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Angular distance

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Angle between the two sightlines or two objects as viewed from an observer

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Angular distance orangular separation is the measure of theangle between theorientation of twostraight lines,rays, orvectors inthree-dimensional space, or thecentral angle subtended by theradii through two points on asphere. When the rays arelines of sight from an observer to two points in space, it is known as theapparent distance orapparent separation.

Angular distance appears inmathematics (in particulargeometry andtrigonometry) and allnatural sciences (e.g.,kinematics,astronomy, andgeophysics). In theclassical mechanics of rotating objects, it appears alongsideangular velocity,angular acceleration,angular momentum,moment of inertia andtorque.

Use

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The termangular distance (orseparation) is technically synonymous withangle itself, but is meant to suggest the lineardistance between objects (for instance, a pair ofstars observed fromEarth).

Measurement

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Since the angular distance (or separation) is conceptually identical to an angle, it is measured in the sameunits, such asdegrees orradians, using instruments such asgoniometers or optical instruments specially designed to point in well-defined directions and record the corresponding angles (such astelescopes).

Formulation

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Angular separation $\theta$ between points A and B as seen from O

{\displaystyle \theta } — Angular separation $\theta$ 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 twoastronomical objects $A {\displaystyle A}$ and $B {\displaystyle B}$ observed from the Earth. The objects $A {\displaystyle A}$ and $B {\displaystyle B}$ are defined by their celestial coordinates, namely theirright ascensions (RA), $(\alpha _{A},\alpha _{B})\in [0,2\pi ]$ ; anddeclinations (dec), $(\delta _{A},\delta _{B})\in [-\pi /2,\pi /2]$ . Let $O {\displaystyle O}$ indicate the observer on Earth, assumed to be located at the center of thecelestial sphere. Thedot product of the vectors $\mathbf {OA}$ and $\mathbf {OB}$ is equal to:

\mathbf {OA} \cdot \mathbf {OB} =R^{2}\cos \theta

which is equivalent to:

\mathbf {n_{A}} \cdot \mathbf {n_{B}} =\cos \theta

In the $(x,y,z)$ frame, the two unitary vectors are decomposed into: $\mathbf {n_{A}} ={\begin{pmatrix}\cos \delta _{A}\cos \alpha _{A}\\\cos \delta _{A}\sin \alpha _{A}\\\sin \delta _{A}\end{pmatrix}}\mathrm {\qquad and\qquad } \mathbf {n_{B}} ={\begin{pmatrix}\cos \delta _{B}\cos \alpha _{B}\\\cos \delta _{B}\sin \alpha _{B}\\\sin \delta _{B}\end{pmatrix}}.$ Therefore, $\mathbf {n_{A}} \cdot \mathbf {n_{B}} =\cos \delta _{A}\cos \alpha _{A}\cos \delta _{B}\cos \alpha _{B}+\cos \delta _{A}\sin \alpha _{A}\cos \delta _{B}\sin \alpha _{B}+\sin \delta _{A}\sin \delta _{B}\equiv \cos \theta$ then:

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

Small angular distance approximation

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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 theSolar System, etc. In the case where $\theta \ll 1$ radian, implying $\alpha _{A}-\alpha _{B}\ll 1$ and $\delta _{A}-\delta _{B}\ll 1$ , we can develop the above expression and simplify it. In thesmall-angle approximation, at second order, the above expression becomes:

\cos \theta \approx 1-{\frac {\theta ^{2}}{2}}\approx \sin \delta _{A}\sin \delta _{B}+\cos \delta _{A}\cos \delta _{B}\left[1-{\frac {(\alpha _{A}-\alpha _{B})^{2}}{2}}\right]

meaning

1-{\frac {\theta ^{2}}{2}}\approx \cos(\delta _{A}-\delta _{B})-\cos \delta _{A}\cos \delta _{B}{\frac {(\alpha _{A}-\alpha _{B})^{2}}{2}}

hence

1-{\frac {\theta ^{2}}{2}}\approx 1-{\frac {(\delta _{A}-\delta _{B})^{2}}{2}}-\cos \delta _{A}\cos \delta _{B}{\frac {(\alpha _{A}-\alpha _{B})^{2}}{2}}

Given that $\delta _{A}-\delta _{B}\ll 1$ and $\alpha _{A}-\alpha _{B}\ll 1$ , at a second-order development it turns that $\cos \delta _{A}\cos \delta _{B}{\frac {(\alpha _{A}-\alpha _{B})^{2}}{2}}\approx \cos ^{2}\delta _{A}{\frac {(\alpha _{A}-\alpha _{B})^{2}}{2}}$ , so that

\theta \approx {\sqrt {\left[(\alpha _{A}-\alpha _{B})\cos \delta _{A}\right]^{2}+(\delta _{A}-\delta _{B})^{2}}}

Small angular distance: planar approximation

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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 $y {\displaystyle y}$ -axis pointing up, parallel to the meridian of right ascension $\alpha$ , and the $x {\displaystyle x}$ -axis along the parallel of declination $\delta$ , the angular separation can be written as: