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How do we detect them?

X-Rays.

Black holes emit a lot of x-rays. X-rays are light rays that occur only in extremes of heat and on the jets perpendicular to the accretion disc there is that heat. So, if an astronomer detects x-rays and can follow them back to their source, there is a good chance they have found a black hole.

Red-shift.

If astronomers find an accretion disc they can analyze the light from either side of disc and using the Doppler effect (red shift) determine the speed of rotation. Through more complicated physics it is possible to determine the size of the disc. If it's bigger than three solar masses, it could easily be a black hole.

Binary Stars

Sometimes two stars rotate around each other and one of those can become a black hole. After analyzing light frequencies of binary systems interesting conclusions can be made using the speed of rotation, whether or not the two are sharing material, the red shift of either object. These deduction methods can tell us whether or not the binary system includes a black hole.