A central premise on which most current theories of core formation in terrestrial planets are based is that core-forming metals segregated from mantle-forming silicates during the planet accretion process (Rubie and Jacobson, 2016). In addition, isotopic evidence indicates that prior to segregation, the core-forming metals partially or fully equilibrated with silicate melt (Kleine and Walker, 2017), implying that the present-day mantle composition holds clues as to the physical conditions under which that segregation took place.
The most widely-considered theory of core segregation is that it took place in discrete steps, each segregation step a consequence of melting produced either by radioactive heating (Monteux et al., 2009) or by a large, energy-dissipating impact (Rubie et al., 2015;Kendall and Melosh, 2016). Upon melting, core-forming metals equilibrated with, and then segregated from, molten silicates within proto-Earth's mantle, near the base of a magma ocean and at a depth roughly corresponding to the intersection of the geotherm and the silicate liquidus (Wood et al., 2006).
Support for these events comes from several lines of evidence. Application of metal-silicate partitioning of siderophile elements to magma oceans has yielded geophysically consistent models of the present-day composition of the core (Badro et al., 2015;Fischer et al., 2017). In addition, laboratory fluid dynamics experiments and calculations (Wacheul and Le Bars, 2018;Clesi et al., 2020;Landeau et al., 2021) reveal that small and medium-sized impacting metal cores are expected to disperse and partially equilibrate with molten silicates on descent through a magma ocean. Theoretical considerations (Stevenson, 1990) and lab experiments (Fleck et al., 2018) indicate that if the magma ocean has a solid base, dispersed metals accumulate into large diapirs there, implying that little further equilibration occurs as these diapirs sink into the core.
Recently, however, a competing theory of planet formation has emerged, in which large impacts are not the primary accretion mechanism. Astronomical observations of protoplanetary disks (Pérez et al., 2015;Andrews, 2015;Carasco-González et al., 2019) have determined that a substantial portion of the solid material surrounding very young stars resides in millimeter-to-centimetersized condensates, collectively referred to as pebbles. The aggregate pebble mass orbiting some young stars has been estimated to exceed one hundred Earth masses (Powell et al., 2019), with total solids (dust, pebbles, and larger bodies) amounting to between one and ten percent of the mass of the surrounding nebular gas (Ansdell et al., 2016).
Other astronomical observations indicate that pebble and dust abundances decrease on a time scale of a few million years (Tychoniec et al., 2018), presumably due to radial drift toward the central star. The orbital velocity of the nebular gas is sub-Keplerian because it is subject to a radial pressure force. Solids orbiting within a protoplanetary disk therefore experience a headwind from the slower rotating gas; the aerodynamic drag from this headwind decreases the pebble angular momentum, forcing them to drift radially inward. Inward drifting pebbles are then gravitationally captured by planetesimals (Eriksson et al., 2020) and by higher mass protoplanets (Voelkel et al., 2012), particularly those that have acquired extensive nebular atmospheres of their own (Popovas et al., 2018).
Calculations show that the rate of inward pebble drift and their settling rate (the rate of gravitational capture by a protoplanet) is high around young solar-mass stars (Johansen and Lambrechts, 2017), such that protoplanets can acquire Earth-like masses within the lifetime of the stellar nebula gas, that is, within a few million years. The accretion energy released by incoming pebbles is expected to heat the nebular atmosphere, leading to very high temperatures on the protoplanet surface, and in conjunction with heating by short-lived radioactive isotopes and gravitational energy released during segregation, still higher temperatures in its interior. Consequently, core segregation processes are predicted to have occurred under high temperature conditions in the proto-Earth during pebble accretion.
Quantitative models of pebble accretion were first applied to gas giant planet formation (Lambrechts and Johansen, 2012), because other mechanisms, such as random planetesimal accretion (Bottke et al., 2010), typically fail to grow a critical mass core 1 within the short nebula lifetime (Nimmo et al., 2018). Pebble accretion has now been extended to terrestrial planets (Levison et al., 2015;Morbidelli et al., 2015), with some success in rationalizing the masses, compositions, and water contents of Earth and Mars (Ida et al., 2019;Johansen et al., 2021) Other calculations show that spatially heterogeneous pebble accretion can, under some circumstances, produce rapid protoplanet rotation (Visser et al., 2020).
Here we apply a simplified one-dimensional, time dependent model of pebble accretion for terrestrial planet formation to metalsilicate segregation and core formation processes in the proto-Earth and its Moon-forming impactor Theia. Our model includes an idealized nebular atmosphere above an idealized accreting protoplanet, the protoplanet consisting of silicate and metal components derived from pebbles. We calculate the metal segregation temperature and pressure as a function of time and final protoplanet mass. Two-body combinations, simulating the merger of the proto-Earth and Theia, yield average segregation temperatures and pressures compatible with core formation conditions inferred from multistage impact-based models constrained by metal-silicate partitioning of siderophile elements.
The efficiency of pebble accretion depends critically on two dimensionless parameters: the Stokes number of the pebbles (sometimes called the dimensionless stopping time), the product of the timescale for deceleration of a pebble due to aerodynamic drag τ d and the local Keplerian orbital angular velocity :
and the headwind number,
Here v hw is the headwind velocity, the velocity of the protoplanet relative to pebbles on the same orbit, and r Hill is the radius of the Hill sphere, 1 For giant planets, the term core usually refers to all dense solids near the planet center; here it refers only to iron-nickel alloys.
where M S is the stellar mass and M and R are the mass and the orbital radius of the protoplanet.
An important criterion for pebbles to settle onto a protoplanet is that they be aerodynamically small, i.e., St cannot be too large (Johansen and Lambrechts, 2017). However, if the Stokes number is too small, say St << 10 -3 , the capture cross-section for pebble settling diminishes and pebble accretion becomes less efficient (Ormel and Klahr, 2010). The most efficient regime for pebble accretion is the so-called Hill regime, where St lies in the range 10 -3 -10 -1 , roughly corresponding to millimeter-to-centimeter sizes. In the Hill regime, pebble settling occurs through a combination of headwind (pebble approach), crosswind (pebble drift toward the star), Keplerian shear (variations in pebble angular velocity with orbital distance), and gravitational attraction.
Fig. 1 shows pebble trajectories relative to an orbiting protoplanet in the Hill regime with St = Z hw = 0.1, calculated using methods described in the Appendix. Arrows indicate the relative pebble motion directions, and colors distinguish settling versus escaping pebbles. With these parameters the pebble stopping time is approximately 6 days, the characteristic pebble diameter is a few centimeters, the headwind and crosswind are approximately 20 m/s and 4 m/s, respectively, and a terrestrial-size protoplanet on Earth's orbit captures 5-10% of inward drifting pebbles. Because of Keplerian shear, pebbles settle from inside as well as outside the protoplanet orbit, and the pebble capture cross-section is large for these parameters, approximately the diameter of the Hill sphere.
The protoplanet mass increases with time t according to
where F is the pebble settling rate (mass flux). Provided the pebbles are concentrated within r Hill of the orbital plane, twodimensional accretion is valid. As demonstrated in the Appendix, the two-dimensional pebble settling rate with
where is the column density of pebbles near the protoplanet orbit.
Access to pebbles is expected to have changed with time, and essentially vanished with the dissipation of the solar nebula. Accordingly, we adopt a time-dependence for the pebble column density at the proto-Earth orbit of the form
where 0 is the initial column density and τ n is the nebula dissipation timescale. Observations suggest a dissipation timescale τ n 3 Myr (Tychoniec et al., 2018), so that the pebble column density given by ( 6) becomes vanishingly small after ∼6 Myr, consistent with the upper limit on chondrule ages (Villeneuve et al., 2009). Because Jupiter and Saturn may have starved the terrestrial protoplanets of incoming pebbles (Morbidelli et al., 2015), we adopt relatively small initial pebble densities, 0 =0.2-0.3 kg/m 2 . The pebble settling radius increases like St 2/3 in the Hill regime, so the quantity St 2/3 is a measure of the abundance of captured pebbles. In our model, 0 St 2/3 =0.043-0.065 kg/m 2 . For comparison, Johansen et al. (2021) assume fixed pebble size but larger initial pebble density, which combine to yield 0 St 2/3 0.1 kg/m 2 for their terrestrial protoplanets. Unlike Johansen et al. (2021), we assume fixed orbits for proto-Earth and Theia, and consequently a fixed nebular temperature.
The pebble mass is divided into mantle-forming and coreforming components, referred to as "silicate" and "metal", respectively. We assume a constant proportionality between the masses of these two pebble types, with μ=0.32 for the metal fraction.
Although this binary mixture does not reflect the actual compositions of the solids in the protoplanetary disk (which are better characterized as carbonaceous and noncarbonaceous), it is a convenient idealization for tracking core segregation processes. We further assume that pebble settling is independent of μ, so that the influxes of mantle-forming silicates and core-forming metals remain in constant proportion and the accreted metal and silicate masses are given by μM and (1μ)M, respectively.
We assign uniform densities to the silicate and metal components, denoted by ρ m (the subscript m for mantle-forming) and ρ c (the subscript c for core-forming), respectively. We ignore the effects of temperature, pressure, and redox reactions on these densities, in order to keep our results as transparent as possible. Because the densities of differentiated silicates and metals in the protoplanet are set equal to their respective component densities, the bulk density of the undifferentiated composite (denoted by subscript u) is related to the accreted protoplanet mass and the metal and silicate volumes by
Fig. 2. Surface temperature versus protoplanet mass calculated for pebble accretion under uniform equilibrium radiative and convective nebular atmospheres. Dashed lines denote the model rheological transition temperature for surface magma ocean behavior and nebula temperature at proto-Earth's orbit, respectively.
As the masses of the protoplanets increase, they gravitationally attract atmospheres composed of nebula gas. Previous investigations (Ikoma and Genda, 2006) have shown that nebular atmospheres develop a two-layer or three-layer structure, consisting of an outermost nearly isothermal layer, inside of which there are one or two layers with either radiative or convective temperature profiles. For our purposes it is sufficient to model only the lower part of the inner layer of the atmosphere, the region in direct contact with the protoplanet surface.
With uniform luminosity and opacity, the equilibrium atmosphere temperature varies inversely with radial distance from the protoplanet center r according to (see Appendix)
where G is the gravitational constant, γ is either the atmospheric specific heat C a for a convective profile or 4R for a radiative profile (R is the modified gas constant) and T n is the temperature at the base of the nearly isothermal outer layer, approximated here as the nebula temperature. The basal atmosphere temperature at the radius of the protoplanet r p , is therefore
Hereafter, we identify T p as the surface temperature, ignoring effects of boundary layers and departures from thermal equilibrium between atmosphere and the protoplanet surface. Assuming the lower layer of the atmosphere has a solar-type composition at all times, its major constituents being hydrogen and helium with the physical properties given in Table 1, surface temperatures during pebble accretion according to (9) are shown in Fig. 2 as functions of protoplanet mass.
The surface temperatures in Fig. 2 have large uncertainties (of order 25%), for several reasons. First, they are based on thermal equilibrium, which might not apply at times of very rapid accretion. Second, they assume uniform atmosphere luminosity. Consideration of pebble energetics (see Appendix) indicates that most of the pebble accretion energy is deposited in the lower atmosphere, rather than at the surface as uniform luminosity implies. Third, interactions between settling pebbles, the atmosphere, and the protoplanet interior modify the atmosphere composition, and thereby modify its near-surface thermal structure. Possible modifications include a silicate vapor phase from pebble ablation (Brouwers et al., 2018), water vapor (Ida et al., 2019), and carbon dioxide (Johansen et al., 2021). The influence of these additional constituents on the surface temperature depends on their concentrations, which increase with increasing protoplanet mass. For example, the partial pressure of silicate vapor at 3400 K is approximately 0.16 bar according to Visscher and Fegley (2013), barely 1% of the surface pressure in the radiative atmosphere above the 0.7M E protoplanet (M E denotes the present-day Earth mass). However, because of higher surface temperatures, the silicate vapor partial pressure becomes comparable to the surface atmosphere pressure on the 1M E protoplanets in Fig. 2, for either atmosphere type. Accordingly, in this study our model Earths are constructed by merging protoplanets with final masses of 0.7M E and smaller, in order to limit the effects of departures from thermal equilibrium and changes in atmosphere composition during the accretion process.
Fig. 3 shows internal structures of a hypothetical terrestrial protoplanet at four stages of pebble accretion: undifferentiated, two partially differentiated stages, and fully differentiated. All three densities are involved, ρ u , ρ m and ρ c , arranged as shown in the figure. In the undifferentiated Stage 1, the protoplanet is a solid homogeneous mixture of metal and silicate components, and its temperature is dictated primarily by heating from 26 Al decay. The temperature at depth is higher than near the surface in this stage because radioactive heat has accumulated at depth over a longer time interval, so much so that the protoplanet begins to melt from the center outward.
Deep melting leads to the transient Stage 2 structure shown in Fig. 3, consisting of a small, segregated metallic core underlying a layer of partially molten silicates drained of metals, above which is undifferentiated material, both partially molten and solid. The density stratification at this stage is unstable because ρ u > ρ m , signifying that an overturn of this structure is inevitable. Our model formulation does not include the dynamics of this overturn. Instead, we specify transition from Stage 2 at the time when the surface temperature reaches the silicate rheological transition temperature T rt , indicating surface magma ocean conditions. At that point, metals accumulate in the surface magma ocean. Their excess density, coupled with the density deficit in the deep partially molten layer, implies overturn of the silicate-bearing layers and an adiabatic temperature variation with depth. The outcome is depicted in Stage 3 of Fig. 3. It consists of silicate melt (a surface magma ocean) overlying differentiated solid or partially molten silicate layers, plus a metallic core. This arrangement is similar to the structure predicted for the aftermath of a giant impact on a protoplanet without a blanketing atmosphere (Kendall and Melosh, 2016).
Stage 3 is also transient. The surface temperature continues to rise with time in proportion to the mass of the protoplanet. This temperature rise causes the depth of the magma ocean to increase, until it occupies the entire silicate part of the protoplanet. From that point on, the protoplanet consists of two differentiated, entirely molten layers: a global magma ocean overlying a liquid metallic core, as depicted in Stage 4 of Fig. 3. This configuration lasts as long as the thermal blanketing effect of the atmosphere remains and the protoplanet continues to grow through pebble accretion. Whether or not a terrestrial protoplanet passes through all four stages in Fig. 3 depends on the total mass of pebbles it acquires. As we show in the following Section, our calculations indicate that Stage 4 is accessed once the protoplanet mass reaches approximately 0.35M E .
The pressure in the protoplanet interior P is determined from the hydrostatic balance dP dr = -ρ g, (10) in which ρ is the density appropriate to each region in Fig. 3, and g is internal gravity, given by
where M i is the total mass inside radius r. For reference, using Table 1 values of ρ m =4×10 3 kg/m 3 , ρ c =10×10 3 kg/m 3 , and μ = 0.32, a spherical 1M E protoplanet with these densities and metal fraction has a 6605 km surface radius, 9.14 m/s 2 surface gravity, and 108 GPa hydrostatic pressure at its 3574 km coremantle boundary radius. In comparison to present-day Earth values, the model surface and core-mantle boundary radii are slightly higher and the model gravity and core-mantle boundary pressure are slightly lower, all qualitatively consistent with the very high temperature conditions implied by pebble accretion.
Differentiated and undifferentiated regions in the interior are identified by comparing model temperatures to melting temperatures at each depth. We use linear silicate solidus and liquidus temperature variations with the forms
where the subscripts s and l refer to solidus and liquidus, respectively, T ms0 and T ml0 are standard (1 bar) pressure values, and T ms and T ml are their pressure derivatives. The values of the pressure derivatives in Table 1 are chosen so that (12) match the Andrault et al. (2011) chondritic melting curves at 70 GPa. Within partial melt regions, we assume that the silicate melt fraction f m varies linearly between the solidus and liquidus, so that
and f m =1 for T > T ml . We also specify the pressure dependence of the rheological transition temperature, above which the silicate partial melt loses strength and behaves like a low viscosity liquid in terms of its dynamics:
in which T rt0 = (T ms0 + T ml0 )/2 and T rt = (T ms + T ml )/2. According to ( 13) and ( 14), the silicate melt fraction is f m = 0.5 at the rheological transition. For core-forming metals, we adopt a linear melting law of the form
The values for T cl0 and T cl in Table 1 are derived from iron melting temperatures by Anzellini et al. (2013), in which we have included a small melting point reduction due to the presence of light elements. Effects of eutectic melting are not considered here.
The thermal structures assigned to the interior regions in Fig. 3 depend on the dominant mode of heat transfer in each region. In undifferentiated material, where T < T rt , heat transfer is too slow in the first few million years to balance the heat produced by 26 Al decay. Accordingly, for this material we assume it accretes at the surface temperature and its temperature increases with time according to
where
is the heat production rate from 26 Al decay, H 0 is its initial (t=0) value, τ Al is its decay rate, and C u is the specific heat of the undifferentiated solid. The partial derivatives in ( 16) signify that the temperature increase applies at fixed radius. The seed temperature profile is calculated the same way, assuming the seed radius increased linearly with time starting at t=0. Differentiated regions are identified by temperatures having exceeded the rheological transition, that is, where T > T rt . At the point in time and depth where this inequality is first met, the density is reduced from ρ u to ρ m , its metal content is transferred to the core, and the core radius is increased by the appropriate amount. In all such differentiated regions we apply adiabatic thermal profiles, on the assumption that thermal advection is the dominant mode of heat transfer. Justification for this assumption is given in the Appendix. Adiabatic temperature profiles are determined in solid and liquid silicate and in liquid metal regions using
where α is thermal expansivity, ρ is density, C is specific heat for each material. For liquid metal and solid silicate we use constant values of α given in Table 1. For liquid silicate regions, where T > T ml , there is a strong pressure effect on thermal expansion (de Koker and Stixrude, 2009). Accordingly, we adopt a pressure variation of the form
where α m0 and α m are constant factors, their values given in Table 1. In silicate partial melt regions, where T ms < T < T ml , the adiabatic temperature gradient is anomalous because of phase changes (Solomatov, 2015). In those regions we adopt an adiabatic temperature variation given by dT dP
T ms + T ml 3 .
(20)
Temperature continuity with the atmosphere is enforced at the protoplanet surface and between regions in the interior. Thermal boundary layers and stratified regions are ignored on the assumption they are thin. Some regions classified as silicate partial melt may at the same time be classified as undifferentiated for purposes of density, while others may be differentiated yet solid. However, because temperatures rise very quickly during pebble accretion, such regions are both localized in depth and short-lived in time, existing only briefly over limited depth regions during portions of Stages 2 and 3. Consequently, they have only minor effects on overall model behavior.
Important diagnostics for pebble accretion are the pressures and temperatures where core-forming metals segregate from silicates. Here, we define core segregation as the final contact between metal and silicates. If metal pebbles are free to descend through molten or partially molten silicate without aggregating into a larger mass, then according to our definition, metal-silicate segregation occurs at the temperature and pressure of the coremantle boundary. Alternatively, if the silicate has the strength to arrest pebble descent, such as at the base of a magma ocean, metal-rich layers will accumulate at those locations. These layers then undergo Rayleigh-Taylor instability (Stevenson, 1990), forming metal diapirs large enough to sink through partially molten or solid silicate and merge with the core. The significance here is that, once large metal diapirs form, most of that metal has effectively segregated, at least insofar as its ability to interact with silicates as it descends toward the core.
To enforce our definition of core segregation, we adopt the following logic. If temperatures in the radial interval separating undifferentiated material and the core-mantle boundary are everywhere equal to or higher than the rheological transition, then the segregation conditions are the core-mantle boundary pressure and temperature at that time. This is the situation for pebbles entering a whole-mantle magma ocean or a basal magma ocean. Alternatively, if the temperature profile lies below the rheological transition somewhere in that radial interval, then the segregation conditions are those corresponding to the pressure and temperature at the transition. This is the situation for a magma ocean that extends downward only partway through the silicate portion of the protoplanet, which is a commonly assumed configuration during Earth accretion (Elkins-Tanton, 2012).
Fig. 4 shows the results of building a 0.7M E protoplanet by pebble accretion, using the properties from Table 1 in equations ( 4)-( 20). Panel a in Fig. 4 shows the time history of the pebble column density , the pebble settling rate F and the protoplanet mass M. The pebble column density is normalized by its initial value (0.205 kg/m 2 ), the pebble settling rate by its maximum value, and protoplanet mass by M E . The vertical dotted line indicates the time when the seed, with initial mass 10 -4 M E , begins to acquire pebbles in the Hill regime. The pebble column density decreases rapidly with time, falling to a few percent of its initial value by 6 Myr. In contrast, the pebble settling rate first increases with time, reaching its maximum around 2.5 Myr, then decreases rapidly. The initial rise in pebble settling is due to the rapid increase in protoplanet mass, which greatly enlarges the settling cross-section. Only after the pebble column density falls below 50% of its initial value does the settling rate decrease substantially. The second panel, Fig. 4b, shows temperatures the lower portion of the nebular atmosphere, assuming a convective thermal profile. Starting from the nebula temperature, the atmosphere rapidly heats up as it deepens, with the surface atmosphere temperature approaching 3800 K toward the end of pebble accretion. At this temperature the equilibrium silicate vapor pressure is about 2 bars (Visscher and Fegley, 2013) compared to the surface hydrostatic pressure of about 30 bars.
The third and fourth panels in Fig. 4 show the thermal evolution of the protoplanet interior. Fig. 4c shows the distribution of temperature throughout the interior, whereas Fig. 4d shows the distributions of solid, melt, and partial melt in the region 0.55 < r/r p < 1, the post-accretion silicate portion of the protoplanet. Together these two panels illustrate the four stages of protoplanet growth depicted in Fig. 3. In the initial Stage 1 the protoplanet is solid throughout, with a thermal gradient governed by the heat produced by 26 Al decay. The partially differentiated Stage 2 begins near 0.9 Myr, when the central temperature exceeds the rheological transition at that pressure. This transition is marked by the inflection points of the dashed and dash-dot curves in Fig. 4c, which show the relative depths of the metallic core (labeled CMB) and the differentiated portion of the protoplanet, respectively. Stage 2 lasts approximately 1 Myr in Fig. 4, that is, between 0.9 and 1.9 Myr. During this time, melting progresses upward, with differentiation approaching 0.54r p , the final core-mantle boundary radius ratio, around 1.6 Myr and reaching the surface around 1.9 Myr. Once the surface temperature exceeds T rt , the model transitions from Stage 2 to Stage 3, marked in Fig. 4d by an inversion of the mantle structure, with silicate liquid overlying differentiated silicate partial melt overlying a basal silicate layer that is also differentiated but solidified by adiabatic compression. From the beginning of Stage 3 on, the relative depth of the core-mantle boundary remains constant and no undifferentiated material remains in the interior, as indicated by the flat segment of the CMB curve in Fig. 4c. However, internal temperatures continue to increase during Stage 3 because the adiabatic thermal profiles in silicate and metallic regions are coupled to the rising surface temperature. In consequence, the surface magma ocean rapidly deepens, the rheological transition reaching the core-mantle boundary around 2.4 Myr. From 2.6 Myr to the end of pebble accretion the entire protoplanet is molten.
Figs. 5 and 6 show the results of building smaller protoplanets by pebble accretion, using the same nebular environment and pebble properties as in Fig. 4. In Fig. 5 a 10 -4 M E seed mass begins accreting pebbles at 0.7 Myr and reaches a final mass of 0.31M E , whereas in Fig. 6 pebble accretion onto the same seed mass begins at 0.35 Myr and the final mass is 0.505M E These starting times were chosen in order to produce two-body masses comparable to the present-day Earth-Moon system. Specifically, by merging the accreted protoplanets in Figs. 4 and5 we model the collision of a larger proto-Earth and a smaller Theia ( Ćuk and Stewart, 2012), yielding the approximate Earth-Moon system mass. Similarly, the same final system mass is obtained by merging two of the pro- toplanets in Fig. 6, a model for the equal mass collision scenario (Canup, 2012).
Panel a in Fig. 5 shows the same general behavior in terms of pebble settling and protoplanet mass as in Fig. 4, except that pebble settling starts later in time in Fig. 5 and the final mass is less. The reduced mass limits the growth of the nebular atmosphere, and as shown in Fig. 5b, the blanketing effect in this case is weaker, such that the final surface temperature barely exceeds 2400 K. The weaker thermal blanketing by the atmosphere strongly affects the evolution of the protoplanet interior, as shown in Figs. 5c and5d. Differentiation and core segregation are delayed in this case, and the internal temperatures are relatively low, such that the core-mantle boundary temperature is only slightly above 3000 K at the end of pebble accretion. More significantly, Fig. 5d shows that the protoplanet never entirely melts in this case (i.e., the evolution does not reach Stage 4). At the end of pebble accretion magma ocean includes a basal a partial melt, although its temperature is everywhere above the rheological transition.
The evolution shown in Fig. 6 for the 0.505M E protoplanet is intermediate between Figs. 4 and5. All four evolutionary stages are present in this case, but Stages 2-4 occur later in time compared to Fig. 4. Likewise, the surface and core-mantle boundary temperatures are more moderate compared to Fig. 4. In short, in spite of some differences in their overall evolution, in each of these cases the core is predicted to fully segregate while pebble accretion is active, although under somewhat different temperature and pressure conditions. M c here denoting the (final) core mass at 6 Myr. The average segregation pressure and temperature are about 60% of their final values, consistent with other continuous or multi-stage core formation models (Siebert et al., 2012).
Fig. 7b shows average segregation pressure and temperature versus final protoplanet mass, for both convective and radiative atmospheres. As expected, the average segregation pressures and temperatures are somewhat higher beneath a convective atmosphere, compared to beneath a radiative atmosphere, although for most final masses these differences are not particularly large. Exceptions are found around 0.2M E in Fig. 7b, corresponding to the mass interval in which pebble accretion ends while the protoplanet lies in evolution Stages 2 and 3 of Fig. 3. As the protoplanet evolves into Stage 4, metal-silicate segregation occurs at the core-mantle boundary, and the average segregation pressure and temperature increase systematically with protoplanet mass for both atmosphere types, as indicated by the curves to the right of the vertical dotted line in Fig. 7b.
Fig. 8 compares core segregation conditions from our pebble accretion model to the segregation conditions from accretion models based on large impacts and constrained by metal-silicate partitioning of siderophile elements. The two dashed curves are our model silicate solidus and liquidus. The solid blue curves are our model segregation conditions for radiative and convective atmospheres (labeled R and C, respectively), and the triangles and circles are averages and final (maximum) values, respectively, of these conditions for various protoplanets. The orange triangles in Fig. 8 are two-body averages of the 0.7M E and 0.31M E protoplanets (i.e., averages of the red and yellow triangles). This simulates the merger of a larger proto-Earth and a smaller Theia, in which the cores of Theia and the proto-Earth are combined without interacting with the silicates. In the same way, the green triangles in Fig. 8 simulate the merger of proto-Earth and Theia with identical masses.
At lower pressure and temperature, the trajectory of our segregation curve in Fig. 8 follows the rheological transition, where we have assumed that metals freely segregate. The average segregation conditions for the 0.31M E protoplanet lie below our model silicate liquidus, as do their final segregation conditions, indicating that most of the metal segregated from partially molten silicate at the rheological transition in these cases. In contrast, for the 0.7M E protoplanets, the final and the average segregation conditions lie above the liquidus. Although some segregation occurs at the rheological transition in these cases, additional segregation occurs at higher temperatures, including above our model silicate liquidus. These high temperature conditions are indicated by the segregation curves labeled C and R in Fig. 8, and correspond to evolution Stage 4 in which small metal volumes fall through silicate magma directly into the core.
The gray symbols in Fig. 8 are a sampling of results from impact-based accretion models constrained by metal-silicate partitioning data on siderophile elements, from Wood et al. (2006), Siebert et al. (2012), Badro et al. (2015), and Rubie et al. (2015).
These models assume either a sequence or a continuous distribution of large impacts, producing a magma ocean whose variable depth defines a segregation pressure curve. Segregation temperatures are usually assumed to lie along the mantle liquidus, with a variety of mantle liquidus profiles being considered. Final segregation conditions from these models are denoted by gray circles in Fig. 8; the corresponding average segregation conditions (gray triangles) were calculated using 60% of the final segregation pressures.
The average segregation conditions from our pebble accretion model are in broad agreement with the average segregation conditions from the impact-based models, especially the mergers of protoplanets pebble-accreted under convective atmospheres. This is perhaps surprising, given the fundamental differences in the two accretion mechanisms. However, some of these differences complement each other. For example, our pebble accretion model segregates at the core-mantle boundary, whereas impact-based models typically segregate at mid-mantle depths. But because segregation under pebble accretion operates when protoplanets are relatively small, the pressure-temperature conditions at the coremantle boundary are similar to the mid-mantle in a larger body built by impacts.
Pebble accretion is a relatively straightforward planet formation mechanism, in that only three main ingredients are needed: a large population of pebbles, nebular gas, and a seed mass. Once these are in place, core segregation proceeds deterministically, on a schedule dictated by the evolution of the protoplanetary disk.
Pebble accretion has multiple implications for core formation in the Earth. It is global in scale, drawing solids and gas from diverse parts of the protoplanetary environment. It synchronizes core segregation to the accretion rate on timescales of a few million years. And it implies pervasive melting and high temperature metal-silicate segregation, thereby offering new ways to interpret mantle siderophile abundances.
In terms of modeling core formation under pebble accretion, there is much room for extensions and improvements. Better constraints are needed on disk properties such as pebble composition and density, Stokes and headwind numbers, and orbital variations. In addition, more realistic models should include full compressibility, redox reactions, better melting laws, atmosphere constituents such as silica and water vapor, as well as the effects of oxygen fugacity, density stratification, and larger impacts.
Peter Olson: Conceptualization, Theory, Software Development, Graphics, Writing. Zachary Sharp: Conceptualization, Writing-Reviewing-Editing, Literature Search. Susmita Garai: Methods, Software Development, Graphics.