Initial thermal conditions¶

A PROTEUS run starts the planet as a hot magma ocean and follows it as it cools. The initial thermal conditions fix the mantle's starting temperature and entropy profile, which is the state the interior solver evolves forward in time. This page explains what that starting state controls, how to set it through the [planet] section of the configuration file, and which option to favour.

The parameters described here are listed with their types and defaults in the planet and volatiles reference. For the physics of the interior modules that consume this state, see the model description.

What the initial conditions do¶

PROTEUS does not model planet formation. Instead it begins from a chosen thermal state and integrates the coupled interior-atmosphere system forward. The initial conditions therefore set:

The starting melt fraction. A hot enough profile starts the mantle fully molten; a cooler profile starts it partially crystallised. The magma-ocean stage only exists while melt is present, so a fully molten start is the usual intent.
The thermal energy budget. The hotter the initial mantle, the more energy has to be radiated away before the planet solidifies, and the longer the cooling track.
The initial atmosphere. A hotter mantle outgasses more vigorously, so the starting surface pressure and composition depend on the initial temperature.

The chosen mode is converted into an initial entropy (or temperature) profile that the interior solver (Aragog or SPIDER) carries forward. What the initial conditions do not do is move the long-term endpoint: the planet still cools toward its solidus or toward radiative balance regardless of where it started. The initial state sets the transient and the total cooling time, not the destination.

How to set the initial state¶

The initial profile is selected by planet.temperature_mode. Each mode anchors the profile at a different reference point and reads a different companion parameter:

Mode	Sets the profile by	Companion parameter(s)
`liquidus_super` (default)	a fully molten adiabat, superheated above the liquidus	`delta_T_super`
`adiabatic_from_cmb`	an adiabat anchored at a fixed CMB temperature	`tcmb_init`
`adiabatic`	an adiabat anchored at the surface	`tsurf_init`
`isothermal`	a uniform temperature	`tsurf_init`
`linear`	a surface-to-centre gradient	`tsurf_init`, `tcenter_init`
`accretion`	accretion energetics	`f_accretion`, `f_differentiation`
`isentropic`	the specific entropy directly	`ini_entropy`, `ini_dsdr`

The default liquidus_super solves for the single adiabat that is fully molten throughout the mantle with a controlled superheat margin (see below). The adiabatic_from_cmb mode anchors an adiabat at the base of the mantle and integrates it upward to the surface; the surface modes anchor at the surface and integrate downward; the remaining modes set the profile from accretion energetics (White & Li, 2025) or from the specific entropy itself.

The default needs nothing beyond the mode name, because delta_T_super already defaults to 500 K:

[planet]
    temperature_mode = "liquidus_super"   # the default; shown here for clarity
    delta_T_super    = 500.0              # [K] above the liquidus at the core-mantle boundary

What to favour¶

Use the default liquidus_super for most runs. It starts the mantle on the coolest single adiabat that is fully molten everywhere, with at least \(\Delta T_\mathrm{super}\) (delta_T_super, in K) of superheat above the silicate liquidus:

\[\min_{P}\,\bigl\lbrack\,T_\mathrm{ad}(P) - T_\mathrm{liq}(P)\,\bigr\rbrack = \Delta T_\mathrm{super}\]

where \(T_\mathrm{ad}\) is the (isentropic) initial adiabat and \(T_\mathrm{liq}\) is the configured silicate liquidus. PROTEUS solves for the surface temperature, and hence the uniform initial entropy, that satisfies this at the most-constraining depth, checking the superheat against the liquidus actually in use. Where the tables reach it, this gives a fully molten initial state with a known margin, without you choosing a surface temperature or entropy by hand. Because the binding depth is shallow, the solved entropy is essentially independent of planet mass, so a mass grid starts on a common adiabat.

Whether the default delta_T_super = 500 K is reachable depends on the P-S tables, the core-mantle boundary pressure and planet.ini_dsdr. On PALEOS-generated tables it is reached at 1 and 10 Earth masses. On the Wolf & Bower (2018) tables from FWL_DATA, with ini_dsdr = 0, it is reached up to a core-mantle boundary pressure of about 355 GPa. Above that the solve clamps to a smaller superheat (443 K at 360 GPa, 85 K at 390 GPa), and above about 397 GPa the table liquidus entropy exceeds the table maximum, so no fully molten state exists and PROTEUS raises. With the Noack & Lasbleis (2020) pressure and radius estimates and a core mass fraction of 0.325, this means 500 K up to 2.75 Earth masses, 194 K at 3 Earth masses, and a raise above about 3.1 Earth masses (398 GPa at 3.15 Earth masses). With the default ini_dsdr = -4.698e-6 J/kg/K/m the usable entropy is lower by |ini_dsdr| times the mantle thickness (about 19.5 J/kg/K at 3 Earth masses): 2.75 Earth masses clamp to 417 K, 3 Earth masses to 73 K, 3.05 Earth masses to 3 K, and 3.1 Earth masses (392 GPa) raise.

The molten check uses the uniform entropy of the initial adiabat. With ini_dsdr < 0 the initial profile is hotter at depth, so the check is conservative: it can report a smaller superheat, or raise, where the deepest node is molten. The clamp warnings also give the superheat at the deepest node (194 K instead of 73 K at 3 Earth masses above). When the check raises but the adiabat at the deepest-node entropy is molten, the message names the ini_dsdr allowance as the cause and gives that margin. Setting delta_T_super = 0 makes the mantle marginally molten, just touching the liquidus at the binding depth.

Requires a silicate liquidus

For every structure module, the initial entropy is solved on the interior P-S tables against their own liquidus (liquidus_P-S.dat), the same curve the interior solver uses for its melt fraction. Where those tables come from depends on the structure module:

"zalmoxis" with a PALEOS mantle EOS (interior_struct.zalmoxis.mantle_eos, default "PALEOS:MgSiO3"): Zalmoxis generates the tables from that EOS, with a liquidus derived from the PALEOS (Fei et al. 2021) curve, and Aragog builds its phase-property tables from PALEOS too. interior_struct.melting_dir is not used. A mixture (a + in mantle_eos) follows its MgSiO\(_3\) component, so a mixture with PALEOS:MgSiO3 gets the tables of a PALEOS:MgSiO3 mantle; a PALEOS mixture without an MgSiO\(_3\) component gets the MgSiO\(_3\) two-phase tables of its PALEOS family.
"spider" and "dummy", or "zalmoxis" when no PALEOS table set is generated, which includes a mixture whose MgSiO\(_3\) component is not PALEOS (for example "WolfBower2018:MgSiO3:0.9+PALEOS:H2O:0.1"): the tables come from FWL_DATA (SPIDER also falls back to its own lookup data), Aragog also takes its phase-property files from that Wolf & Bower (2018) set or interior_struct.eos_dir, interior_struct.melting_dir must be set (the run stops when it is unset, except for interior_energetics.module = "spider" with const_properties = true), and PROTEUS derives liquidus_P-S.dat from that P-T curve.

Aragog takes its thermodynamics and melt fraction from the P-S tables. With "zalmoxis", which gives the core-mantle boundary pressure, Aragog logs a warning when that pressure is above the highest pressure of the P-S table set in use (about 1 TPa for the fetched Wolf & Bower (2018) set).

With "zalmoxis", the structure solve also anchors its temperature profile on a P-T adiabat that is delta_T_super above the P-T liquidus, while the initial entropy is solved on the P-S tables. The initial entropy solves this anchor again at the core-mantle boundary pressure of the current structure, before and after the structure equilibration. When it raises there, no molten state exists and the run stops at the initial entropy; the same error is raised when the anchor integration fails numerically. With a PALEOS mantle and "spider" or "aragog" energetics, a failed anchor in a structure solve or in the adiabat structure re-solve, at the estimated or an intermediate pressure, only logs a warning: the structure solve uses the last solved anchor for the same delta_T_super and mantle EOS, or tcmb_init otherwise, as its core-mantle boundary temperature, and the re-solve keeps the linear-guess structure. The initial entropy then decides at the core-mantle boundary pressure of that fallback structure. In other cases no later step solves the anchor again, so a failed anchor stops the run there. A resumed run that restores the entropy snapshot solves neither the anchor nor the initial entropy again. When Aragog cannot restore the snapshot (no snapshot, or a snapshot length that does not match the mesh), it sets a fresh initial entropy, which solves the anchor again at the current core-mantle boundary pressure and can raise. When the anchor clamps at the PALEOS table below delta_T_super, the initial entropy is capped at the anchor entropy, and a warning names the superheat the anchor reached. The generated P-S tables fill cells where PALEOS has no valid state, so on their own they can report a superheat that PALEOS does not support. For 10 Earth masses (core-mantle boundary at 1474 GPa), delta_T_super = 5000 K and ini_dsdr = 0, the capped initial adiabat matches the anchor within 0.1 K at the surface and at the core-mantle boundary, and is up to 38 K colder near 1.25 GPa: at these entropies the first P-S table row above 1 bar (1.26 GPa) holds cells filled with the 1 bar values. With ini_dsdr < 0 the deepest node sits at the cap and the rest of the profile is colder.

The capped initial entropy is the anchor entropy less the ini_dsdr allowance, and the molten check uses that uniform value. If the P-S adiabat at that entropy is below the P-S liquidus somewhere, PROTEUS raises instead of going above the anchor entropy, where the P-S tables hold filled cells. When the P-S adiabat at the anchor entropy itself is molten, the message names the ini_dsdr allowance as the cause; when it is not, the message gives its P-S superheat and the anchor superheat against the P-T liquidus.

This cap has two known limits:

The cap applies only when the anchor clamps. The anchor measures the superheat against the P-T liquidus and the initial entropy against the P-S table liquidus, and the two curves differ. For 10 Earth masses the anchor reaches a requested delta_T_super of up to about 1181 K, but at the anchor edge entropy the P-S tables give only about 1135 K of superheat (measured with ini_dsdr = 0). A request in between puts the initial entropy up to about 53 J/kg/K above the anchor edge (at 1180 K the surface is 35 K and the core-mantle boundary 113 K hotter than the anchor at its edge).
For 10 Earth masses the anchor edge comes from a narrow band near 1 GPa (0.99 to 1.08 GPa) where the PALEOS liquid table has no valid state at these temperatures. Whether an adiabat is marked valid depends on which pressures the adiabat samples in that band, so the edge, and the cap with it, can shift with the core-mantle boundary pressure.

The superheat is checked exactly at every pressure where the margin can change slope: the table pressure nodes, the liquidus-file nodes, and the pressures where the liquidus entropy crosses a table entropy node. Between two such pressures the bilinear table can let the margin sag by up to about 1 K on the PALEOS tables.

The core-mantle boundary pressure must lie inside the table; PROTEUS raises if it is above the table maximum or if the table liquidus is undefined at any evaluated pressure. If the requested superheat exceeds what the tables reach, the entropy is clamped to the highest usable table entropy (the table maximum, lowered by the entropy a negative ini_dsdr adds at the core-mantle boundary) and a warning reports the achieved superheat. A positive ini_dsdr is rejected with liquidus_super. For a run built only from placeholder modules, use adiabatic_from_cmb instead, which needs no melting-curve lookup.

Very deep mantles

A sufficiently deep mantle cannot be made molten with an arbitrarily large superheat: past a point the adiabat would exceed the equation-of-state table. If the requested delta_T_super cannot be reached, PROTEUS clamps to the largest achievable superheat and logs a warning that reports it. If no adiabat inside the table is fully molten at all, PROTEUS raises a RuntimeError rather than start from a partially molten mantle; lower the planet mass or change the equation of state in that case.

Use adiabatic_from_cmb for a fixed CMB temperature. This mode anchors the adiabat at a user-set core-mantle-boundary temperature tcmb_init and integrates it upward to the surface:

[planet]
    temperature_mode = "adiabatic_from_cmb"
    tcmb_init        = 6000.0   # [K] adiabat anchor at the core-mantle boundary

It needs no melting curve, so it is also the mode used by the all-dummy quick-start configuration, which runs without any external structure solver.

Avoid the surface-anchored modes unless you have a specific reason. Under the current equation of state, an adiabat pinned at the surface (adiabatic, isothermal) can drop the deep mantle below its liquidus at t = 0, leaving a partially solid base that is not a clean magma-ocean start. The default liquidus_super avoids this by solving for full melt directly, and adiabatic_from_cmb avoids it when you supply a hot enough tcmb_init. The linear mode is intended for controlled tests where you set the surface and centre temperatures directly.

Matching a published interior protocol

The isentropic mode sets the initial specific entropy directly through ini_entropy and ini_dsdr, bypassing the melting-curve lookup. Use it when reproducing a reference protocol that specifies the entropy IC, such as the Solar System CHILI intercomparison.