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6. The statistics

6.1. Spectral content

The spectral index at 1.4 GHz, tex2html_wrap_inline2722, besides the total flux density at this frequency, is one of the best determined parameters of the GB/GB2 sources.

The distribution of tex2html_wrap_inline2722, for four different morphological types is shown in Fig. 1 (click here). The distribution of tex2html_wrap_inline2722 for different optical classes: galaxies, QSOs, and undetected (EF) sources, is shown in Fig. 2 (click here).

Figure 2: The same as in Fig. 1 (click here) but for sources of different optical type

The mean values and standard deviations of these distributions are given on the relevant histograms. The standard deviations of tex2html_wrap_inline2722 are always smaller than the corresponding deviation of any two-point spectral index.

The distributions of tex2html_wrap_inline2722 obtained for the FRII and FRI as well as C sources are highly symmetrical. Momental skewness of the tex2html_wrap_inline2722 distributions of FRII and FRI sources taken altogether is 0.036; the same of the C sources is only 0.007. In turn, the large skewness of 1.089 for the CSS sources is evidently caused by different morphological types (small projected doubles, very compact self-absorbed sources, steep-spectrum core-jet structures, and complex sources which do not fit into any of the above categories) constituting the entire CSS class (e.g. Sanghera et al. 1995). The highly symmetrical spectral-index distributions of the lobe-dominated and core-dominated sources support the assumption about a Gaussian functional form which were frequently used to describe these distributions in cosmological evolutionary models (cf. Petrosian & Dickey 1973; Kulkarni 1978; Condon 1984).

6.2. Optical identification content

For some GB/GB2 sources the optical type is uncertain. This is the case with the faintest objects barely visible on the POSS prints or detected by the CCD imaging beyond the POSS limit. Nevertheless for the statistical purpose, taking also into account their radio morphology and spectrum, one can include them into one of the two main categories: galaxy or quasar. Such a simplified optical identification content of the sample is given in Table 4 (click here). For each flux density range, the first row gives the fractions of all available identifications; the second - the corresponding fractions if optical identification is limited to tex2html_wrap_inline2842 (the POSS limit).


  Table 4: The optical type vs. flux density range, cf. the text

The fraction of QSO identifications is similar in the different flux-limited subsamples; a decrease of the fraction with flux density is statistically unsignificant. Limiting the optical identification to tex2html_wrap_inline2842, this fraction decreases by about 4 per cent only in each flux range. The above is consistent with the independence of the radio properties of radio-loud quasars on their optical luminosities (e.g. Peacock et al. 1986). Some enlargement of the QSO fraction is expected after further identifications of the EF sources. Basing on the optical luminosity function of quasars and their radio-to-optical luminosity ratio function (cf. Machalski 1996), about tex2html_wrap_inline2904 more QSOs with tex2html_wrap_inline2906 mag and S1.4>200 mJy can be expected. Furthermore, a decrease of the galaxy identifications with decreasing flux density limit is evident. Therefore, most of the EF sources should be distant powerful galaxies. A distribution of the optical type for different radio morphologies is shown in Table 5 (click here).


Morph. Galaxy QSO EF sum
FRII tex2html_wrap_inline2910 tex2html_wrap_inline2912 tex2html_wrap_inline2914 203
FRI tex2html_wrap_inline2916 tex2html_wrap_inline2918 tex2html_wrap_inline2920 34
CSS tex2html_wrap_inline2922 tex2html_wrap_inline2924 tex2html_wrap_inline2926 68
C tex2html_wrap_inline2928 tex2html_wrap_inline2930 tex2html_wrap_inline2928 66
Disk 2 2
Table 5: The optical type vs. radio morphology


A majority of FRI type emission is connected to radio galaxies, while most of C types, frequently with one-sided (1s) or two-sided (2s) emission detected besides the bright core, are related to QSOs. However, there is no clear distinction between FRI and large C+2s radio sources. Usually a linear extent of C+1s or C+2s structure does not exceed the size of a parent optical object, i.e. about 10-15 kpc. However, in some quasars two-sided emission extends over much larger distances from the core; these are classified here as FRI (e.g. 0827+378, 1148+387). Concerning the FRII sources, we estimate that no more than 10 per cent of EF sources in the sample can be quasars; the remaining ones should be distant galaxies. The compact steep-spectrum (CSS) sources are found both in galaxies and QSOs. The relatively large fraction of CSS EF sources suggests that they are very distant.

6.3. Source counts

6.3.1. The spectrum-dependent counts

The counts of all radio sources at 1.4 GHz are very well established down to a sub-mJy level (e.g. Windhorst et al. 1985). The first limited spectral counts at this frequency were published by Machalski (1978b), but over almost two decades had remained unimproved, and were not used, for example, to constrain the cosmological models. However, such constraints at 2.7 and 5 GHz were successfully applied to the evolutionary models of Condon (1984) and Dunlop & Peacock (1990).

Although dividing of sources into "flat-spectrum" and "steep-spectrum" populations, in the face of "unified models", is now by large unjustified; spectrum-dependent counts can be still useful for cosmological purposes. For these purposes, the source population can be separated into two subpopulations with an arbitrarily chosen spectral index. Such differential counts of the sample sources with tex2html_wrap_inline2936 and tex2html_wrap_inline2938, normalized to the Euclidean ones, are shown in Fig. 3 (click here). Numerical data of these counts, extended to a lower flux density limit on the basis of other 1.4-GHz samples, will be published in a forthcoming paper (Machalski & Jamrozy, in preparation).

Figure 3: Normalized differential counts of "steep-spectrum" and "flat-spectrum" GB/GB2 sources at 1.4 GHz

6.3.2. The radio type-dependent counts

There are already indications that the spatial distributions of sources of different morphological type are not identical, i.e. these sources could evolve differently in cosmic time. While there is no doubt that the powerful radio galaxies and quasars show a strong evolution; the amount of evolution of low-power sources (mJy-level radio galaxies, Seyferts, etc.) is still controversial.

Figure 4: Normalised differential counts of sources of different morphological type at 1.4 GHz

Even among powerful sources the amount of cosmological evolution may not be the same for separate types (e.g. FRI and FRII).

In order to enable the use of the GB/GB2 data for exploring the implications of unified-model schemes, the normalised differential counts of the FRII, FRI, CSS, and C sources are provided in Fig. 4 (click here).

After extending these counts to the flux density of 100 Jy (taking into account the sources with tex2html_wrap_inline2942 in the sky area of 4.22 sr; tex2html_wrap_inline2944, tex2html_wrap_inline2946) it can be seen that the counts of FRI sources to the flux limit of 0.2 Jy are much flatter than the corresponding counts of FRII sources. Also the counts of the CSS and C sources probably differ between themselves, although the statistics available here is not sufficient to prove this. A further study of the above counts over larger sky areas is in progress (Machalski & Jamrozy, in preparation).

6.4. Redshift distributions

Complete redshift data of a given sample of radio sources are highly required for calculations of their intrinsic power, linear size, etc. Such data are also crucial for the observational verification of cosmological evolutionary models. Unfortunately, completing spectroscopic redshift content of most of the flux-limited samples is very difficult for the well known reasons. However, for some objects, e.g. elliptical radio galaxies, redshift can be reliably estimated from their apparent magnitude (especially K mag) and/or angular size. The latter method was used by Vigotti et al. (1989) to estimate galaxy redshifts in the B3-VLA sample.

In this paper, another possibility of estimating a redshift distribution of the subsamples of FRII sources is employed. The estimation is based on the empirical correlation between the power and surface brightness (logP vs. logB) of FRII-type sources; the latter parameter being dependent on the apparent flux density and angular separation between hot spots in their lobes. It can be shown (Machalski, in preparation) that for a given logB (a range of logB, in practice), the distribution of logP has a definite functional form. Therefore, for each FRII source in a sample one can calculate the probability of having logP in a certain power range (bin), and hence a redshift range. A sum of these probabilities in each redshift range, divided by the number of sources involved, gives the normalised distribution of redshift. If the sample is unbiased (the implicit assumption about random sampling of the redshift is present), the estimated z-distribution is statistically consistent with the distribution of true, spectroscopic redshift.

Using this method, the redshift distributions obtained for 88 per cent of the sources in Subsample 1 is shown in Fig. 5 (click here)a. The sources for which the redshift cannot be estimated are exclusively compact ones and mostly optically unidentified. Similarly, the estimated redshift distribution for 63 per cent of the sources in Subsamples 2 and 3 is shown in Fig. 5 (click here)b. These distributions suggest that about 8-10 FRII-type sources in our sample can be expected at redshift z>3. The number of objects above this redshift may be even greater if the unidentified compact sources (especially CSS ones) are taken into account.

Figure 5: Estimated redshift distributions of sources: a) in Subsample 1; b) in Subsample 2+3. The fraction of redshifts measured and/or estimated in each Subsample is given

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