Age, illumination and phase are three different things
The Moon's age is the time elapsed since the last new moon, in days. Its illuminated fraction is the proportion of the disc you see that is sunlit. Its phase name is a label applied to a range of ages. These three quantities are related but not interchangeable, and treating them as one number is the most common error in lunar arithmetic.
Illumination is not proportional to age. It follows a cosine, so it moves fastest around the quarters and slowest around new and full. Between age 7 and age 8 the lit fraction changes by roughly seven percentage points; between age 14 and 15, straddling full moon, it changes by less than one. That is why a full moon looks full for two or three nights while a first quarter looks unmistakably half for only one.
Neither quantity runs on a fixed clock. The mean synodic month is 29.530589 days, but individual lunations vary by up to about half a day either side of that because the Moon's orbit is elliptical and the Earth–Moon system's motion around the Sun is not uniform. A calculator that divides by 29.53 and takes a remainder will therefore drift within each month and be worst exactly where people care most — near new and full. This one finds the actual new moon instant first.
One thing this page does not tell you is where the Moon is in your sky. Age and illumination are the same for every observer on Earth at a given instant, because they depend only on the Sun–Earth–Moon geometry. Rise and set times are entirely local; use the moonrise and moonset calculator for those.
The formula, one variable at a time
The illuminated fraction comes from the phase angle i, the angle at the Moon between the directions to the Sun and to the Earth. When i = 180° the Sun is behind you and the Moon is new; when i = 0° the Moon is full. The disc is a circle with a terminator that is half an ellipse, and the lit area works out as k = (1 + cos i)/2 exactly.
The phase angle itself is approximated here by Meeus's low-precision expression i = 180° − D − 6.289 sin M′ + 2.100 sin M − 1.274 sin(2D − M′) − 0.658 sin 2D − 0.214 sin 2M′ − 0.110 sin D. The leading term is the mean elongation D, the angular distance of the Moon from the Sun as seen from Earth. The 6.289° term is the equation of centre for the Moon's own elliptical orbit; the 1.274° term is evection, the largest solar perturbation of the lunar orbit; the 0.658° term is variation. Meeus states this expression gives k to about 0.0015, which is finer than anyone can judge by eye.
The age needs the exact new moon instant. That comes from the mean phase expression JDE = 2451550.09766 + 29.530588861k plus twenty-five periodic corrections and fourteen small additional terms, all in Meeus chapter 49. The largest correction, −0.40720 sin M′, is worth almost ten hours on its own, which is exactly the amount by which a naive fixed-cycle model goes wrong.
All of this works in dynamical time, which runs ahead of Universal Time by ΔT — currently a little over a minute, and growing as the Earth's rotation slows irregularly. The calculator applies the Espenak–Meeus ΔT polynomials to convert, which matters at the level of the phase instant but is invisible in the illuminated fraction. Julian Day and Julian Ephemeris Day are explained further in the Julian date calculator.
Worked example: the total lunar eclipse of 21 January 2000
A lunar eclipse can only happen at full moon, so this well-documented event is a good check on the arithmetic. The new moon that opened the lunation occurred on 2000 January 6 at 18:14 UT and the full moon on 2000 January 21 at 04:40 UT.
- Put both instants on the same scale. January 6.7597 and January 21.1944, both in days of the month with a decimal fraction.
- Subtract. 21.1944 − 6.7597 = 14.435 days. That is the Moon's age at the moment of the eclipse.
- Compare with the mean. Half a mean synodic month is 29.530589 / 2 = 14.765 days. This lunation reached full 0.330 days — nearly eight hours — earlier than the average, which is the size of the error a fixed-cycle model would have made here.
- Illumination. Evaluating the phase-angle expression at that instant gives i = 359.92°, essentially zero, so k = (1 + cos 359.92°)/2 = 0.9999995, or 100.0%.
- Lunation number. The preceding new moon is Meeus lunation 0, the reference epoch of the whole scheme.
Now take a case away from the extremes: Meeus's own example 48.a, 1992 April 12.0 in dynamical time. The full rigorous method gives an illuminated fraction of 0.6786. The low-precision expression used here returns 0.6801, a difference of 0.0015 — exactly the accuracy Meeus quotes for it, and about a fifth of the change that occurs in a single hour near the quarters. The Moon was waxing gibbous, four days past first quarter.
Age, phase name and illumination through a mean lunation
| Age (days) | Phase | Illumination | Where to look |
|---|---|---|---|
| 0.0 | New Moon | 0% | With the Sun; invisible |
| 3.7 | Waxing Crescent | 15% | Western sky after sunset |
| 7.4 | First Quarter | 50% | Due south at sunset |
| 11.1 | Waxing Gibbous | 85% | Eastern sky in the afternoon |
| 14.8 | Full Moon | 100% | Rises at sunset, sets at sunrise |
| 18.5 | Waning Gibbous | 85% | Rises late evening, up at dawn |
| 22.1 | Last Quarter | 50% | Due south at sunrise |
| 25.8 | Waning Crescent | 15% | Eastern sky before sunrise |
| 29.5 | New Moon | 0% | Cycle restarts |
The quarter ages are the mean synodic month divided into quarters: 29.5306/4 = 7.383, and the eighths at 3.69 and 11.07 give (1 − cos 45°)/2 = 14.6% and (1 − cos 135°)/2 = 85.4%.
How to read the result
Illumination is the number to use for anything about brightness or visibility; age is the number to use for anything about timing. A 50% Moon at age 7.4 and a 50% Moon at age 22.1 look equally lit but sit on opposite sides of the sky and are up at opposite times of day, so quoting only the percentage loses half the information.
The waxing or waning label resolves that ambiguity. Waxing means the lit fraction is growing and the Moon is an evening object, setting after the Sun; waning means it is shrinking and the Moon is a morning object, rising after midnight. The illuminated limb points toward the Sun in both cases, which is why a crescent lit on its right in the northern hemisphere is waxing and one lit on its left is waning — and why the reverse holds south of the equator.
The phase names have no official boundaries, and this calculator applies a defensible convention: the four named instants get a window of half a day either side, and the four intermediate names fill the gaps. A source that calls a Moon of age 15.2 days “full” and another that calls it “waning gibbous” are both defensible. The age and illumination figures, unlike the names, are unambiguous.
Being full does not mean an eclipse. The Moon's orbit is inclined about 5.1° to the ecliptic, so at most full moons it passes above or below the Earth's shadow. An eclipse needs the Moon to be full and near one of the two nodes where the orbits cross, which is why lunar eclipses cluster into seasons rather than happening monthly.
Assumptions and limits
- Age and illumination are geocentric. They are the same for every observer at a given instant; only rise and set times depend on where you stand.
- Illumination is the geometric lit fraction, not brightness. The Moon at first quarter is far less than half as bright as at full, because shadows cast by lunar terrain darken the disc away from full and the surface backscatters strongly.
- The phase-angle expression is Meeus's low-precision form, accurate to about 0.0015 in k — more than sufficient for any visual or planning purpose, and not intended for astrometry.
- Times are in Universal Time after a ΔT correction fitted to 1900–2150. Outside that range the phase instants degrade, though the illuminated fraction does not.
- Phase names are a convention. The half-day windows around the four exact instants are a choice, not a standard.
- Nothing here is a tide model. Spring tides follow new and full moon with a lag that is local; the lunitidal interval calculator handles that.
Why lunations are not all the same length
The synodic month — new moon to new moon — averages 29.530589 days but individual intervals range over roughly half a day either side of that. Two effects dominate. The Moon's orbit is an ellipse, so it moves fastest at perigee and slowest at apogee, and where perigee falls within the cycle changes the interval by hours. On top of that the Earth's own orbit is elliptical, so the Sun's apparent motion — which the Moon has to catch up with to reach new moon again — is faster in January than in July. The Meeus chapter 49 corrections model both, and the largest single term, −0.40720 sin M′, is worth 0.407 days at its extreme: nearly ten hours. Any calculator that works from a fixed 29.53-day cycle inherits that error.
Related calculations and when to use them
This page answers “what did the Moon look like on this date”. For “when is the next one”, the next full moon and new moon calculator finds the exact instants of each phase and lists a whole year of them. For “when will it be up”, the moonrise and moonset calculator needs your coordinates, because those times are local.
Astrophotographers should pair the illumination figure with the twilight times calculator: a night with genuine astronomical darkness is worthless for faint targets if a gibbous Moon is up throughout it, and the two calculations together tell you the actual usable window. Anglers and coastal walkers want the lunitidal interval calculator, which converts the Moon's transit time into a local high-water time.
Several calendars are built on this cycle rather than on the solar year. The Islamic calendar is purely lunar, so its months drift through the seasons at about eleven days a year. The Hebrew and Chinese calendars are lunisolar and insert a whole extra month periodically to stay in step with the Sun. The date of Easter is fixed by an ecclesiastical lunar cycle that approximates rather than tracks the real Moon — the Easter date calculator works through that computus.
Key terms
- Synodic month
- New moon to new moon, averaging 29.530589 days. It is longer than the sidereal month of 27.32 days because the Earth–Moon system has moved along its orbit in the meantime.
- Phase angle
- The Sun–Moon–Earth angle, 180° at new moon and 0° at full. The illuminated fraction is (1 + cos of it)/2.
- Terminator
- The line dividing lit from unlit on the Moon's disc. It appears as half an ellipse, which is why the lit area follows a cosine rather than growing linearly.
- Lunation number
- A running count of new moons. Meeus numbers from the new moon of 2000 January 6 as zero; the older Brown lunation number for the same event is 953.
- ΔT
- The difference between dynamical time, in which the orbital theory is written, and Universal Time, which follows the Earth's irregular rotation.