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Question: Are there any kinds of binary pairings that haven't been discovered yet? Any that are particularly significant, or that might shed some light on binary stellar evolution or theories of capture processes, that are sought, but so far no examples have been found or at least suspected?

By "pairing" I mean a binary object where each is a type of star or a black hole.

Stars can be anything from brown dwarfs to neutron stars. Black holes should be very roughly comparable to stars in size. (I'm not asking about a star in orbit around a supermassive black hole in the center of a galaxy)

For example, recent observations of gravitational wave events have suggested a pair of black holes merging, and a pair of neutron stars merging.

HDE 226868
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uhoh
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As far as I am aware, there are no known examples of black hole+neutron star (BHNS) binaries (e.g. this paper from 2018 notes that "No BHNS binaries (BHNSs) have been discovered so far by radio surveys"). BHNS mergers should be observable with gravitational wave astronomy, so this situation may not last for too much longer. There is a report of a possible BHNS merger detected by LIGO and Virgo (event S190426c). This event has yet to be confirmed, follow-up searches by Hosseinzadeh et al. (2019) and Goldstein et al. (2019) for transients have failed to find a counterpart to the event.

Binaries containing an sdB star and a neutron star have also not been found, though theoretical models predict they should exist in our galaxy (Wu et al., 2019). Sufficiently close sdB+NS binaries should be strong gravitational wave sources, so LISA may discover them.

Depending on how you want to define various "types" of star, you can probably come up with more examples. For example, there is the concept of singleton stars which have never been members of binary systems or had close encounters with other stars, which by definition have not been found in binary pairings.

There are also some weird categories of objects with few known members. For example there are the LP 40-365 stars, which are runaway stars with atmospheres dominated by oxygen and neon, likely the partially-burnt remnants of peculiar type Iax supernovae. None of the four known examples of this class are binary systems, plus the circumstances that lead to their creation makes it seem fairly unlikely that we'll find one in a binary.

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tl;dr Yes there have been theorized, and extremely possible binary systems that have not been observed. One such thing is a TZO, or a Thorne–Żytkow_object. This is a neutron star-red giant binary.

There are many kinds of binary pairings that have not been discovered yet, if they exist. A hypothetical star is the quasi-star, which you could say it is a binary between a black hole and an extremely massive star. You can also consider a Wolf–Rayet star, which can occur from a binary system. Other binary systems would include theoretical stars, also known as exotic stars.

However, I do know of a binary system that is composed of real observed types of stars or stellar remnants. This system is composed of a red giant and a neutron star, with the neutron star spiraling in towards the red giant's core. It starts off with a binary system with a star and a neutron star. When the star becomes a red giant, its atmospheric friction on the neutron star(the red giant envelops the neutron star) makes their orbit deteriorate, and the neutron star and the red giant's core will spiral into each other. After the neutron star meets the core, if the mass of the core does not overcome neutron degeneracy pressure, then the core is replaced with a neutron star. If it does, then a supernova resulting in a black hole occurs. This binary system is particularly significant because this can shed light onto how fusion works, and how binary systems develop, because in this model the neutron star does not absorb a lot of the red giant's mass. The surface of the neutron star would also be so hot that fusion can occur, demonstrating a new way of fusion.

Astronomers can observe these stars by their emitted wavelengths because the fusion creates heavy elements that may find their way to the surface of the giant:

To look for a TZO, the team searched the spectrums for some of the elements predicted to be produced in high quantities in these objects: lithium, rubidium, strontium, vanadium, zirconium, and molybdenum. Not all of these are easy to image, so they focused on lithium, rubidium, and molybdenum. They compared the levels of these elements with elements that create spectral features nearby but aren't expected to be enhanced in TZOs: potassium, calcium, iron, and nickel.

https://arstechnica.com/science/2014/06/red-supergiant-replaced-its-core-with-a-neutron-star/

I would say that no true examples of this kind of binary pairing has been found. However, there are three possible candidates that are observed, though recently, these stars are under serious questioning on their validity(that may not answer part of your question on "at least suspected :("), according to

https://en.wikipedia.org/wiki/Thorne–Żytkow_object

There is another one mentioned by mistertribs(all credits to him), that a black hole and neutron star binary is not observed to exist. These type of systems are possibly the most interesting from a gravitational point of view. They should produce gravitational waves, which could be detectable by LIGO. LIGO has yet to observe such a system and confirm such, which seems a bit puzzling, although something like the blue supergiant in Cygnus X-3 could evolve into a neutron star in the future. This is a case where more data and observations are needed.

However, scientists were only able to constrain the merger rate enough to eliminate the most optimistic models, and that it would take several more observing runs with no results before a continued failure became problematic.

Max0815
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    Excellent, this is just the kind of thing I was thinking of, thanks! – uhoh Mar 01 '19 at 01:11
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    @uhoh updated the answer – Max0815 Mar 01 '19 at 19:21
  • @uhoh what do you think? Possible to mark an answer as correct? – Max0815 Mar 03 '19 at 00:40
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    Your answer is long and interesting, but you don't really propose a binary object at all. A TZO is "...a red giant or supergiant contains a neutron star at its core.". If you would like to call this a binary system, I think you'll have to make an argument for that. However, if you want to make it a lot clearer that you are naming a "neutron star (in the process of) spiraling in towards the red giant's core" as the object, and say that this has never been observed, then it may qualify as a type of binary object. – uhoh Mar 03 '19 at 00:48
  • @uhoh yeah that's how they form I'll add it. – Max0815 Mar 03 '19 at 00:49
  • I often add a tl;dr to the beginning of longer answers in lieu of an abstract, to make it easier for readers to know where the answer is going (e.g. https://space.stackexchange.com/search?q=user%3A12102+%22tl%3Bdr%22). btw some questions have more than one excellent answer and occasionally an OP will be so torn that they will accept neither. It should not happen most of the time, but it does happen. Is a TZO clearly more correct than a BHNS? – uhoh Mar 03 '19 at 00:52
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    @uhoh... you make a great point! – Max0815 Mar 03 '19 at 05:33
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    @uhoh I added the tl;dr. Well, lets first see, your first question, "Are there any kinds of binary pairings that haven't been discovered yet?" is answered in both posts. "Any that are particularly significant, or that might shed some light on binary stellar evolution or theories of capture processes" is answered in mine, but not his, and "so far no examples have been found or at least suspected?" is answered in his, but not mine. I see why you would say this is a stalemate! Cheers! – Max0815 Mar 05 '19 at 23:37
  • There has been some news about BH + NS pairs being detected (posthumously) by gravitational wave detection, maybe you can add a little "update note" at the beginning or end if you have a chance. – uhoh Jul 04 '21 at 00:23