4.2. X-ray luminosities and luminosity functions
The number of clusters per unit volume with X-ray luminosities in the range Lx to Lx + dLx is defined as f(Lx)dLx, where f(Lx) is the X-ray luminosity function. In general, the luminosity function will depend on the method used to select the clusters. One can begin with a statistically complete catalog of optically detected clusters (such as the 'statistical sample' of Abell clusters; see Section 2.1), which is surveyed for X-ray emission. Alternatively, a complete catalog of X-ray sources can be examined optically to determine which sources are associated with clusters of galaxies. In addition to reproducing the observed statistics of X-ray cluster identifications, the X-ray luminosity function is subject to the additional constraints that the total number of X-ray clusters not exceed the total number of all clusters, and the total emissivity of X-ray clusters not produce a larger X-ray background than is observed. Most data on cluster luminosities have been fit to either an exponential or a power-law form of the luminosity function. Thus we define
| (4.1) |
All luminosities in this section are given for the photon energy range of 2 - 10 keV. It is convenient to define L44 Lx / 1044 ergs / s.
Schwartz (1978) derived an estimate of the luminosity function for a sample of 14 Abell clusters in distance class 3 or less, which were detected with the Uhuru, Ariel 5, or SAS-C satellites. More distant clusters are used only to give an upper limit to the luminosity function at high luminosities. This sample is only expected to be complete for 1 L44 10. Schwartz found that the best fit exponential luminosity function has Ae = 4.5 × 10-7 and Lxo = 2.0 × 1044 h50 ergs / s, although a power-law with Ap = 7.9 × 10-7 and p 2.45 would also fit the data if suitably truncated at high and low luminosities.
McHardy (1978a) derived an X-ray luminosity function from the Ariel 5 fluxes for Abell clusters of distance class 3 or less; he argued that the Uhuru fluxes are unreliable for weak sources. While he did not fit his numerical luminosity function to any analytic expression, a suitable fit is given by a power law with Ap = 2.5 × 10-7 and p 2 for 0.2 L44 20.
The HEAO-1 satellite provided a much more extensive data base for determining the luminosity function of clusters. Both the A-1 and A-2 experiments were used to survey the Abell clusters. A luminosity function was derived for a significant portion of the statistical sample of Abell clusters (Section 2.1), using the A-1 data by Ulmer et al. (1981). Exponential and power-law fits to these data gave Ae = 0.49 × 10-7, Lxo = 2.9 × 1044 h50 - 2 ergs / s, and Ap = 1.1 × 10-8, p = 1.7, respectively. When the sample was extended to all Abell clusters and luminosities, the normalization Ae and the characteristic luminosity Lxo both were roughly doubled.
The HEAO-1 A-2 data were used to derive a luminosity function both by surveying the Abell clusters (McKee et al., 1980; Hintzen et al., 1980) and by identifying a complete sample of X-ray sources in a flux-limited survey at high galactic latitude (Piccinotti et al., 1982). The Abell cluster survey included all richness classes and all distance classes less than five. The luminosity function could be fit adequately with either an exponential or power-law form, and the coefficients in equations (4.1) and (4.2) were Ae 2.5 × 10-7, Lxo 1.8 × 1044 h50 - 2 ergs / s, Ap 3.8 × 10-7, and p 2.2. The cluster luminosity from the high latitude survey was not well represented by an exponential; a power law fit gave Ap 3.6 × 10-7 and p 2.15, which agrees well with the A-2 Abell survey result.
Bahcall (1979b) has attempted to predict the X-ray luminosity function of clusters from their optical luminosity function by assuming a one-to-one correspondence between the optical and X-ray luminosity of clusters. She predicts that clusters have a luminosity function that can be represented by two intersecting power laws with p 2.5 for L44 1 and p 1.3 for L44 1. While the HEAO-1 data do not show any clear evidence for a change in the slope of the luminosity function at L44 1, they are probably not inconsistent with such a change because they do not extend much below this luminosity.
One important application of cluster luminosity functions is in determining the contribution of clusters to the hard X-ray background (see Field, 1980, for a review of its properties). From the estimates of the luminosity function of clusters discussed above, it appears that clusters probably provide only about 3 to 10% of the X-ray background in the 2 - 10 keV photon energy band, assuming they do not evolve rapidly with time (Rowan-Robinson and Fabian, 1975; Gursky and Schwartz, 1977; McKee et al., 1980; Hintzen et al., 1980; Piccinotti et al., 1982; Ulmer et al., 1981).