# Angular distance

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Angular distance $\theta$ (also known as angular separation, apparent distance, or apparent separation) is the angle between the two sightlines, or between two point objects as viewed from an observer.

Angular distance shows up in mathematics (in particular geometry and trigonometry) and all natural sciences (e.g. astronomy and geophysics). In the classical mechanics of rotating objects, it appears alongside angular velocity, angular acceleration, angular momentum, moment of inertia and torque.

## 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 $\tan \left({\frac {a}{D}}\right)$ :

$\theta \approx {\dfrac {a}{D}}$ 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,

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