A&A 404, 57-62 (2003)
DOI: 10.1051/0004-6361:20030476
X. Zhang1,2 - W. Reich2 - P. Reich2 - R. Wielebinski2
1 -
National Astronomical Observatories, CAS, Beijing 100012, PR China
2 -
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn,
Germany
Received 30 July 2002 / Accepted 24 March 2003
Abstract
We present new statistical results on the spectral
index-flux density relation for large
samples of radio sources using archival data
of the most sensitive surveys, such as WENSS and NVSS.
Instrumental selection effects and the completeness of the catalogs
used in this study are discussed. Our main results are based on
the spectral indices calculated for about 185 800 sources
from the WENSS (327 MHz) and the NVSS (1.4 GHz)
catalogs and are summarized as follows:
(1) The median spectral index increases from
to
(
)
for S327 flux densities decreasing from 0.1 Jy
down to 23 mJy. The median spectral indices nearly show no variation within the error bars
in the flux density range above 100 mJy up to several Jy. The median spectral
index slightly increases again for S327 above several Jy.
The new results confirm published models of the radio luminosity
function (RLF) for sources with
S327 > 0.1 Jy and give constraints to the
models for sources of
,
respectively.
(2) A dependence of the fractions of ultra-steep-spectrum
sources (USS,
),
steep-spectrum sources (SSS,
)
and
flat-spectrum sources (FSS,
)
is partly responsible for the spectral flattening. Another contribution to
the spectral flattening comes from the variation of
of steep-spectrum sources (
)
themselves which increases with decreasing
flux densities. (3) The spectral flattening for faint sources
(down to
)
with steep spectra (
)
suggests that
is correlated with luminosity rather
than redshift according to the source evolution model of Condon (1984).
Key words: methods: statistical - radio continuum: galaxies - galaxies: evolution
Models of the radio luminosity function (RLF) published in the past years
describe the cosmological evolution of radio source samples. So far most RLF-models attempt
to fit observed source counts, luminosity or redshift distributions of source samples,
which, however, are restricted to relatively strong sources (Peacock & Gull 1981;
Wall et al. 1980, 1981; Subrahmanya & Kapahi 1983; Condon 1984;
Dunlop & Peacock 1990; Jackson & Wall 1999). In general the
-S
relation predicted by RLF-models were just partially confirmed by available data.
However, the source samples used so far were rather limited in size and sensitivity.
Differences between the observed and the predicted
-S
relation provide constraints to RLF-models, which are of particular interest
for the faint sources' properties.
This needs observed
-Srelations of high quality for a wide range of flux densities in order to confirm
RLF-models or set constraints for improvements.
Several investigations on the
-S relation
came to rather different conclusions based on small or compound samples.
Vigotti et al. (1989) reported that the
median spectral index decreases from
to
for flux densities ranging from 1 Jy to
0.1 Jy at 408 MHz (Fig. 1). They used a sample of 1103 sources selected
from the B3 catalog at 408 MHz (Ficarra et al. 1985)
and calculated spectral indices using 1465 MHz data measured with the VLA.
Steppe & Gopal-Krishna (1984) reported a contrary result
for a radio source sample in the same flux density range at 408 MHz:
was found to be about -0.9 at the
1 Jy and -0.75 at the 0.1 Jy flux density
level (Fig. 2). Their sample of 1009 sources was taken from several
catalogs including 5C12 (Benn et al. 1982), B2 (Grueff & Vigotti 1979),
MC1 (Davies et al. 1973), MC2 and MC3 (Large et al. 1981),
All-sky Survey (Robertson 1973). Kapahi & Kulkarni (1986),
Kulkarni & Mantovani (1985, 1985) and Kulkarni et al. (1990) claimed that
the median spectral index is constant at
in the same flux density range from 1 Jy to
0.1 Jy. A constant
differs from the results of Vigotti et al. (1989)
and Steppe & Gopal-Krishna (1984), as mentioned above.
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Figure 1: Result of Vigotti et al. (1989) on the relation between flux density and spectral index for frequencies of 408 MHz and 1465 MHz. The data are from B3 (open circles, Ficarra et al. 1985) and Benn et al. (1988) (filled circles). |
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Figure 2: The result of Steppe & Gopal-Krishna (1984) of the flux density-spectral index relation for frequency of 408 MHz (except sample g which is defined at 611 MHz) and a higher frequency which is 1.4, 2.7 or 5.0 GHz as indicated by crosses, open circles and dots, respectively. |
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A few radio source surveys with large sky coverage and high sensitivities at different frequencies have recently been published providing a new basis to study the spectral index-flux density relation, especially for sources with S327 < 0.1 Jy. In Sect. 3 we give the results of the spectral index-flux density relation using WENSS (Rengelink et al. 1997) and NVSS (Condon et al. 1998) data. Investigations based on the other large catalogs are mainly used for comparison which will be given elsewhere (Zhang et al. 2003). Selection effects and completeness of radio source samples for both statistics and observations are discussed in Sect. 2. The results of an analysis of the WENSS and NVSS data are presented in Sect. 3. In Sect. 4 a discussion and a comparison of the new results with luminosity evolution models are made.
Table 1:
The fractions of radio sources in different
ranks and flux density bins at 327 MHz. The radio sources are from the WENSS
and NVSS catalogs.
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Figure 3: The variation of the source fraction for FSS sources (bottom), SSS sources (top) and USS sources (middle) at 327 MHz. |
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To ensure statistical completeness a flux limitation of the
statistical study (FLSS) is defined first.
Most radio sources selected at low frequencies such as 327 MHz
have spectral indices larger than
.
This holds for a large flux density range.
Figure 3 shows the fractional variation of radio sources whose
spectral indices fall into the ranges of
(USS),
(SSS),
and
(FSS) respectively. Table 1 lists the
fractional distribution of sources in
detail.
Figure 3 confirms the result mentioned above, i.e. the statistical study is complete
up to 99% if the adopted flux limitation of the sample corresponds to a spectral
index of
.
A flux density of 23 mJy at 327 MHz
corresponds for
to the NVSS limit of
S1400 (2.5 mJy). We take this value as
the "flux limitation of the statistical study'' (FLSS), which means that
less than 2% of the steep-spectrum sources recorded by WENSS are
missing in the NVSS. However, all flat-spectrum sources recorded by WENSS
being stronger than its sensitivity limit of 18 mJy cannot
be missed by the NVSS. Although about 23 000 sources fainter than 23 mJy were
cataloged in the WENSS, with most of them having NVSS
counterparts, we only use sources stronger than 23 mJy in this study to ensure that the
statistical results are based on a unbiased sample.
The resolutions of WENSS and NVSS are
and
respectively. They are quite similar in size. This means that
source intensities can be directly used to calculate spectral indices for
most sources.
In this section the main results from the cross identification between the WENSS and NVSS are given. The criteria and procedure employed for the source identification are: 1) to search for all NVSS sources within a diameter of 50'' centered at every WENSS source; 2) to choose the nearest NVSS source to the central WENSS source; 3) to create a first identification table; 4) to do an inverse search, i.e. within a diameter of 50'' centered at each NVSS source identify the nearest WENSS source; 5) to create a second identification table and compare the two tables in order to exclude source pairs with complex corresponding relation. With these criteria and procedure the final cross-identification table used in this study was obtained and spectral indices were calculated accordingly.
Table 2: Statistical results on the relation of spectral index and flux density using WENSS and NVSS samples for sources selected at 327 MHz.
Table 2 shows the statistical results on the relationship
of flux density and median spectral index, which are also displayed in Fig. 4.
Integrated flux densities were used for the spectral index calculations.
In Table 2 the first row gives the median flux
densities and the boundary flux densities in Jy
of each bin at 327 MHz. In the second row of
the table the two numbers of each table element represent the numbers of sources
found in both catalogs and the medians of the spectral indices respectively.
The errors of
in the third line
were determined as described in Yule & Kendall (1950).
Figure 4 displays this result.
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Figure 4: Statistical results of the flux density-spectral index relation using the WENSS-NVSS catalogs. |
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Figure 5:
The contour/grey scale map of the
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To settle a statistical relationship of flux density and median spectral index,
the completeness is very important. There are two kinds of completeness. One is
the completeness set by the noise limit
of each individual catalog, which for example is 18 mJy for the WENSS and 2.5 mJy
for the NVSS. This completeness is important when using each survey individually.
Another completeness is related to the statistics of the cross-identification,
which reflects how well both catalogs match each other. In general,
flat-spectrum sources recorded
by the low frequency survey could not been missed by the high
frequency survey, whereas some faint
steep-spectrum sources near the sensitivity limit of the low frequency survey
may be lost by the high frequency survey. This statistical incompleteness, or selection
effects, is a more serious limitation for frequency-pairs which have either a relatively large
frequency span or the sensitivities of the two surveys do not match each other well.
The frequency difference between WENSS and NVSS is not too large and the sensitivities of them are
high that they match each other quite well. Our statistical spectral index results
using WENSS-NVSS data are complete and reliable for flux
densities larger than 23 mJy at 327 MHz (corresponding to
).
A discussion of selection effects with
cross-identifications using different catalogs will be given elsewhere (Zhang et al., in press).
It is interesting to compare the present results of the
-S relation with
predictions made by multi-frequency models of the evolution of the radio
luminosity function of extragalactic sources. Kulkarni & Mantovani (1985)
used the models of Peacock & Gull (1981, henceforth
referred to as PG) for a comparison. These models use separate luminosity and redshifty functions
for steep- and flat-spectrum sources. Four models are considered: model 1 without
and model 2
with a cut-off for the radio luminosity function at z=5, q0=0.5 and
models 3 and 4 as models 1 and 2, but with q0=0.
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Figure 6:
The
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Figure 7:
Predicted
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Figure 7 gives the comparison between our results with the PG model 1 and
Condon's model predictions. A good agreement
between the present
-S relation and the PG model 1 is found
for S327 sources stronger than 50 mJy. Our results are also in agreement with
Condon's model for sources
S327 > 0.2 Jy.
The fitted range of flux density is much wider than that from previously
published results. PG model 1 predicts a significant spectral steepening
with decreasing flux density from
S408 < 0.05 Jy
to 0.01 Jy (Kapahi & Kulkarni 1986). Although our result strongly supports
the PG model 1 for sources in a wide flux density range,
there is no evidence for a spectral steepening of very faint sources. Instead,
we found a continuous spectral flattening for faint sources. The situation for
sources
S327 < 23 mJy remains unsettled because of the
completeness limit of the samples.
For sources
S327 < 0.1 Jy, the
-S relation
for steep-spectrum sources (
)
is displayed in Fig. 6.
The mean spectral index of
steep-spectrum sources (USS and SSS) increases (spectral flattening).
According to Condon's source evolution model (1984) this behaviour suggests
that
is correlated with luminosity rather then with redshift.
However, more detailed studies of faint steep-spectrum sources are needed.
A more recent investigation of extragalactic radio-source evolution published by
Jackson & Wall (1999) predicts a population mix at 1.4 GHz according to
their dual-population unification scheme. Figure 8 gives the prediction
of the percentage of flat-spectrum sources (PFSS, solid line)
obtained from their paper together with the
new statistical results from the WENSS and NVSS catalogs. This new
result is based on a source selection at
1.4 GHz of the frequency pair of WENSS-NVSS, whereas the source
selection for Fig. 3 and Table 1 is for 327 MHz. Figure 8
confirms the prediction by Jackson & Wall (1999) in the
sense of a varying PFSS with flux density.
There is a nearly constant difference (3%) between their
prediction and the observed results. Table 3 lists source numbers recorded
by NVSS and WENSS in each bin respectively, the ratio of total identification,
and the PFSS (
). Table 3 shows that the
completeness of the cross-identification
between WENSS and NVSS with source selection at the high frequency end is about
90% on average. The "missing'' sources usually have quite a complex structure. Their
spectral indices could not be determined with the same accuracy as for
compact sources therefore they were excluded from the statistics. This
restriction may contribute to the difference between
prediction and statistics as shown in Fig. 8.
Comparing Fig. 8 with Fig. 3, two differences should be mentioned. One is that the PFSS from source selection at the low frequency (Fig. 3) is obviously smaller when compared to the PFSS from source selection at the high frequency. This agrees with the result by Jackson & Wall, i.e. the FSS contribution to source counts is smaller at low frequencies (<1 GHz) than at higher frequencies (>1 GHz). Another difference is that the PFSS is decreasing or constant with increasing flux density if the source selection is made at the low frequency (Fig. 3), whereas PFSS is increasing with increasing flux density for a source selection at high frequency (Fig. 8). This effect was previously mentioned by Steppe & Gopal-Krishna (1984) and Jackson & Wall (1999).
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Figure 8: The predicted percentage of flat-spectrum sources at 1.4 GHz (Jackson & Wall 1999) (solid line) and the statistical result (dashed line) using the WENSS and NVSS catalogs. |
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Table 3: Cross-identification of the WENSS and NVSS catalogs with source selection at 1.4 GHz.
Acknowledgements
X. Zhang thanks for the support from NAO, CAS and the hospitality from MPIfR during his visits in Bonn. This research is also supported by the NSFC.