1 Introduction

During the last decade, new discoveries in the Jupiter and Saturn systems have compelled us to revisit the estimates of tidal dissipation in giant gaseous planets of our solar system. Figure 1 illustrates how dissipation leads to a phase lag between the tidal bulge and the satellite, creating a torque that drags the satellite forward or backward. When the planet rotates faster than the moon orbits (\(\Omega _{\mathrm{p}} > \Omega _{\mathrm{m}}\), as in the case of solar system moon systems), angular momentum is transferred from the planet to the moon, dragging the moon forward and making it migrate outwards. The initial predictions for the tidal quality factor of Jupiter and Saturn were computed by Goldreich and Soter (1966), who solved the evolution equation for the semi-major axis of the moons assuming a frequency-independent tidal dissipation rate, constant over the evolution, in order to place them at their current positions at the age of the solar system. This procedure predicts tidal quality factors of \(10^{4} \lesssim Q \lesssim 10^{6}\) for Saturn and Jupiter.

Fig. 1
figure 1

Cartoon showing a tidally distorted planet of mass \(M_{\mathrm{p}}\) and radius \(R_{\mathrm{p}}\), due to a moon of mass \(M_{\mathrm{m}}\) at semi-major axis \(a_{\mathrm{m}}\). The planet’s spin frequency is \(\Omega _{\mathrm{p}}\) and the moon’s orbital frequency is \(\Omega _{\mathrm{m}}\). The tidal quality factor of the planet, \(Q\), can be visualized as a lag angle of \(\delta = 1/Q\) between the tidal bulge and the moon’s position. In reality, the effective value of \(Q\) can involve more complicated tidal distortion within the planet due to dynamical tides

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Using high precision astrometry, a series of measurements (Lainey et al. 2009, 1983), i.e., a tidal quality factor \(Q \lesssim 10^{3}\) instead of \(Q \sim 10^{4}\) for Saturn. Moreover, the dissipation in Saturn has a complex variation as a function of the tidal frequency: we observe a weak (and potentially smooth) dependence on the frequency for Enceladus, Tethys and Dione, and a much stronger dissipation for Rhea and Titan. Observations of faster moon migration than expected (and hence stronger dissipation) may require revisions of our understanding of the formation and evolution of giant planets systems (Charnoz et al. 2011), and of tidal dissipation in planetary interiors.

These results are also of major interest for exoplanetary systems since giant gaseous planets have been the first type of discovered exoplanets. Indeed, the discovery of 51 Pegasi b by Mayor and Queloz (1995) opened the path to the exploration of giant planets outside our Solar system. The occurrence of giant gaseous planets close to their host stars, the so-called “hot” Jupiters, demonstrated that the orbital architecture of our solar system may not be common. It also spawned new challenging questions about the formation, the dynamical evolution, and the stability of exoplanetary systems. In this context, the study of tidal (and magnetic) star-planet interactions between low-mass stars and close giant planets came to the forefront (e.g., Ahuir et al. 2021).

For example, tidal interactions between a host star and orbiting giant exoplanets were proposed as a candidate to explain the “bloated” hot Jupiter phenomenon (Bodenheimer et al. 2001; Guillot and Showman 2002; Baraffe 2005). The proposed idea is that tidal dissipation can sustain intense internal heating in giant exoplanets, modifying their internal structure with an increase of their radii. Another important but unsolved question is the effects of tides on the orbital distribution of hot Jupiters around their host stars, for which dissipation in both the stars and the planets can play a key role. Let us finally point that these major questions are among the key objectives of ongoing and forthcoming major space missions like CHEOPS, TESS, JWST, PLATO, and ARIEL (Benz et al. 2017; Ricker et al. 2015; Lagage 2015; Rauer et al. 2014; Tinetti et al. 2018) and ground-based instruments like SPIRou (Moutou et al. 2015).

While tidal dissipation is crucial for orbital evolution of both giant planet moon systems and exoplanet systems, the underlying physics is complex. In this chapter, we review our current understanding of giant planet structure, and we discuss how planetary structure and internal dynamics affect various tidal dissipation mechanisms. Comparison between models and measured migration rates are now allowing us to uncover the complex and interrelated histories of giant planet interiors and their moon systems.

2 Giant Planet Structure

2.1 Interiors of Jupiter, Saturn, Uranus and Neptune

In the solar system, four planets have acquired enough hydrogen and helium during the early formation phase of the system to grow rapidly in size and mass and earn the qualification of “giant”. With radii of about 10 times our Earth, Jupiter and Saturn are commonly known as “gas giants” while Uranus and Neptune, which are about 4 times larger than Earth, are usually called “ice giants”. Both denominations are convenient, but we should be aware of their limitations: most of Jupiter and Saturn’s interiors behave as a liquid rather than a gas (they are in fact fluid, e.g., Guillot 2005), while the compositions of Uranus and Neptune are highly uncertain: they could well be more rocky than icy (Helled and Fortney 2020; Kunitomo et al. 2018).

An important property of giant planets in our solar system is that they radiate about twice as much energy as they receive from the Sun (Pearl and Conrath 1991, and references therein). The difference is due to their progressive contraction and cooling (Hubbard 1968). A notable exception is Uranus, whose intrinsic heat luminosity is about 5 to 10 times lower than the other three giant planets and compatible with zero (Pearl and Conrath 1991). Several hypotheses have been put forward (see Helled and Fortney 2020), but the recent 30% upward revision of Jupiter’s intrinsic luminosity (Li et al. 2018) calls for a reanalysis of available data, and ultimately, in situ measurements at Uranus (e.g., Fletcher et al. 2020; Guillot 2022).

In most of the interiors of these four planets, the high radiative opacities and low heat conductivities imply that any non-negligible intrinsic luminosity should ensure convection (Stevenson 1976; Guillot et al. 1994, 2004). The atmospheres of all planets, including Uranus, indeed exhibit significant convective activity (Hueso et al. 2020). At photospheric levels, their temperatures are remarkably uniform in spite of the strongly varying insulation, also a sign of efficient heat transport and hence convection (Ingersoll and Porco 1978). In planets, only very slightly super-adiabatic temperature gradients are required for convection to carry the planetary heat flux, so convective regions are thought to have a nearly adiabatic density profile. These facts motivate the widely used hypothesis that interiors are convective and hence nearly adiabatically stratified.

The structures of our four giant planets as obtained from interior models using the latest gravity measurements by Juno for Jupiter (Durante et al. 2020), gravity and seismology measurements by Cassini for Saturn (Iess et al. 2019; Hedman et al. 2019), and by much less accurate gravity measurements for Uranus (Jacobson 2014) and Neptune (Jacobson 2009), are shown in Fig. 2. For Jupiter, the models predict the existence of a fluid envelope separated into a helium-poor upper part and a helium-rich lower part, and a deep dilute core with a higher proportion of heavy elements mixed with hydrogen (Wahl et al. 2017; Debras and Chabrier 2019; Miguel et al. 2022; Militzer et al. 2022).

Fig. 2
figure 2

Slices of the internal structures of Jupiter, Saturn, Uranus and Neptune highlighting compositional gradients in helium (orange) and heavy elements (blue). Jupiter and Saturn are characterized by a phase separation of helium in metallic hydrogen starting near Mbar pressures. Juno and Cassini measurements also indicate the presence of a dilute heavy element core in these planets, implying that heavy elements are partially mixed with hydrogen and helium rather forming separate pure layers. In this diagram, each layer corresponds to a 2% increase of the helium/hydrogen ratio (orange) or of the heavy element mass fraction \(Z\) (blue). For Uranus and Neptune, a structure dominated by well-separated layers, including a solid superionic water layer, is presented. However, solutions with compositions that evolve more continuously are also highly plausible. Image reproduced with permission from Guillot et al. (2023), copyright by ASP

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For Saturn, models predict that the transition from helium-poor to helium-rich region occurs at much larger depths, and combines with a progressive increase in heavy elements (Mankovich and Fuller 2021). Models of Uranus and Neptune have considerable uncertainties (Nettelmann et al. 2013; Helled et al. 2020) and point to a small, 0.5 to 4 M⊕ fluid hydrogen-helium envelope overlaying a core made of ices and rocks which could either be layered or mixed, possibly including hydrogen and helium as well. In these models, the envelopes are believed to be largely convective and of uniform composition, except for Jupiter and Saturn, in a region in which helium separates from hydrogen and rains out (e.g., Schöttler and Redmer 2018; Brygoo et al. 2021). The cores, however, may be highly stratified and exhibit large composition gradients with depth (see Sect. 2.3).

Although hydrogen and helium are definitely fluid, the state of other elements is more difficult to assess. In Jupiter and Saturn, the temperature profile lies above the estimated solidification curves for most elements shown in Fig. 3, except possibly iron and magnesium oxide (Mazevet et al. 2019; Guillot et al. 2023). Even in this case, these elements are believed to be soluble in metallic hydrogen (González-Cataldo et al. 2014). Whether a solid compact core remains will depend on the availability of energy to mix elements upward (Guillot et al. 2004; Vazan et al. 2018) and on how high-pressure elemental interactions between different species affect phase diagrams.

Fig. 3
figure 3

Phase diagrams of key elements with pressure-temperature profiles of relevant astrophysical objects (the Sun, brown dwarfs CoRoT-15b and Gl229 B, giant exoplanet HD 209458 b, and Jupiter, Saturn, Uranus and Neptune). Pressures are in bars (\(1\text{ bar}=10^{5}\text{ Pa}=10^{6}\text{ dyn/cm}^{2}\)). (a): Location of the hydrogen-helium phase separation leading to helium rain-out in Jupiter and Saturn (yellow). The critical demixing temperatures obtained from ab-initio simulations are shown in orange. Those obtained from high-pressure experiments are shown as two triangles connected by a red curve. (b): Phase diagram of H2O, including the fluid molecular, ionic and plasma phases, the solid ice phases and the superionic phase, as labelled. (c): Region of iron solidification, relevant for the central regions of Jupiter and possibly Saturn. (d): Regions of silicate solidification, relevant in the interiors of Uranus, Neptune, Saturn and Jupiter. Note that MgSiO3 decomposes at high pressures into MgO (which should be solid in Jupiter and Saturn) and SiO2 (which may still be liquid in the deep interior of Jupiter and Saturn). Image reproduced with permission from Guillot et al. (2023), copyright by ASP; see references therein

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In Uranus and Neptune, gravitational moments indicate that the hydrogen-helium envelope should transition to a denser fluid (Helled et al. 2020). For models assuming a three layer structure (hydrogen-helium overlying ices overlying rocks), water in the middle layer forms an ionic fluid which transitions to being superionic at Mbar pressures (French et al. 2009; Redmer et al. 2011, see Fig. 3). Ab-initio calculations indicate that superionic water should behave as a solid (Millot et al. 2019), with strong consequences for the planetary evolution (Stixrude et al. 2021). This situation corresponds to the structure shown in Fig. 2. However, in the case of a non-layered structure, the mixing of other elements may affect the picture, perhaps considerably (Guarguaglini et al. 2019), so that whether Uranus and Neptune are partially solid remains an open question (see Guillot et al. 2023, for details).

2.2 Interiors of Giant Exoplanets

So far, the primary measurements that we can rely on to constrain the global composition of exoplanets are their size and mass, coupled to theoretical expectations drawn from our understanding of solar system giant planets and planetary evolution. The known sample of giant exoplanets is extremely diverse, and includes planets with surprisingly large radii which cannot be reproduced by conventional evolution models as well as very dense planets requiring the presence of tens to hundreds of Earth masses of heavy elements in their interior (e.g., Guillot et al. 2006; Thorngren et al. 2016).

Many of the giant exoplanets known today orbit much closer to their parent star than planets in our solar system. Because of observational biases, some of the planets that can be best characterized (in particular from transit photometry and radial velocimetry) also endure extremely high level of irradiation, orders of magnitude higher than Jupiter. This implies the existence of an outer radiative region that grows as the planet cools and contracts (Guillot et al. 1996), potentially to kbar depths for mature hot Jupiters (Guillot and Showman 2002). Thus, these planets could be made of an inner convective envelope and an outer radiative shell.

Complications arise however: a large fraction of exoplanets are bloated beyond what is predicted by standard evolution models, implying that some missing physics is slowing their contraction (see Guillot et al. 2006). A number of explanations have been proposed (e.g., Fortney et al. 2021). Although tidal interactions were initially identified as a possible explanation (Bodenheimer et al. 2001; Guillot and Showman 2002; Baraffe 2005), the small eccentricities of hot Jupiters indicate tidal heating could only account for a small fraction of the anomalously large planets (Fortney et al. 2021). More likely mechanisms such as Ohmic dissipation or downward energy transport (Batygin and Stevenson 2010; Fortney et al. 2021, and references therein) alter the simple picture of a thick radiative shell. It could be much narrower (Sarkis et al. 2021), or more generally, several radiative/convective zones could be present. In the deeper interior, the probable existence of compositional gradients would also affect static stability in the interior, as well as the cooling of the giant planets themselves (Chabrier and Baraffe 2007).

Finally, giant exoplanets which are more irradiated than those in the solar system (i.e., most of the ones which are currently observed with large instruments) should be warmer and therefore even less likely of being partially solid in their interior. This is particularly obvious when considering the case of HD 209458 b in Fig. 3.

2.3 Static Stability in Giant Planets

While simple models of giant planets are fully convective and adiabatically stratified, several signs point toward stable (non-adiabatic) stratification due to compositional gradients at large depths. The existence of a largely stable deep interior in Saturn had been previously proposed in order to account for Saturn’s high luminosity relative to predictions of adiabatic models (e.g., Leconte and Chabrier 2012, 2013). Fuller (2014) also appealed to stable stratification in order to account for the splitting of normal modes as detected in the rings by Cassini (Hedman and Nicholson 2013). The mode splitting is interpreted as the interaction between f-modes present in the whole planet and g-modes which are present only in stably stratified regions. The presence of a dilute core both in Jupiter and in Saturn (see Fig. 2) seems to indicate that the existence of such a stable region may be the norm rather than the exception. However, for giant planets more massive than Jupiter, more energy is available to mix any primordial core more efficiently (see Guillot et al. 2004), raising the possibility that these massive giant planets might have fully eroded their core and be fully convective, erasing any primordial composition gradients.

The question of stability also arises in other contexts: in the hydrogen-helium phase separation region (see Fig. 3), the formation of helium-rich droplets may or may not lead to the inhibition of convection (Stevenson and Salpeter 1977; Mankovich et al. 2016). In the presence of compositional gradients, double-diffusive convection may arise, possibly leading to a small-scale staircase structure of convective regions sandwiched between conductive interfaces (Rosenblum et al. 2011; Leconte and Chabrier 2012). However, double-diffusive staircases may not always form for realistic conditions in giant planets (Rosenblum et al. 2011; Mirouh et al. 2012; Fuentes et al. 2022), and they may become unstable over long time scales (Moore and Garaud 2016).

In Jupiter, the existence of stable regions deep in the molecular envelope (between 95% and 80% of the planet’s total radius) have been invoked to account for the inferred zonal wind decay (Christensen et al. 2020) and measured magnetic field properties (Moore et al. 2022; Connerney et al. 2022; Sharan et al. 2022). But even features in the convective part of the atmosphere require that the atmosphere is at least partially stable. This is shown in Fig. 4 by measurements of the Brünt-Väisälä frequency, \(N\), defined by

$$ N^{2} = \frac{g}{T}\left (\frac{dT}{dz} + \Gamma _{\mathrm{ad}}\right ), $$
(1)

where \(g\) is the gravitational acceleration, \(T\) temperature, \(z\) altitude, \(\Gamma _{\mathrm{ad}}\equiv -(dT/dz)_{S}\) is the dry adiabatic lapse rate (at constant entropy \(S\)), and a perfect gas of uniform composition has been assumed for simplicity (Wong et al. 2011). Thus, \(N^{2}>0\) only in stable regions. The values of \(N\) shown in Fig. 4 directly measured range from high values of around \(0.02\text{ s}^{-1}\) in the radiative upper atmosphere, to about \(2\times 10^{-3}\text{ s}^{-1}\) in the troposphere near 10 bars, as measured by the Galileo probe. Intermediate values are required to model the observed vortexes. In these regions, latent heat release due to the condensation of H2O, NH4SH and NH3 can lead to subadiabatic temperature gradients accounting for the observed static stability. However, the penetration of cyclones and anticyclones to great depths as seen by Juno (Bolton et al. 2021; Guillot et al. 2023, and references therein) indicates that the stable region is not limited solely to the water condensation zone near 5 bar.

Fig. 4
figure 4

Estimates of static stability in Jupiter’s atmosphere as measured from its Brünt-Väisälä frequency \(N\). Shaded bars (to the left of the dashed line) are from measurements and models of Jupiter’s vertical structure; black and white bars (to the right of the dashed line) are for vortex surroundings as determined by numerical models of the Great Red Spot, and of Jupiter’s anticyclones. Image reproduced with permission from Wong et al. (2011), copyright by Elsevier

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3 Tidal Dissipation Mechanisms

3.1 Tides: General Principles

3.1.1 Tidal Force and Potential

Gaseous giant planets of our solar system have strong tidal interactions with their numerous moons, with tides raised on the moons by the host planet, and vice versa. Tides are for instance responsible for the volcanism of Io (Johnson et al. 1984; Ojakangas and Stevenson 1986) and for the heating of Enceladus (Lainey et al. 2012). The same is hypothesized to occur between close-in giant exoplanets and their host stars (e.g. Winn and Fabrycky 2015, and references therein). In this section, we focus on tidal interactions with moons, but many of the same processes could happen in gas giant exoplanet systems where the tidal force is produced by the planet’s host star.

Giant planets (hereafter the primary) are large, so the gravitational force exerted by the companions (e.g., a moon, hereafter the secondary) varies non-negligibly between the center and the surface. The resulting differential force is the tidal force (\({\mathbf{f}}_{\mathrm{T}}\)), whose dominant quadrupolar component is derived from the tidal potential (\(U_{\mathrm{T}}\)) as:

$$ {\mathbf{f}}_{\mathrm{T}}=-{\boldsymbol{\nabla}}U_{\mathrm{T}}\quad \text{with}\quad U_{ \mathrm{T}}=-\frac{GM_{\mathrm{m}}r^{2}}{d^{3}}P_{2}\left [ \frac{{\mathbf{d}}\cdot{\mathbf{r}}}{d \, r} \right ], $$
(2)

where \(G\) is the gravitational constant, \(M_{\mathrm{m}}\) the mass of the companion (treated here as a point mass), \(d\) (\({\mathbf{d}}\)) the distance (vector) between the centres of mass of the primary and the secondary, \({\mathbf{r}}\) the position vector in the primary relative to its center of mass, and \(P_{2}\) the quadrupolar Legendre polynomial. \(U_{\mathrm{T}}\) is often expanded as a function of the Keplerian orbital parameters (e.g., Murray and Dermott 1999; Kaula 1962; Mathis and Le Poncin-Lafitte 2009; Ogilvie 2014). We also define the tidal frequency in the rotating frame of the planet:

$$ \omega \equiv l \Omega _{\mathrm{m}}-m\Omega _{\mathrm{p}}, $$
(3)

where \(\Omega _{\mathrm{m}}\) and \(\Omega _{\mathrm{p}}\) are the angular velocities of the moon’s orbit and of the mean rotation of the planet, respectively, and \(l\) and \(m\) are integers.

3.1.2 Equilibrium and Dynamical Tides and Their Dissipation

The perturbation of the hydrostatic balance of celestial bodies by tidal forces results in mass redistribution and gravity and pressure perturbations. This mass redistribution creates a tidal bulge approximately in the direction of the companion (Fig. 1) and a large-scale velocity or an elastic displacement in fluid/solid layers, respectively, the so-called fluid/elastic equilibrium tide (e.g. Zahn 1966; Remus et al. 2012; Love 1911; Tobie et al. 2005; Remus et al. 2012). Because the equilibrium tide is not a solution of the full momentum equation, it is completed by the so-called dynamical tide. In fluid bodies, the dynamical tide is composed of inertial, gravity, Alfvén, and acoustic waves excited by the tidal force (e.g. Mathis et al. 2013). Their restoring forces are the Coriolis acceleration, buoyancy (in convectively stable layers or at the free surface), magnetic tension, and pressure, respectively (e.g. Rieutord 2015, and Fig. 5). In solid bodies, the dynamical tide includes tidally-excited shear waves (e.g. Alterman et al. 1959; Tobie et al. 2005).

Fig. 5
figure 5

Type of waves constituting the dynamical tide in fluid planetary layers. We introduce their characteristic frequencies: the Alfvén frequency \(\omega _{\mathrm{A}}\), where \(\vec{k}\) is the wave vector and \({\vec{V}}_{\mathrm{A}}={\vec{B}}/\sqrt{\mu _{0}\rho}\) is the Alfvén velocity (\(\vec{B}\) being the magnetic field, \(\rho \) the density, \(\mu _{0}\) the vacuum permeability), the inertial frequency \(2\Omega _{\mathrm{p}}\) (\(\Omega _{\mathrm{p}}\) being the angular velocity of the planet), the Brünt-Väisälä frequency \(N\) defined in Eq. (1), and the Lamb frequency \(f_{\mathrm{L}}\). For tidally excited waves in stably stratified regions of giant planets, we typically expect \(\omega _{\mathrm{A}} < 2 \Omega _{\mathrm{p}} < N < f_{\mathrm{L}}\), but this may not always be true. Image adapted from Mathis and de Brye (2011)

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In an ideal case without any friction in the primary, the equilibrium and dynamical tides would be in phase with the tidal potential, with the resulting tidal bulges (one per Fourier mode of \(U_{\mathrm{T}}\)) being aligned along the line of centres. In this adiabatic situation, the time-averaged tidal torques on the orbit would vanish (e.g. Zahn 1966). However, the equilibrium and dynamical tides are submitted to dissipation because of viscous or turbulent friction, the diffusion of heat and chemicals, Ohmic heating from magnetic fields, and viscoelastic dissipation in solid regions. Because of the induced losses of energy, the direction of the tidal bulge lags behind the secondary’s position (Fig. 1), resulting in a tidal torque. In addition, the conversion of the kinetic and potential energies of tides into heat impacts the structure and the evolution of the primary.

In the simplest case of a circular co-planar system with semi-major axis \(a_{\mathrm{m}}\), the torque applied to the spin of the primary can be approximated as (Zahn 2013)

$$ \Gamma =- \frac{\left (\Omega _{\mathrm{p}}-\Omega _{\mathrm{m}}\right )}{t_{\mathrm{friction}}} \left (\frac{M_{\mathrm{m}}}{M_{\mathrm{p}}}\right )^{2} M_{\mathrm{p}} R_{\mathrm{p}}^{2} \left (\frac{R_{\mathrm{p}}}{a_{\mathrm{m}}}\right )^{6}, $$
(4)

where \(M_{\mathrm{p}}\) and \(R_{\mathrm{p}}\) are the mass and the radius of the primary and \(t_{\mathrm{friction}}\) is the time characterising tidal dissipation. This expression shows the importance of understanding the physics of tidal dissipation from first principles to obtain a robust evaluation of \(t_{\mathrm{friction}}\) and a prediction of the evolution of the system.

It is common to express \(t_{\mathrm{friction}}\) as a function of a corresponding tidal quality factor \(Q\) (MacDonald 1964), in analogy with the theory of forced oscillators (e.g. Greenberg 2009). A short friction time corresponds to a strong dissipation and a low quality factor (and vice-versa). To evaluate tidal dissipation, we compute the ratio of \(Q\) with the quadrupolar Love number \(k_{2}\), defined as the ratio of the primary’s perturbed gravitational potential to the tidal potential at the primary’s surface, which depends on the primary’s mass concentration. The tidal torque is proportional to \(k_{2}/Q\) (which is equal to the imaginary component of \(k_{2}\), Im[\(k_{2}\)]), i.e., the component of the tidal bulge’s gravitational field that is misaligned with respect to the moon.

3.1.3 Tidal Orbital and Rotational Evolution

For moon orbits that are nearly circular and aligned with the planet’s rotation (as is the case for most of the solar system’s major moons), tidal evolution is relatively simple. In Eq. (4), one can identify the crucial importance of the so-called co-rotation radius at which \(\Omega _{\mathrm{p}}=\Omega _{\mathrm{m}}\). If \(\Omega _{\mathrm{m}} >\Omega _{\mathrm{p}}\), then the companion migrates inwards and the primary rotation accelerates. If \(\Omega _{\mathrm{m}} < \Omega _{\mathrm{p}}\), the companion migrates outwards and the primary rotation slows down. The latter is the configuration of the Earth-Moon system where the Moon migrates outwards by ∼3.8 centimetres per year. When \(\Omega _{\mathrm{m}} >\Omega _{\mathrm{p}}\), the Darwin instability results if \(L_{\mathrm{orb}}\le 3L_{\mathrm{p}}\), where \(L_{\mathrm{orb}}\) and \(L_{\mathrm{p}}\) are the angular momentum of the orbit and of the primary’s rotation respectively. In this case, the companion spirals towards the primary until it is tidally disrupted at the Roche limit (Hut 1980). If \(L_{\mathrm{orb}}\ge 3L_{\mathrm{p}}\), the system evolves towards an equilibrium state where the orbit is circular and the primary’s rotation is aligned and synchronized with the orbit.

We can see that the time characterising dissipation (\(t_{\mathrm{friction}}\)) allows one to predict orbital migration and circularisation time scales, and spin alignment and synchronisation time scales. Each dissipation mechanism leads to a specific frequency-dependence of \(t_{\mathrm{friction}}\) (i.e. smooth or resonant; we refer to Mathis et al. 2013; Ogilvie 8) will be broader and shallower, such that the blue line no longer intercepts the black line, and the resonance locking fixed point no longer exists. When this occurs, the moon will not be able to migrate outward as fast as the resonance location is moving, so resonance locks cannot be sustained. This can also occur if Saturn evolves too quickly (i.e., the black line moves downward in Fig. 8), or if a moon has a large semi-major axis (causing the blue line to shift upward). Hence, we expect resonance locks to eventually break as the moons migrate outwards, though it is not clear when exactly this will occur.

Resonance locks also cannot occur if the resonant locations are moving inward, because then there is no resonance locking fixed point. In this scenario, the moons simply pass through the resonances on short time scales, so we do not expect them to be in resonance today.

To determine whether resonance locking can occur, we must examine the resonance condition,

$$ \sigma _{\alpha }\simeq m \Omega _{\mathrm{m}} \, . $$
(5)

Differentiating the resonance criterion of equation (5) with respect to time leads to the locking criterion

$$ {\dot{\sigma}}_{\alpha }\simeq m {\dot{\Omega}}_{\mathrm{m}} \, . $$
(6)

Since the moons migrate outward, \(\dot{\Omega}_{\mathrm{m}}\) is negative, so the mode frequency must decrease in the inertial frame in order for resonance locking to occur.

In Saturn’s rotating frame the mode frequency is

$$ \omega _{\alpha }= \sigma _{\alpha }- m \Omega _{\mathrm{p}}, $$
(7)

which is negative for all of Saturn’s moons (using \(m\geq 2\)), such that resonant modes have negative frequencies and are retrograde modes. Resonance locking thus requires

$$ \dot{\omega}_{\alpha }\simeq m ( \dot{\Omega}_{\mathrm{m}} - \dot{\Omega}_{ \mathrm{p}} ) \, . $$
(8)

For a constant rotation rate of Saturn, or if Saturn’s rotation rate increases with time, the mode frequency \(\omega _{\alpha}\) must decrease in the rotating frame, i.e., become more negative and such that \(|\omega _{\alpha}|\) increases. Increasing rotation frequency and \(|\omega _{\alpha}|\) are what we expect if Saturn is slowly contracting, but explicit verification of this requirement has not been demonstrated with evolutionary models of Saturn.

Finally, resonance locking may not occur if non-linear effects (which are neglected in most calculations of tidally excited oscillations) become important. A tidally driven parent mode can non-linearly couple to two daughter modes, as described in detail in works such as Weinberg et al. (2021), but non-linear coupling calculations have not been performed for tidally excited modes of giant planets. While we can be sure that resonance locks will eventually break as moons move to larger semi-major axis, it is not clear when this non-linear saturation will occur.

4.2 Consequences of Resonance Locking

If a moon becomes caught in a resonance lock, its tidal migration time scale is dictated by the evolution of the resonant mode frequency, which is determined by the structure of the planet. Hence, the moon’s tidal migration time scale is directly linked to the time scale on which the planet’s structure changes. For gas giant planets undergoing slow contraction, we expect this time scale to be comparable to (or somewhat larger than) the age of the solar system. Hence, to order of magnitude, we expect moon migration time scales of billions to tens of billions of years.

Given a model for the evolution of Saturn’s internal structure, one can calculate the resonance locking-driven migration rates exactly. We defined the mode frequency evolution time scale \(t_{\alpha} = \omega _{\alpha}/\dot{\omega}_{\alpha}\), and the planetary spin evolution timescale \(t_{\mathrm{p}} = \Omega _{\mathrm{p}}/\dot{\Omega}_{\mathrm{p}}\). Equation (8) becomes

$$ \dot{\Omega}_{\mathrm{m}} = \frac{\Omega _{\mathrm{p}}}{t_{\mathrm{p}}} + \frac{\omega _{\alpha}}{m t_{\alpha}} \, . $$
(9)

Since the moon’s orbital frequency changes with semi-major axis as \(\dot{a}_{\mathrm{m}}/a_{\mathrm{m}} = -(2/3) \dot{\Omega}_{\mathrm{m}}/ \Omega _{ \mathrm{m}}\), we have

$$ \frac{\dot{a}_{\mathrm{m}}}{a_{\mathrm{m}}} = -\frac{2}{3} \bigg[ \frac{\omega _{\alpha}}{m \Omega _{\mathrm{m}} t_{\alpha}} + \frac{\Omega _{\mathrm{p}}}{\Omega _{\mathrm{m}} t_{\mathrm{p}}} \bigg] \, . $$
(10)

Since \(\omega _{\alpha}\) is negative, we see that outward migration requires that \(\omega _{\alpha}\) be decreasing with time (\(t_{\alpha}\) is positive) or that the planetary spin frequency be decreasing with time (\(t_{ \mathrm{p}}\) is negative).

For moons at large semi-major axis, resonant modes have \(\omega _{\alpha }\simeq - m \Omega _{\mathrm{p}}\), in which case we obtain

$$ \frac{\dot{a}_{\mathrm{m}}}{a_{\mathrm{m}}} \approx \frac{2}{3} \frac{\Omega _{\mathrm{p}}}{\Omega _{\mathrm{m}}} \bigg[ \frac{1}{t_{\alpha}} - \frac{1}{t_{\mathrm{p}}} \bigg] \, . $$
(11)

Hence, if \(t_{\alpha}\) is not a strong function of mode frequency, we expect outer moons (which have smaller \(\Omega _{\mathrm{m}}\)) to migrate faster than inner moons. This is obviously a very different prediction from viscoelastic dissipation or constant \(Q\) models. Using the definition

$$ t_{\mathrm{tide}}^{-1} = \frac{\dot{a}_{\mathrm{m}}}{a_{\mathrm{m}}} \, , $$
(12)

we can also write

$$ t_{\mathrm{tide}} \approx \frac{3}{2} \frac{\Omega _{\mathrm{m}}}{\Omega _{\mathrm{p}}} \bigg[ \frac{1}{t_{\alpha}} - \frac{1}{t_{\mathrm{p}}} \bigg]^{-1} \, . $$
(13)

If resonance locks are driven by planetary spin evolution such that the \(t_{\mathrm{p}}\) term dominates, outward migration requires that \(t_{p} < 0\), i.e., the planet’s spin frequency must be decreasing with time. Hence, if giant planets are gradually spinning up as they contract, resonance locking cannot occur unless mode frequencies evolve such that \(t_{\alpha }< t_{\mathrm{p}}\).

Using the definition of the tidal quality factor \(Q\), the moon migration rate is given by

$$ \frac{\dot{a}_{\mathrm{m}}}{a_{\mathrm{m}}} = \frac{3 k_{2}}{Q} \frac{M_{m}}{M_{\mathrm{p}}} \left (\frac{R_{\mathrm{p}}}{a_{\mathrm{m}}}\right )^{5} \Omega _{\mathrm{m}}. $$
(14)

Equating this with equation (10) yields the effective quality factor due to a resonance lock, \(Q_{\mathrm{RL}}\)

$$ Q_{\mathrm{RL}} = \frac{9 k_{2}}{2} \frac{M_{\mathrm{m}}}{M_{\mathrm{p}}} \bigg( \frac{R}{a}\bigg)^{\!5} \bigg[ \frac{\omega _{\alpha}}{m \Omega _{\mathrm{m}}^{2} t_{\alpha}} - \frac{\Omega _{\mathrm{p}}}{\Omega _{\mathrm{m}}^{2} t_{\mathrm{p}}} \bigg]^{-1} \, . $$
(15)

The effective tidal quality factor therefore depends on many parameters of the system and is expected to be different for each moon.

4.3 Flavors of Resonance Locking

Depending on the type of oscillation mode that is resonantly locked with a moon, and depending on how a planet is evolving, migration driven by resonance locking could exhibit different behaviors. Let us examine equation (13) above.

Let us start with the limit of constant planetary spin such that \(t_{p} \rightarrow \infty \). In this case, \(t_{\mathrm{tide}} \approx (3 \Omega _{\mathrm{m}}/2 \Omega _{\mathrm{p}}) t_{\alpha}\). If the mode evolution time scale \(t_{\alpha}\) is the same for different moons locked in resonance, then this entails that \(t_{\mathrm{tide}} \propto \Omega _{\mathrm{m}}\). Outer moons with smaller orbital frequencies would migrate faster than inner moons, as mentioned above. This also entails that pairs of moons would cross mean-motion resonances divergently, if both moons are driven outwards by resonance locking. Obviously this cannot always be the case for the Saturn system, because Tethys and Dione are in mean-motion resonance with Mimas and Enceladus, respectively. It is possible that Mimas and Enceladus are in resonance locks with Saturn, while Tethys and Dione are not.

Another possibility is that outer moons (e.g., Rhea) were previously in a mean-motion resonance with one of the inner moons. If Rhea then encountered a resonance lock with Saturn, it could start migrating outwards faster than the inner moon, esca** the mean-motion resonance and migrating outwards as we see it today.

However, it is important to remember that the value of \(t_{\alpha}\) is not the same for each oscillation mode. If we consider pure gravity modes trapped in a stably stratified region, then we expect the g mode frequencies to scale as \(\omega _{\alpha }\propto \int N dr/r\). For a contracting planet with increasing \(N\), we would expect the value of \(t_{\alpha}\) to be positive (allowing for resonance locking) and similar for each mode. However, if modes are partially trapped in different regions of the planet, their values of \(t_{\alpha}\) could be different. For instance, in models of evolving stars (Fuller et al. \(Q\) or viscoelastic models. As mentioned above, the resonance locking time scale is proportional to the planetary evolution time scale, regardless of the planet’s mass and orbital period. Hence, whereas constant \(Q\) models predict that higher mass moons migrate otward faster, the resonance locking migration timescale is independent of mass. Most importantly, whereas constant \(Q\) models predict a very strong dependence on semi-major axis (with migration time scale proportional to \(a_{\mathrm{m}}^{13/2}\), equation (14)), the resonance lock migration time scale is only weakly dependent on semi-major axis. Hence, resonance locking typically predicts much faster migration for outer moons (and correspondingly low effective \(Q\) values) relative to constant \(Q\) models.

Figure 9 shows an example of the qualitative differences between migration driven by resonance locking vs. migration driven by constant \(Q\) models. The biggest difference is that with resonance locking, outer moons migrate relatively faster, and inner moons migrate slower. In Saturn, the consequence is that Rhea and Titan may have migrated by a large fraction of their current semi-major axes, whereas constant \(Q\) models predict almost no migration for these moons. Additionally, Mimas, Enceladus, and Tethys migrate slower and hence may be much older than inferred from constant \(Q\) models.

Fig. 9
figure 9

An example of the differing orbital evolution between resonance locking (colored lines) and constant \(Q\) models (dashed lines). For the latter, \(Q\) values for each moon of Saturn are taken from Lainey et al. (2020) and assumed to be constant in time. For resonance locking, we use \(t_{\mathrm{tide}}=3 t\), which matches measured moon migration rates fairly well, and the color of the lines corresponds to the effective quality factor (Eq. (15)). Neither evolution accounts for mean-motion resonances, which will significantly affect the orbital evolution of the inner moons. Image reproduced from Lainey et al. (2020)

Full size image

However, detailed predictions are plagued by the uncertainties mentioned above. While present day migration rates and hence \(t_{\alpha}\) values can be inferred, it is not clear how those values have evolved over time. Figure 9 assumes that \(t_{\mathrm{tide}} \sim t\), where \(t\) is the age of Saturn, but the constant of proportionality has not been predicted from theoretical models. Current moon migration rates require \(t_{\mathrm{tide}} \sim 3 t\) (Lainey et al. 2020) In reality, the form of \(t_{\mathrm{tide}}(t)\) and \(t_{\alpha}(t)\) is determined by the very uncertain evolution of Saturn’s structure and oscillation mode frequencies. Even the sign of \(t_{\alpha}\) has not been calculated with realistic models. Extrapolation from the present day to small values of \(t\) becomes increasingly uncertain.

Another major uncertainty is whether moons pass through mean-motion resonances, and what happens when they do so. If two moons are each caught in a resonance lock, their relative migration rates are determined by the value of \(t_{\alpha}\) for each resonantly locked mode, which could be quite different from one another. Hence it is certainly possible for either the inner or outer moon to migrate faster, i.e., the migration may be either convergent or divergent. In the latter case (the outer moon migrates faster), the system will simply pass through the mean-motion resonance, and each moon will presumably remain resonantly locked with its respective planetary oscillation.

In the case of convergent migration, the moons may get locked into mean motion resonance, but the details of this process are highly uncertain. At present, nobody has ever computed an orbital integration for this case, including the changing tidal torque, which is an extremely sensitive function of orbital period near resonance (Fig. 8). Hence, the mean-motion resonance could potentially disrupt the resonance lock by pushing the moon out of resonance.

Alternatively, the system may be able to adjust so that the inner moon stably locks into mean motion resonance with the outer moon, driving it outwards. In this case, the inner moon would need to be pushed deeper into resonance with the planetary oscillation, so that the tidal torque on it can be stronger to account for the extra energy and angular momentum needed to drive out the outer moon. If this can occur, the entire system would be driven outwards at the same resonance locking time scale that the inner moon had before the mean-motion resonance was established.

It is very unlikely that both an inner and outer moon could simultaneously be in resonance locks with planetary oscillations, and in mean-motion resonance with each other. The mean-motion resonances occur near integer ratios of the orbital frequencies, but there is no reason to suspect planetary oscillation mode frequencies to obey such a relationship. Even if they did at one point in time, it is unlikely that each oscillation mode frequency would evolve together to sustain this integer ratio for an extended period of time.

Consequently, if the migration of Saturn’s inner moons are driven by resonance locking, it is most likely that the inner moons of mean-motion pairs (i.e., Mimas and Enceladus) are caught in resonance locks while the outer moons (Tethys and Dione) are not. Hence, we would expect to measure a strong tidal torque (small planet \(Q\)) on Mimas and Enceladus, and a weak tidal torque (large planet \(Q\)) on Tethys and Dione. The measured values (Lainey et al. 2020) may indeed be larger for those outer moons, but current measurement uncertainties are very large so it is difficult to claim anything with confidence. Continued monitoring of moon migration rates will help distinguish between different tidal dissipation mechanisms and hence differentiate between possible evolutionary histories.

5 Conclusions

Our understanding of tidal dissipation in giant planets and its consequences for moon migration remain incomplete, but they have improved greatly over the last decade. A major advance has been in our picture of giant planet structure. Both Jupiter and Saturn show evidence for a large and “diffuse” core, likely containing a smooth and gradual compositional change from ice/rock in the core to the gaseous envelope. In Jupiter, this diffuse core helps explain gravity data from Juno, while in Saturn it is revealed in part by ring seismology data from Cassini. A diffuse core is also predicted by modern planetary formation models.

Stably stratified regions between the core and envelope have major implications for tidal dissipation in giant planets. They allow for the excitation of gravity waves in these regions, which can contribute greatly to tidal dissipation. Additionally, thick stably stratified shells increase the excitation rate of inertial waves in the overlying convective envelope, enhancing their contribution to tidal dissipation. Recent models accounting for these effects may explain the rapid moon migration rates in Saturn that have been measured using long-baseline astrometry.

Finally, the continually evolving structures of giant planets may feedback onto tidal migration through a resonance locking process. As a planet’s structure evolves (e.g., due to cooling, helium rain, compositional settling, or core erosion), the frequencies of gravito-inertial modes evolve as well. This may allow moons to lock into resonance with these modes, such that the energy dissipation rate is enhanced over long periods of time. This process could be necessary to explain the rapid migration of Saturn’s outer moons Rhea and Titan.

Although recent progress is promising, there is plenty of room for improvement in future work. More sophisticated planetary formation and evolution models matched to thermal/gravity/seismology data can be developed to provide more realistic planetary internal structures. These will allow for more reliable calculations of tidal energy dissipation rates, including the feedback between the evolving planetary structure and expanding orbits of the moons. Such work will also shed light on the physics of ice giant systems, exoplanet systems, tidal heating of moons, and even the possibility of life in the oceans of Enceladus and Europa.