Angular distance
(also known as angular separation, apparent distance, or apparent separation) is the
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
between the two
sightlines
In architecture, sightlines are a particularly important consideration in the design of civic structures, such as a stage, arena, or monument. They determine the configuration of such items as theater and stadium design, road junction layout an ...
, 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. ...
) 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 celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
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'' so ...
). 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 classi ...
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 object ...
,
angular acceleration
In physics, angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular accelera ...
,
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
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
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 sur ...
).
Measurement
Since the angular distance (or separation) is conceptually identical to an angle, it is measured in the same
units, 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 ...
s, using instruments such as
goniometer
A goniometer is an instrument that either measures an angle or allows an object to be rotated to a precise angular position. The term goniometry derives from two Greek words, γωνία (''gōnía'') 'angle' and μέτρον (''métron'') ' m ...
s 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 obse ...
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 and
observed from the Earth. The objects
and
are defined by their
celestial coordinates, namely their
right ascensions (RA),