Celestial Body moves along Parallel of Declination
Theorem
Consider the celestial sphere $C$ whose observer is $O$.
Let $B$ be a celestial body on $C$.
Let $\delta$ be the declination of $B$.
Let $P_\delta$ be the parallel of declination whose declination is $\delta$.
Then $B$ appears to move along the path of $P_\delta$, in the direction from the north to east to south to west.
It takes $1$ sidereal day for $B$ to travel around $P_\delta$.
Proof
The apparent movement of $C$ is caused by the rotation of Earth about its axis.
This rotation takes $1$ sidereal day, by definition, for the principal vertical circle to be in the same position again.
As the Earth rotates, the observer $O$ moves in an easterly direction.
Hence the celestial sphere $C$ appears to move in the opposite direction, so from north to east and so on.
$B$ remains at the same distance from the north celestial pole as it did.
Hence its declination $\delta$ stays the same.
Hence the result.
$\blacksquare$
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text {II}$. The Celestial Sphere: $19$. Declination and hour angle.