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 ${\displaystyle R'=R\cos \delta _{A}}$ (see Figure).