Metal-Diffusion-in-Galaxy-clusters

Simulation of metal diffusion in the ICM for the Perseus Cluster based on the paper 'Impact of stochastic gas motions on galaxy cluster abundance profiles' by Rebusco et., al 2005. The aim of the project is to show the possible parameters for efficient metal distribution under reasonable diffusion timescales(reproduce observed abundance profile).

Introduction to the relevant astrophysical scenario

Galaxy clusters are the most massive virialized systems in the universe. they consist of $10^2 - 10^3$ galaxies which orbit inside the gravitational well of the gas and Dark matter halo which takes up more then $80 %$ of the total mass of the system. Given that the total mass is up to $10^{15} M_\odot$, the velocity dispersion of galaxies within the cluster is of the order of $\sigma \sim 10^3 km/s$ and the gravitational radius ranges from 1 to 3 Mpc. The aim of this project is the creation of a reliable model which can numerically recreate the observational data related to the iron abundance profile in the Perseus Cluster. Iron is injected from the BCG by supernova activity and stellar winds and subsequently diffused by stochastic gas motions in the ICM. In order to archive such goal the model is modified accordingly and its parameters vary to generate a progressively more realistic scenario. Figure 1 shows the deprojected iron abundance of the Perseus cluster extracted from deep XMM-Newton data, which is expressed as:

$$ Z_{Fe}(r)=0.3 \frac{2.2 + \left( \frac{r_{kpc}}{80}\right)^3}{1+ \left(\frac{r_{kpc}}{80}\right)^3} Z_{Fe\odot}$$

in the code it is multiplied by a factor 1.4 and the background abundance $Z_{Fe,out}=0.4 ,Z_{Fe\odot}$ is subtracted from all of the plots.

Hydrostatic equilibrium

The first requirement for the model is a condition of hydrostatic equilibrium:

$$ \frac{dP}{dr}=-\frac{GM(r)}{r^2}\rho(r)$$

The term $M(r)$ represents the mass of the cluster within the radius r, given that we assume the structure to be spherical; P and $\rho$ are the pressure and density respectively. This equation is a first order ODE that we will need to integrate numerically to obtain the density profile of the gas within the cluster.

Staggered grid

In order to integrate the Hydrostatic equilibrium equation we first need to discretize the radial distance $r$ from the center of the cluster, in fact we are considering the system to be in a spherical symmetry reducing the spacial dimensions of the problem from 3 to 1. I use a system of staggered grids. The first grid is defined as

$$ r_j=r_{min}+(j-1)\frac{r_{max}}{j_{max}-1}$$

where $r_{min}=0kpc$ and $r_{max}=3000kpc$ (just above the virial radius) while $j_{max}$ is the parameter that sets the smallest possible scale $\Delta r$ of the system. I decided to set $j_{max}=5000$. I used a second, superimposed grid in the middle of the first one

$$ r_{j+\frac{1}{2}}=r_j+\frac{1}{2}(r_{j+1}-r_j)$$

as to set an average value for each shell of volume

$$ \Delta V_{j+\frac{1}{2}}=\frac{4}{3}\pi (r_{j+1}-r_j)^3$$

The values of the density will be computed in this second grid. This is useful for numerical methods, density values are defined at the center($r_{j+\frac{1}{2}}$ grid) of small 'control volumes' ($r_j$ grid) and are interpreted as the mean value inside that cell. This 'more physical' way of thinking actually provides for better approximations of the PDEs.

Dark matter

To integrate the Hydrostatic equilibrium equation we need a mass profile $M(r)$ that is generating the gravitational potential. Since the cluster's mass is dominated by the dark matter halo contribution, as a first approximation DM is the only component generating the gravitational potential. We therefore consider the density profile for DM to be that of Navarro-Frenk-White (NFW profile):

$$ \rho_{DM}(r)=\frac{\rho_{DM,0}}{(\frac{r}{r_s})(1+\frac{r}{r_s})^2}$$

where the central dark matter density is $\rho_{DM,0}=7.35\cdot 10^{-26} g/cm^3$ and the scale radius is $r_s=435.7 kpc$. The dark matter mass value for each shell is then given by

$$ \Delta M_{DM,j+\frac{1}{2}}=\rho_{j+\frac{1}{2}} \cdot \Delta V_{j+\frac{1}{2}}$$

which can be compared to the analytical solution

$$ M_{DM}(r)=4\pi \rho_{DM,0} r_s^3 \left[ ln \left( 1+\frac{r}{r_s} \right) -\frac{r}{r+r_s}\right]$$

Gas density profile

Once we have $M_{DM}(r)$ we can integrate the Hydrostatic equilibrium equation to obtain the density profile of the intra cluster medium gas. For an ideal gas the pressure can be expressed as

$$ P=\frac{k_b\rho T}{\mu m_p}$$

where $k_b$ is the Boltzmann constant, $\mu$ is the mean molecular weight and $m_p$ is the mass of the proton. Given that the gas is at constant temperature ($T=8.9 \cdot 10^7 K$ ) we can insert the Equation of state in equation 1 to obtain an expression for the gas density

$$ \frac{dln\rho_g}{dr}=-\frac{G M(r)}{r^2}\frac{\mu m_p}{k_b T}.$$

such ODE can be solved as an FDE - Finite Differences Equation:

$$ ln \rho_{j+\frac{1}{2}}=ln \rho_{j-\frac{1}{2}} -\Delta r \frac{G M_j}{r_j^2}\frac{\mu m_p}{k_b T}$$

setting the initial value for $\rho_j$ becomes extremely important in order to reach the cosmological barion fraction of $f_b \sim 0.16$ at the virial radius, where

$$ f_b=\frac{M_{gas}(r)}{M_{gas}(r)+M_{DM}(r)}.$$

In this first approximation I set:

$$ \rho_0=4.46\cdot 10^{-26} g/cm^3. $$

In order to get a barion fraction of $f_b \sim 0.16$.

The green line in the plot above stands for the corresponding analytical solution:

$$ \rho_g(r)=\rho_0 e^{-27b/2} \left(1+\frac{r}{r_s} \right)^{27b/(2r/r_s)}$$

where $b=\frac{8\pi G \mu m_p \rho_{DM,0} r_s^2}{27k_b T}$ .

addition of a BCG and a temperature gradient

I add the density contribution of a central elliptical galaxy whose stellar mass profile follows the \textbf{1990 Hernquist profile} defined as

$$ M_\ast (r)=M_{BCG}\frac{r^2}{(r+a)^2}$$

where $(1+\sqrt{2})a=12 kpc$ and $M_{BCG}= 10^{12} M_{\odot}$ is the Mass of the brightest cluster galaxy(BCG) which is thought to have the highest impact when it comes to the stellar mass of the cluster.

In an effort to upgrade the model to a more realistic version I introduced the following temperature profile:

$$ \frac{T(r)}{T_{mg}}=1.35\frac{(\frac{x}{0.045})^{1.9}+0.45}{(\frac{x}{0.045})^{1.9}+1}\frac{1}{(1+(\frac{x}{0.6})^2)^{0.45}}$$

where $x=\frac{r}{r_{500}}$ , $r_{500} \sim 1.4 Mpc$ and $T_{mg}=8.9\cdot 10^7 K$ are normalization constants. Since now temperature is no more constant we need to account for it in the Hydrostatic equilibrium equation which becomes

$$ ln \rho_{j+\frac{1}{2}}=ln \rho_{j-\frac{1}{2}} -\Delta r \frac{G M_j}{r_j^2}\frac{\mu m_p}{k_b T_{j+\frac{1}{2}}}-(lnT_{j+1}-lnT_{j-1})$$

I set the initial density :

$$ \rho_0=2.07 \cdot 10^{-25}$$

in order to reach the cosmological barion fraction $f_b \sim 0.16$. The overall trend for the initial gas density given the baryon fraction condition throughout all the possible cases is an increase; this is due to the addition of a central object, which causes an intrinsic baryon fraction decrease, and a temperature gradient which, in virtue of it being lower in the central regions, requires a higher initial gas density.

The temperature and density profiles that are being used are in accordance to those presented in the Rebusco et. al (2005) paper which are the ones based on the deprojected XMM-Newton data.

Diffusion of metals in the ICM

Peaked abundance profiles are a characteristic feature of clusters with cool cores and abundance peaks are likely associated with the brightest cluster galaxies which dwell in cluster cores. The width of the abundance peaks is however significantly broader than the BCG light distribution, suggesting that some gas motions are transporting metals originating from within the BCG.

This model simulates the metal diffusion that occurs in a cool core cluster, more specifically the Perseus cluster, for which the observed iron abundance profile is featured in equation 1. To compute the diffusion of metals in the ICM we apply the standard diffusion equation:

$$ \frac{\partial f}{\partial t}=D \frac{\partial^2 f}{\partial x^2}$$

to the iron density profile $\rho_{Fe}(r)$ , which in radial coordinates becomes:

$$ \frac{\partial \rho_{Fe}}{\partial t}=\frac{1}{1.4} \frac{1}{r^2}\frac{\partial(r^2 D \rho \frac{\partial Z_{Fe}}{\partial r})}{\partial r} +S_{Fe}(r,t)$$

Where $D\sim v_t l_t/3$ is the diffusion coefficient, $v_t$ and $l_t$ are the turbulent velocity and turbulent length respectively, D can be assumed to be constant (in the code, $D=1.3 \cdot 10^{-29} cm^2/s$). Also $Z_{Fe}=1.4\frac{\rho_{Fe}}{\rho}$ and $S_{Fe}(r,t)$ is the source term accounting for the injection of metals in the system.

For the analysis of this problem I focused on three different scenarios, which were accounted for in three separate portions of the code:

• Diffusion term only, where I neglected the source term and solely focused on the diffusion PDE at hand and its effects on an initial iron abundance;
• Source term only, where I only took into account the injection of metals caused by supernova events and stellar winds;
• Diffusion $+$ Source which constitutes the most realistic scenario out of the three; I will later provide some extra considerations about the implications of different choices for the supernova metal injection.

The time integration requires boundary conditions at the extreme of the grid: $\rho_{Fe}(1)=\rho_{Fe}(2)$, $\rho_{Fe}(jmax)=\rho_{Fe}(jmax-1)$ . The time step $\Delta t$ is chosen specifically as to not incur into instability issues; the condition for stability of the method used (FTCS) is $\Delta t \leq \frac{\Delta x^2}{2D}$ which in the code:

$$ \Delta t=C\cdot \left(\frac{(r(5)-r(4))^2}{2D} \right)$$

where C's value is set at 0.4 and the values for computing $\Delta x$ are chosen arbitrarily since the spacial grid is uniform.

Diffusion only term

To solve the diffusion equation, a Forward-Time Centered-Space (FTCS) method is used. This method is stable for diffusion equations because the dominant behaviour in diffusion processes is governed by the second-order spatial derivative, which tends to smooth out fluctuations. without explicitly describing its derivation, the final initial-value problem looks like:

$$ \rho_{Fe,j+\frac{1}{2}}^{n+1}=\rho_{Fe,j+\frac{1}{2}}^n+\frac{\Delta t}{1.4} \frac{ \left[(r^2 D \rho_{j+1} \nabla Z_{Fe,j+1}) - (r^2 D \rho_j \nabla Z_{Fe,j})\right] }{r_{j+1}^3-r_j^3} $$

The j index stands for a value on the spatial grid, while n indicates a value on a time grid. The gradient of the iron abundance is defined as follows:

$$ \nabla Z_{Fe,j}^n=\frac{\nabla Z_{Fe,j+\frac{1}{2}}^n - \nabla Z_{Fe,j-\frac{1}{2}}^n}{r_{j+\frac{1}{2}}-r_{j-\frac{1}{2}}}$$

The equation is then integrated; the initial values for $Z_{Fe}$ are given by equation 1, while the initial values for $\rho$ have already been computed in the previous section. $\rho_{Fe}$ is converted into $Z_{Fe}$ and a background value of $0.4 Z_{Fe,\odot}$ is subtracted from it to isolate the peak generated by supernova injection. The metallicity gradient can be computed over any time span, of which three were chosen: 1, 2, 5 $Gyr$.

For the computation of the diffusion time scale I chose it to be the time needed to halve the iron mass at 80 kpc, which was found to be 4.8 Gyr. This value was found by integrating the system for 8 Gyr and an IF statement checks whether the iron mass up to 80 kpc in the current time cycle has reached half of the initial mass.

Source only term

I introduced a source term given by stellar mass loss and supernova activity:

$$ S_{Fe}(r)=\rho_\ast(r) \left[\alpha_\ast (t)\frac{Z_{Fe}(r)}{1.4} +\alpha_{1a}(t) \frac{M_{Fe,1a}}{M_{1a}}\right]$$

where $Z_{Fe}(r)$ is the stellar iron abundance which I set at solar metallicity as an initial condition. In this equation the two parameters $\alpha_\ast$ and $\alpha_{1a}$ describe stellar mass loss and supernova activity rate. For now I neglect the time dependency of these 2 parameters setting them at constant values. $\alpha_\ast$ depends on the evolution of single stellar populations and i set it at $\sim 4.7 \cdot 10^{-20}$. $\alpha_{1a}$ is more complicated to determine but it is dependent on the rate of supernova type 1a that are happening in the BCG; we relate such dependency via the supernova unit, SNu, the full treatment of how SNu is derived is omitted and for now it is set at $SNu = 0.15$. The value of $\alpha_{1a}= 5.91\cdot 10^{-21} \cdot SNu$. Finally $\rho_\ast$ is the Hernquist density profile which is the density contribution of stars in the BCG:

$$ \rho_\ast(r)=\frac{M_{BCG}}{2\pi}\frac{a}{r}\frac{1}{(r+a)^3}.$$

We re-integrate the diffusion equation this time only considering the source term, starting from a null initial metal abundance.

Diffusion $+$ Source terms

In order to best reproduce the observed Fe-abundance profile taken from Rebusco et al. (2005) (which is the Perseus cluster) to the best accuracy possible, both the source term and the diffusion term were numerically integrated in the code. During every time step Fe is produced and subsequently distributed into the ICM.

The combination of the two different terms, starting from a null metal abundance and performed over the usual three time spans, returns Figure 8. The observed data (red dashed line) is not adequately recreated by the model.

Changes to the parameters

In order to get closer to the best possible model one has to tweak with its parameters. To reproduce the observed slope I try to increase SNu to $SNu=0.5$. The results for this model can be seen if figure 9.

Altough the slope of the observed profile is well reproduced, the metal is not diffused properly. As a final attempt we implement in our model, time varying parameters and a greater diffusion coefficient $D=2.4\cdot 10^{29}$ :

$$ \alpha_\ast(t)=4.7\cdot 10^{-20}\left(\frac{t}{t_{now}}\right)^{-1.26}$$

and:

$$ \alpha_{1a}(t)=4.436\cdot 10^{-20}\left(\frac{SNu(t)}{7.5}\right)\left(\frac{t}{t_{now}}\right)^{-s}$$

where the time dependent supernova unit is

$$ SNu(t)=0.7 \left( \frac{t}{t_{now}}\right)^{-s}$$

where we set the slope $s=1.1$. Estimates of the slope are not precise values as the time dependency of supernova rate is a poorly understood subject. By using these parameters equation 18 is integrated for a time of 8 Gyr, the resulting abundance profile is shown in Figure 10. The Observed profile is well reproduced with a high central peak.

An excessive tweaking of the parameters might return unphysical results as not much is known with respect to the SN rate. As suggested in the Rebusco et. al (2005) paper AGN driven outflows in the center of the BCG could explain diffusion coefficient of the order of $D \sim 2 \cdot 10^{29} cm^2/s$ but many uncertainties are present with regards to the age of the system and the rate of metal ejection by SNe.