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.

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations, examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.