# angles on the celestial sphere

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:

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

• I think you are correct, but suggest using spherical coordinates to confirm.
– user21
Oct 10 '18 at 17:16

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