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How do planetesimals grow to form ~terrestrial mass cores?. Ge/Ay133. For now, let’s ignore the gas. This means we can just worry about gravity. For the pairwise interaction of two bodies, we have:. r=a 1. b. r=a 2. For collisions that are grazing, the

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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/aEarththen 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 dispersion-limited accretion, but the process cannot continue forever…

Three growth regimes can be discerned, depending on the escape velocity/swarm 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.