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What we see here is a star, weighing about 25 times the mass of our Sun, moving periodically( every 10 days or so) around something' invisible,' that we can not see in the data, the analysis tells us that this other' thing' must be at least 9 times more massive than our Sun. The main part of the analysis is elimination : what can weigh nine solar masses, and not emit any light ? A black hole is the only possibility we have got left( this, or a fat invisible alien...).