Lunar phase
Lunar phases are the cyclical changes in the visible shape of the Moon as observed from Earth, resulting from the Moon's orbit around Earth and the relative positions of the Earth, Moon, and Sun.[1] These phases occur because the Sun illuminates only one half of the Moon at a time, and the portion visible from Earth varies as the Moon completes its approximately 29.5-day synodic cycle.[2] The cycle begins with the new moon, when the Moon is positioned between the Earth and Sun, making its illuminated side face away from Earth and rendering it invisible to observers.[1] As the Moon orbits Earth, the phases progress through eight primary stages: following the new moon comes the waxing crescent, a thin sliver of light visible on the right side (from the Northern Hemisphere perspective); then the first quarter, where half the Moon is illuminated and it rises around noon; the waxing gibbous, with more than half but less than fully illuminated; the full moon, when the entire Earth-facing side is lit and it rises at sunset; the waning gibbous, shrinking from full; the third quarter, half-illuminated on the left and rising around midnight; and finally the waning crescent, a diminishing sliver before returning to new.[2] This sequence is influenced by the Moon's tidal locking to Earth, meaning the same side always faces our planet, and the orbital period of about 27.3 days for one rotation relative to the stars, extended to 29.5 days due to Earth's own motion around the Sun.[1] Lunar phases have been observed and documented across cultures for millennia, influencing calendars, navigation, and rituals, while modern astronomy uses them to study the Moon's surface and orbital dynamics.[1] They are distinct from eclipses, which occur only when the Sun, Earth, and Moon align precisely during new or full phases.[2] The average Earth-Moon distance of 238,855 miles (384,399 km) ensures these phases are visible worldwide, though the exact appearance varies slightly by latitude.[1]Fundamentals
Definition and Phenomenon
Lunar phases refer to the cyclical variations in the portion of the Moon's illuminated disk that is visible from Earth, arising from the changing angle between the Sun, Earth, and Moon.[1] These phases manifest as the apparent shape of the Moon shifting over time, creating a sequence of distinct appearances observable in the night sky.[3] The Moon produces no light of its own but reflects sunlight from its surface, with only the half facing the Sun illuminated at any time.[1] From Earth's perspective, the visible illuminated fraction changes because the Moon orbits Earth, altering the alignment of the three bodies; this cycle repeats approximately every 29.5 days, known as the synodic month.[4] During this period, the Moon's position relative to the Sun determines how much of its lit side observers see, from none to fully illuminated. Records of lunar phases date back to Stone Age peoples, who tracked the cycle to measure days and predict seasonal changes.[5] Ancient civilizations, including the Babylonians and Egyptians, observed these phases for practical purposes such as agriculture, navigation, and developing calendars.[6][7] The visual progression begins with the new moon, when the Moon is nearly invisible as it aligns between Earth and the Sun.[1] It then transitions to the waxing crescent, a thin illuminated sliver growing toward the first quarter, where half the disk is lit.[3] This continues through the waxing gibbous phase to the full moon, when the entire visible disk glows brightly opposite the Sun.[1] The cycle then reverses with the waning gibbous, last quarter, waning crescent, returning to new moon.[3]Cause of Lunar Phases
The lunar phases result from the geometry of the Sun-Earth-Moon system, where the Moon reflects sunlight but appears to change shape due to the varying portion of its illuminated hemisphere visible from Earth. The Moon is always half-illuminated by the Sun, similar to how half of Earth is lit during daytime, but the observer on Earth sees different fractions depending on the relative positions. This visibility is determined by the phase angle, defined as the elongation between the Moon and the Sun as viewed from Earth.[1][8] When the phase angle is 0°, the Moon and Sun share the same ecliptic longitude from Earth's perspective, positioning the Moon's illuminated side toward the Sun and away from Earth, resulting in a new moon that is invisible or nearly so. Conversely, at a phase angle of 180°, the Moon is directly opposite the Sun, with its fully illuminated hemisphere facing Earth, producing a full moon. Intermediate angles yield partial illuminations, such as crescent or gibbous appearances.[1][8] The Moon's orbit around Earth is prograde—counterclockwise when viewed from the north side of the ecliptic plane—and nearly coplanar with the ecliptic, inclined by a mean of 5.145° relative to Earth's orbital plane around the Sun. This configuration, combined with Earth's simultaneous revolution around the Sun, defines the synodic month of 29.53059 days as the period for the Moon to return to the same phase, longer than the sidereal orbital period of 27.32166 days because the reference point (the Sun) advances during that time.[8][9] Tidal interactions have caused the Moon to become tidally locked, with its rotational period synchronized to its orbital period, always showing the same face to Earth; however, the phases themselves arise independently from the illumination geometry and would occur even without this locking. The terminator—the great circle boundary separating the Moon's sunlit and shadowed hemispheres—shifts across the visible disk based on the observer's line of sight, appearing as a straight line at quarter phases (90° angle) and curving at other elongations. In diagrams of the system, the terminator is depicted as the edge where incoming sunlight grazes the lunar surface tangent to Earth's viewpoint, highlighting how the changing alignment alters the shadowed fraction.[1][8]Types of Phases
Principal Phases
The principal lunar phases consist of four key stages in the Moon's cycle as observed from Earth: the New Moon, First Quarter, Full Moon, and Last Quarter. These phases mark the moments when the Moon's ecliptic elongation—the angular separation between the Sun and Moon as seen from Earth—is at 0°, 90°, 180°, and 270°, respectively. They represent the primary divisions of the synodic month, which averages 29.53 days, with each principal phase separated by roughly one-quarter of this period.[1][10] The New Moon begins the cycle, occurring when the Moon lies directly between the Earth and the Sun in conjunction, with its illuminated side facing away from Earth. At this phase, the Moon appears invisible from Earth due to 0% illumination on the side facing our planet, though it may be visible as a dark silhouette during a total solar eclipse. It rises and sets with the Sun, making it unobservable against the daytime sky.[1][11] Approximately 7.4 days after the New Moon, the First Quarter phase arrives, with the Moon at a 90° elongation east of the Sun. From the Northern Hemisphere, the right half of the Moon's disk appears illuminated at 50%, as sunlight illuminates the side facing Earth while the Moon is positioned to the east in its orbit. This half-lit Moon rises around noon and sets around midnight, becoming prominent in the evening sky.[12][10] The Full Moon occurs about 14.8 days after the New Moon, when the Moon reaches 180° elongation in opposition to the Sun. The entire visible disk is illuminated at 100%, with the fully lit side facing Earth as the Moon is on the opposite side of our planet from the Sun. It rises at sunset and sets at sunrise, providing bright nighttime illumination.[1][12] Roughly 22.1 days into the cycle, the Last Quarter (also known as Third Quarter) phase takes place at 270° elongation, with the Moon 90° west of the Sun. In the Northern Hemisphere, the left half of the disk is illuminated at 50%, reflecting the waning portion of the cycle. This phase rises around midnight and sets around noon, visible primarily in the morning sky.[10][12] The terms "First Quarter" and "Last Quarter" derive from the Moon's position in its orbit, dividing it into quadrants relative to the Sun-Earth line, rather than indicating a 25% illumination fraction—these phases actually show 50% of the disk lit due to the geometry of illumination.[1][11]Intermediate Phases
The intermediate phases of the Moon occur between the principal phases and are characterized by gradual changes in the visible illuminated portion of the lunar disk, transitioning from less than 50% to more than 50% illumination and vice versa. These phases are divided into crescent and gibbous categories based on the fraction of the Moon's Earth-facing hemisphere that is illuminated by the Sun: crescent phases feature less than 50% illumination, while gibbous phases exceed 50% but fall short of 100%. The principal quarter phases mark exact boundaries at 50% illumination.[13][1] Following the new moon, the waxing crescent phase emerges as a thin, illuminated sliver on the Moon's right side (as viewed from the Northern Hemisphere), with illumination progressively increasing but remaining below 50%. This phase becomes visible shortly after sunset in the western sky, as the angle between the Sun, Earth, and Moon allows a small portion of the sunlit lunar surface to face Earth.[3][1][14] After the first quarter phase, the Moon enters the waxing gibbous stage, where more than 50% but less than 100% of the disk is illuminated, appearing as a bulging, humpbacked shape that continues to grow brighter each night. The term "gibbous" derives from the Latin word for "hunchbacked," reflecting the convex form of the illuminated region. This phase is prominent in the evening sky, rising in the southeast and remaining visible for most of the night.[3][1][15] Symmetrically, after the full moon, the waning gibbous phase mirrors the waxing gibbous, with illumination decreasing from over 50% toward 50% as the Moon's position shifts. The illuminated portion appears on the left side (Northern Hemisphere view), gradually shrinking while still dominating more than half the disk, and the Moon rises later each night after sunset.[3][1][14] Finally, the waning crescent phase precedes the new moon, presenting a thin, fading sliver of light on the left side with less than 50% illumination, often barely discernible except near dawn. It becomes visible low in the eastern sky before sunrise, as the Moon approaches alignment with the Sun from Earth's perspective.[3][1][14]Waxing and Waning Cycles
The lunar phase cycle, known as a lunation or synodic month, begins at the new moon, when the Moon is in conjunction with the Sun as viewed from Earth, and progresses through a sequence of increasing and decreasing illumination over an average duration of 29.53 days.[11] During the first half, the illuminated portion of the Moon's visible disk "waxes," or grows, from a thin crescent to the full moon at opposition, approximately 14.77 days later on average.[11] In the second half, the illumination "wanes," or diminishes, symmetrically in reverse through gibbous and crescent stages back to the new moon, completing the cycle at the next conjunction.[11] The term "waxing" derives from the Old English verb weaxan, meaning "to grow" or "increase," reflecting the apparent expansion of the lit area, while "waning" comes from wanian, meaning "to decrease" or "become smaller," describing the subsequent shrinkage.[16] Although the waxing and waning phases mirror each other in the progression of illumination—from 0% to 100% and back to 0%—the cycle lacks perfect symmetry due to the Moon's elliptical orbit around Earth, which causes variations in its orbital speed. The Moon moves faster near perigee (its closest point to Earth) and slower near apogee (farthest point), altering the time required to traverse equal angular separations relative to the Sun; as a result, the interval from new moon to full moon can differ from the return interval by up to about 1.5 days, typically ranging from 13.9 to 15.2 days for each half.[11] This orbital eccentricity, with an average value of 0.0549, shifts the timing slightly each month, ensuring the waning phase often lags or leads the waxing by 1 to 2 days depending on the Moon's position at conjunction.[8] The overall length of the synodic month also varies seasonally due to this eccentricity and the combined motion of Earth and Moon around the Sun, fluctuating between approximately 29.27 and 29.83 days.[17] At perigee, the Moon's increased speed hastens phase changes, shortening the cycle, while at apogee, slower motion extends it, with extremes occurring when conjunction aligns near these orbital points. These variations, though small, influence the precise timing of phases and have been accounted for in astronomical calculations since ancient times.[11]Calculation Methods
Determining Phase Angle
The lunar phase angle is defined as the angle between the ecliptic longitudes of the Moon and the Sun as observed from Earth, representing the geocentric elongation that determines the Moon's apparent illumination cycle.[11] This angle, denoted E, ranges from 0° at new moon, when the Moon and Sun share nearly the same ecliptic longitude, to 180° at full moon, when they are separated by half a circle along the ecliptic; it then increases from 180° to 360° over the subsequent half-cycle, completing the synodic month.[11] The full elongation (0° to 360°) distinguishes waxing from waning phases, while the principal value (0° to 180°) serves as a fundamental parameter for identifying the Moon's position in its orbital cycle relative to the Sun.[18] To calculate the phase angle, astronomers first convert the desired date and time to Julian date, a continuous count of days since a fixed epoch, which standardizes temporal computations in celestial mechanics. Using this, the mean anomaly of the Moon (its angular position relative to its last perigee) and the mean anomaly of the Sun are derived through low-precision approximations or higher-order ephemerides; these anomalies, combined with orbital elements like eccentricity and inclination, yield the ecliptic longitudes λ_moon and λ_sun. The phase angle E is then computed as the difference E = λ_moon - λ_sun, adjusted by adding or subtracting 360° if necessary to obtain a value between 0° and 360°. A widely adopted algorithm for these calculations is the Meeus method, outlined in Astronomical Algorithms, which provides step-by-step polynomial approximations for solar and lunar positions accurate to about 0.1° over centuries without requiring full ephemeris tables. The process begins with the Julian date to compute the number of centuries past J2000.0, then applies series expansions: for the Sun, longitude is approximated using terms involving the mean anomaly and Earth's orbital eccentricity; for the Moon, it incorporates the mean anomaly, the longitude of the ascending node, and perturbations from the Sun and planets, truncated for practicality to a few dozen terms. The resulting longitudes are differenced to yield the elongation E; this method underpins many computational tools and achieves sufficient precision for phase determination except near eclipses. Historically, ancient civilizations determined lunar phases primarily through direct observation rather than mathematical calculation, tracking the Moon's nightly position relative to the Sun and fixed stars to predict cycles for calendars and agriculture.[19] Stone Age peoples etched phase sequences on bones and cave walls as early as 30,000 years ago, while Mesopotamians and Egyptians around 2000 BCE maintained observational records to align festivals with new moons, relying on visibility thresholds without quantitative angular measures.[19] In modern practice, phase angles are computed using astronomical software that integrates Meeus-style algorithms or precise ephemerides from sources like the Jet Propulsion Laboratory's DE430 series, enabling real-time predictions for any date. Tools such as the U.S. Naval Observatory's data services or open-source libraries implement these routines, outputting phase angles to arcminute accuracy for applications in astronomy and navigation.[20]Calculating Illuminated Fraction
The illuminated fraction of the Moon's disk, denoted as k, represents the proportion of the visible lunar surface directly lit by the Sun, ranging from 0 (completely dark at new moon) to 1 (fully illuminated at full moon). This value is derived from the geocentric elongation E (as calculated above), with the selenocentric phase angle \phi approximately equal to $180^\circ - E under the assumption of parallel solar rays (valid since the Sun-Earth distance greatly exceeds the Earth-Moon distance). The standard simple formula for k is k = \frac{1 - \cos E}{2} for $0^\circ \leq E \leq 180^\circ, or equivalently using the selenocentric \phi, k = \frac{1 + \cos \phi}{2} for $0^\circ \leq \phi \leq 180^\circ.[21] This expression arises from the projected geometry of the illuminated hemisphere onto the observer's line of sight, where the terminator (boundary between light and shadow) divides the disk such that the lit portion's area fraction equals the average over the spherical surface projection. At the extremes, E = 0^\circ yields \cos 0^\circ = 1, so k = 0 (0% illuminated); E = 180^\circ gives \cos 180^\circ = -1, resulting in k = 1 (100% illuminated). For intermediate values, such as the first or last quarter phase where E = 90^\circ, \cos 90^\circ = 0, yielding k = 0.5 (50% illuminated).[21] The visible disk area illuminated is exactly proportional to k in this spherical model. The full moon appears about 6–9 times brighter than at quarter phase due to both this fraction and the opposition effect (enhanced backscattering near full phase). While the terminator traces an ellipse in projection (with eccentricity depending on E), the integrated illuminated area fraction simplifies exactly to k without requiring elliptic integrals, as the projection symmetry preserves the linear relation for a uniform sphere.[11] To compute k for a specific date, first obtain the geocentric elongation E (as detailed in the phase angle determination), then substitute into the formula; for more precision near the limb or accounting for finite distances, use the full selenocentric calculation with ephemeris data. For example, on a date with E \approx 120^\circ (waxing gibbous), \cos 120^\circ = -0.5, so k = (1 - (-0.5))/2 = 0.75 (75% illuminated). A simple step-by-step calculation or pseudocode implementation might proceed as follows:- Input the geocentric elongation E in degrees (e.g., from astronomical software or ephemeris).
- Convert to radians if needed: E_{\text{rad}} = E \times \pi / 180.
- Compute k = (1 - \cos E_{\text{rad}}) / 2.
- Multiply by 100 for percentage.
This yields 50.0% for E = 90^\circ.[21] This geocentric approximation assumes a perfect sphere and neglects libration, which causes apparent rocking of the Moon and can alter the visible illuminated portion by up to a few percent; for higher precision, especially near the limb, use selenocentric coordinates that incorporate the Moon's instantaneous orientation relative to the observer.[11]E_deg = 90 # example geocentric elongation E_rad = E_deg * pi / 180 k = (1 - cos(E_rad)) / 2 print(f"Illuminated fraction: {k * 100:.1f}%")E_deg = 90 # example geocentric elongation E_rad = E_deg * pi / 180 k = (1 - cos(E_rad)) / 2 print(f"Illuminated fraction: {k * 100:.1f}%")