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Tidal interactions play an important role in many astrophysical systems, but uncertainties regarding the tides of rapidly rotating, centrifugally distorted stars and gaseous planets remain. We have developed a precise method for computing the dynamical, nondissipative tidal response of rotating planets and stars, based on summation over contributions from normal modes driven by the tidal potential. We calculate the normal modes of isentropic polytropes rotating at up to ≃90% of their critical breakup rotation rates, and tabulate fits to mode frequencies and tidal overlap coefficients that can be used to compute the frequency-dependent, nondissipative tidal response (via potential Love numbers k ℓm ). Although fundamental modes (f-modes) possess dominant tidal overlap coefficients at (nearly) all rotation rates, we find that the strong coupling of retrograde inertial modes (i-modes) to tesseral (ℓ > ∣m∣) components of the tidal potential produces resonances that may be relevant to gas giants like Jupiter and Saturn. The coupling of f-modes in rapid rotators to multiple components of both the driving tidal potential and the induced gravitational field also affect the tesseral response, leading to significant deviations from treatments of rotation that neglect centrifugal distortion and high-order corrections. For very rapid rotation rates (≳70% of breakup), mixing between prograde f-modes and i-modes significantly enhances the sectoral (ℓ = ∣m∣) tidal overlap of the latter. The tidal response of very rapidly rotating, centrifugally distorted planets or stars can also be modified by resonant sectoral f-modes that are secularly unstable via the Chandrasekhar–Friedman–Schutz mechanism.

The equation of state of hydrogen–helium (H–He) mixtures plays a vital role in the evolution and structure of gas giant planets and exoplanets. Recent equations of state that account for H–He interactions, coupled with H–He immiscibility curves, can now produce more physical evolutionary models, such as accounting for helium rain with greater fidelity than in the past. In this work, we present a set of tools for planetary evolution that provides a Python interface for existing tables of useful thermodynamic quantities, state-of-the-art H–He equations of state, and pressure-dependent H–He immiscibility curves. In particular, for a collection of independent variable choices, we provide scripts to calculate the variety of thermodynamic derivatives used to model convection and energy transport. These include the chemical potential derived from the internal energy, which is a modeling necessity in the presence of composition gradients when entropy is the other primary variable. Finally, an entropy-based convection formalism is presented and fully described that highlights the physical differences between adiabatic and isentropic interior models. This centralized resource is meant to facilitate both giant planet structural and evolutionary modeling and the entry of new research groups into the field of giant planet modeling. All tables of thermodynamic quantities and derivatives are available at https://github.com/Rob685/hhe_eos_misc, along with a unified Python interface. Tutorials demonstrating the interface are also available in the repository.

We study how stably stratified or semiconvective layers alter tidal dissipation rates associated with the generation of inertial, gravito-inertial, interfacial, and surface gravity waves in rotating giant planets. We explore scenarios in which stable (nonconvective) layers contribute to the high rates of tidal dissipation observed for Jupiter and Saturn in our solar system. Our model is an idealized spherical Boussinesq system incorporating Coriolis forces to study effects of stable stratification and semiconvective layers on tidal dissipation. Our detailed numerical calculations consider realistic tidal forcing and compute the resulting viscous and thermal dissipation rates. The presence of an extended stably stratified fluid core significantly enhances tidal wave excitation of both inertial waves (due to rotation) in the convective envelope and gravito-inertial waves in the dilute core. We show that a sufficiently strongly stratified fluid core enhances inertial wave dissipation in a convective envelope much like a solid core does. We demonstrate that efficient tidal dissipation rates (and associated tidal quality factors Q′ )—sufficient to explain the observed migration rates of Saturn's moons—are predicted at the frequencies of the orbiting moons due to the excitation of inertial or gravito-inertial waves in our models with stable layers (without requiring resonance locking). Stable layers could also be important for tidal evolution of hot and warm Jupiters and hot Neptunes, providing efficient tidal circularization rates. Future work should study more sophisticated planetary models that also account for magnetism and differential rotation, as well as the interaction of inertial waves with turbulent convection.

Computed using the APPLE planetary evolution code, we present updated evolutionary models for Jupiter and Saturn that incorporate helium rain, nonadiabatic 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 Rρ, which allows one to explore in an approximate way the efficiency of semiconvection. For both Jupiter and Saturn, we construct models spanning from Rρ = 1 (Ledoux) to Rρ = 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 Rρ 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–Väisälä 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.

We study how stably stratified or semi-convective layers alter the tidal dissipation rates associated with the generation of internal waves in planetary interiors. We consider if these layers could contribute to the high rates of tidal dissipation observed for Jupiter and Saturn in our solar system. We use an idealized global spherical Boussinesq model to study the influence of stable stratification and semi-convective layers on tidal dissipation rates. We carry out analytical and numerical calculations considering realistic tidal forcing and measure how the viscous and thermal dissipation rates depend on the parameters relating to the internal stratification profile. We find that the strongly frequency-dependent tidal dissipation rate is highly dependent on the parameters relating to the stable stratification, with strong resonant peaks that align with the internal modes of the system. The locations and sizes of these resonances depend on the form and parameters of the stratification, which we explore both analytically and numerically. Our results suggest that stable stratification can significantly enhance the tidal dissipation in particular frequency ranges. Analytical calculations in the low-frequency regime give us scaling laws for the key parameters, including the tidal quality factor Q′ due to internal gravity waves. Stably stratified layers can significantly contribute to tidal dissipation in solar and extrasolar giant planets, and we estimate substantial tidal evolution for hot Neptunes. Further investigation is needed to robustly quantify the significance of the contribution in realistic interior models, and to consider the contribution of inertial waves.

We present updated nonadiabatic and inhomogeneous evolution models for Uranus and Neptune, employing an interior composition of methane, ammonia, water, and rocks. Following the formation trends of the gas giants, Uranus and Neptune formation models are applied, where both planets begin with layers stable to convection. Both planets are subject to convective mixing throughout their evolution. Consistent with past work on this subject, the interior heat of Uranus evolution models is preserved by the stability of an outer composition gradient at lower initial entropy, where convective mixing is inhibited over evolutionary timescales. In contrast, if Neptune’s initial entropy is enough to convectively mix its envelope, it undergoes homogenization and adiabatic cooling of the outer 40% of its envelope. The subsequent release of internal energy during Neptune’s evolution, driven by the convective instability of its primordial outer compositional gradient, accounts for its higher luminosity relative to Uranus. This work proposes that the observed luminosity differences between Uranus and Neptune could be explained by the convective stability of their outer envelopes. The extensive convective mixing in Neptune can lead to a higher metallicity in its outer region compared to Uranus, a feature seen in atmospheric measurements and shown in past interior models of Neptune. Due to Neptune’s more pronounced cooling, our models predict favorable conditions for hydrogen–water immiscibility in its envelope.

Juno and Cassini have shown that Jupiter and Saturn likely contain extended gradients of heavy elements. Yet, how these gradients can survive over billions of years remains an open question. Classical convection theories predict rapid mixing and homogenization, which would erase such gradients on timescales far shorter than the planets’ ages. To address this, we estimate the energy required to erode both dense and fuzzy cores, and compare it to what the planet can realistically supply. If the entire cooling budget is available to drive mixing, then even a compact core can, in principle, be destroyed. But if mixing is limited to the thermal energy near the core, which is another plausible scenario, the energy falls short. In that case, Jupiter can erode a fuzzy core by up to approximately 10 M⊕, but a compact one remains intact. Saturn’s core is more robust. Even in the fuzzy case, only about 1 M⊕ is lost, and if the core is compact, erosion is negligible. The outcome depends sensitively on the assumed initial temperature and entropy profiles. Hotter and more superadiabatic interiors are more prone to mixing. We suggest that 3D simulations of convection driven from above, with realistic stratification and enough depth (i.e., many density scale heights) would be of great interest to further constrain the energy budget for core erosion.

Recent measurements of Jupiter’s gravitational moments by the Juno spacecraft and seismology of Saturn’s rings suggest that the primordial composition gradients in the deep interior of these planets have persisted since their formation. One possible explanation is the presence of a double-diffusive staircase below the planet’s outer convection zone, which inhibits mixing across the deeper layers. However, hydrodynamic simulations have shown that these staircases are not long-lasting and can be disrupted by overshooting convection. In this Letter, we suggests that planetary rotation could be another factor for the longevity of primordial composition gradients. Using rotational mixing-length theory and 3D hydrodynamic simulations, we demonstrate that rotation significantly reduces both the convective velocity and the mixing of primordial composition gradients. In particular, for Jovian conditions at t ∼ 108 yr after formation, rotation reduces the convective velocity by a factor of 6, and in turn, the kinetic energy flux available for mixing gets reduced by a factor of 63 ∼ 200. This leads to an entrainment timescale that is more than 2 orders of magnitude longer than without rotation. We encourage future hydrodynamic models of Jupiter and other gas giants to include rapid rotation because the decrease in the mixing efficiency could explain why Jupiter and Saturn are not fully mixed.

Observations from Juno and Cassini suggest that Jupiter and Saturn may possess fuzzy cores—central regions where the abundance of heavy elements varies smoothly with depth. Such gradients pose a longstanding puzzle for models of planetary evolution and formation, which predict that vigorous convection would homogenize the interiors of gas giants within the first ∼106–108 yr of cooling. Previous 3D simulations and analytic predictions for the propagation of a convection zone into a stable region have demonstrated that the rapid rotation of gas giants can significantly slow convective mixing, but not enough to stop it. Another piece of the puzzle is luminosity. Gas giants cool as they age, and with that comes a declining heat flux over time. Recent ideas suggest that when this declining luminosity is combined with rotational effects, convection may stall. We explore this possibility using 3D hydrodynamic simulations that include both rotation and a surface cooling flux that decreases as 1/t. Our results demonstrate that, even without rotation, a declining luminosity can suppress mixing sufficiently to preserve an initial compositional gradient in the deep interior of gas giants. If confirmed by more realistic simulations, this may help to explain the long-term survival of fuzzy cores.