<p>The condition that a binary eclipses is (approximately) just a piece of trigonometry, assuming the stars are spherical and in circular orbits (this is quite likely for close binaries).</p>
<p>The condition that an eclipse will occur is that the inclination angle <span class="math-container">$i$</span> is given by
<span class="math-container">$$ \cos i  &lt; \frac{r_1 + r_2}{a}\ , $$</span>
where <span class="math-container">$r_1$</span> and <span class="math-container">$r_2$</span> are the stellar radii and <span class="math-container">$a$</span> is the binary separation.</p>
<p>If one allows for eccentricity then this is modified to
<span class="math-container">$$ \cos i &lt; \frac{\left(r_1+r_2\right)\left(1 \pm e\sin \omega\right)}{a(1 - e^2)}\ ,$$</span>
where <span class="math-container">$e$</span> is the eccentricity, <span class="math-container">$\omega$</span> is the argument of periastron and the <span class="math-container">$\pm$</span> sign depends whether we are considering transits or occultations of the secondary (see for example <a href="https://arxiv.org/abs/1001.2010" rel="nofollow noreferrer">Winn 2010</a>).</p>

CC BY-SA 4.0
2021-09-01T12:15:59.833Z
2021-09-01T12:15:59.833Z
ProfRob
<p>What is the definition of eclipsing binaries based on the inclination of their orbit and the sum of relative radii? Are close eclipsing binaries defined by an inclination greater than 60 degrees?</p>

CC BY-SA 4.0
2021-09-01T07:07:32.760Z
Eclipsing binaries are defined by their eclipses. The inclination needs to be large enough to make the eclipses visible.
ProfRob
CC BY-SA 4.0
2021-09-01T09:04:25.733Z
It depends on the size of the orbits *and* the radii of the stars.
Peter Erwin
<observational-astronomy><binary-star><eclipsing-star-systems>
Eclipsing binaries - definition with inclination
Anna-Kat
# Eclipsing binaries - definition with inclination

(Asked by: username_0 on 2021-09-01)

What is the definition of eclipsing binaries based on the inclination of their orbit and the sum of relative radii? Are close eclipsing binaries defined by an inclination greater than 60 degrees?

**Tags:** observational-astronomy, binary-star, eclipsing-star-systems
> **Comments:**
> **username_1**: Eclipsing binaries are defined by their eclipses. The inclination needs to be large enough to make the eclipses visible.  
> **username_2**: It depends on the size of the orbits *and* the radii of the stars.  

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## 1 answers

### Answer by username_1 on 2021-09-01. Score: 3 | Accepted answer

The condition that a binary eclipses is (approximately) just a piece of trigonometry, assuming the stars are spherical and in circular orbits (this is quite likely for close binaries).

The condition that an eclipse will occur is that the inclination angle $i$ is given by
$$ \cos i < \frac{r\_1 + r\_2}{a}\ , $$
where $r\_1$ and $r\_2$ are the stellar radii and $a$ is the binary separation.

If one allows for eccentricity then this is modified to
$$ \cos i < \frac{\left(r\_1+r\_2\right)\left(1 \pm e\sin \omega\right)}{a(1 - e^2)}\ ,$$
where $e$ is the eccentricity, $\omega$ is the argument of periastron and the $\pm$ sign depends whether we are considering transits or occultations of the secondary (see for example [Winn 2010](https://arxiv.org/abs/1001.2010)).


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