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An optical double is two stars which appear very close together because they chance to be lined up as seen from Earth but are actually tens, hundreds or thousands of light years apart in space.

A real double star is a binary, where the two stars are gravitationally bound and orbit around each other.

A naked eye binary star is a binary were both of the stars are bright enough to be seen with the naked eye from each and where the physical separation between them is far enough enough for them to be seen from Earth as two separate stars by the unaided human eye instead of being blurred together into a single dot of light. Seeing the two stars is called "splitting" the binary star.

In my opinion there are no binary stars which can be "split" with the naked eye that have orbital periods less than 100,000 years.

Stars I have checked for orbital periods include Mizar and Alcor, which may or may not be gravitationally bound and I haven't found a period listed for them; Epsilon Lyrae with a period of hundreds of thousands of year; Alpha Librae with a period over 200,000 years, Beta Capricornii 700,000 years; and Zeta Corvi and HR 4691 that may or may not be a binary with a period of about 3,500,000 years.

See Summer's Best Naked-Eye Double Stars - Sky & Telescope

20 Fun Naked-Eye Double Stars

So I am doubtful that any naked eye binary stars have orbital periods less than 100,000 years.

So does anyone know of any naked eyes binary stars with orbital periods less than 100,000 years?

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I queried the 4th Catalog of Orbits of Visual Binary Stars for systems with both components having $V<6$ and period $<10^5$ years. There were 77 examples, but the widest separation is 17.5 arcseconds, which would not be resolvable to the human eye. Although there is an updated (sixth) catalog, I could not find any straightforward way of querying it - I doubt it has much more information on bright, naked-eye stars.

Leaving aside whether someone can identify an odd example, the fact remains that such objects are clearly rare. So, why is that?

Kepler's third law can be written as $$ a^3 = M P^2\ , $$ with $a$ in au, the mass in solar masses and the orbital period $P$ in years. Thus you are demanding that $$ a < (10^{10} M)^{1/3}\ . $$

If the angular resolution of the eye is $\theta$, then the apparent separation on the sky must exceed this and thus $$ a > \theta d\ , $$ where $d$ is the distance.

Putting these things together, the distance to the binary would need to be less than some mass-dependent threshold $$ d < (10^{10} M)^{1/3}/\theta\ .$$ Putting some numbers in and using a value of maybe 1 arcminute for a barely resolved pair, $$d < 50 \left(\frac{M}{M_\odot}\right)^{1/3}\left(\frac{P}{10^5 {\rm yr}}\right)^{2/3}\left(\frac{\theta}{\rm arcmin}\right)^{-1}\ {\rm pc}\ . $$ Note that this optimistically assumes the system is at maximum elongation and not projected to a smaller separation.

What this shows, is that for a system mass of order a solar mass, to satisfy your requirements, it does need to be one of the closest naked eye stars in the sky. Another way of looking at it, is that even for the very nearest stars to be resolved as naked eye binaries, their orbital periods would need to be larger than about $5000$ years.

There are only a couple of hundred naked eye stars closer than 50 pc and most of those are at the faint end of what is visible. Of those, maybe half are binary systems, maybe a bit less than half again have a companion that is also a naked eye star. This might suggest that there might be some tens of naked eye binary systems that we can see. But the main issue is that the distribution of orbital periods for those binaries is strongly peaked at much shorter periods (1-100 years) and only perhaps a few-10 percent have orbital periods longer than $5000$ years.

Therefore, the chances of having a naked eye binary visible from Earth, with an orbital period shorter than 10,000 years is, a priori, such that we might expect there to be only one or two examples. But these would most likely be on the edge of resolvability and at the edge of being a naked eye system.

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  • $\begingroup$ So to the actual question "Are there naked eye binary stars with periods less than 100,000 years?" your answer is "Maybe, maybe not."? $\endgroup$
    – uhoh
    Nov 17, 2023 at 10:22
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    $\begingroup$ @uhoh hard to respond to that. I am of the view that the questions and answers that I find the most uninteresting on Astronomy SE are those asking for a simple factual answer to trivia, that could be written on one line. The objects in question are obviously rare, whether somebody can dig up an example or not. And any example would then just be an accident of cosmic coincidence for our particular epoch. In my view, the interesting subtext is why they are rare. Clearly my answer should not be "accepted" as THE answer to the question posed. $\endgroup$
    – ProfRob
    Nov 17, 2023 at 10:53
  • $\begingroup$ I think it's a great, wonderful post; I don't mean to cast shade on it at all. I haven't a clue how to find a list of binaries that would have apparent magnitude and orbital period listed explicitly - that's the kind of thing astronomers know (or are likely able to readily find) and the rest of us don't. $\endgroup$
    – uhoh
    Nov 17, 2023 at 13:29
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    $\begingroup$ A couple of negative results. 61 Cygni is too close to separate 36 Oph is wide enough, but has a period of over 100 000 years. $\endgroup$
    – James K
    Nov 18, 2023 at 22:23
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Yes, Algol is not far away. The two stars orbit each other in 2.867328 days. Further examples.

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  • $\begingroup$ Algol cannot be separated with the naked eye (nor even with conventional telescopes, though they can be separated with an interferometer) $\endgroup$
    – James K
    Nov 17, 2023 at 21:07

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