Zenith Distance of North Celestial Pole equals Colatitude of Observer

Theorem

Let $O$ be an observer of the celestial sphere.

Let $P$ be the position of the north celestial pole with respect to $O$.

Let $z$ denote the zenith distance of $P$.

Let $\psi$ denote the (terrestrial) colatitude of $O$.

Then:

$z = \psi$

Proof

Let $Z$ denote the zenith.

The zenith distance $z$ of $P$ is by definition the length of the arc $PZ$ of the prime vertical.

This in turn is defined as the angle $\angle POZ$ subtended by $PZ$ at $O$.

This is equivalent to the angle between the radius of Earth through $O$ and Earth's axis.

This is by definition the (terrestrial) colatitude of $O$.

Hence the result.

$\blacksquare$

Sources

• 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text {II}$. The Celestial Sphere: $18$. Altitude and azimuth.