov 2007).

In this case, the melting curves employed are joint data to cover the entire mantle pressure range.

At high pressures, the solidus and liquidus are derived from experiments on synthetic chondritic compositions (Andrault et al. 2011), while at lower pressures the melting relations follow the peridotite-based parametrization compiled by Hamano et al. (2013).

These datasets are smoothly joined to provide continuous melting curves across the mantle.
2.6.

Effect of fO2 on mantle solidus Building on the thermodynamic framework provided by the adopted equations of state, we now describe how variations in oxygen fugacity influence mantle melting relations.

The melting temperature of the mantle depends sensitively on its composition and thermodynamic state, which are strongly regulated by redox conditions.