Chapter 1 Introduction
3.7 The Intra-Cluster Medium
66 3. Theoretical Considerations
and
I0 =
1
Z
0
dxln(x) x2−s
c1(1−x)7/2 +c2exp(−18x)
I1 =
1
Z
0
dx x2−s
c1(1−x)7/2+c2exp(−18x)
I2 =
1
Z
0
dxln(x) x3/2−s
c1(1−x)7/2 +c2exp(−18x) .
We found that the values given for the numeric integralsI0, I1 and I2 in Brunetti & Blasi (2005) are not accurate enough to give a stable result.
3.7 The Intra-Cluster Medium 67
with central density n0, core radius rc, and spectral index β. Consider the equation of hydrostatic equilibrium and ideal gas law:
dp
dr =−GρM(r)
r2 (3.90)
p=nkBT (3.91)
and ρ = nµmp, where µ is the effective mass per proton (to account for He fraction).
Given observations of the thermal emission from the ICM, i.e. a spectral fit from X-ray telescopes, one can derive the mass profile:
M(r) = −kBT r Gµmp
d lnn
d lnr +d lnT d lnr
(3.92) This gives a direct handle on cluster mass, radius and temperature from X-ray observations.
This model is strictly valid only in the isothermal, static and undisturbed case. This however does not hinder X-ray observers to fit its projected surface brightness to everything that looks like a cluster. Surprisingly they are quite successful with this approach - this might mean that cluster atmospheres rarely experience major disturbance once the object reaches a mass of a few 1014M.
From the theory of cosmological structure formation one can derive the slope of the M - T relation for clusters. The mass density in an isothermal sphere can be written in terms of the velocity dispersion σ:
ρ(r) = σ2
2πGr2. (3.93)
At the overdensity δ= 200 one usually defines the virial radius:
M = 4
3πr200,vir3 ρ200 (3.94)
so the velocity dispersion is:
σ=M13 H(z)2δG2/1613
. (3.95)
This allows us to write down the M-T relation, assuming the ideal gas law:
M =
16G2 µmpδ3
32
H(z)−1T3/2, (3.96)
which predicts a slope of 1.5, as observed in the X-rays, see section 2.1. This is the self-similar model.
This self-similar model is very successful. Weak lensing, X-ray, and SZ16 data are pre-scribed well with these models. Additionally the theory of structure formation prescribes
16The Sunyaev-Zeldovich effect describes the increase in microwave flux through up-scattering of CMB photons of the thermal gas in the ICM.
68 3. Theoretical Considerations
the hierarchical build-up of clusters from a cosmological model (Press & Schechter, 1974) and lets us understand the statistical properties of the cluster distribution.
To this end cosmological simulations of structure formation reproduce most of these properties surprisingly well. Apart from this, problems still remain, one of which is the cooling-flow problem (Sijacki et al., 2007).
3.7.1 Extending our View of the ICM
As mentioned before, simulations of clusters from cosmological initial conditions are very successful in reproducing thermal properties of the ICM, globally and locally. These sim-ulations usually assume a fluid model for the baryonic matter in the universe - the most advanced ones also follow magnetic fields.
However consider that in a galaxy cluster nth ≈10−3particles/cm3 and TICM ≈108K.
What does this imply for the thermal gas ?
Let us assume that the main interaction of charged particles in a plasma is Coulomb scattering. Following section 3.1.1 we can rewrite the total energy loss per particle from Coulomb collisions in the thermal gas (eq. 3.5) for protons and electrons as:
∆v⊥ = 2e2
bvme, (3.97)
which now gives the change in velocity by Coulomb scattering in the ICM. To get the mean square of velocity perpendicular to the motion of the incoming particle distribution per second we have to integrate over impact parameter again, which gives:
∆v2⊥
=
bmax
Z
bmin
2e2 bvme
2
2πbN vdb (3.98)
= 4e4N m2ev ln Λ.
This means for the total velocity that, approximately:
∆v⊥2
tc=v2
= 3kBT
me . (3.99)
We can now combine 3.98 and 3.99 to estimate the self-collision time of protons and electrons in the ICM (ln Λ≈40):
tc=
√me(3kBT)3/2
2πe4Nln Λ . (3.100)
3.7 The Intra-Cluster Medium 69
10-5 10-4
nth [cm-3]
10-8 10-7 10-6 10-5 hsml/λcoul
Figure 3.14: Left: Gas density projection of the ICM in a galaxy cluster (g72) from a cosmological simulation. The image size is 1.4×1.4 Mpc. Right: particle scale 2hsml over collisional mean free path from Coulomb interactions of the thermal plasma. Data from Dolag et al. (2006).
More importantly the mean free path is then (Spitzer, 1956; Sarazin, 1988):
λe= 33/2(kBT)2 4√
πnthe4ln Λ (3.101)
≈23 kpc
TICM 108K
2
nth 10−3cm−3
−1
(3.102) This result is most surprising: Coulomb scattering is not sufficient to establish collisionality in the ICM on scales of even below a kpc.
However without collisionality every fluid description must break down. Without col-lisional relaxation there is no temperature in a fluid element, no well defined pressure, etc.
We will now explore what that means for simulations.
In figure 3.14 we show an example of a simulated galaxy cluster from cosmological initial conditions. On the left we show a projection of the number density distribution with a side length of 1.4 Mpc. The right panel shows the SPH particle scale over the collisional mean free path (eq. 3.102). The SPH smoothing length is a rough proxi of the resolved scale in the simulation. The fraction reaches values of<8×104. This is equivalent to the mean free path being 1000 times larger than the numerical resolution.
This shows that, considering only Coulomb interactions, the fluid model is formally not applicable on the scales of the simulation. In this case the equilibration of the thermal gas particles, a fluid element on scales of the simulation would not feel pressure, temperature
70 3. Theoretical Considerations
would not be definable, etc. On these scales the gas particles should be collisionless, as dark matter particles are.
We want to stress here that this is not a problem of the simulation or numerics, but of the physics assumed. In the derivation of the fluid equations from the kinetic equations, one assumes that in a fluid element17 collisional timescales and lengths are sufficiently smaller than in the macrophysical system of interest.
However numerical simulations of clusters are a huge success, the fluid description is valid in clusters. This has to mean that Coulomb scattering is not the dominant process of equilibration in the ICM. Note that this is known for a long time (Sarazin, 1988).
Considering this quick investigation it is not surprising that the cooling problem in clusters exists in simulations. One has to ask, what does this mean to the thermal con-ductivity of the ICM (Dolag et al., 2004)? What is the actual mechanism that establishes collisionality on scales below the Spitzer value? What is the role of magnetic fields in this process?
Non-thermal cluster physics probably holds the answer. A recent paper points out collective plasma effects might be responsible for the reduced mean free path (Brunetti &
Lazarian, 2011b). With the upcoming telescopes we will be able to probe the ICM and its non-thermal components with unprecedented precision. This will open doors to test and model plasma physics in a regime not accessible before.
17An SPH particle should be made up of more than one of these elements
Chapter 4
Preamble to the Papers
In this section we will provide an overview of the theoretical articles published on radio halos in the past decade. We consider reacceleration models, followed by hadronic mod-els. We keep the discussion of reacceleration models relatively short because a detailed discussion is beyond the scope of this work.