Moon Phase Calculator

Lunar phase is a function of elongation, the Sun-Moon angle seen from Earth. If that angle is called the phase angle, the illuminated fraction of the visible disk is exactly (1 minus the cosine of the phase angle) divided by two, which is the relation this tool applies after converting moon age into an angle. The counter runs on the mean synodic month of 29.530589 days, the interval from new moon to new moon, which is longer than the 27.321662 day sidereal month because Earth advances roughly 27 degrees along its own orbit during each lunation. The standard eight-phase naming scheme splits the cycle at ages near 7.38, 14.77, and 22.15 days, with the crescent and gibbous intervals filling the space between the four principal instants. The practical limitation is that lunations are not uniform. Real intervals between successive new moons range from about 29.27 to 29.83 days because the lunar orbit has an eccentricity near 0.0549 and is strongly perturbed by the Sun. A linear mean-lunation model can therefore misplace a phase instant by up to roughly 14 hours, which is fine for planning a dark-sky observing window but not for computing occultation timings. Illuminated fraction is also a poor proxy for sky brightness: because of the opposition surge and shadow hiding in the rough regolith, the full Moon is about 11 to 12 times brighter than the first quarter Moon rather than twice as bright. Deep-sky observers generally look for illumination under about 25 percent.

Tracking the lunation is one of the oldest quantitative practices in human record keeping, and it structured the Babylonian, Hebrew, Islamic, Hindu, and Chinese calendars long before any physical model of the Moon existed. Meton of Athens introduced the 19-year cycle in 432 BCE after noticing that 19 tropical years contain almost exactly 235 synodic months, an agreement good to about two hours; Callippus refined the scheme around 330 BCE. Hipparchus, working roughly between 150 and 125 BCE with Babylonian eclipse records, derived a value for the synodic month accurate to within a second of the modern figure. The same arithmetic was later inherited by the Christian computus for dating Easter, codified by Dionysius Exiguus in 525 CE and adjusted in the Gregorian reform of 1582. Analytic lunar theory matured with Isaac Newton's treatment of the three-body problem, then through the work of Leonhard Euler, Charles-Eugene Delaunay, and finally Ernest W. Brown, whose tables of 1919 contained roughly 1,650 periodic terms and served national almanacs until 1984. Brown's series was superseded by the ELP-2000 semi-analytic theory of Michelle Chapront-Touze and Jean Chapront and by the JPL Development Ephemerides, which are numerically integrated and calibrated against lunar laser ranging to the retroreflectors left by Apollo 11, 14, and 15. Jean Meeus popularized compact polynomial approximations for phase instants in Astronomical Algorithms in 1991, the approach most software still uses.