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Minute and second of arc
View on Wikipedia | This article needs additional citations for verification. (March 2022) |
| Arcminute | |
|---|---|
 An illustration of the size of an arcminute (not to scale). A standard association 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 (2,480 ft). | |
| General information | |
| Unit system | Non-SI units mentioned in the SI |
| Unit of | Angle |
| Symbol | ′, arcmin |
| In units | Dimensionless with an arc length of approx. ≈ 0.2909/1000 of the radius, i.e. 0.2909 mm/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 = 0.0185g |
| turns | 1/21600 turn |
A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol ′, is a unit of angular measurement equal to 1/60 of a degree.[1] Since one degree is 1/360 of a turn, or complete rotation, one arcminute is 1/21600 of a turn. The nautical mile (nmi) was originally defined as the arc length of a minute of latitude on a spherical Earth, so the actual Earth's circumference is very near 21600 nmi. A minute of arc is π/10800 of a radian.
A second of arc, arcsecond (abbreviated as arcsec), or arc second, denoted by the symbol ″,[2] is a unit of angular measurement equal to 1/60 of a minute of arc, 1/3600 of a degree,[1] 1/1296000 of a turn, and π/648000 (about 1/206264.8) 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.
Symbols and abbreviations
[edit]The prime symbol ′ (U+2032) designates the arcminute,[2] though a single quote ' (U+0027) is commonly used where only ASCII characters are permitted. One arcminute is thus written as 1′. It is also abbreviated as arcmin or amin.
Similarly, double prime ″ (U+2033) designates the arcsecond,[2] 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.
| 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 | ||
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′.[3][4] 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.[5]
Common examples
[edit]In general, by simple trigonometry, it can be derived that the angle subtended by an object of diameter or length at a distance is given by the following expression:
One arcminute (1′) is the approximate distance two contours can be separated, and still be distinguished, by a person with 20/20 vision. The average apparent diameter of the full Moon is about 31′, or 0.52°.
One arcsecond (1″) is the angle subtended by:
- a U.S. dime coin (0.705 in; 17.9 mm) at a distance of 3.7 kilometres (2.3 mi)[6]
- an object of diameter 725.27 km at a distance of one astronomical unit (149597870.7 km)
- an object of diameter 45866916 km at one light-year (9460730472580.8 km)
- an object of diameter one astronomical unit at a distance of one parsec, per the definition of the latter.[7]
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.[8]
- At crescent phase, Venus measures between 60.2 and 66 seconds of arc.[8]
One milliarcsecond (1 mas) is about the size of a half dollar (1.205 in; 30.6 mm), seen from a distance equal to that between the Washington Monument and the Eiffel Tower (around 6,300 km or 3,900 mi).
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.[9]
One nanoarcsecond is about the size of a nickel (0.835 in; 21.2 mm) on the surface of Neptune as observed from Earth.
History
[edit]The concepts of degrees, minutes, and seconds—as they relate to the measure of both angles and time—derive from Babylonian astronomy and time-keeping. Influenced by the Sumerians, the ancient Babylonians divided the Sun's perceived motion across the sky over the course of one full day into 360 degrees.[10][failed verification] Each degree was subdivided into 60 minutes and each minute into 60 seconds.[11][12] Thus, one Babylonian degree was equal to four minutes in modern terminology, one Babylonian minute to four modern seconds, and one Babylonian second to 1/15 (approximately 0.067) of a modern second.
Uses
[edit]Astronomy
[edit]
Since antiquity, the arcminute and arcsecond have been used in astronomy: in the ecliptic coordinate system as latitude (β) and longitude (λ); in the horizon system as altitude (Alt) and azimuth (Az); and in the equatorial coordinate system as declination (δ). All are 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.
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 astrometric satellite Gaia, launched in 2013, can approximate star positions to 7 microarcseconds (μas).[13]
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 telescopes 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″.[14] 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″.
Cartography
[edit]Minutes (′) and seconds (″) of arc are also used in cartography and navigation. At sea level. one minute of arc along the equator equals exactly one geographical mile (not to be confused with international mile or statute mile) along the Earth's equator or approximately one nautical mile (1,852 metres; 1.151 miles).[15] A second of arc, one sixtieth of this amount, is roughly 30 metres (98 feet). The exact distance varies along meridian arcs or any other great circle 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 (1/1000 of a degree) have about 1/4 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 50°39′44.2″N 1°35′30.5″W.[16]
Property cadastral surveying
[edit]Related to cartography, property boundary surveying using the metes and bounds system and cadastral surveying 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.
Firearms
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The arcminute is commonly found in the firearms industry and literature, particularly concerning the precision of rifles, though the industry refers to it as minute of angle (MOA). It is especially popular as a unit of measurement with shooters familiar with the imperial measurement system because 1 MOA subtends a circle with a diameter of 1.047 inches (which is often rounded to just 1 inch) at 100 yards (2.66 cm at 91 m or 2.908 cm at 100 m), a traditional distance on American target ranges. The subtension is linear with the distance, for example, at 500 yards, 1 MOA subtends 5.235 inches and, at 1,000 yards, 1 MOA subtends 10.47 inches. Since many modern telescopic sights are adjustable in half (1/2), quarter (1/4) or eighth (1/8) 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, 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:
- 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 (mrad). Zeroing an mrad based scope is easy for users familiar with base ten systems. The most common adjustment value in mrad based scopes is 1/10 mrad (which approximates 1⁄3 MOA).
- To adjust a 1/10 mrad scope 0.9 mrad down and 0.4 mrad 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 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 1,000 yards,[17] 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 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 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 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.[18][19]
The metric system counterpart of the MOA is the milliradian (mrad or 'mil'), being equal to 1⁄1000 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 spanning 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).[20] 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.
| 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 |
- 1′ at 100 yards is about 1.047 inches[21]
- 1′ ≈ 0.291 mrad (or 29.1 mm at 100 m, approximately 30 mm at 100 m)
- 1 mrad ≈ 3.44′, so 1/10 mrad ≈ 1/3′
- 0.1 mrad equals exactly 1 cm at 100 m, or exactly 0.36 inches at 100 yards
Human vision
[edit]In humans, 20/20 vision is the ability to resolve a spatial pattern separated by a visual angle of one minute of arc, from a distance of twenty feet. A 20/20 letter subtends 5 minutes of arc total.
Materials
[edit]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.
Manufacturing
[edit]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".
See also
[edit]References
[edit]- ^ a b Weisstein, Eric W. "Arc Second". mathworld.wolfram.com. Retrieved 31 August 2020.
- ^ a b c "Minutes of Arc to Degree Conversion". Inch Calculator. Retrieved 25 July 2021.
- ^ "CELESTIAL NAVIGATION COURSE". International Navigation School. Retrieved 4 November 2010.
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
- ^ "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. Archived from the original on 24 January 2008. Retrieved 4 November 2010.
- ^ Filippenko, Alex, Understanding the Universe (of The Great Courses, on DVD), Lecture 43, time 12:05, The Teaching Company, Chantilly, VA, US, 2007.
- ^ "Cosmic Distance Scales - The Milky Way". Imagine the Universe!.
- ^ a b "What is the diffraction limit of a telescope?". Cornell University. Archived from the original on 17 September 2016.
- ^ Schmitt, Matthias (1 July 2018). "The Kessel Run and the Parsec". Amateur Astronomers Association. Archived from the original on 17 March 2025. Retrieved 11 July 2025.
- ^ Lombardi, Michael A. (5 March 2008). "Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?". Scientific American. SCIENTIFIC AMERICAN, a Division of Springer Nature America, Inc. Retrieved 25 July 2021.
- ^ Correll, Malcolm (November 1977). "Early Time Measurements". The Physics Teacher. 15 (8): 476–479. Bibcode:1977PhTea..15..476C. doi:10.1119/1.2339739.
- ^ Stephenson, F. Richard; Fatoohi, Louay J. (May 1994). "The Babylonian Unit of Time". Journal for the History of Astronomy. 25 (2): 99–110. Bibcode:1994JHA....25...99S. doi:10.1177/002182869402500203.
- ^ Amos, Jonathan (14 September 2016). "Celestial mapper plots a billion stars". BBC News. Retrieved 31 March 2018.
- ^ "Pluto Fact Sheet". nssdc.gsfc.nasa.gov. Retrieved 29 August 2022.
- ^ Kaplan, George H. (1 January 2003). "Nautical mile approximates an arcminute". Ocean Navigator. Navigator Publishing. Retrieved 22 March 2017.
- ^ The Corporation of Trinity House (10 January 2020). "1/2020 Needles Lighthouse". Notices to Mariners. Retrieved 24 May 2020.
- ^ Mann, Richard (18 February 2011). "Mil, MOA or inches?". Shooting Illustrated. Archived from the original on 10 November 2013. Retrieved 13 April 2015.
- ^ Wheeler, Robert E. "Statistical notes on rifle group patterns" (PDF). Archived from the original (PDF) on 26 September 2006. Retrieved 21 May 2009.
- ^ Bramwell, Denton (January 2009). "Group Therapy The Problem: How accurate is your rifle?". Varmint Hunter. 69. Archived from the original on 7 October 2011. Retrieved 21 May 2009.
- ^ Fouad Sabry (2022). Precision Guided Firearm. One Billion Knowledgeable.
- ^ Dexadine Ballistics Software – ballistic data for shooting and reloading. See Talk
External links
[edit]- MOA/ mils By Robert Simeone
- A Guide to calculate distance using MOA Scope by Steve Coffman
Minute and second of arc
View on GrokipediaFundamentals
Definition
A minute of arc, commonly referred to as an arcminute and denoted by the symbol ′, is a unit of angular measurement equal to one-sixtieth (1/60) of a degree.[6] This subdivision allows for finer resolution in measuring angles beyond the degree scale. Exactly, one arcminute corresponds to radians, which is approximately 0.000290888 radians.[7] A second of arc, or arcsecond and denoted by ″, is one-sixtieth (1/60) of an arcminute, thereby equaling one-3600th (1/3600) of a degree.[6] Precisely, it equates to radians, or roughly radians.[7] These units stem from the sexagesimal (base-60) system employed in angular measurement, where 60 serves as the divisor due to its exceptional divisibility—being evenly shared by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60—which simplifies fractional divisions and arithmetic operations in practical computations.[8] Within the framework of a complete circle spanning 360 degrees, arcminutes and arcseconds enable the precise quantification of minute angular separations, such as the apparent diameters of celestial bodies or fine positional differences.[9]Relation to Degrees and Radians
The minute and second of arc form a sexagesimal subdivision of the degree, a unit historically derived from the Babylonian base-60 system. Specifically, one degree equals 60 arcminutes, and one arcminute equals 60 arcseconds.[10][11] Consequently, a full circle of 360 degrees comprises 21,600 arcminutes or 1,296,000 arcseconds.[7][12] To convert between these units and degrees, an arcminute value is divided by 60 to obtain degrees, while an arcsecond value is divided by 3,600 (or equivalently, divided by 60 twice).[10][13] For example, 30 arcminutes equals 0.5 degrees, and 1,800 arcseconds also equals 0.5 degrees. These conversions maintain the hierarchical precision inherent to the system. In relation to radians, the SI-derived unit for plane angles, the connections follow from the established equivalence of 180 degrees to radians. Thus, one arcminute corresponds exactly to degrees or radians, while one arcsecond corresponds exactly to degrees or radians.[14][7] Approximately, this yields 1 arcminute radians and 1 arcsecond radians.[13] Arcseconds provide the necessary precision for sub-degree angular measurements, such as resolving fine details in astronomical observations where objects subtend angles smaller than one degree, whereas decimal degrees suffice for coarser scales like broad geographic coordinates.[15][16] This distinction ensures appropriate granularity without excessive computational overhead in applications requiring high resolution.Notation and Symbols
Standard Symbols
The standard symbols for denoting the minute and second of arc are the single prime (′) for arcminutes and the double prime (″) for arcseconds, as established in the International System of Quantities and Units per the ISO 80000 series and aligned with NIST guidelines.[17] These symbols derive from the sexagesimal angular divisions originating in ancient Babylonian cuneiform notations around 2000 BCE, where positional systems without dedicated glyphs represented subdivisions of the circle into 360 degrees, evolving through Greek astronomical texts using accent marks for fractions, and transmitted through medieval scholarship, culminating in the adoption of prime symbols by the 16th century for clarity in printed scientific literature.[8][18] To prevent ambiguity with quotation marks or linear measurements such as feet (′) and inches (″), modern standards recommend using the distinct typographic prime symbols rather than straight quotes (' and ") in technical writing.[19] In scientific texts, these primes are rendered upright (roman typeface) to distinguish them from mathematical variables, which may appear italicized.[19] No space is placed between a numerical value and the unit symbols °, ′, or ″, but spaces separate the components of a compound angle—e.g., 45° 12′ 30″—ensuring consistent readability.[17] The Unicode encodings for these symbols are U+2032 (PRIME) for the single prime and U+2033 (DOUBLE PRIME) for the double prime, facilitating precise digital representation across platforms. In some older astronomical texts or specialized notations, variations such as superscripted forms (e.g., ¹ or ²) for minutes and seconds appear, reflecting pre-standardization practices before the widespread adoption of primes.[18]Abbreviations and Usage Conventions
The minute of arc is commonly abbreviated in text as arcmin or amin, and sometimes as MOA (minute of angle) in specialized contexts like ballistics, while the full form "minute of arc" is preferred in formal scientific writing for precision.[20] Similarly, the second of arc is abbreviated as arcsec or asec, with the full form "second of arc" used to maintain clarity in technical documents.[21] These textual abbreviations help distinguish angular measurements from time units, and when primes are represented textually, a single apostrophe (') denotes arcminutes and a double quotation mark (") denotes arcseconds.[22] According to guidelines from the National Institute of Standards and Technology (NIST), in technical papers involving the International System of Units (SI), unabbreviated forms like "minute of arc" are recommended for non-SI angular units to avoid ambiguity, particularly with time measurements; specifying "of arc" is advised in mixed-unit contexts.[23] For instance, to prevent confusion with minutes and seconds of time, explicit phrasing such as "30 arcminutes" is standard in interdisciplinary fields.[24] In astronomy, field-specific variations include the milliarcsecond (mas), equivalent to 0.001 arcseconds, which is widely used for measuring tiny angular separations like those in stellar parallax.[23] In surveying, decimal approximations—such as expressing angles in decimal degrees (e.g., 45.5° instead of 45° 30′)—are conventional for ease of computation and data storage, often converting arcminutes and arcseconds to decimal fractions.[25] Best practices for notation emphasize combining units with the degree symbol, as in 45° 30′ 15″, to convey hierarchical precision without excessive verbosity.[22] However, in software and digital rendering, the prime (′) and double prime (″) symbols may be misinterpreted or poorly displayed as apostrophes or quotation marks, potentially confusing them with feet and inches; thus, textual abbreviations like arcmin are recommended for compatibility across platforms.[26]Historical Development
Ancient Origins
The origins of the minute and second of arc trace back to the ancient Mesopotamian civilizations, particularly the Babylonians, who developed the sexagesimal (base-60) numeral system around 2000 BCE for astronomical calculations. This system facilitated the division of the full circle into 360 degrees, with each degree further subdivided into 60 arcminutes and each arcminute into 60 arcseconds, allowing precise measurement of celestial positions along the ecliptic.[27][28] Babylonian astronomical tablets, such as the cuneiform records from the late 2nd millennium BCE, exemplify these divisions in practice, documenting planetary motions and stellar alignments using sexagesimal fractions of degrees for angular extents. In Mesopotamian astronomy, this framework intertwined with timekeeping, as the same base-60 units measured both temporal intervals—dividing the day into 24 hours of 60 minutes each—and angular separations, reflecting the integrated nature of observing heavenly bodies over time.[28][29] The Greeks later adopted and refined this sexagesimal angular system; notably, Hipparchus in the 2nd century BCE employed degrees subdivided into arcminutes and arcseconds to catalog the positions of approximately 850 stars, enabling accurate equatorial coordinates for the first time.[30][31]Standardization in the Modern Era
During the Renaissance, Danish astronomer Tycho Brahe advanced the precision of angular measurements in astronomy, achieving accuracies as fine as 27 arcseconds in observations of stellar positions through the use of large, fixed instruments like quadrants and sextants at his Uraniborg observatory.[32] These refinements built on sexagesimal divisions, enabling Brahe to record planetary and stellar data with unprecedented detail for the era, which later informed Kepler's laws of planetary motion.[33] In the 18th and 19th centuries, minutes and seconds of arc became integral to nautical almanacs, such as the British Nautical Almanac first published in 1767, which tabulated celestial body positions to arcseconds for longitude determination via lunar distances and chronometer comparisons.[34] This adoption extended to international astronomy, where observatories like Greenwich standardized ephemerides using these units for precise ephemeris calculations.[35] Despite French Revolutionary efforts to decimalize angles into grades (each 0.9 sexagesimal degrees) and centesimal minutes under the metric system from 1795, the sexagesimal system persisted in astronomy and navigation due to entrenched traditions and practical utility in tables and instruments.[36] The 20th century saw formal institutional standardization by the International Astronomical Union (IAU), which at its 1928 General Assembly in Leiden established consistent reference frames and measurement protocols for celestial positions, implicitly affirming the arcsecond as the fundamental unit for high-precision astrometry.[37] Subsequent IAU updates, including resolutions on reference systems, reinforced these practices. In 2009, the International Organization for Standardization (ISO) codified the symbols ′ for the minute of arc and ″ for the second of arc in ISO 80000-2, defining them as non-decimal submuliples of the degree for plane angles while aligning with SI conventions.[38][39] The 2019 redefinition of the SI base units, effective May 20, 2019, fixed the values of fundamental constants like the speed of light, indirectly enhancing angular precision by establishing the radian (rad) as an exactly defined coherent unit derived from these constants, thereby providing a more stable relation between sexagesimal subunits and metric angular measures in scientific computations.[40] This shift emphasizes the radian's role without altering the definitions of arcminutes or arcseconds, but it supports higher accuracy in conversions for modern applications like satellite geodesy.[41]Applications in Science and Technology
Astronomy and Celestial Navigation
In astronomy, arcminutes and arcseconds are essential for specifying the positions of celestial objects in the sky, particularly through coordinates of right ascension and declination. The Hipparcos Catalogue, produced by the European Space Agency's astrometry mission, provides positions for over 118,000 stars with a precision of approximately 1 milliarcsecond (0.001 arcseconds) in both right ascension and declination, enabling highly accurate mapping of stellar locations relative to the celestial sphere.[42] This level of detail has revolutionized the reference frame for stellar astrometry, allowing astronomers to track proper motions and parallaxes with unprecedented accuracy.[43] Arcminutes and arcseconds also quantify the apparent sizes of celestial bodies as seen from Earth, which is crucial for understanding visibility and observational requirements. The Moon's angular diameter varies between about 29 and 34 arcminutes, averaging around 30 arcminutes, while the Sun's is similarly about 31 to 32 arcminutes on average.[44] These scales highlight why both appear roughly the same size in the sky, facilitating phenomena like solar eclipses. Telescope resolution limits further emphasize the role of arcseconds; for instance, the Hubble Space Telescope achieves an angular resolution of about 0.05 arcseconds at visible wavelengths, allowing it to resolve fine details in distant galaxies and star clusters that ground-based observatories cannot.[45] In celestial navigation, particularly for maritime use, arcminutes are the primary unit for sextant measurements to determine latitude by observing the altitude of celestial bodies above the horizon. A typical marine sextant measures angles to a precision of 0.1 arcminutes, enabling navigators to compute their position with an accuracy of about 1 nautical mile. Historical marine chronometers complemented these sights by providing precise timekeeping, essential for longitude calculation, as each second of time error corresponds to 15 arcseconds of longitude displacement due to Earth's rotation.[46] This integration of angular measurements and time allowed sailors to achieve positional accuracies on the order of arcminutes to arcseconds during the age of sail. Modern astronomical applications leverage arcseconds for parallax measurements to estimate stellar distances. Stellar parallax is the apparent shift in a star's position against background stars over six months, with the distance in parsecs defined as the reciprocal of the parallax angle in arcseconds; thus, a star with a 1 arcsecond parallax is 1 parsec away, equivalent to 206,265 astronomical units (the number of arcseconds in a radian).[47] Missions like Hipparcos measured parallaxes to milliarcsecond precision, yielding distances for thousands of nearby stars and establishing the cosmic distance scale.[42]Cartography and Geodesy
In cartography and geodesy, minutes and seconds of arc form the basis of the degrees-minutes-seconds (DMS) notation for geographic coordinate systems, which specify positions on Earth's surface using latitude and longitude. This system is integral to GPS receivers, nautical charts, and topographic mapping, where latitude is measured north or south from the equator and longitude east or west from the Prime Meridian, both in angular units subdivided into 60 minutes per degree and 60 seconds per minute. For example, USGS topographic maps and NOAA nautical charts routinely display coordinates in DMS to achieve positional accuracy suitable for navigation and land management.[48][49][50] Geodetic datums like the World Geodetic System 1984 (WGS84) incorporate arcseconds to precisely define the reference ellipsoid's parameters and orientation, ensuring consistency in global positioning. WGS84 orients its Z-axis (Earth's rotation axis) and X-axis (equatorial plane) with an uncertainty of 0.005 arcseconds relative to the International Terrestrial Reference Frame, while transformations between datums often specify coordinate shifts in arcseconds for latitude and longitude. This high angular precision supports arcminute-level accuracy in map projections, reducing errors in ellipsoidal height and horizontal positioning for applications such as satellite geodesy and continental-scale surveys.[51][52] Map scales in cartography leverage angular measurements to relate spherical distances to linear ones on flat representations; at the equator, one arcminute of latitude equates to approximately 1,852 meters, defining the international nautical mile and aiding in the creation of uniform grid systems. This approximation holds for latitude globally due to parallel spacing, but longitude varies, narrowing to zero at the poles, which informs scale calculations in projections like the Universal Transverse Mercator (UTM). Such conversions are essential for deriving representative fractions in maps, where 1 arcminute provides a practical benchmark for mid-latitude distances around 1.85 kilometers.[53][48] In historical cartography, the Mercator projection, developed in 1569 for nautical use, utilized arcminutes to assess and mitigate scale distortions inherent in cylindrical mappings of the globe. The projection preserves rhumb lines as straight paths for constant-bearing navigation but introduces angular scale variations with latitude, quantified in early charts through differences in arcminute equivalents to ensure reliable distance estimations for mariners. For instance, distortions in Mercator-based maps were evaluated to keep errors within a few arcminutes near the equator, influencing the design of 16th- and 17th-century sea atlases by cartographers like Edward Wright, who refined the projection's graticule for improved angular fidelity.[54][55]Surveying and Engineering
In surveying and engineering, theodolites and total stations serve as primary instruments for angular measurements, enabling precise determination of property boundaries and land configurations. These devices integrate optical theodolites with electronic distance measurement (EDM) capabilities, allowing surveyors to record horizontal and vertical angles with accuracies typically ranging from 0.5 to 5 arcseconds, depending on the model and application. For property boundary establishment, such precision translates to positional errors of less than 1 millimeter per kilometer of distance, essential for resolving legal disputes and defining parcel perimeters in urban and rural settings. High-end total stations, like those used in professional land surveys, achieve 1 arcsecond accuracy for routine boundary work, ensuring compliance with jurisdictional standards for demarcation.[56][57][58] C cadastral surveying, which focuses on legal land parcel definitions, frequently employs minutes of arc for angular specifications in systems such as the U.S. Public Land Survey System (PLSS). Under the PLSS, established by the Bureau of Land Management, township and section boundaries are delineated using bearings recorded to the nearest arcminute in field notes and legal descriptions, providing a standardized framework for dividing public lands into rectangular plots of 640 acres per section. This arcminute-level resolution accommodates the grid-based layout while accounting for geodetic convergency over large areas, facilitating accurate retracement of historical surveys for modern property transfers and subdivisions.[59][60] In civil engineering contexts, arcseconds are integral to specifying alignments for infrastructure like bridges and tunnels, where even minor angular deviations can affect structural stability and operational efficiency. Bridge alignments, for example, require angular controls within 2-5 arcseconds to maintain uniform load paths across spans exceeding 100 meters, as deviations could lead to differential settlements. Tunnel gradients, critical for drainage and vehicle clearance, are similarly surveyed to arcsecond precision using total stations to establish longitudinal slopes, ensuring the final bore aligns with entry and exit portals without excessive curvature. These applications draw on theodolite-based setups for initial control networks, transitioning to robotic total stations for ongoing monitoring during construction.[61][62] Error tolerances in these fields emphasize arcsecond-level precision to meet safety and regulatory requirements, with high-accuracy surveys typically limiting angular errors to 5 arcseconds or less for critical measurements. This standard, outlined in U.S. Army Corps of Engineers guidelines for structural deformation monitoring, applies to engineering projects involving boundaries and alignments, where exceeding such tolerances could invalidate surveys or necessitate costly rework. For instance, in bridge and tunnel projects, total station observations are adjusted to achieve relative angular accuracies better than 5 arcseconds, balancing instrument capabilities with environmental factors like temperature and vibration.[61][63]Optics and Human Vision
In human vision, normal visual acuity, often denoted as 20/20, corresponds to the ability to resolve fine details subtending approximately 1 arcminute of angular separation at the eye, such as the stroke width of letters on a standard eye chart viewed at 20 feet.[64] This resolution limit arises from the combined effects of optical aberrations, retinal sampling by photoreceptors, and neural processing in the visual pathway. For more demanding tasks, such as vernier acuity—where alignment offsets between line segments are detected—human observers can achieve hyperacuity thresholds as fine as 3 to 5 arcseconds under optimal conditions, surpassing the spacing of individual retinal receptors.[65][66] The angular resolution of optical instruments like microscopes and cameras is fundamentally constrained by diffraction, as described by the Rayleigh criterion, which defines the minimum resolvable angle θ between two point sources as θ ≈ 1.22 λ / D radians, where λ is the wavelength of light and D is the aperture diameter; converting to arcseconds yields θ ≈ (1.22 λ / D) × 206265 arcseconds.[67] For visible light (λ ≈ 550 nm) and a typical camera lens aperture of D = 50 mm, this limit is around 2.8 arcseconds, enabling resolutions far beyond human capabilities when magnification is applied. In microscopy, similar diffraction limits apply, with high-numerical-aperture objectives achieving effective angular resolutions on the order of several arcseconds for specimen features, though practical limits are often higher due to aberrations and illumination.[68] Anatomically, the fovea centralis of the human retina, responsible for high-acuity vision, features a mosaic of cone photoreceptors with average spacing of about 30 arcseconds in the central foveola, providing the neural substrate for resolving details near the 1 arcminute limit under photopic conditions.[69] This cone density gradient decreases with eccentricity, but the one-to-one wiring of foveal cones to ganglion cells preserves spatial precision for angular subtenses in this range.[70] In clinical optometry, visual acuity deficits are quantified using charts like the Snellen or ETDRS, where letter optotypes are designed such that the critical details (e.g., stroke width) subtend 1 arcminute at the specified testing distance for normal vision, allowing measurement of impairments in arcminute units.[64] For instance, a 20/40 acuity indicates resolution limited to 2 arcminutes, guiding diagnoses of refractive errors, amblyopia, or retinal pathologies.[4] These assessments emphasize angular metrics to standardize evaluation across patients and instruments.[71]Precision Manufacturing and Materials
In precision manufacturing, angular measurements in minutes and seconds of arc are essential for achieving micron-level tolerances in fabricating small components, particularly in semiconductor production where misalignment can lead to defects. Wafer alignment systems in semiconductor fabrication routinely attain rotational accuracies of less than 2 arcseconds to position silicon wafers precisely before lithographic exposure and etching processes. Similarly, advanced positioning stages, such as hybrid hexapods, deliver angular repeatability of 0.1 arcseconds, enabling sub-micron alignments in the assembly of microelectronic devices and photonic components.[72][73] Photolithography exemplifies the critical role of arcsecond-level precision, where mask-to-wafer alignment errors directly impact overlay accuracy and yield rates. In scanning beam electron-beam lithography systems, beam centroids are aligned to within 0.4 arcseconds to ensure seamless stitching of patterns across large areas, minimizing distortion in high-resolution features below 10 nm. Such tolerances prevent misalignment-induced defects, as even sub-arcsecond errors can shift features by tens of nanometers on a 300 mm wafer, significantly reducing manufacturing efficiency.[74] For testing material properties, angular measurements down to micro-arcminutes quantify subtle deformations under stress, providing insights into elasticity and fatigue. Electronic autocollimators, often integrated with strain measurement setups, resolve angular displacements to 0.01 arcseconds (equivalent to 0.6 micro-arcminutes), allowing detection of torsional or bending deformations in composite materials and thin films during tensile or fatigue tests. These tools complement linear strain gauges by capturing rotational components of deformation, essential for validating models in advanced materials like carbon fiber reinforced polymers.[75][76] In laser-based and CNC machining systems, beam divergence and pointing stability are specified in arcseconds to maintain focus over working distances, ensuring clean cuts in delicate materials. High-precision CO2 or fiber lasers for micromachining achieve beam divergences below 1 arcsecond through optimized optics, enabling kerf widths under 50 μm in metals and ceramics without thermal distortion. This angular control is vital for applications like drilling micro-vias in printed circuit boards, where divergence exceeding a few arcseconds would broaden the spot size and compromise edge quality.[77]Ballistics and Firearms
In ballistics and firearms, the minute of arc (MOA) serves as a fundamental unit for measuring angular deviations in projectile trajectories and for calibrating sighting systems. Rifle scopes and sights often feature reticle adjustments calibrated in MOA increments, allowing shooters to compensate for factors like bullet drop, wind drift, and environmental variables by dialing the turrets to shift the point of aim. Typically, these adjustments are made in quarter-MOA clicks (0.25 MOA), providing precise corrections where each full MOA subtends approximately 1.047 inches at 100 yards.[78][79] Trajectory calculations in ballistics frequently express bullet drop in terms of MOA to determine the necessary elevation adjustments for long-range shots. For instance, a .308 Winchester round zeroed at 100 yards may experience a drop equivalent to about 12 MOA (roughly 64 inches) at 500 yards for a typical 168-grain load at 2700 fps muzzle velocity, requiring the shooter to elevate the scope accordingly to align the line of sight with the bullet's curved path under gravity.[80][81][82] This angular compensation is derived from ballistic software or tables that model the projectile's arc, emphasizing MOA's role in translating linear drop into actionable scope adjustments without needing complex trigonometric computations during field use. Military standards for precision firearms, particularly sniper rifles, demand high accuracy expressed in sub-MOA terms to ensure reliable hits at extended ranges. The U.S. Army's M24 Sniper Weapon System, for example, is required to achieve 0.35 MOA extreme spread from a machine rest at 100 yards, enabling consistent performance out to 800 meters or more with match-grade ammunition. Many modern tactical scopes paired with these rifles incorporate 0.25 MOA adjustments to fine-tune for such precision, supporting missions where angular errors must be minimized to fractions of a minute.[83] Historically, during World War II, artillery firing solutions relied on angular measurements in minutes and seconds of arc for accurate targeting. British field artillery manuals specified gun orientation and laying in degrees and minutes of arc, with firing tables providing elevation and deflection data to account for projectile trajectories over varying distances and conditions. These tables, computed using ballistics labs, incorporated arcsecond-level precision in spotter corrections to refine barrages, reflecting the era's transition toward standardized angular units for coordinated fire control.[84]Mathematical and Computational Aspects
Conversions and Formulas
The minute of arc (arcminute) is defined as one-sixtieth of a degree, and the second of arc (arcsecond) is one-sixtieth of an arcminute, equivalent to one-3600th of a degree.[7] These relationships allow for straightforward conversions: to express degrees in arcminutes, multiply the degree value by 60; to express degrees in arcseconds, multiply by 3600. The reverse conversions involve division: arcminutes to degrees requires dividing by 60, and arcseconds to degrees requires dividing by 3600.[85] For integration with radian measures, which are the standard unit in many scientific calculations, angular values in degrees are first converted using the factor . Subdivisions follow proportionally: an arcminute corresponds to radians, while an arcsecond is exactly radians, derived from the full circle of radians equaling 1296000 arcseconds.[86] In scenarios involving small angular separations, such as stellar parallax, the small-angle approximation simplifies relating angular size to linear extent. For an angle in radians much less than 1, the transverse linear distance at a given distance from the observer is approximately .[87] This holds because and under the approximation, enabling estimates of physical sizes from observed angles without trigonometric computation. The degrees-minutes-seconds (DMS) notation, common in precise angular specifications, converts to decimal degrees for computational ease via the compound formula where is the degrees integer, the arcminutes, and the arcseconds.[88] This yields a single decimal value suitable for further trigonometric or radian-based operations.Usage in Programming and Calculations
In programming and computational applications, minutes and seconds of arc are frequently handled as angular units in libraries and algorithms that require precise coordinate transformations, particularly in astronomy, geospatial analysis, and graphics rendering. Python's Astropy library provides robust support for these units through itsastropy.units module, which defines arcminute (u.arcmin) and arcsecond (u.arcsec) as standard angle units and enables seamless conversions to radians or other formats for numerical computations.[89] For instance, in astronomical simulations, developers can convert arcseconds to radians for trigonometric operations like calculating stellar positions, as shown in the following example:
from astropy import units as u
# Convert 1 arcsecond to radians
arcsec = 1 * u.arcsec
radians = arcsec.to(u.[radian](/page/Radian))
print(radians) # Output: 4.84813681109536e-06 rad
from astropy import units as u
# Convert 1 arcsecond to radians
arcsec = 1 * u.arcsec
radians = arcsec.to(u.[radian](/page/Radian))
print(radians) # Output: 4.84813681109536e-06 rad
Math object are adapted to handle degrees-minutes-seconds (DMS) formats by first parsing to decimal degrees and then to radians using the formula radians = degrees * (Math.PI / 180). Libraries like dms-conversion facilitate this by providing functions to parse DMS strings directly into decimal degrees for further processing in graphics or coordinate systems.[91] An example for converting a DMS coordinate in a mapping context:
// Using a simple parser (or dms-conversion library)
function dmsToRadians(dms) {
const parts = dms.split(/°|′|″/).map(Number);
const degrees = parts[0] + (parts[1] / 60) + (parts[2] / 3600);
return degrees * (Math.PI / 180);
}
console.log(dmsToRadians("45°30′0″")); // Output: 0.7908982308135293 radians
// Using a simple parser (or dms-conversion library)
function dmsToRadians(dms) {
const parts = dms.split(/°|′|″/).map(Number);
const degrees = parts[0] + (parts[1] / 60) + (parts[2] / 3600);
return degrees * (Math.PI / 180);
}
console.log(dmsToRadians("45°30′0″")); // Output: 0.7908982308135293 radians
to_dms() function formats decimal degrees to DMS, while inverse operations use string parsing combined with degrees_to_radians() for angular calculations in vector geometries or raster projections. This supports high-volume data processing in open-source GIS environments.
Precision challenges arise in these calculations due to floating-point arithmetic limitations, where standard double-precision (IEEE 754) can introduce errors on the order of milliarcseconds in iterative astronomical simulations, such as orbital predictions or image alignments.[92] For instance, repeated trigonometric evaluations in double precision may accumulate rounding errors exceeding 1 arcsecond over long integrations, as seen in celestial mechanics models. To mitigate this, high-precision libraries like Python's mpmath are employed, which support arbitrary decimal places for angular functions such as sin and cos, ensuring sub-arcsecond accuracy in sensitive applications like exoplanet transit timing. An example using mpmath for a precise arcsecond-to-radian conversion in an astronomy simulation:
from mpmath import mp, mpf, [radians](/page/rad)
mp.dps = 50 # Set 50 decimal places precision
arcsec = mpf('1')
rad = [radians](/page/rad)(arcsec / 3600) # Convert to degrees first, then radians
print(rad) # Output: 4.8481368110953599358991410235794797595635330237270e-6
from mpmath import mp, mpf, [radians](/page/rad)
mp.dps = 50 # Set 50 decimal places precision
arcsec = mpf('1')
rad = [radians](/page/rad)(arcsec / 3600) # Convert to degrees first, then radians
print(rad) # Output: 4.8481368110953599358991410235794797595635330237270e-6