Differentiation of Vesta and the parent bodies of other achondrites

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009je003525, 2010 Differentiation of Vesta and the parent bodies of other achondrites G. Gupta 1 and S. Sahijpal 1 Received 10 October 2009; revised 18 February 2010; accepted 16 April 2010; published 7 August [1] Numerical simulations have been performed to understand the planetary differentiation of Vesta and other differentiated asteroids. The short lived nuclides 26 Al and 60 Fe were used as the heat sources for the differentiation of asteroids of sizes km. The essential aim was to simulate the two proposed differentiation scenarios associated with the formation of the basaltic achondrites. One of these scenarios deals with their origin by partial melting of silicates. The other scenario involves the origin of basalts from the residual melt left subsequent to crystallization in a convective and cooling magma ocean. The core mantle differentiation in both the scenarios was commenced subsequent to 40% silicate melting. The partial melt scenario seems to contradict the general trend in the chronological sequence of core mantle and mantle crust differentiations observed in various differentiated meteorites unless the irons meteorite parent bodies differentiated rapidly compared to prolonged accretion and differentiation of distinct parent bodies of achondrites. Even though the geochemical records of siderophile elements in eucrites favor the residual melt scenario, based on our simulations it is difficult to rapidly and globally cool the magma oceans to produce residual basaltic melts subsequent to crystallization within initial 10 million years. This is in contradiction with the chronological records of the basalts that suggest their early origin. In order to make this scenario viable by invoking rapid and global cooling of the magma ocean it would be essential to substantially remove the insulating crust of the planetesimal by continuous bombardment of small planetesimals in the early solar system. Citation: Gupta, G., and S. Sahijpal (2010), Differentiation of Vesta and the parent bodies of other achondrites, J. Geophys. Res., 115,, doi: /2009je Introduction [2] The planetary differentiation of the differentiated meteorite parent bodies [Haack and McCoy, 2005;McCoy et al., 2006] commenced within the initial few million years of the early solar system. The chronometric records based on 26 Al 26 Mg, 53 Mn 53 Cr, 182 Hf 182 W, and 207 Pb 206 Pb isotopic systematic of the majority of the differentiated meteorites are consistent with the differentiation time spans inferred from the numerical simulation of the differentiation of planetesimals [Hevey and Sanders, 2006; Sahijpal et al., 2007; Kleine et al., 2009]. The core mantle differentiation in the case of iron meteorite parent bodies probably initiated along with the early accretion of the planetesimals, perhaps within the initial million years after the formation of Ca Al rich inclusions. [e.g., Kleine et al., 2002, 2005, 2009]. The mantlecrust differentiation in general seems to have initiated afterwards [e.g., Nyquist et al., 2003, 2009; Wadhwa et al., 2009]. A recent compilation of the published whole rock ages and mineral isochrone ages of the basaltic meteorites [Sanders and Scott, 2007; Kleine et al., 2009; Nyquist et al., 2009; Wadhwa et al., 2009] suggest that the basaltic 1 Department of Physics, Panjab University, Chandigarh, India. Copyright 2010 by the American Geophysical Union /10/2009JE achondrites formed mainly within 4 10 million years (Ma) in the early solar system, with some exceptions. [3] The major collection of the achondrites, the Howardite Eucrite Diogenite (HED) meteorites is considered to have been derived from a single asteroid Vesta [Binzel and Xu, 1993; Drake, 2001; Usui and McSween, 2007; Scott et al., 2009]. Recent studies, however, indicate that some of the eucrites could have also originated on other planetesimals [Yamaguchi et al., 2002, 2006; Mittlefehldt, 2005]. Based on the elemental composition, the eucrites are classified into four main groups: the main group eucrites, the Stannern group, the Nuevo Laredo group, and the high Mg eucrites [Reid et al., 1979]. Broadly, two distinct planetary differentiation scenarios have been proposed for the origin of the eucrites [Drake, 2001; Holzheid and Palme, 2007]. One of these scenarios argue that the eucrites are the result of distinct degree of equilibrium partial melting of chondritic material [Consolmagno and Drake, 1977; Stolper, 1977; Jurewicz et al., 1991, 1993, 1995; Jones et al., 1996]. Stolper [1977] showed that the main group eucrites (e.g., Sioux County) could be produced by 15 20% partial melting of the olivine, pyroxene, plagioclase, Cr rich spinel, and metal. Consolmagno and Drake [1977] showed that the main group eucrites (e.g., Juvinas, Sioux County) could have been derived from a source with chondritic REE abundances by 10 15% equilibrium partial melting. Jurewicz et al. [1991, 1of15

2 1993] showed that eucritic magma could be produced by the partial melting of anhydrous Murchison (CM) under reducing conditions. [4] The alternative source of the eucrites was proposed to be the residual melt left at the end of fractional crystallization in a cooling magma ocean [Ikeda and Takeda, 1985; Warren, 1985; Longhi and Pan, 1988]. The depletion of siderophile elements in eucrites provides evidence that the metal had segregated into the iron core before the formation of eucrites. Newsom [1985] estimated that the depletion of siderophile element Mo in eucrites is consistent with the core formation prior to the formation of eucrites [Hewins and Newsom, 1988]. Presently, the most favored magma ocean model, proposed by Righter and Drake [1997], provides several advantages over the fractional crystallization and the partial melting model. According to this model, the residual melt left subsequent to equilibrium crystallization, at 80 85% crystallization, in a magma ocean would produce the observed composition of the main group of eucrites. Subsequent fractional crystallization of the isolated main group eucrite melt could have produced the Nuevo Laredo group and cumulate eucrites. The Stannern group eucrites have been recently proposed to have been produced by partial crustal melting above the magma ocean [Barrat et al., 2007]. [5] Apart from the eucrites, the angrites constitutes another group of basaltic achondrites. Angrites are considered to be the most primitive group of achondrites that probably formed from the partial melting on a differentiated body of primitive refractory composition [Taylor et al., 1993; Mikouchi et al., 2008; Weiss et al., 2008]. Jurewicz et al. [1991, 1993] demonstrated that the composition of the Angra dos Reis can be explained by a 20% partial melting of a parent body with CV3 (Allende) chondritic composition. Besides the angrites and eucrites, two distinct groups of achondrites, namely, lodranites and ureilites [Mittlefehldt, 2007; Downes et al., 2007] are considered to be the leftovers residues of a low degree of partial melting and basaltic melt extrusion [Taylor et al., 1993]. The acapulcoites constitutes the extreme class of achondrites that have undergone the least amount of partial melting to result in any appreciable amount of melt segregation from the rock [Taylor et al., 1993]. [6] In general, in order to produce the wide range of achondrites among the meterorite collection, the partial melting and/or the equilibrium/fractional crystallization of a large number of planetesimals in the early solar system would be required. The radiogenic decay energy of the short lived radionuclide was proposed as a possible heat source for the heating and the melting of the parent bodies of these achondrites [Urey, 1955]. The evidence for the presence of once live 26 Al in CAIs (Calcium Aluminum rich inclusions) from Allende meteorite was found by Lee et al. [1976]. Wide spread distribution of 26 Al was experimentally confirmed in CAIs from different classes of chondritic meteorites [e.g., MacPherson et al., 1995]. A canonical value of for 26 Al/ 27 Al in the early solar system was measured by the refinements in the analytic techniques [Jacobsen et al., 2008]. However, it should be mentioned that the canonical value of 26 Al/ 27 Al inferred from CAIs does not rule out the possibility of heterogeneous distribution of 26 Al in the early solar system. There could be 26 Al free nebula reservoirs in the early solar system on spatial or temporal scales [Sahijpal and Soni, 2006]. The possible implications of the evolutionary history of asteroids formed in the 26 Al free nebula reservoirs is discussed by Sunshine et al. [2008] [7] The recent evidence for the presence of high excesses of 60 Ni in the grains of trolite in Semarkona meteorite [Mostefaoui et al., 2005] suggested that 60 Fe was also responsible as a heat source for the differentiation of the planetesimals along with 26 Al. The initial 60 Fe/ 56 Fe of the solar system is still uncertain. The estimated values of the initial 60 Fe/ 56 Fe are both high >10 6 [Moynier et al., 2005; Quitte et al., 2005] and low (1 5) 10 7 [Tachibana and Huss, 2003; Regelous et al., 2008]. However, it has been indicated that the extent of heterogeneity of 60 Fe was less than 10% in the formation region of the planetary bodies [Dauphas et al., 2008]. We can anticipate an almost identical upper limit for the heterogeneity of 26 Al in case the two short lived nuclides 26 Al and 60 Fe were produced by a single massive star [Sahijpal and Gupta, 2009]. [8] Several thermal models of asteroids have been developed in order to explain their thermal evolution [Sahijpal, 1997,2006;Sahijpal et al., 1995, 2007; Ghosh and McSween, 1998; Merk et al., 2002; Hevey and Sanders, 2006;Gupta and Sahijpal, 2007, 2009]. Sahijpal [1997] developed thermal models for the radiogenic heating of asteroids with the temperature dependent thermal diffusivity. Ghosh and McSween [1998] developed a model with instantaneous accretion of planetesimal. Hevey and Sanders [2006] modeled the melting of planetesimals heated by 26 Al using finite difference method and incorporated thermal convection. Sahijpal [2006] and Sahijpal et al. [2007] have recently numerically simulated the planetary differentiation of linearly accreting planetesimals of radii in the range of km, with the gradual growth of an iron core due to the flow of (Fe Ni) metal FeS melt toward the center of the planetesimals. This comprehensive work deals with the numerical simulation of the core mantle differentiation in asteroids during the melting of Fe FeS. The mantle crust differentiation in these models was numerically initiated subsequent to an appreciable ( 20%) amount of silicate melting. As discussed elaborately by Sahijpal et al. [2007], and recently reviewed by Kleine et al. [2009], there is a general agreement among the results obtained from the thermal models developed by Merk et al. [2002], Hevey and Sanders [2006], Sahijpal et al. [2007], and Kleine et al. [2009]. [9] In the present work, we have numerically simulated two distinct differentiation scenarios that are relevant for our understanding of the origin of achondrites as elaborately discussed above. The primary concern is to numerically simulate the planetary differentiation of planetesimals with the origin of basaltic achondrites explained by two distinct scenarios. These scenarios involve (1) the partial melt origin of basalts, and (2) the origin of basalts from the residual melts of equilibrium crystallization in a cooling magma ocean. We make an attempt to understand the thermal evolution and differentiation in Vesta and other differentiated asteroids within these two distinct scenarios. The preliminary results of this work were presented in two conferences [Gupta and Sahijpal, 2007, 2009]. The present work is mostly focused on the initial H chondrite compo- 2of15

3 sition of the planetesimals as the melting of these chondrites are well understood both theoretically and experimentally [Taylor et al., 1993; McCoy et al., 2006]. In order to understand the influence of the initial bulk composition [Jurewicz et al., 1991, 1993, 1995; Taylor et al., 1993; Righter and Drake, 1997] on differentiation, we have also considered six different sets of composition. 2. Methodology [10] The numerical simulations performed in the present work involve solving the heat conduction partial differential equation of a planetesimal undergoing a linear accretion growth. The growth resulted in the formation of a porous planetesimal that subsequently suffered thermal sintering and volume loss on account of compaction. The core mantle and the mantle crust differentiations in the body were performed gradually according to the thermal evolution of the body. The differentiation was executed by the inward and outward flows of the melt plumes of Fe FeS and basalts Heat Conduction Equation [11] The radial heat conduction partial differential equation (equation (1)) for a spherically symmetry planetesimal (1 D) with the radionuclides 26 Al and 60 Fe as the heat source was used in order to determine the thermal profiles as a function 2 þ Q C ðexpð tþþ ð1þ This equation was solved numerically using the finite difference method with the classical explicit approximation [Lapidus and Pinder, 1982] by modifying the basic code developed by Sahijpal [1997]. We used the temporal and spatial grids of sizes 1 year and 300 m (for consolidated body), respectively, in order to acquire the numerical stability during thermal evolution, sintering, and differentiation. The simulations were performed in double precision, and, as discussed below, proper precautions were taken to avoid singularities and oscillations in the various thermodynamical quantities that were thoroughly monitored during the simulations. An initial 26 Al and 60 Fe power generated per unit mass for an undifferentiated planetesimal was considered to be Wkg 1 and ( ) 10 8 Wkg 1, respectively [Sahijpal et al., 2007]. The latter range in the case of 60 Fe corresponds to the uncertain value of ( 60 Fe/ 56 Fe) initial in the range of (0.5 2) Simulation Parameters [12] We have considered a linear growth in the radii [Merk et al., 2002] of the unconsolidated (porous) planetesimals of sizes 26 km, 65 km, 130 km, and 351 km from the nebular dust of 55% porosity. The equation that governs the linear accretion is R(t) = Ro + at, where Ro is the initial radius of the unconsolidated planetesimal considered at the time of the initiation of the accretion of the planetesimal. It was taken to be 390 m. This would eventually be reduced to 300 m on account of compaction (consolidation) during sintering. The accretion growth of the planetesimals was performed by modifying the spatial grid array of the finite difference code. This was achieved by the addition of un sintered (porous) 390 m shells to the pre existing planetesimal, thereby resulting in the gradual growth of the planetesimals in an equal incremental size step [Sahijpal et al., 2007]. Since the accretion of the planetesimal was considered to be occurring in an unconsolidated (porous) state of matter we did not execute any thermal re equilibration of the pre existing planetesimal surface with the newly accreted shell as the thermal diffusivity of the unconsolidated body is assumed to be three orders of magnitude less than that of the consolidated body [Yomogida and Matsui, 1984]. The accretion of the body initiates at a specific time interval t Onset (0 3 Ma) after the formation of the Ca Al rich inclusions (CAIs) with the canonical value of for 26 Al/ 27 Al [MacPherson et al., 1995]. The accretion duration (t Duration ) of the planetesimals in the present work was considered in the range of to 1 Ma for different sizes of the planetesimals [Sahijpal et al., 2007]. The lower time interval corresponds to almost instantaneous accretion. During the growth of the body the average temperature of the accreting bodies on the planetesimals were assumed to be identical to the ambient temperature of the nebula that is considered to be 250 K. A constant surface temperature of 250 K was maintained throughout the simulation [Hevey and Sanders, 2006]. A moving boundary condition during the accretion growth of planetesimals was adopted by redefining the planetesimal surface each time the spatial grid array was enlarged [Sahijpal et al., 2007]. [13] The majority of the simulations were performed with H chondrite composition. The uniformly distributed abundances of 1.22% and 27.8% were assumed for the Al and Fe, respectively, in the undifferentiated planetesimals. We have carried out most of the simulations with a value of for ( 60 Fe/ 56 Fe) initial ratio at the time of the formation of CAIs with the canonical value of 26 Al/ 27 Al. Due to the uncertainty in defining the ( 60 Fe/ 56 Fe) initial ratio, we have carried out additional simulations with the ( 60 Fe/ 56 Fe) initial ratio of and The simulation parameters, e.g., the decay energies of 26 Al and 60 Fe, the temperature dependent specific heat and thermal diffusivity, the latent heat of the [(Fe Ni) metal FeS] and silicate melting, the densities of silicate, [(Fe Ni) metal FeS], the solidus and liquidus temperature range of silicate and [(Fe Ni) metal FeS], etc., were adopted from Sahijpal et al. [2007, Table 1]. [14] The porous planetesimal was gradually sintered in the assumed temperature range of K [Sahijpal et al., 2007]. The porosity of the body was reduced from 55 to 0% on account of the compaction of the body. The thermal diffusivity of the unconsolidated (porous) body ( m 2 s 1 ) for the H chondrite composition was assumed to be three orders of magnitude lower than that for the consolidated body m 2 s 1 [Yomogida and Matsui, 1984; Hevey and Sanders, 2006]. The increase in the thermal diffusivity of the body along with the decrease in the volume on account of sintering [Yomogida and Matsui, 1984; Hevey and Sanders, 2006] in the temperature range was executed by adopting a three parameter Sigmoidal function (equation (2)). ¼ i þ f f ¼ A=½1 þðexpð ðtðm; nþ BÞ=CÞÞŠ ð2þ 3of15

4 Here, represents the thermal diffusivity during sintering. i is the thermal diffusivity of the un sintered body, and T(m,n) represents the temperature of a particular spatial grid element m at a temporal grid n. The values of the constants were adopted as A = m 2 s 1 ; B = K; C = K for the numerical stability of the simulation. In the absence of any suitable standard prescription in the literature for the numerical implementation of sintering [see, e.g., Hevey and Sanders, 2006], this assumption was made since among all the standard mathematical functions that deal with three orders of magnitude variations, the Sigmoidal function allows for a well defined and gradual variation of a thermodynamical function between two extreme well defined initial and final states. In the present case these two states are associated with the thermal diffusivity of the consolidated and un consolidated body. Sahijpal et al. [2007] demonstrated that this technique would yield identical results as obtained by Hevey and Sanders [2006]. The sintering effects are resolvable over the temporal scales monitored during the simulation runs and quoted in the present work. It was generally observed that the planetesimal underwent consolidation over a timescale of 8000 years. This timescale is quite sufficient to bring in the drastic volume change in the planetesimal in uniform manner and does not seem to be un realistic. The radii of the planetesimals were gradually reduced during sintering to their final values corresponding to the consolidated bodies. The thermal profiles and the numerical values of various thermodynamical functions were monitored during sintering in order to avoid any singularity and discontinuity. In general, with the present available knowledge, the process of sintering is too complex to be numerically simulated and that too for a planetesimal undergoing accretion. Nonetheless, the presently adopted phenomenological approach will serve as a guideline for the future researchers to quantitatively understand the process of sintering. However, it should be mentioned that in the present case we are dealing with the planetesimals that attain temperatures exceeding their melting points; the choice of the sintering criteria will not significantly influence the final outcome of the thermal evolution and differentiation of planetesimal. The thermal models that deal with either aqueous alteration or low temperature thermal metamorphism demand greater considerations regarding the sintering issue as the temperature generated in these models would be comparable to the sintering temperature. This could pose serious numerical difficulties and uncertainties in such cases regarding the most plausible course of thermal evolution. [15] The porous planetesimals of radii 26 km, 65 km, 130 km, and 351 km acquire their final radii of 20 km, 50 km, 100 km, and 270 km, respectively, subsequent to sintering. We adopted the final radii of the planetesimals to represent the nomenclature of the simulations Differentiation of the Planetesimals [16] We have considered the melting of [(Fe Ni) metal FeS] and silicates in the temperature range of K and K, respectively. A nonlinear temperature dependence of the melt fraction of the bulk silicate generated at a particular temperature [Taylor et al., 1993] was considered in the present work instead of the assumed simplified linear temperature dependence considered earlier by Sahijpal et al. [2007]. The latent heats of the melting of Jkg 1 and Jkg 1 were incorporated into the specific heat during the solidus liquidus temperature range of [(Fe Ni) metal FeS] and silicate, respectively, according to the method adopted by Merk et al. [2002]. The specific heat of 2000 J kg 1 K 1 was assumed for the silicate and [(Fe Ni) metal FeS] melts. The specific heat at a particular spatial grid interval was estimated by taking the weighted average of the specific heat of the melt and the solid mass fractions [Sahijpal et al., 2007]. The density of [(Fe Ni) metal FeS] was assumed to be 7800 Kg m 3. The schematic of the algorithm of the numerical simulation of the planetesimal accretion, followed by sintering and initiation of melting is presented in Figure 1. [17] Numerical and experimental studies of the iron melt segregation from silicate infer complexity that is at present difficult to quantitatively incorporate in the numerical simulations of planetary differentiation [Yoshino et al., 2003; Rushmer et al., 2005; Petford et al., 2006; McCoy et al., 2006]. Sahijpal et al. [2007] performed detailed numerical simulations of the planetary differentiation scenario involving the initiation of the iron core formation prior to the bulk silicate melting. The core mantle differentiation in the present work was triggered subsequent to 40% bulk silicate melting in accordance with the depletion of siderophile elements in eucrite parent body [Newsom, 1985; Hewins and Newsom, 1988; Drake, 2001]. The segregation of iron melt at substantial silicate melting is compatible with the numerical and experimental studies of iron melt segregation. We estimated the mass of [(Fe Ni) metal FeS] and silicates in both melt and un melt forms in each 0.3 km wide spatial grid interval adopted for the finite difference method. Subsequent to 40% melting of the bulk silicates, the dense [(Fe Ni) metal FeS] melt plumes within a specific spatial grid were gradually moved toward the planetesimal center at an assumed rate of one spatial grid (i.e., 300 m) transfer of the melt per temporal grid interval of one year. This assumed rate of descend was numerically achieved by taking into account mass balance calculations. The schematic of the algorithm involved in the planetary differentiation is presented in Figure 2. Within a specific spatial grid interval, the iron melt plumes arriving from the upper layer were appropriately added to the pre existing iron melt plumes in the spatial grid interval. The net momentum inflow of iron melt depends upon two factors, namely, the velocity of the inflow and the bulk mass transfer. We assumed a velocity of 300 m per year on the basis of our choice of the spatial and temporal grid intervals. This could be varied depending upon the choice. However, as mentioned in the beginning of this section we preferred the present combination of the spatial and temporal grid intervals to avoid complexities originating from the presently employed finite difference method with the classical explicit approximation. The present technique allows for a limited range of spatialtemporal grid size combinations. However, the use of Crank Nicholson approximation [Lapidus and Pinder, 1982] in solving the finite difference method would relax the constraints on the choice of temporal grid interval for a specific spatial grid interval as the approximation leads to unconditionally stable solutions. We could develop an elementary numerical code using Crank Nicholson approximation to decipher the thermal profile of the planetesimals 4of15

5 Figure 1. The schematic representation of the simulation algorithm prior to differentiation. The accretion of the porous planetesimal initiates at a time T Onset subsequent to the condensation of CAIs. The linear accretion occurs over a time span, T Duration. The sintering occurs in the temperature range K and results in a volume loss. The melting is initiated at 1213 K and 1450 K in the case of Fe FeS and bulk silicate, respectively. but it is difficult for us to incorporate the planetary differentiation processes in this code in contrary to our code employing classical explicit approximation. Any future attempts to incorporate planetary differentiation processes in the finite difference method with the Crank Nicholson approximation would allow the choice of any desired descend velocity of the iron melt plumes in the differentiating planetesimals. However, it should be mentioned that the present choice of spatial and temporal grid sizes are appropriate to simulate the core mantle differentiation in a realistic manner. This aspect is discussed in details afterwards in this section. [18] During the segregation of iron and silicate, a set of free parameters were used to transfer a fraction of the bulk accumulated mass of iron melt from a specific spatial grid interval to the next spatial grid interval, toward the planetesimal center. These parameters are essential to control the mass flow of the melt in a systematic manner by avoiding any excess of melt accumulation/depletion in any spatial grid. The procedure was executed by monitoring the net density of matter in the melt and un melt forms for all the spatial grids. [19] An identical approach was also followed for the ascend of silicate melt or un melt pockets coming from the interior of the planetesimal. The specific heat, thermal diffusivity, the elemental abundances of Al and Fe were estimated within all the spatial grid intervals at each temporal step. The dense iron melt plumes, subsequent to reaching the central spatial grid, replace the silicates and push the silicate pockets in an upward direction due to buoyancy. This process was executed by using the mass balance calculations and logical statements in the numerical code. The [(Fe Ni) metal FeS] melt upon reaching the center initiated the formation of the dense iron core. The differentiation was executed from inside out, i.e., starting from the inner most spatial grid, the consecutive spatial grid intervals acquired [(Fe Ni) metal FeS] saturation. The gradual descend of the various Fe FeS melt plumes from different spatial grids toward the center of the planetesimal eventually led to the formation of a dense [(Fe Ni) metal FeS] core. The replaced silicate that was pushed upward constituted the aggregate mantle and crust composition. In order to avoid discontinuity in any thermodynamical quantity and clumping in mass and density within any spatial grid interval the movement of the various pockets were monitored and controlled by a suitable set of free parameters as discussed above. These free parameters control the rate of descend and ascend of the various melt and un melt pockets. An identical set of numerical values were chosen for these parameter in all the simulations spanning over the planetesimal accretion time spans of to 1 Ma. Along with the major elemental redistribution during the core mantle and the mantlecrust differentiation, the redistribution of the heat sources 26 Al and 60 Fe was also implemented in the simulations. The silicate matrix melt in all simulations with the H chondrite composition retained 8.8% Fe as FeO along with a proportionate amount of 60 Fe after core mantle differentiation. [20] It should be noted that the numerical criteria adopted for the planetary differentiation by Sahijpal et al. [2007], and in the present work is a phenomenological approach as it is extremely difficult to pursue the problem starting from the first principle. The phenomenology is the only opted scientific approach in circumstances where it is difficult either to work out a theory starting from the first principle, or the associated mathematical complexities are so huge that a parameter based model, like the one opted in the present work, seems to be a better alternative. It should be noted that some of the most fundamental concepts in physical sciences 5of15

6 Figure 2. The schematic representation of the process of planetary differentiation (see text for details). The entire planetesimal was divided into 0.3 km wide concentric shells. The [(Fe Ni) metal FeS] and silicate masses in both melt and un melt forms were estimated in each concentric shell. Subsequent to 40% bulk silicate melting, the [(Fe Ni) metal FeS] melt plumes in the various shells were moved toward the central spatial grids to form the core. The replaced silicate melt and un melt pockets were moved in an upward direction to form the mantle. The core mantle differentiation was performed from inside out. The basaltic melt extrusion from a specific spatial grid interval was carried out by the ascend of the basaltic melt plumes toward the planetesimal surface. The basaltic melt plumes moved outwards by one spatial interval of 0.3 km in a time step of 1 and 1000 year, respectively, in two sets of simulations corresponding to distinct basaltic melt percolation velocities. till date are phenomenological, e.g., the Newton laws, numerous concepts in electrodynamics, high energy physics, cosmology, and even fluid dynamics. Further, the most fundamental Darcy s law in the fluid dynamics, relevant in the present case [Taylor et al., 1993; Yoshino et al., 2003], is in fact a phenomenological approach. In the presently developed numerical code it is easier to move the fluids across an asteroid in a parameterized manner. The fluid flow rates adopted in the present work and by Sahijpal et al. [2007] are in fact not distinct from the rates deduced by the previous fluid dynamics model [Yoshino et al., 2003]. Even though we can parametrically control the fluid flow rates to an extent as discussed previously, yet it would be difficult to precisely reproduce the thermal evolution of planetesimals that involves a complex dependence of the fluid flow rates as a function of fluid viscosity, porosity, and thesizeofthedifferentiating planetesimals [Taylor et al., 1993; Yoshino et al., 2003]. However, the timescales associated with the core mantle differentiation, quoted with the least count of 10 kilo years in the present work, would not be significantly influenced by the differentiation process that occurs on the timescales of 10 kilo years subsequent to the onset of the differentiation. These timescales are within the range of the migration timescales of metal melts based on Darcy s law [Yoshino et al., 2003]. However, the uncertainties in the melt percolation velocities of the basaltic melt subsequent to partial melting [Taylor et al., 1993] could significantly influence the deduced timescales of the mantle crust differentiation. The uncertainties in the timescale involved in the basaltic melt migration to the planetesimal surface could be of the order of the mean life of 26 Al. We have appropriately incorporated this issue in our simulations. [21] As mentioned earlier the temperature dependence of thermal diffusivity and specific heat [Sahijpal, 1997; Sahijpal et al., 2007] were considered in the simulations. The thermal diffusivity of the molten [(Fe Ni) metal FeS] was assumed to be m 2 s 1. The influence of convection as an efficient mode of heat transfer was incorporated in an analogous manner by increasing the thermal diffusivity of the [(Fe Ni) metal FeS] melt core and the molten silicate magma ocean by three orders of magnitude (i.e., m 2 s 1 ) compared to the sintered rock. The convection is a complex process [Marsh, 1989; Hort et al., 1999; Elkins Tanton et al., 2008], and it is difficult at present to incorporate in the model [Hevey and Sanders, 2006; Sahijpal et al., 2007]. In the presently used finite difference method with the classical explicit approximation it is not possible to raise the thermal diffusivity beyond the assumed value of m 2 s 1 for the magma ocean as it causes numerical instabilities [Lapidus and Pinder, 1982]. The Crank Nicholson approximation would be better in handling such instabilities without compromising the use of still higher thermal diffusivity [Lapidus and Pinder, 1982]. Nonetheless, it should be mentioned that the results obtained by Sahijpal et al. [2007] using classical explicit approximations matches exactly with those obtained by Hevey and Sanders [2006] using Crank Nicholson approximation. Hence, we do not anticipate major influence of the use of an even higher thermal diffusivity on the outcome of the present work. We were able to achieve almost uniform temperature within the spatial extent of the magma ocean. [22] The immediate concern regarding the convection in the present work is the behavior of the cooling convective region at the base of the outer crustal layer of the planetesimal. This is the region through which the planetesimal losses thermal energy. The convection shows strong dependence on the fluid viscosity that in turn depends upon the temperature. A variation in the viscosity by more than an order of magnitude can bring down the extent of convection [Marsh, 1989; Hort et al., 1999]. The resulting margin of the magma ocean that constitutes rigid crust, mush and suspension would provide an insulating blanket to the cooling magma ocean. This thermal insulation along with the insulation provided by any survived chondritic crustal layer would eventually control the cooling and the crystallization in the magma ocean. In this present work it is not possible to numerically simulate the behavior of a realistic magma ocean by considering all these aspects. Nonetheless, the survived chondritic crustal layer of thickness 2 3kmin the present setup controls the magma ocean cooling. [23] In order to numerically implement the high thermal diffusivity of the molten magma ocean and the molten [(Fe Ni) metal FeS] core, the thermal diffusivity was gradu- 6of15

7 Figure 3. The schematic of the algorithm of the numerical simulation for the scenario involving partial melt (PM) origin of the basalts. The basaltic volcanism was initiated at 20% bulk silicate melting. The core mantle differentiation was commenced subsequent to 40% bulk silicate melting. ally increased subsequent to 0.5 fraction of bulk silicate melting [Hevey and Sanders, 2006] by using a three parameter Sigmoidal function in the assumed temperature range of K. A distinct set of the constants were adopted for equation (2). These are A = m 2 s 1,B= 1725 K, and C = 10 K, with i as the thermal diffusivity of the sintered body. The temperature range from 1680 to 1820 K corresponds to 0.44 to 0.71 fraction of bulk silicate melting. As mentioned earlier the use of the Sigmoidal function was made to avoid singularity or discontinuity in any simulation parameter. The identical but reverse trend was followed during the cooling of the magma ocean, i.e., the thermal diffusivity was gradually decreased from m 2 s 1 to m 2 s 1 as the temperature dropped from 1820 to 1680 K upon cooling of the magma ocean. Another set of constants for the Sigmoidal function was adopted for equation (2) during the cooling. These are A = m 2 s 1, B = 1600 K, and C = 20 K, with i as the temperature dependent thermal diffusivity of the sintered body. The variations in these parameters were found to have no major influence on the outcome of the present work. [24] In the present work, we have modeled two distinct differentiation scenarios for the origin of basaltic achondrites in order to understand the differentiation of the Vesta and other differentiated asteroids. One of these scenario deals with the origin of the basalts during the partial melting of the silicates, and the other deals with the residual melt origin of the basaltic achondrites subsequent to crystallization in the cooling magma ocean on the planetesimals Partial Melt Origin of the Basaltic Achondrites: PM Model [25] In this case, the mantle crust differentiation occurs prior to the core mantle differentiation due to partial melting of the bulk silicate that commences at 1450 K. In these simulations, the [(Fe Ni) metal FeS] melt moves toward the center at 0.4 fraction of bulk silicate melting and gradually forms the core. The 26 Al rich basaltic melt plumes generated in the different regions of the planetesimals were gradually moved upward toward the surface at 0.2 fraction of bulk silicate melting [Stolper, 1977; Jurewicz et al., 1991, 1993, 1995; Jones et al., 1996]. The systematic of the algorithm for this scenario is shown in Figure 3. The migration of the 26 Al rich basaltic melt plumes generated from different spatial grid intervals was executed distinctly. A set of logical statements were used to execute the systematic upward movement of the plumes. The radioactive heating due to the transit of the various basaltic melt plumes through various spatial grid intervals were appropriately considered. In order to address the issue of the uncertainties of the basaltic melt percolation velocities subsequent to partial melting, we performed two set of simulations with distinct velocities of the basaltic melt migration. Since the uncertainties in the timescales could be of the order of the mean life of 26 Al [Taylor et al., 1993], the basaltic melt plumes in the two set of simulations were moved at the rate of 300 m year 1 and 0.3 m year 1, respectively. The plumes moved outwards by one spatial interval of 300 m in a time step of 1 and 1000 year, respectively, for the two sets of simulations Residual Melt Origin of the Basaltic Achondrites: RM Model [26] The [(Fe Ni) metal FeS] melt generated in this scenario moves toward the center to form the core at 0.4 fraction of bulk silicate melting. Further in this scenario, we have simulated the convective magma ocean beyond 0.5 fraction of bulk silicate melting [Righter and Drake, 1997; Drake, 2001]. We maintained the convection until the temperature dropped gradually below 1600 K, where at least appreciable extent of equilibrium/fractional crystallization is complete to produce a residual melt for basalt [Righter and Drake, 1997; Drake, 2001]. Even though we have not numerically simulated the gravitational settling of crystals in the cooling molten magma subsequently followed by basaltic volcanism, the essential features of this scenario can be well understood based on the present simulations. The schematic of the algorithm of this scenario is presented in Figure 4. [27] Finally, in order to explore the dependence of the differentiation on the initial bulk composition of the planetesimals [Jurewicz et al., 1991, 1993, 1995; Taylor et al., 1993; Righter and Drake, 1997] apart from the presently used H chondritic composition we have considered six different sets of the initial bulk compositions. These are L, LL, CI, and CV chondritic compositions [Jarosewich, 1990], and the two binary combinations of compositions; one with 70% of LL and 30% of CI compositions and the other with 70% of L and 30% of CV compositions [Righter and Drake, 1997]. The temperature dependence of the thermal diffusivity and the specific heat in these simulations 7of15

8 Figure 4. The schematic of the algorithm of the numerical simulation for the scenario involving residual melt (RM) origin of the basalts. The core mantle differentiation was commenced subsequent to 40% bulk silicate melting. The residual melts subsequent to the crystallization in the cooling magma ocean would be the source of basalts in this scenario. were also included [Yomogida and Matsui, 1983]. We assumed an identical set of the remaining thermodynamical functions that were used in the case of H chondrites. 3. Results [28] The numerical simulations of the planetary differentiation of Vesta and other differentiated asteroids have been performed in the present work. The choice of the simulation parameters, specifically the time of the initiation of the planetesimal accretion (t Onset ) subsequent to the formation of CAIs, the accretion duration (t Duration ) and the ( 60 Fe/ 56 Fe) initial was made to substantially explore the parametric space in order to understand the extent of melting and differentiation of planetesimals within distinct scenarios. In general, a rapid accretion rate was chosen for small bodies of radii km, whereas the larger bodies of radii 100 and 270 km were accreted gradually over a time span of one million years. [29] The temporal growth of the [(Fe Ni) metal FeS] core and the extrusion of the basaltic melt during the partial melting of silicate have been presented for a representative set of simulations in Table 1. A representative set of simulations corresponding to the residual melt origin of the basalts have been also considered in the table. The nomenclature of a specific simulation was chosen according to the nature of the adopted differentiation scenario and the general attributes of the planetesimal. We have used the model name PM (Partial Melt scenario) orrm(residual Melt scenario), followed by the radius of the sintered planetesimals ( km), the onset time of the accretion of the planetesimals (0 3 Ma), the accretion time of the planetesimals ( Ma) and the initial ratio of 56 Fe/ 60 Fe in the range In the case of the partial melt scenario two set of simulations were run with the basaltic melt percolation velocities of 300 m year 1 and 0.3 m year 1. The simulations with the slower melt percolation velocity are marked by the suffix s with the radius of the sintered planetesimal. The details of the various simulation parameters are also provided in the footnote of the Table 1. The thermal profiles along with the growth of the iron core have been graphically presented in Figure 5 for the partial melt (PM) scenario and Figure 6 for the residual melt (RM) scenario. [30] The Table 2 presents the dependence of the initiation of the basaltic melt extrusion for different onset times of the accretion of the planetesimals on the initial bulk composition of a 100 km sized planetesimal. The basaltic melt can move to the planetesimal surface over the timescale with uncertainty comparable to the mean life of 26 Al [Taylor et al., 1993]. The Table 3 presents the dependence of the onset time of the initiation of core mantle differentiation on the initial bulk composition of a 100 km sized planetesimal. We have considered both scenarios in order to study the effect of bulk initial compositions on the differentiation of the planetesimals. The results for the planetesimals with four distinct chemical compositions L, LL, CI, and CV and a sets of two binary combinations of compositions, one with (0.7LL + 0.3CI) and the other with (0.7L + 0.3CV) [Jurewicz et al., 1991, 1993, 1995; Taylor et al., 1993; Righter and Drake, 1997] have been compared with the H chondritic composition. 4. Discussion [31] Numerical simulations for the planetary differentiation of Vesta and other differentiated asteroids have been performed. With the present work, along with the earlier study [Sahijpal et al., 2007], a systematic theoretical framework has been developed to critically examine the various proposed planetary differentiation scenarios that differ from each other in terms of the relative timings between the core mantle and mantle crust differentiation besides the geochemical differences associated with various differentiation scenarios [Taylor et al., 1993; Righter and Drake, 1997]. [32] The dependence of the thermal evolution of the planetesimals on the various simulation parameters, specifically on the time of the initiation of the planetesimal accretion (t Onset ) subsequent to the formation of the CAIs, the accretion duration (t Duration ), the size of the planetesimals and the choice of ( 60 Fe/ 56 Fe) initial has been extensively studied in the 8of15

9 Table 1. Growth of the (Fe Ni) metal FeS Core and the Extrusion of the Basaltic Melt in Model PM for the Differentiation of Planetesimals Simulations a Figure References Fe FeS Core b Radius = 20 km 0 + km 2 km 4 km 6 km 8 km Initiation of the Extrusion of Basaltic Melt c PM ( 6) none 1.35 PM ( 6) 5a none none none none none 2.95 PM ( 6) none 1.37 PM ( 6) none none none none none 2.98 Radius = 50 km 0 + km 5km 10km 15km 20km PM ( 6) PM ( 6) PM ( 6) 5b PM ( 6) 5c PM ( 6) none 3.04 Radius = 100 km 0 + km 10 km 20 km 30 km 40 km PM ( 6) RM ( 6) PM ( 6) RM ( 6) PM ( 6) 5d PM100s 1 1 2( 6) (1.55) RM ( 6) PM ( 6) 5e PM ( 7) 5f none none none 1.48 PM ( 6) 5g PM100s 2 1 2( 6) 5h (3.33) RM ( 6) Radius = 270 km 0 + km 27 km 54 km 81 km 108km PM ( 6) PM270s 1 1 2( 6) (2.25) RM ( 6) PM ( 6) 5i PM270s 2 1 2( 6) 5j (3.87) RM ( 6) a The simulations are titled according the choice of the various parameters. These parameters are separated by hyphens. In order these parameters are (1) the simulations types PM and RM and the radius of the planetesimals subsequent to complete sintering. In the case of the partial melt scenario the simulations with the slower melt percolation velocity of 0.3 m year 1 aredesignatedwithasuffix s with the radius of the planetesimal. The remaining simulations were run with the melt percolation velocity of 300 m year 1. (2) The onset time, t Onset (Ma, Million year), to initiate the accretion of a planetesimal from the time of the formation of the CAIs with the canonical value of ( 26 Al/ 27 Al) initial = (3) The accretion duration, t Duration (Ma), of the planetesimal. (4) An initial value [represented as 5( 7)], [represented as 1( 6)]and [represented as 2( 6)] for the 60 Fe/ 56 Fe ratio at the time of formation of the CAIs with the canonical value of ( 26 Al/ 27 Al) initial. b Time (Ma) taken for the Fe FeS core to grow to a specific size. Five different arbitrary choices of the core sizes have been considered. With respect to the final radius of the completely sintered planetesimal, these core sizes are expressed in percentage. In order these are 0 + % (initiation of core formation), 10%, 20%, 30% and 40%. All time spans are measured with respect to the formation of the CAIs with the canonical value of ( 26 Al/ 27 Al) initial = c Time obtained for the initiation of the extrusion of the basaltic melt for PM simulations with respect to the formation of the CAIs with the canonical value of ( 26 Al/ 27 Al) initial. The values in the bracket represent the time for the extrusion of the basaltic melt to the surface of the planetesimals. earlier studies [Merk et al., 2002; Hevey and Sanders, 2006; Sahijpal et al., 2007]. The initiation and the growth of the Fe FeS core, and the initiation of the melting of the bulk silicate critically depend upon these parameters, specifically on the onset time of the accretion of the planetesimals in the time interval 2 3 Ma. We avoid detailed discussion on these aspects. The Figures 5 and 6 along with the Tables 1 3 rather provide insight into these aspects for the presently adopted differentiation scenarios of the planetesimals with different compositions. In the present work, we focus on the outcome of the two presently simulated differentiation scenarios regarding the origin of the main group of basaltic achondrites along with the possible implications on the evolution of differentiated planetesimals. We have parametrically incorporated the influence of thermal convection in both the models. Due to thermal convection the heat transfers efficiently from the molten interiors to the overlying partially melted and un melted exteriors of the planetesimal. This leads to a rapid heating and melting in the exterior regions (Figures 5 and 6). In the following text, we discuss the two differentiation scenarios and finally make assessments regarding the likely hood of these scenarios to be the cause of planetary differentiation of Vesta and other differentiated asteroids Partial Melt Origin of the Basalts: PM Model [33] In the case of the PM simulations, the mantle crust differentiation occurs prior to the core mantle differentiation 9of15