Luminosity volume test for compact steep spectrum radio sources

The V/Vm test for a sample of compact steep-
spectrum sources
(August 1983)

V. K. Kapahi & D. G. Banhatti
Radio Astronomy Centre (TIFR), P. O. Box 1234,
Bangalore 560 012, India

Summary / Abstract. We apply the V/Vm test to a
subsample of compact steep-spectrum sources from a
complete sample of radio sources selected at 2.7 GHz. We
find that the <V/Vm> has a value intermediate between
those found for samples of extended steep-spectrum
sources and those of compact flat-spectrum sources. If the
sample is split into two further classes of sources having
more steep and less steep spectra, the <V/Vm> values for
these then tally roughly with those found for the extended
steep-spectrum and compact flat-spectrum classes of
sources. Implications of this result are discussed.

Keywords: V/Vm test – compact steep-spectrum sources

Introduction
The earliest surveys of radio sources showed that the
extended sources had a steep spectrum and the unresolved
compact sources had a flat (or inverted or complex)
spectrum. It has recently been realized that there are also
compact sources having a steep spectrum. These show up
prominently in surveys at frequencies >≈ 1 GHz. The
evolutionary properties of various samples of extended

steep-spectrum (ESS) and compact flat-spectrum (CFS)
sources have been extensively examined through applying
the V/Vm (i.e., luminosity-volume) test to them, among
other methods like the source count test. It has been found
that the values of <V/Vm> for these two classes are ≈ 0.74
and ≈ 0.60 respectively (see, e.g., Wills & Lynds 1978) for
uniform relativistic world-models. If there is no evolution
of the source population with cosmic epoch in a world-
model, <V/Vm> is expected to be ½.. A larger value
indicates that the sources in the population under
consideration were either more numerous and/or more
luminous in the past. So these <V/Vm> values show that the
CFS sources evolve cosmologically and that the ESS
sources evolve even more strongly. The evolutionary
properties of compact steep-spectrum (CSS) sources have
not been examined partly because they have been
recognized as a separate class only recently and also
because a large enough sample of such sources has only
recently become available (Peacock & Wall 1981; 1982).

The Sample
We have selected the 34 unresolved (with a resolution of 2
to 4 arcsec) sources with steep spectra (spectral index –d
log S / d log ν ≡ α > 0.5) from Peacock & Wall’s (1981) 2.7
GHz sample complete down to 1.5 Jy. Thirteen of the
sources do not have a measured redshift, but 5 of them have
a magnitude estimated for the optically identified galaxy or
quasar. We have estimated the redshifts of these sources
from the straight line fits (Lang et al 1975) to the m - log z
relation for radio galaxies and quasars. The remaining 8
sources are either empty fields to the limit of the PSS prints

or have faint objects associated with them. We have
calculated two sets of V/Vm values for these, taking the
redshifts to be 0.5 for one set and 1.0 for the other.

Results
The V/Vm value for each source was calculated using the
spectral index α between 2.7 GHz and 5 GHz (Peacock &
Wall 1981) in the Einstein-de Sitter world-model. The
mean value for all the 34 sources is intermediate between
the values ≈ 0.60 for CFS sources and ≈ 0.74 for ESS
sources, as found from several studies (see, e.g., Wills &
Lynds 1978). If we split the sample into the 16 sources with
α ≥ 0.7 and the remaining 18 with α < 0.7 we find that the
<V/Vm> values for these two classes are similar to those for
ESS and CFS sources respectively. The various <V/Vm>
values are tabulated in Table 1. These values, however, are
based on small samples and need to be confirmed by
formulating larger independent samples of CSS sources.

------------------------------------------------------------------
Table 1. Values of <V/Vm>
---------------------------------------------------------------
Sample No. of sources# <V/Vm>*
--------------- -------------------- -------------------
α ≥ 0.7 16 (5) 0.69 +/- 0.069
0.71 +/- 0.067
α < 0.7 18 (3) 0.58 +/- 0.063
0.59 +/- 0.064
All CSS 34 (8) 0.63 +/- 0.047
0.65 +/- 0.047
------------------------------------------------------------------------
# The numbers in parentheses indicate the number of sources for
which the redshift has been assumed.
* The upper line gives the value assuming a redshift of 0.5 for the
unidentified and optically very faint objects. The lower line
corresponds to an assumed redshift of 1.0 for these.
------------------------------------------------------------------------
Discussion
Our result that the value of <V/Vm> for CSS sources is
intermediate between those for CFS and ESS sources is
consistent with Peacock & Walls’s (1982) observation that
the percentage of such sources in surveys at many
frequencies between 408 MHz and 5000 MHz down to
various flux density levels (the lowest being 0.015 Jy at
1400 MHz) is approximately the same. Since these surveys
are an admixture of flat-spectrum and steep-spectrum
sources, this indication of the CSS source count being
roughly the same as their count is in accordance with the
<V/Vm> value we have found for the CSS sources.

It has been a general practice to divide extragalactic
radio sources into the steep-spevtrum (α > 0.5) and flat-
spectrum (α < 0.5) classes rather arbitrarily at the value α =
0.5. No satisfactory physical explanation has been given of
this division. Our result shows that, whatever be the
physical explanation for any such spectral division, the
demarcation at α = 0.5 may not be meaningful in at least
one respect, viz., cosmological evolution; and that a
division at a higher value of α may be more meaningful.
To see if the division of CSS (α > 0.5) sources into
sources with more steep (α ≥ 0.7) and less steep (α < 0.7)
spectra shows up in other properties of the sources, we
have calculated the 2.7 GHz luminosities of these sources
(in the Einstein-de Sitter world-model with H0 = 50 km.s-
1.Mpc-1). If the higher <V/Vm> value for sources with
steeper spectra is because these are more luminous than the
other class in accordance with the log P – α correlation
found for steep-spectrum sources (MacLeod & Doherty
1972), this should show in the mean luminosity of these
classes. We find that there is no significant difference
between the mean values of log P for the two classes. We
also note that the identification content (proportion of
galaxies, quasars and empty fields) is not significantly
different for the two classes. Thus, though the samples are
rather small, we conclude that the difference in the
evolutionary behaviour of these two classes of radio
sources cannot be explained by the log P – α correlation.

Conclusion
The V/Vm test applied to a sample of compact steep-
spectrum sources (α > 0.5) indicates that these sources

evolve at a rate intermediate between that for compact flat-
spectrum sources and extended steep-spectrum sources. A
division of the sample into more steep (α ≥ 0.7) and less
steep (α < 0.7) sources shows that the former class evolves
about as much as the extended steep-spectrum sources and
the latter about as much as the compact flat-spectrum
sources. This means that the phenomenological division of
extragalactic radio source spectra at α = 0.5 into flat and
steep may not be as useful as a division at a higher value
(0.7?) of α. The log P – α correlation observed in steep-
spectrum sources cannot explain the difference in
evolutionary behaviour. It should be stressed that these
conclusions rest on results for rather small samples and
should be viewed with caution.

Acknowledgments
We thank Vasant Kulkarni for critical comments.

References
Lang, K R et al (1975) ApJ 202 583-90.
MacLeod, J M & Doherty, L H (1972) Nat 238 88-9.
Peacock, J A & Wall, J V (1981) MN 194 331-49.
-------do------- (1982) MN 198 843-60.
Wills, D & Lynds, R (1978) ApJSup 36 317-58.