Question

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

(a) If the orbital speed of a star in this galaxy, v, is constant at any radius, how does the mass density rho(r) within the halo vary with radius r?

(b) Suppose the rotation curve is fitted by a single relation v x r, answer the same question as in part (a).

(c) If the total mass interior to any radius r is a constant, M(r) M, as if it is dominated by a central point mass, find out how the orbital speed v of a stair varies with raditus r.

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{(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,

`(4\pi G\rho r)/(3) = (v^2)/(r)`

`(4\pi G\rho r)/(1) = (3v^2)/(r)`

`(\rho)/(1) = (3v^2)/(r^2 4\pi G)`

`\rho = (1)/(r^2)`

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

`(4\pi G\rho r)/(3) = (v^2)/(r)`

`(4\pi G\rho r)/(3) = ((\omega r)^2)/(r)`

`\rho = (3\omega r)/(4\pi Gr)`

`\rho = (3\omega^2)/(4\pi G) \propto constant`

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

`(4\pi G\rho r)/(3) = (v^2)/(r)`

`(GM)/(r^2)=(v^2)/(r)`

`v = \sqrt{(GM)/(r)}`

`v= \sqrt{(1)/(r)}`