Problem B.4: Distribution of Dark Matter The most mass of our Milky Way is contained in an inner region close to the core with radius Ro. 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 zo): p(r) = 0, r> Ro (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 different behaviour: 5/2 1+e-dr/Ro Draw the expected and observed rotational velocity into the plot below: 2.0 1.5 1.0 0.5 0.0 2 4 6 10 Radius from Center r [Ra) (d) Scientists believe the reasons for the difference to be dark matter: Determine the rotational velocity due to dark matter upar (r) from Ro and draw it into the plot above. (e) Derive the dark matter mass MpM (r) enclosed in r and explain its distributed. (f) Explain briefly three theories that provide explanations for dark matter. Rotational Velocity v ly Gap1
Problem B.4: Distribution of Dark Matter The most mass of our Milky Way is contained in an inner region close to the core with radius Ro. 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 zo): p(r) = 0, r> Ro (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 different behaviour: 5/2 1+e-dr/Ro Draw the expected and observed rotational velocity into the plot below: 2.0 1.5 1.0 0.5 0.0 2 4 6 10 Radius from Center r [Ra) (d) Scientists believe the reasons for the difference to be dark matter: Determine the rotational velocity due to dark matter upar (r) from Ro and draw it into the plot above. (e) Derive the dark matter mass MpM (r) enclosed in r and explain its distributed. (f) Explain briefly three theories that provide explanations for dark matter. Rotational Velocity v ly Gap1
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Distribution of Dark matter
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.
Transcribed Image Text:Problem B.4: Distribution of Dark Matter
The most mass of our Milky Way is contained in an inner region close to the core with radius Ro.
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 zo):
p(r) =
0, r> Ro
(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 different behaviour:
5/2
1+e-dr/Ro
Draw the expected and observed rotational velocity into the plot below:
2.0
1.5
1.0
0.5
0.0
2
4
6
10
Radius from Center r [Ra)
(d) Scientists believe the reasons for the difference to be dark matter: Determine the rotational
velocity due to dark matter upar (r) from Ro and draw it into the plot above.
(e) Derive the dark matter mass MpM (r) enclosed in r and explain its distributed.
(f) Explain briefly three theories that provide explanations for dark matter.
Rotational Velocity v ly Gap1
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