Lunar Libration Diagram (2D): Kepler Orbit & Wobble Trace

Orbit & Spin

Eccentricity e 0.055

Axial tilt of spin axis 6.7°

Time speed 1.0×

Show libration trail / limb markers

Live Readouts

Libration, longitude

0.0°

Libration, latitude

0.0°

Earth–Moon distance

384,400 km

How it works

This is a flat, top-down companion to the 3D libration simulator: every frame it solves Kepler's equation directly in the orbital plane for the Moon's true position, rather than rendering a 3D globe. A tidally-locked moon spins at a constant rate that matches its mean orbital motion, but Kepler's second law makes its true angular speed non-uniform whenever the orbit is eccentric — faster at perigee, slower at apogee. The gap between the two produces libration in longitude, plotted here as the horizontal offset of the sub-Earth marker on the moon disc.

Kepler's equation:  M = E − e·sin(E)      (solved by Newton's method each frame)
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)

Libration in latitude comes from the spin axis staying fixed in space while the Moon's orbital longitude ν sweeps around Earth. Reworking that 3D geometry into a plane-trigonometry identity (the axis' projection along the north–south line depends only on tilt and ν, since the east–west component always cancels) gives an exact closed form used directly here — no 3D vectors needed:

Libration (lat.):   b = 90° − arccos( sin(tilt)·cos(ν) )

Switch to the libration diagram to watch the sub-Earth point trace a closed loop in (longitude, latitude) space over one orbit — the same kind of wobble chart lunar cartographers have used since the 17th century to prove roughly 59% of the surface is visible from Earth over time, not just 50%.

Eccentricity — departure from a circular orbit; drives libration in longitude. The Moon's real value (0.0549) gives roughly ±6–7° of wobble.
Axial tilt — angle between the spin axis and the orbit normal; drives libration in latitude. Real value ≈ 6.7°.
The dashed ring on the moon disc marks the mean near-side limb (±90° from the sub-Earth meridian); watch the sub-Earth marker swing past it as the Moon rocks.

Orbit path Sub-Earth marker Libration trace