A pebble accretion model for the formation of the terrestrial planets in the Solar System

Abstract

Pebbles of millimeter sizes are abundant in protoplanetary discs around young stars. Chondrules inside primitive meteorites-formed by melting of dust aggregate pebbles or in impacts between planetesimals-have similar sizes. The role of pebble accretion for terrestrial planet formation is nevertheless unclear. Here, we present a model where inward-drifting pebbles feed the growth of terrestrial planets. The masses and orbits of Venus, Earth, Theia (which later collided with Earth to form the Moon), and Mars are all consistent with pebble accretion onto protoplanets that formed around Mars' orbit and migrated to their final positions while growing. The isotopic compositions of Earth and Mars are matched qualitatively by accretion of two generations of pebbles, carrying distinct isotopic signatures. Last, we show that the water and carbon budget of Earth can be delivered by pebbles from the early generation before the gas envelope became hot enough to vaporize volatiles.

Fig. 1. Sketch of the physical processes involved in our pebble accretion model for the formation of terrestrial planets.

Stage (A): The protoplanetary disc is formed consisting of material 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. The remaining solids carry now a noncarbonaceous (NC) signature (red). In stage (B), the disc expands outward due to angular momentum transport from the inner to the outer disc, carrying the NC material along with the gas. Planetesimal belts form at the water ice line (red) and by pileups of pebbles in the outer regions of the protoplanetary disc (blue); this outer planetesimal belt is envisioned here as the birth region of the giant planets (14, 72). In stage (C), protoplanets representing Earth, Venus, and Theia migrate out of the inner planetesimal belt. In the outer Solar 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 planets shift their compositions more toward the CI meteorites. The CC form outside the orbits of Jupiter and Saturn. Last, in stage (E), the protoplanetary disc clears, and the planetesimals of NC and CC composition are scattered into the asteroid belt.

Fig. 2. Temperature maps of the inner 10 AU of an evolving protoplanetary disc.

The left plot shows the temperature when viscous heating is applied everywhere in the protoplanetary disc. In the right plot, we assume that viscous heat is only released when the magnetorotational instability is active above a temperature of 800 K—the remaining disc is magnetically dead and hence only heated by the stellar irradiation. Three contour lines for the Toomre Q parameter are indicated in yellow; values above ≈1 imply gravitational stability and hence validate our neglect of heating from gravitoturbulence. We overplot the sublimation lines of water (H₂O) and the refractory minerals troilite (FeS) and fayalite (Fe₂Si0₄). In the more realistic case where viscous heating is provided only where the magnetorotational instability is active (right plot), the primordial water ice line sits in the region between 1.2 and 2.0 AU in the first million years of disc evolution. This is the likely site for formation of the first generation of planetesimals in the inner Solar System.

Fig. 3. Thermal structure of the inner regions of the protoplanetary disc model.

(A) The temperature at 1-AU distance from the star as a function of time and the temperature of the water ice line. The temperature becomes dominated by stellar irradiation already after 0.3 Ma of disc evolution; the temperature then continues to drop more slowly as the stellar luminosity falls with time. The temperature falls below water vapor saturation after approximately 2 Ma of disc evolution. (B) The maximum planetary mass for growth tracks starting at r₀ = 1.6 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 gas flux rate, $\mathrm{\xi }={\mathit{F}}_{\mathrm{p}}/{\stackrel{·}{\mathit{M}}}_{*·}$. We give the temperature a passive irradiation profile with fixed value of 140 K at 1 AU. The Stokes number at 1 AU for millimeter-sized pebbles is indicated at four different times. The maximum mass that leads to a good match for Venus’ orbit and mass is indicated with a dashed line. This is obtained for a range of Stokes numbers between 0.001 and 0.1, consistent with the evolution of the Stokes number of millimeter-sized pebbles from the early, dense protoplanetary disc to the late, dilute disc (pluses). The pebble-to-gas flux ratio lies in the range between 0.004 and 0.008; these values are similar to the solar ratio of refractory material relative to hydrogen/helium gas.

Fig. 4. Growth tracks and compositions of terrestrial planets.

(A) Numerical growth tracks of protoplanets growing by pebble accretion and planetesimal accretion, starting at r₀ = 1.6 AU. The parameters of the growth track were chosen with Venus and Mars as end-members. Earth obtains a final mass of 0.6 M_E, which we augment by creating Theia slightly later to reach a final mass of 0.4 M_E. (B) Masses of the four planets as a function of time. We show here also the contribution from planetesimal accretion in the birth belt. (C) The fraction of mass accreted from outer Solar System material, as a function of the time when the drifting pebbles change to CI composition. We find agreement with the estimated CI contribution of 42% to Earth and 36% to Mars (36) for a transition time from ureilite composition to CI composition in the range 3.5 Ma to 4.0 Ma.

Fig. 5. Monte Carlo sampling of 1000 protoplanets starting with initial masses of M₀ = 10⁻³ M_E at random times between 0.5 Ma and 5 Ma.

The black dots show results for a planetesimal belt of width W = 0.05 AU, while the green dots show results of a width of W = 0.5 AU. The protoplanets are started at random positions within the planetesimal belt. (A) Results for a pebble-to-gas flux ratio of ξ = 0.0036, as in Fig. 4. The narrow belt case faithfully reproduces the characteristic masses and orbits of Venus and Mars. Considering instead a wider planetesimal belt leads to the formation of massive planets that migrate to the inner edge of the protoplanetary disc. (B) Results for a higher pebble-to-gas flux ratio of ξ = 0.01. Venus is now well below the growth track, while planets in the Mars region grow to ≈ 0.5 M_E before migration becomes important. The high pebble flux allows super-Earths of up to 10 M_E to grow and migrate to the inner edge of the protoplanetary disc.

Fig. 6. Temperature and density structure of the gaseous envelopes of Earth and Mars.

(A) Envelope structure of our Earth analog and (B) our Mars analog immediately before the dissipation of the protoplanetary disc at t = 5 Ma. The top panels show the gas temperature, and the bottom panels the gas density. Three different opacity levels are considered: κ₀ = 0.01, 0.1, and 1.0 m²kg⁻¹. The temperature of Earth’s envelope directly over the magma ocean core reaches 2000 to 3000 K. The saturated vapor pressure of silicates (assumed here to be Forsterite) dominates over the ambient pressure only in a tiny region above the surface that reaches temperatures above approximately 2600 K. We mark the relative entropy level at the water ice line. For the high-opacity case, the relative entropy approaches 50% of the disc entropy; flows from the protoplanetary disc easily reach this entropy level and cleanse the isothermal region of water vapor and ice particles. Mars remains colder than 1000 K throughout the envelope and avoids melting, but the water ice line lies at a similar entropy level to Earth. The envelopes of both Earth and Mars reach temperatures in the range 325 to 425 K slightly below the water ice lines; here, organic molecules undergo pyrolysis and sublimation to form volatile gas species (CH₄, CO, and CO₂). These species cannot recondense and will freely diffuse back to the protoplanetary disc.

Fig. 7. Fraction of water and carbon that survives the passage through the planetary envelope.

The fraction is shown as a function of the mass of the growing Earth. The results are shown for two different values of the ice fraction of the pebbles. The estimated water and carbon fractions of Earth are indicated with dotted lines. Water is delivered by icy pebbles until the temperature in the envelope reaches ice sublimation (assumed here to be at 160 K). The water fraction falls steeply after the protoplanet reaches 0.02 M_E. Carbon is vaporized in two steps—organics vaporize in the temperature range 325 to 425 K, and carbon dust burns at 1100 K. Planetesimals in the birth belt as well as the early-accreted pebbles define an initial carbon fraction of 3000 ppm. The infall of pebbles of CI composition (5% carbon, assumed here to occur at t_CI = 3.8 Ma) when Earth reaches a mass of 0.3 M_E coincides with the increased heating of the planetary envelope, so that the overall carbon fraction actually decreases with increasing mass. The final value lands around 600 ppm, similar to estimates for the bulk Earth composition including a potential carbon reservoir in the core (51, 73).