Dark Matter — Question 1

Question 1

A flat rotation curve implies increasing enclosed mass

Assume a spherical mass distribution and circular speed 200 km/s at r=20 kpc. Use $G=4.30\times10^{-6}$ kpc (km/s)$^2$ per solar mass.

1. Derive and calculate enclosed mass.

2. If speed remains 200 km/s at 40 kpc, find the added mass between the two radii.

Question 1 - Solution

A flat rotation curve implies increasing enclosed mass

(a) Method and calculation

$$v_c^2/r=GM(<r)/r^2\Rightarrow\boxed{M(<r)=v_c^2r/G}.$$$$\boxed{M(<20)=200^2(20)/(4.30\times10^{-6})=1.86\times10^{11}M_\odot}.$$

(b) Interpretation and further calculation

$$\boxed{M(<40)=3.72\times10^{11}M_\odot}.$$$$\boxed{\Delta M=1.86\times10^{11}M_\odot}.$$

Physical check

A flat speed is inconsistent with a fixed enclosed point mass, which predicts speed proportional to the inverse square root of radius. Disk geometry changes detailed inference.