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Stochastic accretion of the Earth

Nature Astronomy volume 6, pages 951–960 (2022)Cite this article

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Abstract

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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Fig. 1: Fits to the bulk compositions of the mantles of the Earth and Vesta.
Fig. 2: Timing of element accretion to the Earth as a function of volatility.
Fig. 3: Elemental abundances modelled in the fully accreted Earth analogue relative to observations.
Fig. 4: Stable isotope fractionation modelled in the fully accreted Earth relative to observations.

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Data availability

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.

Code availability

The code described in the Methods is available upon reasonable request from the corresponding author.

References

  1. Wetherill, G. W. Formation of the Earth. Annu. Rev. Earth Planet. Sci. 18, 205–256 (1990).

    Article  ADS  Google Scholar 

  2. Wetherill, G. W. Provenance of the terrestrial planets. Geochim. Cosmochim. Acta 58, 4513–4520 (1994).

    Article  ADS  Google Scholar 

  3. Chambers, J. E. Making more terrestrial planets. Icarus 152, 205–224 (2001).

    Article  ADS  Google Scholar 

  4. Drążkowska, J. & Alibert, Y. Planetesimal formation starts at the snow line. Astron. Astrophys. 608, A92 (2017).

    Article  ADS  Google Scholar 

  5. Johansen, A. et al. A pebble accretion model for the formation of the terrestrial planets in the Solar System. Sci. Adv. 7, eabc0444 (2021).

    Article  ADS  Google Scholar 

  6. Palme, H. & O’Neill, H. S. C. in Treatise on Geochemistry Vol. 3 (ed Carlson, R. W.) 1–39 (Elsevier, 2014).

  7. Sossi, P. A., Nebel, O., O’Neill, H. S. C. & Moynier, F. Zinc isotope composition of the Earth and its behaviour during planetary accretion. Chem. Geol. 477, 73–84 (2018).

    Article  ADS  Google Scholar 

  8. Wang, K. & Jacobsen, S. B. Potassium isotopic evidence for the origin of the Moon. Nature 538, 487–490 (2016).

    Article  ADS  Google Scholar 

  9. Sharp, Z. D. et al. The chlorine isotope composition of chondrites and Earth. Geochim. Cosmochim. Acta 107, 189–204 (2013).

    Article  ADS  Google Scholar 

  10. Pringle, E. A. & Moynier, F. Rubidium isotopic composition of the Earth, meteorites, and the Moon: evidence for the origin of volatile loss during planetary accretion. Earth Planet. Sci. Lett. 473, 62–70 (2017).

    Article  ADS  Google Scholar 

  11. Humayun, M. & Clayton, R. N. Potassium isotope cosmochemistry: genetic implications of volatile element depletion. Geochim. Cosmochim. Acta 59, 2131–2148 (1995).

    Article  ADS  Google Scholar 

  12. Albarède, F. Volatile accretion history of the terrestrial planets and dynamic implications. Nature 461, 1227–1233 (2009).

    Article  ADS  Google Scholar 

  13. Larimer, J. W. Chemical fractionations in meteorites–I. Condensation of the elements. Geochim. Cosmochim. Acta 31, 1215–1238 (1967).

    Article  ADS  Google Scholar 

  14. Sossi, P. A., Klemme, S., O’Neill, H. S. C., Berndt, J. & Moynier, F. Evaporation of moderately volatile elements from silicate melts: experiments and theory. Geochim. Cosmochim. Acta 260, 204–231 (2019).

    Article  ADS  Google Scholar 

  15. Campbell, I. H. & O’Neill, H. S. C. Evidence against a chondritic Earth. Nature 483, 553–558 (2012).

    Article  ADS  Google Scholar 

  16. Braukmüller, N., Wombacher, F., Funk, C. & Münker, C. Earth’s volatile element depletion pattern inherited from a carbonaceous chondrite-like source. Nat. Geosci. 12, 564–569 (2019).

    Article  ADS  Google Scholar 

  17. Jacobson, S. A. & Morbidelli, A. Lunar and terrestrial planet formation in the Grand Tack scenario. Phil. Trans. R. Soc. A 372, 20130174 (2014).

    Article  ADS  Google Scholar 

  18. Rubie, D. C. et al. Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water. Icarus 248, 89–108 (2015).

    Article  ADS  Google Scholar 

  19. Bell, K. R., Cassen, P. M., Klahr, H. H. & Henning, T. The structure and appearance of protostellar accretion disks: limits on disk flaring. Astron. J. 486, 372–387 (1997).

    Article  Google Scholar 

  20. Dullemond, C. P. & Monnier, J. D. The inner regions of protoplanetary disks. Annu. Rev. Astron. Astrophys. 48, 205–239 (2010).

    Article  ADS  Google Scholar 

  21. Bitsch, B., Johansen, A., Lambrechts, M. & Morbidelli, A. The structure of protoplanetary discs around evolving young stars. Astron. Astrophys. 575, A28 (2015).

    Article  ADS  Google Scholar 

  22. Lodders, K. Solar system abundances and condensation temperatures of the elements. Astrophys. J. 591, 1220–1247 (2003).

    Article  ADS  Google Scholar 

  23. Wood, B. J., Smythe, D. J. & Harrison, T. The condensation temperatures of the elements: a reappraisal. Am. Mineral. 104, 844–856 (2019).

    Article  ADS  Google Scholar 

  24. Jeans, J. H. The Dynamical Theory of Gases (Univ. Cambridge, 1925).

  25. Fischer, R. A. & Ciesla, F. J. Dynamics of the terrestrial planets from a large number of N-body simulations. Earth Planet. Sci. Lett. 392, 28–38 (2014).

    Article  ADS  Google Scholar 

  26. Pierrehumbert, R. T. Principles of Planetary Climate (Cambridge Univ. Press, 2010).

  27. Genda, H. & Abe, Y. Enhanced atmospheric loss on protoplanets at the giant impact phase in the presence of oceans. Nature 433, 842–844 (2005).

    Article  ADS  Google Scholar 

  28. Schönbächler, M., Carlson, R. W., Horan, M. F., Mock, T. D. & Hauri, E. H. Heterogeneous accretion and the moderately volatile element budget of Earth. Science 328, 884–887 (2010).

    Article  ADS  Google Scholar 

  29. Mezger, K., Maltese, A. & Vollstaedt, H. Accretion and differentiation of early planetary bodies as recorded in the composition of the silicate Earth. Icarus 365, 114497 (2021).

    Article  Google Scholar 

  30. Norris, C. A. & Wood, B. J. Earth’s volatile contents established by melting and vaporization. Nature 549, 507–510 (2017).

    Article  ADS  Google Scholar 

  31. Woolum, D. S. & Cassen, P. Astronomical constraints on nebular temperatures: implications for planetesimal formation. Meteorit. Planet. Sci. 34, 897–907 (1999).

    Article  ADS  Google Scholar 

  32. Morbidelli, A. et al. Contemporary formation of early Solar System planetesimals at two distinct radial locations. Nat. Astron. 6, 72–79 (2022).

    Article  ADS  Google Scholar 

  33. Lichtenberg, T., Drazkowska, J., Schönbächler, M., Golabek, G. J. & Hands, T. O. Bifurcation of planetary building blocks during Solar System formation. Science 371, 365–370 (2021).

    Article  ADS  Google Scholar 

  34. Hans, U., Kleine, T. & Bourdon, B. Rb–Sr chronology of volatile depletion in differentiated protoplanets: BABI, ADOR and ALL revisited. Earth Planet. Sci. Lett. 374, 204–214 (2013).

    Article  ADS  Google Scholar 

  35. Anand, A., Pape, J., Wille, M., Mezger, K. & Hofmann, B. Early differentiation of magmatic iron meteorite parent bodies from Mn-Cr chronometry. Geochem. Perspect. Lett. 20, 6–10 (2021).

    Article  Google Scholar 

  36. Trinquier, A., Birck, J.-L., Allègre, C. J., Göpel, C. & Ulfbeck, D. 53Mn–53Cr systematics of the early Solar System revisited. Geochim. Cosmochim. Acta 72, 5146–5163 (2008).

    Article  ADS  Google Scholar 

  37. Dauphas, N., Poitrasson, F., Burkhardt, C. & Kobayashi, H. Planetary and meteoritic Mg/Si and δ30Si variations inherited from solar nebula chemistry. Earth Planet. Sci. Lett. 427, 236–248 (2015).

    Article  ADS  Google Scholar 

  38. Sossi, P. A., Nebel, O., Anand, M. & Poitrasson, F. On the iron isotope composition of Mars and volatile depletion in the terrestrial planets. Earth Planet. Sci. Lett. 449, 360–371 (2016).

    Article  ADS  Google Scholar 

  39. Hin, R. C. et al. Magnesium isotope evidence that accretional vapour loss shapes planetary compositions. Nature 549, 511–515 (2017).

    Article  ADS  Google Scholar 

  40. Young, E. D. et al. Near-equilibrium isotope fractionation during planetesimal evaporation. Icarus 323, 1–15 (2019).

    Article  ADS  Google Scholar 

  41. Tian, Z. et al. Potassium isotopic compositions of howardite-eucrite-diogenite meteorites. Geochim. Cosmochim. Acta 266, 611–632 (2019).

    Article  ADS  Google Scholar 

  42. Paniello, R. C., Day, J. M. D. & Moynier, F. Zinc isotopic evidence for the origin of the Moon. Nature 490, 376–379 (2012).

    Article  ADS  Google Scholar 

  43. Stephant, A. et al. The chlorine isotopic composition of the Moon: insights from melt inclusions. Earth Planet. Sci. Lett. 523, 115715 (2019).

    Article  Google Scholar 

  44. Grimm, R. E. & McSween, H. Y. Heliocentric zoning of the asteroid belt by aluminum-26 heating. Science 259, 653–656 (1993).

    Article  ADS  Google Scholar 

  45. Moskovitz, N. & Gaidos, E. Differentiation of planetesimals and the thermal consequences of melt migration. Meteorit. Planet. Sci. 46, 903–918 (2011).

    Article  ADS  Google Scholar 

  46. Schiller, M., Bizzarro, M. & Siebert, J. Iron isotope evidence for very rapid accretion and differentiation of the proto-Earth. Sci. Adv. 6, eaay7604 (2020).

    Article  ADS  Google Scholar 

  47. Lambrechts, M. & Johansen, A. Rapid growth of gas-giant cores by pebble accretion. Astron. Astrophys. 544, A32 (2012).

    Article  ADS  Google Scholar 

  48. Kurokawa, H. & Tanigawa, T. Suppression of atmospheric recycling of planets embedded in a protoplanetary disc by buoyancy barrierc. Mon. Not. R. Astron. Soc. 479, 635–648 (2018).

    Article  ADS  Google Scholar 

  49. Tian, Z. et al. Potassium isotope composition of Mars reveals a mechanism of planetary volatile retention. Proc. Natl Acad. Sci. USA 118, e2101155118 (2021).

    Article  Google Scholar 

  50. Humayun, M. & Cassen, P. in The Origin of the Earth and Moon (eds Canup, R. M. & Righter, K.) 3–23 (Univ. Arizona Press, 2000).

  51. Charnoz, S. et al. Planetesimal formation in an evolving protoplanetary disk with a dead zone. Astron. Astrophys. 627, A50 (2019).

    Article  Google Scholar 

  52. Beckwith, S. V. W. & Sargent, A. I. Molecular line emission from circumstellar disks. Astrophys. J. 402, 280–291 (1993).

    Article  ADS  Google Scholar 

  53. Ahrens, T. J. & O’Keefe, J. D. Shock melting and vaporization of lunar rocks and minerals. Moon 4, 214–249 (1972).

    Article  ADS  Google Scholar 

  54. Pierazzo, E. & Melosh, H. J. Understanding oblique impacts from experiments, observations, and modeling. Annu. Rev. Earth Planet. Sci. 28, 141–167 (2000).

    Article  ADS  Google Scholar 

  55. Melosh, H. J. in Origin of the Earth (eds Newsom, H. E. & Jones, J. H.) 69–83 (Oxford Univ., 1990).

  56. Lesher, C. E. & Spera, F. J. in The Encyclopedia of Volcanoes 113–141 (Elsevier Inc., 2015).

  57. Bouhifd, M. A. et al. Thermochemistry and melting properties of basalt. Contrib. Mineral. Petrol. 153, 689–698 (2007).

    Article  ADS  Google Scholar 

  58. Connolly, J. A. D. Liquid-vapor phase relations in the Si-O system: A calorically constrained van der Waals-type model. J. Geophys. Res. Planets 121, 1641–1666 (2016).

    Article  ADS  Google Scholar 

  59. Xiao, B. & Stixrude, L. Critical vaporization of MgSiO3. Proc. Natl Acad. Sci. USA 115, 5371–5376 (2018).

    Article  ADS  Google Scholar 

  60. Nakajima, M. & Stevenson, D. J. Melting and mixing states of the Earth’s mantle after the Moon-forming impact. Earth Planet. Sci. Lett. 427, 286–295 (2015).

    Article  ADS  Google Scholar 

  61. Bower, D. J. et al. Linking the evolution of terrestrial interiors and an early outgassed atmosphere to astrophysical observations. Astron. Astrophys. 631, A103 (2019).

    Article  Google Scholar 

  62. Nikolaou, A. et al. What factors affect the duration and outgassing of the terrestrial magma ocean? Astrophys. J. 875, 11 (2019).

    Article  ADS  Google Scholar 

  63. Lebrun, T. et al. Thermal evolution of an early magma ocean in interaction with the atmosphere. J. Geophys. Res. Planets 118, 1155–1176 (2013).

    Article  ADS  Google Scholar 

  64. Sossi, P. A. & Fegley, B. Thermodynamics of element volatility and its application to planetary processes. Rev. Mineral. Geochem. 84, 393–459 (2018).

  65. Grossman, L., Beckett, J. R., Fedkin, A. V., Simon, S. B. & Ciesla, F. J. Redox conditions in the solar nebula: observational, experimental, and theoretical constraints. Rev. Mineral. Geochem. 68, 93–140 (2008).

    Article  Google Scholar 

  66. Visscher, C. & Fegley, B. Chemistry of impact-generated silicate melt-vapor debris disks. Astrophys. J. 767, L12 (2013).

    Article  ADS  Google Scholar 

  67. Solomatov, V. in Treatise on Geophysics 2nd edn (ed Schubert, G.) Vol. 9, 81–104 (Elsevier B.V., 2015).

  68. Corgne, A., Liebske, C., Wood, B. J., Rubie, D. C. & Frost, D. J. Silicate perovskite-melt partitioning of trace elements and geochemical signature of a deep perovskitic reservoir. Geochim. Cosmochim. Acta 69, 485–496 (2005).

    Article  ADS  Google Scholar 

  69. Hunten, D. M. The escape of light gases from planetary atmospheres. J. Atmos. Sci. 30, 1481–1494 (1973).

    Article  ADS  Google Scholar 

  70. Charnoz, S. et al. Tidal pull of the Earth strips the proto-Moon of its volatiles. Icarus 364, 114451 (2021).

    Article  Google Scholar 

  71. Gullbring, E., Hartmann, L., Briceño, C. & Calvet, N. Disk accretion rates for T Tauri stars. Astron. J. 492, 323–341 (1998).

    Article  Google Scholar 

  72. O’Neill, H. S. C. & Palme, H. Collisional erosion and the non-chondritic composition of the terrestrial planets. Phil. Trans. R. Soc. A 366, 4205–4238 (2008).

    Article  ADS  Google Scholar 

  73. Kaib, N. A. & Cowan, N. B. The feeding zones of terrestrial planets and insights into Moon formation. Icarus 252, 161–174 (2015).

    Article  ADS  Google Scholar 

  74. Johansen, A. & Lambrechts, M. Forming planets via pebble accretion. Annu. Rev. Earth Planet. Sci. 45, 359–387 (2017).

    Article  ADS  Google Scholar 

  75. Mittlefehldt, D. W. Asteroid (4) Vesta: I. The howardite-eucrite-diogenite (HED) clan of meteorites. Chem. Erde 75, 155–183 (2015).

    Article  Google Scholar 

  76. Fegley, B. & Schaefer, L. in Principles and Perspectives in Cosmochemistry (eds Goswami, A. & Reddy, B. E.) 347–377 (Springer, 2010).

  77. Siebert, J. et al. Chondritic Mn/Na ratio and limited post-nebular volatile loss of the Earth. Earth Planet. Sci. Lett. 485, 130–139 (2018).

    Article  ADS  Google Scholar 

  78. Wanke, H. & Dreibus, G. Chemical composition and accretion history of terrestrial planets. Phil. Trans. R. Soc. A 325, 545–557 (1988).

    ADS  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Institute of Geochemistry and Petrology, ETH Zurich, Zurich, Switzerland

    Paolo A. Sossi

  2. Institut für Geophysik, Ludwig-Maximilians-Universität München, Munich, Germany

    Ingo L. Stotz

  3. Department of Earth and Environmental Sciences, Michigan State University, East Lansing, MI, USA

    Seth A. Jacobson

  4. Laboratoire Lagrange, UMR7293, Université de Nice Sophia-Antipolis, CNRS, Observatoire de la Côte d’Azur, Nice, France

    Alessandro Morbidelli

  5. School of Earth, Atmosphere and Environment, Monash University, Melbourne, Victoria, Australia

    Hugh St. C. O’Neill

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Contributions

P.A.S. conceived the study, developed the governing equations and assisted with writing the Python code. I.L.S. developed the Python code and ran simulations. S.A.J. performed the N-body simulations. A.M. performed the N-body and pebble accretion simulations. H.S.C.O. contributed to the conceptual development of the study. P.A.S. wrote the paper with input from A.M. and H.S.C.O. All authors contributed to the discussion of the results.

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Correspondence to Paolo A. Sossi.

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Extended data

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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Supplementary Figs. 1–5 and Tables 1 and 2.

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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

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