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Wikipedia's Thorne–Żytkow object begins:

A Thorne–Żytkow object (TŻO or TZO), also known as a hybrid star, is a conjectured type of star wherein a red giant or red supergiant contains a neutron star at its core, formed from the collision of the giant with the neutron star. Such objects were hypothesized by Kip Thorne and Anna Żytkow in 1977.1 In 2014, it was discovered that the star HV 2112, located in the Small Magellanic Cloud (SMC), was a strong candidate.2 Another possible candidate is the star HV 11417, also located in the SMC.3

1Thorne, Kip S.; Żytkow, Anna N. (15 March 1977). "Stars with degenerate neutron cores. I - Structure of equilibrium models".

2Levesque, Emily M.; Massey, Philip; Zytkow, Anna N.; Morrell, Nidia (2014). "Discovery of a Thorne–Żytkow object candidate in the Small Magellanic Cloud". Monthly Notices of the Royal Astronomical Society: Letters. 443: L94–L98 (also in arXiv also Phys.org's Astronomers discover first Thorne-Zytkow object, a bizarre type of hybrid star

3Beasor, Emma; Davies, Ben; Cabrera-Ziri, Ivan; Hurst, Georgia (2 July 2018). "A critical re-evaluation of the Thorne-Żytkow object candidate HV 2112". Monthly Notices of the Royal Astronomical Society. 479 (3): 3101–3105. (also in arXiv)

Presumably any measurements that could be done with current astronomical instrumentation to further confirm that these (or similar) may be TZ objects have already been done, so I'd like to ask instead about what kinds of future observations would be most likely to better confirm the proposed Thorne–Żytkow objects in the small Magellanic cloud.

The next say 50 years of technology development will likely bring completely new instruments and capabilities. I'm not an astronomer but I'm guessing that those may include gravitational wave detectors that are far more sensitive to low-level signals and their direction and in new frequency ranges, space-based optical interferometry for new levels of spatial resolution, new EM wavelength ranges for high resolution imaging, even far larger and more sensitive neutrino telescopes, etc.

Question: Is there a "holy grail" for some future observation that could help to confirm the presence of neutron stars in the centers of massive, otherwise conventional stars? (Thorne–Żytkow objects)

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Primarily it is thought we will confirm/deny them based on detecting unusually large amounts of certain nuclei in the spectra. Things like lithium and calcium which are hard to get in "normal" stars and heavier things like Rubidium.

The problem here is that they are hard things to measure intrinsically in a spectrum and the candidates are claimed to be in the Small Magellanic Cloud (there is debate over its distance) which makes it even harder to detect. The models for TZO's are also quite old now (~30 years), so the expected abundance patterns may have changed.

So on the composition front I think we simply need more and better observations and updated models. I don't think there are too many limitations per se, they just need more observing time.

There is a claim that we might be able to see the gravitational wave signal, if the neutron star is rotating inside the core of the TZO (which is likely). They claim we could see them within 10Kpc this way, but I expect more detailed calculations of the gravitational wave strain may lower that limit. Whether there are any within 10Kpc is also unknown (they are probably rare objects).

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