Dark Matter — Question 2

Question 2

Differentiating a flat-curve mass profile yields an inverse-square density

For an ideal spherical halo with constant circular speed v0, use $M(<r)=v_0^2r/G$ and $dM/dr=4\pi r^2\rho(r)$.

1. Derive the density profile.

2. Find the density ratio at radii 10 and 30 kpc and explain why the model needs inner and outer limits.

Question 2 - Solution

Differentiating a flat-curve mass profile yields an inverse-square density

(a) Method and calculation

$$4\pi r^2\rho(r)=d(v_0^2r/G)/dr=v_0^2/G.$$$$\boxed{\rho(r)=v_0^2/(4\pi Gr^2)}.$$

(b) Interpretation and further calculation

$$\boxed{\rho(30)/\rho(10)=(10/30)^2=1/9}.$$$$\rho\to\infty\ (r\to0),\quad M(<r)\to\infty\ (r\to\infty).$$

Physical check

The ideal profile is a local description over the observed flat part, not a globally physical halo with finite total mass.