Cyril Sturtz

Angela Limare

Marc Chaussidon

Édouard Kaminski

Structure of differentiated planetesimals: a chondritic fridge on top of a magma ocean

Keywords:

come

Introduction

Meteorites are key witnesses of the processes that occurred during the first stages of planetary evolution. In particular, achondritic meteorites sample planetesimals differentiated during the first 5 Myr of the Solar system. Differentiated achondrites are often basaltic and most of them originate from a few parent bodies: the ureilites parent body, the angrites parent body, Vesta 4 -the eucrites parent body [START_REF] Greenwood | Linking asteroids and meteorites to the primordial planetesimal population[END_REF]. Achondrites with andesitic or trachyandesitic compositions are very rare, e.g., GRA 06128 and 06129 [START_REF] Day | Early formation of evolved asteroidal crust[END_REF][START_REF] Shearer | Non-basaltic asteroidal magmatism during the earliest stages of solar system evolution: A view from antarctic achondrites graves nunatak 06128 and 06129[END_REF], Almahata Sitta ALM-A [START_REF] Bischoff | Trachyandesitic volcanism in the early solar system[END_REF], NWA 11119 [START_REF] Srinivasan | Silica-rich volcanism in the early solar system dated at 4.565 ga[END_REF] and NWA 11575 [START_REF] Agee | Northwest africa 11575: Unique ungrouped trachyandesite achondrite[END_REF]. Recently, a new achondrite, Erg Chech 002 (EC002), has been recognized as the oldest andesitic meteorite known so far [START_REF] Barrat | A 4,565-my-old andesite from an extinct chondritic protoplanet[END_REF].

Early Solar system relative chronology is anchored to the Ca-Al-rich inclusions (CAIs) from CV3 chondrites which are the oldest components of primitive chondrites according to U-Pb dating [START_REF] Allègre | The age of the earth[END_REF][START_REF] Bouvier | The age of the solar system redefined by the oldest pb-pb age of a meteoritic inclusion[END_REF][START_REF] Connelly | The absolute chronology and thermal processing of solids in the solar protoplanetary disk[END_REF]. The formation of CAIs is generally considered to date the episode of high-temperature condensation, or reprocessing of these condensates, which produced the first Solar system dust in the accretion disk. A crystallization age of 2.255 ± 0.015 Myr after CAIs have been inferred for EC002 from 26 Al -26 Mg chronology [START_REF] Barrat | A 4,565-my-old andesite from an extinct chondritic protoplanet[END_REF], i. e. ≈ 1.2 Myr older than all other known achondrites, and contemporaneous with the formation of the metallic cores of the parent bodies of iron Structure of differentiated planetesimals meteorites [START_REF] Kruijer | Protracted core formation and rapid accretion of protoplanets[END_REF][START_REF] Kruijer | Age of jupiter inferred from the distinct genetics and formation times of meteorites[END_REF]. Petrological and geochemical data on the two major mineral phases in EC002 (pyroxene and plagioclase) indicate that (i) the parent melt of EC002 was produced by 25% partial melting of a source of chondritic composition at a temperature of 1224 ± 20 o C, (ii) the pyroxenes crystallized at 1186 ± 25 o C and re-equilibrated chemically at around 960 o C, (iii) the cooling of the rock or mush (crystal + melt) took place at ≈ 5 o C yr -1 between 1200 o C and 1000 o C, no melt being present anymore at 1000 o C, and (iv) finally the rock was quenched below ≈ 900 o C at a rate 0.1 -1 o C day -1 [START_REF] Barrat | A 4,565-my-old andesite from an extinct chondritic protoplanet[END_REF]. Although the 26 Al -26 Mg systematics does not allow to date precisely the episode of melting, a model age for melting between 0.95 Myr and 2.2 Myr (mean 1.4 Myr) after CAIs has been proposed [START_REF] Barrat | A 4,565-my-old andesite from an extinct chondritic protoplanet[END_REF]. Note that because the 26 Al -26 Mg age dates physically the closure temperature of Mg diffusion in EC002 plagioclases, the true crystallization age of EC002 can even be slightly earlier than 2.2 Myr, depending on the cooling rate before the closure of the chemical exchanges. In fact, recent Mg isotopic mesurements on mineral separates yielded a 26 Al -26 Mg crystallization age of 1.80 Myr. [START_REF] Fang | Half-life and initial solar system abundance of 146sm determined from the oldest andesitic meteorite[END_REF].

Early differentiation of planetesimals, within the first ≈ 2 Myr of the Solar system, as witnessed by iron meteorites [START_REF] Kruijer | Protracted core formation and rapid accretion of protoplanets[END_REF][START_REF] Kruijer | Age of jupiter inferred from the distinct genetics and formation times of meteorites[END_REF] and EC002 [START_REF] Barrat | A 4,565-my-old andesite from an extinct chondritic protoplanet[END_REF], stems from partial melting caused by the heat released by the radioactive decay of short lived 26 Al (and possibly 60 Fe). Intense radioactive internal heating can indeed induce several phenomena such as partial melting of chondritic materials (Elkins-Tanton, [START_REF] Elkins-Tanton | Chondrites as samples of differentiated planetesimals[END_REF], solidstate convection [START_REF] Tkalcec | Solid-state plastic deformation in the dynamic interior of a differentiated asteroid[END_REF][START_REF] Kaminski | Early accretion of planetesimals unraveled by the thermal evolution of parent bodies of magmatic iron meteorites[END_REF], iron melt segregation (Neumann, Breuer andSpohn, 2012, 2014), core-mantle differentiation [START_REF] Sahijpal | Numerical simulations of the differentiation of accreting planetesimals with 26al and 60fe as the heat sources[END_REF], magma ocean episodes [START_REF] Greenwood | Widespread magma oceans on asteroidal bodies in the early solar system[END_REF], and volcanism [START_REF] Mandler | The origin of eucrites, diogenites, and olivine diogenites: Magma ocean crystallization and shallow magma chamber processes on vesta[END_REF][START_REF] Fu | The fate of magmas in planetesimals and the retention of primitive chondritic crusts[END_REF]. Although the production mechanism of achondritic basaltic or andesitic magmas is still the object of debate, it must rely on the presence of a stable crust, either a primary or a secondary one, at the surface of the planetesimal [START_REF] Taylor | Growth of planetary crusts[END_REF][START_REF] Condie | Chapter 10 -comparative planetary evolution[END_REF]. Primary crusts are formed at the surface of a cooling magma ocean through the flotation of positively buoyant crystals from the magma ocean such as silica [START_REF] Faure | Early silica crust formation in planetesimals by metastable silica-rich liquid immiscibility or cristobalite crystallisation: the possible origin of silica-rich chondrules[END_REF] or plagioclase that formed the anorthosite crust of the Moon [START_REF] Wood | Lunar anorthosites and a geophysical model of the moon[END_REF][START_REF] Warren | The magma ocean concept and lunar evolution[END_REF].

Partial melting of this primary crust can further lead to the formation of a basaltic secondary crust [START_REF] Mandler | The origin of eucrites, diogenites, and olivine diogenites: Magma ocean crystallization and shallow magma chamber processes on vesta[END_REF]. However, the petrology and geochemistry of some differentiated meteorites, such as EC002, indicate that their parent liquids formed directly from chondritic material at shallow depth in planetesimals. The metasomatic minerals found in the CO and CV chondrites [START_REF] Ganino | Reduced and unstratified crust in cv chondrite parent body[END_REF]) could be considered as products resulting from the first step of the thermal heating that could have later induced this partial melting.

An additional argument for the presence of a stable chondritic crust in planetesimals is the paleomagnetic record of a magnetic field in the parent body of primitive meteorites which implies both a corehence a differentiated bodyand preserved, undifferentiated, chondritic material. Thermal models of planetesimals available in the literature fail to account for the preservation of a chondritic crust in differentiated planetesimals [START_REF] Weiss | Differentiated planetesimals and the parent bodies of chondrites[END_REF]. Only few studies specifically considered the link between the thermal evolution of a planetesimal and its accretion history advocating that chondrites and achondrites can originate from the same parent body. For instance, [START_REF] Neumann | Differentiation and core formation in accreting planetesimals[END_REF] considered accretion by radial growth and modeled the differentiation of planetesimals by Darcy flow. They showed that the degree of differentiation varies significantly as a function of the accretion history. [START_REF] Kaminski | Early accretion of planetesimals unraveled by the thermal evolution of parent bodies of magmatic iron meteorites[END_REF] introduced a new accretion model for parent bodies of iron meteorites accounting both for planetesimals differentiation and chondritic crust preservation. Their thermal model took into account solid-state convection, but did not consider explicitly the possibility of generating a magma ocean episode. [START_REF] Dodds | The thermal evolution of planetesimals during accretion and differentiation: Consequences for dynamo generation by thermally-driven convection[END_REF] also proposed that crust constituents can be supplied by accretion of undifferentiated material, and considered the possibility of solid-state convection, followed by a magma ocean episode. But their model still predicts metamorphism of the chondritic material in the crust even though they allowed accretion to start relatively late (0.8 Myr) in order to mitigate heating.

In the present paper, we introduce and model a refined accretion and differentiation scenario that explains the preservation of a chondritic crust, and produces a thermal record consistent with geochemical and petrological evidences provided by some meteorites. As an illustration of the model predictions, we consider the particular case of the andesite EC002.

Thermal regimes and radial structures of accreting planetesimals

As inferred from petrological data, EC002 was formed by ≈ 25% partial melting of a chondritic source [START_REF] Barrat | A 4,565-my-old andesite from an extinct chondritic protoplanet[END_REF]. In the following, we establish in which conditions chondritic material can be preserved in a differentiated planetesimal.

Accretion

Planetesimals are supposed to form by the accretion of cold dust and pebbles in the mid-plane of the accretion disk [START_REF] Visser | On the growth of pebble-accreting planetesimals[END_REF]; [START_REF] Johansen | Forming planets vis pebble accretion[END_REF]). Processes such as turbulence in the disk can produce locally strong concentrations reaching thresholds that trigger quasi-instantaneous accretion of ≈ 100 km size planetesimals [START_REF] Johansen | Rapid planetesimal formation in turbulent circumstellar disks[END_REF][START_REF] Cuzzi | Toward planetesimals: Dense chondrule clumps in the protoplanetary nebula[END_REF] thus bypassing the so-called meter-size barrier [START_REF] Weidenschilling | Aerodynamics of solid bodies in the solar nebula[END_REF]. In such a scenario, most planetesimals should be born big, as also suggested by simulations of the formation of embryos from planetesimals [START_REF] Morbidelli | Asteroids were born big[END_REF]. These accretion thresholds can theoretically be reached at various times and locations in the disk, depending on the availability and concentration of dust and pebbles. Hf -W ages of iron meteorites indicate that their parent bodies accreted over the first 1.5 Myr of the solar system, whereas the accretion of the parent bodies of achondrites and chondrites was probably completed within the first 4 -5 Myrs [START_REF] Kleine | The non-carbonaceous-carbonaceous meteorite dichotomy[END_REF]. In the inner regions of the accretion disk, within the orbit of Jupiter, primordial dust can be rapidly exhausted, within a Myr or so, because of the barrier to dust migration from the outer disk created by the early formation of Jupiter [START_REF] Morbidelli | The great dichotomy of the solar system: Small terrestrial embryos and massive giant planet cores[END_REF][START_REF] Haugbølle | Probing the protosolar disk using dust filtering at gaps in the early solar system[END_REF]. However, accretion of planetesimals can continue after 1 Myr since the flux of pebbles is never totally suppressed even for a fully-grown Jupiter.

In the following, we consider that growth of planetesimals can, in some cases, continue after the quasi-instantaneous accretion of a nucleus of several tens of km size [START_REF] Kaminski | Early accretion of planetesimals unraveled by the thermal evolution of parent bodies of magmatic iron meteorites[END_REF]. We further choose a Weibull accretion law to parametrize protracted accretion. The radius of the rocky body ( ) grows from 0 to the terminal radius according to:

( ) = 1 -1 - 0 3 -( ∕ acc ) 1∕3 , (1)
with acc the characteristic accretion timescale and a constant. Equation (1) will be referred as the "protracted accretion" law hereafter. Three families of accretion scenarios can be defined as a function of the value of (Figure 1). First, = 1 corresponds to a specific case described simply by an exponential law. For < 1, the planetesimal accretes material more rapidly than an exponential law at early times ( < ) and supplied material more slowly at Each solid line represents an iso-value of 99% . "Quasi instantaneous accretion" stands for planetesimals whose accretion timescale 99% is smaller than the lifetime of 26 Al ( 1∕2 /ln 2=1 Myr). "Overlong accretion" represents accretion which could take place in debris disk since 99% is greater than the dissipation of the accretion disk (≈ 10 Myr). The red star corresponds to the typical parameters of the accretion law used in the model ( =0.4Myr and =0.5). The orange points correspond to the extreme values of the accretion law used in the sensitivity study, in section 3.5.

>

, so that the terminal radius is reached later in the history of the planetesimal. For > 1, the early accretion is slower than an exponential law at early time but material is brought more efficiently for > and the terminal radius is reached earlier in the planetesimal history.

We define the characteristic timescale of this law 99% as the time at which the planetesimal radius reaches 99% of the terminal radius. After some developments, 99% writes:

99% = ln 1 -( 0 ∕ ) 3 0.03 1∕ (2)
The accretion timescale 99% is a function of the three parameters of the Weibull law. In figure 2 (a), the variation of 99% with 0 ∕ shows that for 0 ∕ in the range 0 -80%, the accretion timescale remains stable. For 0 ∕ > 80%, the accretion timescale decreases dramatically, and the planetesimal undergoes a quasi-instantaneous accretion. Within this domain (0% < 0 ∕ < 80%) only and have an influence on the accretion history.

Moreover, we can add two constraints on 99% . First, accretion has to be consistent with the lifetime of protoplanetary disk. Gas giants like Jupiter may have formed during the first Myr after CAI [START_REF] Kruijer | Age of jupiter inferred from the distinct genetics and formation times of meteorites[END_REF], and the dissipation of the solar nebula can range from a few to 10 Myr (Karl E. [START_REF] Haisch | Disk frequencies and lifetimes in young clusters[END_REF]. We thus consider that 99% has to be smaller than 10 Myr. Second, we compare the accretion timescale to the characteristic timescale of radioactive decay of 26 Al. If 99% ≪ 1∕2 ∕ ln 2, accretion can be considered instantaneous ( ( ) ≈ )

and the content of radioactive 26 Al, hence the heating rate in the planetesimal are maximal. This case happens for 0 ∕ → 1 and for small values of . Following these two constrains, we provide an abacus showing three families of accretion history in Figure 2 (b) : (i) when 99% < 1 Myr, accretion is considered quasi instantaneous, (ii) when 99% > 10 Myr, the accretion is too long compared to the lifetime of the disk, (iii) when 99% is between these two end-members, the accretion is called "protracted".

Composition

We consider that the accreted undifferentiated chondritic material is a binary mixture of iron and silicates, with a mass fraction Fe and Sil , respectively. The average density of the undifferentiated material is:

1 = Fe Fe + Sil Sil , ( 3
)
with the density of phase . We consider that melt crystallization can generate two types of crystals, one that is negatively buoyant relative to the melt (called hereafter) and one that is positively buoyant relative to the melt (called hereafter). The later produces a flotation crust at the top of the magma ocean, whereas the first induces the formation of a cumulate at the core-mantle boundary (CMB). The relative proportion of the two families of crystal is L,0 ∕ H,0 = 10∕90. This ratio is based on the proportion of plagioclase crystals relative to olivine and pyroxene crystals measured in some chondrites [START_REF] Weisberg | EH3 and EL3 Chondrites: A Petrologic-Oxygen Isotopic Study[END_REF]. Crystals and are effective "mineralogical phases" that provide a simplified description of the more complex mineral assemblages present in chondrites. As a consequence, and minerals are characterized by a solidus and a liquidus temperature. Between the solidus and the liquidus, the solid fraction in the melt, (in vol%), is given by:

= , + , , (4)
, = L,0 L - L -L , ( 5
) , = H,0 H - H -H , ( 6
)
where is the volume averaged bulk temperature in the planetesimal, and are the liquidus and the solidus temperature of mineral , respectively, and ,0 its initial proportion (in vol%), and , the proportion of crystal .

Thermal regime: from conduction regime to solid-state convection

Conduction regime

The bulk energy budget of undifferentiated planetesimal of radius ( ) leads to the following equation [START_REF] Kaminski | Early accretion of planetesimals unraveled by the thermal evolution of parent bodies of magmatic iron meteorites[END_REF] :

dT dt = 0 - 3 - 3 dR dt , (7)
with = , + , the planetesimal average heat capacity, 0 the total rate of internal heating, and the surface heat flux. In equation ( 7), the dR∕dt term corresponds to the effect of accretion of cold material on the thermal evolution of the planetesimal. Internal heating rates by radioactive elements 26 Al and 60 Fe are given by: where is the chondritic abundance of , [ ∕ ] 0 is the initial abundance of radioactive and stable isotopes i and j,  is the decay energy per atom, 1∕2 the half-life.

As the planetesimal melts, latent heat is consumed, and likewise, when it crystalizes, latent heat is released. Outside the solidus to liquidus temperature interval, latent heat is zero. When the temperature falls within the solidus and the liquidus, latent heat contributes to the energy budget as an additional internal heat source quantified as:

LH,Fe,0 = -Fe Fe dT dt 1 Fe -Fe , ( 10
) LH,Sil,0 = -Sil Sil dT dt 1 Sil -Sil , ( 11
)
with the latent heat of component (iron or silicate). The total rate of internal heating is thus 0 = 0,Al + 0,Fe + LH,Sil,0 + LH,Fe,0 . Our model considers the temperature at the surface of the body imposed by the temperature of the disk, which is a reasonably good approximation for planetesimals heated by 26 Al [START_REF] Dodds | The thermal evolution of planetesimals during accretion and differentiation: Consequences for dynamo generation by thermally-driven convection[END_REF] the early stages of the thermal evolution of the planetesimal, heat is evacuated by conduction and the surface heat flux can be calculated analytically [START_REF] Carslaw | Conduction of heat in solids[END_REF]. For an exponential internal heat generation source given by ( ) = 0 exp(-) with zero initial and surface temperatures, the surface heat flux is:

50 W m -1 K -1 (4) Silicate thermal conductivity Sil 3 W m -1 K -1 (4) Thermal expansion (Fe) Fe 7.7 10 -5 K -1 (1) Thermal expansion (Silicate) Sil 2 10 -5 K -1 (1) Specific heat (Silicate) ,Sil 1168 J K -1 kg -1 (3) Specific heat (Fe) ,Fe 622 J K -1 kg -1 (3) Latent heat (Fe) Fe 250 kJ kg -1 (
( ) = - 0 √ tan -1∕2 √ - 1 exp(-) - 2 0 +∞ ∑ =1 exp - 2 2 2 2 - 2 . ( 12
)
Here we consider only the heat production by 26 Al with = ln 2∕ Al 1∕2 . Because conduction is not a very efficient heat transfer, the bulk temperature of the planetesimal increases, until new processes occur in the interior of the planetesimal.

Onset of solid-state convection

As heating proceeds, the planetesimal is bound to melt. Partial melting implies a strong decrease of viscosity and can generate instabilities of pure thermal origin or related to liquid iron percolation that trigger solid-state convection at planetary scale [START_REF] Kaminski | Early accretion of planetesimals unraveled by the thermal evolution of parent bodies of magmatic iron meteorites[END_REF]. Convection is characterized by two dimensionless numbers [START_REF] Roberts | Convection in horizontal layers with internal heat generation. theory[END_REF], the Rayleigh-Roberts number * that accounts for the vigor of convection, and the Prandtl number that quantifies the importance of inertia:

* = * 5 , (13) = , ( 14
)
where is the thermal expansion coefficient, = 4∕3  is the surface gravity with  the gravitational constant, is the thermal conductivity, is the thermal diffusivity, is the dynamic viscosity, = ∕ is the kinematic viscosity, and * is the effective rate of internal heating that takes into account secular cooling [START_REF] Limare | Convection in an internally heated stratified heterogeneous reservoir[END_REF][START_REF] Limare | Transient convection experiments in internally-heated systems[END_REF]

): * = - dT dt . ( 15
)
Convection starts if * is larger than the critical value , = 5758 [START_REF] Schubert | Mantle convection in the Earth and planets[END_REF]). In the case of a material whose viscosity is strongly temperature dependent, convection will develop under a stagnant lid after a delayed onset time SL [START_REF] Davaille | Thermal convection in lava lakes[END_REF]. In the regime of solid-state convection, the heat budget of the convecting planetesimal is still given by equation ( 7) but with a surface heat flux now given by the scaling law [START_REF] Davaille | Thermal convection in lava lakes[END_REF][START_REF] Choblet | 3d thermal convection with variable viscosity: can transient cooling be described by a quasi-static scaling law?[END_REF]:

= 1∕3 Δ 4∕3 , ( 16
) Δ = - ∕ d dT , ( 17
)
where Δ is the characteristic viscous temperature scale for convection with a stagnant lid, = 2.24 [START_REF] Davaille | Thermal convection in lava lakes[END_REF], and a constant that depends on the mechanical boundary condition (here we will use = 3.31

following [START_REF] Limare | Transient convection experiments in internally-heated systems[END_REF]). The continuity of the heat flux through the stagnant lid leads to the determination of its thickness :

= 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 - √ √ √ √ √ 1 - 4 -Δ - ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , ( 18
)
where is the surface temperature.

The scaling laws introduced above emphasizes the key influence of the viscosity on the dynamics of convection.

In planetesimals, this issue is of peculiar importance because of the large variation of viscosity expected during the progressive heating of the material and its partial melting. For the silicate component we use a modified Arrhenius law to account for these dependencies [START_REF] Scott | The effect of large melt fraction on the deformation behavior of peridotite[END_REF][START_REF] Kaminski | Early accretion of planetesimals unraveled by the thermal evolution of parent bodies of magmatic iron meteorites[END_REF]:

( , ) = 0 ( ) exp ,Sil 1 - 1 0 , ( 19
) ( ) = exp -Sil (1 - ) . ( 20
)
where ( ) encompasses the influence of partial melting on the viscosity, 0 is the reference viscosity at 0 = 1000 o C, ,Sil the activation energy for silicates and the ideal gas constant. Because the amount of melt is related to the temperature, the variation of viscosity with temperature is amplified, and the convection may further transition to a magma ocean episode, as explained below.

Magma ocean episode

Rheological transition: magma ocean and core formation

At a certain degree of melting, the solid matrix becomes unstable and the behavior of the mixture becomes that of a suspension of solid crystals in a liquid magma [START_REF] Solomatov | Fluid dynamics of a terrestrial magma ocean[END_REF]. The type of convection changes at rheological transition RT = 60% (Elkins-Tanton, 2012; [START_REF] Guazzelli | Rheology of dense granular suspensions[END_REF], from laminar solid state convection to liquid-like, turbulent convection. In the so called "magma ocean" regime, the viscosity law ( 19) is modified so that ( ) is now expressed by a Krieger-Dougherty law [START_REF] Krieger | A mechanism for non-newtonian flow in suspensions of rigid spheres[END_REF]:

Structure of differentiated planetesimals r R R MO R c δ H δ L δ c
( ) = 1 - -2.5 , ( 21
)
where stands for the volume fraction of crystals in suspension, whereas is the total amount of crystals in the mantle, either in suspension or settled. We set at 60.002% to ensure the continuity of the viscosity at = RT and to reach a viscosity of 14 Pa s for purely liquid magma, consistent with the estimates of Rubie, Melosh, Reid, Liebske and [START_REF] Rubie | Mechanisms of metal-silicate equilibration in the terrestrial magma ocean[END_REF]. The variation of the viscosity as a function of the temperature is given in Appendix 2.4.1 assuming an equilibrium melting.

Beyond the rheological transition, the viscosity is reduced by a few orders of magnitude and opens the way to metal extraction [START_REF] Sturtz | Birth and decline of magma oceans. part 2: wobbling thermal history of early accreted planetesimals[END_REF]. As a matter of fact, before the rheological transition, the planetesimal is a mush with a low degree of partial melting and iron migration to the center of the planetesimal occurs by porous flow [START_REF] Neumann | Differentiation and core formation in accreting planetesimals[END_REF]. The timescale for iron to reach the center of the planetesimal is 1 -10 Myr, which is long compared to the time required for the onset of the magma ocean (less than 1 Myr, as it will be discussed later). Hence we assume that no metal-silicate segregation happened before the rheological transition. On the other hand, once the rheological transition has been reached, there is no more matrix resistance to the iron migration and we consider that it happens instantaneously (Höink, Schmalzl and Hansen, 2006).

A second important consequence of the low viscosity of the convective fluid once the rheological transition has been reached is the potential development of relative motions between melt and crystals, and ultimately the possibility of forming layers of segregated crystals: cumulate of thickness can grow by deposition of the heavy crystals at the core-mantle boundary (CMB), whereas a layer of light crystals can accumulate below the chondritic material brought at the surface of the planetesimal, the two contributions forming a crust of total thickness , as drawn in Figure 3 (a).

Thermal state of the bulk magma ocean and the core

The magma ocean is treated as a convecting shell. Through the introduction of a geometrical factor for the silicate, convecting mantle:

= MO = Fe Fe 1∕3 , ( 22
)
where is the core radius and is the distance between the top of the magma ocean and the center of the planetesimal (figure 3(a)). With parameters summarized in Table 1, = 0.42 right after the core differentiation.

The energy budget for the magma ocean and for the core are:

Sil ,Sil dT MO dt = MO + LH,Sil + CMB - , (23)
Fe ,Fe

dT c dt = + LH,Fe -CMB , ( 24
)
where MO and are the bulk average temperatures of the magma ocean and of the core, respectively, CMB is the heat flux at the CMB, = 4 2 1 -( ∕ ) 2 the top surface area of the magma ocean, = 4∕3 3 (1 -∕ ) 3 -( {1 -∕ } + ∕ ) 3 the volume of the magma ocean, = 4 2 2 (1 -∕ ) 2 the surface area of the core and = 4∕3 3 3 (1 -) 3 its volume.

As Al is highly lithophile, we assume that all 26 Al is fully partitioned into the magma ocean.

On the other hand, we consider that all iron is partitioned into the core, hence that no 60 Fe is present in the magma ocean.

Accordingly, the rate of internal heating in each reservoir can be written by modulating ( 8)-( 11) by a concentration factor:

= 0,Fe -3 1 - -3 , ( 25
) MO = 0,Al 1 - 3 - 1 - + 3 -1 , ( 26
)
whereas the release of latent heat in the two reservoirs is given by:

LH,Fe = -Fe Fe dT c dt 1 Fe -Fe , ( 27
) LH,Sil = -Sil Sil dT MO dt 1 Sil -Sil . ( 28
)
The potential presence of a crust at the surface of the magma ocean and at the CMB yields expressions for the heat fluxes that are more complex than the classical scaling laws used for convective systems. The crust acts like a thermal insulator and the temperature at the base of the crust , is linked to the convective heat flux through:

= Sil Sil Sil Sil 1∕3 MO -, 4∕3 . ( 29
)
Furthermore, heat transfer occurs by conduction in the crust and follows Fourier's law [START_REF] Jaupart | Heat Generation and Transport in the Earth[END_REF], pp.36):

= Sil , - ( ) ( ) ( ) -( ) . ( 30
)
At the beginning of the magma ocean episode, the crust is only composed of undifferentiated material brought by accretion. The presence of radioactive elements in the magma ocean later induces high temperatures close to the liquidus, which in turn lead to partial melting of the base of the crust. As silicates are more refractory than iron, the base of the crust is hot enough for iron melting to occur. Due to the high density contrast, liquid iron segregates from the base of the crust, and the crust is composed of a chondritic upper part, and an iron-depleted lower part (Figure 3 (b)).

This consideration has two main consequences. First, when iron is extracted, the crust thickness decreases. Second, as the crustal thinning occurs at a temperature below the liquidus of silicate, the heat flux increases so the magma ocean cools at a sub-liquidus temperature. As in the previous section when dealing with the massive iron/silicate segregation, we consider the temperature at which the iron is supposed to be extracted from the crust as being the rheological temperature of silicates (1366 o C).

Sturtz, Kaminski, Limare and Tait (2021a) developed a physical model establishing whether crystals formed during magma ocean cooling can be entrained by convection or not, based on a dimensionless parameter defined as the ratio between the convective shear stress and the buoyancy of crystals. This dimensionless parameter is a modified Shields number that combines the convection parameters and the crystals characteristics:

= ℎ 2 Δ * 3∕8 , ( 31
)
with the crystal radius, Δ the density difference between the crystals and the melt, and ℎ = [1 -∕ -∕ -(1 -∕ )] the thickness of the magma ocean. If is below a critical value crystals are not entrained by the convection and they can deposit, whereas if > , convective flow can keep crystals in suspension [START_REF] Shields | Anwendung der aehnlichkeitsmechanik und der turbulenzforschung auf die geschiebebewegung, mitteilung preussischen versuchsanstalt wasserbau schiffbau [application of similarity mechanics and turbulence research on shear flow[END_REF][START_REF] Charru | Erosion and deposition of particles on a bed sheared by a viscous flow[END_REF]. Sturtz et al. (2021a) validated this criterion experimentally for the formation of both cumulates and floating lids, and obtained = 0.29 ± 0.17. In the present case, because the viscosity of the magma ocean is very low (i.e. liquid like), the Shields number falls well below the critical value.

At the CMB, the way heat is transported depends on the magma ocean temperature. In the initial stage of the magma ocean, its temperature increases by internal heating up to maximum values close to the silicate liquidus. In these conditions there is no crystal cumulate at the CMB, and we consider that the core is always well mixed and characterized by an average temperature. On the magma ocean side, a thermal boundary layer forms at the CMB. In the stagnant lid regime, the TBL thickness at the CMB CMB is linked to the stagnant lid thickness SL by [START_REF] Solomatov | Scaling of temperature-and stress-dependent viscosity convection[END_REF][START_REF] Bryson | Constraints on asteroid magnetic field evolution and the radii of meteorite parent bodies from thermal modeling[END_REF][START_REF] Dodds | The thermal evolution of planetesimals during accretion and differentiation: Consequences for dynamo generation by thermally-driven convection[END_REF]:

CMB = SL Δ MO - , ( 32
)
with Δ = -∕ if MO > , and Δ = -Fe ∕ Fe if MO < . The CMB heat flux can then be expressed as a function of the surface heat flux as:

CMB = MO - Δ . ( 33
)
As the magma ocean cools down, a cumulate of "heavy silicate crystals" can form at the CMB. As long as the cumulate thickness is thinner than the TBL thickness at the CMB, heat transfer between the core and the mantle occurs by convection and equation (33) remains valid. If the cumulate is thicker than the TBL, the conductive heat flux CMB in given by:

CMB = Sil ,MO -CMB ( ) ( ) + ( ) ( ) , ( 34
)
where ,MO is the temperature at the base of the magma ocean, and CMB is the temperature at the surface of the core.

If the magma ocean is hotter than the core, we assume that the core acts like a thermostat, and sets CMB = , whereas

,MO is obtained from the heat flux using the scaling law:

CMB = Sil Sil Sil 1∕3 MO -,MO 4∕3 . ( 35
)
Similarly, if the the core is hotter than the magma ocean, we assume ,MO = MO , whereas the temperature CMB is obtained from the heat flux at the CMB given by:

CMB = Fe Fe Fe Fe 1∕3 c -CMB 4∕3 . ( 36
)
Within this framework, the thermal evolution of the system is totally determined by ( 23)-( 24). However, two parameters are missing to solve this set of equations: the thickness of the crust and the thickness of the heavy cumulate .

Mechanical evolution of the crust and of the cumulate

At the surface, the crust is fed both by the accretion of chondritic material and by the flotation of light crystals from the magma ocean. The growth of the chondritic part of the crust follows the protracted accretion law (1) while the thickening due to flotation of light crystals follows the law given by [START_REF] Sturtz | Birth and decline of magma oceans. part 1: erosion and deposition of crystal layers in evolving magmatic reservoirs[END_REF].

The total growth rate is then given by:

d c dt = dR dt + , ( 37
)
where is the volume fraction of light crystals in suspension in the magma ocean, is the sedimentation velocity of these light crystals and is a constant of order unity. Experiments in different types of convective systems have shown that for diluted suspensions scales with Stokes' velocity (Martin andNokes, 1988, 1989;[START_REF] Lavorel | Sedimentation of particles in a vigorously convecting fluid[END_REF][START_REF] Sturtz | Birth and decline of magma oceans. part 1: erosion and deposition of crystal layers in evolving magmatic reservoirs[END_REF]. To take into account the influence of crystals concentration on the settling velocity , we use the viscosity described in section 2.4.1, that takes into account the effect of temperature and the effect of the suspension on the viscosity:

= Δ 2 ( , ) , ( 38
)
with Δ the density difference between the melt and crystals, the crystal radius, and the volume fraction of crystals in suspension.

When the MO temperature decreases below the silicate liquidus, heavy crystals nucleate, sink and form a cumulate at the CMB. The rate of thickening of this layer is given by:

d H dt = , ( 39
)
with the volume fraction of heavy crystal in suspension in the magma ocean. Note that equations ( 37) and ( 39) are valid only when the magma ocean temperature is below the light and heavy silicate crystals melting temperature, respectively. If the magma ocean temperature is higher than their liquidus, the layers undergo melting, i.e., are thermally eroded.

Results

-what was the size of EC002 parent body?

Necessity to take accretion into account

A magma ocean episode is consistent both with an efficient core formation by metal-silicate differentiation and with the preservation of an undifferentiated crust at the surface. Because EC002 analyses indicate an early accretion, we first consider a scenario in which its parent body accreted instantaneously at 0 [START_REF] Li | Demographics of planetesimals formed by the streaming instability[END_REF]. Within the framework developed above, this scenario simply implies dR∕dt = 0 in equation ( 7). We solved numerically equation (7) using a finite difference scheme. As the conservation of energy is considered for the entire planetesimal, the physical parameters do not have a radial dependence. They are all volume averaged, for each reservoir, as underlined in equations 23 and 24. As a consequence, we only need a temporal discretization, and we use an order one explicit scheme. In the parameter space ( , 0 ) we track the conditions required to produce a magma ocean episode. As shown in Figure 4 (a), a magma ocean occurs in a planetesimal of radius if it accreted early enough. If the accretion takes place too late, the radioactive source is not sufficient to supply the required energy to reach the rheological transition.

Once the magma ocean episode has started (at ≈ 0.45 Myr for a 100 km body, accreted instantaneously at 0 =0), a chondritic crust remains at the surface as part of the stagnant lid with an initial thickness spanning from 0.5 to 1.5 km depending on the size of the planetesimal. However, as illustrated in Figure 4 (b), this crust is bound to undergo a first melting at ≈ 0.64 Myr that decreases its thickness. This is due to the fact that the crust insulates thermally the magma ocean where all 26 Al is essentially stored. Note that the crust also contains 26 Al, but the effects are negligible on the global budget. After this first melting episode, the chondritic reservoir that can be preserved at the surface has a very small thickness. For instance, for a in the range of 50 to 500 km, the chondritic material can only be preserved in a 100 -600 m thick crust. Although the crust thickens at the end of the magma ocean episode by addition of light crystals segregated from the magma ocean, its chondritic part is not thick enough to survive foundering triggered by impacts [START_REF] Hevey | A model for planetesimal meltdown by 26al and its implications for meteorite parent bodies[END_REF][START_REF] Elkins-Tanton | Chondrites as samples of differentiated planetesimals[END_REF]. We thus conclude that EC002 cannot be explained by a scenario of instantaneous accretion and we now consider a scenario of protracted accretion.

Structure evolution during planetesimals accretion

To illustrate the typical thermal history of an accreting planetesimal that undergoes a magma ocean episode, we consider a parent body with a final radius = 100 km. To start with, we consider the following parameters of the accretion law, consistent with the "protracted accretion" domain: 0 ∕ = 0.5, = 0.4 Myr and = 0.5. The time evolution of the internal structure, the surface heat flux, the bulk temperature, and the Shields number are shown in Figure 5. During the first 0.7 Myr, heat is dissipated by conduction only. At 0.7 Myr the rheological transition is reached and instantaneous and complete core-mantle differentiation occurs thanks to the low viscosity of the medium. Only a relatively thin crust, relic of the stagnant lid, remains at the surface. As chondritic material keeps being accreted, the thickness of this crust grows, with an increasing insulating effect on the magma ocean. Consequently, the bulk magma ocean temperature becomes larger than the silicate rheological transition temperature which induces the basal melting and iron segregation from the lower crust. Note that this increasing temperature episode also produces the melting to some extent of the first deposited cumulate. As the melting of the chondritic material proceeds, its iron component is incorporated into the core. Protracted accretion keeps bringing new material that globally thickens the crust and contributes to its insulation effect. As 26 Al is still active, albeit with a decreasing power, the crust continues to melt and to segregate iron at its base. This feedback effect causes successive melting episodes spanning from 0.7 to 2.3 Myr with several local maxima visible both on the heat flux and bulk temperature. The magma ocean temperature is buffered just above the rheological transition temperature after the first maximum. The value of this absolute maximum can be as high as the silicate liquidus temperature, provided the planetesimal terminal radius is large enough (see Appendix C).

After 2.3 Myr, 26 Al source has been largely exhausted, and there is no more crust melting. At high temperatures only heavy crystals are formed and sediment, but magma ocean continuous cooling starts to produce light crystals forming a flotation crust beneath the chondritic crust. The rheological transition is reached in the bulk of the silicate mantle at 4 Myr, and the magma ocean freezes. At this point the convection regime transitions back to solid-state convection.

Two key elements should be emphasized in this scenario. First, as the protracted accretion occurs on a longer timescale than the half-life of the radioactive elements, the primordial heat released in the magma ocean is moderate enough to enable the preservation of a km size crust even early in the thermal history. Second, internal heat production induces crustal melting episodes over a few Myr period. The intensity of these phenomena is a function of the amount of primordial heat stored in the magma ocean, and, correspondingly, by the size of the planetesimal. In the following, we discuss how this kind of model can be used to interpret meteoritical data, taking EC002 as an example.

Lower bound of parent body size: important partial melting episodes in the lower crust

The first constraint we consider deals with the genesis of the parent magma. EC002 is though to have been formed by 25% partial melting of chondritic material that constitutes the early crust of the parent body. We have found in our model (i) that this type of crust is likely to exist, (ii) that partial melting of the lower part of the crust is usual. To obtain more detailed information on EC002 parent body, we can evaluate the time at which the crustal melting episode occurs as a function of the planetesimal size. Results are displayed in Figure 6 (a). In the case of protracted accretion, Magma ocean episode begins at the rheological transition (red vertical dashed line), and yields core/mantle differentiation. The crust is stable and thickens due to accretion supply. Insulation effect of the crust leads to iron extraction at its base between 0.7 to 2.3 Myr. When magma ocean temperature is sub-liquidus, crystals nucleate and deposits form, first a cumulate at CMB, then a floating deposit. Note that the early formed cumulate remelts partially during the crustal thinning episode. We also display the surface heat flux (b), the temperature of the magma ocean and the core (c) and the evolution of the Shields number in the magma ocean (d) during the planetesimal evolution.

a magma ocean episode occurs in planetesimals with final radius > 55 km. The larger the planetesimal, the earlier the melting begins and the longer this magma ocean episode lasts. This can be explained by the ratio between the surface heat losses over the heat generated by the radioactive source that scales with the surface over volume ratio, and decreases with the planetesimal size. Consequently, the larger the planetesimal, the more heat from radioactive decay is stored.

We assume that the parent magma of EC002 has been formed during this massive partial melting episode of the crust. As a consequence, the melting period has to encompass the date derived from geochronology. According to Figure 6 (a), we deduce that the planetesimal final radius has to be larger than 80 km in order to undergo a crustal melting episode consistent with the time at which EC002 parent magma was produced (≈ 1.8 Myr).

Upper bound of the parent body size: EC002 has to be preserved in a "chondritic fridge"

A cooling rate of 5 o C yr -1 has been inferred for EC002 from petrological data. Such a rate is consistent with the thermal evolution of a few tens of meters size magmatic reservoirs buried about 1 km below the surfacesee Appendix B. The depth of storage of the magma corresponds to its neutral buoyancy relative to the crustal material.

In the shallowest layers of the planetesimal crust, the density is mainly controlled by the porosity, which itself results from the brecciation induced by the impacts. It is thus likely that the magma reservoirs form at the base of the regolith. Figure 7: Zoom on the upper layers of the planetesimal showing constraints on the time at which a magma produced by 25% of melting in the chondritic crust and that has risen into the crust (dashed purple line) can be preserved form subsequent remelting. Only magma produced after can be preserved from remelting (i.e. their temperature remains below the solidus, solid dashed line). Here, we use the following parameters:

= 100 km, 0 ∕ = 0.5, = 0.4 Myr, = 0.5.

The scenario that models the evolution of the parent magma of EC002 is displayed in Figure 7. The magma parent forms by 25% partial melting of chondritic material. At 1.8 Myr, the magma is stored at the base of the regolith, at a depth where it must be preserved from remelting (horizontal dashed line in Figure 7). We assume that this transfer happened by melt channeling hence is quasi instantaneous [START_REF] Spiegelman | Causes and consequences of flow organization during melt transport: The reaction infiltration instability in compactible media[END_REF][START_REF] Chauveau | Porous compaction in transient creep regime and implications for melt, petroleum, and co2 circulation[END_REF]. The depth at which the pluton is preserved from remelting is materialized by the horizontal dashed line in Figure 7. This depth can be translated into a corresponding time at which the chondritic material supplied by accretion is preserved as an undifferentiated crust ( = "chondritic fridge" time). We combine the two constraints (magma ascent occurring during a crustal melting episode, and no subsequent re-melting) to build an abacus relating the size of the parent body to the time at which the magma crystallizes (Figure 6 (c)). This abacus gives the size of the parent body of a meteorite as a function of the date at which the parent magma crystallizes. For instance, for a crystallization time of 1.8 Myr, as it seems to be the case for EC002, the parent bodies should have had a final radius between 80 and 130 km. The thermal equilibration of the so-formed pluton is quasi-instantaneous (≈ 40 yr) compared to the characteristic timescale of crustal thermal evolution (several Myr). Then, the magma evolves as a pluton stored in the crust where it further cools down at a rate of a few o C Myr -1 . [START_REF] Barrat | A 4,565-my-old andesite from an extinct chondritic protoplanet[END_REF] proposed that EC002 further experienced a high cooling rate of 0.1 -1 o C∕day when it had cooled down below a temperature of 900 o C, which would be consistent with cooling subsequent to an excavation by impact. The genesis and the complete thermal history of EC002 can then be summarized schematically in Figure 8.

Sensitivity to the accretion law parameters

In the previous section we estimated the final radius of the parent body of EC002 between 80 and 130 km, and we obtained that its chondritic crust was a few km thick. To obtain these results we used a protracted accretion law based on three parameters ( 0 ∕ , acc and ) that presumably influence the thermal evolution of the planetesimal.

We already highlighted that for 0 ∕ ranging from 0 to 0.8, the accretion history remains largely unchanged. As a consequence, our results are not affected by 0 ∕ within this range. Parameters and induce a characteristic timescale for accretion 99% that (i) has to be consistent with the duration of the dissipation of the protoplanetary disk, (ii) must be greater that the decay of 26 Al. In the following, we explore the sensitivity of in the range 0.5 and 1.5 for = 0.4 Myr, and we test from 0.07 Myr to 0.87 Myr with = 0.5.

Influence of

We first consider the influence of on the onset of the magma ocean episode and of the crustal melting. We fix the parameters 0 ∕ = 0.5 and = 0.4 Myr and we present results for three values of (0.38, 0.5, 1.5 Myr), resulting from the intersection with the limits of the "protracted accretion" domain in Figure 2(b). Figure 9 shows that the magma ocean characteristics depends only slightly on . Planetesimals with final radius > 55 km trigger a magma ocean with an onset time always ranging from 0.5 to 1.5 Myr. The duration of the magma ocean is also robust, spanning over 2 to 10 Myr. The episode of crustal thinning and melting depends on . The larger the value of , the shorter the duration of crust melting. almost identical. In this case the accretion is much slower than the decay of radioactive elements and consequently, after the first crustal melting episode, the chondritic crust is thicken by accretion, reaching a few km thick chondritic crust (see figure 11 (a), purple and red curves), which will restore the insulation of the magma ocean. Consequently, heat released by radioactivity is not efficiently evacuated and the magma ocean temperature rises, along with the lower crust temperature, which, in turn, is able to produce partial melting of the crust. This induces cycles of crustal thinning and chondritic crust growing, which lengthens the episode of crustal melting. High values of corresponds to quasi-instantaneous accretion (grey zone in Figure 10(a)). In this case, when the crust is thinned by basal melting and iron segregation, accretion is nearly finished, and thus the crust cannot grow though the external supply of chondritic material, and less than 1 km crust remains at the end of the accretion (see also figure 11 (a), grey curve). In this case, the first crustal thinning and melting episode increases the heat losses, which rapidly balances the heat released by radioactive elements.

We conclude that is an important parameter for the history of the accreting planetesimal. Smaller values of correspond to parent bodies with a thicker "chondritic fridge" (a few km). The crust melting episode spans the time range between 1 and 4 Myr. EC002 data are therefore consistent with this family of planetesimals. Higher values of correspond to planetesimals that are unable to preserve a substantial undifferentiated crust, as undifferentiated material is brought early compared to the decay of radioactive elements (in other words they have a small 99% ).

Influence of

We also evaluate the influence of on the size of EC002 parent body (Figure 10 Based on this sensitivity study, the parent body of EC002 radius can be estimated thanks to the Figure 10. If we exclude cases lying too close to the "quasi instantaneous accretion", the parent body of EC002 has a final radius between 70 and 130 km. We highlighted that the characteristics of the preserved chondritic crust and the thermal history are linked to the comparison between the characteristic accretion timescale 99% and the half-life of 26 Al ( 1∕2 =0.717 Myr), which should be a factor of about 3 times larger ( 99% > 2Myr).

The onset time of crustal melting and the "chondritic fridge" time are the two conditions that set bounds to the accretion history of a planetesimal consistent with EC002. They both should be lower than the crystallization time measured for EC002 (1.8 Myr); the first condition ensures the existence of the parent magma and the second one the preservation of the pluton in the crust. These conditions limit the parameters of the accretion law to the shaded zone in Figure 12.

Other geophysical implications

Sustainable "chondritic fridge" at the surface of planetesimals

Our model, applied in the previous section to the thermal history of EC002, actually describes general features in the evolution of planetesimals. In particular, considering a protracted accretion has an important impact on the planetesimal structure. Early instantaneous accretion induces very intense internal heating, due to the large amount of 26 Al that is stored. As a consequence, the peak bulk temperature is so high that any crust present at the surface is bound to melt and to become very thin (below 1 km). Protracted accretion will balance internal heating by the addition of cold material and will thus help to preserve a relatively thick crust that may later undergo melting episodes (Figure 11). These phenomena favor the formation of evolved rocks in planetesimals that preserve information from the early times of their history. For instance, some meteorites bear traces of early dynamos on their parent bodies, as the Kaba meteorite that recorded a dynamo as early as 10 Myr after CAI [START_REF] Gattacceca | New constraints on the magnetic history of the cv parent body and the solar nebula from the kaba meteorite[END_REF]. Dodds et al.

(2021) calculated that planetesimal cores with radius larger than 400 km (i.e., planetesimals with radius higher than ≈ 918 km) are able to sustain a dynamo at these early times. But the crust in the case of instantaneous accretion remains only ≈ 1 km thick so that it is unlikely it could register a remanent magnetization. To avoid crustal melting, the authors made the accretion begin at 0.8 Myr after CAIs, which corresponds roughly to a decay of 26 Al to half of its initial value. In the case of a protracted accretion that begins at = 0 and a planetesimal with = 920 km, our model predicts at 10 Myr a 49 km thick crust including (i) a 15 km thick chondritic upper crust, (ii) a 1 km thick layer and (iii) a 33 km thick differentiated layer of floating crystals. The undifferentiated part of the crust lies below the Curie temperature (≈ 800 o C) which would provide suitable conditions for the record of the early magnetic activity. This cold zone is a natural consequence of protracted accretion, in contrast with the runaway accretion scenario used by [START_REF] Dodds | The thermal evolution of planetesimals during accretion and differentiation: Consequences for dynamo generation by thermally-driven convection[END_REF] in which accretion ends early. A more detailed modeling of the possibility to sustain a dynamo in this type of body is required [START_REF] Monteux | Why might planets and moons have early dynamos?[END_REF][START_REF] Bryson | Constraints on asteroid magnetic field evolution and the radii of meteorite parent bodies from thermal modeling[END_REF][START_REF] Dodds | The thermal evolution of planetesimals during accretion and differentiation: Consequences for dynamo generation by thermally-driven convection[END_REF] in order to provide a precise information on the time at which the dynamo is triggered.

Geochronology of iron meteorites

The Hf -W systematics established for iron meteorites bring some constraints to refine their accretion scenario.

Hf -W age is usually interpreted as the date of instantaneous core formation [START_REF] Kruijer | Protracted core formation and rapid accretion of protoplanets[END_REF]. However, if the contribution of iron from crustal melting occurs over a long period of time, ages inferred from Hf -W systematics integrate the entire process of differentiation [START_REF] Neumann | Multistage core formation in planetesimals revealed by numerical modeling and hf-w chronometry of iron meteorites[END_REF]. We calculated the proportion of the core that is supplied by core melting (Figure 13 (a)) and the apparent age of the core inferred from

Hf -W systematics using our model, and we display it as a function of the time of the first differentiation in Figure 13 (b). In order to illustrate this effect we show in these figures only the results obtained for accretion histories characterized by 99% =5 Myr (red curves) or 99% =10 Myr (purple curves). The proportion of core added during accretion and crust melting is less important for lower values of the accretion timescale 99% (data not shown). Within the displayed values (Figure 13 (a)) the volume of core added depends on the values of the group of parameters ( and ), the larger they are, the bigger the effect. The first differentiation occurs at the rheological transition that happens at 0.5 -1.5 Myr, while the apparent age of the core is always younger than the age of the first differentiation, and can be very different, depending on the values of the accretion parameters we are ready to admit. There is also a dependence of the volume of core added with the final planetesimal radius which reaches a plateau at large values. The apparent age of the core does not reach a similar plateau and actually continues to strongly increase with .

These considerations can be used to interpret the age dichotomy in the iron meteorites. Earlier Hf -W inferred ages range from 0.5 to 1.5 Myr for non-carbonaceous (NC) magmatic iron meteorites, and from 2 to 3 Myr for carbonaceous (CC) magmatic iron meteorites [START_REF] Kruijer | Protracted core formation and rapid accretion of protoplanets[END_REF][START_REF] Kruijer | Age of jupiter inferred from the distinct genetics and formation times of meteorites[END_REF]. Recent revised Hf -W model ages corrected in Table 1, properly describes only the formation of dry, NC bodies.

Degassing despite a "chondritic fridge"?

The presence of a "chondritic fridge" on top of planetesimals rises the question of volatile behavior in such systems.

The surface temperature is taken as a constant, which is a reasonably good approximation (less than 10 K variation)

for small radius planetesimals heated by 26 Al [START_REF] Dodds | The thermal evolution of planetesimals during accretion and differentiation: Consequences for dynamo generation by thermally-driven convection[END_REF]. More rigorously, the surface temperature results from the balance between the emitted radiative flux, the absorbed solar radiative flux and the heat flux at the surface of the body [START_REF] Henke | Thermal evolution model for the h chondrite asteroid-instantaneous formation versus protracted accretion[END_REF]. The (quasi)isothermal boundary condition and low temperature at the surface will determine the preservation of a stagnant lid. A planetary body with a sub-surface magma ocean can be subject to partial or total crust melting and therefore go through volatile loss and EC002 meteorite can be taken as a proof of such magmatic activity. The amplitude and timescale of this magmatic activity should depend mainly on the body radius and the distance to the sun. We cannot rule out a transitory disruption of the crust during the first few Myr. A transitory disruption of the stagnant lid and the exposure of molten rock at the surface would lead to a transitory increase of the radiative heat flux which will cool down the surface and restore the isothermal boundary condition with the aid of the newly accreted, cold material. This transitory episode should be very short with respect to the duration of the magma ocean, provided planetesimals cannot preserve an atmosphere. This should be the case for the small radius bodies like the ones considered in the present study. Degassing is hence possible even though the planetesimal is covered by a "chondritic fridge" as established in a large number of studies [START_REF] Norris | Earth's volatile contents established by melting and vaporization[END_REF][START_REF] Young | Near-equilibrium isotope fractionation during planetesimal evaporation[END_REF][START_REF] Sossi | Evaporation of moderately volatile elements from silicate melts: experiments and theory[END_REF][START_REF] Collinet | Widespread production of silica-and alkali-rich melts at the onset of planetesimal melting[END_REF][START_REF] Hirschmann | Early volatile depletion on planetesimals inferred from c-s systematics of iron meteorite parent bodies[END_REF].

Conclusion

We highlighted in this paper how geochemical and petrological data obtained from differentiated achondrites can be used to constrain the formation and evolution of their parent bodies. Our results are based on a thermal model that takes into account protracted accretion, core-mantle differentiation, magma ocean episode and crustal thickening by continuous accretion of chondritic material at the surface. This model predicts that the reference internal structure of a planetesimal formed early with a size ≳ 55 km is a differentiated interior (with a liquid metallic core and a crystallizing magma ocean) covered by an undifferentiated chondritic crust that is stable and will undergo localized basal partial melting. We show how cooling rates provide information about the depth and the size of the reservoir where the parent magma of achondrites crystallized in the chondritic crust. The datation of parent magma formation can then be used to provide some bounds on the parent body size and on the parameters of its accretion laws. This approach is general and could be used to study other parent bodies than that of EC002. We assume an equilibrium melting, so that the liquid content is directly given by (4). The rheological transition occurs for a liquid content of 40% (i.e.: a solid content of 60%). The cooling rates measured in some crystals provide constraints on the condition of magma storage in the crust. For a cooling rate of 5 o C yr -1 , like the one measure in EC002, a simple model for the cooling of a magma reservoir can predict the size and the depth at which magma is stored. We consider a magma body of typical length at a depth 0 bearing a thermal anomaly Δ compared to the surface temperature . During a quasi-static evolution of a period of time dt, the variation of internal energy is d(ΔT) LS, where is the surface of the system through which heat losses occur. The heat flux can be written dt, where is assumed to be conductive [START_REF] Kaminski | Lithosphere structure beneath the phanerozoic intracratonic basins of north america[END_REF]

B. Cooling model for a magmatic reservoir buried at a certain depth

Figure 1 :

 1 Figure 1: Accretion curves giving the evolution of the normalized radius of planetesimals (with 0 ∕ = 0.5 and = 0.4 Myr) for three values of the parameter : = 0.5 (red line), the typical value used in the model, = 1 (blue line) corresponding to an exponential law, and = 1.5 (purple line). The grey vertical line corresponds to t= = 0.4 Myr.

Figure 2 :

 2 Figure 2: (a): Influence of 0 ∕ on the accretion timescale 99% . (b): Diagram showing the three families of accretion history as a function of the parameters and .Each solid line represents an iso-value of 99% . "Quasi instantaneous accretion" stands for planetesimals whose accretion timescale 99% is smaller than the lifetime of 26 Al ( 1∕2 /ln 2=1 Myr). "Overlong accretion" represents accretion which could take place in debris disk since 99% is greater than the dissipation of the accretion disk (≈ 10 Myr). The red star corresponds to the typical parameters of the accretion law used in the model ( =0.4Myr and =0.5). The orange points correspond to the extreme values of the accretion law used in the sensitivity study, in section 3.5.

Figure 3 :

 3 Figure 3: (a) Illustration of the stratification inside a planetesimal during a magma ocean episode. (b) Illustration of the iron-silicate differentiation that occurs at the base of the crust. Between ( ) and seg , the crust remains cold and chondritic. Between seg and, the crust temperature is high enough to trigger iron-silicate segregation. Iron is drained, and the remaining crust is made of silicate component only. As iron leaves, the lower part of the crust thins.

Figure 4 :

 4 Figure 4: Instantaneous accretion. (a): Diagram showing the conditions (radius and accretion time 0 ) of for a planetesimal to differentiate. (b): Evolution of the internal structure evolution of a 100 km radius planetesimal that accreted instantaneously at 0 = 0 Myr.

Figure 5 :

 5 Figure5: Typical evolution of a 100 km terminal radius planetesimal that begins its protracted accretion at = 0 Myr ( 0 ∕ = 0.5, = 0.4 Myr, = 0.5). (a): Evolution of the internal structure of the accreting planetesimal. Magma ocean episode begins at the rheological transition (red vertical dashed line), and yields core/mantle differentiation. The crust is stable and thickens due to accretion supply. Insulation effect of the crust leads to iron extraction at its base between 0.7 to 2.3 Myr. When magma ocean temperature is sub-liquidus, crystals nucleate and deposits form, first a cumulate at CMB, then a floating deposit. Note that the early formed cumulate remelts partially during the crustal thinning episode. We also display the surface heat flux (b), the temperature of the magma ocean and the core (c) and the evolution of the Shields number in the magma ocean (d) during the planetesimal evolution.

Figure 6 :

 6 Figure 6: (a) Duration of the crustal melting episode in accreting planetesimal as a function of the planetesimal final radius. (b) Time at which the chondritic material supplied by accretion is preserved as a undifferentiated crust as a function of the planetesimal final radius. (c) Abacus showing the time at which the magma stored in the chondritic crust is preserved as a function of the planetesimal final radius. All simulations are performed with = 0.5, = 0.4 Myr and 0 ∕ = 0.5. In the three sub-panels, the horizontal line gives the time at which the parent magma of EC002 rises in the crust according to petrological data (≈ 1.8 Myr).

  Figure 6 (b) displays this time as a function of the planetesimal final radius.

Figure 8 :

 8 Figure 8: Proposed genesis and thermal history of EC002. (a) Partial melting of the chondritic lower crust generates the parent magma. (b) Magma rises to a shallow storage depth. (c) The parent magma crystallizes quickly and forms a pluton. (d) The pluton is excavated by an impact. Note that the crust thickens continuously during the thermal history, due to accretion of cold material. For sake of simplicity, this figure does not show this effect.

Figure 10 Figure 9 :Figure 10 :

 10910 Figure10(a) contains the abacus showing the time at which magma stored in the chondritic crust is preserved for the three values of . For relatively small values of , that correspond to high values of 99% , the two zones are

Figure 11 :

 11 Figure 11: Thickness of the chondritic crust as a function of planetesimal final radius for different values of (a) with = 0.4 Myr and of (b) with = 0.5. 0 ∕ = 0.5. The color of the curves are set by the value of 99% : purple (10Myr), red (5Myr), blue (2Myr), grey (1Myr).

  (b)). The red zones in Figures 10(a) and (b) are identical and correspond to the values of parameters = 0.5 and = 0.4 Myr, represented by the red star in Figure 2 (b). We take the two end-members for = 0.5 corresponding to the intersection with the limits of the "protracted accretion" domain in Figure 2 (b): ranges from 0.07 to 0.87 Myr. Small values of correspond to an accretion history that is close to an instantaneous one, whereas high values of correspond to a longer accretion timescale. The parent body of EC002 should have a final radius in the range from 70 to 100 km for acc = 0.87 Myr, from 80 to 130 km for acc = 0.4 Myr and from 210 to 400 km for = 0.07 Myr (Figure 10 (b)). The thickness of the preserved chondritic crust increases with , and small values of are not consistent with the preservation of a pluton in an undifferentiated crust (crust thickness < 1km), and hence, cannot explain EC002 formation (Figure 11(b), grey curve). Figure 11 also displays two curves that correspond to values of and from the intersection with 99% =2Myr (blue curves). They indicate that a substantial crust, of 2-3 km, can be produced and later preserved only for larger values of 99% . Figures 10 and 11 point out that apparently the most important parameter for the accretion history is 99% , and not the individual and producing it.

Figure 12 :

 12 Figure 12: Parameters of the accretion law consistent with EC002.

Figure 13 :

 13 Figure13: (a) Proportion of the core that is brought by iron supply due to progressive melting of the chondritic crust as a function of the planetesimals final radius. (b) Apparent age of the core inferred from Hf -W systematics as a function of the age of the first differentiation that corresponds to the onset of the magma ocean. Black line represent a 1:1 dependence between the two ages. Red curves represent values of 99% =5 Myr and purple curves 99% =10 Myr, full lines =0.5, dashed lines =1 and dashed and dotted lines =1.5.

Figure 14 :

 14 Figure 14: Rheological law (blue) of the planetesimal and liquid fraction (red) in the body as a function of the temperature.We assume an equilibrium melting, so that the liquid content is directly given by (4). The rheological transition occurs for a liquid content of 40% (i.e.: a solid content of 60%).

Figure 15 :

 15 Figure15: Evolution of the size of a magmatic reservoir that undergoes a cooling rate of 5 o C∕yr as a function of the depth at which it is stored. The area colored in blue stands for temperature anomalies spanning from 1000 to 1200 o C. The forbidden zone represents non-physical solutions that implies reservoirs that are larger than the crust thickness.

Table 1 Parameters

 1

	4)

used in the thermal model. References are the following : Neumann et al. (2014) (1), Tang and Dauphas (2012) (2), Neumann et al. (2012) (3), Kaminski et al. (

2020

) (4),

[START_REF] Solomatov | Fluid dynamics of a terrestrial magma ocean[END_REF]

(5), adapted and simplified from

Righter and Drake (1997) (6)

.

Acknowledgments

This paper is part of Cyril Sturtz's PhD thesis (Université Paris Cité, Institut de Physique du Globe de Paris).

The authors would like to thank the editor Doris Breuer for her editorial handling and two anonymous reviewers for their fruitful comments that improve this manuscript. This study contributes to the IdEx Université de Paris ANR-18-IDEX-0001. This work was supported by the Programme National de Planétologie (PNP) of CNRS/INSU, co-funded by CNES.

⋆

This study contributes to the IdEx Université de Paris ANR-18-IDEX-0001. ⋆⋆ This work was supported by the Programme National de Planétologie (PNP) of CNRS/INSU, co-funded by CNES. sturtz@ipgp.fr (C. Sturtz

A. Rheological law and composition as a function of the temperature and the conservation of energy yields

where is the thermal conductivity of the enclosing rock, is its density and its specific heat. Results of magma reservoir size as a function of its depth to get the required cooling rate are displayed in Figure 15. To attain 5 o C yr -1 , the magma has to be stored in small reservoir, which means that the reservoir that cools is a few meter size diapir or a magma lens stored at a few km depth.

Furthermore, this approach shows that the shallower the reservoir the larger its size for a given cooling rate. This calculation predicts also that the crust of the parent body has to be thicker than 100 m in order to sustain a reservoir whose size is smaller than the crust thickness and at least 300 m in order to have reservoirs that represents 10% of the crust thickness (Figure 15).