The origin of titanium-rich basaltic magmatism on the Moon remains enigmatic. Ilmenite-bearing cumulates in the lunar mantle are often credited as the source, but their partial melts are not a compositional match and are too dense to enable eruption. Here we use petrological reaction experiments to show that partial melts of ilmenite-bearing cumulates react with olivine and orthopyroxene in the lunar mantle, shifting the melt composition to that of the high-Ti suite. New high-precision Mg isotope data confirm that high-Ti basalts have variable and isotopically light Mg isotope compositions that are inconsistent with equilibrium partial melting. We employ a diffusion model to demonstrate that kinetic isotope fractionation during reactive flow of partial melts derived from ilmenite-bearing cumulates can explain these anomalously light Mg isotope compositions, as well as the isotope composition of other elements such as Fe, Ca and Ti. Although this model does not fully replicate lunar melt-solid interaction, we suggest that titanium-rich magmas erupted on the surface of the Moon can be derived through partial melting of ilmenite-bearing cumulates, but melts undergo extensive modification of their elemental and isotopic composition through reactive flow in the lunar mantle. Reactive flow may therefore be the critical process that decreases melt density and allows high-Ti melts to erupt on the lunar surface.

This report contains a progress update on the major achievements and challenges experienced so far for the Ministero dell'Università e delle Ricerche (MUR) regarding my Young Reseachers project.

This report contains updates for the European Research Council's Executive Agency on the major achievements and challenges experienced during the first half of my ERC Starting Grant project.

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The terrestrial planets are depleted in moderately volatile elements (e.g., K and Zn) relative to chondritic material. This depletion could be due to incomplete condensation or evaporative loss, either in precursor material or in accreting bodies. Potassium isotopes may distinguish between these different processes as they correlate with body mass, the smaller bodies being isotopically lighter (Tian et al., 2021), implying evaporative loss from the fully-formed bodies. However, the correlation of isotopic fractionation with elemental concentration is weak, and evaporative loss from a body as large as the Earth is challenging. In this work, we investigate how K loss and isotopic fractionation proceed during planetary growth, using a quantitative model of evaporative loss and a N-body accretion model. We consider adiabatic conditions for mass flux and equilibrium at the melt-vapour interface with a temperature-dependent partition coefficient and a constant isotope fractionation factor ?=0.99913. First, we model mass loss as a consequence of heating events (e.g. impacts) by elevating temperatures, and find that mass loss does not occur for bodies exceeding roughly 1023 kg (~1.5 lunar masses). Second, we study the potential for K loss driven by 26Al heating. Contrasting with previous work (Young et al, 2019), we find that temperatures buffer near the solidus with negligible evaporative loss and thus negligible isotopic fractionation, because once liquid-supported convection initiates, cooling rates exceed 26Al heating rates. Additional, rapid heating by e.g. impacts is thus required for significant evaporative loss from planetesimals. In ongoing work, we track potassium loss and isotopic fractionation over the course of N-body simulations of the runway and oligarchic stages of accretion. Preliminary results show a rough inverse correlation between body mass and isotope anomaly, implying that the observed correlation (Tian et al., 2021) could be a result of early evaporative loss followed by accretion, as well as mixing and dilution after overall mass loss ceases.

Introduction: The terrestrial planets are depleted in moderately volatile elements (MVEs) relative to chondritic material [1]. This depletion could be due to either incomplete condensation [2] or partial mass loss, either in precursor material or in accreting bodies. The Earth has an excess of the heavier Mg and Si isotopes [3], which may be due to evaporative loss caused by impact-driven melting during accretion [3] or by heating due to 26Al decay [4]. MVEs like potassium and zinc also show isotopic anomalies [5,6]. Potassium isotopes show a correlation with body mass [6], with the smaller bodies being isotopically lighter. This observation could imply evaporative loss from the fully-formed bodies. However, the correlation of isotopic fractionation with elemental concentration is weak, and evaporative loss from a body as large as the Earth is challenging [4]. In this work we apply a quantitative model of evaporative loss to potassium isotopes, coupling it with an N-body accretion model [7] to investigate how fractionation and loss proceed as planetesimals grow. Here we focus on the potential for loss driven by 26Al heating [4] rather than by impacts [3,8]. Evaporative Loss: We first explore mass loss for an array of initial accreting body conditions, with initial masses in the range 0.01-1.38x1023 kg corresponding to a range of radii from ~408 to ~2112 km. Evaporative mass loss from initially molten bodies occurs via hydrodynamic escape [3,4] with the surface pressure determined by the temperature of the molten interior. We consider both isothermal and adiabatic conditions for mass flux and elemental escape. The adiabatic mass loss rate is solved following [9]. The effective surface temperature of the molten body is determined by balancing the isoviscous convective and radiative heat fluxes [3] and the interior temperature decreases as the model run proceeds. To calculate the loss of potassium we assume a temperature-dependent partition coefficient. The isotopic evolution is then tracked assuming equilibrium fractionation at the melt-vapour interface [4] and a constant fractionation factor ?. Each simulation ends under one of two potential conditions. Either the magma ocean cools to its solidus, set at 1400 K, or the interior temperature drops to the point at which adiabatic escape is no longer energetically possible [10]. Either scenario results in mass loss ceasing. Realizations at each initial temperature were run for increasing body masses until negligible mass loss was achieved after the first time step. Figure 1. A. Fraction of potassium lost via hydrodynamic escape as a function of initial interior temperature and body mass. Here an adiabatic atmosphere with n=0.2 is assumed. B. Potassium isotope anomaly, assuming ?=0.99913. Single-body Results: Figure 1A shows the fraction of potassium lost for various initial body masses and internal temperatures. For high initial temperatures (above ~2750 K), all the potassium is lost. Mass loss does not occur for bodies exceeding roughly 1023 kg. We observe a maximum fraction lost at intermediate masses. This is because large bodies have a high gravity, impeding escape, while small bodies cool rapidly, limiting the time for escape to occur. Figure 1B shows that the ??41/39K is generally higher for higher initial temperatures and smaller masses, with again a maximum at intermediate masses where mass loss is most efficient. N-body Model: Planetary bodies grow by collisions, so that the final elemental and isotopic composition of an object is a mixture of the starting bodies' compositions. We use the approach outlined above and track the evolution of potassium as bodies collide, assuming simple mixing. That is, we assume all 54th Lunar and Planetary Science Conference 2023 (LPI Contrib. No. 2806) 1350.pdf potassium loss and fractionation happens early, as the bodies form, and that impact vaporization or mechanical erosion during subsequent collisions is negligible. For our accretion model we use output from [7] in which planetesimal growth through the runaway and oligarchic stages of accretion is tracked, and the influence of migrating giant planets is included. We assume that each starting body has an initial temperature set by heating due to 26Al decay. The accretion time is varied randomly from 0 to 2 Myr after CAI and the peak temperature is calculated accordingly, based on a decay constant of 0.99 Myr-1 and a temperature at 1 Myr of 1486 K. Mass loss for each body is then calculated assuming this initial temperature and an isothermal atmosphere. For the isotopic calculations a constant fractionation factor of ?=0.99913 is assumed. N-body Results: Figure 2. Potassium isotope anomaly vs. K/U ratio for bodies growing by collision according to [7]. Dot size is proportional to final mass; color indicates number of collisions. EPB=eucrite parent body. The dotted line indicates the expected results for pure Rayleigh fractionation. Figure 2 shows the calculated ??41/39K against the K/U ratio of the final objects, where the colors indicate the number of impacts. Bodies that suffer no impacts follow the Rayleigh fractionation line (dotted), where more significant K loss results in larger isotopic anomalies. Larger bodies typically show more subdued isotopic anomalies due to mixing and dilution. Figure 3. Potassium isotope anomaly as a function of final body mass. Color scheme as for Figure 2. Figure 3 shows the calculated ??41/39K against the mass of the final objects. The upper envelope of possibilities shows a rough inverse correlation (ellipse) between ??41/39K and mass, similar to the trend observed by [6]. Initial mass loss and fractionation is easier for smaller bodies, and smaller bodies are also less subject to subsequent dilution and mixing by later impacts. Thus, the trend observed by [6] does not necessarily imply that evaporative mass loss had to take place after accretion had finished; this signature could instead be a result of early evaporative loss followed by accretion, mixing and dilution. Future Work: Rather than simply specifying an initial temperature, a more realistic approach would be to track the internal temperature evolution due to the combined effects of 26Al heating, mass loss and radiative cooling. Tracking additional MVEs, such as Zn, would provide additional constraints. Experimental determination of the relevant ? values is desirable. Conclusions: Small, initially molten bodies develop large potassium anomalies via evaporative loss (Fig 1). Subsequent mixing and dilution during accretion can retain an inverse correlation between body mass and isotope anomaly (Fig 3), as observed.

Several studies suggested that the Mo isotope composition of Earth's mantle may be subtly sub-chondritic [1,2]. This observation cannot be reconciled with a likely barely detectable enrichment in heavy Mo isotopes in Earth's mantle following core-mantle differentiation [3]. A study of the Mo isotope composition of Earth's crust suggested it may be super- chondritic [4]. Complementarity between Mo isotopes in Earth's crust and mantle implies that Mo isotopes can provide valuable insights into the evolution of Earth's mantle-crust system. However, a sub-chondritic Mo isotope composition of the accessible mantle is debated. Given the incompatibility of Mo, mid-ocean ridge basalts (MORB) are arguably the most obvious type of rocks to study the Mo isotope composition of the mantle. Two previous studies yielded variably sub-chondritic to super-chondritic Mo isotope compositions in MORB [1,2], with no obvious systematics to explain the variability. More recently, a study focussed on enriched MORB, i.e. (La/Sm)N > 1, suggested they obtained higher Mo isotope ratios following metasomatism of mantle lithosphere caused by low degree partial melts derived from the mantle [5], thus explaining some of the variability in MORB. We analysed depleted MORB, i.e. (La/Sm)N < 1, from the Pacific, Indian and Atlantic oceans to determine their Mo isotope compositions and estimate a value for the bulk mantle. Our samples are characterised by sub-chondritic Mo isotope compositions on average, and none of the individual depleted MORB display super-chondritic values. Combined with literature data, we find that the bulk, accessible mantle is on average slightly sub-chondritic. Modelling suggests that >1 billion years of plate tectonic cycling of dehydrated, subducted oceanic crust into the mantle can explain the evolution of the mantle Mo isotope and Ce/Pb ratios in tandem, which is not the case for extraction of mantle partial melts. Our results thus add to the notion that the depleted mantle has been extensively modified by subduction-processed, oceanic crust.

In order to better constrain the Mg isotopic composition of the mantle, we have analysed twenty-eight samples of both oceanic and continental peridotite using a high-precision, critical mixture double spiking approach. The unaltered samples show no variability ?Mg in outside analytical uncertainty and yield a value of -0.236 ± 0.006? (2 s.e.) for the accessible mantle, substantiating its non-chondritic composition. We have also determined inter-mineral Mg isotopic fractionations for a sub-set of samples. We document small but significant differences in ?Mg between olivine and pyroxenes, ?Mg = -0.118 ± 0.018? and ?Mg = -0.056 ± 0.018?, in excellent agreement with ab initio calculations for temperatures ~1000 °C, as recorded by mineral thermometry in the peridotites. The differences in ?Mg between olivine and spinel (?Mg ) are more variable and generally higher than theoretical calculations at corresponding temperatures, likely due to incomplete Fe-Mg diffusive exchange during post-eruptive cooling of the xenoliths. Using these data, together with a recently determined olivine-melt fractionation factor for Mg isotopes, we show that partial melting has a negligible influence on the ?Mg of residual peridotites. This helps account for the minimal variability of ?Mg in fresh, mantle peridotites. However, the ?Mg of primary mantle melts are predicted to be discernibly higher than their sources (?Mg ~ 0.06? and ~0.123? for representative partial melts of peridotitic and pyroxenitic sources respectively) across a wide range of melting conditions. Such elevated ?Mg values are not generally observed in the current dataset of mantle derived melts. We propose that this inconsistency is likely a consequence of diffusive fractionation during partial re-equilibration between low Mg/Fe melts migrating through high Mg/Fe mantle en route to the surface.

We determined equilibrium Mg isotope fractionation between olivine and melt (?26/24MgOl/melt) in five, naturally quenched, olivine-glass pairs that were selected to show clear textural and chemical evidence of equilibration. We employed a high-precision, critical mixture double-spiking approach to obtain a weighted mean of ?26/24MgOl/melt = -0.071 ± 0.010 , for values corrected to a common olivineglass temperature of 1438 K. As function of temperature, the fractionation can be expressed as ?26/24MgOl/melt = (-1.46 ± 0.26) × 105/T2. The samples analysed have variable H2O content from 0.1 to ~1.2 wt. %, yet no discernible difference in ?26/24MgOl/melt was evident. We have used this ?26/24MgOl/melt to revisit the puzzling issue of elevated Mg isotope ratios in arc lavas. In new Mg isotope data on sample suites from the Lesser Antilles and Mariana arcs, we show that primitive samples have MORB-like Mg isotope ratios while the evolved samples tend to have isotopically heavier compositions. The magnitude of this variability is well explained by olivine fractionation during magmatic differentiation as calculated with our new equilibrium ?26/24MgOl/melt.

Mid-ocean ridge basalts (MORB) reveal large mantle compositional heterogeneity, whose origin remains debated. Here we present a systematic study of molybdenum isotopes on well-characterized MORB glass samples from the East Pacific Rise (EPR) and near-EPR seamounts. Our analyses show significant Mo isotope variations with ?Mo (relative to NIST SRM3134) ranging from -0.23? to -0.06?. We argue that these Mo isotope variations are not caused by processes of MORB melt generation and evolution but reflect mantle isotopic heterogeneity. Taking together with the literature data, we show that MORB Mo isotope compositions vary systematically with geochemical parameters indicating mantle enrichment. These observations are best explained by two-component mixing between an incompatible element depleted endmember (e.g., low La/Sm, Nb/La, Nb/Zr and Th/Yb, and high Sm/Nd and Nd/Nd) with low ?Mo (~-0.21?) and an incompatible element enriched endmember (e.g., high La/Sm, Nb/La, Nb/Zr and Th/Yb, and low Sm/Nd and Nd/Nd) with high ?Mo (~-0.05?). The association of heavier Mo isotope compositions with the geochemically more enriched MORB is inconsistent with recycled ocean crust with or without sediment being the enriched endmember. Instead, this is consistent with the enriched endmember being of magmatic origin, most likely lithologies of low-degree melt metasomatic origin dispersed in the more depleted peridotite matrix in the MORB mantle. Thus, with MORB Mo isotope systematics, we confirm that recycled oceanic mantle lithosphere metasomatized by low degree melt plays a key role in the formation of E-MORB source lithologies. Our study also highlights Mo isotopes as an effective tool for studying upper mantle processes.

Several studies have suggested that the Earth's upper mantle is slightly enriched in light molybdenum isotopes relative to bulk Earth, defined by chondrites, but there is no consensus on the presence of this subtle but potentially notable signature. To establish better whether or not the Mo/Mo of Earth's upper mantle is indeed sub-chondritic, we have analysed hand-picked glasses of depleted (i.e. chondrite normalised La/Sm<1) mid-ocean ridge basalts (MORB) from the Pacific, Atlantic and Indian ocean basins. The mean Mo isotope composition of our depleted MORB relative to reference NIST SRM 3134 (?Mo) is -0.22±0.03? (95% confidence interval, c.i.) compared to a value of -0.15±0.01? (95% c.i.) for bulk Earth. Our high precision analyses of the U/U activity ratios of these samples are within uncertainty of unity, which rules out the effect of possible secondary, sea-floor processes as the dominant cause of their low ?Mo. We further report experimental data showing that sulphide liquid has ?Mo 0.25±0.01? lower than basaltic silicate liquid at 1400 °C. This fractionation is too small to significantly alter the Mo isotope composition of basalts relative to their sources during melting or differentiation. Our MORB data show that resolvably sub-chondritic Mo isotope compositions are common in the upper mantle. Moreover, an appropriately weighted average ?Mo of depleted and enriched MORB, taken from this study and the literature, yields an estimated mantle value of -0.20±0.01?, indicating that the upper mantle as a whole is sub-chondritic. Since prior work demonstrates that core formation will not create a residual silicate reservoir with a sub-chondritic ?Mo, we propose that this feature is a result of recycling oceanic crust with low ?Mo because of Mo isotope fractionation during subduction dehydration. Such an origin is in keeping with the sub-chondritic Th/U and low Ce/Pb of the depleted mantle, features which cannot be explained by simple melt extraction. We present mass balance models of the plate tectonic cycle that quantitatively illustrate that the ?Mo of the Earth's mantle can be suitably lowered by such oceanic crustal recycling. Our Mo isotope study adds to the notion that the depleted mantle has been substantially modified by geodynamic cycling of subduction-processed oceanic crust.

The abundances of water and highly to moderately volatile elements in planets are considered critical to mantle convection, surface evolution processes, and habitability. From the first flyby space probes to the more recent "Perseverance" and "Tianwen-1" missions, "follow the water," and, more broadly, "volatiles," has been one of the key themes of martian exploration. Ratios of volatiles relative to refractory elements (e.g., K/Th, Rb/Sr) are consistent with a higher volatile content for Mars than for Earth, despite the contrasting present-day surface conditions of those bodies. This study presents K isotope data from a spectrum of martian lithologies as an isotopic tracer for comparing the inventories of highly and moderately volatile elements and compounds of planetary bodies. Here, we show that meteorites from Mars have systematically heavier K isotopic compositions than the bulk silicate Earth, implying a greater loss of K from Mars than from Earth. The average "bulk silicate" ?41K values of Earth, Moon, Mars, and the asteroid 4-Vesta correlate with surface gravity, the Mn/Na "volatility" ratio, and most notably, bulk planet H2O abundance. These relationships indicate that planetary volatile abundances result from variable volatile loss during accretionary growth in which larger mass bodies preferentially retain volatile elements over lower mass objects. There is likely a threshold on the size requirements of rocky (exo) planets to retain enough H2O to enable habitability and plate tectonics, with mass exceeding that of Mars.

Previous work has shown that Mo isotopes measurably fractionate between metal and silicate liquids, even at temperatures appropriate for core formation. However, the effect of variations in the structural environment of Mo in the silicate liquid, especially as a function of valence state, on Mo isotope fractionation remained poorly explored. We have investigated the role of valence state in metal-silicate experiments in a gas-controlled furnace at 1400 °C and at oxygen fugacities between 10 and 10, i.e. between three and 0.2 log units below the iron-wüstite buffer. Two sets of experiments were performed, both with a silicate liquid in the CaO-AlO-SiO system. One set used molybdenum metal wire loops as the metal source, the other liquid gold alloyed with 2.5 wt% Mo contained in silica glass tubes. X-ray absorption near-edge spectroscopy analysis indicates that Mo/?Mo in the silicate glasses varies between 0.24 and 0.77 at oxygen fugacities of 10 and 10 in the wire loop experiments and between 0.15 and 0.48 at 10 and 10 in the experiments with Au-Mo alloys. Double-spiked analysis of Mo isotope compositions furthermore shows that Mo isotope fractionation between metal and silicate is a linear function of Mo/?Mo in the silicate glasses, with a difference of 0.51? in Mo/Mo between purely Mo-bearing and purely Mo-bearing silicate liquid. The former is octahedrally and the latter tetrahedrally coordinated. Our study implies that previous experimental work contained a mixture of Mo and Mo species in the silicate liquid. Our refined parameterisation for Mo isotope fractionation between metal and silicate can be described as ?Mo=[Formula presented] Molybdenum isotope ratios therefore have potential as a proxy to constrain the oxygen fugacity during core formation on planetary bodies if the parameterisation of Mo/?Mo variation with oxygen fugacity is expanded, for instance to include iron-bearing systems. On Earth literature data indicate that the upper mantle is depleted in heavy Mo isotopes relative to the bulk Earth, as represented by chondrites. As previously highlighted, this difference is most likely not caused by core formation, which either enriches the mantle in heavy Mo isotopes or causes no significant fractionation, depending on temperature and, as we determined here, Mo content. We reaffirm that core formation does not account for the Mo isotope composition of the modern upper mantle, which may instead reflect the effect of fractionation during subduction as part of global plate recycling.

We have determined ?Mg, the mass-independent variations in Mg/Mg, of primitive, bulk meteorites to precisions better than ±3 ppm (2se). Our measurements of samples from 10 different chondrite groups show ?Mg that vary from -5 to 22 ppm. Our data define an array with a positive slope in a plot of ?Mg against Al/Mg, which can be used to determine (Al/Al), i.e. initial Al/Al, and (?Mg), i.e. initial ?Mg. On such an isochron plot, the best fit of our new measurements combined with literature data implies (Al/Al) of (4.67±0.78)×10 and (?Mg) of -31.6 ± 5.7 ppm (2se) for ordinary and carbonaceous chondrites, other than CR chondrites, which have anomalously low ?Mg. These parameters are within uncertainty of those defined by previous measurements of bulk calcium-, aluminium-rich inclusions (CAIs) that set canonical (Al/Al)~05×10. The most straightforward interpretation of all these observations is that differences in the Al/Mg of bulk ordinary and carbonaceous chondrites are dominantly controlled by variable contributions of early-formed refractory and major silicate components derived from a common, canonical reservoir. The ?Mg of enstatite chondrites are slightly more radiogenic (~3 ppm) at similar Al/Mg to the ordinary chondrites. We speculate that this is related to the timing of removal of a refractory component from the source reservoirs of these different meteorite groups; the higher ?Mg of the enstatite chondrites suggests later (~0.5 Ma post CAIs) condensation and loss of this refractory component. Despite inferred consistency of (Al/Al) and (?Mg) across most chondrite groups, some nebular heterogeneity is required to account for the compositions of CR chondrites. Our preferred interpretation is that the CR source region has lower (?Mg). As the most appropriate isotopic reference for the Earth, our new mean enstatite chondrite composition allows us to assess possible ingrowth of Mg from live Al during accretion of the Earth. The Earth has ?Mg within uncertainty of enstatite chondrites, despite its higher Al/Mg. This requires that the terrestrial increase in Al/Mg, which we attribute to vapour loss during accretion, must have happened >1.5 Ma post CAI formation, in an instantaneous fractionation model.

Fluids liberated from subducting slabs are critical in global geochemical cycles. We investigate the behaviour of Mo during slab dehydration using two suites of exhumed fragments of subducted, oceanic lithosphere. Our samples display a positive correlation of ?Mo with Mo/Ce, from compositions close to typical mantle (-0.2? and 0.03, respectively) to very low values of both ?Mo (-1?) and Mo/Ce (0.002). Together with new, experimental data, we show that molybdenum isotopic fractionation is driven by preference of heavier Mo isotopes for a fluid phase over rutile, the dominant mineral host of Mo in eclogites. Moreover, the strongly perturbed ?Mo and Mo/Ce of our samples requires that they experienced a large flux of oxidised fluid. This is consistent with channelised, reactive fluid flow through the subducted crust, following dehydration of the underlying, serpentinised slab mantle. The high ?Mo of some arc lavas is the complement to this process.

The study of siderophile element isotope compositions in planetary mantles offers a new methodology to constrain the temperatures of core formation, provided there is an appropriate calibration of the temperature-dependence and possibly pressure-dependence of isotope fractionation between metal and silicate and of the metal-silicate partitioning for these elements. In this review, we examine recent studies that have shown that Si, Fe, Mo, Cr, Cu, Ni, N and C could potentially be used to constrain the temperature of metal-silicate equilibration using single stage or continuous models of core formation, yielding contrasted results. Such an approach requires assumptions about the building blocks of the Earth and it is generally considered that the composition of some chondrites is representative of bulk Earth. This is obviously more complex for volatile elements such as Cu, N or C, as the isotope composition of the building blocks of the Earth could have been affected by thermal processing. On the basis of a chondritic bulk composition, one can estimate a temperature of core formation assuming a model for this process. If the metal-silicate equilibration is incomplete, as is likely the case for giant impacts, then the composition of the mantle of the impactor and the fraction of metal that equilibrates needs to be assessed carefully. It has been shown recently that the degree of equilibration will be a function of the metal-silicate partition coefficient and will be hence very different for Si, Cr, or Mo, an aspect that has not been considered in previous studies and may help explain differences in interpretation. In this context, the expected temperatures of equilibration are quite variable and are a function of the impactor's conditions of metal-silicate segregation. Another complication arises when considering continuous models of core formation: the most siderophile elements will be sensitive to the last episodes of core formation, while the budget of less siderophile elements will reflect its integrated accretion history (e.g. Cr or Si). A model including Si, Cr and Mo isotope data that takes into account these aspects has been constructed and shown to be consistent with scenarii that were derived from siderophile element data.

It has long been recognized that Earth and other differentiated planetary bodies are chemically fractionated compared to primitive, chondritic meteorites and, by inference, the primordial disk from which they formed. However, it is not known whether the notable volatile depletions of planetary bodies are a consequence of accretion or inherited from prior nebular fractionation. The isotopic compositions of the main constituents of planetary bodies can contribute to this debate. Here we develop an analytical approach that corrects a major cause of measurement inaccuracy inherent in conventional methods, and show that all differentiated bodies have isotopically heavier magnesium compositions than chondritic meteorites. We argue that possible magnesium isotope fractionation during condensation of the solar nebula, core formation and silicate differentiation cannot explain these observations. However, isotopic fractionation between liquid and vapour, followed by vapour escape during accretionary growth of planetesimals, generates appropriate residual compositions. Our modelling implies that the isotopic compositions of magnesium, silicon and iron, and the relative abundances of the major elements of Earth and other planetary bodies, are a natural consequence of substantial (about 40 per cent by mass) vapour loss from growing planetesimals by this mechanism.

Double spiking is conventionally used to make accurate determinations of natural mass-dependent isotopic fractionations for elements with four or more stable isotopes. Here we document a methodology which extends the effective application of double spiking to three isotope systems. This approach requires making a mixture with isotope ratios that lie on a 'critical curve' where the sample - double-spike mixing line and the tangent to the instrumental mass-bias curve are coincident. Inversion of the mixing equations for such a mixture leads to a solution for the sample fractionation which is independent (to first order) of the uncertainty in the instrumental mass-bias and, hence, independent of any mass-dependent artefacts in the measurement such as those produced by residual matrix not completely removed by prior chemical purification. In practice, mixtures can be made which yield an accuracy conservatively estimated to be ~ 0.005?/amu. The precision of the method is explored as a function of double-spike composition for Mg, Si and K isotope systems. We show that for Mg and Si measurement precision is not compromised by the compositions of viable critical mixtures nor by uncertainty magnification during inversion of the equations. Thus, double spiking provides a valuable means to obtain robust, high precision isotopic measurements of Mg and Si. For K, however, the low abundance of K in the optimal critical mixture places a significant practical limitation on the application of double spiking to analyses of this element.