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 $O'$

enter image description here

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:

$\phi = \theta*cos(\delta)$

  • $\begingroup$ I think you are correct, but suggest using spherical coordinates to confirm. $\endgroup$
    – user21
    Commented Oct 10, 2018 at 17:16

1 Answer 1


This answer in math.stackexchange provides the correct relationship between $\theta$ and $\phi$. And this answer supplies the proof.


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