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Note: just because you can't see a black hole doesn't mean it's not there.
You see stars because they emit radiation; however, a black hole does not emit light. So how do we know it's there?
It's like knowing that a sweaty basketball player is in the room. You don't need to see him. You can notice his effect (i.e. stink) on other people by their reactions.
Two often used methods of finding black holes is:
The atmosphere of a red giant is very loosely held by the giant star. As a result, it can easily be transferred to a companion black hole. This process is called accretion. As the matter falls into the black hole, it heats up and gives off radiation of various wavelengths.
Here's a press release that describes how the Chandra X-ray observatory was used to find a binary system of black holes at the center of a galaxy.
The second method of finding black holes is by their gravitational interaction with a companion. If you see a single star and use the Doppler shift of its light to measure the wobble in its orbit, then you can calculate the mass of its companion. If you calculate a companion of mass greater than 3 solar masses, then it is likely a black hole.
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