Radial Dependency of Navarro-Frenk-White profile

[spaceid]

Homework Statement

Determine the radial dependence of gas density for an isothermal cluster of galaxies with temperature T that has a total mass density profile given by the Navarro, Frenk, & White form:

ρ(r) = ρcδc , (r/rs)(1 + r/rs)2

where ρc is the critical density, rs is a characteristic radius and δc is a dimensionless parameter. You should assume that the cluster is in hydrostatic equilibrium.

Homework Equations

ρc = 3H2/8πG

The Attempt at a Solution

I am a bit confused at how to start this problem. I am thinking that I need to take an integral where the bounds refer to the size of the radius of the cluster. However, I am not sure where to take this integral. I thought maybe I should use ρ=m/V and do m = ∫ 4∏r^2ρ(r) dr, but I am not sure why I would do that. It is more of a guess to get myself started. I am also confused where T comes in since it is mentioned in the problem. Any hints to get me started in the right direction wold be very much appreciated.

 

[phyzguy]

Think of a shell of gas at radius r. Gravity is pulling the shell inward, and pressure is pushing it outward. Since the shell is in hydrostatic equilibrium, these two forces must balance. Since you know rho as a function of r, you can compute the mass inside r as a function of r, which should allow you to compute the gravitational force pulling the shell inward. From this, you can compute the pressure gradient which you can integrate to find the pressure as a function of r. Since you know the pressure and the density, you should be able to compute the temperature.