Solar System Formation & Dynamics

The rate at which planets grow can be estimated from a statistical argument. Let us consider a protoplanet of mass , radius and surface escape speed , orbiting within a very large number of smaller planetesimals. The planetesimals have a surface density and a velocity dispersion .

The idea for calculating the rate of growth is to assume that the planetesimals, rather than being treated individually, can be considered as a “gas” of very many “molecules” that is accreted by the planet. The first step is thus to calculate the mid plane density of the planetesimal swarm. If we consider a single planetesimal, most of its velocity is circular motion about the star. However, it also has a small out-of-plane velocity, of the order of , which results in an inclination,

.

The vertical excursions of the planetesimal above and below the mid plane are thus,

,

with the Keplerian angular velocity. We conclude that the disk of planetesimals should have a vertical thickness , and a volume density,

.

The statistical estimate of the growth rate of the planet is then just the density of the planetesimal disk, multiplied by the velocity at infinity and by the collision cross-section. Accounting for gravitational focusing, we have,

.

Substituting for the density, we find,

.

This calculation already reveals a number of important points, some obvious and some not so obvious:

• The growth rate depends linearly on the surface density of the planetesimal population
• Growth is faster where the angular velocity is higher, i.e. in the inner disk
• The velocity dispersion enters only via the gravitational focusing term

We can get an idea for how growth proceeds by considering the limit where the velocity dispersion of the planetesimals (recall that this is the random component of their motion) is small compared to the escape speed of a growing protoplanet. In this limit the gravitational focusing enhancement to the cross section is large, and the term in parenthesis simplifies to,

.

(Here we have substituted for the escape velocity in terms of the mass and radius.) We can simplify this by noting that , so that . The formula for the rate of planetary growth then takes the form (considering only the mass and radius dependent terms),

.

The predicted rate of growth increases more rapidly than linearly with the mass! This result, which follows directly from the role that gravitational focusing plays in increasing the collision cross section, is important. It implies that if we start from a uniform population of small bodies, one object that happens to get “lucky” and grow faster initially is thereafter able to consistently accrete more rapidly than the average (smaller) object. This “rich get richer” mode of growth is described as runaway growth.