This is a somewhat simplified, and drawn / formulated in a manner more targeted to astronomy, version of a diagram and a question that I've also posted in math.stackexchange.com.
In the diagram below I depict:
- the celestial sphere with the North Celestial Pole (NCP), the South Celestial Pole (SCP) and the celestial equator
- a latitude circle at latitude $\delta$ (declination)
- two points $A$ and $B$ on that latitude circle
- the center of the celestial sphere $O$ and the center of the latitude circle $O'$
My question is:
Given angle $\phi$ between the rays $OA$ and $OB$, what is the angle $\theta$ between the rays $O'A$ and $O'B$ ? I am trying to derive $\theta$ given $\phi$ and $\delta$. Observe that if $A$ is the first point of Aries, then the angle $\theta$ is the Right Ascension of a star located at point $B$.
My assumption is that the following formula holds:
$\phi = \theta*cos(\delta)$