Angular distance

Angular distance appears in mathematics (in particular geometry and trigonometry) and all natural sciences (e.g., kinematics, astronomy, and geophysics). In the classical mechanics of rotating objects, it appears alongside angular velocity, angular acceleration, angular momentum, moment of inertia and torque. As an angle, the angular separation is measured using angular 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). 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 {\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 their right ascensions (RA), ( α A , α B ) ∈ [ 0 , 2 π ] {\displaystyle (\alpha _{A},\alpha _{B})\in [0,2\pi ]} ; and declinations (dec), ( δ A , δ B ) ∈ [ − π / 2 , π / 2 ] {\displaystyle (\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 the celestial sphere. The dot product of the vectors O A {\displaystyle \mathbf {OA} } and O B {\displaystyle \mathbf {OB} } is equal to: which is equivalent to: In the ( x , y , z ) {\displaystyle (x,y,z)} frame, the two unitary vectors are decomposed into: n A = ( cos ⁡ δ A cos ⁡ α A cos ⁡ δ A sin ⁡ α A sin ⁡ δ A ) a n d n B = ( cos ⁡ δ B cos ⁡ α B cos ⁡ δ B sin ⁡ α B sin ⁡ δ B ) . {\displaystyle \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, n A ⋅ n B = cos ⁡ δ A cos ⁡ α A cos ⁡ δ B cos ⁡ α B + cos ⁡ δ A sin ⁡ α A cos ⁡ δ B sin ⁡ α B + sin ⁡ δ A sin ⁡ δ B ≡ cos ⁡ θ {\displaystyle \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: 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 θ ≪ 1 {\displaystyle \theta \ll 1} radian, implying α A − α B ≪ 1 {\displaystyle \alpha _{A}-\alpha _{B}\ll 1} and δ A − δ B ≪ 1 {\displaystyle \delta _{A}-\delta _{B}\ll 1} , we can develop the above expression and simplify it.