Minute and second of arc
An illustration of the size of an arcminute (not to scale.) A standard association football ball (22 cm diameter) subtends an angle of 1 arcminute at a distance of approximately 775 meters.
|Unit system||Non-SI units mentioned in the SI|
|Symbol||′ or arcmin|
|In units||Dimensionless with an arc length of approx. ≈ 0.2908/ of the radius, i.e. 0.2908 mm/|
|1 ′ in ...||... is equal to ...|
|degrees||1/° = 0.016°|
|radians||π/ ≈ 0.000290888 rad|
|milliradians||≈ 0.2908 mil|
|gons||600/g = 66.6g|
A minute of arc, arcminute (arcmin), arc minute, or minute arc is a unit of angular measurement equal to 1/ of one degree. Since one degree is 1/ of a turn (or complete rotation), one minute of arc is 1/ of a turn – it is for this reason that the Earth's circumference is almost exactly 21,600 nautical miles. A minute of arc is π/ of a radian.
A second of arc, arcsecond (arcsec), or arc second is 1/ of an arcminute, 1/ of a degree, 1/ of a turn, and π/ (about 1/) of a radian. These units originated in Babylonian astronomy as sexagesimal subdivisions of the degree; they are used in fields that involve very small angles, such as astronomy, optometry, ophthalmology, optics, 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.
The number of square arcminutes in a complete sphere is 148510660 square arcminutes (the surface area of a unit sphere in square units divided by the solid angle area subtended by a square arcminute, also in square units – so that the final result is a dimensionless number).
The names "minute" and "second" have nothing to do with the identically named units of time "minute" or "second". The identical names reflect the ancient Babylonian number system, based on the number 60.
Symbols and abbreviations
The standard symbol for marking the arcminute is the prime (′) (U+2032), though a single quote (') (U+0027) is commonly used where only ASCII characters are permitted. One arcminute is thus written 1′. It is also abbreviated as arcmin or amin or, less commonly, the prime with a circumflex over it ().
The standard symbol for the arcsecond is the double prime (″) (U+2033), though a double quote (") (U+0022) is commonly used where only ASCII characters are permitted. One arcsecond is thus written 1″. It is also abbreviated as arcsec or asec.
|Unit||Value||Symbol||Abbreviations||In radians, approx.|
|Degree||1/ turn||° (degree)||deg||17.4532925 mrad|
|Arcminute||1/ degree||′ (prime)||arcmin, amin, am, , MOA||290.8882087 μrad|
|Arcsecond||1/ arcminute = 1/ degree||″ (double prime)||arcsec, asec, as||4.8481368 μrad|
|Milliarcsecond||0.001 arcsecond = 1/ degree||mas||4.8481368 nrad|
|Microarcsecond||0.001 mas = 0.000001 arcsecond||μas||4.8481368 prad|
In celestial 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′. This notation has been carried over into marine GPS receivers, which normally display latitude and longitude in the latter format by default.
An arcminute is approximately the resolution of the human eye.
- an object of diameter 725.27 km at a distance of one astronomical unit,
- an object of diameter 45866916 km at one light-year,
- an object of diameter one astronomical unit (149597871 km) at a distance of one parsec, by definition.
A 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.
Also notable examples of size in arcseconds are:
- Hubble Space Telescope has calculational resolution of 0.05 arcseconds and actual resolution of almost 0.1 arcseconds, which is close to the diffraction limit.
- crescent Venus measures between 60.2 and 66 seconds of arc.
Since antiquity the arcminute and arcsecond have been used in astronomy. In the ecliptic coordinate system, latitude (β) and longitude (λ); in the horizon system, altitude (Alt) and azimuth (Az); and in the equatorial coordinate system, declination (δ), are all 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.
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 in the course of a year or of a solar system body as the Earth rotates. These small angles may also be written in milliarcseconds (mas), or thousandths of an arcsecond. The unit of distance, the parsec, named from the parallax of one arc second, was developed for such parallax measurements. It is the distance at which the mean radius of the Earth's orbit would subtend an angle of one arcsecond.
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 arcsecond.[a] Because of the effects of atmospheric seeing, ground-based telescopes will smear the image of a star to an angular diameter of about 0.5 arcsecond; in poor seeing conditions this increases to 1.5 arcseconds or even more. The dwarf planet Pluto has proven difficult to resolve because its angular diameter is about 0.1 arcsecond.[clarification needed]
Space telescopes are not affected by the Earth's atmosphere but are diffraction limited. For example, the Hubble Space Telescope 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 arcsecond on a 10 m class telescope.
Minutes (′) and seconds (″) of arc are also used in cartography and navigation. At sea level one minute of arc along the equator or a meridian (indeed, any great circle) equals exactly one geographical mile along the Earth's equator or approximately one nautical mile (1852 meters, or ≈1.15078 statute miles). A second of arc, one sixtieth of this amount, is roughly 30 meters or 100 feet. The exact distance varies along meridian arcs because the figure of the Earth 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, 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 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 (1/ of a degree) have about 1/ the precision of degrees-minutes-seconds (1/ of a degree) and specify locations within about 120 meters or 400 feet.
Property cadastral surveying
Related to cartography, property boundary surveying using the metes and bounds system 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.
The arcminute is commonly found in the firearms industry and literature, particularly concerning the accuracy of rifles, though the industry refers to it as minute of angle (MOA). It is especially popular with shooters familiar with the imperial measurement system because 1 MOA is subtended by a sphere with a diameter of 1.047 inches at 100 yards (2.908 cm at 100 m), a traditional distance on U.S. target ranges. The subtension is linear with the distance, for example, at 500 yards, 1 MOA is subtended by a sphere with a diameter of 5.235 inches, and at 1000 yards 1 MOA is subtended by a sphere with a diameter of 10.47 inches. Since many modern telescopic sights are adjustable in half (1/), quarter (1/), or eighth (1/) 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 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 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 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
Another common system of measurement in firearm scopes is the milliradian or mil. Zeroing a mil based scope is easy for users familiar with base ten systems. The most common adjustment value in mil based scopes is 1/ mil (which approximates 1⁄3 MOA).
- To adjust a 1/ mil scope 0.9 mil down and 0.4 mil 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". 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(m/) × distance. In the example previously given, for 1 minute of arc, and substituting 3,600 inches for 100 yards, 3,600 tan(1/) ≈ 1.047 inches. In metric units 1 MOA at 100 meters ≈ 2.908 centimeters.
Sometimes, a precision firearm's accuracy will be measured in MOA. This simply means that under ideal conditions i.e. no wind, match-grade ammo, clean barrel, and 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 shooting an average 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.
Rifle manufacturers and gun magazines often refer to this capability as sub-MOA, meaning it shoots 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 counterpart of the MOA is the milliradian or mil, being equal to one 1000th of the target range, laid out on a circle that has the observer as centre and the target range as radius. The number of mils on a full such circle therefore always is equal to 2 × π × 1000, regardless the target range. Therefore, 1 MOA ≈ 0.2908 mil. This means that an object which spans 1 mil on the reticle is at a range that is in meters equal to the object's size in millimeters (e.g. an object of 100 mm @ 1 mrad is 100 meters away). So there is no conversion factor required, contrary to the MOA system. A reticle with markings (hashes or dots) spaced with a one mil apart (or a fraction of a mil) are collectively called a mil reticle. If the markings are round they are called mil-dots.
In the table below conversions from mil to metric values are exact (e.g. 0.1 mil equals exactly 1 cm at 100 meters), while conversions of minutes of arc to both metric and imperial values are approximate.
|in at |
|1⁄12′||0.083′||0.024 mil||2.42 mm||0.242 cm||0.0958 in||0.087 in|
|0.25⁄10 mil||0.086′||0.025 mil||2.5 mm||0.25 cm||0.0985 in||0.09 in|
|1⁄8′||0.125′||0.036 mil||3.64 mm||0.36 cm||0.144 in||0.131 in|
|1⁄6′||0.167′||0.0485 mil||4.85 mm||0.485 cm||0.192 in||0.175 in|
|0.5⁄10 mil||0.172′||0.05 mil||5 mm||0.5 cm||0.197 in||0.18 in|
|1⁄4′||0.25′||0.073 mil||7.27 mm||0.73 cm||0.29 in||0.26 in|
|1⁄10 mil||0.344′||0.1 mil||10 mm||1 cm||0.39 in||0.36 in|
|1⁄2′||0.5′||0.145 mil||14.54 mm||1.45 cm||0.57 in||0.52 in|
|1.5⁄10 mil||0.516′||0.15 mil||15 mm||1.5 cm||0.59 in||0.54 in|
|2⁄10 mil||0.688′||0.2 mil||20 mm||2 cm||0.79 in||0.72 in|
|1′||1.0′||0.291 mil||29.1 mm||2.91 cm||1.15 in||1.047 in|
|1 mil||3.438′||1 mil||100 mm||10 cm||3.9 in||3.6 in|
(Values in bold face are exact. All mil fractions are given in tenths, which is more convenient for practical use.)
- 1′ at 100 yards equals 22619/ 21600 = 1.04717593 in ≈ 1.047 inches
- 1′ ≈ 0.291 mil (or 2.91 cm at 100 m, approximately 3 cm at 100 m)
- 1 mil ≈ 3.44′, so 1/ mil ≈ 1/′
- 0.1 mil equals exactly 1 cm at 100 m, or approximately 0.36 inches at 100 yards
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.
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".
- Some studies have shown a larger angular diameter for Betelgeuse. Various studies have produced figures of between 0.042 and 0.069 arcseconds for the star's diameter. The variability of Betelgeuse and difficulties in producing a precise reading for its angular diameter make any definitive figure conjectural.
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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
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[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.
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- MOA / mils By Robert Simeone