Position angle

Under position angle astronomers understand a direction indication in the equatorial coordinate system ( right ascension and declination ), which refers to the direction to the north pole of the sky.

Definition

The position angle of an object 1 by object 2 is the angle that the radiation emanating from object 2 to object 1 with connecting line emanating from object 2 connection line to the north celestial pole. He is from north through east (ie anticlockwise) and counted from 0 ° to 360 °.

The lines above are always great circle sections on the celestial sphere. It is to consider each of the shorter of the two possible leading to the destination point great circle sections.

Calculation

Do the objects 1 and 2, the equatorial coordinates α1, α2 and δ1, δ2, so the position angle of object 1, with respect to object 2 are 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 range between 90 ° and 270 °.

If required can always integral multiples are subtracted to get the result in a desired range of 360 ° or added. In particular, if the arctangent provides a negative angle, a positive angle of equivalent can be achieved by the addition of 360 °.

Applications

The position angle is used to describe the relative position of two objects or directions of movement in the night sky and is mainly used for the following:

• Relative positions of celestial bodies, such as double stars or nearby galaxies
• Direction of motion of comets, asteroids and meteors
• Direction of the proper motions of stars.
• Places in close conjunctions (apparent encounters ) of stars
• Orientation of the path axes, such as binary stars and exoplanets
• Orientation of the axes of rotation of planets

Examples

• The two rear box star of the Big Dipper are known to have the Pole Star. The upper box star, alpha UMa, has the coordinates α1 = 165.93 ° and δ1 = 61.75 °. The lower box star, beta UMa has the coordinates α2 = 165.46 ° and δ2 = 56.38 °. Therefore, the position angle alpha with respect to beta UMa is 2.4 °; the connecting line shows, as expected, almost due north and differs only slightly from the east. And vice versa, with respect to alpha beta UMa at a position angle of 182.8 °. Note that the two position angles do not differ by exactly 180 °.

• All fixed stars move in the course of the diurnal rotation of the celestial sphere exactly in the direction of a position angle of 270 °.

• The star Algieba is a double star. The attendant is present at a distance of 4.4 " from the primary star and a position angle of 125 degrees.

Derivation

To derive the formula, consider the spherical triangle whose corners of object 1 ( with the coordinates α1, δ1 ), Object 2 ( with coordinates α2, δ2 ) and the celestial north pole N are formed. The acting on the object 2 interior angle P is the desired position angle ( see figure).

The sine law of spherical trigonometry gives the relationship

,

So

This formula could be resolved after the P sought. Through the knowledge of sin ( P), if P is not uniquely determined. P can 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 clear from the prior art sine value. The usual implementations of the arcsine provide an angle 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

Provides division of the two equations

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 declination δ1 from the range -90 ° .. 90 ° originates and its cosine therefore can not be negative, the shortening so the quadrant determination not affected.

To convince yourself that the calculation formula remains valid even when the angle P is greater than 180 ° in the spherical triangle, consider the complementary triangle that contains the angle 360 ° -P. The resulting occurring negative signs stand out away at Formelherleitung and the resulting formula is identical to that given at the outset.

Vertical position angle

If the position angle relative to the direction to the zenith instead of toward the north celestial pole are determined, so q is to subtract from the angle P for the object 2 calculated parallactic angle.

Example: 7 August 2011 culminated in Munich the half-full moon ( α2 = 239.1 °, δ2 = -23.2 ° ) at 20:06 CEST at an altitude of 18.8 °, while the sun ( α1 = 137.4 °, δ1 = 16.4 ° ) in the NW was 4.8 ° altitude shortly before the downfall. The position angle of the sun with respect to the moon was P = arctan ( -5.137 ) = 281.0 °. Since the moon peaked, was q = 0, and the direction to the sun closed not only with the northern direction, but also with the vertical angle 281.0 °. Although the sun was low was as the moon, the connecting line lunisolar therefore not left the moon disc to the bottom right ( the horizontal equivalent to 270 °), but by 11 ° upwards to the upper right, and the terminator of the moon was correspondingly by 11 ° to the tilted to the left, although you would expect that he would have to be tilted to the right towards the setting sun.