The Evolution of Jupiter and Saturn as a function of the Parameter R$_ρ$

TL;DR

This work tackles the problem of interior stratification in Jupiter and Saturn by incorporating helium rain, fuzzy heavy-element cores, and non-adiabatic processes into evolutionary models. It introduces a global convective-criterion parameter $R_ ho$ to bracket the Ledoux ($R_ ho=1$) and Schwarzschild ($R_ ho=0$) limits and systematically explores a wide range of interior configurations with the APPLE code, updated EOS, and atmosphere boundary conditions to reproduce observables at 4.56 Gyr. The main findings show that lower $R_ ho$ increases interior mixing, producing higher atmospheric metallicities and cooler deep interiors, while reducing the total heavy-element mass needed and lowering initial entropies; the final states remain consistent with measured $T_{ m eff}$, $R_{ m eq}$, and gravitational moments. Saturn, in particular, displays Brunt-Väisälä frequency profiles that align with Cassini seismology, supporting a compositionally stratified interior with helium rain and a diluted core. Overall, the study provides a practical framework bridging extreme convective regimes and highlights the need for a self-consistent semiconvection theory in giant-planet evolution, with important implications for interpreting interior structure from gravity and seismology data.

Abstract

Computed using the APPLE planetary evolution code, we present updated evolutionary models for Jupiter and Saturn that incorporate helium rain, non-adiabatic thermal structures, and "fuzzy" extended heavy-element cores. Building on our previous Ledoux-stable models, we implement improved atmospheric boundary conditions that account for composition-dependent effective temperatures and systematically explore the impact of varying the parameter $R_ρ$, which allows one to explore in an approximate way the efficiency of semiconvection. For both Jupiter and Saturn, we construct models spanning from $R_ρ=1$ (Ledoux) to $R_ρ=0$ (Schwarzschild), and identify best-fit solutions that match each planet's effective temperature, equatorial radius, lower-order gravitational moments, and atmospheric composition at 4.56 Gyr. We find that lower $R_ρ$ values lead to stronger convective mixing, resulting in higher surface metallicities and lower deep interior temperatures, while requiring reduced heavy-element masses and lower initial entropies to stabilize the dilute inner cores. Our Saturn models also broadly agree with the observed brunt frequency profile inferred from Cassini ring seismology, with stable layers arising from both the helium rain region and the dilute core. These findings support the presence of complex, compositionally stratified interiors in both gas giants.

Figures (5)

• Figure 1: Evolutionary model for Jupiter with an initial fuzzy core that experiences Schwarzschild convection ($R_{\rho}=0$). This model reproduces, within $\sim0.7$% of the various measured quantities, the current values of the effective temperature, equatorial radius, atmospheric helium abundance, outer envelope metallicity, and $J_2$ and $J_4$ gravitational moments. The initial outer entropy is 7.8 k$_{\rm B}$/baryon, while the interior entropy is 5.8 k$_{\rm B}$/baryon. The panels display: (top left) the evolution of the temperature profile; (top middle) the evolution of the helium mass fraction ($Y$) profile; (top right) the evolution of the effective temperature and equatorial radius; (bottom left) the evolution of the entropy profile; (bottom middle) the evolution of the $Z$ profile; and (bottom right) the evolution of the gravitational moments $J_2$ and $J_4$. This Jupiter model contains 38 M$_{\oplus}$ of heavy elements, including a compact core of 1 M$_{\oplus}$ ($Z = 1$). The outer metallicity starts at 1.9 Z$_{\odot}$ and increases to 3.66 Z$_{\odot}$. Helium rain begins at $\sim 3.5$ Gyrs, based on the LHR0911 miscibility curves with a -75 K temperature shift, resulting in outer helium depletion to $Y = 0.233$, consistent with Galileo entry probe measurements vonZahn1998.
• Figure 2: Best-fit Saturn evolutionary model assuming an initially fuzzy core and Schwarzschild convection ($R_{\rho}=0$). This model reproduces Saturn’s present-day effective temperature, equatorial radius, outer envelope metallicity, and gravitational moment, $J_2$ within 0.5% of their measured values. The initial entropy is 7.0 k$_\mathrm{B}$/baryon at the outer boundary and $\sim$4.0 k$_\mathrm{B}$/baryon in the deep interior. The six panels mirror those shown in Figure \ref{['fig:jupiter_schw_best_fit']}: (top left) temperature profile evolution; (top middle) helium mass fraction ($Y$) profile; (top right) evolution of effective temperature and equatorial radius; (bottom left) entropy profile; (bottom middle) heavy-element mass fraction ($Z$) profile; and (bottom right) gravitational moments $J_2$ and $J_4$. The model contains a total of 24.5 $M_\oplus$ in heavy elements, including a compact 1.5 $M_\oplus$ core. The outer envelope metallicity rises from an initial value of 2.0 $Z_\odot$, to 9 $Z_\odot$ due to interior mixing. The LHR0911 hydrogen–helium miscibility curve shifted by -75 K (equivalent to the corresponding Jupiter model) leads to an atmospheric helium mass fraction of $Y_{\rm atm} \sim 0.168$, in agreement with constraints from Conrath2000 and Koskinen2018 (pink shaded region in the top middle panel). The miscibility boundary and helium rain region for the current epoch of the best-fit model are highlighted using a blue field in the top left panel.
• Figure 3: Best fit evolutionary models for Jupiter with varying $R_{\rho}$. Only the initial and final (4.56 Gyr.) timestamps are shown for comparison. The dark blue curve corresponds to our updated Ledoux model ($R_{\rho} = 1$), while the red curve shows the Schwarzschild model ($R_{\rho} = 0$). The light blue, green, and orange curves represent intermediate cases with $R_{\rho} = 0.75$, $0.50$, and $0.25$. All quantities are within 1$\sigma$ of their measured central values.
• Figure 4: Same as Figure \ref{['fig:jupiter_rho_comp']}, but for Saturn: best-fit evolutionary models with varying $R_{\rho}$ values. As $R_{\rho}$ decreases, the models exhibit higher effective temperatures during early evolution, enhanced final envelope metallicities, broader helium rain regions, and more helium depletion at the surface. The total heavy-element mass and initial entropies are adjusted accordingly to reproduce Saturn’s present-day structure, with the Schwarzschild model requiring lower initial central temperatures and a more compact core.
• Figure 6: Brunt-Väisälä frequency ratios for Jupiter and Saturn at 4.56 Gyr for models with varying $R_{\rho}$. The shaded region in panel b indicates the seismologically inferred stratified zone from Mankovich2021, with a characteristic ratio of 2 extending to 60% of Saturn’s radius. Our models reveal comparable stable regions—one associated with the dilute core and another arising from the helium rain layer in both planets. The location of the peaks of our distributions for Saturn matches well with the predictions of MF2021. The raw brunt data have been smoothed out for better visualization in this plot.