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N-BODY SIMULATIONS OF PLANET FORMATION: VARYING
THE NUMBER OF PLANETARY EMBRYOS
J.E. CHAMBERS
Armagh Observatory,
College Hill, Armagh, Northern Ireland, BT61 9DG, UK.
1. Introduction
The formation of planets like Earth probably took place in three stages
(Wetherill, 1990). Initially, grains of dust in the protoplanetary nebula un-
derwent gentle collisions, sticking together to form larger agglomerates.
These gained mass until they became large enough to gravitationally per-
turb other objects in the nebula. These perturbations gave the largest bod-
ies the most favourable orbits for accreting more material, which allowed
them to sweep up most of the solid material in the nebula. Finally, these
\planetary embryos" collided with one another to form a handful of planets
with long-lived orbits.
Chambers and Wetherill (1998) have simulated the nal stage of this
process using N-body integrations. Their simulations started with a few
tens of planetary embryos, moving on circular nearly-coplanar orbits in
the region now occupied by the terrestrial planets. Each simulation ended
with 1{4 planets moving on widely-spaced orbits. However, the mass and
orbital distributions of these objects dier signicantly from those of the
inner planets in the Solar System. At the time it was unclear whether
these dierences were signicant, or merely re ected the choice of initial
conditions. In this paper, I present the results of new simulations, and
compare these with those of Chambers and Wetherill (1998).
2. The Simulations
The simulations of Chambers and Wetherill (1998) began with a disk of
24{40 planetary embryos with semi-major axes 0:55 < a < 1:8 au. The
embryo masses ranged from  Mercury-sized at the inner edge of the disk,
to  1:5 times the mass of Mars at the outer edge. The orbits were initially

2 J.E. CHAMBERS
circular, with inclinations of 0:1 ф , and the remaining orbital elements were
distributed randomly.
The new simulations dier principally by having more planetary
embryos. Each integration started with 80{120 objects, with semi-major
axes 0:5 < a < 2:0 au. Unlike the earlier simulations, the initial mass dis-
tribution was either at, or chosen so that the largest embryos were in the
inner part of the disk, rather than the outer part as before. The integra-
tions were performed using a mixed-variable symplectic integrator, using a
Bulirsch-Stoer algorithm to follow close encounters between embryos.
3. The Results
Figure 1 shows the masses, m, and semi-major axes, a, of all the surviving
objects at the end of the simulations. The upper panel is a composite of
results from 9 of the old calculations, while the lower panel combines data
from 12 new simulations. The square symbols represent the 4 inner planets
of the Solar System. In the old integrations, there is a trend towards smaller
planets with increasing a, but little similarity with the terrestrial planets.
However, for the new integrations the m-a distribution is much closer to
that of the inner planets. In particular, the largest objects have 0:6 < a <
1:2 au, similar to Earth and Venus. Bodies outside this region tend to be
less massive, which is also true of Mercury and Mars, although most of the
objects produced by the simulations are larger than these planets.
Figure 2 shows the time-averaged eccentricities, e, versus mass for the
nal objects. In the old simulations, almost all the nal objects have orbits
that are more eccentric than the terrestrial planets. Eccentricities of 0.1 or
0.2 are common|much larger than the time-averaged values for Earth and
Venus. In the new simulations, the range of nal e values is similar, but,
the mean value is signicantly smaller. This is especially true for the more
massive bodies. Several of the objects have e similar to Earth and Venus,
although most of the objects from the simulations have more eccentric
orbits than these planets. There is also a weak trend towards decreasing e
with increasing mass, which is seen clearly in the terrestrial planets.
In conclusion, increasing the number of planetary embryos, N , at the
start of the nal stage of accretion produces a much better t to the ob-
served masses and semi-major axes of the terrestrial planets. In addition,
several of the objects produced by the simulations with N = 80{120 have
eccentricities comparable to Earth and Venus.
References
Chambers, J.E. and Wetherill, G.W. (1998) Making the Terrestrial Planets: N-Body
Integrations of Planetary Embryos in Three Dimensions, Icarus, 136, 304{327.
Wetherill, G.W. (1990) Formation of the Earth, Annu. Rev. Earth Planet. Sci., 18, 205{
256

N-BODY SIMULATIONS OF PLANET FORMATION 3
Figure 1. Mass versus semi-major axis, for the surviving objects at the end of 9 old
simulations using 24{40 initial bodies (upper) and 12 new simulations using 80{120 bodies
(lower). The square symbols show the 4 terrestrial planets for comparison.

4 J.E. CHAMBERS
Figure 2. Time-averaged eccentricity versus mass, for the surviving objects at the end of
9 old simulations using 24{40 initial bodies (upper) and 12 new simulations using 80{120
bodies (lower). The square symbols show the 4 terrestrial planets for comparison.