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
0
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

The rotation velocity is written by the gravitational potential as.

V = √ R ∂ϕ / ∂ R

Explanation:

The rotation velocity is written by the gravitational potential as.

V = √ R ∂ϕ / ∂ R

Where ϕ = Σϕi

ϕ is the potential of the "i" component of the mass.

We know that  

Vi (R) = R ∂ Φi / ∂ R

V(R) = √ΣVi^2

For multiple components.

VR = √ V(BH)R^2 + Vb (R^2) + Vd (R)^2 + Vh (r)^2

The circular velocity is given by

Vb (R) = √GM R / R