Question

5. According to current theory of galaxy disk formation, galaxy disks are

   embedded in extended dark matter halos whose density profiles can

   roughly be written as ρ(r) = V_vir 2 /(4πGr2), where V_vir is a constant.

(1) What is the total mass of the halo within a radius R, M(< R)?

(2) What is the rotation curve, Vc(R), of the disk embedded in such a

halo (assuming disk mass can be neglected)?

(3) What is the mean mass density of the halo within radius R?

(4) Suppose the virial radius of a halo, Rvir, is defined so that the mean

density within Rvir is a constant, ρ. How does Rvir changes with Vvir?

(5) If disk luminosity, Ld, is proportional to M(< Rvir), show that the 3

Tully-Fisher relation is Ld ∝ Vmax.

Answer 1

given
density profile, rho(r) = Vvir^2 /4*pi*Gr^2

1. for a disc of radius R
mass inside an annulus of radius r < R is dM
dM = rho(r)*2*pi*r*dr
hence
integrating this we get
integral(dM) = integral Vvir^2 *2*pi*r*dr/ 4*pi*G*r^2
M = Vvir^2 * ln(R/Ro)/ 2G
where Ro is inner radius of the halo

2. for rotation curve
from force balance at radius r
(dM)v^2/r = G*M(r)*dM/r^2
here dM is mass of annulus at radius r and v is tangential velocity of rotation of the galaxy
also, M(r) is mass of disc enclosed in radius r
hence
for velocity curve
v/r has to be found

(dM)v^2/r = G*M(r)*dM/r^2
v^2 = G*M(r)/r

also
M(r) = Vvir^2 * ln(r/Ro)/ 2G
hence
v^2*r = G*(Vvir^2 * ln(r/Ro)/ 2G)
v^2/r^2 = (Vvir^2 * ln(r/Ro))/2r^3
v/r = [sqrt([Vvir^2 * ln(r/Ro)]/2r)]/r

3. mean mass density = rho
rho = M/4*pi*(R^2 - Ro)^2 = Vvir^2 * ln(R/Ro)/ 8G*pi*(R^2 - Ro^2)
where Ro is inner radius of halo or
if rho'is central density of halo, then
rho'= Vvir^2/4*pi*G*Ro^2

d. virial radius of halo, Rvir such that mean density is constant within the virial radius
hence, on increasing Vvir, the density profile of the halo changes, and for the mean density to remain the same, Rvir has to increase accordingly to keep rho'constant