That’s quite hard to answer, because my project doesn’t look at black holes directly. Instead, we observe explosions and outbursts of stars. Some of these explosions form a black hole: technically, only those that are “core-collapse supernovae” (which is something you could look up on Wikipedia) and then only if the exploding star is big enough; smaller stars form neutron stars instead of black hole). Also, to identify the right kind of explosion we need extra observations from the ground and not all our observations from space have been followed up in this way.
Anyway, I can estimate the number. We – Gaia Photometric Science Alerts – have found and reported 4505 star explosions so far. Of these, we think about half are some kind of supernova, based on the follow-up observations. Among the supernovae, perhaps a third are the core-collapse kind; the rest are violent events in binary star-systems and don’t involve black holes. At a *very* rough guess, possibly a quarter of the core-collapse supernovae might be heavy enough form black holes instead of neutron stars. Working that out, Gaia has seen about 185 black holes that were actually being forming in supernovae.
We also see a few events in “active galactic nuclei”. These things are the really big black holes at the centres of galaxies, and the hot gas around the black holes (we see the gas, not the black holes directly). They wake up and get much brighter whenever stars fall in.
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