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We have computed, using 17 infrared radial velocities, the first set of orbital elements for the M giant in the symbiotic binary V2116 Ophiuchi. The giant's companion is a neutron star, the bright X-ray source GX 1+4. We rule out the previously proposed period of 304 days and instead find an orbital period of 1161 days, by far the longest of any known X-ray binary. The orbit has a modest eccentricity of 0.10, with an orbital circularization time of ≲5 × 106 yr. The large mass function of the orbit significantly restricts the mass of the M giant. Adopting a neutron star mass of 1.35 M☉, the maximum mass of the M giant is 1.22 M☉, making it the less massive star. Spectrum synthesis analysis of several infrared spectral regions results in slightly subsolar abundances for most metals. Carbon and nitrogen are in the expected ratio resulting from the red-giant first dredge-up phase. The lack of 17O suggests that the M giant has a mass less than 1.3 M☉, consistent with our maximum mass. The surface gravity and maximum mass of the M giant result in a radius of 103 R☉, much smaller than its estimated Roche lobe radius. Thus, the mass loss of the red giant is via a stellar wind. These properties argue that the M giant is near the tip of the first-ascent giant branch. Although the M-giant companion to the neutron star has a mass similar to the late-type star in low-mass X-ray binaries, its near-solar abundances and apparent runaway velocity are not fully consistent with the properties of this class of stars. Thus, in many ways this symbiotic X-ray binary system is unique, and various scenarios for its possible evolution are discussed.