  [Abridged] We present an extensive suite of terrestrial planet formation
simulations that allows quantitative analysis of the stochastic late stages of
planet formation. We quantify the feeding zone width, Delta a, as the
mass-weighted standard deviation of the initial semi-major axes of the
planetary embryos and planetesimals that make up the final planet. The size of
a planet's feeding zone in our simulations does not correlate with its final
mass or semi-major axis, suggesting there is no systematic trend between a
planet's mass and its volatile inventory. Instead, we find that the feeding
zone of any planet more massive than 0.1M_Earth is roughly proportional to the
radial extent of the initial disk from which it formed: Delta
a~0.25(a_max-a_min), where a_min and a_max are the inner and outer edge of the
initial planetesimal disk. These wide stochastic feeding zones have significant
consequences for the origin of the Moon, since the canonical scenario predicts
the Moon should be primarily composed of material from Earth's last major
impactor (Theia), yet its isotopic composition is indistinguishable from Earth.
In particular, we find that the feeding zones of Theia analogs are
significantly more stochastic than the planetary analogs. Depending on our
assumed initial distribution of oxygen isotopes within the planetesimal disk,
we find a ~5% or less probability that the Earth and Theia will form with an
isotopic difference equal to or smaller than the Earth and Moon's. In fact we
predict that every planetary mass body should be expected to have a unique
isotopic signature. In addition, we find paucities of massive Theia analogs and
high velocity moon-forming collisions, two recently proposed explanations for
the Moon's isotopic composition. Our work suggests that there is still no
scenario for the Moon's origin that explains its isotopic composition with a
high probability event.
