In a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude.^[1] Here Southern latitudes are defined to be negative, and as a result the colatitude is a non-negative quantity, ranging from zero at the North pole to 180° at the South pole.

The colatitude corresponds to the conventional 3D polar angle in spherical coordinates, as opposed to the latitude as used in cartography.

Examples

Latitude and colatitude sum up to 90°.

Astronomical use

The colatitude is most useful in astronomy because it refers to the zenith distance of the celestial poles. For example, at latitude 42°N, Polaris (approximately on the North celestial pole) has an altitude of 42°, so the distance from the zenith (overhead point) to Polaris is 90 − 42 = 48°.

Adding the declination of a star to the observer's colatitude gives the maximum latitude of that star (its angle from the horizon at culmination or upper transit). For example, if Alpha Centauri is seen with a latitude of 72° north (108° south) and its declination is known (60°S), then it can be determined that the observer's colatitude is 108° − 60° = 48° (i.e. their latitude is 90° − 48° = 42°S).

Stars whose declinations exceed the observer's colatitude are called circumpolar because they will never set as seen from that latitude. If an object's declination is further south on the celestial sphere than the value of the colatitude, then it will never be seen from that location. For example, Alpha Centauri will always be visible at night from Perth, Western Australia because the colatitude is 90° − 32° = 58°, and 60° is greater than 58°; on the other hand, the star will never rise in Juneau because its declination of −60° is less than −32° (the negation of Juneau's colatitude). Additionally, colatitude is used as part of the Schwarzschild metric in general relativity.

References