Editing Angular diameter

{{Short description|How large a sphere or circle appears}}
{{More citations needed|date=September 2009}}
[[File:Angular diameter.jpg|thumb|300px|Angular diameter: the angle subtended by an object]]

The '''angular diameter''', '''angular size''', '''apparent diameter''', or '''apparent size''' is an [[angular separation]] (in [[units of angle]]) describing how large a [[sphere]] or [[circle]] appears from a given point of view. In the [[vision sciences]], it is called the ''[[visual angle]]'', and in [[optics]], it is the ''[[angular aperture]]'' (of a [[lens (optics)|lens]]). The angular diameter can alternatively be thought of as the [[angular displacement]] through which an eye or camera must rotate to look from one side of an apparent circle to the opposite side.

A person can [[Angular resolution|resolve]] with their [[naked eye]]s diameters down to about 1&nbsp;[[arcminute]] (approximately 0.017° or 0.0003 radians).<ref name="Yanoff2009">{{cite book |  title=Ophthalmology 3rd Edition  | author1-first=Myron | author1-last=Yanoff | author2-first=Jay S. | author2-last=Duker | publisher=MOSBY Elsevier | year=2009 | isbn=978-0444511416 | page=54 | url=https://books.google.com/books?id=u43MTFr7-m8C&pg=PA54 }}</ref> This corresponds to 0.3&nbsp;m at a 1&nbsp;km distance, or to perceiving [[Venus]] as a disk under optimal conditions.

==Formulation==
[[File:Angular diameter formula.svg|thumb|400px|right|Diagram for the formula of the angular diameter]]

The angular diameter of a [[circle]] whose plane is perpendicular to the displacement vector between the point of view and the center of said circle can be calculated using the formula<ref>This can be derived using the formula for the length of a chord found at {{cite web |url=http://mathworld.wolfram.com/CircularSegment.html |title=Circular Segment|access-date=2015-01-23 |url-status=live |archive-url=https://web.archive.org/web/20141221042937/http://mathworld.wolfram.com/CircularSegment.html |archive-date=2014-12-21 }}</ref><ref>{{Cite web |title=Angular Diameter {{!}} Wolfram Formula Repository |url=https://resources.wolframcloud.com/FormulaRepository/resources/Angular-Diameter |access-date=2024-04-10 |website=resources.wolframcloud.com}}</ref>
:<math>\delta =  2\arctan \left(\frac{d}{2D}\right),</math>
in which <math>\delta</math> is the angular diameter (in units of angle, normally radians, sometimes in degrees, depending on the [[arctangent]] implementation), <math>d</math> is the linear diameter of the object (in units of length), and <math>D</math> is the distance to the object (also in units of length). When <math>D \gg d</math>, we have:<ref>{{Cite web |title=7A Notes: Angular Size/Distance and Areas |url=https://w.astro.berkeley.edu/~casey_lam/7A_Angular_Distance_and_Area.pdf}}</ref>
:<math>\delta \approx d / D</math>,
and the result obtained is necessarily in [[radians]].

===For a sphere===
For a spherical object whose linear diameter equals <math>d</math> and where <math>D</math> is the distance to the {{em|center}} of the sphere, the angular diameter can be found by the following modified formula{{Citation needed|date=April 2024}}
:<math>\delta =  2\arcsin \left(\frac{d}{2D}\right)</math>

Such a different formulation is because the apparent edges of a sphere are its tangent points, which are closer to the observer than the center of the sphere, and have a distance between them which is smaller than the actual diameter. The above formula can be found by understanding that in the case of a spherical object, a right triangle can be constructed such that its three vertices are the observer, the center of the sphere, and one of the sphere's tangent points, with <math>D</math> as the hypotenuse and <math>\frac{d_\mathrm{act}}{2D}</math> as the sine.{{Citation needed|date=April 2024}}

The formula is related to the [[Horizon#Zenith angle|zenith angle to the horizon]], 
:<math>\delta = \pi - 2\arccos\left(\frac{R}{R+h}\right)</math>
where ''R'' is the radius of the sphere and ''h'' is the distance to the near {{em|surface}} of the sphere.

The difference with the case of a perpendicular circle is significant only for spherical objects of large angular diameter, since the following [[small-angle approximation]]s hold for small values of <math>x</math>:<ref>{{cite web |url=http://www.mathstat.concordia.ca/faculty/rhall/mc/arctan.pdf |title=A Taylor series for the functionarctan |access-date=2015-01-23 |url-status=dead |archive-url=https://web.archive.org/web/20150218190328/http://www.mathstat.concordia.ca/faculty/rhall/mc/arctan.pdf |archive-date=2015-02-18 }}</ref>
:<math>\arcsin x \approx \arctan x \approx x.</math>

==Estimating angular diameter using the hand==
[[File:Estimating angular size with hand.gif|thumb|Approximate angles of 10°, 20°, 5°, and 1° for the hand outstretched at arm's length]]

Estimates of angular diameter may be obtained by holding the hand at right angles to a [[Angular mil#Use|fully extended arm]], as shown in the figure.<ref>{{cite web |url=https://dept.astro.lsa.umich.edu/ugactivities/Labs/coords/index.html |title=Coordinate Systems|access-date=2015-01-21 |url-status=dead |archive-url=https://web.archive.org/web/20150121044615/https://dept.astro.lsa.umich.edu/ugactivities/Labs/coords/index.html |archive-date=2015-01-21 }}</ref><ref>{{cite web |url=https://www.bartbusschots.ie/s/2013/06/08/photographing-satellites/ |title=Photographing Satellites |date=8 June 2013 |url-status=live |archive-url=https://web.archive.org/web/20150121045706/https://www.bartbusschots.ie/s/2013/06/08/photographing-satellites/ |archive-date=21 January 2015 }}</ref><ref>[[v:Physics and Astronomy Labs/Angular size|Wikiversity: Physics and Astronomy Labs/Angular size]]</ref>

==Use in astronomy==
[[File:Elements of astronomy- accompanied with numerous illustrations, a colored representation of the solar, stellar, and nebular spectra, and celestial charts of the northern and the southern hemisphere (14804687203).jpg|thumb|right|A 19th century depiction of the apparent size of the Sun as seen from the Solar System's planets (incl. [[72 Feronia]] and the then  most outlying known asteroid, here called ''Maximiliana'').]]

In [[astronomy]], the sizes of [[celestial object]]s are often given in terms of their angular diameter as seen from [[Earth]], rather than their actual sizes. Since these angular diameters are typically small, it is common to present them in [[arcsecond]]s ({{pprime}}). An arcsecond is 1/3600th of one [[degree (angle)|degree]] (1°) and a radian is 180/''π'' degrees. So one radian equals 3,600 × 180/<math>\pi</math> arcseconds, which is about 206,265 arcseconds (1 rad ≈ 206,264.806247"). Therefore, the angular diameter of an object with physical diameter ''d'' at a distance ''D'', expressed in arcseconds, is given by:<ref>{{cite book |title=Stars and Galaxies |author=Michael A. Seeds |author2=Dana E. Backman |publisher=Brooks Cole |date=2010 |edition=7 |page=39 |isbn=978-0-538-73317-5}}</ref>
:<math>\delta = 206,265 ~ (d / D) ~ \mathrm{arcseconds}</math>.

These objects have an angular diameter of 1{{pprime}}:
*an object of diameter 1&nbsp;cm at a distance of 2.06&nbsp;km
*an object of diameter 725.27&nbsp;km at a distance of 1 [[astronomical unit]] (AU)
*an object of diameter 45&nbsp;866&nbsp;916&nbsp;km at 1 [[light-year]]
*an object of diameter 1 AU (149&nbsp;597&nbsp;871&nbsp;km) at a distance of 1 [[parsec]] (pc)

Thus, the angular diameter of [[Earth's orbit]] around the [[Sun]] as viewed from a distance of 1 pc is 2{{pprime}}, as 1 AU is the mean radius of Earth's orbit.

The angular diameter of the Sun, from a distance of one [[light-year]], is 0.03{{pprime}}, and that of [[Earth]] 0.0003{{pprime}}. The angular diameter 0.03{{pprime}} of the Sun given above is approximately the same as that of a human body at a distance of the diameter of Earth.

This table shows the angular sizes of noteworthy [[astronomical object|celestial bodies]] as seen from Earth:

{| class="wikitable"
! Celestial object
! Angular diameter or size
! align="left"| Relative size
|-
| [[Magellanic Stream]] 
| align="center"|over 100°
| 
|-
| [[Gum Nebula]] 
| align="center"|36°
| 
|-
| [[Milky Way]]
| align="center"|30° (by 360°)
| 
|-
| Width of spread out hand with arm stretched out
|align="center"|20°
| covering 353 meter of something viewed from a distance of 1&nbsp;km
|-
| [[Serpens-Aquila Rift]]
| align="center"|20° by 10°
| 
|-
| [[Canis Major Overdensity]]
|align="center"|12° by 12°
|
|-
| [[Smith's Cloud]]
|align="center"|11°
| 
|-
| [[Large Magellanic Cloud]]
| align="center"|10.75° by 9.17°
| Note: brightest [[galaxy]], other than the Milky Way, in the [[night sky]] (0.9 [[apparent magnitude]]&nbsp;(V))
|-
| [[Barnard's loop]]
| align="center"|10°
| 
|-
| [[Zeta Ophiuchi]] Sh2-27 nebula
|align="center"|10°
| 
|-
| Width of fist with arm stretched out
|align="center"|10°
| covering 175 meter of something viewed from a distance of 1&nbsp;km
|-
| [[Sagittarius Dwarf Spheroidal Galaxy]]
|align="center"|7.5° by 3.6°
|
|-
| [[Northern Coalsack Nebula]]
| align="center"|7° by 5°<ref name="OMeara 2019">{{cite web | last=O'Meara | first=Stephen James | title=The coalsacks of Cygnus | website=Astronomy.com | date=2019-08-06 | url=https://astronomy.com/magazine/stephen-omeara/2019/08/the-coalsacks-of-cygnus | access-date=2023-02-10}}</ref>
| 
|-
| [[Coalsack nebula]]
| align="center"|7° by 5°
| 
|-
| [[Cygnus OB7]]
| align="center"|4° by 7°<ref name="Dobashi Matsumoto Shimoikura Saito 2014 p=58">{{cite journal | last1=Dobashi | first1=Kazuhito | last2=Matsumoto | first2=Tomoaki | last3=Shimoikura | first3=Tomomi | last4=Saito | first4=Hiro | last5=Akisato | first5=Ko | last6=Ohashi | first6=Kenjiro | last7=Nakagomi | first7=Keisuke | title=Colliding Filaments and a Massive Dense Core in the Cygnus Ob 7 Molecular Cloud | journal=The Astrophysical Journal | publisher=American Astronomical Society | volume=797 | issue=1 | date=2014-11-24 | issn=1538-4357 | doi=10.1088/0004-637x/797/1/58 | page=58| s2cid=118369651 | arxiv=1411.0942 | bibcode=2014ApJ...797...58D }}</ref>
| 
|-
| [[Rho Ophiuchi cloud complex]]
| align="center"|4.5° by 6.5°
| 
|-
| [[Hyades (star cluster)|Hyades]]
| align="center"|5°30{{prime}}
| Note: brightest [[star cluster]] in the night sky, 0.5 apparent magnitude&nbsp;(V)
|-
| [[Small Magellanic Cloud]]
| align="center"|5°20{{prime}} by 3°5{{prime}}
|
|-
| [[Andromeda Galaxy]]
| align="center"|3°10{{prime}} by 1°
| About six times the size of the Sun or the Moon. Only the much smaller core is visible without [[long-exposure photography]].
|-
|[[Charon (moon)|Charon]] (from the surface of [[Pluto]])
|align="center"|3°9’
|
|-
| [[Veil Nebula]]
|align="center"|3°
| 
|-
| [[Heart Nebula]]
|align="center"|2.5° by 2.5°
| 
|-
| [[Westerhout 5]]
|align="center"|2.3° by 1.25°
| 
|-
| [[Sh2-54]]
|align="center"|2.3°
| 
|-
| [[Carina Nebula]]
|align="center"|2° by 2°
| Note: brightest [[nebula]] in the night sky, 1.0 apparent magnitude&nbsp;(V)
|-
| [[North America Nebula]]
|align="center"|2° by 100{{prime}}
|
|-
| Earth in the [[Extraterrestrial sky#The Moon|Moon's sky]] 
|align="center"|2° - 1°48{{prime}}<ref name="Gorkavyi Krotkov Marshak 2023 pp. 1527–1537">{{cite journal | last1=Gorkavyi | first1=Nick | last2=Krotkov | first2=Nickolay | last3=Marshak | first3=Alexander | title=Earth observations from the Moon's surface: dependence on lunar libration | journal=Atmospheric Measurement Techniques | publisher=Copernicus GmbH | volume=16 | issue=6 | date=2023-03-24 | issn=1867-8548 | doi=10.5194/amt-16-1527-2023 | pages=1527–1537| bibcode=2023AMT....16.1527G | doi-access=free }}</ref>
| Appearing about three to four times larger than the Moon in Earth's sky
|-
| Moon as it appeared in Earth's sky 3.9 billion years ago
|align="center"|1.5°
| The Moon appeared 3.9 billion years ago 2.8 times larger than it does today.<ref name="b319">{{cite web | title=Earth-Moon Dynamics | website=Lunar and Planetary Institute (LPI) | url=https://www.lpi.usra.edu/exploration/training/illustrations/earthMoon/ | access-date=2025-04-07}}</ref>
|-
| The [[Sun]] in the sky of [[Mercury (planet)|Mercury]]
|align="center"|1.15° - 1.76°
|<ref name="o116">{{cite web | title=The Sun and Transits as Seen From the Planets | website=RASC Calgary Centre | date=2018-11-05 | url=https://calgary.rasc.ca/sun_and_transits.htm | access-date=2024-08-23}}</ref>
|-
| [[Orion Nebula]]
|align="center"|1°5{{prime}} by 1°
|
|-
| Width of little finger with arm stretched out
|align="center"|1°
| covering 17.5 meter of something viewed from a distance of 1&nbsp;km
|-
| The Sun in the sky of [[Venus (planet)|Venus]]
|align="center"|0.7°
|<ref name="o116"/><ref name="k092">{{cite web | title=How large does the Sun appear from Mercury and Venus, as compared to how we see it from Earth? | website=Astronomy Magazine | date=2018-05-31 | url=https://www.astronomy.com/observing/how-large-does-the-sun-appear-from-mercury-and-venus-as-compared-to-how-we-see-it-from-earth/ | access-date=2024-08-23}}</ref>
|-
|[[Io (moon)|Io]] (as seen from the “surface” of Jupiter)
|35’ 35”
|
|-
| [[Moon]]
| align="center"|34{{prime}}6{{pprime}} – 29{{prime}}20{{pprime}}
| 32.5–28 times the maximum value for Venus (orange bar below) / 2046–1760{{pprime}} the Moon has a diameter of 3,474&nbsp;km
|-
| [[Sun]]
| align="center"| 32{{prime}}32{{pprime}} – 31{{prime}}27{{pprime}}
| 31–30 times the maximum value for Venus (orange bar below) / 1952–1887{{pprime}} the Sun has a diameter of 1,391,400&nbsp;km
|-
|[[Triton (moon)|Triton]] (from the “surface” of Neptune)
|28’ 11”
|
|-
| Angular size of the distance between Earth and the Moon as viewed from [[Mars]], at [[inferior conjunction]]
| align="center"| about 25{{prime}}
|
|-
|[[Ariel (moon)|Ariel]] (from the “surface” of Uranus)
|24’  11”
|
|-
|[[Ganymede (moon)|Ganymede]] (from the “surface” of Jupiter)
|18’ 6”
|
|-
|[[Europa (moon)|Europa]] (from the “surface” of Jupiter)
|17’ 51”
|
|-
|[[Umbriel]] (from the “surface” of Uranus)
|16’ 42”
|
|-
| [[Helix Nebula]]
| align="center"| about 16{{prime}} by 28{{prime}}
|
|-
| [[Jupiter]] if it were as close to Earth as [[Mars]]
| align="center"| 9.0{{prime}} – 1.2{{prime}}
| 
|-
| Spire in [[Eagle Nebula]]
| align="center"| 4{{prime}}40{{pprime}}
| length is 280{{pprime}}
|-
| [[Phobos (moon)|Phobos]] as seen from [[Mars]]
| align="center"| 4.1{{prime}}
| 
|-
| [[Venus]]
| align="center"| 1{{prime}}6{{pprime}} – 0{{prime}}9.7{{pprime}}
| <hr style="width:660px;height:8px;background:orange" /><hr style="width:97px;height:8px;" />
|-
| [[International Space Station]] (ISS)
| align="center"| 1{{prime}}3{{pprime}}
| <ref name="Angular size">{{cite web | title=Problem 346: The International Space Station and a Sunspot: Exploring angular scales | website=Space Math @ NASA ! | date=2018-08-19 | url=https://spacemath.gsfc.nasa.gov/weekly/7Page1.pdf | access-date=2022-05-20}}</ref> the ISS has a width of about 108 m
|-
| Minimum resolvable diameter by the [[human eye]] 
| align="center"| 1{{prime}}
| <ref name="Wong 2016"/> 0.3 meter at 1&nbsp;km distance<ref name="Science in School – scienceinschool.org 2016"/>
<br>
For visibility of objects with smaller apparent sizes see [[apparent magnitude#List of apparent magnitudes|the necessary apparent magnitudes]].
|-
| About 100&nbsp;km on the surface of the [[Moon]] 
| align="center"| 1{{prime}}
| Comparable to the size of features like large lunar craters, such as the [[Copernicus (lunar crater)|Copernicus crater]], a prominent bright spot in the eastern part of [[Oceanus Procellarum]] on the waning side, or the [[Tycho (lunar crater)|Tycho crater]] within a bright area in the south, of the [[Near side of the Moon|lunar near side]].
|-
| [[Jupiter]]
| align="center"|50.1{{pprime}} – 29.8{{pprime}}
| <hr style="width:501px;height:8px;" /><hr style="width:298px;height:8px;" />
|-
| Earth as seen from Mars
| align="center"|48.2{{pprime}}<ref name="o116"/> – 6.6{{pprime}}
| <hr style="width:481px;height:8px;" /><hr style="width:66px;height:8px;" />
|-
| Minimum resolvable gap between two lines by the human eye
| align="center"|40{{pprime}}
| a gap of 0.026&nbsp;mm as viewed from 15&nbsp;cm away<ref name="Wong 2016">{{cite web | last=Wong | first=Yan | title=How small can the naked eye see? | website=BBC Science Focus Magazine | date=2016-01-24 | url=https://www.sciencefocus.com/the-human-body/how-small-can-the-naked-eye-see/ | access-date=2022-05-23}}</ref><ref name="Science in School – scienceinschool.org 2016">{{cite web | title=Sharp eyes: how well can we really see? | website=Science in School – scienceinschool.org | date=2016-09-07 | url=https://www.scienceinschool.org/article/2016/sharp-eyes-how-well-can-we-really-see/ | access-date=2022-05-23}}</ref>
|-
| [[Mars]]
| align="center"| 25.1{{pprime}} – 3.5{{pprime}}
| <hr style="width:251px;height:8px;" /><hr style="width:35px;height:8px;" />
|-
|Apparent size of Sun, seen from [[90377 Sedna]] at aphelion
|20.4"
|
|-
| [[Saturn]]
| align="center"| 20.1{{pprime}} – 14.5{{pprime}}
| <hr style="width:207px;height:8px;" /><hr style="width:149px;height:8px;" />
|-
| [[Mercury (planet)|Mercury]]
| align="center"| 13.0{{pprime}} – 4.5{{pprime}}
| <hr style="width:130px;height:8px;" /><hr style="width:45px;height:8px;" />
|-
| Earth's Moon as seen from Mars
| align="center"| 13.27{{pprime}} – 1.79{{pprime}}
| <hr style="width:133px;height:8px;" /><hr style="width:18px;height:8px;" />
|-
| [[Uranus]]
| align="center"| 4.1{{pprime}} – 3.3{{pprime}}
| <hr style="width:41px;height:8px;" /><hr style="width:33px;height:8px;" />
|-
| [[Neptune]]
| align="center"| 2.4{{pprime}} – 2.2{{pprime}}
| <hr style="width:24px;height:8px;" /><hr style="width:22px;height:8px;" />
|-
| [[Ganymede (moon)|Ganymede]]
| align="center"| 1.8{{pprime}} – 1.2{{pprime}}
| <hr style="width:18px;height:8px;" /><hr style="width:12px;height:8px;" /> Ganymede has a diameter of 5,268&nbsp;km
|-
| An [[astronaut]] (~1.7 m) at a distance of 350&nbsp;km, the average altitude of the ISS
| align="center"| 1{{pprime}}
| 
|-
| Minimum resolvable diameter by [[Galileo Galilei]]'s largest [[Galileo's objective lens|38mm refracting telescope]]s
| align="center"| ~1{{pprime}}
| <ref name="Graney 2006">{{cite journal | last=Graney | first=Christopher M. | title=The Accuracy of Galileo's Observations and the Early Search for Stellar Parallax | date=Dec 10, 2006 | doi=10.1007/3-540-50906-2_2 | arxiv=physics/0612086 }}</ref> Note: 30x<ref name="Esposizioni on-line - Istituto e Museo di Storia della Scienza">{{cite web | title=Galileo's telescope - How it works | website=Esposizioni on-line - Istituto e Museo di Storia della Scienza | url=https://brunelleschi.imss.fi.it/esplora/cannocchiale/dswmedia/esplora/eesplora2.html | language=it | access-date=May 21, 2022}}</ref> magnification, comparable to very strong contemporary terrestrial [[binoculars]]
|-
| [[Ceres (dwarf planet)|Ceres]]
| align="center"| 0.84{{pprime}} – 0.33{{pprime}}
| <hr style="width:8.4px;height:8px;" /><hr style="width:3.3px;height:8px;" />
|-
| [[4 Vesta|Vesta]]
| align="center"| 0.64{{pprime}} – 0.20{{pprime}}
| <hr style="width:6.4px;height:8px;" /><hr style="width:2px;height:8px;" />
|-  
| [[Pluto]]
| align="center"| 0.11{{pprime}} – 0.06{{pprime}}
| <hr style="width:1.1px;height:8px;" /><hr style="width:0.6px;height:8px;" />
|-
| [[Eris (dwarf planet)|Eris]]
| align="center"| 0.089{{pprime}} – 0.034{{pprime}}
| <hr style="width:0.89px;height:8px;" /><hr style="width:0.34px;height:8px;" />
|-
| [[R Doradus]]
| align="center"| 0.062{{pprime}} – 0.052{{pprime}}
| <hr style="width:0.62px;height:8px;" /><hr style="width:0.52px;height:8px;" />Note: R Doradus is thought to be the extrasolar star with the largest apparent size as viewed from Earth
|-
| [[Betelgeuse]]
| align="center"| 0.060{{pprime}} – 0.049{{pprime}}
| <hr style="width:0.6px;height:8px;" /><hr style="width:0.49px;height:8px;" />
|-
| [[Alphard]]
| align="center"| 0.00909{{pprime}}
| <hr style="width:0.09px;height:16px;" />
|-
| [[Alpha Centauri A]]
| align="center"| 0.007{{pprime}}
| <hr style="width:0.07px;height:16px;" />
|-
| [[Canopus]]
| align="center"| 0.006{{pprime}}
| <hr style="width:0.06px;height:16px;" />
|-
| [[Sirius]]
| align="center"| 0.005936{{pprime}}
| <hr style="width:0.06px;height:16px;" />
|-
| [[Altair]]
| align="center"| 0.003{{pprime}}
| <hr style="width:0.03px;height:16px;" />
|-
| [[Rho Cassiopeiae]]
| align="center" | 0.0021{{pprime}}<ref>{{Cite arXiv |last1=Anugu |first1=Narsireddy |last2=Baron |first2=Fabien |last3=Monnier |first3=John D. |last4=Gies |first4=Douglas R. |last5=Roettenbacher |first5=Rachael M. |last6=Schaefer |first6=Gail H. |last7=Montargès |first7=Miguel |last8=Kraus |first8=Stefan |last9=Bouquin |first9=Jean-Baptiste Le |date=2024-08-05 |title=CHARA Near-Infrared Imaging of the Yellow Hypergiant Star $\rho$ Cassiopeiae: Convection Cells and Circumstellar Envelope |class=astro-ph.SR |eprint=2408.02756v2 |language=en}}</ref>
| <hr style="width:0.07px;height:16px;" />
|-
| [[Deneb]]
| align="center"| 0.002{{pprime}}
| <hr style="width:0.02px;height:16px;" />
|-
| [[Proxima Centauri]]
| align="center"| 0.001{{pprime}}
| <hr style="width:0.01px;height:16px;" />
|-
| [[Alnitak]]
| align="center"| 0.0005{{pprime}}
|
|-
| [[Proxima Centauri b]]
| align="center"| 0.00008{{pprime}}
|
|-
|Event horizon of black hole [[M87*]] at center of the M87 galaxy, imaged by the [[Event Horizon Telescope]] in 2019.
| align="center"| 0.000025{{pprime}}
({{val|2.5|e=-5}})
|Comparable to a tennis ball on the Moon
|-
|A star like [[Alnitak]] at a distance where the [[Hubble Space Telescope]] would just be able to see it<ref>800 000 times smaller angular diameter than that of Alnitak as seen from Earth. Alnitak is a blue star so it gives off a lot of light for its size. If it were 800 000 times further away then it would be magnitude 31.5, at the limit of what Hubble can see.</ref>
| align="center"| {{val|6|e=-10}} arcsec
|
|}

[[Image:Diffraction limit diameter vs angular resolution.svg|thumb|Log-log plot of [[aperture]] diameter vs [[angular resolution]] at the diffraction limit for various light wavelengths compared with various astronomical instruments. For example, the blue star shows that the [[Hubble Space Telescope]] is almost diffraction-limited in the visible spectrum at 0.1 arcsecs, whereas the red circle shows that the human eye should have a resolving power of 20 arcsecs in theory, though normally only 60 arcsecs.]]
[[File:Comparison angular diameter solar system.svg|thumb|300px|Comparison of angular diameter of the Sun, Moon and planets. 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 5&nbsp;cm wide on your monitor, view it from 5.15&nbsp;m away.]]
[[File:Jupiter.mit.Io.Ganymed.Europa.Calisto.Vollmond.10.4.2017.jpg|thumb|250px|This photo compares the apparent sizes of [[Jupiter]] and its four [[Galilean moons]] ([[Callisto (moon)|Callisto]] at maximum [[elongation (astronomy)|elongation]]) with the apparent diameter of the [[full moon|full Moon]] during their [[conjunction (astronomy)|conjunction]] on 10&nbsp;April 2017.]]

The angular diameter of the Sun, as seen from Earth, is about 250,000 times that of [[Sirius]]. (Sirius has twice the diameter and its distance is 500,000 times as much; the Sun is 10<sup>10</sup> times as bright, corresponding to an angular diameter ratio of 10<sup>5</sup>, so Sirius is roughly 6 times as bright per unit [[solid angle]].)

The angular diameter of the Sun is also about 250,000 times that of [[Alpha Centauri A]] (it has about the same diameter and the distance is 250,000 times as much; the Sun is 4×10<sup>10</sup> times as bright, corresponding to an angular diameter ratio of 200,000, so Alpha Centauri A is a little brighter per unit solid angle).

The angular diameter of the Sun is about the same as that of the [[Moon]]. (The Sun's diameter is 400 times as large and its distance also; the Sun is 200,000 to 500,000 times as bright as the full Moon (figures vary), corresponding to an angular diameter ratio of 450 to 700, so a celestial body with a diameter of 2.5–4{{pprime}} and the same brightness per unit solid angle would have the same brightness as the full Moon.)

Even though Pluto is physically larger than Ceres, when viewed from Earth (e.g., through the [[Hubble Space Telescope]]) Ceres has a much larger apparent size.

Angular sizes measured in degrees are useful for larger patches of sky. (For example, the three stars of [[Orion's Belt|the Belt]] cover about 4.5° of angular size.) However, much finer units are needed to measure the angular sizes of galaxies, nebulae, or other objects of the [[night sky]].

Degrees, therefore, are subdivided as follows:
* 360 [[degree (angle)|degree]]s (°) in a full circle
* 60 [[arc-minute]]s ({{prime}}) in one degree
* 60 [[arc-second]]s ({{pprime}}) in one arc-minute

To put this in perspective, the [[full moon|full Moon]] as viewed from Earth is about {{frac|1|2}}°, or 30{{prime}} (or 1800{{pprime}}). The Moon's motion across the sky can be measured in angular size: approximately 15° every hour, or 15{{pprime}} per second. A one-mile-long line painted on the face of the Moon would appear from Earth to be about 1{{pprime}} in length.

{{wide image|Moon_distance_range_to_scale.svg|800px|Minimum, mean and maximum distances of the Moon from Earth with its angular diameter as seen from Earth's surface, to scale}}

In astronomy, it is typically difficult to directly measure the distance to an object, yet the object may have a known physical size (perhaps it is similar to a closer object with known distance) and a measurable angular diameter. In that case, the angular diameter formula can be inverted to yield the [[angular diameter distance]] to distant objects as 
:<math>d \equiv 2 D \tan \left( \frac{\delta}{2} \right).</math>

In non-Euclidean space, such as our expanding universe, the angular diameter distance is only one of several definitions of distance, so that there can be different "distances" to the same object.  See [[Distance measures (cosmology)]].

===Non-circular objects===
Many [[deep-sky object]]s such as [[galaxies]] and [[nebula]]e appear non-circular and are thus typically given two measures of diameter: major axis and minor axis. For example, the [[Small Magellanic Cloud]] has a visual apparent diameter of {{DEC|5|20}} × {{DEC|3|5}}.

===Defect of illumination===
Defect of illumination is the maximum angular width of the unilluminated part of a celestial body seen by a given observer. For example, if an object is 40{{pprime}} of arc across and is 75% illuminated, the defect of illumination is 10{{pprime}}.

==See also==
* [[Angular diameter distance]]
* [[Angular resolution]]
* [[Apparent magnitude]]
* [[List of stars with resolved images]]
* [[Moon illusion]]
* [[Perceived visual angle]]
* [[Solid angle]]
* [[Visual acuity]]
* [[Visual angle]]

==References==
{{reflist}}

== External links ==
*[https://web.archive.org/web/19971007100829/http://ceres.hsc.edu/homepages/classes/astronomy/fall97/Mathematics/sec9.html Small-Angle Formula] (archived 7 October 1997)
*[http://www.astronomynotes.com/solarsys/s2.htm Visual Aid to the Apparent Size of the Planets]
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[[Category:Elementary geometry]]
[[Category:Astrometry]]
[[Category:Angle]]
[[Category:Equations of astronomy]]