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A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of Angular unit measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a turn. The nautical miles (nmi) was originally defined as the meridian arc of a minute of latitude on a spherical Earth, so the actual Earth's circumference is very near . A minute of arc is of a radian.
A second of arc, arcsecond (abbreviated as arcsec), or arc second, denoted by the symbol , is a unit of Angular unit measurement equal to of a minute of arc, of a degree, of a turn, and (about ) of a radian.
These units originated in Babylonian astronomy as sexagesimal (base 60) 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. For a two-dimensional area such as on (the surface of) a sphere, square arcminutes or seconds may be used.
Similarly, double prime (U+2033) designates the arcsecond, though a double quote (U+0022) is commonly used where only ASCII characters are permitted. One arcsecond is thus written as 1″. It is also abbreviated as arcsec or asec.
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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 and aviation GPS receivers, which normally display latitude and longitude in the latter format by default.
One arcminute () is the approximate distance two contours can be separated by, and still be distinguished by, a person with visual acuity. The average angular diameter of the full moon is about , or .
One arcsecond () is the angle subtended by:
One milliarcsecond () is about the size of a half dollar (), seen from a distance equal to that between the Washington Monument and the Eiffel Tower (around ).
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 nickel () on the surface of Neptune as observed from Earth.
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 systems; 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 parallax angle of one arc second, was developed for such parallax measurements. The distance from the Sun to a celestial object is the reciprocal of the angle, measured in arcseconds, of the object's apparent movement caused by parallax.
The European Space Agency's astrometry satellite Gaia mission, launched in 2013, can approximate star positions to 7 microarcseconds (μas).
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 blurring, ground-based 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″. 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 are diffraction limited. For example, the Hubble Space Telescope can reach an angular size of stars down to about 0.1″.
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-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 ( of a degree) have about the precision of degrees-minutes-seconds ( of a degree) and specify locations within about . 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.
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:
Another common system of measurement in firearm scopes is the milliradian (mrad). Zeroing an mrad based scope is easy for users familiar with decimal systems. The most common adjustment value in mrad based scopes is mrad (which approximates 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, 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() × distance. In the example previously given, for 1 minute of arc, and substituting 3,600 inches for 100 yards, 3,600 tan() ≈ 1.047 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 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, 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 (mrad or 'mil'), being equal to 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 × × 1000, regardless the target range. Therefore, 1 MOA ≈ 0.2909 mrad. This means that an object which spans 1 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 mm subtending 1 mrad is 100 metres away). 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 mrad equals exactly 10 mm at 100 metres), while conversions of minutes of arc to both metric and imperial values are approximate.
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