Angular distance

Angular distanceorangular separationis the measure of theanglebetween theorientationof twostraight lines,rays,orvectorsinthree-dimensional space,or thecentral anglesubtended by theradiithrough two points on asphere.When the rays arelines of sightfrom an observer to two points in space, it is known as theapparent distanceorapparent separation.

Angular distance appears inmathematics(in particulargeometryandtrigonometry) and allnatural sciences(e.g.,kinematics,astronomy,andgeophysics). In theclassical mechanicsof rotating objects, it appears alongsideangular velocity,angular acceleration,angular momentum,moment of inertiaandtorque.

Use

The termangular distance(orseparation) is technically synonymous withangleitself, but is meant to suggest the lineardistancebetween objects (for instance, a couple ofstarsobserved fromEarth).

Measurement

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

Formulation

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 $A$ and $B$ observed from the Earth. The objects $A$ and $B$ are defined by theircelestial 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 $O$ indicate the observer on Earth, assumed to be located at the center of thecelestial sphere.Thedot productof 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

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 $\theta \ll 1$ 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

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

\theta \approx {\sqrt {\delta x^{2}+\delta y^{2}}}

where $\delta x=(\ Alpha _{A}-\ Alpha _{B})\cos \delta _{A}$ and $\delta y=\delta _{A}-\delta _{B}$ .

Note that the $y$ -axis is equal to the declination, whereas the $x$ -axis is the right ascension modulated by $\cos \delta _{A}$ because the section of a sphere of radius $R$ at declination (latitude) $\delta$ is $R'=R\cos \delta _{A}$ (see Figure).

References