Saturn's Evolutionary History and Seismology: Survival of Deep Stably Stratified Regions in Evolutionary Models of Saturn Consistent with Ring Seismology

Abstract

With recent advances in the modeling of the solar system giant planets, rapid progress has been made in understanding the remaining questions pertaining to their formation and evolution. However, this progress has largely neglected the significant constraints on the interior of Saturn's structure imposed by the observed oscillation frequencies in its rings. Here, we study initial conditions for Saturn's evolution that, after $4.56\;\mathrm{Gyr}$ of evolution, give rise to planetary structures admitting oscillation frequencies consistent with those observed via Saturn's ring seismology. Restricting our attention to models without compact rocky cores, we achieve simultaneous good agreement with most observed properties of Saturn at the level of current evolutionary models and with key frequencies in the observed oscillation spectrum. Our preliminary work suggests that Saturn's interior stably stratified region may be moderately less extended ($\sim 0.4$--$0.5R_{\rm Sat}$) than previously thought, which is important for reconciling the seismic constraints with evolutionary models. We also tentatively find that the deep helium gradients inferred by previous, static structural modelling of Saturn's ring seismology may not be required to reproduce the observed seismology data.

Figures (8)

• Figure 1: Evolution of the best-fitting Saturn model from sur2025_fuzzycores with its $4M_{\oplus}$ compact core removed and restributed uniformly throughout the envelope. The left four panels show the profiles of the internal temperature, entropy, heavy element mass fraction ($Z$), fractional helium mass fraction ($Y' \equiv Y / (1 - Z)$), and the Brunt-Väisälä frequency $N$ normalized to $\Omega_{\rm dyn} \equiv \sqrt{G M_{\rm Sat} / R_{\rm eq}^3}$ as functions of the normalized mass coordinate of Saturn at four times (legend in upper middle panel). The large variations in $N$ are due to the discontinuities in the composition and thermal profiles of the planet ("stair steps"), which are commonly seen in 1D planetary evolution models vazan2018jupitertejadaarevalo2025_jupiterfuzzy. The smoothed $N$ profile that is used to compute oscillation modes is shown as the faint, thick red line. The right two panels show the temporal evolution of the effective temperature $T_{\rm eff}$, the equatorial radius (in units of the measured equatorial radius of Saturn given in Table \ref{['tab:params']}), and the gravitational zonal harmonics $J_2$ and $J_4$. The vertical grey dashed line in the right two panels shows $t = 4.56\;\mathrm{Gyr}$, the current age of Saturn. The final rotation rate is $1.01 P_{\rm Sat}$. For comparison, the faint dashed lines in the right two panels show the evolution for the original model from sur2025_fuzzycores with a rocky core. We see that the presence of the rocky core reduces $J_2$ and $J_4$ by $\sim 20$--$30\%$, while the effect on $T_{\rm eff}$ and $R_{\rm eq}$ is much smaller.
• Figure 2: The top panel shows the surface gravitational potential perturbations for identified oscillation mode eigenfunctions for evolutionary model shown in Fig. \ref{['fig:sat_rotnocore']} evaluated at $t=4.56\;\mathrm{Gyr}$, where the normalization is arbitrary but the relative amplitudes are correct under energy equipartition. The best candidate for the W84.64 frequency (vertical blue dashed line) is shown as the blue dot, the best candidate for the W76.44 frequency (vertical red dashed line) is shown as the red dot, and the other identified modes are shown as black crosses. It is clear that the identified $g$ mode is too low in frequency to match the observed W76.44 frequency. Note that many modes are detected but have surface gravitational potential perturbations below the y-axis cutoff; we denote these with black triangles. The bottom two panels show color plots of the 2D gravitational potential perturbations ($\delta \Phi$) of the two identified modes, color-coded and labeled.
• Figure 3: Same as Fig. \ref{['fig:sat_rotnocore']} but for a model with a deep linear gradient in $Y' \equiv Y / (1 - Z)$mankovich2021diffuse and similar $Z$ and entropy profiles. An elevated Brunt-Väisälä frequency compared to the model shown in Fig. \ref{['fig:sat_rotnocore']} can be seen due to the additional composition stratification, resulting in better agreement with the ring seismology constraints (see Fig. \ref{['fig:sat_ygrad11_surf']}). Similar to Fig. \ref{['fig:sat_rotnocore']}, the $J_2$ and $J_4$ values are elevated compared to their observed values. The final spin period is $0.995 P_{\rm Sat}$.
• Figure 4: Same as Fig. \ref{['fig:sat_rotnocore_surf']}, but for the model shown in Fig. \ref{['fig:sat_ygrad11']} containing a $Y'$ gradient. Both the W84.64 and W76.44 frequencies are well-matched. The eigenfunctions shown in the bottom row do not show rosette-like features, and we find that the W84.64 mode appears $f$-mode-like, while the W76.44 mode appears $g$-mode-like, in agreement with previous work mankovich2021diffuse. The mode with which the $g$ mode near the W76.44 frequency undergoes an avoided crossing is shown in the third row, with the dashed borders; its corresponding mode frequency and amplitude are shown with the colored cross in the top panel.
• Figure 5: Similar to Fig. \ref{['fig:sat_rotnocore']} but for a model with initially uniform $Y' \equiv Y / (1 - Z)$ and a steeper $Z$ profile. The final spin period is $0.994 P_{\rm Sat}$. Comparable agreement with observational constraints is seen in all panels.