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If it's lack interaction keeps dark matter from behaving like matter and ending up distributed like matter. Then why did it gather at the galactic gravitational well in the first place and why end up with a distribution that causes constant rotational speeds. Does this constant rotational speeds increase the likelihood of stable solar system formation. Would a galactic rotation profile like out solar system or Saturn's rings cause such frequent solar interaction and mean time between interactions would be so low a stable planetary system would rarely occur and given enough time never occur.

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Dark matter does not interact electromagnetically. It does interact gravitationally. That would lead to it being distributed in a large, nearly spherical halo in which our Galaxy is embedded. This leads to a constant rotation velocity with radius about the Galactic centre.

Any distribution of dark matter which consists of a very large quantity of gravitating matter distributed on much larger scales than the visible (baryonic) matter would result in a nearly constant rotation velocity with radius.

I think I understand the last part of your question. The lack of differential rotation with radius has very little bearing on interaction between stars. Once dispersed from their denser birth sites, the rate of close gravitational interactions between star systems is very low. But this is not because they all move in circular orbits at the same speed. Even at the same galactocentric radius stars have a dispersion in their kinematics of a few tens of km/s in each direction (about 10-20% of the orbital speed) due to perturabtions of their orbits and the fact that their orbits are in general non-circular.

Even if there were significant differential rotation with radius it would still be the case that stars at a particular radius would all orbit with roughly the same speed with a similar dispersion about this average.

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