Suppose that we have a single population of evolving radio sources.
Further, suppose we form a flux-limited sample of these sources. Assuming
a Euclidean universe, the number of sources in the luminosity range
is 
, where 
is
the volume density of sources in that luminosity range. Since
is proportional to the time that a given source spends in this luminosity
range, we have
implying that
The evolutionary model of Section 2 allows us to determine the functions
and 
. Thus, eqn. (4) is an explicit expression for
as a function of 
. We have chosen to examine the
distribution of since it is the cocoon radius that will be
identified observationally as the half-size of the radio source.
Figure: (a) Theoretical and observed size distributions. The dotted line
shows the fine-grained (i.e. unbinned) theoretical model which, for
clarity, has not been displayed beyond 
. The filled squares show
the binned theoretical distribution using bins of size 
.
The data from Fig. 10 of OB97 (with 1- 
errors) are shown as open
squares. (b) radio luminosity as a function of for these same
(theoretical) intermittent sources.
Figure 2a shows a comparison of the observed size distribution of OB97 with
our theoretical size distribution for a single population of periodically
intermittent sources. To facilitate this comparison, we have binned the
theoretical size distribution using bins of . In order to
match the distribution of OB97, we set 
(determined by the
slope of the distribution at large sizes) and assume a burst duration of
30,000yr. All other parameters have the values of Section 2.2.
The intermittency of the sources allows the qualitative features of the
OB97 size distribution to be reproduced. In particular, there is a plateau
in the size distribution resulting from sources that are still undergoing
their first few bursts of activity. If the break in the OB97 distribution
at small sizes ( 
) is real, this could be identified as
being due to sources that are still undergoing their first burst of
activity. In this idealized case of a single evolving population, there is
fine structure within the size distribution corresponding to the distinct
cycles of activity. In practice, the stochastic nature of the parameters
in any real source population will wash out this fine structure. In
particular, the disagreement between our model and the data at 
can be resolved if we consider realistic source populations. Note that we
have not included the largest size bin of the OB97 distribution since this
is probably affected by the complete turning-off of old sources.
Figure 2b shows the radio luminosity Q as a function of the total source
size 
. This is to be compared with Fig. 9 of OB97. An important
feature of Fig. 2b is the dramatic decline in radio luminosity between the
first burst and all subsequent bursts. In other words, if one were to
assume a constantly fed source and extrapolate from large sources to small
sources, then one would substantially underestimate the small source
luminosity.