Angular diameter

Jump to navigationJump to search

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.

• 
  

Formula[edit]

Diagram for the formula of the angular diameter

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 formula^[1]

${\displaystyle \delta =2\arctan \left({\frac {d}{2D}}\right),}$

in which ${\displaystyle \delta }$ is the angular diameter, and ${\displaystyle d}$ and ${\displaystyle D}$ are the actual diameter of and the distance to the object. When ${\displaystyle D\gg d}$, we have ${\displaystyle \delta \approx d/D}$, and the result obtained is in radians.

For a spherical object whose actual diameter equals ${\displaystyle d_{\mathrm {act} },}$ and where ${\displaystyle D}$ is the distance to the centre of the sphere, the angular diameter can be found by the formula

${\displaystyle \delta =2\arcsin \left({\frac {d_{\mathrm {act} }}{2D}}\right)}$

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 ${\displaystyle x\ll 1}$:^[2]

${\displaystyle \arcsin x\approx \arctan x\approx x}$ .