Angular distance in the context of Geophysics

Celestial navigation, also known as astronavigation, is the practice of position fixing using stars and other celestial bodies that enables a navigator to accurately determine their actual current physical position in space or on the surface of the Earth without relying solely on estimated positional calculations, commonly known as dead reckoning. Celestial navigation is performed without using satellite navigation or other similar modern electronic or digital positioning means.

Celestial navigation uses "sights," or timed angular measurements, taken typically between a celestial body (e.g., the Sun, the Moon, a planet, or a star) and the visible horizon. Celestial navigation can also take advantage of measurements between celestial bodies without reference to the Earth's horizon, such as when the Moon and other selected bodies are used in the practice called "lunars" or the lunar distance method, used for determining precise time when time is unknown.

Right ascension (abbreviated RA; symbol α) is the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the (hour circle of the) point in question above the Earth. When paired with declination, these astronomical coordinates specify the location of a point on the celestial sphere in the equatorial coordinate system.

An old term, right ascension (Latin: ascensio recta) refers to the ascension, or the point on the celestial equator that rises with any celestial object as seen from Earth's equator, where the celestial equator intersects the horizon at a right angle. It contrasts with oblique ascension, the point on the celestial equator that rises with any celestial object as seen from most latitudes on Earth, where the celestial equator intersects the horizon at an oblique angle.

A sextant is a doubly reflecting navigation instrument that measures the angular distance between two visible objects. The primary use of a sextant is to measure the angle between an astronomical object and the horizon for the purposes of celestial navigation.

The estimation of this angle, the altitude, is known as sighting or shooting the object, or taking a sight. The angle, and the time when it was measured, can be used to calculate a position line on a nautical or aeronautical chart—for example, sighting the Sun at noon or Polaris at night (in the Northern Hemisphere) to estimate latitude (with sight reduction). Sighting the height of a landmark can give a measure of distance off and, held horizontally, a sextant can measure angles between objects for a position on a chart. A sextant can also be used to measure the lunar distance between the moon and another celestial object (such as a star or planet) in order to determine Greenwich Mean Time and hence longitude.

In celestial navigation, lunar distance, also called a lunar, is the angular distance between the Moon and another celestial body. The lunar distances method uses this angle and a nautical almanac to calculate Greenwich time if so desired, or by extension any other time. That calculated time can be used in solving a spherical triangle. The theory was first published by Johannes Werner in 1524, before the necessary almanacs had been published. A fuller method was published in 1763 and used until about 1850 when it was superseded by the marine chronometer. A similar method uses the positions of the Galilean moons of Jupiter.

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry.

The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another.

Polaris is a star in the northern circumpolar constellation of Ursa Minor. It is designated α Ursae Minoris (Latinized to Alpha Ursae Minoris) and is commonly called the North Star. With an apparent magnitude that fluctuates around 1.98, it is the brightest star in the constellation and is readily visible to the naked eye at night. The position of the star lies less than 1° away from the north celestial pole, making it the current northern pole star. The stable position of the star in the Northern Sky makes it useful for navigation.

Although appearing to the naked eye as a single point of light, Polaris is a triple star system, composed of the primary, a yellow supergiant designated Polaris Aa, in orbit with a smaller companion, Polaris Ab; the pair is almost certainly in a wider orbit with Polaris B. The outer companion B was discovered in August 1779 by William Herschel, with the inner Aa/Ab pair only confirmed in the early 20th century.

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry.

The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance. For example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another.

A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called spherical distance.

The size of a central angle Θ is 0° < Θ < 360° or 0 < Θ < 2π (radians). When defining or drawing a central angle, in addition to specifying the points A and B, one must specify whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°). Equivalently, one must specify whether the movement from point A to point B is clockwise or counterclockwise.

Appulse is the least apparent distance between one celestial object and another, as seen from a third body during a given period. Appulse is seen in the apparent motion typical of two planets together in the sky, or of the Moon to a star or planet while the Moon orbits Earth, as seen from Earth. An appulse is an apparent phenomenon caused by perspective only; the two objects involved are not necessarily near in physical space.

An appulse is related to a conjunction, but the definitions differ in detail. While an appulse occurs when the apparent separation between two bodies is at its minimum, a conjunction occurs at the moment when the two bodies have the same right ascension or ecliptic longitude. In general, the precise time of an appulse will be different from that of a conjunction.

Reflecting instruments are those that use mirrors to enhance their ability to make measurements. In particular, the use of mirrors permits one to observe two objects simultaneously while measuring the angular distance between the objects. While reflecting instruments are used in many professions, they are primarily associated with celestial navigation as the need to solve navigation problems, in particular the problem of the longitude, was the primary motivation in their development.

In astrology, an aspect is an angle that planets make to each other in the horoscope; as well as to the Ascendant, Midheaven, Descendant, Lower Midheaven, and other points of astrological interest. As viewed from Earth, aspects are measured by the angular distance in degrees and minutes of ecliptic longitude between two points. According to astrological tradition, they indicate the timing of transitions and developmental changes in the lives of people and affairs relative to the Earth.

For example, if an astrologer creates a Horoscope that shows the apparent positions of the celestial bodies at the time of a person's birth (Natal Chart), and the angular distance between Mars and Venus is 92° ecliptic longitude, the chart is said to have the aspect "Venus Square Mars" with an orb of 2° (i.e., it is 2° away from being an exact Square; a Square being a 90° aspect). The more exact an aspect, the stronger or more dominant it is said to be in shaping character or manifesting change.

A milliradian (SI-symbol mrad, sometimes also abbreviated mil or mils) is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Milliradians are used in adjustment of firearm sights by adjusting the angle of the sight compared to the barrel (up, down, left, or right). Milliradians are also used for comparing shot groupings, or to compare the difficulty of hitting different sized shooting targets at different distances. When using a scope with both mrad adjustment and a reticle with mrad markings (called an "mrad/mrad scope"), the shooter can use the reticle as a ruler to count the number of mrads a shot was off-target, which directly translates to the sight adjustment needed to hit the target with a follow-up shot. Optics with mrad markings in the reticle can also be used to make a range estimation of a known size target, or vice versa, to determine a target size if the distance is known, a practice called "milling".

Milliradians are generally used for very small angles, which allows for very accurate mathematical approximations to more easily calculate with direct proportions, back and forth between the angular separation observed in an optic, linear subtension on target, and range. In such applications it is useful to use a unit for target size that is a thousandth of the unit for range, for instance by using the metric units millimeters for target size and meters for range. This coincides with the definition of the milliradian where the arc length is defined as ⁠1/1,000⁠ of the radius. A common adjustment value in firearm sights is 1 cm at 100 meters which equals ⁠10 mm/100 m⁠ = ⁠1/10⁠ mrad.

A sun dog (or sundog) or mock sun, also called a parhelion (plural parhelia) in atmospheric science, is an atmospheric optical phenomenon that consists of a bright spot to one or both sides of the Sun. Two sun dogs often flank the Sun within a 22° halo.

The sun dog is a member of the family of halos caused by the refraction of sunlight by ice crystals in the atmosphere. Sun dogs typically appear as a pair of subtly colored patches of light, around 22° to the left and right of the Sun, and at the same altitude above the horizon as the Sun. They can be seen anywhere in the world during any season, but are not always obvious or bright. Sun dogs are best seen and most conspicuous when the Sun is near the horizon.

A moon dog (or moondog) or mock moon, also called a paraselene (plural paraselenae or paraselenes) in meteorology, is an atmospheric optical phenomenon that consists of a bright spot to one or both sides of the Moon. They are exactly analogous to sun dogs.

A member of the halo family, moon dogs are caused by the refraction of moonlight by hexagonal-plate-shaped ice crystals in cirrus clouds or cirrostratus clouds. They typically appear as a pair of faint patches of light, at around 22° to the left and right of the Moon, and at the same altitude above the horizon as the Moon. They may also appear alongside 22° halos.

A polar stratospheric cloud (PSC) is a cloud that forms in the winter polar stratosphere at altitudes from 15,000 to 25,000 m (49,000 to 82,000 ft). They are best observed during civil twilight, when the Sun is between 1° and 6° below the horizon, as well as in winter and in more northerly latitudes. One main type of PSC is composed of mostly supercooled droplets of water and nitric acid and is implicated in the formation of ozone holes. The other main type consists only of ice crystals, which are not harmful. This type of PSC is also called nacreous (/ˈneɪkriəs/; from nacre, or mother of pearl), due to its iridescence.