The Origin of the Solar System 43
is time for their mutual gravitational attraction to focus their
trajectories toward each other, soFgis large, and the chance
of a collision is high. Fast moving bodies typically do not
collide unless they are traveling directly toward each other
becauseFg∼=1 in this case. The relative velocities of plan-
etesimals depend on their orbits about the Sun. Objects
with similar orbits are the most likely to collide with each
other. In particular, planetesimals moving on nearly circular,
coplanar orbits have high collision probabilities while ones
with highly inclined, eccentric (elliptical) orbits do not.
Most close encounters between planetesimals did not
lead to a collision, but bodies often passed close enough
for their mutual gravitational tug to change their orbits.
Statistical studies show that after many such close encoun-
ters, high-mass bodies tend to acquire circular, coplanar
orbits, while low-mass bodies are perturbed onto eccen-
tric, inclined orbits. This is called dynamical friction and is
analogous to the equipartition of kinetic energy between
molecules in a gas. Dynamical friction means that, on av-
erage, the largest bodies in a particular region experience
the strongest gravitational focusing; therefore, they grow
the fastest (Fig. 17). This state of affairs is calledrunaway
growthfor obvious reasons. Most planetesimals remained
small, while a few objects, calledplanetary embryos, grew
much larger.
Runaway growth continued as long as interactions be-
tween planetesimals determined their orbital distribution.
However, once embryos became more than about a thou-
sand times more massive than a typical planetesimal, gravi-
FIGURE 17 Runaway growth of a few large planetesimals takes
place due to a combination of dynamical friction (which gives
large planetesimals circular and coplanar orbits), and
gravitational focusing (which increases the chance of a collision
between bodies moving on similar orbits).
tational perturbations from the embryos became more im-
portant. The evolution now entered a new phase called
oligarchic growth. The relative velocities of planetesi-
mals were determined by a balance between perturba-
tions from nearby embryos and damping due to gas drag.
Embryos continued to grow faster than planetesimals, but
growth was no longer unrestrained. Large embryos stirred
up nearby planetesimals more than small embryos did,
weakening gravitational focusing and slowing growth. As
a result, neighboring embryos tended to grow at similar
rates. Embryos spaced themselves apart at regular radial
intervals, with each one staking out an annular region of
influence in the nebula called a feeding zone.
As embryos became larger, they perturbed planetesimals
onto highly inclined and eccentric orbits. The planetes-
imals began to collide with one another at high speeds,
causing fragmentation and breakup. A huge number of
sub-kilometer-sized collision fragments were generated, to-
gether with a second generation of fine dust particles. Gas
drag operates efficiently on small fragments, so their or-
bits rapidly became almost circular and coplanar. As a re-
sult, many fragments were quickly swept up by embryos,
increasing the embryos’ growth rates still further.
Numerical calculations show that embryo feeding zones
were typically about 10 Hill radii in width, where the Hill
radius of an embryo with massMand orbital radiusais
given byrh=a(
M
3 Msun) (^1) / 3
(10)
If an embryo were to accrete all of the solid material in its
feeding zone it would stop growing when its mass reached
a value called theisolation mass, given by
Miso∼=
(
8 b^3 π^3 ^3 solida^6
3 Msun
)^1 / 2
(11)
whereis the surface (column) density of solid material
in that region of the disk, and b≈10 is the width of a
feeding zone in Hill radii. The surface density in the Sun’s
protoplanetary nebula is not known precisely, but for plau-
sible values, the isolation masses would have been about
0.1 Earth masses at 1 AU, and around 10 Earth masses
in the outer solar system. Calculations suggest that bodies
approached their isolation mass in the inner solar system
roughly 10^5 years after planetesimals first appeared in large
numbers. Growth was slower in the outer solar system, but
bodies were probably nearing their isolation mass at 5 AU
after 10^6 years.
Large embryos significantly perturbed nearby gas in the
nebula forming spiral waves. Gas passing through these
waves had a higher density than that in the surrounding
region. Gravitational interactions between an embryo and
its spiral waves transferred angular momentum between