A pebble accretion model for the formation of the terrestrial planets in the Solar System
Anders Johansen, Thomas Ronnet, Martin Bizzarro, Martin Schiller, Michiel Lambrechts, ?ke Nordlund, Helmut Lammer
TTitle
A pebble accretion model for the formation of the terrestrial planets in the Solar System
Authors
Anders Johansen ∗ , , Thomas Ronnet , Martin Bizzarro , Martin Schiller , Michiel Lambrechts ,˚Ake Nordlund , & Helmut Lammer Affiliations Center for Star and Planet Formation, GLOBE Institute, University of Copenhagen, ØsterVoldgade 5-7, 1350 Copenhagen, Denmark Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box43, 22100 Lund, Sweden Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen,Denmark Space Research Institute, Austrian Academy of Sciences, Schmiedlstr. 6, 8042 Graz, Austria
Abstract
Pebbles of millimeter sizes are abundant in protoplanetary discs around young stars. Chondrulesinside primitive meteorites – formed by melting of dust aggregate pebbles or in impacts betweenplanetesimals – have similar sizes. The role of pebble accretion for terrestrial planet formation isnevertheless unclear. Here we present a model where inwards-drifting pebbles feed the growthof terrestrial planets. The masses and orbits of Venus, Earth, Theia (which later collided withthe Earth to form the Moon) and Mars are all consistent with pebble accretion onto protoplanetsthat formed around Mars’ orbit and migrated to their final positions while growing. The isotopiccompositions of Earth and Mars are matched qualitatively by accretion of two generations ofpebbles, carrying distinct isotopic signatures. Finally, we show that the water and carbon budgetof Earth can be delivered by pebbles from the early generation before the gas envelope becamehot enough to vaporise volatiles. 1 a r X i v : . [ a s t r o - ph . E P ] F e b ntroduction The long formation time-scale of gas giants and ice giants in the outer regions of protoplan-etary discs by traditional planetesimal accretion (
1, 2 ) instigated the development of the peb-ble accretion theory in which the pebbles drifting through the protoplanetary disc are accretedrapidly by the growing protoplanets (
3, 4 ). While pebble accretion clearly aids the formation ofgas giants ( ), the formation of terrestrial planets has so far, with a few exceptions ( ), mainlybeen explored in classical N -body simulations where terrestrial planets grow by successive im-pacts between increasingly massive protoplanets ( ).Observations of protoplanetary discs reveal that very young stars are orbited by severalhundred Earth masses of pebbles ( ), embedded in a gaseous disc of 100 times higher mass.This pebble population vanishes on a characteristic time-scale of a few million years ( ), likelydue to a combination of radial drift and planetesimal formation outside gaps in the gas causedby the gravity of the growing planets ( ). The orbital speed of the gas in a protoplanetarydisc is set by a balance between the gravity from the central star and the outwards-pointingpressure force. The resulting sub-Keplerian orbital motion of the gas acts as a headwind on thepebbles, draining them of their orbital angular momentum and causing them to drift radiallytowards the star ( ). The inner regions of protoplanetary discs, where terrestrial planets form,therefore witness a flow of hundreds of Earth masses of pebbles throughout the life-time of theprotoplanetary disc.The millimeter-sized chondrules found within primitive meteorites likely represent peb-bles that formed around the young Sun; the leading mechanism for heating and melting thesepebbles to form igneous chondrules is shock waves in the solar protoplanetary disc ( ). Al-ternatively, chondrules may have formed from the molten debris of collisions between massiveprotoplanets ( ). Chondrules formed over the first 3 Myr of the evolution of the protoplanetarydisc (
19, 20 ) and dominated the mass budget when the parent bodies of the ordinary chondrites2eteorite class formed in the inner regions of the solar protoplanetary disc ∼
7, 21 ). The largest planetesimals then continueto accrete pebbles and grow to protoplanets with masses between the Moon and Mars.The radial flux of pebbles towards the star was recently demonstrated to determine the out-come of planetary growth in the inner regions of the protoplanetary disc ( ): high pebble fluxesfrom the outer protoplanetary disc lead to formation of migrating chains of super-Earths, whileany reduction in the pebble flux, e.g. by the emergence of giant planets in the outer Solar Systemwhose gravity on the gas disc acts as a pressure barrier for pebble drift, strands the protoplan-ets at Mars-masses. This was followed by a phase of giant impacts akin to classical terrestrialplanet formation.In this paper we explore the possibility that the radial drift of pebbles continues to drivethe growth from protoplanets to masses comparable to Earth and Venus by pebble accretion.The role of giant impacts for terrestrial planet formation is here reduced to the Moon-formingimpact between Earth and Theia, an additional terrestrial planet that collided with Earth afteran instability in the system of primordial terrestrial planets that orbited our young Sun ( ). Weshow that a pebble accretion scenario for terrestrial planet formation provides explanations forseveral properties of the terrestrial planets in the Solar System, including (1) the masses andorbits of Venus, Earth and Mars, (2) the isotopic composition of Earth and Mars and (3) thedelivery of carbon and water to Earth in amounts that are comparable to the inferred reservoirs.3ur model, depicted as a sketch in Figure 1, provides important constraints into the timingand flux of primitive outer disk material to the inner Solar System. In particular, we concludethat Jupiter did not halt inward mass transport of outer Solar System dust aggregates to the ac-cretion region of terrestrial planets, in contrast to models that invoke Jupiter as an impermeablebarrier that separated the inner and outer Solar System ( ). Results
We develop and present here a model for terrestrial planet formation driven dominantly bypebble accretion, but we include also self-consistently the contribution from planetesimal ac-cretion. We start by identifying the water ice line as the most likely location for the formationof a first generation of planetesimals that acted as seeds for pebble accretion in the solar proto-planetary disc. As the stellar luminosity decreases with time, the ice line moves interior to theregion of terrestrial planet formation. This clearly has implications for the delivery of water toterrestrial planets that form by pebble accretion.
Location of the water ice line in the solar protoplanetary disc
Planetesimal formation by the streaming instability requires that the ratio of the surfacedensity of pebbles relative to that of the gas is elevated above a threshold metallicity, higherthan the nominal solar value, in order to trigger the formation of dense filaments that collapse toform planetesimals ( ). Such an increased pebble density has been demonstrated to occur earlyin the evolution of the protoplanetary disc by pile up of dust released by icy pebbles interior ofthe water ice line ( ) or by the deposition of water vapor transported by diffusion to exterior ofthe ice line ( ). We therefore assume in our model that an early generation of planetesimalsformed at the water ice line and that the planetesimals that grew to form Venus, Earth, Theiaand Mars were among them.The orbital distance of the water ice line depends on the luminosity of the young Sun as wellas on the efficiency of viscous heating by the gas that is accreted. We present here both a model4here viscous heating is provided at all distances from the star as well as a more realistic modelwhere viscous heating is only provided when the magnetorotational instability is active above T = 800 K ( ). This “dead zone” model is described in details in Materials and Methods.We assume that the protoplanetary disc column density evolves as a viscous α -disc from aninitial accretion rate of ˙ M ∗ = 10 − M (cid:12) yr − at t = 0 down to ˙ M ∗ = 10 − M (cid:12) yr − after 5Myr. This decrease in the accretion rate implies that the heating becomes more dominated bythe stellar irradiation with time. We show the calculated temperature of the solar protoplanetarydisc in Figure 2. Additional details of the structure of the protoplanetary disc model with a deadzone are shown in Supplementary Figure S1.The ice line in the more realistic dead zone model sits initially in the region between 1.2and 2.0 AU, whereas the model with viscous heating everywhere has a primordial ice line at7 AU. As the luminosity of the Sun decreases with the contraction of the protostar, the waterice line moves interior of 1 AU after around 1.5–2 Myr in both models. Since the limitingcase where viscous heating is applied everywhere is not physically realistic, as demonstrated inmagnetohydrodynamical simulations ( ), we adopt the dead zone model when integrating thegrowth tracks of planets. Overall we infer from the dead zone model that the first planetesimalsin the inner Solar Systems likely formed in a region devoid of magnetically-driven turbulencebetween 1.2 and 2.0 AU. Analytical growth tracks of terrestrial planets
Protoplanets in the protoplanetary disc will grow at a rate ˙ M given by the accretion ofpebbles and planetesimals and migrate towards the star at a rate ˙ r . The combination of thesetwo rates yields the growth track equation d M/ d r = ˙ M / ˙ r . This equation can be integratedanalytically (
5, 31 ) for the case of growth by 2-D pebble accretion (relevant when the scale-height of the pebbles is smaller than the pebble accretion radius) and standard, inwards type I5lanetary migration to yield the analytical growth track expression M ( r ) = M max (cid:34) − (cid:18) rr (cid:19) − ζ (cid:35) / . (1)We express the growth track here in the limit where M (cid:29) M , where M is the starting mass.The gas temperature of the protoplanetary disc is assumed to depend on the distance from theSun as T ∝ r − ζ and we take a constant value of ζ = 3 / , valid for a protoplanetary discthat intercepts stellar irradiation at a grazing angle ( ). This power law approximation for thetemperature holds in the dead zone of the protoplanetary disc where the terrestrial planets growand migrate. Equation (1) has two free parameters: r is the starting position of the protoplanetand M max is the mass of the protoplanet when it has migrated to r = 0 . We fit now the growthtrack to Mars and Venus, assuming that their current masses reflect their primordial masseswhen the solar protoplanetary disc dissipated. Earth has suffered a giant impact that led tothe formation of the Moon, so the current mass of Earth does not represent its pre-impact mass.Matching the masses and locations of Mars (index 1) and Venus (index 2) in equation (1) allowsus to divide out the M max parameter to yield r ζ − (cid:104) r − ζ − ( M /M ) / r − ζ (cid:105) = 1 − ( M /M ) / . (2)We see how the starting position, r , is dominantly set by the planet with the smallest mass( r → r for M (cid:28) M ). For Mars and Venus we obtain r = 1 . AU, slightly exteriorof Mars’ current orbit at r = 1 . AU. This location is broadly consistent with the likelylocation of the water ice line during the first million years of protoplanetary disc evolution whenconsidering that the terrestrial planet zone was heated mainly by the radiation from the centralstar. The maximum mass is then set by applying equation (1) to Venus, giving M max = 1 . M E .Our successful fit of a single growth track to Mars and Venus is always mathematicallypossible when the outer planet (Mars) is less massive than the inner planet (Venus). How physi-cally plausible this growth track is lies in the value of the maximum mass M max . This maximum6ass, or migration mass, depends on a number of parameters of the protoplanetary disc ( ). Wefocus here on the pebble Stokes number, St , a dimensionless number that characterizes the fric-tional stopping time of the pebbles, and the ratio of the radial flux of pebbles relative to the fluxof the gas, ξ = F p / ˙ M (cid:63) , where ˙ M (cid:63) is the gas accretion rate onto the star. The maximum massfurthermore depends on starting position r , which we fix here to 1.6 AU, and the temperatureprofile for which we take a power law with index ζ = 3 / and temperature T = 140 K at 1 AU;we neglect any temporal dependence of the stellar luminosity. We illustrate the dependence ofthe maximum mass on these parameters in Figure 3. The maximum mass that yields the best fitto the orbit and mass of Venus is obtained for a range of St from 0.001 to 0.1 and a range of ξ from 0.004 to 0.008. These ranges represent nominal values of the Stokes number of millimeter-sized pebbles and radial pebble fluxes ( ). From this we conclude that the masses and orbits ofVenus and Mars are consistent with growth by accretion of millimeter-sized pebbles combinedwith standard type I planetary migration. Numerical growth tracks of terrestrial planets
After the successful application of the analytical growth tracks to Venus and Mars, we nowturn to numerical integration of the masses and orbits of protoplanets undergoing pebble ac-cretion, planetesimal accretion and inwards type I migration, using the code developed andpresented in ( ) and ( ). The protoplanetary disc model is described in Materials and Methods.Planetesimals with fixed radii of 100 km are present from the beginning of the simulation asa Gaussian belt of width 0.05 AU centred at 1.6 AU. The total planetesimal mass is approxi-mately 0.5 M E ; this mass is broadly consistent with models of early planetesimal formation atthe water ice line in protoplanetary discs of solar metallicity ( ).We start protoplanets representing Venus, Earth, Theia and Mars after t = 0 . Myr, . Myr, . Myr and . Myr, respectively, after the formation of the protoplanetary disc. Theprotoplanets are all given initial masses of M = 1 . × − M E . The starting times are chosen7o be consistent with the growth time from an initial mass function of planetesimals peaking at M ∼ − – − M E to our starting mass of M = 10 − M E by planetesimal collisions andpebble accretion ( ); the difference in growth time-scales for the different bodies is essentiallyset by their different levels of eccentricity, which controls the pebble accretion rate, until thebodies are circularized by the transition to rapid pebble accretion at M ∼ − M E ( ), approx-imately the mass of the largest asteroid Ceres. We consider for simplicity pebble accretion onlyin the Hill regime, valid from a factor 10 times lower than our starting mass ( ). We ignore thegravitational interaction between the planets as they grow; in the Supplementary Material wepresent the results of a full N -body simulation showing that the single-planet growth tracks arereproduced when including the gravity between the growing planets.In Figure 4 we show growth tracks starting at r = 1 . . The different starting timesof the protoplanets make a hierarchy of final masses and orbits. We confirm from the numer-ical integrations that Venus and Mars belong to the same growth track, although Mars growsfaster than estimated because of accreting a significant mass fraction (approximately 60%) inplanetesimals from its birth belt. The other planets accrete a similar contribution from planetes-imals, but their total mass budget is dominated by pebble accretion and hence they follow theanalytical growth track better than Mars. Earth obtains a final mass of . M E , but this massis augmented by our assumption that Theia was a fifth terrestrial planet with an original orbitbetween Earth and Mars. We construct Theia to have a mass of . M E . This mass is broadlyconsistent with models for Moon formation that invoke a very massive impactor to vaporize theEarth-Theia collision debris ( ). Isotopic composition of the planets
The mass-independent isotopic compositions of various elements such as O, Ti, Cr and Cahave been measured for Earth as well as for Mars (from martian meteorites) and the major me-teorite classes ( ), and shows that variability exists between the various classes of meteorites8nd terrestrial planets. This nucleosynthetic variability likely reflects either a time-dependentsupply of grains with different nucleosynthetic heritage to the solar protoplanetary disc or, alter-natively, a temperature-dependent destruction of either pre-solar grains or grains condensed inthe local interstellar medium ( ), resulting in a composition dichotomy between the inner,hot regions and the outer, cold regions of the solar protoplanetary disc.Broadly speaking, the isotopic abundances of meteorites fall in two distinct groups whenplotting any two elements against each other. One group is represented by the non-carbonaceous(NC) meteorites, including ordinary chondrites, and the other group by the carbonaceous chon-drites (CC). Earth sits on a mixing line between the two components, together with the enstatitechondrites ( ), establishing that a compositional gradient exists between the NC and CC reser-voirs. Therefore, Earth could have formed either from material akin to enstatite chondrites toyield the measured isotopic fingerprint ( ) or from a combination of material akin to ordinarychondrites mixed with material akin to carbonaceous chondrites (
36, 38 ). The latter picturenevertheless opens the question of how carbonaceous chondrites, which likely formed beyondthe ice line in the solar protoplanetary disc, could have been a major source of material for theterrestrial planets.Based on the mean isotopic abundances of the non-carbonaceous chondrite group on theone hand and of the carbonaceous chondrite group on the other hand ( ), it has been estimatedthat Earth and Mars accreted between 30%–43% and 15%–30% mass from the latter group,respectively, when considering simultaneously the isotopes Cr and Ti. Based on the isotopes Cr and O, Earth and Mars should have accreted between 18%–32% and 2.6%–18% fromthe carbonaceous chondrite group, respectively. Using instead the abundance of Ca in theureilites and CI chondrites as end members for the mixing, Earth requires a contribution of 42%from pebbles of CI composition and Mars 36% ( ). We used Ca (with end members ureilitesand CI chondrites) to match the composition of Earth and Mars in Figure 4, but choosing other9nd members from the non-carbonaceous and the carbonaceous groups would give qualitativelysimilar results.To calculate the contribution of material from the outer regions of the solar protoplanetarydisc to the terrestrial planets formed in our simulations, we assume that the planetesimals in thebirth belt as well as the pebbles (and chondrules) that drift through the disc for the first t CI (avariable time) have the isotopic composition of the ureilites meteorites (an end member of thenon-carbonaceous meteorite group understood to represent the initial inner disc composition( )), followed by an influx of dust-aggregate pebbles of composition similar to CI chondritesfor the last . − t CI , before the protoplanetary disc finally dissipates after 5 Myr. Thetime t CI can be chosen to fit best the measured compositions of Earth and Mars. In panel(C) of Figure 4 we match ( ) in getting approximately 42% contribution of pebbles with CIcomposition to Earth for a transition time t CI = 3 . . In Supplementary Figure S2 we showthat the CI material resided beyond 10 AU after yr of evolution of the protoplanetary disc,having a total gas and dust mass of 0.027 M (cid:12) out of the total disc mass of . M (cid:12) at thisstage, and was subsequently pushed outwards by the viscous expansion of the disc before fallingback through the terrestrial planet region at t CI = 3 . . At this time the gas flux through theprotoplanetary disc is ˙ M ∗ ≈ × − M (cid:12) yr − and the radial pebble flux is F p ≈ . M E Myr − .Venus and Theia obtain very similar compositions to Earth, since all three planets accrete fromthe same drifting material.Mars would get a similar fraction as the other planets of non-carbonaceous relative to car-bonaceous material from pebble accretion; however Mars has an isotopic composition that liescloser to the ordinary chondrites than Earth. This could indicate that Mars terminated its growthearlier than Venus, Earth and Theia and hence avoided the incorporation of pebbles of CI com-position that drifted in later ( ). However, that raises the question of how Mars’ eccentricityand inclination could have been excited during the gaseous disc phase. Instead, we assume here10hat all the four planets accrete planetesimals within their birth belt. This contribution of mate-rial with inner Solar System ureilite-like composition affects Mars the most, since it accretes thelargest fraction of its mass in the planetesimal birth belt. Hence, we can qualitatively explainthe compositional difference between Earth and Mars as a natural consequence of Mars beingless massive than Earth and thus having a larger planetesimal accretion contribution to its mass.The compositions of Earth and Mars are broadly reproduced by a pebble composition transitiontime t CI in the range between 3.5 and 4.0 Myr.The isotopic signature of iron in the mantle of the Earth is very similar to that of iron inCI chondrites ( ). This agreement is consistent with our model. The early-accreted iron fromthe first generation of pebbles came from the NC reservoir and was hence largely reduced, withiron present as metallic iron nuggets as in the ordinary chondrites. This iron accreted togetherwith ice until the envelope reached the ice sublimation temperature at a mass of . M E afterapproximately 2 Myr (see sections on volatile delivery below). The iron arriving together withice would have oxidized in contact with liquid water in the protoplanet, melted by the heating by Al, but this early oxidized iron only constitutes a small fraction of the total oxidized iron foundin the mantle of the Earth today. Material accreted later contained too little Al to melt andhence the iron remained in solid form in a primitive mantle overlying the earlier differentiatedlayers. The bulk of the metal of NC composition separated from the silicates and entered thecore as the protoplanet finally melted by the accretion heat. The second generation of pebblesof CI composition contained a large fraction of oxidised iron that remains in the mantle today.This is in agreement with the abundances of siderophile elements in the Earth’s mantle whichrequire increasingly oxidizing conditions during silicate-metal separation with increasing massof the growing Earth ( ). The oxidised iron in the Earth’s mantle constitutes around 16% ofthe total iron on Earth, but the outer core likely contains a significant fraction of oxidised iron aswell ( ). Given our approximately 40% contribution from pebbles of CI composition to Earth,11hese pebbles should be approximately 60% oxidised in order to account for the iron budgetin the Earth’s mantle. The iron in the CI chondrites is in fact nearly 100% oxidised, but theoxidation of primordial FeS and metallic iron could have occurred by liquid water flow in theparent body ( ). The total mass of the oxidized iron the Earth’s mantle implies furthermorethat the iron in the Earth’s core consists of 25% iron derived from CI-like solids and 75% ironderived from NC material.Jupiter has been proposed to play an important role in the separation of the NC and CCreservoirs in the solar protoplanetary disc ( ). However, hydrodynamical simulations showthat the gap formed by the combined action of Jupiter and Saturn in the protoplanetary disc ispermeable to small dust aggregates ( ), particularly if the gap edge develops turbulence e.g.through the Rossby wave instability ( ). Larger pebbles will become trapped in the peak ofthe gas pressure exterior to the gap – but fragmenting pebble collisions continuously create dustthat can pass through the gap ( ). Hence, in our view Jupiter did not present an efficient barrierto the supply of dust particles from the outer regions of the protoplanetary disc to the terrestrialplanet zone. Jupiter could nevertheless have acted to reduce the flux of pebbles, consistent withthe low pebble flux needed to explain the masses and orbits of Venus, Earth, Theia and Mars inour model. Free parameters in our model
It is remarkable that it is possible to choose a starting position and four starting times of theprotoplanets and match the orbits, masses and compositions of Venus, Earth, Theia and Mars bypebble accretion, including a contribution from planetesimal accretion in the birth belt, usingrealistic parameters for the pebble sizes and the radial pebble flux. Our model nevertheless has anumber of parameters that are to a varying degree free to choose. We briefly discuss the choiceof those here.We distinguish between three classes of parameters: (a) free parameters that are uncon-12trained from observations, (b) free parameters that are constrained from observations and (c)fixed parameters that are constrained by observations and set according to the most reasonablevalue. The truly free parameters in class (a) are t (the four starting times of the protoplanetswhen they have the starting mass M ), r (the starting position of the protoplanets), t CI (timeof infall of pebbles of CI composition) and M pla (the total planetesimal mass in the birth belt)and ξ (the pebble mass flux relative to the gas mass flux, which we fine tune around the solarcomposition to fit the growth tracks). The only free parameter in class (b) is R p (the pebblesize, we set it here to be 1 mm, similar to chondrule sizes and to pebble sizes inferred fromobservations of protoplanetary discs). The fixed parameters in class (c) are ζ = 3 / (the tem-perature gradient in the protoplanetary disc), ˙ M ∗ = 10 − M (cid:12) yr − and ˙ M ∗ = 10 − M (cid:12) yr − (the initial and final accretion rate of the protoplanetary disc), t disc = 5 Myr (the life-time of theprotoplanetary disc), α = 10 − (the turbulent viscosity in the protoplanetary disc) and δ = 10 − (the diffusion coefficient of the gas, which governs the degree to which the pebbles sediment tothe mid-plane). We chose reasonable values, based on observations of protoplanetary discs, forall parameters in class (c) and did not attempt to vary them in order to obtain better fits to theobserved properties of the terrestrial planets.The combined set of parameters describes a model that has as outcome the masses, orbitsand compositions of the planets Venus, Earth, Theia and Mars. Here Theia is a special case,since its mass is only constrained to add up with the mass of proto-Earth to obtain the currentmass of the Earth. We therefore consider the real prediction of our model to be M V (the massof Venus), M M (the mass of Mars), M E + M T (the total mass of proto-Earth and Theia), r V (theorbit of Venus), r M (the orbit of Mars), f CI , E (the composition of Earth), f CI , T (the compositionof Theia, must be equal to that of Earth) and f CI , M (the composition of Mars). These eightmodel predictions are constructed by essentially choosing three starting times of the protoplan-ets (considering the starting times of Earth and Theia to be degenerate), the starting position of13he protoplanets, the radial the pebble flux relative to the gas flux, the time of infall of pebblesof CI composition and the total mass of planetesimals in the birth belt. This gives a total ofseven truly free parameters that are used to fit the data. The fact that the number of parame-ters is smaller than the number of successful predictions implies that the model has a genuineexplanation power for the properties of the terrestrial planets in the Solar System. Monte Carlo populations
We now explore the effect of varying the birth positions and times of the protoplanets as wellas the width of the planetesimal belt. We start 1,000 protoplanets with masses M = 10 − M E and give them random starting times between 0.5 Myr and 5.0 Myr. The positions are drawnfrom a Gaussian belt centred at r = 1 . and with a width of either W = 0 .
05 AU or W = 0 . . The growth tracks of Figure 4 were integrated using the narrow belt. The widerbelt has ten times more mass in planetesimals than the narrow belt, so a total of approximately M E .In Figure 5 we show the final positions and masses of the considered protoplanets. Wedemonstrate results for two values of the pebble flux: ξ = 0 . as in Figure 4 and a highervalue of ξ = 0 . . The narrow belt case faithfully reproduces the characteristic masses andorbits of Venus and Mars. Widening the planetesimal belt to 0.5 AU increases the maximallyreached planetary masses to between M E and M E (super-Earths). The mass hierarchy withincreasing planetary mass with decreasing distance from the star is maintained as in the narrowbelt case, but the terrestrial planets no longer occupy characteristic positions in mass versussemi-major axis.The lower panel of Figure 5 shows the results when increasing the pebble-to-gas flux ratioto ξ = 0 . . This is closer to the nominal flux in the absence of drift barriers, such as planets,further out in the disc. The narrow belt case now fits Earth better, but Venus is well below thecharacteristic growth track and planets at the location of Mars now reach much higher masses14efore they start to migrate towards the star. Super-Earths of masses up to M E pile up atthe inner edge of the protoplanetary disc. The decisive role of the pebble flux in determiningwhether protoplanets in the inner regions of a protoplanetary disc grow to super-Earths or “only”to Mars masses was already discovered in ( ); here we demonstrate that the combination ofa low pebble flux and a narrow planetesimal belt leads to the formation of analogues of theterrestrial planets in the Solar System in terms of both their masses and orbits. Sublimation of volatiles in the planetary envelope
The accreted pebbles will carry molecules that are in solid form at the temperature levelof the surrounding protoplanetary disc. Since the water ice line is interior of the accretionregion of Earth and Venus during the main accretion phase of the terrestrial planets, this impliesthat the pebbles will be rich in volatiles. However, the growing planets attract gas envelopesthat are heated by the high pebble accretion luminosity and this heat will in turn process thepebbles thermally while they are falling towards the surface of the protoplanet. Our methods forcalculating the temperature and density of the envelope in hydrostatic and energy equilibriumare described in Materials and Methods.In Figure 6 we show the temperature and density of the envelopes of our Earth and our Marsanalogues at t = 5 Myr as a function of the distance from the center of protoplanet, for three dif-ferent values of the opacity κ (see Materials and Methods). The density of the hydrogen/heliumenvelope rises exponentially inwards from the Bondi radius (defined as R B = GM/c where c s is the sound speed of the gas in the protoplanetary disc). For Earth, the envelope temperaturelies above the 2,600 K of silicate sublimation close to the planetary core. This high surfacetemperature will lead to extensive melting of the protoplanet and to the formation of a magmaocean of at least 1,000 km in depth ( ). Our Mars analogue does not reach silicate meltingtemperatures at the surface, but trapped heat released by Al within the protoplanet could heatthe interior above the melting temperature. 15e indicate in Figure 6 the location of the water ice line with blue pluses labelled with theentropy, relative to the entropy level of the surrounding protoplanetary disc, at the depth of thewater ice line. Hydrodynamical simulations have demonstrated that flows from the protoplane-tary disc reach down to a relative entropy level of s/s ≈ . ( ). Hence, for the high-opacitycases, with κ = 0 . kg − and . kg − , the water vapor that is released by sublimatingthe ice layers from the pebbles will be recycled back to the protoplanetary disc. Such a highopacity is easily reached for a dust-to-gas ratio around the solar value or higher.The recycling flows will first transport water vapor back across the ice line, where the vapornucleates on tiny dust particles that carry the dominant surface area. Whether these ice-coveredparticles eventually reach the protoplanetary disc depends on the time-scale to coagulate tolarge enough sizes to sediment back through the gas flow. The coagulation time-scale of the iceparticles can be calculated from the approximate expression τ c ∼ a ice ρ • (∆ v ) ρ ice . (3)Here a ice is the radius of the particles, ρ • is their material density, ∆ v is the particle collisionspeed and ρ ice is the spatial density of the nucleated ice particles. We then assume that thecollision speed is dominated by brownian motion. That yields τ c ∼ a ice ρ • (cid:112) k B T /mρ g (cid:15) − ∼ . (cid:18) a ice µ m (cid:19) / (cid:18) T
170 K (cid:19) − / (cid:18) ρ g − kg m − (cid:19) − (cid:16) (cid:15) . (cid:17) − . (4)We defined here (cid:15) = ρ ice /ρ g as the ratio of the density of ice particles to the gas. The coagulationtime-scale should now be compared to the time-scale for the recycling flow to transport theparticles from the ice line at R ice out to the Hill radius with the characteristic speed v rec ∼ ΩR ice .That gives τ rec ∼ ( R H /R ice ) Ω − ∼ Ω − ∼ . . (5)This time-scale is comparable to the coagulation time-scale around the ice line, but the coagu-lation time-scale increases steeply in the decreasing gas density exterior of the ice line. Hence,16he freshly nucleated ice particles do not coagulate appreciably during their transport from theice line to the Hill radius with the recycling flows. Water delivery by pebble accretion
Figure 7 shows the water fraction of our growing Earth analogue, assuming that water canonly be delivered to the protoplanet in the form of ice accreted before the envelope reaches thetemperature to sublimate water ice. We assume that the accreted planetesimals are dry (sincethey formed early enough to dry out due to the heat from Al, see reference ( )) and that theaccreted pebbles carry either the nominal 35% ice by mass, which assumes that some oxygenis bound in the volatile CO molecule ( ), or a much dryer 10%. The water fraction peaksat a protoplanet mass just below . M E (approximately the mass of the Moon) and then fallssteeply for higher masses as water ice is sublimated in the hot envelope. The final water fractionof our Earth analogue is 3,000 ppm for the nominal ice fraction (35%) and 1,000 ppm for thelow ice fraction (10%). The nominal case lands thus at the high end of the 2,000-3,000 ppmestimated for Earth ( ). Theia with a final mass of . M E acquires 4,000 ppm of water andwould increase the total water fraction of Earth to 3,500 ppm after the giant impact, but thisvalue could be reduced by loss of volatiles in energetic the impact.All the water is delivered as ice in the early stages of protoplanet growth when the proto-planet is still solid. The protoplanet will melt later by accretion heat and radioactive decay andhence the magma ocean inherits a significant fraction of water. Parts of this water will laterbe released from the magma during crystallisation ( ) and likely mixed with water formedfrom hydrogen ingassed from the protoplanetary disc ( ). Any water that is outgassed duringthe growth of the protoplanet will be retained as a high-density primary atmosphere below thehydrogen/helium envelope, protected against loss to diffusive convection by the gradient in themean molecular weight between the atmosphere and the envelope. Our model thus shows thatwater could have been delivered to Earth from “pebble snow” accreted during the early growth17tages when the protoplanet was still solid and the envelope cold enough for water ice to survivethe passage to the surface. Carbon delivery by pebble accretion
In Figure 6 we also indicate the temperatures where organic compounds are vaporised. Wetake the range between 325 K and 425 K for the vaporisation of 90% of the carbon by pyrolysisand sublimation ( ). Pyrolysis and sublimation of organics converts these molecules to veryvolatile carbon-bearing molecules such as CH , CO and CO. These species will not condensein the envelope and can diffuse freely to the recycling zone. We assume that protoplanetary discflows penetrate into the Hill sphere with the characteristic speed ΩR B at the Bondi radius R B .This results in a mass loss rate of carbon-bearing molecules as ˙ M C = 4 πR ρ C ΩR B . (6)We now equate this expression with the mass accretion rate of carbon, f C ˙ M , and isolate thedensity of carbon-bearing molecules, ρ C , at the Bondi radius to be ρ C = f C ˙ M πR Ω = 1 . × − kg m − (cid:18) f C . (cid:19) (cid:32) ˙ M . M E Myr − (cid:33) (cid:18) M . M E (cid:19) − × (cid:18) c s . × m s − (cid:19) (cid:16) r AU (cid:17) / . (7)This density is much lower than the gas density in the protoplanetary disc, ρ g ∼ − kg m − ,and hence the equilibrium ratio of volatile carbon molecules relative to the gas is much lowerthan the ambient density of CO, the main carbon-bearing molecule in inner regions of the pro-toplanetary disc. The equilibrium mixing ratio at the Bondi radius, ρ C /ρ g , will be maintainedin the entire envelope by turbulent diffusion with coefficient D . This diffusion equilibrium ismaintained after a characteristic time-scale of τ = R D . (8)18he characteristic value of the gas mass in the envelope of terrestrial planets that grow bypebble accretion is M g ∼ − M E . The time-scale to match this gas mass with carbon-bearingmolecules released in the organics sublimation/pyrolysis region is τ C = M g f C ˙ M = 1 × yr (cid:18) M g − M E (cid:19) (cid:18) f C . (cid:19) − (cid:32) ˙ M . M E Myr − (cid:33) . (9)The turbulent diffusion time-scale must be shorter than this value to avoid building up a layerin the envelope dominated by carbon-bearing molecules, which would lead to an increase in themean molecular weight of the gas and hence potentially protect the carbon-bearing moleculesfrom turbulent diffusion. Setting the turbulent diffusion coefficient D = v turb × R B we get aturbulent speed of v turb = R B τ = 1 . × − m s − (cid:18) M . M E (cid:19) (cid:18) c s . × m s − (cid:19) − (cid:18) τ × yr (cid:19) − . (10)This is an extremely modest speed compared to the sound speed and hence we conclude thatthe carbon-bearing species released by vaporisation of organics will become transported to therecycling flows of the protoplanetary disc before they can build up a substantial mass fractionin the envelope.Only approximately 10% of the carbon provided by the pebbles will therefore make it belowthe region where organics are vaporised. The remaining pure carbon dust burns at 1,100 K ( ).This temperature is reached approximately 10,000 km above the surface of our Earth analogue.Assuming that the carbon is only retained if it makes it to the protoplanet’s surface, we estimatethe total carbon delivery to our Earth analogue as follows. We assume that the carbon contentsof the early generation of pebbles is similar to ordinary chondrites (approximately 0.3%) andsimilar to CI chondrites for the late generation (approximately 5%). The result is shown inFigure 7. The fraction of carbon is initially constant at 3,000 ppm (set by the carbon contentsof ordinary chondrites), but falls steadily after carbon vaporisation starts in the envelope. OurEarth analogue ends up with 600 ppm of carbon.19he total amount of carbon in our planet is relatively poorly constrained. Earth’s bulk carbonmay reside in the core, adding up to a total of (3 − × kg or 500-1,000 ppm ( ). Thisis within range of our estimate for the carbon delivery integrated over the growth of the Earth.Our Mars analogue does not reach high enough temperatures to burn carbon dust in its envelopeand therefore obtains a carbon fraction as high as 2,000 ppm. The more massive planets (Venus,Earth and Theia) accrete a substantial amount of carbon in their earlier growth phases, beforeentering a phase of burning carbon dust in the envelope after growing beyond . M E . Thiscarbon-free accretion phase lowers the resulting carbon fraction by approximately a factor ofthree. Discussion
We built our model for terrestrial planet formation by pebble accretion around fundamentalphysical processes that have been explored within the context of modern planet formation the-ory. The water ice line has been demonstrated as a likely site for early planetesimal formation;the location of the primordial water ice line in the 1–2 AU region is a direct consequence ofthe elevated luminosity of the young Sun, compared to its current value, and fits broadly withthe location of Mars. Taking Mars as a planet that underwent only modest growth by pebbleaccretion as one end member and Venus that experienced substantial growth and migration asthe other end member, we demonstrated that the masses and orbits of these two planets areconsistent with growth by accretion of millimeter-sized pebbles in a protoplanetary disc with arelatively low radial flux of solid material. The radial pebble flux needed to match the massesand orbits of the terrestrial planets corresponds to approximately M E Myr − at a stellar ac-cretion rate of − M (cid:12) yr − (after 1 Myr of evolution in our model) and M E Myr − whenthe protoplanetary disc dissipates at an accretion rate of − M (cid:12) yr − after 5 Myr. These lowpebble fluxes correspond well to the estimated − M E of solids left in protoplanetary discsat the 3–5 Myr evolution stages ( ). Although such evolved protoplanetary discs may not be20ble to form gas-giant planets any longer, they could presently be in their main phase of formingterrestrial planets by pebble accretion.The mass, orbit and composition of Mercury do not fit within this picture. Instead, wepropose that Mercury formed by accretion of metallic pebbles and planetesimals outside of thesublimation front of Fayalite (Fe SiO ), which we demonstrate in Figure 2 crosses Mercury’scurrent orbit after 0.5 Myr of disc evolution. This process is similar to how icy pebbles triggerthe streaming instability exterior of the water ice line ( ). Growth by pebble and planetesimalaccretion is very rapid so close to the Sun, so the current mass of Mercury is reached within afew hundred thousand years. Mercury then ceased to accrete as the inner regions of the solarprotoplanetary disc underwent depletion by disc winds ( ), stranding Mercury with a dominantmetallic core and a mantle whose silicates were reduced by accretion of pebbles rich in sulphurreleased at the Troilite (FeS) sublimation front ( ). Mercury could thus be the oldest of theterrestrial planets, having compiled its bulk mass within a million years of the formation of theSun.Within our model framework, Earth reaches 60% of its current mass at the dissipation ofthe protoplanetary disc after 5 Myr. This mass of the proto-Earth agrees well with constraintsfrom matching the Ar/ Ar, Ne/ Ne and Ar/ Ne ratios by drag from the hydrodynamicescape of the primordial hydrogen/helium envelope ( ). We augment the mass of Earth withthe introduction of an additional planet Theia (40% Earth mass) between the orbits of Earth andMars, to later collide with the Earth to form our Moon. The radial drift of pebbles naturally pro-vides Earth and Theia with very similar compositions, even when the isotopic signature of thedrifting pebbles changes with time. Hence, the Earth and the Moon inherit the indistinguishableisotopic compositions of their parent planets.Models of protoplanetary discs show that the water ice line inevitably passes interior to 1AU as the stellar luminosity decreases with time ( ). This implies that protoplanets growing21n the terrestrial planet zone will accrete a significant mass fraction of ice. However, ice issublimated in the hot gas envelope once the protoplanet mass reaches . M E and the waterthen escapes back to the protoplanetary disc with the recycling flows that penetrate into the Hillsphere. Volatile delivery filtered through a hot envelope gives a good match to the water contentsof Earth. The early-accreted water ice becomes incorporated into the magma ocean after theEarth melts by the accretion heat and the water will later degas as a dense vapor atmosphereupon crystallisation of the Earth ( ), forming the first surface water masses on our planet.The water on our Earth analogue is accreted as ice in the first 2 Myr of the evolution ofthe protoplanetary disc, before the planetary envelope becomes hot enough to sublimate the icefrom the accreted pebbles. Since water is delivered with pebbles from the early generation toVenus, Theia and Mars as well, we predict that the primordial water delivered to Mars shouldhave the same D/H ratio as water on Earth. The early generation of pebbles in our model isrepresented by the non-carbonaceous meteorite group, which includes the ordinary chondrites.Water in the ordinary chondrites is heavier than Earth’s water ( ), although a recent studyfound that enstatite chondrites have a D/H ratio similar to water in the Earth’s mantle ( ). TheD/H ratio of the ordinary chondrites could have been increased significantly by oxidation ofiron on the parent body, a process that releases light hydrogen ( ). The relatively dry parentbodies of the ordinary chondrites could have formed at the warm side of the water ice line after3 Myr, when the ice line was interior of the current orbit of Venus. Some of these would laterhave been dynamically implanted into the asteroid belt by scattering, now seen as the S-typeasteroids in the inner regions of the asteroid belt ( ).Carbon, in the form of organics and carbon dust, is more refractory than water and canreach the planetary surface for envelope temperatures up to 1,100 K (where carbon dust burns).Carbon is thus delivered to the growing protoplanets at masses up to . M E , after which theenvelope becomes hot enough to vaporize all the incoming carbon-bearing molecules. Our Earth22nalogue accretes carbon at up to a time of 3.5 Myr. As with the water, the isotopic compositionof the carbon will therefore reflect material in the inner regions of the solar protoplanetary disc.The overall carbon delivery filtered through the planetary envelopes gives a good match to theestimated carbon reservoirs on Earth, including a carbon component residing in the core.Hence both water and carbon – essential ingredients for life – may have been delivered toEarth in the form of “pebble snow” in the early phases of the growth of our planet. This impliesthat volatile delivery need not be a stochastic effect of a few giant impacts ( ) but may insteadfollow predictable patterns that can be calculated from the evolution of the protoplanetary discand the radial drift of pebbles. If millimeter-sized pebbles are the main providers of volatilesto planets in the inner regions of protoplanetary discs, then we predict that super-Earths (moremassive counterparts of terrestrial planets found frequently around solar-like stars) would havethe same overall volatile budgets as terrestrial planets, being unable to accrete any additionalvolatiles due to the intense heat in their gas envelopes. The accretion of dense atmospheres ofhydrogen and helium poses an additional challenge to the habitability of super-Earths.We believe that our pebble-driven model of terrestrial planet formation has several advan-tages over the classical models that employ collisions between planetary embryos as the maindriver of planetary growth. While the latter successfully produce planetary systems similar tothe Solar System’s terrestrial planets in terms of masses and orbits, incorporating outer SolarSystem material into the growing planets by giant impacts is very challenging. Simulations ofwater-delivery to terrestrial planets employ collisions with wet protoplanets originating beyond2.5 AU – the assumed ice line based on the dichotomy between the S-type asteroids, associ-ated with the ordinary chondrites, that dominate the inner regions of the asteroid belt and theC-type asteroids, associated with the carbonaceous chondrites, that dominate the outer regions.This position of the ice line is in clear conflict with models of irradiated protoplanetary discsshowing that the ice line was likely situated interior of 1 AU for most of the life-time of the23isc. Notwithstanding this discrepancy, the classical models provide the terrestrial planets witha water fraction between 0.1% and 1%. This fraction can then be considered the maximumcontribution from volatile-rich outer Solar System material in the classical model – and henceit stands in contrast to the 40% contribution needed to explain the isotopic composition of theEarth based on Ca ( ) and particularly to the match of the iron isotopic composition of theEarth’s mantle with the CI chondrites ( ). Models invoking the inwards migration of Jupiterthrough the asteroid belt to explain the small mass of Mars yield similarly low fractions of outerSolar System volatile-rich material in the terrestrial planets ( ).As an alternative view, the Earth could have formed from a local reservoir of material thatcarried the exact same elemental and isotopic composition as measured on Earth. The enstatitechondrites have been proposed as candidate material matching Earth’s isotopic composition,although the iron isotopes stand out as an exception ( ). Such an approach nevertheless neces-sitates that all the terrestrial planets formed from local reservoirs of distinct composition, per-haps a remnant of the condensation sequence in the initially hot solar protoplanetary disc ( )although this view conflicts with the volatile-rich mantle of Mercury ( ). However, explainingthe isotopic differences between the terrestrial planets by local reservoirs highlights the problemof the similarity of the Earth and the Moon, as Theia would thus be expected to have a distinctcomposition different from that of Earth. In the pebble-driven model, Earth and Theia naturallyget very similar compositions, since they accrete from the same radial flux of drifting pebbles. Materials and Methods
Protoplanetary disc model
We consider a protoplanetary disc model where the temperature is set by both irradiationfrom the host star and by viscous heating from the magnetorotational instability. The latter isactive when the temperature is above 800 K ( ). We therefore increase the turbulent viscositythat controls viscous heating from a background level of α v = 10 − to α v = 10 − , relevant for24ully active turbulence caused by the magnetorotational instability, over the temperature rangefrom 800 K to 1,000 K. We calculate the temperature structure iteratively by solving the tran-scendental equation T = T ( α v ) ( ) where α v is a function of T . We follow the database ofI. Baraffe ( ) to calculate the luminosity evolution of the young Sun. We fit the Sun’s luminos-ity evolution as L ( t ) = 2 L (cid:12) ( t/ Myr) − . for the times 0.5-5 Myr, where L (cid:12) is the luminosityof the modern Sun. We then multiply this value by a factor exp {− / [ t/ (0 .
15 Myr)] } to repre-sent the increase in stellar luminosity (including emission from accretion and from deuteriumburning, but not including the luminosity of the disc itself) in the early stages of the stellarcollapse phase ( ); this effectively gives a peak in the protostellar luminosity of L ≈ L (cid:12) at t = 0 . − . Myr, followed by a power law decline ending at L ≈ . L (cid:12) at t = 5 Myr. Theprotoplanetary disc evolves via the α -viscosity prescription and the pebble mass flux throughthe disc is set to a fixed fraction ξ of the radial gas flux; this assumption of a constant flux ratiois justified by growth bottlenecks in the viscously expanding outer regions of the protoplanetarydisc ( ). The temporal evolution of the structure of the protoplanetary disc is shown in FigureS1. The pebble column density Σ p follows from setting the pebble flux, F p = 2 πrv r Σ p , equalto ξ ˙ M ∗ where ˙ M (cid:63) is the gas accretion rate through the disc. Here v r is the radial drift speedof the pebbles, which is calculated self-consistently from the temperature and pressure gradi-ent of the gas in the protoplanetary disc. The pebbles have a fixed size of 1 millimeter; thischoice of pebble size is motivated by the characteristic size of chondrules, particle growth tothe bouncing barrier ( ) and observations of protoplanetary discs ( ). We have checked thatthe fragmentation-limited particle size is larger than 1 mm at 1 AU for all the evolution stagesof the adopted protoplanetary disc, for a turbulence strength of δ = 10 − and a critical fragmen-tation speed of 1 m/s ( ). The pebble column density for ξ = 0 . is indicated in Figure S1.The pebbles have Stokes numbers in the range 0.001-0.01 (see Figure 3); for these small sizes25he drift speed is low enough that ξ approximately represents the ratio of the column density ofpebbles relative to the gas. The pebble accretion radius is calculated based on the expressionsprovided in ( ) and are integrated in either 3-D (when the pebble scale-height is larger thanthe accretion radius) or 2-D (when the pebble scale-height is smaller than the accretion radius).To keep the model simple, we do not decrease the pebble accretion rate when the ice layers aresublimated in the envelope at protoplanet masses above . M E . Calculating the equilibrium structure of the envelope
We solve the structure of the planetary envelope by integrating the standard equations ofhydrostatic balance and outwards luminosity transport from the Hill sphere down to the centerof the protoplanet. The boundary conditions at the Hill sphere are the temperature and densityof the protoplanetary disc. We set the temperature gradient to the minimum of the radiativetemperature gradient and the convective temperature gradient. Our consideration of the equilib-rium structure of the envelope ignores the time needed to heat the material in the envelope to itsequilibrium value. The accretion luminosity is generally high enough to validate this approach.The radiative temperature gradient is defined by the opacity; we assume here that the opacityfollows a power law with the temperature, κ = κ ( T /
100 K) . with κ = 0 . , . , or . kg − ( ). The mean molecular weight of the gas is set to a weighted average of a solarmixture of hydrogen/helium and silicate vapor at its saturated vapor pressure.The total luminosity generated by the accretion, the latent heat of sublimation of silicatesand the decay of Al is L = GM ˙ MR sil − Q sil ˙ M + L . (11)Here R sil is the either the location of the silicate sublimation front (which is calculated self-consistently by equating the saturated vapor pressure of silicates with the ambient pressure) orthe surface of the planet (if the temperature does not reach silicate sublimation). The latent heatof sublimation of silicate rock ( Q sil ) reduces the accretion energy when the temperature reaches26ilicate sublimation in the envelope. We nevertheless ignore the latent heat of water and silicatesin the calculations, since silicate sublimation only happens at planetary masses approaching M E while water sublimation is followed immediately by nucleation of the outwards transportwater vapour. The luminosity generated by the decay of Al is set to L = r E M . (12)Here r = 1 . × kg − s − exp( − t/τ ) is the decay rate of Al per total silicate mass ( ), t is the time, τ = 1 .
03 Myr is the decay constant, E = 3 .
12 MeV is the energy released perdecay ( ) and M is the mass of the silicate core. We assume that all the heat from Al isreleased in the core, since the total mass of silicate vapor in the envelope is much lower than thecore mass for the planetary masses that we consider.
Supplementary Materials
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Acknowledgments
General
This project started after discussions held at the Europlanet and International Space Science In-35titute Workshop “Reading Terrestrial Planet Evolution in Isotopes and Element Measurements”held in Bern in October 2018.
Funding
A.J. acknowledges funding from the European Research Foundation (ERC Consolidator Grant724687-PLANETESYS), the Knut and Alice Wallenberg Foundation (Wallenberg AcademyFellow Grant 2017.0287) and the Swedish Research Council (Project Grant 2018-04867). M.B.acknowledges funding from the Carlsberg Foundation (CF18 1105) and the European ResearchCouncil (ERC Advanced Grant 833275-DEEPTIME). M.S. acknowledges funding from theVillum Foundation (grant number
Author contributions
The original idea for the project resulted from discussions between all of the co-authors, withequal contributions. A.J. performed the computer simulations in the main paper, analysedthe data and wrote the manuscript. T.R. developed and performed the N -body simulationspresented in the Supplementary Material. All authors contributed to finalising the originalmanuscript written by A.J. Competing interests
The authors declare that they have no competing interests
Data and materials availability
All data needed to evaluate the conclusions in the paper are present in the paper and/or theSupplementary Materials. Additional data related to this paper may be requested from theauthors. 36 igures and Tables
Figure 1:
Sketch of the physical processes involved in our pebble accretion model for theformation of terrestrial planets.
Stage (A): the protoplanetary disc is formed consisting ofmaterial with solar composition (blue), represented in the meteoric record by the CI meteorites.Thermal processing in the inner disc vaporizes presolar grains carrying isotopic anomalies. Theremaining solids carry now a non-carbonaceous (NC) signature (red). In Stage (B) the discexpands outwards due to angular momentum transport from the inner to the outer disc, carryingthe NC material along with the gas. Planetesimal belts form at the water ice line (red) and bypile ups of pebbles in the outer regions of the protoplanetary disc (blue); this outer planetesimalbelt is envisioned here as the birth region of the giant planets (
14, 72 ). In Stage (C) protoplanetsrepresenting Earth, Venus and Theia migrate out of the inner planetesimal belt. In the outerSolar System Jupiter, Saturn, Uranus and Neptune grow by pebble accretion and gas accretion.In Stage (D) the CI material has drifted past the terrestrial planet zone and the terrestrial planetsshift their compositions more towards the CI meteorites. The carbonaceous chondrites (CC)form outside of the orbits of Jupiter and Saturn. Finally, in stage (E) the protoplanetary discclears and planetesimals of NC and CC composition are scattered into the asteroid belt.37 .1 1.0 10.0 r [AU]0.11.0 t [ M y r] F e S F e S F a F a H O Q Viscous heatingeverywhere1.01.52.02.53.03.54.0 l og ( T / K ) r [AU]0.11.0 t [ M y r] F e S F a F a H O Q Dead zone1.01.52.02.53.03.54.0 l og ( T / K ) Figure 2:
Temperature maps of the inner 10 AU of an evolving protoplanetary disc . The leftplot shows the temperature when viscous heating is applied everywhere in the protoplanetarydisc. In the right plot we assume that viscous heat is only released when the magnetorotationalinstability is active above a temperature of 800 K – the remaining disc is magnetically dead andhence only heated by the stellar irradiation. Three contour lines for the Toomre Q parameterare indicated in yellow; values above ≈ O)and the refractory minerals Troilite (FeS) and Fayalite (Fe SiO ). In the more realistic casewhere viscous heating is provided only where the magnetorotational instability is active (rightplot), the primordial water ice line sits in the region between 1.2 AU and 2.0 AU in the firstmillion years of disc evolution. This is the likely site for formation of the first generation ofplanetesimals in the inner Solar System. 38 t [Myr]050100150200250300 T [ K ] A −> Heating dominated bystellar irradiation Ice line1 AU M max [ M E ] ξ = F p / M * . . M E . M E . M E B St @ 2 3 4 5 Myr Figure 3: (A) The temperature at 1 AU distance from the star a function of time and thetemperature of the water ice line.
The temperature becomes dominated by stellar irradiationalready after 0.3 Myr of disc evolution; the temperature then continues to drop more slowly asthe stellar luminosity falls with time. The temperature falls below water vapour saturation afterapproximately 2 Myr of disc evolution. (B) The maximum planetary mass for growth tracksstarting at r = 1 . AU.
The mass is shown as a function of the pebble Stokes number, St ,and the ratio of the radial pebble flux rate through the protoplanetary disc relative to the gasflux rate, ξ = F p / ˙ M (cid:63) . We give the temperature a passive irradiation profile with fixed value of140 K at 1 AU. The Stokes number at 1 AU for millimeter-sized pebbles are indicated at fourdifferent times. The maximum mass that leads to a good match for Venus’ orbit and mass isindicated with a dashed line. This is obtained for a range of Stokes numbers between 0.001and 0.1, consistent with the evolution of the Stokes number of millimeter-sized pebbles fromthe early, dense protoplanetary disc to the late, dilute disc (pluses). The pebble-to-gas flux ratiolies in the range between 0.004 to 0.008; these values are similar to the solar ratio of refractorymaterial relative to hydrogen/helium gas. 39 .0 0.5 1.0 1.5 2.0 r [AU]0.010.101.00 M [ M E ] A t [Myr]0.0010.0100.1001.000 M [ M E ] B Total massPlanetesimal contribution ME,TV t CI [Myr]0.00.20.40.60.81.0 f C I VenusEarthTheiaMarsUr−CI C Figure 4: (A) Numerical growth tracks of protoplanets growing by pebble accretion andplanetesimal accretion, starting at r = 1 . . The parameters of the growth track werechosen with Venus and Mars as end members. Earth obtains a final mass of . M E , which weaugment by creating Theia slightly later to reach a final mass of . M E . (B) Masses of the fourplanets as a function of time . We show here also the contribution from planetesimal accretionin the birth belt. (C) The fraction of mass accreted from outer Solar System material , as afunction of the time when the drifting pebbles change to CI composition. We find agreementwith the estimated CI contribution of 42% to Earth and 36% to Mars ( ) for a transition timefrom ureilite composition to CI composition in the range 3.5–4.0 Myr.40 .0 0.5 1.0 1.5 2.0 r [AU]0.0010.0100.1001.00010.000 M [ M E ] A ξ = 0.0036 r [AU]0.0010.0100.1001.00010.000 M [ M E ] B ξ = 0.01 W = 0.05 AU W = 0.5 AU Figure 5:
Monte Carlo sampling of 1,000 protoplanets starting with initial masses of M =10 − M E at random times between 0.5 Myr and 5 Myr. The black dots show results fora planetesimal belt of width W = 0 .
05 AU while the green dots show results of a width of W = 0 . . The protoplanets are started at random positions within the planetesimal belt.(A) Results for a pebble-to-gas flux ratio of ξ = 0 . , as in Figure 4. The narrow beltcase faithfully reproduces the characteristic masses and orbits of Venus and Mars. Consideringinstead a wider planetesimal belt leads to the formation of massive planets that migrate to theinner edge of the protoplanetary disc. (B) Results for a higher pebble-to-gas flux ratio of ξ =0 . . Venus is now well below the growth track, while planets in the Mars region grow to ≈ . M E before migration becomes significant. The high pebble flux allows super-Earths of upto M E to grow and migrate to the inner edge of the protoplanetary disc.41 arth T [ K ] A Silicate vaporH /He gas B ond i r a d i u s H ill r a d i u s r [m]10 −8 −6 −4 −2 ρ [ kg m − ] κ = 0.01, s ice / s = 0.09 κ = 0.1, s ice / s = 0.22 κ = 1.0, s ice / s = 0.54 Ice sublimationOrganics pyrolysis/sublimation
Earth
Mars T [ K ] B Silicate vaporH /He gas B ond i r a d i u s H ill r a d i u s r [m]10 −8 −6 −4 −2 ρ [ kg m − ] κ = 0.01, s ice / s = 0.09 κ = 0.1, s ice / s = 0.22 κ = 1.0, s ice / s = 0.54 Ice sublimationOrganics pyrolysis/sublimation
Mars
Figure 6: (A) Envelope structure of our Earth analogue and (B) our Mars analogue im-mediately prior to the dissipation of the protoplanetary disc at t = 5 Myr . The top panelsshow the gas temperature and the bottom panels the gas density. Three different opacity levelsare considered: κ = 0 . , . , . m kg − . The temperature of the Earth’s envelope directlyover the magma ocean core reaches 2,000–3,000 K. The saturated vapor pressure of silicates(assumed here to be Forsterite) dominates over the ambient pressure only in a tiny region abovethe surface that reaches temperatures above approximately 2,600 K. We mark the relative en-tropy level at the water ice line. For the high-opacity case the relative entropy approaches50% of the disc entropy; flows from the protoplanetary disc easily reach this entropy level andcleanse the isothermal region of water vapor and ice particles. Mars remains colder than 1,000K throughout the envelope, and avoids melting, but the water ice line lies at a similar entropylevel to Earth. The envelopes of both Earth and Mars reach temperatures in the range 325–425K slightly below the water ice lines; here organic molecules undergo pyrolysis and sublimationto form volatile gas species (CH , CO and CO ). These species can not recondense and willfreely diffuse back to the protoplanetary disc. 42 .001 0.010 0.100 1.000 M [ M E ]0.00010.00100.01000.1000 M W / M , M C / M CarbonWater (10%)Water (35%)
Figure 7:
Fraction of water and carbon that survives the passage through the planetaryenvelope.
The fraction is shown as a function of the mass of the growing Earth. The resultsare shown for two different values of the ice fraction of the pebbles. The estimated water andcarbon fractions of Earth are indicated with dotted lines. Water is delivered by icy pebbles untilthe temperature in the envelope reaches ice sublimation (assumed here to be at 160 K). Thewater fraction falls steeply after the protoplanet reaches . M E . Carbon is vaporised in twosteps – organics vaporise in the temperature range 325-425 K and carbon dust burns at 1,100K. Planetesimals in the birth belt as well as the early-accreted pebbles define an initial carbonfraction of 3,000 ppm. The infall of pebbles of CI composition (5% carbon, assumed here tooccur at t CI = 3 . ) when Earth reaches a mass of . M E coincides with the increasedheating of the planetary envelope, so that the overall carbon fraction actually decreases withincreasing mass. The final value lands around 600 ppm, similar to estimates for the bulk Earthcomposition including a potential carbon reservoir in the core (
51, 73 ).43 upplementary Materials1 Supplementary Text
N-body simulations of terrestrial planet formation by pebble accretion
To verify the calculations of single growth tracks presented in the main text we present here N -body simulations of terrestrial planet formation by pebble accretion. The N -body simula-tions are performed with the rebound code, available at http://github.com/hannorein/ .We use the hybrid MERCURIUS integrator and a time-step of × − yr / (2 π ) . The pebbleaccretion efficiency is calculated based on ( ) and ( ). For the damping of eccentricity, in-clination and semi-major axis by gravitational torques from the gas disc, we follow ( ) with k mig = 1 .The gas surface density is identical to ( ) with Σ g = 610 g cm − ( r/ AU) − / exp( − t/τ ) ,aspect ratio h = 0 . r/ AU) / and the pebble flux F peb = 40 M E Myr − exp( − t/τ ) . Here τ = 1 . is the accretion time-scale of the protoplanetary disc. The pebbles are givenStokes numbers St = 10 − exp( t/τ ) . This exponential model is different from the α -discmodel presented in the main text, but we recover similar masses and orbits of the terrestrialplanets.The protoplanets are started at the positions r = (1 . , . , . , .
65) AU , to avoid over-lapping orbits from the beginning. The starting times are t = (0 . , . , . , .
78) Myr andthe starting masses are M = (10 − , − , − , − ) M E . We ignore in the N -body simu-lations the accretion of planetesimals from the birth belt. Therefore we start Mars with a tentimes higher mass than we use for the other protoplanets, in order to mimic the substantialcontribution of planetesimal accretion to Mars’ mass that we find in the main text.The results are shown in Figure S3. The growth tracks largely follow the single-planetgrowth tracks presented in the main text. There is only little interaction between Venus, Earth44nd Theia while they grow, exciting eccentricities up to e ∼ − . The planets thus remainon fairly circular orbits, which validates our assumption of pebble accretion on circular orbitsin the main text. Overall we conclude that the N -body simulations give similar results to thesingle-planet growth tracks presented in the main text.45 Supplementary Figures t [Myr]10 -9 -8 -7 -6 M * [ M O • y r - ] A R d i s c [ AU ] R disc M * .. 0.1 1.0 10.0 100.0 1000.0 r [AU]10 -4 -2 B -4 -2 Σ g , Σ p [ kg m - ] Σ p Σ g MMSN t = 5.0 Myr t = 3.0 Myr t = 1.0 Myr t = 0.2 Myr α v = 10 -4 for T < 800 K α v = 10 -2 for T > 1000 K Figure S1:
Details of the protoplanetary disc model that includes heating by the magne-torotational instability above T = 800 K. Panel (A) shows the stellar accretion rate as afunction of time (left axis) and the characteristic disc size (right axis). Panel (B) shows the col-umn density profile of gas and pebbles at four different evolution times, as well as the minimummass solar nebula for comparison (dotted line). The inner regions are viscously heated by themagnetorotational instability and hence the column density is lowered to maintain a constantaccretion rate onto the star. We show the pebble column density only interior of 2 AU, since Σ p is constructed for the terrestrial planet zone using a constant ratio ( ξ = 0 . ) of the pebbleflux relative to the gas flux from the outer protoplanetary disc.46 .1 1.0 10.0 100.0 r [AU]0.0010.0100.100 M > [ M O • ] t [ M y r] r CI,0 M CI,0 t CI r CI (t) r CI, max for α =0.005 r CI, max for α =0.002 r CI for α =0.01 Figure S2:
The cumulative gas mass in the protoplanetary disc as a function of distancefrom the star at a time of t = 10 yr (left axis) and the location of the transition from NCto CI composition as a function of time (right axis). The CI material is required here to enterthe terrestrial planet region at a time of t = 3 . . This gives an initial transition line atapproximately 10 AU at t = 0 (which has already expanded to 20 AU at t = 10 yr where theplot starts). The initial mass of gas and dust in the CI region is 0.027 M (cid:12) of the 0.044 M (cid:12) total mass residing in the protoplanetary disc at t = 10 yr . The plus and the asterisk symbolsmark the maximum distance of the NC-CI interface curve for two lower values of α : AU for α = 0 . and AU for α = 0 . . 47 .0 0.5 1.0 1.5 2.0 a [AU] -3 -2 -1 M [ M E ] A t [Myr] -3 -2 -1 M [ M E ] B t [Myr] a , q , Q [ AU ] C t [Myr] -5 -4 -3 -2 -1 e D Figure S3:
Results of N -body simulations of terrestrial planet formation from protoplanetsthat start around r = 1 . in the protoplanetary disc.in the protoplanetary disc.