# Maurizio Paolillo » 19.Clusters of Galaxies - Part II

### Contents (Lecture prepared in collaboration with Elisabetta de Filippis)

• Galaxy Cluster masses
• The spatial distribution of the Intra Cluster Medium
• Spectroscopy of the Intra Cluster Medium
• Cool cores
• Shock fronts
• Cold fronts

### Cluster Mass

In the previous lecture we have shown that for most clusters we can assume that:

• the ICG is in hydrostatic equilibrium:

(1) $\nabla P_{ICG}=-\rho_{ICG}\nabla \Phi(r)$

• and that the ICG is an ideal gas:

(2) $P_{ICG}=-\frac{\rho_{ICG} k T_{ICG}} {\mu m_p}$

If we make one further assumption, that its distribution is spherically symmetric, Eq.(1) can be written as :

(3) $\frac{1} {\rho_{ICG}}\frac{dP_{ICG}} {dr}=\frac{d\Phi(r)} {dr}-\frac{G M_{cl}k T_{ICG}} {r^2}$

### Cluster Mass (cont.ed)

Combining equations (1), (2) & (3) we obtain:

(4) $M_{tot}(r)=-\frac{k_B T_{ICG}(r)} {\mu m_p G} \left( \frac {d \ln \rho_{ICG}(r)} {d \ln r} + \frac {d \ln T_{ICG}(r)} {d \ln r} \right)$

which represents the total gravitating mass in the cluster within a radius r.

X-ray observations of galaxy clusters allow us to measure:

1. the volume density distribution of intra-cluster gas $\rho_{gas}(r)$ from X-ray observations
2. the temperature radial profile of the ICG TICG (r)

and therefore, from Eq.(4) to obtain an estimate not only of the mass of the intra-cluster gas, but also of the total gravitating mass within the cluster.

### Cluster Mass (cont.ed)

Since we can measure the volume density distribution of intra-cluster gas ρICG(r) form X-ray observations, the mass of the gas will simply be given by its integral over the whole cluster volume:

(5) $M_{gas}(r ) = \int \rho_{gas} dV$

We know that the cluster mass is given mainly by dark matter and by the ICG: Mtot~ MDM+ MICG.
Eqs.(4) and (5) provide us the means to measure both the total mass of the cluster and its gas mass. We can hence also write:

(6) MDM(r) = Mtot(r) – Mgas(r)

From X-ray observations alone, we can therefore measure not only the mass of the intra-cluster gas which is emitting in X-rays, but also the radial distributions of both the total gravitating mass and the dark matter.

### Cluster Mass (cont.ed)

At distances of about 1 Mpc from the cluster centre, typical mass values are:

Mtot ≈1014-1015 Msol and $M_{gas} \approx 10^{12}-10^{14} M_{sol}$

For approximately isothermal clusters Eq.(4) transforms in the following, much simpler expression for the total cluster mass: $M_{tot}(r)=-\frac{k_B T_{ICG}} {\mu m_p G} \cdot \frac {d \ln \rho_{ICG}(r)} {d \ln r}$

The assumption of spherical symmetry can be relaxed to more realistic ellipsoidal or triaxial shapes, but at the high price of extremely more complicated expressions for the total cluster mass.

### Intra-Cluster Gas Spatial Distribution

One model that has been extensively used to describe the ICG radial distribution is the so called “β-model” (Cavaliere & Fusco-Femiano 1976). In this model, both the galaxies and the intra-cluster gas are assumed to be isothermal, bound to the cluster, and in equilibrium.
The galaxies are assumed to have an isotropic velocity dispersion (σr), and hence their spatial density is given by:
(7) $\frac {d \ln \rho_{gal}} {dr} = \frac {1} {\sigma_r^2}\frac {d \Phi(r)} {dr}$

The same can be written for the gas distribution:
(8) $\frac {d \ln \rho_{ICG}} {dr} = \frac {\mu m_p} {k T_{ICG}} \frac {d \Phi(r)} {dr}$

Then, eliminating the potential from Eqs. (7) and (8) yields:
(9) $\rho_{gas} \propto \rho_{gal}^\beta$

Where $\beta$ is the ratio of the galaxy-to-gas velocity dispersion: $\beta \equiv \frac {\mu m_p \sigma_r^2} {k T_{ICG}}$

### Intra-Cluster Gas Spatial Distribution (cont.ed)

If the galaxy distribution can be described by the King approximation to a self-gravitating isothermal sphere:

(10) $\rho (r) = \rho_0 [ 1+x^2 ]^{-3/2}$

From eqs.(9) and (10) directly follows the β-model: $\rho_{ICG}(r)=\rho_{ICG_{0}} \left[ 1+ \left ( \frac {r} {r_c} \right)^2 \right]^{-3 \beta /2}$

which describes the gas density spatial distribution.

For typical clusters: $\beta \equiv \frac {\mu m_p \sigma_r^2} {k T_{ICG}} = 0.76 \left ( \frac{\sigma_r} {10^3 km/s} \right )^2 \left ( \frac {T_{ICG}} {10^8 K} \right )^{-1}$ More complicated models, such as a double β-model (given by the sum of two single β-models) often provide a more accurate description of the data.

### Spectroscopy

Spectroscopy of the intra-cluster gas provides information on its temperature and chemical composition (see discussion on Coronal approximation in Lecture 5).

Observed spectra show:

1. A continuum component
produced by thermal bremsstrahlung, the predominant X-ray emission process in clusters at $T_g>10^8 K$ and an atomic density of n=10-3 cm-3
2. Line emission
produced by several processes, such as: collisional excitation of valence or inner shell electrons, radiative and dielectronic recombination, inner shell collisional ionization, and radiative cascades following any of these processes. In addition to the 7 keV Fe line complex, X-ray spectrum of solar abundance plasma also contain lines form C, N, O, Ne, Mg, Si, S, Ar, Ca, and L lines of Fe and Ni.
All of these emission processes (continuum and line emission) give emissivities that increase in proportion with the ion and electron densities, and otherwise depend only on the temperature XMM spectrum of the galaxy cluster Abell 851. Grey and black points show the spectra for the different detectors on board of the satellite.

### Observations

Until the last years of the past century all clusters in the X-rays looked regular and spherical.

In 1999, the advent of the Chandra and XMM-Newton satellites has shown that most clusters are instead strongly irregular and that violent physical processes are taking place inside these large structures.

### Observations – Cool Cores

The central 100 kpc of most clusters show strong peaks in the surface brightness distribution, which correspond to decrease drops by a factor of three or more in the gas temperatures.
As a results these clusters have cooling times (the time for a sound wave to cross the entire cluster) tcool~108 yrs, and hence shorter than the age of the Universe. $t_{cool}=\frac {5/2 nKT} {n_H^2 \Lambda(T)} = \frac {2.5 \cdot 2.3 n_H kT[ergs]} {n_H^2 \Lambda(T)} =$ $= \frac {2.5 \cdot 2.3 \cdot 1.6 \cdot 10^{-9} kT[ergs]} {n_H \Lambda(T)} = \frac {9.2 \cdot 10^{-9} kT[ergs] 1.2} {n_e \Lambda(T) 3.156 \cdot 10^7 } [yrs]$

As a consequence, the stronger the X-ray emission from the ICG, the more the gas cools.
Some kind of heating phenomenon has to balance the cooling, which would otherwise cause a catastrophe in the innermost regions of these clusters.

### Observations – Cool Cores (cont.ed)

Among the most accredited heating mechanisms that could counterbalance the cooling, are:

• Turbulence, shocks, merging. Hα filaments and metal abundance radial profiles should be destroyed, and hence not observed, in cool core clusters. This is though not always the case.
• Heating by Supernovae. It has been estimated that SNe can justify up to 10% of the energy required to stop the cooling.
• Thermal conduction acting on the temperature gradient within the cluster cores. Conduction is more efficient at high temperatures (T>5 keV), while in the innermost, cooler regions, its strength is expected to be almost negligible. Magnetic fields could further reduce its effect.
• AGN. Gas simulations have shown that radio cavities, caused by the presence of a central AGN, can distribute up to 30% of their internal energy into the ICG. One point in favor of this last hypothesis is the high frequency of central AGN in cool core clusters.

### Observations – Shock Fronts

Galaxy clusters form via mergers of smaller subunits. Such mergers dissipate a large fraction of the subclusters’ vast kinetic energy through gas dynamic shocks, heating the intra-cluster gas and probably accelerating high energy particles.
Shocks contain information on the velocity and geometry of the merger. They also provide a unique laboratory for studying the intra-cluster plasma, including processes such as thermal conduction.

Three types of phenomena can generate shocks in the ICG.

1. In the cluster central regions, powerful AGNs often blow bubbles in the ICG, which may drive shocks within the central few hundred kpc region. These shocks have M(*) ˜ 1. It is difficult to derive accurate temperature and density profiles for such low-contrast events, so they are poorly suited for using shocks as a diagnostic tool.

(*) M (Mach number) is the speed of an object moving through air, or any fluid substance, divided by the speed of sound in that fluid. It is commonly used to represent an object’s speed when it is travelling at the speed of sound or faster.

### Observations – Shock Fronts (cont.ed)

2. At very large off-center distances (several Mpc), cosmological simulations predict that intergalactic medium should continue to accrete onto the clusters through a system of shocks that separate the intergalactic medium from the hot, mostly virialized inner regions. The intergalactic medium is much cooler than the ICM, so these accretion (or infall) shocks should be strong, with M ˜ 10 – 100. As such, they are likely to be the sites of effective cosmic ray acceleration, with consequences for the cluster energy budget and the cosmic γ-ray. However, these shocks have never been observed in X-rays or at any other wavelengths, and may not be in the foreseeable future, because they are located in regions with very low X-ray surface brightness.

3. If an infalling subcluster has a deep enough gravitational potential to retain at least some of its gas when it enters the dense, X-ray bright region of the cluster into which it is falling, we may observe spectacular merger shocks. These last discontinuities provide the only way to measure the gas bulk velocities in the sky plane and the gas compression ratio.”

### Observations – Shock Fronts (cont.ed)

The figures show the best example of shock due to merging ever observed, in which the fast-moving substructure A (the “bullet”) has had a recent merger with a second substructure B.

The cluster shows a steep surface brightness edge at the tip of the bullet, whose shape indicates spherical gas density discontinuities in projection.
The temperature at the tip of the bullet is low (~7 keV) and is likely to have been the temperature of the subcluster. The region ahead of the bullet is instead much hotter (~20-30 keV), while the temperature eventually decreases outside the shock feature.
The strong pressure increase at the shock front provides the final proof that this is indeed a shock front.

### Observations – Shock Fronts (cont.ed)

Let us consider a fast moving dense gas cloud, with semi axes a and b, moving from right to left (as show in the figure) within a hot plasma. If the dense gas cloud were at rest relative to the hot gas, the pressures should not show discontinuities. However, a jump is observed on both sides of a dense front. The most natural explanation would be that the cloud moves through the ambient gas and is subject to its ram pressure in addition to thermal pressure.
From our data, we can determine the velocity required to produce the observed additional pressure.
If the velocity of the body exceeds the speed of sound, a bow shock forms at some distance upstream from the body. The gas parameters just inside the bow shock will be denoted with an index 2. The gas parameters inside the body will have index of 0′. Far upstream from the body, the gas is undisturbed and flows freely (“free stream”; region 1). Near the outer edge of the body, the gas decelerates approaching zero velocity (“stagnation point”; region 0).

### Observations – Shock Fronts (cont.ed)

Following Landau and Lifshitz (“Fluid mechanics”, 1959), the ratio of pressures in the free stream and at the stagnation point is a function of the cloud speed v. $\frac{p_0}{p_1}=\frac{\gamma+1}{2}^{\frac{(\gamma+1)}{(\gamma-1)}}M_1^2 \left[\gamma-\frac {\gamma -1} {2 M_1^2} \right ]^{-1/(\gamma-1)}$
where γ=5/3 is the adiabatic index of a monatomic gas.

The above formula is valid if M1≡v/c1>1, and hence if the gas cloud is moving faster than the sound speed of the hotter gas. If this discontinuity is interpreted as a shock front, it is straightforward to derive the expected temperature and density jump, the shock propagation velocity, and the velocity of the gas behind the shock, using the Rankine-Hugoniot shock equations (Landau and Lifshitz , 1959): $\frac {\rho_2} {\rho_1} = \frac {(1+\gamma)M_1^2}{2+(\gamma-1) M_1^2}$ $\frac {T_2} {T_1} = \frac {2\gamma M_1^2-\gamma+1}{ (\gamma+1)} \frac{\rho_1} {\rho_2}$

If the observed density and temperature discontinuities agree with the above expected values, one can confidently assume that the hypothesis of a dense gas cloud moving at supersonic speed v is a fair one.

### Observations – Cold Fronts

Among the first Chandra cluster results was the discovery of “cold fronts” in merging clusters Abell 2142 and Abell 3667, as density discontinuities in the gas distribution across sharp edges.
At the time they were interpreted as shocks fronts. If these features were shocks, the gas on the denser, inside of the density jump would have to be hotter than that observed in the less dense gas outside the edge. These clusters show instead opposite temperature changes, ruling out the shock interpretation.
What are these sharp edges then? One hint was given by the gas pressure profiles across the edges (simply the product of the density and the measured temperatures), which show that there is approximate pressure equilibrium across the density discontinuity, as opposed to a large pressure jump expected in presence of a shock.

### Observations – Cold Fronts (cont.ed)

The most accredited model for the origin of cold fronts is that these are post-shock events, originating during a merger observed after the passage of the point of minimum separation, and presently moving apart. The model, initially proposed for Abell 2142, is shown schematically in the figure.
Dense gas clouds are remnants of the cool cores of the two merging sub clusters that have survived shocks and mixing. The preceding stage of the merger is shown in upper panel. In upper panel, shaded circles depict dense cores of the two colliding sub clusters. The less dense outer subcluster gas has been stopped by the collision shock.
Shock fronts 1 and 2 in the central region of top panel have propagated to the cluster outskirts in lower panel, failing to penetrate the dense cores that continue to move through the shocked gas. The dense cores (or, more precisely, regions of the sub clusters where the pressure exceeded that of the shocked gas in front of them, which prevented the shock from penetrating them) continued to move ahead through the shocked gas, pulled along by their host dark matter clumps.

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