  We have investigated the planetesimal accretion rate onto giant planets that
are growing through gas accretion, using numerical simulations and analytical
arguments. We derived the condition for gap opening in the planetesimal disk,
which is determined by a competition between the expansion of the planet's Hill
radius due to the planet growth and the damping of planetesimal eccentricity
due to gas drag. We also derived the semi-analytical formula for the
planetesimal accretion rate as a function of ratios of the rates of the Hill
radius expansion, the damping, and planetesimal scattering by the planet. The
predicted low planetesimal accretion rate due to gap opening in early gas
accretion stages quantitatively shows that "phase 2," which is a long slow gas
accretion phase before onset of runaway gas accretion, is not likely to occur.
In late stages, rapid Hill radius expansion fills the gap, resulting in
significant planetesimal accretion, which is as large as several $M_{\oplus}$
for Jupiter and Saturn. The efficient onset of runaway gas accretion and the
late pollution may reconcile the ubiquity of extrasolar giant planets with
metal-rich envelopes of Jupiter and Saturn inferred from interior structure
models. These formulae will give deep insights into formation of extrasolar gas
giants and the diversity in metallicity of transiting gas giants.
