Minute and second of arc

A minute of arc, arcminute, or minute arc (MOA), is a unit of angular measurement equal to one sixtieth (1⁄60) of one degree (circle⁄21,600), or (π⁄10,800) radians. In turn, a second of arc or arcsecond is one sixtieth (1⁄60) of one minute of arc. Since one degree is defined as one three hundred and sixtieth (1⁄360) of a rotation, one minute of arc is 1⁄21,600 of a rotation. It is used in those fields which require a unit for the expression of small angles, such as astronomy, navigation and marksmanship.

The number of square arcminutes in a complete sphere is

$4\pi \left({\frac {1}{\pi }}10\,800\right)^{2}={\frac {1}{\pi }}466\,560\,000,$

or approximately 148,510,660.498 square arcminutes.

The arcsecond is 1⁄3,600 of a degree, or 1⁄1,296,000 of a circle, or (π⁄648,000) radians, which is approximately 1⁄206,265 radian.

To express even smaller angles, standard SI prefixes can be employed; in particular, the milliarcsecond, abbreviated mas, is 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 (${\hat {'}}$).

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.

The sexagesimal system of angular measurement

Unit	Value	Symbol	Abbreviations	In radians (approx.)
Degree	1⁄360 circle	°	deg	17.4532925 mrad
Arcminute	1⁄60 degree	′ (prime)	arcmin, amin, am, ${\hat {'}}$, MOA	290.8882087 µrad
Arcsecond	1⁄60 arcminute	″ (double prime)	arcsec, asec, as	4.8481368 µrad
Milliarcsecond	1⁄1,000 arcsecond		mas	4.8481368 nrad
Microarcsecond	10−6 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.

Uses

Firearms

The arcminute is commonly found in the firearms industry and literature, particularly concerning the accuracy of rifles, though the industry tends to refer 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 an 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 10 inches or more. In varmint hunting or competitive target shooting, this can easily mean the difference between a hit and a miss.

Calculating the physical equivalent group size equal to one minute of arc can be done using the equation: equivalent group size = tan(MOA⁄60) × distance. In the example previously given and substituting 3,600 inches for 100 yards, 3,600 tan(1 MOA⁄60) inches = 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 is 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.

Cartography

Minutes of angle (and its subunit, seconds of angle or SOA—equal to a sixtieth of a MOA) are also used in cartography and navigation. At sea level, one minute of angle (around the equator or a meridian) equals about 1.86 kilometres (1.16 mi)), approximately one nautical mile (approximately, because the Earth is slightly oblate); a second of angle is one sixtieth of this amount: about 30 meters or 100 feet.

Traditionally positions are given using degrees, minutes, and seconds of angles in two measurements: one for latitude, the angle north or south of the equator; and one for longitude, the angle east or west of the Prime Meridian. Using this method, any position on or above the Earth's reference ellipsoid can be precisely given. However, because of the somewhat clumsy base-60 nature of MOA and SOA, many people now prefer to give positions using 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 as degrees-minutes-seconds (1⁄3,600 of a degree), and so identify 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.

Astronomy

Comparison of angular diameter of the Sun, Moon, planets and the International Space Station. To get a true representation of the sizes, view the image at a distance of 103 times the width of the "Moon: max." circle. For example, if this circle is 10 cm wide on your monitor, view it from 10.3 m away.

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 quadrillion 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 millarcseconds, 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.

Human vision

In humans, 20/20 vision is the ability to resolve a spatial pattern separated by a visual angle of one minute of arc. A 20/20 letter subtends 5 minutes of arc total.

For raster graphics, Apple Inc asserts that a display of approximately 300 ppi at a distance of 12 inches (305 mm) from one's eye, or 57 arcseconds per pixel is the maximum amount of detail that the human retina can perceive. Raymond Soneira, president of DisplayMate Technologies, however, stated that the resolution of the human retina is higher than claimed by Apple, working out to 477 ppi at 12 inches (305 mm) or 36 arcseconds per pixel.

Materials

The deviation from parallelism between two surfaces, for instance in optical engineering, is usually measured in arcminutes or arcsecond.