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Advanced PhysicsQ&A LibraryThe most mass of our Milky Way is contained in an inner region close to the core with radius R0. 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.Start your trial now! First week only $4.99!*arrow_forward*

Question

The most mass of our Milky Way is contained in an inner region close to the core with radius R0.

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.

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