Chapter i
The Moon
The oldest clock that anyone can read without an instrument, and one that refuses to divide evenly into anything else.
From one new moon to the next takes, on average, 29.53059 days: 29 days, 12 hours, 44 minutes and 3 seconds. This is the synodic month, the month of phases. Single months stray from that mean. The shortest run to about 29.27 days and the longest to about 29.83, because the Moon’s orbit is an ellipse and the Earth’s is too.1
The Moon goes once round the Earth, measured against the stars, in only 27.32 days. The two days’ difference come from the Earth itself. While the Moon makes its circuit, the Earth carries it about a thirteenth of the way round the Sun, and the Moon has to travel a little further to stand in the same line with the Sun again. The two periods are bound by a simple sum:
This month
The phase drawn for each day is the one at noon, local time, as seen from the northern hemisphere. South of the equator the picture is mirrored: the waxing Moon is lit on the left.
The coming full moons
| Full moon | Time (yours) | Lunations from now |
|---|
These are the Moon’s true phases, worked out below, and should fall within a minute or two of the times printed in published almanacs.2
The true Moon and the mean
The simplest way to find a phase is to count whole average months from one known new moon. Meeus takes the mean new moon of 6 January 2000 and adds 29.530588861 days at a time. That gives the mean phase, the moment an imaginary Moon would reach it if it went round at an even pace. The real Moon does not, and its true phase can fall well over half a day either side.
Two terms do most of the work. The Moon’s orbit is an ellipse, and the Moon runs fastest near perigee, its nearest point. A phase that falls in the fortnight after perigee comes early, and one after apogee comes late, by up to 0.407 of a day at new and full moon, about 9¾ hours. The Earth’s orbit is an ellipse too, so the Sun seems to move fastest across the sky in early January, and the Moon has further to go to catch it. That term is worth up to 0.173 of a day, about 4 hours. At the quarters the Moon’s term is larger, 0.628 of a day, about 15 hours, because the evection, a stretching of the Moon’s orbit by the Sun’s pull, adds to it there. Two dozen smaller terms follow, and fourteen smaller still for the pull of the planets.2
The next full moon, worked out
| Term | What it is | Hours |
|---|
Nineteen years
Twelve lunar months fall about eleven days short of a year, and thirteen overshoot it, so the phases drift through the calendar. But 235 months come within hours of nineteen years.3 After nineteen years the Moon shows the same face on very nearly the same dates. Meton of Athens introduced the cycle to the Athenian calendar in 432 BC. It still underlies the Hebrew calendar and the calculation of Easter.
In the Easter calculation, the first step, a = year mod 19, is this cycle. It finds the year’s place in the nineteen, which medieval calendars called its golden number (a + 1). This year’s golden number is . See it at work →
Sources
- Wikipedia, “Lunar month”, for the mean synodic and sidereal months and the range of single months.
- Jean Meeus, Astronomical Algorithms, 2nd ed. (Willmann-Bell, 1998), chapter 49, for the mean new moon of 2000 January 6 (JDE 2451550.09766) and the mean month of 29.530588861 days used here. The true phases use the chapter’s periodic and planetary terms, checked by the site’s build against the chapter’s worked example (the new moon of February 1977) and against the U.S. Naval Observatory’s phases of the Moon for early 2024, a U.S. government work in the public domain, all of which it matches to within a minute.
- Wikipedia, “Metonic cycle”, for Meton and the 235-month cycle.
- Fred Espenak and Jean Meeus, “Polynomial expressions for Delta T”, NASA Goddard Space Flight Center, for ΔT.