How do astronomers detect black holes if they cannot be observed with any kind of instrument?

The tale-tell sign that a black hole exists is the x-rays they emit from their outskirts, though from within their own realm they don’t emit anything, not even light. This is how the first black hole Cygnus-X1 (see the graphic below) was ‘smelled out’ by instruments aboard a suborbital Aerobe rocket. (There were no X-ray satellites or space observatories in those days). The exact location of the X-ray source could not be determined and more then five years passed before astronomers found a visible star whose motions were being influenced by an invisible nearby companion. In tune with the star’s varying motion, the intensity of the x-rays showed rapid fluctuations,  which meant the invisible companion is extremely compact, having a mass at least 10 times the Sun or even more.
Cygnus X-1
Since the object, exerting tremendous gravitational force on the shiny neighbor, is not itself a star and since its mass is well above the limit of a neutron star (about 3.2 solar masses), it must be a black hole. This is in accord with the General Theory of Relativity which predicts existence of black holes.

Here’s what is going on in the case of Cygnus-X1 black hole. The visible blue star, designated HDE 226868, is continuously losing a portion of its gaseous mass to the nearby black hole. There is a vast stream of hot gas, which does not flow directly into the black hole. The torrent spirals around it and gradually works its way inward toward the boundary of no return called the event horizon. It is this superhot gas that emits flicking X-rays as its last groan and it is this radiation that reliably indicated the presence of Cygnus-X1 black hole. Many such cosmic demons have been detected since then. In fact, there are enough reasons to believe that all galaxies, including our milky way harbor a super massive black hole at their centre. 

Additional reading:
Black hole (Wikipedia)
Cygnus X-1 (Wikipedia)

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