The angular diameter, angular width, angular size, apparent diameter, or apparent size is an angular separation (in units of angle) 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 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 (approximately 0.017° or 0.0003 radians). 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
δ = 2 arctan ( d 2 D ) , {\displaystyle \delta =2\arctan \left({\frac {d}{2D}}\right),}
in which δ {\displaystyle \delta } is the angular diameter (in units of angle, normally radians, sometimes in degrees, depending on the arctangent implementation), d {\displaystyle d} is the linear diameter of the object (in units of length), and D {\displaystyle D} is the distance to the object (also in units of length). When D ≫ d {\displaystyle D\gg d} , we have:
δ ≈ d / D {\displaystyle \delta \approx d/D} , and the result obtained is necessarily in radians.
For a sphere For a spherical object whose linear diameter equals d {\displaystyle d} and where D {\displaystyle D} is the distance to the center of the sphere, the angular diameter can be found by the following modified formula
δ = 2 arcsin ( d 2 D ) {\displaystyle \delta =2\arcsin \left({\frac {d}{2D}}\right)}
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 D {\displaystyle D} as the hypotenuse and d a c t 2 D {\displaystyle {\frac {d_{\mathrm {act} }}{2D}}} as the sine. The formula is related to the zenith angle to the horizon,
δ = π − 2 arccos ( R R + h ) {\displaystyle \delta =\pi -2\arccos \left({\frac {R}{R+h}}\right)}
where R is the radius of the sphere and h is the distance to the near surface 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 x {\displaystyle x} :
arcsin x ≈ arctan x ≈ x . {\displaystyle \arcsin x\approx \arctan x\approx x.}
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 (1°) and a radian is 180/π degrees. So one radian equals 3,600 × 180/ π {\displaystyle \pi } arcseconds, which is about 206,265 arcseconds (1 rad ≈ 206,264.806247"). Therefore, the angular diameter of an object with physical diameter d at a distance D, expressed in arcseconds, is given by:
δ = 206 , 265 ( d / D ) a r c s e c o n d s {\displaystyle \delta =206,265~(d/D)~\mathrm {arcseconds} } . These objects have an angular diameter of 1″:
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