HOME

TheInfoList



OR:

Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) is the
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
between the two sightlines, or between two point objects as viewed from an observer. Angular distance appears in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
(in particular
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
) and all
natural science Natural science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatab ...
s (e.g.
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
and
geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
). In the
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
of rotating objects, it appears alongside
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
, angular acceleration,
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, moment of inertia and
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
.


Use

The term ''angular distance'' (or ''separation'') is technically synonymous with ''angle'' itself, but is meant to suggest the linear
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between objects (for instance, a couple of
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s observed from
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
).


Measurement

Since the angular distance (or separation) is conceptually identical to an angle, it is measured in the same
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
, such as degrees or
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s, using instruments such as goniometers or optical instruments specially designed to point in well-defined directions and record the corresponding angles (such as
telescope A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to observe ...
s).


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 An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often us ...
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 , 2\pi/math>; and declinations (dec), (\delta_A, \delta_B) \in \pi/2, \pi/2/math>. Let O indicate the observer on Earth, assumed to be located at the center of the
celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, ...
. The
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of the vectors \mathbf and \mathbf is equal to: :\mathbf\cdot\mathbf= R^2 \cos\theta which is equivalent to: :\mathbf.\mathbf = \cos\theta In the (x,y,z) frame, the two unitary vectors are decomposed into: :\mathbf = \begin \cos\delta_A \cos\alpha_A\\ \cos\delta_A \sin\alpha_A\\ \sin\delta_A \end \mathrm \mathbf = \begin \cos\delta_B \cos\alpha_B\\ \cos\delta_B \sin\alpha_B\\ \sin\delta_B \end . Therefore, :\mathbf\mathbf = \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^\left sin\delta_A \sin\delta_B + \cos\delta_A \cos\delta_B \cos(\alpha_A - \alpha_B)\right/math>


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 \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 The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians: : \begin \sin \theta &\approx \theta \\ \cos \theta &\approx 1 - \ ...
, at second order, the above expression becomes: :\cos\theta \approx 1 - \frac \approx \sin\delta_A \sin\delta_B + \cos\delta_A \cos\delta_B \left - \frac\right/math> meaning :1 - \frac \approx \cos(\delta_A-\delta_B) - \cos\delta_A\cos\delta_B \frac hence :1 - \frac \approx 1 - \frac - \cos\delta_A\cos\delta_B \frac. 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 \approx \cos^2\delta_A \frac, so that :\theta \approx \sqrt


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 \alpha, and the x-axis along the parallel of declination \delta, the angular separation can be written as: : \theta \approx \sqrt where \delta x = (\alpha_A - \alpha_B)\cos\delta_A and \delta y=\delta_A-\delta_B. Note that the y-axis is equal to the declination, whereas the x-axis is the right ascension modulated by \cos\delta_A because the section of a sphere of radius R at declination (latitude) \delta is R' = R \cos\delta_A (see Figure).


See also

*
Milliradian A milliradian ( SI-symbol mrad, sometimes also abbreviated mil) is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Milliradians are used in adjustment of firearm sights by adjusting ...
*
Gradian In trigonometry, the gradian, also known as the gon (from grc, γωνία, gōnía, angle), grad, or grade, is a unit of measurement of an angle, defined as one hundredth of the right angle; in other words, there are 100 gradians in 90 degree ...
*
Hour angle In astronomy and celestial navigation, the hour angle is the angle between two planes: one containing Earth's axis and the zenith (the '' meridian plane''), and the other containing Earth's axis and a given point of interest (the ''hour circle ...
*
Central angle A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc le ...
*
Angle of rotation In mathematics, the angle of rotation is a measurement of the amount, of namely angle, that a figure is rotated about a fixed point, often the center of a circle. A clockwise rotation is considered a negative rotation, so that, for instance ...
*
Angular diameter The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is ...
*
Angular displacement Angular displacement of a body is the angle (in radians, degrees or revolutions) through which a point revolves around a centre or a specified axis in a specified sense. When a body rotates about its axis, the motion cannot simply be analyzed ...
*
Great-circle distance The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...
*


References


CASTOR, author(s) unknown. "The Spherical Trigonometry vs. Vector Analysis"
* {{DEFAULTSORT:Angular Distance Angle Astrometry Trigonometry