Tidal interactions play a key role in the dynamics and evolution of icy worlds. The intense tectonic activity of Europa and the eruption activity on Enceladus are clear examples of the manifestation of tidal deformation and associated dissipation. While tidal heating has long been recognized as a major driver in the activity of these icy worlds, the mechanism controlling how tidal forces deform the different internal layers and produce heat by tidal friction still remains poorly constrained. As tidal forcing varies with orbital characteristics (distance to the central planet, eccentricity, obliquity), the contribution of tidal heating to the internal heat budget can strongly change over geological timescales. In some circumstances, the tidally-produced heat can result in internal melting and surface activity taking various forms. Even in the absence of significant heat production, tidal deformation can be used to probe the interior structure, the tidal response of icy moons being strongly sensitive to their hydrosphere structure. In the present paper, we review the methods to compute tidal deformation and dissipation in the different layers composing icy worlds. After summarizing the main principle of tidal deformation and the different rheological models used to model visco-elastic tidal response, we describe the dissipation processes expected in rock-dominated cores, subsurface oceans and icy shells and highlight the potential effects of tidal heating in terms of thermal evolution and activity. We finally anticipate how data collected by future missions to Jupiter's and Saturn's moons could be used to constrain their tidal response and the consequences for past and present activities.

Faculty of Mathematics and Physics Department of Geophysics Charles University 5 Holesšovičkách 2 Praha Praha 8 180 00 Czech Republic

Faculty of Mathematics and Physics Mathematical Institute Charles University Sokolovská 83 Praha Praha 8 186 75 Czech Republic

IMCCE CNRS Observatoire de Paris PSL University Sorbonne Université Paris France

Laboratoire de Planétologie et Géosciences UMR 6112 CNRS Nantes Université Université d'Angers Le Mans Université Nantes France

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A G, Wahr J, Zhong S (2014) The effects of laterally varying icy shell structure on the tidal response of Ganymede and Europa. J Geophys Res, Planets 119(3):659–678

Adams CJ, Iverson NR, Helanow C, Zoet LK, Bate CE (2021) Softening of temperate ice by interstitial water. Front Earth Sci 9:702761

Alterman Z, Jarosch H, Pekeris C (1959) Oscillations of the Earth. Proc R Soc Lond Ser A, Math Phys Sci 252(1268):80–95

Amit H, Choblet G, Tobie G, Terra-Nova F, Čadek O, Bouffard M (2020) Cooling patterns in rotating thin spherical shells—application to Titan’s subsurface ocean. Icarus 338:113509

Amorim DO, Gudkova T (2024) Constraining Earth’s mantle rheology with Love and Shida numbers at the M2 tidal frequency. Phys Earth Planet Inter 347:107144

Andrade ENDC (1910) On the viscous flow in metals, and allied phenomena. Proc R Soc Lond Ser A, Contain Pap Math Phys Character 84(567):1–12

Arzi AA (1978) Critical phenomena in the rheology of partially melted rocks. Tectonophysics 44(1–4):173–184

Ashkenazy Y, Tziperman E (2021) Dynamic Europa ocean shows transient Taylor columns and convection driven by ice melting and salinity. Nat Commun 12(1):6376 PubMed PMC

Auclair-Desrotour P, Mathis S, Laskar J, Leconte J (2018) Oceanic tides from Earth-like to ocean planets. Astron Astrophys 615:A23. 10.1051/0004-6361/201732249

Auclair-Desrotour P, Leconte J, Bolmont E, Mathis S (2019) Final spin states of eccentric ocean planets. Astron Astrophys 629:A132. 10.1051/0004-6361/201935905

Aygün B, Čadek O (2023) Impact of the core deformation on the tidal heating and flow in Enceladus’ subsurface ocean. Authorea Preprints

Bagdassarov N, Dingwell DB (1993) Frequency dependent rheology of vesicular rhyolite. J Geophys Res, Solid Earth 98(B4):6477–6487

Bagheri A, Khan A, Al-Attar D, Crawford O, Giardini D (2019) Tidal response of Mars constrained from laboratory-based viscoelastic dissipation models and geophysical data. J Geophys Res, Planets 124(11):2703–2727

Bagheri A, Efroimsky M, Castillo-Rogez J, Goossens S, Plesa A-C, Rambaux N, Rhoden A, Walterová M, Khan A, Giardini D (2022a) Tidal insights into rocky and icy bodies: an introduction and overview. In: Schmelzbach C, Stahler SC (eds) Geophysical exploration of the Solar System. Advances in geophysics, vol 63. Elsevier, Amsterdam, pp 231–320

Bagheri A, Khan A, Deschamps F, Samuel H, Kruglyakov M, Giardini D (2022b) The tidal–thermal evolution of the Pluto-Charon system. Icarus 376:114871

Beddingfield CB, Cartwright RJ, Leonard E, Nordheim T, Scipioni F (2022a) Ariel’s elastic thicknesses and heat fluxes. Planet Sci J 3(5):106

Beddingfield CB, Leonard E, Cartwright RJ, Elder C, Nordheim TA (2022b) High heat flux near Miranda’s inverness corona consistent with a geologically recent heating event. Planet Sci J 3(7):174

Behounková M, Cadek O (2014) Tidal dissipation in heterogeneous bodies: Maxwell vs Andrade rheology. In: European Planetary Science Congress, vol 9, EPSC2014–409

Běhounková M, Tobie G, Choblet G, Čadek O (2010) Coupling mantle convection and tidal dissipation: applications to Enceladus and Earth-like planets. J Geophys Res, Planets 115(E9):E09011

Běhounková M, Tobie G, Choblet G, Čadek O (2012) Tidally-induced melting events as the origin of south-pole activity on Enceladus. Icarus 219(2):655–664. 10.1016/j.icarus.2012.03.024

Běhounková M, Tobie G, Choblet G, Čadek O (2013) Impact of tidal heating on the onset of convection in Enceladus’s ice shell. Icarus 226(1):898–904

Běhounková M, Tobie G, Čadek O, Choblet G, Porco C, Nimmo F (2015) Timing of water plume eruptions on Enceladus explained by interior viscosity structure. Nat Geosci 8:601–604. 10.1038/ngeo2475

Běhounková M, Souček O, Hron J, Čadek O (2017) Plume activity and tidal deformation on Enceladus influenced by faults and variable ice shell thickness. Astrobiology 17:941–954. 10.1089/ast.2016.1629 PubMed PMC

Běhounková M, Tobie G, Choblet G, Kervazo M, Melwani Daswani M, Dumoulin C, Vance SD (2021) Tidally induced magmatic pulses on the Oceanic floor of Jupiter’s moon Europa. Geophys Res Lett 48(3):e2020GL090077. 10.1029/2020GL090077

Benjamin D, Wahr J, Ray RD, Egbert GD, Desai SD (2006) Constraints on mantle anelasticity from geodetic observations, and implications for the J2 anomaly. Geophys J Int 165(1):3–16

Berckhemer H, Kampfmann W, Aulbach E, Schmeling H (1982) Shear modulus and Q of forsterite and dunite near partial melting from forced-oscillation experiments. Phys Earth Planet Inter 29(1):30–41

Berne A, Simons M, Keane JT, Park RS (2023) Inferring the mean thickness of the outer ice shell of Enceladus from diurnal crustal deformation. J Geophys Res, Planets 128(6):e2022JE007712. 10.1029/2022JE007712

Berne A, Simons M, Keane JT, et al. (2024) Jet activity on Enceladus linked to tidally driven strike-slip motion along tiger stripes. Nat Geosci 17:385–391. 10.1038/s41561-024-01418-0

Beuthe M (2008) Thin elastic shells with variable thickness for lithospheric flexure of one-plate planets. Geophys J Int 172(2):817–841. 10.1111/j.1365-246X.2007.03671.x

Beuthe M (2013) Spatial patterns of tidal heating. Icarus 223(1):308–329. 10.1016/j.icarus.2012.11.020

Beuthe M (2015a) Tides on Europa: the membrane paradigm. Icarus 248:109–134. 10.1016/j.icarus.2014.10.027

Beuthe M (2015b) Tidal Love numbers of membrane worlds: Europa, Titan, and co. Icarus 258:239–266. 10.1016/j.icarus.2015.06.008

Beuthe M (2016) Crustal control of dissipative ocean tides in Enceladus and other icy moons. Icarus 280:278–299. 10.1016/j.icarus.2016.08.009

Beuthe M (2018) Enceladus’s crust as a non-uniform thin shell: I tidal deformations. Icarus 302:145–174

Beuthe M (2019) Enceladus’s crust as a non-uniform thin shell: II tidal dissipation. Icarus 332:66–91

Beuthe M, Rivoldini A, Trinh A (2016) Enceladus’s and Dione’s floating ice shells supported by minimum stress isostasy. Geophys Res Lett 43(19):10088–10096. 10.1002/2016GL070650

Bierson CJ (2024) The impact of rheology model choices on tidal heating studies. Icarus 414:116026

Bierson C, Nimmo F (2016) A test for Io’s magma ocean: modeling tidal dissipation with a partially molten mantle. J Geophys Res, Planets 121:2211–2224

Biot MA (1954) Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena. J Appl Phys 25(11):1385–1391

Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J Acoust Soc Am 28(2):179–191

Bire S, Kang W, Ramadhan A, Campin J-M, Marshall J (2022) Exploring ocean circulation on icy moons heated from below. J Geophys Res, Planets 127(3):e2021JE007025

Bland MT, Showman AP, Tobie G (2009) The orbital–thermal evolution and global expansion of Ganymede. Icarus 200(1):207–221. 10.1016/j.icarus.2008.11.016

Bolmont E, Breton SN, Tobie G, Dumoulin C, Mathis S, Grasset O (2020) Solid tidal friction in multi-layer planets: application to Earth, Venus, a super Earth and the TRAPPIST-1 planets. Potential approximation of a multi-layer planet as a homogeneous body. Astron Astrophys 644:A165. 10.1051/0004-6361/202038204

Brennan A, Thusyanthan N, Madabhushi S (2005) Evaluation of shear modulus and damping in dynamic centrifuge tests. J Geotech Geoenviron Eng 131(12):1488–1497

Budiansky B, O’Connell RJ (1976) Elastic moduli of a cracked solid. Int J Solids Struct 12(2):81–97

Čadek O, Tobie G, van Hoolst T, Masse M, Choblet G, Lefevre A, Mitri G, Baland R-M, Běhounková M, Bourgeois O, Trinh A (2016) Enceladus’s internal ocean and ice shell constrained from Cassini gravity, shape, and libration data. Geophys Res Lett 46:5653–5660. 10.1002/2016GL068634

Čadek O, Souček O, Běhounková M, Choblet G, Tobie G, Hron J (2019) Long-term stability of Enceladus’ uneven ice shell. Icarus 319(2018):476–484. 10.1016/j.icarus.2018.10.003

Cappuccio P, Hickey A, Durante D, Di Benedetto M, Iess L, De Marchi F, Plainaki C, Milillo A, Mura A (2020) Ganymede’s gravity, tides and rotational state from JUICE’s 3GM experiment simulation. Planet Space Sci 187:104902

Cappuccio P, Di Benedetto M, Durante D, Iess L (2022) Callisto and Europa gravity measurements from JUICE 3GM experiment simulation. Planet Sci J 3(8):199

Cassen P, Reynolds RT, Peale SJ (1979) Is there liquid water on Europa? Geophys Res Lett 6(9):731–734. 10.1029/GL006i009p00731

Castillo-Rogez JC, Efroimsky M, Lainey V (2011) The tidal history of iapetus: spin dynamics in the light of a refined dissipation model. J Geophys Res, Planets 116(E9):E09008

Castillo-Rogez J, Hemingway D, Rhoden A, Tobie G, McKinnon W (2018) Origin and evolution of Saturn’s mid-sized moons. In: Schenk PM (eds) Enceladus and the icy moons of Saturn. University of Arizona Press, Tucson, pp 285–306

Charnay B, Tobie G, Lebonnois S, Lorenz RD (2022) Gravitational atmospheric tides as a probe of Titan’s interior: application to dragonfly. Astron Astrophys 658:A108

Chen EMA, Nimmo F (2011) Obliquity tides do not significantly heat Enceladus. Icarus 214(2):779–781. 10.1016/j.icarus.2011.06.007

Chen EMA, Nimmo F (2016) Tidal dissipation in the lunar magma ocean and its effect on the early evolution of the Earth-Moon system. Icarus 275:132–142. 10.1016/j.icarus.2016.04.012

Chen EMA, Nimmo F, Glatzmaier GA (2014) Tidal heating in icy satellite oceans. Icarus 229:11–30. 10.1016/j.icarus.2013.10.024

Choblet G, Tobie G, Sotin C, Běhounková M, Čadek O, Postberg F, Souček O (2017) Powering prolonged hydrothermal activity inside Enceladus. Nat Astron 1(12):841–847

Choblet G, Tobie G, Buch A, Čadek O, Barge LM, Běhounková M, Camprubi E, Freissinet C, Hedman M, Jones G, et al (2021) Enceladus as a potential oasis for life: Science goals and investigations for future explorations. Exp Astron, 1–39

Čížková H, van den Berg AP, Spakman W, Matyska C (2012) The viscosity of Earth’s lower mantle inferred from sinking speed of subducted lithosphere. Phys Earth Planet Inter 200:56–62

Cole DM (1998) Modeling the cyclic loading response of sea ice. Int J Solids Struct 35(31–32):4067–4075

Cole DM, Durell GD (1995) The cyclic loading of saline ice. Philos Mag A 72(1):209–229

Costa A, Caricchi L, Bagdassarov N (2009) A model for the rheology of particle-bearing suspensions and partially Molten rocks. Geochem Geophys Geosyst 10(3)

Ćuk M, El Moutamid M (2022) Three-body resonances in the Saturnian system. Astrophys J Lett 926(2):L18

Ćuk M, El Moutamid M, Lari G, Neveu M, Nimmo F, Noyelles B, Rhoden A, Saillenfest M (2024) Long-term evolution of the Saturnian system. Space Sci Rev 220:20. 10.1007/s11214-024-01049-2 PubMed PMC

Daher H, Arbic BK, Williams JG, Ansong JK, Boggs DH, Müller M, Schindelegger M, Austermann J, Cornuelle BD, Crawford EB, Fringer OB, Lau HCP, Lock SJ, Maloof AC, Menemenlis D, Mitrovica JX, Green JAM, Huber M (2021) Long-term Earth-Moon evolution with high-level orbit and ocean tide models. J Geophys Res, Planets 126(12):e06875. 10.1029/2021JE006875 PubMed PMC

Davies AG, Perry JE, Williams DA, Nelson DM (2024) Io’s polar volcanic thermal emission indicative of magma ocean and shallow tidal heating models. Nat Astron 8(1):94–100

de Kleer K, Nimmo F, Kite E (2019) Variability in Io’s volcanism on timescales of periodic orbital changes. Geophys Res Lett 46(12):6327–6332

De La Chapelle S, Milsch H, Castelnau O, Duval P (1999) Compressive creep of ice containing a liquid intergranular phase: rate-controlling processes in the dislocation creep regime. Geophys Res Lett 26(2):251–254

De Marchi F, Di Achille G, Mitri G, Cappuccio P, Di Stefano I, Di Benedetto M, Iess L (2021) Observability of Ganymede’s gravity anomalies related to surface features by the 3GM experiment onboard ESA’s JUpiter ICy moons Explorer (JUICE) mission. Icarus 354:114003. 10.1016/j.icarus.2020.114003

De Marchi F, Cappuccio P, Mitri G, Iess L (2022) Frequency-dependent Ganymede’s tidal Love number k2 detection by JUICE’s 3GM experiment and implications for the subsurface ocean thickness. Icarus 386:115150. 10.1016/j.icarus.2022.115150

Denny KE, Hedman MM, Bockelée-Morvan D, Filacchione G, Capaccioni F (2024) Constraining time variations in Enceladus’s water-vapor plume with near-infrared spectra from Cassini’s visual and infrared mapping spectrometer. Planet Sci J 5(6):144. 10.3847/PSJ/ad4c69

Ding H, Chen Z, Pan Y, Zou C (2021) The complex Love numbers of long-period zonal tides retrieved from global GPS displacements: applications for determining mantle anelasticity. J Geophys Res, Solid Earth 126(9):e2021JB022380

Dingwell DB, Webb SL (1989) Structural relaxation in silicate melts and non-Newtonian melt rheology in geologic processes. Phys Chem Miner 16:508–516

Dirkx D, Gurvits L, Lainey V, Lari G, Milani A, Cimò G, Bocanegra-Bahamon T, Visser P (2017) On the contribution of PRIDE-JUICE to Jovian system ephemerides. Planet Space Sci 147:14–27. 10.1016/j.pss.2017.09.004

Doggett T, Greeley R, Figueredo P, Tanaka K (2009) Geologic stratigraphy and evolution of Europa’s surface. In: Pappalardo RT, McKinnon WB, Khurana KK (eds) Europa. University of Arizona Press, Tucson, pp 137–160

Dombard AJ, Sessa AM (2019) Gravity measurements are key in addressing the habitability of a subsurface ocean in Jupiter’s moon Europa. Icarus 325:31–38. 10.1016/j.icarus.2019.02.025

Dougherty M, Khurana K, Neubauer F, Russell C, Saur J, Leisner J, Burton M (2006) Identification of a dynamic atmosphere at Enceladus with the Cassini magnetometer. Science 311(5766):1406–1409 PubMed

Durante D, Hemingway D, Racioppa P, Iess L, Stevenson D (2019) Titan’s gravity field and interior structure after Cassini. Icarus 326:123–132

Durek JJ, Ekström G (1996) A radial model of anelasticity consistent with long-period surface-wave attenuation. Bull Seismol Soc Am 86(1A):144–158

Efroimsky M (2012) Tidal dissipation compared to seismic dissipation: in small bodies, Earths, and super-Earths. Astrophys J 746(2):150

Efroimsky M, Makarov VV (2014) Tidal dissipation in a homogeneous spherical body. I. Methods. Astrophys J 795(1):6

Egbert GD, Ray RD (2001) Estimates of M2 tidal energy dissipation from TOPEX/Poseidon altimeter data. J Geophys Res 106(C10):22475–22502. 10.1029/2000JC000699

Egbert GD, Bennett AF, Foreman MGG (1994) TOPEX/POSEIDON tides estimated using a global inverse model. J Geophys Res 99(C12):24821–24852. 10.1029/94JC01894

Egbert GD, Ray RD, Bills BG (2004) Numerical modeling of the global semidiurnal tide in the present day and in the last glacial maximum. J Geophys Res, Oceans 109(C3):C03003. 10.1029/2003JC001973

Ermakov AI, Park RS, Roa J, Castillo-Rogez JC, Keane JT, Nimmo F, Kite ES, Sotin C, Lazio TJW, Steinbrügge G, et al. (2021) A recipe for the geophysical exploration of Enceladus. Planet Sci J 2(4):157

Farhat M, Auclair-Desrotour P, Boué G, Laskar J (2022) The resonant tidal evolution of the Earth-Moon distance. Astron Astrophys 665:L1. 10.1051/0004-6361/202243445

Faul UH, Jackson I (2005) The seismological signature of temperature and grain size variations in the upper mantle. Earth Planet Sci Lett 234(1–2):119–134

Faul U, Jackson I (2015) Transient creep and strain energy dissipation: an experimental perspective. Annu Rev Earth Planet Sci 43:541–569

Faul UH, Fitz Gerald JD, Jackson I (2004) Shear wave attenuation and dispersion in melt-bearing olivine polycrystals: 2. Microstructural interpretation and seismological implications. J Geophys Res, Solid Earth 109(B6):B06202

Figueredo PH, Greeley R (2004) Resurfacing history of Europa from pole-to-pole geological mapping Icarus 167:287–312. 10.1016/j.icarus.2003.09.016

Gaeman J, Hier-Majumder S, Roberts JH (2012) Sustainability of a subsurface ocean within Triton’s interior. Icarus 220(2):339–347

Genova A, Parisi M, Gargiulo AM, Petricca F, Andolfo S, Torrini T, Del Vecchio E, Glein CR, Cable ML, Phillips CB, et al. (2024) Gravity investigation to characterize Enceladus’s ocean and interior. Planet Sci J 5(2):40

Gerkema T, Shrira VI (2005) Near-inertial waves in the ocean: beyond the “traditional approximation”. J Fluid Mech 529:195–219. 10.1017/S0022112005003411

Gerkema T, Zimmerman J (2008) An introduction to internal waves. Lecture notes. Royal NIOZ, Texel

Gerkema T, Zimmerman JTF, Maas LRM, van Haren H (2008) Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Rev Geophys 46(2):RG2004. 10.1029/2006RG000220

Goldsby D, Kohlstedt DL (2001) Superplastic deformation of ice: experimental observations. J Geophys Res, Solid Earth 106(B6):11017–11030

Goossens S, van Noort B, Mateo A, Mazarico E, van der Wal W (2024) A low-density ocean inside Titan inferred from Cassini data. Nat Astron, 1–10

Grasset O, Dougherty M, Coustenis A, Bunce E, Erd C, Titov D, Blanc M, Coates A, Drossart P, Fletcher L, et al. (2013) JUpiter ICy moons Explorer (JUICE): an ESA mission to orbit Ganymede and to characterise the Jupiter system. Planet Space Sci 78:1–21

Green JAM, Huber M, Waltham D, Buzan J, Wells M (2017) Explicitly modelled deep-time tidal dissipation and its implication for lunar history. Earth Planet Sci Lett 461:46–53. 10.1016/j.epsl.2016.12.038

Hamilton CW, Beggan CD, Still S, Beuthe M, Lopes RM, Williams DA, Radebaugh J, Wright W (2013) Spatial distribution of volcanoes on Io: implications for tidal heating and magma ascent. Earth Planet Sci Lett 361:272–286

Hansen CJ, Esposito L, Stewart A, Colwell J, Hendrix A, Pryor W, Shemansky D, West R (2006) Enceladus’ water vapor plume. Science 311(5766):1422–1425 PubMed

Hansen C, Esposito L, Colwell J, Hendrix A, Portyankina G, Stewart A, West R (2020) The composition and structure of Enceladus’ plume from the complete set of Cassini UVIS occultation observations. Icarus 344:113461

Hartkorn O, Saur J (2017) Induction signals from Callisto’s ionosphere and their implications on a possible subsurface ocean. J Geophys Res Space Phys 122(11):11677–11697. 10.1002/2017JA024269

Hay HCFC, Matsuyama I (2017) Numerically modelling tidal dissipation with bottom drag in the oceans of Titan and Enceladus. Icarus 281:342–356. 10.1016/j.icarus.2016.09.022

Hay HCFC, Matsuyama I (2019) Nonlinear tidal dissipation in the subsurface oceans of Enceladus and other icy satellites. Icarus 319:68–85. 10.1016/j.icarus.2018.09.019

Hay HCFC, Trinh A, Matsuyama I (2020) Powering the Galilean satellites with moon-moon tides. Geophys Res Lett 47(15):e88317. 10.1029/2020GL088317

Hay HCFC, Matsuyama I, Pappalardo RT (2022) The high-frequency tidal response of ocean worlds: application to Europa and Ganymede. J Geophys Res, Planets 127(5):e07064. 10.1029/2021JE007064

Hedman MM, Gosmeyer CM, Nicholson PD, Sotin C, Brown RH, Clark RN, Baines KH, Buratti BJ, Showalter MR (2013) An observed correlation between plume activity and tidal stresses on Enceladus. Nature 500(7461):182–184. 10.1038/nature12371 PubMed

Hemingway DJ, Mittal T (2019) Enceladus’s ice shell structure as a window on internal heat production. Icarus 332:111–131. 10.1016/j.icarus.2019.03.011

Hendershott MC (1972) The effects of solid Earth deformation on global ocean tides. Geophys J 29(4):389–402. 10.1111/j.1365-246X.1972.tb06167.x

Hendershott MC (1981) Long waves and ocean tides. In: Warren BA, Wunsch C (eds) Evolution of physical oceanography. MIT Press, Cambridge, pp 292–341

Henning WG, Hurford T (2014) Tidal heating in multilayered terrestrial exoplanets. Astrophys J 789(1):30

Henning WG, O’Connell RJ, Sasselov DD (2009) Tidally heated terrestrial exoplanets: viscoelastic response models. Astrophys J 707(2):1000

Hirth G, Kohlstedt DL (1995a) Experimental constraints on the dynamics of the partially molten upper mantle: deformation in the diffusion creep regime. J Geophys Res, Solid Earth 100(B2):1981–2001

Hirth G, Kohlstedt DL (1995b) Experimental constraints on the dynamics of the partially molten upper mantle: 2. Deformation in the dislocation creep regime. J Geophys Res, Solid Earth 100(B8):15441–15449

Hoppa G, Tufts BR, Greenberg R, Geissler P (1999) Strike-slip faults on Europa: global shear patterns driven by tidal stress. Icarus 141(2):287–298. 10.1006/icar.1999.6185

Hough SS (1898) On the application of harmonic analysis to the dynamical theory of the tides. Part II: on the general integration of Laplace’s dynamical equations. Philos Trans R Soc Lond Ser A 191:139–185 10.1098/rsta.1898.0005

Howell SM (2021) The likely thickness of Europa’s icy shell. Planet Sci J 2(4):129

Howell SM, Pappalardo RT (2020) Nasa’s Europa Clipper—a mission to a potentially habitable ocean world. Nat Commun 11(1):1311 PubMed PMC

Howett CJA, Spencer JR, Pearl J, Segura M (2011) High heat flow from Enceladus’ south polar region measured using 10-600 cm−1 Cassini/CIRS data. J Geophys Res, Planets 116(E3):E03003. 10.1029/2010JE003718

Howett C, Miles G, Quick LC (2025) Constraining the mass and energy of dissipation systems. Space Sci Rev 221

Hulbe CL, Klinger M, Masterson M, Catania G, Cruikshank K, Bugni A (2016) Tidal bending and strand cracks at the kamb ice stream grounding line, West Antarctica. J Glaciol 62(235):816–824

Hurford TA, Helfenstein P, Hoppa GV, Greenberg R, Bills BG (2007) Eruptions arising from tidally controlled periodic openings of rifts on Enceladus. Nature 447(7142):292–294. 10.1038/nature05821 PubMed

Hussmann H, Spohn T (2004) Thermal-orbital evolution of Io and Europa. Icarus 171:391–410. 10.1016/j.icarus.2004.05.020

Hussmann H, Choblet G, Lainey V, Matson DL, Sotin C, Tobie G, van Hoolst T (2010) Implications of rotation, orbital states, energy sources, and heat transport for internal processes in icy satellites. Space Sci Rev 153:317–348. 10.1007/s11214-010-9636-0

Hussmann H, Rodríguez A, Callegari N Jr, Shoji D (2019) Early resonances of Tethys and Dione: implications for Ithaca Chasma. Icarus 319:407–416

Idini B, Nimmo F (2024) Resonant stratification in Titan’s global ocean. Planet Sci J 5:15. 10.3847/PSJ/ad11ef

Iess L, Jacobson RA, Ducci M, Stevenson DJ, Lunine JI, Armstrong JW, Asmar SW, Racioppa P, Rappaport NJ, Tortora P (2012) The tides of Titan. Science 337(6093):457. 10.1126/science.1219631 PubMed

Iess L, Stevenson D, Parisi M, Hemingway D, Jacobson R, Lunine J, Nimmo F, Armstrong J, Asmar S, Ducci M, et al. (2014) The gravity field and interior structure of Enceladus. Science 344(6179):78–80 PubMed

Ingersoll AP, Ewald SP, Trumbo SK (2020) Time variability of the Enceladus plumes: orbital periods, decadal periods, and aperiodic change. Icarus 344:113345. 10.1016/j.icarus.2019.06.006

Ishibashi I, Zhang X (1993) Unified dynamic shear moduli and damping ratios of sand and clay. Soil Found 33(1):182–191

Jackson I (2000) Laboratory measurement of seismic wave dispersion and attenuation: recent progress. In: Karato S-I et al. (eds) Earth’s deep interior: mineral physics and tomography from the atomic to the global scale. Geophysical Monograph, vol 117. AGU, Washington DC, pp 265–289. 10.1029/GM117p0265

Jackson I, Fitz Gerald JD, Faul UH, Tan BH (2002) Grain-size-sensitive seismic wave attenuation in polycrystalline olivine. J Geophys Res, Solid Earth 107(B12):ECV-5

Jacobson RA (2022) The orbits of the main Saturnian satellites, the Saturnian system gravity field, and the orientation of Saturn’s pole. Astron J 164(5):199

Jara-Orué H, Vermeersen B (2016) Tides on Jupiter’s moon Ganymede and their relation to its internal structure. Geol Mijnb 95(2):191–201

Jayne SR, St. Laurent LC (2001) Parameterizing tidal dissipation over rough topography. Geophys Res Lett 28(5):811–814. 10.1029/2000GL012044

Journaux B, Kalousová K, Sotin C, Tobie G, Vance S, Saur J, Bollengier O, Noack L, Rückriemen-Bez T, Van Hoolst T, et al. (2020) Large ocean worlds with high-pressure ices. Space Sci Rev 216:7. 10.1007/s11214-019-0633-7

Kalousová K, Sotin C (2018) Melting in high-pressure ice layers of large ocean worlds—implications for volatiles transport. Geophys Res Lett 45(16):8096–8103

Kalousová K, Sotin C (2020) Dynamics of Titan’s high-pressure ice layer. Earth Planet Sci Lett 545:116416

Kalousová K, Souček O, Tobie G, Choblet G, Čadek O (2016) Water generation and transport below Europa’s strike-slip faults. J Geophys Res, Planets 121(12):2444–2462

Kalousová K, Sotin C, Choblet G, Tobie G, Grasset O (2018) Two-phase convection in Ganymede’s high-pressure ice layer—implications for its geological evolution. Icarus 299:133–147

Kamata S (2023) Poroviscoelastic gravitational dynamics. J Geophys Res, Planets 128(7):e2022JE007700

Kamata S, Matsuyama I, Nimmo F (2015) Tidal resonance in icy satellites with subsurface oceans. J Geophys Res, Planets 120(9):1528–1542. 10.1002/2015JE004821

Kang W, Bire S, Marshall J (2022a) The role of ocean circulation in driving hemispheric symmetry breaking of the ice shell of Enceladus. Earth Planet Sci Lett 599:117845

Kang W, Mittal T, Bire S, Campin J-M, Marshall J (2022b) How does salinity shape ocean circulation and ice geometry on Enceladus and other icy satellites? Sci Adv 8(29):eabm4665 PubMed PMC

Karato S-i (2010) Rheology of the Earth’s mantle: a historical review. Gondwana Res 18(1):17–45

Karato S, Spetzler HA (1990) Defect microdynamics in minerals and solid state mechanisms of seismic wave attenuation and velocity dispersion in the mantle. Rev Geophys 28(4):399–421. 10.1029/RG028i004p00399

Karato S-i, Wu P (1993) Rheology of the upper mantle: a synthesis. Science 260(5109):771–778 PubMed

Kervazo M, Tobie G, Choblet G, Dumoulin C, Běhounková M (2021) Solid tides in Io’s partially molten interior-contribution of bulk dissipation. Astron Astrophys 650:A72

Kervazo M, Tobie G, Choblet G, Dumoulin C, Běhounková M (2022) Inferring Io’s interior from tidal monitoring. Icarus 373:114737

Khan A, Liebske C, Rozel A, Rivoldini A, Nimmo F, Connolly J, Plesa A-C, Giardini D (2018) A geophysical perspective on the bulk composition of Mars. J Geophys Res, Planets 123(2):575–611

Khurana K, Kivelson M, Stevenson D, Schubert G, Russell C, Walker R, Polanskey C (1998) Induced magnetic fields as evidence for subsurface oceans in Europa and Callisto. Nature 395(6704):777–780 PubMed

Khurana KK, Jia X, Kivelson MG, Nimmo F, Schubert G, Russell CT (2011) Evidence of a global magma ocean in Io’s interior. Science 332(6034):1186–1189 PubMed

Kite ES, Rubin AM (2016) Sustained eruptions on Enceladus explained by turbulent dissipation in tiger stripes. Proc Natl Acad Sci 113(15):201520507. 10.1073/pnas.1520507113 PubMed PMC

Kivelson MG, Khurana KK, Russell CT, Volwerk M, Walker RJ, Zimmer C (2000) Galileo magnetometer measurements: a stronger case for a subsurface ocean at Europa. Science 289(5483):1340–1343. 10.1126/science.289.5483.1340 PubMed

Kivelson MG, Khurana KK, Volwerk M (2002) The permanent and inductive magnetic moments of Ganymede. Icarus 157(2):507–522. 10.1006/icar.2002.6834

Koh Z-W, Nimmo F, Lunine JI, Mazarico E, Dombard AJ (2022) Assessing the detectability of Europa’s seafloor topography from Europa Clipper’s gravity data. Planet Sci J 3(8):197. 10.3847/PSJ/ac82aa

Kohlstedt DL, Zimmerman ME (1996) Rheology of partially molten mantle rocks. Annu Rev Earth Planet Sci 24(1):41–62

Kuchta M, Tobie G, Miljković K, Běhounková M, Souček O, Choblet G, Čadek O (2015) Despinning and shape evolution of Saturn’s moon iapetus triggered by a giant impact. Icarus 252:454–465

Kvorka J, Čadek O (2022) A numerical model of convective heat transfer in Titan’s subsurface ocean. Icarus 376:114853

Lainey V, Arlot J-E, Karatekin Ö, Van Hoolst T (2009) Strong tidal dissipation in Io and Jupiter from astrometric observations. Nature 459:957–959 PubMed

Lainey V, Jacobson RA, Tajeddine R, Cooper NJ, Murray C, Robert V, Tobie G, Guillot T, Mathis S, Remus F, Desmars J, Arlot J-E, De Cuyper J-P, Dehant V, Pascu D, Thuillot W, Le Poncin-Lafitte C, Zahn J-P (2017) New constraints on Saturn’s interior from Cassini astrometric data. Icarus 281:286–296. 10.1016/j.icarus.2016.07.014

Lainey V, Casajus LG, Fuller J, Zannoni M, Tortora P, Cooper N, Murray C, Modenini D, Park RS, Robert V, et al. (2020) Resonance locking in giant planets indicated by the rapid orbital expansion of Titan. Nat Astron 4(11):1053–1058

Lainey V, Rambaux N, Tobie G, Cooper N, Zhang Q, Noyelles B, Baillié K (2024) A recently formed ocean inside Saturn’s moon Mimas. Nature 626(7998):280–282. 10.1038/s41586-023-06975-9 PubMed

Lamb H (1993) Hydrodynamics, 6th edn. Cambridge University Press

Le Gall A, Leyrat C, Janssen MA, Choblet G, Tobie G, Bourgeois O, Lucas A, Sotin C, Howett C, Kirk R, Lorenz RD, West RD, Stolzenbach A, Massé M, Hayes AH, Bonnefoy L, Veyssière G, Paganelli F (2017) Thermally anomalous features in the subsurface of Enceladus’s south polar terrain. Nat Astron 1:0063. 10.1038/s41550-017-0063

Lee U, Saio H (1997) Low-frequency nonradial oscillations in rotating stars. I. Angular dependence. Astrophys J 491(2):839–845. 10.1086/304980

Liao Y, Nimmo F, Neufeld JA (2020) Heat production and tidally driven fluid flow in the permeable core of Enceladus. J Geophys Res, Planets 125(9):e2019JE006209

Lindzen RS, Chapman S (1969) Atmospheric tides. Space Sci Rev 10(1):3–188. 10.1007/BF00171584

Lobo AH, Thompson AF, Vance SD, Tharimena S (2021) A pole-to-equator ocean overturning circulation on Enceladus. Nat Geosci 14(4):185–189

Longuet-Higgins MS (1966) Planetary waves on a hemisphere bounded by meridians of longitude. Philos Trans R Soc Lond Ser A 260(1111):317–350. 10.1098/rsta.1966.0053

Longuet-Higgins MS (1968) The eigenfunctions of Laplace’s tidal equations over a sphere. Philos Trans R Soc Lond Ser A 262(1132):511–607. 10.1098/rsta.1968.0003

Longuet-Higgins MS, Pond GS (1970) The free oscillations of fluid on a hemisphere bounded by meridians of longitude. Philos Trans R Soc Lond Ser A 266(1174):193–223. 10.1098/rsta.1970.0006

Lopes RM, Kamp LW, Smythe WD, Mouginis-Mark P, Kargel J, Radebaugh J, Turtle EP, Perry J, Williams DA, Carlson R, et al. (2004) Lava lakes on Io: observations of Io’s volcanic activity from Galileo nims during the 2001 fly-bys. Icarus 169:140–174

MacKenzie SM, Neveu M, Davila AF, Lunine JI, Craft KL, Cable ML, Phillips-Lander CM, Hofgartner JD, Eigenbrode JL, Waite JH, et al. (2021) The Enceladus orbilander mission concept: balancing return and resources in the search for life. Planet Sci J 2(2):77

Marusiak AG, Vance S, Panning MP, Běhounková M, Byrne PK, Choblet G, Daswani MM, Hughson K, Journaux B, Lobo AH, et al. (2021) Exploration of icy ocean worlds using geophysical approaches. Planet Sci J 2(4):150

Matsuyama I (2014) Tidal dissipation in the oceans of icy satellites. Icarus 242:11–18. 10.1016/j.icarus.2014.07.005

Matsuyama I, Beuthe M, Hay HCFC, Nimmo F, Kamata S (2018) Ocean tidal heating in icy satellites with solid shells. Icarus 312:208–230. 10.1016/j.icarus.2018.04.013

Mavko GM (1980) Velocity and attenuation in partially molten rocks. J Geophys Res, Solid Earth 85(B10):5173–5189

Mazarico E, Buccino D, Castillo-Rogez J, Dombard AJ, Genova A, Hussmann H, Kiefer WS, Lunine JI, McKinnon WB, Nimmo F, et al. (2023) The Europa Clipper gravity and radio science investigation. Space Sci Rev 219:30. 10.1007/s11214-023-00972-0

McCarthy C, Cooper RF (2016) Tidal dissipation in creeping ice and the thermal evolution of Europa. Earth Planet Sci Lett 443:185–194

McCarthy C, Takei Y, Hiraga T (2011) Experimental study of attenuation and dispersion over a broad frequency range: 2. The universal scaling of polycrystalline materials. J Geophys Res, Solid Earth 116(B9):B09207

McCarthy C, Nielson MA, Coonin AN, Minker JS, Domingos AA (2019) Acoustic and microstructural properties of partially molten samples in the ice–ammonia system. Geosciences 9(8):327

McKenzie D (1984) The generation and compaction of partially molten rock. J Petrol 25(3):713–765

Melwani Daswani M, Vance SD, Mayne MJ, Glein CR (2021) A metamorphic origin for Europa’s ocean. Geophys Res Lett 48(18):e2021GL094143 PubMed PMC

Meyer J, Wisdom J (2008) Tidal evolution of Mimas, Enceladus, and Dione. Icarus 193(1):213–223

Meyer J, Elkins-Tanton L, Wisdom J (2010) Coupled thermal-orbital evolution of the early Moon. Icarus 208(1):1–10. 10.1016/j.icarus.2010.01.029

Mitri G, Meriggiola R, Hayes A, Lefevre A, Tobie G, Genova A, Lunine JI, Zebker H (2014) Shape, topography, gravity anomalies and tidal deformation of Titan. Icarus 236:169–177

Miyazaki Y, Stevenson DJ (2022) A subsurface magma ocean on Io: exploring the steady state of partially molten planetary bodies. Planet Sci J 3(11):256. 10.3847/PSJ/ac9cd1

Monteux J, Tobie G, Choblet G, Le Feuvre M (2014) Can large icy moons accrete undifferentiated? Icarus 237:377–387

Monteux J, Qaddah B, Andrault D (2023) Conditions for segregation of a crystal-rich layer within a convective magma ocean. J Geophys Res, Planets 128(5):e2023JE007805. 10.1029/2023JE007805

Moore WB (2001) The thermal state of Io. Icarus 154(2):548–550

Moore WB, Hussmann H (2009) Thermal evolution of Europa’s silicate interior. In: Pappalardo RT, McKinnon WB, Khurana K (eds) Europa. University of Arizona Press, Tuscon, pp 369–380. 10.2307/j.ctt1xp3wdw.21

Morabito L, Synnott S, Kupferman P, Collins SA (1979) Discovery of currently active extraterrestrial volcanism. Science 204(4396):972–972 PubMed

Motoyama M, Tsunakawa H, Takahashi F (2020) Tidal resonance of eigenmode oscillation in the early Earth’s ocean and its acceleration effect on the Moon’s orbital evolution. Icarus 335:113382. 10.1016/j.icarus.2019.07.016

Mura A, Adriani A, Tosi F, Lopes R, Sindoni G, Filacchione G, Williams D, Davies A, Plainaki C, Bolton S, et al. (2020) Infrared observations of Io from Juno. Icarus 341:113607

Murase T, McBirney AR (1973) Properties of some common igneous rocks and their melts at high temperatures. Geol Soc Am Bull 84(11):3563–3592

Musseau Y, Tobie G, Dumoulin C, Gillmann C, Revol A, Bolmont E (2024) The viscosity of Venus’ mantle inferred from its rotational state. Icarus 422:116245

Nakajima M, Ingersoll AP (2016) Controlled boiling on Enceladus. 1. Model of the vapor-driven jets. Icarus 272:309–318. 10.1016/j.icarus.2016.02.027

Nakajima A, Ida S, Kimura J, Brasser R (2019) Orbital evolution of Saturn’s mid-sized moons and the tidal heating of Enceladus. Icarus 317:570–582

Neveu M, Rhoden AR (2019) Evolution of Saturn’s mid-sized moons. Nat Astron 3:543–552. 10.1038/s41550-019-0726-y PubMed PMC

Nimmo F (2004) What is the young’s modulus of ice? Europa’s icy shell meeting. Europa’s Icy Shell, LPI Contribution 1195, 7005

Nimmo F (2023) Strong tidal dissipation at Uranus? Planet Sci J 4(12):241

Nimmo F, Brown ME (2023) The internal structure of Eris inferred from its spin and orbit evolution. Sci Adv 9(46):eadi9201 PubMed PMC

Nimmo F, Spencer JR, Pappalardo RT, Mullen ME (2007) Shear heating as the origin of the plumes and heat flux on Enceladus. Nature 447(7142):289–291. 10.1038/nature05783 PubMed

Nimmo F, Porco C, Mitchell C (2014) Tidally modulated eruptions on Enceladus: Cassini ISS observations and models. Astron J 148(3):46. 10.1088/0004-6256/148/3/46

Nimmo F, Neveu M, Howett C (2023) Origin and evolution of Enceladus’s tidal dissipation. Space Sci Rev 219:57. 10.1007/s11214-023-01007-4 PubMed PMC

Noyelles B, Baillie K, Charnoz S, Lainey V, Tobie G (2019) Formation of the Cassini division - II. Possible histories of Mimas and Enceladus. Mon Not R Astron Soc 486(2):2947–2963

Ogawa M (2014) A positive feedback between magmatism and mantle upwelling in terrestrial planets: implications for the Moon. J Geophys Res, Planets 119(11):2317–2330

Ogawa M, Yanagisawa T (2011) Numerical models of Martian mantle evolution induced by magmatism and solid-state convection beneath stagnant lithosphere. J Geophys Res, Planets 116(E8):E08008

Ogilvie GI (2009) Tidal dissipation in rotating fluid bodies: a simplified model. Mon Not R Astron Soc 396(2):794–806. 10.1111/j.1365-2966.2009.14814.x

Ogilvie GI (2013) Tides in rotating barotropic fluid bodies: the contribution of inertial waves and the role of internal structure. Mon Not R Astron Soc 429(1):613–632. 10.1093/mnras/sts362

Ogilvie GI (2014) Tidal dissipation in stars and giant planets. Annu Rev Astron Astrophys 52:171–210. 10.1146/annurev-astro-081913-035941

Ogilvie GI, Lin DNC (2004) Tidal dissipation in rotating giant planets. Astrophys J 610(1):477–509. 10.1086/421454

Ojakangas GW, Stevenson DJ (1989) Thermal state of an ice shell on Europa. Icarus 81(2):220–241. 10.1016/0019-1035(89)90052-3

Park R, Mastrodemos N, Jacobson R, Berne A, Vaughan A, Hemingway D, Leonard E, Castillo-Rogez J, Cockell C, Keane J, et al. (2024) The global shape, gravity field, and libration of Enceladus. J Geophys Res, Planets 129(1):e2023JE008054

Pauer M, Musiol S, Breuer D (2010) Gravity signals on Europa from silicate shell density variations. J Geophys Res 115:E12005. 10.1029/2010JE003595

Peale SJ, Cassen P, Reynolds RT (1979) Melting of Io by tidal dissipation. Science 203(4383):892–894 PubMed

Peale S, Cassen P, Reynolds R (1980) Tidal dissipation, orbital evolution, and the nature of Saturn’s inner satellites. Icarus 43(1):65–72

Peltier WR (1974) The impulse response of a Maxwell Earth. Rev Geophys Space Phys 12:649–669. 10.1029/RG012i004p00649

Pleiner Sládková K, Souček O, Běhounková M (2021) Enceladus’ tiger stripes as frictional faults: effect on stress and heat production. Geophys Res Lett 48(19):e2021GL094849. 10.1029/2021GL094849

Poirier JP, Boloh L, Chambon P (1983) Tidal dissipation in small viscoelastic ice moons: the case of Enceladus. Icarus 55(2):218–230

Porco CC, Helfenstein P, Thomas P, Ingersoll A, Wisdom J, West R, Neukum G, Denk T, Wagner R, Roatsch T, et al. (2006) Cassini observes the active south pole of Enceladus. Science 311(5766):1393–1401 PubMed

Porco C, DiNino D, Nimmo F (2014) How the geysers, tidal stresses, and thermal emission across the south polar terrain of Enceladus are related. Astron J 148(3):45. 10.1088/0004-6256/148/3/45

Postberg F, Clark RN, Hansen CJ, Coates AJ, Ore CMD, Scipioni F, Hedman MM, Waite JH, Dotson R (2018) Plume and surface composition of Enceladus University of Arizona Press, Tucson, pp 129–162. http://www.jstor.org/stable/j.ctv65sw2b.16

Pou L, Nimmo F, Rivoldini A, Khan A, Bagheri A, Gray T, Samuel H, Lognonné P, Plesa A-C, Gudkova T, et al. (2022) Tidal constraints on the Martian interior. J Geophys Res, Planets 127(11):e2022JE007291

Preblich B, Greenberg R, Riley J, O’Brien D (2007) Tidally driven strike–slip displacement on Europa: viscoelastic modeling. Planet Space Sci 55(10):1225–1245. 10.1016/j.pss.2007.01.018

Ray RD, Eanes RJ, Lemoine FG (2001) Constraints on energy dissipation in the Earth’s body tide from satellite tracking and altimetry. Geophys J Int 144(2):471–480

Reeh N, Christensen EL, Mayer C, Olesen OB (2003) Tidal bending of glaciers: a linear viscoelastic approach. Ann Glaciol 37:83–89

Rekier J, Trinh A, Triana SA, Dehant V (2019) Internal energy dissipation in Enceladus’s subsurface ocean from tides and libration and the role of inertial waves. J Geophys Res, Planets 124(8):2198–2212. 10.1029/2019JE005988

Renaud JP, Henning WG (2018) Increased tidal dissipation using advanced rheological models: implications for Io and tidally active exoplanets. Astrophys J 857(2):98

Renner J, Evans B, Hirth G (2000) On the rheologically critical melt fraction. Earth Planet Sci Lett 181(4):585–594

Reynard B, Sotin C (2023) Carbon-rich icy moons and dwarf planets. Earth Planet Sci Lett 612:118172

Rhoden AR (2023) Mimas: frozen fragment, ring relic, or emerging ocean world? Annu Rev Earth Planet Sci 51:367–387

Rhoden AR, Walker ME, Rudolph ML, Bland MT, Manga M (2024) The evolution of a young ocean within Mimas. Earth Planet Sci Lett 635:118689

Ricard Y, Bercovici D, Schubert G (2001) A two-phase model for compaction and damage: 2. Applications to compaction, deformation, and the role of interfacial surface tension. J Geophys Res, Solid Earth 106(B5):8907–8924

Rieutord M, Valdettaro L (2010) Viscous dissipation by tidally forced inertial modes in a rotating spherical shell. J Fluid Mech 643:363–394. 10.1017/S002211200999214X

Roberts JH (2015) The fluffy core of Enceladus. Icarus 258:54–66

Roberts JH, Nimmo F (2008) Tidal heating and the long-term stability of a subsurface ocean on Enceladus. Icarus 194(2):675–689

Roberts JH, McKinnon WB, Elder CM, Tobie G, Biersteker JB, Young D, Park RS, Steinbrügge G, Nimmo F, Howell SM, et al. (2023) Exploring the interior of Europa with the Europa Clipper. Space Sci Rev 219:46. 10.1007/s11214-023-00990-y PubMed PMC

Rollins KM, Evans MD, Diehl NB, Daily III WD (1998) Shear modulus and damping relationships for gravels. J Geotech Geoenviron Eng 124(5):396–405

Romanowicz B, Mitchell B (2015) 1.25—deep Earth structure: Q of the Earth from crust to core. Treatise Geophys 1:789–827

Rosier SH, Marsh OJ, Rack W, Gudmundsson GH, Wild CT, Ryan M (2017) On the interpretation of ice-shelf flexure measurements. J Glaciol 63(241):783–791

Rovira-Navarro M, Rieutord M, Gerkema T, Maas LRM, van der Wal W, Vermeersen B (2019) Do tidally-generated inertial waves heat the subsurface oceans of Europa and Enceladus? Icarus 321:126–140. 10.1016/j.icarus.2018.11.010

Rovira-Navarro M, Gerkema T, Maas LRM, van der Wal W, van Ostayen R, Vermeersen B (2020) Tides in subsurface oceans with meridional varying thickness. Icarus 343:113711. 10.1016/j.icarus.2020.113711

Rovira-Navarro M, Katz RF, Liao Y, van der Wal W, Nimmo F (2022) The tides of Enceladus’ porous core. J Geophys Res, Planets 127(5):e2021JE007117 PubMed PMC

Rovira-Navarro M, Matsuyama I, Hay HCFC (2023) Thin-shell tidal dynamics of ocean worlds. Planet Sci J 4(2):23. 10.3847/PSJ/acae9a

Sabadini R, Yuen DA, Boschi E (1982) Polar wandering and the forced responses of a rotating, multilayered, viscoelastic planet. J Geophys Res, Solid Earth 87(B4):2885–2903

Sabadini R, Vermeersen B, Cambiotti G (2016) Global Dynamics of the Earth: Applications of Viscoelastic Relaxation Theory to Solid-Earth and Planetary Geophysics. 10.1007/978-94-017-7552-6

Sagan C, Dermott SF (1982) The tide in the seas of Titan. Nature 300(5894):731–733. 10.1038/300731a0

Saito M (1974) Some problems of static deformation of the Earth. J Phys Earth 22(1):123–140

Saur J, Duling S, Roth L, Jia X, Strobel DF, Feldman PD, Christensen UR, Retherford KD, McGrath MA, Musacchio F, Wennmacher A, Neubauer FM, Simon S, Hartkorn O (2015) The search for a subsurface ocean in Ganymede with Hubble Space Telescope observations of its auroral ovals. J Geophys Res Space Phys 120(3):1715–1737. 10.1002/2014JA020778

Schenk PM, McKinnon WB (1989) Fault offsets and lateral crustal movement on Europa: evidence for a mobile ice shell. Icarus 79(1):75–100. 10.1016/0019-1035(89)90109-7

Schindelegger M, Green JAM, Wilmes SB, Haigh ID (2018) Can we model the effect of observed sea level rise on tides? J Geophys Res, Oceans 123(7):4593–4609. 10.1029/2018JC013959

Schmeling H (1985) Numerical models on the influence of partial melt on elastic, anelastic and electric properties of rocks. Part I: elasticity and anelasticity. Phys Earth Planet Inter 41(1):34–57

Schmeling H, Kruse JP, Richard G (2012) Effective shear and bulk viscosity of partially molten rock based on elastic moduli theory of a fluid filled poroelastic medium. Geophys J Int 190(3):1571–1578

Schwarz J, Weeks W (1977) Engineering properties of sea ice. J Glaciol 19(81):499–531

Scott T, Kohlstedt D (2006) The effect of large melt fraction on the deformation behavior of peridotite. Earth Planet Sci Lett 246(3–4):177–187

Scott DR, Stevenson DJ (1986) Magma ascent by porous flow. J Geophys Res, Solid Earth 91(B9):9283–9296

Sears WD (1995) Tidal dissipation in oceans on Titan. Icarus 113(1):39–56

Seed HB, Wong RT, Idriss I, Tokimatsu K (1986) Moduli and damping factors for dynamic analyses of cohesionless soils. J Geotech Eng 112(11):1016–1032

Segatz M, Spohn T, Ross M, Schubert G (1988) Tidal dissipation, surface heat flow, and figure of viscoelastic models of Io. Icarus 75(2):187–206

Shaw HR (1969) Rheology of basalt in the melting range. J Petrol 10(3):510–535

Shibuya S, Mitachi T, Fukuda F, Degoshi T (1995) Strain rate effects on shear modulus and damping of normally consolidated clay. Geotech Test J 18(3):365–375

Sinha NK (1989) Elasticity of natural types of polycrystalline ice. Cold Reg Sci Technol 17(2):127–135. 10.1016/S0165-232X(89)80003-5

Sládková K, Souček O, Kalousová K, Bĕhounková M (2020) Tidal walking on Europa’s strike slip faults - insight from numerical modeling. J Geophys Res, Planets. 10.1029/2019je006327

Smith BA, Soderblom LA, Beebe R, Boyce J, Briggs G, Carr M, Collins SA, Cook AF, Danielson GE, Davies ME, et al. (1979) The Galilean satellites and Jupiter: Voyager 2 imaging science results. Science 206(4421):927–950 PubMed

Soderlund KM (2019) Ocean dynamics of outer Solar System satellites. Geophys Res Lett 46(15):8700–8710

Soderlund KM, Rovira-Navarro M, Le Bars M, Schmidt BE, Gerkema T (2024) The physical oceanography of ice-covered moons. Annu Rev Mar Sci 16(1):25–53 PubMed

Sohl F, Sears WD, Lorenz RD (1995) Tidal dissipation on Titan. Icarus 115(2):278–294

Sotin C, Head JW, Tobie G (2002) Europa: tidal heating of upwelling thermal plumes and the origin of lenticulae and chaos melting. Geophys Res Lett 29(8):1233. 10.1029/2001GL013844

Sotin C, Tobie G, Wahr J, McKinnon WB, McKinnon W, Khurana K (2009) Tides and tidal heating on Europa. In: Pappalardo RT, McKinnon WB, Khurana KK (eds) Europa. University of Arizona Press, Tucson, pp 85–118. 10.2307/j.ctt1xp3wdw.10

Sotin C, Kalousová K, Tobie G (2021) Titan’s interior structure and dynamics after the Cassini-Huygens mission. Annu Rev Earth Planet Sci 49:579–607

Souček O, Hron J, Běhounková M, Čadek O (2016) Effect of the tiger stripes on the deformation of Saturn’s moon Enceladus. Geophys Res Lett 43(14):7417–7423. 10.1002/2016GL069415

Souček O, Běhounková M, Čadek O, Hron J, Tobie G, Choblet G (2019) Tidal dissipation in Enceladus’ uneven, fractured ice shell. Icarus 328:218–231. 10.1016/j.icarus.2019.02.012. ISSN 0019-1035

Souček O, Běhounková M, Lanzendörfer M, Tobie G, Choblet G (2024) Variations in plume activity reveal the dynamics of water-filled faults on Enceladus. Nat Commun 15(1):7405 PubMed PMC

Spada G (2008) Alma, a fortran program for computing the viscoelastic Love numbers of a spherically symmetric planet. Comput Geosci 34(6):667–687

Sparks R, Annen C, Blundy J, Cashman K, Rust A, Jackson M (2019) Formation and dynamics of magma reservoirs. Philos Trans R Soc A 377(2139):20180019 PubMed

Spencer JR, Rathbun JA, Travis LD, Tamppari LK, Barnard L, Martin TZ, McEwen AS (2000) Io’s thermal emission from the Galileo photopolarimeter-radiometer. Science 288(5469):1198–1201 PubMed

Spencer JR, Pearl JC, Segura M, Flasar FM, Mamoutkine A, Romani P, Buratti BJ, Hendrix AR, Spilker LJ, Lopes RMC (2006) Cassini encounters Enceladus: background and the discovery of a south polar hot spot. Science 311(5766):1401–1405. 10.1126/science.1121661 PubMed

Spencer JR, Howett CJA, Verbiscer A, Hurford TA, Segura M, Spencer DC (2013) Enceladus heat flow from high spatial resolution thermal emission observations. Eur Planet Sci Congr 8:1–2

Spencer J, Nimmo F, Ingersoll AP, Hurford T, Kite E, Rhoden A, Schmidt J, Howett C (2018) Plume origins and plumbing: from ocean to surface. In: Schenk PM et al. (eds) Enceladus and the icy moons of Saturn. University of Arizona Press, Tucson, pp 163–174

Spencer DC, Katz RF, Hewitt IJ (2020) Magmatic intrusions control Io’s crustal thickness. J Geophys Res, Planets 125(6):e2020JE006443

Spencer DC, Katz RF, Hewitt IJ (2021) Tidal controls on the lithospheric thickness and topography of Io from magmatic segregation and volcanism modelling. Icarus 359:114352

Spetzler H, Anderson DL (1968) The effect of temperature and partial melting on velocity and attenuation in a simple binary system. J Geophys Res 73(18):6051–6060

Squyres SW, Reynolds RT, Cassen PM, Peale SJ (1983) The evolution of Enceladus. Icarus 53(2):319–331

Steinbrügge G, Stark A, Hussmann H, Sohl F, Oberst J (2015) Measuring tidal deformations by laser altimetry. A performance model for the Ganymede laser altimeter. Planet Space Sci 117:184–191

Steinbrügge G, Schroeder DM, Haynes MS, Hussmann H, Grima C, Blankenship DD (2018) Assessing the potential for measuring Europa’s tidal Love number h2 using radar sounder and topographic imager data. Earth Planet Sci Lett 482:334–341

Steinbrügge G, Hussmann H, Tobie G, Castillo-Rogez J, Lainey V, Stark A (2023) The tidal deformation and tidal evolution of Ganymede. Planet Space Sci 28:46

Steinke T, van Sliedregt D, Vilella K, Van der Wal W, Vermeersen B (2020) Can a combination of convective and magmatic heat transport in the mantle explain Io’s volcanic pattern? J Geophys Res, Planets 125(12):e2020JE006521

Sundberg M, Cooper RF (2010) A composite viscoelastic model for incorporating grain boundary sliding and transient diffusion creep; correlating creep and attenuation responses for materials with a fine grain size. Philos Mag 90(20):2817–2840

Tackley PJ, Schubert G, Glatzmaier GA, Schenk P, Ratcliff JT, Matas J-P (2001) Three-dimensional simulations of mantle convection in Io. Icarus 149(1):79–93

Tajeddine R, Rambaux N, Lainey V, Charnoz S, Richard A, Rivoldini A, Noyelles B (2014) Constraints on Mimas’ interior from Cassini ISS libration measurements. Science 346(6207):322–324 PubMed

Tajeddine R, Soderlund KM, Thomas PC, Helfenstein P, Hedman MM, Burns JA, Schenk PM (2017) True polar wander of Enceladus from topographic data. Icarus 295:46–60

Takei Y (1998) Constitutive mechanical relations of solid-liquid composites in terms of grain-boundary contiguity. J Geophys Res, Solid Earth 103(B8):18183–18203

Takei Y, Holtzman BK (2009) Viscous constitutive relations of solid-liquid composites in terms of grain boundary contiguity: 1. Grain boundary diffusion control model. J Geophys Res, Solid Earth 114(B6):B06205

Takei Y, Karasawa F, Yamauchi H (2014) Temperature, grain size, and chemical controls on polycrystal anelasticity over a broad frequency range extending into the seismic range. J Geophys Res, Solid Earth 119(7):5414–5443

Takeuchi H, Saito M (1972) Seismic surface waves. Methods Comput Phys, Adv Res Appl 11:217–295. 10.1016/B978-0-12-460811-5.50010-6

Tatibouet J, Perez J, Vassoille R (1986) High-temperature internal friction and dislocations in ice ih. J Phys 47(1):51–60

Tatibouet J, Perez J, Vassoille R (1987) Study of grain boundaries in ice by internal friction measurement. J Phys, Colloq 48(C1):C1-197

Taylor GI (1936) The oscillations of the atmosphere. Proc R Soc Lond Ser A 156(888):318–326. 10.1098/rspa.1936.0150

Thomas PC, Tajeddine R, Tiscareno MS, Burns JA, Joseph J, Loredo TJ, Helfenstein P, Porco C (2016) Enceladus’s measured physical libration requires a global subsurface ocean. Icarus 264:37–47. 10.1016/j.icarus.2015.08.037

Tobie G, Choblet G, Sotin C (2003) Tidally heated convection: constraints on Europa’s ice shell thickness. J Geophys Res, Planets 108(E11):5124. 10.1029/2003JE002099

Tobie G, Grasset O, Lunine JI, Mocquet A, Sotin C (2005a) Titan’s internal structure inferred from a coupled thermal-orbital model. Icarus 175(2):496–502

Tobie G, Mocquet A, Sotin C (2005b) Tidal dissipation within large icy satellites: applications to Europa and Titan. Icarus 177(2):534–549. 10.1016/j.icarus.2005.04.006

Tobie G, Lunine JI, Sotin C (2006) Episodic outgassing as the origin of atmospheric methane on Titan. Nature 440(7080):61–64 PubMed

Tobie G, Čadek O, Sotin C (2008) Solid tidal friction above a liquid water reservoir as the origin of the south pole hotspot on Enceladus. Icarus 196(2):642–652

Tobie G, Lunine J, Monteux J, Mousis O, Nimmo F (2014) The origin and evolution of Titan. In: Müller-Wodarg I et al. (eds) Titan: interior, surface, atmosphere, and space environment. Cambridge University Press, pp 29–62

Tobie G, Grasset O, Dumoulin C, Mocquet A (2019) Tidal response of rocky and ice-rich exoplanets. Astron Astrophys 630:A70. 10.1051/0004-6361/201935297

Tort M, Dubos T, Bouchut F, Zeitlin V (2014) Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography. J Fluid Mech 748:789–821. 10.1017/jfm.2014.172

Tyler RH (2008) Strong ocean tidal flow and heating on moons of the outer planets. Nature 456(7223):770–772. 10.1038/nature07571 PubMed

Tyler RH (2009) Ocean tides heat Enceladus. Geophys Res Lett 36(15):L15205. 10.1029/2009GL038300

Tyler R (2011) Tidal dynamical considerations constrain the state of an ocean on Enceladus. Icarus 211(1):770–779. 10.1016/j.icarus.2010.10.007

Tyler R (2014) Comparative estimates of the heat generated by ocean tides on icy satellites in the outer Solar System. Icarus 243:358–385. 10.1016/j.icarus.2014.08.037

Tyler R (2019) TROPF (v1.0). Zenodo. 10.5281/zenodo.3523274

Tyler RH (2021) On the tidal history and future of the Earth-Moon orbital system. Planet Sci J 2(2):70. 10.3847/PSJ/abe53f

Tyler RH, Henning WG, Hamilton CW (2015) Tidal heating in a magma ocean within Jupiter’s moon Io. Astrophys J Suppl Ser 218(2):22. 10.1088/0067-0049/218/2/22

Vallis GK (2017) Atmospheric and Oceanic fluid dynamics: fundamentals and large-scale circulation. 10.1017/9781107588417

Van Hoolst T, Baland R-M, Trinh A (2016) The diurnal libration and interior structure of Enceladus. Icarus 277:311–318

Van Hoolst T, Tobie G, Vallat C, et al. (2024) Geophysical characterization of the interiors of Ganymede, Callisto and Europa by ESA’s JUpiter ICy moons Explorer. Space Sci Rev 220:54. 10.1007/s11214-024-01085-y

Vance S, Harnmeijer J, Kimura J, Hussmann H, DeMartin B, Brown JM (2007) Hydrothermal systems in small ocean planets. Astrobiology 7(6):987–1005 PubMed

Vance SD, Hand KP, Pappalardo RT (2016) Geophysical controls of chemical disequilibria in Europa. Geophys Res Lett 43(10):4871–4879. 10.1002/2016GL068547

Varshalovich DA, Moskalev AN, Khersonskii VK (1988) Quantum theory of angular momentum. 10.1142/0270

Vaughan DG (1995) Tidal flexure at ice shelf margins. J Geophys Res, Solid Earth 100(B4):6213–6224

Vigneresse JL, Barbey P, Cuney M (1996) Rheological transitions during partial melting and crystallization with application to felsic magma segregation and transfer. J Petrol 37(6):1579–1600

Wahr JM, Zuber MT, Smith DE, Lunine JI (2006) Tides on Europa, and the thickness of Europa’s icy shell. J Geophys Res, Planets 111(E12):E12005. 10.1029/2006JE002729

Wahr J, Selvans ZA, Mullen ME, Barr AC, Collins GC, Selvans MM, Pappalardo RT (2009) Modeling stresses on satellites due to nonsynchronous rotation and orbital eccentricity using gravitational potential theory. Icarus 200(1):188–206

Waite JH Jr, Combi MR, Ip W-H, Cravens TE, McNutt RL Jr, Kasprzak W, Yelle R, Luhmann J, Niemann H, Gell D, et al. (2006) Cassini ion and Neutral Mass Spectrometer: Enceladus plume composition and structure. Science 311(5766):1419–1422 PubMed

Walterová M, Běhounková M, Efroimsky M (2023) Is there a semi-Molten layer at the base of the lunar mantle? J Geophys Res, Planets 128(7):e2022JE007652

Wang H, Boyd JP, Akmaev RA (2016) On computation of Hough functions. Geosci Model Dev 9(4):1477–1488. 10.5194/gmd-9-1477-2016

Webb DJ (1980) Tides and tidal friction in a hemispherical ocean centred at the equator. Geophys J 61(3):573–600. 10.1111/j.1365-246X.1980.tb04833.x

Webb DJ (1982) Tides and the evolution of the Earth-Moon system. Geophys J 70:261–271. 10.1111/j.1365-246X.1982.tb06404.x

White OL, Schenk PM, Bellagamba AW, Grimm AM, Dombard AJ, Bray VJ (2017) Impact crater relaxation on Dione and Tethys and relation to past heat flow. Icarus 288:37–52

Widmer R, Masters G, Gilbert F (1991) Spherically symmetric attenuation within the Earth from normal mode data. Geophys J Int 104(3):541–553

Wild CT, Marsh OJ, Rack W (2017) Viscosity and elasticity: a model intercomparison of ice-shelf bending in an Antarctic grounding zone. J Glaciol 63(240):573–580

Williams JG, Boggs DH (2015) Tides on the Moon: theory and determination of dissipation. J Geophys Res, Planets 120(4):689–724

Wisdom J (2008) Tidal dissipation at arbitrary eccentricity and obliquity. Icarus 193(2):637–640

Wisdom J, Dbouk R, Militzer B, Hubbard WB, Nimmo F, Downey BG, French RG (2022) Loss of a satellite could explain Saturn’s obliquity and young rings. Science 377(6612):1285–1289 PubMed

Wulff A-M, Hashida T, Watanabe K, Takahashi H (1999) Attenuation behaviour of tuffaceous sandstone and granite during microfracturing. Geophys J Int 139(2):395–409

Zahn JP (1975) The dynamical tide in close binaries. Astron Astrophys 41:329–344

Zahnle K, Walker JCG (1987) A constant daylength during the Precambrian era? Precambrian Res 37(2):95–105. 10.1016/0301-9268(87)90073-8 PubMed

Zandanel A, Truche L, Hellmann R, Myagkiy A, Choblet G, Tobie G (2021) Short lifespans of serpentinization in the rocky core of Enceladus: implications for hydrogen production. Icarus 364:114461

Zhong S, Qin C, Wahr J (2012) Can tidal tomography be used to unravel the long-wavelength structure of the lunar interior? Geophys Res Lett 39(15)

Zhou W, Chen Y, Ma G, Yang L, Chang X (2017) A modified dynamic shear modulus model for rockfill materials under a wide range of shear strain amplitudes. Soil Dyn Earthq Eng 92:229–238

Zimmer C, Khurana KK, Kivelson MG (2000) Subsurface oceans on Europa and Callisto: constraints from Galileo magnetometer observations. Icarus 147(2):329–347. 10.1006/icar.2000.6456

Zou C, Ding H, Luan W (2024) Anelasticity of the lower mantle inferred from the pole and lunar monthly tides using global doris coordinate time series. Glob Planet Change 236:104415