Angular distance

Angular distance (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 appears 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.

UseEdit

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).

MeasurementEdit

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).

EquationEdit

General caseEdit

 
Angular separation   between points A and B as seen from O

To derive the equation that describes the angular separation of two points located on the surface of a sphere as seen from the center of the sphere, we use the example of two astronomical objects   and   observed from the Earth. The objects   and   are defined by their celestial coordinates, namely their right ascensions (RA),  ; and declinations (dec),  . Let   indicate the observer on Earth, assumed to be located at the center of the celestial sphere. The dot product of the vectors   and   is equal to:

 

which is equivalent to:

 

In the   frame, the two unitary vectors are decomposed into:

 

Therefore,

 

then:

 

Small angular distance approximationEdit

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:

 

meaning

 

hence

 .

Given that   and  , at a second-order development it turns that  , so that

 

Small angular distance: planar approximationEdit

 
Planar approximation of angular distance on sky

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).

See alsoEdit

ReferencesEdit

  • CASTOR, author(s) unknown. "The Spherical Trigonometry vs. Vector Analysis".
  • Weisstein, Eric W. "Angular Distance". MathWorld.