Certain spiral density waves in Saturn's rings are generated through resonances with planetary normal modes, making them valuable probes of Saturn's internal structure. Previous research has primarily focused on the rotation rates of these waves. However, other characteristics of these waves also contain valuable information about the planet's interior. In this work, we investigate the amplitudes of the waves across the C-ring by analyzing high signal-to-noise profiles derived from phase-corrected averages of occultation profiles obtained by Cassini's Visual and Infrared Mapping Spectrometer (VIMS). By fitting these wave profiles to linear density wave models, we estimate the ring's surface mass density, mass extinction coefficient and effective kinematic viscosity at 34 locations in the C-ring, as well as the amplitude of the gravitational potential perturbations associated with 6 satellite resonances and 28 planetary normal mode resonances. Our estimates of the C-ring's mass extinction coefficient, indicate that the typical particle mass density is around 0.3 g/cm^3 interior to 84,000 km, but can get as low as 0.03 g/cm^3 exterior to 84,000 km. We also find the ring's viscosity is reduced in the outer C-ring, which is consistent with the exceptionally high porosity of the particles in this region. Meanwhile, we find the amplitudes of Saturn's normal modes are complex functions of frequency, l and m, implying that multiple factors influence how efficiently these modes are excited. This analysis identified two primary sources of these normal-mode oscillations: a deep source located close to Saturn's core, and a shallow source residing near the surface.

The detection of Uranus normal mode oscillations would provide hugely valuable diagnostics for ice giant interior structure. Since oscillation modes are generally sensitive to fluid stability, and low angular degree modes reach deep into the interior, their frequencies can deliver powerful information independent of that from the improved gravity field measurements we expect from radio tracking of the spacecraft1. Here we explore two methodologies for seismology of Uranus by an orbiting spacecraft. First, ring seismology uses stellar occultation or imaging data to uncover resonances between non-radial planetary normal modes and ring orbits. This has been successful at Saturn, where the planet's oscillating gravity field generates waves near epicyclic (Lindblad) or vertical resonances2-8. For Uranus's system of predominantly narrow (1-10 km scale) rings, ring seismology would entail the search for m-lobed standing patterns on Uranus's narrow rings that betray the influence of gravitational forcing near a resonance. The resonant shepherding of the Epsilon ring by the moons Cordelia and Ophelia is a striking example of confinement of a narrow Uranian ring9, but the mechanism confining up to 8 other rings is unknown10, and a rich spectrum of Uranian oscillations may be partly responsible. Uranus interior models that satisfy all available data predict that the narrow rings overlap with Uranian fundamental modes and internal gravity modes11, as well as inertial modes. We model the mode spectrum in a set of new Uranus interior models to quantify the constraining power than one or two ring seismology detections in the Uranian rings would have for Uranus interior structure. Second, Doppler imaging seismology would produce a time series of radial velocity maps of the Uranian photosphere, from which frequencies of normal modes could be extracted. The radial velocity signal is likely dominated by the higher frequency pressure (p) modes, i.e., trapped sound waves, the type of mode responsible for the 5-minute oscillation in the sun. We show how frequency measurements for a set of p modes with consecutive radial order can be used to construct an échelle diagram, where deviations from constant frequency spacing are sensitive diagnostics of composition or sound speed interfaces in the planetary interior. The modest requirements of ring seismology stand in contrast to Doppler imaging's requirement for dedicated instrumentation (in the form of an interferometer or magneto-optical filter design) and more substantial payload. This will need to be weighed against the advantage that p modes offer for localizing features in the planetary interior (especially when combined with inversion techniques12) and synergies with other science areas, especially atmospheric dynamics13. Further insights into the Uranian normal mode spectrum may be obtainable from direct gravitational seismology, i.e., the detection of modes through their gravitational influence on the trajectory of the spacecraft itself14,15. This builds on the tentative evidence for Jupiter and Saturn seismicity in the Doppler tracking of the Juno and Cassini spacecraft16,17. References:[1] Parisi et al. 2024, PSJ; doi:10.3847/PSJ/ad4034 [2] Marley & Porco 1993, Icarus; doi:10.1006/icar.1993.1189 [3] Hedman & Nicholson 2013, AJ; doi:10.1088/0004-6256/146/1/12 [4] Hedman & Nicholson 2014, MNRAS; doi:10.1093/mnras/stu1503 [5] French et al. 2019, Icarus; doi:10.1016/j.icarus.2018.10.013 [6] Hedman et al. 2019, AJ; doi:10.3847/1538-3881/aaf0a6 [7] French et al. 2021, Icarus; doi:10.1016/j.icarus.2021.114660 [8] Hedman et al. 2022, PSJ; doi:10.3847/PSJ/ac4df8 [9] Porco & Goldreich 1987, AJ; doi:10.1086/114354 [10] French et al. 2024, Icarus; doi:10.1016/j.icarus.2024.115957 [11] A'Hearn et al. 2022; doi:10.3847/PSJ/ac82bb [12] Jackiewicz et al. 2012, Icarus; doi:10.1016/j.icarus.2012.06.028 [13] Schmider et al. 2024, PSJ; doi:10.3847/PSJ/ad3066 [14] Friedson 2020, Ph. Tr. R. Sc. A; doi:10.1098/rsta.2019.0475 [15] Parisi et al. 2024, in preparation [16] Durante et al. 2022, Nat. Co.; doi:10.1038/s41467-022-32299-9 [17] Markham et al. 2020, PSJ; doi:10.3847/PSJ/ab9f21

Determining the internal structure of Uranus is a key objective for planetary science. Knowledge of Uranus's bulk composition and the distribution of elements is crucial to understanding its origin and evolutionary path. In addition, Uranus represents a poorly understood class of intermediate-mass planets (intermediate in size between the relatively well studied terrestrial and gas giant planets), which appear to be very common in the Galaxy. As a result, a better characterization of Uranus will also help us to better understand exoplanets in this mass and size regime. Recognizing the importance of Uranus, a Keck Institute for Space Studies (KISS) workshop was held in September 2023 to investigate how we can improve our knowledge of Uranus's internal structure in the context of a future Uranus mission that includes an orbiter and a probe. The scientific goals and objectives of the recently released Planetary Science and Astrobiology Decadal Survey were taken as our starting point. We reviewed our current knowledge of Uranus's interior and identified measurement and other mission requirements for a future Uranus spacecraft, providing more detail than was possible in the Decadal Survey's mission study and including new insights into the measurements to be made. We also identified important knowledge gaps to be closed with Earth-based efforts in the near term that will help guide the design of the mission and interpret the data returned.

We investigate the prospect of using ring seismology to probe the interiors of the ice giants Uranus and Neptune. We produce normal mode spectra for different interior models of Uranus using the program GYRE. These normal mode spectra provide predictions of where in the rings of Uranus we might see effects of interior oscillations. The inner rings of Uranus look to be a promising location for identifying planetary normal mode resonances. The diversity of normal mode spectra implies that identification of even one or two modes in the rings of Uranus would eliminate a variety of interior models, and thus aid in the interpretation of Voyager observations and future spacecraft measurements. In addition, these calculations should show what aspects of the planets’ internal structure can be probed with ring seismology.

Recently, the Juno and Cassini spacecraft shed light on the interior of both Jupiter and Saturn, the two gas giants of the Solar System. Juno is currently orbiting Jupiter in a highly elliptical 53.5-day orbit, with a perijove altitude of about 4000 km. After the 33rd passage in April 2021 (labeled PJ33), the mission ended its nominal mission and entered its extended mission. On the contrary, the Cassini spacecraft ended its mission on September 15th, 2017 with a deliberate plunge into Saturn’s atmosphere. In its final phase, the Grand Finale, Cassini provided insights on Saturn’s rings, atmosphere, and interior. Out of the 22 proximal orbits, six pericenter passes have been devoted to the determination of the gravity field of the planet.The gravity science experiments on board Juno and Cassini precisely measured, respectively, Jupiter and Saturn zonal gravitational fields. The measured gravity harmonics have been used to constrain the interior structure and atmospheric zonal flow on both planets.The Cassini data analysis have shown the need to include unknown accelerations to properly fit the data to the expected noise (Iess, 2019). Similar unexplained accelerations have been observed also on Juno gravity data (Durante, 2020). Since Jupiter and Saturn are gas giants, unconventional phenomena can be at play, including normal modes, non-zonal atmospheric dynamics, or flows in the dynamo region.The analysis of the accelerations acting on Cassini provided evidence for p-modes on the planet (Markham, 2020), while the analysis of Juno gravity data revealed p-modes on Jupiter and provided upper bounds on lower frequency f-modes. Here, we show the results for normal modes on both Jupiter and Saturn obtained with the analysis of gravity data of both Juno and Cassini.

We investigate the material properties of a mixture of hydrogen, helium, and oxygen representative of Saturn’s interior at pressure–temperature conditions of a recent Saturn model (see Mankovich & Fortney) with molecular dynamics simulations based on density functional theory. Their model considers the demixing of hydrogen and helium and predicts a He-rich layer above a diluted core. We calculate the thermodynamic and transport properties and discuss the impact on Saturn’s evolution and interior structure. We find a significant impact of the He-rich layer on the specific heat capacity, speed of sound, viscosity, diffusion coefficients, thermal and electrical conductivity, Lorenz number, and magnetic and thermal diffusivities.

We present a review of Saturn's interior structure and thermal evolution, with a particular focus on work in the past 5 years. Data from the Cassini mission, including a precise determination of the gravity field from the Grand Finale orbits, and the still ongoing identification of ring wave features in Saturn's C-ring tied to seismic modes in the planet, have led to dramatic advances in our understanding of Saturn's structure. Models that match the gravity field suggest that differential rotation, as seen in the visible atmosphere, extends down to at least a depth of 10,000 km (1/6$^{\rm th}$ the planet's radius). At greater depths, a variety of different investigations all now point to a deep Saturn rotation rate of 10 hours and 33 minutes. There is very compelling evidence for a central heavy element concentration (``core''), that in most recent models is 12-20 Earth masses. Ring seismology strongly suggests that the core is not entirely compact, but is dilute (mixed in with the overlying H/He), and has a substantial radial extent, perhaps out to around one-half of the planet's radius. A wide range of thermal evolution scenarios can match the planet's current luminosity, with progress on better quantifying the helium rain scenario hampered by Saturn's poorly known atmospheric helium abundance. We discuss the relevance of magnetic field data on understanding the planet's current interior structure. We point towards additional future work that combines seismology and gravity within a framework that includes differential rotation, and the utility of a Saturn entry probe.

Normal mode seismology is a promising means of measuring rotation in gas giant interiors, and ring seismology presents a singular opportunity to do so at Saturn. We calculate Saturn’s normal modes of oscillation and zonal gravity field, using nonperturbative methods for normal modes in the rigidly rotating approximation, and perturbative methods for the shifts that Saturn’s deep winds induce in the mode frequencies and zonal gravity harmonics. The latter are calculated by solving the thermogravitational wind equation in an oblate geometry. Comparing many such models to gravity data and the frequencies of ring patterns excited by Saturn’s normal modes, we use statistical methods to estimate that Saturn’s cloud-level winds extend inward along cylinders before decaying at a depth 0.125–0.138 times Saturn’s equatorial radius, or 7530–8320 km, consistent with analyses of Cassini’s gravity and magnetic field data. The seismology is especially useful for pinning down Saturn’s poorly constrained deep rotation period, which we estimate at 2 π /Ω _S = 634.7 minutes (median) with a 5/95% quantile range of 633.8–635.5 minutes. Outstanding residuals in mode frequencies at low angular degree suggest a more complicated deep interior than has been considered to date. Smaller but still significant residuals at high angular degrees also show that our picture for the thermal, composition, and/or rotation profile in Saturn’s envelope is not yet complete.

The Juno spacecraft has been collecting data to shed light on the planet's origin and characterize its interior structure. The onboard gravity science experiment based on X-band and Ka-band dual-frequency Doppler tracking precisely measured Jupiter's zonal gravitational field. Here, we analyze 22 Juno's gravity passes to investigate the gravity field. Our analysis provides evidence of new gravity field features, which perturb its otherwise axially symmetric structure with a time-variable component. We show that normal modes of the planet could explain the anomalous signatures present in the Doppler data better than other alternative explanations, such as localized density anomalies and non-axisymmetric components of the static gravity field. We explain Juno data by p-modes having an amplitude spectrum with a peak radial velocity of 10-50 cm/s at 900-1200 μHz (compatible with ground-based observations) and provide upper bounds on lower frequency f-modes (radial velocity smaller than 1 cm/s). The new Juno results could open the possibility of exploring the interior structure of the gas giants through measurements of the time-variable gravity or with onboard instrumentation devoted to the observation of normal modes, which could drive spacecraft operations of future missions.

The excitation of density and bending waves in Saturn's C ring by planetary oscillation modes presents a unique opportunity to learn about gas giant interiors and rotation. However, theoretical complications related to Saturn's rapid and differential rotation pose a barrier to the full utilization of ring wave detections. We calculate oscillation modes using a complete, non-perturbative treatment of differential rotation modelled after Saturn's zonal winds in self-consistently computed, polytropic equilibria. We find that previous, approximate treatments of the effects of differential rotation in Saturn overestimate shifts in the frequencies of fundamental modes (f-modes) thought to be responsible for the majority of the waves detected in the C ring, due to an omitted modification of the equilibrium shape and structure of the planet by differential rotation. The bias introduced by these frequency overestimates is small, but significant relative to the uncertainties afforded by Cassini data. We additionally consider the non-perturbative effects of Saturn-like differential rotation on the rotational mixing of f-modes and internal gravity modes (g-modes), which is relevant to detections of multiple density waves with very closely split pattern speeds. We find that higher order rotational effects can produce orders-of-magnitude enhancements in the surface gravitational perturbations of g-modes dominated by large spherical harmonic degrees ℓ, regardless of frequency separation from the sectoral f-mode. Despite this enhancement, we find that the observed fine-splitting of density waves is unlikely to involve g-modes dominated by ℓ≳ 10. This restriction may aid in the inference of possible internal structures for Saturn.

We assess the prospect of using ring seismology to probe the interiors of the ice giants Uranus and Neptune. We do this by calculating normal-mode spectra for different interior models of Uranus and Neptune using the stellar oscillation code GYRE . These spectra provide predictions of where in these planets’ ring systems the effects of interior oscillations might be detected. We find that f -mode resonances with azimuthal order m = 2 or 7 ≤ m ≤ 19 fall among the inner rings (6, 5, 4, α , and β ) of Uranus, while f -mode resonances with 2 ≤ m ≤ 12 fall in the tenuous ζ ring region. In addition, f -mode resonances with m = 2 or 6 ≤ m ≤ 13 may give azimuthal structure to Neptune’s tenuous Galle ring. We also find that g -mode resonances may fall in the middle to outer rings of these planets. Although an orbiter is most likely required to confirm the association between any waves in the rings and planetary normal modes, the diversity of normal-mode spectra implies that identification of just one or two modes in the rings of Uranus or Neptune would eliminate a variety of interior models and thus aid in the interpretation of Voyager observations and future spacecraft measurements.

The Juno spacecraft measured Jupiter's gravity field and determined the even and odd zonal harmonics, J ( n ), with unprecedented precision. However, interpreting these observations has been a challenge because it is difficult to reconcile the unexpectedly small magnitudes of the moments J (4) and J (6) with conventional interior models that assume a large, distinct core of rock and ice. Here we show that the entire set of gravity harmonics can be matched with models that assume an ab initio equation of state, wind profiles, and a dilute core of heavy elements that are distributed as far out as 63% of the planet's radius. In the core region, heavy elements are predicted to be distributed uniformly and make up only 18% by mass because of dilution with hydrogen and helium. Our models are consistent with the existence of primary and secondary dynamo layers that will help explain the complexity of the observed magnetic field.

14 Interior modeling of Jupiter and Saturn has advanced to a state where thousands 15 of models are generated that cover the uncertainty space of many parameters. This 16 approach demands a fast method of computing their gravity field and shape. Moreover, 17 the Cassini mission at Saturn and the ongoing Juno mission delivered gravitational 18 harmonics up to J12. Here, we report the expansion of the Theory of Figures, which is 19 a fast method for gravity field and shape computation, to the 7th-order (ToF7), which 20 allows for computation of up to J14. We apply three different codes to compare the 21 accuracy using polytropic models. We apply ToF7 to Jupiter and Saturn interior models 22 in conjunction with CMS-19 H/He-EOS. For Jupiter, we find that J6 is best matched 23 by a transition from He-depleted to He-enriched envelope at 2–2.5 Mbar. However, the 24 atmospheric metallicity reaches 1× solar only if the adiabat is perturbed toward lower 25 densities, or if the surface temperature is enhanced by ∼ 14K from the Galileo value. 26 Our Saturn models imply a largely homogeneous-in-Z envelope at 1.5–4× solar atop 27 a small core. Perturbing the adiabat yields metallicity profiles with extended, heavy28 element enriched deep interior (diffuse core) out to 0.4 RSat, as for Jupiter. Classical 29 models with compact, dilute, or no core are possible as long as the deep interior is 30 enriched in heavy-elements. Including a thermal wind fitted to the observed wind 31 speeds, representative Jupiter and Saturn models are consistent with all observed Jn 32 values. 33

Interior modeling of Jupiter and Saturn has advanced to a state where thousands of models are generated that cover the uncertainty space of many parameters. This approach demands a fast method of computing their gravity field and shape. Moreover, the Cassini mission at Saturn and the ongoing Juno mission delivered gravitational harmonics up to J12. Here, we report the expansion of the Theory of Figures, which is a fast method for gravity field and shape computation, to the 7th-order (ToF7), which allows for computation of up to J14. We apply three different codes to compare the accuracy using polytropic models. We apply ToF7 to Jupiter and Saturn interior models in conjunction with CMS-19 H/He-EOS. For Jupiter, we find that J6 is best matched by a transition from He-depleted to He-enriched envelope at 2-2.5 Mbar. However, the atmospheric metallicity reaches 1xtimes solar only if the adiabat is perturbed toward lower densities, or if the surface temperature is enhanced by 14 K from the Galileo value. Our Saturn models imply a largely homogeneous-in-Z envelope at 1.5-4x solar atop a small core. Perturbing the adiabat yields metallicity profiles with extended, heavy-element enriched deep interior (diffuse core) out to 0.4 RSat, as for Jupiter. Classical models with compact, dilute, or no core are possible as long as the deep interior is enriched in heavy-elements. Including a thermal wind fitted to the observed wind speeds, representative Jupiter and Saturn models are consistent with all observed Jn values.

The best constraints on the internal structures of giant planets have historically originated from measurements of their gravity fields1–3. These data are inherently mostly sensitive to a planet’s outer regions, stymieing efforts to measure the mass and compactness of the cores of Jupiter2,4,5 and Saturn6,7. However, studies of Saturn’s rings have detected waves driven by pulsation modes within the planet8–11, offering independent seismic probes of Saturn’s interior12–14. The observations reveal gravity-mode pulsations, which indicate that part of Saturn’s deep interior is stable against convection13. Here, we compare structural models with gravity and seismic measurements from Cassini to show that the data can only be explained by a diffuse, stably stratified core–envelope transition region in Saturn extending to approximately 60% of the planet’s radius and containing approximately 17 Earth masses of ice and rock. This gradual distribution of heavy elements constrains mixing processes at work in Saturn, and it may reflect the planet’s primordial structure and accretion history. Like a seismograph, Saturn’s rings are sensitive to oscillations coming from the planet’s interior. State-of-the-art modelling shows that Cassini’s measurements of ring waves point to a convectively stable diffuse core within Saturn, which extends for 60% of its radius and contains 17 Earth masses of ice and rock.

Saturn's rings act as a system of innumerable test particles that are remarkably sensitive to periodic disturbances in the planet's gravitational field. We identify 15 additional density and bending waves in Saturn's C ring driven by the planet's internal normal mode oscillations. Taking advantage of a highly accurate absolute radius scale for the rings, we are able to detect weak, high-wavenumber (up to m=14) waves with km-scale radial wavelengths. From a systematic scan of the entire C ring, we report the discovery and identification of 11 new Outer Lindblad Resonances (OLRs), two counterpart inner Lindblad resonances (ILRs), and two new Outer Vertical Resonances (OVRs). The close agreement of the observed resonance locations and wave rotation rates with the predictions of models of Saturn's interior suggests that all of the new waves are driven by Saturnian f-mode oscillations. As classified by their spherical harmonic shapes, the modes in question range in azimuthal wavenumber from m=8 to 14, with associated resonance orders l-m ranging from 0 to 8, where l is the overall angular wavenumber of the mode. Our suite of detections for l-m=4 is now complete from m=8 to m=14. Curiously, detections with l-m=2 are less common. These newly-identified non-sectoral waves sample latitudinal as well as radial structure within the planet and may thus provide valuable constraints on Saturn's differential rotation. Allowing for the fact that the two ILR-type waves appear to be due to the same normal modes as two of the OLR-type waves, the 13 additional modes identified here bring to 34 the number of distinct f-modes suitable for constraining interior models.

Here we highlight recent advances in our knowledge about Saturn's ring system and bring forward the outstanding science issues that could be addressed by studying the ring systems of the ice giants.We focus on interactions between planetary rings and other elements in the system, including the moons, host planet, and its magnetosphere, and conclude that ring science investigations, in accordance with magnetospheric and atmospheric science disciplines, are essential in advancing our knowledge of solar system evolution, the origin and evolution of the moons and Ocean Worlds, as well as contemporary phenomena observed in the ice giant systems.We request that the study of ice giant ring systems to be considered a top priority for all future ice giant explorations.

Normal mode oscillations in Saturn excite density and bending waves in the C Ring, providing a valuable window into the planet's interior. Saturn's fundamental modes (f modes) excite the majority of the observed waves, while gravito-inertial modes (rotationally modified g modes) associated with stable stratification in the deep interior provide a compelling explanation for additional density waves with low azimuthal wavenumbers m. However, multiplets of density waves with nearly degenerate frequencies, including an m=3 triplet, still lack a definitive explanation. We investigate the effects of rapid and differential rotation on Saturn's oscillations, calculating normal modes for independently constrained interior models. We use a non-perturbative treatment of rotation that captures the full effects of the Coriolis and centrifugal forces, and consequently the mixing of sectoral f modes with g modes characterized by very different spherical harmonic degrees. Realistic profiles for differential rotation associated with Saturn's zonal winds can enhance these mode interactions, producing detectable oscillations with frequencies separated by less than 1%. Our calculations demonstrate that a three-mode interaction involving an f mode and two g modes can feasibly explain the finely split m=3 triplet, although the fine-tuning required to produce such an interaction generally worsens agreement with seismological constraints provided by m=2 density waves. Our calculations additionally demonstrate that sectoral f mode frequencies are measurably sensitive to differential rotation in Saturn's convective envelope. Finally, we find that including realistic equatorial antisymmetry in Saturn's differential rotation profile couples modes with even and odd equatorial parity, producing oscillations that could in principle excite both density and bending waves simultaneously.

Gravity field measurements only weakly constrain the deep interiors of Jupiter and Saturn, stymieing efforts to measure the mass and compactness of these planets' cores, crucial properties for understanding their formation pathways and evolution. However, studies of Saturn's rings by Cassini have revealed waves driven by pulsation modes within Saturn, offering independent seismic probes of Saturn's interior. The observations reveal gravity mode (g mode) pulsations that indicate that a part of Saturn's interior is stably stratified by composition gradients, and the g mode frequencies directly probe the buoyancy frequency within the planet. We compare structure models with gravity and new seismic measurements from Cassini to show that the data can only be explained by a diffuse, stably stratified core-envelope transition region in Saturn extending to approximately 60% of the planet's radius. This predominantly stable interior imposes significant constraints on Saturn's intrinsic magnetic field generation. The gradual distribution of heavy elements required by the seismology constrains mixing processes at work in Saturn, and it may reflect the planet's primordial structure and accretion history.