# Culmination (astronomy)

In astronomy, the culmination (Latin culmen = summit) is the passage of an astronomical object through the highest ( upper culmination ) or the lowest ( lower culmination ) daily position on its apparent circular path in the sky . The same name is also used for the height and time of the two rounds.

For the position measured with the elevation angle h, the time at which this position was passed is given. The elevation angle is negative when the culmination occurs below the horizon and is not visible. This generally applies to the lower culmination.

## Elevation angle at culmination

The elevation angle of the object is given by ${\ displaystyle h}$

• the declination of the object (northern half of the sky:; southern half of the sky:) and${\ displaystyle \ delta}$${\ displaystyle \ delta> 0}$${\ displaystyle \ delta <0}$
• the latitude of the observation location ( northern hemisphere : ; southern hemisphere : )${\ displaystyle \ varphi}$${\ displaystyle \ varphi> 0}$${\ displaystyle \ varphi <0}$

according to the following formulas (these are only exact if the culmination point is on the meridian ):

Elevation angle
upper culmination ${\ displaystyle h _ {\ mathrm {OK}} = + 90 ^ {\ circ} - | \ delta - \ varphi |}$
lower culmination ${\ displaystyle h _ {\ mathrm {UK}} = - 90 ^ {\ circ} + | \ delta + \ varphi |}$

## Culminating height and visibility

• The circumpolar stars never set, their lower culmination is always above the horizon:${\ displaystyle h _ {\ mathrm {UK}}> 0}$
• Conversely, stars near the opposite pole in the sky can never be seen from the other half of the earth; the upper culmination also has a negative elevation angle:${\ displaystyle h _ {\ mathrm {OK}} <0}$
Examples are the stars of the Southern Cross ( stars in the southern hemisphere), which can only be observed up to about 25 ° north latitude in the upper culmination.${\ displaystyle \ delta \ approx -35 ^ {\ circ} <0 \ \ to}$
• For objects with a declination between the two above. Values, only the upper culmination lies above the horizon; these objects rise and fall.${\ displaystyle \ delta}$

It follows:

Hemisphere Visibility of stars that meet the following condition
Northern hemisphere
${\ displaystyle \ varphi> 0}$
circumpolar: always ${\ displaystyle \ delta> + (90 ^ {\ circ} - \ varphi)}$
not always ${\ displaystyle - (90 ^ {\ circ} - \ varphi) <\ delta <+ (90 ^ {\ circ} - \ varphi)}$
never ${\ displaystyle \ delta <- (90 ^ {\ circ} - \ varphi)}$
Southern hemisphere
${\ displaystyle \ varphi <0}$
never ${\ displaystyle \ delta> + (90 ^ {\ circ} + \ varphi)}$
not always ${\ displaystyle - (90 ^ {\ circ} + \ varphi) <\ delta <+ (90 ^ {\ circ} + \ varphi)}$
circumpolar: always ${\ displaystyle \ delta <- (90 ^ {\ circ} + \ varphi)}$

In the northern hemisphere of the earth, the upper culmination of a star is south of the zenith when its declination is less than the latitude, and north of the zenith (between the zenith and the north pole) when its declination is greater than the latitude; the lower culmination, if it is visible, is always north of the zenith (beyond the north pole).

## Culmination and meridian

In the case of an astronomical object with constant declination, both culmination points lie on the meridian of the observation location (exactly in the direction of the south or north point of the horizon). The time of culmination and the passage of the meridian are then identical.

In the case of celestial bodies with their own motion ( sun , moon , planets , asteroids , satellites , etc.), the culmination points usually do not lie exactly on the meridian because their declination changes constantly.

The sun z. B. something rises or falls as it passes the meridian; The sum of these two movements causes the upper culmination of the sun between winter and summer solstice to take place slightly after , in the second half of the year, before the meridian passage. However, the deviation of the sun culmination from the meridian is so small that the designation midday height for the upper culmination contains only an insignificant error. The times for the upper culmination and the true noon are almost identical, the time difference is typically a few seconds.

Satellites and the moon, on the other hand, have relatively large movements of their own, so that the deviations from the meridian can be considerable here. With the moon, the time difference between culmination and passage of the meridian is several minutes and can be calculated approximately as follows: ${\ displaystyle \ Delta t}$

${\ displaystyle \ Delta t \ approx (\ tan \ varphi - \ tan \ delta) \ cdot {\ frac {\ mathrm {d} t} {\ mathrm {d} \ delta}}}$

## Culmination and sidereal time

The upper culmination of a celestial body plays a role in the sidereal time measurement of its right ascension angle, which is specified in the measure of time (angle) : the moment of the upper culmination of the vernal equinox (reference point for the right ascension angle) is assigned the sidereal time 00:00 . If any celestial body culminates, it has since moved over a right ascension angle to which the meanwhile valid sidereal time corresponds. The specification of the right ascension as sidereal time depends on the observation location, i.e. H. 00:00 o'clock sidereal time is not everywhere at the same time, since the vernal equinox culminates at a different time on every degree of longitude on earth.

The time between two culminations of the vernal equinox is a sidereal day , which in the same way as a sunny day is divided into (Sternzeit-) hours, minutes and seconds. The right ascension of the fixed stars and thus the sidereal time is unchangeable (meaning of the word fix ), the right ascension of the sun, on the other hand, increases daily by about 1 °, the angle of the earth's orbit around the sun. A sidereal day is therefore about 4 sidereal minutes shorter than a solar day (see also sidereal period , synodic period ). In this ratio, all sidereal time units are smaller than those of solar time:

${\ displaystyle {\ frac {t _ {\ mathrm {star}}} {t _ {\ mathrm {sun}}}} \ approx {\ frac {0 {,} 99727} {1}}}$