Editing Minute and second of arc (section)

==Uses==
===Astronomy===
[[File:Comparison angular diameter solar system.svg|thumb|upright=1.5|Comparison of angular diameter of the Sun, Moon, planets and the International Space Station. True represent&shy;ation of the sizes is achieved when the image is viewed at a distance of 103 times the width of the "Moon: max." circle. For example, if the "Moon: max." circle is 10&nbsp;cm wide on a computer display, viewing it from {{convert|10.3|m|yd|abbr=in}} away will show true representation of the sizes.]]

Since antiquity, the arcminute and arcsecond have been used in [[astronomy]]: in the [[ecliptic coordinate system]] as latitude (β) and longitude (λ); in the [[horizontal coordinate system|horizon system]] as altitude (Alt) and [[azimuth]] (Az); and in the [[equatorial coordinate system]] as [[declination]] (δ). All are measured in degrees, arcminutes, and arcseconds. The principal exception is [[right ascension]] (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 system]]s; 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 '''par'''allax angle of one arc '''sec'''ond, was developed for such parallax measurements. The distance from the Sun to a celestial object is the [[Multiplicative inverse|reciprocal]] of the angle, measured in arcseconds, of the object's apparent movement caused by parallax.

The [[European Space Agency]]'s [[astrometry|astrometric]] satellite [[Gaia mission|Gaia]], launched in 2013, can approximate star positions to 7 microarcseconds (μas).<ref>{{cite news |url = https://www.bbc.com/news/science-environment-37355154 |title=Celestial mapper plots a billion stars|last=Amos|first=Jonathan|date=2016-09-14|work=BBC News|access-date=2018-03-31|language=en-GB }}</ref>

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 [[astronomical seeing|blurring]], ground-based [[telescope]]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]] has proven difficult to resolve because its [[angular diameter]] is about 0.1″.<ref>{{Cite web |title=Pluto Fact Sheet |url=https://nssdc.gsfc.nasa.gov/planetary/factsheet/plutofact.html |access-date=2022-08-29 |website=nssdc.gsfc.nasa.gov}}</ref> Techniques exist for improving seeing on the ground. [[Adaptive optics]], for example, can produce images around 0.05″ on a 10&nbsp;m class telescope.

Space telescopes are not affected by the Earth's atmosphere but are [[Diffraction limit#Diffraction limit of telescopes|diffraction limited]]. For example, the [[Hubble Space Telescope]] can reach an angular size of stars down to about 0.1″.

===Cartography===

Minutes (′) and seconds (″) of arc are also used in [[cartography]] and [[navigation]]. At [[sea level]] one minute of arc along the [[equator]] equals exactly one [[geographical mile]] (not to be confused with international mile or statute mile) along the Earth's equator or approximately {{convert|1|nmi|m mi|sigfig=4|spell=in|abbr=off|lk=on}}.<ref>{{cite web |url=http://www.oceannavigator.com/January-February-2003/Nautical-mile-approximates-an-arcminute/ |title=Nautical mile approximates an arcminute |date=1 January 2003 |first=George H. |last=Kaplan |magazine=Ocean Navigator |publisher=Navigator Publishing |access-date=2017-03-22}}</ref> A second of arc, one sixtieth of this amount, is roughly {{convert|30|m|abbr=off}}. The exact distance varies along [[meridian arc]]s or any other [[great circle]] arcs because the [[figure of the Earth]] is slightly [[Oblate spheroid|oblate]] (bulges a third of a percent at the equator).

Positions are traditionally given using degrees, minutes, and seconds of arcs for [[latitude]], the arc north or south of the equator, and for [[longitude]], the arc east or west of the [[Prime Meridian]]. Any position on or above the Earth's [[reference ellipsoid]] can be precisely given with this method. However, when it is inconvenient to use  [[radix|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 ({{sfrac|1|{{val|1000}}}} of a degree) have about {{sfrac|1|4}} the precision of degrees-minutes-seconds ({{sfrac|1|{{val|3600}}}} of a degree) and specify locations within about {{convert|120|m|abbr=off}}.  For navigational purposes positions are given in degrees and decimal minutes, for instance, the [[Needles Lighthouse]] is at 50°39′44.2″N 1°35′30.5″W.<ref>{{cite web|author= The Corporation of Trinity House |title=1/2020 Needles Lighthouse|date=10 January 2020|series=Notices to Mariners|url=https://www.trinityhouse.co.uk/notice-to-mariners/1/2020-needles-lighthouse|access-date=24 May 2020}}</ref>

===Property cadastral surveying===

Related to cartography, property boundary [[surveying]] using the [[metes and bounds]] system and [[cadastral surveying]] relies on fractions of a degree to describe property lines' angles in reference to [[cardinal direction]]s. 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&nbsp;65°&nbsp;39′ 18″ West&nbsp;85.69&nbsp;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===
[[File:Ballistic table for 7.62x51 mm NATO (mrad and moa).png|thumb|right|Example ballistic table for a given [[7.62×51mm NATO]] load. Bullet drop and wind drift are shown both in [[milliradian|mrad]] and minute of angle.]]

The arcminute is commonly found in the [[firearm]]s industry and literature, particularly concerning the [[Accuracy and precision|precision]] of [[rifle]]s, 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&nbsp;MOA [[subtension|subtends]] a circle with a diameter of 1.047 [[inch]]es (which is often rounded to just 1 inch) at 100 [[yard]]s ({{convert|1.047|in|cm|abbr=on|disp=out}} at {{convert|100|yd|m|disp=out|abbr=on}} or 2.908&nbsp;cm at 100&nbsp;m), a traditional distance on American [[Shooting range|target ranges]].  The [[subtension]] is linear with the distance, for example, at 500 yards, 1&nbsp;MOA subtends 5.235 inches, and at 1000 yards 1&nbsp;MOA subtends 10.47 inches.
Since many modern [[telescopic sight]]s are adjustable in half ({{sfrac|1|2}}), quarter ({{sfrac|1|4}}) or eighth ({{sfrac|1|8}}) 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, or a  target delineated for such purposes), 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 {{frac|1|2}} 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 {{frac|1|4}} 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 {{frac|1|8}} 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

[[File:MOA and mrad comparison.png|thumb|right|Comparison of minute of arc (MOA) and [[milliradian]] (mrad).]]
Another common system of measurement in firearm scopes is the [[milliradian]] (mrad). Zeroing an mrad based scope is easy for users familiar with [[decimal|base ten]] systems. The most common adjustment value in mrad based scopes is {{sfrac|1|10}}&nbsp;mrad (which approximates {{frac|1|3}} MOA).
* To adjust a {{sfrac|1|10}}&nbsp;mrad scope 0.9&nbsp;mrad down and 0.4&nbsp;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,<ref>{{cite web |last=Mann |first=Richard |url=http://www.shootingillustrated.com/index.php/6227/mil-moa-or-inches/ |title=Mil, MOA or inches? |publisher=Shooting Illustrated |date=2011-02-18 |access-date=2015-04-13 |archive-url=https://web.archive.org/web/20131110204817/http://www.shootingillustrated.com/index.php/6227/mil-moa-or-inches/ |archive-date=10 November 2013 |url-status=dead }}</ref> 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({{sfrac|''m''|60}})&nbsp;×&nbsp;distance. In the example previously given, for 1 minute of arc, and substituting 3,600&nbsp;inches for 100 yards, 3,600 tan({{sfrac|1|60}}) ≈ 1.047&nbsp;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 [[shot grouping|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 interval|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.<ref>{{cite web|first=Robert E. |last=Wheeler |title=Statistical notes on rifle group patterns |url=http://www.bobwheeler.com/guns/GroupStat.pdf |archive-url=https://web.archive.org/web/20060926154900/http://www.bobwheeler.com/guns/GroupStat.pdf |url-status=dead |archive-date=26 September 2006 |access-date=21 May 2009 }}</ref><ref>{{cite journal |first=Denton |last=Bramwell |date=January 2009 |title=Group Therapy The Problem: How accurate is your rifle? |journal=Varmint Hunter |volume=69 |url=http://www.longrangehunting.com/articles/accurate-rifle-groups-1.php |access-date=21 May 2009 |archive-url=https://web.archive.org/web/20111007225056/http://www.longrangehunting.com/articles/accurate-rifle-groups-1.php |archive-date=7 October 2011 |url-status=dead }}</ref>

The [[metric system]] counterpart of the MOA is the [[milliradian]] (mrad or 'mil'), being equal to {{fraction|1000}} 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 × {{pi}} × 1000, regardless the target range. Therefore, 1 MOA ≈ 0.2909&nbsp;mrad. This means that an object which spans 1&nbsp;mrad on the [[reticle]] is at a range that is in metres equal to the object's linear size in millimetres (e.g. an object of 100&nbsp;mm subtending 1 mrad is 100 metres away).<ref>{{cite book |author=Fouad Sabry |url=http://google.co.uk/books/edition/Precision_Guided_Firearm/RdmTEAAAQBAJ?pg=PT220&gbpv=1 |date=2022 |title=Precision Guided Firearm |publisher=One Billion Knowledgeable}}</ref> 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&nbsp;mrad equals exactly 10&nbsp;mm at 100 metres), while conversions of minutes of arc to both metric and imperial values are approximate.

{{Conversion between common sight adjustments based on milliradian and minute of arc}}
* 1′ at 100 yards is about 1.047 inches<ref>[http://dexadine.com/WhatMOA.htm Dexadine Ballistics Software – ballistic data for shooting and reloading]. See [[Talk:Minute and second of arc#Stop quibbling|Talk]]</ref>
* 1′ ≈ 0.291&nbsp;mrad (or 29.1&nbsp;mm at 100&nbsp;m, approximately 30&nbsp;mm at 100&nbsp;m)
* 1&nbsp;mrad ≈ 3.44′, so {{sfrac|1|10}}&nbsp;mrad ≈ {{sfrac|1|3}}′
* 0.1&nbsp;mrad equals exactly 1&nbsp;cm at 100&nbsp;m, or exactly 0.36 inches at 100 yards

===Human vision===
In humans, [[Normal vision|20/20 vision]] is the ability to resolve a [[spatial pattern]] separated by a [[visual angle]] of one minute of arc, from a distance of twenty [[Foot (unit)|feet]].
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 [[epitaxy|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".