Terrestrial Planet Formation and Delivery of Water: Theory and Simulations
Primary work cited:
Raymond, S.N., Quinn, T., Lunine, J.I. 2004. Icarus 168:1-17

The authors present the results of simulations of late-stage planetary
accretion in order to motivate the formation terrestrial
planets of varying size and with varying volatile content,
most notably water.  The intent is not to specifically model
the planetary formation and end result of our own Solar System
but rather to vary initial parameters of a
post-runaway-accretion stage system in order to motivate the
possibility, and relative commonality, of a variety of
terrestrial planet systems.  In 44 simulations, a total of 111
terrestrial planets (a<2 AU, M>0.2MEarth) were formed, 11 in
the habitable zone between 0.9 and 1.1 AU out of 43 total
planets between 0.8 and 1.5 AU.  These 11 varied from dry to
100+ oceans (with 1 ocean = 1.5 x 1024 g of water, the amount
of water in Earth’s hydrosphere).

Since the purpose was not to specifically achieve the positions and
compositions of the four terrestrial planets of this Solar System
under the influence of Jupiter at its true mass and in its true orbit,
different initial conditions were given to the simulation; the
parameters of a Jupiter-like planet’s mass, eccentricity, and
semimajor axis were varied along with the time of formation, density
of solids in the protoplanetary disk, and location of the snow line.
The simulation began at the onset of late-stage dynamical evolution,
which was preceded by the initial condensation of grains from the
solar nebula, the early stage of accretion to km-sized planetesimals,
and the middle stage of runaway or oligarchic growth characterized by
the “rich get richer” gravitational accretion.  The late stage
is typified by the eventual perturbations of the otherwise relatively
stable orbits of large planetesimals, orbiting the central star at
regular intervals.  These perturbations induce eccentricities, even
the slightest of which can generate crossing orbits and collisions to
create terrestrial planets (Lissauer).

Specifically problematic in the application of terrestrial planet
formation theory to our own system is the fact that the snow line- the
line at which H20 sublimates, and inside which solids cannot have H20-
is believed to have been anywhere from roughly 2 to 5 AU in the solar
nebula, significantly outside the orbital distance of Earth.  One
hypothesis, the “late veneer,” in which the Earth was formed out
of materials originating at approximately its current distance from
the Sun and then subsequently became hydrated with a large number of
cometary impacts, is inconsistent with the observed D/H ratio of three
comets, which is twice the Earth’s oceanic value.  The alternative is
that hydration occurred when planetesimals accreted ice during the
late stage of evolution, during which high eccentricities were induced
in bodies formed outside the snow line; these bodies were eventually
perturbed into crossing orbits with the inner planetesimals, and their
anhydrous minerals were heated radioactively, frictionally, or through
collisions and released their H20 as liquid.  This proved to be a
reasonable theory, as terrestrial planets were formed ranging from
completely dry to having greater than 300 oceans of water.

Every one of the 44 simulations resulted in terrestrial planet
formation.  The parameter whose variation produced the largest effect
was that of planetesimal mass; a large (~0.1 MEarth) planetesimal mass
and a high surface density outside of the snow line gave terrestrial
planets with large masses and high water contents in comparison to the
results of simulations at the lower (0.01 MEarth) initial masses and
surface densities.  Jupiter’s eccentricity also proved significant, as
a higher eJ of 0.1 and 0.2 (similar to values observed in extrasolar
systems) (exoplanets.org) increased both the number of ejected bodies
and the number that collided with the Sun, and in this depleted system
the terrestrial planets were both smaller and volatile-poor.

Other references
http://exoplanets.org/almanacframe.html (October 28, 2004)
Lissauer, J.J. 1993. Annu. Rev. Astron. Astrophys. 31:129-74.