Angular distance

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

General case

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 $A$ and $B$ observed from the Earth. The objects $A$ and $B$ are defined by their celestial coordinates, namely their right ascensions (RA), $(\alpha _{A},\alpha _{B})\in [0,2\pi ]$ ; and declinations (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 the celestial sphere. The dot 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}} .\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}} \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 ofthe 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 the small-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}}

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

If we consider a detector imaging a small sky field (dimension much less than one radian) with $z$ pointing up and $x$ parallel to the equator (ecliptic plane), 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 fact that the $y$ -axis is directly given by the declination and that $x$ is modulated by $\cos \delta _{A}$ is just the consequence that the section of a sphere of radius $R$ at declination (latitude) $\delta$ is $R'=R\cos \delta _{A}$ (see Figure).

References

Use