Angular diameter

# Angular diameter

Overview
The angular diameter or apparent size of an object as seen from a given position is the “visual diameter” of the object measured as an angle. In the vision sciences it is called the visual angle
Visual angle
The visual angle is the angle a viewed object subtends at the eye, usually stated in degrees of arc.It also is called the object's angular size....

. The visual diameter is the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

of the perspective projection
Projection (mathematics)
Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...

of the object on a plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

through its centre that is perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to the viewing direction. Because of foreshortening, it may be quite different from the actual physical diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

for an object that is seen under an angle.
Discussion

Encyclopedia
The angular diameter or apparent size of an object as seen from a given position is the “visual diameter” of the object measured as an angle. In the vision sciences it is called the visual angle
Visual angle
The visual angle is the angle a viewed object subtends at the eye, usually stated in degrees of arc.It also is called the object's angular size....

. The visual diameter is the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

of the perspective projection
Projection (mathematics)
Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...

of the object on a plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

through its centre that is perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to the viewing direction. Because of foreshortening, it may be quite different from the actual physical diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

for an object that is seen under an angle. For a disk-shaped object at a large distance, the visual and actual diameters are the same.

## Formulas

The angular diameter of an object can be calculated using the formula:
in which is the angular diameter, and and are the visual diameter of and the distance to the object, expressed in the same units. When is much larger than , may be approximated by the formula , in which case the result is in radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s.

For a spherical object whose actual diameter equals , the angular diameter can be found with the formula:
For practical use, the distinction between the visual diameter and the actual diameter only makes a difference for spherical objects that are relatively close.

## Use in astronomy

In astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

the sizes of objects in the sky are often given in terms of their angular diameter as seen from Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

, 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
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

, and a radian is 180/ degrees, so one radian equals 3600.180/ arcseconds, which is about 206265 arcseconds. Therefore, the angular diameter of an object with visual diameter d at a distance D, expressed in arcseconds, is given by: = 206265 d / D arcseconds.

The angular diameter of Earth's orbit around the Sun, from a distance of one parsec
Parsec
The parsec is a unit of length used in astronomy. It is about 3.26 light-years, or just under 31 trillion kilometres ....

, is 2″ (two arcseconds).

The angular diameter of the Sun, from a distance of one light-year
Light-year
A light-year, also light year or lightyear is a unit of length, equal to just under 10 trillion kilometres...

, is 0.03″, and that of the 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 the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

This table shows the angular sizes of noteworthy celestial bodies
Astronomical object
Astronomical objects or celestial objects are naturally occurring physical entities, associations or structures that current science has demonstrated to exist in the observable universe. The term astronomical object is sometimes used interchangeably with astronomical body...

as seen from the Earth:
Celestial body Angular diameter Relative size (10 pixels per arcsecond)
Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

31.6′ – 32.7′ 28.7 – 29.7 times the maximum value for Venus (orange bar below)
Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

29.3′ – 34.1′ 26.6 – 31.0 times the maximum value for Venus (orange bar below)
Venus
Venus
Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...

9.565″ – 66.012″ />
Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...

29.800″ – 50.115″ />
Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...

14.991″ – 20.790″ />
Mars
Mars
Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...

3.492″ – 25.113″ />
Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...

4.535″ – 13.019″ />
Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

3.340″ – 4.084″ />
Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...

2.179″ – 2.373″ />
Ceres 0.330″ – 0.840″ />
Vesta 0.20" – 0.64" />
Pluto
Pluto
Pluto, formal designation 134340 Pluto, is the second-most-massive known dwarf planet in the Solar System and the tenth-most-massive body observed directly orbiting the Sun...

0.063″ – 0.115″ />
R Doradus is the name of a red giant Mira variable star in the far-southern constellation Dorado, although visually it appears more closely associated with the constellation Reticulum. Its distance from Earth is 204 ± 9 light-years...

0.052″ – 0.062″ />
Betelgeuse
Betelgeuse
Betelgeuse, also known by its Bayer designation Alpha Orionis , is the eighth brightest star in the night sky and second brightest star in the constellation of Orion, outshining its neighbour Rigel only rarely...

0.049″ – 0.060″ />
Eris
Eris (dwarf planet)
Eris, formal designation 136199 Eris, is the most massive known dwarf planet in the Solar System and the ninth most massive body known to orbit the Sun directly...

0.034" – 0.089″ />
Alphard
Alphard
Alphard is the brightest star in the constellation Hydra. The name Alphard is from the Arabic الفرد , "the solitary one", there being no other bright stars near it. It was also known as the "backbone of the Serpent" to the Arabs. In ancient China it formed part of an asterism called the "red bird"...

0.00909″ />
Alpha Centauri A 0.007″ />
Canopus
Canopus
Canopus |Alpha]] Carinae) is the brightest star in the southern constellation of Carina and Argo Navis, and the second brightest star in the night-time sky, after Sirius. Canopus's visual magnitude is −0.72, and it has an absolute magnitude of −5.53.Canopus is a supergiant of spectral...

0.006″ />
Sirius
Sirius
Sirius is the brightest star in the night sky. With a visual apparent magnitude of −1.46, it is almost twice as bright as Canopus, the next brightest star. The name "Sirius" is derived from the Ancient Greek: Seirios . The star has the Bayer designation Alpha Canis Majoris...

0.005936″ />
Altair 0.003″ />
Deneb
Deneb
Deneb is the brightest star in the constellation Cygnus and one of the vertices of the Summer Triangle. It is the 19th brightest star in the night sky, with an apparent magnitude of 1.25. A blue-white supergiant, Deneb is also one of the most luminous nearby stars...

0.002″ />
Proxima Centauri
Proxima Centauri
Proxima Centauri is a red dwarf star about 4.2 light-years distant in the constellation of Centaurus. It was discovered in 1915 by Robert Innes, the Director of the Union Observatory in South Africa, and is the nearest known star to the Sun, although it is too faint to be seen with the naked eye...

0.001″ />

The table shows that the angular diameter of Sun, when seen from Earth is approximately 32 arcminutes (1920 arcseconds or 0.53 degrees), as illustrated above.

This meaning the angular diameter of the Sun is about 250,000 that of Sirius (it has twice the diameter and the distance is 500,000 times as much; the Sun is 10,000,000,000 times as bright, corresponding to an angular diameter ratio of 100,000, so Sirius is roughly 6 times as bright per unit solid angle
Solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

).

The angular diameter of the Sun is also about 250,000 that of Alpha Centauri A (it has the same diameter and the distance is 250,000 times as much; the Sun is 40,000,000,000 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 diameter is 400 times as large and the 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 to 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
Hubble Space Telescope
The Hubble Space Telescope is a space telescope that was carried into orbit by a Space Shuttle in 1990 and remains in operation. A 2.4 meter aperture telescope in low Earth orbit, Hubble's four main instruments observe in the near ultraviolet, visible, and near infrared...

, Ceres has a much larger apparent size.

While angular sizes measured in degrees are useful for larger patches of sky (in the constellation of Orion, for example, the three stars of the belt cover about 3 degrees of angular size), we need much finer units when talking about the angular size of galaxies, nebulae or other objects of the night sky.

Degrees, therefore, are subdivided as follows:
• 360 degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

s (°) in a full circle
• 60 arc-minutes (′) in one degree
• 60 arc-seconds (″) in one arc-minute

To put this in perspective, the full moon viewed from earth is about ½ degree, or 30 arc minutes (or 1800 arc-seconds). The moon's motion across the sky can be measured in angular size: approximately 15 degrees every hour, or 15 arc-seconds per second. A one-mile-long line painted on the face of the moon would appear to us to be about one arc-second in length.

In astronomy, it is typically difficult to directly measure the distance to an object. But 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)
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 to another quantity that is not directly...

.

## Non-circular objects

Many deep sky objects such as galaxies and nebula
Nebula
A nebula is an interstellar cloud of dust, hydrogen gas, helium gas and other ionized gases...

s are non-circular and are thus typically given two measures of diameter: Major Diameter and Minor Diameter. For example, the Small Magellanic Cloud
Small Magellanic Cloud
The Small Magellanic Cloud is a dwarf galaxy. It has a diameter of about 7,000 light-years and contains several hundred million stars. It has a total mass of approximately 7 billion times the mass of our Sun....

has a visual apparent diameter of × .

## 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 seconds of arc across and is 75 percent illuminated, the defect of illumination is 10 seconds of arc.

• Angular diameter distance
• Angular resolution
Angular resolution
Angular resolution, or spatial resolution, describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object...

• Solid angle
Solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

• Visual acuity
Visual acuity
Visual acuity is acuteness or clearness of vision, which is dependent on the sharpness of the retinal focus within the eye and the sensitivity of the interpretative faculty of the brain....

• Visual angle
Visual angle
The visual angle is the angle a viewed object subtends at the eye, usually stated in degrees of arc.It also is called the object's angular size....

• Visual Angle Illusion
Visual angle illusion
In human visual perception, the visual angle, denoted θ, subtended by a viewed object sometimes looks larger or smaller than its actual value. One approach to this phenomenon posits a subjective correlate to the visual angle: the perceived visual angle or perceived angular size...