Research ArticleASTROPHYSICS

Growth of asteroids, planetary embryos, and Kuiper belt objects by chondrule accretion

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Science Advances  17 Apr 2015:
Vol. 1, no. 3, e1500109
DOI: 10.1126/sciadv.1500109
  • Fig. 1 The maximum particle density versus time for streaming instability simulations without self-gravity at resolutions 643 (black line), 1283 (yellow line), and 2563 (blue line).

    Particle density is measured in units of the mid-plane gas density ρg and time in units of the orbital period Torb. The first 30 orbits are run with a solids-to-gas ratio of Z = 0.01, with only modest overdensities seen in the particle density. Half of the gas is then removed over the next 10 orbits, triggering strong particle concentration, of up to 10,000 times the local gas density at the highest resolution. Doubling the resolution leads to a more than a quadrupling of the maximum particle density.

  • Fig. 2 The maximum particle density versus the length scale (measured in gas scale heights, H).

    We have taken spheres with diameters from one grid cell up to the largest scale of the simulation and noted the maximum value of the density over all simulation snapshots. The results are relatively converged on each scale. The increase in maximum particle density with increasing resolution is an effect of resolving ever smaller-scale filaments. Blue dotted lines mark the Roche density at three values of the gas column density at 2.5 AU relative to the minimum mass solar nebula (MMSN). The red dotted line indicates the characteristic length scale of the streaming instability. The black line shows a power law of slope −2, which shows that the maximum density follows approximately the inverse square of the length scale.

  • Fig. 3 Birth size distribution of planetesimals forming by the streaming instability in 25-cm-sized particles.

    The differential number of planetesimals (per 1022 g) is calculated with respect to the nearest size neighbors in the simulation. Yellow, blue, and red circles indicate individual planetesimals forming in computer simulations at 1283, 2563, and 5123 grid cells, respectively. The size distribution of the highest-resolution simulation is fitted with a power law dN/dMMqM (red line; the fit is based on the cumulative size distribution shown in Fig. 4, but does not include the exponential tapering). Simulations with lower values of the particle column density Σp yield successively smaller radii for the largest planetesimals, down to below 100 km in radius for a column density similar to the Minimum Mass Solar Nebula model (Σp = 4.3 g/cm2 at r = 2.5 AU). Binary planetesimals (marked with circles) appear only in the highest-resolution simulation because binary survival requires sufficient resolution of the Hill radius. The differential size distribution of the asteroid belt (dashed line) shows characteristic bumps at 60 and 200 km. The planetesimal birth sizes from the simulations are clearly not in agreement with main belt asteroids.

  • Fig. 4 Cumulative birth size distribution of planetesimals forming by the streaming instability in our highest-resolution simulation (5123).

    The cumulative size distribution is fitted with an exponentially tapered power law N>M−0.6 exp[−(M/Mexp)(4/3)] (red line), with exponential tapering at Mexp = 1.2 × 1023 g (dotted line). Shallower or steeper power laws yield poorer fits to the populations of small and large planetesimals, respectively. We choose to fit the cumulative size distribution rather than the differential size distribution to avoid the noise inherently present in the latter. The fit can be translated to dN/dMM−1.6 exp[−(M/Mexp)(4/3)] (the power law part of this fit is indicated in Fig. 3) as well as to dN/dRR−2.8 exp[−(R/Rexp)4] in the differential size distribution per unit radius.

  • Fig. 5 The size distribution of asteroids and embryos after accreting chondrules for 5 My (left panel) and selected masses and inclinations as a function of time (right panel).

    The nominal model (black line) closely matches the steep size distribution of main belt asteroids (red line, representing the current asteroid belt multiplied by a depletion factor of 540) from 60 to 200 km in radius. The size distribution becomes shallower above 200 km; this is also seen in the asteroid belt, although the simulations underproduce Ceres-sized planetesimals by a factor of about 2 to 3. A simulation with no chondrules (blue line) produces no asteroids larger than 300 km. Inclusion of chondrules up to centimeter sizes (pink line in insert) gives a much too low production of Ceres-sized asteroids, whereas setting the exponential cutoff radius of planetesimals to 50 km (green line) leads to a poorer match to the bump at 60 km. The right plot shows that the formation of the first embryos after 2.5 My quenches chondrule accretion on the smaller asteroids by exciting their inclinations i (right axis). The dotted lines indicate the mass contribution from planetesimal-planetesimal collisions. Asteroids and embryos larger than 200 km in radius owe at least two-thirds of their mass to chondrule accretion.

  • Fig. 6 Mean chondrule sizes (〈d〉, upper panel) as a function of layer radius R and size distribution width (σφ, lower panel) as a function of mean chondrule size.

    Yellow lines in the upper panel indicate the chondrule size evolution in individual asteroids and embryos, whereas red lines indicate mean accreted chondrule sizes at different times. The accreted chondrule size increases approximately linearly with planetesimal size at t = 0 My. Asteroids stirred by the growing embryos over the next 2 My accrete increasingly larger chondrules because asteroid velocities align with the sub-Keplerian chondrule flow at aphelion. Finally, asteroids accrete surface layers of mainly very small chondrules, down to below 0.1 mm in diameter, at late times when their large inclinations decouple the asteroid orbits from the large chondrules in the mid-plane. The width of the chondrule size distribution in the lower panel is given in terms of σφ, the base-2-logarithmic half-width of the cumulative mass distribution of chondrules (σφ = 1 implies that two-thirds of the chondrules have diameters within a factor 21 = 2 from the mean). Dots indicate chondrule layers inside asteroids and embryos. Chondrules are aerodynamically sorted by the accretion process to values of σφ in agreement with the chondrules found in ordinary chondrites (hashed region). Unfiltered accretion from the background size distribution of chondrules (blue dot, size distribution of unsedimented particles; yellow dot, size distribution in the mid-plane) yields specific pairs of 〈d〉 and σφ that are not consistent with constraints from ordinary chondrites.

  • Fig. 7 Growth of embryos and terrestrial planets at 1 AU.

    Left panel shows the size distribution at four different times, and right panel shows the mass of the most massive body in the simulation as a function of time. The growth up to 1000-km-sized embryos is mainly driven by planetesimal accretion because chondrule-sized pebbles are tightly coupled to the gas and hence hard to accrete. However, chondrule accretion gradually comes to dominate the accretion as the embryos grow. The largest body reaches Mars size after 3 My, with more than 90% contribution from chondrule accretion. A giant impact occurs just before 4 My, wherein the largest body accretes the third largest body in the population. The continued accretion of chondrules leads to the formation of an Earth-mass body after 5 My. A simulation with no chondrules evolves very differently: a large number of embryos form with masses just below the isolation mass of Miso ≈ 0.01 ME.

  • Fig. 8 Growth of embryos and terrestrial planets at 1 AU, with chondrule sizes up to 1 cm.

    The increase in chondrule size compared to the previous figure enhances the chondrule accretion rate substantially. The initial growth to 1000-km-sized embryos now has approximately equal contribution from planetesimal accretion and chondrule accretion. An Earth-mass terrestrial planet forms already after 2 My, driven by a combination of chondrule accretion and giant impacts.

  • Fig. 9 Growth of icy planetesimals at 10 AU.

    The growth rate by pebble accretion is as high as in the asteroid belt because the lower column density of pebbles is counteracted by the increased sedimentation in the more dilute gas. The annulus of 0.2-AU width produces in the end an Earth-sized protoplanet and a single Moon-sized embryo. Pebble accretion overwhelmingly dominates the growth (right panel). The icy protoplanet that forms has only a few parts in a thousand mass contribution from collisions. Large Ceres-sized planetesimals have a contribution from collisions of less than 5%, whereas the 200-km planetesimal owes about 1/10 of its growth from 130 km to planetesimal collisions. Note how the Ceres-sized planetesimal (blue line in the right panel) got a head start for efficient accretion of pebbles by experiencing a significant collision after 700,000 years.

  • Fig. 10 Planetesimal growth at 25 AU.

    Two models are considered: a low density model where the internal density is set to ρ = 0.5 g/cm3, similar to comets and binary Kuiper belt objects, and a high density model where the internal density is set to ρ = 2 g/cm3, similar to the dwarf planet Ceres. The low density model has turbulent stirring α = 10−6, whereas the high density model has α = 10−5. Both models display ordered growth up to 300-km radii, with a steep size distribution beyond 100-km sizes. This is followed by a runaway growth of a single, massive body. The right panel shows the size of the largest body as a function of time as well as the speed relative to a circular orbit. The runaway growth is facilitated by a steep decline in the eccentricity of the orbit because the high pebble accretion rate damps the eccentricity.

  • Fig. 11 The maximum planetesimal radius in the asteroid belt versus time for three different values of the turbulent viscosity α.

    Here, α = 2 × 10−6 represents the strength of turbulence caused by streaming instabilities and Kelvin-Helmholtz instabilities in a sedimented mid-plane layer of chondrules, α = 10−4 represents the turbulence strength in a dead zone stirred by active surface layers, and α = 10−3 represents the turbulence strength caused directly by the magnetorotational instability. Turbulent stirring of chondrules sets the scale height and mid-plane density of the chondrule layer and hence dictates the planetesimal growth rate. The formation time of the first embryo depends strongly on the degree of stirring.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/1/3/e1500109/DC1

    Materials and Methods

    Text

    Fig. S1. The nondimensionless planetesimal masses μ = ρpx)3 (here ρp is the particle density represented by the planetesimal in a grid cell and δx is the size of the cell) and corresponding contracted radii as a function of time after self-gravity is turned on.

    Fig. S2. The accretion radius Racc (that is, the impact parameter required for accretion, given in units of the Bondi radius RB) versus the particle friction time tf (in units of the Bondi time-scale tB), for planetesimal radii R between 10−4 RB and 104 RB.

    Fig. S3. The evolution of eccentricity e and inclination i of 1000 planetesimals with mass M = 1024 g located at 1 AU with a surface density of 10 g/cm2, for three values of the time-step.

    Fig. S4. The evolution of eccentricity e and inclination i of 800 planetesimals with mass M = 1024 g located at 1 AU with a surface density of 10 g/cm2.

    Fig. S5. Phase points covering planetesimal orbits with inclinations i = 0, i = 0.0016, i = 0.0032 and i = 0.0064.

    Fig. S6. The effect of eccentricity and inclination on the chondrule accretion rate, at an orbital distance of 2.5 AU.

    Fig. S7. The accretion rate of planetesimals at 25 AU, as a function of the planetesimal size.

    Fig. S8. The stratification integral (that is, the mean particle density over the accretion cross section) as a function of the accretion radius of the planetesimal.

    Fig. S9. The number of remaining bodies versus time for the constant kernel test.

    Fig. S10. Damping of eccentricities and inclinations of 10,000 1-cm–sized planetesimals located between 30 and 35 AU, emulating a test problem defined in Morbidelli et al. (21).

    Fig. S11. Cumulative size distribution after 3 Myr of coagulation within a population of planetesimals of initial diameters 100 km (50 km in radius), as shown by the cross.

    Fig. S12. Formation of planetesimals from centimeter-sized particles in a 2D shearing sheet simulation with the same spatial extent as the streaming instability simulations presented in the main text.

    References (5677)

  • Supplementary Materials

    This PDF file includes:

    • Materials and Methods
    • Text
    • Fig. S1. The nondimensionless planetesimal masses μ = ρpx)3 (here ρp is the particle density represented by the planetesimal in a grid cell and δx is the size of the cell) and corresponding contracted radii as a function of time after self-gravity is turned on.
    • Fig. S2. The accretion radius Racc (that is, the impact parameter required for accretion, given in units of the Bondi radius RB) versus the particle friction time tf (in units of the Bondi time-scale tB), for planetesimal radii R between 10−4 RB and 104 RB.
    • Fig. S3. The evolution of eccentricity e and inclination i of 1000 planetesimals with mass M = 1024 g located at 1 AU with a surface density of 10 g/cm2, for three values of the time-step.
    • Fig. S4. The evolution of eccentricity e and inclination i of 800 planetesimals with mass M = 1024 g located at 1 AU with a surface density of 10 g/cm2.
    • Fig. S5. Phase points covering planetesimal orbits with inclinations i = 0, i = 0.0016, i = 0.0032, and i = 0.0064.
    • Fig. S6. The effect of eccentricity and inclination on the chondrule accretion rate, at an orbital distance of 2.5 AU.
      Fig. S7. The accretion rate of planetesimals at 25 AU, as a function of the planetesimal size.
    • Fig. S8. The stratification integral (that is, the mean particle density over the accretion cross section) as a function of the accretion radius of the planetesimal.
    • Fig. S9. The number of remaining bodies versus time for the constant kernel test.
    • Fig. S10. Damping of eccentricities and inclinations of 10,000 1-cm–sized planetesimals located between 30 and 35 AU, emulating a test problem defined in Morbidelli et al. (21).
    • Fig. S11. Cumulative size distribution after 3 Myr of coagulation within a population of planetesimals of initial diameters 100 km (50 km in radius), as shown by the cross.
    • Fig. S12. Formation of planetesimals from centimeter-sized particles in a 2D shearing sheet simulation with the same spatial extent as the streaming instability simulations presented in the main text.
    • References (56–77)

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