Angular distance

Celestial coordinates

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 ${\displaystyle A}$ and ${\displaystyle B}$ observed from the Earth. The objects ${\displaystyle A}$ and ${\displaystyle B}$ are defined by their celestial coordinates, namely their right ascensions (RA), ${\displaystyle ({\alpha }_A,{\alpha }_B)\in [0,2\pi ]}$; and declinations (dec), ${\displaystyle ({\delta }_A,{\delta }_B)\in [-\pi /2,\pi /2]}$. Let ${\displaystyle O}$ indicate the observer on Earth, assumed to be located at the center of the celestial sphere. The dot product of the vectors ${\displaystyle \mathbf{O}\mathbf{A}}$ and ${\displaystyle \mathbf{O}\mathbf{B}}$ is equal to:

${\displaystyle \mathbf{O}\mathbf{A}\cdot \mathbf{O}\mathbf{B}=R^2\cos \theta }$

which is equivalent to:

${\displaystyle {\mathbf{n}}_{\mathbf{A}}\cdot {\mathbf{n}}_{\mathbf{B}}=\cos \theta }$

In the ${\displaystyle (x,y,z)}$ frame, the two unitary vectors are decomposed into: $${\displaystyle {\mathbf{n}}_{\mathbf{A}}=\left(\begin{array}{c}\cos {\delta }_A\cos {\alpha }_A\\ \cos {\delta }_A\sin {\alpha }_A\\ \sin {\delta }_A\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d}{\mathbf{n}}_{\mathbf{B}}=\left(\begin{array}{c}\cos {\delta }_B\cos {\alpha }_B\\ \cos {\delta }_B\sin {\alpha }_B\\ \sin {\delta }_B\end{array}\right).}$$ Therefore, $${\displaystyle {\mathbf{n}}_{\mathbf{A}}\cdot {\mathbf{n}}_{\mathbf{B}}=\cos {\delta }_A\cos {\alpha }_A\cos {\delta }_B\cos {\alpha }_B+\cos {\delta }_A\sin {\alpha }_A\cos {\delta }_B\sin {\alpha }_B+\sin {\delta }_A\sin {\delta }_B\equiv \cos \theta }$$ then:

${\displaystyle \theta ={\cos }^{-1}\left[\sin {\delta }_A\sin {\delta }_B+\cos {\delta }_A\cos {\delta }_B\cos ({\alpha }_A-{\alpha }_B)\right]}$

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 ${\displaystyle \theta \ll 1}$ radian, implying ${\displaystyle {\alpha }_A-{\alpha }_B\ll 1}$ and ${\displaystyle {\delta }_A-{\delta }_B\ll 1}$, we can develop the above expression and simplify it. In the small-angle approximation, at second order, the above expression becomes:

${\displaystyle \cos \theta \approx 1-\frac{{\theta }^2}{2}\approx \sin {\delta }_A\sin {\delta }_B+\cos {\delta }_A\cos {\delta }_B\left[1-\frac{({\alpha }_A-{\alpha }_B{)}^2}{2}\right]}$

meaning

${\displaystyle 1-\frac{{\theta }^2}{2}\approx \cos ({\delta }_A-{\delta }_B)-\cos {\delta }_A\cos {\delta }_B\frac{({\alpha }_A-{\alpha }_B{)}^2}{2}}$

hence

${\displaystyle 1-\frac{{\theta }^2}{2}\approx 1-\frac{({\delta }_A-{\delta }_B{)}^2}{2}-\cos {\delta }_A\cos {\delta }_B\frac{({\alpha }_A-{\alpha }_B{)}^2}{2}}$.

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

${\displaystyle \theta \approx \sqrt{{\left[({\alpha }_A-{\alpha }_B)\cos {\delta }_A\right]}^2+({\delta }_A-{\delta }_B{)}^2}}$

Small angular distance: planar approximation

If we consider a detector imaging a small sky field (dimension much less than one radian) with the ${\displaystyle y}$-axis pointing up, parallel to the meridian of right ascension ${\displaystyle \alpha }$, and the ${\displaystyle x}$-axis along the parallel of declination ${\displaystyle \delta }$, the angular separation can be written as:

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

where ${\displaystyle \delta x=({\alpha }_A-{\alpha }_B)\cos {\delta }_A}$ and ${\displaystyle \delta y={\delta }_A-{\delta }_B}$.

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