Radial Dependency of Navarro-Frenk-White profile

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In summary, the problem is to determine the radial dependence of gas density for an isothermal cluster of galaxies with temperature T, using the Navarro, Frenk, & White form for the total mass density profile. This involves computing the mass inside a shell at radius r, and using this to find the gravitational force and pressure gradient, which can be integrated to find the pressure and temperature as functions of r.
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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.
 
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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.
 

1. What is the Navarro-Frenk-White (NFW) profile?

The Navarro-Frenk-White (NFW) profile is a mathematical function commonly used to describe the density distribution of dark matter halos in the universe. It was proposed by Julio Navarro, Carlos Frenk, and Simon White in 1996 and has since become a widely accepted model for the distribution of dark matter in simulations of galaxy formation.

2. What does "Radial Dependency" mean in relation to the NFW profile?

In the context of the NFW profile, radial dependency refers to how the density of dark matter changes as a function of distance from the center of a galaxy or dark matter halo. The NFW profile assumes that the density of dark matter increases towards the center and then decreases towards the outskirts of a galaxy, following a specific mathematical form.

3. How is the NFW profile related to the distribution of dark matter in the universe?

The NFW profile is often used to describe the distribution of dark matter in the universe because it accurately fits the data from simulations of galaxy formation. The profile predicts that the density of dark matter increases towards the center of a galaxy, which is consistent with observations of rotation curves of galaxies.

4. What are the parameters used in the NFW profile?

The NFW profile has two main parameters: the scale radius (rs) and the characteristic density (ρs). The scale radius represents the distance at which the density of dark matter drops to half of its maximum value, while the characteristic density is a measure of the overall density of dark matter in the halo.

5. How does the NFW profile compare to other density profiles?

The NFW profile is just one of many density profiles used to describe the distribution of dark matter. Other commonly used profiles include the Einasto profile, the Burkert profile, and the Hernquist profile. Each profile makes different assumptions about the shape of the dark matter halo and has its own set of parameters. The NFW profile is often preferred because it accurately fits the data and has only two parameters, making it relatively simple to use.

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