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I understand that we can infer much about the parameters, such as the radius and mass, of stars in eclipsing binary systems. But how can eclipsing binaries be used to gauge distances? The current Wikipedia entry, for example, for binary star mentions that eclipsing binaries can be used to gauge distances, even to nearby galaxies, but I don't understand exactly how.

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Most distance methods are luminosity-based: you measure an object's flux, assume it has a particular luminosity, and then determine the distance from that using the inverse-square law. So the question becomes: how do you determine the luminosity?

The eclipsing-binary method uses the idea that a star's luminosity can be defined as $L = 4 \pi f_{s} R^{2}$, where $f_{s}$ is the flux per unit area from the star's surface and $R$ is the radius of the star. So how do we determine those?

The surface flux of the star can be estimated from stellar-atmosphere models combined with detailed spectroscopy of the star; alternately, there are tight empirical relations between the surface flux and the color for late-type stars (known as the surface-brightness--color relationship [SBCR]). The latter seem to be preferred for recent eclipsing-binary work (where the binary consists of red giant stars).

The radius of the star is where the eclipsing-binary method comes in. By detailed photometric monitoring, you can determine the period of the orbit, the timing and duration of the eclipses, and the duration of the eclipse onsets and exits (i.e., the partial eclipse phases before and after the full eclipse). And with detailed spectroscopic monitoring, you can determine the radial velocities of the stars, which lets you determine the orbital velocities (from the observed radial velocities of the stars, via the Doppler shifts of their spectra). From the period of the orbit and the orbital velocities, you can determine the size of the orbits. Given the size of the system and the velocities of the stars, you can work out the radii of the stars (e.g., for the same velocity and orbit size, a star with a small radius will go into full eclipse later, but more rapidly, than a large star will).

Note that knowing the geometry and timing of the orbit also allows you to better determine the colors or spectroscopy of the individual stars, and thus their individual surface fluxes. (E.g., when the system is not in eclipse, you know you are seeing light from both stars at once, while if one star completely eclipses the other, you know you are only seeing light from it, and can then work out how much of the combined light is due to the other star.)

Things are simplest if the orbit is circular and the inclination to our line of sight is $90^{\circ}$ (i.e., the binary is edge-on); but as long as there are at least partial eclipses and monitoring of the radial velocity, you can still work out the geometry of the system and use the method.

So the key advantage of the eclipsing-binary method is that you can work out the geometry and timing of the system, so that you can determine the radii of the two stars, and then combine that with their surface fluxes to get their luminosities -- and then their distance.

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The key here is mass.

In nonbinary systems (or other external factors) it can be a bit tough to determine mass of a star. You just see a point source of light; we can get spectral features which can in some instances be used to give us some idea, but it’s rather tough to otherwise get mass.

If you do have mass (which you can get because binary stars are moving and move according to their masses) then you can use the mass-luminosity relationship or other theories of special objects in combination with the color to determine the luminosity and thus the distance, since knowing the output of flux on the surface and the flux you’re measuring in your telescope can then easily be turned into a distance (Flux by definition is luminosity per a spherical square surface, the radius of the sphere being your distance from the object. With flux and luminosity, you just solve for the distance.)

For eclipsing binaries, you can determine the mass one of two ways. If you take the spectra of the stars while the two stars are not eclipsing, you can determine their speeds through the redshirting of one object’s lines and the blue shifting of the others; you can then determine the period of the orbit by the light curve, and using both, be able to find the size of the orbit. With this information, you can find the collective mass of the two stars.

Alternatively, you can carefully look at the light curves of the stars, and determine the size based off of how long ingress (the star to go from partially eclipsing to totally eclipsing) takes in conjunction with the speed of the orbit. With size and color, you can then determine luminosity and mass, although mass in this case is less important, as luminosity is the end goal here.

The former method fails with two objects of significantly different brightnesses, such that the spectral lines of one are washed out by the other.

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    $\begingroup$ This is incomplete, because you aren’t considering the eclipsing nature of the binary and how that is used to determine the sizes of both stars, in addition to their masses. $\endgroup$ Commented Sep 15, 2021 at 5:16
  • $\begingroup$ You’re right, I failed to mention the way mass was determined in the system, so I made an edit for that. However I’m not sure I understand the need for size; while it’s good to know, don’t you need to know the speed of the object to find the size, and if you know the speed of the object from spectra and it’s period from the light curve, could you not then determine the semi major axis and use Newton’s version of Kepler law to figure out the masses without the sizes? $\endgroup$
    – Justin T
    Commented Sep 15, 2021 at 6:47
  • $\begingroup$ After doing bit more reading: the eclipsing binary method does not rely on determine the masses of the stars at all. I’ll see if I can write a short answer summarizing it. $\endgroup$ Commented Sep 16, 2021 at 8:05
  • $\begingroup$ Note that the "mass-luminosity" relation is only valid for main sequence stars, so you have to verify a star is main sequence in order to use it. $\endgroup$ Commented Sep 16, 2021 at 11:03

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