A phase is an instant, not a night
New, first quarter, full and last quarter are moments, not periods. Full moon is the instant the Moon's ecliptic longitude is exactly 180° from the Sun's; new moon is the instant they coincide. That instant falls at a specific second of a specific day in Universal Time, and which calendar date it lands on depends on where you are — a full moon at 02:00 UT is the previous evening across the Americas.
Almanacs and apps that publish only a date have collapsed that instant into whichever day it fell on at their own reference longitude. That is why two sources sometimes disagree by a day about when a full moon occurred, and neither is wrong. Working from the instant removes the ambiguity, which is why this calculator gives both a Universal Time and a local time and asks for your offset explicitly.
The four phases divide the cycle into quarters by elongation, not by time. Because the Moon's angular speed varies through its elliptical orbit, the four intervals within one lunation are not equal: the stretch containing perigee runs noticeably shorter than the stretch containing apogee. Quoting a quarter as “about 7.4 days” after the previous phase is an average, not a rule.
Lunations are numbered so that events can be referred to unambiguously. Meeus counts from the new moon of 2000 January 6 as lunation 0, so k is negative before that date. The older Brown lunation number, still used in some almanacs, counts from 1923 January 17 and is simply k + 953 for the same event.
The formula, one variable at a time
The starting point is the mean phase expression JDE = 2451550.09766 + 29.530588861k + 0.00015437T² − 0.00000015T³ + 0.00000000073T⁴, with T = k/1236.85 in Julian centuries. The constant is the mean new moon nearest the start of 2000, the linear coefficient is the mean synodic month, and the higher terms account for the slow secular change in the Moon's mean motion.
To get a quarter phase you add a fraction to k: 0.25 for first quarter, 0.50 for full and 0.75 for last quarter. To find which k you want, the approximation k ≈ (year − 2000) × 12.3685 gets you within one lunation, and the calculator then steps forward until it finds the first event on or after your reference instant.
The mean value alone is not good enough for anything. Twenty-five periodic corrections are added, built from the Moon's mean anomaly M′, the Sun's mean anomaly M, the Moon's argument of latitude F and the longitude of the ascending node Ω, with an eccentricity factor E that accounts for the slow decrease in the Earth's orbital eccentricity. Fourteen further small terms, each a sine of a linear function of k, mop up perturbations worth up to about half a minute each.
Everything above is in dynamical time. Converting to Universal Time means subtracting ΔT, the accumulated difference caused by the irregular slowing of the Earth's rotation, which this calculator takes from the Espenak–Meeus polynomial fits. ΔT is currently a little over a minute, was near zero around 1900, and is not predictable far ahead — which is the real limit on quoting a phase time for a distant future year. The Julian date calculator covers the day-numbering scheme itself.
Worked example: the last quarter of January 2044
This is Meeus's own example 49.b, which makes it a clean check on the whole chain.
- Estimate k. The date is about 2044.04, so k ≈ (2044.04 − 2000) × 12.3685 = 544.7. For a last quarter you need k + 0.75, so take k = 544 and work with 544.75.
- Julian centuries. T = 544.75 / 1236.85 = 0.44045.
- Mean instant. 2451550.09766 + 29.530588861 × 544.75 = 2451550.09766 + 16086.78828 = 2467636.88594, plus 0.00015437T² = 0.00003, giving 2467636.88597.
- Periodic corrections. The quarter-phase series, dominated by −0.62801 sin M′ and +0.17172E sin M, together with the W term that separates first from last quarter, contributes −0.39411 days.
- True instant. 2467636.88597 − 0.39411 = JDE 2467636.49186.
- Convert to a calendar date. That is 2044 January 21.99186, or 23h48m17s in dynamical time on 21 January 2044.
Notice the size of the correction: 0.394 days is 9.5 hours. Had you stopped at the mean instant you would have placed the event at 09:16 on 22 January rather than 23:48 on the 21st. Run the same chain for Meeus's example 49.a, the new moon of February 1977 at k = −283, and the mean instant 2443192.94101 is corrected by −0.28983 days to JDE 2443192.65118 — nearly seven hours earlier, moving the event from mid-morning to 03h37m40s.
The periodic corrections and how much each can move the instant
| Term | Coefficient (days) | Maximum effect | Physical cause |
|---|---|---|---|
| sin M′ | −0.40720 | 9 h 46 m | Equation of centre of the Moon's own elliptical orbit |
| E sin M | +0.17241 | 4 h 08 m | Equation of centre of the Earth's orbit, shifting the Sun's apparent position |
| sin 2M′ | +0.01608 | 23.2 min | Second-order term of the lunar equation of centre |
| sin 2F | +0.01039 | 15.0 min | Inclination of the lunar orbit to the ecliptic |
| E sin(M′ − M) | +0.00739 | 10.6 min | Evection: solar perturbation of lunar eccentricity |
| E sin(M′ + M) | −0.00514 | 7.4 min | Companion evection term |
| E² sin 2M | +0.00208 | 3.0 min | Second-order solar equation of centre |
Each maximum effect is the coefficient times 24 hours, since a sine reaches 1. The first two terms alone can displace a phase by nearly fourteen hours, which is why a fixed 29.53-day cycle cannot give a usable time of day.
How to read the result
Take the Universal Time as the authoritative answer and treat the local column as a convenience. The local column applies the fixed offset you typed and knows nothing about daylight saving, so add the summer hour yourself when it applies — or convert the UT figure with the time zone converter, which asks about the rule explicitly.
The interval to the following event of the same phase is worth reading. It varies by up to about half a day either side of the mean synodic month of 29.530589 days, and the calculator says by how much and in which direction. Long intervals cluster where the Moon passes apogee near the phase; short ones where it passes perigee.
Times are good to about a minute over the range this calculator covers, and the limit is ΔT rather than the orbital theory. Because the Earth's rotation slows irregularly, ΔT is measured rather than predicted, so a phase time quoted for 2140 carries an uncertainty of minutes in Universal Time even though its dynamical instant is solid. That is why the Julian Ephemeris Day is reported alongside the clock time.
A full moon date is not an eclipse date. Every lunar eclipse happens at full moon, but most full moons pass clear of the Earth's shadow because the lunar orbit is tilted about 5.1°. Use the phase instant here as the starting point and check an eclipse catalogue for whether the Moon was also near a node.
Assumptions and limits
- The result is a geocentric instant. It is the same moment worldwide; only the local date and clock reading change with your offset.
- ΔT is modelled, not known, outside the observed record. The Espenak–Meeus polynomials are fitted to 1900–2150 here; beyond that the Universal Time degrades while the dynamical instant does not.
- No daylight saving is applied. The local column adds exactly the offset you entered.
- Phase instants say nothing about visibility. Whether the Moon is above your horizon at that moment needs the moonrise and moonset calculator.
- Eclipses are not predicted. A phase instant is a necessary condition for an eclipse, never a sufficient one.
- The quarters are defined by elongation, not by elapsed time, so the four intervals inside one lunation are unequal.
Why the mean expression is not enough
Meeus chapter 49 separates the problem into a mean phase, which is a simple linear function of the lunation index, and a set of periodic corrections. The mean part is exact in the long run by construction — it reproduces the mean synodic month of 29.530589 days — but any individual event can sit up to about fifteen hours away from it. Two terms account for most of that: the Moon's own equation of centre at 0.40720 days and the Earth's at 0.17241 days. Adding the full series brings the residual to a few seconds in dynamical time, at which point the accuracy of the answer in civil time is set entirely by ΔT.
Related calculations and when to use them
This calculator answers “when”. For “what does it look like right now”, the moon phase calculator gives the Moon's age and illuminated fraction for an arbitrary instant using the same underlying phase instants. For “will I be able to see it”, you need local rise and set times from the moonrise and moonset calculator.
Observers planning around the Moon should combine the full moon dates here with the twilight times calculator. The dark-sky window that matters is the intersection of astronomical night with the Moon being below the horizon, and around full moon that intersection is empty however dark the twilight figures look. Coastal users want the lunitidal interval calculator, since spring tides follow new and full moon by a locally fixed lag.
The same cycle drives several calendars. The Islamic calendar begins each month at the first sighting of the young crescent, typically a day or so after the new moon instant computed here, which is why its dates cannot be fixed by calculation alone in every jurisdiction. The date of Easter uses an ecclesiastical Moon that approximates rather than follows the real one; the Easter date calculator works through that computus, and the equinox and solstice calculator supplies the solar half of the same problem.
Key terms
- Lunation
- One complete cycle of the Moon's phases, new moon to new moon. Indexed by k, with 2000 January 6 as k = 0 in Meeus's numbering.
- Julian Ephemeris Day
- A Julian Day expressed in dynamical time rather than Universal Time. Orbital theories are written in this scale.
- ΔT
- Dynamical time minus Universal Time, caused by the irregular slowing of the Earth's rotation. Measured, not derivable, so it limits how precisely a future phase can be quoted in civil time.
- Mean anomaly
- The angle a body would have swept from perihelion if it moved at constant speed. Its sine drives the largest correction terms in the phase series.
- Argument of latitude
- The Moon's angular distance from the ascending node of its orbit, written F. It appears in the corrections and also governs whether an eclipse is possible.