Biggest digression

Astronomical triangle with the 3 sides ( width 90 ° -  B , pole distance 90 ° - dec., Zenith distance z ) and the 3 angles ( azimuth Az , hour angle t , parallactic angle q ). The star enters the largest digression about an hour later, when its direction of movement points exactly to the zenith .

The greatest digression refers to those two points or points in time of the daily star movement at which a star moves exactly vertically upwards or downwards ( eastern and western digression). Digression means the momentary angle difference, related to the horizon, of a star to the local meridian or a corresponding mire . On the daily star orbit , this angle is greatest when the star moves perpendicular to the horizon (i.e. parallel to the meridian). The issue is only relevant for circumpolar stars .

Basics

At this moment - which, however, in the measuring telescope a few seconds and freiäugig takes a few minutes - is the parallax angle  q exactly 90 ° or 90 ° (see picture), and the star reaches its greatest eastern and western azimuth , so the greatest angular distance ( Digression ) from the north point.

These two positions only occur in circumpolar stars whose upper culmination lies between the pole and the zenith. In the northern hemisphere of the earth, the declination δ of the star must therefore be greater than the geographical latitude B of the observation location, e.g. B. for Munich or Vienna δ> + 48 ° . In the southern hemisphere , the declination must be smaller than B (i.e. more southerly), e.g. for Cape Town δ <-34 ° .

All other (more southern) stars in the northern sky move monotonously to the right , i.e. H. always in the sense of east → south → west. If one disregards the rising or setting of the stars, their azimuth increases continuously from 0 ° (lower culmination deep in the north ) through 90 ° (east, first vertical ) ascending to 180 ° (south, upper culmination ), and then falling over 270 ° (west) again to 360 ° (= 0 °) in the north.

calculation

Celestial coordinates in astronomical coordinate systems : The spherical triangle to be considered has the corner points North Pole (blue) in the northern hemisphere . Zenith (black) and star (purple). the azimuth in the horizontal system is measured from the meridian (black) and in the equatorial system the declination (red) is measured from the equatorial plane and the hourly angle (cyan) is also measured from the meridian. The geographical latitude of the observation site is identical to the pole height . ${\displaystyle a}$ ${\displaystyle \delta }$ ${\displaystyle \tau }$ ${\displaystyle \phi }$

Since the astronomical triangle ( pole - zenith - star ) becomes right-angled for the moment of greatest digression , the spherical formulas are considerably simplified. If the azimuth of the star is designated with , its declination with , its hour angle with and the geographical latitude of the location with , then the sine or tangent rate is reduced to ${\displaystyle a}$ ${\displaystyle \delta }$ ${\displaystyle \tau }$ ${\displaystyle \phi }$

${\displaystyle \sin a={\frac {\cos \delta }{\cos \phi }}}$ (positive for eastern digression, negative in western)

${\displaystyle \cos \tau ={\frac {\tan \phi }{\tan \delta }}}$ (Hour angle in the fourth or first quadrant)

Application in geodesy

According to W. Embacher, the first formula can be used for precise latitude and azimuth measurements if you combine pairs of stars in the northeast and west:
Because the largest digression is interesting from an observation point of view and advantageous because the vertical star passage enables a particularly precise setting on the vertical thread of a theodolite or passage instrument. The air turbulence can also be determined visually, and an exact time is not required ( time error ). These three advantages make z. B. use the Embacher method of azimuth and latitude determination .

See also

• Greatest elongation

literature

• Karl Ramsayer : Geodetic Astronomy . Volume IIa. In: Handbook of surveying. JB Metzler-Verlag, Stuttgart 1969