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Theangular diameter,angular size,apparent diameter, orapparent size is anangular separation (inunits of angle) describing how large asphere orcircle appears from a given point of view. In thevision sciences, it is called thevisual angle, and inoptics, it is theangular aperture (of alens). The angular diameter can alternatively be thought of as theangular displacement through which an eye or camera must rotate to look from one side of an apparent circle to the opposite side.
A person canresolve with theirnaked eyes diameters down to about 1 arcminute (approximately 0.017° or 0.0003 radians).[1] This corresponds to 0.3 m at a 1 km distance, or to perceivingVenus as a disk under optimal conditions.
The angular diameter of acircle 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[2][3]
in which is the angular diameter (in units of angle, normally radians, sometimes in degrees, depending on thearctangent implementation), is the linear diameter of the object (in units of length), and is the distance to the object (also in units of length). When, we have:[4]
and the result obtained is necessarily inradians.
For a spherical object whose linear diameter equals and where is the distance to thecenter of the sphere, the angular diameter can be found by the following modified formula[citation needed]
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 as the hypotenuse and as the sine.[citation needed]
The formula is related to thezenith angle to the horizon,
whereR is the radius of the sphere andh is the distance to the nearsurface of the sphere.
The difference with the case of a perpendicular circle is significant only for spherical objects of large angular diameter, since the followingsmall-angle approximations hold for small values of:[5]
Estimates of angular diameter may be obtained by holding the hand at right angles to afully extended arm, as shown in the figure.[6][7][8]
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Inastronomy, the sizes ofcelestial objects are often given in terms of their angular diameter as seen fromEarth, rather than their actual sizes. Since these angular diameters are typically small, it is common to present them inarcseconds (″). An arcsecond is 1/3600th of onedegree (1°) and a radian is 180/π degrees. So one radian equals 3,600 × 180/ arcseconds, which is about 206,265 arcseconds (1 rad ≈ 206,264.806247"). Therefore, the angular diameter of an object with physical diameterd at a distanceD, expressed in arcseconds, is given by:[9]
These objects have an angular diameter of 1″:
Thus, the angular diameter ofEarth's orbit around theSun as viewed from a distance of 1 pc is 2″, as 1 AU is the mean radius of Earth's orbit.
The angular diameter of the Sun, from a distance of onelight-year, is 0.03″, and that ofEarth 0.0003″. The angular diameter 0.03″ 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 noteworthycelestial bodies as seen from Earth:
| Celestial object | Angular diameter or size | Relative size |
|---|---|---|
| Magellanic Stream | over 100° | |
| Gum Nebula | 36° | |
| Milky Way | 30° (by 360°) | |
| Width of spread out hand with arm stretched out | 20° | 353 meter at 1 km distance |
| Serpens-Aquila Rift | 20° by 10° | |
| Canis Major Overdensity | 12° by 12° | |
| Smith's Cloud | 11° | |
| Large Magellanic Cloud | 10.75° by 9.17° | Note: brightestgalaxy, other than the Milky Way, in thenight sky (0.9apparent magnitude (V)) |
| Barnard's loop | 10° | |
| Zeta Ophiuchi Sh2-27 nebula | 10° | |
| Width of fist with arm stretched out | 10° | 175 meter at 1 km distance |
| Sagittarius Dwarf Spheroidal Galaxy | 7.5° by 3.6° | |
| Northern Coalsack Nebula | 7° by 5°[10] | |
| Coalsack nebula | 7° by 5° | |
| Cygnus OB7 | 4° by 7°[11] | |
| Rho Ophiuchi cloud complex | 4.5° by 6.5° | |
| Hyades | 5°30′ | Note: brighteststar cluster in the night sky, 0.5 apparent magnitude (V) |
| Small Magellanic Cloud | 5°20′ by 3°5′ | |
| Andromeda Galaxy | 3°10′ by 1° | About six times the size of the Sun or the Moon. Only the much smaller core is visible withoutlong-exposure photography. |
| Charon (from the surface ofPluto) | 3°9’ | |
| Veil Nebula | 3° | |
| Heart Nebula | 2.5° by 2.5° | |
| Westerhout 5 | 2.3° by 1.25° | |
| Sh2-54 | 2.3° | |
| Carina Nebula | 2° by 2° | Note: brightestnebula in the night sky, 1.0 apparent magnitude (V) |
| North America Nebula | 2° by 100′ | |
| Earth in theMoon's sky | 2° - 1°48′[12] | Appearing about three to four times larger than the Moon in Earth's sky |
| TheSun in the sky ofMercury | 1.15° - 1.76° | [13] |
| Orion Nebula | 1°5′ by 1° | |
| Width of little finger with arm stretched out | 1° | 17.5 meter at 1 km distance |
| The Sun in the sky ofVenus | 0.7° | [13][14] |
| Io (as seen from the “surface” of Jupiter) | 35’ 35” | |
| Moon | 34′6″ – 29′20″ | 32.5–28 times the maximum value for Venus (orange bar below) / 2046–1760″ the Moon has a diameter of 3,474 km |
| Sun | 32′32″ – 31′27″ | 31–30 times the maximum value for Venus (orange bar below) / 1952–1887″ the Sun has a diameter of 1,391,400 km |
| Triton (from the “surface” of Neptune) | 28’ 11” | |
| Angular size of the distance between Earth and the Moon as viewed fromMars, atinferior conjunction | about 25′ | |
| Ariel (from the “surface” of Uranus) | 24’ 11” | |
| Ganymede (from the “surface” of Jupiter) | 18’ 6” | |
| Europa (from the “surface” of Jupiter) | 17’ 51” | |
| Umbriel (from the “surface” of Uranus) | 16’ 42” | |
| Helix Nebula | about 16′ by 28′ | |
| Jupiter if it were as close to Earth asMars | 9.0′ – 1.2′ | |
| Spire inEagle Nebula | 4′40″ | length is 280″ |
| Phobos as seen fromMars | 4.1′ | |
| Venus | 1′6″ – 0′9.7″ | |
| International Space Station (ISS) | 1′3″ | [15] the ISS has a width of about 108 m |
| Minimum resolvable diameter by thehuman eye | 1′ | [16] 0.3 meter at 1 km distance[17]
|
| About 100 km on the surface of theMoon | 1′ | Comparable to the size of features like large lunar craters, such as theCopernicus crater, a prominent bright spot in the eastern part ofOceanus Procellarum on the waning side, or theTycho crater within a bright area in the south, of thelunar near side. |
| Jupiter | 50.1″ – 29.8″ | |
| Earth as seen from Mars | 48.2″[13] – 6.6″ | |
| Minimum resolvable gap between two lines by the human eye | 40″ | a gap of 0.026 mm as viewed from 15 cm away[16][17] |
| Mars | 25.1″ – 3.5″ | |
| Apparent size of Sun, seen from90377 Sedna at aphelion | 20.4" | |
| Saturn | 20.1″ – 14.5″ | |
| Mercury | 13.0″ – 4.5″ | |
| Earth's Moon as seen from Mars | 13.27″ – 1.79″ | |
| Uranus | 4.1″ – 3.3″ | |
| Neptune | 2.4″ – 2.2″ | |
| Ganymede | 1.8″ – 1.2″ | Ganymede has a diameter of 5,268 km |
| Anastronaut (~1.7 m) at a distance of 350 km, the average altitude of the ISS | 1″ | |
| Minimum resolvable diameter byGalileo Galilei's largest38mm refracting telescopes | ~1″ | [18] Note: 30x[19] magnification, comparable to very strong contemporary terrestrialbinoculars |
| Ceres | 0.84″ – 0.33″ | |
| Vesta | 0.64″ – 0.20″ | |
| Pluto | 0.11″ – 0.06″ | |
| Eris | 0.089″ – 0.034″ | |
| R Doradus | 0.062″ – 0.052″ | Note: R Doradus is thought to be the extrasolar star with the largest apparent size as viewed from Earth |
| Betelgeuse | 0.060″ – 0.049″ | |
| Alphard | 0.00909″ | |
| Alpha Centauri A | 0.007″ | |
| Canopus | 0.006″ | |
| Sirius | 0.005936″ | |
| Altair | 0.003″ | |
| Rho Cassiopeiae | 0.0021″[20] | |
| Deneb | 0.002″ | |
| Proxima Centauri | 0.001″ | |
| Alnitak | 0.0005″ | |
| Proxima Centauri b | 0.00008″ | |
| Event horizon of black holeM87* at center of the M87 galaxy, imaged by theEvent Horizon Telescope in 2019. | 0.000025″ (2.5×10−5) | Comparable to a tennis ball on the Moon |
| A star likeAlnitak at a distance where theHubble Space Telescope would just be able to see it[21] | 6×10−10 arcsec |
The angular diameter of the Sun, as seen from Earth, is about 250,000 times that ofSirius. (Sirius has twice the diameter and its distance is 500,000 times as much; the Sun is 1010 times as bright, corresponding to an angular diameter ratio of 105, so Sirius is roughly 6 times as bright per unitsolid angle.)
The angular diameter of the Sun is also about 250,000 times that ofAlpha Centauri A (it has about the same diameter and the distance is 250,000 times as much; the Sun is 4×1010 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 theMoon. (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″ 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 theHubble 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 ofthe 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 thenight sky.
Degrees, therefore, are subdivided as follows:
To put this in perspective, thefull Moon as viewed from Earth is about1⁄2°, or 30′ (or 1800″). The Moon's motion across the sky can be measured in angular size: approximately 15° every hour, or 15″ per second. A one-mile-long line painted on the face of the Moon would appear from Earth to be about 1″ in length.
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 theangular diameter distance to distant objects as
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. SeeDistance measures (cosmology).
Manydeep-sky objects such asgalaxies andnebulae appear non-circular and are thus typically given two measures of diameter: major axis and minor axis. For example, theSmall Magellanic Cloud has a visual apparent diameter of 5° 20′ × 3° 5′.
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″ of arc across and is 75% illuminated, the defect of illumination is 10″.
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