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A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...
. Since one degree is of a
turn Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * Turn, ...
(or complete rotation), one minute of arc is of a turn. The nautical mile (nmi) was originally defined as the arc length of a minute of latitude on a spherical Earth, so the actual Earth circumference is very near . A minute of arc is of a radian. A second of arc, arcsecond (arcsec), or arc second, denoted by the symbol , is of an arcminute, of a degree, of a turn, and (about ) of a radian. These units originated in
Babylonian astronomy Babylonian astronomy was the study or recording of celestial objects during the early history of Mesopotamia. Babylonian astronomy seemed to have focused on a select group of stars and constellations known as Ziqpu stars. These constellations ...
as sexagesimal subdivisions of the degree; they are used in fields that involve very small angles, such as
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
,
optometry Optometry is a specialized health care profession that involves examining the eyes and related structures for defects or abnormalities. Optometrists are health care professionals who typically provide comprehensive primary eye care. In the Un ...
,
ophthalmology Ophthalmology ( ) is a surgical subspecialty within medicine that deals with the diagnosis and treatment of eye disorders. An ophthalmologist is a physician who undergoes subspecialty training in medical and surgical eye care. Following a med ...
,
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultra ...
,
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
, land surveying, and marksmanship. To express even smaller angles, standard SI prefixes can be employed; the milliarcsecond (mas) and microarcsecond (μas), for instance, are commonly used in astronomy. For a three-dimensional area such as on a sphere, '' square arcminutes'' or ''seconds'' may be used.


Symbols and abbreviations

The prime symbol () designates the arcminute, though a single quote (U+0027) is commonly used where only
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
characters are permitted. One arcminute is thus written as 1′. It is also abbreviated as arcmin or amin. Similarly, double prime (U+2033) designates the arcsecond, though a double quote (U+0022) is commonly used where only
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
characters are permitted. One arcsecond is thus written as 1″. It is also abbreviated as arcsec or asec. In
celestial navigation 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 ...
, seconds of arc are rarely used in calculations, the preference usually being for degrees, minutes, and decimals of a minute, for example, written as 42° 25.32′ or 42° 25.322′. This notation has been carried over into marine GPS receivers, which normally display latitude and longitude in the latter format by default.


Common examples

The average apparent diameter of the
full Moon The full moon is the lunar phase when the Moon appears fully illuminated from Earth's perspective. This occurs when Earth is located between the Sun and the Moon (when the ecliptic longitudes of the Sun and Moon differ by 180°). This mean ...
is about 31 arcminutes, or 0.52°. One arcminute is the approximate resolution of the human eye. One arcsecond is the approximate angle subtended by a U.S. dime coin (18 mm) at a distance of . An arcsecond is also the angle subtended by * an object of diameter at a distance of one
astronomical unit The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbi ...
, * an object of diameter at one light-year, * an object of diameter one astronomical unit () at a distance of one parsec, per the definition of the latter. One milliarcsecond is about the size of a half dollar, seen from a distance equal to that between the
Washington Monument The Washington Monument is an obelisk shaped building within the National Mall in Washington, D.C., built to commemorate George Washington, once commander-in-chief of the Continental Army (1775–1784) in the American Revolutionary War and ...
and the
Eiffel Tower The Eiffel Tower ( ; french: links=yes, tour Eiffel ) is a wrought-iron lattice tower on the Champ de Mars in Paris, France. It is named after the engineer Gustave Eiffel, whose company designed and built the tower. Locally nickname ...
. One microarcsecond is about the size of a period at the end of a sentence in the Apollo mission manuals left on the Moon as seen from Earth. One nanoarcsecond is about the size of a penny on
Neptune Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 time ...
's moon Triton as observed from Earth. Also notable examples of size in arcseconds are: *
Hubble Space Telescope The Hubble Space Telescope (often referred to as HST or Hubble) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. It was not the first space telescope, but it is one of the largest and most ver ...
has calculational resolution of 0.05 arcseconds and actual resolution of almost 0.1 arcseconds, which is close to the diffraction limit. * At crescent phase,
Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never f ...
measures between 60.2 and 66 seconds of arc.


History

The concepts of degrees, minutes, and seconds—as they relate to the measure of both angles and time—derive from
Babylonian Babylonian may refer to: * Babylon, a Semitic Akkadian city/state of ancient Mesopotamia founded in 1894 BC * Babylonia, an ancient Akkadian-speaking Semitic nation-state and cultural region based in central-southern Mesopotamia (present-day Iraq) ...
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
and time-keeping. Influenced by the Sumerians, the ancient Babylonians divided the Sun's perceived motion across the sky over the course of one full day into 360 degrees. Each degree was subdivided into 60 minutes and each minute into 60 seconds. Thus, one Babylonian degree was equal to four minutes in modern terminology, one Babylonian minute to four modern seconds, and one Babylonian second to (approximately 0.067) of a modern second.


Uses


Astronomy

Since antiquity, the arcminute and arcsecond have been used in
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
: in the ecliptic coordinate system as latitude (β) and longitude (λ); in the horizon system as altitude (Alt) and
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematicall ...
(Az); and in the equatorial coordinate system as declination (δ). All are measured in degrees, arcminutes, and arcseconds. The principal exception is
right ascension Right ascension (abbreviated RA; symbol ) 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 pai ...
(RA) in equatorial coordinates, which is measured in time units of hours, minutes, and seconds. Contrary to what one might assume, minutes and seconds of arc do not directly relate to minutes and seconds of time, in either the rotational frame of the Earth around its own axis (day), or the Earth's rotational frame around the Sun (year). The Earth's rotational rate around its own axis is 15 minutes of arc per minute of time (360 degrees / 24 hours in day); the Earth's rotational rate around the Sun (not entirely constant) is roughly 24 minutes of time per minute of arc (from 24 hours in day), which tracks the annual progression of the Zodiac. Both of these factor in what astronomical objects you can see from surface telescopes (time of year) and when you can best see them (time of day), but neither are in unit correspondence. For simplicity, the explanations given assume a degree/day in the Earth's annual rotation around the Sun, which is off by roughly 1%. The same ratios hold for seconds, due to the consistent factor of 60 on both sides. The arcsecond is also often used to describe small astronomical angles such as the angular diameters of planets (e.g. the angular diameter of Venus which varies between 10″ and 60″); the proper motion of stars; the separation of components of binary star systems; and parallax, the small change of position of a star or solar system body as the Earth revolves about the Sun. These small angles may also be written in milliarcseconds (mas), or thousandths of an arcsecond. The unit of distance called the parsec, abbreviated from the parallax angle of one arc second, was developed for such parallax measurements. The distance from the Sun to a celestial object is the reciprocal of the angle, measured in arcseconds, of the object's apparent movement caused by parallax. The
European Space Agency , owners = , headquarters = Paris, Île-de-France, France , coordinates = , spaceport = Guiana Space Centre , seal = File:ESA emblem seal.png , seal_size = 130px , image = Views in the Main Control Room (120 ...
's
astrometric Astrometry is a branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. It provides the kinematics and physical origin of the Solar System and this galaxy, the Milky Way. Histor ...
satellite
Gaia In Greek mythology, Gaia (; from Ancient Greek , a poetical form of , 'land' or 'earth'),, , . also spelled Gaea , is the personification of the Earth and one of the Greek primordial deities. Gaia is the ancestral mother—sometimes parthe ...
, launched in 2013, can approximate star positions to 7 microarcseconds (µas). Apart from the Sun, the star with the largest angular diameter from Earth is R Doradus, a red giant with a diameter of 0.05″. Because of the effects of atmospheric blurring, ground-based
telescope A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to obse ...
s will smear the image of a star to an angular diameter of about 0.5″; in poor conditions this increases to 1.5″ or even more. The dwarf planet
Pluto Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the Sun. It is the largest k ...
has proven difficult to resolve because its angular diameter is about 0.1″. Space telescopes are not affected by the Earth's atmosphere but are diffraction limited. For example, the
Hubble Space Telescope The Hubble Space Telescope (often referred to as HST or Hubble) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. It was not the first space telescope, but it is one of the largest and most ver ...
can reach an angular size of stars down to about 0.1″. Techniques exist for improving seeing on the ground. Adaptive optics, for example, can produce images around 0.05″ on a 10 m class telescope.


Cartography

Minutes (′) and seconds (″) of arc are also used in
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an ...
and
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
. At
sea level Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardise ...
one minute of arc along the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can al ...
equals exactly one geographical mile along the Earth's equator or approximately . A second of arc, one sixtieth of this amount, is roughly . The exact distance varies along
meridian arc In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. The purpose of measuring meridian arcs is to ...
s or any other great circle arcs because the
figure of the Earth Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A sphere is a well-known historical approxima ...
is slightly oblate (bulges a third of a percent at the equator). Positions are traditionally given using degrees, minutes, and seconds of arcs for
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north po ...
, the arc north or south of the equator, and for
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
, the arc east or west of the
Prime Meridian A prime meridian is an arbitrary meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian in a 360°-system) form a great ...
. Any position on or above the Earth's reference ellipsoid can be precisely given with this method. However, when it is inconvenient to use base-60 for minutes and seconds, positions are frequently expressed as decimal fractional degrees to an equal amount of precision. Degrees given to three decimal places ( of a degree) have about the precision of degrees-minutes-seconds ( of a degree) and specify locations within about . For navigational purposes positions are given in degrees and decimal minutes, for instance The Needles lighthouse is at 50º 39.734’N 001º 35.500’W.


Property cadastral surveying

Related to cartography, property boundary surveying using the metes and bounds system and
cadastral surveying Cadastral surveying is the sub-field of cadastre and surveying that specialises in the establishment and re-establishment of real property boundaries. It involves the physical delineation of property boundaries and determination of dimensions, are ...
relies on fractions of a degree to describe property lines' angles in reference to cardinal directions. A boundary "mete" is described with a beginning reference point, the cardinal direction North or South followed by an angle less than 90 degrees and a second cardinal direction, and a linear distance. The boundary runs the specified linear distance from the beginning point, the direction of the distance being determined by rotating the first cardinal direction the specified angle toward the second cardinal direction. For example, ''North 65° 39′ 18″ West 85.69 feet'' would describe a line running from the starting point 85.69 feet in a direction 65° 39′ 18″ (or 65.655°) away from north toward the west.


Firearms

The arcminute is commonly found in the firearms industry and literature, particularly concerning the precision of rifles, though the industry refers to it as minute of angle (MOA). It is especially popular as a unit of measurement with shooters familiar with the imperial measurement system because 1 MOA subtends a circle with a diameter of 1.047
inch Measuring tape with inches The inch (symbol: in or ″) is a unit of length in the British imperial and the United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelft ...
es (which is often rounded to just 1 inch) at 100 yards ( at or 2.908 cm at 100 m), a traditional distance on American target ranges. The subtension is linear with the distance, for example, at 500 yards, 1 MOA subtends 5.235 inches, and at 1000 yards 1 MOA subtends 10.47 inches. Since many modern telescopic sights are adjustable in half (), quarter () or eighth () MOA increments, also known as ''clicks'', zeroing and adjustments are made by counting 2, 4 and 8 clicks per MOA respectively. For example, if the point of impact is 3 inches high and 1.5 inches left of the point of aim at 100 yards (which for instance could be measured by using a spotting scope with a calibrated reticle), the scope needs to be adjusted 3 MOA down, and 1.5 MOA right. Such adjustments are trivial when the scope's adjustment dials have a MOA scale printed on them, and even figuring the right number of clicks is relatively easy on scopes that ''click'' in fractions of MOA. This makes zeroing and adjustments much easier: * To adjust a MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 × 2 = 6 clicks down and 1.5 x 2 = 3 clicks right * To adjust a MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 x 4 = 12 clicks down and 1.5 × 4 = 6 clicks right * To adjust a MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 x 8 = 24 clicks down and 1.5 × 8 = 12 clicks right Another common system of measurement in firearm scopes is the milliradian (mrad). Zeroing an mrad based scope is easy for users familiar with base ten systems. The most common adjustment value in mrad based scopes is  mrad (which approximates MOA). * To adjust a  mrad scope 0.9 mrad down and 0.4 mrad right, the scope needs to be adjusted 9 clicks down and 4 clicks right (which equals approximately 3 and 1.5 MOA respectively). One thing to be aware of is that some MOA scopes, including some higher-end models, are calibrated such that an adjustment of 1 MOA on the scope knobs corresponds to exactly 1 inch of impact adjustment on a target at 100 yards, rather than the mathematically correct 1.047 inches. This is commonly known as the Shooter's MOA (SMOA) or Inches Per Hundred Yards (IPHY). While the difference between one true MOA and one SMOA is less than half of an inch even at 1000 yards, this error compounds significantly on longer range shots that may require adjustment upwards of 20–30 MOA to compensate for the bullet drop. If a shot requires an adjustment of 20 MOA or more, the difference between true MOA and SMOA will add up to 1 inch or more. In competitive target shooting, this might mean the difference between a hit and a miss. The physical group size equivalent to ''m'' minutes of arc can be calculated as follows: group size = tan() × distance. In the example previously given, for 1 minute of arc, and substituting 3,600 inches for 100 yards, 3,600 tan() ≈ 1.047 inches. In metric units 1 MOA at 100 metres ≈ 2.908 centimetres. Sometimes, a precision-oriented firearm's performance will be measured in MOA. This simply means that under ideal conditions (i.e. no wind, high-grade ammo, clean barrel, and a stable mounting platform such as a vise or a benchrest used to eliminate shooter error), the gun is capable of producing a group of shots whose center points (center-to-center) fit into a circle, the average diameter of circles in several groups can be subtended by that amount of arc. For example, a ''1 MOA rifle'' should be capable, under ideal conditions, of repeatably shooting 1-inch groups at 100 yards. Most higher-end rifles are warrantied by their manufacturer to shoot under a given MOA threshold (typically 1 MOA or better) with specific ammunition and no error on the shooter's part. For example, Remington's M24 Sniper Weapon System is required to shoot 0.8 MOA or better, or be rejected from sale by quality control. Rifle manufacturers and gun magazines often refer to this capability as ''sub-MOA'', meaning a gun consistently shooting groups under 1 MOA. This means that a single group of 3 to 5 shots at 100 yards, or the average of several groups, will measure less than 1 MOA between the two furthest shots in the group, i.e. all shots fall within 1 MOA. If larger samples are taken (i.e., more shots per group) then group size typically increases, however this will ultimately average out. If a rifle was truly a 1 MOA rifle, it would be just as likely that two consecutive shots land exactly on top of each other as that they land 1 MOA apart. For 5-shot groups, based on 95% confidence, a rifle that normally shoots 1 MOA can be expected to shoot groups between 0.58 MOA and 1.47 MOA, although the majority of these groups will be under 1 MOA. What this means in practice is if a rifle that shoots 1-inch groups on average at 100 yards shoots a group measuring 0.7 inches followed by a group that is 1.3 inches, this is not statistically abnormal. The
metric system The metric system is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the Intern ...
counterpart of the MOA is the milliradian (mrad or 'mil'), being equal to of the target range, laid out on a circle that has the observer as centre and the target range as radius. The number of milliradians on a full such circle therefore always is equal to 2 × × 1000, regardless the target range. Therefore, 1 MOA ≈ 0.2909 mrad. This means that an object which spans 1 mrad on the reticle is at a range that is in metres equal to the object's size in millimetres (e.g. an object of 100 mm subtending 1 mrad is 100 metres away). So there is no conversion factor required, contrary to the MOA system. A reticle with markings (hashes or dots) spaced with a one mrad apart (or a fraction of a mrad) are collectively called a mrad reticle. If the markings are round they are called mil-dots. In the table below conversions from mrad to metric values are exact (e.g. 0.1 mrad equals exactly 10 mm at 100 metres), while conversions of minutes of arc to both metric and imperial values are approximate. * 1′ at 100 yards is about 1.047 inchesDexadine Ballistics Software – ballistic data for shooting and reloading
See Talk
* 1′ ≈ 0.291 mrad (or 29.1 mm at 100 m, approximately 30 mm at 100 m) * 1 mrad ≈ 3.44′, so  mrad ≈ ′ * 0.1 mrad equals exactly 1 cm at 100 m, or approximately 0.36 inches at 100 yards


Human vision

In humans,
20/20 vision Visual acuity (VA) commonly refers to the clarity of vision, but technically rates an examinee's ability to recognize small details with precision. Visual acuity is dependent on optical and neural factors, i.e. (1) the sharpness of the retinal ...
is the ability to resolve a spatial pattern separated by a
visual angle Visual angle is the angle a viewed object subtends at the eye, usually stated in degrees of arc. It also is called the object's angular size. The diagram on the right shows an observer's eye looking at a frontal extent (the vertical arrow) t ...
of one minute of arc. A 20/20 letter subtends 5 minutes of arc total.


Materials

The deviation from parallelism between two surfaces, for instance in optical engineering, is usually measured in arcminutes or arcseconds. In addition, arcseconds are sometimes used in rocking curve (ω-scan) x ray diffraction measurements of high-quality epitaxial thin films.


Manufacturing

Some measurement devices make use of arcminutes and arcseconds to measure angles when the object being measured is too small for direct visual inspection. For instance, a toolmaker's optical comparator will often include an option to measure in "minutes and seconds".


See also

*
Centesimal minute and second of arc In trigonometry, the gradian, also known as the gon (from grc, γωνία, gōnía, angle), grad, or grade, is a unit of measurement of an angle, defined as one hundredth of the right angle; in other words, there are 100 gradians in 90 degree ...
* * *
Square minute In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...
*
Square second In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...
*
Steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radi ...
* Milliradian


Notes


References


External links


MOA/ mils
By Robert Simeone *A Guide t
calculate distance using MOA Scope
by Steve Coffman {{Portal bar, Mathematics, Physics, Astronomy, Science
Arcminute A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The na ...