Angular diameter
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. 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 of the angular diameter.
Formula
The angular diameter of a circle whose plane is perpendicular to the displacement vector between the point of view and the centre of said circle can be calculated using the formulain which is the angular diameter, and is the actual diameter of the object, and is 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 :
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/ 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
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:
| 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 / 1887-1952″ |
| Moon | 29′20″ - 34′6″ | 28-32.5 times the maximum value for Venus / 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″ | |
| Alphard | 0.00909″ | |
| Alpha Centauri A | 0.007″ | |
| Canopus | 0.006″ | |
| Sirius | 0.005936″ | |
| Altair | 0.003″ | |
| Deneb | 0.002″ | |
| Proxima Centauri | 0.001″ | |
| Alnitak | 0.0005″ | |
| Event horizon of black hole M87* at center of the M87 galaxy, imaged by the Event Horizon Telescope in 2019. | 0.000025″ | |
| A star like Alnitak at a distance where the Hubble Space Telescope would just be able to see it | arcsec |
and its four Galilean moons with the apparent diameter of the full Moon during their conjunction on 10 April 2017.
The table shows that the angular diameter of Sun, when seen from Earth is approximately 32′, as illustrated above.
Thus the angular diameter of the Sun 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
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.