Parallactic angle
As parallactic angle q which denotes spherical astronomy those angle of the astronomical triangle (the triangle celestial - zenith object) which abuts on the object. It indicates the angle at which the direction to the celestial north pole differs from the direction to the zenith of the observer.
An object on the observer's meridian has the parallactic angle q = 0 °; for this object, the celestial north pole and zenith lie in the same direction. According to usual convention, the object has a negative parallactic angle before its culmination and a positive parallactic angle after its culmination. The parallactic angle is undefined for an object at the north celestial pole or in the zenith.
Fixed star tracks
The angle is related to the apparent direction of movement of the stars on their daily orbit in the sky. It changes its value constantly because the star's orbit appears curved in relation to the horizon, and it becomes zero in the meridian : there every star reaches its highest level (without its own movement ) and moves horizontally at this moment .
For locations on the earth's equator, the following applies when each star rises and sets , because there all the stars cross the horizon vertically ; the rotation of the earth “rolls” the observer towards the star when it rises or away from it when it sets.
While in higher northern latitudes the stars seem to move to the right when looking to the south - those of the eastern half of the sky upwards and those of the western half downwards - other directions of movement can also be noticed when looking to the north. Circumpolar stars also have a vertical direction of movement twice a day - namely for . This position is also called the eastern or western largest digression because the angular distance from the north, the azimuth , reaches its maximum value there.
Field rotation
While a celestial object wanders across the sky in the course of its "daily movement" (i.e. the movement from the rise through the culmination to the setting), its parallactic angle changes continuously. A constellation, for example, which extends in a north-south direction in the sky (such as Orion ) will always remain aligned with the north celestial pole during this movement. However, it will only be aligned with the zenith at the moment of culmination and thus be perpendicular to the observer ( q = 0). When rising it appears to an observer (in the northern hemisphere) to be inclined to the left ( q <0), when it is set it appears to be inclined to the right ( q > 0). While the orientation of the constellation in relation to the north-south direction in the sky remains unchanged, its orientation to the vertical direction of the observer is constantly changing.
The vertical axis of an equatorial camera is aligned with the celestial north pole. During a tracked long exposure, the constellation for the film always remains "upwards" (ie towards the North Pole) and can be photographed without any problems. The vertical axis of a camera mounted azimuthally on an ordinary photo tripod , on the other hand, is aligned with the zenith. For their film, the constellation does not stay “upwards” (ie in the direction of the zenith), since its orientation with respect to the zenith changes constantly. For this camera, the image section rotates during a long exposure so that the images of the stars are pulled apart to form star trails.
calculation
For an observer at latitude B and for a point on the celestial sphere which has the declination δ and the hour angle t , the parallactic angle q can be calculated by
If the denominator of the fraction is negative, then 180 ° must be added to the result to bring the angle into the correct quadrant.
Derivation
To derive the calculation formula, consider the spherical triangle , the corners of which are formed by the point under consideration as well as the north celestial pole and the zenith (see figure). The interior angle at the point under consideration is the parallactic angle q .
The sine law of spherical trigonometry provides the relationship
so
This formula could already be solved for the sought q . However, knowing sin ( q ) does not yet uniquely determine q . q can come from all four quadrants of the full circle and there are usually two angles in the full circle from different quadrants that have the same sine value, so that the determination of the angle from the known sine value is not unambiguous. The usual implementations of the arc sine supply that of the two angles in question, which is in the range −90 ° .. + 90 °, so that a subsequent correction in another quadrant may be necessary.
Instead of cumbersome geometrical considerations, one usually uses the fact that an angle can be clearly determined if its sine and cosine values are known. The correct quadrant can be clearly identified from their combination of signs .
The sine-cosine theorem provides the relationship
Division of the two equations gives
The correct quadrant can be determined by considering the signs of the denominator and numerator separately . Some programming languages have a variant of the arcus tangens function, which does this automatically (often referred to as arctan2 ). If only the usual arc tangent function is available, it takes the sign of the total fraction into account. The user then has to add 180 ° as a quadrant correction if the denominator of the fraction is negative.
Shortening the factor does not change the determination of the quadrant. The geographical latitude B comes from the range −90 °… + 90 ° and its cosine is therefore positive.
