Radiating Bondi Flows II: Giant Planet Accretion Models

Abstract

In the core accretion model of giant planet formation, the late stages of runaway growth are regulated by the hydrodynamic infall of gas from the protoplanetary disk. For a subset of planet-disk pairings, this scenario is analogous to the classical Bondi problem, which has motivated a Bondi-like parameterization of accretion in some population synthesis models. Existing models and the associated classical Bondi rate however, are predicated upon an adiabatic equation of state. In reality, the planet and its associated accretion shock supply a luminosity that substantially heats the accretion flow. In Paper I of this series, we demonstrate that such radiative feedback can dramatically suppress accretion by orders of magnitude. Here we quantify this effect under realistic planet-forming conditions. We find that for planets forming in an unperturbed disk, accretion is suppressed by 1-2 orders of magnitude interior to $\sim 10$ AU. For planets that open a gap, this feedback is less dramatic and the effect is $\sim$ 1 order in magnitude. We investigate the effect of various assumptions regarding dust opacities, shock efficiency, and planet radius and find this radiative suppression mechanism to be fairly insensitive to these effects. We also perform full time-dependent simulations demonstrating that the associated adverse entropy profiles are accurate and stable to convection. A simple and flexible set of open-source tools are provided to incorporate this radiative feedback into existing accretion models and population synthesis frameworks.

Figures (7)

• Figure 1: Steady-state accretion rates relative to the adiabatic Bondi rate as a function of planet mass and orbital distance. Left column shows models with dust opacities that adopt a maximum particle size $a_{\rm max}=1$ cm. Right column shows models from the grid testing a different opacity by changing the maximum particle size to $a_{\rm max}=10\mu$m. Upper rows select models with a luminosity assuming all accretion energy is converted to radiation at a planetary boundary layer shock. Lower rows assume this energy conversion is $50\%$ efficient (or a doubling of shock radius which is degenerate with efficiency). Regions where the model grid of this work found no steady-state transonic accretion solution are left blank. Contours of $f_{\rm acc}$ are also overlaid at the $[10^{-2}, 10^{-1}, 10^0]$ with solid lines. At the same levels, dashed contours estimating the accretion rates using the analytic formula (B7) from Paper I are shown for comparison.
• Figure 2: Contours of the characteristic optical depth $\tau_B$ ($\sim \rho_\infty$) across the fiducial parameter space for gap and no-gap models. Their ratio is the gap depth $D$.
• Figure 3: Mass accretion rates in the style of Figure \ref{['fig:fiducial']}, but for models incorporating the density perturbation induced by a gap. Dotted lines are drawn where $t_{\rm gap} = t_{\rm acc}$ and $t_{\nu} = t_{\rm gap}$.
• Figure 4: Comparison of the Mach number (left), entropy (middle), and luminosity (right) returned by solving the steady-state equations (dotted lines), 1D simulations (dashed lines) and 2D simulations (solid lines). Each set of curves correspond to the solution for a $50M_\oplus$ planet placed at one of 2, 10, or 100 AU. Symbols labeling each set of curves are placed at the sonic point. For 2D simulations, the radial profiles along all angles are plotted with our "forcing" at the outer boundary setting the spread in the ordinate.
• Figure 5: Radial variation of timescales associated with each 1D simulation.