Answer to Question #122902 in Astronomy | Astrophysics for SODIQ ADIO KAREEM

Question

Because the mass outside this inner region is almost constant, the density distribution can be

written as following (assume a flat Milky Way with height z0):

ρ(r) = (

ρ0, r ≤ R0

0, r > R0

(a) Derive an expression for the mass M(r) enclosed within the radius r.

(b) Derive the expected rotational velocity of the Milky Way v(r) at a radius r.

(c) Astronomical observations indicate that the rotational velocity follows a dierent behaviour:

vobs(r) = p

Gπρ0z0R0

5/2

1 + e

−4r/R0

5

4

Draw the expected and observed rotational velocity into the plot below:

(d) Scientists believe the reasons for the dierence to be dark matter: Determine the rotational

velocity due to dark matter vDM (r) from R0 and draw it into the plot above.

(e) Derive the dark matter mass MDM (r) enclosed in r and explain its distributed.

(f) Explain briefly three theories that provide explanations for dark matter.

Answer 1

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