Infall of planetesimals onto growing giant planets: onset of runaway gas accretion and metallicity of their gas envelopes

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.