![Position angle](https://upload.wikimedia.org/wikipedia/commons/9/9c/Position_angle.svg)
In
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 ...
, position angle (usually abbreviated PA) is the convention for measuring angles on the sky. The
International Astronomical Union
The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) is a nongovernmental organisation with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreac ...
defines it as the angle measured relative to the
north celestial pole
The north and south celestial poles are the two points in the sky where Earth's axis of rotation, indefinitely extended, intersects the celestial sphere. The north and south celestial poles appear permanently directly overhead to observers at ...
(NCP), turning positive into the direction of the
right ascension
Right ascension (abbreviated RA; symbol ) is the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the (hour circle of the) point in question above the earth.
When paired w ...
. In the standard (non-flipped) images, this is a counter
clockwise
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
measure relative to the axis into the direction of positive
declination
In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of the ...
.
In the case of observed
visual binary stars, it is defined as the angular offset of the secondary star from the primary relative to the
north celestial pole
The north and south celestial poles are the two points in the sky where Earth's axis of rotation, indefinitely extended, intersects the celestial sphere. The north and south celestial poles appear permanently directly overhead to observers at ...
.
As the example illustrates, if one were observing a hypothetical binary star with a PA of 135°, that means an imaginary line in the eyepiece drawn from the north celestial pole to the primary (P) would be offset from the secondary (S) such that the angle would be 135°.
When graphing visual binaries, the NCP is, as in the illustration, normally drawn from the center point (origin) that is the Primary downward–that is, with north at bottom–and PA is measured counterclockwise. Also, the direction of the
proper motion
Proper motion is the astrometric measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the center of mass of the Solar System, compared to the abstract background of the more dista ...
can, for example, be given by its position angle.
The definition of position angle is also applied to extended objects like galaxies, where it refers to the angle made by the major axis of the object with the NCP line.
Nautics
The concept of the position angle is inherited from nautical navigation on the oceans, where the optimum
compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...
course is the course from a known position to a target position with minimum effort. Setting aside the influence of winds and ocean currents, the optimum course is the course of smallest distance between the two positions on the ocean surface. Computing the compass course is known as the ''inverse problem'' of
geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
.
This article considers only the abstraction of minimizing the distance between and traveling on the surface of a sphere with some radius : In which direction angle relative to North should the ship steer to reach the target position?
Global geocentric coordinate system
![PositionAngle](https://upload.wikimedia.org/wikipedia/commons/f/f2/PositionAngle.svg)
Detailed evaluation of the optimum direction is possible if the sea surface is approximated by a sphere surface. The standard computation places the ship at a geodetic
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
and geodetic
longitude
Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...
, where is considered positive if north of the equator, and where is considered positive if east of
Greenwich
Greenwich ( , ,) is a town in south-east London, England, within the ceremonial county of Greater London. It is situated east-southeast of Charing Cross.
Greenwich is notable for its maritime history and for giving its name to the Greenwich ...
. In the global coordinate system centered at the center of the sphere, the Cartesian components are
::
and the target position is
::
The North Pole is at
::
The
minimum distance is the distance along a great circle that runs through and . It is calculated in a plane that contains the sphere center and the
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
,
::
where is the angular distance of two points viewed from the center of the sphere, measured in
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. The cosine of the angle is calculated by 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 two vectors
::
If the ship steers straight to the North Pole, the travel distance is
::
If a ship starts at and swims straight to the North Pole, the travel distance is
::
Brief Derivation
The ''cosine formula'' of
spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
yields for the
angle between the great circles through that point to the North on one hand and to on the other hand
::
::
The ''sine formula'' yields
::
Solving this for and insertion in the previous formula gives an expression for the tangent of the position angle,
::
::
Long Derivation
Because the brief derivation gives an angle between 0 and which does not reveal the sign (west or east of north ?), a more explicit derivation is desirable which yields separately the sine and the cosine of such that use of the
correct branch of the inverse tangent allows to produce an angle in the full range .
The computation starts from a construction of the great circle between and . It lies in the plane that contains the sphere center, and and is constructed rotating by the angle around an axis . The axis is perpendicular to the plane of the great circle and computed by the normalized vector
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
of the two positions:
::
A right-handed tilted coordinate system with the center at the center of the sphere is given by the
following three axes: the
axis , the axis
::
and the axis .
A position along the great circle is
::
The compass direction is given by inserting the two vectors and and computing the gradient of the vector with respect to at .
::
The angle is given by splitting this direction along two orthogonal directions in the plane tangential to the sphere at the point . The two directions are given by the partial derivatives of with respect to and with respect to , normalized to unit length:
::
::
::
points north and points east at the position .
The position angle projects
into these two directions,
::
,
where the positive sign means the positive position angles are defined to be north over east. The values of the cosine and sine of are computed by multiplying this equation on both sides with the two unit vectors,
::
::
Instead of inserting the convoluted expression of , the evaluation may employ that the
triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
is invariant under a circular shift
of the arguments:
::
If
atan2
In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive is used to compute the value, one can reduce both expressions by division through
and multiplication by ,
because these values are always positive and that operation does not change signs; then effectively
::
See also
*
Parallactic angle In spherical astronomy, the parallactic angle is the angle between the great circle through a celestial object and the zenith, and the hour circle of the object. It is usually denoted ''q''. In the triangle zenith—object—celestial pole, the par ...
*
Angular distance
Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) is the angle between the two sightlines, or between two point objects as viewed from an observer.
Angular distance appears in mathematics (in pa ...
Further reading
*
References
{{Reflist
External links
The Orbits of 150 Visual Binary Stars, by Dibon Smith (Accessed 2/26/06)
Astronomical coordinate systems
Angle
Observational astronomy