<p>This is a relatively straightforward application of spherical trigonometry.  (See also: <a href="https://astronomy.stackexchange.com/questions/40943/about-coordinate-systems-and-angle-differences/48571#48571">About coordinate systems and angle differences</a> and <a href="http://astrophysicsformulas.com/astronomy-formulas-astrophysics-formulas/angular-distance-between-two-points-on-a-sphere/" rel="nofollow noreferrer">Angular Distance Between Two Points on a Sphere</a>)</p>
<p>The expression to compute the angle on the celestial sphere between to points is:
<span class="math-container">$$\Psi = \arccos\left(\sin\theta_1\sin\theta_2 + \cos\theta_1\cos\theta_2\cos(\phi_1-\phi_2)\right)$$</span>
where <span class="math-container">$\Psi$</span> is the angular separation, <span class="math-container">$\phi_1$</span> and <span class="math-container">$\phi_2$</span> are the right ascensions of the first and second direction, and <span class="math-container">$\theta_1$</span> and <span class="math-container">$\theta_2$</span> are the declinations of the first and second direction.</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th style="text-align: left;">Star Pair</th>
<th style="text-align: right;">Separation (deg)</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: left;">Alnitak - Saiph</td>
<td style="text-align: right;">7.9203</td>
</tr>
<tr>
<td style="text-align: left;">Alnitak - Sirius</td>
<td style="text-align: right;">21.6568</td>
</tr>
<tr>
<td style="text-align: left;">Saiph - Sirius</td>
<td style="text-align: right;">15.6351</td>
</tr>
</tbody>
</table>
</div>
<p>I am including my conversion of the RA and DEC to degrees in case I have made a mistake if anyone wants to check my computations.</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th style="text-align: left;">Star</th>
<th style="text-align: right;">RA (HH MM SS)</th>
<th style="text-align: right;">RA (deg)</th>
<th style="text-align: right;">DEC (DD MM SS)</th>
<th style="text-align: right;">DEC (deg)</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: left;">Alnitak</td>
<td style="text-align: right;">05 40 45.527</td>
<td style="text-align: right;">85.1897</td>
<td style="text-align: right;">-01 56 33.26</td>
<td style="text-align: right;">-1.9426</td>
</tr>
<tr>
<td style="text-align: left;">Saiph</td>
<td style="text-align: right;">05 47 45.389</td>
<td style="text-align: right;">86.9391</td>
<td style="text-align: right;">-09 40 10.58</td>
<td style="text-align: right;">-9.6696</td>
</tr>
<tr>
<td style="text-align: left;">Sirius</td>
<td style="text-align: right;">06 45 08.917</td>
<td style="text-align: right;">101.2872</td>
<td style="text-align: right;">-16 42 58.02</td>
<td style="text-align: right;">-16.7161</td>
</tr>
</tbody>
</table>
</div>
CC BY-SA 4.0
2022-02-20T13:11:37.870Z
2022-02-20T17:34:46.163Z
2022-02-20T17:34:46.163Z
GrapefruitIsAwesome
<p>I am using the <a href="http://simbad.u-strasbg.fr/simbad/" rel="nofollow noreferrer">SIMBAD database</a> for <a href="https://en.wikipedia.org/wiki/Alnitak" rel="nofollow noreferrer">Alnitak</a>, <a href="https://en.wikipedia.org/wiki/Saiph" rel="nofollow noreferrer">Saiph</a>, and <a href="https://en.wikipedia.org/wiki/Sirius" rel="nofollow noreferrer">Sirius</a>:</p>
<pre><code>Alnitak ; FK5 coord. (J2000): RA = 05 40 45.527 DEC = -01 56 33.26; 
Saiph   ; FK5 coord. (J2000): RA = 05 47 45.389 DEC = -09 40 10.58; 
Sirius  ; FK5 coord. (J2000): RA = 06 45 08.917 DEC = -16 42 58.02; 
</code></pre>
<p>What is the trigonometric relationship in degrees (length of sides and inclusive angles to four decimal places) between these three stars when they are used to form a triangular aterism?</p>

CC BY-SA 3.0
2014-05-09T01:03:56.957Z
I guess, to measure the length of sides, you should also give the distance.
Py-ser
CC BY-SA 3.0
2014-05-09T06:01:04.153Z
Question is not clear. Do you want spherical triangle formulas? Or real space ones?
Envite
<star><distances><mathematics>
Triangular Asterism Trigonometry between Alnitak, Saiph, and Sirius?
Odys
# Triangular Asterism Trigonometry between Alnitak, Saiph, and Sirius?

(Asked by: username_0 on 2014-05-08)

I am using the SIMBAD database for Alnitak, Saiph, and Sirius:
Alnitak ; FK5 coord. (J2000): RA = 05 40 45.527 DEC = -01 56 33.26; 
Saiph   ; FK5 coord. (J2000): RA = 05 47 45.389 DEC = -09 40 10.58; 
Sirius  ; FK5 coord. (J2000): RA = 06 45 08.917 DEC = -16 42 58.02; 

What is the trigonometric relationship in degrees (length of sides and inclusive angles to four decimal places) between these three stars when they are used to form a triangular aterism?

**Tags:** star, distances, mathematics
> **Comments:**
> **username_1**: I guess, to measure the length of sides, you should also give the distance.  
> **username_2**: Question is not clear. Do you want spherical triangle formulas? Or real space ones?  

---

## 1 answers

### Answer by username_3 on 2022-02-20. Score: 1

This is a relatively straightforward application of spherical trigonometry. (See also: [About coordinate systems and angle differences](https://astronomy.stackexchange.com/questions/40943/about-coordinate-systems-and-angle-differences/48571#48571) and [Angular Distance Between Two Points on a Sphere](http://astrophysicsformulas.com/astronomy-formulas-astrophysics-formulas/angular-distance-between-two-points-on-a-sphere/))

The expression to compute the angle on the celestial sphere between to points is:
$$\Psi = \arccos\left(\sin\theta\_1\sin\theta\_2 + \cos\theta\_1\cos\theta\_2\cos(\phi\_1-\phi\_2)\right)$$
where $\Psi$ is the angular separation, $\phi\_1$ and $\phi\_2$ are the right ascensions of the first and second direction, and $\theta\_1$ and $\theta\_2$ are the declinations of the first and second direction.

| Star Pair | Separation (deg) |
| --- | --- |
| Alnitak - Saiph | 7.9203 |
| Alnitak - Sirius | 21.6568 |
| Saiph - Sirius | 15.6351 |

I am including my conversion of the RA and DEC to degrees in case I have made a mistake if anyone wants to check my computations.

| Star | RA (HH MM SS) | RA (deg) | DEC (DD MM SS) | DEC (deg) |
| --- | --- | --- | --- | --- |
| Alnitak | 05 40 45.527 | 85.1897 | -01 56 33.26 | -1.9426 |
| Saiph | 05 47 45.389 | 86.9391 | -09 40 10.58 | -9.6696 |
| Sirius | 06 45 08.917 | 101.2872 | -16 42 58.02 | -16.7161 |


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