The nearby late M star Gliese 569B was recently found by adaptive optics imaging to be a double with separation ~1 AU. To explore the orbital motion and masses, we have undertaken a high resolution (~0.05 arcsec) astrometric study. Images were obtained over 1.5 years with bispectrum speckle interferometry at the 6.5m MMT and 6m SAO telescopes. Our data show motion corresponding to more than half the orbital period, and constrain the total mass to be > 0.115 M_solar, with a most probable value of 0.145 M_solar. Higher masses cannot be excluded without more extended observations, but from statistical analysis we find an 80% probability that the total mass is less than 0.21 M_solar. An infrared spectrum of the blended B double obtained with the MMT has been modeled as a blend of two different spectral types, chosen to be consistent with the measured J and K band brightness difference of a factor ~2. The blended fit is not nearly as good as that to a pure M8.5+ template. Therefore we hypothesize that the brighter component likely has two unresolved components with near equal masses, each the same as the fainter component. If Gl 569B is a triple our dynamical limits suggest each component has a mass of 50 (+23/-4) M_jupiter. We infer an age for the system of 300 Myr, from its kinematic motion which places its as a member of the Ursa Major moving group. All the above parameters are consistent with the latest DUSTY evolutiuon models for brown dwarfs.
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