Moon Phases

Why the Moon changes shape every night

Junior level — plain language, no maths

Look at the Moon across a month and it seems to grow and shrink — a thin sliver, a half, a fat disc, a full circle, then back again. But the Moon never actually changes shape. The secret is that the Moon makes no light of its own. Like a giant ball hanging in space, it simply catches sunlight and reflects it back to us — and the Sun can only ever light up one half of it at a time, the half that happens to be facing the Sun.

As the Moon circles the Earth once a month, we get to look at that lit half from different angles. When the Moon sits between us and the Sun, its lit side faces away from us and we see darkness — a new moon. When it's on the far side, the whole lit face turns toward us — a full moon. In between we catch it side-on and see just a slice: a crescent or a half. The shape you see is simply how much of the sunlit half is pointing your way.

The full cycle — new, to full, and back to new — takes about 29 and a half days, which is roughly where the idea of a "month" comes from. And here's a lovely quirk: the Moon always keeps the same face pointed at Earth, so we never see its far side from home. In the simulation below, move the Moon around its orbit and watch its phase change.

The geometry of phases, and why eclipses are rare

Student level — the core equations

A moon phase is pure geometry. Sunlight always illuminates exactly half the Moon; what changes is the angle between the Sun, Earth and Moon. Call that the phase angle: at 0° the Moon is roughly between us and the Sun (new), at 180° it is opposite the Sun (full), and the lit fraction we see follows \(k = \tfrac{1}{2}(1 - \cos\theta)\). The waxing half of the cycle (growing) and the waning half (shrinking) are mirror images, lit on opposite sides.

There is a subtlety in the timing. The Moon takes 27.3 days to complete one orbit against the background stars — the sidereal month — but 29.5 days to return to the same phase — the synodic month. The gap exists because the Earth has moved along its own orbit in the meantime, so the Moon must swing a little further to line up with the Sun again. The phase cycle we actually see is the synodic one.

So why isn't there an eclipse at every new and full moon? Because the Moon's orbit is tilted about 5° to the plane of Earth's orbit. Most months the Moon passes a little above or below the direct Sun-Earth line, and its shadow misses. Only when a new or full moon happens to fall near the two points where the orbits cross — the nodes — do we get a solar or lunar eclipse. That tilt is the reason eclipses are occasional treats, not monthly events.

Tidal locking, orbital recession and long cycles

Scholar level — full mathematical depth

01Why one face is hidden forever

The Moon's rotation period exactly equals its orbital period, so it is tidally locked. This is no coincidence: Earth's gravity raises a slight bulge on the Moon, and any mismatch between spin and orbit drags on that bulge, applying a torque that brakes the rotation until the two rates match. Almost every large moon in the Solar System has arrived at the same 1:1 spin-orbit resonance by the same route — synchronous rotation is the natural endpoint of tidal friction.

02Tides steal the Moon's orbit outward

Tidal coupling runs both ways. The Moon raises ocean bulges on Earth, and Earth's faster rotation drags them ahead of the Earth-Moon line; their gravitational tug adds angular momentum to the Moon's orbit, pushing it outward by about \(3.8\ \text{cm}\) per year — a rate now measured to millimetres by bouncing lasers off reflectors the Apollo astronauts left behind. The same transfer slows Earth's spin, lengthening the day by roughly 1.7 milliseconds per century.

03Conservation bookkeeping

All of it is one conserved quantity being shuffled around. The total angular momentum of the Earth-Moon system is fixed, so as Earth's rotational share bleeds away, the Moon's orbital share must grow — which is precisely why a receding Moon and a lengthening day go hand in hand. Run the clock forward and the process ends only when Earth's day and the lunar month both stretch to the same value, a mutual tidal lock tens of billions of years off.

04The tilted, twisting orbit

The 5.14° tilt is not fixed in space: the line of nodes where the Moon's orbit crosses the ecliptic precesses all the way around in 18.6 years, and the orbit's long axis precesses in 8.85 years. These slow swings modulate when and where eclipses can occur and thread them into the famous Saros cycle of 18 years 11⅓ days, after which the Sun-Earth-Moon geometry nearly repeats and a near-twin eclipse returns.

05Reading a phase quantitatively

The illuminated fraction \(k = \tfrac{1}{2}(1+\cos\alpha)\), written in terms of the Sun-Moon elongation, is only the geometric skeleton. The Moon is a startlingly poor mirror — its albedo is about 0.12, darker than worn asphalt — and its brightness does not scale linearly with lit area. Near full moon it surges disproportionately bright, the opposition surge, as surface shadows vanish and glass beads in the regolith backscatter sunlight straight at the source.

06A calendar that never quite fits

The synodic month is not a whole number of days, nor does a whole number of them fit a solar year — twelve synodic months fall about 11 days short. Every lunar calendar in history has wrestled with this mismatch, from the 19-year Metonic cycle that realigns Moon and Sun (the basis of computing Easter) to the leap months of the Hebrew and Chinese calendars. The Moon's tidy-looking rhythm is, at root, gloriously incommensurate with the year.