Resizing the giants: How modelling adiabatic interiors impacts predicted planetary radii

Abstract

The interiors of giant planets are commonly assumed to be convective and therefore adiabatic, making the adiabatic temperature gradient a key ingredient in interior and evolution models. While there are multiple numerically distinct ways to compute this gradient, their impact on inferred planetary structure and radius has not been systematically assessed. We investigate how the numerical treatment of adiabatic temperature profiles affects inferred planetary radii and internal structure, quantifying the impact of different methods for calculating the adiabatic gradient and different forms of the temperature differential equation on static interior models. We computed static interior models of a one Jupiter mass planet using a state-of-the-art hydrogen-helium equation of state, comparing five methods for evaluating the adiabatic gradient against a ground-truth isentropic baseline, for both the logarithmic and non-logarithmic forms of the temperature equation. The choice of numerical method significantly impacts the inferred interior structure and the radius. With the logarithmic temperature equation, central temperatures deviate by several thousand Kelvin and surface radii differ by up to 3.4%, exceeding the ~1% precision of current giant exoplanet radius measurements threefold. The non-logarithmic form reduces deviations to below ~1% for most methods. We therefore recommend spline derivatives to evaluate the adiabatic gradient via the triple-product rule, combined with the non-logarithmic temperature equation. Finite differencing and direct use of tabulated gradients or derivatives should be avoided.

Figures (5)

• Figure 1: A comparison of the resulting adiabatic temperature and density profiles for different methods of calculating $\nabla_{\rm{ad}}$ (see legend and text for details). The dashed lines show the baseline profile calculated by enforcing an isentropic interior. The top two rows show the resulting temperature profiles and their relative difference compared to the baseline. The bottom two rows show $\rho/\sqrt{P}$ and the relative difference in the density compared to the baseline.
• Figure 2: Same as Figure \ref{['fig:planet_solution_nabla_ad']}, but using the non-logarithmic version of the differential equation to determine the adiabatic temperature profile.
• Figure 3: Comparison of interior profiles for different integration methods of Equation \ref{['eq:diff_eq_log']}. The baseline isentropic profiles are shown in dashed black lines. See text for details.
• Figure 4: Comparison of interior profiles when solving the logarithmic temperature equation (Eq. \ref{['eq:diff_eq_log']}) for different method of calculating the adiabatic gradient (see legend and Section \ref{['sec:results']}). The baseline isentropic profiles are shown in dashed black lines. There is no clear best-performing method since it changes depending on the surface temperature.
• Figure 5: Comparison of interior profiles when solving the non-logarithmic temperature equation (Eq. \ref{['eq:diff_eq']}) for different method of calculating the adiabatic gradient (see legend and Section \ref{['sec:results']}). In contrast to the results in Figure \ref{['fig:nabla_ad_comparison']}, there are now methods that clearly perform worse independent of the surface temperature.