Zenith Distance is Complement of Celestial Altitude
Jump to navigation
Jump to search
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.
$\blacksquare$
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text {II}$. The Celestial Sphere: $18$. Altitude and azimuth.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): zenith distance (coaltitude)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): zenith distance (coaltitude)