Is the barycenter of the Milky Way Galaxy located within the supermassive black hole, or its event horizon, at the center of our galaxy?
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5$\begingroup$ Actually, it occurs to me to wonder whether "barycenter" works well in a situation where relativistic considerations are significant. E.g., if a naive computation/conception says the barycenter is "inside a black hole", aren't there problems with the geometry? $\endgroup$– paul garrettCommented Jul 18, 2022 at 19:34
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1$\begingroup$ Similar situation: the barycenter of the Earth/Sun system is only a few hundred km from the center of the sun, well inside the Schwarzschild radius of the black hole the Sun would form. $\endgroup$– chepnerCommented Jul 19, 2022 at 18:29
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2$\begingroup$ @chepner That's not right, since the sun would have a schwatzchld radius of about 3km, anyway the barycentre of the solar system is often outside the sun's photosphere, $\endgroup$– James KCommented Jul 19, 2022 at 20:18
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$\begingroup$ :facepalm: I misread m as km on the Wikipedia page for Schwartzschild Radius. $\endgroup$– chepnerCommented Jul 19, 2022 at 20:26
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$\begingroup$ @chepner FWIW, the Sun's Schwarzschild radius is ~2953.250 metres. (The Earth's is only 8.870056 mm). The mean Sun : Earth-Moon barycentre distance is just under 455 km. $\endgroup$– PM 2RingCommented Jul 20, 2022 at 1:17
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Probably not, but only because the black hole is so small, compared to the galaxy.
It is a bit hard to define the exact boundaries of the galaxy. Unlike the solar system, which is composed of a small number of particle-like bodies, most of the mass of the galaxy is in the form of a vast invisible cloud of dark matter that forms a "halo". The visible disc of the galaxy sits within this halo. The dark matter halo doesn't have sharply defined edge, and interacts with the mass of nearby galaxies, like the Magellanic clouds and even Andromeda galaxy.
With this in mind, and so not knowing the exact distribution of mass in the galaxy, or even where the galaxy ends, defining the "barycentre" is rather problematic, and locating it is even more so. Dynamic friction will cause supermassive blackholes to migrate towards the centre, but with such a large amount of mass, it is unlikely that the barycentre (however it is defined) falls right at the location of the black hole, because the black hole is so small compared to the rest of the galaxy.
For a two body problem, the barycentre is very important in understanding orbits, as both bodies will move in ellipses with the barycentre at the focus. For systems with more than two bodies, the barycentre is of less importance, as the orbits of particles will not be elliptical, and the barycentre doesn't have such a significant role in the description of the kinematics of the system.
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3$\begingroup$ I'd be very interested in your comments about problems with the geometry in "computing" a barycenter that might be inside a black hole... Thinking that barycenter is a sort of flat-Euclidean idea, to begin with... But I am very naive about these things, physics-wise. Math-wise, I'm ok. :) $\endgroup$ Commented Jul 18, 2022 at 23:17