Minute and second of arc
A minute of arc (MOA), arcminute (arcmin) or minute arc is a unit of angular measurement equal to one-sixtieth (1⁄60) of one degree. As one degree is 1⁄360 of a circle, one minute of arc is 1⁄21,600 of a circle (or, in radians, π⁄10,800). It is used in fields that involve very small angles, such as astronomy, optometry, ophthalmology, optics, navigation, land surveying and marksmanship.
The number of square arcminutes in a complete sphere is = approximately 148,510,660 square arcminutes.
A second of arc (arcsecond, arcsec) is 1⁄60 of an arcminute, 1⁄3,600 of a degree, 1⁄1,296,000 of a circle, and π⁄648,000 (about 1⁄206,265) of a radian. This is approximately the angle subtended by a U.S. dime coin (18mm) at a distance of 4 kilometres (about 2.5 mi).
To express even smaller angles, standard SI prefixes can be employed; the milliarcsecond (mas), for instance, is commonly used in astronomy.
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||1⁄360 circle||°||deg||17.4532925 mrad|
|Arcminute||1⁄60 degree||′ (prime)||arcmin, amin, am, , MOA||290.8882087 µrad|
|Arcsecond||1⁄60 arcminute||″ (double prime)||arcsec, asec, as||4.8481368 µrad|
|Milliarcsecond||0.001 arcsecond||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, written for example 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.
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. It is especially popular with shooters familiar with the Imperial measurement system because 1 MOA subtends approximately one inch at 100 yards, a traditional distance on target ranges. Since most modern rifle scopes are adjustable in half (1⁄2), quarter (1⁄4), or eighth (1⁄8) MOA increments, also known as clicks, this makes zeroing and adjustments much easier. For example, if the point of impact is 3" high and 1.5" left of the point of aim at 100 yards, 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.
One thing to be aware of is that some scopes, including some higher-end models, are calibrated such that an adjustment of 1 MOA corresponds to exactly 1 inch, rather than 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⁄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. 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" groups on average at 100 yards shoots a group measuring .7" followed by a group that is 1.3" this is not statistically abnormal.
The Metric System counterpart of the MOA is the MilRad, 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 MilRads on a full such circle therefore always is equal to 2 x π x 1000, regardless the target range. Therefore, 1 MOA = 0.2908 MilRad. This means that an object which spans 1 MilRad 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 Milrad is 100 meters away). So there is no conversion factor required, contrary to the MOA system. The markings on a reticle that mark MilRads are called MilDots. Such reticle is called a MilDot Reticle.
Minutes of arc (and its subunit, seconds of arc or SOA—equal to a sixtieth of a MOA) 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 approximately one Nautical mile (1.852 km or 1.151 mi). A second of arc, one sixtieth of this amount, is about 30 meters or roughly 100 feet. The exact distance varies along meridian arcs because the figure of the Earth is slightly oblate.
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, because of the somewhat clumsy base-60 nature of minutes and seconds, positions are frequently expressed in fractional degrees only, expressed in decimal form to an equal amount of precision. Degrees given to three decimal places (1⁄1,000 of a degree) have about 1⁄4 the precision of degrees-minutes-seconds (1⁄3,600 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 and arcsecond are also used in astronomy. Degrees (and therefore arcminutes) are used to measure declination, or angular distance north or south of the celestial equator. The arcsecond is also often used to describe parallax, due to very small parallax angles for stellar parallax, and tiny angular diameters (e.g. Venus varies between 10′′ and 60′′). The parallax, proper motion and angular diameter of a star may also be written in milliarcseconds (mas), or thousandths of an arcsecond. The parsec gets its name from "parallax second", for those arcseconds.
The ESA astrometric space probe Gaia is hoped to measure star positions to 20 microarcseconds (µas) when it begins producing catalog positions sometime after 2016. There are about 1.3 trillion µas in a circle. As seen from Earth, one µas is about the size of a period at the end of a sentence in the Apollo mission manuals left on the moon. Currently the best catalog positions of stars actually measured are in terms of milliarcseconds, by the U.S. Naval Observatory. A milliarcsecond is about the size of a dime atop the Eiffel Tower as seen from New York City.
Apart from the Sun, the star with the largest angular diameter from Earth is R Doradus, a red supergiant with a diameter of 0.05 arcsecond. 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. This is roughly equivalent to a (40 mm) ping-pong ball viewed at a distance of 50 miles (80 km).
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.
The deviation from parallelism between two surfaces, for instance in optical engineering, is usually measured in arcminutes or arcseconds.
Notes and references
- Filippenko, Alex, Understanding the Universe (of The Great Courses, on DVD), Lecture 43, time 12:05, The Teaching Company, Chantilly, VA, USA, 2007
- "CELESTIAL NAVIGATION COURSE". International Navigation School. Retrieved 4 November 2010.
It is a straight forward method [to obtain a position at sea] and requires no mathematical calculation beyond addition and subtraction of degrees and minutes and decimals of minutes
- "Astro Navigation Syllabus". Retrieved 4 November 2010.
[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.
- "Shipmate GN30". Norinco. Retrieved 4 November 2010.
- Mann, Richard (2011-02-18). "Mil, MOA or inches?". Shooting Illustrated. Retrieved 2015-04-13.
- Wheeler, Robert E. "Statistical notes on rifle group patterns" (PDF). Retrieved 21 May 2009.
- Bramwell, Denton (January 2009). "Group Therapy The Problem: How accurate is your rifle?". Varmint Hunter 69. Retrieved 21 May 2009.
- 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.
- NASA.gov Pluto Fact Sheet
MOA / mils By Robert Simeone