11institutetext: Institut für Theoretische Physik and Astrophysik, Julius-
Maximilians-Universität Würzburg, Emil-Fischer-Straße 31, D-97074 Würzburg,
Germany 11email<EMAIL_ADDRESS>22institutetext: Max-
Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

# Dual-high-frequency VLBI study of blazar-jet brightness-temperature
gradients and collimation profiles

P. R. Burd 11 M. Kadler 11 K. Mannheim 11 A.-K. Baczko 22 J. Ringholz 11 E.
Ros 22

(Version: Received 04/10/2021; accepted 06/12/2021)

###### Abstract

Context. On the kiloparsec scale, extragalactic radio jets show two distinct
morphologies related to their power: collimated high-power jets ending in a
bright termination shock and low-power jets opening up close to the core and
showing a more diffuse surface brightness distribution. The emergence of this
morphological dichotomy on the parsec scale at the innermost jet regions can
be studied with very-long-baseline interferometry (VLBI) radio observations of
blazars in which the jet emission is strongly Doppler boosted due to
relativistic bulk motion at small angles between the jet direction and the
line of sight.

Aims. We seek to characterize the geometry and emission profiles of the
parsec-scale radio jets of flat-spectrum radio quasars (FSRQs) and BL Lacertae
objects (BL Lacs) on parsec scales to derive properties of the magnetic field,
environment, and energetics for different classes of extragalactic jets.

Methods. We analyze the VLBI radio data of 15 FSRQs, 11 BL Lacs, and two radio
galaxies contained in both the Monitoring Of Jets in Active galactic nuclei
with VLBA Experiments (MOJAVE) data archive and the Boston University (BU)
blazar group sample archive at 15 GHz and 43 GHz, repectively. We derived the
brightness-temperature and jet-width gradients along the jet axis from
parameterizations of the jets using 2D Gaussian brightness distributions.

Results.  In most BL Lac objects, the diameter and brightness-temperature
gradients along the jet axis can generally be described well by single power
laws, while the jets of FSRQs show more complex behavior and remain more
strongly collimated on larger physical scales. We find evidence for a
transition of the global jet geometry from a parabolic to a conical shape in
the BL Lac objects 3C 66A, Mrk 421 and BL Lacertae, the radio galaxy 3C 111
and the FSRQs CTA 26, PKS 0528+134, 4C +71.07, 4C +29.45, and 3C 279 outside
the Bondi sphere.

Conclusions.  Our results combined with findings from kinematic VLBI studies
that the jets of FSRQs exhibit larger bulk Lorentz factors than BL Lacs are in
agreement with relativistic magnetohydrodynamical jet-disk simulations in
which the flattening of the jet magnetization profile due to magnetic fields
from the accretion disk leads to a more persistent collimation in high-
accretion-rate blazars.

###### Key Words.:

Galaxies:active – Galaxies:jets– Jets:parsec-scale

##  1 Introduction

Parsec-scale properties of jets in radio-loud active galactic nuclei (AGN)
have been studied in much detail using the radio-astronomical technique of
very long baseline interferometry (VLBI). In particular, the monitoring
program MOJAVE (Lister et al., 2009) has been highly successful in exploring
the jet kinematics of sources at a frequency of $15\leavevmode\nobreak\
\mathrm{GHz}$ (Lister et al., 2016). These data have been used to study the
acceleration and collimation of parsec-scale jets (Homan et al., 2015), their
opening angles (Pushkarev et al., 2017b), polarization properties (Lister et
al., 2018; Pushkarev et al., 2017a), spectral distributions (Hovatta et al.,
2014), and other applications.
By employing the model of a freely expanding relativistic jet (Blandford &
Königl, 1979; Königl, 1981), the underlying physical properties such as the
electron density, magnetic field strength, or power of the jets can be
inferred from such observations. The strong observational biases due to
relativistic bulk motion, such as aberation and Doppler boosting, can be
corrected for by using statistical methods (Cohen et al., 2007). Beyond
scaling models, VLBI observations can also be compared with detailed
relativistic magnetohydrodynamical simulations (Porth et al., 2011; Fendt,
2006), which describe the launching of jets from the black-hole accretion disk
(Blandford & Payne, 1982) and ergosphere (Blandford & Znajek, 1977), as well
as their acceleration and collimation. The simulations describe the fluid-
dynamical evolution of the Poynting-flux dominated jets on subparsec scales
all the way to the scales where they become visible on (multi) parsec scales
as typical kinetic-energy-dominated jets to VLBI and connected radio
interferometers. On VLBI scales, shock waves are commonly believed to
accelerate particles to ultrarelativistic energies by the Fermi mechanism,
causing moving knots in the jet flow and multiwavelength flares (e.g.,
Marscher et al., 2008).

In addition to brightness temperature ($T_{\rm B}$) observations of the VLBI
cores of AGN jets (Homan et al., 2021), the measurement of brightness-
temperature gradients along their jets is a powerful independent observational
tool. This approach can probe the internal and external physical conditions in
AGN jets and has been employed in several single-source jet studies. For 3C
111, Kadler et al. (2008) measured brightness-temperature gradients and their
evolution to study the particle and magnetic field density associated with
individual jet components during flares. In the case of NGC 1052 (a twosided
source), Kadler et al. (2004) and Baczko et al. (2019) studied the $T_{\rm B}$
gradients along both jets. Kadler et al. (2004) find a break of the $T_{\rm
B}$-gradients near the cores which they attribute to possible free-free
absorption associated with a dusty torus. Baczko et al. (2019) further find
that single power-law fits do not sufficiently describe the measured $T_{\rm
B}$ behavior and adopt a two-zone model with a break in the brightness-
temperature and diameter gradient of the jet. Studies of the brightness
temperatures of VLBI jets and their gradients with distance from the core are
still rare for larger samples.

In order to characterize jets close to their collimation zone, Pushkarev &
Kovalev (2012) carried out dual-frequency observations at $2\,\mathrm{GHz}$
and $8\,\mathrm{GHz}$, respectively, using a sample of 30 AGN jets. They found
that at $8\,\mathrm{GHz}$ the jet diameter gradient shows flatter slopes than
expected for freely expanding jets, while at $2\,\mathrm{GHz}$ the jet
geometry is consistent with a conical geometry. They found brightness
temperature gradients of $T_{\rm B}\propto r^{-2.2}$. Furthermore, Pushkarev &
Kovalev (2012) found indications for the presence of a transition zone in
blazar jets where the jet flow switches from a confined, accelerating jet
(parabolic geometry) to a freely expanding jet (conical geometry). In the case
of M87, Asada & Nakamura (2012) showed that such a geometry transition can be
observed in the vicinity of the Bondi-sphere (Russell et al., 2015) where the
external medium is expected to change its pressure gradient. More recent works
also found these transitions in several different sources, NGC 6251 (Tseng et
al., 2016), NGC 4261 (Nakahara et al., 2018), NGC 1052 (Nakahara et al., 2020;
Baczko et al., 2021), NGC 315 (Boccardi et al., 2021; Park et al., 2021) and
1H0323+342 (Hada et al., 2018). In the cases of M87 (Mertens et al., 2016;
Hada et al., 2018; Park et al., 2019), NGC 315 (Park et al., 2021) and
1H0323+342 (Hada et al., 2018) the jets also show accelerating jet components
near the geometry-transition zones. Kovalev et al. (2020) systematically
searched for these geometry transitions in 331 MOJAVE sources at
$15\,\mathrm{GHz}$ and $1.4\,\mathrm{GHz}$. They demonstrated the presence of
geometry transitions from a parabolic to a conical geometry in 10 nearby
sources on scales of several thousand Schwarzschild radii $R_{\textrm{S}}$.
Potter & Cotter (2015) applied parameters, obtained from 42 simultaneous
blazar spectral energy distributions (SED) to a 1d-relativistic fluid-
dynamical jet model to predict a region where the jets experience a transition
between a magnetically-dominated region (closer to the central engine), where
the jet geometry is parabolic and a particle-dominated region of the jet
(farther downstream) where the jet geometry is conical. Potter & Cotter (2015)
predict a correlation between the radii of the jets in the transition zone and
the jet power as well as the maximum bulk Lorentz factors. Consequently, they
also predict a dichotomy of the blazar classes, BL Lacertae objects (BL Lacs)
and flat-spectrum radio quasars (FSRQs), regarding the jet radii in the
transition zone, where BL Lacs are expected to show smaller jet radii than
FSRQs. Kovalev et al. (2020) argue that external pressure gradients most
likely are not solely responsible for the geometry transitions within the VLBI
jets and that the mechanism might involve a transition of the jets from a
Poynting-flux to a particle-dominated configuration.
In order to investigate the $T_{\rm B}$-gradients and morphological changes of
parsec-scale jets at higher frequencies, we conducted a systematic dual-high-
frequency ($15\,\mathrm{GHz}$ and $43\,\mathrm{GHz}$) analysis of a sample of
blazars using data obtained with the Very Long Baseline Array (VLBA) data as
part of the MOJAVE111http://www.physics.purdue.edu/astro/MOJAVE/ (15 GHz) and
Boston University222https://www.bu.edu/blazars/VLBAproject.html (BU, 43 GHz)
jet monitoring programs. The following cosmological constants were adopted
throughout the paper: $h=0.71,\Omega_{\Lambda}=0.73,\,\Omega_{M}=0.27$.

##  2 Sample

The objects and their observation epochs (i.e., characterizing the time
interval during which the observation was carried out) for this study were
selected by the following criteria:

  * a)

The sources are in the MOJAVE data archive (Lister et al., 2018) and the BU
sample (Jorstad & Marscher, 2016).

  * b)

Data exists for epochs between 2003 and 2013

  * c)

The imaged jet brightness distribution can be sufficiently represented and
parametrized with at least three 2D-Gaussian jet components in at least 5
epochs333We exclude the peculiar, complex-structure radio galaxy 0316+413
(3C84).

We chose the earliest epoch considered in our study to be at the onset of the
new MOJAVE program (Lister & Homan, 2005) at which the overall image fidelity
and typical image noise levels improved systematically over earlier data sets
from the progenitor 2cm-Survey program (Kellermann et al., 1998). The
resulting sample consists of 15 flat-spectrum radio quasars (FSRQs), 11 BL
Lacertae objecs (BL Lacs) and two radio galaxies (RGs), in total 28 objects.
The object names, their classification, redshifts. and black hole masses are
listed in Table 1.

Table 1: Source sample and source properties

Name | Alt. name | Class. | $z$ |  $\mathrm{log}_{10}\left(M_{\textrm{BH}}/M_{\odot}\right)$ |  $\frac{\mathrm{pc}}{\mathrm{mas}}$ | Epochs
---|---|---|---|---|---|---
0219+428 | 3C 66A | B | $0.34^{\textrm{a}}$ |  $(8.47-8.65)^{\textrm{I}}$ |  $4.8$ | $19/38$
0336$-$019 | CTA 26 | Q | $0.852^{\textrm{b}}$ |  $(7.55-7.73)^{\textrm{I}}$ |  $7.7$ | $18/15$
0415+379 | 3C 111 | G | $0.0491^{\textrm{c}}$ |  $(8.18-8.38)^{\textrm{II}}$ |  $0.95$ | $59/20$
0430+052 | 3C 120 | G | $0.033^{\textrm{d}}$ |  $(7.08-7.58)^{\textrm{III}}$ |  $0.65$ | $59/24$
0528+134 | PKS 0528+134 | Q | $2.07^{\textrm{e}}$ |  $(8.35-8.52)^{\textrm{I}}$ |  $8.5$ | $18/79$
0716+714 | S5 0716+71 | B | $0.803^{\textrm{f}}$ |  $(7.51-7.68)^{\textrm{I}}$ |  $7.5$ | $59/71$
0735+178 | OI 158 | B | $0.45^{\textrm{g}}$ |  $(8.35-8.51)^{\textrm{I}}$ |  $5.8$ | $14/60$
0827+243 | OJ 248 | Q | $0.94^{\textrm{h}}$ |  $(6.94-7.12)^{\textrm{I}}$ |  $7.9$ | $14/14$
0829+046 | OJ 049 | B | $0.174^{\textrm{i}}$ |  $(8.55-8.71)^{\textrm{I}}$ |  $2.9$ | $14/64$
0836+710 | 4C +71.07 | Q | $2.218^{\textrm{j}}$ |  $(7.8-7.97)^{\textrm{I}}$ |  $8.4$ | $14/14$
0851+202 | OJ 287 | B | $0.306^{\textrm{k}}$ |  $(7.12-7.31)^{\textrm{I}}$ |  $4.5$ | $26/28$
0954+658 | S4 0954+65 | B | $0.45^{\textrm{l}}$ |  $(8.6-8.77)^{\textrm{I}}$ |  $5.7$ | $21/60$
1101+384 | Mrk 421 | B | $0.0308^{\textrm{m}}$ |  $(7.28-7.49)^{\textrm{I}}$ |  $0.61$ | $27/40$
1127$-$145 |  PKS 1127$-$14 | Q | $1.184^{\textrm{n}}$ |  $(7.77-7.94)^{\textrm{I}}$ |  $8.3$ | $15/44$
1156+295 | 4C +29.45 | Q | $0.725^{\textrm{i}}$ |  $(8.2-8.38)^{\textrm{I}}$ |  $7.3$ | $20/14$
1219+285 | W Comae | B | $0.102^{\textrm{o}}$ |  $(7.17-7.37)^{\textrm{I}}$ |  $1.9$ | $11/47$
1222+216 | 4C +21.35 | Q | $0.433^{\textrm{i}}$ |  $(8.38-8.55)^{\textrm{I}}$ |  $5.6$ | $23/17$
1226+023 | 3C 273 | Q | $0.1583^{\textrm{p}}$ |  $(8.13-8.3)^{\textrm{I}}$ |  $2.7$ | $32/13$
1253$-$055 | 3C 279 | Q | $0.6^{\textrm{q}}$ |  $(8.65-8.83)^{\textrm{I}}$ |  $6.7$ | $24/69$
1308+326 | OP 313 | Q | $0.997^{\textrm{i}}$ |  $(7.64-8.56)^{\textrm{IV}}$ |  $8.0$ | $19/28$
1633+382 | 4C +38.41 | Q | $1.814^{\textrm{h}}$ |  $(8.36-8.54)^{\textrm{I}}$ |  $8.5$ | $27/79$
1652+398 | Mrk 501 | B | $0.0337^{\textrm{r}}$ |  $(8.02-8.22)^{\textrm{I}}$ |  $0.66$ | $12/6$
1730$-$130 | NRAO 530 | Q | $0.902^{\textrm{s}}$ |  $(8.35-8.52)^{\textrm{I}}$ |  $7.8$ | $24/71$
1749+096 | OT 081 | B | $0.322^{\textrm{t}}$ |  $(8.66-8.66)^{\textrm{V}}$ |  $4.6$ | $26/49$
2200+420 | BL Lac | B | $0.0686^{\textrm{u}}$ |  $(8.02-8.17)^{\textrm{I}}$ |  $1.3$ | $64/80$
2223$-$052 | 3C 446 | Q | $1.404^{\textrm{v}}$ |  $-$ |  $8.5$ | $15/14$
2230+114 | CTA 102 | Q | $1.037^{\textrm{w}}$ |  $(8.4-8.57)^{\textrm{I}}$ |  $8.1$ | $18/13$
2251+158 | 3C 454.3 | Q | $0.859^{\textrm{x}}$ |  $(7.21-7.4)^{\textrm{I}}$ |  $7.7$ | $22/14$
444Columns: (1) IAU B 1950 name, (2) Alternative (common) name, (3)
Classification, (4) Redshift, (5) Black-hole mass, (6) Angular to linear scale
conversion factor, (7) Number of epochs ($15\leavevmode\nobreak\
\mathrm{GHz}/43\leavevmode\nobreak\ \mathrm{GHz}$)

a) Torres-Zafra et al. (2018), b) Wills & Lynds (1978), c) Eracleous & Halpern
(2004), d) Michel & Huchra (1988), e) Hunter et al. (1993), f) Stadnik &
Romani (2014), g) Nilsson et al. (2012), h) Pâris et al. (2017), i) Schneider
et al. (2010), j) Stickel & Kuehr (1993), k) Stickel et al. (1989), l) Wills
et al. (1992), m) Ulrich et al. (1975), n) Wilkes (1986), o) Paiano et al.
(2017), p) Strauss et al. (1992), q) Marziani et al. (1996), r) Stickel et al.
(1993), s) Junkkarinen (1984), t) Stickel et al. (1988), u) Vermeulen et al.
(1995), v) Wright et al. (1983), w) Falomo et al. (1994), x) Jackson & Browne
(1991), I) Fan et al. (2008), II) Chatterjee et al. (2011) III) Kaspi et al.
(2000) IV) Gupta et al. (2012) V) Falomo et al. (2003)

##  3 Method

###  3.1 Parameter fitting

The optically thin synchrotron emission regions in the jet were fitted with
circular Gaussians using
difmap555https://science.nrao.edu/facilities/vlba/docs/manuals/oss2013a/post-
processing-software/difmap (Shepherd, 1997). Initially, the visibility map is
opened and a starting model consisting of a circular Gaussian component in the
center is fitted to best match the data. If the residual map still shows
nonzero intensity after subtracting the established model, further Gaussians
are added and fitted until the residual map shows only (white) noise. As an
example, the corresponding jet model resulting from a single epoch of 4C
+71.07 (0836+710) observations is shown in Fig.1 for both frequencies. Our
model fits are optimized to represent accurately the local surface brightness
and jet width along the jet rather than individual moving knots within the
jet. For this reason, our model fits differ from the ones used in standard
MOJAVE kinematics studies. The diameter $d$, flux $F=I\times\Omega_{G}$ ($I$
is the intensity and $\Omega_{G}$ the solid angle of the fit Gaussian
component) , and the (angular) distance from the core $r$ are measured for
each component. Unresolved components are filtered according to

$r_{\mathrm{lim}}=b_{\Psi}\sqrt{\frac{4\,\mathrm{ln}(2)}{\pi}\mathrm{ln}\left(\frac{\mathrm{S/N}}{\mathrm{S/N}-1}\right)}$
(1)

which describes the minimal radius a resolved fitted component can achieve for
a given signal-to-noise ratio ($\mathrm{S/N}$) and beam size ($b_{\Psi}$),
(cf. Kovalev et al., 2005). The uncertainties of the parameters of each
component are approximated with the post-fit RMS $\sigma_{rms}$ information
(Fomalont, 1999). The uncertainty of the flux density is further altered by
also geometrically adding a systematic value of $5\%$ of the flux density to
account for absolute flux-scale uncertainties:

$\displaystyle\sigma_{F}=\sqrt{\sigma_{\mathrm{rms}}^{2}+\sigma_{\mathrm{rms}}^{2}\mathrm{S/N}+\left(0.05F\right)^{2}}$
(2) $\displaystyle\sigma_{d}=d\frac{\sigma_{F}}{F}\quad.$ (3)

The diameters of the remaining components from each epoch are plotted together
into a single graph against the distance from the radio core, at a 1-$\sigma$
level (Figs. 19 ff.). The radio core itself is the unresolved component
closest to the (0/0)-position in the map. All distances are measured against
the position of the respective core component. This ensures that the set of
model components characterizing the jet is representative of the long-term
time-averaged intensity of the evolving jet. By contrast, individual model
components at a given epoch are representative of the local jet brightness and
width at a given time. The components in these sources typically have
considerable proper motions of up to several milliarcseconds along the jet per
year (e.g., Lister et al., 2016), evolving in brightness and in width during
this time. Our long-term multi-epoch data base allows us to parameterize the
jet geometry and brightness temperature for each source in our sample at two
separate high radio frequencies and at submilliarcsecond resolution. To this
end, we fit the jet diameter with $d\propto r^{l}$, that is as a power-law of
distance $r$ with index $l$. The brightness temperature is also fitted as a
power law $T_{\rm B}\propto r^{s}$, characterized by the index $s$. This
procedure is performed for all sources at both frequencies, $15\,\mathrm{GHz}$
and $43\,\mathrm{GHz}$, along the respective entire jet length and in the
overlap regions of both frequencies. To determine a physical overlap region,
the core shift (Pushkarev et al., 2012) has to be taken into account since the
turnover in the synchrotron spectrum from optically thick to optically thin
emission depends on the frequency, that is, the radio core appears closer to
the jet apex at higher frequencies. The core shift $\Delta r_{cs}$ usually has
to be calculated on a source-by-source basis between two frequencies. At lower
frequencies ($\sim 8-15\,\mathrm{GHz}$), Pushkarev et al. (2012) studied the
nuclear opacity of radio jets on a larger sample of 191 sources. For higher
frequencies, O’Sullivan & Gabuzda (2009) analyzed the core shift between $\sim
15$ and $43\,\mathrm{GHz}$ for three sources, $1418+546$, $2007+777$ and
$2200+420$. The core shifts are calculated assuming conical jet geometries and
are measured to be between $0.04\lesssim\Delta r_{cs}/\mathrm{mas}\lesssim
0.1$, which corresponds to $5-10\%$ of the respective beam size of the used
data. Typical beam sizes for the data used in this work are $\sim
0.5\,\mathrm{mas}$ for $15\,\mathrm{GHz}$ and $\sim 0.3\,\mathrm{mas}$ for
$43\,\mathrm{GHz}$, and thus by at least a factor of 3 larger than reported
core shifts between these frequencies. The effect of such small core shifts in
this study are negligible, however, in the case of 2200+420 we tested the
impact of this core shift on the results which will be discussed in the
corresponding paragraph in Sect. 6.

Figure 1: Fitted Gaussian-component representation of 4C +71.07 (0836+710) at
$15\,\mathrm{GHz}$ (top) and at $43\,\mathrm{GHz}$ (bottom). The contours for
both frequencies are shown at a $5\leavevmode\nobreak\ \sigma$ level and are
separated by a factor of 2 . In the $15\,\mathrm{GHz}$ image the map noise
level is $\sim 0.2\leavevmode\nobreak\ \mathrm{mJy}$. The contour levels are
$[-0.85,0.85,1.7,3.4,6.7,13.7,27.5,54.9,109.8,219.6,430.2,878.5]\leavevmode\nobreak\
\mathrm{mJy}$). At $43\,\mathrm{GHz}$ the map noise level is $\sim
0.8\leavevmode\nobreak\ \mathrm{mJy}$. The contour levels are
$[-3.8,3.8,7.5,15.1,30.2,60.3,120.6,241.2,284.5,482.5,964.9,\\\
1929.8,3859.6]\leavevmode\nobreak\ \mathrm{mJy}$.

###  3.2 Scaling model for the jets

For the parameterization of the jet diameter and brightness temperature, we
employ the relativististic jet model introduced by Blandford & Königl (1979)
and Königl (1981). The basic assumptions of the model are:

  * a)

The jet has a semi-opening angle $\Phi$ and an inclination angle $\vartheta$
to the line of sight. The diameter of the jet scales along the jet-axis
$\boldsymbol{r}$, $d_{\mathrm{jet}}\propto r^{l}$. The corresponding jet
morphologies relate to a free expansion with constant advance speed $(l=1)$,
accelerated motion $(l<1)$ or jet deceleration $(l>1)$.

  * b)

Electrons are injected continuously into the jet and the electron energy
density ($n_{e}$) decreases as a power law along the jet axis according to
$n_{e}\propto r^{n}$.

  * c)

The magnetic field scales with a power law along the jet axis according to
$B\propto r^{b}$. A purely toroidal (azimuthal) magnetic field scales with the
jet transverse radius $R$ as $B\propto R^{-1}$, resulting in a scaling along
the jet axis of $B\propto r^{-l}$. A purely poloidal magnetic field (parallel
to the jet axis) would scale with the jet radius according to $B\propto
R^{-2}$ which translates to scaling along the jet-axis according to $B\propto
r^{-2\,l}$.

The observed flux $F_{\nu}$ of a circular component with a diameter
$d_{\mathrm{jet}}$ and radiating area $A\propto d_{\mathrm{jet}}^{2}$ is
proportional to

$F_{\nu}\propto A\cdot d_{\mathrm{jet}}\cdot j_{\nu}$ (4)

where $j_{\nu}\propto n_{e}\cdot
B\cdot\left(\frac{\nu}{\nu_{B}}\right)^{\alpha}\,,(\nu_{B}\propto B),$ is the
emissivity of the synchrotron emission (cf. Krolik, 1999, Sect. 9.2.1). The
properties of the jet geometry, magnetic field, and the electron energy
density a)-c), Eq. (4) implies that the flux $F_{\nu}$ of a model component
has a power law dependency, scaling along the jet axis

$F_{\nu}\propto r^{3l+n+b(1-\alpha)}$ (5)

The brightness temperature $T_{\rm B}$ of a Gaussian-shaped component is given
by

$T_{\rm B}=\frac{2\,\mathrm{ln}(2)}{\pi
k}\frac{F_{\nu}\lambda^{2}(1+z)}{R^{2}}$ (6)

where $k$ denotes the Boltzmann constant, $\lambda$ the observed wavelength,
and $r=d_{\mathrm{jet}}/2$ the radius of the component666Note: If an emission
region is fitted with an elliptical Gaussian component, $r^{2}$ must be
replaced by $\theta_{\mathrm{maj}}\theta_{\mathrm{min}}$, where
$\theta_{\mathrm{maj}}$ and $\theta_{\mathrm{min}}$ are the major and minor
axis of the elliptical Gaussian, respectively.. Eq.6 implies that $T_{\rm
B}\propto F_{\nu}/R^{2}$ ($R^{2}\propto r^{2l}$) and therefore

$T_{\rm B}\propto r^{l+n+b(1-\alpha)}\quad.$ (7)

For convenience, we define the index $s$ that describes the scaling of $T_{\rm
B}\propto r^{s}$ by

$s=l+n+b(1-\alpha)\quad.$ (8)

Now, the complete set of scaling indices is given by the shape index $l$, the
density index $n$, the magnetic field index $b$, and the brightness-
temperature index $s$. Their interdependence depends on the degree of
fulfillment of adiabatic expansion, particle number conservation, and
magnetic-field equipartition.

####  3.2.1 Comparing the modelfit method with the stacked-image analysis

We compared the method of estimating the jet widths at a given position,
described above with the stacked-image analysis as employed by Pushkarev et
al. (2017b) and Kovalev et al. (2020). It is not a-priori clear that both
methods should yield the same results, since it is known that stacked images
map out a broad profile while at any given time traveling jet features do not
fill out this entire stacked-profile range (Pushkarev et al., 2017b; Lister et
al., 2021). On small scales, a varying jet nozzle can lead to strong
variations of component position angles in the inner jet, especially in high-
resolution data at high frequencies, which biases the apparent jet geometry in
stacked images. In individual sources, further differences can arise when a
source exhibits a strongly resolved wide jet emission region far downstream
from the core. The modelfit method will typically yield one single large
component to represent such an extended region in each individual epoch while
image stacking and image-plane fitting of stacked images will benefit from
enhanced signal-to-noise ratio and cover a large area. In result, a diameter
gradient fit with the modelfit method will put much stronger weight on the
(typically many) components from the compact jet and little weight on the
single extended Gaussian component representing the diffuse emission. The
extended jet regions in the image-stacking method are also weighted less
because the method is sensitive to the signal to noise ratio (e.g., Kovalev et
al., 2020; Casadio et al., 2021). However, since the jet is split into
equidistant sections along the jet axis, the larger area of extended emission
will yield more data points to be fit in these regions as the modelfit method.
For such cases the modelfit method can derive flatter diameter gradients than
the stacked image-plane method. We emphasize that a modelfit component in a
single epoch at a given position in some cases will not fill the physical jet
width. Therefore it is important to not derive jet parameters from single
epoch fits. Therefore when studýing the gradients all epochs are considered at
once. In consideration of the arguments made above, we consider the modelfit
method in general as the most robust way to characterize the jet geometry.

A further difference between our analysis and the approach of Kovalev et al.
(2020) is that we are not attempting to fit an additional parameter $r_{0}$ to
consider the separation of the VLBI jet core from the true jet base. Median
value of this parameter determined on large samples are of the order of 0.1
mas (e.g., Kravchenko et al., in prep.) but the generalized fitting of both
$r_{0}$ and $l$ does not converge in case of all of our data sets. We
therefore choose to neglect this offset, understanding that this tends to lead
to somewhat flattened apparent jet gradients. We tested the strength of this
effect in case of 18 sources for which a generalized fitting was possible and
found differences in $|l|$ in the range of $1.4\times 10^{-3}$ to $0.27$. In
case of sources with well-determined and realistically small $r_{0}$ values,
we found that both methods, model fitting and image stacking, yield comparable
results and are overall in good agreement.

####  3.2.2 Changes in the geometrical shape

To measure possible changes in the geometrical shape of the jets, we
subdivided each jet along the distance from the core into $10^{4}$ slices. The
center of each slice represents a possible break point $x_{b}$ in the jet. One
of these break points is picked randomly, then two power-law fits are
performed left and right of $x_{b}$, respectively. If the fit parameters are
plausible, namely $\chi_{\mathrm{red}}^{2}\sim 1$, the errors of the inner and
outer jet geometry fits ($l_{in},l_{out}$) are of the same order of magnitude
and also if $l_{in}<l_{out}$, the solution is added to the sum of solutions.
This is done 5000 times. To then find the best ($l_{in},l_{out},x_{b}$), the
mean value of the respective parameter is calculated over all runs and
weighted with the respective $\chi_{\mathrm{red}}^{2}$ value. This set of
parameters ($l_{in},l_{out},x_{b}$) is used as initial values for a smooth
broken-power-law fit of the form

$f(x)=A\leavevmode\nobreak\ 2^{(l_{in}-l_{out})/\Delta}\leavevmode\nobreak\
\left(\frac{x}{x_{b}}\right)^{l_{in}}\leavevmode\nobreak\
\left[1+\left(\frac{x}{x_{b}}\right)^{\Delta}\right]^{(l_{out}-l_{in})\Delta}$
(9)

(Nakahara et al., 2020; Astropy Collaboration et al., 2013; Price-Whelan et
al., 2018), where $A=d_{j}(x_{b})$ is the jet diameter at the break point
$x_{b}$, $l_{1}$ is the diameter gradient index for the inner jet (closer to
the radio core) and $l_{2}$ is the diameter gradient of the outer jet. This
procedure turned out to be necessary because the fit of the smoothly broken
power law is sensitive to properly guessed initial values. In order to not
having to find them by visual inspection, we performed the search algorithm
described above. There are different ways this function can be fit to data.
Baczko et al. (2021) deploy the orthogonal distance regression while Casadio
et al. (2021) use a gnuplot implementation of the Levenberg-Marquardt
algorithm. In this work we use scipy (Virtanen et al., 2019) in order to fit
the smoothly broken power law with a classical least square method. We used
the trust-region reflective algorithm rather than the Levenberg-Marquardt
algorithm as in Boccardi et al. (2021). The problem at hand is a bound problem
in the sense that we let the fit algorithm vary the initial conditions. These
initial conditions are obtained from the numerical splitting of the jet. A
comparison between the least square fit results and the orthogonal distance
regression can be found in Appendix B. We tested the parameter range of
$\Delta$ between 10 (Baczko et al., 2021) and 100 (Boccardi et al., 2021).
Within the errors the results are not distinguishable, therefore we set the
parameter constant to 100.

##  4 Results

The results of the diameter and brightness-temperature gradient fits are shown
in the appendix A in Figs. 19-46 and summarized in Table 2. Results were
obtained for the full jet length at each frequency and for the overlap regions
probed at both frequencies.

Table 2: Fitting results of jet geometry and brightness-temperature gradients

.  Name $l_{\mathrm{15}}$ $l_{\mathrm{43}}$ $s_{\mathrm{15}}$
$s_{\mathrm{43}}$ 0219+428 $1.02\pm 0.17$ $(1.0,151)$ $0.57\pm 0.16$
$(0.9,179)$ $-2.05\pm 0.43$ $(2.1,151)$ $-1.6\pm 0.35$ $(1.1,179)$ $0.63\pm
0.12$ $(0.4,97)$ $0.56\pm 0.17$ $(1.1,149)$ $-2.08\pm 0.38$ $(1.1,97)$
$-1.45\pm 0.36$ $(1.2,149)$ 0336-019 $0.7\pm 0.17$ $(3.5,68)$ $0.81\pm 0.25$
$(2.7,55)$ $-1.12\pm 0.38$ $(3.8,68)$ $-1.87\pm 0.48$ $(3.1,55)$ $0.68\pm
0.18$ $(3.9,58)$ $0.84\pm 0.26$ $(3.0,49)$ $-1.18\pm 0.36$ $(3.1,58)$ $-1.7\pm
0.51$ $(3.6,49)$ 0415+379 $0.62\pm 0.19$ $(2.5,515)$ $0.53\pm 0.2$ $(1.2,151)$
$-1.83\pm 0.59$ $(5.3,515)$ $-1.51\pm 0.55$ $(2.5,151)$ $0.58\pm 0.2$
$(2.8,420)$ $0.6\pm 0.18$ $(1.3,131)$ $-1.57\pm 0.63$ $(6.0,420)$ $-1.56\pm
0.53$ $(2.7,131)$ 0430+052 $0.56\pm 0.23$ $(5.6,491)$ $0.46\pm 0.17$
$(1.0,112)$ $-1.54\pm 0.54$ $(6.5,491)$ $-1.67\pm 0.48$ $(2.3,112)$ $0.56\pm
0.19$ $(3.0,223)$ $0.43\pm 0.18$ $(1.2,89)$ $-1.42\pm 0.54$ $(5.0,223)$
$-1.44\pm 0.51$ $(2.5,89)$ 0528+134 $0.79\pm 0.21$ $(1.8,98)$ $0.45\pm 0.19$
$(1.5,242)$ $-2.65\pm 0.46$ $(2.5,98)$ $-1.23\pm 0.43$ $(2.4,242)$ $0.6\pm
0.18$ $(1.8,51)$ $0.51\pm 0.19$ $(1.5,194)$ $-2.34\pm 0.59$ $(4.5,51)$
$-1.94\pm 0.43$ $(2.4,194)$ 0716+714 $0.84\pm 0.16$ $(0.6,248)$ $0.78\pm 0.21$
$(1.8,194)$ $-3.0\pm 0.35$ $(0.3,248)$ $-2.6\pm 0.52$ $(1.0,194)$ $0.79\pm
0.14$ $(0.6,185)$ $0.78\pm 0.19$ $(1.5,178)$ $-2.84\pm 0.35$ $(0.3,185)$
$-2.56\pm 0.46$ $(0.7,178)$ 0735+178 $0.85\pm 0.08$ $(0.6,55)$ $0.77\pm 0.21$
$(1.2,132)$ $-2.46\pm 0.37$ $(2.3,55)$ $-1.75\pm 0.36$ $(1.0,132)$ $0.83\pm
0.08$ $(0.7,46)$ $0.75\pm 0.21$ $(1.2,110)$ $-1.85\pm 0.33$ $(1.6,46)$
$-1.77\pm 0.29$ $(0.8,110)$ 0827+243 $0.82\pm 0.19$ $(1.2,57)$ $0.56\pm 0.17$
$(0.7,37)$ $-2.61\pm 0.47$ $(1.6,57)$ $-2.55\pm 0.48$ $(1.6,37)$ $0.83\pm
0.17$ $(1.0,36)$ $0.54\pm 0.14$ $(0.6,30)$ $-3.11\pm 0.5$ $(2.2,36)$ $-2.74\pm
0.44$ $(1.4,30)$ 0829+046 $0.98\pm 0.13$ $(1.0,76)$ $0.7\pm 0.23$ $(1.7,176)$
$-2.56\pm 0.3$ $(0.9,76)$ $-1.87\pm 0.54$ $(1.8,176)$ $0.7\pm 0.12$ $(0.6,51)$
$0.79\pm 0.24$ $(1.8,144)$ $-2.0\pm 0.33$ $(0.9,51)$ $-1.89\pm 0.56$
$(2.0,144)$ 0836+710 $0.77\pm 0.18$ $(2.7,99)$ $0.37\pm 0.22$ $(3.9,44)$
$-1.96\pm 0.49$ $(14.2,99)$ $-1.97\pm 0.7$ $(15.8,44)$ $0.46\pm 0.18$
$(1.2,58)$ $0.3\pm 0.22$ $(4.7,40)$ $-1.98\pm 0.52$ $(14.1,58)$ $-1.63\pm
0.66$ $(15.9,40)$ 0851+202 $0.88\pm 0.19$ $(1.5,110)$ $0.82\pm 0.34$
$(5.1,263)$ $-2.87\pm 0.46$ $(2.9,110)$ $-2.79\pm 0.81$ $(8.3,263)$ $0.69\pm
0.17$ $(1.3,75)$ $0.94\pm 0.33$ $(5.7,234)$ $-2.1\pm 0.49$ $(3.0,75)$
$-3.05\pm 0.77$ $(8.7,234)$ 0954+658 $1.2\pm 0.21$ $(1.2,102)$ $1.09\pm 0.25$
$(2.8,184)$ $-3.39\pm 0.47$ $(1.3,102)$ $-2.97\pm 0.52$ $(2.4,184)$ $1.37\pm
0.21$ $(1.2,77)$ $1.17\pm 0.26$ $(3.1,161)$ $-3.98\pm 0.5$ $(1.6,77)$
$-3.32\pm 0.52$ $(2.7,161)$ 1101+384 $0.89\pm 0.19$ $(1.8,151)$ $0.61\pm 0.26$
$(1.9,106)$ $-2.19\pm 0.26$ $(0.9,151)$ $-1.57\pm 0.46$ $(1.5,106)$ $0.75\pm
0.15$ $(1.9,97)$ $0.65\pm 0.27$ $(2.0,92)$ $-2.2\pm 0.2$ $(1.1,97)$ $-1.57\pm
0.45$ $(1.5,92)$ 1127-145 $0.48\pm 0.24$ $(6.3,95)$ $0.35\pm 0.26$ $(7.1,96)$
$-1.14\pm 0.66$ $(10.7,95)$ $-1.52\pm 0.62$ $(6.1,96)$ $0.41\pm 0.27$
$(7.5,38)$ $0.71\pm 0.25$ $(5.8,74)$ $-1.03\pm 0.61$ $(9.2,38)$ $-2.18\pm 0.6$
$(6.1,74)$ 1156+295 $0.76\pm 0.16$ $(3.6,61)$ $0.73\pm 0.18$ $(1.6,27)$
$-2.47\pm 0.39$ $(1.1,61)$ $-2.24\pm 0.49$ $(2.5,27)$ $0.55\pm 0.14$
$(0.5,31)$ $0.73\pm 0.18$ $(1.6,26)$ $-2.21\pm 0.47$ $(1.4,31)$ $-2.44\pm
0.49$ $(2.7,26)$ 1219+285 $0.77\pm 0.2$ $(2.1,84)$ $0.73\pm 0.23$ $(1.3,104)$
$-1.78\pm 0.38$ $(1.3,84)$ $-1.76\pm 0.5$ $(1.9,104)$ $0.98\pm 0.18$
$(0.8,36)$ $0.74\pm 0.26$ $(1.7,71)$ $-2.58\pm 0.32$ $(0.5,36)$ $-1.56\pm
0.52$ $(2.2,71)$ 1222+216 $0.85\pm 0.19$ $(2.2,173)$ $0.55\pm 0.17$ $(0.7,66)$
$-2.17\pm 0.58$ $(3.5,173)$ $-2.5\pm 0.58$ $(2.3,66)$ $0.93\pm 0.14$
$(0.4,50)$ $0.53\pm 0.18$ $(0.8,51)$ $-3.47\pm 0.52$ $(2.1,50)$ $-2.89\pm
0.61$ $(2.7,51)$ 1226+023 $0.68\pm 0.21$ $(12.9,267)$ $0.5\pm 0.23$
$(4.7,134)$ $-1.86\pm 0.47$ $(9.1,267)$ $-1.65\pm 0.47$ $(3.6,134)$ $0.81\pm
0.24$ $(9.1,172)$ $0.48\pm 0.23$ $(5.2,119)$ $-1.93\pm 0.5$ $(6.8,172)$
$-1.63\pm 0.46$ $(3.8,119)$ 1253-055 $0.85\pm 0.3$ $(10.1,123)$ $0.44\pm 0.24$
$(3.7,271)$ $-2.33\pm 0.63$ $(9.2,123)$ $-1.95\pm 0.69$ $(4.6,271)$ $0.92\pm
0.22$ $(4.0,49)$ $0.6\pm 0.24$ $(3.6,210)$ $-2.96\pm 0.5$ $(4.2,49)$ $-2.53\pm
0.69$ $(4.3,210)$ 1308+326 $0.9\pm 0.25$ $(2.8,118)$ $0.87\pm 0.25$ $(2.6,81)$
$-2.58\pm 0.61$ $(3.6,118)$ $-2.81\pm 0.66$ $(3.7,81)$ $0.69\pm 0.21$
$(1.9,46)$ $0.87\pm 0.25$ $(2.6,81)$ $-2.88\pm 0.56$ $(3.1,46)$ $-2.81\pm
0.66$ $(3.7,81)$ 1633+382 $0.74\pm 0.22$ $(4.8,149)$ $0.71\pm 0.22$
$(2.9,351)$ $-2.16\pm 0.47$ $(5.3,149)$ $-1.9\pm 0.45$ $(4.2,351)$ $0.75\pm
0.23$ $(5.2,139)$ $0.72\pm 0.22$ $(3.0,300)$ $-1.84\pm 0.49$ $(7.1,139)$
$-1.92\pm 0.44$ $(4.6,300)$ 1652+398 $0.64\pm 0.14$ $(6.5,60)$ $0.51\pm 0.22$
$(1.4,17)$ $-1.72\pm 0.18$ $(1.2,60)$ $-1.33\pm 0.39$ $(2.0,17)$ $0.53\pm
0.15$ $(0.6,22)$ $0.81\pm 0.16$ $(0.8,13)$ $-1.37\pm 0.17$ $(0.3,22)$
$-1.51\pm 0.16$ $(0.4,13)$ 1730-130 $0.68\pm 0.2$ $(9.4,140)$ $1.04\pm 0.26$
$(2.8,165)$ $-1.63\pm 0.46$ $(6.5,140)$ $-2.78\pm 0.48$ $(2.9,165)$ $0.85\pm
0.21$ $(5.8,66)$ $1.05\pm 0.26$ $(2.9,149)$ $-2.64\pm 0.39$ $(3.7,66)$
$-2.79\pm 0.44$ $(2.8,149)$ 1749+096 $0.98\pm 0.16$ $(0.9,82)$ $0.79\pm 0.26$
$(2.5,145)$ $-3.1\pm 0.34$ $(1.0,82)$ $-3.19\pm 0.64$ $(3.7,145)$ $0.99\pm
0.17$ $(1.0,65)$ $0.95\pm 0.27$ $(2.5,107)$ $-3.34\pm 0.38$ $(1.2,65)$
$-3.47\pm 0.54$ $(2.6,107)$ 2200+420 $1.59\pm 0.3$ $(5.1,446)$ $0.84\pm 0.18$
$(3.4,616)$ $-2.98\pm 0.46$ $(6.0,446)$ $-2.4\pm 0.49$ $(3.6,616)$ $0.91\pm
0.17$ $(3.6,354)$ $0.96\pm 0.17$ $(3.2,530)$ $-2.45\pm 0.4$ $(3.5,354)$
$-2.58\pm 0.46$ $(3.5,530)$ 2223-052 $0.61\pm 0.24$ $(2.8,76)$ $0.89\pm 0.24$
$(2.5,54)$ $-2.49\pm 0.42$ $(2.5,76)$ $-2.45\pm 0.45$ $(3.7,54)$ $0.63\pm
0.25$ $(3.7,41)$ $0.96\pm 0.23$ $(2.3,46)$ $-2.75\pm 0.48$ $(3.8,41)$
$-2.44\pm 0.41$ $(3.2,46)$ 2230+114 $0.76\pm 0.23$ $(6.8,161)$ $0.58\pm 0.26$
$(2.3,49)$ $-1.91\pm 0.41$ $(4.3,161)$ $-1.7\pm 0.53$ $(2.5,49)$ $0.71\pm
0.13$ $(1.2,41)$ $0.69\pm 0.27$ $(2.8,42)$ $-2.48\pm 0.35$ $(2.1,41)$
$-1.83\pm 0.54$ $(2.8,42)$ 2251+158 $0.77\pm 0.2$ $(13.6,132)$ $0.48\pm 0.26$
$(3.6,52)$ $-1.94\pm 0.39$ $(8.3,132)$ $-1.96\pm 0.63$ $(5.5,52)$ $0.88\pm
0.2$ $(9.7,72)$ $0.48\pm 0.28$ $(4.0,46)$ $-2.46\pm 0.4$ $(6.4,72)$ $-1.75\pm
0.67$ $(5.9,46)$

777For each source, the top row refers to the full jet length probed and the
bottom row refers to the overlap region probed at both frequencies; Columns:
(1) IAU B 1950 name, (2) Fit parameter of jet-geometry at 15 GHz $l$ (reduced
$\chi^{2}$, degrees of freedom), (3) Fit parameter of jet-geometry at 43 GHz
$l$ (reduced $\chi^{2}$, degrees of freedom), (4) Fit parameter of brightness-
temperature gradient at 15 GHz $l$ (reduced $\chi^{2}$, degrees if freedom),
(3) Fit parameter of brightness-temperature gradient at 43 GHz $l$ (reduced
$\chi^{2}$, degrees of freedom)

A statistical comparison was carried out to find whether the scaling
parameters show systematic differences between the FSRQ and BL Lac classes.
The two RGs in the sample, 3C 111 and 3C 120, show a behavior similar to the
FSRQs with a pronounced complex substructure consisting of bright radio knots.
They will be discussed in more detail in Sect. 6. The distributions of the
scaling parameters are shown in Figs. 4-7. All histograms are shown with a
standard binning and a binning method based on Bayesian block analysis
(Scargle et al., 2013), implemented in astropy (Astropy Collaboration et al.
2013, Price-Whelan et al. 2018). The Bayesian binning method allows us to
pinpoint significant differences between histograms at a glance. For all plots
the indices (Q,B,G) denote FSRQs, BL Lacs and radio galaxies, respectively.
The index ”ol” denotes values that are calculated in the overlap region in
contrast to the entire jet length. These were introduced in order to make the
often densely packed plots more readable. In cases of uneven sample sizes,
only the Kolmogorov-Smirnov (KS) test was performed. For samples of equal
size, viz. the comparison of both frequencies, also the Spearman, Kendall and
Pearson correlation coefficients were calculated. In cases where a difference
between two classes are indicated by the KS-test, we checked the robustness of
the difference by repeatedly performing the KS test while randomly excluding
sources from each class (15 FSRQs, 11 BL Lacs) to find out which and how many
sources are driving the possible differences. This was done by randomly
excluding up to ten two sources from each sample. The results of the KS-test
analysis is shown in Fig. 2 and Fig. 3.

Figure 2: The p-values for each pair of parameters involving the jet geometry
compared with the two-sided KS test. The fields are color coded ranging from
$p<5\,\%$ (dark blue) to $p\geq 10\,\%$ (yellow). The lower triangle, where
all $l$ values are compared and the upper triangle where all
$\chi^{2}_{\mathrm{red}}$-values are compared, in the plot have to be read
separately. The indices (Q,B,G) denote FSRQs, BL Lacs and radio galaxies,
respectively. The index ”ol” denotes values that are calculated in the overlap
region in contrast to the entire jet length. These were introduced in order to
make the plots more readable  Figure 3: The p-values for each pair of
parameters involving the $T_{\mathrm{B}}$ gradients compared with the two-
sided KS test. The fields are color coded ranging from $p<5\,\%$ (dark blue)
to $p\geq 10\,\%$ (yellow). The lower triangle, where all $s$ values are
compared and the upper triangle where all $\chi^{2}_{\mathrm{red}}$-values are
compared, in the plot have to be read separately.

###  4.1 Diameter gradients

For the diameter gradients the results of the statistical comparisons between
the frequencies are listed in Table 3 and the results of the comparison of the
source classes are shown in Fig 2. We find typical values of $l$ in the range
between $0.5$ to $1$, i.e., collimated jet geometries. Uncertainties range
between $\sim 0.08$ in case of the apparently most uniform jet geometries to
$\sim 0.3$ for some complex cases where deviations from a simple $r^{l}$
power-law behavior are most pronounced. The most common $l$ value at 43 GHz is
around $0.5$ with typically somewhat larger values at 15 GHz.

##### Comparison of both frequencies:

Figure 4 shows the $l$-value distribution at both frequencies along the entire
jet detected at the respective frequency (top) and the overlap region
(bottom). Along the entire jets there is a hint that the jets at
$43\,\mathrm{GHz}$ are more collimated than the jets at $15\,\mathrm{GHz}$
(KS: $p\sim 2.6\%$ ). This tentative difference vanishes when comparing the
jets in the overlap region (KS: $p\sim 94.4\%$).

##### Comparison of the source classes at $15\,\mathrm{GHz}$:

Figure 5 shows the $l-$value distribution for the 15 GHz jets. The
measurements along the entire jet (first row) indicate a difference in jet
geometry between the FSRQs and the BL Lac objects (KS: $p\sim 0.4\%$). The
FSRQs show smaller $l$-values and thus stronger collimated jets than the BL
Lac objects. The two radio galaxies 3C 111 and 3C 120 show collimated jets in
the same regime as the FSRQs. As before in the case of the 15 GHz to 43 GHz
comparison, however, when considering the overlap regions (third row), the
differences between FSRQs and BL Lacs in terms of $l$ become insignificant
(KS: $p\sim 62.1\%$). When studying the distribution of the
$\chi^{2}_{\mathrm{red}}$ values for the jet-geometry power-law fits (Fig. 5,
top row, right column) we find a pronounced difference between the FSRQs and
the BL Lacs (KS: $p\sim 0.2\%$). A simple power law fit is a systematically
worse description for the FSRQs than for the BL Lacs, regarding the
$\chi^{2}_{\mathrm{red}}$ values. This difference does not disappear when
comparing the overlap regions in the jet (third row right column, KS: $p\sim
3.2\%$).

##### Comparison of the source classes at $43\,\mathrm{GHz}$:

At $43\,\mathrm{GHz}$ the $l$-values measured along the entire jet (Fig. 5,
second row, left column) yield no significant distinction between FSRQs and BL
Lacs (KS: $p\leavevmode\nobreak\ \sim 11.9\%$) nor between FSRQs and RGs (KS:
$p\leavevmode\nobreak\ \sim 41.2\%$). Also the comparison of BL Lacs and RGs
yield no hint for a difference between the source classes (KS:
$p\leavevmode\nobreak\ \sim 7.7\%$). In the overlap region, however, BL Lacs
behave differently than FSRQs (KS: $p\sim 2.7\%$), showing a tendency for
larger $l$-values. Regarding the $\chi^{2}_{\mathrm{red}}$-distributions, no
difference between FSRQs and BL Lacs can be constituted along the entire jet
length and the overlap region, see the right column in Fig 4, second and
fourth row.

Table 3: Correlation coefficients for different statistical tests Pair of | Statistical | Correlation | p-value
---|---|---|---
parameters | test | Coefficient |
diameter gradients |  |  |
$l_{15}-l_{43}$ | KS |  | $0.026$
| Pearson | 0.42 | 0.027
| Spearman | 0.37 | 0.056
| $\tau_{\mathrm{kendall}}$ | 0.27 | 0.044
$l_{15,\mathrm{overlap}}-l_{43,\mathrm{overlap}}$ | KS |  | $0.94$
| Pearson | 0.43 | $0.021$
| Spearman | 0.25 | $0.20$
| $\tau_{\mathrm{kendall}}$ | 0.15 | $0.27$
$T_{\mathrm{B}}$ gradients |  |  |
$s_{15}-s_{43}$ | KS |  | $0.55$
| Pearson | 0.61 | $5.3\times 10^{-4}$
| Spearman | 0.59 | 0.0011
| $\tau_{\mathrm{kendall}}$ | 0.45 | $7.8\times 10^{-4}$
$s_{15,\mathrm{overlap}}-s_{43,\mathrm{overlap}}$ | KS |  | $0.35$
| pearson | 0.73 | $1.0\times 10^{-5}$
| spearman | 0.70 | $4.0\times 10^{-5}$
| $\tau_{\mathrm{kendall}}$ | 0.51 | $1.5\times 10^{-4}$

Figure 4: Distribution of $l$-values along the entire jet (top panel) and
along the overlap-region (bottom). Values at 15 GHz are shown in blue and 43
GHz in red. The histograms are shown for equal bin width of $\Delta l=0.2$
(dashed lines) and for Bayesian binning (solid lines; see text)

Figure 5: Distributions of the geometry scaling parameter $l$ (left) and its
variance $\chi^{2}$ (right) shown with Bayesian (solid lines) and uniform
binning (dashed lines) for FSRQs (red) BL Lacs (bue), and RGs (green). Upper
four panels: Distributions for the entire jet at both frequencies; Lower four
panels: Distributions for the overlap-region at both frequencies. Bin sizes
were chosen as 0.2 for the $l$ distributions and 1.0 for the variance
distributions.

###  4.2 Brightness-temperature gradients

For the power-law brightness-temperature gradient index $s$ the results of the
statistical comparisons between the frequencies are listed in Table 3 and the
results of the comparison of the source classes are shown in Fig 3. Most
sources show $T_{b}$ power law indices in the range of $-3\lesssim
s\lesssim-1.5$ with the width of the distribution somewhat increasing when
only the overlap region at both frequencies is considered. Uncertainties of
individual $s$-values range from 0.2 to 0.7, reflected in the large Bayesian-
block widths. Most commonly, the slopes along the full jet widths lie in the
range between $-2$ and $-1.5$.

##### Comparison of both frequencies:

Figure 6 shows the $s$-value distribution of both frequencies evaluated along
the entire detected jet at the respective frequency (top) and along the
overlap region (bottom). The $s$-values are indistinguishable for both
frequencies in the entire jet and overlap regions, respectively.

##### Comparison of the source classes at $15\,\mathrm{GHz}$:

Figure 7 (top row) shows the $s$-value distribution at $15\,\mathrm{GHz}$
along the full jet length and the corresponding overlap region, where the
colors indicate the source classes. Regarding the distribution of the
brightness temperature power law indices, the different source classes are
indistinguishable, both along the entire jet length and the overlap region.
The $\chi^{2}_{\mathrm{red,s_{15}}}$ distribution (Fig. 7, top row), however,
indicates a difference between FSRQs and BL Lacs along the entire jet length.
The BL Lac objects show a tendency for smaller
$\chi^{2}_{\mathrm{red,s_{15}}}$ values than FSRQs, indicating that a single
power law is a better description for the brightness-temperature gradient for
BL Lacs than for FSRQs (KS: $p\sim 0.2\%$). With reference to the overlap
region, this difference also becomes apparent (KS: $p\sim 0.1\%$).

##### Comparison of the source classes at $43\,\mathrm{GHz}$:

Figure 7 shows the $s$-value distribution at $43\,\mathrm{GHz}$ along the full
jet length (second row) and the corresponding overlap region (fourth row). The
$s$-value distributions are statistically indistinguishable both along the
entire jet and the overlap region. As in the case of the jet-geometry power-
law fits, the $\chi^{2}_{\mathrm{red,s_{43}}}$ value distributions (Fig. 7)
indicate smaller values for BL Lac object than FSRQs along the entire jet
lengths and the overlap regions (KS: $p\sim 1.4\%$ in both cases).

Figure 6: Distribution of power-law slopes $s$ of the brightness-temperature
gradient along the entire jet (top panel) and along the overlap-region
(bottom) at 15 GHz (blue) and 43 GHz (red). The histograms are shown for
uniform bin width $\Delta s=0.5$ (dashed lines) and for Bayesian binning
(solid lines; see text)

Figure 7: Bayesian binning (solid lines) and uniform binning (dashed lines) of
the $T_{\rm B}$ power-law slope distributions. The colors indicate the source
classes (red: FSRQs, blue: BL Lacs, green: RGs). Top two rows: Distributions
along the entire jet for both frequencies; Bottom two lines: Distributions
along the overlap-region of both frequencies. Left column: Distribution of
$s$-values. For both frequencies, the bin size is 0.5. Right column
$\chi^{2}_{\mathrm{red}}$ value distribution describing the brightness
temperature. For both frequencies, a unit bin size was chosen.

##  5 Discussion

In this section, we discuss the general implications of the gradient fits at
$15$ GHz and $43$ GHz that we have obtained in the previous section for our
sample of 28 blazars and radio galaxies. A discussion of our findings
regarding the individual sources will follow in the next section.

###  5.1 Brightness-temperature gradients

Pushkarev & Kovalev (2012) analyzed the $T_{\rm B}$-gradients for 30 sources
at $2\,\mathrm{GHz}$ and $8\,\mathrm{GHz}$ and found a mean value of $\langle
s\rangle\sim-2.2$. Furthermore Kravchenko et al (in prep., 2021 priv. com.)
find a similar value analyzing the entire MOJAVE data archive at
$15\leavevmode\nobreak\ \mathrm{GHz}$. This is consistent with our smaller
dual-frequency sample results (cf. Sect. 4.2). The $T_{\rm B}$-gradients are
determined by the jet geometry (see below), particle and magnetic-field
gradients. We conclude that studies at different frequencies in general lead
to consistent results for these physical quantities. Specifically, for both
our frequencies, the $s$ distributions are not clearly distinguishable along
the entire jets and the overlap regions. Moreover, the different source
classes of BL Lacs and FSRQs do not show any strong differences in terms of
the average $T_{\rm B}$-gradient trends. However, the
$\chi^{2}_{\mathrm{red}}$-distribution at both frequencies show that a single
power law fit is more appropriate for BL Lacs than for FSRQs, indicating that
the latter show a more complex $T_{\rm B}$-distribution along the jet axis due
to local surface brightness enhancements, leading to more scatter in our
global fits. This is consistent with the findings of Jorstad et al. (2017) who
find high $T_{\rm B}$-values to be more common in knots of FSRQs as compared
to BL Lacs. This gives support to the idea of systematically higher Doppler
factors and stronger shocks in FSRQ jets and/or possibly more rapid expansion
in the jets of BL Lacs.

###  5.2 Jet geometry

Measured along the entire jet lengths there is a systematic difference in
collimation between the jets measured at $15\,\mathrm{GHz}$ and
$43\,\mathrm{GHz}$. The jets at $43\,\mathrm{GHz}$ appear more collimated
(tend to show smaller $l$-values) than at $15\,\mathrm{GHz}$. In the overlap
region, this difference is not significant. It is therefore plausible, that
the differences arise from the geometrical properties of the jet at larger
scales, i.e. that the $l$-values change with $r$. Indeed, Kovalev et al.
(2020) showed that in many cases, it is possible to measure a transition from
parabolic to conical jet geometries.

At $15\,\mathrm{GHz}$, the source classes show a notable difference when
comparing them along the entire jet. FSRQs systematically show lower
$l$-values than BL Lacs indicating a stronger jet collimation. This is also
found by Kravchenko et al. (in prep., 2021 priv. com.) on a larger 15 GHz AGN-
jet sample. In our study, this difference is again not significantly found in
the (smaller) overlap region of the images at both frequencies indicating
again that the differences arise further down the jet and reinforcing the idea
of a geometry transition on parsec scales. Analyzing the
$\chi^{2}_{\mathrm{red}}$ of the single power-law fits to the jet geometry
also yields interesting results. The FSRQs tend to show larger
$\chi^{2}_{\mathrm{red}}$ values, both along the respective entire jet and in
the overlap regions compared to BL Lacs. This indicates that the geometric
structure in FSRQs is more complicated than the smooth power law observed in
BL Lacs. The complexity could arise from regions where the full jets
recollimate and widen again locally. Alternatively, some of the model
components in FSRQ jets might indeed be representative of local brightness
enhancements that are smaller than the full jet width at this distance. An
example where this seems to be the case can be seen in Fig. 1 (top panel) at
the relative coordinates ($-6,-7$).

At $43\,\mathrm{GHz}$ the difference between the source classes is not
significantly detected along the entire jet nor in the overlap region. This is
true for both, the $l$-values distribution and the $\chi^{2}_{\mathrm{red}}$
\- distribution. This could be simply a result of reduced image quality at the
higher frequency. Alternatively, the high-frequency emission could be
associated with plasma closer to the inner jet axis (the spine of the jet)
while the lower-frequency emission might be dominated by outer layers
(sheath). In such a stratified-jet scenario, the difference between FSRQ and
BL Lac jet geometry that we find at 15 GHz can be interpreted as a process
that is largely associated with the sheath while the spine remains unaffected.

###  5.3 Measurement constraints on jet-internal gradients

Figure 8 illustrates the allowed parameter space for the jet-internal
gradients of the electron density $n$, the magnetic field strength $b$, and
the spectral index $\alpha$ constrained by the measured values of the power-
law slopes of the brightness temperature $s$ and jet diameter $l$.

To investigate these dependencies in detail, we first reconsider some
theoretical standard scenarios (cf. Kadler et al., 2004) in order to estimate
plausible $s$-values, depending on $l,n,b$ and $\alpha$. We assume
equipartition between the electron energy density and magnetic field strength
density, thus $n_{e}\leavevmode\nobreak\ \propto B^{2}$ and therefore the
scaling parameter of the electron energy density $n$ depends on the scaling of
the magnetic field along the jet axis ($n=2b$).

  * •

Conical jet ($l=1$):

    * –

torodial magnetic field ($b=-1$),equipartition ($n=-2$), $\alpha=-0.5$
$\rightarrow$ $s=-2.5$

    * –

polodial magnetic field ($b=-2$), equipartition ($n=-4$), $\alpha=-0.5$
$\rightarrow$ $s=-6$

In the case of a conical jet, equipartition, and a reasonable spectral index
of $\alpha=-0.5$ the $s$-values thus have a domain of reasonable values of
$-6\leq s\leq-2.5$

  * •

Parabolic jet ($l=0.5$):

    * –

torodial magnetic field ($b=-0.5$),equipartition ($n=-1$), $\alpha=-0.5$
$\rightarrow$ $s=-1.25$

    * –

polodial magnetic field ($b=-1$), equipartition ($n=-2$), $\alpha=-0.5$
$\rightarrow$ $s=-3$

In the case of a parabolic jet, equipartition, and a reasonable spectral index
of $\alpha=-0.5$ the $s$-values have a domain of reasonable values of $-3\leq
s\leq-1.25$

The measured $s$ and $l$ values for both frequencies seen in Fig. 8 show an
anticorrelation ($r_{\mathrm{spearman}}\sim-0.73$,
$r_{\mathrm{pearson}}\sim-0.69$) in the sense that larger $l$ values lead to
more negative $s$ values. The $r_{\mathrm{pearson}}$ even indicates a simple
linear connection of both parameters. This can also be constituted for the
frequencies separately
$(r_{\mathrm{spearman,15}},r_{\mathrm{pearson,15}},r_{\mathrm{spearman,43}},r_{\mathrm{pearson,43}})\sim(-0.73,-0.69,-0.64,-0.69)$.
Further splitting the values into source classes, ignoring the radio galaxies
by number within the sample also keeps the anti-correlation measurable
$(r_{\mathrm{spearman,15,Q}},r_{\mathrm{pearson,15,Q}},r_{\mathrm{spearman,43,Q}},r_{\mathrm{pearson,43,Q}})\sim(-0.6,-0.62,-0.56,-0.66)$
and
$(r_{\mathrm{spearman,15,B}},r_{\mathrm{pearson,15,B}},r_{\mathrm{spearman,43,B}},r_{\mathrm{pearson,43,B}})\sim(-0.56,-0.58,-0.86,-0.79)$.
The anti-correlation of the $s$ and $l$ values suggests that more collimated
jets imply flatter brightness-temperature gradients. It shows that the main
impact of a resulting $s$-value comes from the jet geometry because other wise
an uncorrelated data cloud would be expected. However the anti-correlation is
not as clear, as the $s$ values seem not to depend solely on $l$. The rank
correlation indices show ranges of $-0.8\lesssim r_{s/p}\lesssim-0.5$, which
hints to a possible impact of the parameters $n$ and $b$ to the resulting
$s$-values. This will be discussed source by source in the next section. The
data are fit with a linear model (gray dashed lines) to measure the slope of
the anti correlation which is $\sim-3.4$. Equation 7 however suggests that
this slope should be $\leavevmode\nobreak\ -1$ if $n,b$ and $\alpha$ where
completely independent of the jet geometry. However as seen in the standard
scenarios above and Sect.3.2, at least the scaling of the magnetic field
strength depends on the jet geometry. Depending on how $n$ and $b$ are de-
coupled, the scaling of the electron energy density also depends on the jet
geometry. The black dashed lines indicate the parameter space of the
parameters $(n,b,\alpha)_{min}=(-2.5,-1.25,-1)$ and
$(n,b,\alpha)_{max}=(-1,-0.5,0)$. This can be concluded to be the parameter
space of jet-internal $l$, $b$, and $n$ gradients that is in agreement with
the observational constraints for our sample analysis of VLBI jets.

Figure 8: The parameter space of $n,b$ and $\alpha$, spanned by the measured
$s$ and $l$ values at both frequencies.

###  5.4 Tests for redshift biases

The redshifts of the sources in our sample range from 0.0308 (Mrk 421) to
2.218 (4C +71.07). Moreover, the angular-to-linear scale conversion also
strongly depends on the orientation of the jets to the line of sight. Thus,
the angular scales probed with the VLBA in different sources can correspond to
very different linear scales. To test whether our conclusions are affected by
these effects and whether the overlap regions are comparable for the different
sources, we first compute the apparent length of the jet from its maximum
observed angular distance from the radio core. Second, we deproject the jet
length from the plane of the sky using the inclination angle $\theta$ between
the jet direction and the line of sight given in Pushkarev et al. (2009) to
obtain its radial length in parsecs. For sources for which Pushkarev et al.
(2009) does not list an inclination angle, we searched the literature in order
to obtain an inclination angle, which is discussed on a source by source basis
in Sect. 6. Assuming that the distance between the radio core and the jet base
is small compared to the jet length, we find the overlap region typically at
jet lengths $5\lesssim d_{c}/\mathrm{pc}\lesssim 100$. Along the entire jet
length and the overlap region, respectively, the brightness temperature and
diameter gradients for each jet at $15\,\mathrm{GHz}$ were fitted. We tested
the interesting pairs from table 3 and and Fig. 2,3, regarding the jet
collimation and complexity. The results from these tests are consistent with
the findings from the projected $\mathrm{mas}$-scale jets. Along the entire
jet length, FSRQs are more collimated than BL Lac objects (KS: $p\sim 1.7\%$).
Within the overlap region this difference is not significant (KS: $p\sim
28.7\%$). The interpretation is the same as before. The jets close to the base
are similar while a possible difference in collimation can be measured further
down stream. Regarding the $\chi^{2}_{\mathrm{red}}$ distribution for the
$l_{15}$ fits, FSRQs show a more complex structure along the entire jet length
than BL Lac objects (KS: $p\sim 0.44\%$) which is again not found significant
in the overlap region (KS: $p\sim 25.9\%$). Regarding the
$\chi^{2}_{\mathrm{red}}$ distribution from the $s_{15}$-fits, FSRQs
consistently show larger $\chi^{2}_{\mathrm{red}}$-values than BL Lacs, both
along the entire jet length (KS: $p\sim 2.4\%$) and the overlap region (KS:
$p\sim 0.04\%$). We conclude that using the lengths of the deprojected jets to
describe the scaling behavior rather than their angular diameters does not
alter our findings from the previous sections.

###  5.5 Possible biases form geometry transition sources

With the mean value and the variance of the $l$ and $s$-value and the
$\chi^{2}_{\mathrm{red}}$ distributions, we simulated normal distributions
with different class sizes. We tested, how much each class can be reduced in
size in order for the KS-test to yield reliable results and therefore find the
differences between the known distributions. We find, that each class should
contain at least 8-10 values in order for the KS test to reliably find
differences in known distributions. Knowing this, we excluded all sources that
show a geometry transition (3 BL Lacs, 5 FSRQs and one radio galaxy). This
reduces the sample sizes to 8 BL Lacs, 10 FSRQs and one radio galaxy. With the
reduced sample, the difference in jet collimation at $15\,\mathrm{GHz}$
between FSRQs and BL Lacs become less significant along the entire jet length
(KS: $p\sim 7\%$), however the complexity of the jet is still larger for FSRQs
than for BL Lacs, for both the diameter and brightness temperature gradient
fit, when comparing the $\chi^{2}_{\mathrm{red}}$-values of the source classes
for the different fits (KS: p-values in the range of $(0.1-0.5)\%$). We want
to emphasize, that this has to be taken with caution, since the sample sizes
are reduced to an extent to where it is not entirely clear whether the KS-test
is even able to yield interpretable results. Overall these findings however do
not contradict the found differences between the source classes.

###  5.6 Geometry transition within the VLBI jets

A number of VLBI jets are known to exhibit transitions from a parabolic
($l\sim 0.5$) to a conical ($l\sim$ 1) geometry, for example for M87 (Asada &
Nakamura, 2012), NGC 6251 (Tseng et al., 2016), NGC 4261 (Nakahara et al.,
2018), NGC 1052 (Nakahara et al., 2020), NGC 315 (Boccardi et al., 2021; Park
et al., 2021) and 1H0323+342 (Hada et al., 2018). M87 (Mertens et al., 2016;
Hada et al., 2018; Park et al., 2019), NGC 315 (Park et al., 2021) and
1H0323+342 (Hada et al., 2018) also feature accelerating jet components near
the geometry transition zones. By comparing $2\leavevmode\nobreak\
\mathrm{GHz}$ and $8\leavevmode\nobreak\ \mathrm{GHz}$ data, Pushkarev &
Kovalev (2012) found hints for a transition in the diameter scaling in jets.
Kovalev et al. (2020) showed that geometry transitions from a parabolic to a
conical geometry can be observed on scales of several thousand Schwarzschild
radii $R_{\textrm{S}}$ in individual AGN jets.

According to Potter & Cotter (2015), the four-momentum of a relativistic jet
is composed of i) the internal, kinetic and rest mass energy, ii) the magnetic
energy and iii) the synchrotron, inverse Compton, and adiabatic losses.
Depending on which term is larger than the magnetic term, the jet is said to
be particle dominated or magnetically dominated, respectively. Transitions
from parabolic to conical geometry as observed in the cases listed above may
be explained by the jet switching from a magnetically dominated to a
kinetically dominated flow. In the following, we use our systematic modeling
of the 28 sources in our sample to put constraints on possible geometry breaks
seen with this dual-frequency approach in AGN jets.

####  5.6.1 Geometry breaks

Figure. 10-18, show the geometry breaks inferred from the fit procedure
described above along with estimated positions of the Bondi radius $r_{b}$ for
each source. In order to obtain $r_{b}$, the mass of the SMBH and gas
temperature of the matter accreting onto the SMBH has to be known: (Bondi,
1952; Russell et al., 2015).

$\frac{r_{b}}{\mathrm{pc}}=31\times\left(\frac{k_{b}T}{\mathrm{keV}}\right)^{-1}\times\left(\frac{M}{10^{9}M_{\odot}}\right)\quad.$
(10)

In the case of M87, the black-hole mass was determined by Gebhardt et al.
(2011) ($M=(6.6\pm 4.4)\times 10^{9}M_{\odot}$) and Walsh et al. (2013)
($M=3.5^{+0.9}_{-0.7}\,10^{9}M_{\odot}$). Later the Event Horizon Telescope
Collaboration et al. (2019) found the black-hole mass to be $M=(6.5\pm
0.9)\,10^{9}M_{\odot}$ in accordance with Gebhardt et al. (2011). Russell et
al. (2015) determined a gas temperature of $T\sim 0.9\,\mathrm{keV}$ in the
center of M87 resulting in a Bondi radius of $r_{b}\sim 224\,\mathrm{pc}$ from
X-ray observations. For blazars, the temperature of the coronal plasma near
the SMBH must be estimated taking into account that the observed X-ray
emission is dominated by the nonthermal emission from the jet. For a sample
low-power radio galaxies, Balmaverde et al. (2008) measured gas temperatures
between $0.4\lesssim E/\mathrm{keV}\lesssim 1.7$. For high-power sources,
typical gas temperatures near the SMBH are up to $\sim 7\,\mathrm{keV}$, (Yaji
et al., 2010; Gaspari et al., 2013, e.g.). Since we did not measure the Bondi
radius for the sample, discussed in this work, we allow the gas temperature to
vary within the bounds $0.4\lesssim E/\mathrm{keV}\lesssim 7$ for each source
and compare the region of plausible Bondi radii with the geometry transition
region. The respective jet-inclination (viewing) angle is taken into account
by using the measured angles from Pushkarev et al. (2009) for the cases of CTA
26 (0336$-$019, $\theta\sim 3.2\,\mathrm{deg}$), PKS 0528+134 (0528+134,
$\theta\sim 1.7\,\mathrm{deg}$), 4C +71.07 (0836+710, $\theta\sim
3.2\,\mathrm{deg}$), 4C +71.07 (1156+295, $\theta\sim 2\,\mathrm{deg}$) and BL
Lacertae (2200+420, $\theta\sim 7.5\,\mathrm{deg}$). In the other cases, we
took values of the inclination angles from the references cited in Sect. 6.
The results of the broken geometry fits are listed in Table 4. In each case we
compare the $\chi^{2}_{\mathrm{red}}$ of smoothly broken power law fit with a
single power law fit to both the $15\,\mathrm{GHz}$ and $43\,\mathrm{GHz}$
data. In all cases an improvement is achieved by using the smoothly broken
power law fit to both frequencies. For each source we bracket the Bondi radius
by the minimum/maximum black hole mass from Table 1, the minimum/maximum gas
plausible temperature and the maximum/minimum inclination angle. For 3C 66A
(0219+428), CTA 26 (0336-019), PKS 0528+134 (0528+134), 4C +71.07 (0836+710),
Mrk 421 (1101+384), 4C +29.45 (1156+295), 3C 279 (1255-055) and BL Lac
(2200+420) the transition zone lies beyond plausible Bondi radius ranges. In
such cases Kovalev et al. (2020) argue that the geometry transition could mark
the transition of a magnetically dominated to a particle-dominated jet. The
transition zone of 3C 111 (0415+379) lies within the plausible Bondi radius
ranges. In this case, the lowest gas temperatures has to be assumed in order
for the Bond sphere to extent beyond the transition region. For CTA 26
(0336$-$019) the jet features a bright, stationary component in the brightness
temperature gradient between $d_{c}\sim 1-2\leavevmode\nobreak\ \mathrm{mas}$
during the ten year observation period, see Fig. 20. This possibly indicates
stationary shocks or jet curvature, where the jet in parts crosses the line of
sight, creating the seemingly stationary features. The distribution of the
break points shown in Fig. 9 (top) tentatively suggests relatively low values
for BL Lacs as compared to a wider distribution ranging to about a factor of
ten larger maximum values in the case of FSRQs. This is found to be
significant by a two-sample KS test ($p\sim 3.7\%$). Furthermore, a KS-test
with the $H_{0}$-hypothesis, that $x_{b,Q}<x_{b,B}$ yields $p\sim 1.7\%$ which
suggests, that the break point of the geometry transition is further down the
jet stream for FSRQs than for BL Lacs. This is also found on the entire MOJAVE
data sample at $15\leavevmode\nobreak\ \mathrm{GHz}$ by Kravchenko et al. (in
prep., 2021 priv. com.) A difference might indeed be expected according to
Potter & Cotter (2015) based on the assumption that the geometry breaks result
from the magnetic-to-kinetic transition. Figure 9 (bottom) shows the
distribution of the jet radii at the transition zone (break point $x_{b}$).
Again, we do not find a significant dichotomy between the two source classes
(KS: $p\sim 14.3\%$). Again an alternative KS-test with
$H_{0}:R_{j}(x_{b,Q})<R_{j}(x_{b,B})$ yields $p\sim 7\%$. This suggests that
the jet radii at the break points of FSRQs and BL Lacs are likely drawn from
the same distribution however FSRQs show a tendency for larger radii at the
break point. It must be emphasized, however, that the sample contains only ten
sources that do show a geometry break, of which only three are of the BL Lac
type. Furthermore, we cautiously remark that different methods can lead to
somewhat different results. For example, Boccardi et al. (2021) analyzed the
jet of NGC315 in the visibility domain at different epochs and frequencies
finding a transition from parabolic to conical jet geometry already at a
distance of about half a parsec. Park et al. (2021) performing a
multifrequency single-epoch study in the image plane, found the transition to
occur at a distance about 10 times larger. In order to further shed light onto
the issue of systematic jet geometry transitions, it would be interesting to
study more sources from even higher ($>43\leavevmode\nobreak\ \mathrm{GHz}$)
frequencies to lower frequencies ($\sim 1\leavevmode\nobreak\ \mathrm{GHz}$)
in order to study the collimation of a jet across a large scale and determine
whether geometry breaks might happen on all scales or whether the transitions
found are interesting physical points along the physical flow of the jet.

Figure 9: FSRQs are shown in red, BL Lacs are shown in blue. Top: Distribution
of the break points in linear scale and deprojected.Bottom: The distribution
of the jet radii at the respective geometry break point.  Table 4: Geometry
transition properties of individual sources

.  Name Class $x_{b}/\mathrm{mas}$ ($x_{b}/\mathrm{pc}$)
$R_{j}(x_{b})/\mathrm{mas}$ $l_{\mathrm{in}}$ $l_{\mathrm{out}}$
$\theta/\mathrm{deg}$ $\chi^{2}_{\mathrm{red,broken}}$
$\chi^{2}_{\mathrm{red,single}}$ $r_{B}/\mathrm{(}mas)\,(r_{B}/\mathrm{(}pc))$
3C 66A B $2.31\pm 0.32$ $(65)$ $0.42\pm 0.14$ $0.56\pm 0.13$ $1.33\pm 0.11$
$9.8^{\textrm{a}}$ $4.0$ $(326)$ $8.1(328)$ $0.046-1.255$ $(1.31-35.51)$ CTA
26 Q $2.12\pm 0.26$ $(293)$ $0.68\pm 0.15$ $0.62\pm 0.12$ $1.78\pm 0.27$
$3.2^{\textrm{b}}$ $16.8$ $(119)$ $21.3$ $(121)$ $0.0011-0.0310$ $(0.16-4.27)$
3C 111 G $5.42\pm 0.29$ $(16)$ $0.56\pm 0.09$ $0.60\pm 0.08$ $1.69\pm 0.24$
$19.0^{\textrm{c}}$ $12.1$ $(662)$ $12.8$ $(664)$ $0.23-6.55$ $(0.67-19.07)$
PKS 0528+134 Q $2.10\pm 0.38$ $(596)$ $0.56\pm 0.16$ $0.54\pm 0.11$ $1.24\pm
0.25$ $1.7^{\textrm{b}}$ $10.0$ $(336)$ $10.9$ $(338)$ $0.0035-0.0925$
$(0.99-26.32)$ 4C +71.04 Q $2.75\pm 0.41$ $(413)$ $0.35\pm 0.13$ $0.48\pm
0.14$ $1.15\pm 0.15$ $3.2^{\textrm{b}}$ $15.7$ $(139)$ $19.0$ $(141)$
$0.0018-0.0495$ $(0.28-7.42)$ Mrk 421 B $1.93\pm 0.38$ $(13-34)$ $0.63\pm
0.19$ $0.54\pm 0.13$ $1.02\pm 0.13$ $(2-5)^{\textrm{d}}$ $10.1$ $(253)$ $11.5$
$(255)$ $0.0048-0.3522$ $(0.084-2.456)$ 4C +29.45 Q $2.52\pm 0.60$ $(522)$
$0.91\pm 0.37$ $0.73\pm 0.17$ $1.23\pm 0.21$ $2.0^{\textrm{b}}$ $8.5$ $(84)$
$9.3$ $(86)$ $0.0034-0.0918$ $(0.084-2.456)$ 3C 279 Q $0.61\pm 0.22$ $(97)$
$0.15\pm 0.09$ $0.49\pm 0.15$ $0.78\pm 0.09$ $2.4^{\textrm{b}}$ $33.1$ $(390)$
$33.8$ $(392)$ $0.012-0.337$ $(1.98-53.74)$ BL Lac B $1.97\pm 0.13$ $(20)$
$0.37\pm 0.07$ $0.78\pm 0.09$ $1.97\pm 0.08$ $7.5^{\textrm{b}}$ $9.6$ $(1058)$
$17.0$ $1060$ $0.046-1.184$ $(0.46-11.76)$

888Columns: (1) Common name, (2) Classification, (3) geometry break point in
angular scale (geometry break point in linear scale deprojected), (4) Jet
radius at the break point, (5) $l_{\mathrm{in}}$, (5) $l_{\mathrm{out}}$, (6)
inclination angle, (7) $\chi^{2}_{\mathrm{red}}$ for the broken power law fit,
(8) $\chi^{2}_{\mathrm{red}}$ for the single power law fit.(9) range of
plausible Bondi radii in angular scale (range of plausible Bondi radii in
linear scale deprojected)

a) Hovatta et al. (2009), b) Pushkarev et al. (2009), c) Kadler et al. (2008),
d) Lico et al. (2012),

##  6 Comments on individual sources

#####  $0219+428$ (3C 66A):

The brightness-temperature along the jet of this source shows a local excess
over the general power-law decrease between $2\lesssim
d_{\mathrm{core}}/\mathrm{mas}\lesssim 3$, which is visible at both
frequencies (Fig. 19). This excess indicates a local deviation from the simple
jet model (see Eq. 7). The jet-diameter does decrease as a power law as
described by the model but shows a dip at the same distance from the radio
core where the $T_{\mathrm{B}}$ excess is observed. The dip is persistent over
the epochs, indicated by the different colors, represented within the dip.
This might hint at a stationary region where the jet recollimates and the
local surface brightness temperature rises.

The $s$-values at both frequencies are indistinguishable within the errors,
however the $l$-values differ along the respective entire jet length, while
remaining indistinguishable in the overlap region. We find a transition of the
geometry in this jet, which is shown in Fig. 10, featuring a parabolic jet
shape up to $d_{c}\sim 2.3\,\mathrm{mas}$ that transitions to a conical shape
jet downstream.

To estimate the angle to the line of sight, the variability Doppler factor can
be used to estimate $\theta\sim 9.8\,\mathrm{deg}$ as discussed in Hovatta et
al. (2009). The scaled Bondi radius is shown in Fig. 10 projected for the
derived angle, respective SMBH mass and the gas plausible gas temperature
range (dotted lines). The break point $x_{b}$ lies outside the Bondi-sphere,
which is located close to the possible recollimation region.

Figure 10: Jet diameter vs. core distance for 3C 66A (0219+428). The jet shows
a transition from a parabolic shape to a conical shape at $x_{b}\sim
2.3\,\mathrm{mas}$ (dashed line). The dotted lines indicate the estimated
location of the Bondi radius.

##### 0336$-$019 (CTA 26):

The jet diameter along the jet axis in this source is characterized by strong
scattering around the best-fit model (Fig. 20). This behavior is typical for a
FSRQ (see discussion above). When combining the information from both
frequencies as described in Sect.3.2.2, it is possible to constrain a geometry
break point in this source (Fig. 11). The jet is best described through a
parabolic shape out to $d_{c}\sim 2.1\,\mathrm{mas}$, which opens up to a cone
afterwards. The Bondi radius is smaller than the location of this break by two
orders of magnitude and is therefore unlikely to be related in any way. The
$T_{\rm B}$ profile shows an excess at $d_{c}\sim 1-2\,\mathrm{mas}$ which is
persistent across the epochs (indicated by the different colored components in
Fig. 20), which might be associated with a standing shock at this location in
the jet.

Figure 11: Jet diameter vs. core distance for CTA 26 (0336-019). The jet shows
a transition from a parabolic shape to a conical shape at $x_{b}\sim
2.1\,\mathrm{mas}$ (dashed line). The dotted line indicate the estimated
location of the Bondi radius.

##### 0415$+$379 (3C 111):

This is one out of only two radio galaxies in our sample. Model-fitting
results for this source are shown in Fig. 21. The morphology and kinematics
are rather complex and best described with a parabolic-jet model at both
frequencies and relatively flat brightness temperature gradients $-1.8\lesssim
s\lesssim-1.5$. Kadler et al. (2004) fit a brightness temperature gradient of
with a power law index of $s\sim-2.4$ by fitting individual tracked
components. The difference can be explained by the fitting of more epochs in
this work and not excluding single components for the power law fit. A break
in the jet geometry from parabolic to hyperbolic shape can be constrained at
$\sim 5.4\,\mathrm{mas}$ from the radio core, see Fig. 12. We adopt the
inclination angle of $19\,\mathrm{deg}$ from Kadler et al. (2008) to project
the position of the Bondi radius, which may coincide with the transition
region for the smallest plausible gas temperatures near the SMBH. Beuchert et
al. (2018) fit the brightness temperature and the diameter along the jet axis
for this source with two separate power laws in the region
$<3\leavevmode\nobreak\ \mathrm{mas}$ and $>3\leavevmode\nobreak\
\mathrm{mas}$ from the radio core. They find a similar switching of the jet
geometry consistently with our findings. Kovalev et al. (2020) also find a
transition of geometry at $x_{b}=(7.03\pm 0.5)\,\mathrm{mas}$. Boccardi et al.
(2021) find a transition in this region.Taking the systematic errors at a
1-$\sigma$-limit into account, there is still a difference between the found
break point in this study and the break point found in Kovalev et al. (2020)
and Boccardi et al. (2021). We tested the impact of different established
fitting methods (least square and orthogonal distance regression) in Append B
and found that different methods yield the same results with the given data
within the errors. We therefore attribute this difference between our result
and the Kovalev et al. (2020) and Boccardi et al. (2021) results to the use of
different data sets, especially the different epochs with different time
coverage of the dynamically evolving sources and with different observational
setups and image noise levels (cf. Sect. 2) and the different sets of model
fits (cf. Sect. 3) parameterizing this highly complex source.

Figure 12: Jet diameter vs. core distance for 3C 111 (0415+379). The jet shows
a transition from a parabolic shape to a conical shape at $x_{b}\sim
5.4\,\mathrm{mas}$ (dashed line). The dotted line indicates the estimated
location of the Bondi radius.

##### 0430$+$052 (3C 120):

The model fitting results for second radio galaxy in our sample are shown in
Fig. 22. We find a parabolic jet shape on all scales tested along the entire
jet length. The $s$-values are flatter than expected from a freely expanding
jet which is consistent with the narrowly confined jet geometry. Kovalev et
al. (2020) and Boccardi et al. (2021) find a transition of the jet shape from
parabolic to conical at $\sim 1.5\leavevmode\nobreak\ \mathrm{mas}$ from the
radio core. This region is also resolved by the data used in this work,
however Kovalev et al. (2020) probe somewhat larger scales in their analysis
of $15\,\mathrm{GHz}$ and $1.4\,\mathrm{GHz}$ data. Examining Fig. 1 in
Kovalev et al. (2020) it becomes clear, that the geometry transition is seen
primarily due to the use $1.4\leavevmode\nobreak\ \mathrm{GHz}$ data. Overall
our findings are consistent with Kovalev et al. (2020) in the sense that
closer to the physical jet base they find a parabolic shape consistent with
our results. Even when rigorously searching for a geometry transition, it is
not possible to clearly constrain a significant break point on the scales
covered by our higher-frequency data. This is due to the large scatter of the
data in the region where Kovalev et al. (2020) and Boccardi et al. (2021) find
the geometry transition.

##### 0528$+$134 (PKS 0528+134):

This FSRQ shows a prominent excess in brightness temperature between
$0.3\lesssim d_{\mathrm{core}}/\mathrm{mas}\lesssim 1$ distance from the radio
core at $43\,\mathrm{GHz}$ which is not seen in the $15\,\mathrm{GHz}$ data
(Fig. 23). Since this excess in the $43\leavevmode\nobreak\ \mathrm{GHz}$ is
seen in epochs with are close in time to each other (see color coding in Fig.
23) this could be due to a radio flare in this source. This excess can be
associated with a dip in the jet-diameter profile of the $43\,\mathrm{GHz}$
jet. This indicates a region where the jet locally contracts. The observations
seem to probe the jet in the region where its geometry is still parabolic. The
geometry fits at both frequencies are indistinguishable within their errors:
$l_{43}=0.45\pm 0.19,\,l_{15}=0.79\pm 0.21$. This also holds in the overlap
region of both frequencies. The brightness temperature gradients however seem
to be significantly flatter at $43\,\mathrm{GHz}$ than at $15\,\mathrm{GHz}$,
where the latter shows the canonical value, expected from a simple Blandford &
Königl (1979)-jet ($s_{43}=-1.23\pm 0.43,\,s_{15}=-2.65\pm 0.46$), when
considering the respective entire jet lengths. The difference of the
$s$-values can be attributed to a variation of one of the parameters $n$
and/or $b$ on the different angular scales probed. Due to the excess in
$T_{b}$ at $43\,\mathrm{GHz}$ and the subsequent steepening of the gradient’s
fit, this difference vanishes in the overlap zone of both frequencies. At high
frequencies above $\sim 30\,\mathrm{GHz}$ (Marscher, 2016), the radio core is
not necessarily the region where the synchrotron radiation becomes optically
thin as in the Blandford & Königl (1979) jet model but might rather represent
a standing recollimation shock which is backed by polarization measurements in
the radio core of 1803+784 at $43\,\mathrm{GHz}$ (Cawthorne et al., 2013),
which they link to conical recollimation shocks (Daly & Marscher, 1988;
Cawthorne & Cobb, 1990; Cawthorne, 2006). Jorstad et al. (2001) studied 42
$\gamma$-ray detected blazars at $22\,\mathrm{GHz}$ and $43\,\mathrm{GHz}$ of
which 27 showed a sum of 45 nonmoving components besides the radio core which
they also link to standing recollimation shocks if the component is close to
the radio core and to jet bending accompanied by maximized Doppler boosting of
the component is farther down to jet. We find a geometry break from parabolic
to conical jet geometry in this source at $x_{b}\sim 2.1\,\mathrm{mas}$. The
radius of the Bondi sphere is closer to the radio core than $x_{b}$ in this
source. Combining $15\,\mathrm{GHz}$ and $1.4\,\mathrm{GHz}$, Kovalev et al.
(2020), do not find a geometry break in this source. We explain this by the
additional use of $43\leavevmode\nobreak\ \mathrm{GHz}$ data in this work
which allow to resolve components closer to the physical jet base due to
following reasons: With the single power law fit, Kovalev et al. (2020) find a
diameter gradient power law index of $\sim 0.8$ which is consistent with the
value we find at $15\leavevmode\nobreak\ \mathrm{GHz}$ ($l_{15}=0.79\pm 0.21$)
along the entire jet length. In the overlap region between the
$43\,\mathrm{GHz}$ and $15\,\mathrm{GHz}$ the jet shape is parabolic. This
reveals that the jet features a geometry transition which can be constrained
closer to the jet based, by combining the high frequency observations.

Figure 13: Jet diameter vs. core distance for PKS 0528$+$134 (0528+134). The
jet shows a transition from a parabolic shape to a conical shape at $x_{b}\sim
2.1\,\mathrm{mas}$ (dashed line). The dotted line indicates the estimated
location of the Bondi radius.

##### 0716+714 (S5 0716+71):

This source features a jet profile with $0.5\leq l\leq 1$ at both frequencies
and a quasi-canonical $s$-value in the range of the simple Blandford & Königl
(1979) jet model at both frequencies along the entire jet lengths and the
overlap region (Fig. 24). No strong excesses in the brightness-temperature or
the jet-diameter profiles are found and no geometry break can be constrained
in this source.

##### 0735$+$178 (OI 158):

The jet in this BL Lac object is slightly collimated at both frequencies,
along the entire jet lengths and the overlap region. Combining
$15\,\mathrm{GHz}$ and $1.4\,\mathrm{GHz}$ data Kovalev et al. (2020) also
find this slight collimation in this source ($l\sim 0.93$).

The $s$-values cannot be distinguished in the overlap region, however along
the entire jet length the $15\,\mathrm{GHz}$ jet shows the canonical Blandford
& Königl (1979)-expected value while the $43\,\mathrm{GHz}$ jet shows a
flatter $T_{b}$-gradient ($s_{43}=-1.75\pm 0.36,\,s_{15}=-2.46\pm 0.37$; see
Fig. 25). According to Eq. 7 this might be attributed to a variation in the
magnetic field strength gradient between the frequencies..

##### 0827$+$243 (OJ 248):

At each of the two frequencies, this quasar jet features $l$ and $s$-values
which are indistinguishable within their errors, along the entire jet length
and the overlap region (Fig. 26). In comparison, the $l_{43}$-value is
consistent with a parabolic jet structure while the $l_{15}$-value tends to be
closer to a conical jet geometry. However, due to the large uncertainties and
the overall consistency of the values with each other, a geometry transition
zone cannot be constrained in this source. Combining $15\,\mathrm{GHz}$ and
$1.4\,\mathrm{GHz}$ the jet geometry can be fit with a cone (Kovalev et al.,
2020) which is consistent with the findings in this work in the
$15\leavevmode\nobreak\ \mathrm{GHz}$ jets.

##### 0829+046 (OJ 049):

This BL Lac object shows a slightly colliamted jet at $43\,\mathrm{GHz}$ (Fig.
27). The $l$-values here are consistent with the $15\,\mathrm{GHz}$ results in
the overlap region. Considering the entire jet length, the $15\,\mathrm{GHz}$
data shows a larger $l$ value indicative of a conical jet on larger scales.
However, a consistent geometry break cannot be constrained due to the large
scatter of the data at $43\,\mathrm{GHz}$. This scatter is largely caused by a
$T_{b}$-excess accompanied by a recollimation of the jet which can be seen at
$43\,\mathrm{GHz}$ between $0.2\lesssim d_{c}/\,\mathrm{mas}\lesssim 1$, which
is less pronounced at $15\,\mathrm{GHz}$. This can be understood as a result
of jet recollimation similar to the case of PKS 0528+134. Investigating the
$43\leavevmode\nobreak\ \mathrm{GHz}$ light curve of this source, provided on
the website of thwe BU monitoring
program999https://www.bu.edu/blazars/VLBA_GLAST/0829.html, reveals, that the
source showed flaring activity within the time period, analyzed in this wortk
between 2008 and 2009 and in the time period between 2012 and 2013. In these
time periods the MOJAVE data base provides sparse data with about one epoch
per year which might be the reason that this excess is not seen in the
$15\leavevmode\nobreak\ \mathrm{GHz}$ data.

##### 0836$+$710 (4C +71.07):

Model-fitting results for this distant ($z=2.218$) quasar are shown in Fig.
28. The jet is more collimated at $43\,\mathrm{GHz}$ than at
$15\,\mathrm{GHz}$ if the full jet lengths in both data sets are considered.
However, the $l$-value in the overlap region of the $15\,\mathrm{GHz}$ jet is
smaller and consistent with the 43 GHz geometry, which indicates a more
collimated jet on small scales. This indicates that there is a geometry
transition from parabolic to conical within this jet. The $s$-values are
consistent with the canonical value of the simple Blandford & Königl (1979)
model except for the overlap region at $43\,\mathrm{GHz}$. The fact that the
$s$-values are consistent with the canonical value for a freely expanding jet
while the jet itself features a parabolic shape at $43\,\mathrm{GHz}$,
suggests that either $n$ or $b$ are causing the flattening the $s$-values.
Interestingly, Orienti et al. (2020) find limb-brightened polarized flux up to
$1\,\mathrm{mas}$ from the core which indicates a complex jet structure and
that the magnetic field gradient changes at this point.

A two-zone jet geometry model is shown in Fig. 14. The jet features a
parabolic shape up to $d_{c}\sim 2.7\,\mathrm{mas}$, transitioning to a
conical shape further downstream. The projected Bondi radius along the jet
axis is smaller than the location of the transitioning zone by almost two
orders of magnitude. The change in the magnetic-field gradient, suggested by
the results of Orienti et al. (2020) might indicate that the jet changes from
magnetically dominated to particle dominated which in turn can explain the
geometry transition (Kovalev et al., 2020). The complexity of this source is
also well studied regarding growing Kelvin-Helmholtz instabilities within the
jet at multiple scales (Perucho et al., 2012; Vega-García et al., 2019, 2020).
They also show that the jet is likely particle dominated over large scales out
to $\sim 100\,\mathrm{mas}$ in jet length.

Figure 14: Jet diameter vs. core distance for 4C $+$71.07 (0836+710). The jet
shows a transition from a parabolic shape to a conical shape at $x_{b}\sim
2.7\,\mathrm{mas}$ (dashed line). The dotted line indicate the estimated
location of the Bondi radius.

##### 0851$+$202 (OJ 287):

This well known BL Lac object has been suggested as a likely candidate for a
binary supermassive black hole system Sillanpaa et al. (1988); Lehto &
Valtonen (1996). In this context, it is interesting to note that this source
is among the objects with the absolute highest scatter in the jet-diameter
fits of our whole sample (Fig. 29). It features $l$ and $s$-values which are
indistinguishable within the uncertainties, along the entire jet length and
the overlap region. On larger scales at 15 GHz, the average jet geometry
appears quasi conical, which can be interpreted as a freely expanding jet,
which is consistent with the findings at lower frequencies from Kovalev et al.
(2020).

##### 0954$+$658 (S4 0954+65):

This BL Lac shows a jet geometry which in both frequencies is consistent with
a conical jet along the entire jet length and the overlap region (Fig. 30).
Also the $s$-values are indistinguishable from each other, however they tend
to be steeper than the canonical value of the simple Blandford & Königl (1979)
model, especially at $43\,\mathrm{GHz}$ in the overlap region, and at
$15\,\mathrm{GHz}$. As the jet overall seems to feature a conical shape, this
must be understood in the context of the parameters $n$ and $b$. Indeed when
allowing not only toroidal magnetic field components (Blandford & Königl,
1979) but also poloidal magnetic field components (Königl, 1981), even steeper
$T_{b}$-gradients can be obtained.

##### 1101$+$384 (Mrk 421):

This is a nearby ($z=0.0308$) high-peaked BL Lac object in which we are
probing the smallest linear scales within our sample (0.61 pc/mas). The source
features $s$-values which are not distinguishable within their uncertainties
at a given frequency (Fig. 31). However $s_{43}$ overall shows a clear
tendency of being flatter than $s_{15}$. This is also resembled in that fact,
that $l_{43}$ is consistent with a parabolic jet shape while $l_{15}$ is
consistent with a conical jet shape. This hints to a possible geometry
transition within this source which is indeed found by our algorithm (see
Sect. 3.2.2). A two-zone jet geometry model is shown in Fig. 15 and features a
parabolic jet shape out to $d_{c}\sim 1.9\,\mathrm{mas}$ where it transitions
to a conical shape jet further downstream. On kpc-scales, the inclination
angle of this source was estimated to be $\theta\sim 19\,\mathrm{deg}$, based
on the jet to counter-jet flux density ratio (Giroletti et al., 2006). On pc-
scales, inclination angles for this source were estimated based on the Doppler
factors obtained from variability analysis (Hovatta et al., 2015) to be
between $0.5\lesssim\theta/\mathrm{deg}\lesssim 4$, however they address a
point made by Lico et al. (2012), that the maximum measured component velocity
$\beta_{\mathrm{app,max}}=0.218\pm 0.026$ (Lister et al., 2019) is most likely
not the flow velocity of the jet because the beaming properties predicted by
the flux ratio of the jet and the counter jet require viewing angles close to
zero. Lico et al. (2012) estimate the inclination angle to be
$2\lesssim\theta/\mathrm{deg}\lesssim 5$ based on the argument that the jet
shows limb brightening (Piner et al., 2010), an indication for a transverse
velocity structure across the jet axis. For our analysis we adopt the angle
range given by Lico et al. (2012). The resulting estimated location of the
Bondi radius is well below the distance of the break.

Figure 15: Jet diameter vs. core distance for Mrk 421 (1101+384). The jet
shows a transition from a parabolic shape to a conical shape at $x_{b}\sim
1.9\,\mathrm{mas}$ (dashed line). The dotted line indicates the estimated
location of the Bondi radius.

##### 1127$-$145 (PKS 1127$-$14):

Model-fitting results for this quasar jet are shown in Fig. 32. It features a
parabolic jet shape at both frequencies. The $s$-values are flatter than
expected for a freely expanding jet. The relatively large
$\chi^{2}_{\mathrm{red}}$ in the gradient fits, indicate a complex jet
structure. Taking both frequencies together, a systematic geometry transition
cannot be found in this source. The lower frequency data from Kovalev et al.
(2020) indicate a steepening of the diameter gradient. Cmbining the
$43\leavevmode\nobreak\ \mathrm{GHz}$, $15\leavevmode\nobreak\ \mathrm{GHz}$
and $1.4\leavevmode\nobreak\ \mathrm{GHz}$ data of this source might reveal a
systematic geometry transition.

##### 1156$+$295 (4C 29.45):

This quasar features brightness temperature gradients which are consistent
with a freely expanding jet (Fig. 33). The diameter gradients, however,
indicate a collimation of the jet at both frequencies. A possible jet geometry
transition is indicated when comparing the overlap region in the
$15\,\mathrm{GHz}$ jet with the entire jet length. The two-zone jet geometry
model is shown in Fig. 16. The jet features a parabolic shape out to
$d_{c}\sim 2.5\,\mathrm{mas}$, transitioning to a conical shape further out.
The projected Bondi radius along the jet axis is smaller than the location of
the transitioning zone by almost two orders of magnitude. In this source no
steady bright component can be seen in the $T_{b}$-gradient, which is
consistent with the low scatter atypical for quasars (cf. Figs. 5 and 7).At
lower frequencies Kovalev et al. (2020) do not find a geometry transition. The
single power law fit they find is consistent with $l_{\mathrm{out}}\sim 1.3$
from our two zone model.

Figure 16: Jet diameter vs. core distance for 4C 29.45 (1156+295). The jet
shows a transition from a parabolic shape to a conical shape at $x_{b}\sim
2.5\,\mathrm{mas}$ (dashed line). The dotted line indicate the estimated
location of the Bondi radius.

##### 1219$+$285 (W Comae):

This source features $l$ and $s$-values which are indistinguishable within
their uncertainties, along the entire jet length and the overlap region (Fig.
34). No significant break is found in the jet geometry of this source. The jet
geometry overall is consistent both with a conical and parabolic jet, due to
the scattering of the data. The $s$-values are consistent with the canonical
parameter values of the simple Blandford & Königl (1979) model. Allowing for
the extensions of Königl (1981) and in combination with the conical jet shape,
this can be interpreted as a freely expanding jet.

##### 1222$+$216 (4C +21.35):

This is a quasar that features $T_{b}$ gradients which are consistent with the
$s$-value expected in a canonical Blandford & Königl (1979) jet model, at both
frequencies, as seen in Fig. 35. In the overlap region, the data suggest a
steeper $T_{\mathrm{B}}$ gradient, especially at 15 GHz. When applying the
search, discussed in Subsect. 5.6 it is not possible to constrain a break
point between two different slopes for the geometry scaling. This might hint
to a more complex, transverse jet structuring, where at $43\,\mathrm{GHz}$ in
the overlap region, a more collimated substructure of the jet is measured than
at $15\,\mathrm{GHz}$.

##### 1226$+$023 (3C 273):

The $T_{b}$ gradients in 3C 273 at both frequencies are flatter than expected
in a freely expanding jet. Indeed, the diameter gradient at both frequencies
is consistent with a parabolic jet shape (Fig. 36), although the data show
large scatter as it is typical for quasar jets. As it is apparent in Fig. 8,
low $l$-values tend to be associated with flat $s$ gradients. The complex
structure of the 3C 273 jet has been analyzed in great detail by several
authors. In particular, a double-helix structure out to kpc-scales was found
(see Lobanov & Zensus (2001)).

##### 1253$-$055 (3C 279):

Similarly to 3C 273, the $s$-values are consistent with a canonical Blandford
& Königl (1979) jet model. However also in this source, the $43\,\mathrm{GHz}$
jet diameter gradient is consistent with a parabolic jet shape and is more
collimated than the $15\,\mathrm{GHz}$ jet diameter gradient which is
consistent with a conical jet, in both the overlap region and the respective
entire jet lengths. Akiyama et al. (2018) find a transition from a parabolic
to a conical jet shape however they also make use of VLBA data at
$1.4\,\mathrm{GHz}$ and MERLIN data at $1.6\,\mathrm{GHz}$, which probes the
jet further downstream out to kpc scales. They constrain a transition zone
which in Fig. 37 would appear at $\sim 1\,\mathrm{mas}$. Our higher-frequency
VLBA data are more sensitive to smaller scales and we find a transition zone
at $x_{b}\sim 0.6\,\mathrm{mas}$. In the outer fit component, we still find an
$l$-value smaller than unity and a possible second break at or above 1 mas is
not probed by our data. The radius of the Bondi sphere for this source is well
below the location of both the geometry break point reported by Akiyama et al.
(2018), however also may coincide with the transition region, we found for the
lowest gas temperatures. With the lower frequency VLBA data, Kovalev et al.
(2020) did not constrain a geometry break. The single power law fit of the jet
geometry is consistent with the $l_{15}$-value we find.

Figure 17: Jet diameter vs. core distance for 3C 279 (1253-055). The jet shows
a transition from a parabolic shape to a conical shape at $x_{b}\sim
0.6\,\mathrm{mas}$ (dashed line). The dotted line indicates the estimated
location of the Bondi radius.

##### 1308$+$326 (OP 313):

This quasar features $l$ and $s$-values which are indistinguishable within
their uncertainties, along the entire jet length and the overlap region (Fig.
38). The jet geometry overall appears conical and the $s$-values are
consistent with the canonical parameter values from the Blandford & Königl
(1979) model, which can be interpreted as a freely expanding jet. This is also
consistent with Kovalev et al. (2020).

##### 1633$+$382 (4C +38.41):

This quasar jet features $l$ and $s$-values which are indistinguishable within
their uncertainties, along the entire jet length and the overlap region (Fig.
39). In spite of strong scatter in the data, the jet geometry appears conical
and the $s$-values are consistent with the canonical parameter values the
Blandford & Königl (1979) model which can be interpreted as a freely expanding
jet. Lower frequency data suggest an opening of the jet with a diameter
gradient index of $l\sim 1.3$ (Kovalev et al., 2020). Also for this source it
would be interesting to combine the high and low frequency data in order to
constrain a possible geometry transition.

##### 1652$+$398 (Mrk 501):

This is the second closest source in our sample and compared to most other
sources is covered by only a relatively small number of observing epochs. It
features diameter gradients at both frequencies which are consistent with a
parabolic jet shape, as seen in Fig. 40. This is suggestive of a jet that is
observed on scales where it is still accelerating rather than freely
expanding. This is also consistent with the $T_{b}$-gradients which are
flatter than those expected for a freely expanding jet.

##### 1730$-$130 (NRAO 530):

This is a quasar jet that features $l$ and $s$ values which are consistent
with a freely expanding jet, when analyzing the overlap region at both
frequencies (Fig. 41). Along the entire jet length, however, the
$15\,\mathrm{GHz}$ jet seem to be more collimated and features a flatter
$T_{b}$-gradient. This is dominated by the influence of a rather distant
component between $20\lesssim d_{c}/\mathrm{mas}\lesssim 30$ from the core
which appears in epochs between $\sim 2006$ and $\sim 2012$. This component is
relatively compact and features comparatively high $T_{b}$-values with respect
to the distance from the core. This might be suggestive of a shocked region
where the jet recollimates. Lu et al. (2011) find this component to be
stationary. At lower frequenceie Kovalev et al. (2020) also do not constrain a
geometry transition. The scaling of the jet diameter is consisntent with what
is found in this study.

##### 1749$+$096 (OT 081):

Model-fit results for this BL Lac object are shown in Fig. 42. Its diameter
gradients are consistent with a conical jet shape while the brightness
temperature gradients are steeper than expected within the simple Blandford &
Königl (1979) jet model. This suggests that the parameters $n$ and $b$ are
responsible for the steep $T_{\mathrm{B}}$ gradients. A poloidal magnetic
field component could explain a steepening of the brightness temperature
gradients while maintaining a conical jet shape.

##### 2200$+$420 (BL Lacertae):

The inner region of this prototypical BL Lac object features an overall jet
shape which is consistent with a cone which is also confirmed by Casadio et
al. (2021) by additionally taking GMVA data at $86\,\mathrm{GHz}$ into
account. If fit with a simple power-law model (Fig. 43), the $s$-values are
consistent with the canonical value expected for a simple Blandford & Königl
(1979) jet model. At $\sim 1.5\,\mathrm{mas}$ from the respective cores, we
find a region where the jet locally shows a dip in its diameter and excess
$T_{b}$ values which is also consistent with the findings of Casadio et al.
(2021). They associate this region with a recollimation shock which is
connected to the changing external pressure gradient close to the Bondi radius
(Kovalev et al., 2020).The large $\chi^{2}_{\mathrm{red}}$ values suggest that
the overall jet structure is complex and is not sufficiently represented by a
single power law. The two-zone diameter-gradient model is shown in Fig. 18
(top) and features a parabolic jet shape out to $d_{c}\sim 2.0\,\mathrm{mas}$
transitioning to a hyperbolic shape jet further downstream which is consistent
with the findings of Kovalev et al. (2020). Analyzing the jet in more detail
it becomes clear that the jet of BL Lacertae is even more complex and still
not fully described by a two-zone diameter-gradient model. To emphasize this
we also show Fig. 18 (bottom). Here the break points are not found rigorously
but are a ’fit by eye’. In this representation, the jet has a parabolic region
up to roughly the point where the transition was fitted in the single-
transition zone model, opening up to an extreme jet geometry ($l\sim 2.5$),
before transitioning again at $\sim 5\,\mathrm{mas}$ into a quasi-conical jet.
The latter transition is roughly coinciding with the estimated location of the
Bondi radius. This seems to contradict the findings of Casadio et al. (2021)
where they do not find a geometry transition, additionally analyzing
$86\,\mathrm{GHz}$ GMVA data. Casadio et al. (2021) argue that the region
between $0.5\lesssim r/\mathrm{mas}\lesssim 1.5$ should be excluded from a fit
because in this region the jet shows a different expansion rate accompanied by
a decrease in brightness which we also find in the brightness-temperature data
(Fig. 43). Not excluding this region may yield a parabolic geometry of this
jet. This tentative contradiction can be resolved by the jet’s complex
structure where clearly different regions with different behavior in the jet
can be found. This would not affect the overall jet shape, measured over
several decades in the distance from the respective radio cores, but would
allow local geometry transitions of the jet.
We also tested the effect of the measured core shift of $\sim
0.1\,\mathrm{mas}$, reported by O’Sullivan & Gabuzda (2009). The resulting jet
geometry and brightness temperature gradients along the entire jet length are
indistinguishable within the errors from the case where the core shift is not
considered. Also the found geometry break position and the inner and outer
diameter gradient do not differ.

Figure 18: Jet diameter vs. core distance for BL Lac (2200+420). Top: Single
transition zone model (rigorous fit). The shape transitions at $x_{b}\sim
1.8\,\mathrm{mas}$ from parabolic to conical. Bottom: Two-transition model
where the break points are ’fit by eye’ parametrizes the complex structure.

##### 2223$-$052 (3C 446):

This quasar jet features $s$ values which are indistinguishable regarding the
two frequencies and are well described by the Blandford & Königl (1979) jet
model. The diameter gradients are also indistinguishable, however there seems
to be peculiarity within the $15\,\mathrm{GHz}$ data. While the
$43\,\mathrm{GHz}$ jet shape is consistent with a cone, the $15\,\mathrm{GHz}$
jet shape is suggestive of a parabolic shape. This might be explained by the
large scatter (as it is found to be characteristic for quasars) and the
relatively low number of observing epochs for this source. A systematic jet
shape transition however cannot be found in this source. The overall parabolic
jet shape is consistent with the findings of Kovalev et al. (2020).

##### 2230$+$114 (CTA 102):

This quasar (see e.g., Fromm et al., 2011, 2013a, 2013b) features $s$-values
which are consistent with a parabolic jet in equipartition at both
frequencies. The jet geometry can consistently be fit with a parabola at both
frequencies, however along the entire jet axis at $15\,\mathrm{GHz}$, the
$l$-value becomes also consistent with a cone due to the larger uncertainties
introduced by the larger scatter of the data beyond $d_{c}\sim
2\,\mathrm{mas}$. A transition in this source cannot be constrained.

##### 2251$+$158 (3C 454.3):

Similarly to 1222$+$216 (4C +21.35) and 1253$-$055 (3C 279), the $s$-vlaues
are consistent with a Königl (1981) jet model. However also in this source,
the $43\,\mathrm{GHz}$ jet diameter gradient is consistent with a parabolic
jet shape and is more collimated than the $15\,\mathrm{GHz}$ jet diameter
gradient which is consistent with a conical jet, in both the overlap region
and the respective entire jet lengths. Also here when rigorously searching for
possible break points and an inner and outer slope, it cannot be constrained
properly, due to the large scatter of the data points between $0.2\lesssim
d_{c}/\mathrm{mas}\lesssim 3$ in the $43\,\mathrm{GHz}$ jet. This jet also
might be a candidate where at higher frequencies, more collimated, inner
structures of a multilayered jet are observed. The overall parabolic jet shape
is consistent with the findings of Kovalev et al. (2020).

##  7 Summary and conclusions

Using dual-frequency VLBI observational data for a sample of 15 FSRQs, 11 BL
Lacs, and 2 radio galaxies at $15\,\mathrm{GHz}$ and $43\,\mathrm{GHz}$, the
geometry and brightness temperature gradients along the jet axes has been
determined and interpreted. Our main observational findings are the following:

  * •

The geometry of the VLBI jets in our sample can generally be described as
power laws with indices $l$ in the range between 0.5 and 1, i.e., collimated
jet geometries.

  * •

At $43\,\mathrm{GHz}$, the jets tentatively show a stronger collimation than
at $15\,\mathrm{GHz}$. This is a significant difference if the full jet extent
seen at both frequencies is considered, but not significant when the analysis
is restricted to the overlap region where the jet is imaged at both
frequencies (Table 3)

  * •

FSRQs show stronger collimation than BL Lacs at both frequencies, but again
this is not significant if restricted to the overlap region (Figs. 5 and 7).

  * •

FSRQs show a more complex structure than BL Lac objects (with more local
variations of the brightness temperature gradients leading to higher
$\chi^{2}$ values of the power law fits in Figs. 5 and 7). This is true for
the jet geometry, indicating regions where the jets collimate and widen
locally, as well as for the brightness temperature gradients indicating
regions in the jet where the flux density is locally higher than in other
regions.

  * •

We found parabolic-to-conical geometry transitions in 5 FSRQs: CTA 26
(0336$-$019), PKS 0528$+$134, 4C +71.07 (0836$+$710), 4C +29.45 (1156$+$295),
3C 279 (1253$-$055). In addition, we found parabolic-to-conical geometry
transitions in 3 BL Lacs: 3C 66A (0219$+$428), Mrk 421 (1101$+$384), and BL
Lacertae (2200$+$420); and in a radio galaxy: 3C 111 (0415$+$379).

  * •

The jet radius in the transition zone is smaller in BL Lac objects than in
FSRQs and the geometry transitions occurs further down the jet for FSRQs which
is likely due to systematically smaller SBMH masses in BL Lacs.

  * •

The FSRQs CTA 26 (0336$-$019) and 4C +71.07 (0836$+$710) show a stationary
component downstream the jet past the Bondi radius.

  * •

BL Lacertae (2200$+$420) shows a jet structure of substantially greater
complexity than a simple transition from a parabolic to a conical shape due to
local effects like pressure gradient change and locally different expansion
rates between $0.5\lesssim r/\mathrm{mas}\lesssim 1.5$, (Casadio et al., 2021)

The high-frequency VLBI observations apparently resolve jets on physical
scales where in many cases their shapes change from parabolic (collimated) to
conical (free) morphologies. This observational finding is in agreement with
relativistic magnetohydrodynamical jet models, e.g., Fendt (2006); Porth et
al. (2011). These models show that the accretion disk in blazars with high
accretion rates (FSRQs) flattens the magnetization profile at the jet base
compared to jets from disks with low accretion rates (BL Lacs), either by
advection of magnetic flux with the disk or by generating it with a disk
dynamo. The radially flattened magnetization profile and the strong disk wind
envelope are predicted to lead to (i) a collimation of the toroidal-field
dominated jet extending to large distances from the core, (ii) morphological
complexity as these jets are prone to produce axially propagating knots, and
(iii) high terminal speeds. By contrast, jets in BL Lacs with steeper
magnetization profiles at their base and weaker wind envelopes are expected
more uniform, slower, and widen closer to the core (Potter & Cotter, 2015),
which is in good agreement with our observational results. The acceleration of
the jets is driven by the conversion of Poynting flux to kinetic energy flux
of particles entrained from the surrounding disk wind or injected by
reconnection events or electrostatic gaps at the jet base (Wendel et al.,
2021). The breakdown of the collimation (due to the kink instability) first
affects the low-frequency sheath of the jet, in accord with the observational
results reported here. On the kinetic level of description, the energy
conversion is likely to be associated with turbulence, anomalous resistivity,
and ultrarelativistic particle acceleration. These dissipative processes
involve complex 3D-substructures in the jet plasma leading to fast flux
variability at high energies. In this scenario, no direct correlation with the
extent of the Bondi radius around the central supermassive black hole is
predicted, which is in agreement with our results. Further improvements of
this type of studies are possible: adding polarimetric analyses and increasing
the sample size, the dynamical range, and the accessible bandwidth of VLBI
observations, holds the promise to further advance our understanding of the
physical processes involved in the blazar jet phenomenon.

###### Acknowledgements.

We are grateful to Till Steinbring, who did important preliminary work on
which this study partly has been based. We appreciate valuable scientific
discussions with the MOJAVE team, especially Matthew L. Lister, Yuri Y.
Kovalev, and Evgeniya V. Kravchenko, which helped us to improve this paper.
This research has made use of data from the MOJAVE database that is maintained
by the MOJAVE team (Lister et al., 2009, 2018). This study makes use of VLBA
data from the VLBA-BU Blazar Monitoring Program (BEAM-ME and VLBA-BU-BLAZAR;
http://www.bu.edu/blazars/BEAM-ME.html), funded by NASA through the Fermi
Guest Investigator Program. The VLBA is an instrument of the National Radio
Astronomy Observatory. The National Radio Astronomy Observatory is a facility
of the National Science Foundation operated by Associated Universities, Inc.

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##  Appendix A Single power law fits

Figure 19: 0219+428 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 20: 0336-019 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 21: 0415+379 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 22: 0430+052 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 23: 0528+134 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 24: 0716+714 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 25: 0735+178 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 26: 0827+243 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 27: 0827+243 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 28: 0836+710 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 29: 0851+202 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 30: 0954+658 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 31: 1101+384 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 32: 1127-145 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 33: 1156+295 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 34: 1219+285 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 35: 1222+216 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$

. The color coding for both frequencies indicates the epoch, where lighter
colors indicate more recent epochs.

Figure 36: 1226+023 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 37: 1253-055 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 38: 1308+326 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 39: 1633+382 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as
function of $d_{\rm c}$. The color coding for both frequencies indicates the
epoch, where lighter colors indicate more recent epochs.

Figure 40: 1652+398 jet diameter $d_{\rm j}$ as a function of the distance
from the core $d_{\rm c}$ and the brightness temperature $T_{\rm B}$ as