M080 · Spin meets orbit

Tidal Locking & Libration

Spin an elongated satellite, add dissipation and follow its marked face around an orbit. Compare circulation, libration and synchronous rotation against an explicit energy account.

Orientation from radial line $\text{Ready}$

Inertial spin / orbital rate $\text{Ready}$

Normalized gravity torque $\text{Ready}$

Relative mechanical energy $\text{Ready}$

Accumulated dissipation $\text{Ready}$

Current energy-budget residual $\text{Ready}$

GRAVITY / GEOMETRY / MOTION

Turn it. Then let it settle.

Initial radial offset

\delta_0

55.00\,

Initial spin / orbit rate

\omega_0/n

2.40\,

Inertia asymmetry

\eta=(B-A)/C

0.60\,

Dissipation coefficient

\gamma

0.60\,

Half integration step

b_h

0.00\,

Circular orbit, fixed spin axis normal to the orbital plane. Time is orbital angle, not seconds. Positive rotation is counterclockwise from above. The gold spot is a material point; the dashed radial line rotates with the orbit. Satellite dimensions are enlarged for visibility.

Leading quadrupole gravity-gradient torque for a small homogeneous ellipsoid. Inertia asymmetry controls its axis ratio. Linear damping relative to the orbit is a chosen teaching model, proportional to asymmetry; it vanishes for a sphere. Orbit backreaction, eccentricity, obliquity, deformation, material response and real capture times are omitted.

Fourth-order Runge–Kutta over four orbits; the half-step option doubles sample count. The energy account is in the co-orbiting frame, scaled by spin inertia times orbital rate squared. Dissipation plus relative mechanical energy stays constant up to the displayed numerical residual. This is not total orbital-system energy.

Planar rigid-body gravity-gradient approximation on a prescribed circular orbit. Phenomenological damping, normalized time and enlarged satellite; no orbit migration, material rheology or lunar capture timescale.

Start with the essentials

Focus question: Why does facing the planet require continued rotation?
One-sentence intuition: Synchronous rotation means one inertial turn per orbit. Zero relative spin is not zero inertial spin.

Core mathematical model

Time and orientation

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

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

Torques and motion

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

The gravity-gradient term follows a rigid-body quadrupole expansion. The damping term is phenomenological.

Relative energy and dissipation

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

All terms are normalized by spin inertia times orbital angular rate squared. The chosen potential is zero at radial alignment.

Shape and small libration

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

The displayed homogeneous ellipsoid has this axis ratio. Its small-amplitude period gives an independent numerical check.

Common difficulties

A torque does not guarantee capture

Typical misconceptionGravity automatically stops relative rotation.

Better mental modelSwitch off dissipation: relative mechanical energy is 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 Orbit and Radial views.
What to observe: The body turns in inertial space while staying fixed relative to the radial line.
02
Compare equal starts
Run With dissipation, then No dissipation. Scrub to the end and compare the energy chart.
What to observe: Dissipation transfers relative mechanical energy into the accumulated loss account. Without it the oscillation or circulation persists.
03
Check the model
Try Spherical body, then halve the integration step in a nonspherical case.
What to observe: A sphere has neither modeled torque nor damping. A smaller step reduces the numerical energy-budget residual.

Primary sources: MIT Press · Spin–orbit coupling NASA · Tidal locking