M080 · A deformable moon

Tidal Deformation, Locking & Breakup

How can tides lock a moon or tear it apart? Unequal gravity can tear a weak moon apart. Internal friction can let its spin settle to one turn per orbit.

Follow a deformable moon as spin and orbit evolve. Compare tidal locking with conservative motion, then test whether weak and strong materials survive.

Body axis ratio $1.25$

Bulk spin / orbital turn $1.20$

Surviving connections $234$

Viscous heat $0$

Gravity torque on the body $0$

Model assumptions

A coarse elastic node network evolves under gravity and internal viscosity. Breaking removes overstretched springs; the fragments still attract each other. Orbit and spin respond together. The material is schematic, and the run stops at the primary surface; real capture times, collisions and accretion are outside this model.

Start with the essentials

Focus question: When does tidal stretching lead to synchronous rotation, and when does the body break?
One-sentence intuition: The same unequal gravitational pull that deforms the body can exchange angular momentum between its spin and orbit; material dissipation removes mechanical energy.

Core mathematical model

Acceleration relative to the moving centre

\begin{aligned}\mathbf a_i^{(p)}&=\frac{GM_p(\mathbf x_p-\mathbf x_i)}{|\mathbf x_p-\mathbf x_i|^3}\\\Delta\mathbf a_i&=\mathbf a_i^{(p)}-\frac1M\sum_jm_j\mathbf a_j^{(p)}\end{aligned}

Subtract the mass-weighted mean acceleration to see the gravitational stretching. The integrator applies the full gravity to every mass and includes the reaction on the primary.

A spring and a viscous damper act along each connection

\begin{aligned}\mathbf F_{ij}&=[k_{ij}(\ell_{ij}-\ell_{ij,0})+c_{ij}\dot\ell_{ij}]\hat{\mathbf n}_{ij}\\\dot Q_{ij}&=c_{ij}\dot\ell_{ij}^{\,2}\ge0\end{aligned}

The direction points from the first node to the second. The nodes receive equal and opposite forces. A spring fails in tension at the selected strain; its stored elastic energy becomes fracture energy.

Angular momentum and energy of the complete system

\begin{aligned}L_z&=L_{\rm orbit}+L_{\rm internal}\\E_{\rm mechanical}+Q+E_{\rm fracture}&=E_{\rm initial}\end{aligned}

All forces are internal and central. Angular momentum stays constant; mechanical energy can become viscous heat or fracture energy. The displayed residuals show numerical integration error.

A separate fluid Roche reference

\begin{aligned}\mathcal R&=\frac{\pi G\rho}{n^2}\simeq11.10\\\frac{r_{\rm Roche}}R&\simeq\left(\frac{4\times11.10}{3}\frac{M_{\rm p}}M\right)^{1/3}\end{aligned}

This reference assumes an incompressible homogeneous fluid, synchronous circular motion and a dominant primary. The elastic, breakable network has different assumptions and need not fail at this circle.

Common difficulties

A synchronous moon still spins

Typical misconceptionThe same face points at the planet because the moon has stopped spinning.

Better mental modelWatch the gold material marker in Whole orbit. The body turns in inertial space as it goes around the primary. Near synchronism, its bulk spin and orbital angular rate agree. After breakup this ratio describes the whole collection of fragments, not a rigid moon.

Run the experiment

01
Watch it settle
Choose Settle into synchronism and let it play. Scrub toward the end and compare the spin and angular-momentum charts.
What to observe: The initially faster spin approaches the orbital turn. Its lost angular momentum appears in the orbit, and viscous heat grows.
02
Remove the viscosity
Choose Keep the energy. Compare it with Settle into synchronism at the same replay time.
What to observe: Gravity can still transfer angular momentum, but the mechanical energy remains in the system. The motion does not settle the same way.
03
Change the material on the same passage
Run Tidal breakup, then Stronger material. Inspect Deforming body, Whole orbit and Body edge-on.
What to observe: The weaker network loses connections and stretches into fragments. The stronger one stays connected. A node hitting the primary ends the weak run at the model boundary.

Model, units and sources

A self-gravitating, 43-node elastic network represents one deformable body. Each central spring has a parallel viscous damper. Its initial rest lengths support the chosen isolated rotating shape; changing the initial spin prepares a different prestress. The primary responds to every node.

\begin{gathered}G=M=R=1,\quad t_g=\sqrt{\frac{R^3}{GM}}\\E_g=\frac{GM^2}{R},\quad L_g=M\sqrt{GMR}\end{gathered}

Length uses the initial volume-equivalent radius. Positive rotation is counterclockwise viewed from the north orbital pole. The orbit starts at apoapsis. Internal gravity uses Plummer softening of 0.08 radii; primary gravity is unsoftened. The primary has mass 30 and a stopping radius of 2.2. A node reaching that surface ends the run. Contact, accretion, plastic flow, fracture healing and planetary spin are omitted.

Breaking occurs when a spring exceeds the chosen tensile strain relative to its rest length. Its stored elastic energy becomes fracture energy, separately from viscous heat. Disconnected nodes still gravitate. After breakup, bulk spin is angular momentum divided by polar inertia, not the rotation rate of a rigid body. The two diagnostics below use the entire surviving mass.

The network stiffness is an isotropic estimate applied to the initial bond lengths, not a calibrated continuum modulus. This coarse, anisotropic lattice demonstrates mechanisms; it does not predict a real moon’s capture time or disruption distance. The dashed fluid Roche circle is a separate circular, synchronous, homogeneous-fluid reference in the dominant-primary approximation. It is not this elastic network’s failure threshold.

Replay selects actual stored integration frames. Conservative forces use velocity Verlet; symmetric pairwise damping transfers its exact kinetic loss to heat. Failure events subdivide a step up to 1024 times. Use Half integration step to compare numerical errors; it does not remove the coarse-material approximation.