thumb|250px|An illustration of how position angle is estimated through a telescope eyepiece; the primary star is at center.
In astronomy, position angle (usually abbreviated PA) is the convention for measuring angles on the sky.
The International Astronomical Union defines it as the angle measured relative to the north celestial pole (NCP), turning positive into the direction of the right ascension.
In the standard (non-flipped) images, this is a counterclockwise measure relative to the axis into the direction of positive declination.
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.
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 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 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.
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
thumb|400px|The position angle of the point t at the point s is the angle at which the green and the dashed great circles intersect at s.
The unit directions ,  and the rotation axis  are marked by arrows.
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  and geodetic longitude , where  is considered positive if north of the equator, and where  is considered positive if east of Greenwich.
In the global coordinate system centered at the center of the sphere, the Cartesian components are
{\mathbf s}=R\left(\begin{array}{c} \cos\varphi_s \cos\lambda_s \\ \cos\varphi_s \sin\lambda_s \\ \sin\varphi_s \end{array}\right)
and the target position is
{\mathbf t}=R\left(\begin{array}{c} \cos\varphi_t \cos\lambda_t \\ \cos\varphi_t \sin\lambda_t \\ \sin\varphi_t \end{array}\right).
The North Pole is at
{\mathbf N}=R\left(\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right).
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,
d_{s,t}=R\theta_{s,t}
where  is the angular distance of two points viewed from the center of the sphere, measured in radians.
The cosine of the angle is calculated by the dot product  of the two vectors
\mathbf{s}\cdot \mathbf{t} = R^2\cos \theta_{s,t} = R^2(\sin\varphi_s\sin\varphi_t+\cos\varphi_s\cos\varphi_t\cos(\lambda_t-\lambda_s))
If the ship steers straight to the North Pole, the travel distance is
d_{s,N} = R\theta_{s,N} = R(\pi/2-\varphi_s)
If a ship starts at  and swims straight to the North Pole, the travel distance is
d_{t,N} = R\theta_{t,n} =R(\pi/2-\varphi_t)
Brief Derivation
The cosine formula of spherical trigonometry yields for the  angle  between the great circles through  that point to the North on one hand and to  on the other hand
\cos\theta_{t,N} = \cos\theta_{s,t}\cos\theta_{s,N}+\sin\theta_{s,t}\sin\theta_{s,N}\cos p.
\sin\varphi_t = \cos\theta_{s,t}\sin\varphi_s +\sin\theta_{s,t}\cos\varphi_s\cos p.
The sine formula yields
\frac{\sin p}{\sin \theta_{t,N}} = \frac{\sin(\lambda_t-\lambda_s)}{\sin\theta_{s,t}}.
Solving this for  and insertion in the previous formula gives an expression for the tangent of the position angle,
\sin\varphi_t = \cos\theta_{s,t}\sin\varphi_s +\frac{\sin(\lambda_t-\lambda_s)}{\sin p}\cos\varphi_t\cos\varphi_s\cos p;
\tan p = \frac{\sin(\lambda_t-\lambda_s)\cos\varphi_t\cos\varphi_s}{\sin\varphi_t-\cos\theta_{s,t}\sin\varphi_s}.
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 of the two positions:
\mathbf{\omega} = \frac{1}{R^2\sin \theta_{s,t}}\mathbf{s}\times \mathbf{t}  = \frac{1}{\sin \theta_{s,t}}\left(\begin{array}{c} \cos\varphi_s\sin\lambda_s\sin\varphi_t -\sin\varphi_s\cos\varphi_t\sin\lambda_t \\ \sin\varphi_s\cos\lambda_t\cos\varphi_t -\cos\varphi_s\sin\varphi_t\cos\lambda_s \\ \cos\varphi_s\cos\varphi_t\sin(\lambda_t-\lambda_s) \end{array}\right).
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
\mathbf{s}_\perp = \omega \times \frac{1}{R}\mathbf{s} = \frac{1}{\sin\theta_{s,t}} \left(\begin{array}{c} \cos\varphi_t\cos\lambda_t(\sin^2\varphi_s+\cos^2\varphi_s\sin^2\lambda_s)-\cos\lambda_s(\sin\varphi_s\cos\varphi_s\sin\varphi_t+\cos^2\varphi_s\sin\lambda_s\cos\varphi_t\sin\lambda_t)\\ \cos\varphi_t\sin\lambda_t(\sin^2\varphi_s+\cos^2\varphi_s\cos^2\lambda_s)-\sin\lambda_s(\sin\varphi_s\cos\varphi_s\sin\varphi_t+\cos^2\varphi_s\cos\lambda_s\cos\varphi_t\cos\lambda_t)\\ \cos\varphi_s[\cos\varphi_s\sin\varphi_t-\sin\varphi_s\cos\varphi_t\cos(\lambda_t-\lambda_s)] \end{array}\right)
and the axis .
A position along the great circle is
\mathbf{s}(\theta) = \cos\theta \mathbf{s}+\sin\theta \mathbf{s}_\perp,\quad 0\le\theta\le 2\pi.
The compass direction is given by inserting the two vectors  and  and computing the gradient of the vector with respect to  at .
\frac{\partial}{\partial\theta}\mathbf{s}_{\mid \theta=0}=\mathbf{s}_\perp.
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:
\mathbf{u}_N = \left( \begin{array}{c} -\sin\varphi_s\cos\lambda_s\\ -\sin\varphi_s\sin\lambda_s\\ \cos\varphi_s \end{array}\right);
\mathbf{u}_E = \left(\begin{array}{c} -\sin\lambda_s\\ \cos\lambda_s\\ 0 \end{array} \right);
\mathbf{u}_N\cdot \mathbf{s} = \mathbf{u}_E\cdot \mathbf{u}_N =0
points north and  points east at the position .
The position angle  projects  into these two directions,
\mathbf{s}_\perp = \cos p \,\mathbf{u}_N+\sin p\, \mathbf{u}_E,
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,
\cos p = \mathbf{s}_\perp \cdot \mathbf{u}_N =\frac{1}{\sin\theta_{s,t}}[\cos\varphi_s\sin\varphi_t - \sin\varphi_s\cos\varphi_t\cos(\lambda_t-\lambda_s)];
\sin p = \mathbf{s}_\perp \cdot \mathbf{u}_E =\frac{1}{\sin\theta_{s,t}}[\cos\varphi_t\sin(\lambda_t-\lambda_s)].
Instead of inserting the convoluted expression of , the evaluation may employ that the triple product is invariant under a circular shift of the arguments:
\cos p = (\mathbf{\omega}\times \frac{1}{R}\mathbf{s})\cdot \mathbf{u}_N = \omega\cdot(\frac{1}{R}\mathbf{s}\times \mathbf{u}_N).
If atan2 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
\tan p = \frac{\sin(\lambda_t-\lambda_s)}{\cos\varphi_s\tan\varphi_t -\sin\varphi_s\cos(\lambda_t-\lambda_s)}.
See also
Parallactic angle
Angular distance
Further reading
References
External links
The Orbits of 150 Visual Binary Stars, by Dibon Smith (Accessed 2/26/06)
