How do planetesimals grow to
form ~terrestrial mass cores?
For now, let’s ignore the gas. This means we can just worry about gravity.
For the pairwise interaction of two bodies, we have:
For collisions that are grazing, the
velocity at “impact” can be shown to be
Plugging this into the equation above and
setting q = p/2 allows us to solve for the
impact parameter b. The cross section is
not geometric (true for any central force):
Thus, the ratio of the encounter velocity to the escape velocity from the
planetesimal(s) is critical. You need SOME V to get collisions in the
first place, just not “too much”! This was first worked out by Safronov,
and so now bears his name, with the definitions:
Safronov Number
The next step in this analytical approach is to assume V characterizes the
random velocity distribution (w.r.t. the background Keplerian field) of a
“swarm” of planetesimals. In this gravitational focusing scenario:
Note that for q >> 1,
the growth rate goes
like a4 so larger bodies
grow faster.
Planetesimal surface density.
For such “orderly growth,” numerically you find the timescale to be:
More precisely, if x ≡ a/aEarth then the equations become:
dx/dt = (1-x3)/t
For x  1,
x ≈ 1 – e-3t/t
Under these conditions, 99% of a planet is built after 2t,
Thus the estimated timescales for the assembly of Earth
mass planets is >>nebular gas lifetime even at 1 AU, and
it gets worse fast as you move out!
What do you find numerically?
Numerical studies have,
for the most part, shown
that the assumptions in
the Safronov model are
reasonably valid. Growth
in a given zone tends to
lead to a single body
whose mass is >> than
that of the remaining
swarm of planetesimals.
As a function of radius…
How might we speed things up? The next key idea, called “runaway
growth,” follows from the concept of dynamical friction you have
been thinking about for Problem Set #3. Here’s a numerical version:
In this picture, the
larger bodies grow
most rapidly under
accretion, but the
process cannot
continue forever…
Three growth
regimes can be
depending on the
dispersion and disk
thickness, as
summarized nicely
by Armitage (2009):
This leads to the concept of an “isolation mass” once growth is well
into the shear dominated regime. Mathematically:
Where B ≈ 4 and rH = Hill Sphere:
Relatively evenly spaced, hard to excite to orbit
crossing eccentricities…. Growth slows dramatically.
The details of the growth of these oligarchs again depends on the
relative sizes of the planetesimals & their random velocities, the
escape velocities from the oligarchs, and the Keplerian field.
Goldreich, Lithwick & Sari (2004, GLS) define two limiting regimes:
Dispersion Dominated
Shear Dominated
Shear dominated oligarchy only operates over a fairly small size distribution
of the swarm, but can be very fast. Note also that the isolation mass can be
much larger in the outer solar system (if you assume that ice is available at
3-6×MMSN), which is good for gas accretion!
The planetary core end game:
What happens next? A few things…. For sufficiently large cores, of
order 10 MEarth, gas accretion can occur. We’ll cover that next. Smaller
radius objects? As GLS note, the key is again the relative sizes of the
escape velocity from the body to the escape velocity from the solar system:
Note again the timescales are >> gaseous disk lifetimes from astronomers.
We will talk more about ejection when we discuss the Kuiper Belt/Oort
Cloud and debris disks in future lectures. Numerically for a~1-few AU?
Is gas critical to the orbits of terrestrial planets?
(Gas surface density
runs from high to
low in these plots.)
Kominami & Ida 2002,
Icarus 157, 43-56; ibid
2004, 167, 231-243.