How To Calculate The Moon Phase

The phase of the Moon is the fraction of its Earth-facing side that the Sun lights up, and it changes because the Moon orbits the Earth roughly every 29.53 days. That period is the synodic month, the time from one new moon to the next, and it is the cycle the calendar of phases is built on.

The Moon does not emit light; it reflects sunlight. Half of the Moon is always lit, and the phase we see is the portion of that lit half facing us. At new moon the lit half faces away, at full moon it faces us, and the intermediate phases are the geometry in between.

The synodic month is longer than the time the Moon takes to orbit the Earth once, which is about 27.32 days. The extra two days exist because the Earth moves around the Sun while the Moon goes around the Earth, so the Moon has to travel a little further to line up with the Sun again.

A Julian Day Number turns a calendar date into a single running count of days, which makes the arithmetic of a repeating cycle simple. The Julian Day Number for 18 October 2026 is 2461332. Subtracting a reference new moon gives the number of days since that new moon, and the remainder after dividing by 29.53 gives the Moon's age.

The reference new moon most often used is the one on 6 January 2000, at Julian Day Number 2451550.1. That decimal part matters: the new moon happened partway through the day, and treating it as exactly midnight would shift every later result by a fraction of a day.

Dividing the age by the synodic month gives the phase fraction, which runs from 0 at new moon through 0.25 at first quarter and 0.5 at full moon, back to 1 at the next new moon. The phase name is just a band of that fraction, and the eight familiar names divide the cycle into equal eighths.

Illumination follows from the phase fraction through a cosine. The fraction is turned into an angle by multiplying by a full turn, and one minus the cosine of that angle, halved, gives the illuminated percentage. It is 0 at new moon, 1 at full moon, and close to 0.5 at the quarters.

This is a mean-phase model. It treats the Moon's orbit as a circle travelled at a constant rate, and the real orbit is an ellipse, so the true phase can differ from the calculated one by a few hours to roughly half a day. For planning an evening walk that is irrelevant; for timing an eclipse it is not.

The phase is a global property of the Earth-Moon-Sun geometry, so everyone on Earth sees essentially the same phase on the same date. What differs with location is the Moon's position in the sky, its rise and set times, and whether the disc is tilted, which is why the crescent can look like a smile from one hemisphere and a bowl from the other.

Because the cycle is close to but not exactly 29.5 days, the phases drift through the calendar rather than landing on the same dates each month. Every so often that drift squeezes two full moons into a single calendar month, and the second of the pair is what is loosely called a blue moon.