Angular distance

 ( Angle ) 

Angular distance or angular separation is the measure of the angle between the orientation of two , rays, or vectors in three-dimensional space, or the central angle subtended by the radius through two points on a sphere. When the rays are lines of sight from an observer to two points in space, it is known as the apparent distance or apparent separation.

Angular distance appears in mathematics (in particular geometry and trigonometry) and all (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.

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Use The term angular distance (or separation) is technically synonymous with angle itself, but is meant to suggest the linear distance between objects (for instance, a pair of observed from Earth).

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Measurement Since the angular distance (or separation) is conceptually identical to an angle, it is measured in the same units, such as degrees or , using instruments such as or optical instruments specially designed to point in well-defined directions and record the corresponding angles (such as ).

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Formulation 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 ascension, $(\backslash alpha\_A,\; \backslash alpha\_B)\backslash in\; 0,$; and Declination, $(\backslash delta\_A,\; \backslash delta\_B)\; \backslash in\; -\backslash pi/2,$. Let $O$ indicate the observer on Earth, assumed to be located at the center of the celestial sphere. The dot product of the vectors $\backslash mathbf\{OA\}$ and $\backslash mathbf\{OB\}$ is equal to:

$\backslash mathbf\{OA\}\backslash cdot\backslash mathbf\{OB\}=\; R^2\; \backslash cos\backslash theta$

which is equivalent to:

$\backslash mathbf\{n\_A\}\; \backslash cdot\; \backslash mathbf\{n\_B\}\; =\; \backslash cos\backslash theta$

In the $(x,y,z)$ frame, the two unitary vectors are decomposed into: $$\backslash mathbf\{n\_A\}\; =\; \backslash begin\{pmatrix\}$$

\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, $$\backslash mathbf\{n\_A\}\; \backslash cdot\; \backslash mathbf\{n\_B\}\; =\; \backslash cos\backslash delta\_A\; \backslash cos\backslash alpha\_A\; \backslash cos\backslash delta\_B\; \backslash cos\backslash alpha\_B\; +\; \backslash cos\backslash delta\_A\; \backslash sin\backslash alpha\_A\; \backslash cos\backslash delta\_B\; \backslash sin\backslash alpha\_B\; +\; \backslash sin\backslash delta\_A\; \backslash sin\backslash delta\_B\; \backslash equiv\; \backslash cos\backslash theta$$ then:

$\backslash theta\; =\; \backslash cos^\{-1\}\backslash left\backslash sin\backslash delta\_A$

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

$\backslash cos\backslash theta\; \backslash approx\; 1\; -\; \backslash frac\{\backslash theta^2\}\{2\}\; \backslash approx\; \backslash sin\backslash delta\_A\; \backslash sin\backslash delta\_B\; +\; \backslash cos\backslash delta\_A\; \backslash cos\backslash delta\_B\; \backslash left1$

meaning

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

hence

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

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

$\backslash theta\; \backslash approx\; \backslash sqrt\{\backslash left(\backslash alpha\_A^2\; +\; (\backslash delta\_A-\backslash delta\_B)^2\}$

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Small angular distance: planar approximation 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 $\backslash alpha$, and the $x$-axis along the parallel of declination $\backslash delta$, the angular separation can be written as:

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

where $\backslash delta\; x\; =\; (\backslash alpha\_A\; -\; \backslash alpha\_B)\backslash cos\backslash delta\_A$ and $\backslash delta\; y=\backslash delta\_A-\backslash delta\_B$.

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

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See also

• Milliradian
• Gradian
• Hour angle
• Central angle
• Angle of rotation
• Angular diameter
• Angular displacement
• Great-circle distance

• CASTOR, author Michael A. Earl. "The Spherical Trigonometry vs. Vector Analysis".

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