11institutetext:  National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, China

# Doppler boosting effect and flux evolution of superluminal components in QSO
3C345

S.J. Qian 11

(Compiled by using A&A latex)

###### Abstract

Context. The precessing jet-nozzle scenario previously proposed was applied to
interpret the VLBI-measured kinematics of five superluminal components (C4,
C5, C9, C10 and C22) and their flux density evolution in blazar 3C345.

Aims. It is shown that in the inner-trajectory sections their kinematic
properties, including trajectory, coordinates, core separation and apparent
velocity can be well model-simulated by using the scenario with a precession
period of 7.30 yr (4.58 yr in the source frame) and a precessing common
trajectory, which produces the individual knot-trajectories at their
corresponding precession phases.

Methods. Through the model-simulation of their kinematic behavior their bulk
Lorentz factor, viewing angle and Doppler factor were derived as functions of
time. These anticipatively determined Lorentz /Doppler factors were used to
investigate the knots’ Doppler-boosting effect and interpret their flux
evolution.

Results. It was found that the light-curves of the five superluminal
components observed at 15, 22 and 43 GHz were extraordinarily well coincident
with their Doppler boosting profiles ($[\delta(t)/\delta_{max}]^{3+\alpha}$,
$\alpha$–spectral index). Additionally, some flux fluctuations on shorter
time-scales could be due to variations in knots’ intrinsic flux and spectral
index.

Conclusions. The close relation between the flux evolution and the Doppler-
boosting effect not only firmly validates the precessing jet-nozzle scenario
being fully appropriate to explain the kinematic and emission properties of
superluminal components in QSO 3C345, but also strongly supports the
traditional common point-view: superluminal components are physical entities
(shocks or plasmoids) participating relativistic motion toward us with
acceleration/deceleration along helical trajectories.

###### Key Words.:

galaxies: active – galaxies: jets – galaxies: nucleus – galaxies: individual
3C345

##  1 Introduction

3C345 (z=0.595) is a prototypical quasar emanating emission over the entire
electromagnetic spectrum from radio, infrared, optical, UV and X-rays to high-
energy $\gamma$ rays. It is also one of the best-studied blazars (e.g.,
Biretta et al. Bi86 (1986), Hardee et al. Ha87 (1987), Steffen et al. St96
(1996), Unwin et al. Un97 (1997), Zensus et al. Ze97 (1997), Klare Kl03
(2003), Klare et al. Kl05 (2005), Lobanov & Roland Lo05 (2005), Jorstad et al.
Jo05 (2005), Jo13 (2013), Jo17 (2017), Qian et al. 1991a , Qi96 (1996), Qi09
(2009), Schinzel et al. Sc10 (2010), Schinzel 2011a , Schinzel et al. 2011b ,
Homan et al. Ho14 (2014)). Its remarkable and violent variations in all these
wavebands and the spectral energy distribution have been extensively monitored
and studied, leading to many important results on the emission properties of
the source. Studies of the correlation between the variabilities at multi-
frequencies (from radio to $\gamma$ rays) play an important role.
3C345 is a remarkable compact flat-spectrum radio source which was one of the
firstly discovered quasars to have a relativistic jet, emanating superluminal
components steadily.
VLBI-observations have revealed the parsec structure of its jet and monitored
the kinematic behavior of its superluminal components ejected from the radio
core over a quite long time-interval since $\sim$1979\.
It has been shown that flaring activities of the source in multi-frequencies
(from radio to $\gamma$-rays) are closely connected with the jet-activity and
ejection of superluminal components. In addition, VLBI-monitoring observations
have shown that the swing of the knot’s ejection position angle and the
apparent direction of its relativistic jet could be produced by jet-precession
(or jet-nozzle precession) with a precession period of $\sim$5-10 years. The
determination of its jet-nozzle precession may be very important for
understanding the kinematics of superluminal components and the properties of
the central energy engine in its nucleus.
Since 1991 (Qian et al. 1991a , Qi09 (2009)), we have tried to interpret the
VLBI-kinematics of superluminal components in 3C345 in terms of a precessing
jet-nozzle scenario. In the recent work (Qian Qi22 (2022)) the kinematics of
twenty-seven superluminal components observed during a 38 yr period was
explained in detail. It was found that these superluminal knots could be
divided into two groups, which were suggestively ejected by their respective
precessing jet nozzle. Based on this division the precession period of both
nozzles was derived to be 7.3$\pm$0.36 yr in the same direction.
Our precessing jet-nozzle scenario proposed to interpret the VLBI-measured
kinematics in 3C345 and other blazars has been based on two assumptions:

  * •

(1) jet-nozzle precesses with a certain period and ejects superluminal
components and magnetized plasmas at different precession phases, which form
the entire jet (or jet-body); thus each component has its corresponding
ejection precession phase.

  * •

(2) Inner sections of knots’ trajectories along which the superluminal
components move are assumed to follow a common helical pattern which precesses
to produce the individual knot trajectories at their corresponding precession
phases.

By applying the precessing nozzle scenario, the swing of the ejection position
angle of the superluminal components could be investigated and the VLBI-
kinematics of the superluminal components could be fitted by using model-
simulation methods, including the observed features: their trajectory, core
separation, coordinates and apparent velocity versus time.
Most importantly, the bulk Lorentz factor, viewing angle and Doppler factor of
superluminal components as continuous functions of time could be derived. Thus
the acceleration/deceleration in the motion of superluminal components and the
relation between their flux variations and Doppler boosting effect could be
investigated. This may be regarded as the distinct advantage of our scenario
and method with respect to other scenarios. For example, in the recent work
(Qian Qi22 (2022)), the flux evolution of the superluminal knot C9 in 3C345
was investigated and for the first time found that its 15GHz and 43 GHz light-
curves were very well coincident with its Doppler boosting profile, suggesting
the radio flux variations (flaring events) observed in knot C9 being fully due
to its Doppler boosting effect. This strongly supports the traditional or
common viewpoint: superluminal components are physical entities (shocks or
plasmoids) moving with relativistic speeds toward us along helical
trajectories at small viewing angles. At the same time this finding also
strongly supports the electromagnetic mechanisms for the acceleration of
superluminal components. It seems that the observed trajectories of
superluminal components could not result from an underlying jet-pattern
(induced by hydrodynamic/magneto-hydrodynamic instabilities) lit-up by
plasmoids ejected from the nuclear activities.
The precessing nozzle scenario has been applied to investigate the VLBI-
kinematic behavior of superluminal knots in several QSOs, e.g., 3C279,
B1308+326, PG1302+202, NRAO150, 3C454.3 and OJ287 (Qian et al. Qi14 (2014),
Qi17 (2017), 2018a , 2019a , Qian Qi13 (2013), Qi16 (2016), 2018b , Qian et
al. Qi21 (2021)). Through model-simulation of the VLBI-kinematics of their
superluminal components, we have shown that all these quasars could have
precessing jet-nozzles with certain precession periods. For the four blazars
(3C279, OJ287, 3C454.3 and 3C345) we have tentatively shown that they could
have double precessing-jet structures and some precessing common trajectory
patterns could exist, along which the superluminal components move according
to their precession phases. However, we found that only within the inner
sections of observed trajectories the superluminal components followed the
precessing common trajectory pattern, while in the outer sections they
followed their own individual trajectory patterns. Thus as for their whole
trajectories concerned different trajectory patterns should be chosen to make
model-fits to the entire kinematics of individual knots.
Interestingly, both jets (designated as jet-A and jet-B) were tentatively
found to precess with the same precession period in the same direction. Thus
the precession of both jet-nozzles might be due to the keplerian motion of the
putative binary black holes in their nuclei (e.g., see Begelman et al. Be80
(1980)).
Especially in the case of blazar OJ287, its quasi-periodic optical variability
has been suggested to be connected with its double-jet activity (Villata et
al.Vi98 (1998)). Qian (2018b ) has summarized both the theoretical and
observational evidence for the possible existence of a double-jet structure in
its nucleus, and interpreted the VLBI-kinematics of its superluminal
components in detail in terms of precessing nozzle scenario within the
framework of double-jet structure (Qian 2019b , 2019c , Qi20 (2020); also Qian
et al. Qi07 (2007)).
Search for periodicities in optical and radio light-curves (e.g. Sillanpää et
al. Si88 (1988), Babadzhanyants et al.Ba95 (1995), Kudryavtseva et al.Ku06
(2006), Qian et al. Qi07 (2007)) are important and could provide key
information on the nature of the central engine in blazars.
In the case of blazar 3C345, in our earlier studies we found that the observed
tracks of knots C4 and C5 could be reproduced by the rotation of a common
helical trajectory (Qian et al. 1991a , 1991b , Qian & Zhang Qi99 (1999)).
Qian et al. (Qi09 (2009)) analyzed the distribution of the position angles for
seven superluminal components (C4 to C10) at different core separations of
0.15 mas, 0.20 mas and 0.25 mas, and found that their inner trajectories
(within core separation $r_{n}{\stackrel{{\scriptstyle<}}{{{}_{\sim}}}}$0.4
mas) could be explained in terms of the precession of a common trajectory and
a precession period of its jet-nozzle of $\sim$7.36 yr was derived. Based on
the position angle swing of its superluminal components, some authors argued
for the existence of a jet precession period: e.g., $\sim$8–10 yr (Lobanov &
Zensus Lo99 (1999), Klare et al. Kl05 (2005), Klare Kl03 (2003), Lobanov &
Roland Lo05 (2005)).
We would like to point out that our precessing jet-nozzle scenario is well
consistent with the magneto-hydrodynamic theories for the
formation/collimation/acceleration of relativistic jets in active galactic
nuclei (e.g., Artymoviz Ar98 (1998), Blandford & Payne Bl82 (1982), Blandford
& Znajek Bl77 (1977), Camenzind Ca86 (1986), Ca87 (1987), Ca90 (1990), Li et
al. Li92 (1992), Lovelace et al. Lo86 (1986),Meier & Nakamura Me06 (2006),
Nakamura & Asada Na13 (2013), Shi et al. Sh12 (2012), Sh15 (2015), Valhakis &
Königl Vl03 (2003), Vl04 (2004)).
In this paper we investigate the relation between the Doppler boosting effect
and the flux density evolution for five superluminal components (C4, C5, C9,
C10 and C22 of jet-A) in blazar 3C345 within the framework of our precessing
jet-nozzle scenario. We confirm for the first time that the radio flux density
light-curves of these knots were closely coincident with their Doppler
boosting profiles which had been anticipatively derived through the model-
fitting of their VLBI-kinematics. 111Interpretation of the flux evolution in
terms of Doppler-boosting effect for more superluminal components in blazar
3C345 will be presented elsewhere (Qian, in preparation). These results may
strongly justify the validity of our precessing jet-nozzle scenario and the
correctness or appropriateness of the chosen model-parameters and the modeled
functions.
As in the previous work (Qian Qi22 (2022)) the observational data for knots
C4, C5, C9, C10 and C22 we adopted from Schinzel (2011a ) and some data were
re-calculated to make the compact core as the unified origin of coordinates.
Due to core-shift effects only 43GHz and 22GHz observations were used (except
for knot C5, for which only 15GHz data are available). Generally, we would not
mark the errors of measurements in the figures obtained from the model-
fittings for clarity, but one should keep in mind that the errors in
measurements of knot’s position were in the range of $\sim$0.05–0.1 mas.
We apply the concordant cosmological model (Spergel et al.Sp03 (2003), Hogg
Hog99 (1999)) with ${\Omega}_{\lambda}$=0.73 and ${\Omega}_{m}$=0.27, and
$H_{0}$=71 km${s^{-1}}{{Mpc}^{-1}}$. Thus the luminosity distance of 3C345 is
3.49Gpc, angular-diameter distance $D_{a}$=1.37 Gpc, 1 mas=6.65 pc, 1
mas/yr=34.6c.

Figure 1: Geometry of the precessing jet-nozzle scenario for 3C345. The jet-
axis is defined in the $(X,Z)$-plane by parameters ($\epsilon$, $\psi$) and
function $\it{x_{0}(z_{0})}$. The common helical trajectory pattern is defined
by functions A(Z) and $\phi$(Z) given in section 2 (Figure 2).

Figure 2: Model parameters of the helical trajectory for knot C4: amplitude
A(Z) and phase $\phi$(Z).

##  2 Geometry of the model

The precessing jet-nozzle model has been previously described (Qian et al.
2019a ,Qi09 (2009), 1991a ) for investigating the VLBI-kinematics and
trajectory-distribution of superluminal components on parsec scales in the QSO
3C345. The formalism of the geometry is recapitulated as follows (referring to
Qian et al. Qi21 (2021), Qian Qi22 (2022)).
A special geometry consisting of four coordinate systems is as shown in Figure
1. We assume that the superluminal components move along helical trajectories
around the curved jet axis (i.e., the axis of the helix).
We use coordinate system ($X_{n},Y_{n},Z_{n}$) to define the plane of the sky
($X_{n},Z_{n}$) and the direction of observer ($Y_{n}$), with $X_{n}$-axis
pointing toward the negative right ascension and $Z_{n}$-axis toward the north
pole.
The coordinate system ($X,Y,Z$) is used to locate the curved jet-axis in the
plane ($X,Z$), where $\epsilon$ represents the angle between $Z$-axis and
$Y_{n}$-axis and $\psi$ the angle between $X$-axis and $X_{n}$-axis. Thus
parameters $\epsilon$ and $\psi$ are used to define the plane where the jet-
axis locates relative to the coordinate system ($X_{n},Y_{n},Z_{n}$).
We use coordinate system
(${\it{x}}^{\prime}$,${\it{y}}^{\prime}$,${\it{z}}^{\prime}$) along the jet-
axis to define the helical trajectory pattern for a knot, introducing
parameters $A(s_{0})$ (amplitude) and $\phi(s_{0})$ (phase), where $s_{0}$
represents the arc-length along the axis of helix (or curved jet-axis).
${\it{z}}^{\prime}$-axis is along the tangent of the axis of helix.
${\it{y}}^{\prime}$-axis is parallel to the $Y$-axis and $\eta$ is the angle
between ${\it{x}}^{\prime}$-axis and $X$-axis (see Figure 1).
In general we assume that the jet-axis can be defined by a function
$x_{0}(z_{0})$ in the $(X,Z)$-plane as follows.

${x_{0}}=p({z_{0}}){{z_{0}}^{\zeta}}$ (1)

where

$p({z_{0}})={p_{1}}+{p_{2}}[1+\exp(\frac{{z_{t}}-{z_{0}}}{{z_{m}}})]^{-1}$ (2)

$\zeta$, $p_{1}$, $p_{2}$, $z_{t}$ and $z_{m}$ are constants. The exponential
term is devised for describing the jet-axis gradually curving toward the
north, as the trajectory of knot C4 shows on large-scales.

${s_{0}}=\int_{0}^{z_{0}}{\sqrt{1+(\frac{d{x_{0}}}{d{z_{0}}})^{2}}}\,{d{z_{0}}}$
(3)

Therefore, the helical trajectory of a knot can be described in the (X,Y,Z)
system as follows.

$X({s_{0}})=A({s_{0}}){\cos{\phi({s_{0}})}}{\cos{\eta({s_{0}})}}+{x_{0}}$ (4)
$Y({s_{0}})=A({s_{0}}){\sin{\phi({s_{0}})}}$ (5)
$Z({s_{0}})=-A({s_{0}}){\cos{\phi({s_{0}})}}{\sin{\eta({s_{0}})}}+{z_{0}}$ (6)

where $\tan{\eta({s_{0}})}$=$\frac{d{x_{0}}}{d{z_{0}}}$. The projection of the
helical trajectory on the sky-plane (or the apparent trajectory) is
represented by

${X_{n}}={X_{p}}{\cos{\psi}}-{Z_{p}}{\sin{\psi}}$ (7)
${Z_{n}}={X_{p}}{\sin{\psi}}+{Z_{p}}{\cos{\psi}}$ (8)

where

${X_{p}}=X({s_{0}})$ (9)
${Z_{p}}={Z({s_{0}})}{\sin{\epsilon}}-{Y({s_{0}})}{\cos{\epsilon}}$ (10)

(All coordinates and amplitude (A) are measured in units of mas). Introducing
the functions

${\Delta}=\arctan[(\frac{dX}{dZ})^{2}+(\frac{dY}{dZ})^{2}]^{-\frac{1}{2}}$
(11) ${{\Delta}_{p}}=\arctan(\frac{dY}{dZ})$ (12)
${{\Delta}_{s}}=\arccos[(\frac{dX}{d{s_{0}}})^{2}+(\frac{dY}{d{s_{0}}})^{2}+(\frac{dZ}{d{s_{0}}})^{2}]^{-\frac{1}{2}}$
(13)

we can then calculate the viewing angle $\theta$, apparent transverse velocity
${\beta}_{a}$, Doppler factor $\delta$ and the elapsed time T, at which the
knot reaches distance $z_{0}$ as follows:

${\theta}=\arccos[{\cos{\epsilon}}(\cos{\Delta}+\sin{\epsilon}\tan{{\Delta}_{p}})]$
(14) ${\Gamma}=(1-{\beta}^{2})^{-\frac{1}{2}}$ (15)
${\delta}=[{\Gamma}(1-{\beta}{\cos{\theta}})]^{-1}$ (16)
${{\beta}_{a}}={{\beta}{\sin{\theta}}/(1-{\beta}{\cos{\theta}})}$ (17)
$T=\int^{{s_{0}}}_{0}{\frac{(1+z)}{{\Gamma\delta}{v}{\cos{{\Delta}_{s}}}}}{d{s_{0}}}$
(18)

The amplitude and phase of the helical trajectory for superluminal knots are
defined as follows.

${A({Z})}={{A_{0}}[\sin({\pi}{Z}/{Z_{1}})]^{1/2}}$ (19)
${\phi}({Z})={{\phi}_{0}}-{({Z}/{Z_{2}})^{1/2}}$ (20)

$A_{0}$ represents the amplitude coefficient of the common helical trajectory
pattern and ${\phi}_{0}$ is the precession phase of an individual knot.
The aim of our model fitting of the kinematics of the superluminal components
observed in 3C345 was to show that most components have their inner
trajectories following the precessing common trajectory and their kinematics
can be interpreted in terms of our precessing jet nozzle scenario, indicating
the possible presence of a supermassive black hole binary in its nucleus (for
details see Qian Qi22 (2022) where a double-jet structure (jet-A and jet-B) in
3C345 was proposed).
The main subject of this paper is to investigate the flux evolution of five
superluminal knots (C4, C5, C9 C10 and C22) of jet-A and shows that their flux
evolution could well be interpreted in terms of the Doppler boosting effect
during their accelerated/decelerated motion along helical trajectories. Thus
the kinematic and emission properties of the five components can be unitedly
explained within the framework of our precessing nozzle scenario.
As shown above, the precessing common helical trajectory which the
superluminal components follow is defined by parameters ($\epsilon$, $\psi$) ,
and formulas (1)-(2) and (19)-(20) and the corresponding parameters in them.
For jet-A we assume the following values:
$\epsilon$=0.0349 rad=$2^{\circ}$; $\psi$=0.125 rad=$7.16^{\circ}$;
$\zeta$=2.0, $p_{1}$=0; $p_{2}$=1.34$\times{10^{-4}}$; $z_{t}$=66 mas;
$z_{m}$=6 mas; $A_{0}$=0.605 mas, $Z_{1}$=396 mas and $Z_{2}$=3.58 mas. The
precession phase $\phi_{0}$ is related to the knot’s ejection time:

${\phi_{0}}=4.28+{\frac{2\pi}{T_{0}}}({T_{0}}-1979.00)$ (21)

where $T_{0}$=7.30 yr–precession period of the jet-nozzle. The functions
modeled for the amplitude and phase of the precessing helical trajectory
pattern is shown in Figure 2 (for C4, $\phi_{0}$=4.28 rad).
We would like to indicate here that all the five superluminal components move
along the precessing common trajectory only in their inner trajectory
sections. In the outer trajectory sections they move along their own
individual trajectories. In order to model-fit their outer trajectories we
shall introduce changes in parameters $\epsilon$ and $\psi$ (which define the
modeled jet-axis direction) only, without introducing any changes in the
common helical trajectory pattern.

###  2.1 A note on the model-parameters

During the model-simulation of the kinematics of superluminal components in
3C345 the values chosen for the model parameters and the forms for the
associated functions were not statistical samples and not unique. They were
only specific and physically applicable sets of working ingredients which were
obtained through trial and error methods over the past few years. But it was
shown that they could be used to analyze the distribution of the observed
trajectories and kinematics of superluminal components in blazar 3C345 on
VLBI-scales, especially discovering the possible division of its superluminal
knots into two groups with different kinematic properties. Our methods aimed
at: (1) seeking for the possible jet-precession and determining the precession
period; (2) searching for a double-jet structure; (3) disentangling the
observed superluminal components into two groups attributed to respective
jets; (4) investigating the relation between knots’ flux evolution and Doppler
boosting effect; (5) studying the properties of the putative supermassive
binary black holes in its nucleus.
Similar methods have also been applied to blazar 3C279 (Qian et al. 2018a ),
OJ287 (Qian 2018b ) and 3C454.3 (Qian et al. Qi21 (2021)). Interestingly, we
tentatively found that all the four blazars seemed to have double-jet
structures.
Since in our works model-simulations of the kinematics of superluminal
components involved multi-parameters and multi-functions, we introduced a new
criterion to judge the validity of the model-fitting results, instead of
ordinarily used statistical errors estimated for the model parameters. That
is, a reasonable and effective model-fitting of the kinematics of a
superluminal knot was required to satisfy the condition that its observed
inner-trajectory had to be fitted to follow the precessing common trajectory
predicted by the scenario within $\pm$5$\%$ of the precession period. A good
example is presented in Figure 15 for knot C9 below.
In addition, since the outer-trajectory of the superluminal components
deviated from the precessing common trajectory pattern the model-fits to their
outer-trajectories were performed through changing the parameters ($\epsilon$,
$\psi$) which defined the modeled jet-axis direction, while the common helical
pattern was kept unchanged.
As shown in the previous paper (Qian Qi22 (2022)) the kinematics of the
superluminal components of both jets in 3C345 could be well model-fitted and a
precession period of 7.30$\pm$0.36 yr for both the jet-nozzles were derived.
In Figure 3 are shown the distribution of the precessing common trajectory for
jet-A of 3C345 at different precession phases (left panel) and the observed
trajectories of the superluminal knots C4, C5 and C13 (right panel) for
comparison.

##  3 Basic equations for Doppler boosting effect and flux evolution

In order to investigate the relation between the flux evolution of
superluminal components and their Doppler boosting effect during their
accelerated/decelerated motion along helical trajectories, the observed flux
density $S_{obs}(\nu,t)$ of superluminal components can be defined as the
Doppler-boosted flux density:

${S_{obs}}(\nu,t)={S_{int}}(\nu){\times}{\delta(t)}^{3+\alpha}$ (22)

$S_{int}(\nu)$=$S_{0}$ ${({\nu/{\nu_{0}})}^{-\alpha}}$–the intrinsic flux
density, $S_{0}$–the flux density at the fiducial frequency $\nu_{0}$, and
$\alpha$–spectral index for the whole range of the observing frequencies
(10–43 GHz), $\delta(t)$–Doppler factor. 222In most general case both
$S_{int}$ and $\alpha$ could be defined as functions of $\nu$ and t , and thus
Doppler-boosting profiles varying with time and frequency, and complex flux
evolution could be investigated. But here in this paper we assume that
$S_{0}(t)$ and $\alpha(\nu,t)$ are constants, not depending on time and
frequency.
We also use the normalized flux density ${S_{obs,N}}(\nu,t)$ which is defined
as

${S_{obs,N}}(\nu,t)={S_{obs}}(\nu,t)/{S_{obs,max}(\nu)}$ (23)

$S_{obs,max}(\nu)$–the observed maximum flux density.
The Doppler-boosting profile (a normalized profile) is defined as:

${S_{D}}(t)=[\delta(t)/\delta_{max}]^{3+\alpha}$ (24)

Here $\delta_{max}$ – the modeled maximum Doppler factor.
We also use the intrinsic flux density, approximately defined as:
${S_{int}}(\nu)$=${S_{obs,max}(\nu)}$/$[\delta_{max}]^{3+\alpha}$.
In the precessing nozzle scenario the Doppler factor $\delta(t)$ of
superluminal knots depends on their motion along helical trajectories which
are produced by the precessing common trajectory at corresponding precession
phases (or ejection epochs).
As already shown in the previous paper, the kinematics of the superluminal
components was well model-simulated, and their bulk Lorentz factor $\Gamma(t)$
and Doppler factor $\delta(t)$ as functions of time were predictively derived,
which could be used to investigate the relation between the Doppler boosting
effect and their flux evolution. We found that in order to correctly model-fit
their flux evolution, the bulk Lorentz factor and Doppler factor previously
derived should be appropriately modified. Moreover, in order to investigate
the flux evolution at multi-frequency on the whole trajectory (in both inner
and outer trajectory sections), variations in parameters $\epsilon$ and
$\psi$, intrinsic flux density $S_{int}$ and spectral index $\alpha$ should
also be taken into consideration.

Figure 3: Distribution of the precessing common trajectory for jet-A in 3C345
(left panel). The jet axis is at position angle $\sim{-97.2^{\circ}}$ with its
cone aperture $\sim{42.5^{\circ}}$ (at core separation 0.5 mas). The opening
angle of the jet is $\sim{1.12^{\circ}}$ in space. The observed trajectories
of knots C4, C5 and C13 are presented in the right panel for a comparison with
the model distribution.

Figure 4: Knot C4. The traveled distance Z(t) along the Z-axis (left panel)
and the model parameters $\epsilon$ and $\psi$ as functions of time. Within
Z=40.0 mas (t$\leq$1987.08) $\epsilon$=$2.0^{\circ}$ and
$\psi$=$7.16^{\circ}$, and knot C4 moved along the precessing common
trajectory, while beyond Z=40.0 mas $\epsilon$ increased and $\psi$ decreased
and knot C4 moved along its own individual track. Figure 5: Knot C4: Model-fit
to its inner trajectory within $X_{n}{\simeq}$1.14 mas. The black line
describes the precessing common trajectory-section with the precession phase
$\phi_{0}$=4.28 rad and the corresponding ejection time $t_{0}$=1979.0. The
lines in magenta and blue show the precessing common trajectories for
precession phases $\phi_{0}$+0.31 rad and $\phi_{0}$-0.31 rad (i.e., $\pm$5%
of the precession period), respectively. The curve in violet represents the
model-fit to the whole trajectory.

Figure 6: Knot C4: Model fits to the entire kinematics of knot C4 (within
$r_{n}{\simeq}$8 mas), including its whole trajectory, core separation and
coordinate $X_{n}$ (upperthree panels), and coordinate $Z_{n}$, the model-
derived apparent speed and viewing angle, and the model-derived bulk Lorentz
factor and Doppler factor (bottom three panels).

##  4 Interpretation of kinematics and flux evolution for knot C4

As shown in previous paper (Qian Qi22 (2022)), the entire kinematics of knot
C4 with its observed core separation extended to $\sim$8 mas could be well
model-simulated in terms of our precessing nozzle model. However, only its
inner trajectory within a core-separation of $\sim$1.8 mas followed the
precessing common trajectory pattern. Thus in order to model-fit its entire
kinematics and investigate its flux evolution we had to appropriately modify
the previously derived bulk Lorentz factor and model-parameters $\epsilon$ and
$\psi$.

Figure 7: Knot C4: Light curves observed at 10 GHz and 22 GHz with a few data-
points observed at 15 and 43 GHz.

Figure 8: Knot C4: coincidence of the 10 GHz and 22 GHz light-curves with the
Doppler boosting profiles: the observed light-curves (left two panels) and the
normalized light-curves (right two panels).

###  4.1 Model simulation of kinematics for knot C4

In the present model the inner trajectory of knot C4 within $X_{n}{\leq}$1.14
mas ($r_{n}{\leq}$1.15 mas, traveled distance $Z_{c}{\leq}$40.0 mas$\sim$266
pc) could be well fitted by the precessing common trajectory pattern (Fig.5).
Its precession phase was modeled as $\phi_{0}$=4.28 rad and ejection epoch
$t_{0}$=1979.0.
In Figure 4 are presented the traveled distance Z(t) along the Z-axis, and the
curves of model parameters $\epsilon(Z)$ and $\psi(Z)$ (right panel), which
indicate that within Z$\leq$40.0 mas (t$\leq$1987.08) $\epsilon$=$2.0^{\circ}$
and $\psi$=$7.16^{\circ}$, and knot C4 moved along the precessing common
trajectory, while beyond Z=40.0 mas $\epsilon$ started to increase and $\psi$
stated to decrease, and knot C4 started to move along its own individual
trajectory (in its outer trajectory-section). The model-fit to its inner
trajectory section is shown in Figure 5: the curves in black, magenta and blue
represents the precessing common trajectories for precession phases
$\phi_{0}$=4.28 rad and $\phi_{0}{\pm}$0.31 rad, respectively. The curve in
violet represents the model-fit to the whole trajectory.
Correspondingly, its bulk Lorentz factor was adjusted as follows in order to
take its deceleration after 1982.67 into consideration: for Z$\leq$0.5 mas
$\Gamma$=6.0; for Z=0.5-2.0 mas $\Gamma$ increased from 6 to 8; for Z=2-18 mas
$\Gamma$ increased from 8 to 13 ; for Z=18-35 mas $\Gamma$ decreased from 13
to 10; for Z=35-40 mas $\Gamma$ decreased from 10 to 5.5; for Z$>$40 mas
$\Gamma$=5.5. $\Gamma_{max}$=13.0 at 1982.67, and the corresponding maximum
Doppler factor $\delta_{max}$=23.5.
It can be seen from Figure 6 that the entire kinematics of knot C4 could be
well explained in terms of our precessing nozzle scenario, including the whole
trajectory $Z_{n}(X_{n})$, core separation $r_{n}(t)$ and coordinate
$X_{n}(t)$ (upper three panels), and coordinate $Z_{n}(t)$, the derived
apparent velocity $\beta_{app}(t)$/viewing angle $\theta(t)$ and the derived
bulk Lorentz factor $\Gamma(t)$/Doppler factor $\delta(t)$ (bottom three
panels).
The apparent speed derived showed two bumps (Figure 6, bottom/middle panel):
one occurred before 1987.08 in its inner-trajectory section where knot C4
followed the precessing common trajectory with its maximum speed of $\sim$8c
at $\sim$1982.7, while the other occurred during the period of 1995–2005 in
its outer trajectory section where knot C4 moved along its own individual
trajectory. The derived viewing angle showed variations along a concave curve
in the inner trajectory section, while it showed a bump structure in its outer
trajectory section. This is the first time to obtain a continuous curve for
the apparent-velocity/viewing-angle of knot C4 during a time-interval of
$\sim$20 years.
Both the derived bulk Lorentz factor and Doppler factor showed only one-bump
structure occurred in its inner trajectory section before 1987.08, i.e., it
occurred when its motion followed the precessing common trajectory (Figure 6,
bottom/right panel), showing its intrinsic acceleration/deceleration along the
precessing common helical trajectory.

###  4.2 Doppler boosting effect and flux evolution of knot C4

The derived Doppler-factor curve shown in Figure 6 (bottom/right panel)
reveals a distinct bump structure with its peak at $\sim$1982.6. This is a
very good case for yielding the Doppler boosting profile (normalized at its
peak, $[{\delta(t)}/{\delta_{max}}]^{3+\alpha}$) and studying the relation
between its flux-density evolution and the Doppler boosting effect.
The observed 10 GHz and 22 GHz light-curves of knot C4 are shown in Figure 7.
Both the light-curves showed a peak flux density at $\sim$1983, closely
corresponding to the peak in the Doppler factor curve ($\delta_{max}$=23.5 at
$\sim$1982.64). A spectral index $\alpha$(22-10GHz)=0.18 was assumed. The
intrinsic flux densities were assumed as: 3.54$\times{10^{-4}}$Jy (10 GHz) and
3.28$\times{10^{-4}}$Jy (22 GHz).
The model-fits to the 10 GHz and 22 GHz light-curves in terms of Doppler-
boosting effect are shown in Figure 8, respectively. It can be seen that the
10GHz and 22GHz light-curves were extremely well coincident with the Doppler
boosting profiles (for both the observed flux light-curves and the normalized
flux light-curves). This implies that the flux-density variations observed at
10 GHz and 22 GHz were fully induced by its Doppler boosting effect, and the
variations in its intrinsic flux density were not detectable, i.e., the
emission of the relativistic shock producing the superluminal component C4 was
very stable.
The domination of Doppler boosting effect in the emission of knot C4 strongly
justifies the traditional point-view which is supported by most blazar
physicists and VLBI-observers: superluminal components participate
relativistic motion toward us along helical trajectories with
acceleration/deceleration. This result strongly proves that our precessing
nozzle scenario can interpret the entire VLBI-kinematics of knot C4 and
correctly derive its Doppler factor as a continuous function of time which can
be used to interpret its flux density evolution. The explanation of the entire
kinematics and emission properties for knot C4 in terms of our precessing
nozzle scenario is very encouraging and instructive.

Figure 9: Knot C5. The traveled distance Z(t) along the Z-axis (left panel),
and the curves of ${\epsilon}$(t) and ${\psi}$(t) (right panel). Within Z=44.8
mas (t=1990.4) $\epsilon$=$2^{\circ}$ and $\psi$=$7.16^{\circ}$, and knot C5
moved along the precessing common trajectory. Beyond Z=44.8 mas it moved along
its own individual trajectory. Figure 10: Knot C5: Model-fit to its inner
trajectory section (within $r_{n}$=1.25 mas (or $X_{n}$=1.21 mas, t=1990.4).
The curve in black represents the precessing common trajectory for precession
phase $\phi_{0}$=5.83 rad (corresponding ejection time $t_{0}$=1980.80). The
curves in magenta and blue represent the precessing common trajectories for
precession phases $\phi_{0}$+0.31 rad and $\phi_{0}$-0.31 rad, respectively.

Figure 11: Knot C5. Model-fitting results: trajectory $Z_{n}$($X_{n}$), core
separation $r_{n}$ and coordinate $X_{n}$(t) (upper three panels), and
coordinate $Z_{n}$(t), the modeled apparent velocity ${\beta}_{a}$(t) and
viewing angle $\theta$(t), bulk Lorentz factor $\Gamma$(t) and Doppler factor
$\delta$(t) (bottom three panels). Precession phase $\phi_{0}$=5.83 rad and
ejection time $t_{0}$=1980.80.

##  5 Interpretation of kinematics and flux evolution for knot C5

###  5.1 Model simulation of kinematics for knot C5

For the interpretation of the kinematics of superluminal components in jet-A
of 3C345 in terms of the precessing nozzle scenario the kinematic behavior of
knot C5 is quite significant, because its ejection epoch $t_{0}$=1980.80 and
corresponding precession phase $\phi_{0}$=5.83 rad were different from those
of knot C9 ($t_{0}$=1995.06 and $\phi_{0}$=5.54+4$\pi$) by about two
precession periods (Comparing Figure 10 with Figure 15 in the next section for
knot C9). Moreover, its apparent trajectory was very similar to that of knot
C9, demonstrating the recurrence of the precessing common trajectory and
confirming the basic assumption in our scenario: the superluminal components
move along the common trajectory which precesses to give rise to the
individual trajectories of the superluminal knots at corresponding precession
phases.
In Figure 9 are shown the traveled distance of knot C5 along the Z-axis (left
panel) and the curves of parameters $\epsilon$ and $\psi$, which indicate:
during the period 1980.80-1990.42 (Z$\leq$44.8 mas, $X_{n}{\leq}$1.21 mas,
$r_{n}{\leq}$1.25 mas) knot C5 moved along the precessing common trajectory.
After 1990.42 $\epsilon$ started to increase and $\psi$ started to decrease,
and knot C5 started to move along its own individual trajectory (in its outer
trajectory section).
The model-fit to its inner trajectory section is shown in Figure 10: the
curves in black, magenta and blue represent the precessing common trajectories
for precession phases $\phi_{0}$=5.83 rad and 5.83$\pm$0.31 rad ($\pm$5% of
the precession period), respectively. The curve in violet represents the fit
to the whole trajectory.
The model fits to its whole kinematic behavior are shown in Figure 11. It can
be seen that its entire trajectory, core separation, two coordinates are well
model-simulated. The derived apparent speed showed a distinct bump during the
period $\sim$1983–1990, while the viewing angle varied along a depressed
curve. The derived curves of bulk Lorentz factor and Doppler factor had
distinct bumps during 1986.2–1988.3. Thus the intrinsic
acceleration/deceleration and Doppler boosting effect occurred in its inner
trajectory section where knot C5 moved along the precessing common trajectory.

Figure 12: Knot C5: the light-curves observed at 10 GHz and 22 GHz with some
data-points observed at 15 GHz and 43 GHz.

Figure 13: Knot C5. The 10 GHz and 22 GHz light-curves were well coincident
with the Doppler boosting profiles for both the observed flux light-curve
(left two panels) and the normalized flux light-curve (right two panels).

Bulk acceleration and deceleration were required and the Lorentz factor was
modeled as: for Z$\leq$4.0 mas $\Gamma$=5.5; for Z=4.0-15 mas $\Gamma$
increased from 5.5 to 13.2 ($\Gamma_{max}$=13.2 at 1986.21); for Z=15–34 mas,
$\Gamma$=const.=13.2; for Z=34-45 mas $\Gamma$ decreased from 13.2 to 7.0
(1990.53); for Z$>$45 mas $\Gamma$=7.0.
It can be seen that the entire kinematic behavior within core separation
${r_{n}}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}$1.25 mas can be well fitted,
implying that its observed precessing common trajectory could extend to a
spatial distance of $Z_{c}$=44.8 mas (or $\sim$298 pc) from the core.

###  5.2 Doppler boosting effect and flux evolution of knot C5

The bump structure derived for the Doppler factor of knot C5 provided an very
good explanation of the flux evolution of knot C5 during its motion along the
precessing common trajectory. The observed 10 GHz and 22 GHz light-curves are
shown in Figure 12, clearly revealing a flare event during $\sim$1985–1991
with a peaking flux at $\sim$1988.0, which was closely coincident with the
Doppler-boosting profile. We assumed the spectral index
$\alpha$(10–22GHz)=0.30, the intrinsic flux densities were assumed to be
1.5$\times{10^{-4}}$Jy (10 GHz) and 7.7$\times{10^{-5}}$Jy (22 GHz). The
model-derived maximum Doppler factor $\delta_{max}$=24.6 at $\sim$1987.4.
The model-fits to the observed 10 GHz and 22 GHz light-curves are shown in
Figure 13 (left two panels), while their normalized light-curves were fitted
by the normalized Doppler-boosting profiles
($[\delta(t)/\delta_{max}]^{3+\alpha}$) in the right two panels. It can be
seen that the 15 GHz and 22 GHz light-curves were very well coincident with
the Doppler boosting profiles which were calculated by using the model-
predicted Lorentz factor curve $\Gamma(t)$ and the model-predicted viewing
angle curve $\theta(t)$ (Fig.11, bottom/right and bottom/middle panels,
respectively).
The shorter time-scale variations at 22 GHz during $\sim$1983–1984 could be
due to the intrinsic flux variations of knot C5 (see the discussion on the
flux variations of knots C9 and C10 below).

Figure 14: Knot C9. Left panel: the traveled distance Z(t) along the Z-axis.
The right panel shows $\epsilon$=const.=$2^{\circ}$ and $\psi$=$7.16^{\circ}$
before 1999.94 ($r_{n}{\leq}$1.25 mas or $X_{n}{\leq}$1.22 mas), when it moved
along the precessing common trajectory (in its inner-trajectory section).
After 1999.94 $\psi$ increased and knot C9 started to move along its own
individual trajectory (in its outer trajectory section).  Figure 15: Knot C9.
Model fit to the inner trajectory section ($Z_{c}{\leq}$ 44.8 mas,
$X_{n}{\leq}$1.21 mas, t$\leq$1999.94) where it moved along the precessing
common trajectory. The black curve represents the precessing common trajectory
for precession phase $\phi_{0}$=5.54+4$\pi$ (ejection time $t_{0}$=1995.06),
while curves in magenta and blue represent those for precession phases
$\phi_{0}{\pm}$0.31 rad ($\pm$5% of the precession period). The curve in
violet represents the model-fit to its whole trajectory.

Figure 16: Knot C9. Model-fitting results for the whole trajectory
$Z_{n}$($X_{n}$), core separation $r_{n}(t)$ and coordinate $X_{n}$(t) (upper
three panels); model-fitting results for coordinate $Z_{n}$(t), model-derived
apparent-velocity $\beta_{app}(t)$ and viewing-angle $\theta(t)$, and model-
derived bulk Lorentz-factor $\Gamma(t)$ and Doppler-factor $\delta(t)$ (bottom
three panels). $\Gamma(t)$ has a platform of maximum ($\simeq$18.5–18.0)
during 1997.4-2001.3.

##  6 Interpretation of kinematics and flux evolution for knot C9

As shown in the previous paper (Qian Qi22 (2022)), The results of model-
fitting of its kinematics and the explanation of its flux evolution in terms
of Doppler boosting effect are very important and encouraging for the
application of our precessing nozzle scenario to study the phenomena in 3C345
and other blazars. Knot C9 was a typical and exceptionally instructive example
of applying the scenario for a satisfying interpretation of its VLBI-
kinematics and flux evolution. Here we recapitulate the main results obtained
in Qian (Qi22 (2022)) and supplement some new results on its flux evolution
and spectral features.

###  6.1 Model simulation of kinematics for knot C9

In the left panel of Figure 14 the traveled distance Z(t) along the Z-axis
from the core of knot C9 is shown. In the right panel is shown the parameters
$\epsilon$ and $\psi$ as functions of time. It can be seen that $\epsilon(t)$
is a constant (0.0349 rad=$2^{\circ}$), but $\psi(t)$ varied with time: before
1999.94 $\psi$=0.125 rad=$7.16^{\circ}$ and its trajectory followed the
precessing common trajectory for the precession phase $\phi_{0}$=5.54+4$\pi$
(corresponding to its ejection time $t_{0}$=1995.06). After 1999.94 $\psi$
increased and its trajectory deviated from the precessing common trajectory
and knot C9 moved along its own individual trajectory (in its outer trajectory
section).
The model-fit to its inner trajectory section (or precessing common
trajectory) is shown in Figure 15, where the curve in black represents the
precessing common trajectory section for the precession phase $\phi_{0}$=
5.54+4$\pi$. The curves in magenta and blue represent the model-trajectories
for precession phases $\phi_{0}$$\pm$0.31rad, showing its ejection epoch being
fitted with an accuracy of $\pm$0.36 yr (5% of the precession period). The
curve in violet represents the model-fit to its entire trajectory. Its
precessing common trajectory section extended to $r_{n}{\sim}$1.25 mas or the
traveled distance $Z_{c}{\sim}$44.8 mas=298 pc.
In Figure 16 the model-fitting results of its entire kinematics are presented:
the entire trajectory $Z_{n}(X_{n})$, core separation $r_{n}(t)$, coordinate
$X_{n}(t)$ (upper three panels), and coordinate $Z_{n}(t)$, the modeled
apparent velocity $\beta_{app}(t)$, viewing angle $\theta(t)$, bulk Lorentz
factor $\Gamma(t)$ and Doppler factor $\delta(t)$ (bottom three panels).
It can be seen that the kinematic properties of knot C9 during the entire
observing period $\sim$1997–2008 (within core separation $r_{n}{\sim}$4.1 mas)
were very well model-fitted.
Its apparent speed, viewing angle, bulk Lorentz factor and Doppler factor as
functions of time were derived through the model-fitting of its entire
kinematics. The accelerated motion observed in knot C9 during 1986–1988 could
be well explained by the increase in its Lorentz factor and its motion along
the helical trajectory.
In order to explain its apparent motion and flux evolution its bulk
acceleration and deceleration were required and its bulk Lorentz factor was
modeled as (bottom/right panel in Figure 16): For Z$\leq$1.0 mas $\Gamma$=4;
for Z=1–6 mas $\Gamma$ increased from 4 to 18.5 ($\Gamma_{max}$=18.5 at
1997.44); for Z=6–14 mas $\Gamma$=18.5; For Z=14–20 mas $\Gamma$ decreased
from 18.5 to 18.0; For Z=20-61.7 mas (t=2001.28) $\Gamma$=18.0; For Z=61.7–80
mas (t=2003.84) $\Gamma$ decreased from 18.0 to 13.0; for Z$>$80 mas
$\Gamma$=13.0.
The modeled Doppler-factor curve showed a smooth bump structure during
1996–2002, providing a very determinative Doppler-boosting effect to explain
its flux variations observed at 43 GHz, 22 GHz and 15 GHz.
The derived apparent velocity $\beta_{a}(t)$ showed two peaks while the
viewing angle $\theta(t)$ varied along a depressed curve (bottom/middle panel
in Figure 16).

Figure 17: Knot C9. Lght-curves observed at 15 GHz, 22 GHz and 43GHz.

Figure 18: Knot C9. Light-curves observed at 15 GHz and 43 GHz (left panel)
and the reformed light curves with an assumed spectral index
$\alpha$(15-22-43GHz)=0.80.

Figure 19: Knot C9. Top two panels: the observed 43GHz light-curve and its
normalized light-curve fitted by the Doppler-boosting profiles. Middle two
panels: the observed 22 GHz light-curve and its normalized light-curve fitted
by the Doppler-boosting profiles. Bottom two panels: the observed 15 GHz
light-curve and its normalized light-curve matched by the Doppler-boosting
profiles.

Figure 20: Knot C9. Upper panels: the reformed 43GHz light-curve and its
normalized light-curve well matched with the Doppler boosting profiles. Bottom
panels: the reformed 15 GHz light-curve and its normalized version well
coincident with its Doppler boosting profiles. These reformed curves more
appropriately matched the Doppler-boosting profiles.

###  6.2 Doppler-boosting effect and flux evolution of knot C9

The modeled Doppler factor $\delta$(t) as a continuous function of time shown
in Figure 16 (bottom/right panel) had a smooth bump structure during
1997–2002, thus providing a distinctly smooth Doppler-boosting profile to
study the Doppler-boosting effect in the flux variations of knot C9. This was
a rare and extremely valuable opportunity to test our precessing nozzle
scenario and investigate the relation between its flux evolution and Doppler-
boosting effect.
The light-curves observed at 43 GHz, 22 GHz and 15 GHz of knot C9 are shown in
Figure 17. In Figure 18 are presented the reformed 15GHz and 43GHz light-
curves: the rising phase of the observed 43GHz light-curve was reformed by
adding the 43GHz flux calculated from the 15 GHz rising-phase data-points with
a spectral index $\alpha$(15-43GHz)=0.80. Similarly, the decaying phase of the
observed 15GHz light-curve was reformed by adding the 15GHz flux calculated
from the 43 GHz decaying-phase data-points.
It can be seen that the observed and reformed light-curves reveal a flaring
behavior during $\sim$1997.50-2000.25, having a close correlation by visual
inspection.
Based on the datasets observed at 43 GHz, 22 GHz and 15 GHz we obtained
$\alpha$(22-43 GHz)$\simeq$0.70$\pm$0.59 and $\alpha$(15-43
GHz)$\simeq$0.81$\pm$0.25. In the model-fits to the flux evolution the
spectral index was assumed $\alpha$(15-22-43 GHz)=0.80.
The modeled maximum Doppler factor $\delta_{max}$=31.86 occurred at
$t_{max}$=1998.78. The intrinsic flux densities were assumed to be:
3.82$\times{10^{-6}}$Jy (15 GHz), 2.58$\times{10^{-6}}$Jy (22 GHz) and
1.65$\times{10^{-6}}$Jy (43 GHz), corresponding the observed maximum flux
densities: 1.97 Jy (1999.38), 1.33 Jy (1999.12) and 0.85 Jy (1999.60),
respectively.
In Figure 19 are presented the model-fits to the light-curves observed at 43
GHz, 22 GHz and 15 GHz (left three panels) and their normalized light-curves
(right three panels).
In Figure 20 are presented the model-fits to the reformed light-curves
observed at 43 GHz and 15 GHz (left two panels) and their normalized light-
curves (right two panels). Obviously, the reformed light-curves were fitted
more appropriately by the Doppler-boosting profiles.
It can be seen from Figures 19 and 20 that both the observed and reformed
light-curves during 1997.5–2000.3 were very well coincident with the Doppler-
boosting profiles which was anticipatively-determined through the model-
simulation of the kinematics of knot C9.
However, there were some shorter time-scale flux variations during
2000.5–2008.5 remained to be explained. Since these variations were not
related to the Doppler-boosting effect induced by the accelerated/decelerated
motion of knot C9, they could be due to: (i) the variations in intrinsic flux
density $S_{int}$ of knot C9 and (ii) the variations in its spectral index
$\alpha$ (see equation (22) in section 3). For example, assuming that the
shorter time-scale variations at 15 GHz were due to the variations in its
intrinsic flux density with the spectral index unchanged ($\alpha$(15
GHz)=0.80), then the variations at 22 GHz and 43 GHz could be due to the
variations in spectral index ($\alpha$(22 GHz) and $\alpha$(43 GHz)) at the
two frequencies which should be different from that at 15 GHz.
Obviously, the interaction of the traveling relativistic shocks (superluminal
components) with the surrounding evironments could produce these intrinsic
variations.

##  7 Interpretation of kinematics and flux evolution for knot C10

The flux evolution of knot C10 is a very instructive one, providing another
valuable opportunity to test our precessing nozzle scenario. Its flux
variations consisted of two main flaring events caused by the Doppler boosting
effect, involving its complex behavior of bulk acceleration/deceleration and
change in its trajectory pattern: one occurred in the inner precessing common
trajectory section, while the other in the outer individual trajectory
section.
We first discuss the model-simulation of its kinematics and then its Doppler-
boosting effect and flux evolution.
Its precession phase was modeled as $\phi_{0}$(rad)=6.14+4$\pi$, corresponding
to the ejection time $t_{0}$=1995.76. Its precessing common trajectory
apparently extended to $X_{n}{\sim}$0.35 mas (or the traveled distance
${Z_{c}}\sim$6.13 mas along the Z-axis, or $\sim$40.8 pc from the core).

###  7.1 Model simulation of kinematics for knot C10

In Figure 21 are presented the curves for the following functions: (1) Z(t)
describes its traveled distance along the Z-axis from the core as a function
of time (left panel); (2) functions $\epsilon(t)$ and $\psi(t)$ (right panel)
describe the jet-axis direction of inner and outer trajectory sections,
showing only in the inner trajectory section (during the time-interval of
1995.76-1999.14 (corresponding to $X_{n}{\leq}$0.35 mas) where knot C10 moved
along the precessing common trajectory.
Figure 22 shows the transition from the precessing common trajectory-section
to the outer trajectory-section at $X_{n}\sim$0.35 mas. The curves in black,
magenta and blue describe the precessing common trajectories for precession
phases $\phi_{0}$=6.14+4$\pi$ and ${\phi_{0}}\pm$0.31 rad, respectively. The
curve in violet describes the model-fit to the whole trajectory.
The results of model-fitting of the whole kinematics for knot C10 are
presented in Figure 23, including the model-fits of entire trajectory
$Z_{n}(X_{n})$, core separation $r_{n}(t)$, coordinates $X_{n}(t)$ and
$Z_{n}(t)$, and the derived apparent velocity $\beta_{app}(t)$ and viewing
angle $\theta(t)$, the derived bulk Lorentz factor $\Gamma(t)$ and Doppler
factor $\delta(t)$. It can be seen by visual inspection that the trajectory,
core separation, coordinates are well fitted by our precessing nozzle model.
As a distinct feature we noticed that, at $\sim$1999.14 its core separation
$r_{n}$ suddenly increased (Fig.23, top/middle panel), which was possibly due
to the change in the modeled jet-axis direction (see Fig.21, right panel),
where its motion along the precessing common trajectory ended and transited to
its own individual trajectory (Fig.22).

###  7.2 Doppler-boosting effect and flux evolution of knot C10

In order to consider the Doppler-boosting effect in the flux-evolution of knot
10 we needed to know its spectral index in radio bands. Based on the data
observed at three frequencies (15, 22 and 43 GHz) we obtained: spectral index
$\alpha$(22-43GHz)$\simeq$0.64$\pm 0.4$ and
$\alpha$(15-43GHz)$\simeq$0.90$\pm$0.27. Thus in the following model-fitting
of the Doppler-boosting effect we adopted a spectral index $\alpha$=0.80 in
the whole band of 15 to 43 GHz. The Doppler-boosting profile was then defined
by $[\delta(t)/{\delta}_{max}]^{3.80}$ (Figs.25 and 26), where
$\delta_{max}$=14.11 was the maximum Doppler factor of the first flare event
at 1998.36 in our fitting model.
The radio light-curves observed at 15, 22 and 43 GHz are shown in Figure 24.
The observed light-curves at 15, 22 and 43 GHz were normalized by using the
observed maximum fluxes of the first flare, taken as 2.17, 1.60 and 0.94 Jy,
respectively. Thus the corresponding intrinsic fluxes at 15, 22 and 43 GHz
were chosen as $S_{in}$=9.3$\times{10^{-5}}$Jy, 6.8$\times{10^{-5}}$Jy and
4.0$\times{10^{-5}}$Jy, respectively.
As shown in Figure 23 the anticipatively derived bulk Lorentz factor showed
two-bumps: one occurred in the inner trajectory-section (or the precessing
common trajectory before 1999.14) with its $\Gamma_{max}$=8.5 at epoch
1998.36), while the other occurred in the outer trajectory section with its
$\Gamma_{max}$=17.0 at epoch 2000.92. Specifically, its bulk Lorentz factor
was modeled to indicate its behavior of acceleration/deceleration to re-
acceleration/deceleration : for Z$\leq$0.6 mas (1995.76-1997.00) $\Gamma$=3.0;
for Z=0.6-4.2 mas (1997.00-1998.38) $\Gamma$ increased from 3.0 to 8.5; for
Z=4-6.4 mas (1998.38-1999.41) $\Gamma$ decreased from 8.5 to 4.0; for
Z=6.4-12.1 mas (1999.41-2000.92) $\Gamma$ increased from 4.0 to 17.0; for
Z=12.1-23.2 mas (2000.92-2003.87) $\Gamma$ decreased from 17.0 to 7.0 and then
kept to be const.=7.0 (Fig.23, bottom/right panel).
Correspondingly, the derived apparent velocity $\beta_{app}$(t) also showed a
double-bump structure as shown in Figure 23 (bottom/middle panel).
However, the derived Doppler factor showed three bumps, the first one occurred
in the precessing common trajectory section ($\delta_{max}$=14.11 at epoch
1998.36), while the other two occurred in the outer trajectory section with
their $\delta_{max}$=11.35 at 1999.98 and $\delta_{max}$=13.89 at 2002.00,
respectively.
It can be seen from Figures 25 and 26 that the observed flux light-curves were
similar to the Doppler-boosting profiles, but had much complex structures.
Most interestingly, the 22 GHz light-curves (both observed and normalized)
were extremely well fitted by the Doppler-boosting profiles as shown in Figure
25.
The 15 GHz and 43 GHz light-curves shown in Figure 26 were also fitted quite
well by the Doppler-boosting profiles. However, at the two frequencies a
number of data-points largely deviated from the modeled Doppler-boosting
profiles. This phenomenon is quite similar to that observed in knot C9
(Figs.19 and 20).
As an example, during $\sim$2000.0-2001.5 the data-points observed at 15 and
43 GHz distributed much higher than the modeled Doppler-boosting profiles
(Fig. 26). This could be due to the following effects: (1) at 15 GHz knot C10
had its spectral index to be larger than 0.80 or its intrinsic flux density
being larger than the adopted value 9.3${\times}10^{-5}$Jy; (2) at 43 GHz knot
C10 had its spectral index to be smaller than 0.80 or its intrinsic flux being
larger than the adopted value 4.0${\times}10^{-5}$Jy. Additionally, both the
spectral indexes and intrinsic fluxes at 15 and 43 GHz were variable with
time.333No such behavior was clearly observed at 22 GHz: knot C10 has a
spectral index 0.80 and a steady intrinsic flux 6.8${\times}10^{-5}$Jy at this
frequency.
It worths noting that for knot C10 the selection of the model-parameters was
quite difficult, because the model-fits involved multiple functions
simultaneously (requiring model-fits to its trajectory, coordinates, core
separation, apparent velocity and flux evolution (complex light-curves) at
multi-frequencies. All these observed properties were functions of time. Thus
the successful interpretation of its kinematics and flux evolution seems very
encouraging for validating our precessing nozzle scenario.

Figure 21: Knot C10. Left panel: the traveled distance Z(t) from the core (or
spatial distance) along the Z-axis. Right panel: parameters $\epsilon(t)$ and
$\psi(t)$ which define the modeled direction of jet-axis. Before 1999.14
($X_{n}{<}$0.35 mas) $\epsilon$=$2.0^{\circ}$ and $\psi$=$7.16^{\circ}$, and
knot C10 moved along the precessing common trajectory. After 1999.14
($X_{n}{>}$0.35 mas) $\epsilon$ started to increase and knot C10 started to
move along its own individual trajectory in its outer trajectory-section.
Parameter $\psi$ started to decrease from 2000.92 ($X_{n}{>}$1.19 mas) and its
outer trajectory pattern changed once again.  Figure 22: Knot C10. Model-fit
to the inner trajectory section in terms of the precessing common trajectory
which apparently extended to $X_{n}{\sim}$0.35 mas (corresponding spatial
distance ${Z_{c}}\sim$6.13 mas or 40.8 pc). The curve in violet represents the
model-fit to its whole trajectory, while the curves in black, magenta and blue
represent its precessing common trajectories for precession phases
$\phi_{0}$(rad)=6.14+4$\pi$ and 6.14$\pm$0.31+4$\pi$, respectively.

Figure 23: Knot C10: Model simulation of the VLBI-kinematics. Upper panels:
apparent trajectory $Z_{n}(X_{n})$ , core separation $r_{n}(t)$ and coordinate
$X_{n}(t)$. Bottom panels: coordinate $Z_{n}(t)$, apparent speed
$\beta_{app}(t)$/viewing angle $\theta(t)$ and bulk Lorentz factor
$\Gamma(t)$/Doppler factor $\delta(t)$. It is noted that the sudden increase
in core separation $r_{n}$ at 1999.14 was due to a sudden change in the
modeled jet-axis direction and its trajectory started to deviate from the
precessing common trajectory. Both apparent velocity and bulk Lorentz factor
show a two-bump structure, while the modeled Doppler factor shows a three-bump
structure with peaks at 1998.36, 1999.98 and 2002.00, respectively. Figure 24:
Knot C10: light-curves observed at 15 GHz, 22 GHz and 43 GHz.

Figure 25: Knot C10: The 22 GHz variations were extremely well fitted by the
Doppler boosting profile with an adopted spectral index $\alpha$=0.80 for the
observed flux light-curve (left panel) and the normalized flux light-curve
(right panel).

Figure 26: Knot C10: The 15GHz and 43GHz light-curves well fitted by the
Doppler-boosting profiles. Variations in its intrinsic flux density and
spectral index at the two frequencies probably gave rise to the shorter time-
scale flux fluctuations.

Figure 27: Knot C22: the traveled distance along the Z-axis (left panel) and
the curves of $\epsilon$ and $\psi$ (right panel), describing the change in
the modeled jet-axis direction. Before 2009.98 (Z$\leq$2.3 mas)
$\epsilon$=$2.0^{\circ}$ and $\psi$=$7.16^{\circ}$, knot C22 moved along the
precessing common trajectory in its inner trajectory-section. After 2009.98,
$\epsilon$ and $\psi$ started to increase and knot C22 started to move along
its own individual trajectory in its outer trajectory-section. Figure 28: Knot
C22: Model-fit to its inner trajectory-section by the precessing common
trajectory. Black curve represents the precessing common trajectory-section
for the precession phase 5.28 rad+8$\pi$, the curves in magenta and blue
represent the precessing common trajectory sections for precession phases
5.28+8$\pi$$\pm$0.31 rad, respectively. the curve in violet represents the
model fit to the whole trajectory.

Figure 29: Knot C22: Model fits to its kinematic properties with its
precession phase $\phi_{0}$=5.28 rad+8$\pi$, $t_{0}$=2009.36, including the
fits to the whole trajectory, core separation and coordinate $X_{n}$ (upper
three panels); coordinate $Z_{n}$, the derived apparent-velocity/viewing-angle
and bulk Lorentz-factor/Doppler-factor (bottom three panels).

Figure 30: Knot 22: The observed 43 GHz flux light-curve (left panel) and the
normalized flux light-curve (right panel) were very well fitted by the
Doppler-boosting profiles.

##  8 Interpretation of kinematics and flux evolution for knot C22

As shown in the previous paper (Qian Qi22 (2022)), the model-fit to the
kinematic behavior of knot C22 was quite important, because: (1) the ejection
time of knot C22 (2009.36) was approximately equal to that (1995.06) of knot
C9 plus $\sim$two precession periods (14.6 yr) and that (1980.80) of knot C5
plus $\sim$4 precession periods. Thus our precessing nozzle scenario could
explain its kinematic behavior during a time-interval of four precession
periods ($\sim$29.2 years); (2) the observed trajectories of knot C5, C9 and
C22 quite clearly indicated the recurrence of the precessing common trajectory
pattern, validating this helical pattern adopted in our precessing nozzle
scenario; (3) the three knots all revealed the intrinsic
acceleration/deceleration in their motion; (4) their Doppler factors derived
from the model-simulation of their kinematics showed bump-structures, implying
the existence of Doppler-boosting effect.
Here in this paper we shall confirm that the flux variations observed in knot
C22 could be successfully interpreted in terms of its Doppler-boosting effect
as for the flux evolution observed in knots C4, C5, C9 and C10.
All these distinct features listed above may actually justify our precessing
nozzle scenario for investigating the VLBI-phenomena in QSO 3C345.

###  8.1 Model simulation of kinematics for knot C22

As shown in the previous paper (Qian Qi22 (2022)) the model-fitting of the
kinematics for knot C22 of jet-A in 3C345 was very important because the VLBI-
observations of knot C22 extended the periodic behavior of jet-A to 4
precession periods relative to the ejection of knot C4 (from $\sim$1979 to
2009, about 30 years).
According to the precessing nozzle scenario the kinematics of C22 could be
model-fitted by using precession phase $\phi_{0}$(rad)=5.28+8$\pi$ and
ejection epoch $t_{0}$=2009.36.
In Figure 27 are shown its traveled distance Z(t) along the Z-axis (left
panel) and the curves of functions $\epsilon(t)$ and $\psi(t)$. During the
period 2009.36–2009.98 in its inner trajectory-section (Z$\leq$2.3 mas,
$X_{n}{\leq}$0.156 mas, spatial distance $Z_{c}{\leq}$15.3 pc),
$\epsilon$=$2.0^{\circ}$ and $\psi$=$7.16^{\circ}$, and knot C22 moved along
the precessing common trajectory. After 2009.98 $\epsilon$ and $\psi$ started
to increase and knot C22 started to move along its own individual trajectory,
which deviated from the precessing common trajectory in its outer trajectory-
section.
The model-fit to its inner trajectory-section in terms of the precessing
common trajectory is shown in Figure 28.
The model-fitting results for its entire kinematics are shown in Figure 29. It
can be seen that the entire trajectory, core separation, two coordinates were
very well fitted in terms of our precessing nozzle scenario.
The model-derived apparent velocity/viewing angle and the bulk Lorentz
factor/Doppler factor are shown in the bottom/middle and bottom/right panels
of Figure 29, respectively.
Specifically, its bulk Lorentz factor was modeled as: for Z$\leq$0.20 mas
(2009.51) $\Gamma$=5.0; for Z=0.2–1.2 mas(2009.51-2009.80) $\Gamma$ increased
form 5.0 to 15.0; for Z=1.2–2.3 mas (2009.80-2009.98) $\Gamma$ decreased from
15.0 t0 10.2; For Z=2.3–3.6 mas (2009.98-2010.41) $\Gamma$ decreased from 10.2
to 8.0 and then kept constant beyond Z=3.6 mas (after 2010.41).
As shown in Figure 29 (bottom/right panel) that the model-predicted Doppler
factor showed a bump-structure during the period $\sim$2009.5-2010.2 with its
peak $\delta_{max}$=16.90 at 2009.87, which should induce Doppler-boosting
effect in the flux evolution of knot C22 and it really occurred as described
below.

###  8.2 Doppler-boosting effect and flux evolution of knot C22

The close relation between the flux variations and the Doppler-boosting effect
found in knot C22 provided one more valuable and successful test of our
precessing nozzle scenario.
It can be seen that both the observed 43 GHz light-curve and its normalized
light-curve were extraordinarily well fitted by the Doppler-boosting profile,
as shown in Figure 30: a flaring event during 2009.6–2010.2 with its peak of
$\sim$1.49Jy at $\sim$2009.91 was well coincident with the Doppler boosting
profile $[{\delta}(t)/{\delta_{max}}]^{3.5}$. 444Here we arbitrarily assumed a
spectral index 0.50. The intrinsic flux density was then derived to be
5.72$\times{10^{-5}}$Jy. That is, The flux-density evolution of knot C22 can
be fully interpreted in terms of its Doppler boosting effect. Thus our
precessing nozzle scenario gets through one more valuable test.

##  9 Conclusive remarks

We can make some conclusions as follows.
(1) We have applied our precessing jet-nozzle scenario previously proposed to
interpret the VLBI-kinematics and flux evolution for five superluminal
components (C4, C5, C9, C10 and C22) in QSO 3C345. It is shown that the VLBI-
measured kinematic properties of all these superluminal knots (including
trajectory, core separation, coordinates, apparent velocity as functions of
time) can be well model-simulated and explained, adopting a precession period
of 7.30$\pm$0.36 yr and a precessing common trajectory pattern. The model-
simulation methods dealing with the model-fitting of multiple functions with
multiple parameters were now validated.
(2) The apparent trajectories observed for the five superluminal knots could
be divided into two sections: inner section and outer section. In the inner
sections the superluminal components moved along the precessing common
trajectory (or common helical-pattern trajectory), while in the outer sections
they moved along their own individual trajectories which deviated from the
common precessing trajectory pattern.
(3) Through the model-simulation of their kinematic properties their bulk
Lorentz factor and Doppler factor as functions of time were naturally derived
and thus the relation between their Doppler boosting effect and flux evolution
could be investigated. It is shown that the knots’ radio light-curves (radio
flaring events) occurred during their motion in the inner regions could be
successfully explained in terms of their Doppler-boosting effect. The
simultaneous determination of the multiple parameters as functions of time
describing the kinematic and dynamic properties may be regarded as the
advantage of our precessing jet-nozzle scenario and model-simulation methods.
(4) Using the anticipatively-determined Doppler factors, for the first time we
were able to study the flux evolution of superluminal components in QSO 3C345.
We found that the radio light-curves of the five components were fully
coincident with the Doppler boosting profiles 555Shorter time-scale flux
variations could be understood as due to the fluctuations in the intrinsic
emission and spectral index of the knots, which could be produced by the
interaction of the traveling relativistic shocks with the complex physical
environments.. Most importantly, for all the five superluminal components (C4,
C5, C9, C10 and C22) the radio flaring events produced by the Doppler-boosting
effect occurred during their motion in the inner trajectory regions (i.e., the
precessing common trajectory sections).
Obviously, these results summarized above are significant for our precessing
jet-nozzle scenario and we can now make an overall review of our scenario to
provide a full-view for studying the VLBI-phenomena in QSO 3C345 as follows.

  * •

The plasmoids and magnetized plasmas ejected by the precessing nozzle was
assumed to form the entire jet of 3C345, thus VLBI-observations could find
many superluminal components moving along different trajectories, forming a
complex distribution of superluminal components in the jet.

  * •

In fact these observed trajectories were formed at different precession phases
by the precession of a common helical trajectory pattern associated with the
nozzle-precession, although for different superluminal components they had
outer trajectory-sections where they moved along their own individual
trajectory patterns.

  * •

The observed swing of the ejection position angle of superluminal components
and its quasi-periodicity could be understood in terms of the nozzle-
precession.

  * •

The observed kinematics (including apparent trajectory, core distance,
coordinates, apparent speed) can be well fitted consistently for all the
superluminal components, if a double precessing-nozzle structure (or double-
jet structure) was assumed. 666Possible evidence of a double-jet structure has
been discussed in Qian (Qi22 (2022)). The close relation between the flux
evolution and Doppler-boosting effect for the superluminal components in jet-B
of 3C345 (in addition to jet-A) will be presented elsewhere, further
justifying the double-nozzle structure in 3C345.

  * •

The double-jet structure as an assumption was based on the model-simulation
results: jet-A and jet-B have different kinematic and dynamic properties,
including different jet direction, precessing common trajectory pattern and
opening angle of precession cone, but having a similar precession period and
direction of precession. This type of precession could originate from
keplerian motion in the putative binary black hole in the nucleus of 3C345.

  * •

The precession period 7.3$\pm$0.36 yr for both the jet-A and jet-B seems
model-simulated appropriately, since for both jets the kinematic behaviors
have been well explained and model-fitted.

  * •

Since the inner trajectory-sections observed for all the superluminal
components of both jet-A and jet-B in 3C345 are well model-fitted within an
accuracy of $\pm$5% of the precession period, the precessing common trajectory
patterns derived for both jet-A and jet-B should be regarded to be physically
appropriate, although they are not unique. They should not deviate from the
actual helical patterns too far.

  * •

The modeled bulk Lorentz factor and Doppler factor derived for the
superluminal components by model-fits of their kinematics clearly revealed the
acceleration/deceleration in their motion. These anticipatively-derived
Doppler factors provide valuable opportunities to study the relation between
their flux-evolution and the Doppler-boosting effect: a key test for our
precessing nozzle scenario.

  * •

The flare events (flux variations) observed to occur in their inner
trajectory-sections for all the five superluminal components (C4, C5, C9, C10
and C22) could be fully interpreted in terms of their Doppler-boosting
effects. The kinematics, dynamics and emission properties of all the five
superluminal components were consistently well interpreted, thus validating
the whole scenario and the model-simulation methods. 777The shorter time-scale
flux variations had no relation to the Doppler-boosting effect induced by the
intrinsic acceleration/deceleration of the knots could be due to the
variations in the intrinsic flux density and spectral index of the knots at
different frequencies (Equation (22) in Sect.4).

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##  Appendix A Doppler boosting effect and flux evolution of a superluminal
component in blazar 3C279

3C279 is one of the most prominent blazars which have been intensively
observed and studied on VLBI-scales. The VLBI-kinematic behaviors observed at
43 GHz of its superluminal components have been analyzed in detail and
successfully interpreted and model-simulated in terms of the precessing nozzle
scenario with a double-jet structure for 31 components (Qian et al. 2019a ).
Here as a supplement, we introduce the results of model-fitting of the VLBI-
kinematics for one superluminal component C13 in 3C279 and the interpretation
of its flux evolution in terms of its Doppler-boosting effect during its
accelerated and decelerated motion.
In the previous paper (Qian et al. 2019a ) the superluminal components in
3C279 was assumed to move along a common parabolic trajectory which precesses
to give rise to the individual trajectories of the knots at their
corresponding precession phases. In this framework, knot C13 had its ejection
time $t_{0}$=1998.88 and corresponding precession phase $\omega$=3.65 rad. Its
bulk Lorentz factor $\Gamma(t)$ and Doppler factor $\delta(t)$ were derived as
functions of time, showing its accelerated motion during the period
1999.2–1999.5. Its modeled Lorentz factor and Doppler factor increased. The
anticipatively-derived Doppler factor implied that knot C13 should have flux
variations induced by the Doppler boosting effect. However, in the previous
model-simulation of the kinematics for knot C13 we did not take its
deceleration after $\sim$1999.5 into consideration.
Here in this appendix, for analyzing the relation between its flux evolution
and Doppler boosting effect during the period $\sim$1999.0-2001.0, we take
both its accelerated and decelerated motion into consideration in order to fit
the rising and decaying phases of its 15 GHz flux light curve. The 15GHz flux
density data was adopted from Roland et al. (Ro13 (2013)), where the knot was
designated as C5.

Figure 31: Knot C13 in 3C279. Top panel: the travelled distance Z(t) along the
Z-axis. The curves of $\epsilon(t)$ and $\Psi(t)$ (middle panel) shows that
for Z$>$5.0 mas (or after 1999.25) parameter $\psi$ started to increase and
knot C13 started to move along its own individual trajectory. The model fit to
its inner trajectory section ($X_{n}{\leq}$0.10 mas, $t{\leq}$1999.25) in
terms of the precessing common trajectory is shown in the bottom panel: the
curves in black, magenta and blue represent the precessing common trajectories
for the precession phases $\omega$=3.65 rad and 3.65$\pm$0.63 rad ($\pm$10% of
the precession period), respectively.

Figure 32: Knot C13 in 3C279: Model-fits to the whole kinematics,including
both inner trajectory-section (within $r_{n}{\leq}$0.15 mas, where knot C13
moved along the precessing common trajectory) and outer trajectory section
(beyond $r_{n}$=0.15 mas, where knot C13 moved along its own individual
trajectory): the whole trajectory, core separation and coordinate $X_{n}$ (top
three panels); coordinate $Z_{n}$, apparent-velocity/viewing-angle and bulk
Lorentz-factor/Doppler-factor (bottom three panels).

In Figure A.1 are shown the travelled distance Z(t) along the Z-axis (top
panel), the curves of parameters $\epsilon(t)$ and $\psi(t)$, and the model-
fit of the inner trajectory-section ($X_{n}{\leq}$0.1 mas).
The curves of $\epsilon$ and $\psi$ (middle panel of Fig.A.1) show that
$\epsilon$=$1.32^{\circ}$=constant, but parameter $\psi$ changed with time:
before 1999.25 (Z$\leq$5.02 mas, $r_{n}{\leq}$0.15 mas and $X_{n}{\leq}$0.10
mas,) $\psi$=$28.70^{\circ}$, and after 1999.25 $\psi$ increased to
$34.38^{\circ}$ at 1999.37 (Z=7.0 mas). Thus knot C13 moved along the
precessing common trajectory within $X_{n}{\leq}$0.10 mas and its precessing
common trajectory section extended to $\sim$33.4 pc. Beyond $X_{n}$=0.10 mas
it moved along its own individual trajectory in its outer trajectory section.
In the bottom panel of Fig.A.1 the model-fit to its inner trajectory section
is shown, where knot C13 moved along the precessing common trajectory: the
dashed lines in black, magneta and blue represent the precessing common
trajectories corresponding to the precession phases $\omega$=3.65 rad and
3.65$\pm$0.63 rad ($\pm$10% of the preecssion period), respectively.
The model-fits to its kinematics are shown in Figure A.2, including its whole
trajectory $Z_{n}(X_{n})$, core separation $r_{n}(t)$ and coordinate
$X_{n}(t)$ (upper three panels), and coordinate $Z_{n}(t)$, the derived
apparent-speed $\beta_{app}(t)$/viewing-angle $\theta(t)$ and the derived
Lorentz-factor $\Gamma(t)$ and Doppler-factor $\delta(t)$ are shown in the
bottom three panels. Specifically, its bulk Lorentz factor was modeled as: for
Z$\leq$5.0 mas $\Gamma$=16.0; for Z=5.0-13.0 mas $\Gamma$ increased from 16.0
t0 21.0 (maximum Lorentz factor at 1999.65); for Z=13.0-40.0 mas $\Gamma$
decreased from 21.0 t0 15.0 and then kept to be constant. Correspondingly, the
Doppler factor had similar variations with a maximum value
$\delta_{max}$=33.68 at Z=13.0 mas (t=1999.65). During the period
1999.25-2001.0 the derived apparent speed had a bump structure similar to that
of Lorentz/Doppler factor, but the derived viewing angle changed in a very
small range [$1.39^{\circ}$-$1.34^{\circ}$-$1.56^{\circ}$].
Obviously, the bump structure of its Doppler factor as a function of time
which was anticipatively-determined for knot C13 indicated the existence of
Doppler boosting effect in its flux evolution. Using the derived Doppler
factor $\delta(t)$ and arbitrarily assuming a spectral index $\alpha$=0.5, we
performed the model-fits to the observed 15 GHz light curves (one for the
observed flux density light-curve and the other for the normalized flux light
curve) in Figure A.3. It can be seen that both the flux light-curves were very
well coincident with its Doppler boosting profile. Thus we see that the
kinematic behavior and the emission properties observed for superluminal
component C13 in 3C279 could be consistently and successfully interpreted in
terms of our precessing jet-nozzle scenario (Qian et al. 2019a ).

Figure 33: Knot C13 in 3C279: The observed 15 GHz light curve (left panel) and
the normalized flux light curve are well interpreted in terms of its Doppler
boosting effect ($\propto{[\delta(t)/\delta_{max}]^{3.5}}$).

We have made detailed studies of the close relation between Doppler-boosting
effect and flux evolution for the superluminal components in 3C345 and 3C279,
and shown that their kinematic/dynamic behaviors and emission properties can
consistently and successfully interpreted in terms of our precessing nozzle
scenario. Thus the investigation of the relation between Doppler-boosting
effect and flux evolution for the superluminal components in other blazars or
AGNs (e.g., 3C454.3, OJ287 and B1308+326) would be important. We would like to
try to do some work in the near future.