Lunar Libration — Why We See 59% of the Moon

Orbit & Spin

Eccentricity e 0.055

Axial tilt of spin axis 6.7°

Time speed 1.0×

Show far-side / limb markers

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Libration, longitude

0.0°

Libration, latitude

0.0°

Earth–Moon distance

384,400 km

How it works

A tidally locked moon spins at a constant rate that matches its mean orbital motion, but Kepler's second law says its true orbital speed is not constant — it moves faster near perigee and slower near apogee. The two rates only agree on average, so the sub-Earth point drifts back and forth: libration in longitude.

Kepler's equation:  M = E − e·sin(E)
True anomaly:       tan(ν/2) = √((1+e)/(1−e))·tan(E/2)
Spin angle:         ψ = M   (uniform rate, tidally locked)
Libration (long.):  ℓ ≈ ψ − ν = M − ν   (≈ 2e·sin M for small e)
Distance:           r = a(1 − e·cos E)

A second, independent wobble comes from the spin axis: it stays nearly fixed in space while the Moon's orbital position sweeps around Earth, so the angle between the axis and the Earth–Moon line changes over one orbit — libration in latitude. Combined with a small east–west parallax from an observer's position on Earth's surface (diurnal libration, not modeled here), these effects let Earth-based observers see about 59% of the lunar surface over time, not just the 50% visible from a single fixed viewpoint.

Eccentricity — sets the orbit's departure from a circle; drives libration in longitude. The Moon's real value (0.0549) gives roughly ±6–8° of wobble.
Axial tilt — angle between the spin axis and the orbit normal; drives libration in latitude. Real value ≈ 6.7°.
The dark equatorial band marks the mean near-side limb (±90° from the sub-Earth meridian); watch it swing past the front edge as the Moon rocks.

Near side (mean) Far side (mean) Orbit path

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