EARTH FORMATION: SMALL DEBRIS DISKS IN THE TERRESTRIAL ZONE
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.
Figure 1: Model Grid
We performed two sets of calculations in 32 annuli at 1 AU.
The figure below shows the small annulus in green. The Sun is the orange dot in the center of the annulus.
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.
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.
The evolution of the planetesimals follows a standard pattern.
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:
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
Several aspects of the calculations are uncertain
We plan to address these issues in future studies
To measure the sensitivity of our results to the size of the annulus, we also plan to make calculations in larger annuli.