Angular distance

In mathematics (in particular geometry and trigonometry) and all natural sciences (e.g. astronomy and geophysics), the angular distance (angular separation, apparent distance, or apparent separation) between two point objects, as viewed from a location different from either of these objects, is the angle of length between the two directions originating from the observer and pointing toward these two objects.

Use

The term angular distance (or separation) is technically synonymous with angle itself, but is meant to suggest the (often vast, unknown, or irrelevant) linear distance between these objects (for instance, stars as observed from Earth).

Measurement

Since the angular distance (or separation) is conceptually identical to an angle, it is measured in the same units, such as degrees or radians, using instruments such as goniometers or optical instruments specially designed to point in well-defined directions and record the corresponding angles (such as telescopes).

Equation

In order to calculate the angular distance $\theta$ in arcseconds for binary star systems, extrasolar planets, solar system objects and other astronomical objects, we use orbital distance (semi-major axis), $a$ , in AU divided by stellar distance $D$ in parsecs, per the small-angle approximation for $\tan({\frac {a}{D}})$ :

$\theta \approx {\dfrac {a}{D}}$ Given two angular positions, each specified by a right ascension (RA), $\alpha \in [0,2\pi ]$ ; and declination (dec), $\delta \in [-\pi /2,\pi /2]$ , the angular distance between the two points can be calculated as,

$\theta =\cos ^{-1}\left[\sin(\delta _{1})\sin(\delta _{2})+\cos(\delta _{1})\cos(\delta _{2})\cos(\alpha _{1}-\alpha _{2})\right]$ 