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How to estimate upper and lower limits for the period of a binary that is not eclipsing? What parameters are necessary, please?

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    $\begingroup$ This is a bit unclear. Do you mean you know something is a binary and it isn't eclipsing. What other information do you have? The lack of an eclipse doesn't constrain the orbital period strongly. $\endgroup$
    – ProfRob
    Oct 17, 2022 at 15:00

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A binary system could have an orbital period anything from the period the object would have if the two stars were almost touching (I'm assuming a contact binary would give a noticeable light curve modulation) to being so wide that it can just survive being broken up by the Galactic tidal field.

Both of these limits will depends on the mass of the binary components. The former limit will depend on the radii of the components.

The lack of eclipses does not place very strong prior contraints on the probability of the binary having any particular orbital period, except at very close separations where the lack of any eclipses becomes unlikely. The probability of eclipse for a given separation is something like $P(a) \sim (R_1 +R_2)/a$, where $R_{1,2}$ are the radii of the components and $a$ is their separation, which approaches 1 when the stars are almost touching. Thus the probability of not eclipsing is $1 -P(a)$ and this reduces the a-priori probability that your non-eclipsing object is a very close (short period) binary.

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    $\begingroup$ Thank you and is there a formula for the period? $\endgroup$
    – Elena Greg
    Oct 18, 2022 at 6:15
  • $\begingroup$ @ElenaGreg There is Kepler's third law that relates period to mass and separation. Otherwise I 'm not sure what you mean. $\endgroup$
    – ProfRob
    Oct 18, 2022 at 6:36
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Some basics, hope it helps.

What do we know about a binary - it depends on method the binary was found.

Wiki binary star - see "Classification" part.

Eclipses are the cheapest approach to search for binaries, but for non-eclipsing pairs other methods are available:

  1. Visual binary.

We see two stars close one to another. By parallax measurements we can know are the stars on similar distance to us, or not (just line of sight projection). In the first case the stars are likely gravitationally bound and are orbiting each other. If this is true, we can measure their projected separation - and calculate their minimum distance to each other (knowing distance to us from parallax). If we can estimate the stars masses (from their spectra, astroseismology, etc) - than we have the lower bound of their orbital period.

If the period is not very long (up to decades) - we can just observe for years to find the real period. But many visual binary stars have minimal period estimates in range of centuries or millenia, so the lower bound from projected separation is the best we can have.

  1. Spectroscopic binaries

In this case the system don't need to be resolved as two distinct stars. Also it's enough to see the light from just one component (the other can be too faint to see). If we see in spectra that a star system has regular velocity variations "to us, than from us" - it means there is an orbital motion.

If we have observed several rotation periods with good "phase coverage" - than the period is constrained very precisely. But if the period is too long and haven't been observed in full yet - than all we can say is "we see some non-linear acceleration term that is probably caused by gravitationally bound massive companion". Estimates of the orbital period will be very uncertain in this case.

The caveat with this method - it's much more expensive "per object" compared to other methods of binary stars study.

  1. Photometric non-eclipsing binary

If stars in a binary close enough to each other - than their sides directed to companion will be some hotter because of radiation of the companion. The pair can be non-eclipsing in our line of sight, but still we can observe a slight periodic brightness variation, so the rotation period can be found.

PS: it's not an exhaustive list of binary detection methods. There are more.

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