Parallactic angle

As parallactic angle q the spherical astronomy called that corner of the astronomical triangle ( the triangle celestial pole - zenith - object) that is applied to the object. It specifies the angle by which the direction is different to the north celestial pole from the direction to the zenith of the observer on the object.

An object on the meridian of the observer has the parallactic angle q = 0 °; are available for this object north celestial pole and the zenith in the same direction. According to the usual convention, the object has a negative before its culmination and its culmination after a positive parallactic angle. For an object at the north celestial pole at the zenith or the parallactic angle is undefined.

Stellar orbits

The angle is related to the apparent direction of motion of celestial bodies together on their daily path in the sky. He constantly changes its value, because the star path appears curved towards the horizon, and is in the meridian to zero: there reaches each star (without self-motion ) and its peak moves horizontally at this moment.

For positions on the Earth's equator is considered in the establishment and demise of each star, because there all the stars perpendicular cross the horizon ( the Earth's rotation "rolls" the observer virtually the star and away from him).

While moving with us most of the stars to the upper right ( eastern half of the sky) or to the bottom right ( west half ), circumpolar stars have twice daily also vertical motion - namely. This position is called eastern and western Largest digression, because there ( the azimuth) reaches the angular distance north of its maximum value.

Field rotation

While a celestial object as part of its " daily motion " (ie, the movement from the east over the culmination of the downfall ) moves across the sky, is constantly changing its parallactic angle. A constellation, for example, which extends in a north-south direction in the sky (such as the Orion ), will always remain aligned unchanged during this movement to the north celestial pole. It will be, however, directed only at the moment of culmination on the zenith and thus perpendicular to the observer (q = 0). At sunrise it appears to an observer ( in the Northern Hemisphere ) tilted to the left (q <0), in the sinking, it seems inclined to the right (q > 0). While the orientation of the star image with respect to the north -south direction in the sky thus remains unchanged, constantly changes its orientation to the vertical direction of the observer.

The vertical axis of an equatorially mounted camera is aligned with the north celestial pole. During a tracking long-term exposure, the constellation for the film always remains ( ie towards the North Pole ) aligned "up" and can be photographed easily. The vertical axis on a common a tripod -mounted camera, however, is azimuthally oriented on the zenith. For the film, the constellation is not "up" ( ie towards the zenith ) aligned because its orientation is constantly changing for the zenith. For this camera, the image section rotates during a long exposure, so that the images of the stars are pulled apart to star trails.

Calculation

For an observer on the latitude and B for a point on the celestial sphere, which the declination δ and has the hour angle t, the parallactic angle q can be calculated by

If the denominator of the fraction is negative, need to be added to the result of 180 ° to bring the angle in the correct quadrant.

Derivation

To derive the formula, consider the spherical triangle whose corners are formed by the considered point and the north celestial pole and the zenith ( see figure). The fitting at the point of consideration is the interior angle parallactic angle q.

The sine law of spherical trigonometry gives the relationship

So

This formula could be resolved after the desired q. However, by the knowledge of sin ( q) q is not uniquely determined. q may all four quadrants of the full circle and stem are in full circle, usually two angle different quadrants, which have the same sine value, so that the determination of the angle is not clearly known from the sine value. The usual implementations of the arcsine provide those of the two eligible angle, which is in the range -90 ° .. 90 °, so that may have a subsequent correction in a different quadrant is required.

Instead of complicated geometrical considerations one usually uses in such cases, the fact that an angle can be determined uniquely if its sine and cosine are known. At the sign combination, the correct quadrant can be clearly seen.

The sine - cosine theorem gives the relationship

Division of the two equations gives

By separately considering the sign of the numerator and denominator will determine the correct quadrant. Some programming languages ​​have a variant of the arc tangent function, which this is done automatically ( often referred to atan2 ). If only the usual arc tangent function is available, take into account this the sign of the total fraction. The user must then add more than 180 ° quadrant correction if the denominator of the fracture is negative.

The factor could be reduced on a break, because the latitude B .. 90 ° comes from the range -90 ° and its cosine therefore can not be negative, thus shortening the quadrant determination not affected.