Position angle

The position angle is an indication of direction in the sky, based on the direction to the north celestial pole.

Under position angle understand astronomers an indication of direction in the equatorial coordinate system ( right ascension and declination ), based on the direction to the north pole of the sky relates.

definition

The position angle of an object 1, based on object 2, is the angle that the connecting line from object 2 to object 1 includes with the connecting line from object 2 to the north celestial pole . It is counted from north to east (i.e. counterclockwise ) and from 0 ° to 360 °.

The mentioned connecting lines are always great circle sections on the celestial sphere . The shorter of the two possible great circle sections leading to the target point must be considered.

calculation

If the objects 1 and 2 have the equatorial coordinates α ₁ , δ ₁ and α ₂ , δ ₂ , the position angle of object 1 with respect to object 2 can be calculated by

{\ displaystyle P = \ arctan \ left ({\ frac {\ sin (\ alpha _ {1} - \ alpha _ {2})} {\ cos (\ delta _ {2}) \ tan (\ delta _ { 1}) - \ sin (\ delta _ {2}) \ cos (\ alpha _ {1} - \ alpha _ {2})}} \ right)}

If the denominator of the fraction is negative, add 180 ° to the result to bring the angle into the correct range between 90 ° and 270 °.

If necessary, whole-number multiples of 360 ° can always be added or subtracted in order to bring the result into the desired range. If, in particular, the arctangent produces a negative angle, an equivalent positive angle can be achieved by adding 360 °.

Applications

Star coverage of the Pleiades by the moon with indication of the respective position angles of entry and exit

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

mutual position of celestial bodies , e.g. B. from binary stars or neighboring galaxies
Direction of movement of comets , asteroids and meteors
Direction of proper motions of stars .
Location information for close conjunctions (apparent encounters) of stars
Orientation of orbital axes , for example in binary stars and exoplanets
Orientation of the axes of rotation of planets

Examples

The two rear box stars of the Big Dipper are known to point to the North Star . The upper box star, Dubhe , has the coordinates α ₁ = 165.93 ° and δ ₁ = 61.75 °. The lower box star, Merak , has the coordinates α ₂ = 165.46 ° and δ ₂ = 56.38 °. Dubhe's position angle with respect to Merak is therefore 2.4 °; As expected, the connecting line points almost exactly to the north and deviates only slightly to the east. Conversely, Merak is at a position angle of 182.8 ° with respect to Dubhe. Note that the two position angles do not differ by exactly 180 °.

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

The star Algieba is a double star. The companion is currently 4.4 " from the main star and at a positional angle of 125 degrees.

Derivation

The spherical triangle used to derive the formula for calculating the position angle

To derive the calculation formula, consider the spherical triangle , the corners of which are formed by object 1 (with the coordinates α ₁ , δ ₁ ), object 2 (with the coordinates α ₂ , δ ₂ ) and the north celestial pole N. The interior angle P applied to object 2 is the position angle sought (see figure).

The sine law of spherical trigonometry provides the relationship

${\ displaystyle {\ frac {\ sin (P)} {\ sin (90 ^ {\ circ} - \ delta _ {1})}} = {\ frac {\ sin (\ alpha _ {1} - \ alpha _ {2})} {\ sin (d)}}}$ ,

so

${\ displaystyle {\ begin {aligned} \ sin (d) \ sin (P) & = \ sin (\ alpha _ {1} - \ alpha _ {2}) \ sin (90 ^ {\ circ} - \ delta _ {1}) \\ & = \ sin (\ alpha _ {1} - \ alpha _ {2}) \ cos (\ delta _ {1}) \ end {aligned}}}$

This formula could already be solved for the P sought . However, knowing sin ( P ) does not yet uniquely determine P. P can come from all four quadrants of the full circle and there are usually two angles in the full circle from different quadrants which 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 arcsine supply an angle 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

${\ displaystyle {\ begin {aligned} \ sin (d) \ cos (P) & = \ cos (90 ^ {\ circ} - \ delta _ {1}) \ sin (90 ^ {\ circ} - \ delta 2) - \ sin (90 ^ {\ circ} - \ delta _ {1}) \ cos (90 ^ {\ circ} - \ delta _ {2}) \ cos (\ alpha _ {1} - \ alpha _ {2}) \\ & = \ sin (\ delta _ {1}) \ cos (\ delta _ {2}) - \ cos (\ delta _ {1}) \ sin (\ delta _ {2}) \ cos (\ alpha _ {1} - \ alpha _ {2}) \ end {aligned}}}$

Division of the two equations yields

${\ displaystyle {\ begin {aligned} \ tan (P) = {\ frac {\ sin (d) \ sin (P)} {\ sin (d) \ cos (P)}} & = {\ frac {\ sin (\ alpha _ {1} - \ alpha _ {2}) \ cos (\ delta _ {1})} {\ sin (\ delta _ {1}) \ cos (\ delta _ {2}) - \ cos (\ delta _ {1}) \ sin (\ delta _ {2}) \ cos (\ alpha _ {1} - \ alpha _ {2})}} \\ & = {\ frac {\ sin (\ alpha _ {1} - \ alpha _ {2})} {\ cos (\ delta _ {2}) \ tan (\ delta _ {1}) - \ sin (\ delta _ {2}) \ cos (\ alpha _ {1} - \ alpha _ {2})}} \ end {aligned}}}$

The correct quadrant can be determined by considering the signs of the denominator and numerator separately . Some programming languages have a variant of the arctangent function which does this automatically (often referred to as arctan2 or atan2). If only the usual arctangent 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.

The factor could be shortened in the fraction because the declination δ ₁ comes from the range −90 ° ... + 90 ° and its cosine can therefore not become negative, so shortening does not affect the quadrant determination. ${\ displaystyle \ cos (\ delta _ {1})}$

To ascertain that the calculation formula is also valid if the angle P in the spherical triangle is greater than 180 °, consider the complementary triangle, the angle 360 ° - P contains. The resulting negative signs are canceled out in the formula derivation and the resulting formula is identical to the one given at the beginning.

Vertical position angle

If the position angle with respect to the direction to the zenith is to be determined instead of the direction to the celestial north pole , the parallactic angle q calculated for object 2 must be subtracted from the angle P.

Example:
On August 7, 2011 , the waxing crescent moon (α ₂ = 239.1 °, δ ₂ = −23.2 °) culminated in Munich at 8:06 p.m. CEST at an altitude of 18.8 °, while the sun ( α ₁ = 137.4 °, δ ₁ = + 16.4 °) in the north-northwest with an altitude of 4.8 ° was shortly before the sinking. The position angle of the sun with respect to the moon was P = arctan (−5.137) = 281.0 °. Since the moon was culminating, q = 0, and the direction to the sun included the angle 281.0 ° not only with the north, but also with the vertical. Although the sun was lower than the moon, the connecting line between the moon and the sun did not leave the lunar disc to the lower right (the horizontal would correspond to 270 °), but rose by 11 ° to the upper right, and the terminator of the moon was accordingly 11 ° behind tilted left , although you'd expect it to have tilted right toward the setting sun.

Individual evidence

^ Jean Meeus : Astronomical Algorithms . 2nd ed., Willmann-Bell, Richmond 1998, ISBN 0-943396-61-1 , chap. 17th
^ Jean Meeus: Astronomical Algorithms . 2nd ed., Willmann-Bell, Richmond 1998, ISBN 0-943396-61-1 , chap. 48

[1] Jean Meeus : Astronomical Algorithms . 2nd ed., Willmann-Bell, Richmond 1998, ISBN 0-943396-61-1 , chap. 17th

[2] Jean Meeus: Astronomical Algorithms . 2nd ed., Willmann-Bell, Richmond 1998, ISBN 0-943396-61-1 , chap. 48