Marine Paquet
email: paquet@ipgp.fr

Paolo A Sossi

Frédéric Moynier
email: moynier@ipgp.fr

Origin and abundances of volatiles on Mars from the zinc isotopic composition of Martian meteorites

Keywords: Mars, zinc isotopes, volatile elements, meteorites, chondrites

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Barring a sample return mission to Mars, Martian meteorites are the sole tangible records of its crust available for laboratory investigation. These samples provide insights into the composition of Mars, as well as the processes that led to its accretion and differentiation (e.g., [START_REF] Mcsween | What we have learned about Mars from SNC meteorites[END_REF][START_REF] Nyquist | Ages and geologic histories of Martian meteorites[END_REF][START_REF] Treiman | The SNC meteorites are from Mars[END_REF][START_REF] Halliday | The accretion, composition and early differentiation of Mars[END_REF][START_REF] Mcsween | Elemental composition of the Martian crust[END_REF][START_REF] Taylor | The bulk composition of Mars[END_REF][START_REF] Tait | Chondritic late accretion to Mars and the nature of shergottite reservoirs[END_REF][START_REF] Udry | What Martian meteorites reveal about the interior and surface of Mars[END_REF][START_REF] Paquet | Highly siderophile elements in shergottite sulfides and the sulfur content of the Martian mantle[END_REF][START_REF] Tian | Potassium isotope composition of Mars reveals a mechanism of planetary volatile retention[END_REF].

Volatile elements play an essential role in the evolution of planetary bodies. However, the timing, budget, nature and extent of volatile-depletion in planetary bodies during the earliest stages of solar system evolution remain poorly constrained. The terrestrial planets are depleted in moderately volatile elements (MVEs) relative to the Sun or CI-chondrites, (e.g., [START_REF] Anders | Mars and Earth: Origin and Abundance of Volatiles: Mars has only 3 percent of Earth's share of volatiles, but got them from the same meteoritic source[END_REF][START_REF] Dreibus | Volatiles on Earth and Mars: A comparison[END_REF][START_REF] Palme | Cosmochemical Estimates of Mantle Composition[END_REF][START_REF] Khan | Geophysical and cosmochemical evidence for a volatile-rich Mars[END_REF]. Recent work on K isotopes showed that Martian meteorites have systematically heavier K isotopic compositions than the Bulk Silicate Earth (BSE), suggesting that volatile depletion proceeded to a greater extent, producing stronger K isotope fractionation by evaporation relative to their potential precursor materials [START_REF] Tian | Potassium isotope composition of Mars reveals a mechanism of planetary volatile retention[END_REF]. However, this interpretation contrasts with the observation that the abundances of volatile elements, such as Zn, Rb and K, are higher in the estimated composition of the Martian mantle than in the BSE [START_REF] Yoshizaki | The composition of Mars[END_REF][START_REF] Khan | Geophysical and cosmochemical evidence for a volatile-rich Mars[END_REF].

Zinc is an ideal element to investigate volatile delivery and loss during planetary accretion [START_REF] Day | Evaporative fractionation of volatile stable isotopes and their bearing on the origin of the Moon[END_REF] because it is lithophile and moderately volatile (with a half-condensation temperature (Tc) of 704 K), more so than K (Tc = 993 K) under solar nebula conditions [START_REF] Wood | The condensation temperatures of the elements: A reappraisal[END_REF]. [START_REF] Wood | Accretion and core formation: The effects of sulfur on metal-silicate partition coefficients[END_REF] showed Zn also exhibits chalcophile behavior to some extent, resulting in Zn partitioning into metal slightly increasing with the S content in the metal. Recent studies suggested that the Martian core may be richer in S than that of the Earth, with ~10 wt.% of S (e.g., [START_REF] Wang | Earth's moderately volatile element composition may not be chondritic: Evidence from In, Cd and Zn[END_REF][START_REF] Steenstra | A synthesis of geochemical constraints on the inventory of light elements in the core of Mars[END_REF][START_REF] Yoshizaki | The composition of Mars[END_REF][START_REF] Khan | Geophysical and cosmochemical evidence for a volatile-rich Mars[END_REF]. For 10 wt.% of S in the metal, the partition coefficient for Zn is 0.06 × DFe [START_REF] Wood | Accretion and core formation: The effects of sulfur on metal-silicate partition coefficients[END_REF], or DZn ~ 0.06 for DFe ~10 [START_REF] Righter | Effect of water on metal-silicate partitioning of siderophile elements: a high pressure and temperature terrestrial magma ocean and core formation[END_REF] -value that decreases with increasing pressure [START_REF] Mann | Evidence for high-pressure core-mantle differentiation from the metal-silicate partitioning of lithophile and weakly-siderophile elements[END_REF]. Limited to negligible Zn isotope fractionation is expected due to metal-silicate and sulfide-silicate partitioning [START_REF] Bridgestock | Unlocking the zinc isotope systematics of iron meteorites[END_REF][START_REF] Mahan | Elemental partitioning and isotopic fractionation of Zn between metal and silicate and geochemical estimation of the S content of the Earth's core[END_REF][START_REF] Xia | The effect of core segregation on the Cu and Zn isotope composition of the silicate Moon[END_REF]) with respect to the temperatures required for the formation of the Martian core (~2300 K; e.g., [START_REF] Righter | Core formation in Earth's moon, Mars, and Vesta[END_REF][START_REF] Rai | Core-mantle differentiation in Mars[END_REF], independently of the S content of the experiments. Thus, it is reasonable to assume that the Bulk Mars and the Bulk Silicate Mars have nearly identical Zn isotopic composition. Furthermore, Zn exhibits limited isotope fractionation (<0.1 ‰ per atomic mass unit) during igneous processes on Earth, namely partial melting and fractional crystallization (e.g., [START_REF] Chen | Zinc isotope fractionation during magmatic differentiation and the isotopic composition of the bulk Earth[END_REF][START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF]. In terrestrial and chondritic samples, variations of δ 66 Zn (the per mille deviation of the 66 Zn/ 64 Zn ratio from the JMC-Lyon standard) larger than 1‰ have been observed mostly in association with volatility-related processes (e.g., [START_REF] Moynier | Isotopic composition of zinc, copper, and iron in lunar samples[END_REF][START_REF] Moynier | Isotopic fractionation of zinc in tektites[END_REF][START_REF] Moynier | Nature of volatile depletion and genetic relationships in enstatite chondrites and aubrites inferred from Zn isotopes[END_REF][START_REF] Kamber | Evaporative loss of moderately volatile metals from the superheated 1849 Ma Sudbury impact melt sheet inferred from stable Zn isotopes[END_REF][START_REF] Mathur | Fingerprinting the Cretaceous-Paleogene boundary impact with Zn isotopes[END_REF]. Thus, Zn isotopic fractionation, combined with variations in Zn abundances in igneous rocks from various differentiated planetary bodies, such as the Earth, the Moon or Mars, can reveal significant differences in volatile depletion and recondensation events for these bodies.

Previous Zn isotopic data for Martian meteorites show a range of δ 66 Zn from +0.13 to +0.35‰, defining a mean value (+0. 25 ± 0.11 ‰, 2SD n=11;[START_REF] Paniello | Zinc isotopic evidence for the origin of the Moon[END_REF] that is not resolvable from the isotopic composition of the BSE (+0.16 ± 0.06 ‰ to +0.20 ± 0.03 ‰; [START_REF] Mccoy-West | The Fe and Zn isotope composition of deep mantle source regions: Insights from Baffin Island picrites[END_REF][START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF][START_REF] Doucet | Archean lithospheric differentiation: Insights from Fe and Zn isotopes[END_REF][START_REF] Fang | Zinc isotope fractionation in mantle rocks and minerals, and a revised δ 66 Zn value for the Bulk Silicate Earth[END_REF]) within error. Thus, it is necessary to re-evaluate the Zn isotopic composition of Mars, to test whether the hypothesis that volatile depletion and associated isotopic fractionation should be more pronounced on smaller bodies, based on K isotopes [START_REF] Tian | Potassium isotope composition of Mars reveals a mechanism of planetary volatile retention[END_REF], also holds for Zn. To better estimate the Zn isotopic composition of the Bulk Silicate Mars (BSM), we present new high precision Zn isotope data for ten Martian meteorites that represent the major petrologic types. Combined with previous measurements, this new dataset allows assessment of the Zn isotope composition of the Martian mantle. After accounting for the effects of shallow-level crystallization and partial melting, it is possible to derive a δ 66 Zn value for the BSM. We interpret these data in terms of the accretionand differentiation history of Mars.

In addition, nucleosynthetic anomalies provide essential information to identify the origin of the material accreted to planetary bodies, as well as early Solar System conditions and processes (e.g., [START_REF] Clayton | Oxygen isotope studies of achondrites[END_REF][START_REF] Trinquier | Widespread 54 Cr heterogeneity in the inner solar system[END_REF][START_REF] Trinquier | Origin of nucleosynthetic isotope heterogeneity in the solar protoplanetary disk[END_REF][START_REF] Burkhardt | Terrestrial planet formation from lost inner solar system material[END_REF][START_REF] Savage | Zinc isotope anomalies in primitive meteorites identify the outer solar system as an important source of Earth's volatile inventory[END_REF][START_REF] Steller | Nucleosynthetic zinc isotope anomalies reveal a dual origin of terrestrial volatiles[END_REF][START_REF] Martins | Nucleosynthetic isotope anomalies of zinc in meteorites constrain the origin of Earth's volatiles[END_REF]. Zinc isotopic anomalies, defined as e 66 Zn (deviation in parts per 10,000 of the 66 Zn/ 64 Zn from the terrestrial standard JMC-Lyon, normalized to 68 Zn/ 64 Zn), exhibit a difference between non-carbonaceous chondrite (NC) and carbonaceous chondrite (CC) material [START_REF] Savage | Zinc isotope anomalies in primitive meteorites identify the outer solar system as an important source of Earth's volatile inventory[END_REF][START_REF] Steller | Nucleosynthetic zinc isotope anomalies reveal a dual origin of terrestrial volatiles[END_REF][START_REF] Paquet | Contribution of Ryugu-like material to Earth's volatile inventory by Cu and Zn isotopic analysis[END_REF], as also observed in more refractory elements [START_REF] Trinquier | Widespread 54 Cr heterogeneity in the inner solar system[END_REF][START_REF] Trinquier | Origin of nucleosynthetic isotope heterogeneity in the solar protoplanetary disk[END_REF][START_REF] Budde | Molybdenum isotopic evidence for the origin of chondrules and a distinct genetic heritage of carbonaceous and non-carbonaceous meteorites[END_REF][START_REF] Kruijer | Age of Jupiter inferred from the distinct genetics and formation times of meteorites[END_REF][START_REF] Nanne | Origin of the non-carbonaceous-carbonaceous meteorite dichotomy[END_REF]. The recognition that these meteorite groups define two distinct isotopic populations permits the origin of volatile elements on Earth to be quantified, because Earth has a mass-independent Zn isotopic composition intermediate between CC and NC material [START_REF] Savage | Zinc isotope anomalies in primitive meteorites identify the outer solar system as an important source of Earth's volatile inventory[END_REF][START_REF] Steller | Nucleosynthetic zinc isotope anomalies reveal a dual origin of terrestrial volatiles[END_REF][START_REF] Paquet | Contribution of Ryugu-like material to Earth's volatile inventory by Cu and Zn isotopic analysis[END_REF]. Such modeling based on Zn nucleosynthetic anomalies indicates that ~30% of Earth's Zn budget originates from carbonaceous chondrites, which can be explained by the accretion of ~5 to 12% by mass of CI/Ryugu-like or CV-like material, respectively [START_REF] Savage | Zinc isotope anomalies in primitive meteorites identify the outer solar system as an important source of Earth's volatile inventory[END_REF][START_REF] Steller | Nucleosynthetic zinc isotope anomalies reveal a dual origin of terrestrial volatiles[END_REF][START_REF] Paquet | Contribution of Ryugu-like material to Earth's volatile inventory by Cu and Zn isotopic analysis[END_REF], with the remainder coming from NC material. Because the different elements used to define the NC-CC dichotomy record different stages of a planet's accretion history owing to their different siderophilicities and volatilities (e.g., [START_REF] Dauphas | The isotopic nature of the Earth's accreting material through time[END_REF], the information contained in Zn anomalies differs from that in other elements. Following the same approach, we use the mass-independent Zn isotopic composition of Mars to better understand the provenance of volatile elements on Mars.

Samples and methods

Petrological characteristics

Martian meteorites are the only samples currently available from Mars and are traditionally divided into three main groups: shergottites, nakhlites and chassignites, collectively termed SNCs.

They are igneous rocks and have mafic to ultramafic compositions (~4 to 30 wt.% MgO; e.g., Udry et al., 2020 and references therein). Shergottites, broadly basaltic rocks derived from partial melting of distinct regions of the Martian mantle, are the most abundant class of Martian meteorites (~89% of the total collection by number; [START_REF] Udry | What Martian meteorites reveal about the interior and surface of Mars[END_REF]. Shergottites are usually divided into three petrographically defined groups (basaltic, olivine-phyric and poikilitic [ultramafic cumulates]) but also into three geochemical groups based on their incompatible trace element abundances and lithophile isotope ratios (enriched, intermediate and depleted; [START_REF] Borg | A petrogenetic model for the origin and compositional variation of the Martian basaltic meteorites[END_REF]. Nakhlites and chassignites are cumulates: nakhlites are clinopyroxenites to olivineclinopyroxenites that comprise cumulus olivine and pyroxene with minor glass, plagioclase, phosphate minerals, fayalite-rich olivine in the mesostasis, titanomagnetite, and sulfides, whereas chassignites are dunites and contain cumulus olivine with chromite inclusions and interstitial plagioclase, orthopyroxene, phosphate and sulfide minerals (e.g., [START_REF] Treiman | The nakhlite meteorites: Augite-rich igneous rocks from Mars[END_REF]Udry et al., 6 2020). Nakhlites and chassignites have similar crystallization (~1.3 Ga) and ejection (~11 Ma) ages, which suggests that they likely originate from the same location on Mars [START_REF] Nyquist | Ages and geologic histories of Martian meteorites[END_REF][START_REF] Cohen | Taking the pulse of Mars via dating of a plume-fed volcano[END_REF][START_REF] Udry | 1.34 billion-year-old magmatism on Mars evaluated from the co-genetic nakhlite and chassignite meteorites[END_REF]. For each sample, ~100 mg of homogenized powder was dissolved using a mixture of concentrated HNO3/HF and heated at 130°C in PTFE beakers. Following evaporation of the HNO3/HF, samples were subsequently treated with 6 mol.L -1 HCl and heated again to dissolve remaining fluoride complexes. Samples were then evaporated to dryness. Chemical purification was conducted for some samples at the Institut de Physique du Globe de Paris and for others at the Australian National University, Canberra. In Paris, the analytical methodology followed that described by van Kooten and Moynier (2019), while in Canberra it followed the procedure described in [START_REF] Sossi | Combined separation of Cu, Fe and Zn from rock matrices and improved analytical protocols for stable isotope determination[END_REF]. The Zn isotope compositions were then measured using a Thermo Scientific Neptune Plus Multi-Collector Inductively-Coupled-Plasma Mass Spectrometer (MC-ICP-MS) at either IPGP or at ANU, following the methods described in van Kooten and Moynier (2019) and [START_REF] Sossi | Combined separation of Cu, Fe and Zn from rock matrices and improved analytical protocols for stable isotope determination[END_REF], respectively.

Zinc purification and measurements

Zinc isotope compositions are reported in Table 1, as the deviation in parts per 1000 from the terrestrial standard JMC-Lyon:

d ! Zn = % " !" # !" $% #$%&'() " !" # !" $% #*+,-./01 -1( × 10 2 (1)
where x=66 or 68. Uncertainties are given as the 2 times standard deviation (2SD) of replicate measurements. Standard reference powders were analyzed together with the SNCs, yielding values of δ 66 Zn = ±0.29 ± 0.01 (2SE, n=4; IPGP) and +0.28 ± 0.03 (2SE, n=2; ANU) for BHVO-2 and +0.21 ± 0.04 (2SE, n=2; ANU) for Allende, respectively, which are consistent with recommended literature values [START_REF] Moynier | The isotope geochemistry of zinc and copper[END_REF]. The external reproducibility was therefore better than 0.04 on the δ 66 Zn value.

Zinc isotope measurements were also corrected for mass-dependent fractionation using the exponential law (Russel et al., 1978), with the internal normalizing ratio of 68 Zn/ 64 Zn of 0.3856 [START_REF] Rosman | A survey of the isotopic and elemental abundance of zinc[END_REF]. Zinc isotopic anomalies represent the deviation from the mass-dependent fractionation line (Figure S1) and are quantified using the epsilon notation, which corresponds to the deviation in parts per 10,000 from the terrestrial standard JMC-Lyon, as follows:

e 33 Zn , = % " !" $$ !" $% #$%&'() " !" $$ !" $% #*+,-./01 -1( × 10 4 (2)
where c is the normalizing ratio 68 Zn/ 64 Zn.

Results

All Martian meteorites fall slightly above the mass-dependent fractionation line in a δ 68 Zn versus δ 66 Zn plot (Figure S1), which is consistent with previous studies on Mars (e.g., [START_REF] Paniello | Zinc isotopic evidence for the origin of the Moon[END_REF]. Martian meteorites span a range of δ 66 Zn from +0.18 ± 0.02 (Tissint) to +0.52 ± 0.02 (ALHA 77005), with an average value of +0.36 ± 0.08 (Figure 1, Table 1). Five of the Martian meteorites analyzed in [START_REF] Paniello | Zinc isotopic evidence for the origin of the Moon[END_REF] are also presented herein (Figure 2). It is worth noting that data from this study tend to exhibit slightly higher δ 66 Zn values than those from [START_REF] Paniello | Zinc isotopic evidence for the origin of the Moon[END_REF], which could be attributed to heterogeneities in the measured Martian meteorites.

Nakhlites, enriched and depleted shergottites are, on average, indistinguishable within uncertainty (+0.36 ± 0.00 ‰ [n=2, 2SE], +0.36 ± 0.07 ‰ [n=3, 2SE] and +0.35 ± 0.12 ‰ [n=4, 2SE], respectively). This assertion is confirmed by a two-sample Kolmogorov-Smirnov test that returned D values of 0.29 (depleted vs. enriched shergottites), 0.71 (depleted shergottites vs. nakhlites) and 0.71 (enriched shergottites vs. nakhlites), where D is the maximum absolute value of the difference between the cumulative distribution functions for the different groups of Martian meteorites [START_REF] Miller | Statistics and chemometrics for analytical chemistry[END_REF]. For a confidence level of α = 0.05, the D values need to be greater than Dcriteria = *-𝑙 𝑛(𝛼/2) /2 × *(𝑛 5 + 𝑛 6 )/𝑛 5 /𝑛 6 = 0.67, where 𝑛 5 and 𝑛 6 are the numbers of samples in each population.

The Martian samples with the lowest δ 66 Zn values tend to exhibit lower MgO contents (Figure 2a), opposite to what is observed for terrestrial basalts [START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF], however there is no correlation with their Zn abundances (Figure 2b). Our study also provides evidence for the first mass-independent Zn isotope variations (e 66 Zn) in Martian meteorites (Figure 4), ranging from -0.25 to +0.07, except for one MIL 03346 replicate (+0.42) (Table 1), with an average value for Mars of -0.07 ± 0.08 (2SE, n = 10).

Discussion

As demonstrated above, Martian meteorites exhibit a range of δ 66 Zn values. These variations may arise from terrestrial late-stage and alteration processes, magmatic differentiation, volatile depletion processes prior or during accretion, and core formation on Mars. In the following discussion, we will address the potential contribution of these different processes to the Zn isotopic compositions of the Martian meteorites individually.

Potential disturbance of Zn during late-stage processes

All Martian meteorites considered in this study were recovered from hot (Sahara) or cold (Antarctica) deserts, and therefore may have undergone terrestrial alteration and weathering, potentially perturbing the Zn isotope compositions in these rocks from their primary igneous values. Such alteration processes can affect refractory lithophile isotope compositions by the formation of secondary minerals in hot desert environments (e.g., [START_REF] Crozaz | Chemical alteration and REE mobilization in meteorites from hot and cold deserts[END_REF][START_REF] Symes | The age of the Martian meteorite Northwest Africa 1195 and the differentiation history of the shergottites[END_REF], or remobilized lanthanides, mostly in the presence of phosphates, due to prolonged ice exposure in cold desert environments (e.g., [START_REF] Crozaz | Chemical alteration and REE mobilization in meteorites from hot and cold deserts[END_REF]. Even though isotopic disturbances have been observed in Antarctic and hot desert meteorites, initial 87 Sr/ 86 Sr ratios for most samples investigated here show little variation relative to the 4.5 Ga line (e.g., [START_REF] Tait | Chondritic late accretion to Mars and the nature of shergottite reservoirs[END_REF], supporting the hypothesis that these Martian meteorites underwent limited terrestrial alteration. In terrestrial rocks, Zn is mostly hosted in silicates (olivine, orthopyroxene) and oxides (ilmenite, spinel) (e.g., Le [START_REF] Le Roux | Mineralogical heterogeneities in the Earth's mantle: Constraints from Mn, Co, Ni and Zn partitioning during partial melting[END_REF][START_REF] Fang | Zinc isotope fractionation in mantle rocks and minerals, and a revised δ 66 Zn value for the Bulk Silicate Earth[END_REF], and is mobile only when the minerals are heavily altered [START_REF] Debret | Iron and zinc stable isotope evidence for open-system high-pressure dehydration of antigorite serpentinite in subduction zones[END_REF], which is not the case here. Moreover, normalizing Zn abundances to another element, such as Fe, that behaves similarly during petrogenetic processes, provided it is predominantly ferrous (e.g., Le [START_REF] Le Roux | Zn/Fe systematics in mafic and ultramafic systems: Implications for detecting major element heterogeneities in the Earth's mantle[END_REF][START_REF] Davis | Experimentally determined mineral/melt partitioning of first-row transition elements (FRTE) during partial melting of peridotite at 3 GPa[END_REF], represents a test of the degree to which the original magmatic abundances of Zn are preserved. Thus, if undisturbed, Martian meteorites owing to the relatively reduced fO2 (~IW; [START_REF] Wadhwa | Redox state of Mars' upper mantle and crust from Eu anomalies in shergottite pyroxenes[END_REF] under which they crystallized, should lie on olivine-and/or pyroxene control lines producing near constant Zn/Fe ratios. Samples that experienced secondary weathering/alteration should then diverge from the constant Zn/Fe value. All Martian meteorites have comparable Zn/Fe ratios, ranging between 2.6 × 10 -4 and 5.1 × 10 -4 , even though they are not representative of a single differentiation suite, with the exception of ALH 84001 which plots off the array (7.3 × 10 -4 , Table 1). ALH 84001 is an orthopyroxenite that preserves evidence of interaction with aqueous fluids while on Mars in the form of microscopic carbonate disks [START_REF] Thomas-Keprta | Origins of magnetite nanocrystals in Martian meteorite ALH84001[END_REF]. Despite this, its δ 66 Zn is +0.31 ± 0.01 and is consistent with the other Martian samples. Thus, the Zn isotope composition of the Martian meteorites can be regarded as reflecting the primary signature of magma from which they derived.

Zinc fractionation during magmatic processes

Magmatic processes such as partial melting and fractional crystallization have been shown to fractionate Zn isotopes in terrestrial environments (e.g., [START_REF] Mccoy-West | The Fe and Zn isotope composition of deep mantle source regions: Insights from Baffin Island picrites[END_REF]. Strikingly, the array defined by Martian magmas in MgO -δ 66 Zn space is a mirror image of that expressed in terrestrial basalts or komatiites (e.g., [START_REF] Chen | Zinc isotope fractionation during magmatic differentiation and the isotopic composition of the bulk Earth[END_REF][START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF], with decreasing δ 66 Zn with decreasing MgO (Figure 2a). This behavior was also noted by [START_REF] Paniello | Zinc isotopic evidence for the origin of the Moon[END_REF], but the precision of Zn isotopic data was not sufficient to define a trend.

Mineral proportions in shergottites are similar to those in terrestrial basalts, with olivine constituting 7 -29 % of olivine-phyric shergottites and 35 -60 % of poikilitic shergottites [START_REF] Bridges | The SNC meteorites: basaltic igneous processes on Mars[END_REF][START_REF] Sarbadhikari | Petrogenesis of olivinephyric shergottite Larkman Nunatak 06319: Implications for enriched components in Martian basalts[END_REF]. The bulk of the remainder of the minerals is comprised of pyroxenes (largely pigeonite), in which Mg 2+ and Fe 2+ are also octahedrally coordinated [START_REF] Cameron | High-temperature crystal chemistry of acmite, diopside, hedenbergite jadeite, spodumene and ureyite[END_REF]. Therefore, as in terrestrial basalts, the Zn budget of the solid phases is dominated by ferromagnesian silicates. Since VI Zn 2+ is the prevailing bonding environment owing to its substitution for VI Fe 2+ in these phases, hypotheses for differences in the terrestrial and Martian arrays must appeal to changes in melt composition.

In natural terrestrial silicate melts, Zn is expected to occur in tetrahedral coordination, owing to its lack of octahedral site preference energy, and by analogy with Zn-bearing minerals and in aluminosilicate glasses (e.g., [START_REF] Neumann | Notes on the mineralogy and geochemistry of zinc[END_REF][START_REF] Dumas | EXAFS study of titanium and zinc environments during nucleation in a cordierite glass[END_REF]. However, the distinctive feature of Martian magmas is their high FeOt contents, which acts predominantly as a network modifier, as Fe is almost exclusively Fe 2+ [START_REF] Righter | Redox systematics of Martian magmas with implications for magnetite stability[END_REF]. For primary Martian magmas, such as EETA 79001A [START_REF] Filiberto | Fe 2+ -Mg partitioning between olivine and basaltic melts: Applications to genesis of olivine-phyric shergottites and conditions of melting in the Martian interior[END_REF], NBO/T is ~ 2, which is higher than typical terrestrial magmas (up to ~1), but similar to komatiites. Increasing proportions of network modifiers tend to increase the coordination number, such that a higher proportion of Zn would be expected to be in VI-fold coordination in such melts. This behavior is corroborated by the tendency for other divalent first-row transition metals, such as Co and Ni, to exist in octahedral complexes in more basic compositions (e.g., [START_REF] Farges | Transition elements in water-bearing silicate glasses/melts. Part I. A high-resolution and anharmonic analysis of Ni coordination environments in crystals, glasses, and melts[END_REF]. However, while this shift will reduce isotopic fractionation between the two phases, the melt becoming enriched in heavy isotopes in terrestrial melts [START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF], it is unknown whether it would reverse the trend to the extent observed for Martian magmas.

Since shergottites and nakhlites/chassignites are likely not cogenetic magmas [START_REF] Wadhwa | Redox state of Mars' upper mantle and crust from Eu anomalies in shergottite pyroxenes[END_REF][START_REF] Borg | A petrogenetic model for the origin and compositional variation of the Martian basaltic meteorites[END_REF][START_REF] Bridges | The SNC meteorites: basaltic igneous processes on Mars[END_REF], their chemical and isotopic signatures could also derive from differences in their source composition. However, given the degree of homogeneity shown for the Zn isotopic composition of terrestrial rocks (e.g., [START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF], this is unlikely. Another possibility is that a fraction of the Zn is hosted in oxides (e.g., spinel), in which it exists in IV-fold co-ordination [START_REF] O'neill | Simple spinels; crystallographic parameters, cation radii, lattice energies, and cation distribution[END_REF]) and should therefore concentrate the heavier isotopes, though this would also be the case on Earth. Volatile loss of Zn during magma emplacement is also precluded [START_REF] Toutain | Evidence for Zn isotopic fractionation at Merapi volcano[END_REF] because it would result in the preferential loss of the lighter isotopes of Zn, causing δ 66 Zn to increase towards lower MgO contents, which is the opposite of what is observed for Martian meteorites (Figure 2). Additional experiments on Zn isotope fractionation in igneous systems are required to understand the drivers of Zn isotope variations in natural rocks, and in particular the role of its co-ordination environment, both in minerals and melts.

Zinc composition of the Bulk Silicate Mars

The high Fe content of Bulk Silicate Mars as calculated from the SNC suite is consistent with spectral data of the Martian surface, which reveals similarly enriched FeO contents relative to Earth's mantle or its basaltic crust [START_REF] Taylor | Bulk composition and early differentiation of Mars[END_REF][START_REF] Mcsween | Elemental composition of the Martian crust[END_REF][START_REF] Khan | Geophysical and cosmochemical evidence for a volatile-rich Mars[END_REF].

This observation is true for other divalent cations, including Mg 2+ and Mn 2+ , where Mg/Fe and Mn/Fe ratios of shergottites and in-situ measurements of Martian samples fall on liquid lines of descent (e.g., [START_REF] Mcsween | Elemental composition of the Martian crust[END_REF]. Such similarity arises from the near-absence of Fe 3+ in the Martian mantle (e.g., [START_REF] Righter | Redox systematics of Martian magmas with implications for magnetite stability[END_REF], and suggests that Zn 2+ behaves analogously to Fe during magmatic differentiation. Their concordance increases the likelihood that the concentrations, and potentially isotopic compositions of Zn in the SNCs are reflections of that of the Martian mantle.

Therefore, although the sample size is small, the SNCs span a broad range of MgO contents that allow for the characterization of trends likely controlled by igneous processes (Figure 2a). The fact that these appear to be preserved also supports closed-system transport of the meteorites to Earth with respect to Zn. Extrapolation of the shergottite array (with the exception of Tissint which plots within the BSE range) to the MgO content of the Martian mantle of 32.81 ± 0.22 wt.% [START_REF] Khan | Geophysical and cosmochemical evidence for a volatile-rich Mars[END_REF] yields a δ 66 Zn = +0.50 ± 0.18, distinctly heavier than that of the Earth (from +0.16 ± 0.06 ‰ to +0.20 ± 0.03 ‰; [START_REF] Mccoy-West | The Fe and Zn isotope composition of deep mantle source regions: Insights from Baffin Island picrites[END_REF][START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF][START_REF] Doucet | Archean lithospheric differentiation: Insights from Fe and Zn isotopes[END_REF][START_REF] Fang | Zinc isotope fractionation in mantle rocks and minerals, and a revised δ 66 Zn value for the Bulk Silicate Earth[END_REF]. Alternatively, using the average of the shergottites results in a value of +0.38 ± 0.05 ‰ (2SE, n = 6), which lies within uncertainty of our reported estimate based on the δ 66 Zn -MgO correlation (R 2 = 0.65, excluding Tissint). Thus, for the following discussion we report our initial estimate of +0.50 ± 0.18 ‰ for the BSM.

Behavior of Zn and other moderately volatile elements during accretion of Mars

In δ 66 Zn -Zn/Mg space, the Earth's estimate plots at the intersection of 1) the carbonaceous chondrites trend following the equation δ 66 Zn = 0.363 log(Zn/Mg) +1.40 [START_REF] Luck | Zn and Cu isotopic variations in chondrites and iron meteorites: early solar nebula reservoirs and parent-body processes[END_REF][START_REF] Pringle | The origin of volatile element depletion in early solar system material: Clues from Zn isotopes in chondrules[END_REF][START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF] and 2) the path of fractionation induced by evaporation or condensation, for a fractionation factor between liquid/solid and gas yielding the composition of the Moon [START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF]. The Earth's composition is interpreted as having been derived from the accretion of material that experienced chemical and isotopic fractionation in the solar nebula, similar to that experienced by carbonaceous chondrites but extended to more volatile-depleted extremes. Our estimate of the bulk Zn/Mg ratio of Mars, 3.34 × 10 -4 (MgO = 32.81 ± 0.22 wt.%, Znmantle = 65.9 ± 8.5 ppm), would result in a composition of +0.14 ‰, similar to that of the Earth (Figure 3), were Mars to have also inherited the majority of its Zn from carbonaceous chondritelike components. However, with an estimated composition of +0.50 ± 0.18 ‰ for the BSM, the δ 66 Zn of Mars is greater than that defined by the carbonaceous chondrite array.

Accounting for a similar proportion of the total Zn residing in the Martian core (which corresponds to a mass fraction of 25%, with a higher S content; e.g., [START_REF] Khan | Geophysical and cosmochemical evidence for a volatile-rich Mars[END_REF] that in the Earth's core [START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF], we obtain Znbulk = 99.8 ppm. Assuming that no Mg partitioned into the core (e.g., [START_REF] Ringwood | Solubilities of mantle oxides in molten iron at high pressures and temperatures: implications for the composition and formation of Earth's core[END_REF], the Zn/Mg ratio of Mars is 5.05 × 10 -4 . Notably, Zn isotopic fractionation at pressure and temperature conditions relevant for the formation of the Martian core is negligible (Brigestock et al., 2014;[START_REF] Mahan | Elemental partitioning and isotopic fractionation of Zn between metal and silicate and geochemical estimation of the S content of the Earth's core[END_REF]. Thus, it would require [Zn] = 651 ppm for Mars to plot on the carbonaceous chondrite array, corresponding to a partition coefficient of DZn ~ 45, which is not compatible with experimental data (e.g., [START_REF] Wood | Accretion and core formation: The effects of sulfur on metal-silicate partition coefficients[END_REF]. Alternatively, we obtain [Zn] = 305 ppm and DZn ~ 19 if we consider a δ 66 Zn of +0.38.

However, the average of SNC meteorites indicates that Bulk Silicate Mars exhibits a e 66 Zn value (-0.08 ± 0.07, 2SE, n = 10) intermediate between that of ordinary and carbonaceous chondrites (non-carbonaceous (ordinary + enstatite) chondrites: -0.20 ± 0.04, 2SE, n = 20; carbonaceous chondrites: +0.31 ± 0.05, 2SE, n = 22, and Ryugu asteroid materials: +0.33 ± 0.03, n=4 [START_REF] Savage | Zinc isotope anomalies in primitive meteorites identify the outer solar system as an important source of Earth's volatile inventory[END_REF][START_REF] Steller | Nucleosynthetic zinc isotope anomalies reveal a dual origin of terrestrial volatiles[END_REF][START_REF] Paquet | Contribution of Ryugu-like material to Earth's volatile inventory by Cu and Zn isotopic analysis[END_REF]), similar within error to that of the Earth (+0.015 ± 0.075, 2SE [START_REF] Savage | Zinc isotope anomalies in primitive meteorites identify the outer solar system as an important source of Earth's volatile inventory[END_REF]; -0.07 ± 0.013, 2SE [START_REF] Steller | Nucleosynthetic zinc isotope anomalies reveal a dual origin of terrestrial volatiles[END_REF]), but closer to that of the NC endmember (Figure 4). Because there are no known terrestrial processes which can mass-independently fractionate Zn isotopes, the intermediate e 66 Zn of Mars between carbonaceous chondrites and ordinary chondrites suggests a potential mixing between these two reservoirs to explain Mars' isotopic compositions, which is consistent with other nucleosynthetic anomalies such as in Ti, Cr, O, Ca or Fe (e.g., [START_REF] Clayton | Oxygen isotope studies of achondrites[END_REF][START_REF] Trinquier | Widespread 54 Cr heterogeneity in the inner solar system[END_REF][START_REF] Trinquier | Origin of nucleosynthetic isotope heterogeneity in the solar protoplanetary disk[END_REF][START_REF] Schiller | Isotopic evolution of the protoplanetary disk and the building blocks of Earth and the Moon[END_REF][START_REF] Schiller | Iron isotope evidence for very rapid accretion and differentiation of the proto-Earth[END_REF]. Moreover, since the mass independent isotope composition of Zn indicates it was largely inherited from an NC-like reservoir on Mars, rather than from CCs, there is no reason to expect that bulk Mars lies on the CC array, strongly arguing against significant quantities of Zn in the Martian core. Using the e 66 Zn values from Mars, carbonaceous and ordinary chondrites, together with their Zn abundances (see Table 2 for details), we calculated the proportions of CC and NC required to reproduce Mars' e 66 Zn, which yields to the following proportions: 4 % Ryugu-like material + 96 % NC (6 % CI + 94 % NC, or 14 % CV + 86 % NC), representing ~22 % of Mars' Zn delivered by outer solar system material, which is slightly lower than the estimation for the Earth's (~30 %; [START_REF] Savage | Zinc isotope anomalies in primitive meteorites identify the outer solar system as an important source of Earth's volatile inventory[END_REF][START_REF] Steller | Nucleosynthetic zinc isotope anomalies reveal a dual origin of terrestrial volatiles[END_REF][START_REF] Paquet | Contribution of Ryugu-like material to Earth's volatile inventory by Cu and Zn isotopic analysis[END_REF]. This observation contrasts with the notion that planets further from the Sun should have higher contribution from outer Solar System (e.g., Ryugu-like or CC-like) materials.

It is then possible to estimate the δ 66 Zn and Zn/Mg of such a mixture (assuming the Mg content of Mars, ~19.8 wt.% [START_REF] Khan | Geophysical and cosmochemical evidence for a volatile-rich Mars[END_REF], is not affected by evaporation), and calculate the isotopic fractionation induced by evaporation, assuming a Rayleigh evaporation law and a fractionation factor ∆ 66 Znliq-vap. The scenario modeled by [START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF] used ∆ 66 Znliq-vap = 1‰ to account for the composition of the Moon derived from the Earth (Figure 3). We consider two endmembers: the first in which all the non-carbonaceous chondrites are ordinary chondrites (OC), and the second in which all the non-carbonaceous chondrites are enstatite chondrites (EH).

Elemental abundances for OC and EH are from [START_REF] Lodders | The planetary scientist's companion[END_REF] and average δ 66 Zn for OC is from Paniello (2013, PhD thesis Washington University Saint Louis) and [START_REF] Creech | Tin and zinc stable isotope characterisation of chondrites and implications for early Solar System evolution[END_REF], and average δ 66 Zn for EH is from [START_REF] Moynier | Nature of volatile depletion and genetic relationships in enstatite chondrites and aubrites inferred from Zn isotopes[END_REF], respectively (Table 2).

In this scenario, the isotopic fractionation induced by ~20-25 % evaporation (with ∆ 66 Znliq-vap = 1‰) from a mixture of 4 % Ryugu-like material (or from 6 % of CI to 14 % of CV) and 96 % OC (94 % and 86 %, for CI and CV, respectively) reproduces both the Zn mass-dependent andindependent BSM composition (Figure 3). Starting compositions calculated from a mixture of outer Solar System material and EH are too enriched in Zn to reproduce the BSM. These values are consistent with previous studies which argued in favor of 9 to 18 % of carbonaceous chondritelike material based on Cr anomalies and ∆ 17 O [START_REF] Warren | Stable-isotopic anomalies and the accretionary assemblage of the Earth and Mars: A subordinate role for carbonaceous chondrites[END_REF], and up to 36 % of CI-like material based on µ 48 Ca anomalies [START_REF] Schiller | Isotopic evolution of the protoplanetary disk and the building blocks of Earth and the Moon[END_REF] to account for Mars' composition.

Assuming that equilibrium isotope fractionation between liquid and vapor prevailed (∆ 33 Zn (78-9%' =

:;<.25×5< $ ? @ &
, where T is in K; [START_REF] Ducher | Equilibrium zinc isotope fractionation in Zn-bearing minerals from first-principles calculations[END_REF], then the temperature required to achieve a 66 Zn/ 64 Zn fractionation factor of 1‰, consistent with the Zn/Mg and δ 66 Zn composition of Mars, is constrained between 557 and 704 K. At such low temperatures, it is difficult to engender significant isotope fractionation in the condensed phase during vapor loss from a Mars-mass planet because the temperatures are well below the solidi of the silicate minerals that comprise the Martian mantle, meaning mass transport is not sufficiently rapid to cause global Zn depletion. Therefore, should equilibrium isotope fractionation during evaporation be responsible for its Zn isotopic composition, then this would have had to have occurred on small (e.g., cm-sized) precursor grains from which gas-solid separation is facilitated. Curiously, the temperatures calculated above bracket the 50 % condensation temperature of Zn from the solar nebula at 10 -4 bar (~700 K; [START_REF] Wood | The condensation temperatures of the elements: A reappraisal[END_REF]. Equilibrium isotope fractionation factors at higher temperatures are not able to reproduce the combined Zn/Mg ratio and isotopic composition for the BSM by evaporation in a CC-like precursor or during Mars' history (Figure 5). A more accurate picture of the Zn isotope composition of Mars is precluded due to the spread of Zn isotope values in OCs, ranging from -1.31 ‰ to + 1.75 ‰ (Figure 3). Nevertheless, the general conclusion holds that, should the Zn isotope composition of the BSM have been subsequently modified by equilibrium evaporation, then it must have occurred at low temperatures that could only have existed in the solar nebula. Alternatively, evaporation did not proceed under equilibrium conditions, but was rather kinetic in nature to induce larger isotopic fractionation, with a ∆ 66 Znliqvap factor ~1 ‰ that reflected the local evaporation environment. It is worth noting that the maximum effect on e 66 Zn expected for a fractionation (kinetic or under equilibrium) of 0.5‰ is ~0.1e, which is within the error of the estimate of the BSM composition. Because Mars' e 66 Zn is close to that of NC, it is also possible that Mars' isotopic signature is inherited from its precursors (mainly OC) from the solar nebula, obviating the need for the addition of CC material.

The apparent isotope differences between CC and NC meteorites, which was first identified for O and Cr isotopes [START_REF] Clayton | Oxygen isotope studies of achondrites[END_REF][START_REF] Trinquier | Widespread 54 Cr heterogeneity in the inner solar system[END_REF], and later with Ti, Ni, Mo and also Zn anomalies [START_REF] Trinquier | Origin of nucleosynthetic isotope heterogeneity in the solar protoplanetary disk[END_REF][START_REF] Warren | Stable-isotopic anomalies and the accretionary assemblage of the Earth and Mars: A subordinate role for carbonaceous chondrites[END_REF][START_REF] Budde | Molybdenum isotopic evidence for the origin of chondrules and a distinct genetic heritage of carbonaceous and non-carbonaceous meteorites[END_REF][START_REF] Kruijer | Age of Jupiter inferred from the distinct genetics and formation times of meteorites[END_REF][START_REF] Nanne | Origin of the non-carbonaceous-carbonaceous meteorite dichotomy[END_REF][START_REF] Burkhardt | Terrestrial planet formation from lost inner solar system material[END_REF][START_REF] Savage | Zinc isotope anomalies in primitive meteorites identify the outer solar system as an important source of Earth's volatile inventory[END_REF][START_REF] Steller | Nucleosynthetic zinc isotope anomalies reveal a dual origin of terrestrial volatiles[END_REF], has been ascribed to the spatial location at which these meteorite groups formed, separated by a physical barrier that prevented their mixing over their formation timescales, 1 -4 Myr after CAI [START_REF] Kruijer | Age of Jupiter inferred from the distinct genetics and formation times of meteorites[END_REF]. In this model, the CCs are presumed to have formed in the outer Solar System (i.e., beyond the orbit of Jupiter) while NCs are derived from the inner Solar System. Physical accretion models for the terrestrial planets, in the context of N-body simulations, both in the Grand Tack- [START_REF] Jacobson | Lunar and terrestrial planet formation in the Grand Tack scenario[END_REF] and standard models [START_REF] Wetherill | Provenance of the terrestrial planets[END_REF] indicate that planets that formed further from the Sun should, on average, contain higher fractions of outer Solar System materials (e.g., [START_REF] Wetherill | Provenance of the terrestrial planets[END_REF][START_REF] Cuzzi | Material enhancement in protoplanetary nebulae by particle drift through evaporation fronts[END_REF][START_REF] Rubie | Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water[END_REF][START_REF] Sossi | Stochastic accretion of the Earth[END_REF], though the Grand Tack model results in a more random distribution of material in the inner disk from different heliocentric distances than does the standard model. Although Mars has a marginally higher Zn content than the Earth, that the Zn isotope composition of Mars contains a higher fraction of nominally inner Solar System OC-like material than the Earth suggests that the dichotomy of CC-vs. NC-material may not exclusively relate to their location in the protoplanetary disk. In addition, masses and orbits of terrestrial planets, including Mars, are also consistent with pebble accretion onto protoplanets accreted around Mars' orbit and then migrated to their final position while growing (e.g., [START_REF] Johansen | A pebble accretion model for the formation of the terrestrial planets in the Solar System[END_REF]; Mars' isotopic composition closer to the NC than is the Earth could indicate an earlier growth termination for Mars (as indicated in its Hf-W age of 9 Myr compared to ~30 Myr for the Earth; [START_REF] Dauphas | Hf-W-Th evidence for rapid growth of Mars and its status as a planetary embryo[END_REF][START_REF] Kleine | Tungsten isotopes in planets[END_REF]. Mars therefore may not have accreted pebbles of CI composition that drifted later towards the inner part of the protoplanetary disk (e.g., [START_REF] Schiller | Isotopic evolution of the protoplanetary disk and the building blocks of Earth and the Moon[END_REF]. In future, isotopic data for other volatile systems with more than 3 isotopes, such as Cd or Te, should be acquired to further investigate the volatile accretion history of Mars by combining mass-dependent isotopic fractionation and nucleosynthetic anomalies to better understand the nature and distribution of volatile-bearing material in the early Solar System.

Conclusions

We measured the Zn isotopic composition of ten Martian meteorites from the main petrologic groups to investigate the origin of volatile depletion on Mars. Extrapolating the shergottite array to a MgO value representative of the Martian mantle, our new dataset allows assessment of its Zn isotope composition, representative of the BSM (bulk silicate Mars) (δ 66 Zn = +0.50 ± 0.18), which is shown to be distinctly heavier than that of the Earth. This composition can be extended to that of the Bulk Mars insofar as negligible Zn isotope fractionation is observed between metal and silicate at pressure and temperature conditions relevant for Martian core segregation. The BSM plots above the carbonaceous chondrite array defined in a δ 66 Zn vs log(Zn/Mg) plot, arguing in favor of isotopic fractionation during volatile loss by evaporation/condensation, or the provenance of Zn from largely OC-like precursors. The mass-dependent and mass-independent Zn isotope compositions of Mars provide new constraints on the origin of volatile elements, suggesting that between ~4 to 6 % of outer Solar System material (Ryugu-like or CI-like, respectively) were delivered to Mars to account for its Zn budget. Sossi et al. (2018). Diamonds: this study;circles: Paniello et al., (2012) [START_REF] Paniello | Zinc isotopic evidence for the origin of the Moon[END_REF] for Zn isotopes and elemental compositions; MgO content from [START_REF] Tait | Chondritic late accretion to Mars and the nature of shergottite reservoirs[END_REF]. Bulk Silicate Earth composition range for δ 66 Zn (gray field) from [START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF] Literature data for the Zn isotope and elemental compositions from: carbonaceous chondrites [START_REF] Luck | Zn and Cu isotopic variations in chondrites and iron meteorites: early solar nebula reservoirs and parent-body processes[END_REF][START_REF] Pringle | The origin of volatile element depletion in early solar system material: Clues from Zn isotopes in chondrules[END_REF][START_REF] Mahan | A history of violence: Insights into post-accretionary heating in carbonaceous chondrites from volatile element abundances, Zn isotopes and water contents[END_REF][START_REF] Paquet | Contribution of Ryugu-like material to Earth's volatile inventory by Cu and Zn isotopic analysis[END_REF], CR [START_REF] Mahan | A history of violence: Insights into post-accretionary heating in carbonaceous chondrites from volatile element abundances, Zn isotopes and water contents[END_REF] and ordinary chondrites (Paniello, 2013, PhD thesis Washington University Saint Louis;[START_REF] Creech | Tin and zinc stable isotope characterisation of chondrites and implications for early Solar System evolution[END_REF]; see SourceData table for Mg contents. Bulk Silicate Earth composition from [START_REF] Palme | Cosmochemical Estimates of Mantle Composition[END_REF] and [START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF]; carbonaceous chondrite array from [START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF]. The dashed lines represent the fraction of Zn evaporated from a starting composition deriving from mixing between carbonaceous (4 % Ryugu-like material, 6 % CI or 14 % CV) and non-carbonaceous materials (96 %, 94 % or 86 %, respectively) (fractionation factor of ∆ 66 Znliq-vap = +1 ‰, with a 5% increment step (see Table 2 and text for details)). The Bulk Silicate Mars plots away from the carbonaceous chondrite array. The error for the BSM estimate represents the 95 % confidence interval for MgO =32.8 wt.% [START_REF] Khan | Geophysical and cosmochemical evidence for a volatile-rich Mars[END_REF] derived from the linear regression in Figure 2c. Earth. Literature data for carbonaceous chondrites (Savage et al., 2022 (large symbols);Steller et al., 2022 (small symbols); Paquet et al., 2022 (open circle)), ordinary and enstatite chondrites, carbonaceous and non-carbonaceous iron meteorites and ureilites [START_REF] Savage | Zinc isotope anomalies in primitive meteorites identify the outer solar system as an important source of Earth's volatile inventory[END_REF][START_REF] Steller | Nucleosynthetic zinc isotope anomalies reveal a dual origin of terrestrial volatiles[END_REF]. Bulk Silicate Earth value for e 66 Zn is +0.015 ± 0.075 (2SE; [START_REF] Savage | Zinc isotope anomalies in primitive meteorites identify the outer solar system as an important source of Earth's volatile inventory[END_REF] and -0.07 ± 0.013 (2SE; [START_REF] Steller | Nucleosynthetic zinc isotope anomalies reveal a dual origin of terrestrial volatiles[END_REF]. Error bars are 2SE.

Figures and figure captions

  Zinc isotopic compositions were measured in ten Martian meteorites, including seven shergottites (depleted: Allan Hills [ALHA] 77005, Dar al Gani [DaG] 476, Elephant Moraine [EETA] 79001, Tissint; enriched: Larkman Nunatak [LAR] 06319, Roberts Massif [RBT] 04262, Zagami), two nakhlites (Miller Range [MIL] 0346, MIL090032) and one orthopyroxenite (ALH 84001).

Figure 1 .

 1 Figure 1. Zinc isotope composition for all Martian samples. Bulk Silicate Earth (BSE -gray field) from[START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF]. Diamonds: this study; circles:[START_REF] Paniello | Zinc isotopic evidence for the origin of the Moon[END_REF]. Error bars:

  Figure 2. δ 66 Zn as a function of the (a) MgO content (wt.%) and (b) Zn content (ppm) for all Martian meteorites (diamonds: this study). (c) Average δ 66 Zn for each Martian meteorite as a function of the MgO content (wt.%) for the shergottites from this study. Literature data (circles) are from Paniello et al. (2012) for Zn isotopes and elemental compositions; MgO content from

  The solid line in (c) is the linear regression of the dataset (R 2 =0.65 excluding Tissint), and the dashed lines represent the 95% confidence error envelope. BSM is the Bulk Silicate Mars estimate. Error bars: 2SD.

Figure 3 .

 3 Figure 3. Zinc isotope compositions of planetary materials (modified from[START_REF] Sossi | Zinc isotope composition of the Earth and its behaviour during planetary accretion[END_REF]).

Figure 4 .

 4 Figure 4. Variations of e 66 Zn among different groups of meteorites, Mars (this study) and the

Figure 5 .

 5 Figure 5. Equilibrium evaporation trajectories for different isotope fractionation factors a, which depend on the temperature T (in K). The starting point for the models was based upon the assumption that the original building blocks of Mars had 96 % OC and 4 % Ryugu-like material contribution.

Table captions Table 1 .

 captions1 Zinc stable isotope and major element compositions of Martian meteorites.

Table 2 .

 2 Mixing models.

Acknowledgements

This work was supported by the IPGP analytical platform PARI, Region Ile-de-France SESAME Grants no. 12015908, and DIM ACAV +, the ERC grant agreement No. 101001282 (METAL) (F.M.) the UnivEarthS Labex program (numbers: ANR-10-LABX-0023 and ANR-11-IDEX-0005-02) (F.M.). P.A.S. was supported by an Australian Postgraduate Award and SNSF Ambizione Fellowship #180025. We thank K. Righter of NASA for providing the MWG samples.

We thank Herbert Palme and Klaus Mezger for their constructive comments that improved the manuscript, and Editor Rajdeep Dasgupta for efficient editing of the paper. The authors declare no conflicts of interest. All data referred to in this article can be found in the tables or cited references.

Supplementary figures and figure captions


Patty Coupeau

Jean-Baptiste Fasquel

Josselin Demas

Lucie Hertz-Pannier

Mickaël Dinomais

DETECTING CEREBRAL PALSY IN NEONATAL STROKE CHILDREN: GNN-BASED DETECTION CONSIDERING THE STRUCTURAL ORGANIZATION OF BASAL GANGLIA

Keywords: Cerebral palsy, neonatal arterial ischaemic stroke, basal ganglia structural organization, graph neural network, graph classification

As a long-term consequence of neonatal arterial ischaemic stroke (NAIS), the presence of cerebral palsy (CP) depends on the structural integrity of brain areas, especially of basal ganglia. Yet, it remains challenging to establish an early diagnosis of CP from a conventional structural MRI. In this study, we introduce a graph neural network-based classification for the recognition of NAIS children and mainly for the detection of children with CP among the NAIS ones. From the structural MRI of 68 children aged 7 years old and their corresponding segmentation of basal ganglia, we construct graphs where nodes represent structures, carrying on node and edge attributes structural information (volumes, distances). The classification accuracy achieved by the proposed method is of 86% for the detection of NAIS and of 89% for the detection of CP among neonatal stroke children.

INTRODUCTION

With a birth prevalence of 1/4000, neonatal arterial ischaemic stroke (NAIS) is the most common and frequent of childhood stroke [START_REF] Kirton | Paediatric stroke: pressing issues and promising directions[END_REF]. NAIS is defined as a cerebro-vascular event occurring between birth and 28 postnatal days with evidence of focal arterial infarction [START_REF] Chabrier | New insights (and new interrogations) in perinatal arterial ischemic stroke[END_REF]. Around 20-30% of term infants with NAIS will develop unilateral cerebral palsy (CP) which is the most frequent motor impairment in children [START_REF] Chabrier | New insights (and new interrogations) in perinatal arterial ischemic stroke[END_REF]. As a rule, infants with the most extensive brain lesions are at highest risk for long-term disabilities [START_REF] Lee | Predictors of outcome in perinatal arterial stroke: a population-based study[END_REF]. However, some infants in whom brain lesions are relatively small but specifically located along the motor pathway in the cerebrum may also develop CP [START_REF] Dinomais | Long term motor function after neonatal stroke: lesion localization above all[END_REF]. Apart from the most severe cases, the motor outcome is difficult to predict from neonatal conventional MRI. Prediction of the motor developmental trajectory remains challenging and new clinical and/or radiological tools are under development to better predict long-term motor outcome [START_REF] Husson | Motor outcomes after neonatal arterial ischemic stroke related to early MRI data in a prospective study[END_REF]. Establishing an early diagnosis of CP and understanding the occurrence of CP are important as this can expedite motor rehabilitation intervention. In previous work [START_REF] Chabrier | Multimodal outcome at 7 years of age after neonatal arterial ischemic stroke[END_REF], we demonstrate that, as a long-term consequence of NAIS, presence or absence of CP depends on the structural integrity of brain areas anatomically remote from the infarct site (ispilesional thalamus). Broadly, the basal ganglia appears as a major component of preserving motor function following early brain lesion [START_REF] Hassett | Bihemispheric developmental alterations in basal ganglia volumes following unilateral perinatal stroke[END_REF].

⋆ Corresponding author: patty.coupeau@univ-angers.fr Thus, we present a method of diagnosing CP from a structural MRI not only based on volume relations as in [START_REF] Hassett | Bihemispheric developmental alterations in basal ganglia volumes following unilateral perinatal stroke[END_REF] but also on the spatial organization of the basal ganglia. For that, we propose to work with graphs where each node represents a brain structure extracted from the MRI and the edges carry the distance between structures. The detection of CP is then done using a graph neural network (GNN) [START_REF] Bacciu | A gentle introduction to deep learning for graphs[END_REF] trained to perform binary classification. Working with graphs allows to represent more explicitly spatial relationships between structures than machine learning classifiers where volumes and distances would be covariates. The use of graphs to diagnose pathology from MRI has recently been proven (neuropsychiatric disorders classification [START_REF] Liu | An enhanced multimodal brain graph network for classifying neuropsychiatric disorders[END_REF], graph-based diagnosis of autism spectrum disorder [START_REF] Yang | Collaborative learning of graph generation, clustering and classification for brain networks diagnosis[END_REF]). Our proposal has the advantage of working with small graphs (only 8 nodes) and simple attributes (volumes, distances), thus requiring fewer training parameters than a direct CNNbased classification from the image. Our main contribution lies in the original proposed GNN-based method to recognize children with NAIS but also to diagnose the presence of CP among these children from a structural T1-weighted MRI.

The proposal is described in Section 2. Experiments are detailed in Section 3 and discussed in Section 4.

METHOD

Figure 1 provides an overview of the proposed approach. From the segmentation of basal ganglia on MRI, we construct a graph where only symmetrical structures are linked. Beforehand, from the segmentation of control subjects, we build the matrix Dref of the average distance between the barycenters of each brain structure, being required to compute edge attributes of graphs for both training (A) and inference (B). A GNN (C) is then trained with these graphs to predict the class y of the subject.

Graph construction

We define the graph G = (V, E, X, L) where V is the set of nodes (each node v ∈ V corresponds to a basal ganglia structure) and E is the set of edges. We propose to connect only the supposedly symmetrical structures (i.e. left caudate with right caudate, etc.) since we want to highlight an asymmetry between hemispheres. X is a node attribute assignment function X : V → R regarding the volume of the corresponding structure normalized to the total brain volume. L is an edge attribute assignment function L : E → R such as: ref,k indicates the average distance between structures i and j in the direction k for control children. Dref is previously constructed, from the MRI of control subjects, as the average on all these children of the distance between the barycenters of every segmented structures (basal ganglia) according to the 3 axes (x: anterior-posterior, y: inferior-superior and z: right-left). Thus, in Figure 1, the red box in Dref indicates the control average distance between the right and left caudate along the 3 axes. The edge attributes are computed to be greater than 1 in order to avoid too small values (close to 0) which would cancel the node attributes during the convolution operation described in Eq.2 (hereafter detailed).

L(i, j) = 1 3 3 k=1 1 + |1 - d i,j k D i,j ref,k | (1)

Graph neural network

As illustrated in Figure 1-C, the architecture of the proposed GNN is composed of 2 graph convolution layers (with output channel dimensions c being an hyperparameter except for the 2 nd layer where c = 2 in this work) followed by affine ReLU activation, a global max pooling and a fully-connected layer with two output channels corresponding to the two classes. The aggregation of the neighborhood information takes place within the convolution operations. We consider the graph convolution operator ECConv [START_REF] Simonovsky | Dynamic edgeconditioned filters in convolutional neural networks on graphs[END_REF] due to its ability to manage scalar but also multi-dimensional edge attributes. Note that, we could have considered other convolution operators managing edge attributes such as GATConv [START_REF] Veličković | Graph Attention Networks[END_REF]. Let X l (i) be the attribute of node i at layer l ∈ [1, 2] and L(i, j) the scalar attribute of the edge connecting nodes i and j. The message passing at layer l can be defined as [START_REF] Simonovsky | Dynamic edgeconditioned filters in convolutional neural networks on graphs[END_REF]:

X l+1 (i) = σ(W l+1 X l (i) + j∈N (i) F l+1 (L(j, i))X l (j)) (2)
where σ denotes the ReLU function, W l+1 ∈ ℜ c×1 is a matrix of trainable weights and F l+1 : ℜ → ℜ c×1 is a differentiable function (a multi-layer perceptron). N (i) represents all nodes j neighboring i (connected to i). Then, the global max pool operator returns a graphlevel output by taking the channel-wise maximum across the node dimension, so that its output is computed by: yg = maxi∈V X 2 (i). yg (2-dimensional vector) then passes through a linear layer (FC) whose output ŷ is of dimension 2. The class y of the graph is computed as the argmax of ŷ.

EXPERIMENTS

We divide the experiments into two parts. First, we perform a binary classification to recognize children with NAIS among healthy controls. This task is considered to evaluate whether the proposal is able to distinguish two visually distinct populations. Then, we present a classification to detect children with CP among children with NAIS.

Dataset

Our method is evaluated with a cohort of 68 children aged 7 years old. These children include healthy controls and children with injury in the right or left hemisphere. Each child got an MRI scan (3D T1-weighted volumes of 256x256x176 voxels with a spatial resolution of 1x1x1mm 3 ). A first atlas-based segmentation is performed using the Hammersmith atlas [START_REF] Hammers | Threedimensional maximum probability atlas of the human brain, with particular reference to the temporal lobe[END_REF]. Then, each MRI is manually corrected slice by slice with ITKSnap [START_REF] Yushkevich | ITK-SNAP: An interactive tool for semi-automatic segmentation of multi-modality biomedical images[END_REF]. Each segmentation is visually checked and validated by consensus with a specialist.

Concerning the detection of children with cerebral palsy, we kept in our dataset only the 37 children with NAIS. All children were examined during the 7-year assessment by an experienced clinician (paediatric neurologist, paediatric rehabilitation specialist). The definition of cerebral palsy provided by the Surveillance of CP in Europe Network (SCPE) [START_REF]Surveillance of cerebral palsy in Europe: a collaboration of cerebral palsy surveys and registers[END_REF] was used. Among the 37 children, 13 children were diagnosed as having CP.

Evaluation protocol

All the experiments are carried out in a Python environment using the PyTorch Geometric library [START_REF] Fey | Fast Graph Representation Learning with PyTorch Geometric[END_REF]. The GNN architecture pre-sented in Section 2.2 is used by setting the output dimension c of the first convolution layer to 5. The model is trained with Adam (Adaptive Moment Estimation), on 300 epochs with a learning rate lr = 0.001. A binary cross-entropy loss function is considered. Distribution of the dataset for both experiments in training and test data is presented in Table 1. A cross-validation strategy is implemented to test our method on all the children while having sufficient training data. To deal with the imbalanced dataset (e.g. only 13 children with CP vs 24 without), 3 stratified random samplings (all with the distribution indicated in Table 1) are conducted in both cases (results are averaged over the 3 draws). Twenty training control subjects are used to compute the reference distances (Dref in Figure 1). To assess the performance of the method, we measure the balanced accuracy of classification [START_REF] Brodersen | The Balanced Accuracy and Its Posterior Distribution[END_REF]. We also study the complementarity of both node and edge attributes.

Results

No attr. Edges Nodes Nodes+Edges NAIS 0.5 0.63 0.82 0.86 PC 0.5 0.67 0.85 0.89 Table 2. Complementarity of node and edge attributes for NAIS and PC detection (in terms of balanced accuracy). no attr: no attributes.

Recognition of neonatal stroke children

Table 2 (1 st line) reports the classification performances obtained. We observe that our method reaches an accuracy of 86%. This is a simple task since the difference between the two classes can be done visually with an MRI by detecting the lesion and evaluating the symmetry between each hemisphere (Figure 2). Nevertheless, it validates the ability of the method to classify populations based on the structural information of basal It also demonstrates the complementarity of each attribute. Indeed, in Table 2, we observe that the node attributes increase the balanced accuracy from 0.5 to 0.82. Even if the combination of both attributes little improves the results (0.82 to 0.86), we observe that edge attributes help for the classification (balanced accuracy of 0.63 vs 0.5 without any attribute).

Detection of cerebral palsy among children with NAIS

Table 2 (2 nd line) reports the classification performances obtained. Proposal reaches an accuracy of 89% on the 37 children. Details of the classification are shown in Table 3.

Truth

Predicted No CP CP

No CP 22 2 CP 2 11
Table 3. Confusion matrix for the classification of NAIS children with and without cerebral palsy.

Of the 24 children without CP, 22 are correctly classified. In the same way, our method allows to detect 11 children with CP over the 13 in the cohort. In total, the proposal is wrong on 4 children which is a rather good result considering the difficulty of the problem mentioned previously and illustrated in Figure 3. Indeed, it is difficult to distinguish the populations since both have a lesion. The proximity of the lesion to the basal ganglia plays a role but it is visually difficult to make a decision (i.e. does the child has a PC?). We observe the complementarity of node and edge attributes, each improving the accuracy in the classification task (Table 2).

DISCUSSION

This work illustrates a relationship between the structural organization of basal ganglia and the motor function of children. The combination of the structure volumes and of the distances between "symmetrical" structures in a graph allows to recognize children with neonatal stroke with a balanced accuracy of 86% but also to recognize in most cases (89%) children with NAIS who have developed cerebral palsy. Classification of the subjects is done through a GNN with a small architecture (79 trainable parameters). These good performances need to be compared to competing methods for classification problems like non-graph machine learning classifiers (e.g. SVM, TabFPN [START_REF] Hollmann | TabPFN: A Transformer That Solves Small Tabular Classification Problems in a Second[END_REF]) or deep learning methods operating directly from the image (CNN-based classification) to confirm the relevance of such a simple graph-based method. Results are obtained with a dataset size similar to those considered for such studies: 40 children in Lee et al [START_REF] Lee | Predictors of outcome in perinatal arterial stroke: a population-based study[END_REF], 64 children in Hassett et al [START_REF] Hassett | Bihemispheric developmental alterations in basal ganglia volumes following unilateral perinatal stroke[END_REF] including 20 controls and 44 with neonatal stroke while we consider 68 children including 31 controls and 37 with NAIS. However, they need to be confirmed on other populations. Indeed, the study is performed on 7-year-old children but we should wonder if the relationship between structures and motor functions remains true at other ages, especially to diagnose CP in the early years of the child. A deepening of this work would be to study a possible correlation between structural organization of the basal ganglia and clinical scores describing children's motor skills. This regression problem might require more attributes on nodes and edges but also other modality of MRI like functional MRI as proposed by [START_REF] Yang | Collaborative learning of graph generation, clustering and classification for brain networks diagnosis[END_REF]. Such a prompt detection of motor deficit after NAIS appears necessary to implement early motor interventions.

The method lies in the need of a segmentation of the basal ganglia. However, in the case of stroke brains, this is a complex operation [START_REF] Hassett | Bihemispheric developmental alterations in basal ganglia volumes following unilateral perinatal stroke[END_REF], not yet automated, and thus operator-dependent. To make our method really efficient, it would be necessary to automate this preliminary step of basal ganglia segmentation.

CONCLUSION

We propose a GNN-based strategy to detect cerebral palsy in neonatal stroke children considering the structural organization of the basal ganglia (volumes, distances between structures), this having never been studied yet. Perspectives for this work are numerous. We aim to study a possible correlation between the structural organization of the basal ganglia and various clinical scores in order to detect a motor deficit after NAIS from an MRI. This would help to set up early motor interventions. Furthermore, we will apply the method to younger children to attempt to diagnose CP early in children with NAIS. This could also be applied to children who have begun rehabilitation in order to evaluate the effectiveness of care. Finally, we will reinforce the efficiency and robustness of the method by automating the segmentation of basal ganglia on injured brains and by comparing the use of other graph convolution operators in the GNN.

COMPLIANCE WITH ETHICAL STANDARDS

The AVCnn study (PHRC regional n°0308052 and PHRC interrégional n°1008026; eudract number 2010-A00329-30; Clinical trial NCT02511249) was done in accordance with the international ethical standards and the Declaration of Helsinki. The current evaluation at 7 years was approved by the regional ethics committee (Comité de protection des personnes Sud-Est 1) on May 25 th 2010. Informed consent was obtained from each participant.

2 F (L) 1 Fig. 1 .

 211 Fig. 1. Overview of the proposed GNN-based method for NAIS and PC detection

Fig. 2 .

 2 Fig. 2. NAIS detection: example of MRI of both classes. Left: control brain with a good symmetry between hemispheres. Right: injured brains (the lesion is circled in red).

Fig. 3 .

 3 Fig. 3. PC detection: Example of MRI of both classes. Left: injured brain without cerebral palsy. Right: injured brain of a child with cerebral palsy. Lesion is circled in red.

Table 1 .

 1 Training and test dataset sizes for both experiments i.e.

		Train		Test
	NAIS detection Control NAIS Control NAIS
	Distribution	20	25	11	12
	CP detection	No CP	CP	No CP	CP
	Distribution	16	9	8	4

detection of neonatal stroke (NAIS) and of cerebral palsy (CP). 3 stratified draws with the indicated distribution are considered in both cases.

ACKNOWLEDGMENTS

No funding was received for conducting this study. The authors have no relevant financial or non-financial interests to disclose.