**4.2. X-ray luminosities and luminosity functions**

The number of clusters per unit volume with X-ray luminosities in the
range *L*_{x}
to *L*_{x} + *dL*_{x} is defined as
*f*(*L*_{x})*dL*_{x}, where
*f*(*L*_{x}) 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) (4.2) |

All luminosities in this section are given for the photon energy range
of 2 - 10 keV. It is convenient to define *L*_{44}
*L*_{x} /
10^{44} 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
*L*_{44}
10. Schwartz found that
the best fit exponential luminosity function has *A*_{e} =
4.5 × 10^{-7} and *L*_{xo} = 2.0 ×
10^{44} *h*_{50} ergs / s, although a
power-law with *A*_{p} = 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 *A*_{p} = 2.5 × 10^{-7} and
*p* 2 for 0.2
*L*_{44}
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 *A*_{e} =
0.49 × 10^{-7},
*L*_{xo} = 2.9 × 10^{44}
*h*_{50}^{ - 2} ergs / s, and *A*_{p}
= 1.1 × 10^{-8}, *p* = 1.7, respectively.
When the sample was extended to all Abell clusters and luminosities, the
normalization *A*_{e} and the characteristic luminosity
*L*_{xo} 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 *A*_{e}
2.5 ×
10^{-7}, *L*_{xo}
1.8 ×
10^{44} *h*_{50}^{ - 2} ergs / s,
*A*_{p}
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
*A*_{p}
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
*L*_{44}
1 and *p*
1.3 for
*L*_{44}
1.
While the HEAO-1 data do not show any clear evidence for a change in
the slope of the luminosity function at *L*_{44}
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).