<p>This answer in <a href="https://math.stackexchange.com/a/2949040/50256">math.stackexchange</a> provides the correct relationship between <span class="math-container">$\theta$</span> and <span class="math-container">$\phi$</span>. And <a href="https://math.stackexchange.com/a/2950640/50256">this answer</a> supplies the proof.</p>

CC BY-SA 4.0
2018-10-10T22:13:59.217Z
2018-10-10T22:13:59.217Z
Marcus Junius Brutus
<p>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 <a href="https://math.stackexchange.com/q/2949028/50256">math.stackexchange.com</a>.</p>

<p>In the diagram below I depict:</p>

<ul>
<li>the celestial sphere with the North Celestial Pole (NCP), the South Celestial Pole (SCP) and the celestial equator</li>
<li>a latitude circle at latitude <span class="math-container">$\delta$</span> (declination)</li>
<li>two points <span class="math-container">$A$</span> and <span class="math-container">$B$</span> on that latitude circle</li>
<li>the center of the celestial sphere <span class="math-container">$O$</span> and the center of the latitude circle <span class="math-container">$O'$</span></li>
</ul>

<p><a href="https://i.sstatic.net/x03mQ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/x03mQ.png" alt="enter image description here"></a></p>

<p>My question is:</p>

<p>Given angle <span class="math-container">$\phi$</span> between the rays <span class="math-container">$OA$</span> and <span class="math-container">$OB$</span>, what is the angle <span class="math-container">$\theta$</span> between the rays <span class="math-container">$O'A$</span> and <span class="math-container">$O'B$</span> ? I am trying to derive <span class="math-container">$\theta$</span> given <span class="math-container">$\phi$</span> and <span class="math-container">$\delta$</span>. Observe that if <span class="math-container">$A$</span> is the first point of Aries, then the angle <span class="math-container">$\theta$</span> is the Right Ascension of a star located at point <span class="math-container">$B$</span>.</p>

<p>My assumption is that the following formula holds:</p>

<p><span class="math-container">$\phi = \theta*cos(\delta)$</span></p>

<coordinate><declination><mathematics><right-ascension>
angles on the celestial sphere
Marcus Junius Brutus
# angles on the celestial sphere

(Asked by: username_0 on 2018-10-10)

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](https://math.stackexchange.com/q/2949028/50256).

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](https://i.sstatic.net/x03mQ.png)](https://i.sstatic.net/x03mQ.png)

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:

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

**Tags:** coordinate, declination, mathematics, right-ascension


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## 1 answers

### Answer by username_0 on 2018-10-10. Score: 1

This answer in [math.stackexchange](https://math.stackexchange.com/a/2949040/50256) provides the correct relationship between $\theta$ and $\phi$. And [this answer](https://math.stackexchange.com/a/2950640/50256) supplies the proof.


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