The term angular distance (or separation) is technically synonymous with angle itself, but is meant to suggest the linear distance between objects (for instance, a couple of stars observed from Earth).
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).
The above expression is valid for any position of A and B on the sphere. In astronomy, it often happens that the considered objects are really close in the sky: stars in a telescope field of view, binary stars, the satellites of the giant planets of the solar system, etc. In the case where radian, implying and , we can develop the above expression and simplify it. In the small-angle approximation, at second order, the above expression becomes:
Given that and , at a second-order development it turns that , so that
If we consider a detector imaging a small sky field (dimension much less than one radian) with the -axis pointing up, parallel to the meridian of right ascension , and the -axis along the parallel of declination , the angular separation can be written as:
where and .
Note that the -axis is equal to the declination, whereas the -axis is the right ascension modulated by because the section of a sphere of radius at declination (latitude) is (see Figure).