M080 + AC12 · Deformation, lag and heat

Tidal Heating and Spin–Orbit Locking

How can repeated deformation warm an icy world? An eccentric orbit changes the tide throughout each revolution. If the material deforms with a delay, part of the mechanical work becomes heat. A liquid layer can change that response, while sufficiently soft solids can produce similar external signals. Sustaining an ocean also requires the heat supply to balance cooling.

Drag an ice-shell boundary and compare deformation, phase lag and tidal power. Explore the original rigid spin and capture model in its own mode.

Peak radial deformation $u_{r,\max}=3.08\,\mathrm{m}$

Gravity Love amplitude $|k_2|=0.0522$

Linear response lag $\delta=16.9^{\circ}$

Tidal heating power $\mathcal P=100\,\mathrm{GW}$

Change the shell, then follow its lag

Ice-shell thickness

50.0\,\mathrm{km}

d\;[\mathrm{km}]

Orbital eccentricity

0.00500

e

Drag the liquid boundary to change shell thickness. In Orbit view, drag periapsis to change eccentricity; arrow keys also work.

Material response and ideal elastic limit

Viscosity

\eta=3.16\times10^{14}\,\mathrm{Pa\,s}

Elastic shear modulus

\mu_E=3.50\,\mathrm{GPa}

Ideal elastic response without dissipation

The solid preset rescales stiffness and viscosity to reproduce the shell’s external response within this equal-density model. Actual ocean inference needs additional structural or material information.

Follow the derivation and rerun the conditional inference

Model assumptions

The default is a synthetic uniform-density, incompressible water world with an inviscid liquid interior and a homogeneous Maxwell shell of the same density. Finite-thickness degree-two elasticity includes self-gravity; the full-radius shell is its oceanless homogeneous-solid limit. A prescribed synchronous orbit drives leading-order eccentricity tides, including radial and optical-libration terms. All power is shear dissipation in the shell; no ocean dynamics, rocky core, depth-varying rheology, convection or coupled orbital/thermal evolution is integrated. Displayed deformations are linearly magnified by the stated factor, and body sizes and orbital distances are schematic. The conditional solid/shell preset demonstrates non-identifiability of this model, not detection of a real ocean. The original rigid spin mode retains a circular orbit, phenomenological damping and normalized spin-energy units; its accumulated dissipation uses those normalized energy units.

Phase within one physical orbit

t/P=0.175

Forcing and displacement, each divided by its own peak

Lag in one surface component

Complex gravity response to harmonic probes

Power and a conductive reference across shell thickness

Units, complex Love numbers and approximation boundaries

$h_2=0.0832-0.0253i,\qquad k_2=0.0499-0.0152i$

\mathcal P=-\frac{21}{2}\frac{GM_p^2R^5}{a^6}n e^2\operatorname{Im}k_2

Time dependence uses a positive imaginary exponential, so a passive response has negative imaginary Love number. The Love phase characterizes the linear harmonic response even when the actual eccentricity is zero. The loop shows lag of one radial surface component; total power integrates every degree-two component and is not a local temperature map.

Radius is 500 km and both layers have density 1000 kilograms per cubic metre. The orbit and eccentricity are prescribed; orbital damping and thermal feedback are not evolved. The thickness sweep covers liquid-bearing cases from 10 to 450 km, at the current fixed rheology. Its conductive reference assumes constant conductivity 2.5 watts per metre per kelvin and boundary temperatures 100 and 273 kelvin, with no convection. The harmonic probe range ends at twice the orbital frequency; internal inertia is omitted.

Start with the essentials

Focus question: How do shell thickness, material memory and eccentricity connect deformation to power?
One-sentence intuition: The same complex Love response sets the deformation, its lag and the total tidal work. Under the declared positive-time convention, passive Love loss has negative imaginary part.

Core mathematical model

A material with a relaxation time

\mu^*=\mu_E\frac{i\omega\tau_M}{1+i\omega\tau_M},\qquad\tau_M=\eta/\mu_E

A homogeneous Maxwell shell has one relaxation time. The synthetic world supplies explicit material properties and forcing frequency.

One finite-shell response

h_2=\left[\frac25+z_h\frac{\mu^*}{\rho gR}\right]^{-1},\qquad k_2=\frac35h_2

The geometric factor comes from a traction boundary solve. Equal density in shell and liquid gives the fixed Love-number ratio.

Positive power from a passive lag

\mathcal P=-\frac{21}{2}\frac{GM_p^2R^5}{a^6}n e^2\operatorname{Im}k_2

This is the leading eccentricity tide of a prescribed synchronous orbit, including optical libration. The orbit and material remain prescribed during the replay.

Common difficulties

Bending alone does not imply heating

Typical misconceptionA moving shape must dissipate energy.

Better mental modelIdeal elasticity returns its stored energy each cycle. A nonzero lag and the full work integral determine net heating.

The same external signal can hide different interiors

Typical misconceptionA strong Love response uniquely establishes an ocean.

Better mental modelIn the equal-density example, rescaling modulus and viscosity with geometric stiffness preserves all external response frequencies. Real structural inference requires additional information.

Run the experiment

01
Predict a larger driver
In Orbit view increase eccentricity from 0.005 to 0.010. Predict displacement and power before reading them.
What to observe: Leading-order displacement doubles and total power quadruples at fixed material and orbital scale.
02
Remove loss, then remove forcing
Toggle ideal elasticity, then restore Maxwell response and choose the circular preset.
What to observe: Elastic bending remains without heating. Circular synchronous motion removes this changing tide altogether.
03
Move a physical boundary
Drag the liquid boundary toward a thicker shell. Compare inner radius, lag and physical displacement; note the stated display gain.
What to observe: A thicker shell changes both the boundary geometry and the finite-shell restoring factor. The metre-valued readout gives the absolute amplitude alongside the magnified shape.
04
Compare a shell with a soft solid
Compare delayed-shell and rescaled-solid presets, then export both predictions.
What to observe: The hidden liquid radius differs while the complex external response agrees. The tutorial reconstructs this family from frozen synthetic records.

Start with the essentials

Focus question: Why does facing the planet require continued rotation?
One-sentence intuition: A body in synchronous rotation turns once in inertial space on every orbit. Having no spin relative to the planet does not mean having no spin at all.

Core mathematical model

Orbital time and orientation angle

u=nt,\quad\delta=\theta-u,\quad\frac{\omega}{n}=1+\delta'

A prime means a derivative with respect to normalized orbital time.

Equation of motion with tidal torque and damping

\delta''=-\frac32\eta\sin(2\delta)-\gamma\eta\delta',\qquad\eta=\frac{B-A}{C}

The gravity-gradient term comes from a quadrupole expansion for a rigid body. The damping term is phenomenological.

Energy of the relative motion and energy lost

\mathcal E=\frac12(\delta')^2+\frac34\eta[1-\cos(2\delta)],\quad Q'=\gamma\eta(\delta')^2,\quad(\mathcal E+Q)'=0

Every term is divided by the spin moment of inertia times the square of the orbital angular rate. The potential is set to zero when the long axis lies along the radial line.

Shape and the period of small librations

\frac ba=\sqrt{\frac{1-\eta}{1+\eta}},\qquad T_u=\frac{2\pi}{\sqrt{3\eta}}\quad(\gamma=0,\ |\delta|\ll1)

The uniform ellipsoid drawn on screen has this axis ratio. Its small-amplitude period gives an independent check on the numerics.

Common difficulties

A torque alone does not lock the spin

Typical misconceptionGravity will stop the relative rotation on its own.

Better mental modelSwitch off dissipation. The mechanical energy of the relative motion is then conserved, and the body can keep circulating or librating.

Run the experiment

01
Follow one face
Choose Already synchronous and play one orbit. Watch the gold spot in the Orbit and Radial views.
What to observe: The body turns in inertial space, yet it stays fixed relative to the radial line.
02
Compare runs from the same start
Run With dissipation, then No dissipation. Scrub to the end and compare the energy charts.
What to observe: With dissipation, the mechanical energy of the relative motion moves steadily into the accumulated loss. Without it, the libration or circulation goes on.
03
Check the model
Try Spherical body. Then halve the integration step for a body that is not spherical.
What to observe: A sphere feels no modeled torque and no damping. A smaller step shrinks the numerical error left in the energy balance.

Finite-shell mechanical reference · Titan response and conditional interior · Derivations and public experiment

Tidal Heating and Spin–Orbit Locking

The delayed response does net work each cycle

Why does a changing tide leave heat behind?

Core mathematical model

A material with a relaxation time

One finite-shell response

Positive power from a passive lag

Common difficulties

Bending alone does not imply heating

The same external signal can hide different interiors

Run the experiment

Predict a larger driver

Remove loss, then remove forcing

Move a physical boundary

Compare a shell with a soft solid

Losing relative spin

Core mathematical model

Orbital time and orientation angle

Equation of motion with tidal torque and damping

Energy of the relative motion and energy lost

Shape and the period of small librations

Common difficulties

A torque alone does not lock the spin

Run the experiment

Follow one face

Compare runs from the same start

Check the model

Related experiments and articles

Tidal response and ocean worlds