Tidal Locking: Spin-Down to Synchronous Rotation (2D)

Orbit

Semi-major axis (a) 7.0 R

Eccentricity (e) 0.12

Spin & Tides

Initial spin (× orbital rate) 3.0×

Tidal dissipation strength (k) 1.0×

Live Readout

Spin rate ω

0.000 rad/s

Orbital rate n

0.000 rad/s

Synchronization

Spin period / Orbital period

— / —

Libration (longitude)

0.0°

Elapsed orbits

0.00

How it works

The moon follows a true Keplerian ellipse around the planet (one focus at the origin). Its true anomaly ν obeys Kepler's second law exactly:

n  = n0 / a^1.5              (mean motion)
r  = a(1 − e²) / (1 + e·cos ν)
dν/dt = n·(1 + e·cos ν)² / (1 − e²)^1.5

The moon's own spin rate ω relaxes exponentially toward the mean motion n under a linear (weak-friction) tidal-damping model, with a time constant that grows sharply with distance — closer moons lock far faster, exactly as with Io, the Moon, or Charon in reality:

τ(a) = τ0 · (a / 7)^4 / k
dω/dt = −(ω − n) / τ(a)

Even after ω locks to n, the moon still visibly wobbles: because ν sweeps faster near periapsis and slower near apoapsis (Kepler's second law) while the spin turns at a constant rate, the sub-planet point oscillates back and forth — this is physical libration in longitude, with peak amplitude ≈ 2e radians, exactly as seen on the real Moon.

Bright marker — the near-side point fixed to the moon's crust; it turns green when it is pointing at the planet within 3°.
Spin kick — injects a sudden impulse into ω (an impact event), restarting the despin so you can watch it relock.
Trail — the last orbit's worth of positions, fading with time.

Orbit path Moon & near-side marker Marker locked on planet