Flux evolution and kinematics of superluminal components in blazar 3C345

3C345 (z=0.595) is a prototypical quasar which has been detected in all wavebands of the entire electromagnetic spectrum from radio/mm, infrared/optical/UV and X-rays to high-energy $\gamma$ rays (Biretta et al. Bi86 (1986), Babadzhanyants et al. Ba95 (1995), Schramm et al. Sch93 (1993), Moore & Stockman Mo84 (1984), Wang et al. Wa04 (2004), Malina et al. Ma94 (1994), Lobanov & Zensus Lo99 (1999), Schinzel et al. 2011b , Zensus Ze97 (1997), Ros et al. Ro00 (2000)). It exhibits remarkable and violent multi-waveband variations and its spectral energy distribution has been extensively monitored and studied, yielding significant results on the physical 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 one of the firstly discovered quasar having a relativistic jet. VLBI-observations have revealed its parsec structure and monitored the kinematic behavior of superluminal components ejected from the radio core (Qian et al. 1991a , 1991b , Qi96 (1996), Qi09 (2009), Klare Kl03 (2003), Klare et al. Kl05 (2005), Steffen et al. St95 (1995), Ros et al. Ro00 (2000), Schinzel et al. 2010a , Lobanov & Roland Lo05 (2005), Lobanov & Zensus Lo94 (1994), Zensus et al. Ze95 (1995), Hardee Ha87 (1987)). The flaring activities of the source at multi-frequencies (from radio to $\gamma$-rays) are closely connected with the ejection of superluminal components (e.g., Schinzel 2010b ).
VLBI-monitoring observations have also shown that the swing of the knot’s ejection position angle could be periodic and caused by its jet-precession. We have tried to interpret the VLBI-kinematics of superluminal components in 3C345 in terms of a precessing jet-nozzle scenario since early years (Qian et al. 1991a , 1991b , Qi96 (1996), Qi09 (2009)). Recently, Qian (2022a ) has tried to make model simulations of the kinematic behavior for twenty-seven superluminal components (measured during a 38 yr period) in detail. It was found that these superluminal knots could be hypothetically separated into two groups, which were ejected from a precessing double nozzle (jet-A and jet-B), respectively. Based on this division a precession period of $\sim$7.3 yr was derived for both the jet-nozzles.
In a recent paper (Qian 2022b ), the flux evolution of five superluminal components (C4, C5, C9, C10 and C22) belonging to jet-A was well explained in terms of their Doppler boosting effect combined with the intrinsic flux variations of the components.
In this paper we present the results of modelfits to the flux evolution for five superluminal components (C19, C20, C21, B5 and B7) belonging to jet-B. Before doing so, we firstly give a brief description of the results obtained in the previous work for the five components belonging to jet-A (Qian 2022b ), taking knot C9 as a representative example.

• •

  Based on an assumed helical pattern of precessing common trajectory for jet-A the trajectory distribution of the superluminal components of jet-A is shown in Figure 1 (left panel) for different knots ejected at different precession phseses. It clearly demonstates both the swing of their ejection position angles and different curved tracks cooresponding to different ejection times, which have been successfully applied to modelfit the observed kinematics of the superluminal components of jet-A (Qian 2022a , 2022b ).

• •

  Generally, the superluminal knots moved along their precessing common trajectories in the inner-jet regions, while they moved along their own individual tracks in the outer jet regions. This transition of trajectory pattern occurred at different core distances for different knots. The apparent trajectory of knot C9 was measured extending to core distance of $\sim$4.16 mas, equivalent to a traveled distance Z$\sim$106 mas=705 pc. Its entire trajectory is well fitted in terms of our precessing nozzle scenario as shown in Figure 1 (right). Particularly, its inner trajectory can be very well fitted by the precessing common trajectory pattern (Figure 2), corresponding to a precession phase $\phi_{0}$=5.54+2$\pi$ (or ejection time 1995.06) and extending to a core distance $\sim$1.25 mas, which is equivalent to a traveled distance Z$\sim$44.8 mas=298 pc. The precessing common trajectory pattern assumed in the scenario for jet-A is also applicable to interpret the trajectories measured for other superluminal knots in jet-A (e.g. C4, C5, C7, C10–C13, C22 and C23; Qian 2022a , 2022b ).

• •

  The curved jet-axis and the helical pattern assumed for the precessing common trajectory of jet-A in our scenario result in prominent curvatures in the apparent trajectories of superluminal components (Fig.1, left panel). The remarkable trajectory curvature measured for knot C9 is very well model-fitted in our precessing nozzle scenario (Figure 2). Within $X_{n}$=1.22 mas knot C9 moved along the precessing common track. Most of the observed data-points are located extremely well within the area delimited by the red and blue lines which indicate the trajectories calculated for the precession phases $\phi_{0}$=5.54${\pm}$0.31 rad (or $\pm$5% of the precession period). This distinctively demonstrates that the helical pattern assumed for the precessing common trajectory is appropriate and valid to describe the kinematic behavior of knot C9 and other knots (e.g., C4, C5 and C22; for knot C4, its common track extends to $X_{n}{\sim}$1.14 mas, equivalent to a traveled distance of $\sim$267 pc). Moreover, the model-fits to the apparent trajectories of multi-knots imply the existence of both a precessing nozzle and a precessing common track of helical pattern.

• •

  For knot C9, its core distance $r_{n}(t)$, coordinates $X_{n}(t)$ and $Z_{n}(t)$ are well fitted and shown in Figure 3 (upper two and bottom left panels). Both the model-derived apparent velocity $\beta_{app}(t)$ and viewing angle $\theta(t)$ as continuous functions of time shown in Figure 3 (bottom right panel) reveal prominent structural details. During the period of $\sim$1997–2005 $\beta_{app}(t)$ exhibits a structure of acceleration/deceleration/re-acceleration/deceleration: (a) at 1997.44 (core separation $r_{n}$=0.33 mas) $\beta_{app}$=17.91 (a maximum) and $\theta$=$2.41^{\circ}$; (b) at 1998.77 ($r_{n}$=0.88 mas) $\beta_{app}$=11.44 (a minimum) and $\theta$=$1.14^{\circ}$ (a minimum, corresponding to the maximal Doppler factor $\delta_{max}$=31.86, see Figure 4 below); (c) at 2001.28 ($r_{n}$=1.77 mas) $\beta_{app}$=17.91 (a maximum) and $\theta$=$2.94^{\circ}$.

• •

  The bulk Lorentz factor $\Gamma(t)$ and Doppler factor $\delta(t)$ derived for knot C9 are shown in Figure 4. During the period of $\sim$1997.5-2001.4 the bulk Lorentz factor $\Gamma$$\simeq$18.5–18.0, while the Doppler factor $\delta$ has a smooth bump structure, which was completely caused by the change in the viewing angle, as shown in Figure 3 (bottom right panel). The maximal Doppler factor is 31.86 at 1998.77 (at $r_{n}$=0.88 mas), coincident with the minimal viewing angle $\theta$=$1.14^{\circ}$ and minimal apparent speed $\beta_{app}$=11.44.

• •

  In order to compare our results with those obtained by Jorstad et al. (Jo05 (2005)) we show the apparent speed $\beta_{app}$ as a function of core distance $r_{n}$ for knot C9 in Figure 5. It exhibits a pronounced oscillating structure with two maxima (both $\beta_{app,max}$=17.91) at core distances $r_{n}$=0.33 mas and 1.77 mas, and a minimum $\beta_{app,min}$=11.4 at $r_{n}$=0.88 mas, corresponding to a minimal viewing angle $\theta_{min}$=$1.14^{\circ}$. In addition, the decreasing part (in the range of $r_{n}$$\sim$1.2–1.8 mas) of the 43 GHz light curve measured by Jorstad et al. (Jo05 (2005)) for knot C9 can be well fitted by the Doppler boosting profile ${S_{int}}[\delta(t)/\delta_{max}]^{3+\alpha}$ derived as a function of core distance in this paper as shown in Figure 6. Thus, we unexpectedly find that the model-simulated results obtained within the framework of our precessing nozzle scenario are fully consistent with the phenomena observed and analyzed by Jorstad et al. (Jo05 (2005)) as follows:
  
  The change in apparent speed and the twisted trajectory are also pronounced for C9. Although the historically extrapolated position of C9 (using the average apparent speed of 15.6c) agrees well with the the measured position at epoch 1996.81 (Ros et al. 2000). It decelerates significantly from $\sim$20c near the core to $\sim$10c at 1 mas and then accelerates beyond 1.5 mas. The deceleration is accompanied by brightening of the knot and rotation of the EVPA by $\sim$$50^{\circ}$, making it almost perpendicular to the jet direction.
  
  It is worth noticing that the rapid variations in its intrinsic flux density (or the steepening of the local spectrum at 43 GHz) resulted in the peaking stage of the flare not observed at 43GHz. Detailed model-fits to the light curves measured by Schinzel (2011a ) at 15, 22 and 43 GHz will be presented below in Figure 7.

• •

  It should be emphasized that the bulk Lorentz factor $\Gamma(t)$, viewing angle $\theta(t)$ and Doppler factor $\delta(t)$ as functions of time were directly derived from the model fitting of the kinematics only, and the Doppler boosting effect ($[\delta(t)]^{3+\alpha}$) is purely predicted by the precesssing nozzle scenario. Thus, whether the predicted Doppler boosting profile ${S_{int}}[\delta(t)/\delta_{max}]^{3+\alpha}$ can explain the flux evolution of superluminal knots in 3C345 would be a significant test for our precessing nozzle scenario. In the model-simulation the intrinsic flux density $S_{int}$ were adopted to be 3.83, 2.58 and 1.65 $\mu$Jy for 15, 22 and 43 GHz, respectively. The spectral index was adopted as $\alpha$(15-22-43GHz)=0.80 (${S_{\nu}}\,{\propto}\,{{\nu}^{-\alpha}}$).

• •

  The light curves observed at 15, 22 and 43 GHz for knot C9 ¹¹1Data are adopted from Schinzel (2011a ) only. are shown in Figure 7 (left panels). They exhibit very complicated structures with flux density fluctuations on short time-scales of $\sim$0.5 years, but the Doppler boosting effect having a time scale of $\sim$5 years (1996.5–2001.5) as predicted by our model-simulation can be explicitly discerned: (1) During both the rising ($\sim$1997.0–1998.5) and decaying ($\sim$1999.5–2001.0) stages the flux densities measured at 15 GHz are well fitted by the Doppler boosting profiles (solid black lines in Fig.7, right panels). Thus the model-predicted Doppler-boosting profiles can be regarded as correctly determined with its maximum at 1998.77, where the apparent speed and viewing angle are at minimum; (2) At both 22 and 43 GHz there are a few data-points are also fitted by the profiles during the rising satge ($\sim$1997–1998); (3) The low flux densities measured at 43 GHz during 1998–1999 and measured at 22 GHz during the decaying stage (1999.5–2000.5) are obviously due to the intrinsic variations of knot C9. For example, during 1998–1999 the measured spectral index $\alpha$(15-43GHz)$\sim$1.8 much larger than that ($\alpha$(15–43GHz)$\sim$0.80) during both rising and decaying stages; (4) The fluctuations on time-scales of 0.5–1 years in flux density measured at 15, 22 and 43 GHz during 2001–2008 can also be interpreted in terms of the intrinsic variations of knot C9, because the Doppler boosting effect has diminished during that period. Thus in order to fully explain the flux evolution of superluminal components in 3C345 both intrinsic variations and Doppler boosting effect should be taken into account; (5) In Figure 7 (right panels) the dashed red and blue lines show the profiles deviating from the model-derived Doppler boosting profiles (solid lines in black) by $\pm$0.15 Jy (for 43 GHz) and 0.30 Jy (for 22 and 15 GHz). It can be seen that most of the measured data-points fall between the two lines, indicating that the model-fits to the measured light curves are successful.

The brief summary described above for the midel-fitting results of the kinematic behavior and flux evolution for the superluminal components of jet-A in 3C345 explicitly shows that our processing nozzle scenario is most appropriate and valid to explain the kinematic, dynamic and emission properties observed in the superluminal components of blazar 3C345.
In the following we shall investigate the flux evolution associated with the Doppler boosting effect for five superluminal knots (C19, C20, C21, B5 and B7) belonging to the jet-B of 3C345. As in the previous work (Qian 2022b ) the observational data are adopted from Schinzel (2011a ) and Jorstad et al. (Jo05 (2005)).

According to the precessing nozzle scenario proposed for 3C345 (Qian 2022a , 2022b ) the superluminal knots could be separated into two groups (group-A and group-B) which were hypothetically assumed to be related to a double-jet structure (jet-A plus jet-B), because the two jets were found to be precessing with the same period of 7.3 yr, but having different patterns of precessing common trajectory.
The geometry and model-parameters describing the precessing nozzle scenario proposed for 3C345 have been discussed in detail in the previous works (Qian 2022a , 2022b ). Here we only list the main points of the scenario related to the investigation of the kinematics of the superluminal knots of group-B as follows:

(1) Jet-axis

The direction of the axis of jet-B in space, around which superluminal knots of group-B move, is defined by parameters $\epsilon$ and $\psi$. The jet-axis is assumed as:

where $\epsilon$=$1.5^{\circ}$, $\psi$=$12.0^{\circ}$, $\zeta$=1.0 and p=1.34$\times$$10^{-4}$ are adopted.

(2) Amplitude and phase of trajectory

The amplitude and phase of the trajectory of superluminal knots are defined as:

where $A_{0}$ represents the amplitude coefficient of the common trajectory pattern. $A_{0}$=1.09 mas and $Z_{1}$=396 mas are adopted. ${\phi}_{0}$ is the precession phase of an individual knot, which is related to its ejection time $t_{0}$:

where $T_{0}$ is the precession period of the jet nozzle.
The distribution of the precessing trajectories of the knots is shown in Figure 8, displaying a bunch of straight-line trajectories, which is completely different from that of the curved trajectories of the knots in jet-A.

(3) Doppler boosting effect

In order to investigate the relation between the flux evolution of superluminal components and their Doppler boosting effect during their accelerated/decelerated motion and model-fitting of the measured light curves, we define the Doppler boosting profile as ${S_{int}}[\delta(t)]^{3+\alpha}$ or $[\delta(t)/{\delta_{max}}]^{3+\alpha}$, where $S_{int}$–the intrinsic flux density of the knot, $\delta(t)$–Doppler factor and $\alpha$–spectral index (in the frequency range of 10.7–43 GHz). The observed flux density $S_{obs}(\nu,t)$ of the superluminal components is Doppler-boosted:

$S_{int}(\nu)$=$S_{0}$ ${({\nu/{\nu_{0}})}^{-\alpha}}$, $S_{0}$–the flux density at the fiducial frequency $\nu_{0}$. ²²2Here in this paper we assume $S_{int}$ and $\alpha$ are constants, not depending on time and frequency. In general case both $S_{int}$ and $\alpha$ could be defined as functions of frequency and time. Thus flux evolution of superluminal knots would be associated with more complex Doppler-boosting effect.

(4) Cosmological model

We will apply the concordant cosmological model (Spergel et al. Sp03 (2003), Hogg Ho99 (1999)) with $\Omega_{\lambda}$=0.73 and $\Omega_{m}$=0.27, and $H_{0}$=71km$s^{-1}$${Mpc}^{-1}$. Thus the luminosity distance of 3C345 $D_{L}$=3.49 Gpc, the angular-diameter distance $D_{a}$=1.37 Gpc, 1 mas=6.65 pc and 1 mas/yr=34.6 c. 1 c is equivalent to an angular speed 0.046 mas/yr in the rest frame.

According to the precessing nozzle scenario for jet-B, knot C19 has its ejection time $t_{0}$=2004.85 and corresponding precession phase $\phi_{0}$=1.77+2$\pi$. Its traveled distance Z(t) and parameters $\epsilon(t)$ and $\psi(t)$ are model-fitted as shown in Figure 9. Before 2008.87 $\epsilon$=$1.50^{\circ}$ and $\psi$=$12.0^{\circ}$, knot C9 moved along the precessing common trajectory. The cooresponding traveled distance Z=14.6 mas=97.1 pc. After 2008.87 (core separation $r_{n}{>}$0.26 mas or $X_{n}{>}$0.25 mas) $\epsilon$ and $\psi$ started to change and knot C19 started to move along its own individual track, deviating from the precessing common track.

According to the precessing nozzzle scenario for jet-B, the ejection time of knot C20 is $t_{0}$=2007.68 and corresponding precession phase $\phi_{0}$=4.20+2$\pi$. Its traveled distance Z(t) and parameters $\epsilon(t)$ and $\psi(t)$ were model-fitted as functions of time as showm in Figure 14. Before 2008.66 $\epsilon$=$1.50^{\circ}$ and $\psi$=$12.0^{\circ}$, knot C20 moved along the precessing common trajectory. The corresponding traveled distance Z=4.0 mas=26.6 pc. After 2008.66 (core separation $r_{n}{>}$0.14 mas or coordinate $X_{n}{>}0.13$ mas) parameter $\epsilon$ started to change (parameter $\psi$ started to change after 2009.4) and its motion started to follow its own individual track, deviating from the precessing common track.

According to the precessing nozzle secnario for jet-B the ejection time of knot C21 is $t_{0}$=2007.32 and the corresponding precession phase $\phi_{0}$=3.90+2$\pi$. Its traveled distance Z(t) and modeled parameters $\epsilon(t)$ and $\psi(t)$ as continuous functions of time are shown in Figure 19. Before 2009.08 $\epsilon$=$1.50^{\circ}$ and $\psi$=$12.0^{\circ}$, knot C21 moved along the precessing common track, which extends to the traveled distance Z=3.40 mas=22.6 pc. After 2009.08 ($r_{n}{\geq}$0.11 mas or $X_{n}{\geq}$0.10 mas) parameter $\epsilon$ started to change (parameter $\psi$ started to change after 2009.40) and knot C21 started to follow its own individual track, deviating from the precessing common track. That is, the transition between the precessing common track and its individual track occurred at $X_{n}{\sim}$0.10 mas.

According to the precessing nozzle scenario for jet-B the ejection time of knot B5 is $t_{0}$=2010.48 and the corresponding precession phase $\phi_{0}$=0.33+4$\pi$. Its traveled distance Z(t) and modeled parameters $\epsilon(t)$ and $\psi(t)$ as functions of time are shown in Figure 24. Before 2010.88 ($X_{n}{\leq}$0.015 mas) $\epsilon$=$1.50^{\circ}$ and $\psi$=$12.0^{\circ}$ knot B5 moved along the precessing common trajectory which extends to the traveled distance Z=0.60 mas=4.0 pc. After that epoch both $\epsilon$ and $\psi$ started to change and knot B5 started to follow its own individual track, deviating from the precessing common track. That is, the transition between the precessing common track and its individual track occurred very near to the core (at $r_{n}{\sim}{X_{n}}$=0.015 mas) and the Doppler-boosting effect occurred completely in the outer jet regions.

According to the precessing nozzle scenario for jet-B the ejection time of knot B7 is $t_{0}$=2011.60 and its corresponding precession phase $\phi_{0}$=1.30+4$\pi$. Its traveled distance Z(t) and the modeled parameters $\epsilon(t)$ and $\psi(t)$ as functions of time are shown in Figure 29. Before 2012.60 ($X_{n}{\leq}$0.036 mas) $\epsilon$=$1.50^{\circ}$ and $\psi$=$12.0^{\circ}$ knot B7 moved along the precessing common trajectory, extending to the traveled distance Z=2.0 mas=13.3 pc. After 2012.60 both parameters started to change and knot B7 started to move along its own individual track, deviating from the precessing common track. That is, the transition from the common track to its individual track occurred at ${r_{n}}{\sim}{X_{n}}$=0.036 mas. Thus its flaring associated with the Doppler boosting effect completely occurred in the outer jet region.

Based on the precessing nozzle scenario proposed for jet-B in 3C345 (Qian 2022a ) we have model-fitted the flux evolution of five superluminal components (C19, C20, C21, B5 and B7) associated with their kinematics (accelerated/decelerated motion). Combining with the results for the five superluminal knots (C4, C5, C9, C10 and C22) of jet-A (Qian 2022b ), we have interpreted the flux evolution as caused by the Doppler boosting effect for ten superluminal knots in 3C345.
In fact, similar studies can be made for more superluminal knots in 3C345: e.g., knots C7, C11, C12 and C23 of jet-A and C15–C18, B8, B11 and B12 of jet-B.
In addition, we have also applied our precessing nozzle scenario to investigate the flux evolution of superluminal components associated with their Doppler boosting effect in QSO B1308+326 (Qian 2023a ) and blazars 3C454.3 (Qian 2023b ) and 3C279 (Qian 2022b ). In all these cases the flux evolution of the superluminal components can be interpreted in terms of their Doppler boosting effect combined with their intrinsic flux variations.
We would like to note that there are signicant differences in the application of the precessing nozzle scenario to QSOs and blazars. For QSOs [e.g., for B1308+326 (Qian et al. Qi17 (2017), Qian 2023a ), PG1302-102 (Qian et al. 2018a ) and NRAO 150 (Qian Qi16 (2016))] the precessing nozzle scenario has been applied to study the model-fitting of the kinematics and flux evolution of superluminal components ejected from a single precessing jet, while for blazars [e.g., 3C345 (Qian 2022a , 2022b and this paper); 3C454.3 (Qian et al. Qi07 (2007), Qi14 (2014), Qi21 (2021), Qian 2023b ); 3C279 (Qian Qi11 (2011), Qi12 (2012), Qi13 (2013), 2022a ); OJ287 (Qian Qi15 (2015), 2018b , Qi19 (2019), Qi20 (2020))] a double jet structure (jet-A plus jet-B) has been hypothetically introduced to study the kinematics and flux evolution of their superluminal components, because their knots could be separated into two groups (group-A and group-B) which need different patterns of precessing common trajectory to fit their kinematics, respectively.
It is worth noting the fact: Analysing the observational data in earlier years (i.e. the observational data on the knots of group-A only), Klare (Kl05 (2005)), Lobanov & Roland (Lo05 (2005)) and Qian et al. (Qi09 (2009)) suggested 3C345 having a precessing jet, while analysing the whole observational data on the knots of both group-A and group-B, Schinzel et al. (2011a ) suggested no precession of the jet in 3C345. Thus the separation of the superluminal knots into two groups may be a plausible assumption for investigating the jet precession in 3C345, otherwise we would have to relinquish the assumption of jet precession for 3C345.
The full interpretation of the kinematics and flux evolution of the superluminal components in 3C345 demonstrates that our precessing nozzle scenario proposed for blazar 3C345 can interpret the kinematic, dynamic and emission properties of its superlumiunal components, although the assumption of a double jet structure is only a working hypothesis.
We have been trying to propose a scenario with a single precessing jet, as an significant alternative, to interpret the kinematics of the superluminal knots in 3C345 as a whole, because there seems a possibility that the knots of group-A and group-B can be combined into one group. For example, we may assume that the trajectory-pattern transition of the knots in the jet-B could occur at core distances $r_{n}{<}$0.1 mas and their innermost tracks could connect with the precessing common trajectory of jet-A at core distances $r_{n}{<}$0.10–0.05 mas. In such a scenario the observed tracks of the knots of group-B are only the outer trajectories beyond their trajectory-transitions, while their innermost tracks within the transitions ($r_{n}{\leq}$0.05–0.1 mas) have not been observed. The trajectory-transition observed in the core-distance range of $\sim$0.1–0.4 mas in knot C8 (Qian 2022a ) is an extremely instructive example. In this case the precession of the jet may be related to the precession of the disk around the black hole in the nucleus of 3C345. This kind of scenario with a single precessing jet nozzle should be tested by the VLBI observations with higher resolutions in the future.