= Lunar phase
{wiki}
= Moon phase
{synonym}
A <lunar phase> is the appearance of the sunlit portion of the <Moon> as seen from <Earth>. Outside a <lunar eclipse>, the distant <Sun> illuminates approximately one hemisphere of the <Moon>, but an observer sees a changing overlap of that hemisphere with the hemisphere facing <Earth>. The <lunar phase> therefore changes even though <synchronous rotation> keeps approximately the same <near side of the Moon> facing <Earth>. https://science.nasa.gov/moon/moon-phases/[NASA: Moon phases and illumination geometry]{ref}.
The cycle runs through new Moon, waxing crescent, first quarter, waxing gibbous, full Moon, waning gibbous, last quarter and waning crescent. “Waxing” means that the <lunar illuminated fraction> is increasing; “waning” means it is decreasing. A crescent has less than half its apparent disk sunlit, and a gibbous <lunar phase> has more than half. First and last quarter each show half a disk: “quarter” refers to progress around the cycle, rather than a quarter of the disk being lit. https://aa.usno.navy.mil/data/MoonFraction[US Naval Observatory: disk fractions and waxing versus waning]{ref}.
An instructive <circular orbit> model puts <Earth> at the origin, the <Sun> far away along $+x$, and the <Moon> at $R(\cos\theta,\sin\theta)$. The sunward position $x>0$, $\theta=0$, gives new Moon; the opposite position $x<0$, $\theta=\pi$, gives full Moon. Counterclockwise motion gives first quarter at $\theta=\pi/2$ and last quarter at $\theta=3\pi/2$. The <Sun>'s direction is fixed during this snapshot of the cycle. Every small globe in the orbital view is illuminated on its $+x$ half; the adjacent apparent disks are separate views from <Earth>, rather than globes viewed from above.
\Image[/lunar-phases-orbit-and-disks.png]
{title=Eight lunar phases in a circular orbit and the corresponding apparent disks from Earth}
The disk views choose north up, with the upward direction perpendicular to the <orbital plane>. In this convention waxing <lunar phases> are bright on the right and waning <lunar phases> on the left. Local horizon orientation and viewing from different locations on <Earth> can rotate the apparent disk; its lit edge always points toward the projected direction of the <Sun>. The <lunar illuminated fraction> is unaffected by this rotation.
The useful <angle> is the <phase angle> $\alpha=\angle SME$, with $S$ the <Sun>, $M$ the <Moon> and $E$ the observer at <Earth>'s center: its vertex is at the <Moon>. Full Moon has $\alpha=0$, quarter has $\alpha=\pi/2$, and new Moon has $\alpha=\pi$ in the ideal model. By contrast, the <elongation> $\epsilon=\angle SEM$ has its vertex at <Earth>. For a distant <Sun>, its rays are parallel, so
$$
\alpha=\pi-\epsilon,\qquad k=\frac{1+\cos\alpha}{2}=\frac{1-\cos\epsilon}{2}.
$$
Here $k$ is the <lunar illuminated fraction>: a fraction of projected disk <area>, not a fraction of the <Moon>'s total <surface area> or its received <radiant flux> relative to full Moon. The <phase angle> and unsigned <elongation> each lie between $0$ and $\pi$, so either alone gives the illuminated amount without distinguishing waxing from waning. The orbital direction resolves that ambiguity.
\Image[/lunar-phase-illuminated-fraction.png]
{title=Projected illuminated fraction versus the Sun–Moon–observer phase angle at the Moon}
The <lunar illuminated fraction> has an elegant geometric derivation. Model the <Moon> as a <sphere> of unit radius and view it along the $z$ direction. Take the <Sun>'s unit direction to be $(\sin\alpha,0,\cos\alpha)$. A visible surface point above apparent disk coordinates $(x,y)$ has $z=\sqrt{1-x^2-y^2}$. Direct illumination requires the outward normal's <dot product> with the sunward direction to be positive. The <day-night terminator> therefore obeys $x\sin\alpha+z\cos\alpha=0$. At a fixed $y$, set $q=\sqrt{1-y^2}$; the visible <day-night terminator> is $x_t=-q\cos\alpha$, and the illuminated strip extends from $x_t$ to the right edge $q$. Integrating its width gives
$$
A_{\mathrm{lit}}=\int_{-1}^{1}(1+\cos\alpha)\sqrt{1-y^2}\,dy
=\frac{\pi}{2}(1+\cos\alpha),\qquad
k=\frac{A_{\mathrm{lit}}}{\pi}.
$$
The <integral> is the <area> of half a unit <circle>. Thus a three-dimensional visibility problem reduces to a <circle>'s <area> and a <cosine>. The derivation assumes parallel sunlight, a distant observer and a <sphere>, and ignores a <lunar eclipse>. Real observed brightness additionally depends on surface <scattering of light> and observing geometry, so $k=1/2$ does not promise half the full-Moon <radiant flux>.
For a worked example, <elongation> $\epsilon=60^\circ$ gives <phase angle> $\alpha=120^\circ$ and $k=(1+\cos120^\circ)/2=1/4$: a crescent with $25\%$ of its apparent disk sunlit. At <elongation> $120^\circ$, the <phase angle> is $60^\circ$ and $k=3/4$: a gibbous <lunar phase>. At $90^\circ$, $k=1/2$. In each case the same $k$ occurs once while waxing and once while waning in the ideal cycle.
Ordinary <lunar phases> are not caused by <Earth>'s shadow. A <lunar eclipse> occurs only when the <Moon> actually enters that shadow near full Moon. The <orbital inclination> of the <Moon> is about $5^\circ$ relative to <Earth>'s <orbital plane> around the <Sun>; most full Moons pass above or below the shadow. This is why full Moon does not imply a <lunar eclipse> every month. https://science.nasa.gov/moon/eclipses/[NASA: lunar eclipses and the Moon's roughly five-degree orbital tilt]{ref}.
The phase cycle is a <synodic month>, about $29.5$ days, whereas the <Moon>'s <orbital period> relative to distant stars is a <sidereal month>, about $27.3$ days. During one <sidereal month>, <Earth> advances around the <Sun>, so the <Moon> must move farther to repeat the sunward alignment. For coplanar <circular orbits> traversed in the same direction, subtract their <mean motions>: $n_{\mathrm{syn}}=n_{\mathrm{sid}}-n_{\mathrm{year}}$, with $n=2\pi/P$. Hence
$$
\frac{1}{P_{\mathrm{syn}}}=\frac{1}{P_{\mathrm{sid}}}-\frac{1}{P_{\mathrm{year}}},\qquad
P_{\mathrm{syn}}=\frac{P_{\mathrm{sid}}P_{\mathrm{year}}}{P_{\mathrm{year}}-P_{\mathrm{sid}}}.
$$
Using $P_{\mathrm{sid}}=27.321661\,\mathrm{days}$ and $P_{\mathrm{year}}=365.256\,\mathrm{days}$ gives $P_{\mathrm{syn}}=29.530591\,\mathrm{days}$, consistent with the mean <synodic month> of $29.53059$ days. After one <sidereal month>, the sunward direction has advanced about $26.93^\circ$; catching up takes about $2.21$ more days. These are mean-period estimates: <orbital eccentricity> and changing geometry make individual <synodic months> vary. https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html[Fred Espenak, NASA: sidereal and synodic months and varying lunations]{ref}. https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html[David R. Williams, NASA: Earth's sidereal orbital period]{ref}.
The <lunar phase> is also a practical clock for observing. Full Moon is opposite the <Sun> and rises roughly at sunset; first quarter is about a <right angle> from the <Sun> and is convenient in the evening, while last quarter is convenient before dawn. These are rough guides, not exact rise-time predictions for every place and season. https://science.nasa.gov/moon/moon-phases/[NASA: approximate visibility times by lunar phase]{ref}. Historically, repeated <lunar phases> provided the month in lunar calendars, while half-lit disks gave <Aristarchus of Samos> a way to infer astronomical distances. https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html[Fred Espenak, NASA: lunations and lunar calendars]{ref}.
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