Hot Jupiters are Inflated Primarily by Shallow Heating

TL;DR

This paper addresses the hot Jupiter radius inflation problem by developing a thermal evolution model that combines deep interior heating with delayed cooling, allowing for substantial shallow heating near the radiative-convective boundary. Using a homogeneous, self-consistent catalog of host-star and planetary properties, the authors apply a two-level hierarchical Bayesian framework to infer population-level heating parameters, marginalizing over planet-specific properties. They find that interior cooling is reduced by roughly a factor of 20–60 compared with deep-heating-only models, implying that shallow heating accounts for the majority of inflation (roughly 95–97% of the heating) and enabling reinflation with modest deep heating. The results favor shallow heating mechanisms such as Ohmic dissipation and advection, with testable predictions for phase curve offsets as a function of equilibrium temperature, and they emphasize the importance of incorporating shallow heating in atmospheric and circulation models for hot Jupiters.

Abstract

The unexpectedly large radii of transiting hot Jupiters have led to many proposals for the physical mechanisms responsible for heating their interiors. While it has been shown that hot Jupiters reinflate as their host stars brighten due to heating deep in planetary interiors, young hot Jupiters also exhibit signs of delayed cooling possibly related to heating closer to their surfaces. To investigate this tension, we enhance our previously published hot Jupiter thermal evolution model by adding a parameter that allows for both deep heating and delayed cooling. We fit our thermal evolution models to a homogeneous, physically self-consistent catalog of accurate and precise hot Jupiter system properties in a hierarchical Bayesian framework. We find that hot Jupiters' interior cooling rates are reduced on average by 95\%--98\% compared to simpler anomalous heating models. The most plausible explanation for this inference is substantial shallow heating just below their radiative--convective boundaries that enables reinflation with much less deep heating. Shallow heating by Ohmic dissipation and/or temperature advection are therefore important components of accurate models of hot Jupiter atmospheres, especially in circulation models. If hot Jupiters are inflated primarily by shallow heating as we propose, then we predict that their observed phase curve offsets should increase with temperature in the range $T_{\text{eq}}~\lesssim1500~\text{K}$, peak in the range $1500~\text{K}~\lesssim~T_{\text{eq}}~\lesssim~1800~\text{K}$, and decrease in the range $T_{\text{eq}}~\gtrsim~1800~\text{K}$.

Figures (4)

• Figure 1: Demonstration of our thermal evolution model's interior heating parameterization and its effects on the modeled planet's radius evolution. In the top panel, we plot as the black line a 1 M$_\odot$ host star's luminosity as a function of age, highlighting the pre- and post-main sequence region as the shaded areas. In the bottom panel, we plot as colored lines the 1 M$_{\text{Jup}}$ planet's radius assuming (1) standard thermal evolution (light blue), (2) only deep interior heating (dark blue), (3) only delayed cooling (light green), and (4) both deep heating and delayed cooling (dark green). Deep heating allows the planet to reinflate as the host star's luminosity increases, while delayed cooling slows the rate of contraction.
• Figure 2: Demonstration of the effects of differing deep heating fractions or initial specific entropies on a planet's radius evolution. We plot as colored lines the radius of a 1 M$_{\text{Jup}}$ planet with a 10 M$_\oplus$ core and envelope metallicity $Z_{\text{env}}=0.1$ orbiting a 1 M$_\odot$ star at 0.05 AU as a function of time, varying the fraction of total heating caused by deep heating f$_{\text{DH}}$ (assuming $s_{\text{i}} = 10~k_{B}/$baryon) in the upper panel and the initial specific entropy $s_{\text{i}}$ (assuming f$_{\text{DH}} = 10^{-1.5}$) in the lower panel. Larger deep heating fractions cause the planet to reach its equilibrium radius quickly, while smaller deep heating fractions keep it inflated for a longer period of time. At larger values of $s_{\text{i}}$, the planet begins its thermal evolution at a larger radius before cooling over several Gyr. Though the effect of changing $s_{\text{i}}$ vanishes after about 500 Myr, different assumptions for it in our individual retrievals could affect the inferred interior structure of the youngest planets in our sample. Our analysis marginalizes over the hot Jupiter population to extract the population-level aggregate f$_{\text{DH}}$, so we experiment with choosing several values of $s_{\text{i}}$ in our hierarchical Bayesian inference to mitigate any bias due to only exploring one value of it.
• Figure 3: Example interior structure retrieval corner plot for WASP-6 b (left, green) and CoRoT-25 b (right, blue), assuming an initial specific entropy of 10 k$_B$ / baryon. The parameters we fit and use as inputs in our model are described in detail in Table \ref{['tab:bayesian']}. Due to degeneracies between $Z^i$ and f$_{\text{L}}$, it is impossible to infer f$_{\text{L}}$ using only a single planet. This degeneracy is only present for younger planets like WASP-6 b, meaning that older planets like CoRoT-25 b are instead useful for constraining differences caused by the use of the updated equation of state.
• Figure 4: Corner plots for the hyperparameters of our population-level intrinsic luminosity reduction factor $f_{\text{L}}$ and population mean heat factor $H$. Each posterior's model assumes a different initial specific entropy for the individual interior structure retrievals it marginalizes over. Not shown are the individual planets' heat factor hyperparameters, which are included to account for small differences in the theoretical equation of state used between this work and Thorngren2018. We find that 2-3% of the heating hot Jupiters experience can be attributed to deep heating, regardless of our assumption for the initial specific entropy.