# Culmination

In observational astronomy, the culmination of a celestial object (the Sun, the Moon, a planet, a star, or a deep-sky object) or constellation is either of two moments when it reaches the observer's meridian[1][2] on the celestial sphere, on which it appears to move along a circular path over the course of a day, due to Earth's rotation.

As viewed from the surface of a terrestrial body, except its geographic poles, a celestial object passes through the meridian twice a day: once at its upper culmination, when it reaches its highest point as viewed from a given location on Earth, and once at its lower culmination, when it reaches its lowest point. The term culmination is often used to mean upper culmination.[1][2][3]

An object's altitude (A) in degrees at its upper culmination is equal to 90 subtracted by the observer's latitude (L) and added by the object's declination (δ): A = 90° − L + δ.

## Cases

Three cases are dependent on the observer's latitude (L) and the declination (δ) of the celestial object:

• The object is above the horizon even at its lower culmination; i.e. if | δ + L | > 90° (i.e. if in absolute value the declination is more than the colatitude, in the corresponding hemisphere)
• The object is below the horizon even at its upper culmination; i.e. if | δL | > 90° (i.e. if in absolute value the declination is more than the colatitude, in the opposite hemisphere)
• The upper culmination is above and the lower below the horizon, so the body is observed to rise and set daily; in the other cases (i.e. if in absolute value the declination is less than the colatitude)

The third case applies for objects in a part of the full sky equal to the cosine of the latitude (at the equator it applies for all objects, because the sky turns around the horizontal north-south line; at the poles it applies for none, because the sky turns around the vertical line). The first and second case each apply for half of the remaining sky.

## Period of time

The period between one upper culmination and the next is about 24 hours, while the period between an upper one and a lower one is 12 hours. The orbital motion and proper motion of Earth affect the period between successive upper culminations. Due to the proper and improper motions of the Sun, one solar day (the interval between like culminations of the Sun) is slightly longer than one sidereal day (the interval between like culminations of any reference star).[citation needed] The mean difference is 1/365.24219, since Earth takes 365.24219 days to complete one orbit around the Sun.

## The Sun

From the tropics and middle latitudes, the Sun is visible in the sky at its upper culmination (at solar noon) and invisible (below the horizon) at its lower culmination (at solar midnight). When viewed from the region within either polar circle around the winter solstice of that hemisphere (the December solstice in the Arctic and the June solstice in the Antarctic), the Sun is below the horizon at both of its culminations.

Supposing that the declination of the Sun is +20° on a given summer day, the complementary angle of 70° (from the Sun to the pole) is added to and subtracted from the observer's latitude to find the solar altitudes at upper and lower culminations, respectively.

• From 52° north, the upper culmination is at 58° above the horizon due south, while the lower is at 18° below the horizon due north. This is calculated as 52° + 70° = 122° (the supplementary angle being 58°) for the upper, and 52° − 70° = −18° for the lower.
• From 80° north, the upper culmination is at 30° above the horizon due south, while the lower is at 10° above the horizon (midnight sun) due north.

## Circumpolar stars

From most of the Northern Hemisphere, Polaris (the North Star) and the other stars of the constellation Ursa Minor appear to circle counterclockwise around the north celestial pole and remain visible at both culminations (as long as the sky is clear and dark enough). Such objects that always remain above the horizon, as viewed from the observer's latitude, are described as circumpolar.