The angular diameter, angular size, apparent diameter, or apparent size is an angular measurement 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). The angular diameter can alternatively be thought of as the angle through which an eye or camera must rotate to look from one side of an apparent circle to the opposite side. Angular radius equals half the angular diameter.
in which is the angular diameter, and and are the actual diameter of and the distance to the object. When , we have , and the result obtained is in radians.
For a spherical object whose actual diameter equals and where is the distance to the centre of the sphere, the angular diameter can be found by the formula
The difference is due to the fact that the apparent edges of a sphere are its tangent points, which are closer to the observer than the centre of the sphere. For practical use, the distinction is only significant for spherical objects that are relatively close, since the small-angle approximation holds for :
In astronomy, the sizes of celestial objects 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 arcseconds (″). An arcsecond is 1/3600th of one degree (1°), and a radian is 180/ degrees, so one radian equals 3,600*180/ arcseconds, which is about 206,265 arcseconds. Therefore, the angular diameter of an object with physical diameter d at a distance D, expressed in arcseconds, is given by:
These objects have an angular diameter of 1″:
The angular diameter of the Sun, from a distance of one light-year, is 0.03″, and that of Earth 0.0003″. The angular diameter 0.03″ of the Sun given above is approximately the same as that of a person at a distance of the diameter of Earth.
This table shows the angular sizes of noteworthy celestial bodies as seen from Earth:
|Celestial body||Angular diameter or size||Relative size|
|Andromeda Galaxy||3°10′ by 1°||About six times the size of the Sun or the Moon. Only the much smaller core is visible without long-exposure photography.|
|Sun||31′27″ – 32′32″||30–31 times the maximum value for Venus (orange bar below) / 1887–1952″|
|Moon||29′20″ – 34′6″||28–32.5 times the maximum value for Venus (orange bar below) / 1760–2046″|
|Helix Nebula||about 16′ by 28′|
|Spire in Eagle Nebula||4′40″||length is 280″|
|Venus||9.7″ – 1′6″|
|Jupiter||29.8″ – 50.1″|
|Saturn||14.5″ – 20.1″|
|Mars||3.5″ – 25.1″|
|Mercury||4.5″ – 13.0″|
|Uranus||3.3″ – 4.1″|
|Neptune||2.2″ – 2.4″|
|Ceres||0.33″ – 0.84″|
|Vesta||0.20″ – 0.64″|
|Pluto||0.06″ – 0.11″|
|R Doradus||0.052″ – 0.062″|
|Betelgeuse||0.049″ – 0.060″|
|Eris||0.034″ – 0.089″|
|Alpha Centauri A||0.007″|
|A star like Alnitak at a distance where the Hubble Space Telescope would just be able to see it||6×10−10 arcsec|
The table shows that the angular diameter of Sun, when seen from Earth is approximately 32′ (1920″ or 0.53°), as illustrated above.
Thus the angular diameter of the Sun 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 1010 times as bright, corresponding to an angular diameter ratio of 105, 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 4×1010 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 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″ 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 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:
To put this in perspective, the full Moon as viewed from Earth is about 1⁄2°, 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 the angular 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. See Distance measures (cosmology).
Many deep-sky objects such as galaxies and nebulae 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 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″.
119 Tauri (also known as CE Tauri) is a red supergiant star in the constellation Taurus. It is a semiregular variable and its angular diameter has been measured at about 10 mas.88 Aquarii
88 Aquarii (abbreviated 88 Aqr) is a star in the equatorial constellation of Aquarius. 88 Aquarii is the Flamsteed designation, though it also bears the Bayer designation c2 Aquarii. In dark conditions it is visible to the naked eye with an apparent visual magnitude of +3.68. Based upon parallax measurements, this star is at a distance of around 271 light-years (83 parsecs) from Earth.The spectrum of 88 Aquarii matches an evolved giant star with a classification of K1 III. Its measured angular diameter is 3.24 ± 0.20 mas, which, at the estimated distance of Delta Ophiuchi, yields a physical size of about 29 times the radius of the Sun. The cool, orange hued glow of this star comes from the outer atmosphere's effective temperature of 4,430 K.98 Aquarii
98 Aquarii (abbreviated 98 Aqr) is a star in the equatorial constellation of Aquarius. 98 Aquarii is the Flamsteed designation, although it also bears the Bayer designation b1 Aquarii. It is visible to the naked eye with an apparent visual magnitude of +3.97. The distance to this star, 163 light-years (50 parsecs), is known from parallax measurements made with the Hipparcos spacecraft.With over double the mass of the Sun, this is an evolved giant star that has a stellar classification of K0 III. The measured angular diameter of this star is 2.54 ± 0.13 mas. At the estimated distance of 98 Aquarii, this yields a physical size of about 14 times the radius of the Sun. The expanded outer envelope has an effective temperature of 4,630 K, giving it the orange glow of a K-type star.99 Aquarii
99 Aquarii (abbreviated 99 Aqr) is a star in the equatorial constellation of Aquarius. 99 Aquarii is the Flamsteed designation, although it also bears the Bayer designation b2 Aquarii. It is visible to the naked eye with an apparent visual magnitude of 4.38; according to the Bortle Dark-Sky Scale this is bright enough to be seen even from city skies under ideal viewing conditions. Based upon parallax measurements, the distance to this star is around 283 light-years (87 parsecs).This is a giant star with a stellar classification of K4 III. It is a suspected variable star that apparently ranges in magnitude between 4.35 and 4.45. The measured angular diameter of this star is 3.55 ± 0.21 mas. At the estimated distance of Delta Ophiuchi, this yields a physical size of about 33 times the radius of the Sun. The outer atmosphere has an effective temperature of 3980 K, giving it the orange-hued glow of a cool, K-type star.This star was a candidate member of the Ursa Major Moving Group based on the work of American astronomer Nancy Roman, but this membership is now in question.Angular diameter distance
The angular diameter distance is a distance measure used in astronomy. It is defined in terms of an object's physical size, , and the angular size of the object as viewed from earth.
The angular diameter distance depends on the assumed cosmology of the universe. The angular diameter distance to an object at redshift, , is expressed in terms of the comoving distance, as:
Where is the FLRW coordinate defined as:
Where is the curvature density and is the value of the Hubble parameter today.
In the currently favoured geometric model of our Universe, the "angular diameter distance" of an object is a good approximation to the "real distance", i.e. the proper distance when the light left the object. Note that beyond a certain redshift, the angular diameter distance gets smaller with increasing redshift. In other words, an object "behind" another of the same size, beyond a certain redshift (roughly z=1.5), appears larger on the sky, and would therefore have a smaller "angular diameter distance".Betelgeuse
Betelgeuse, also designated α Orionis (Latinised to Alpha Orionis, abbreviated Alpha Ori, α Ori), is on average the ninth-brightest star in the night sky and second-brightest in the constellation of Orion. It is a distinctly reddish, semiregular variable star whose apparent magnitude varies between 0.0 and 1.3, the widest range of any first-magnitude star. Betelgeuse is one of three stars that make up the Winter Triangle asterism, and it marks the center of the Winter Hexagon. If the human eye could view all wavelengths of radiation, Betelgeuse would be the brightest star in the night sky.
Classified as a red supergiant of spectral type M1-2, the star is one of the largest stars visible to the naked eye. If Betelgeuse were at the center of the Solar System, its surface would extend past the asteroid belt, completely engulfing the orbits of Mercury, Venus, Earth, Mars, and possibly Jupiter. However, there are several other red supergiants in the Milky Way that could be larger, such as Mu Cephei and VY Canis Majoris. Calculations of its mass range from slightly under ten to a little over twenty times that of the Sun. It is calculated to be 640 light-years away, yielding an absolute magnitude of about −6. Less than 10 million years old, Betelgeuse has evolved rapidly because of its high mass. Having been ejected from its birthplace in the Orion OB1 Association—which includes the stars in Orion's Belt—this runaway star has been observed moving through the interstellar medium at a speed of 30 km/s, creating a bow shock over four light-years wide. Betelgeuse is in a late stage of stellar evolution, and it is expected to explode as a supernova within the next million years.
In 1920, Betelgeuse became the first extrasolar star to have the angular size of its photosphere measured. Subsequent studies have reported an angular diameter (apparent size) ranging from 0.042 to 0.056 arcseconds, with the differences ascribed to the non-sphericity, limb darkening, pulsations, and varying appearance at different wavelengths. It is also surrounded by a complex, asymmetric envelope roughly 250 times the size of the star, caused by mass loss from the star itself. The angular diameter of Betelgeuse is only exceeded by R Doradus and the Sun.Distance measures (cosmology)
Distance measures are used in physical cosmology to give a natural notion of the distance between two objects or events in the universe. They are often used to tie some observable quantity (such as the luminosity of a distant quasar, the redshift of a distant galaxy, or the angular size of the acoustic peaks in the CMB power spectrum) to another quantity that is not directly observable, but is more convenient for calculations (such as the comoving coordinates of the quasar, galaxy, etc.). The distance measures discussed here all reduce to the common notion of Euclidean distance at low redshift.
In accord with our present understanding of cosmology, these measures are calculated within the context of general relativity, where the Friedmann–Lemaître–Robertson–Walker solution is used to describe the universe.Magnitude of eclipse
The magnitude of eclipse is the fraction of the angular diameter of a celestial body being eclipsed. This applies to all celestial eclipses. The magnitude of a partial or annular solar eclipse is always between 0.0 and 1.0, while the magnitude of a total solar eclipse is always greater than or equal to 1.0.
This measure should not be confused with the covered fraction of the apparent area (disk) of the eclipsed body, whereas the magnitude of an eclipse is strictly a ratio of diameters. Neither should it be confused with the astronomical magnitude scale of apparent brightness.Messier 39
Messier 39 or M39, also known as NGC 7092, is an open cluster of stars in the constellation of Cygnus, positioned two degrees to the south of the star Pi Cygni and around 9° east-northeast of Deneb. The cluster was discovered by Guillaume Le Gentil in 1749, then Charles Messier added it to his catalogue in 1764. When observed in a small telescope at low power the cluster shows around two dozen members, but it is best observed with binoculars. It has a total integrated magnitude (brightness) of 5.5 and spans an angular diameter of 29 arcminutes – about the size of the full Moon. M39 is at a distance of about 1,010 light-years (311 parsecs) from the Sun.
This cluster has an estimated mass of 232 M☉ and a linear tidal radius of 8.6±1.8 pc. Of the 15 brightest components, six form binary star systems with one more suspected. HD 205117 is a probable eclipsing binary system with a period of 113.2 days that varies by 0.051 in visual magnitude. Both members appear to be subgiant stars. There are at least five chemically peculiar stars in the cluster and ten suspected short-period variable stars.Minute and second of arc
A minute of arc, arcminute (arcmin), arc minute, or minute arc is a unit of angular measurement equal to 1/ of one degree. Since one degree is 1/ of a turn (or complete rotation), one minute of arc is 1/ of a turn – it is for this reason that the Earth's circumference is almost exactly 21,600 nautical miles. A minute of arc is π/ of a radian.
A second of arc, arcsecond (arcsec), or arc second is 1/ of an arcminute, 1/ of a degree, 1/ of a turn, and π/ (about 1/) of a radian. These units originated in Babylonian astronomy as sexagesimal 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.
The number of square arcminutes in a complete sphere is 510660 square arcminutes (the surface area of a unit sphere in square units divided by the solid angle area subtended by a square arcminute, also in square units – so that the final result is a dimensionless number). 148NGC 2460
NGC 2460 is a spiral galaxy in the constellation Camelopardalis. It was discovered by German astronomer Wilhelm Tempel on August 11, 1882.It is also identified as an active nucleus galaxy. Its redshift of 0.004837 gives an angular diameter distance of 21.501 megaparsecs, or approximately 70 million light-years.NGC 637
NGC 637 is an open cluster of stars in the constellation Cassiopeia. It has a Trumpler classification of I2m and an angular diameter of 4.2′. At a distance of approximately 7,045 light years, this corresponds to a diameter of 9.8 light-years (3.0 pc). The estimated age of this cluster is 5–15 million years.The cluster was discovered on 9 November 1787 by astronomer William Herschel.Solar eclipses on Jupiter
Solar eclipses on Jupiter occur when any of the natural satellites of Jupiter pass in front of the Sun as seen from the planet Jupiter.
For bodies which appear smaller in angular diameter than the Sun, the proper term would be a transit. For bodies which are larger than the apparent size of the Sun, the proper term would be an occultation.
There are five satellites capable of completely occulting the Sun: Amalthea, Io, Europa, Ganymede and Callisto. All of the others are too small or too distant to be able to completely occult the Sun, so can only transit the Sun. Most of the more distant satellites also have orbits that are strongly inclined to the plane of Jupiter's orbit, and would rarely be seen to transit.
When the four largest satellites of Jupiter, the Galilean satellites, occult the Sun, a shadow transit can be seen on the surface of Jupiter which can be observed from Earth in telescopes.
Eclipses of the Sun from Jupiter are not particularly rare, since Jupiter is very large and its axial tilt (which is related to the plane of the orbits of its satellites) is relatively small—indeed, the vast majority of the orbits of all five of the objects capable of occulting the Sun will result in a solar occultation visible from somewhere on Jupiter.
The related phenomenon of satellite eclipses in the shadow of Jupiter has been observed since the time of Giovanni Cassini and Ole Rømer in the mid Seventeenth Century. It was soon noticed that predicted times differed from observed times in a regular way, varying from up to ten minutes early to up to ten minutes late. Rømer used these errors to make the first accurate determination of the speed of light, correctly realizing that the variations were caused by the varying distance between Earth and Jupiter as the two planets moved in their orbits around the Sun.
Spacecraft can be used to observe the solar eclipses on Jupiter; these include Pioneer 10 and Pioneer 11 (1973 and 1974), Voyager 1 and Voyager 2 (1979), Galileo orbiter (1995–2003), Cassini–Huygens (2000) and New Horizons (2007) observed the transits of their moons and its shadows.Solar eclipses on Neptune
Solar eclipses on Neptune occur when any of the natural satellites of Neptune pass in front of the Sun as seen from the planet.
For bodies which appear smaller in angular diameter than the Sun, the proper term would be a transit and bodies which are larger than the apparent size of the Sun, the proper term would be an occultation.
All of Neptune's inner moons and Triton can eclipse the Sun as seen from Neptune.
All other satellites of Neptune are too small and/or too distant to produce an umbra.
From this distance, the Sun's angular diameter is reduced to one and a quarter arcminutes across. Here are the angular diameters of the moons that are large enough to fully eclipse the Sun: Naiad, 7–13'; Thalassa, 8–14'; Despina, 14–22'; Galatea, 13–18'; Larissa, 10–14'; Proteus, 13–16'; Triton, 26–28'.
Just because the moons are large enough to fully eclipse the Sun does not necessarily mean that they will do so. Eclipses of the Sun from Neptune are rare due to the planet's long orbital period and large axial tilt of 28 degrees. In addition, the largest moon, Triton, has an orbital inclination of about 25 degrees to Neptune's equator. This makes eclipses of the Sun by Triton rare. Even when such an eclipse does occur, it passes rather quickly, as Triton moves in the opposite direction of Neptune's spin.Solar eclipses on Uranus
Solar eclipses on Uranus occur when any of the natural satellites of Uranus passes in front of the Sun as seen from Uranus. Eclipses can occur only near a solar ring plane-crossing of Uranus (equinox), occurring approximately every 42 years, with the last crossing being in 2007/2008.For bodies that appear smaller in angular diameter than the Sun, the proper term would be a transit and bodies that are larger than the apparent size of the Sun, the proper term would be an occultation.
Twelve satellites of Uranus—Cressida, Desdemona, Juliet, Portia, Rosalind, Belinda, Puck, Miranda, Ariel, Umbriel, Titania and Oberon—are large enough and near enough to eclipse the Sun.
All other satellites of Uranus are too small or too distant to produce an umbra.
At its distance from the Sun, the Sun's angular diameter is reduced to a tiny disk about 2 arcminutes across. The angular diameters of the moons large enough to fully eclipse the sun are: Cressida, 6–8'; Desdemona, 6–7'; Juliet, 10–12'; Portia, 9–13'; Rosalind, 4–5'; Belinda, 6–8'; Puck, 6–8'; Miranda, 10–15'; Ariel, 20–23'; Umbriel, 15–17'; Titania, 11–13'; Oberon, 8–9'.Tau2 Aquarii
Tau2 Aquarii (τ2 Aqr, τ2 Aquarii) is the Bayer designation for a star in the equatorial constellation of Aquarius. It is visible to the naked eye with an apparent visual magnitude of +4.0. Because the star lies near the ecliptic it is subject to occultations by the Moon.This is an orange-hued giant star with a stellar classification of K5 III. The measured angular diameter, after correction for limb darkening, is 5.12 ± 0.05 mas. At an estimated distance of 318 light-years (97 parsecs) based on parallax measurements, this yields a physical size of about 53 times the radius of the Sun.Total penumbral lunar eclipse
A total penumbral lunar eclipse is a lunar eclipse that occurs when the moon becomes completely immersed in the penumbral cone of the Earth without touching the umbra.The path for the moon to pass within the penumbra and outside the umbra is very narrow. It can only happen on the Earth's northern or southern penumbral edges. In addition, the size of the penumbra is sometimes too small where the moon enters it to contain the moon. The width of the Earth's penumbra is determined by the sun's angular diameter at the time of the eclipse, and the moon's angular diameter is larger than the sun over part of its elliptical orbit, depending on whether the eclipse occurs at the nearest (perigee) or farthest point (apogee) in its orbit around the earth. The majority of the time, the size of the moon and the size of the Earth's penumbra where the moon crosses it mean that most eclipses will not be total penumbral in nature.Transit of Deimos from Mars
A transit of Deimos across the Sun as seen from Mars occurs when Deimos passes directly between the Sun and a point on the surface of Mars, obscuring a small part of the Sun's disc for an observer on Mars. During a transit, Deimos can be seen from Mars as a small dark spot rapidly moving across the Sun's face.
The event could also be referred to as a partial eclipse of the Sun by Deimos. However, since the angular diameter of Deimos is only about 1/10 of the angular diameter of the Sun as seen from Mars, it is more natural to refer to it as a transit. The angular diameter of Deimos is only 2½ times the angular diameter of Venus as seen from Earth during a transit of Venus from Earth.Édouard Stephan
Édouard Jean-Marie Stephan (31 August 1837 – 31 December 1923) was a French astronomer. His surname is sometimes spelled Stéphan in some literature, but this is apparently erroneous.
He was born in Sainte Pezenne (today one of the districts of the town of Niort) and attended the Ecole Normale Superieure, and graduated at the top of his class in 1862.He was the director of the Marseille Observatory from 1864 to 1907 (until 1872 he was subordinate to Urbain le Verrier). In the early part of his career there, he had limited opportunities to do observations because he was preoccupied with improving the observatory. He discovered the asteroid 89 Julia in 1866. In 1867 he used the new telescope to observe a transit of Mercury.Between 1870 and 1875, Stephan systematically studied nebulae, precisely recording their positions and discovering many new ones. His goal was to enable the exact measurement of stellar proper motions by creating a reference system of fixed objects.In 1873, Stephan was the first person to attempt to measure the angular diameter of a star using interferometry, converting the 80 cm telescope at Marseille Observatory into an interferometer. He did this by obscuring the reflector with a mask containing two vertical slits.
The star he chose to perform this experiment was Sirius. He did not succeed in resolving any stellar disks, but by 1874 had obtained an upper limit to stellar diameters of 0.158" (the true angular diameter of Sirius is 0.0059 arcseconds, and for comparison, the angular diameter of Alpha Centauri and Betelgeuse are 0.0145 and 0.05 arcseconds respectively).In 1881 he discovered NGC 5, and he discovered the galaxy NGC 6027 the following year using the 80 cm reflector.Among others, he discovered Stephan's Quintet, also known as "Arp 319", a group of five galaxies. Stephan made this discovery with the first telescope equipped with a reflection coated mirror.In 1884 the French Academy of Sciences awarded him the Valz Prize (Prix Valz). His name is associated with the periodic comet 38P/Stephan-Oterma, although Jérôme Coggia saw it first.
He became a Chevalier of the Légion d'honneur in 1868 and an Officier of the Légion d'honneur in 1879.