Circular Problems
Douglas P. Hamilton

Probably the most striking regularity observed in our Solar System is
that the planets all follow nearly circular, nearly coplanar orbits
about the Sun. This feature lends strong support to the theory that
the planets accreted out of the solar nebula - a flattened disk of gas
and dust that surrounded the young Sun. But do flattened disks of dust
and gas inevitably lead to a few massive widely-spaced planets on
circular uninclined orbits? In the case of the inner Solar System,
results from recent numerical simulations* suggest that this may not
be the case.

The formation of the terrestrial planets - Mercury, Venus, Earth, and
Mars - can be loosely divided into three consecutive
stages. Initially, dust grains move slowly through the solar nebula,
bumping into one another in low velocity "sticking" collisions that
gradually build up kilometer-sized "planetesimals"(1). Next, gravity
from the planetesimals begins to enhance the accretion rate; this
second stage is characterized by a rapid period of "run-away growth"
in which the largest planetesimals outcompete and ultimately absorb
their smaller neighbors(2).  During these first two stages of
planetary formation, gas drag from the solar nebula removes energy
from the orbits of solid bodies, and damps away their radial and
vertical motions as well.  Thus solid bodies are thought to follow
circular uninclined orbits that gradually evolve inward toward the
Sun. The inward migration brings planetesimals together, and their
nearly circular orbits lead to low relative velocities and sticking
collisions, rather than to energetic impacts that can shatter
planetesimals.

This simple picture is altered by the gravitational perturbations of
Jupiter and Saturn, which are traditionally thought to have formed
after 10^6-10^7 years(3) (roughly around the time that there are
several dozen Mercury-sized "protoplanets" in the terrestrial region),
but which alternate theories suggest may have formed as quickly as
100-1000 years due to nebular gas instability(4). Working under the
assumption that the shorter timescale is correct, S. Kortenkamp and
G. Wetherill (Carnegie Institute) perform numerical simulations of the
second stage of planetary formation and find that when both gas drag
and jovian perturbations are important, different-sized planetesimals
follow slightly eccentric orbits which are out of phase with one
another(5). Collisions between these objects are often energetic
enough to prevent accretion, thus hindering the formation of
protoplanets.  The magnitude of the effect depends sensitively on the
orbits of the giant planets; it is diminished by the fact that Jupiter
probably formed further from the Sun than it is now, subsequently
migrating inward over 10^7-10^8 years(6). Nevertheless, if Jupiter
formed quickly, this mechanism slows the runaway growth timescales for
the terrestrial planets, and significantly hinders the formation of
large planetesimals in the asteroid belt. Additionally, it suggests
that if large Jupiter-mass planets in other planetary systems formed
quickly rather than slowly, then the zone in which Earth-like planets
could form might be significantly reduced.

At the beginning of the third and final stage of terrestrial planet
formation the largest objects have attained sizes of small planets
(~3000km). Several dozen to several hundred of these "protoplanets"
perturb one another gravitationally and merge in giant collisions
until ultimately only the four terrestrial planets and the Moon
remain. New numerical simulations of this stage of planetary formation
were reported by two teams: J. Chambers (Armagh Observatory)(7) and
C. Agnor (U. Colorado), R. Canup, H. Levison (SWRI). These and
other(8) groups follow ~50 protoplanets for ~10^8 years. With
reasonable starting conditions, their models produce inner planetary
systems with roughly the correct number, sizes, and orbital spacings
of planets. All groups, however, find that Earth-sized objects have
orbital eccentricities and inclinations that are 5-10 times larger
than observed for the present-day Earth and Venus. The eccentricities
and inclinations are pumped up early in the simulations, primarily by
secular interactions between the orbits of the protoplanets(7).

So how did Venus and Earth end up on nearly circular orbits? There are
only a few possibilities. First, the simulations may not include some
critical physics which damps orbital eccentricities and inclinations
(e.g. dynamical friction with smaller objects or larger than expected
nebular gas drag) which would cause planets on circular orbits to be
formed more readily than the simulations suggest. Second, perhaps the
inner planets in our solar system did form with initially large
eccentricities and inclinations and these have been subsequently
damped by some dissipative force (e.g. solar tides, although these are
thought to be too weak) over 4.5 billion years of Solar System
history. Finally, perhaps elliptical orbits, rather than circular
ones, are indeed the natural product of planetary formation.  Using
the anthropic principal, J. Chambers points out that if nearly
circular orbits are necessary for climatic stability and the
development of intelligent life, then Earth's low eccentricity can be
understood. This argument is somewhat unsatisfying, however, in that
it explains neither the low inclinations of Venus and Earth, nor the
low eccentricity of Venus.

Douglas P. Hamilton is in the Department of Astronomy, University of
Maryland, College Park, Maryland 20742, USA.

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* Reported at the 30th annual DPS meeting, Madison, Wisconsin,
Oct. 11-16, 1998.

(1) Weidenschilling, S.J. & Cuzzi, J.N. Protostars and Planets III,
pp. 1031-1060 (1993).

(2) Wetherill, G.W. & Stewart, G.R. Icarus 106, 190-209 (1993).

(3) Pollack, J.B., Hubickyj, O. & Greenzweig, Y. Icarus 124, 62-85
(1996).

(4) Boss, A.P. Science, 276, 1836-1839 (1997).

(5) Fernandez, J.A. & Ip, W.H. Plan. Spa. Sci. 44, 431-439 (1996).

(6) Kortenkamp, S.J. & Wetherill, G.W. Icarus, Submitted (1998).

(7) Chambers, J.E. & Wetherill, G.W. Icarus, Submitted (1998).

(8) Ito, T. & Tanikawa, K. Icarus, Submitted (1998).