Smithsonian Astrophysical Observatory


S. Kenyon (SAO) & B. Bromley (Univ Utah)



We use a hybrid planet formation code to compute the formation of terrestrial planets from an initial ensemble of 1-10 km planetesimals. Links below describe the codes.  

  • Multiannulus coagulation code for small objects
  • N-body code for large objects
  • Hybrid code connects two types of objects


    Figure 1: Model Grid

    We performed two sets of calculations in 32 annuli at 1 AU.  

  • Small annuli with widths of 0.01 AU
  • Larger annuli with widths of 0.02 AU

    The figure below shows the small annulus in green. The Sun is the orange dot in the center of the annulus.


    Initial Conditions

    We start with 1-10 km planetesimals in nearly circular orbits around the Sun. The input parameters are the initial surface density and the orbital eccentricity and inclination.  

  • e = 10-4
  • i = 6 × 10-5
  • Σ = 15 X g cm-2 at 1 AU

    A minimum mass solar nebula (MMSN), with just enough solid material for the terrestrial planets, has X = 1.

    These calculations include gas drag and velocity evolution from inelastic and elastic collisions. The coagulation code also includes a fragmentation algorithm.

  • Low velocity collisions produce mergers
  • Medium velocity collisions produce mergers and some debris
  • High velocity collisions produce mostly debris



    The evolution of the planetesimals follows a standard pattern.  

  • Small objects collide and merge into large objects
  • Dynamical friction damps the larger objects
  • Gravitational focusing increases collision cross-sections
  • Runaway growth concentrates most of the mass in the largest objects
  • The gravity of the largest objects stirs up the smaller objects
  • Smaller objects have more eccentric orbits and collide at larger velocities
  • Collisions between smaller objects then produce debris

    Copious collisions can produce observable amounts of debris.

    The following figures describe the evolution in more detail.


    Figure 2: Evolution of the largest objects

    Masses of the largest objects for a thin torus model with X = 1. The horizontal error bars indicate the extent of the orbit for each object.

    (a) lower panel: t = 0.1 Myr

    (b) upper panel: t = 1 Myr

    During the first 0.1-0.2 Myr, dynamical friction damps the orbits of the largest objects. The orbital eccentricities of the largest objects are roughly 10 times smaller than the eccentricities of smaller objects. Objects with smaller e grow more rapidly than those with larger e.

    From 0.1 Myr to 1 Myr, viscous stirring becomes more important than dynamical friction. Thus, all orbits become more eccentric. By 1 Myr, all objects have e = 0.01 or larger.



    Figure 3: Evolution of the dust mass

    Evolution of the dust mass in small grains ('S'; 1 μm to 1 mm) and large grains ('L'; 1 mm to 1 m) for models with X = 0.5, 1, and 2 as indicated in the legend. More massive disks produce more debris at earlier times than less massive disks. For all disks, the dust production rate converges to a power-law decline with time.

    When the calculations begin, small collision velocities yield small dust production rates. As larger objects form, viscous stirring and dynamical friction lead to larger collision velocities and more dust production: the largest objects stir up the smallest objects, which collide and fragment. More dust production results in larger total dust masses.

    Eventually, collision velocities become large enough to completely disrupt the colliding bodies. Because the colliding bodies occupy more volume, they have fewer collisions. The dust production rate and the dust mass then fall with time.

    Because collision rates scale with the total mass, calculations with more mass in planetesimals evolve faster than calculations with less mass in planetesimals. These calculations produce more dust earlier and larger planets than calculations with less initial mass.


    Figure 4: Evolution of infrared excess

    Evolution of the broadband 10 μm (N-N0) and 20 μm (Q-Q0) excesses as a function of time. Fragmentation produces large excesses relative to a stellar photosphere at times of 104 yr to 105 yr. By 1 Myr, the 10 μm excess is nearly unobservable; a small excess persists at 20 μm.

    Left panels: 0.84-1.16 AU torus calculation

    Right panels: 0.68-1.32 AU torus calculations

    In both panels:

  • solid lines: X = 0.5
  • dot-dashed lines: X = 1
  • dashed lines: X = 2

    The mid-IR excesses depend on the dust mass and the optical depth. All of the disks have large optical depth, τ = 1-10. The optical depth scales with the initial mass but falls as larger planets form. This behavior leads to a complicated evolution for the mid-IR excess as a function of initial mass and the size of the annulus.

    The double-peaked evolution of some IR excesses is an effect of the optical depth

    The trend of larger excess with larger initial mass occurs for disks with τ of roughly 1. As τ approaches 10, the amount of the excess is not a linear function of dust mass.



    Planet formation in the terrestrial zone leads to copious dust production

    The dusty debris of terrestrial planet formation is detectable

    At 1 AU, the timescale to detect dusty debris is roughly 1-2 Myr

    Inside of 1 AU, debris production will occur on shorter timescales

    Outside of 1 AU, the dusty debris from terrestrial planet formation might last for 10-30 Myr. This timescale probably depends on the timescale for giant planet formation


    The Future

    Several aspects of the calculations are uncertain

  • We do not allow planets to migrate by interactions with the gas
  • We do not include Jupiter formation, which can affect dust lifetime
  • We assume that the gas disappears in 1-10 Myr

    We plan to address these issues in future studies

  • Our code provides a straightforward way to include migration
  • We can also include prescriptions for formation of gas giants
  • We are looking into treatments for evolution of the gas

    To measure the sensitivity of our results to the size of the annulus, we also plan to make calculations in larger annuli.  


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