Zenith Distance is Complement of Celestial Altitude

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Theorem

Let $X$ be the position of a star (or other celestial body) on the celestial sphere.

The zenith distance $z$ of $X$ is the complement of the altitude $a$ of $X$:


$z = 90 \degrees - a$


Proof

The vertical circle through $X$ is defined as the great circle that passes through $Z$.

By definition, the angle of the arc from $Z$ to the horizon is a right angle.

Hence $z + a = 90 \degrees$.

The result follows.

$\blacksquare$


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