Question

Suppose that a spiral galaxy is located at the center of a spherically symmetric dark matter halo

Answer 1

To solve the problem it is necessary to use the concepts of Orbital Speed considering its density, and its angular displacement.

In general terms the Orbital speed is described as,

$V_{orbit} = \sqrt{\frac{G\rho 4\pi r^3}{3}}$

PART A) If the orbital speed of a star in this galaxy is constant at any radius, then,

$\frac{4\pi G\rho r}{3} = \frac{v^2}{r}$

$\frac{4\pi G\rho r}{1} = \frac{3v^2}{r}$

$\frac{\rho}{1} = \frac{3v^2}{r^2 4\pi G}$

$\rho = \frac{1}{r^2}$

PART B) This time we have$v=\omega t$, where $\omega$ is the angular velocity (constant at this case)

$\frac{4\pi G\rho r}{3} = \frac{v^2}{r}$

$\frac{4\pi G\rho r}{3} = \frac{(\omega r)^2}{r}$

$\rho = \frac{3\omega r}{4\pi Gr}$

$\rho = \frac{3\omega^2}{4\pi G} \propto constant$

PART C) If the total mass interior to any radius r is a constant,

$\frac{4\pi G\rho r}{3} = \frac{v^2}{r}$

$\frac{GM}{r^2}=\frac{v^2}{r}$

$v = \sqrt{\frac{GM}{r}}$

$v= \sqrt{\frac{1}{r}}$