Angular diameter

The angular diameter, angular width, angular size, apparent diameter, or apparent size is an angular separation 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. 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 resolve with their naked eyes diameters down to about 1 arcminute. This corresponds to 0.3 m at a 1 km distance, or to perceiving Venus as a disk under optimal conditions.

Formulation

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
in which is the angular diameter, is the linear diameter of the object, and is the distance to the object. When, we have:
and the result obtained is necessarily in radians.

For a sphere

For a spherical object whose linear diameter equals and where is the distance to the of the sphere, the angular diameter can be found by the following modified formula
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.
The formula is related to the zenith angle to the horizon,
where R is the radius of the sphere and h is the distance to the near 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 approximations hold for small values of :

Estimating angular diameter using the hand

Estimates of angular diameter may be obtained by holding the hand at right angles to a fully extended arm, as shown in the figure.

Use in astronomy

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 and a radian is 180/π 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:

an object of diameter 1 cm at a distance of 2.06 km
an object of diameter 725.27 km at a distance of 1 astronomical unit
an object of diameter 45 866 916 km at 1 light-year
an object of diameter 1 AU at a distance of 1 parsec

Thus, the angular diameter of Earth's orbit around the Sun 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 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 human body at a distance of the diameter of Earth.
This table shows the angular sizes of noteworthy celestial bodies as seen from Earth, and various other noteworthy celestial objects:

Celestial object	Angular diameter or size	Relative size
Magellanic Stream	over 100°
Gum Nebula	36°
Milky Way	30°
Width of spread out hand with arm stretched out	20°	covering 353 meter of something viewed from a distance of 1 km
Serpens-Aquila Rift	20° by 10°
Jupiter in the sky of Io	19.5°
Canis Major Overdensity	12° by 12°
Smith's Cloud	11°
Large Magellanic Cloud	10.75° by 9.17°	Note: brightest galaxy, other than the Milky Way, in the night sky
Barnard's Loop	10°
Zeta Ophiuchi Sh2-27 nebula	10°
Width of fist with arm stretched out	10°	covering 175 meter of something viewed from a distance of 1 km
Sagittarius Dwarf Spheroidal Galaxy	7.5° by 3.6°
Northern Coalsack Nebula	7° by 5°
Coalsack Nebula	7° by 5°
Cygnus OB7	4° by 7°
Huya's moon	6° 25”	Largest moon from the perspective of a dwarf planet primary
Rho Ophiuchi cloud complex	4.5° by 6.5°
Hyades	5°30	Note: brightest star cluster in the night sky, 0.5 apparent magnitude
Small Magellanic Cloud	5°20 by 3°5
Saturn in the sky of Titan	5.09°
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.
Charon	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: brightest nebula in the night sky, 1.0 apparent magnitude
North America Nebula	2° by 100
Earth in the Moon's sky	2° - 1°48	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	1.5°	The Moon appeared 3.9 billion years ago 2.8 times larger than it does today.
The Sun in the sky of Mercury	1.15° - 1.76°
Orion Nebula	1°5 by 1°
Width of little finger with arm stretched out	1°	covering 17.5 meter of something viewed from a distance of 1 km
The Sun in the sky of Venus	0.7°
Io	35’ 35”
Moon	346 – 2920	32.5–28 times the maximum value for Venus / 2046–1760 the Moon has a diameter of 3,474 km
Sun	3232 – 3127	31–30 times the maximum value for Venus / 1952–1887 the Sun has a diameter of 1,391,400 km
Triton	28’ 11”
Angular size of the distance between Earth and the Moon as viewed from Mars, at inferior conjunction	about 25
Ariel	24’ 11”
Ganymede	18’ 6”
Europa	17’ 51”
Umbriel	16’ 42”
Helix Nebula	about 16 by 28
Miranda	15’ 30”
Tethys	15’ 30”
Titan	15’ 12”
Titania	13’ 12”
Phobos	12’ 56”
Dione	12’ 5”
Rhea	11’ 12”
Mimas	10’ 42”
Enceladus	9’ 38”
Oberon	9’ 22”
Callisto	9’ 8”
Jupiter if it were as close to Earth as Mars	9.0 – 1.2
Spire in Eagle Nebula	440	length is 280
Deimos	2’ 7”
Iapetus	1’ 26”
Venus	–
International Space Station	13	the ISS has a width of about 108 m
Minimum resolvable diameter by the human eye	1	0.3 meter at 1 km distance For visibility of objects with smaller apparent sizes see the necessary apparent magnitudes.
About 100 km on the surface of the Moon	1	Comparable to the size of features like large lunar craters, such as the Copernicus crater, a prominent bright spot in the eastern part of Oceanus Procellarum on the waning side, or the Tycho crater within a bright area in the south, of the lunar near side.
Jupiter	50.1 – 29.8
Earth as seen from Mars	48.2 – 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
Mars	25.1 – 3.5
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
Apparent size of Sun, seen from 90377 Sedna at aphelion	2.04"
Ganymede	1.8 – 1.2	Ganymede has a diameter of 5,268 km
An astronaut at a distance of 350 km, the average altitude of the ISS	1
Minimum resolvable diameter by Galileo Galilei's largest 38mm refracting telescopes	~1	Note: 30x magnification, comparable to very strong contemporary terrestrial binoculars
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
Deneb	0.002
Proxima Centauri	0.001
Alnitak	0.0005
Proxima Centauri b	0.00008
Event horizon of black hole M87* at center of the M87 galaxy, imaged by the Event Horizon Telescope in 2019.	0.000025	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	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: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 with the apparent diameter of the full Moon during their conjunction on 10 April 2017.
The angular diameter of the Sun, as seen from Earth, is about 250,000 times that of Sirius.
The angular diameter of the Sun is also about 250,000 times that of Alpha Centauri A.
The angular diameter of the Sun is about the same as that of the Moon.
Even though Pluto is physically larger than Ceres, when viewed from Earth Ceres has a much larger apparent size.
Angular sizes measured in degrees are useful for larger patches of sky. 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 degrees 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 as viewed from Earth is about °, or 30. 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 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.