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Antisolar point

The antisolar point is the location on the celestial sphere directly opposite the Sun from an observer's vantage on Earth, situated 180 degrees away from the Sun's position.^[1]^[2] During daylight hours, it lies below the horizon, and its position on the ground is marked by the shadow of the observer's head in an open area.^[1]^[2] In atmospheric optics, the antisolar point acts as the geometric center for a variety of visually striking phenomena caused by the interaction of sunlight with atmospheric particles such as water droplets and ice crystals.^[1] Primary rainbows appear as arcs approximately 42 degrees from the antisolar point, resulting from the refraction, internal reflection, and dispersion of sunlight within raindrops.^[3] Secondary rainbows, fainter and reversed in color order, form about 51 degrees from the same point after two internal reflections.^[3] Fogbows also circle the antisolar point under specific conditions involving diffraction. Higher-order rainbows (tertiary and above), involving more internal reflections, are centered on the Sun rather than the antisolar point.^[1] Glories—concentric, rainbow-like rings produced by diffraction and interference around cloud or fog droplets—manifest directly around the antisolar point and are commonly observed from elevated viewpoints like airplanes.^[4]^[1] The Brocken spectre, an optical illusion featuring an observer's enlarged, often haloed shadow cast on mist or clouds, projects toward the antisolar point, creating a dramatic effect on mountaintops or in valleys.^[1] Additionally, anticrepuscular rays—beams of light appearing to converge opposite the Sun—extend across the sky to meet at the antisolar point, enhancing twilight displays.^[5] These effects underscore the antisolar point's fundamental role in explaining light propagation and scattering in Earth's atmosphere.^[1]

Definition and Geometry

Geometric Definition

The antisolar point is defined as the location on the celestial sphere that lies exactly 180 degrees, or diametrically opposite, from the Sun's position as viewed from the observer's eye.^[6] This point represents an abstract direction in the sky, determined solely by the relative alignment of the observer, the Sun, and the infinite extent of the celestial sphere, rather than any physical object or fixed coordinate.^[7] Conceptually, the antisolar point marks the direction from which solar rays, after undergoing back-scattering or reflection by atmospheric particles or water droplets, appear to the observer to originate.^[8] This apparent origin forms the foundational geometry for phenomena involving light redirected toward the observer from the Sun's incoming path, emphasizing the point's role in the optics of opposition effects.^[9] The antisolar point's recognition traces back to early optical theories, notably in René Descartes' 1637 Discours de la Méthode, where he analyzed rainbow formation through ray tracing in water-filled globes, describing light deviations that concentrate rays toward the direction opposite the Sun—implicitly the antisolar point—yielding a primary bow angle of approximately 42 degrees.^[10] Descartes' quantitative approach, using the correct law of refraction, marked a pivotal advancement in understanding such geometries without modern terminology for the point itself.^[11] Unlike stars or other celestial bodies, the antisolar point is not a fixed entity in space but shifts dynamically with the observer's movement and the Sun's apparent position, always maintaining its oppositional relation.^[6] This observer-centric nature distinguishes it as a perspectival construct essential to atmospheric optics.

Position and Coordinates

The antisolar point is defined in the observer's local horizon coordinate system by transforming the Sun's position such that its azimuth A_{as} is A_s + 180^\circ (modulo $360^\circ), where A_s is the Sun's azimuth measured clockwise from north, and its altitude h_{as} is -h_s, with h_s being the Sun's altitude above the horizon.^[1]^[12] This positioning ensures the antisolar point lies directly opposite the Sun on the celestial sphere, maintaining an angular separation of exactly $180^\circ in ideal geometric models without atmospheric effects.^[1]^[13] In vector terms, the position of the antisolar point can be represented as the negative of the Sun's unit direction vector from the observer: \vec{r}_{as} = -\vec{r}_s, where \vec{r}_s is the unit vector pointing toward the Sun in a coordinate system aligned with the local zenith, north, and east.^[14] This geometric opposition implies that during daylight hours, when the Sun's altitude h_s > 0^\circ, the antisolar point is located below the horizon at altitude h_{as} < 0^\circ, rendering it invisible from ground level unless the observer is elevated.^[1] At night or precisely at sunset and sunrise, when h_s \leq 0^\circ, the antisolar point rises to or above the horizon.^[15] Atmospheric refraction introduces minor deviations from the exact $180^\circ separation, typically on the order of a few arcminutes near the horizon, but these effects are negligible for most positional calculations and are not derived in standard geometric models.^[1] The visibility of the antisolar point is further influenced by the Sun's elevation: when the Sun is at the zenith (h_s = 90^\circ), the antisolar point coincides with the nadir (h_{as} = -90^\circ) directly beneath the observer; conversely, when the Sun is on the horizon (h_s = 0^\circ), the antisolar point lies on the opposite horizon at the same altitude (h_{as} = 0^\circ).^[1]^[15]

Observation Methods

Ground-Level Viewing

From the ground, the antisolar point can be located by observing the shadow cast by the observer's head on a flat surface during clear daylight conditions, as this shadow's direction points directly toward the projection of the antisolar point on the Earth's surface.^[1]^[2] This method works because the antisolar point lies 180 degrees opposite the Sun in the sky, aligning the observer's shadow with the line from the Sun through the observer to the antisolar direction.^[1] This shadow technique also aids in identifying the center of circular rainbows, where extending the line of the observer's shadow intersects the rainbow's arc, confirming the antisolar point as the geometric center approximately 42 degrees away from the observer's eye level.^[16]^[17] However, ground-level observation faces significant limitations, as the antisolar point is typically positioned below the horizon during daytime when the Sun is above it, rendering direct viewing impossible without elevation; it serves primarily as a reference via shadows or associated low-angle atmospheric effects.^[15] At dawn or dusk, when the Sun is low, the antisolar point rises above or near the horizon, allowing indirect visibility through phenomena like the Belt of Venus or Earth's shadow cone.^[15]^[18] For more precise determination of the antisolar point's position, a compass can measure the azimuth of the observer's shadow to find the horizontal direction opposite the Sun, while the altitude angle below the horizon is the negative of the Sun's altitude, which can be measured separately using a clinometer (with appropriate solar filters for safety) or calculated from solar position data.^[19] Smartphone applications that calculate solar position, such as SunCalc or Sun Surveyor, enable inversion to derive the antisolar coordinates by adding 180 degrees to the solar azimuth and negating the solar altitude.^[20]^[21] For instance, if the Sun is at solar noon with an altitude of 60 degrees, the antisolar point would be at an azimuth of 240 degrees and an altitude of -60 degrees, placing it 60 degrees below the horizon in the opposite direction.^[19]

Aerial and Elevated Viewing

Observation of the antisolar point from aerial or elevated positions, such as aircraft, mountains, or high-altitude balloons, allows it to appear above the local horizon under certain conditions, unlike ground-level views where it remains hidden below the Earth's surface. The visibility arises because the observer's elevation causes a dip in the apparent horizon, effectively exposing more of the celestial sphere in the antisolar direction. Specifically, the antisolar point becomes visible above the local horizon when the horizon dip angle δ exceeds the Sun's altitude h_s; the dip is approximated by δ ≈ 0.03 √H degrees, where H is the observer's height in meters.^[22] For low solar altitudes, this requires modest heights—for instance, at H = 10,000 m (typical cruising altitude for commercial aircraft), δ ≈ 3°, enabling visibility when h_s < 3°. This geometric effect shifts the apparent position of the antisolar point slightly due to Earth's curvature, but the adjustment is negligible for altitudes below a few kilometers.^[22] Pilots and elevated observers frequently encounter the antisolar point directly in the sky opposite the Sun, where it serves as the convergence point for atmospheric rays or the center of optical bows visible against clouds or mist below. From aircraft, looking rearward reveals the point as the hub of phenomena like the aircraft's shadow, often surrounded by colorful rings when water droplets are present in the line of sight. Similar sightings occur from mountain summits, where the elevated vantage extends the visible sky, allowing direct alignment with the antisolar direction without ground obstruction.^[23]^[24] Historical observations from high-altitude balloons and early aircraft have documented full circular rainbows centered on the antisolar point, revealing the complete geometry unobscured by the horizon. For example, hot air balloon flights have captured 360° rainbows when mist or rain is positioned below the observer, with the antisolar point at the circle's core. Airplane passengers and pilots have similarly reported these full circles since the advent of commercial aviation, particularly during descents into layered clouds at dawn or dusk, highlighting the point's role as the optical center.^[24]^[1] Direct viewing of the antisolar point from elevated positions necessitates precautions to avoid Sun glare, as locating the direction opposite the Sun may involve brief glances toward the solar position; polarized sunglasses or indirect scanning methods are recommended to protect retinal health.^[23]

Reflection-Based Phenomena

Rainbows

The antisolar point serves as the geometric center for rainbow formation, where sunlight undergoes refraction, internal reflection, and dispersion in spherical water droplets suspended in the atmosphere. For a primary rainbow, incoming rays from the Sun enter a droplet, refract toward the normal, reflect once off the interior surface, refract again upon exiting, and deviate by a minimum angle of approximately 138° from the original direction, resulting in a cone of light with a radius of about 42° around the antisolar point. This deviation angle D for the primary rainbow is given by the equation

D = 180^\circ + 2i - 4r,

where i is the angle of incidence at the droplet surface, r = \sin^{-1}(\sin i / n) is the angle of refraction, and n \approx 1.333 is the refractive index of water for visible light; the minimum D occurs when \sin i = \sqrt{(4 - n^2)/3}. The dispersion of light into colors arises because n varies slightly with wavelength, yielding a red outer edge at ~42° and violet inner edge at ~40° from the antisolar point.^[25] A secondary rainbow forms through two internal reflections within the droplet, producing a total minimum deviation of approximately 231° and a cone radius of about 50° from the antisolar point, with colors reversed (violet outer, red inner) due to the additional reflection. The deviation angle for the secondary rainbow follows a similar form:

D = 360^\circ + 2i - 6r,

with the minimum at \sin i = \sqrt{(9 - n^2)/8}, resulting in a fainter arc positioned above the primary rainbow and sharing the same antisolar center. Supernumerary bows, appearing as faint, closely spaced arcs inside the primary rainbow, result from wave interference between rays exiting the droplet with slightly different path lengths, most prominent in uniform small droplets (~0.5 mm diameter).^[25] The angular geometry of a rainbow forms a cone with its apex at the observer's eye and axis aligned along the line from the eye to the antisolar point, ensuring the phenomenon appears as a circular arc centered on the observer's shadow. From ground level, only partial arcs are visible due to the horizon obstructing the lower portion, but full circular rainbows can be observed from aircraft or high elevations when the observer is above the rain layer. Color separation and bow sharpness depend on droplet uniformity and size; larger, irregular droplets (~1 mm) produce broader, less saturated colors via geometric optics, while uniform smaller droplets enhance interference effects for supernumerary bows. Between the primary and secondary rainbows lies Alexander's dark band, a region of suppressed intensity where no rays deviate to reach the observer, as scattering angles fall between ~42° and ~50° from the antisolar point.^[25]

Glory

The glory is a diffraction-based optical phenomenon that manifests as a series of concentric, iridescent rings centered precisely on the antisolar point, encircling the observer's shadow projected onto clouds or fog. It arises from the back-scattering of sunlight by small, nearly spherical water or ice droplets, where light diffracts around the droplet edges and interferes to produce the ring structure.^[26] The rings form due to constructive and destructive interference of these diffracted waves, with angular radii typically ranging from about 1° to 5°, influenced by the uniformity and size of the droplets involved.^[27] The coloration of the glory features red hues on the outer margins of each ring transitioning to blue nearer the center, a pattern driven by the wavelength dependence of the diffraction scattering angles—shorter blue wavelengths scatter at slightly larger angles than longer red ones.^[28] This chromatic arrangement results from the interference of surface waves propagating around the droplets in opposite directions. For example, the angular radius \theta of the innermost (primary) ring is approximately \theta \approx \frac{24^\circ}{r}, where r is the droplet radius in micrometers; consequently, smaller droplets yield larger, more expansive glories.^[29] Glories are most visible when the observer gazes toward a layer of clouds or fog directly opposite the Sun, ensuring alignment between the light source, observer, and shadow at the antisolar point, a configuration often achieved from elevated vantage points like mountains or, particularly, aircraft where the plane's shadow falls on underlying cloud decks.^[30] At the glory's core, an enhanced brightness occurs due to the opposition effect, wherein shadows within the droplet ensemble suppress multiple scattering, allowing direct back-scattered light to dominate and intensify the central aureole.^[31] This phenomenon was first documented in the 18th century by Spanish explorer Antonio de Ulloa, who observed it during the 1735–1744 French Geodesic Mission in the Andes, publishing his account in 1748 that highlighted its appearance around the observer's shadow, though its precise tie to antisolar geometry emerged from later optical analyses.^[32]

Shadow and Ray Phenomena

Brocken Spectre

The Brocken spectre is an optical illusion in which an observer's shadow is projected onto a layer of clouds or mist, appearing greatly enlarged and directed toward the antisolar point. This phenomenon occurs when the sun is low behind the observer, illuminating water droplets or ice crystals in the intervening clouds, which scatter the light and cast the shadow forward. The shadow's apparent gigantism arises from the perspective effect, as the uniform distance of the cloud layer from the observer creates an illusion of depth and scale without reference points for size.^[33]^[34] The illusion is visible when clouds or fog lie between the observer and the antisolar point, typically requiring elevated positions such as mountaintops with the sun at a low angle. It is frequently observed on peaks like the Brocken in Germany's Harz Mountains, from which the phenomenon derives its name, due to the area's persistent mists and accessibility. The spectre was first formally described in 1780 by Johann Silberschlag, a German naturalist who documented sightings during his visits to the Harz region.^[34]^[35]^[33] Geometrically, the shadow forms along rays parallel to the line connecting the sun and the observer, with the rays appearing to converge perspectivally at the antisolar point due to the observer's viewpoint. Multiple images of the spectre can arise from slight movements that alter the perspective or from light bouncing within the cloud layer, creating overlaid shadows. Sometimes, the shadow is accompanied by a glory, appearing as faint colored rings centered on it.^[33] Examples of the Brocken spectre are common among mountaineers in ranges like the Himalayas, Scottish Highlands, or Canary Islands, where low-lying fog enhances visibility. It can also be observed from aircraft flying above cloud layers with the sun behind, projecting the plane's shadow onto the clouds below. The eerie, humanoid form of the enlarged figure has historically evoked psychological unease, often interpreted as a ghostly apparition before its optical nature was understood.^[33]^[35]^[34]

Anticrepuscular Rays

Anticrepuscular rays, also referred to as antisolar rays, are beams of sunlight that appear to converge toward the antisolar point on the horizon opposite the Sun due to the linear perspective effect. These rays form when parallel shafts of sunlight pass through gaps in clouds or are scattered by atmospheric aerosols and particles, such as water droplets or dust, creating visible contrasts against the darker sky. Although the rays are physically parallel, the geometry of human vision causes them to seem to emanate from or meet at the distant antisolar point, analogous to how parallel railroad tracks appear to merge on the horizon.^[36]^[37] Visibility of anticrepuscular rays is optimal during dawn or dusk, particularly at sunset when the Sun's low altitude allows the beams to span the sky in a dramatic arch toward the opposite horizon. They are frequently observed near mountainous regions or in areas with distant cloud formations, such as towering cumulus or cumulonimbus clouds, which cast long shadows that enhance the rays' contrast. Atmospheric conditions favoring their appearance include light winds, high temperatures, and hazy skies with ample scattering particles; for instance, studies in central Oklahoma documented these rays on 76.5% of clear days under such conditions. To view them, an observer must turn away from the Sun and look toward the antisolar point, often requiring patience as the rays are fainter than their counterparts near the Sun.^[38]^[37] In distinction from crepuscular rays, which appear to fan outward from the Sun due to the same perspective illusion, anticrepuscular rays represent the opposite view of identical parallel beams, converging apparently at the antisolar point rather than diverging from the solar position. The low Sun angle during twilight intensifies their effect, and Mie scattering by atmospheric particles can tint the rays with hues like orange, pink, or blue, adding to their visual appeal. These phenomena have been depicted in 19th-century art, such as works by the British painter John Constable who captured their dramatic convergence, reflecting an emerging scientific understanding of the perspective-based optics involved.^[36]^[37]^[39]

Anthelic Point

The anthelic point is the location on the celestial sphere directly opposite the Sun in azimuth—specifically, at an azimuth angle of A_s + 180^\circ, where A_s is the Sun's azimuth—but at the same elevation as the Sun, h_s, rather than the negative elevation of the antisolar point. This distinguishes it from the antisolar point, which lies below the horizon when the Sun is above it. The anthelic point plays a central role in antisolar halo phenomena, serving as the geometric center for arcs and spots formed by light rays deviated by nearly 180° through atmospheric ice crystals.^[9]^[13] A key feature associated with the anthelic point is the anthelion, a faint, diffuse white patch or bright spot that appears at or very near this location on the parhelic circle. The anthelion forms primarily through two mechanisms involving hexagonal ice crystals: external reflection from the lower (bottom) faces of horizontally oriented plate crystals, which redirects sunlight toward the observer after a near-180° deviation, or low-order refraction (typically single or double internal reflections combined with refractions) in vertically oriented column crystals, also yielding deviations close to 180°. These processes contrast with the smaller deviation angles (around 22° or 46°) that produce solar halos, resulting in the anthelion's position far from the Sun. Seminal analyses confirm that such ray paths in pencil-shaped or plate-like crystals concentrate light at the anthelic point, creating the observed brightness despite the phenomenon's rarity.^[40] Visibility of the anthelic point and anthelion is optimal when the Sun's elevation is low (below about 30°), positioning the anthelic point near the horizon where overlapping rays from oriented crystals in high-altitude cirrus clouds are more discernible against the brighter sky background. At higher solar elevations, the point rises, but the required crystal orientations become less common, dimming the display. Anthelic arcs, such as the Wegener arcs (formed by multiple internal reflections in column crystals) or Tricker arcs (involving plate crystals), often converge toward the anthelic point, enhancing its prominence in complex halo simulations and observations. These arcs typically span 20°–30° angular distances from the point, emphasizing its focal role in 180°-deviation halos.^[41]^[42]

Subanthelion

The subanthelion is the point on the subparhelic circle directly opposite the Sun in azimuth and at an altitude equal to the negative of the Sun's altitude, positioned below the horizon when the Sun is elevated. It corresponds to the projection of the antisolar point onto this lower horizontal circle and serves as the convergence point for various subanthelic arcs or spots in atmospheric halo displays.^[43]^[44] This phenomenon manifests as a rare bright spot resulting from sunlight undergoing multiple internal reflections and refractions in column-oriented ice crystals, producing a total deviation of approximately 180° and directing light back toward the antisolar region. The spot typically appears 10–20° below the horizon when the Sun is at a similar elevation above it, often accompanied by faint subanthelic arcs formed by Parry-oriented crystals with nearly horizontal principal axes and two horizontal prism faces. These arcs emerge from complex ray paths, such as those involving prism face entries and exits, contributing to the overall brightness at the subanthelion.^[44]^[45]^[46] Visibility of the subanthelion is limited due to its subhorizon location, making it prone to obstruction by terrain or atmospheric haze, and it is usually faint unless enhanced by overlapping arcs or optimal crystal orientations. Observations are most feasible from elevated vantage points, such as mountains, or especially from aircraft, where the nadir view avoids ground blockage. It remains elusive in ground-based viewing without such advantages, appearing only intermittently in displays rich with other halos.^[45]^[47] Geometrically, the subanthelion lies along the vertical antisolar meridian—the great circle passing through the zenith, nadir, Sun, and antisolar point—and specifically on the extension of the vertical circle that intersects the anthelic point above the horizon. This alignment toward the nadir underscores its role as a focal point for downward-directed light scattering in the antisolar hemisphere.^[41]^[43] Notable examples include sightings at the South Pole on January 11, 1999, where diamond dust produced a prominent subanthelion amid a complex halo display with sundogs and 120° parhelia, captured via fisheye photography. Aerial observations, such as during a 2009 flight from Helsinki to Prague revealing a subanthelion with subparhelic circle segments, and a 2011 transatlantic flight from Paris to Washington DC showing it with diffuse subanthelic arcs, highlight its occurrence in polar and mid-latitude cirrus layers during favorable ice crystal conditions.^[41]^[45]^[47]

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