Problem 23 - An Introduction to Modern Astrophysics

Question

Using Eq. (52) for the density profile of the dark matter halo, show that (a) \(\rho_{\mathrm{NFW}} \propto r^{-1}\) for \(r \ll a\) and \(\rho_{\mathrm{NFW}} \propto r^{-3}\) for \(r \gg a\). (b) the integral of the mass from \(r=0\) to \(r \rightarrow \infty\) is infinite. $$\rho_{\mathrm{NFW}}(r)=\frac{\rho_{0}}{(r / a)(1+r / a)^{2}}$$

Step-by-Step Solution

Answer

(a) \(\rho_{\mathrm{NFW}} \propto r^{-1}\) for \(r \ll a\), and \(\rho_{\mathrm{NFW}} \propto r^{-3}\) for \(r \gg a\). (b) The total mass is infinite.

Step 1: Analyze the formula for small r

Consider the given density profile:\[ \rho_{\mathrm{NFW}}(r) = \frac{\rho_{0}}{(r / a)(1+r / a)^{2}} \]For \( r \ll a \), \( r / a \) is very small, so the expression \((1 + r/a)^2\) can be approximated by 1. Therefore, the equation simplifies to:\[ \rho_{\mathrm{NFW}}(r) \approx \frac{\rho_{0}}{r/a} = \frac{\rho_{0} a}{r} \]This shows that \( \rho_{\mathrm{NFW}} \propto r^{-1} \) for \( r \ll a \).

Step 2: Analyze the formula for large r

For \( r \gg a \), i.e., \( r / a \) is large, we focus on the behavior \((1 + r/a)^2 \approx (r/a)^2\). Now rewrite the original formula:\[ \rho_{\mathrm{NFW}}(r) = \frac{\rho_{0}}{(r/a)} \left(\frac{1}{(r/a)^2}\right) = \frac{\rho_{0} a^{3}}{r^{3}} \]This reveals that \( \rho_{\mathrm{NFW}} \propto r^{-3} \) for \( r \gg a \).

Step 3: Setup the integral to find the mass

The mass enclosed in the halo up to a radius \( r \) is given by:\[ M(r) = 4\pi \int_{0}^{r} \rho_{\mathrm{NFW}}(r') r'^{2} \, dr' \]We need to investigate the behavior of the integral as \( r \rightarrow \infty \).

Step 4: Evaluate the mass integral for small r

For \( r \ll a \), we found \( \rho_{\mathrm{NFW}} \propto r^{-1} \), leading to the integral:\[ M(r) \approx 4 \pi \rho_{0} a \int_{0}^{r} r' \, dr' \]This integral simplifies to \( M(r) \approx 2\pi \rho_{0} a r^{2} \), which is finite for any finite \( r \).

Step 5: Evaluate the mass integral for large r

For \( r \gg a \), we had \( \rho_{\mathrm{NFW}} \propto r^{-3} \). The integral of interest then is:\[ M(r) \approx 4 \pi \rho_{0} a^{3} \int_{r}^{\infty} r'^{-1} \, dr' \]This integral diverges as it represents \( \ln(r) \) evaluated as \( r \rightarrow \infty \), thus showing that the mass is infinite.

Key Concepts

NFW Density Profile

The NFW (Navarro-Frenk-White) density profile is a mathematical model used to describe the distribution of dark matter within a halo. This profile is characterized by its radial dependence on the distance from the center of the halo. The formula is given by: \[ \rho_{\mathrm{NFW}}(r) = \frac{\rho_{0}}{(r / a)(1+r / a)^{2}} \]where:- \( \rho_{0} \) represents a characteristic density,- \( r \) is the radial distance from the center of the halo,- \( a \) is a scale radius that marks the transition between different behavior regimes in the profile.
For small distances \( r \ll a \), the profile approximates to \( \rho_{NFW} \propto r^{-1} \), indicating a denser core. For large distances \( r \gg a \), it simplifies to \( \rho_{NFW} \propto r^{-3} \), suggesting that density decreases rapidly further out.
These approximations provide insight into the structure of dark matter haloes, and they fit well with observational data from galaxies.

Density Distribution

The density distribution of a dark matter halo describes how dark matter is spread out in space. In the context of the NFW profile, this distribution changes based on how far you are from the center.

• In the central region (\( r \ll a \)), the density is high, following an inverse relationship with radius \( \rho \propto r^{-1} \).
• In contrast, in the outer regions (\( r \gg a \)), the density decreases more sharply \( \rho \propto r^{-3} \).

This variation tells us a lot about the gravitational effects that dark matter can have at different scales within the halo. Since dark matter does not emit light, its distribution is crucial to our understanding of how galaxies form and evolve over time.
By analyzing these distribution patterns, astronomers can infer critical properties of the halo, such as total mass and the gravitational pull exerted on visible matter like stars.

Astrophysical Integrals

Astrophysical integrals are tools used to calculate quantities like mass or luminosity within a given volume of space. For the NFW density profile, one common task is to integrate the density over a volume to find the total mass enclosed within a radius \( r \).
The formula to calculate the mass enclosed in a halo is:\[ M(r) = 4\pi \int_{0}^{r} \rho_{\mathrm{NFW}}(r') \, r'^{2} \, dr' \]
For small \( r \), the density \( \rho_{\mathrm{NFW}} \propto r^{-1} \), which leads to a finite mass when integrated. However, the situation changes for large \( r \) where \( \rho_{\mathrm{NFW}} \propto r^{-3} \). When this is integrated from \( r \) to infinity, it results in a diverging function, indicating an infinite mass contribution from the outer parts. This suggests that in reality, an NFW halo might need to be truncated or modified at large scales to account for this infinity in practical calculations.
Understanding and solving these integrals is key to making accurate models of how dark matter behaves in the universe.