X-ray Properties of Clusters of Galaxies.

The X-ray properties of clusters of galaxies are closely related to the underlying gravitational potential. This makes them ideal probes of cluster evolution. This section shows how certain X-ray properties may be related to the total mass of a cluster, $M_{\rm {total}}$, the ultimate measure of cluster evolution. Most closely related to $M_{{\rm total}}$ is the X-ray temperature, $T_{{\rm X}}$. For example, under the assumption that the ICM consists of an ideal gas in hydrostatic equilibrium it may be shown that,

$$M_{\rm {total}}(<r)=-\frac{kT_{\rm {X}} r}{G\mu m_{\rm {H}}}... ...{{\rm dln}r} + \frac{{\rm dln}T_{\rm {X}}}{{\rm dln}r} \right)$$ (11)

where $M_{\rm {total}}(<r)$ is the total cluster mass within a radius, $r$, $m_{\rm {H}}$ is the mass of a hydrogen atom and $\mu\$ is the mean molecular mass of the gas.

Under the assumption of spherical symmetry and measuring or assuming the temperature and density profile of the cluster, the total gravitating mass may be related to the X-ray temperature of the ICM.

With the advent of X-ray telescopes fitted with CCD cameras yielding arcsecond spatial resolution along with high spectral resolution such as Chandra and XMM such methods of determining cluster masses have become feasible, although they only remain accurate within the assumptions of hydrostatic equilibrium and spherical symmetry.

The density profile of the ICM is often assumed to follow the form,

$$\rho(r)=\rho_{0} \left( 1 + \left( \frac{r}{r_{{\rm c}}} \right)^{2} \right) ^{- \frac{3\beta}{2}}$$ (12)

first determined by King (1962) for the distribution of galaxies in clusters and later found to approximate the ICM density profile by Cavaliere & Fusco-Femiano (1976). The core radius of the cluster is given by $r_{{\rm c}}$ and is typically $\sim 250 h^{-1}$ kpc for clusters. The outer slope, $\beta$, is the ratio of energy in galaxies compared to the energy in the gas,

$$\beta=\frac{\mu m_{{\rm H}} \sigma^{2}}{3kT_{{\rm X}}}$$ (13)

where $\sigma$ is the velocity dispersion of the galaxies. A typical value for clusters is $\beta \sim 0.66$.

The most easily measured observational property of the ICM useful in studying cluster evolution is the X-ray luminosity, $L_{{\rm X}}$. The relation between $L_{{\rm X}}$ and the underlying potential well is not as direct as $T_{{\rm X}}$, and must be related to the total mass of the cluster via the X-ray temperature.

The X-ray luminosity is given by,

$$L_{{\rm X}} \propto \int{n_{{\rm e}}n_{{\rm i}} \sqrt{T} {\rm d}V}$$ (14)

which may be found by integrating the emissivity of a plasma radiating by thermal bremsstrahlung over frequency and volume, where the emissivity is given by

$$\kappa_{\nu}=\frac{1}{3\pi^{2}}\frac{Z^{2}e^{6}}{\epsilon^{3... ...{{\rm e}} {\rm exp} \left( - \frac{h\nu}{kT_{{\rm X}}} \right)$$ (15)

where $n_{{\rm i}}$ is the ion number density, $n_{{\rm e}}$ is the electron number density, $Z$ is the charge of the nuclei and $g(\nu,T_{{\rm X}})$ is the Gaunt factor given by,

$$g(\nu,T_{{\rm X}}) = \frac{\sqrt{3}}{\pi}{\rm ln}\left( \frac{kT_{{\rm X}}}{h\nu} \right)$$ (16)

see Longair (1998), page 88.

If the assumptions, $n_{{\rm e}} \propto n_{{\rm i}} \propto M_{{\rm gas}} \propto M_{{\rm total}} \propto T_{{\rm X}}$, are made then it can be seen from equation 3.4 that $L_{{\rm X}} \propto T_{{\rm X}}^{2.5} \propto M_{{\rm total}}^{2.5}$. In fact measurements show a relation closer to $L_{{\rm X}} \propto T_{{\rm X}}^{3}$ (White et al. 1997, Fairley et al. 2000).

The existence of relations relating $M_{{\rm total}}$ to $T_{{\rm X}}$ and $L_{{\rm X}}$ allows these properties to be used as proxies for the mass when studying cluster evolution. Such uses are discussed in the following section.