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Where's the invisible matter?

Classical Mechanics Level 4

In a region of space are a bunch of stars in circular orbits at different radii and some unknown amount of non-visible matter. You observe the speeds of the stars and notice that the tangential velocity of each star is a constant \(v_0\), independent of radius. If the non-visible matter is distributed spherically symmetrically around the center of the galaxy, what is the ratio of the density of the non-visible matter at some distance 10R to the density at some distance R?

Details and assumptions

  • You may assume the total mass of the stars is negligible compared to the mass of the non-visible matter.

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