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How are galaxies and solar systems similar?

Is it possible that galaxies are nothing more than supermassive proto-planetery discs (with the "bulge" at the center being the beginning of a "sun")? If this might be the case, and it is theorized that we have a black hole at the center of our galaxy, can it also be the case that we have a mini black hole at the center of our sun?

As you mention, there are similarities between spiral galaxies and protoplanetary disks, namely their disk-like appearance. The physical process behind the disk formation is the same in both cases: both the matter that collapses to form a galaxy and that which collapses to form a star or planet must conserve angular momentum. In doing so, disks are formed. In fact, the formation of disks is a ubiquitous feature of astronomical objects: rings around a planet, accretion disks around black holes, protoplanetary disks and spiral galaxy disks are all examples of angular momentum in action.

Unfortunately, this similarity does not imply that there is a mini-black hole at the centre of our Sun. Black holes can be regarded as regions of spacetime with infinite density. Our best solar models, however, indicate that the central density of the Sun is around 150 grams per centimetre cubed, which is far from infinite. Moreover, a black hole smaller than the Sun would exert strong tidal forces on the solar material surrounding it (it turns out that the smaller the black holes, the stronger the tidal forces). We have no evidence of disturbed material on that scale. Finally, our understanding of the formation mechanisms for black holes does not include a means for miniature black holes to form at the centres of stars - such black holes would have to be remnants of the Big Bang (or formed by mechanisms which we haven't yet predicted), in which case a new theory of star formation would have to be developed.

All things considered, it is highly unlikely that there is a miniature black hole in the centre of the Sun.

Astronomers do think, however, that there may be a supermassive black hole at the centre of most galaxies (by "supermassive", they mean millions of times more massive than the Sun). These supermassive black holes may have played a significant role in galaxy formation, and serve to stabilize the galactic disk.

July 2002, Kristine Spekkens (more by Kristine Spekkens) (Like this Answer)

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