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Radial-velocity variation of a tertiary star orbiting a binary black hole in coplanar and non-coplanar triples: short- and long-term anomalous behavior

A number of ongoing surveys are likely to discover star-black hole binaries in our Galaxy in the near future. A fraction of them may be triple systems comprising an inner binary, instead of a single black hole, which might be progenitors of binary black holes (BBHs) routinely discovered now from the gravitational wave. We extend our previous proposal to locate inner BBHs from the short-term radial-velocity (RV) variation of a tertiary star in coplanar triples, and we consider noncoplanar triples and their long-term RV variations as well. Specifically, we assume coplanar and noncoplanar triples with an inner BBH of the total mass $20~M_\odot$, whose outer and inner orbital periods are 80 days and 10 days, respectively. We perform a series of N-body simulations and compare the results with analytic approximate solutions based on quadrupole perturbation theory. For coplanar triples, the pericenter shift of the outer star can be used to detect the hidden inner BBH. For noncoplanar triples, the total RV semi-amplitude of the outer star is modulated periodically on the order of $100$km/s due to its precession over roughly the Kozai-Lidov oscillation timescale. Such long-term modulations would be detectable within a decade, independent of the short-term RV variations on the order of of $100$ m/s at roughly twice the orbital frequency of the inner binary. Thus the RV monitoring of future star-black hole binary candidates offers a promising method for searching for their inner hidden BBHs in optical bands.