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.
Runaway growth concentrates mass in the largest objects
In our models, runaway growth leads to the promotion of 10-40
large objects into the n-body code. These objects interact
dynamically. Some collide and merge. All continue to accrete
from the planetesimal swarm.
Eventually, a few of the largest bodies begin to sweep up the
lower mass n-bodies as well as the remaining planetesimals.
These objects clear out a path along their obits that is relatively
free of other objects.
For models with initial surface densities of 1-2 minimum mass
solar nebulae (X = 1-2), collisions, dynamical interactions,
and mergers lead to several objects with masses of
0.25-1.5 Earth masses. The timescale to produce
Earth mass planets is roughly 10 Myr.
The following figures describe the evolution in more detail.
Figure 2: Evolution of the Number of N-Bodies
Number of n-bodies as a function of time for three calculations.
At time t = 0, all objects have radii of 4-15 km; none are in
the n-body code. After 10,000 yr, the first large objects begin
to form. The number of n-bodies, with masses of 0.2 Lunar masses,
peaks at 30,000 to 100,000 yr, when the largest objects have
each accumulated roughly 10 lunar masses (0.1 Earth masses).
Dynamical interactions then begin to scatter large bodies
throughout the grid. Some of these merge to form larger objects.
Newly-promoted n-bodies merge quickly with the largest bodies.
As mergers continue, the number of n-bodies drops until a few
large objects remain.
Figure 3: Evolution of the largest object
We ran many calculations with the same initial conditions.
In the figure below, each curve plots the spread in the mass
at each time for the ensemble of calculations. The blue and
green curves have Σ = 16 g cm-2 at 1 AU;
the magenta curves have Σ = 32 g cm-2 at 1 AU.
For almost any set of initial conditions, the range in the mass
of the largest object is a factor of 2-3. Models starting with
smaller objects evolve faster. The arrow indicates the mass of
Figure 4: Evolution of eccentricity
The eccentricity of the largest object shows a strong secular
evolution to larger eccentricities plus chaotic changes.
The figure below shows how the eccentricity evolves for the
largest objects in the calculations of Figure 2.
For the first
10,000 yr, viscous stirring by large planetesimals
raises e while dynamical friction by small
planetesimals damps e. After 10,000 yr, interactions
between individual bodies raise e; dynamical friction
tries to lower e.
As objects grow to a significant
fraction of an Earth mass, this interplay between chaotic
scattering and dynamical friction causes e to oscillate
between e = 0.01 and e = 0.1. The arrow indicates
e for the Earth.
Figure 5: Time evolution of N-Body positions #1
The next figure plots the heliocentric distance as a
function of time for all of the n-bodies in a calculation
with &Sigma: = 15 g cm-2.
The dashed lines indicate the inner and outer boundaries of
the planetesimal grid. N-bodies can cross these boundaries.
At the start of each track, the mass is roughly 0.2 Lunar masses.
Bodies accrete mass from the planetesimal swarm and from mergers
with other n-bodies. Color changes along a track indicate an
n-body + n-body merger.
At roughly 100,000 yr, dynamical interactions between n-bodies begin;
gravitational scattering produces chaotic orbits and occasional
mergers. After 10 Myr, three planets have cleared all but one
smaller n-body from the grid. The mass at t = 10 Myr is at the
right of each track for the largest surviving n-bodies. The mass of
the largest survivor is 50% of the mass of the Earth.
Figure 6: Evolution of N-Body positions #2
N-Body positions for a second model with
Σ = 15 g cm-2 at 1 AU. Chaotic interactions last
longer in this model, but by 10 Myr, two planets with
roughly 50% of the mass of the Earth have cleared most
of the smaller n-bodies out of the grid.
Figure 7: Evolution of N-Body positions #3
N-Body positions as a function of time for a model with
Σ = 32 g cm-2. Due to the larger
initial surface density, planets grow faster.
Chaotic interactions and mergers begin sooner.
After 2-3 Myr, two large bodies begin
to clear out smaller n-bodies in the grid.
By 10 Myr, the largest of these is 50% more
massive than the Earth.
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