Exploration of Cluster Structure 2331 Self Similarity of Cluster Structure

If the structure of clusters is characterized by the approach to a dynamical equilibrium state, how much similarity should we expect between the shapes of different galaxy clusters? If cluster formation is approximately modeled by the gravitational collapse of a homogeneous spherical overdensity of noninteracting dark matter, we expect that the collapse process is self-similar as well as the produced "Dark Matter halos." That is, we expect less massive systems to be scaled down versions of the more massive clusters. The cluster dark matter central densities are, for example, expected to be the same for clusters of all masses if they have formed at the same epoch. In general the central density is then proportional to the background density of the Universe at a characteristic formation time of the cluster (e.g., the turn-around time marking the start of the collapse). The characteristic radius depends then on mass and formation time. Inhomogeneities in the initial conditions and in the collapse as well as gas dynamical processes are expected to introduce variations into this self-similar scenario. Galaxy clusters in a stage of a merging of subunits or clusters in another major phase of matter accretion are of course also expected to show deviations from the equilibrium structure. Earlier theoretical considerations of possible equilibrium structures have resulted in the popular King model [86] (based on a guess of the velocity distribution function of the mass carrying particles), which was first developed to describe the structure of globular clusters, but also found to roughly reproduce the galaxy density distribution profile in galaxy clusters. The density distribution for this model is given by the formula r2 \-3/2

where pg is the gravitational mass density and rc is called the core radius. An isothermal ICM, with a temperature scaling with the velocity dispersion of the gravitating mass particles (e.g., galaxies) as given by a parameter ¡, ^T2 = 5, has a gas density profile described by the same form as (2) but with an exponent -3/25. The advantage of this simple description is, that it can easily be integrated along the line-of-sight to give the observed surface brightness in the case of isothermality with the surface brightness SX x PI2CM:

where R is the projected radius. In fact this function, generally called the "5 -model," turned out to provide a good approximate fit to the observed surface brightness profiles, except for cooling flow clusters, which may have an extra central brightness enhancement [81] and in the far cluster outskirts, where the surface brightness profiles tend to steepen [151]. Figure 23.6, shows typical surface brightness and line-of-sight integrated emission measure profiles that can be roughly fit by a ¡-model [4].

Fig. 23.6 Left: Surface brightness profiles of 40 relaxed appearing clusters as observed in deep ROSAT observations [4]. Middle: Emission measure profile determined from the surface brightness profile by accounting for the temperature variation of the emissivity function and the cosmological surface brightness dimming with increasing redshift. Right: Scaled emission measure (integrated along the line-of-sight) profile, SX/(A3?/2(1 + z)9/2T138), as a function of the scaled cluster radius. With this self-similar scaling the emission measure profiles show little dispersion at radii larger than 0.1Rv while the inner profiles vary depending on the cool core structure (Arnaud et al. [4])

Fig. 23.6 Left: Surface brightness profiles of 40 relaxed appearing clusters as observed in deep ROSAT observations [4]. Middle: Emission measure profile determined from the surface brightness profile by accounting for the temperature variation of the emissivity function and the cosmological surface brightness dimming with increasing redshift. Right: Scaled emission measure (integrated along the line-of-sight) profile, SX/(A3?/2(1 + z)9/2T138), as a function of the scaled cluster radius. With this self-similar scaling the emission measure profiles show little dispersion at radii larger than 0.1Rv while the inner profiles vary depending on the cool core structure (Arnaud et al. [4])

Recent N-body simulations with increased resolution suggest an improved description of the dark matter halo shape of the form, pg(r)— r-p (r + rs)p-q where rs is the scale radius. The most popular representations are p = 1 and q = 3 [108] (NFW-profile) and p = 1.5 and q = 3 [103] (Moore et al. profile). Given the mass profile the gas density and surface brightness profile can be determined with the knowledge of the temperature profile. The exact shape of nearly self-similar temperature profiles is still a matter of debate (e.g., [116, 153, 162]). As shown in Sect. 23.2 the NFW model provides a consistent description of the observed relaxed galaxy clusters.

If clusters have approximately self-similar shapes, the global parameters of clusters are also expected to follow tight relations. This was first described as an important tool for the study of the cluster population and its evolution by Kaiser [83]. If we consider clusters formed at the same epoch, it is easy to derive the basic, global relations,

where R* is a characteristic radius and M the cluster mass. If the X-ray emission of a volume element is eXbol - pI2CM tx/2 then we can, for example, obtain a relation between bolometric X-ray luminosity, LXbol, and ICM temperature, TX, of the form

Lxbol - Picm R'3 TX/2 - fb2pg2 R\ TX/2 - f2M2 R-3 TX/2 - TX. (23.5)

While this is the self-similar prediction, the observed relation features an exponent more close to 3, e.g. [3]. This is the most famous relation breaking the self-similar scenario and gave rise to deeper concern.

The resolution to this discrepancy is most probably found in extra heating and cooling of the ICM, which affects the thermal structure of the cluster gas in less and more massive clusters differently [117,118,156]. A large effort is currently made to understand this scenario and the relevant role of heating and cooling in detail. Briefly, the general arguments are the following: while the specific thermal energy gained by the ICM from gravitational heating in the self-similar scenario is proportional to the depth of the gravitational potential and thus proportional to M2/3 according to (4), any heating process that is tied to the galaxy population and will essentially be proportional to the galaxy light will be roughly independent of mass. In practice the energy of supernovae in early starbursts and AGN activity in the galaxies is believed to release one or a few keV per particle as specific thermal energy into the ICM (of the protocluster). While this is almost negligible for the ICM of a massive cluster in which gravitational heating has raised the gas temperatures to 5-10 keV, it will make a big effect in the ICM of a galaxy group with a temperature around 1 keV. Several theoretical works have been devoted to such preheating modeling, e.g. [23,148]. In a similar way, the effect of radiative cooling is different for clusters within the self-similar structure picture. If the central gas density is the same but the temperature rises with mass, the cooling time is shorter in smaller clusters and cooling should have a larger effect there. In some models the predicted effect of cooling is mass condensation and the inflow of adiabatically heated gas

Scaled radius (r/rv) Temperature (keV)

Fig. 23.7 Left: Surface brightness profiles scaled by T1/2 (to account for the different extent of clusters in the line-of-sight) as a function of scaled radius, r/rV « T1/2 [117]. Right: Entropy of the ICM at a radius of r = 0.1r200 as a function of the cluster bulk temperature [118]. The lower line shows the expectation for pure gravitational heating and the upper line the best fit of a power law function to the data (with a slope of -0.65)

Scaled radius (r/rv) Temperature (keV)

Fig. 23.7 Left: Surface brightness profiles scaled by T1/2 (to account for the different extent of clusters in the line-of-sight) as a function of scaled radius, r/rV « T1/2 [117]. Right: Entropy of the ICM at a radius of r = 0.1r200 as a function of the cluster bulk temperature [118]. The lower line shows the expectation for pure gravitational heating and the upper line the best fit of a power law function to the data (with a slope of -0.65)

from the outside. The net effect is an increase in the central entropy with a decrease in the central density and central surface brightness as shown in Fig. 23.7, e.g. [118]. Results from current refined simulations suggest that both effects play a role simultaneously. It turns out that the study of these processes are not only important for the understanding of the thermal structure of the cluster ICM but may also be the clue to understand the heating feedback regulation of galaxy formation.

It has become practice to describe the structure of the ICM in these studies by an entropy parameter, which is defined as S = kBTn-2/3 expressed in units of keV cm2. This definition differs from the classical entropy (see the review by Voit [156] for a profound explanation). In the strictly self-similar model as defined by (4) and (5) the entropy at a given scaled radius R* « M1/3 ^ T~X would be given by S(R*) ^ TX « M2/3. Because of the effect of extra heating and cooling the observed relation as shown in Fig. 23.7, is approximately [118]

This scaling also leads to a modified scaling law for the surface brightness (emission measure) - temperature relation of the form SX « T138. This scaling was used in the right panels of Fig. 23.6. Its application yields nicely self-similar emission measure profiles at radii r > 0.1Rvir.

There has been much recent progress in reproducing the observed ICM thermal structure in simulations (e.g. [22,84]) but the issue is still far from being solved. In particular, the heat input not only of supernovae during star formation episodes but also AGN activity in cluster galaxies is now considered in the modeling (e.g. [140]). There is an important boundary conditions in a comprehensive modeling, including galaxy formation that has to be met. The correct amount of the stellar mass fraction of the order of 10% should be produced. For example, pure cooling models tend to overproduce the stellar mass by a large factor. This remains a very important field of research for observations with Chandra and XMM-Newton last not least because it promises to shed new light on the processes that control galaxy formation.

0 0

Post a comment