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Subsections

Statistical Properties of the Sample

Introduction

Many statistical studies of cluster properties have been carried out but most of them involve samples from the northern sky (eg. [Andernach et al.1988], [Ledlow and Owen1995]). One recent study for southern clusters is presented in [Reid1999]. Of particular interest in this project was to investigate possible correlations between cluster properties and the presence of diffuse sources. If found, these correlations could give some insight into the cause of diffuse radio emission in clusters.

The first step was to establish some standard parameters to be used throughout the project, to allow a statistical study and to facilitate comparison with previous research. It was necessary to ensure the region of each cluster being considered was defined by the same radius. The linear distance $(\frac{1}{3}R_A)$ , was calculated for each cluster using Equation 2.1. One third of the Abell radius was chosen to ensure the region being studied fitted entirely within the image in each case and so that our results could be compared with those of [Ledlow and Owen1995].

Redshift Distribution

The redshift distribution of the cluster sample is shown in Figure 4.1. In this figure, and throughout the rest of this chapter, clusters with possible diffuse sources are represented by shaded boxes.

**Figure 4.1:** The redshift distribution of the cluster sample.
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The mean redshift of our cluster sample is 0.0541, and the mean redshift of the clusters which contain diffuse sources is 0.0572. The detection of diffuse sources follows a similar distribution to the overall redshift distribution.

Radio Source Count

Several studies of individual clusters suggest that diffuse emission may be associated with a higher than average level of radio activity in the cluster (eg. A2256, [Röttgering et al.1994], A3100, [Reid et al.1999]). To test this we counted the number of sources in each cluster, with $S_{peak} \ge 10$ mJy, by examining the images in kview. The results are shown in Figures 4.2 and 4.3. The mean radio source count is 4.7 and the mean for clusters with a possible diffuse source is 4.2.

Without trying to optically identify every source in the field, it is not possible to tell which of the sources definitely belong to the cluster and which just fall within the projected area of the cluster. We calculated the number of background sources that one would expect to see by chance in an equivalent area on the sky, using the method given by [Large1987]. This distribution is shown in Figure 4.4. The mean is slightly lower than the mean source count, showing that there is a small excess of cluster related radio sources. This is confirmed by the fact that the mean ratio of source count to chance coincidence rate is greater than 1, Figure 4.5. The distributions in Figure 4.5 also show that the clusters with candidate diffuse sources do not have a noticeably higher than average radio source count.

**Figure:** Distribution of source counts, within $\frac{1}{3}R_A$ above 10 mJy, for distance class 3 clusters. Mean = 5.7
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**Figure:** Distribution of source counts, within $\frac{1}{3}R_A$ above 10 mJy, for distance class 4 clusters. Mean = 3.7
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**Figure 4.4:** Distribution of the number of radio sources expected due to chance coincidence. Mean = 3.4
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**Figure 4.5:** Ratio of radio source count to chance coincidence rate. Mean = 2.1. All clusters which have no radio sources will give a ratio of zero, leading to a disproportionately high number in the first bin.
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Cluster Richness

In this section we investigate whether the detection rate of radio sources (and specifically diffuse sources) is correlated with the richness class of the cluster. The richness class is a measure of the number of galaxies in a cluster and is also related to the galaxy density near the cluster centre [Ledlow and Owen1995]. Hence this section tests two things; the effect of a higher galaxy density and the effect of looking at a larger number of galaxies.

For each class we have taken the total number of galaxies, N_G, to be the average of the range given by [Abell1958]. Our results are shown in Table 4.1 which is equivalent to Table 2 in [Ledlow and Owen1995].

Table: Detection rate as a function of richness class. R is richness class. Range is the galaxy population of that class. N_G is the average population. N_C is the number of clusters in our sample. N_S is the number of radio sources detected. N_D is the number of diffuse radio sources detected. Fraction(S) and Fraction(D) are given by Equation 4.1

R	Range	N_G	N_C	N_S	Fraction(S)	N_D	Fraction(D)
0	30-49	39	49	278	0.15	6	0.003
1	50-79	65	23	98	0.07	4	0.003
2	80-129	105	16	55	0.03	4	0.002
3	130-199	165	3	14	0.03	0	0
4	200-299	250	1	5	0.02	0	0

The Fraction columns give the number of radio sources (or diffuse sources) divided by the number of clusters surveyed and the average number of galaxies in that richness class:

$\begin{displaymath}\mbox{Fraction(S)} = \frac{N_S}{N_C \times N_G} \qquad \mbox{and} \qquad \mbox{Fraction(D)} = \frac{N_D}{N_C \times N_G} \end{displaymath}$

(4.1)

This normalizes the detection rates, removing the dependence on the number of galaxies. The resulting fractions are consistent across richness classes suggesting that the number of radio sources and diffuse radio sources detected simply scales with the number of galaxies surveyed. This implies that the higher galaxy density has no additional effect (such as increased galaxy-galaxy interactions) on the presence of radio sources which is in agreement with [Ledlow and Owen1995] and [Reid1999].

Bautz-Morgan Cluster Types

The Bautz-Morgan classifications [Bautz and Morgan1970] reflect the degree to which the brightest galaxy stands out against the general cluster background. The original criteria are given in Table 4.2.

Table: The original criteria and standard clusters that define the Bautz-Morgan classifications.

Type	Description	Standard
I	Clusters containing a centrally located cD galaxy	A2199, A2029
I-II	Intermediate
II	Clusters where the brightest galaxy or galaxies are intermediate in appearance between class cD and the Virgo-type giant ellipticals	A194, A2197 A1656(Coma)
II-III	Intermediate
III	Clusters containing no dominant galaxies.	Virgo, A2065

We calculated the number of radio sources and diffuse sources detected in each B-M type, and normalized by dividing by the number of clusters of that type. The results are shown in Table 4.3.

Table: Detection rate as a function of Bautz-Morgan type. N_C, N_S and N_D are as in Table 4.1.

B-M	N_C	N_S	N_S / N_C	N_D	N_D / N_C
I	19	101	5.3	2	0.11
I-II	30	162	5.4	8	0.27
II	25	94	3.8	1	0.04
II-III	8	59	7.4	2	0.25
III	10	34	3.4	1	0.10

The number of radio sources detected per cluster is fairly consistent across B-M type, except for the high value in type II-III which is probably due to poor statistics (there are only 8 clusters of this type). However, the number of diffuse sources per cluster is significantly higher for type I-II clusters (again disregarding type II-III). It is interesting that the number of diffuse sources detected is highest for clusters with this intermediate type, which is not as well defined as types I, II and III. This could suggest that the morphology of the clusters is unclear, possibly due to disruption of the galaxy distributions and ICM. This would be consistent with the model where diffuse emission is caused by turbulence due to cluster mergers.

Abell Cluster Types

The Abell Type classifications [Abell1965] reflect the regularity of the cluster morphology. The regular, R, clusters have a marked degree of spherical symmetry and high central concentration of galaxies. The bright galaxies in regular clusters are nearly all elliptical or S0 galaxies. Examples of regular clusters are A1656 (Coma) and A2065. The irregular, I, clusters do not have marked spherical symmetry of strong (if any) central concentrations. They have an amorphous appearance and their bright galaxies are of all types; spirals, ellipticals and S0's. Examples of irregular clusters are A2151 (Hercules), Virgo and the Local Group. Between regular and irregular clusters there are two intermediate types, RI and IR.

We calculated the number of radio sources and diffuse sources detected with each Abell type, and normalized by dividing by the number of clusters of that type. The results are shown in Table 4.4. The number of radio sources per cluster is fairly consistent across Abell type. The fraction of diffuse sources detected in the more regular clusters (R and RI) is the same as the fraction detected in the more irregular cluster (IR and I).

This result seems to be at variance with the result in Section 4.4, from which we would expect a higher detection rate in the more irregular clusters. However, the Abell types have very general descriptors and hence the B-M classification is a stronger one. We believe that the trend for B-M type is significant and that the same trend is not apparent for Abell type because of the more subjective classifications.

Table: Detection rate as a function of Abell Type. N_C, N_S and N_D are as in Table 4.1.

Type	N_C	N_S	N_S / N_C	N_D	N_S / N_C
R	29	136	4.7	6	0.21
RI	22	101	4.6	2	0.09
IR	19	101	5.3	3	0.16
I	22	112	5.1	3	0.14

X-ray Luminosity

As stated in Section 2.3.3, 37 out of our 92 clusters (40 $\%$ ) are included in the XBACs and REFLEX catalogues. The REFLEX catalogue has a flux density limit of $f_X > 3\times 10^{-12} \mbox{ erg s}^{-1} \mbox{cm}^{-2}$ , and so we can assume that the other 55 of our clusters fall below this limit. Six of our 14 clusters with diffuse sources (43 $\%$ ) have an X-ray flux density above this limit. The relationship between intrinsic luminosity, L_X and observed flux density, f_X, is simply the inverse-square law $L_X = 4\pi d^2 f_X$ where d is given by Equation 3.2. The distribution of X-ray luminosities for our sample is shown in Figure 4.6.

Our results do not show a clear trend between the presence of a diffuse source and the X-ray luminosity. However, it is hard to compare this with previous studies which have found that diffuse sources tend to be found in clusters with higher X-ray luminosity, since our cluster sample has been optically selected, whereas the other studies have all worked with X-ray selected samples.

**Figure 4.6:** Distribution of X-ray luminosities for our sample.
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Summary

We have carried out a statistical study of the properties of clusters that contain diffuse sources. We have found 14 candidate diffuse sources in a 92 cluster sample ( $15\%$ ), a rate comparable with that found in a recent study by [Giovannini et al.1999] who found 29 candidates in a 205 cluster sample ( $14\%$ ) from the NRAO VLA Sky Survey.

The detection rate of radio sources (or diffuse sources) was found to be consistent across Abell type and cluster richness. There appears to be a preference for diffuse sources to be present in B-M Type I-II clusters, which may reflect the fact that these clusters have a disturbed morphology and hence have been given an intermediate class. We found no evidence to support the hypothesis that the presence of diffuse sources is linked to the radio activity or X-ray luminosity of clusters.

Ideally, a statistical study that covers all known diffuse sources needs to be carried out, to give greater significance to the results from the studies done so far.

Next: A Multiwavelength Study of Up: Diffuse Radio Sources in Previous: Diffuse Sources

Tara Murphy
1999-10-31