11institutetext: Astronomical Observatory of the Jagiellonian University, Orla
171 Cracow, Poland

# Studying the highly bent spectra of FR II$-$type radio galaxies with the KDA
EXT model

Elżbieta Kuligowska  e

(Received 10 July 2017)

###### Abstract

Context. The Kaiser, Dennett-Thorpe & Alexander (KDA) (1997) EXT model, that
is, the extension of the KDA model of FR(Faranoff & Riley)II$-$type source
evolution, is applied and confronted with the observational data for selected
FR II$-$type radio sources with significantly aged radio spectra.

Aims. A sample of FR II$-$type radio galaxies with radio spectra strongly bent
at their highest frequencies is used for testing the usefulness of the KDA EXT
model.

Methods. The dynamical evolution of FR II$-$type sources predicted with the
KDA EXT model is briefly presented and discussed. The results are then
compared to the ones obtained with the classical KDA approach, assuming the
source’s continuous injection and self-similarity.

Results. The results and corresponding diagrams obtained for the eight sample
sources indicate that the KDA EXT model predicts the observed radio spectra
significantly better than the best spectral fit provided by the original KDA
model.

###### Key Words.:

galaxies: active – galaxies: evolution – galaxies: jets – radio continuum:
galaxies

††offprints: E. Kuligowska<EMAIL_ADDRESS>
##  1 Introduction

The analytical models of FR II$-$type (Fanaroff & Riley 1974) radio sources
are based on the source’s internal physics and characterise them in terms of
dynamics, energetics, and luminosity evolution. These models originate from
the ”standard model” for double radio sources (Scheuer 1974, Blandford & Rees
1974) in which FR II sources are composed of two opposite jets of plasma
ejected from the vicinity of the supermassive black hole as a result of the
accretion process. A travelling jet encounters the interstellar medium (ISM)
of its host galaxy, the intergalactic medium (IGM), and finally the
environment in the galaxy cluster, the intracluster medium (ICM). Its
interaction with those media results in the formation of the terminal bow
shock and the interface between the shocked and unperturbed external material
and the relativistic electrons reaccelerated in the so-called hot spot
regions.

The more sophisticated models for FR II-type sources such as KDA (Kaiser,
Dennett-Thorpe & Alexander 1997), BRW (Blundell, Rawlings & Willott 1999) and
MK (Manolakou & Kirk 2002) differ from the standard model in the assumptions
on the transport mechanisms of the new particles between the jets, hot spots,
and lobes. While the KDA model assumes a constant injection index for the
particles, in the BRW model this index varies between the different energy
regimes. However, the MK model again assumes a constant injection index, but
also takes into account the reacceleration of the particles in the turbulent
processes occurring during their transport to the lobe. It is worth adding
that Barai and Wiita (2006) performed detailed quantitative tests of three
popular models describing the evolution of FR II radio sources (KDA, BRW and
MK) based on multidimensional Monte Carlo simulations. Their main aim was to
compare the predictions of these models with the observational data (samples
constructed based on low-frequency radio surveys revised Third Cambridge
Catalogue of Radio Sources (3CRR, Laing et al. 1983), revised 6C (6CE,
Rawlings et al. 2001) and 7C Redshift Survey (7CRS, Willot et al. 2001). Barai
and Wiita concluded that none of the examined models produce fully
satisfactory results (acceptable fits to all the properties of the samples).
However, their statistical test suggests that the KDA model still appears to
provide better results than the Manolakou & Kirk and Blundell et al. models.

The KDA EXT model originally published by Kuligowska (2017) arose from the
need to better understand the processes occurring in FR II$-$type radio
sources after the end of their nuclear activity. In contrast to the previously
developed models describing the dynamics of these sources and based on the
”standard model”, such as KDA, BRW and MK, KDA EXT was constructed on the
assumption of the lack of so-called continuous injection (hereafter CI) of new
particles into the lobes at the time at which the given source is observed.
This implies that the radio lobes are no longer powered by the central active
galactic nucleus (AGN), and the whole source’s structure begins to dissipate
at radio wavelengths.

Non-thermal continuum radio emission of the lobes is due to the synchrotron
process and to the inverse-Compton scattering of ambient photons of the cosmic
microwave background. The synchrotron radiation results from the ultra-
relativistic particles interacting with the magnetic field, with the most
energetic ones emitting energy at high radio frequencies and losing it at the
fastest rate. Thus in the case of sources with no injection of new particles
from their nucleus, one should be able to observe a substantial steepening of
their spectra. In fact such FR II-type sources, with radio spectra highly bent
at GHz frequencies, are known. Among them are both very weak radio relics
(Shulevski et al. 2015) and objects with so-called recurrent activity (Saikia
& Jamrozy 2010), as well as the more typical, powerful radio galaxies with
curved spectra (i.e., the ones modeled and discussed by Kuligowska (2017). It
is worth emphasising that the problem of evolution of FR II-type sources after
their jet activity has stopped was previously widely discussed by Kaiser &
Cotter (2002). They argue that, after the jet switches off, the information
about ceasing of the lobes should propagate with the speed of sound. If the
internal sound speed in the lobes is relatively low, the adiabatic evolution
of the cocoon is unchanged for a long period of time. Lobes may still be
overpressured with respect to the ambient gas and continue their expansion
(model A). However, if the sound speed is fast, the entire radio structure
almost immediately transforms the so-called coasting phase and its adiabatic
evolution slows down rapidly (model B).

In the present paper, the suitability of the KDA EXT model for studying the FR
II-type sources with bent spectra is further analysed. However, instead of
finding examples of such sources in a wide range of available literature or
radio catalogues, the present study is focused on the large and representative
sample of 388 bright radio sources observed at 74 MHz (Helmboldt 2008). This
sample was used for the several reasons: It is big enough to ensure the
presence of sources with various spectral shapes; it consists of sources that
are powerful at low radio frequencies, allowing reliable modelling of their
dynamics with KDA or KDA-like models; and it provides ready-to-use flux
densities of the sources, available over a wide range of the radio spectrum.
Here, the analytical formulae for the dynamics and luminosity evolution of the
lobes, given by the KDA EXT model, is applied to the eight FR II-type sources
with presumably terminated jet inflow carefully selected from the Helmboldt
2008 sample.

The organisation of this article is as follows: the summary of the KDA EXT
model and its comparison with the original KDA model is provided in Section 2.
The description of the sample of Helmboldt (2008) is given in Section 3 along
with the methodology for further selecting the sources with strongly curved
spectra, for which the available observations are sufficient to obtain
satisfactory and accurate models. Section 4 presents the method of fitting
both the KDA EXT and KDA models to the observational data of the selected
radio galaxies, as well as the results (radio spectra obtained with the
models). Finally, Section 5 presents a brief summary of these results.

##  2 The summary of KDA and KDA EXT models

###  2.1 KDA and its background

The KDA model is based on a dynamical description developed by Kaiser &
Alexander (1997) and further combined with the radiative processes analysed by
Kaiser et al. (1997). Its most important assumptions are: power-law radial
density distribution (King’s 1972 model) of unperturbed ambient gas
surrounding the radio source, continuous injection of new particles into the
jet, and energy conservation within the jet. The model predicts (among others)
the most evident observational parameters of a radio source, that is, the
total length of its jets (identified with lobes) and its total radio power at
a given frequency. The final equations for these parameters are given below,
and their derivation is presented by Kaiser et al. (1997).

The length of the jet at a given source’s age is given by

$r_{\rm j}(t)=c_{1}\left(\frac{Q_{\rm
j}}{\rho_{0}a_{0}^{\beta}}\right)^{1/(5-\beta)}t^{3/(5-\beta)},$ (1)

where $\beta$ is the exponent of the density profile, $a_{0}$ is the size of
the radio core, $\rho_{0}$ is the central density of this core, $Q_{\rm j}$ is
the jet’s power and $r_{\rm j}$ is identified with one-half of the source’s
linear size $D$, $r_{\rm j}=D/2$. If two of the model parameters, $Q_{\rm j}$
and $\rho_{0}a_{0}^{\beta}$, are specified, the model predicts the time
evolution of the source’s length.

The radio emission of the source is calculated by splitting the cocoon
surrounding the jet into infinitesimal evolving volume elements and tracking
its total adiabatic and radiative losses. The sum of the contributions from
volume elements gives the total emission (at a frequency $\nu$), $P_{\nu}(t)$,
as a integral over the injection time $t_{\rm i}$, which can then be solved
numerically.

$P_{\nu}(t)=\int\limits^{t}_{t_{\rm min}}\,dt_{\rm
i}\frac{\sigma_{T}c\;r}{6{\pi}\nu(r+1)}Q_{\rm j}n_{0}(P_{\rm
hc})^{(1-\Gamma_{\rm c})/\Gamma_{\rm c}}$ $\times\frac{\gamma^{3-p}t_{\rm
i}^{a_{1}/{3(p-2)}}}{[{t^{-a_{1}/3}}-a_{2}(t,t_{\rm
i})\gamma)]^{2-p}}\left(\frac{t}{t_{\rm i}}\right)^{-a_{1}(1/3+\Gamma_{\rm
B})},$ (2)

where $(P_{\rm hc})$, the ratio of the jet head pressure $(p_{\rm h})$ and the
uniform cocoon pressure $(p_{\rm c})$, is a function of $(R_{\rm T})$ \- the
axial ratio of the cocoon described by the empirical formula adopted from
Kaiser (2000), $\Gamma_{\rm c}$ and $\Gamma_{\rm B}$ are the adiabatic indices
in the equation of state of the cocoon material and the magnetic field,
respectively, $\sigma_{T}$ is the Thompson cross-section, $c$ is the speed of
light, $r$ is the ratio of the energy density of the magnetic field
$u{{}_{B}}$, $a_{1}$ and $a_{2}$ are the physical constants defined by Kaiser
(1997), $n_{0}$ is the number density of electrons (Heavens & Drury 1988), $p$
is the exponent of the energy distribution of jet particles, and $\gamma$ is
their Lorentz factor.

###  2.2 KDA EXT model

The idea of developing the KDA EXT model grew out of the need for an
estimation of the age of the source at which its nuclear activity starts to
cease (hereafter referred as $t_{\rm br}$). One can assume that stopping this
activity ceases the inflow of new particles into the jet and eventually
reduces the radio brightness of the entire radio structure. For such sources,
the KDA model may not be reliable (even if it can formally be fitted to the
observed data), because its basic condition of a continuous injection process
is no longer fulfilled. The extended model should then consider both the new
parameter describing the source’s dynamics ($t_{\rm br}$) and the overall
evolution of the source after the jet stops.

The KDA EXT model (Kuligowska 2017) resulting from these theoretical
considerations allows us to predict the lobe (jet) length and radio power of
the source at a given frequency. It is assumed that the termination of the
energy supply propagates from the AGN to the lobes with a relatively low sound
speed (Kaiser et al. 2000) and, after switching off the jets, the adiabatic
evolution of the lobes of the large sources does not change significantly for
a long time. In particular, the lobes may still be overpressured with respect
to the external gaseous environment and therefore continue to expand behind a
bow shock, as described by Kaiser & Cotter (2002). In the paper of Kuligowska
2017, the effective formula for lobe length, describing its adiabatic
evolution after termination of the activity and compared to the analogous
equation given by the KDA model, is based on the work of Kaiser & Cotter and
derived as:

$D(t,t_{\rm br})=\left\\{{c_{1}\left(\frac{Q_{\rm
j}}{\rho_{0}a^{\beta}_{0}}\right)^{1/{(5-\beta)}}t^{{3/{(5-\beta)}}}\hskip
20.0pt{\rm for\,}\,t<t_{\rm br}\atop{D(t_{\rm br})\left(\frac{t}{t_{\rm
br}}\right)^{\frac{2(\Gamma_{\rm c}+1)}{\Gamma_{\rm c}(7+3\Gamma_{\rm
c}-2\beta)}}\hskip 17.0pt{\rm for\,}\,t\geq t_{\rm br}}}\right.,$ (3)

where $D(t_{\rm br})$ is the cocoon length at the time of switching off the
energy supply. Hereafter the Kaiser & Cotter’s approach in the limiting case
when the internal sound speed is fast (their model B) is adopted.

The energy-loss process due to the synchrotron and inverse-Compton scattering
of the Cosmic Microwave Background (CMB) photons is characterised by the
energy break in the energy spectrum comprised of particles with different
energies. In the case of a source supplied by a constant flow of particles,
the highest break frequency (for the pitch angle $\theta=90\degr$) is given
by:

$\nu_{\rm br,CI}=C_{1}B\gamma_{\rm
br}^{2}=\frac{C_{1}}{C_{2}^{2}}\frac{B}{(B^{2}+B_{\rm iC}^{2})^{2}t^{2}},$ (4)

which is assumed to be valid for $t\leq t_{\rm br}$, where $B$ and $B_{\rm
iC}$ are the magnetic field strengths related to the synchrotron and inverse-
Compton losses, and $C_{1}$ and $C_{2}$ are the physical constants defined by
Pacholczyk (1970). However, if the pitch angle of the particle distribution is
highly isotropic (Jaffe & Perola 1973), the break frequency valid for
$t>t_{\rm br}$ is given by:

$\nu_{\rm br,JP}=\frac{C_{1}}{C_{2}^{2}}\frac{B}{\\{(2/3B^{2}+B_{\rm
iC}^{2})(t-t_{\rm br})\\}^{2}},$ (5)

where $C_{1}/C_{2}^{2}=2.51422\times 10^{12}$ and $B_{\rm iC}=0.318(1+z)^{2}$
nT.

Thus the analytical formula for the total radio power of a source at a given
frequency is the simple sum of these two integrals over time $t$. In the
context of the KDA EXT model, the first of them is the total source power
until the time of the jet switch-off, and the second one represents its radio
power emitted from $t_{\rm br}$ until the actual age of the observed source,
$t$. The total power of a source is given by:

$P_{\rm\nu}(t)=\left\\{{P_{\rm\nu}(t_{\rm min},t_{\rm br})+P_{\rm\nu}(t_{\rm
br},t)\hskip 25.0pt{\rm for}\hskip 5.0ptt_{\rm br}>t_{\rm
min}}\atop{P_{\rm\nu}(t_{\rm min},t)\hskip 84.0pt{\rm for}\hskip 5.0ptt_{\rm
br}\leq t_{\rm min}}\right.,$ (6)

where the first top term is the integral given by Equation 2 with the limit of
integration $t_{\rm min}$ changed to $t_{\rm br}$, ($t_{\rm min}$, defined by
Kaiser et al. (1997), is the minimum injection time for which the lobe
material is still radiating at a given frequency $\nu$), and the second top
term corresponds to Equation 9 of Kuligowska (2017). The lower integral refers
to the classical KDA integration given by Equation 2.

##  3 Selected sample of FR II-type sources with aged radio spectra

###  3.1 Helmboldt’s sample of 388 bright radio sources at 74 MHz

The KDA EXT model is expected to predict observational parameters (i.e., jet
length, radio power value at given frequency) of those FR II-type radio
sources in which nuclear activity is terminated or suspended. It is assumed
that the continuum injection KDA model does not provide accurate fits of their
observed radio spectra. While most of the known sources are active and can be
fit with the KDA model, KDA EXT is also able to reproduce the curved spectra
of the dying radio sources. However, it should be emphasised that KDA EXT
model is still useful in the study of active sources with the CI process: it
represents a generalisation of the classical KDA model, and it can be easily
reduced to the KDA model by setting one of its free parameters ($t_{\rm br}$)
to zero.

One can assume that jet termination usually entails a rapid decrease of the
source’s radio flux density which is most significant at the highest
frequencies (depending on the value of $t_{\rm br}$: from about 5 GHz and up
to 20$-$50 GHz or even more) - as previously mentioned in Section 1. In this
work we select sources with extremely bent spectra rather than simply the
sources with very steep radio spectra.

In the previous paper of Kuligowska (2017) six examples of such sources were
presented. This previous ”test” sample for the KDA EXT model was compiled from
literature and composed of a variety of radio galaxies, including giant radio
galaxies (GRGs) and radio sources with restarting activity. Although the final
sample included some relatively young objects, it was constructed with the key
assumption that predominantly old and rather large sources should have their
radio spectra significantly bent towards the highest frequencies. Here this
presupposition is abandoned. It is rather simply assumed that any sources with
bent spectra may be studied with the KDA EXT model, regardless of their other
observational parameters, such as linear size, age, or redshift. The
additional condition to be met by such sources is their relatively high radio
brightness - in particular, the high value of radio flux density at low
frequencies, which gives a better chance of good spectrum coverage over a wide
range of radio frequencies (cf. Sect. 3.3).

For the above reasons, the sample of 388 radio sources compiled by Helmboldt
et al. (2008) should be perfect in the search for those sources. First, it
seems to be big enough to provide a representative subsample of objects of
interest. Second, it was compiled from the available literature for all
sources within the VLA Low Frequency Sky Survey (VLSS) brighter than 15 Jy at
74 MHz, providing only the most powerful radio sources. Moreover, the sample
also contains carefully collected radio frequency spectra for these sources,
provided in the literature and compiled using the corrections needed to put
the flux densities from all the references on the same absolute flux density
scale of Baars et al. (1977). It follows that the sample of Helmboldt gives a
convenient list allowing research on the ready-made radio spectra of these
sources. It is also worth noting that the flux densities measured by VLSS are
more reliable than many others from older low-frequency surveys, and the sky
coverage of VLSS itself is large enough to provide a satisfactory spatial
diversity of the sample sources.

###  3.2 Selection criteria

As mentioned above, the primary criterion is the strong collapse of the
source’s radio spectrum at the highest frequencies. To extract such sources,
firstly all the source spectra provided by Helmboldt were visually inspected,
and all objects without highly bent spectra, typical for sources with
continuous activity, were rejected, as well as those which were evidently not
FR II-type radio sources, due to previous classification or their different
spectral shapes (i.e., compact steep spectrum (CSS) sources with spectra
peaking around 100 MHz or lower, or GHz peaked spectrum (GPS) sources with
inverted spectra). After this initial step, 47 previously selected radio
sources were carefully examined for the completeness of the observational data
required for the application of the KDA EXT model. In particular, this model
needs broad (wide enough to include very low and high frequencies) coverage of
the radio spectrum (this criterion was valid for all cases) and a redshift
value determined for the parent radio galaxy (unknown for almost one third of
the initially selected objects). However, the most important was the
availability of the radio maps. They are not only necessary to correctly
determine the length and thickness of the lobes (input parameters for both KDA
EXT and KDA models), but also to confirm the FR II-type morphology of the
given source, provided by the literature or online data catalogues. In
particular, radio maps usually clearly show the presence of bright radio lobes
and hot spots. Unfortunately, good-quality radio maps are not available in the
case of 15 sources of the preliminary sample. Moreover, despite the shapes of
their radio spectra, the maps of two previously selected sources (3C048 and
NGC1692) do not show the characteristic FR II morphology. After further
verification, 3C048 turns out to be an example of a core-dominated source
(Laing et al. 1983), while NGC1692 (0453$-$20) was classified as a source with
”diffuse and amorphous radio emission” by Morganti et al. (1993). The radio
galaxy 3C217 fulfils all the selection criteria, however it was previously
examined with the same KDA EXT model in the earlier paper by Kuligowska
(2017). For this reason its KDA EXT analysis is not repeated in this work, and
the reader is referred to that publication.

In the next step, the progression of the radio spectrum slope for the
remaining 26 sources was further investigated. The spectral index
$\alpha_{\nu}$, changing along with the spectrum, was measured and its slope
gradient was calculated for every pair of neighbouring radio flux densities.
Here the author assumes that the so-called high curvature (bend) of the given
spectrum must result in: 1) high ($\alpha_{\nu}\sim$ 1.5) spectral indices
occurring in the highest-frequency part of the spectrum, implying a high-
frequency slope significantly exceeding the limiting value of $\alpha_{\rm
inj}+0.5$ expected in the CI process; 2) violent steepening of the spectrum
between the frequencies in the range of 1400 MHz up to 4800, 10000 MHz or even
20000 MHz, giving an increment of $\alpha_{\nu}$ of nearly one between these
data points; and 3) relatively low spectral index (not exceeding 0.6) for the
low-frequency flux densities (26$-$178 MHz).

The above considerations finally led to the further limitation of the initial
sample to nine sources: 3C006.1, 3C032, 3C041, 3C123, 3C247, 3C300, 3C325,
3C401 and PKS0235-19. Further, 3C300 was excluded because its radio spectrum
is very irregular and does not allow for a reliable fit of the KDA EXT model
due to its repeated breaks and flattening at higher frequencies. The final
sample includes almost exclusively 3C sources and consists of objects of
rather mid-scale linear sizes (68$-$386 kpc) and relatively low (z up to 0.84)
redshifts, except 3C325 whose high (z$=$1.135) redshift makes this source
especially interesting from the viewpoint of studying the processes of
interruption of AGN activity.

In summary, the final number of dying FR II-type sources selected from
Helmboldt’s compilation and fulfilling the formal criteria (specific
steepening of their spectra) is of the order of 21, which is 5.4$\%$ of the
initial sample. However, only 8 of these are further investigated here for the
reasons given above (lack of sufficient data, irregular spectrum). At this
point it would be interesting to provide the percentage of all the sources
with terminated jet activity drawn from the complete population of FR II- and
FR I-type sources. Unfortunately, this is rather problematic due to the
difficulty in estimating the total number of such radio sources as well as
detecting the weak radio emission of dying radio galaxies. However, Giovannini
et al. (1988) argue that only a few percent of all radio sources are dying,
based on data from Bologna (B2, Colla et al. 1970) and Third Cambridge
Catalogue of Radio Sources (3C, Edge et al. 1959) catalogues. This result is
in line with the percentage obtained for Helmboldt’s sample. It is also worth
emphasising (Murgia et al. 2011) that dying radio sources are not synonymous
with the radio relics, typically not associated with individual galaxies.

Table 1: Observational data for FR II-type sources selected from the sample of Helmboldt (2008). Name |  | 3C006.1 | 3C32 | 3C41 | 3C123 | 3C247 | 3C325 | 3C401 |  PKS 0235$-$19
---|---|---|---|---|---|---|---|---|---
$z$ |  |  0.840$\pm$ 0.0004 | 0.400 |  0.795$\pm$ 0.001 |  0.2177$\pm$ 0.0003 | 0.7489 | 1.135 | 0.201 | 0.620
$LAS[\arcsec]$ |  |  32$\pm$ 3 |  72$\pm$ 1 |  26$\pm$ 1 |  41$\pm$ 3 |  25$\pm$ 3 |  29$\pm$ 4 |  24$\pm$ 2 |  45$\pm$ 2
$R_{t}$ |  |  3.3$\pm$ 2.2 |  4.6$\pm$ 0.6 |  4.6$\pm$ 1.6 |  4.0$\pm$ 1.5 |  1.5$\pm$ 0.7 |  1.7$\pm$ 1.0 |  3.0$\pm$ 1.7 |  1.5$\pm$ 0.27
$D[kpc]$ |  |  246$\pm$ 23 |  386$\pm$ 5 |  200$\pm$ 8 |  123$\pm$ 11 |  188$\pm$ 22 |  235$\pm$ 33 |  68$\pm$ 7 |  310$\pm$ 14
$S_{74}[Jy]$ |  |  23.93 $\pm$ 2.39 |  38.63 $\pm$ 3.86 |  19.68 $\pm$ 1.97 |  421.96 $\pm$ 42.20 |  17.29 $\pm$ 1.73 |  28.28 $\pm$ 2.83 |  31.38 $\pm$ 3.14 |  35.66 $\pm$ 3.57

###  3.3 The summary of observational data for the sample sources

Table 1 presents the observational data of the sample sources used for testing
the usefulness of the KDA EXT model. Redshifts are given with the highest
known accuracy available in the literature. Despite the undefined values of
some redshift uncertainties, it is assumed that the errors of the given
spectroscopic redshifts are very small and do not affect the result of
modelling. The ‘largest angular sizes’ ($LAS$) and $R_{t}$ values of axial
sources are estimated based on the accessible radio maps (along with the
uncertainties given by their angular resolution). The projected linear sizes
of the sources, $D$, are calculated following the formula $D=4.848\times
10^{-6}L_{\rm A}(z)LAS$ and their measurement uncertainties are calculated
with the partial derivatives. Another model parameter, inclination angle of
the jet axis $\theta$, is not listed in the table because in the case of all
the analysed sources it can be assumed to be close to 90 degrees (sources are
not significantly inclined to the line of sight). Moreover, it can be
demonstrated that a slight deviation of this angle does not significantly
affect the modelling results.

Table 1 also includes the flux density values of the sources at 74 MHz. The
remaining flux densities for a large set of frequencies and their errors are
derived from Helmboldt (2008). It is worth noting here that the original
uncertainties on the measurements are often underestimated (further discussed
in the following Section). Due to the large number and variety of the
available flux densities, these data are not listed in the table, but are
provided in the individual source summaries (Sect. 4).

Before fitting the model parameters (cf. following Section), all the flux
densities derived following Helmboldt were inspected and some clearly
undervalued data points (i.e. high frequency measurements covering the hot
spots only) were removed from the input sets for the certain sources.
Additionally, the 150 MHz TIFR GMRT Sky Survey (TGSS, Intema et al. 2017) flux
measurements available for all eight sources were added and used in the
modelling.

The values of the sources’ radio power $P_{\nu}$ are calculated from the given
flux densities according to the formula $P_{\nu}=S_{\nu}L_{\rm
D}(z)^{2}(1+z)^{(\alpha_{\nu}-1)}$, where $L_{\rm D}(z)$ is the luminosity
distance of the given source determined assuming a flat Universe with Hubble
constant $H_{0}$=71 ${\rm km\,s^{-1}Mpc^{-1}}$ and the Lambda cold dark matter
($\Lambda$CDM) model with cosmological $\Omega_{m}$=0.27 and
$\Omega_{\Lambda}$=0.73, and where $\alpha_{\nu}$ is the spectral index
measured as the spectrum slope gradient calculated separately for every pair
of neighbouring radio flux densities. Moreover, determining the initial
dynamical parameters for both the KDA and KDA EXT models requires at least
five sources’ radio power $P_{\nu}$ values at significantly distant observing
frequencies for the reliable application of the models.

We include recent 150 MHz TGSS flux densities to increase the number of
measurements and accuracy of the calculations. However, for the sources 3C41,
3C123, and PKS 0235-19 the 150 MHz data points clearly stand out from the
trend lines. In these cases, the use of the TGSS flux densities in the
modelling does not improve the accuracy of the fits, and the use of these
values is debatable. Moreover, the TGSS flux densities of some sources are
recognised as having systematically underestimated values (up to even 50%).
Special fixes to the original TGSS, resulting in the revised Rescaled Subset
of the Alternative Data Release 1 (TGSS-RSADR1) catalogue, were recently
proposed by Hurley-Walker (2017). Thus TGSS flux densities for these three
sources have been verified by cross checking bright point-like sources in
their vicinity (comparing their TGSS flux densities with the flux densities
observed at close frequencies). For 3C41, both the target and the sources in
its close neighbourhood have their TGSS flux densities a few percent lower
than the values obtained in the 6CVI survey at 151 MHz. In the case of 3C123
and PKS 0235-19 the situation is more complex. Their TGSS flux densities (and
those of the sources in their vicinity) are a few percent larger than the
corresponding flux densities derived from the GaLactic and Extragalactic All-
sky MWA survey (GLEAM, Hurley-Walker et al. 2017) at 150.5 MHz. Also, the
TGSS-RSADR1 catalogue is not fully conclusive here: for 3C123, it provides a
flux density value even lower than the one obtained with the original TGSS
(271.70 vs. 272.08 Jy), while in the case of PKS 0235-19 the flux is actually
larger and has a value of 24.20 Jy (vs. 24.12 Jy for TGSS). In the case of
3C41 there is no available flux density provided by TGSS-RSADR1 due to its
limited range of declination. Nevertheless it can be clearly proven that these
small differences in the flux densities at 151 MHz do not significantly affect
the results of the modelling and the final statistical tests used to examine
the compared models (cf. Sect. 4). In fact, changing these TGSS values to the
values given by TGSS-RSADR1 only results in a difference in the third decimal
place for the calculated likelihood ratios.

##  4 Application of the KDA EXT model to the sample sources and its
comparison to the KDA fits.

###  4.1 Fitting procedure

In the theoretical considerations, the only unknown parameters are the
source’s physical parameters, such as injection index $\alpha_{\rm inj}$, age
$t$, the jet’s power $Q_{\rm jet}$ , and central density $\rho_{0}$. However,
the original KDA model only enables the prediction of the time evolution of
the source’s observational parameters (jet length $D$, radio luminosity
$P_{\nu}$ at a given frequency, the shape of radio spectrum, and the lobe’s
volume $V$) with the previous assumption about the intrinsic dynamics of this
source, defined by the physical quantities mentioned above. This implies that
it is first necessary to solve the reverse problem of estimating the initial
values of $t$, $\alpha_{\rm inj}$, $Q_{\rm jet}$, and $\rho_{0}$ in order to
subsequently fit the proper KDA or KDA EXT model to the observed source
parameters. This was performed using the DYNAGE algorithm (Machalski et al.
2007). In this step, the age resulting from the KDA model (tKDA) for the low-
frequency parts of the spectra of the sample sources was determined. Here the
high-frequency flux densities mostly contributing to the high curvatures of
the observed source spectra at above 2000 MHz were excluded from the input
data. Additionally, the appropriate correction for the changed length of the
jets was taken into account in the KDA model, neglecting the input flux
densities at these high frequencies results in fitting the younger age of the
source and the lower value of the length of its lobes). This has provided the
estimation of the approximate values of four model parameters: $\alpha_{\rm
inj}$, the source’s dynamical age, $t$, jet power, $Q_{j}$; and the central
core density, $\rho_{0}$. The remaining model parameters were also adopted as
in Kuligowska (2007), for example, $\Gamma_{\rm c}$=$\Gamma_{\rm
a}$=$\Gamma_{\rm B}$=$5/3$ for the ”cold” equation of state, and $\beta$=1.5.

Determining the so-called “best age solution” for the initial KDA fit demanded
testing some model parameters: here it was assumed that the source’s age, $t$,
may vary from $10^{4}$ up to $10^{8}$ years, while the $\alpha_{\rm inj}$ is
from the rather wide range of 0.3 up to 1.0, and is fitted with the accuracy
of 0.01. In the first (fitting the KDA values) and the second (determining
$t_{br}<t$) steps, the best age solutions were fitted with an accuracy of 0.01
million years (Myr). It should be noted that $t_{br}<t$ solutions are
unambiguous: there are clear and single minima of the obtained $\chi^{2}$
values in the function of tested $t_{br}<t$ values.

In the second step, the values of source power $P_{\nu_{em}}$ given by
Equation 6 at the given frequencies for a number of arbitrarily chosen values
for the jet switch-off time $t_{br}$, fulfilling the $t_{br}<t$ condition,
were calculated. Then, the best fit of these model results was determined
using the least-squares method by minimizing the expression:

$\displaystyle\chi^{2}_{red}=\frac{1}{n-r-1}\sum\limits_{n}\left(\frac{S_{\nu_{0}}-S_{MOD}}{\Delta
S_{\nu_{0}}}\right)^{2},$ (7)

where n is the number of the compared values (flux densities), r is the number
of the model’s free parameters, $S_{\nu_{0}}$ and $\Delta S_{\nu_{0}}$ are
flux densities, and $S_{MOD}$ are the model flux densities re-calculated from
the model values of the emitted power $P_{\nu_{em}}$ according to

$\displaystyle S_{\rm
MOD}=P_{\nu_{em}}\left(\frac{1+z}{L_{D}^{2}(z)}\right)=P_{\nu_{0}(1+z)}\left(\frac{1+z}{L_{D}^{2}(z)}\right).$
(8)

The $S_{\rm MOD}$ flux densities, resulting from the KDA (best age solution)
and KDA EXT models, are given in Tables 2$-$9 along with the values of
$\chi^{2}$, determining the goodness of the resulting fits, and the numbers of
their free parameters, r. The accuracy of both the fitted models is compared
to the observed flux densities representing the real radio spectra of the
sample sources. In addition, the results of the likelihood ratio test
($\Lambda$) were calculated for the compared KDA and KDA EXT best model fits,
based on their $\chi^{2}$ distributions and for the assumed level of
significance of 10$\%$ (0.01), in order to obtain the reliable statistical
tool for the model selection (providing the formal statistical preference of
one model over another). The use of this tool is justified here by the fact
that the KDA and the KDA EXT models are nested (KDA EXT is a more general case
of the KDA, meaning all the parameters of the simpler KDA model also occur in
the KDA EXT, and the KDA model may be derived directly from KDA EXT by setting
one of its free parameters, $t_{br}$, to 0). It is worth noting that the KDA
EXT has only one more degree of freedom than the KDA (associated with
$t_{br}$).

The formal values of the flux density uncertainties resulting from the KDA and
KDA EXT models are also provided. These uncertainties include both the
uncertainties of the calculated radio powers ($P_{\rm\nu}$), and the errors of
the redshift values known from the literature. It should be noted, however,
that these uncertainties are rather crude, especially due to the lack of
reliable redshift uncertainties. They are known only for three of the eight
analysed sources (cf. Table 1), and for the remaining ones the rough values of
the redshift error are equal to 0.001 (the higher known redshift uncertainty
from the examined sources sample) have been adopted. As a result, the final
flux density errors may be overvalued or underestimated. Such approximate
uncertainties, however, are only supposed to provide an overall view on the
capabilities and limitations of the KDA and KDA EXT models (cf. Sect. 5).

###  4.2 Fitting results

Table 10 presents the values of the model free parameters and derived physical
parameters of the sample sources resulting from the best fit of the KDA and
KDA EXT models. These derived parameters are the cocoon pressure, $p_{\rm c}$;
the total emitted energy, Uc; the strength of the magnetic field, Beq; and the
radial expansion speed of the cocoon’s head, vh. It is worth noting that their
values are calculated for t $>$ tbr (after the switch-off), analogous to the
formulae given by Kuligowska (2017). Namely, v${}_{\rm h}(t)$ and $p_{\rm c}$
for t¿tbr are calculated with Equations 11 and 12 of Kuligowska 2017,
respectively, and Uc is given by the formula U${}_{\rm c}(t)$=u${}_{\rm
c}(t)$V${}_{\rm c}(t)$=p${}_{\rm c}(t)$V${}_{\rm c}(t)/(\Gamma_{c}-1)$. A
compilation of the results obtained for the individual sources from the sample
is presented below. In this paper the given errors have been enlarged due to
their initial incompleteness, presumably resulting from not taking into
account all of the error components. Following the typical percentage error
values estimated, for example, for VLSS flux densities, it was arbitrarily
assumed that the actual error values can be up to 10% of the given flux
density for the observational frequency range 20$-$750 MHz, and equal to about
5% for the higher frequencies. The $\chi^{2}$ values in Tables 2-9 are derived
with the errors defined in the above way.

####  4.2.1 3C006.1

3C006.1 is an example of classical FR II-type radio source with long lobes and
distinct hot spots. It lies at relatively high redshift (z$=$0.84) and was
previously studied by Neff et al. (1995). It is worth noting that the radio
core of this galaxy was not detected in the VLA maps at 6 cm, and the authors
argue that its nucleus may have changed its spatial orientation. This guess is
based on radio morphology of the source: a faint inner lobe structure is
visible on VLA radio maps at 20 cm.

The radio spectrum coverage for this source extends from 74 MHz up to 21.7
GHz. Such a broad spectrum usually results in reliable KDA EXT or KDA model
fits to the observed flux densities. However, in this case the strong bend
typical for old FR II-type sources is visible only at extremely high
frequencies (starting from 10.7 GHz), and both fitted models are rather poor,
especially in comparison with other objects from the sample. This source also
has a very flat low-frequency part of the spectrum (spectral index between 74
and 178 MHz equal to 0.45), making it difficult to determine the initial
parameters of the models. As a result, spectra obtained from the predicted
flux densities $S_{\rm MOD}\,(KDA)$ and $S_{\rm MOD}\,(KDA\,EXT)$ (Fig. 1)
depart from both low and high-frequency data points, though the KDA EXT
visually fits better to the observed flux densities and the provided result
for the likelihood ratio test clearly indicates a preference for the KDA EXT
over the KDA model.

Table 2: Model flux densities $S_{MOD}$ from the KDA and KDA EXT fitting for 3C006.1 and their goodness of fit to the observed data $S_{\nu_{0}}$. $\nu_{0}$ [MHz] |  $S_{\nu_{0}}\pm\Delta$ [Jy] |  $S_{MOD}$ [Jy] |  $S_{MOD}$ [Jy]
---|---|---|---
|  | KDA EXT | KDA
74 |  23.93 $\pm$ 2.39 |  32.32 $\pm$ 3.24 |  31.44 $\pm$ 3.24
150 |  18.27 $\pm$ 1.83 |  19.21 $\pm$ 1.93 |  18.48 $\pm$ 1.93
178 |  15.13 $\pm$ 1.51 |  16.84 $\pm$ 1.69 |  16.36 $\pm$ 1.69
232 |  13.45 $\pm$ 1.34 |  13.69 $\pm$ 1.37 |  13.23 $\pm$ 1.37
325 |  9.96 $\pm$ 1.00 |  10.46 $\pm$ 1.05 |  9.98 $\pm$ 1.05
1400 |  3.65 $\pm$ 0.36 |  3.06 $\pm$ 0.30 |  3.00 $\pm$ 0.30
2260 |  2.20 $\pm$ 0.22 |  2.01 $\pm$ 0.20 |  1.97 $\pm$ 0.20
2695 |  1.87 $\pm$ 0.19 |  1.73 $\pm$ 0.17 |  1.68 $\pm$ 0.17
3900 |  1.42 $\pm$ 0.14 |  1.25 $\pm$ 0.13 |  1.20 $\pm$ 0.13
4800 |  1.10 $\pm$ 0.06 |  1.03 $\pm$ 0.10 |  1.00 $\pm$ 0.10
4900 |  1.07 $\pm$ 0.05 |  1.02 $\pm$ 0.10 |  0.98 $\pm$ 0.10
5000 |  1.03 $\pm$ 0.05 |  0.99 $\pm$ 0.10 |  0.96 $\pm$ 0.10
7700 |  0.64 $\pm$ 0.03 |  0.66 $\pm$ 0.07 |  0.65 $\pm$ 0.07
8390 |  0.60 $\pm$ 0.03 |  0.61 $\pm$ 0.06 |  0.60 $\pm$ 0.06
10700 |  0.46 $\pm$ 0.02 |  0.47 $\pm$ 0.05 |  0.48 $\pm$ 0.05
11200 |  0.42 $\pm$ 0.02 |  0.45 $\pm$ 0.05 |  0.46 $\pm$ 0.05
14900 |  0.27 $\pm$ 0.01 |  0.31 $\pm$ 0.03 |  0.35 $\pm$ 0.03
21740 |  0.14 $\pm$ 0.01 |  0.18 $\pm$ 0.02 |  0.25 $\pm$ 0.02
$r$ |  | 4 | 3
$\chi^{2}$ |  | 4.97 | 21.39
|  $\Lambda$=0.03 |  ($p$=0.1) |

####  4.2.2 3C32

The radio source 3C32 is the largest source in the examined sample, with its
lobes totalling 385.7 kpc in length. Its central AGN have been classified as
Seyfert type 2 galaxy by Véron-Cetty & Véron (2006). The spectra resulting
from the predicted flux densities $S_{\rm MOD}\,(KDA)$ and $S_{\rm
MOD}\,(KDA\,EXT)$ are shown in Figure 2. In this case, the preference of the
KDA EXT model over KDA model is evident, with the $\Lambda$ value of 0.002 for
a level of significance of 10$\%$. However, the goodness of the fit to data
points ($\chi^{2}$ is not high and some observational data points are very far
away from the trend line (compare Table 3)).

Table 3: Model flux densities $S_{MOD}$ from the KDA and KDA EXT fitting for 3C32 and their goodness of fit to the observed data $S_{\nu_{0}}$. $\nu_{0}$ [MHz] |  $S_{\nu_{0}}\pm\Delta$ [Jy] |  $S_{MOD}$ [Jy] |  $S_{MOD}$ [Jy]
---|---|---|---
|  | KDA EXT | KDA
74 |  38.63 $\pm$ 3.86 |  44.37 $\pm$ 4.47 |  51.89 $\pm$ 5.23
150 |  27.82 $\pm$ 2.78 |  27.30 $\pm$ 2.75 |  27.82 $\pm$ 2.80
160 |  23.60 $\pm$ 2.36 |  25.59 $\pm$ 2.58 |  26.43 $\pm$ 2.66
365 |  14.23 $\pm$ 1.42 |  13.40 $\pm$ 1.35 |  12.36 $\pm$ 1.25
750 |  8.59 $\pm$ 0.86 |  7.28 $\pm$ 0.73 |  6.20 $\pm$ 0.62
1400 |  4.20 $\pm$ 0.21 |  4.14 $\pm$ 0.42 |  3.36 $\pm$ 0.34
2695 |  2.11 $\pm$ 0.11 |  2.19 $\pm$ 0.22 |  1.74 $\pm$ 0.18
2700 |  2.25 $\pm$ 0.11 |  2.19 $\pm$ 0.22 |  1.74 $\pm$ 0.18
4850 |  1.23 $\pm$ 0.06 |  1.18 $\pm$ 0.12 |  0.96 $\pm$ 0.10
5009 |  1.17 $\pm$ 0.06 |  1.14 $\pm$ 0.11 |  0.93 $\pm$ 0.09
10695 |  0.31 $\pm$ 0.02 |  0.43 $\pm$ 0.05 |  0.43 $\pm$ 0.04
$r$ |  | 4 | 3
$\chi^{2}$ |  | 9.80 | 23.37
|  $\Lambda$=0.002 |  ($p$=0.1) |

####  4.2.3 3C41

Radio source 3C41 shows classic extended FR II-type morphology and lies at
relatively high $z$. It is worth noting that its KDA EXT model fit, presented
on Figure 3, seems very reliable, with a resulting $\chi^{2}$ value almost ten
times less than the corresponding value for KDA model. The likelihood ratio
test, however, suggests that there is no significant preference of the KDA EXT
model over KDA, or at least the results of both models are statistically
comparable, with $\Lambda$ slightly higher than the assumed significance
level. The spectra calculated from both fitted models are also given in Table
4. Interestingly, in the case of 3C41, the flat low-frequency part of the KDA
EXT radio spectrum visually fits the observed data well, regardless of the
poor statistical $\Lambda$ result.

Table 4: Model flux densities $S_{MOD}$ from the KDA and KDA EXT fitting for 3C41 and their goodness of fit to the observed data $S_{\nu_{0}}$. $\nu_{0}$ [MHz] |  $S_{\nu_{0}}\pm\Delta$ [Jy] |  $S_{MOD}$ [Jy] |  $S_{MOD}$ [Jy]
---|---|---|---
|  | KDA EXT | KDA
74 |  19.68 $\pm$ 1.97 |  21.47 $\pm$ 2.98 |  29.71 $\pm$ 2.16
112 |  15.51 $\pm$ 1.55 |  17.33 $\pm$ 2.25 |  22.33 $\pm$ 1.74
150 |  12.20 $\pm$ 1.22 |  14.75 $\pm$ 1.82 |  18.11 $\pm$ 1.48
160 |  14.20 $\pm$ 1.42 |  14.32 $\pm$ 1.74 |  17.29 $\pm$ 1.44
232 |  11.21 $\pm$ 1.12 |  11.69 $\pm$ 1.32 |  13.15 $\pm$ 1.18
325 |  9.39 $\pm$ 0.94 |  9.63 $\pm$ 1.03 |  10.21 $\pm$ 0.98
365 |  9.40 $\pm$ 0.94 |  9.01 $\pm$ 0.94 |  9.33 $\pm$ 0.91
1400 |  3.71 $\pm$ 0.19 |  3.82 $\pm$ 0.32 |  3.22 $\pm$ 0.39
1410 |  3.65 $\pm$ 0.18 |  3.80 $\pm$ 0.32 |  3.21 $\pm$ 0.38
1670 |  3.43 $\pm$ 0.17 |  3.37 $\pm$ 0.28 |  2.79 $\pm$ 0.34
2695 |  2.26 $\pm$ 0.11 |  2.36 $\pm$ 0.19 |  1.88 $\pm$ 0.24
2700 |  2.30 $\pm$ 0.12 |  2.35 $\pm$ 0.19 |  1.88 $\pm$ 0.24
4850 |  1.47 $\pm$ 0.07 |  1.47 $\pm$ 0.12 |  1.16 $\pm$ 0.15
5000 |  1.46 $\pm$ 0.07 |  1.44 $\pm$ 0.11 |  1.12 $\pm$ 0.14
10695 |  0.76 $\pm$ 0.04 |  0.73 $\pm$ 0.06 |  0.59 $\pm$ 0.07
14900 |  0.51 $\pm$ 0.03 |  0.53 $\pm$ 0.05 |  0.45 $\pm$ 0.05
31400 |  0.21 $\pm$ 0.01 |  0.24 $\pm$ 0.02 |  0.24 $\pm$ 0.02
$r$ |  | 4 | 3
$\chi^{2}$ |  | 1.62 | 15.37
|  $\Lambda$=0.11 |  ($p$=0.1) |

####  4.2.4 3C123

This next radio source from the 3C catalogue is classified as a FR II-type
galaxy with extended tails of both lobe components (Fanaroff & Riley 1974). It
is also characterised by a relatively low redshift (z$=$0.218, Spinrad et al.
1985) and short lobes (compare Table 1). 3C123 also shows exceptionally high
radio luminosity and non-typical radio structure (Riley & Pooley 1978). Its
hot spots along with the general spectral analysis were studied in detail by
Looney & Hardcastle (2000) with the highest resolution to date (0.6
$\arcsec$). The authors concluded that 3C123 may have two pairs of (primary
and secondary) hot spots, and there is no clear evidence for a continuing
outflow between these hot spots, suggesting that the source may be undergoing
recurrent activity. If this source is actually restarting, one could argue it
would be more effective to use the model of Brocksopp et al. (2011) which also
introduces a free parameter related to the age at which energy supply to the
jet stops, tj. Nevertheless, 3C123 meets the strict selection criteria
described in Section 3.2 and here it is simply assumed that the application of
the KDA EXT model to the case of this source is also both justified and
interesting.

The radio spectrum of this source is very broad thanks to its high radio
brightness. Surprisingly, analysis shows that there is no major difference
between the spectra resulting from the predicted flux densities for KDA EXT
and KDA models (Fig. 4), though the KDA EXT fit is slightly more consistent
with observations. There is also no statistical preference of the KDA EXT
model, according to the obtained $\Lambda$ value for the adopted level of
significance (Table 5).

Table 5: Model flux densities $S_{MOD}$ from the KDA and KDA EXT fitting for 3C123 and their goodness of fit to the observed data $S_{\nu_{0}}$. $\nu_{0}$ [MHz] |  $S_{\nu_{0}}\pm\Delta$ [Jy] |  $S_{MOD}$ [Jy] |  $S_{MOD}$ [Jy]
---|---|---|---
|  | KDA EXT | KDA
26 |  840.10 $\pm$ 84.00 |  981.23 $\pm$ 98.38 |  1020.40 $\pm$ 102.31
38 |  628.26 $\pm$ 62.83 |  882.61 $\pm$ 88.46 |  913.63 $\pm$ 91.61
74 |  421.96 $\pm$ 42.20 |  691.46 $\pm$ 69.33 |  707.57 $\pm$ 70.95
80 |  342.00 $\pm$ 34.20 |  443.37 $\pm$ 44.46 |  446.45 $\pm$ 44.76
150 |  272.08 $\pm$ 27.21 |  274.26 $\pm$ 27.52 |  271.14 $\pm$ 27.12
160 |  226.60 $\pm$ 22.66 |  256.90 $\pm$ 25.78 |  255.13 $\pm$ 25.58
178 |  208.25 $\pm$ 20.83 |  237.56 $\pm$ 23.82 |  235.92 $\pm$ 23.66
318 |  135.45 $\pm$ 13.55 |  153.38 $\pm$ 15.37 |  150.93 $\pm$ 15.13
325 |  151.07 $\pm$ 15.11 |  150.58 $\pm$ 15.09 |  148.17 $\pm$ 14.86
327 |  148.88 $\pm$ 14.89 |  149.89 $\pm$ 15.01 |  147.49 $\pm$ 14.79
408 |  122.34 $\pm$ 12.23 |  126.12 $\pm$ 12.65 |  123.53 $\pm$ 12.39
750 |  80.10 $\pm$ 8.01 |  76.87 $\pm$ 7.71 |  74.95 $\pm$ 7.51
1400 |  49.73 $\pm$ 2.49 |  45.16 $\pm$ 4.53 |  43.63 $\pm$ 4.37
2695 |  27.50 $\pm$ 1.38 |  25.16 $\pm$ 2.052 |  24.14 $\pm$ 2.37
2700 |  23.35 $\pm$ 1.17 |  25.16 $\pm$ 2.052 |  24.09 $\pm$ 2.42
4850 |  16.03 $\pm$ 0.80 |  14.65 $\pm$ 1.48 |  13.89 $\pm$ 1.40
8420 |  10.15 $\pm$ 0.51 |  8.67 $\pm$ 0.87 |  8.12 $\pm$ 0.81
10500 |  7.90 $\pm$ 0.40 |  7.01 $\pm$ 0.70 |  6.53 $\pm$ 0.65
10695 |  8.18 $\pm$ 0.41 |  6.88 $\pm$ 0.69 |  6.41 $\pm$ 0.64
14900 |  5.30 $\pm$ 0.27 |  4.99 $\pm$ 0.50 |  4.60 $\pm$ 0.46
15300 |  5.41 $\pm$ 0.27 |  4.85 $\pm$ 0.49 |  4.47 $\pm$ 0.45
22000 |  3.45 $\pm$ 0.17 |  3.39 $\pm$ 0.34 |  3.10 $\pm$ 0.31
$r$ |  | 4 | 3
$\chi^{2}$ |  | 2.96 | 9.72
|  $\Lambda$=0.72 |  ($p$=0.1) |

####  4.2.5 3C247

3C247 is another example of a FR II-type radio galaxy at rather high redshift.
It was previously studied by Laing et al. (1983) who suggested that its parent
galaxy coincides with a weak radio core. Its Multi Element Radio Linked
Interferometer Network (MERLIN) radio map (Chidi et al. 1994) shows rather
thin and elongated lobes with bright hot spots. The spectra resulting from the
predicted flux densities for KDA EXT and KDA models are presented in Figure 5,
and the exact flux density values, along with the goodness of the fit to the
data and the likelihood ratio analysis for the models, are listed in Table 6.
Although visually the KDA EXT model predicts the observed spectra of 3C247
with better accuracy, the resulting $\Lambda$ value does not confirm a
preference for this model over the KDA model.

Table 6: Model flux densities $S_{MOD}$ from the KDA and KDA EXT fitting for 3C247 and their goodness of fit to the observed data $S_{\nu_{0}}$. $\nu_{0}$ [MHz] |  $S_{\nu_{0}}\pm\Delta$ [Jy] |  $S_{MOD}$ [Jy] |  $S_{MOD}$ [Jy]
---|---|---|---
|  | KDA EXT | KDA
74 |  17.29 $\pm$ 1.73 |  18.64 $\pm$ 1.87 |  23.74 $\pm$ 2.39
82 |  16.50 $\pm$ 1.65 |  17.72 $\pm$ 1.78 |  22.06 $\pm$ 2.22
150 |  13.64 $\pm$ 1.36 |  13.02 $\pm$ 1.31 |  14.08 $\pm$ 1.42
232 |  10.31 $\pm$ 1.03 |  10.22 $\pm$ 1.02 |  10.11 $\pm$ 1.02
325 |  8.40 $\pm$ 0.84 |  8.37 $\pm$ 0.84 |  7.74 $\pm$ 0.78
327 |  8.63 $\pm$ 0.86 |  8.33 $\pm$ 0.84 |  7.70 $\pm$ 0.77
365 |  7.92 $\pm$ 0.79 |  7.77 $\pm$ 0.78 |  7.04 $\pm$ 0.71
408 |  7.01 $\pm$ 0.70 |  7.25 $\pm$ 0.73 |  6.44 $\pm$ 0.65
750 |  4.93 $\pm$ 0.49 |  4.81 $\pm$ 0.48 |  3.89 $\pm$ 0.39
1400 |  2.88 $\pm$ 0.14 |  3.02 $\pm$ 0.30 |  2.28 $\pm$ 0.23
2695 |  1.60 $\pm$ 0.08 |  1.75 $\pm$ 0.18 |  1.29 $\pm$ 0.13
2700 |  1.66 $\pm$ 0.08 |  1.75 $\pm$ 0.18 |  1.29 $\pm$ 0.13
4800 |  0.93 $\pm$ 0.05 |  1.02 $\pm$ 0.10 |  0.77 $\pm$ 0.08
4850 |  1.01 $\pm$ 0.05 |  1.01 $\pm$ 0.10 |  0.77 $\pm$ 0.08
4900 |  0.96 $\pm$ 0.05 |  1.00 $\pm$ 0.10 |  0.76 $\pm$ 0.08
5000 |  0.940 $\pm$ 0.05 |  0.98 $\pm$ 0.09 |  0.74 $\pm$ 0.07
10695 |  0.400 $\pm$ 0.02 |  0.42 $\pm$ 0.04 |  0.38 $\pm$ 0.04
14900 |  0.250 $\pm$ 0.01 |  0.28 $\pm$ 0.03 |  0.28 $\pm$ 0.03
$r$ |  | 4 | 3
$\chi^{2}$ |  | 1.35 | 11.63
|  $\Lambda$=0.34 |  ($p$=0.1) |

####  4.2.6 3C325

With $z=$1.135, 3C325 is the most redshifted galaxy in the sample. It is
characterised by a highly curved radio spectrum and a large radio structure
with bright lobes. Its classical FR II-type morphology is associated with a
parent quasar (Véron-Cetty & Véron 2006). Preference of the KDA EXT over KDA
solution is not evident: the calculated $\Lambda$ value is exactly equal to
the assumed significance level for the adopted statistical test. Figure 6
shows that this model fits the high-frequency part of the radio spectrum
especially well, while departing from the low flux density data points except
for the 150 MHz TGSS flux (see also Table 7).

Table 7: Model flux densities $S_{MOD}$ from the KDA and KDA EXT fitting for 3C325 and their goodness of fit to the observed data $S_{\nu_{0}}$. $\nu_{0}$ [MHz] |  $S_{\nu_{0}}\pm\Delta$ [Jy] |  $S_{MOD}$ [Jy] |  $S_{MOD}$ [Jy]
---|---|---|---
|  | KDA EXT | KDA
74 |  28.28 $\pm$ 2.83 |  37.64 $\pm$ 3.78 |  39.22 $\pm$ 0.65
150 |  24.16 $\pm$ 2.42 |  22.73 $\pm$ 2.28 |  21.88 $\pm$ 0.56
178 |  15.85 $\pm$ 1.59 |  20.07 $\pm$ 2.02 |  18.86 $\pm$ 0.45
232 |  14.08 $\pm$ 1.41 |  16.43 $\pm$ 1.65 |  15.01 $\pm$ 0.33
327 |  12.08 $\pm$ 1.21 |  12.52 $\pm$ 1.26 |  11.11 $\pm$ 0.16
750 |  6.38 $\pm$ 0.64 |  6.23 $\pm$ 0.63 |  5.25 $\pm$ 0.12
966 |  5.09 $\pm$ 0.26 |  4.97 $\pm$ 0.50 |  4.16 $\pm$ 0.08
1400 |  3.56 $\pm$ 0.18 |  3.55 $\pm$ 0.36 |  2.96 $\pm$ 0.05
2700 |  1.89 $\pm$ 0.10 |  1.89 $\pm$ 0.19 |  1.60 $\pm$ 0.03
4850 |  0.98 $\pm$ 0.05 |  1.03 $\pm$ 0.10 |  0.92 $\pm$ 0.01
10695 |  0.42 $\pm$ 0.02 |  0.42 $\pm$ 0.04 |  0.43 $\pm$ 0.01
14900 |  0.25 $\pm$ 0.01 |  0.27 $\pm$ 0.03 |  0.31 $\pm$ 0.01
$r$ |  | 4 | 3
$\chi^{2}$ |  | 3.45 | 10.60
|  $\Lambda$=0.10 |  ($p$=0.1) |

####  4.2.7 3C401

3C401 is a rather small ($D=$67.5 kpc) and low-redshifted ($z=0.201$) FR II-
type radio source previously studied by Laing et al. (1983). It coincides with
a Seyfert galaxy and is characterised by a radio core of medium brightness
equal to 28.54 $\pm$ 0.03 mJy (Hardcastle et al. 1998) and high axial ratio of
the lobes. In this case, there is no statistical preference of the KDA EXT
model fit over the KDA model, with the $\Lambda$ value being almost seven
times larger than the level of significance. The spectra resulting from the
predicted flux densities $S_{\rm MOD}\,(KDA)$ and $S_{\rm MOD}\,(KDA\,EXT)$
are listed in Table 8, along with their corresponding goodness of the fit to
$\chi^{2}$ data, which is evidently lower in the case of the KDA EXT fit.

Table 8: Model flux densities $S_{MOD}$ from the KDA and KDA EXT fitting for 3C401 and their goodness of fit to the observed data $S_{\nu_{0}}$. $\nu_{0}$ [MHz] |  $S_{\nu_{0}}\pm\Delta$ [Jy] |  $S_{MOD}$ [Jy] |  $S_{MOD}$ [Jy]
---|---|---|---
|  | KDA EXT | KDA
74 |  31.38 $\pm$ 3.14 |  42.52 $\pm$ 5.46 |  54.15 $\pm$ 4.29
150 |  28.02 $\pm$ 2.80 |  27.74 $\pm$ 2.65 |  30.32 $\pm$ 2.61
178 |  23.55 $\pm$ 2.36 |  24.52 $\pm$ 2.13 |  26.28 $\pm$ 2.80
232 |  22.31 $\pm$ 2.23 |  20.49 $\pm$ 1.60 |  21.07 $\pm$ 2.47
325 |  15.31 $\pm$ 1.53 |  16.17 $\pm$ 1.45 |  15.91 $\pm$ 1.63
365 |  14.93 $\pm$ 1.49 |  14.85 $\pm$ 0.80 |  14.41 $\pm$ 1.50
740 |  8.66 $\pm$ 0.87 |  8.64 $\pm$ 0.63 |  7.90 $\pm$ 0.78
966 |  7.11 $\pm$ 0.71 |  6.96 $\pm$ 0.46 |  6.28 $\pm$ 0.70
1400 |  5.07 $\pm$ 0.25 |  5.08 $\pm$ 0.26 |  4.55 $\pm$ 0.51
2695 |  2.82 $\pm$ 0.14 |  2.82 $\pm$ 0.15 |  2.54 $\pm$ 0.28
4850 |  1.58 $\pm$ 0.08 |  1.59 $\pm$ 0.15 |  1.48 $\pm$ 0.16
4900 |  1.55 $\pm$ 0.08 |  1.57 $\pm$ 0.14 |  1.47 $\pm$ 0.16
5000 |  1.36 $\pm$ 0.07 |  1.54 $\pm$ 0.14 |  1.44 $\pm$ 0.15
8000 |  0.90 $\pm$ 0.05 |  0.93 $\pm$ 0.09 |  0.93 $\pm$ 0.09
10695 |  0.67 $\pm$ 0.03 |  0.67 $\pm$ 0.07 |  0.70 $\pm$ 0.07
10800 |  0.64 $\pm$ 0.03 |  0.66 $\pm$ 0.07 |  0.69 $\pm$ 0.07
14900 |  0.43 $\pm$ 0.02 |  0.44 $\pm$ 0.05 |  0.51 $\pm$ 0.04
$r$ |  | 4 | 3
$\chi^{2}$ |  | 1.81 | 6.61
|  $\Lambda$=0.69 |  ($p$=0.1) |

####  4.2.8 PKS 0235$-$19

This is the only source in the sample not covered by the 3C catalogue due to
its location in the southern hemisphere. Nonetheless, PKS 0235$-$19 is listed
in the Parkes Radio Catalogue (Wright & Otrupcek 1990). Its nucleus has been
classified as a Seyfert 1 galaxy with broad Balmer lines (Tadhunter et al.
1998). The spectra predicted with the KDA and KDA EXT models are shown in
Figure 8. The relevant $\chi^{2}$ and $\Lambda$ values, along with the
resulting flux densities calculated for these two models, are presented in
Table 9, confirming the preference for the KDA EXT solution.

Table 9: Model flux densities $S_{MOD}$ from the KDA and KDA EXT fitting for PKS 0235$-$19 and their goodness of fit to the observed data $S_{\nu_{0}}$. $\nu_{0}$ [MHz] |  $S_{\nu_{0}}\pm\Delta$ [Jy] |  $S_{MOD}$ [Jy] |  $S_{MOD}$ [Jy]
---|---|---|---
|  | KDA EXT | KDA
74 |  35.66 $\pm$ 3.57 |  45.45 $\pm$ 4.57 |  60.33 $\pm$ 6.31
150 |  24.12 $\pm$ 2.41 |  28.23 $\pm$ 2.84 |  32.30 $\pm$ 3.93
160 |  26.50 $\pm$ 2.65 |  27.01 $\pm$ 2.73 |  30.73 $\pm$ 3.76
365 |  15.60 $\pm$ 1.56 |  14.54 $\pm$ 1.46 |  14.37 $\pm$ 2.06
408 |  13.34 $\pm$ 1.33 |  13.29 $\pm$ 1.34 |  12.96 $\pm$ 1.89
635 |  9.49 $\pm$ 0.95 |  9.22 $\pm$ 0.93 |  8.50 $\pm$ 1.34
1400 |  4.62 $\pm$ 0.23 |  4.55 $\pm$ 0.46 |  3.94 $\pm$ 0.70
2300 |  2.96 $\pm$ 0.15 |  2.81 $\pm$ 0.28 |  2.41 $\pm$ 0.46
2700 |  2.41 $\pm$ 0.12 |  2.39 $\pm$ 0.24 |  2.05 $\pm$ 0.40
3900 |  1.58 $\pm$ 0.08 |  1.62 $\pm$ 0.16 |  1.42 $\pm$ 0.29
4850 |  1.31 $\pm$ 0.07 |  1.27 $\pm$ 0.13 |  1.14 $\pm$ 0.24
10695 |  0.44 $\pm$ 0.02 |  0.47 $\pm$ 0.05 |  0.51 $\pm$ 0.12
11200 |  0.37 $\pm$ 0.02 |  0.44 $\pm$ 0.04 |  0.49 $\pm$ 0.11
$r$ |  | 4 | 3
$\chi^{2}$ |  | 3.71 | 17.59
|  $\Lambda$=0.013 |  ($p$=0.1) |

Figure 1: Best KDA EXT fit (solid line) and KDA fit (dotted line) for radio
galaxy 3C006.1. The values of flux density and frequency are presented in
logarithmic scale. The observed flux densities and their errors are marked
with data points. Figure 2: As in Figure 1, but for 3C 32. Figure 3: As in
Figure 1, but for 3C 41. Figure 4: As in Figure 1, but for 3C 123. Figure 5:
As in Figure 1, but for 3C 247. Figure 6: As in Figure 1, but for 3C 325.
Figure 7: As in Figure 1, but for 3C 401. Figure 8: As in Figure 1, but for
PKS 0235$-$19.

Table 10: Physical parameters of the sources from the sample derived from the best KDA fits for the entire available radio spectrum (upper lines) and KDA EXT fits (bottom lines). Ages are given in Myr. The accuracy of their numerical determination is of the order of $\pm$ 0.01 Myr.  Name | 3C006.1 | 3C32 | 3C41 | 3C123 | 3C247 | 3C325 | 3C401 | PKS 0235$-$19
---|---|---|---|---|---|---|---|---
tKDA [Myr] | 19.09 | 45.71 | 9.12 | 2.31 | 26.55 | 31.61 | 0.20 | 25.69
t${}_{\rm KDA\,EXT}$ [Myr] | 19.09 | 38.02 | 2.00 | 1.91 | 9.15 | 17.54 | 6.27 | 25.59
tbr [Myr] | 18.43 | 35.91 | 1.73 | 1.82 | 7.31 | 16.60 | 5.71 | 23.22
$\alpha_{\rm inj}$ | 0.38 | 0.56 | 0.37 | 0.59 | 0.44 | 0.44 | 0.80 | 0.46
| 0.35 | 0.41 | 0.46 | 0.54 | 0.38 | 0.46 | 0.43 | 0.46
$Q_{\rm jet}$ [W] | 17.1$\times$1038 | 3.6$\times$1038 | 1.9$\times$1039 | 1.8$\times$1039 | 4.1$\times$1039 | 1.5$\times$1039 | 3.9$\times$1039 | 8.9$\times$1038
| 22.7$\times$1038 | 4.9$\times$1038 | 3.8$\times$1039 | 1.5$\times$1039 | 1.3$\times$1039 | 8.11$\times$1038 | 1.01$\times$1038 | 8.1$\times$1038
$\rho_{0}$ [kg/m3] | 4.8$\times$10-22 | 6.0$\times$10-22 | 4.8$\times$10-22 | 1.7$\times$10-23 | 3.5$\times$10-23 | 3.5$\times$10-22 | 1.8$\times$10-25 | 2.5$\times$10-23
| 6.0$\times$10-22 | 5.7$\times$10-22 | 6.9$\times$10-24 | 1.9$\times$10-23 | 7.4$\times$10-24 | 2.9$\times$10-23 | 7.7$\times$10-23 | 2.1$\times$10-23
$p_{\rm c}$ [N/m2] | 2.5$\times$10-11 | 3.9$\times$10-12 | 5.1$\times$10-11 | 3.1$\times$10-11 | 7.1$\times$10-12 | 1.91$\times$10-14 | 4.72$\times$10-11 | 3.01$\times$10-12
| 3.1$\times$10-10 | 5.1$\times$10-12 | 1.6$\times$10-11 | 4.3$\times$10-11 | 8.0$\times$10-12 | 5.11$\times$10-12 | 2.19$\times$10-11 | 2.51$\times$10-12
$U_{\rm c}$ [J] | 1.03$\times$1054 | 5.16$\times$1053 | 5.38$\times$1053 | 1.32$\times$1053 | 3.47$\times$1053 | 1.50 $\times$1054 | 2.45$\times$1052 | 7.30$\times$1053
| 1.36$\times$1054 | 5.87$\times$1052 | 2.40 $\times$1053 | 9.22$\times$1052 | 3.01$\times$1053 | 5.26$\times$1053 | 2.0$\times$1052 | 6.5$\times$1053
$B_{\rm eq}$ [nT] | 6.33 | 2.52 | 9.12 | 7.06 | 3.39 | 5.56 | 8.73 | 2.21
| 7.01 | 2.88 | 5.06 | 8.37 | 3.61 | 2.87 | 5.94 | 2.01
$v_{\rm h}/c$ | 0.018 | 0.012 | 0.026 | 0.074 | 0.01 | 0.015 | 0.38 | 0.017
| 0.017 | 0.014 | 0.013 | 0.071 | 0.033 | 0.019 | 0.015 | 0.018

##  5 Discussion

This work concerns the KDA EXT model describing the dynamics of the FR-II-type
radio sources after recent cessation of their jet activity (i.e. having low
$(t-t_{br})/t_{br}$ ratios not exceeding 0.25). KDA EXT is based on the KDA
model and assumes self-similarity of the particle flow. Thus one may expect
this model not to be applicable in the case of presumably very old radio
sources whose jets terminated in the distant past. Theoretical considerations
along with some earlier modelling results suggest, rather, that, for so-called
radio relics (Murgia et al. 2011) in which aging of the radio structure is
very evident and the self-similarity is manifestly no longer valid, neither
the KDA nor the KDA EXT models can correctly predict their observed radio
spectra. Following this supposition, the presented research was focused on
those sources showing only modest spectral aging.

This paper is an extended illustration of the general predictions and
qualitative analysis of the KDA EXT model, but is restricted due to the very
limited sample of the analysed sources. The number of sources showing both the
FR II structure and strongly curved radio spectra is small. Future work
focussed on proper measurement data for a larger sample of such sources may
help greatly in both constraining the parameters of the examined model and
evaluation of its reliability.

The new results generally confirm those obtained earlier by Kuligowska (2017):
in the majority of the analysed cases, the solutions for individual sources
summarised in Section 4 still show that the KDA EXT model is useful and
suitable for studying the sources with strongly bent radio spectra in which
central activity terminated relatively recently (or formally: with low
($t-t_{br}$)/$t_{br}$) ratios). In the context of the presented results (All
Figures and Table 10), the most controversial cases are radio sources 3C41 and
3C401. Figure 3 (3C41) shows that in the range of the observed flux densities
for this source, the spectrum resulting from the KDA model is initially much
steeper than the one obtained with the KDA EXT model. However, the
corresponding entries in Table 10 clearly indicate that the $\alpha_{\rm inj}$
is much lower in the case of the KDA model. This difference is easy to explain
if only the wider range of the spectrum (adding the extremely low frequencies
ranging from less than 1 MHz to 10 MHz) is presented. Thus the spectra for
both models have been calculated for lower (starting at 0.5 MHz) frequencies.
It should be noted that for the frequencies lower than 10 MHz there is no
technical ability to observe radio flux densities with the present-day ground-
based telescopes, mostly due to the impact of the ionosphere. The analysis of
this low-frequency model spectra confirms that the initial slope is
effectively flatter in the case of the KDA EXT model. On the other hand, in
the case of 3C401 the resulting KDA and KDA EXT plots (Fig. 7) are not very
different in the range of the lower observational frequencies, though the
formal measurement uncertainties for both models are rather large in this
scope. If, however, the KDA and KDA EXT models for this source are
recalculated taking into account the extremely low frequencies lying beyond
the observed data points, it becomes clear that the discrimination between
$\alpha_{\rm inj}$ values (also very evident in Table 10) and the overall
difference between the two models are actually significant.

The source 3C123 is also an interesting case. Here, both model (KDA and KDA
EXT) fits and their resulting $\chi^{2}$ values are quite similar. Also the
applied statistical analysis based on the likelihood ratio test does not
confirm the preference of the KDA EXT model. Once again, the solution
presented for this source proves that very congruous model-predicted spectral
shapes can be obtained starting with not only different models, but also
different values of their input parameters (compare Table 10 and the paper of
Brocksopp et al. (2011).

Regardless of the apparently curved spectra of the sample sources, an attempt
was made to fit their observed spectra with the classical KDA model with high
values of the $\alpha_{\rm inj}$ parameter. Here, conversely, it was assumed
that these source are still active and their jets still supply the radio
structure. However, it is easy to notice that these resulting fits do not
reproduce the observed flux density properly and in the case of 3C401, they
result in a very young source age of 0.2 Myr.

In addition, the result of the likelihood ratio test clearly indicates that
the difference between the KDA and KDA EXT models calculated for this
particular source is not statistically significant at the significance level
p$=$0.10$\%$, despite the large difference in their $\chi^{2}$ values. It may
be concluded that there is no statistical preference for the KDA EXT, though
the subjective visual plot analysis favours this model.

The results of the KDA EXT modelling presented in this work do not always
reproduce the spectra of the sample sources with satisfactory accuracy. There
are numerous points departing from the model lines, especially in the low-
frequency parts of the predicted spectra. It should be noted that for some
sample sources, a good fit of this spectral range was actually achieved, while
in other cases a number of low-frequency points lie far from the model
(compare 3C325 or 3C401). The same situation also applies (to a lesser extent)
to the high-frequency parts of the spectra. Except for the underestimated
values of the observed flux densities and their errors, the above
inconsistency may result from the following factors:

  1. 1.

Variation in the rate of particles transported in the jets (not included in
the KDA EXT).

  2. 2.

The source’s restarting activity, resulting in another variation in the jet
composition.

  3. 3.

General inaccuracy in determining the initial KDA model parameters for the
given source for their further use in fitting the KDA EXT model.

  4. 4.

Inaccuracy in estimating the geometrical parameters of the sources such as
lobe length or axial ratio, possibly due to the lack of sufficiently detailed
radio maps.

The KDA EXT model predicting the evolution of the lobes of FR II-type radio
sources after termination of the jet activity, tested here once again on a new
sample of galaxies, provides a satisfactory solution for at least four of the
eight examined radio sources. Statistically, the KDA EXT model is at least as
good, and often better, than the original KDA model, and as such it is a
useful tool for fitting and studying the steepened spectra of FR II radio
sources.

ACKNOWLEDGEMENTS

This research was supported by the Polish National Science Centre grant No.
2013/09/B/ST9/00599. The authors express sincere gratitude to the anonymous
referee for the useful and essential comments on the manuscript.

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