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| Arcminute | |
|---|---|
 An illustration of the size of an arcminute (not to scale). A standardassociation football (soccer) ball (with a diameter of 22 cm or 8.7 in) subtends an angle of 1 arcminute at a distance of approximately 756 m (827 yd). | |
| General information | |
| Unit system | Non-SI units mentioned in the SI |
| Unit of | Angle |
| Symbol | ′, arcmin |
| In units | Dimensionless with anarc length of approx. ≈0.2909/1000 of the radius, i.e. 0.2909mm/m |
| Conversions | |
| 1 ′in ... | ... is equal to ... |
| degrees | 1/60° = 0.016° |
| arcseconds | 60″ |
| radians | π/10800 ≈ 0.000290888 rad |
| milliradians | 5π/54 ≈ 0.2909 mrad |
| gradians | 3/200g = 0.015g |
| turns | 1/21600 turn |
Aminute of arc,arcminute (abbreviated asarcmin),arc minute, orminute arc, denoted by the symbol′, is a unit ofangular measurement equal to1/60 of adegree.[1] Since one degree is1/360 of aturn, or complete rotation, one arcminute is1/21600 of a turn. Thenautical mile (nmi) was originally defined as thearc length of a minute of latitude on a spherical Earth, so the actualEarth's circumference is very near21600 nmi. A minute of arc isπ/10800 of aradian.
Asecond of arc,arcsecond (abbreviated asarcsec), orarc second, denoted by the symbol″,[2] is a unit ofangular measurement equal to1/60 of a minute of arc,1/3600 of a degree,[1]1/1296000 of a turn, andπ/648000 (about1/206264.8) of a radian.
These units originated inBabylonian astronomy assexagesimal (base 60) subdivisions of the degree; they are used in fields that involve very small angles, such asastronomy,optometry,ophthalmology,optics,navigation,land surveying, andmarksmanship.
To express even smaller angles, standardSI prefixes can be employed; themilliarcsecond (mas) andmicroarcsecond (μas), for instance, are commonly used in astronomy. For a three-dimensional area such as on a sphere,square arcminutes orseconds may be used.
Theprime symbol′ (U+2032) designates the arcminute,[2] though a single quote' (U+0027) is commonly used where onlyASCII characters are permitted. One arcminute is thus written as 1′. It is also abbreviated asarcmin oramin.
Similarly,double prime″ (U+2033) designates the arcsecond,[2] though a double quote" (U+0022) is commonly used where onlyASCII characters are permitted. One arcsecond is thus written as 1″. It is also abbreviated asarcsec orasec.
| Unit | Value | Symbol | Abbreviations | In radians, approx. | |
|---|---|---|---|---|---|
| Degree | 1/360 turn | ° | Degree | deg | 17.4532925 mrad |
| Arcminute | 1/60 degree | ′ | Prime | arcmin, amin, am, MOA | 290.8882087 μrad |
| Arcsecond | 1/60 arcminute =1/3600 degree | ″ | Double prime | arcsec, asec, as | 4.8481368 μrad |
| Milliarcsecond | 0.001 arcsecond =1/3600000 degree | mas | 4.8481368 nrad | ||
| Microarcsecond | 0.001 mas =0.000001 arcsecond | μas | 4.8481368 prad | ||
Incelestial navigation, 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′.[3][4] This notation has been carried over intomarine GPS and aviation GPS receivers, which normally display latitude and longitude in the latter format by default.[5]
The averageapparent diameter of thefull Moon is about 31 arcminutes, or 0.52°.
One arcminute is the approximate distance two contours can be separated by, and still be distinguished by, a person with20/20 vision.
One arcsecond is the approximateangle subtended by aU.S. dime coin (18 mm) at a distance of 4 kilometres (about 2.5 mi).[6] An arcsecond is also the angle subtended by
One milliarcsecond is about the size of a half dollar, seen from a distance equal to that between theWashington Monument and theEiffel Tower.
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 onNeptune's moonTriton as observed from Earth.
Also notable examples of size in arcseconds are:
The concepts of degrees, minutes, and seconds—as they relate to the measure of both angles and time—derive fromBabylonianastronomy and time-keeping. Influenced by theSumerians, the ancient Babylonians divided the Sun's perceived motion across the sky over the course of onefull day into 360 degrees.[9][failed verification] Each degree was subdivided into 60 minutes and each minute into 60 seconds.[10][11] Thus, one Babylonian degree was equal to four minutes in modern terminology, one Babylonian minute to four modern seconds, and one Babylonian second to1/15 (approximately 0.067) of a modern second.
Since antiquity, the arcminute and arcsecond have been used inastronomy: in theecliptic coordinate system as latitude (β) and longitude (λ); in thehorizon system as altitude (Alt) andazimuth (Az); and in theequatorial coordinate system asdeclination (δ). All are measured in degrees, arcminutes, and arcseconds. The principal exception isright 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″); theproper motion of stars; the separation of components ofbinary star systems; andparallax, 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 theparsec, abbreviated from theparallax angle of one arcsecond, was developed for such parallax measurements. The distance from the Sun to a celestial object is thereciprocal of the angle, measured in arcseconds, of the object's apparent movement caused by parallax.
TheEuropean Space Agency'sastrometric satelliteGaia, launched in 2013, can approximate star positions to 7 microarcseconds (μas).[12]
Apart from the Sun, the star with the largestangular diameter from Earth isR Doradus, ared giant with a diameter of 0.05″. Because of the effects of atmosphericblurring, ground-basedtelescopes 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 planetPluto has proven difficult to resolve because itsangular diameter is about 0.1″.[13] Techniques exist for improving seeing on the ground.Adaptive optics, for example, can produce images around 0.05″ on a 10 m class telescope.
Space telescopes are not affected by the Earth's atmosphere but arediffraction limited. For example, theHubble Space Telescope can reach an angular size of stars down to about 0.1″.
Minutes (′) and seconds (″) of arc are also used incartography andnavigation. Atsea level one minute of arc along theequator equals exactly onegeographical mile (not to be confused with international mile or statute mile) along the Earth's equator or approximately onenautical mile (1,852metres; 1.151miles).[14] A second of arc, one sixtieth of this amount, is roughly 30 metres (98 feet). The exact distance varies alongmeridian arcs or any othergreat circle arcs because thefigure of the Earth is slightlyoblate (bulges a third of a percent at the equator).
Positions are traditionally given using degrees, minutes, and seconds of arcs forlatitude, the arc north or south of the equator, and forlongitude, the arc east or west of thePrime Meridian. Any position on or above the Earth'sreference ellipsoid can be precisely given with this method. However, when it is inconvenient to usebase-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 (1/1000 of a degree) have about1/4 the precision of degrees-minutes-seconds (1/3600 of a degree) and specify locations within about 120 metres (390 feet). 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.[15]
Related to cartography, property boundarysurveying using themetes and bounds system andcadastral surveying relies on fractions of a degree to describe property lines' angles in reference tocardinal 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.
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The arcminute is commonly found in thefirearms industry and literature, particularly concerning theprecision ofrifles, though the industry refers to it asminute of angle (MOA). It is especially popular as a unit of measurement with shooters familiar with theimperial measurement system because 1 MOAsubtends a circle with a diameter of 1.047inches (which is often rounded to just 1 inch) at 100yards (2.66 cm at 91 m or 2.908 cm at 100 m), a traditional distance on Americantarget ranges. Thesubtension 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 moderntelescopic sights are adjustable in half (1/2), quarter (1/4) or eighth (1/8) MOA increments, also known asclicks,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 aspotting 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 thatclick in fractions of MOA. This makes zeroing and adjustments much easier:
Another common system of measurement in firearm scopes is themilliradian (mrad). Zeroing an mrad based scope is easy for users familiar withbase ten systems. The most common adjustment value in mrad based scopes is1/10 mrad (which approximates1⁄3 MOA).
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,[16] 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 tom minutes of arc can be calculated as follows: group size = tan(m/60) × distance. In the example previously given, for 1 minute of arc, and substituting 3,600 inches for 100 yards, 3,600 tan(1/60) ≈ 1.047 inches. Inmetric 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 agroup 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, a1 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'sM24 Sniper Weapon System is required to shoot 0.8 MOA or better, or be rejected from sale byquality control.
Rifle manufacturers and gun magazines often refer to this capability assub-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.[17][18]
Themetric system counterpart of the MOA is themilliradian (mrad or 'mil'), being equal to1⁄1000 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 ×π × 1000, regardless the target range. Therefore, 1 MOA ≈ 0.2909 mrad. This means that an object which spans 1 mrad on thereticle is at a range that is in metres equal to the object's linear size in millimetres (e.g. an object of 100 mm subtending 1 mrad is 100 metres away).[19] 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 calledmil-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.
| Increment, or click | (mins of arc) | (milli- radians) | At 100 m | At 100 yd | ||
|---|---|---|---|---|---|---|
| (mm) | (cm) | (in) | (in) | |||
| 1⁄12′ | 0.083′ | 0.024 mrad | 2.42 mm | 0.242 cm | 0.0958 in | 0.087 in |
| 0.25⁄10 mrad | 0.086′ | 0.025 mrad | 2.5 mm | 0.25 cm | 0.0985 in | 0.09 in |
| 1⁄8′ | 0.125′ | 0.036 mrad | 3.64 mm | 0.36 cm | 0.144 in | 0.131 in |
| 1⁄6′ | 0.167′ | 0.0485 mrad | 4.85 mm | 0.485 cm | 0.192 in | 0.175 in |
| 0.5⁄10 mrad | 0.172′ | 0.05 mrad | 5 mm | 0.5 cm | 0.197 in | 0.18 in |
| 1⁄4′ | 0.25′ | 0.073 mrad | 7.27 mm | 0.73 cm | 0.29 in | 0.26 in |
| 1⁄10 mrad | 0.344′ | 0.1 mrad | 10 mm | 1 cm | 0.39 in | 0.36 in |
| 1⁄2′ | 0.5′ | 0.145 mrad | 14.54 mm | 1.45 cm | 0.57 in | 0.52 in |
| 1.5⁄10 mrad | 0.516′ | 0.15 mrad | 15 mm | 1.5 cm | 0.59 in | 0.54 in |
| 2⁄10 mrad | 0.688′ | 0.2 mrad | 20 mm | 2 cm | 0.79 in | 0.72 in |
| 1′ | 1.0′ | 0.291 mrad | 29.1 mm | 2.91 cm | 1.15 in | 1.047 in |
| 1 mrad | 3.438′ | 1 mrad | 100 mm | 10 cm | 3.9 in | 3.6 in |
In humans,20/20 vision is the ability to resolve aspatial pattern separated by avisual angle of one minute of arc, from a distance of twentyfeet.A 20/20 letter subtends 5 minutes of arc total.
The deviation from parallelism between two surfaces, for instance inoptical engineering, is usually measured in arcminutes or arcseconds.In addition, arcseconds are sometimes used inrocking curve (ω-scan)x ray diffraction measurements of high-qualityepitaxial thin films.
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'soptical comparator will often include an option to measure in "minutes and seconds".
It is a straightforward method [to obtain a position at sea] and requires no mathematical calculation beyond addition and subtraction of degrees and minutes and decimals of minutes
[Sextant errors] are sometimes [given] in seconds of arc, which will need to be converted to decimal minutes when you include them in your calculation.
