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?
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$\begingroup$ Eclipsing binaries are defined by their eclipses. The inclination needs to be large enough to make the eclipses visible. $\endgroup$– ProfRobCommented Sep 1, 2021 at 7:07
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$\begingroup$ It depends on the size of the orbits and the radii of the stars. $\endgroup$– Peter ErwinCommented Sep 1, 2021 at 9:04
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1 Answer
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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).