After protoplanets have acquired sufficient mass to open partial gaps in
their natal protostellar disks, residual gas continues to diffuse onto
some horseshoe streamlines under effect of viscous dissipation, and
meander in and out of the planets’ Hill sphere. Inside the Hill sphere,
the horseshoe streamlines intercept gas flow in circumplanetary disks.
The host stars’ tidal perturbation induce a barrier across the
converging streamlines’ interface. Viscous transfer of angular momentum
across this tidal barrier determines the rate of mass diffusion from the
horseshoe streamlines onto the circumplanetary disks, and eventually the
accretion rate onto the protoplanets. We carry out a series of numerical
simulations to test the influence of this tidal barrier on super-thermal
planets. In weakly viscous disks, protoplanets’ accretion rate steeply
decreases with their masses above the thermal limit. As their growth
time scale exceeds the gas depletion time scale, their masses reach
asymptotic values comparable to that of Jupiter. In relatively thick and
strongly viscous disks, protoplanets’ asymptotic masses exceed several
times that of Jupiter. Such massive protoplanets strongly excite the
eccentricity of nearby horseshoe streamlines, destabilize orderly flow,
substantially enhance the diffusion rate across the tidal barrier, and
elevate their growth rate until their natal disk is severely depleted.
Based on the upper fall-off in the observe mass distribution of known
exoplanets, we suggest their natal disks had relatively low viscosity (α
∼ 10−3), modest thickness (H/R ∼ 0.03 − 0.05), and limited masses
(comparable to that of minimum mass solar nebula model).