Moon Illumination Fraction

Because the illuminated fraction k depends only on the magnitude of the phase angle θ, which is always between 0° and 180°, and it is the same whether the Moon is growing or shrinking. A 50% illuminated Moon occurs twice per month — once at First Quarter (waxing) and once at Last Quarter (waning). The two cases are geometrically mirror images: at First Quarter, the right side is lit (Northern Hemisphere); at Last Quarter, the left side is lit. The phase name, the lunar age, and the elongation all differ between the two, even though k and θ are identical. The toggle tells the calculator which side of the cycle you are on.

Because the "quarter" refers to the phase (one quarter of the synodic month has elapsed since New Moon), not to the illuminated fraction. At First Quarter, the Moon has completed roughly 7.4 days of its 29.5-day cycle — one quarter — but the visible disc is exactly half lit, since θ = 90° and k = 0.5. The naming is historical: the quarters of the lunar month are New, First Quarter, Full, and Last Quarter, and the phase names describe the cycle position rather than the appearance. In informal speech, "half moon" usually refers to a quarter-phase Moon, but the technically correct term is "quarter".

Not directly. The Moon is tidally locked to the Earth, so it always shows us the same face — the "near side". What changes is how much of that face is illuminated. At New Moon, the entire near side is dark (night on the Moon's near side), and we see no illuminated disc. But the far side — the "dark side" of popular song — is at that moment fully lit. The far side was not photographed until the Soviet Luna 3 probe in 1959 and not seen by human eyes until the Apollo 8 crew orbited the Moon in December 1968. "Dark side" is really a misnomer: the far side receives just as much sunlight as the near side, averaged over a lunar month; it is simply never visible from Earth.

Earthshine is the faint illumination of the Moon's night side by sunlight reflected off Earth. It is visible to the naked eye as a soft grey glow filling out the otherwise dark disc of a thin crescent Moon, and it was described by Leonardo da Vinci around 1510. It is strongest a day or two before and after New Moon, because at those times the Earth appears nearly "full" as seen from the Moon, and reflects its maximum light back onto the lunar night side. Through a small telescope, earthshine reveals the dark maria and even major craters that are otherwise invisible during the crescent phases.

Yes — the relation k = (1 + cos θ)/2 applies to any spherical body illuminated by a distant source and viewed from a distant observer. It gives the phase of Mercury and Venus as seen from Earth, the phases of Jupiter's Galilean moons as seen from Jupiter, and the phases of distant exoplanets as they orbit their host stars. The only difference is that for planets, the phase angle depends on both the planet's and Earth's orbital positions, so it varies over a synodic period rather than a synodic month. The formula itself is universal — a geometric consequence of shading a sphere with parallel light and projecting the result onto a plane.

Two reasons. First, the Moon's orbit is elliptical, so its distance from Earth varies by about 12% between perigee and apogee; the angular size changes, and the phase at a given orbital angle shifts slightly. Second, the terminator — the line between the lit and dark hemispheres — is not perfectly sharp on the actual Moon; it is softened by the roughness of the lunar surface, by the finite angular size of the Sun as seen from the Moon (about half a degree, the same as from Earth), and by the fact that we see more illuminated area on crater rims and mountain peaks near the terminator. These effects are small — typically a percent or two — but they mean that the formula gives a very close approximation rather than an exact prediction of the visual appearance.

Brightness is roughly proportional to k but with a strong non-linear enhancement near Full Moon, called the opposition surge. Approximate apparent magnitudes: Full Moon −12.7, Gibbous −11.5, First Quarter −10.0, Crescent −8.0, New Moon (dark) around +0 as seen from Earth's night side. The Full Moon is about 12 times brighter than a quarter Moon, even though k is only twice as large. The extra brightening comes from the backscattering of sunlight by the lunar regolith — when the Sun is directly behind the observer, fine dust grains reflect light more efficiently than at any other angle.