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Do quasars collapse and make a galaxy or do they eventually spew out enough matter to make one? Because I heard they are the center of galaxies, and I wonder if they can collapse from their own gravity or from becoming unstable, and if so, what would happen?

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A quasar is a name given to a particular state of activity of an active galactic nucleus - characterised by high levels of accretion onto a supermassive black hole.

i.e. Quasars are not "objects" in their own right and if the accretion stops then so does the quasar-like activity and you end up with a more dormant galactic nucleus, like the one in our own Milky Way galaxy.

Thus quasars do not "make a galaxy", although there are interesting but poorly understood correlations between the properties of galaxies, their supermassive back holes, their history of mass accretion and the properties of the galaxies they reside in. They thus probably play an important role in galaxy formation and evolution, principally through the injection of momentum and energy from their winds, jets and radiation.

There is also no sense in which they can collapse. A supermassive black hole is as compact as it can be. It can of course continue to accrete matter and get bigger, but most supermassive blackholes have a mass that is only a small fraction of the mass of the galaxy they are at the center of.

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    $\begingroup$ Thank you for clearing that up, quasars are a new subject to me. $\endgroup$ Commented Mar 4 at 14:48

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