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I have read, at http://www.thestargarden.co.uk/Black-holes.html for example, that whole stars can be ejected from certain galaxies. "These are thought to have been part of a binary star system that broke apart as it approached a supermassive black hole. As one star was captured, and the other was pushed away at a velocity exceeding the escape velocity of the Galaxy."

My question is about the jets that can form at the poles of the rotating supermassive black hole: "At the rotation axis of the supermassive black hole, matter from the accretion disc can be pushed away at the speed of light, creating jets that can extend for thousands of light-years."

At thousands of light-years away, is this matter still gravitationally bound to the galaxy? (If not, maybe some particles can go around in a big loop that takes a while.)

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Yes, definitely.

While some matter returns to the galaxy as a so-called "galactic fountain" (e.g. Biernacki & Teyssier 2018), some material is ejected at super-escape velocity, becoming part of the intergalactic medium.

This is one of the mechanisms responsible for polluting the intergalactic medium (IGM) with metal-rich gas (i.e. elements heavier than helium). A recent observation of this is presented by Fujimoto et al. (2020).

The IGM itself is too dilute to form stars, and hence metals, itself, but is nevertheless observed to contain quite a lot of metals, usually seen as absorption lines in the spectra of background quasars (e.g. Songaila & Cowie 1996; Aguirre et al. 2008). These must be blown out from the galaxies, either by stellar feedback (through radiation pressure, cosmic rays, and supernova feedback), or by AGN activity (see also Germain et al. 2009).

Note however that although material can reach relativistic velocities, "pushed away at the speed of light" is just a tad too fast. Only massless particles can travel at the speed of light, but if you're massless it isn't really a big achievement — photons do that all the time, which is why we see the galaxies in the first place.

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