Chondritic meteorites are thought to be representative of the material that formed the Earth. However, the Earth is depleted in volatile elements in a manner unlike that observed in any chondrite, and yet these elements retain chondritic isotope ratios. Here we use N-body simulations to show that the Earth did not form only from chondrites, but by stochastic accretion of many precursor bodies whose variable compositions reflect the temperatures at which they formed. Earth’s composition is reproduced when the initial temperatures of planetesimal- to embryo-sized bodies are set by disk accretion rates of (1.08 ± 0.17) × 10−7 solar masses per year, although they may be perturbed by 26Al heating on bodies formed at different times. Our model implies that a heliocentric gradient in composition was present in the protoplanetary disk and that planetesimals formed rapidly within ~1 Myr, consistent with radiometric volatile depletion ages of the Earth.
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Raw data produced by the numerical models are provided via Zenodo at https://doi.org/10.5281/zenodo.6412661. Interested parties may contact S.A.J. and A.M. to access the output of N-body and pebble accretion simulations upon reasonable request. All other data supporting the findings of this study are available within the paper and its Supplementary Information.
The code described in the Methods is available upon reasonable request from the corresponding author.
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We thank D. Ebel for comments on an earlier version of this work. P.A.S. acknowledges support from the Swiss National Science Foundation (SNSF) under an Ambizione Fellowship (number 180025), and appreciates discussions with R. Pierrehumbert, M. Schönbächler, T. Lichtenberg and K. Mezger. I.L.S. recognizes support from Deutsche Forschungsgemeinschaft (DFG) project number STO1271/2-1. A.M. acknowledges support from the European Research Council (ERC) advanced grant HolyEarth N. 101019380.
The authors declare no competing interests.
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Extended Data Fig. 1 The cumulative radial distribution (in AU) of the mass (in Earth masses, ME) of accreted components to the Earth analogue planet.
The lines indicate the mass accreted up to a given heliocentric distance, while the colours represent the different Grand Tack N-body simulations (black = R18, red = R8, green = R7, blue = R2). The heliocentric distances (AU) up to which material is sourced for 10%, 50% and 90% of the planet growth are, respectively, R18–0.72, 1.18, 6.43; R8–0.76, 1.04, 7.13; R7–0.72, 1.04, 2.73; R2–0.77, 0.93, 1.93. The secondary y-axis shows the midplane temperature (Tmidplane) in K calculated with Eq. 1 for three mass accretion rates, 1.5, 1.0 and 0.5 × 10−7 solar masses/yr depicted, respectively, as the upper brown envelope, dashed curve and lower brown envelope.
Extended Data Fig. 2 Initial temperature distributions among planetesimals and embryos in N-body simulation GT/8:1/0.025/R2.
a, Initial temperatures imposed by Eq. (2), according to \(\dot M\) = 1×10−7 solar masses/yr. b, The alternative hypothesis, in which initial temperatures are randomly selected from a normal distribution that is intended to simulate 26Al heating independent of heliocentric distance, here 1000 ± 175 K.
Extended Data Fig. 3 The tendency for atmospheric loss to occur from the proto-Earth analogue as a function of its accretion time for four N-body simulations.
Examples of the evolution of the log(escape parameter) for the growing Earth analogue in each of the four N-body simulations, in which the escape parameter is calculated with mi = 33.5 g/mol (the mean value of the Monte Carlo simulations). The red horizontal line denotes the threshold value at which Jeans (above) and Hydrodynamic escape (below) prevail. Negative spikes record impact events leading to temperature increases, superimposed upon the longer-wavelength trend of increasing λesc as accretion proceeds and the proto-Earth analogue grows to its final mass. This attests to the increasing difficulty of atmospheric loss to occur during progressively later stages of accretion, a result consistent among all N-body simulations. Simulation ‘GT6R2’ represents the most energetic accretion pathway for the Earth analogue, while simulation ‘GT5R7’ is the most tranquil.
Extended Data Fig. 4 Timing of element accretion to the Earth analogue as a function of volatility for four N-body simulations.
Points represent the time in the N-body simulation (ordinate) at which a given element (abscissa) reaches a threshold value, y, (10 %, red; 50 %, green; or 90 %, purple) of its final abundance in the Earth analogue. These points are yi % accreted = 100×(xiM)t/(xiM)f, where the subscripts t and f denote the threshold time and the final time, respectively, x is the element mole fraction and M the mass of the body. Error bars (1 standard deviation, 1σ) denote the range given by Monte Carlo models for different \(\dot M\), σembryo and mi across 1000 simulations. No error bar indicates that the element reaches the given % accreted value at the same time in each MC simulation. The mass of the Earth, Mt/Mf, is denoted by horizontal lines at 10 %, 50 % and 90 %. The offsets between these lines and the coloured points indicate differential accretion times. The scatter in the refractory element accretion times reflects their enrichment/dilution by the absence/presence of more abundant elements with Tc50 around 1350 K, as indicated by their consistent accretion times.
Extended Data Fig. 5 Elemental abundances in the fully accreted Earth for four N-body simulations.
CI-, Mg-normalised elemental abundances in the BSE6 as a function of 50% nebular condensation temperatures (red23; purple22,76), compared with abundances predicted by 1000 Monte Carlo simulations per N-body simulation. The average composition of the Earth analogue over 1000 runs is given by the solid blue line, with its percentile range (25th to 75th, light blue and 10th to 90th, dark blue). The dashed black line shows the single simulation that minimises the Root-Mean-Square (RMS) deviation from the data. The RMS values are listed in the top left-hand corner, along with the values of the three random variables to which they correspond. Only lithophile elements were plotted as siderophile elements are further depleted by core formation, which is not considered in our model.
Extended Data Fig. 6 Variation of the Root-Mean-Square Deviation (RMS) to the observed elemental abundances as a function of the three randomised variables in 1000 Monte Carlo simulations.
The four N-body simulations are organised by row, in which the RMS is plotted on the ordinate axis and the value of the random variable on the abscissa, organised by column (mass accretion rate (\(\dot M\)), mean atmospheric mass (mi) and embryo feeding zone (σembryo). Although the goodness of fit is nearly independent of σembryo and mi, it has a well-defined minimum as a function of mass accretion rate. The mass accretion rate defined by this minimum varies only marginally among the four N-body simulations, with a mean of 1.08(±0.17)×10−7 solar masses/year, demonstrating that the composition of the fully accreted Earth analogue is largely sensitive to the initial compositions of its constituents, which are in turn set by the temperatures (and hence heliocentric distances) at which they were formed.
Extended Data Fig. 7 Stable isotope fractionation for each fictive element in the fully accreted Earth analogue as a function of volatility for four N-body simulations.
The red line indicates the mean isotopic fractionation (in per mille relative to the initial value, that is, CI chondritic) in the fully accreted Earth analogue across 1000 Monte Carlo runs for each N-body simulation. The isotope ratio in g/mol (mi = numerator isotope, Δmi = 2 atomic mass units, amu) is listed in the legend. The vertical dashed lines delimit zones pertaining to Late Accretion (defined here as volatile elements whose isotopes are present in chondritic ratios), Partial Evaporation (elements bear an isotopic signature of partial evaporation during atmospheric loss) and Non-Volatile (elements were not lost and are also chondritic).
Extended Data Fig. 8 Variation of the average stable isotopic fractionation (expressed in log ‰) across all elements as a function of the three randomised variables in 1000 Monte Carlo simulations.
The four N-body simulations are organised by row, in which the logarithm of the mean isotope fractionation is plotted on the ordinate axis and the value of the random variable on the abscissa, organised by column; mass accretion rate (\(\dot M\)) (in solar masses/yr), mean atmospheric mass (mi) (in g/mol) and embryo feeding zone (σembryo) (in K). The degree of isotopic fractionation in the fully accreted Earth analogue is only mildly dependent on \(\dot M\) and σembryo, but shows a strong negative correlation with the mean atmospheric mass, mi, reflecting the scaling between atmospheric loss rate and atmosphere scale height with its molar mass. Changes in slope reflect the participation of different, major atmospheric loss events (hydrodynamic escape) engendered by multiple collisions.
Extended Data Fig. 9 Composite plots of four N-body simulations for accretion time, bulk composition and isotope fractionation.
From left to right, the panels are equivalent to Extended Data Figs. 4, 5 & 7, respectively. The results show that there is no systematic difference in the accretion times of the volatile- and non-volatile elements in the simulations (left-hand side panel). The middle panel illustrates that the bulk composition of the Earth can be reproduced using the initial conditions shown in Extended Data Fig. 2b. Isotopic fractionation (right-hand side panel) is always larger than for the equivalent simulation using the initial conditions shown in Extended Data Fig. 2a (compare with Extended Data Fig. 7). Due to the presence of volatile elements in the proto-Earth throughout its growth, most isotopic fractionation is seen in the most volatile elements, and the bimodal nature of the peaks (for example simulation GT/8:1/0.025/R2) carries the imprint of two separate collisional events.
Extended Data Fig. 10 Escape parameter, isotope composition, elemental abundances and accretion rate of the Earth in a pebble accretion scenario.
a, The escape parameter as a function of accretion time for a fixed mean atmospheric mass of 10 g/mol. Despite the lower mass, the proto-Earth never reaches a regime of hydrodynamic escape, due to the lower peak temperatures for a given mass with respect to planetesimal accretion. Because pebble accretion is continuous in time, the evolution of the escape parameter is devoid of any negative spikes. b, Consequently, evaporation and atmospheric loss is less marked than in the planetesimal accretion models, as attested to by the lesser degree of isotopic fractionation (<2×10−7 ‰) evolved in the Earth. The thick red line represents the mean value, with the red field corresponding to the maximum and minimum isotope fractionation experienced, as a function of nebular condensation temperature. As for the planetesimal accretion simulations, isotope fractionation is most marked for elements with Tc50 around ~700 K, representing those that are sufficiently volatile to record atmospheric loss, but not so volatile so as to have been overprinted by late-accreting material. c, The CI-, Mg-normalised elemental abundances in the BSE6 relative to nebular condensation temperatures (red23; purple22) compared with abundances predicted from the pebble accretion simulation. The dark blue curve represents the mean and the blue field its 1 s uncertainty propagated from Eq. (2). Accretion of pebbles also results in a smooth, concave-down pattern as a function of time. However, volatile element depletion for pebble accretion is less marked than in the planetesimal accretion model, reflecting a lower mean temperature of the accreting pebbles. If the pebble flux is increased and truncated when the Earth reaches its final mass (for example, by the formation of a Jupiter barrier) the Earth would accrete faster in a warmer disk, causing more depletion in elements with higher Tc50. Nevertheless, the transition from undepleted to depleted elements would remain too abrupt in Tc50 relative to observations. d, The accretion rate of Earth (red curve) as a function of time, superimposed on the temperature evolution of the surrounding nebular gas at 1 AU (black curve), valid for accretion rates that scale with M2/3. It shows that the temperature decrease occurs over too short a time interval, declining to ~500 K after ~106 yr, by which point the Earth has accreted only ~15 % of its total mass, meaning minimal volatile depletion is expressed in its final composition.
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Sossi, P.A., Stotz, I.L., Jacobson, S.A. et al. Stochastic accretion of the Earth. Nat Astron 6, 951–960 (2022). https://doi.org/10.1038/s41550-022-01702-2