AC12
Ocean worlds from material memory to tidal heat
A changing tide can warm an icy world when deformation lags behind the forcing. Follow a material law into a finite-shell solution, compare very different interiors with the same response, and ask which measurements could distinguish them.
Start
An icy world can warm as it bends
A changing tide can turn orbital work into heat. Follow the lag, then see why a strong gravity response alone cannot identify an underground ocean.
L1
Material memory turns a tidal cycle into heat
Derive Maxwell’s complex shear modulus, the sign of passive loss and the work per cycle, with a saved dimensional example.
L2
A finite ice shell, from boundary tractions to Love numbers
Solve an incompressible shell over equal-density liquid, add self-gravity and derive the eccentricity-tide power from the same displacement.
L3
What a complex Love number can identify
Derive a shell–solid degeneracy, build a weighted fit from frozen responses and separate tidal power from a conditional thermal budget.
Tutorial
Fit the response, then expose the shell–solid family
Download a complete Python experiment with synthetic complex responses, weighted residuals, a conditional thickness profile and an incompatible record.
L4
Read Titan’s response, interior inference and a proposed next measurement
Read fixed Beuthe, Cassini–Titan and Dragonfly papers by equation and figure, separating measurements, conditional interiors and forecast precision.
Three textbook chapters develop the material law, finite-thickness shell and conditional inference. The public Python experiment preserves synthetic inputs and incompatible records. The final guide separates measured Titan gravity response from model-dependent interiors and a proposed Dragonfly experiment.
Sources and calculation scope
- Beuthe, Tides on Europa: the membrane paradigm, arXiv:1410.4735v1
Fixed 17 October 2014 preprint; Icarus publication DOI 10.1016/j.icarus.2014.10.027. Appendix A equations 105–116 give the finite-thickness homogeneous incompressible shell; Appendix C gives Maxwell correspondence; Appendix E derives work and power; Appendix G fixes the eccentricity/libration tide. Figures 8–9 and 12–13 expose approximation and thermal-model dependence. Full relevant sections and appendices were read. Our uniform-density Maxwell example is much simpler than a real moon.
- Petricca et al., Titan’s strong tidal dissipation precludes a subsurface ocean, Nature 648 (2025)
Version of record, 17 December 2025, DOI 10.1038/s41586-025-09818-x, CC BY 4.0; full publisher-deposited text and Methods, PMC12711566, read. Figure 1 reports gravity-response estimates, Figure 2 compares conditional interior posteriors and Figure 3 is an inferred-interior schematic. Figure 1 error bars are three-sigma; Figure 2 ellipses are one-sigma. Positive reported imaginary response denotes their loss convention; our positive-time-exponential convention uses negative imaginary Love numbers for passive loss.
- Petricca & Buccino, A Radio Science Experiment for Dragonfly to Investigate Titan’s Rotation and Interior Structure, GRL 53 (2026)
Published 20 October 2026, DOI 10.1029/2026GL125753. Full sections 1–5 and Figures 1–3 read. This is a proposed experiment and simulated formal-covariance forecast, not a measured displacement Love number. Figure 2 compares Doppler/range and mission assumptions; Figure 3 contains model-dependent structural sensitivities. Ephemeris and calibration systematics are not all captured by formal uncertainties.