2.2. Cluster baryon count

Eq. (21) implies that for a cluster with an isothermal atmosphere

(22)

where N_e is the total number of electrons in the cluster. Thus for an isothermal cluster, a model-independent estimate of the total number of electrons in the cluster, and hence the total number of baryons, can be deduced if the temperature, T_gas, and metallicity of the cluster gas can be deduced from a good X-ray spectrum.

X-ray data for a cluster can also be used to calculate the cluster's total mass, M_tot, using the assumption of hydrostatic equilibrium and spherical symmetry, eq. (23). If this mass estimate is combined with the baryonic mass, taken from N_e deduced from the SZ effect (or from the electron count deduced from the X-ray data), the baryonic mass fraction in the cluster gas, f_b, can be found. Since the gas in clusters contains most of the baryons (stars and stellar remnants are a relatively small correction), the value of f_b found in this way should be appropriate for the cluster as a whole (and should be a reliable lower limit). If, then, clusters of galaxies are fair samples of the mass content of the Universe, the derived value of f_b should be close to the value 0.12 ± 0.02 (Turner 2002) for the Universe as a whole.

On the other hand, if clusters are not fair samples of the total matter content of the Universe, then the variation of f_b as a function of redshift, derived by this technique, would be a powerful clue to the processes of cluster formation. At present there is no evidence for significant variations in the cluster-based f_b with redshift (Carlstrom et al. 2002), but the errors on the values of f_b are large.

Clearly this measurement can be made only if the calibration of the SZ effect is good (Sec. 2.1.1). The measurement also relies on assumptions (Sec. 2.2.1) about the gas distribution.

2.2.1. Isothermal spherical clusters

The key assumptions about cluster properties in finding the baryon count are

1. the gas must be isothermal -- this is hard to test, since the outer parts of the cluster are X-ray faint, but can make a significant contribution to the SZ effects; and
2. the measurement of the gas temperature must be good -- this depends on excellent calibration of the X-ray telescope, and also on the quality of the spectrum since the derived temperature is somewhat dependent on the metallicity of the cluster gas, and this is not usually well determined. For cooler clusters, the nature of the absorbing gas column in our Galaxy will also be important, since the shape of the X-ray spectrum at low energies depends on this column.

The resulting systematic errors in the gas temperature (and it is assumed that the electron and ion temperatures are the same: this is by no means assured) are probably only (1 - 2)% in the X-ray bright regions, so the appropriately-weighted systematic error in the gas temperature over the population of electrons and baryons is likely to be only ~ 4%. This is not a major issue.

The strong additional assumption that the cluster has a spherical, or almost spherical, shape is needed if the X-ray data are to be used to determine the total cluster mass and so to determine the cluster baryonic content. Projection effects that could affect the measurement for f_b are a major worry in using this method for individual clusters: a preferable technique would be to apply it for clusters selected in a manner independent of their orientation (i.e., using clusters with surface brightnesses far above some X-ray threshold, or using clusters from a blind SZ effect survey). The individual measurements of f_b would then be subject to projection effects, but the average f_b can be extracted from the population if the intrinsic distribution of cluster atmosphere shapes can be recovered.

2.2.2. Large-scale model

The isothermal model of Cavaliere, Fusco-Femiano (1976) is a convenient and frequently-used description of the large-scale structure of the atmosphere of a cluster of galaxies. Its form can be derived from

(23)

(e.g., Fabricant, Lecar, Gorenstein 1980) which describes how the density, _gas, and temperature, T_gas, of a gas in hydrostatic equilibrium in a spherically-symmetric gravitational potential well is related to the mass of the cluster, M_tot(r), within radius r. If we assume that the gas is isothermal, and that the total mass distribution has the form

(24)

where r_c, the core radius, defines a characteristic scale and M_c is the mass within r_c, then a consistent description of the density of the atmosphere is obtained with

(25)

where is a constant which determines the shape of the gas distribution, and depends on the ratio of a characteristic gravitational potential energy and the thermal energy in the gas

(26)

The alternative derivation of this isothermal model by Cavaliere, Fusco-Femiano (1976) brings out the interpretation of in terms of the relative scale heights of gas and dark matter in the potential well. Most uses of eq. (25) seek to determine ₀, r_c, and without considering the detailed properties of the underlying mass distribution.

It should be noted that the physical consistency of this much-used model for the gas distribution depends on radial symmetry (a simple distortion from a spherical to an elliptical model for the gas density would imply a mass distribution which is not necessarily positive everywhere), and on gas at different heights in the atmosphere having come to the same temperature without having necessarily followed the same thermal history. Eq. (25) is not unique in the sense that gas with a different thermal history might follow a significantly different density distribution. Thus, for example, if the gas has the same specific entropy at all heights, then the gas density should be described by

(27)

where is a structure constant with a similar meaning to .

A similar procedure for the mass profile of Navarro, Frenk White (1995), and an isothermal gas, leads to the gas density distribution

(28)

where the new structure constant

(29)

r_s is the scale of the Navarro model and M_s is the mass within radius r_s, so that has a similar physical meaning to . It can be seen that in this solution as r 0. Examples of eq. (25) and (28) profiles are shown in Fig. 12.

Figure 12. Representative isothermal models for cluster atmospheres. Solid line, eq. (28) with rs = 0.2r200 and = 10. Dashed line, eq (25) with rc = 0.1r200 and = 1.

The total masses for both mass distributions in Fig. 12 diverge as r , so both must be truncated at some outer radius. One possibility is to truncate at r₂₀₀, the radius at which the mean enclosed mass density is 200 × the critical density of the Universe at the redshift at which the cluster is seen.

The run of density and temperature in a cluster atmosphere are usually measured from the X-ray image and spectrum, where a density model for the gas of the form of eq. (25) or eq. (28) is fitted to the X-ray surface brightness. The X-ray surface brightness at a point offset by r in projected distance from the centre of the gas distribution (assumed spherical) is (in energy per unit time per unit solid angle per unit frequency)

(30)

where (T_gas) is the X-ray emissivity of the gas. For an isothermal gas with a -model density distribution

(31)

if the density distribution of eq. (25) is taken to extend to infinity, with a constant gas temperature throughout.

This cannot be a fully physically-consistent description of the atmosphere: it is clearly too simple, as would be the corresponding result for the density distribution of eq. (28). Both involve infinite total masses, so an outer cut-off radius (not consistently modelled) is adopted, and eq. (28) also requires an inner cutoff radius. The choice of an isothermal description for the gas is also questionable: what process forces the temperature to become constant, when the gas is accumulated at a number of times from a number of sources (for example, infall and galactic winds), and is imperfectly mixed by galaxy motions or intracluster turbulence?

However, some large-scale gas model like eq. (25) or (28) is essential to relate the X-ray and SZ effects of clusters since X-ray and SZ measurements are sensitive to different parts of the gas distribution. These models also provide a convenient relationship between the atmosphere and the underlying mass distribution. Note that eq. (25) and eq. (28) generally require that the gas constitutes a radially-varying fraction of the total cluster mass.