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 parallactic angle will be the position angle of the zenith at the celestial object. Despite its name, this angle is unrelated with parallax. The parallactic angle is 0° or 180° when the object crosses the meridian.

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 parallactic angle will be the position angle of the zenith at the celestial object. Despite its name, this angle is unrelated with parallax. The parallactic angle is 0° or 180° when the object crosses the meridian.

Derivation

Then, applying the spherical law of sines, and using \cos \theta = \sin(90^\circ - \theta) , we have \frac{\cos \varphi}{\sin q} = \frac{\cos a}{\sin h} And applying the spherical law of cosines: \sin \varphi = \sin a \sin \delta + \cos a \cos \delta \cos q \sin a = \sin \varphi \sin \delta + \cos \varphi \cos \delta \cos h By solving these equations for \sin q and \cos q and using \tan q = \sin q\ / \cos q we obtain the formula recommended by Meeus: \tan q = \frac{\sin h}{\cos \delta \tan \varphi - \sin \delta \cos h} \mathbf{s} = \left(\begin{array}{c} \cos\delta \cos \alpha \\ \cos \delta\sin\alpha\\ \sin\delta\end{array} \right). The North Celestial Pole is at \mathbf{N} = \left(\begin{array}{c}0 \\ 0 \\ 1 \end{array}\right). In this same coordinate system the zenith is found by inserting altitude, , , into the transformation formulas to get \mathbf{z} = \left(\begin{array}{c} \cos\varphi \cos l\\ \cos\varphi \sin l\\ \sin\varphi\end{array} \right), where is the observer's geographic latitude, and the local sidereal time. This also describes a rotating, right-handed, observer coordinate frame, with X-axis aligned to the south, where the local meridian intersects the horizon, Y-axis toward the eastern horizon, and Z-axis toward the zenith. This is the coordinate frame in which altitude and azimuth are measured. For the star, at some moment, , with expected altitude, , define its zenith distance as z = \pi/2-a . Its hour-angle, h = l - \alpha , measures the elapsed sidereal time interval since the star crossed the local Meridian and is negative if the star is east of the meridian and its crossing is pending. The normalized cross product is the rotation axis that turns the star into the direction of the zenith: \mathbf{\omega}_z = \frac{1}{\sin z}\mathbf{s}\times \mathbf{z} \mathbf{u}_\alpha = \left(\begin{array}{c} -\sin\alpha \\ \cos\alpha\\ 0 \end{array}\right). These are orthogonal: \mathbf{u}_\delta \cdot \mathbf{u}_\alpha=0;\quad \mathbf{u}^2_\delta = \mathbf{u}^2_\alpha=1. The parallactic angle is the angle of the initial section of the great circle at s, east of north, \omega_z\times \mathbf{s} = \cos q\, \mathbf{u}_\delta + \sin q \, \mathbf{u}_\alpha. \cos q = (\omega_z\times \mathbf{s})\cdot \mathbf{u}_\delta = \frac{1}{\sin z}(\cos\delta\sin\varphi -\sin\delta \cos\varphi \cos h), \sin q = (\omega_z\times \mathbf{s})\cdot \mathbf{u}_\alpha = \frac{1}{\sin z}\sin h \cos\varphi. (The previous formula is the sine formula of spherical trigonometry. ) The values of and of are positive, so using atan2 functions one may divide both expressions through these without losing signs; eventually \tan q = \frac{\sin h\cos\varphi}{\cos\delta \sin\varphi -\sin\delta \cos\varphi \cos h}= \frac{\sin h}{\cos\delta \tan\varphi-\sin\delta \cos h} yields the angle in the full range . The advantage of this expression is that it does not depend on the various offset conventions of azimuth, ; the uncontroversial offset of the hour angle, , takes care of this. \dot h =\frac{2\pi}{T_s} and the time derivative of the expression is \dot q \frac{1}{\cos^2 q}= \frac{\cos\varphi[\cos h \cos\delta \sin\varphi-\sin\delta\cos\varphi]}{(\cos\delta \sin\varphi-\sin\delta \cos\varphi \cos h)^2} \dot h; \dot q = \frac{\cos \varphi[\cos h \cos\delta \sin \varphi -\sin \delta \cos\varphi]}{\sin ^2z}\dot h = \frac{\cos \varphi \cos a \cos A}{\sin^2 z}\dot h =\frac{\cos\varphi \cos A}{\sin z}\dot h. The value derived above always refers to the north celestial pole as the origin of coordinates even if it is not visible (i.e., if the telescope is south of the equator). Some authors introduce more complicated formulas with variable signs to derive similar angles for telescopes south of the equator that use the south celestial pole as the reference.

See also

Libration Equatorial mount Altazimuth mount

Further reading

References

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