  Standard accretion disk models suggest that the snow line in the solar nebula
migrated interior to the Earth's orbit in a late stage of nebula evolution. In
this late stage, a significant amount of ice could have been delivered to 1 AU
from outer regions in the form of mm to dm-sized pebbles. This raises the
question why the present Earth is so depleted of water (with the ocean mass
being as small as 0.023% of the Earth mass). Here we quantify the amount of icy
pebbles accreted by terrestrial embryos after the migration of the snow line
assuming that no mechanism halts the pebble flow in outer disk regions. We use
a simplified version of the coagulation equation to calculate the formation and
radial inward drift of icy pebbles in a protoplanetary disk. The pebble
accretion cross section of an embryo is calculated using analytic expressions
presented by recent studies. We find that the final mass and water content of
terrestrial embryos strongly depends on the radial extent of the gas disk, the
strength of disk turbulence, and the time at which the snow lines arrives at 1
AU. The disk's radial extent sets the lifetime of the pebble flow, while
turbulence determines the density of pebbles at the midplane where the embryos
reside. We find that the final water content of the embryos falls below 0.023
wt% only if the disk is compact (< 100 AU), turbulence is strong at 1 AU, and
the snow line arrives at 1 AU later than 2-4 Myr after disk formation. If the
solar nebula extended to 300 AU, initially rocky embryos would have evolved
into icy planets of 1-10 Earth masses unless the snow-line migration was slow.
If the proto-Earth contained water of ~ 1 wt% as might be suggested by the
density deficit of the Earth's outer core, the formation of the proto-Earth was
possible with weaker turbulence and with earlier (> 0.5-2 Myr) snow-line
migration.
