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{"ops":[{"insert":{"image":"a3900b3c17fa930fb71a5c6c43a6fdd8"}},{"insert":"Because the mass outside this inner region is almost constant, the density distribution can be\nwritten as following (assume a flat Milky Way with height z0):\n\u03c1(r) = (\n\u03c10, r \u2264 R0\n0, r > R0\n(a) Derive an expression for the mass M(r) enclosed within the radius r.\n(b) Derive the expected rotational velocity of the Milky Way v(r) at a radius r.\n(c) Astronomical observations indicate that the rotational velocity follows a dierent behaviour:\nvobs(r) = p\nG\u03c0\u03c10z0R0\n\n5/2\n1 + e\n\u22124r/R0\n0\n5\n4\n\nDraw the expected and observed rotational velocity into the plot below:\n(d) Scientists believe the reasons for the dierence to be dark matter: Determine the rotational\nvelocity due to dark matter vDM (r) from R0 and draw it into the plot above.\n(e) Derive the dark matter mass MDM (r) enclosed in r and explain its distributed.\n(f) Explain briefly three theories that provide explanations for dark matter.\n"}]}

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