The lunar cycle explained
This is the lunar cycle explained as astronomy rather than as astrology: what physically produces the phases, why one lunation averages 29.53 days rather than 27.3, and how the illuminated fraction is calculated from a single angle.
Two different months
The Moon completes one orbit of Earth in 27.32 days. That is the sidereal month, measured against the fixed stars. But the phases do not repeat on that schedule, because while the Moon orbits, the Earth-Moon system travels around the Sun too, so the Moon has to swing further to return to the same alignment with the Sun. That longer interval is the synodic month, averaging 29.53 days, and it is the cycle the phases follow.
The two-day gap between them is the single most common source of confusion about moon phases, and it is why a phase does not fall on the same calendar date each month.
What actually makes a phase
Half of the Moon is lit at all times, apart from during an eclipse. Nothing about the amount of sunlight changes across a cycle. What changes is how much of the lit half faces Earth, and that depends on one quantity: the angle between the Moon and the Sun as seen from Earth, called the elongation.
- At 0 degrees the Moon lies roughly between Earth and the Sun, its lit face turned away, new moon.
- At 90 degrees we see half the lit face and half the dark face, first quarter.
- At 180 degrees Earth lies between the two and the whole lit face is turned toward us, full moon.
- At 270 degrees we again see half of each, last quarter.
The Moon's orbit is tilted about 5 degrees from the plane of Earth's orbit, which is why the alignment at 0 and 180 degrees is usually imperfect and we do not get an eclipse every month.
The illumination formula
The illuminated fraction of the disc follows directly from elongation:
k = (1 − cos ψ) / 2, where ψ is the elongation in degrees and k is the fraction of the disc that is lit.
Check it against the endpoints. At ψ = 0, cos 0 = 1, so k = 0 and the disc is dark. At ψ = 180, cos 180 = −1, so k = 1 and the disc is fully lit. At ψ = 90, k = 0.5. Because the function is a cosine rather than a straight line, illumination changes fastest around the quarters and slowest around new and full, which is why a full moon looks full for two or three nights while a quarter moon visibly changes overnight.
Why the eight names are unequal in duration but equal in angle
The convention divides the 360-degree cycle into eight 45-degree bins, each centred on its exact moment rather than beginning at it. New Moon therefore runs from 337.5 through 0 to 22.5 degrees. Because illumination does not change linearly with angle, those equal angular bins correspond to unequal illumination bands: New Moon covers 0% to 4% lit, while First Quarter covers roughly 31% to 69%. A date at 36% illumination genuinely belongs to First Quarter, which surprises people who expect the label to track the percentage.
Why no two lunations are the same length
The 29.53-day figure is a mean. Individual lunations run from about 29.27 to 29.83 days, a spread of roughly 13 hours either side, for 2 reasons. The Moon's orbit is an ellipse, so it moves faster near perigee and slower near apogee, and the phase cycle stretches or compresses accordingly. And the Sun's gravity perturbs the orbit by an amount that depends on where in the year the lunation falls.
The practical consequence: any method that counts a fixed 29.53 days forward from one reference new moon drifts, and over several decades the drift is large enough to name the wrong phase. Computing the Moon's actual position for the date avoids the problem, which is the approach documented on the the calculation notes.
Seeing the cycle for yourself
Waxing phases follow the Sun across the sky and are visible in the evening after sunset. Waning phases rise later each night and are best seen before dawn, which is why most people have seen far more waxing moons than waning ones without noticing the bias. In the northern hemisphere the lit limb is on the right while waxing and the left while waning; south of the equator that is reversed.
To find the phase for a particular date, the single-date lookup does the calculation, and the methods guide covers doing it by hand.