define the distributions of physical parameters for all objects.

Instead of analyzing tentative distributions of physical parameters of jets in
our sample, we instead focus on “representative” values. For each reliable
knot, we calculate a metric of overall uncertainty,
$\xi\equiv\sqrt{\sigma_{\delta_{\text{var}}}^{2}+\sigma_{\beta_{\text{app}}}^{2}}$,
where $\sigma_{\delta_{\text{var}}}$ and $\sigma_{\beta_{\text{app}}}$ are the
uncertainties in the variability Doppler factor and proper motion. We focus on
these two parameters, since they factor directly into the determination of
$\Gamma$ and $\Theta_{\circ}$ (both given in Eq. 6), as well as the opening
semi-angle (described below). We then choose as “representative” the knot from
the reliable sample of each jet that has the optimal combination of relatively
high apparent speed and relatively low value of $\xi$. We adopt the criterion
$\xi/(\beta+\delta_{\text{var}})<0.15$ to select knots with low uncertainty.
Choosing the highest apparent speed with a relatively low uncertainty ensures
that the derived physical conditions are consistent with the most extreme, yet
robust, behavior of the jet. The statistics of the chosen knots, one per
source, would not have differed significantly across the sample had we instead
chosen the knots with the highest established Doppler factor and lowest
uncertainty. The typical knots are used to determine differences between the
distributions of parameters for each subclass in Figures 8, 19, 20, and 21.

For each jet, we follow Jorstad et al. (2017) in calculating the opening semi-
angle in two ways. Method A is based on the relation between the intrinsic and
projected opening angles:

$\theta_{\circ}^{\text{A}}=\theta_{\text{p}}\sin{\langle\Theta_{\circ}\rangle}$
(7)

(Jorstad et al., 2005). Here, $\theta_{\text{p}}$ is the projected opening
semi-angle of the jet, and $\langle\Theta_{\circ}\rangle$ is the
representative viewing angle of the jet. We define $\theta_{\text{p}}$ as
twice the standard deviation of the knot position angles about a fixed value,
regardless of whether the jet position angle is found to vary, as described in
§3.2. We adopt the average uncertainties of
$\langle\Theta_{\text{jet}}\rangle$, $\langle\sigma_{\Theta}\rangle$ as the
uncertainty of $\theta_{\text{p}}$.

Method B applies to jets with multiple reliable knots. We take the viewing
angles of the fastest and slowest knot with low values of $\xi$ (see above) to
be the minimum and maximum viewing angles, $\Theta_{\circ}^{\text{min}}$ and
$\Theta_{\circ}^{\text{max}}$. We then calculate the intrinsic opening semi-
angle as

$\theta_{\circ}^{\text{B}}=\frac{1}{2}\left(\Theta_{\circ}^{\text{max}}-\Theta_{\circ}^{\text{min}}\right).$
(8)

This definition would be appropriate if the difference in apparent speeds were
due mainly to different viewing angles of the knots owing to the brightness
centroids of the knots being located at different distances from the jet axis.

Table 13 gives the typical values of the physical parameters of knots in each
source as follows: 1—name of the source; 2—luminosity distance,
$D_{\text{Gpc}}$, in Gpc; 3—the knot selected as the representative high-speed
knot; 4—Doppler factor of the representative knot, $\delta_{\text{var}}$, and
its uncertainty; 5—Lorentz factor of the representative knot, $\Gamma$, and
its uncertainty; 6—viewing angle of the representative knot, $\Theta_{\circ}$,
and its uncertainty; 7—projected viewing opening semi-angle,
$\theta_{\text{p}}$, of the jet and its uncertainty; 8—opening semi-angle of
the jet, $\theta_{\circ}^{\text{A}}$, derived using method A, and its
uncertainty; 9—the knot selected as the representative low-speed knot; and
10—opening semi-angle of the jet, $\theta_{\circ}^{\text{B}}$, derived using
method B, and its uncertainty.

Table 13: Typical Physical Parameters of Jets

Source | $D_{\text{Gpc}}$ | Fast Knot | $\delta_{\text{var}}$ | $\Gamma$ | $\Theta_{\circ}$ | $\theta_{\text{p}}$ | $\theta_{\circ}^{\text{A}}$ | Slow Knot | $\theta_{\circ}^{\text{B}}$
---|---|---|---|---|---|---|---|---|---
| [Gpc] |  |  |  | [deg] | [deg] | [deg] |  | [deg]
0219+428 | 2.458 | B4 | $12.3\pm 2.8$ | $22.8\pm 3.5$ | $\phantom{0}4.1\pm 0.5$ | $18.0\pm\phantom{0}5.5$ | $\phantom{0}1.3\pm 0.4$ | B5 | $\phantom{0}0.4\pm 0.4$
0235+164 | 6.121 | B5 | $37.9\pm 5.4$ | $20.0\pm 2.5$ | $\phantom{0}0.7\pm 0.2$ | $13.2\pm\phantom{0}6.8$ | $\phantom{0}0.2\pm 0.1$ | B3 | $\phantom{0}1.2\pm 0.3$
0316+413aaEstimates of the parameters are discussed in $\S$5.5. | 0.077 | $\ldots$ | $\sim 6.9$ | $\sim 4.8$ | $\sim 7.2$ | $19.2\pm 0.4$ | $\sim 2.4$ | $\ldots$ | $\ldots$
0336–019 | 5.422 | B5 | $\phantom{0}9.0\pm 4.1$ | $\phantom{0}9.6\pm 0.8$ | $\phantom{0}6.3\pm 2.8$ | $12.8\pm\phantom{0}4.6$ | $\phantom{0}1.4\pm 0.8$ | B3 | $\phantom{0}0.8\pm 1.6$
0415+379 | 0.215 | K22 | $\phantom{0}2.8\pm 0.2$ | $\phantom{0}8.5\pm 0.5$ | $15.2\pm 0.4$ | $\phantom{0}9.8\pm\phantom{0}2.2$ | $\phantom{0}2.6\pm 0.6$ | K5 | $11.5\pm 1.3$
0420–014 | 5.928 | B3 | $25.2\pm 3.4$ | $21.2\pm 0.6$ | $\phantom{0}2.2\pm 0.4$ | $35.2\pm\phantom{0}3.7$ | $\phantom{0}1.4\pm 0.3$ | B5 | $\phantom{0}2.2\pm 0.6$
0430+052 | 0.145 | C15 | $\phantom{0}1.9\pm 0.4$ | $10.0\pm 1.8$ | $17.9\pm 0.8$ | $\phantom{0}6.4\pm\phantom{0}2.9$ | $\phantom{0}2.0\pm 0.9$ | C4 | $\phantom{0}4.0\pm 0.6$
0528+134 | 16.109 | C2 | $17.3\pm 2.4$ | $11.7\pm 0.8$ | $\phantom{0}2.9\pm 0.6$ | $26.6\pm\phantom{0}6.0$ | $\phantom{0}1.4\pm 0.4$ | B3 | $\phantom{0}0.4\pm 0.4$
0716+714 | 1.553 | B13 | $23.3\pm 3.6$ | $17.1\pm 1.1$ | $\phantom{0}2.3\pm 0.4$ | $38.6\pm\phantom{0}3.9$ | $\phantom{0}1.5\pm 0.4$ | B4 | $\phantom{0}0.7\pm 0.7$
0735+178 | 2.327 | B3 | $12.2\pm 3.3$ | $\phantom{0}7.2\pm 1.4$ | $\phantom{0}3.4\pm 1.6$ | $27.6\pm 10.7$ | $\phantom{0}1.6\pm 1.0$ | B4 | $\phantom{0}1.1\pm 0.8$
0827+243 | 6.129 | B6 | $10.8\pm 1.4$ | $38.0\pm 3.9$ | $\phantom{0}3.7\pm 0.2$ | $16.2\pm\phantom{0}5.8$ | $\phantom{0}1.0\pm 0.4$ | B7 | $\phantom{0}0.1\pm 0.3$
0829+046 | 0.840 | B5 | $19.6\pm 6.0$ | $13.6\pm 1.9$ | $\phantom{0}2.6\pm 1.2$ | $21.2\pm\phantom{0}5.7$ | $\phantom{0}1.0\pm 0.5$ | B4 | $\phantom{0}4.4\pm 3.0$
0836+710 | 17.185 | B3 | $13.5\pm 0.7$ | $17.8\pm 0.5$ | $\phantom{0}4.1\pm 0.2$ | $18.0\pm\phantom{0}3.3$ | $\phantom{0}1.3\pm 0.2$ | B6 | $\phantom{0}0.1\pm 0.3$
0851+202 | 1.589 | B12 | $\phantom{0}6.4\pm 1.9$ | $11.8\pm 1.7$ | $\phantom{0}7.9\pm 1.3$ | $24.6\pm\phantom{0}1.2$ | $\phantom{0}3.4\pm 0.6$ | B6 | $\phantom{0}2.5\pm 1.5$
0954+658 | 1.969 | B14 | $\phantom{0}9.8\pm 1.6$ | $38.8\pm 4.7$ | $\phantom{0}3.9\pm 0.2$ | $19.4\pm\phantom{0}2.1$ | $\phantom{0}1.3\pm 0.2$ | B19 | $\phantom{0}5.1\pm 0.8$
1055+018 | 5.722 | B3 | $16.9\pm 3.9$ | $12.8\pm 1.0$ | $\phantom{0}3.2\pm 1.0$ | $36.8\pm\phantom{0}6.5$ | $\phantom{0}2.1\pm 0.7$ | B1 | $\phantom{0}1.4\pm 1.1$
1101+384aaEstimates of the parameters are discussed in $\S$5.5. | 0.131 | $\ldots$ | $\sim 65.5$ | $\sim 16.4$ | $\sim 1.4$ | $27.8\pm 14.2$ | $\sim 0.7$ | $\ldots$ | $\ldots$
1127–145 | 8.142 | C1 | $13.6\pm 2.5$ | $12.1\pm 0.5$ | $\phantom{0}4.2\pm 0.9$ | $20.8\pm\phantom{0}4.3$ | $\phantom{0}1.5\pm 0.4$ | B2 | $\phantom{0}3.8\pm 0.6$
1156+295 | 4.446 | B2 | $15.0\pm 2.8$ | $10.1\pm 1.0$ | $\phantom{0}3.4\pm 1.0$ | $22.2\pm\phantom{0}3.8$ | $\phantom{0}1.3\pm 0.4$ | B3 | $\phantom{0}0.2\pm 0.5$
1219+285 | 0.470 | B1 | $\phantom{0}4.5\pm 1.1$ | $\phantom{0}5.3\pm 0.3$ | $12.6\pm 2.6$ | $16.0\pm 10.6$ | $\phantom{0}3.5\pm 2.4$ | B3 | $\phantom{0}4.6\pm 4.1$
1222+216 | 2.379 | B5 | $\phantom{0}7.0\pm 0.6$ | $34.7\pm 2.7$ | $\phantom{0}4.9\pm 0.1$ | $42.8\pm\phantom{0}1.9$ | $\phantom{0}3.7\pm 0.2$ | B3 | $\phantom{0}1.3\pm 0.1$
1226+023 | 0.755 | B2 | $\phantom{0}3.2\pm 0.3$ | $15.7\pm 1.1$ | $10.9\pm 0.2$ | $10.8\pm\phantom{0}0.7$ | $\phantom{0}2.0\pm 0.1$ | B16 | $\phantom{0}1.2\pm 0.3$
1253–055 | 3.080 | C35 | $\phantom{0}8.2\pm 0.9$ | $59.6\pm 5.6$ | $\phantom{0}3.5\pm 0.1$ | $36.8\pm\phantom{0}1.0$ | $\phantom{0}2.3\pm 0.1$ | C31 | $\phantom{0}1.0\pm 0.1$
1308+326 | 6.591 | B3 | $18.5\pm 2.9$ | $14.6\pm 0.7$ | $\phantom{0}3.0\pm 0.6$ | $25.8\pm\phantom{0}4.7$ | $\phantom{0}1.3\pm 0.4$ | B1 | $\phantom{0}7.9\pm 1.5$
1406–076 | 10.854 | B4 | $18.4\pm 2.9$ | $18.0\pm 0.7$ | $\phantom{0}3.1\pm 0.5$ | $29.0\pm\phantom{0}7.2$ | $\phantom{0}1.6\pm 0.5$ | B2 | $\phantom{0}3.1\pm 0.7$
1510–089 | 1.919 | B1 | $37.0\pm 2.7$ | $25.6\pm 1.1$ | $\phantom{0}1.4\pm 0.2$ | $16.0\pm\phantom{0}2.8$ | $\phantom{0}0.4\pm 0.1$ | B10 | $\phantom{0}1.6\pm 0.6$
1611+343bbfootnotemark:  | 10.009 | B1 | $25.7\pm 3.9$ | $14.0\pm 1.8$ | $\phantom{0}1.2\pm 0.4$ | $26.2\pm\phantom{0}3.3$ | $\phantom{0}0.5\pm 0.2$ | $\ldots$ | $\ldots$
1622–297 | 5.133 | B5 | $21.9\pm 2.8$ | $18.0\pm 0.6$ | $\phantom{0}2.6\pm 0.4$ | $23.4\pm\phantom{0}3.3$ | $\phantom{0}1.0\pm 0.2$ | B4 | $\phantom{0}0.7\pm 0.6$
1633+382 | 13.775 | B3 | $23.8\pm 3.9$ | $13.6\pm 1.8$ | $\phantom{0}1.6\pm 0.5$ | $18.2\pm\phantom{0}6.1$ | $\phantom{0}0.5\pm 0.2$ | B5 | $\phantom{0}0.4\pm 0.3$
1641+399 | 3.480 | B4 | $13.2\pm 1.7$ | $20.8\pm 1.0$ | $\phantom{0}4.1\pm 0.3$ | $17.2\pm\phantom{0}2.6$ | $\phantom{0}1.2\pm 0.2$ | B1 | $\phantom{0}8.5\pm 0.3$
1652+398aaEstimates of the parameters are discussed in $\S$5.5. | 0.149 | $\ldots$ | $\sim 31.5$ | $\sim 16.4$ | $\sim 0.7$ | $53.8\pm 11.8$ | $\sim 0.7$ | $\ldots$ | $\ldots$
1730-130 | 5.817 | B6 | $14.8\pm 3.7$ | $21.3\pm 1.9$ | $\phantom{0}3.7\pm 0.7$ | $27.2\pm\phantom{0}1.4$ | $\phantom{0}1.8\pm 0.3$ | B3 | $\phantom{0}2.0\pm 0.5$
1749+096 | 1.685 | B7 | $12.2\pm 1.9$ | $10.8\pm 0.7$ | $\phantom{0}4.6\pm 0.8$ | $31.0\pm\phantom{0}2.3$ | $\phantom{0}2.5\pm 0.5$ | B5 | $\phantom{0}1.9\pm 0.5$
1959+650aaEstimates of the parameters are discussed in $\S$5.5. | 0.209 | $\ldots$ | $\sim 30.3$ | $\sim 16.4$ | $\sim 1.0$ | $38.2\pm 12.9$ | $\sim 0.7$ | $\ldots$ | $\ldots$
2200+420 | 0.311 | B8 | $\phantom{0}4.3\pm 1.2$ | $14.6\pm 3.0$ | $\phantom{0}9.5\pm 0.8$ | $\phantom{0}9.0\pm\phantom{0}1.8$ | $\phantom{0}1.5\pm 0.3$ | B6 | $\phantom{0}0.1\pm 2.0$
2223-052 | 10.053 | B4 | $16.3\pm 2.3$ | $10.3\pm 1.0$ | $\phantom{0}2.3\pm 0.7$ | $29.4\pm\phantom{0}3.2$ | $\phantom{0}1.5\pm 0.4$ | B2 | $\phantom{0}0.7\pm 1.1$
2230+114 | 6.911 | B4 | $37.5\pm 6.7$ | $23.6\pm 2.6$ | $\phantom{0}1.2\pm 0.4$ | $20.2\pm\phantom{0}3.6$ | $\phantom{0}0.4\pm 0.1$ | B2 | $\phantom{0}0.8\pm 0.2$
2251+158 | 5.477 | B10 | $28.8\pm 2.2$ | $15.7\pm 1.0$ | $\phantom{0}1.1\pm 0.2$ | $24.4\pm\phantom{0}1.6$ | $\phantom{0}0.5\pm 0.1$ | B9 | $\phantom{0}0.1\pm 0.2$

bbfootnotemark:

The source contains only one “reliable” knot.

Figure 23 presents a comparison between $\theta_{\circ}^{\text{A}}$ and
$\theta_{\circ}^{\text{B}}$. For the majority of sources, there is good
agreement (within the uncertainty) between the two opening semi-angles derived
via the different methods. There are a small number of sources for which
$\theta_{\circ}^{\text{B}}>\theta_{\circ}^{\text{A}}$, in contrast with the
large number of sources obeying this uncertainty found in Jorstad et al.
(2017). This is likely due to the new definition of
$\theta_{\circ}^{\text{B}}$, in which we take the opening semi-angle to be the
difference between the fastest and slowest high-significance knot. One can
explain this discrepancy for sources that have very small viewing angles,
e.g., the BL 0235+164, where the knots have no preferential direction on the
plane of the sky. Based on geometrical arguments, this cannot be the case for
all sources in our sample.

Figure 23: Comparison between opening semi-angles, $\theta_{\circ}$ of sample
sources, derived using methods A and B (see text for details). The gray dotted
line indicates $\theta_{\circ}^{\text{A}}=\theta_{\circ}^{\text{B}}$.

Larger values of $\theta_{\circ}^{\text{B}}$ are to be expected if the jets of
blazars are structured in a “spine-sheath” manner (e.g., Pelletier & Roland,
1989; Sol et al., 1989; Celotti et al., 2001; Ghisellini et al., 2005;
Tavecchio & Ghisellini, 2008; D’Arcangelo et al., 2009; Xie et al., 2012;
Mimica et al., 2015). Such a structure would consist of two flows: a faster,
lower-density component streaming along the central axis (spine), and a
slower, denser component along the edges of the jet. As the Doppler beaming
would be stronger for the faster flow if viewed at a narrow angle, the spine
would be expected to dominate the overall jet emission, while the sheath would
be more intense at wider viewing angles (e.g., RGs, Bruni et al., 2021).
Evidence for a spine-sheath structure in several blazar jets comes from linear
polarization of VLBI images (e.g., Pushkarev et al., 2005; Asada et al., 2010;
Gabuzda et al., 2014; Mertens et al., 2016), especially when including the
space-based _RadioAstron_ antenna in the array (Bruni et al., 2021).

It is possible that most knots seen in our images flow down the spine of a
jet, (propelled by the rotational energy of the black hole, Blandford &
Znajek, 1977), while a relatively small number of knots form along the slower
sheath (e.g., from a magnetically driven wind from the accretion disk,
Blandford & Payne, 1982). Several sources in our sample have been inferred to
have a spine-sheath structure, such as 0316+413 (Giovannini et al., 2018),
1055+018 (Attridge et al., 1999; MacDonald et al., 2017), 1226+023 (MacDonald
et al., 2017; Bruni et al., 2021), 1510–089 (MacDonald et al., 2015), 1652+398
(Giroletti et al., 2004; Pushkarev et al., 2005), 0836+710, 1253–055, and
2230+114 (MacDonald et al., 2017). Neither 1652+398 nor 0316+413 have enough
reliable superluminal knots for us to determine their jet structure (see
$\S$5.5 to obtain estimates of their physical parameters). The sources
1055+018 and 1226+023 both have
$\theta_{\circ}^{\text{A}}>\theta_{\circ}^{\text{B}}$. The spine-sheath
structure in 1055+018 was identified from the polarization of the emission
rather than the total intensity that we have analyzed here. The inferred
layered structure in 1226+023 was based on images with extremely high spatial
resolution at lower frequencies (1.6 and 4.8 GHz); it may be the case that we
have not observed a slow knot in the sheath of this source. Conclusive
evidence for spine-sheath structure in the jets of our sample can be found
through an analysis of stacked images and maps of polarized intensity (as has
been done for several sources in MacDonald et al., 2015, 2017), but is beyond
the scope of this work.

###  5.4 Comparison with Other Physical Parameter Estimations

Several methods exist that use a variety of data at different frequencies to
estimate the Doppler factor in blazar jets and subsequently infer the physical
parameters of the jets themselves (e.g., Ghisellini et al., 1993; Lähteenmäki
& Valtaoja, 1999; Jorstad et al., 2005; Fan et al., 2013, 2014; Chen, 2018;
Lister et al., 2019; Zhang et al., 2020). Reviews of the methods and their
results can be found in, e.g., Liodakis & Pavlidou (2015); Liodakis et al.
(2017a, b). Here we compare a few of these results with those inferred from
our piece-wise kinematic fitting.

In order to quantify whether the distributions of the physical parameters from
our work differ from those of others, we employ the Wilcoxon rank-sum
test101010The KS test, used elsewhere in this work, is sensitive to
differences in the general shape of two distributions (i.e., dispersion,
skewness, etc.), whereas the WRS test is sensitive to differences in location
(e.g., the median) of the two samples. (WRS, or, alternatively, the Mann-
Whitney $U$-test; Mann & Whitney, 1947). Since we wish to test whether
estimation of the physical parameters depends on the method employed, we
assume that the parameters are drawn from the same distribution, and thus the
WRS test can indicate whether there is a systematic bias in the methodology.
The WRS test operates under the null hypothesis that two distributions are
drawn from the same sample, and for this test smaller $p$-values indicate a
larger likelihood that the distributions are different. We consider two data
sets to be different if their WRS $p$-value $<0.005$ ($\sim 3\sigma$
confidence level). Figure 24 shows a comparison between the results of this
work and those of Jorstad et al. (2017) and Liodakis et al. (2018), to
accompany the indicated $p$-values.

Figure 24: Comparisons of the physical parameters of blazar jets derived in
this work ($x$-axis) with those derived in Jorstad et al. (2017) ($y$-axis,
filled circles) and Liodakis et al. (2018) ($y$-axis, open circles). A gray
dotted line showing a 1:1 relation is shown in each panel, and the $p$-value
of a WRS test between the derived values (from Table 14) is given in the
legend.

The most direct comparison can be made with Jorstad et al. (2017), as the
current work shares data, methodology, and underlying physical assumptions. In
this comparison, we ignore the three sources with no reliable knots for
deriving values of physical parameters. The majority of our estimated values
of ($\delta_{\text{var}},\ \Gamma,\ \Theta_{\circ},\ \theta_{\text{p}},\
\theta_{\circ}^{\text{A}},$ and $\theta_{\circ}^{\text{B}}$) agree within the
sum of the two estimated $1\sigma$ uncertainties. Table 14 gives $p$-values
between the “representative” physical parameters given in Table 13 and the
average values from Jorstad et al. (2017). For most parameters, there is no
statistically significant difference in the distributions. However, there is a
slight tendency for the representative value of $\Gamma$ defined in this work
to be higher than that of Jorstad et al. (2017). This is likely due to how we
define a “representative” knot as the fastest one with a relatively small
$\xi$ value, rather than averaging all knots together as was done in Jorstad
et al. (2017). Similarly, the reported $\theta_{\circ}^{\text{A}}$ values in
this work are slightly larger than those inferred previously, but this is
probably due to $\langle\Theta_{\circ}\rangle$ and $\theta_{\text{p}}$ now
being defined from $>10$ years of observation, which samples a wider range of
knot behaviors.

Table 14: Wilcoxon rank-sum test $p$-values between this work and previous results. Parameter | Comparison Work |  $p$-value
---|---|---
$\delta_{\text{var}}$ | Jorstad et al. (2017) | 0.315
| Liodakis et al. (2018) | 0.423
$\Gamma$ | Jorstad et al. (2017) | 0.002
| Liodakis et al. (2018) | 0.071
$\Theta_{\circ}$ | Jorstad et al. (2017) | 0.028
| Liodakis et al. (2018) | 0.275
$\theta_{\text{p}}$ | Jorstad et al. (2017) | 0.054
$\theta_{\circ}^{\text{A}}$ | Jorstad et al. (2017) | 0.001
$\theta_{\circ}^{\text{B}}$ | Jorstad et al. (2017) | 0.229

Another useful comparison can be made with the work of Liodakis et al. (2018).
These authors modeled the 15 GHz light curves of 1029 blazars and blazar-like
AGNs, as observed with the Owen’s Valley Radio Observatory (OVRO; Richards et
al., 2011), with a series of exponential rise and decay profiles (flares)
superposed on a stochastic background. For each flare, a variability
brightness temperature was calculated. Since each source can have numerous
flares within the light curve, distributions of variability brightness
temperatures can be made for each source. Comparison with an expected
equilibrium brightness temperature then yields distributions of Doppler
factors, $\delta_{\text{var}}$. The authors then used the maximum apparent
speeds of radio knots at 15 GHz from the MOJAVE program (Lister et al., 2016)
to calculate $\Gamma$ and $\Theta_{\circ}$. Agreement between the two methods
would provide important constraints for the Doppler factors and equipartition
states in the jets.

All of the sources in the VLBA-BU-BLAZAR program have been observed by OVRO
and are included in Liodakis et al. (2018), except for the FSRQ 1622$-$297,
which is below the declination limit for the OVRO monitoring program. Two
sources, the BLs 0735+178 and 0954+658, did not have reliable redshift
estimates at the time of Liodakis et al. (2018). These have since been refined
in NED to $z=0.424$ and $0.368$, respectively. Using the same formalism as
Liodakis et al. (2018), we have updated the Lorentz factors and viewing angles
(Eq. 6) to be $\Gamma=8.4\pm 1.6,\ \Theta_{\circ}=3.3\pm 1.6$ for 0735+178 and
$\Gamma=15.4\pm 1.2,\ \Theta_{\circ}=3.5\pm 1.1$ for 0954+658.

Comparing the physical parameters $\delta_{\text{var}}$, $\Gamma$, and
$\Theta_{\circ}$ between our current work and Liodakis et al. (2018), we again
see good agreement between the distributions. The majority of estimates of the
physical parameters of each jet lie within the sum of the uncertainties
between the two samples. WRS $p$-values between the two data sets are given in
Table 14. No comparison meets our significance threshold, indicating that the
distributions are not significantly different. Discrepancies in the values of
parameters calculated by Liodakis et al. (2018) and this work are likely due
to the difference in observing frequencies, resulting in different parts of
the jet being probed.

Finally, we compare our results with modeling of the relation between
$\Theta_{\circ}$ and $\Gamma$. Given that Doppler beaming increases with
smaller viewing angles while geometrical likelihood increases with larger
angle, the most probable viewing angle is somewhat smaller than that which
maximizes the apparent velocity, $\Theta_{\text{max}}=\arcsin{1/\Gamma}\approx
1/\Gamma$ for $\Gamma\gg 1$ (Marscher, 1990). Vermeulen & Cohen (1994) found
that the most likely angle is $\sim 0.5\Theta_{\text{max}}$. Meanwhile, Lister
& Marscher (1997) used Monte Carlo simulations to determine statistical
properties of beamed sources in a flux-limited sample, including the viewing
angle at which a jet will appear in a flux-limited sample given its Lorentz
factor and intrinsic luminosity. They found that the peak of the simulated
distribution is $\Theta_{\circ}\sim 0.4\Theta_{\text{max}}$, with a
considerable number of sources having $\Theta_{\circ}>\Theta_{\text{max}}$.

Figure 25: Observed viewing angle $\Theta_{\circ}$ vs. theoretical viewing
angle $\Theta_{\text{max}}\approx\Gamma^{-1}$ for reliable knots in FSRQs (red
circles), BLs (blue squares), and RGs (green triangles).

To see if these relations hold observationally, we plot the viewing angle of
all reliable knots in our sample against $\Gamma^{-1}$ in Figure 25. The data
appear to fall into several distinct groups. To guide the readers eye, we have
added three lines indicating values for which $\Theta_{\circ}=0.5$, 1, and 2
times $\Theta_{\text{max}}$. The vast majority of knots fall between the 0.5
and 2$\times$ lines and with $\Theta_{\text{max}}\lesssim 0.15$ rad. Two
extended populations appear, roughly following the
$\Theta_{\circ}\sim\Theta_{\text{max}}$ and $2\Theta_{\text{max}}$ lines.
Almost all of the RG knots appear along these extended populations.

To get a statistical sense of which relation most accurately describes the
observations, we have followed Lister & Marscher (1997) by calculating the
percent difference between $\Theta_{\circ}$ and $\Theta_{\text{max}}$, which
we plot as a histogram for each source type in Figure 26. There are 9 FSRQ
knots for which the difference is $>300\%$, usually in cases of very high
values of $\Gamma$. As these represent a small fraction ($\sim 5\%$) of the
knots, we have omitted them to reduce the size of the figure. For FSRQs and
BLs, we recover the simulated distribution of Lister & Marscher (1997). In
these two distributions, the peak appears near $-50\%$ difference, but there
is a long tail showing that many knots can have
$\Theta_{\circ}>\Theta_{\text{max}}$. The two distributions are similar
statistically, with a KS test statistic of $\mathcal{D}=0.72$ and $p=0.073$.
However, we do not see a similar distribution for the RGs
($\mathcal{D}=0.460,\ p=1.5\times 10^{-5}$ for the FSRQ vs RG and
$\mathcal{D}=0.513,\ p=6.3\times 10^{-6}$ for the BL vs RG distributions).
There is a wide spread of the percent difference for these sources, and no
obvious distribution peak due to the small number of observed knots. The wide
spread is an indication that the expected $\Theta_{\text{max}}$ relation with
$\Gamma$ is not a good indicator of the true viewing angle for RGs (Lister &
Marscher, 1997).

Figure 26: Distributions for FSRQs, BLs, and RGs of the percent difference
between the observed viewing angle $\Theta_{\circ}$ and expected viewing angle
$\Theta_{\text{max}}\approx\Gamma^{-1}$. A grey dotted line has been added at
zero difference.

###  5.5 Averages of Physical Parameters of AGN Subclasses

We use the same formalism as Jorstad et al. (2017) to calculate the weighted
averages of the physical parameters for each subclass in our sample. Two
values of the average opening semi-angle values are obtained for the two
different methods of estimating $\theta_{\circ}$,
${\overline{\theta_{\circ}}}^{\text{A}}$ and
${\overline{\theta_{\circ}}}^{\text{B}}$. We calculate the uncertainty in the
average quantities as the weighted standard deviation of the parameters of the
sources in the subclass about these means, where the weights are the inverse
of the variance of the parameter for each source. Table 15 gives the result of
the averaging as follows: 1—Name of the AGN subclass; 2—number of sources in
the subclass, $N_{s}$, used to calculate the averages; 3—Doppler factor,
$\overline{\delta}$, and its weighted standard deviation from the included
knots; 4—bulk Lorentz factor, $\overline{\Gamma}$, and its weighted standard
deviation; 5—viewing angle, $\overline{\Theta_{\circ}}$, and its weighted
standard deviation; 6—opening semi-angle of the jet according to method A,
$\overline{\theta_{\circ}}^{\text{A}}$, and its weighted standard deviation;
and 7—opening semi-angle of the jet according to method B,
$\overline{\theta_{\circ}}^{\text{B}}$, and its weighted standard deviation.

Table 15: Average Physical Parameters of AGN Jets for Different Subclasses Subclass | $N_{s}$ | $\overline{\delta}$ | $\overline{\Gamma}$ | $\overline{\Theta_{\circ}}$ | $\overline{\theta_{\circ}}^{\text{A}}$ | $\overline{\theta_{\circ}}^{\text{B}}$
---|---|---|---|---|---|---
|  |  |  | [deg] | [deg] | [deg]
FSRQ | 22aaBecause the source 1611+343 only has one reliable superluminal knot, there is no estimate for $\theta_{\circ}^{\text{B}}$ for this source, and 21 sources were used to calculate $\overline{\theta_{\circ}}^{\text{B}}$. | $7.1\pm 6.1$ | $16.0\pm 4.7$ | $\phantom{0}4.0\pm 2.3$ | $1.2\pm 0.9$ | $1.2\pm 1.5$
BL | 11 | $3.7\pm 4.3$ | $\phantom{0}5.5\pm 4.1$ | $\phantom{0}2.7\pm 2.2$ | $0.7\pm 0.8$ | $1.3\pm 1.1$
RG | 2 | $2.6\pm 0.5$ | $\phantom{0}8.6\pm 0.6$ | $15.7\pm 1.5$ | $2.4\pm 0.4$ | $5.3\pm 4.0$

While the physical parameters of FSRQs and BLs overlap when the standard
deviations are included, the expected trend of larger Doppler and Lorentz
factors in FSRQs relative to BLs is observed in the current data (Jorstad et
al., 2005). Higher values of $\delta$ in FSRQs are supported by the higher
brightness temperatures of moving components. The relative similarity of the
viewing and opening semi-angles between the two subclasses (compared to
previous results, e.g., Pushkarev et al., 2009; Jorstad et al., 2017) can be
explained by our choice of the fastest “typical” knot to represent particular
sources. In a spine-sheath jet structure, the fastest knots all come from the
spine, resulting in an opening semi-angle that would be smaller than if all
knots were considered together (as would be expected in BLs). The physical
parameters of RGs are also consistent with the predictions of the unified
scheme of AGNs (Urry & Padovani, 1995), with comparable Lorentz factors,
smaller Doppler factors, and much larger viewing angles compared to FSRQs and
BLs. However, as was pointed out in Jorstad et al. (2017), these RGs are the
brightest $\gamma$-ray emitters of their subclass, and may not be
representative of the subclass as a whole.

Three sources in our sample do not have superluminal knots meeting our
reliability criteria described at the beginning of $\S$5: the RG 0316+413 and
the BLs 1652+398 and 1959+650. Thus, we cannot calculate their physical
parameters using their knot motions. Instead, we make a rough estimate for the
physical parameters of these sources based on two assumptions: (1) the
intrinsic opening semi-angle of the jet is equal to
${\overline{\theta_{\circ}}}^{\text{A}}$, the average of the subclass to which
the source belongs, and (2) $\Gamma\theta_{\circ}\sim 0.2$ is universal for
bright $\gamma$-ray AGNs (Clausen-Brown et al., 2013; Jorstad et al., 2017).
The low apparent speeds of the knots in the BL 1101+384 indicate that we are
looking at the sheath of the jet, so we also use the same formalism to
estimate the parameters of this source. The estimated parameters for these
four sources are given in Table 13. The above assumptions result in viewing
angles $\sim 2\times$ smaller than the subclass average for these sources. As
we have only observed 1959+650 at 6 epochs it is possible that the jet does
indeed have physical parameters as estimated above, but more monitoring is
necessary. However, the fact that only slow proper motions have been observed
in 0316+413, 1101+384, and 1652+398 despite intensive monitoring campaigns
(e.g.; this work; Blasi et al., 2013; Lister et al., 2016) creates a “Doppler
crisis” with their high-energy properties, which require high Doppler factors
(and subsequently high proper motions). The decelerations observed in Figure
15 (5-10 pc from the core for parallel acceleration, and 1-2 pc from the core
for perpendicular acceleration) might be able to explain the discrepancy, as
high Doppler factors would only be seen closer to the central engine, beyond
the opacity limit of 43 GHz observations. It is also possible that a mildly
relativistic flow, corresponding to a “sheath” around the faster central
spine, would obscure the faster spine, as seems to be the case in 0316+413
(Nagai et al., 2014).

##  6 Discussion

###  6.1 Brightness Temperatures and Doppler Factors

The brightness temperature distributions presented in $\S$3.1 are similar to
those expected for unbeamed, incoherent synchrotron radiation produced by
relativistic electrons that are in equipartition with the magnetic field
$T_{\mathrm{b,eq}}\sim 5\times 10^{10}$ K (Readhead, 1994). The higher values
of $T_{\text{b,obs}}^{\text{s}}$ found in FSRQs and BLs can be partially
explained by Doppler boosting: $T_{\text{b,obs}}\propto\delta
T_{\text{b,int}}$; the Doppler factor has been inferred to be as high as
$\delta\sim 60$ in blazars (e.g., Liodakis et al., 2018). During high activity
states (such as outbursts) of blazars, it is possible that equipartition is
violated such that the energy density of radiating particles is greater than
that of the magnetic field in the VLBI core (e.g., Homan et al., 2006; Kovalev
et al., 2016). Such non-equipartition states could lead to the extremely high
($T_{\text{b,obs}}^{\text{s}}\gtrsim 10^{13}$ K) brightness temperatures found
in some cores of some sources.

Figure 27: Typical Doppler factor of variability for each source vs. the
maximum brightness temperature of the core, A0, for resolved (closed symbols)
and unresolved (open symbols with right-facing arrows, component size $a\leq
0.02$ mas) epochs. FSRQs are shown in red circles, BLs in blue squares, and
RGs in green triangles.

Figure 27 compares the typical Doppler factor of variability,
$\delta_{\text{var}}$, for each source with the maximum brightness temperature
of the core component _A0_ in the host galaxy frame,
$T_{\text{b,obs}}^{s}=T_{\text{b,obs}}(1+z)$. In the figure, both the maximum
resolved brightness temperature (closed symbols) and lower limit to the
maximum brightness temperature (open symbols, based on the observed size of
the component; see $\S$2.1) are plotted. We have connected the resolved
brightness temperatures and lower limits in each source with a dotted line to
aid the reader. Sources with physical parameters estimated via the method
presented in $\S$5.5 are omitted. In general, we see that sources with higher
maximum brightness temperatures tend to have higher Doppler factors. Also, the
maximum brightness temperatures of FSRQs tend to be higher than those of BLs
(as expected from the distributions in Figure 4). Taking the three subclasses
together, the Doppler factors and maximum brightness temperatures of the cores
have a Pearson’s $r$ correlation coefficient of $0.367$ for the resolved
components. There is more scatter in the data when considering only the lower
limits, which have a correlation coefficient of $0.281$ with the maximum
brightness temperatures of the cores. The maximum core brightness temperatures
range up to $\sim 10^{13}$ K. These could be indicative of non-equipartition
states in the 43 GHz cores of blazars. We see a connection between the
$\gamma$-ray state of blazars and their 43 GHz properties (see $\S$6.3 below),
so these non-equipartition values are expected. However, the moving components
all have $T_{\text{b,obs}}^{s}<5\times 10^{12}$ K, which can be explained by
Doppler boosting with our observed Doppler factor values,
$\delta_{\text{var}}\lesssim 60$.

###  6.2 Flux Variability

As discussed in $\S$5.1, the rise and decay of flux in mm-wave light curves of
blazars can be roughly modeled by exponential profiles (e.g., Terasranta &
Valtaoja, 1994; Lister, 2001; Savolainen et al., 2002). We are thus able to
estimate a timescale of variability, $\tau_{\text{var}}$, from the flux decay
rate, $k$, as $\tau_{\mathrm{var}}=|1/k|$ yr. We show in Figure 18 that these
decay rates are often primarily due to radiative losses, modulated by light-
travel delays, as opposed to adiabatic expansion. As the emission from blazar
jets at 43 GHz is dominated by synchrotron radiation, standard synchrotron
emission formulae (e.g., Pacholczyk, 1970; Longair, 2011) yield an expected
relation between the emitted flux density and distance down the jet $R$,
$S_{43}\propto R^{-a}$. Using the proper motion of a knot, $\mu=dR/dt$, we can
estimate the age of a knot as $T_{\text{age}}=\langle R\rangle/\mu$ in the
observer’s frame. Homan et al. (2002) adopted a decay rate of a knot that is
inversely proportional to $T_{\text{age}}$:

$k=\frac{(dS_{43}/dt)}{S_{43}}=-a\times\frac{(dR/dt)}{R}=-a/T_{\text{age}}.$
(9)

We have already calculated the normalized decay rates of the majority of knots
in our sample as $(dS_{43}/dt)/S$ (see $\S$5.1). We revise the relation in
Equation 9 to a more general form: $k=-aT_{\text{age}}^{-b}$, where $a$ and
$b$ are constants. We transform the component ages to the source (i.e., host
galaxy) frame ($s$) through $T_{\text{age}}^{s}=T_{\text{age}}/(1+z)$. Figure
28 shows the decay rate $k^{s}$ of all knots in our reliable sample versus the
age of the feature, $T_{\text{age}}$, separated by blazar subtype. We have fit
the generalized equation for the decay rate to each subclass, and show the
results by the red dashed, blue dot-dashed, and green dotted curves for FSRQs,
BLs, and RGs, respectively, in the figure.

Figure 28: Flux decay rate $k$ for reliable knots versus the age of the
feature in the host galaxy frame, $T_{\text{age}}^{s}$, for FSRQ (red
circles), BL (blue squares), and RG knots (green triangles). Fits to the model
$k=-a(T_{\text{age}}^{s})^{-b}$ are plotted in the same color as the subtype
of blazar, with values as indicated in the legend.

Correcting the age for redshift results in different phenomenological fits for
all three subtypes. The BLs have a best-fit exponent $b$ near unity, while
FSRQs have $b\sim 0.75$. The exponent for RGs is a factor of 1.6 smaller than
for BLs, with $b=0.65$. The results for BLs and FSRQS are similar to that
found in Homan et al. (2002), despite the authors’ assumption that the flux
decay of knots is linear rather than exponential.

We provide the distributions of knot ages for each subclass in the left panel
of Figure 29. All three distributions are qualitatively similar, with a
substantial peak in the distribution for components that live for $\sim 1$-$2$
years. There are several knots in FSRQs that are long-lived, with
$T_{\text{age}}^{s}>5$ yr, while no such knots exist in BLs and RGs. Of the
three subtypes, the BL distribution has the largest fraction of short-lived
knots, with 70% of knots having $T_{\text{age}}^{s}<1$ yr. A KS test between
the distributions confirms the similarities, with $\mathcal{D}=0.242,\
p=0.003$ for FSRQs vs BLs, $\mathcal{D}=0.177,\ p=0.345$ for FSRQs vs RGs, and
$\mathcal{D}=0.130,\ p=0.786$ for BLs vs RGs.

If the decay of the flux density of a knot is controlled by the light-crossing
time, intrinsically smaller knots should have shorter decay times in the
plasma frame. Since the angular sizes of knots are similar for all of the
sources (see Table 2), knots in objects with lower redshifts — all of the RGs
— should have smaller sizes. Given the lower mean redshift values of the RGs
versus the BLs and FSRQs, this effect must be canceled by the higher Doppler
factors found for the BLs and FSRQs in order to explain the similar 1-2 year
ages of the peaks of the distributions seen in Figure 29. To verify this, we
calculate the age of components in the plasma frame of each jet by multiplying
the redshift-corrected age by the Doppler factor of variability, so that
$T_{\text{age}}^{\text{plasma}}=\delta_{\text{var}}T_{\text{age}}/(1+z)$. The
distributions for each source type are shown in Figure 29 (right). The FSRQ
and BL distributions differ at a $\sim 3\sigma$ level, with
$\mathcal{D}=0.244$ and $p=0.003$. However, both distributions peak at a knot
age $\sim 10$ yr. A large number of BL knots have shorter ages, while there
are more knots in FSRQs with greater ages. Scaling by $\delta_{\text{var}}$
separates the RG distribution from the FSRQ and BL ($\mathcal{D}=0.813,\
p=3.2\times 10^{-16}$ and $\mathcal{D}=0.718,\ p=4.9\times 10^{-11}$
respectively). The typical intrinsic age of a RG knot is $\sim 1$ yr, a factor
of ten lower than for FSRQs and BLs, roughly in accord with the ratios of
physical sizes of their knots.

It is important to point out the effect that our roughly monthly observation
cadence might have on the observed lifetimes of knots. Very bright components
in the jets of blazars are easy to identify in the radio maps and fit with
Gaussian profiles (as in $\S$2), thus would need to have extremely fast decays
($<1$ month) to not be seen at other epochs. Given the paucity of short-decay
knots seen in the distributions in Fig. 29, this scenario is unlikely.
However, faint components are difficult to distinguish from noise, especially
if such knots travel down the jet quickly with short decay times (e.g., if
their decay were $<0.25$ yr, as we require at least 4 observations of a knot
for it to be properly identified). Such knots would be missed in the above
analysis. Fast, faint components have been seen in some sources (e.g., the BL
0219+428 (3C 66A); Jorstad et al., 2005), but to detect them requires denser
temporal sampling.

Figure 29: Distributions of the age of reliable knots in the host galaxy
frame, $T_{\text{age}}^{s}=T_{\text{age}}/(1+z)$ (left), and plasma frame,
$T_{\text{age}}^{\text{plasma}}=\delta_{\text{var}}T_{\text{age}}/(1+z)$
(right), in FSRQs (top), BLs (middle), and RGs (bottom). Values in parantheses
are the number of knots in each distribution. Note that the age scale is
neither linear nor logarithmic.

###  6.3 Gamma-Ray Connection

It is well established that there exists a connection between the $\gamma$-ray
state of blazars and their parsec-scale jet behaviors. Nieppola et al. (2011)
have shown that there is a statistically significant correlation between the
$\gamma$-ray and 37 GHz radio light curves. The brightest $\gamma$-ray flares
tend to coincide with with the early stages of a millimeter-wave outburst
(León-Tavares et al., 2011). Such trends are not universal: for example, at 15
GHz a significant correlation was only found in 4 of 41 analyzed blazars (Max-
Moerbeck et al., 2014). However, the F-GAMMA program has found the radio
variations of 54 $\gamma$-ray bright blazars that lag the $\gamma$-rays, with
delays systematically decreasing as the observed frequency changes from
centimeter to millimeter/submillimeter (Fuhrmann et al., 2014). With regard to
knots traveling down the parsec-scale jet, studies of individual sources have
revealed a wide diversity in the behavior of blazars. The majority of
$\gamma$-ray outbursts appear to be connected with the propagation of a knot
down the jet (e.g., Jorstad et al., 2001a; Marscher et al., 2010; Agudo et
al., 2011a; Jorstad et al., 2013; Morozova et al., 2014; Jorstad et al.,
2017). In several of these cases, the flux profile of the $\gamma$-ray
outburst can be connected with the passage of superluminal knots through the
core or quasi-stationary features (e.g, Agudo et al., 2011a; Weaver et al.,
2019; Liodakis et al., 2020). However, these features are usually located
parsecs from the central black hole, at odds with the small sizes required by
causality arguments from the short timescales of high-energy variability
(Aharonian et al., 2007; Tavecchio et al., 2010; Nalewajko et al., 2014).
Another problem is the origin of high-energy outbursts known as “orphan”
flares (e.g., Krawczynski et al., 2004; Marscher et al., 2010; MacDonald et
al., 2015; Sobacchi et al., 2021; Yang et al., 2021). During such flares, the
$\gamma$-ray flux may increase by a factior $\sim 100$ and yet little
variability is seen at longer wavelengths, with no interaction between moving
and stationary features observed in VLBA images.

While a full analysis of the relationship between the parsec-scale jet
kinematics of the blazars in our sample and their $\gamma$-ray behavior is
beyond the scope of this work, here we present some simple relationships and
correlations. It has been shown by the MOJAVE survey that the jets of
$\gamma$-ray bright quasars have faster average apparent motions and brighter
parsec-scale cores than those with weaker $\gamma$-ray fluxes (e.g., Kovalev
et al., 2009; Lister et al., 2009, 2016). This relation implies that the jets
of brighter $\gamma$-ray quasars have higher Doppler boosting factors. In
order to determine if there is a relationship between the observed physical
parameters of the sources in our sample and their $\gamma$-ray flux, we
calculate the maximum $\gamma$-ray flux, $S_{\gamma,\text{max}}^{\text{obs}}$,
(ignoring upper-limits) of each source according to their 10-yr _Fermi_ -LAT
light curves.

In the analyses that follow, we have corrected the fluxes for redshift effects
through $S=S^{\text{obs}}(1+z)^{1+\alpha}$, where $S^{\text{obs}}$ is the
observed flux, $z$ is the redshift, and $\alpha$ is the spectral index of the
source at the observed frequency. The $(1+z)^{\alpha}$ factor is the
k-correction, while the $(1+z)^{1}$ factor accounts for the change in
frequency interval. In the radio band, $\alpha_{43}\sim 0$. For the
$\gamma$-ray spectral index, we have taken the observed power-law index $p$
from the _Fermi_ 4${}^{\text{th}}$ Catalog of AGN (Ajello et al., 2020),
regardless of which spectral model best-fits each source. Then,
$\alpha_{\gamma}=|p+1|$.

Figure 30 shows the relationship between $S_{\gamma,\text{max}}$ and the
apparent speed $\beta_{\text{app}}$ and Doppler factor $\delta_{\text{var}}$
of the “typical” knot of each source. The sample as a whole has a weak
positive correlation between the physical parameters in the jet and
$S_{\gamma,\text{max}}$, with Pearson’s $r$ correlation coefficients of
$0.361$ and $0.274$ for $\beta_{\text{app}}$ and $\delta_{\text{var}}$,
respectively. The significant scatter is indicative of the diversity in knot
behaviors seen in each jet. Our sample consists of too few sources of each
type to determine if there is a difference between the subclasses. However,
even at the maxima in their $\gamma$-ray light curves BLs and RGs are in
general fainter than FSRQs, with lower speeds and Doppler factors (as expected
in the standard AGN unification scheme; e.g., Urry & Padovani, 1995).

Figure 30: Maximum $\gamma$-ray flux $S_{\gamma,\text{max}}$ (scaled by
$10^{-8}$ ph cm-2 s-1) versus _(a)_ apparent speed $\beta_{\text{app}}$ and
_(b)_ Doppler factor $\delta_{\text{var}}$ of the “typical” knot in FSRQs (red
circles), BLs (blue squares), and RGs (green triangles).

To further investigate a connection between the $\gamma$-ray and 43 GHz radio
behavior, in Figure 31 we show $S_{\gamma,\text{max}}$ compared to the maximum
flux density of the core component _A0_ in each source in Table 13 for which
the physical parameters were calculated rather than estimated from the
$\Gamma\theta_{\circ}\sim 0.2$ relation. We find a $\sim 3\sigma$ positive
correlation between $S_{\gamma,\text{max}}$ and the maximum core flux density
of each source, with a Pearson’s $r$ coefficient of $0.465$. We also see that
the results of the core brightness temperature distributions from $\S$3.1
extend to the maximum core flux densities, with the FSRQ cores generally
brighter than the BL and RG cores.

Figure 31: Maximum $\gamma$-ray flux, $S_{\gamma,\text{max}}$ versus maximum
core flux density at 43 GHz, $S_{43,\text{core\ max}}$, for FSRQs (red
circles), BLs (blue squares), and RGs (green triangles).

The correlations observed here, while weak, support the conclusion of a
connection between the activity states of a blazar at 43 GHz and at
$\gamma$-ray energies. In general, the brighter the 43 GHz core of a source,
and the faster the knots in a source appear to move down the jet, the brighter
the jet can be at $\gamma$-rays. We plan to perform correlation analyses
between the $\gamma$-ray and 43 GHz core light curves and comparisons between
knot time of ejections and $\gamma$-ray outbursts for the whole sample in a
future work.

##  7 Summary

We have presented a new analysis of the jet kinematics in 35 blazars and 3
radio galaxies observed with the VLBA at 43 GHz over ten years under the VLBA-
BU-BLAZAR program. The total observed time period is 2007 June through 2018
December, resulting in 3705 total intensity images of the jets. The first five
years of data were presented in Jorstad et al. (2017), and are included in
this study as well after applying a new analysis procedure. The primary
findings and conclusions of the ten years of data are summarized as follows:

  1. 1.

We have implemented a new piece-wise linear fitting method to derive the
kinematics of moving components in the jets. This algorithm allows us to
estimate the distance down the jet at which acceleration (if any) occurs. We
have identified 559 distinct emission features, of which 38 are the 43 GHz
cores, which are assumed to be stationary. We have further classified 96
components as quasi-stationary, leaving 425 knots that move with respect to
the VLBI core. Among these moving knots, 75 exhibit non-linear motion, for
which we calculate the acceleration parallel and perpendicular to the jet
direction.

  2. 2.

To incorporate the presence of lower limits to the brightness temperature of
10.1% of all components into our analysis, we have used the Kaplan-Meier
estimator for the survival function of the cores and knots to derive a median
value of the brightness temperature for each source. We plan to provide a
deeper analysis of the survival functions of the core and knot brightness
temperatures, covering the temporal properties and whether all sources can be
described with a similar parameterized survival function, in a future work
(Weaver, in prep.).

  3. 3.

We find, despite a low significance in a KS test due to a small number of
sources, that FSRQs and BLs have more intense cores than those of RGs, but the
knots in the extended jets of BLs are significantly less intense than the
knots of RGs and FSRQs. The brightness temperature distributions of the non-
core components can be explained by Doppler boosting of the expected intrinsic
brightness temperature.

  4. 4.

With 10.5 years of data, we have analyzed parsec-scale jet position angle
variations of each source. We have quantitatively placed the jets into four
categories. The majority of jets (23) have jet position angles that are
constant with time, while a set of six sources has a very wide jet that is
constant in position angle. Two sources, the FSRQs 0420$-$014 and 1226+023,
have jet position angles that change linearly with time. The remaining seven
sources have complex variations of position angle, which we have fit with 3rd-
order splines.

  5. 5.

Due to our piece-wise linear fitting method, any individual moving knot can
have multiple motions over different time periods. We therefore have analyzed
a total of 529 knot speeds. These speeds range from $0.01c$ to $78c$. The
FSRQs generally have the fastest knots, with a distribution that peaks between
$8c$-$10c$ and extends to $\sim 78c$. The distribution of apparent speeds for
BLs peak between $0c$-$2c$, as well as $8c$-$10c$, but only extends up to
$\sim 30c$. RGs have a very narrow distribution of speeds, only extending to
$\sim 10c$. We find superluminal motion in at least one knot of the RG
0316+413 of $\beta_{\text{app}}=1.38c\pm 0.03c$, while the remaining
superluminal knots in RGs come from 0415+379 and 0430+052. The moving knots in
FSRQs tend to travel in the same direction, while knots of BLs show larger
variations in the direction of motion.

  6. 6.

On average, there exist $\sim 2$ quasi-stationary features in the jet of each
FSRQ, $\sim 3$ in each BL, and $\sim 6$ in each RG. Almost half of these
features have appeared or disappeared during the 10.5-year observation period,
indicating that they may be transient. The locations of the majority of
stationary features are concentrated within 1 mas of the core (projected),
independent of the subclass. We have characterized the fluctuations in the
positions of these features using confidence ellipses, which show that the
stationary features in RGs tend to “slosh” around the average position in a
direction parallel to the jet axis. However, there is no clear preference of
directional shifts for the stationary features of BLs and FSRQs.

  7. 7.

We have detected 104 different acceleration regions in the sample. Only a few
BLs have knots with discernible acceleration. We find that, statistically,
these accelerations are consistent with changes in the Lorentz factor of the
knots rather than changes in the jet direction. Parallel to the jet axis, we
see an initial positive acceleration close to the core of jet ($<5$ pc), while
beyond $10$ pc the knots tend to decelerate. Perpendicular to the jet axis, we
find an initial positive acceleration for projected distances $0$-$1$ pc from
the core and negative acceleration $1$-$2$ pc from the core. Beyond $\sim 5$
pc, we see very small magnitudes of acceleration. Similar trends are found
farther from the core at longer wavelengths (e.g., Homan et al., 2015).

  8. 8.

To test whether interactions with standing shocks have an effect on the moving
knots, we have paired the acceleration regions with the closest (on average)
stationary features. We have found that few pairs fail to follow an expected
1:1 relation, indicating that accelerations take place at or beyond the
stationary features. Such behavior is expected if the stationary features are
structural components such as recollimation shocks.

  9. 9.

We have derived Doppler factors of variability, $\delta_{\text{var}}$, for a
subset of reliable knots in the sample, which, along with the measured
apparent speeds, have allowed us to calculate the Lorentz factors $\Gamma$ and
viewing angle $\Theta_{\circ}$. The flux decays of the knots in our sample are
statistically likely to be caused by radiative losses (modulated by light-
travel delays) rather than adiabatic expansion.

  10. 10.

The Lorentz factors of knots in the jets of our sample range from $2$-$60$,
with faster knots observed in FSRQs than in BLs and RGs. Our results agree
with the expectation that the jets of RGs lie at larger viewing angles with
respect to the line of sight, with an average $\Theta_{\circ}\sim
15\arcdeg$-$30\arcdeg$, compared to the FSRQ and BL jets with peak viewing
angles $\sim 0\arcdeg$-$4\arcdeg$.

  11. 11.

For most of the jets in our sample, we see a wide diversity in the apparent
speeds and Doppler factors of knots moving down the parsec-scale jet. For
sources with many observed knots (such as the RG 0415+379), we see that this
diversity can be explained by a distribution of physical parameters from which
a knot’s parameters can be drawn statistically. Whether the distributions of
knot parameters are similar for different sources remains unclear, as many of
our sources will likely require a further 5 to 10 years of monitoring for
enough knots to be observed to determine their distributions robustly.

  12. 12.

We have defined the “typical” knot in each jet by choosing the fastest knot
with the smallest uncertainty in $\beta_{\text{app}}$ and
$\delta_{\text{var}}$ to represent the jet as a whole. From these typical
values, we calculate the Lorentz factor and viewing angle of each jet, and
find that, on average, FSRQ jets have higher values of $\delta_{\text{var}}$
and $\Gamma$ compared to BLs, while knots in RGs are slower and have much
wider viewing angles and intrinsic opening semi-angles. The physical
parameters of each jet as calculated using the “typical” knot are
statistically similar to other estimates of the physical parameters determined
through lower-frequency monitoring (e.g., Liodakis et al., 2018), which probes
farther distances from the core.

  13. 13.

In general, the knots of FSRQs are longer lived than the knots of BLs and RGs,
reaching farther projected distances from the core. The knots of FSRQs and BLs
have flux decay rates that are related to their age roughly as
$k\sim-2T_{\text{age}}^{-1}$ despite the differences in their age
distributions, while RG knots follow $k\sim 2.4T_{\text{age}}^{-0.65}$. The
distribution of ages of all subclasses peak in the 1-2 yr range despite large
differenes in the physical sizes of knots of RGs versus BLs and FSRQs. This
can be explained by the lower Doppler factors of the RGs.

  14. 14.

We find weak correlations between the maximum $\gamma$-ray flux and the
typical apparent speeds and Doppler factors of jets in our sample. We also
find that the blazars with brighter 43 GHz cores typically reach higher
maximum $\gamma$-ray fluxes. We plan to perform a temporal analysis comparing
the ejection times of knots in our sample with $\gamma$-ray outbursts in a
future study.

We thank the anonymous referee for comprehensive and constructive feedback
that greatly improved this paper. The research at Boston University was
supported in part by NASA grants 80NSSC17K0649, 80NSSC20K1567, and
80NSSC20K1566 (Fermi Guest Investigator Program), the NRAO Student Observing
Support Program, and Massachusetts Space Grant 316080. I.A. acknowledges
financial support from the Spanish “Ministerio de Ciencia e Innovación”
(MCINN) through grants AYA2016-80889-P and PID2019-107847RB-C44, and through
the “Center of Excellence Severo Ochoa” award for the Instituto de Astrofísica
de Andalucía-CSIC (SEV-2017-0709). The VLBA is an instrument of the National
Radio Astronomy Observatory. The National Radio Astronomy Observatory is a
facility of the National Science Foundation operated by Associated
Universities, Inc. This research has made use of the NASA/IPAC Extragalactic
Database (NED), which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration. The optical data used in this work were obtained using
the 1.8 m Perkins Telescope Observatory in Flagstaff, Arizona, USA, which is
owned and operated by Boston University. This publication makes use of data
obtained at the Metsähovi Radio Observatory, operated by Aalto University in
Finland. This research made use of Astropy,111111https://www.astropy.org, a
community-developed core Python package for Astronomy (Astropy Collaboration
et al., 2013, 2018).

##  Appendix A Survival Analysis of Jet Component Brightness Temperatures

Treatment of data with lower and upper limits (or, more generally, right- and
left-censored data, respectively) in astronomy has been performed in a number
of fashions, of which many are biased or incorrect (Feigelson & Babu, 2012).
For example, if the number of limits is small, then they are unlikely to
change the result of statistical tests, however, they will bias the results as
the number of limits in a sample grows. The most statistically rigorous
procedures consider all the data points, both detected values and censored
data, to model the properties of a parent distribution of the variable in
question under specific mathematical constraints. Such procedures fall under
the umbrella of “survival analysis” (Avni et al., 1980; Avni & Tananbaum,
1986; Feigelson & Nelson, 1985; Schmitt, 1985; Isobe et al., 1986; LaValley et
al., 1992). A few methods for censored data have been developed specifically
for astronomical contexts (Akritas et al., 1995; Akritas & Siebert, 1996).

Most survival analysis techniques are based on estimating the “survival
function” of an entire data set (observations and censored data), where the
survival function of a parameter, $x$, is

$\mathcal{S}(x)=\mathrm{Pr}(X>x).$ (A1)

That is, $\mathcal{S}(x)$ is the probability that the value of $x$, $X$, is
above a specified value. Alternatively, one may estimate the “hazard
function”, $\mathcal{H}(x)$:

$\mathcal{H}(x)=\frac{\mathrm{d\ ln}\mathcal{S}(x)}{\mathrm{d}x}=\lim_{\Delta
x\rightarrow 0}\frac{\mathrm{Pr}(x<X<x+\Delta x|X>x)}{\Delta x}.$ (A2)

The focus on the survival and hazard function of data in survival analysis is
due to their relation to the common statistical concepts of probability
density functions (PDFs), $f(x)$, and cumulative distribution functions
(CDFs), $F(x)$, of data through $f(x)=\mathcal{S}(x)\mathcal{H}(x)$ and
$F(x)=1-\mathcal{S}(x)$. Thus, estimates of $\mathcal{S}(x)$ and
$\mathcal{H}(x)$ can describe the underlying parent distributions of the data,
even in the presence of censored or truncated121212Truncated data occur when
no information, not even an upper or lower limit, is known for a quantity of
interest, such as in the case of surveys of astronomical objects being
truncated at some sensitivity limit. data, depending on the estimator.

In order to determine a “typical” brightness temperature for each source in
our sample, we estimate the survival function of the log10 of the brightness
temperature of cores and knots, $\mathcal{S}(\mathrm{log}_{10}\
T_{\mathrm{b,obs}}^{\mathrm{s}})$, using the Kaplan-Meier estimator
implemented as part of the
lifelines131313https://lifelines.readthedocs.io/en/latest/index.html package
(Davidson-Pilon, 2019). The Kaplan-Meier estimator (Kaplan & Meier, 1958) is a
non-parametric method to estimate $\mathcal{S}(x)$ and was developed to
directly take right-censored data into account:

$\hat{\mathcal{S}}(x)=\prod_{i:x_{i}<x}\left(1-\frac{d_{i}}{n_{i}}\right)$
(A3)

where $x_{i}$ is a discrete value of the parameter of interest $x$ with at
least one observed or censored data point, $d_{i}$ is the number of “events”
(observed data) up to that value, and $n_{i}$ the total number of data points
“known to survive” past $x_{i}$ (have a value $x>x_{i}$ or are censored).

Figure A1: Kaplan-Meier estimated survival functions of the brightness
temperature of cores (left) and knots (right) in the BL 0219+428. The survival
function for components observed between 2013 January and 2018 December are
shown in black, while those observed between 2007 June and 2018 December are
shown in blue. Shaded regions represent 95% confidence intervals to the
survival function. The tables below each panel show the summary statistics of
each survival function in the form of the number of unused (“At risk”, both
observed and censored), censored, and observed (“event”) data points at each
major interval (included per the recommendation by Morris et al., 2019).

Figure A1 shows, as an example, the Kaplan-Meier estimates for the survival
function of the cores (left) and knots (right) of the BL 0219+428 for both the
time period presented here (2013 January - 2018 December) as well as the total
observed time period (2007 June - 2018 December). Figure SET A1 contains
similar plots for all sources in our sample. Doubling the number of brightness
temperatures used by combining those presented in this work with those
presented in Jorstad et al. (2017) yields better estimates for the survival
function, increasing the resolution along $T_{\mathrm{b,obs}}^{\mathrm{s}}$
and resulting in narrower confidence intervals along the curve. The two
sources added after 2013, the BLs 1652+398 and 1959+650, have the same
survival functions presented in the two time periods.

Fig. Set A1. Core and Knot Brightness Temperature Survival Functions

To determine a “typical” brightness temperature for the core and knots of a
source, we calculate the median of the survival function, i.e., where
$\mathcal{S}(\mathrm{log}_{10}\ T_{\mathrm{b,obs}}^{\mathrm{s}})=0.5$, for
each time period. These median brightness temperatures, with 95% confidence
intervals, are given in Table A1. Due to the low number of core components
observed for the BL 1959+650 $(N=6)$, we use the lower confidence interval as
a lower limit to the brightness temperature of the core for the source (among
all 13 BL objects, a single lower limit will not significantly affect the
results).

Table A1: Median Brightness Temperatures of Source Components as Determined Through Survival Analysis Source | Core Median $T_{\mathrm{b,obs}}^{\mathrm{s}}$ | Knot Median $T_{\mathrm{b,obs}}^{\mathrm{s}}$
---|---|---
|  $\times 10^{10}$ [K] |  $\times 10^{8}$ [K]
| 2013-2018 | 2007-2018 | 2013-2018 | 2007-2018
0219+428 | $\phantom{0}22.94^{+\phantom{0}\phantom{0}2.68}_{-\phantom{0}11.66}$ | $\phantom{0}19.08^{+\phantom{0}\phantom{0}8.00}_{-\phantom{0}\phantom{0}5.76}$ | $\phantom{0}\phantom{0}\phantom{0}6.43^{+\phantom{0}\phantom{0}\phantom{0}4.70}_{-\phantom{0}\phantom{0}\phantom{0}3.51}$ | $\phantom{0}\phantom{0}7.68^{+\phantom{0}\phantom{0}2.58}_{-\phantom{0}\phantom{0}2.47}$
0235+164 | $\phantom{0}29.70^{+\phantom{0}15.27}_{-\phantom{0}\phantom{0}9.54}$ | $\phantom{0}52.12^{+\phantom{0}\phantom{0}9.41}_{-\phantom{0}10.35}$ | $\phantom{0}\phantom{0}73.47^{+\phantom{0}\phantom{0}99.32}_{-\phantom{0}\phantom{0}28.91}$ | $178.45^{+\phantom{0}97.42}_{-\phantom{0}89.28}$
0316+413 | $\phantom{0}15.43^{+\phantom{0}\phantom{0}2.29}_{-\phantom{0}\phantom{0}1.36}$ | $\phantom{0}14.74^{+\phantom{0}\phantom{0}0.98}_{-\phantom{0}\phantom{0}0.92}$ | $\phantom{0}108.22^{+\phantom{0}\phantom{0}14.91}_{-\phantom{0}\phantom{0}16.13}$ | $144.69^{+\phantom{0}12.15}_{-\phantom{0}15.45}$
0336-019 | $\phantom{0}50.70^{+\phantom{0}38.27}_{-\phantom{0}16.91}$ | $\phantom{0}43.34^{+\phantom{0}15.41}_{-\phantom{0}\phantom{0}9.55}$ | $\phantom{0}121.16^{+\phantom{0}133.33}_{-\phantom{0}\phantom{0}55.53}$ | $121.16^{+113.61}_{-\phantom{0}59.63}$
0415+379 | $\phantom{0}13.82^{+\phantom{0}\phantom{0}4.07}_{-\phantom{0}\phantom{0}4.52}$ | $\phantom{0}15.87^{+\phantom{0}\phantom{0}4.30}_{-\phantom{0}\phantom{0}2.05}$ | $\phantom{0}\phantom{0}35.55^{+\phantom{0}\phantom{0}\phantom{0}9.01}_{-\phantom{0}\phantom{0}\phantom{0}9.81}$ | $\phantom{0}34.42^{+\phantom{0}\phantom{0}8.72}_{-\phantom{0}\phantom{0}8.68}$
0420-014 | $\phantom{0}62.10^{+\phantom{0}29.36}_{-\phantom{0}19.15}$ | $\phantom{0}62.67^{+\phantom{0}18.46}_{-\phantom{0}11.03}$ | $\phantom{0}\phantom{0}23.75^{+\phantom{0}\phantom{0}21.53}_{-\phantom{0}\phantom{0}14.43}$ | $159.39^{+215.45}_{-\phantom{0}65.80}$
0430+052 | $\phantom{0}14.47^{+\phantom{0}\phantom{0}5.33}_{-\phantom{0}\phantom{0}3.08}$ | $\phantom{0}15.87^{+\phantom{0}\phantom{0}3.75}_{-\phantom{0}\phantom{0}4.48}$ | $\phantom{0}\phantom{0}40.45^{+\phantom{0}\phantom{0}13.63}_{-\phantom{0}\phantom{0}13.43}$ | $\phantom{0}34.42^{+\phantom{0}\phantom{0}8.03}_{-\phantom{0}15.16}$
0528+134 | $\phantom{0}68.72^{+\phantom{0}19.42}_{-\phantom{0}11.04}$ | $102.16^{+\phantom{0}25.30}_{-\phantom{0}27.49}$ | $\phantom{0}\phantom{0}29.29^{+\phantom{0}\phantom{0}15.27}_{-\phantom{0}\phantom{0}14.89}$ | $\phantom{0}45.28^{+\phantom{0}16.25}_{-\phantom{0}11.95}$
0716+714 | $292.24^{+158.50}_{-119.45}$ | $180.94^{+\phantom{0}66.61}_{-\phantom{0}29.07}$ | $\phantom{0}\phantom{0}43.14^{+\phantom{0}132.46}_{-\phantom{0}\phantom{0}16.12}$ | $\phantom{0}49.09^{+\phantom{0}46.03}_{-\phantom{0}18.84}$
0735+178 | $\phantom{0}10.77^{+\phantom{0}\phantom{0}\infty}_{-\phantom{0}\phantom{0}7.37}$ | $\phantom{0}\phantom{0}3.02^{+\phantom{0}\phantom{0}0.89}_{-\phantom{0}\phantom{0}0.46}$ | $\phantom{0}\phantom{0}\phantom{0}4.03^{+\phantom{0}\phantom{0}\phantom{0}3.17}_{-\phantom{0}\phantom{0}\phantom{0}0.71}$ | $\phantom{0}\phantom{0}3.48^{+\phantom{0}\phantom{0}1.57}_{-\phantom{0}\phantom{0}0.47}$
0827+243 | $\phantom{0}38.45^{+\phantom{0}15.63}_{-\phantom{0}\phantom{0}9.82}$ | $\phantom{0}36.71^{+\phantom{0}19.39}_{-\phantom{0}\phantom{0}6.46}$ | $\phantom{0}\phantom{0}54.96^{+\phantom{0}\phantom{0}62.35}_{-\phantom{0}\phantom{0}19.98}$ | $104.78^{+\phantom{0}35.31}_{-\phantom{0}49.82}$
0829+046 | $\phantom{0}19.26^{+\phantom{0}\phantom{0}5.22}_{-\phantom{0}\phantom{0}3.54}$ | $\phantom{0}11.92^{+\phantom{0}\phantom{0}4.69}_{-\phantom{0}\phantom{0}2.54}$ | $\phantom{0}\phantom{0}20.54^{+\phantom{0}\phantom{0}47.24}_{-\phantom{0}\phantom{0}\phantom{0}8.67}$ | $\phantom{0}13.94^{+\phantom{0}\phantom{0}6.60}_{-\phantom{0}\phantom{0}2.07}$
0836+710 | $\phantom{0}70.65^{+\phantom{0}40.35}_{-\phantom{0}16.58}$ | $\phantom{0}63.84^{+\phantom{0}15.81}_{-\phantom{0}10.26}$ | $\phantom{0}\phantom{0}72.30^{+\phantom{0}\phantom{0}35.92}_{-\phantom{0}\phantom{0}21.60}$ | $\phantom{0}43.14^{+\phantom{0}18.39}_{-\phantom{0}\phantom{0}8.72}$
0851+202 | $396.16^{+140.87}_{-139.31}$ | $292.24^{+158.50}_{-\phantom{0}72.65}$ | $\phantom{0}106.49^{+\phantom{0}139.92}_{-\phantom{0}\phantom{0}55.79}$ | $115.43^{+\phantom{0}68.87}_{-\phantom{0}38.31}$
0954+658 | $174.39^{+276.35}_{-\phantom{0}69.36}$ | $104.06^{+\phantom{0}18.78}_{-\phantom{0}25.14}$ | $\phantom{0}\phantom{0}62.53^{+\phantom{0}\phantom{0}25.21}_{-\phantom{0}\phantom{0}23.99}$ | $\phantom{0}63.54^{+\phantom{0}12.34}_{-\phantom{0}20.40}$
1055+018 | $203.98^{+119.45}_{-\phantom{0}75.34}$ | $184.30^{+\phantom{0}56.49}_{-\phantom{0}36.58}$ | $\phantom{0}357.13^{+1494.43}_{-\phantom{0}235.97}$ | $184.30^{+818.63}_{-102.04}$
1101+384 | $\phantom{0}\phantom{0}3.73^{+\phantom{0}\phantom{0}0.71}_{-\phantom{0}\phantom{0}0.66}$ | $\phantom{0}\phantom{0}3.47^{+\phantom{0}\phantom{0}0.63}_{-\phantom{0}\phantom{0}0.39}$ | $\phantom{0}\phantom{0}\phantom{0}1.46^{+\phantom{0}\phantom{0}\phantom{0}0.52}_{-\phantom{0}\phantom{0}\phantom{0}0.42}$ | $\phantom{0}\phantom{0}1.80^{+\phantom{0}\phantom{0}0.45}_{-\phantom{0}\phantom{0}0.34}$
1127-145 | $\phantom{0}\phantom{0}8.55^{+\phantom{0}\phantom{0}9.66}_{-\phantom{0}\phantom{0}3.11}$ | $\phantom{0}28.62^{+\phantom{0}10.90}_{-\phantom{0}\phantom{0}8.08}$ | $\phantom{0}\phantom{0}17.48^{+\phantom{0}\phantom{0}18.07}_{-\phantom{0}\phantom{0}12.82}$ | $\phantom{0}17.76^{+\phantom{0}\phantom{0}6.38}_{-\phantom{0}\phantom{0}9.17}$
1156+295 | $\phantom{0}94.03^{+\phantom{0}23.29}_{-\phantom{0}14.38}$ | $\phantom{0}94.90^{+\phantom{0}23.50}_{-\phantom{0}15.25}$ | $\phantom{0}\phantom{0}26.16^{+\phantom{0}\phantom{0}26.20}_{-\phantom{0}\phantom{0}\phantom{0}9.51}$ | $\phantom{0}35.55^{+\phantom{0}17.66}_{-\phantom{0}11.41}$
1219+285 | $\phantom{0}\phantom{0}5.06^{+\phantom{0}\phantom{0}1.93}_{-\phantom{0}\phantom{0}2.57}$ | $\phantom{0}\phantom{0}4.13^{+\phantom{0}\phantom{0}1.17}_{-\phantom{0}\phantom{0}1.64}$ | $\phantom{0}\phantom{0}13.72^{+\phantom{0}\phantom{0}\phantom{0}9.65}_{-\phantom{0}\phantom{0}\phantom{0}7.60}$ | $\phantom{0}\phantom{0}8.59^{+\phantom{0}\phantom{0}3.47}_{-\phantom{0}\phantom{0}2.94}$
1222+216 | $\phantom{0}14.34^{+\phantom{0}15.64}_{-\phantom{0}\phantom{0}5.62}$ | $\phantom{0}41.39^{+\phantom{0}18.46}_{-\phantom{0}13.29}$ | $\phantom{0}\phantom{0}29.77^{+\phantom{0}\phantom{0}\phantom{0}8.15}_{-\phantom{0}\phantom{0}\phantom{0}8.56}$ | $\phantom{0}39.16^{+\phantom{0}16.69}_{-\phantom{0}\phantom{0}6.89}$
1226+023 | $\phantom{0}36.71^{+\phantom{0}10.38}_{-\phantom{0}10.13}$ | $\phantom{0}39.89^{+\phantom{0}11.75}_{-\phantom{0}\phantom{0}5.47}$ | $\phantom{0}140.09^{+\phantom{0}\phantom{0}56.50}_{-\phantom{0}\phantom{0}33.60}$ | $117.31^{+\phantom{0}25.06}_{-\phantom{0}15.85}$
1253-055 | $317.52^{+340.27}_{-\phantom{0}89.68}$ | $281.66^{+\phantom{0}38.81}_{-\phantom{0}60.03}$ | $\phantom{0}517.59^{+\phantom{0}208.74}_{-\phantom{0}111.26}$ | $485.24^{+195.69}_{-\phantom{0}78.91}$
1308+326 | $\phantom{0}36.04^{+\phantom{0}26.63}_{-\phantom{0}12.46}$ | $\phantom{0}33.79^{+\phantom{0}\phantom{0}9.55}_{-\phantom{0}\phantom{0}6.46}$ | $\phantom{0}127.17^{+\phantom{0}202.28}_{-\phantom{0}\phantom{0}89.25}$ | $485.24^{+163.52}_{-213.78}$
1406-076 | $\phantom{0}37.05^{+\phantom{0}12.72}_{-\phantom{0}\phantom{0}9.47}$ | $\phantom{0}31.10^{+\phantom{0}\phantom{0}6.64}_{-\phantom{0}11.12}$ | $\phantom{0}\phantom{0}44.56^{+\phantom{0}\phantom{0}95.53}_{-\phantom{0}\phantom{0}15.27}$ | $\phantom{0}42.45^{+\phantom{0}19.08}_{-\phantom{0}13.63}$
1510-089 | $\phantom{0}84.96^{+\phantom{0}58.73}_{-\phantom{0}22.86}$ | $\phantom{0}70.65^{+\phantom{0}15.89}_{-\phantom{0}20.42}$ | $\phantom{0}\phantom{0}73.47^{+\phantom{0}\phantom{0}62.17}_{-\phantom{0}\phantom{0}33.67}$ | $\phantom{0}84.96^{+\phantom{0}26.81}_{-\phantom{0}34.26}$
1611+343 | $\phantom{0}22.52^{+\phantom{0}\phantom{0}8.01}_{-\phantom{0}10.93}$ | $\phantom{0}16.92^{+\phantom{0}11.97}_{-\phantom{0}\phantom{0}6.45}$ | $\phantom{0}137.85^{+\phantom{0}156.42}_{-\phantom{0}\phantom{0}73.27}$ | $\phantom{0}15.36^{+\phantom{0}12.10}_{-\phantom{0}\phantom{0}4.93}$
1622-297 | $\phantom{0}\phantom{0}8.32^{+\phantom{0}26.10}_{-\phantom{0}\phantom{0}3.49}$ | $\phantom{0}\phantom{0}8.71^{+\phantom{0}\phantom{0}4.00}_{-\phantom{0}\phantom{0}3.56}$ | $\phantom{0}\phantom{0}90.62^{+\phantom{0}136.69}_{-\phantom{0}\phantom{0}51.46}$ | $109.98^{+\phantom{0}86.61}_{-\phantom{0}36.51}$
1633+382 | $166.53^{+\phantom{0}26.47}_{-\phantom{0}34.28}$ | $156.12^{+\phantom{0}18.26}_{-\phantom{0}22.65}$ | $\phantom{0}\phantom{0}18.64^{+\phantom{0}\phantom{0}\phantom{0}6.69}_{-\phantom{0}\phantom{0}\phantom{0}4.47}$ | $\phantom{0}14.87^{+\phantom{0}\phantom{0}7.04}_{-\phantom{0}\phantom{0}3.19}$
1641+399 | $147.72^{+\phantom{0}50.69}_{-\phantom{0}46.50}$ | $105.02^{+\phantom{0}26.01}_{-\phantom{0}23.90}$ | $\phantom{0}\phantom{0}36.13^{+\phantom{0}\phantom{0}\phantom{0}7.71}_{-\phantom{0}\phantom{0}\phantom{0}7.31}$ | $\phantom{0}27.91^{+\phantom{0}\phantom{0}4.88}_{-\phantom{0}\phantom{0}3.77}$
1652+398 | $\phantom{0}\phantom{0}0.72^{+\phantom{0}\phantom{0}0.38}_{-\phantom{0}\phantom{0}0.22}$ | $\phantom{0}\phantom{0}0.72^{+\phantom{0}\phantom{0}0.38}_{-\phantom{0}\phantom{0}0.22}$ | $\phantom{0}\phantom{0}\phantom{0}2.69^{+\phantom{0}\phantom{0}\phantom{0}0.63}_{-\phantom{0}\phantom{0}\phantom{0}0.61}$ | $\phantom{0}\phantom{0}2.69^{+\phantom{0}\phantom{0}0.63}_{-\phantom{0}\phantom{0}0.61}$
1730-130 | $209.70^{+151.57}_{-\phantom{0}77.45}$ | $153.27^{+\phantom{0}85.32}_{-\phantom{0}52.98}$ | $\phantom{0}\phantom{0}60.54^{+\phantom{0}\phantom{0}24.42}_{-\phantom{0}\phantom{0}18.09}$ | $\phantom{0}61.53^{+\phantom{0}30.56}_{-\phantom{0}14.76}$
1749+096 | $221.62^{+\phantom{0}92.98}_{-\phantom{0}64.06}$ | $247.55^{+\phantom{0}69.97}_{-\phantom{0}74.76}$ | $\phantom{0}\phantom{0}46.02^{+\phantom{0}\phantom{0}69.41}_{-\phantom{0}\phantom{0}17.20}$ | $\phantom{0}55.85^{+111.45}_{-\phantom{0}18.54}$
1959+650 | $\phantom{0}\phantom{0}1.64^{+\phantom{0}\phantom{0}\infty}_{-\phantom{0}\phantom{0}0.00}$ | $\phantom{0}\phantom{0}1.64^{+\phantom{0}\phantom{0}\infty}_{-\phantom{0}\phantom{0}0.00}$ | $\phantom{0}\phantom{0}\phantom{0}1.58^{+\phantom{0}\phantom{0}\phantom{0}3.47}_{-\phantom{0}\phantom{0}\phantom{0}1.27}$ | $\phantom{0}\phantom{0}1.58^{+\phantom{0}\phantom{0}3.47}_{-\phantom{0}\phantom{0}1.27}$
2200+420 | $329.45^{+270.41}_{-138.22}$ | $259.23^{+\phantom{0}85.76}_{-\phantom{0}49.53}$ | $\phantom{0}\phantom{0}\phantom{0}6.64^{+\phantom{0}\phantom{0}\phantom{0}1.68}_{-\phantom{0}\phantom{0}\phantom{0}1.08}$ | $\phantom{0}\phantom{0}9.78^{+\phantom{0}\phantom{0}1.35}_{-\phantom{0}\phantom{0}0.90}$
2223-052 | $\phantom{0}10.77^{+\phantom{0}\phantom{0}2.06}_{-\phantom{0}\phantom{0}1.81}$ | $\phantom{0}12.14^{+\phantom{0}\phantom{0}2.06}_{-\phantom{0}\phantom{0}1.67}$ | $\phantom{0}\phantom{0}19.26^{+\phantom{0}\phantom{0}\phantom{0}8.65}_{-\phantom{0}\phantom{0}\phantom{0}5.32}$ | $\phantom{0}40.45^{+\phantom{0}22.08}_{-\phantom{0}13.43}$
2230+114 | $335.58^{+154.15}_{-\phantom{0}96.99}$ | $145.02^{+\phantom{0}93.57}_{-\phantom{0}23.30}$ | $\phantom{0}\phantom{0}24.53^{+\phantom{0}\phantom{0}15.27}_{-\phantom{0}\phantom{0}\phantom{0}7.88}$ | $\phantom{0}28.36^{+\phantom{0}12.74}_{-\phantom{0}\phantom{0}6.45}$
2251+158 | $149.09^{+\phantom{0}80.86}_{-\phantom{0}55.06}$ | $256.85^{+128.50}_{-\phantom{0}94.86}$ | $1122.85^{+\phantom{0}669.92}_{\phantom{0}-212.45}$ | $774.75^{+212.13}_{-204.55}$

We note that this analysis is based on a few approximations, such as non-
variability of the brightness temperature of components and similar properties
of sources within each subclass. It is not expected _a priori_ that the
survival function of the component brightness temperatures for a source should
be the same for different time periods (e.g., the faster decrease in the
survival function for knots observed from 2013 January to 2018 December versus
knots observed during the entire time period for the BL 0219+428 near
brightness temperatures $\sim 10^{8}$ K, as in Fig. A1). However, temporal
analysis of the brightness temperatures in the jets of blazars as well as an
investigation into the assumptions outlined above are beyond the scope of this
paper, which is focused on the kinematics of the features. We plan to present
such an analysis in a separate publication (Weaver, in prep.).

## References

  * Abdo et al. (2010) Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010, ApJS, 188, 405, doi: 10.1088/0067-0049/188/2/405
  * Acero et al. (2015) Acero, F., Ackermann, M., Ajello, M., et al. 2015, ApJS, 218, 23, doi: 10.1088/0067-0049/218/2/23
  * Agudo (2009) Agudo, I. 2009, in Astronomical Society of the Pacific Conference Series, Vol. 402, Approaching Micro-Arcsecond Resolution with VSOP-2: Astrophysics and Technologies, ed. Y. Hagiwara, E. Fomalont, M. Tsuboi, & M. Yasuhiro, 330.  https://arxiv.org/abs/0803.3388
  * Agudo et al. (2012) Agudo, I., Marscher, A. P., Jorstad, S. G., et al. 2012, ApJ, 747, 63, doi: 10.1088/0004-637X/747/1/63
  * Agudo et al. (2007) Agudo, I., Bach, U., Krichbaum, T. P., et al. 2007, A&A, 476, L17, doi: 10.1051/0004-6361:20078448
  * Agudo et al. (2011a) Agudo, I., Jorstad, S. G., Marscher, A. P., et al. 2011a, ApJ, 726, L13, doi: 10.1088/2041-8205/726/1/L13
  * Agudo et al. (2011b) Agudo, I., Marscher, A. P., Jorstad, S. G., et al. 2011b, ApJ, 735, L10, doi: 10.1088/2041-8205/735/1/L10
  * Aharonian et al. (2007) Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., et al. 2007, ApJ, 664, L71, doi: 10.1086/520635
  * Ajello et al. (2020) Ajello, M., Angioni, R., Axelsson, M., et al. 2020, ApJ, 892, 105, doi: 10.3847/1538-4357/ab791e
  * Akritas et al. (1995) Akritas, M. G., Murphy, S. A., & Lavalley, M. P. 1995, Journal of the American Statistical Association, 90, 170, doi: 10.1080/01621459.1995.10476499
  * Akritas & Siebert (1996) Akritas, M. G., & Siebert, J. 1996, MNRAS, 278, 919, doi: 10.1093/mnras/278.4.919
  * Aleksić et al. (2011) Aleksić, J., Antonelli, L. A., Antoranz, P., et al. 2011, ApJ, 730, L8, doi: 10.1088/2041-8205/730/1/L8
  * Angel & Stockman (1980) Angel, J. R. P., & Stockman, H. S. 1980, ARA&A, 18, 321, doi: 10.1146/annurev.aa.18.090180.001541
  * Asada et al. (2010) Asada, K., Nakamura, M., Inoue, M., Kameno, S., & Nagai, H. 2010, ApJ, 720, 41, doi: 10.1088/0004-637X/720/1/41
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Attridge et al. (1999) Attridge, J. M., Roberts, D. H., & Wardle, J. F. C. 1999, ApJ, 518, L87, doi: 10.1086/312078
  * Avni et al. (1980) Avni, Y., Soltan, A., Tananbaum, H., & Zamorani, G. 1980, ApJ, 238, 800, doi: 10.1086/158040
  * Avni & Tananbaum (1986) Avni, Y., & Tananbaum, H. 1986, ApJ, 305, 83, doi: 10.1086/164230
  * Bach et al. (2005) Bach, U., Krichbaum, T. P., Ros, E., et al. 2005, A&A, 433, 815, doi: 10.1051/0004-6361:20040388
  * Beskin & Kuznetsova (2000) Beskin, V. S., & Kuznetsova, I. V. 2000, Nuovo Cimento B Serie, 115, 795.  https://arxiv.org/abs/astro-ph/0004021
  * Blandford et al. (1977) Blandford, R. D., McKee, C. F., & Rees, M. J. 1977, Nature, 267, 211, doi: 10.1038/267211a0
  * Blandford & Payne (1982) Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883, doi: 10.1093/mnras/199.4.883
  * Blandford & Znajek (1977) Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433, doi: 10.1093/mnras/179.3.433
  * Blasi et al. (2013) Blasi, M. G., Lico, R., Giroletti, M., et al. 2013, A&A, 559, A75, doi: 10.1051/0004-6361/201321858
  * Bowker & Lieberman (1972) Bowker, A., & Lieberman, G. 1972 (Englewood Cliffs NJ: Prentice-Hall, Inc.)
  * Britzen et al. (2019) Britzen, S., Fendt, C., Zajaček, M., et al. 2019, Galaxies, 7, 72, doi: 10.3390/galaxies7030072
  * Britzen et al. (2005) Britzen, S., Witzel, A., Krichbaum, T. P., et al. 2005, MNRAS, 362, 966, doi: 10.1111/j.1365-2966.2005.09369.x
  * Britzen et al. (2017) Britzen, S., Qian, S. J., Steffen, W., et al. 2017, A&A, 602, A29, doi: 10.1051/0004-6361/201629999
  * Britzen et al. (2018) Britzen, S., Fendt, C., Witzel, G., et al. 2018, MNRAS, 478, 3199, doi: 10.1093/mnras/sty1026
  * Bruni et al. (2021) Bruni, G., Gómez, J. L., Vega-García, L., et al. 2021, arXiv e-prints, arXiv:2101.07324.  https://arxiv.org/abs/2101.07324
  * Casadio et al. (2015) Casadio, C., Gómez, J. L., Jorstad, S. G., et al. 2015, ApJ, 813, 51, doi: 10.1088/0004-637X/813/1/51
  * Casadio et al. (2021) Casadio, C., MacDonald, N. R., Boccardi, B., et al. 2021, arXiv e-prints, arXiv:2102.08952.  https://arxiv.org/abs/2102.08952
  * Celotti et al. (2001) Celotti, A., Ghisellini, G., & Chiaberge, M. 2001, MNRAS, 321, L1, doi: 10.1046/j.1365-8711.2001.04160.x
  * Cerruti et al. (2019) Cerruti, M., Zech, A., Boisson, C., et al. 2019, MNRAS, 483, L12, doi: 10.1093/mnrasl/sly210
  * Chatterjee et al. (2019) Chatterjee, K., Liska, M., Tchekhovskoy, A., & Markoff, S. B. 2019, MNRAS, 490, 2200, doi: 10.1093/mnras/stz2626
  * Chatterjee et al. (2011) Chatterjee, R., Marscher, A. P., Jorstad, S. G., & et al. 2011, ApJ, 734, 43, doi: 10.1088/0004-637X/734/1/43
  * Chen (2018) Chen, L. 2018, ApJS, 235, 39, doi: 10.3847/1538-4365/aab8fb
  * Clausen-Brown et al. (2013) Clausen-Brown, E., Savolainen, T., Pushkarev, A. B., Kovalev, Y. Y., & Zensus, J. A. 2013, A&A, 558, A144, doi: 10.1051/0004-6361/201322203
  * Cohen et al. (1971) Cohen, M. H., Cannon, W., Purcell, G. H., et al. 1971, ApJ, 170, 207, doi: 10.1086/151204
  * Cohen et al. (1977) Cohen, M. H., Kellermann, K. I., Shaffer, D. B., et al. 1977, Nature, 268, 405, doi: 10.1038/268405a0
  * Conway & Wrobel (1995) Conway, J. E., & Wrobel, J. M. 1995, ApJ, 439, 98, doi: 10.1086/175155
  * D’Arcangelo et al. (2009) D’Arcangelo, F. D., Marscher, A. P., Jorstad, S. G., et al. 2009, ApJ, 697, 985, doi: 10.1088/0004-637X/697/2/985
  * Davidson-Pilon (2019) Davidson-Pilon, C. 2019, Journal of Open Source Software, 4, 1317, doi: 10.21105/joss.01317
  * Event Horizon Telescope Collaboration et al. (2019) Event Horizon Telescope Collaboration, Akiyama, K., Alberdi, A., et al. 2019, ApJ, 875, L1, doi: 10.3847/2041-8213/ab0ec7
  * Fan et al. (2014) Fan, J.-H., Bastieri, D., Yang, J.-H., et al. 2014, Research in Astronomy and Astrophysics, 14, 1135, doi: 10.1088/1674-4527/14/9/004
  * Fan et al. (2013) Fan, J.-H., Yang, J.-H., Liu, Y., & Zhang, J.-Y. 2013, Research in Astronomy and Astrophysics, 13, 259, doi: 10.1088/1674-4527/13/3/002
  * Feigelson & Babu (2012) Feigelson, E. D., & Babu, G. J. 2012, Modern Statistical Methods for Astronomy
  * Feigelson & Nelson (1985) Feigelson, E. D., & Nelson, P. I. 1985, ApJ, 293, 192, doi: 10.1086/163225
  * Fromm et al. (2013) Fromm, C. M., Ros, E., Perucho, M., et al. 2013, A&A, 557, A105, doi: 10.1051/0004-6361/201321784
  * Fuhrmann et al. (2014) Fuhrmann, L., Larsson, S., Chiang, J., et al. 2014, MNRAS, 441, 1899, doi: 10.1093/mnras/stu540
  * Gabuzda et al. (2014) Gabuzda, D. C., Reichstein, A. R., & O’Neill, E. L. 2014, MNRAS, 444, 172, doi: 10.1093/mnras/stu1381
  * Ghisellini et al. (1993) Ghisellini, G., Padovani, P., Celotti, A., & Maraschi, L. 1993, ApJ, 407, 65, doi: 10.1086/172493
  * Ghisellini et al. (2005) Ghisellini, G., Tavecchio, F., & Chiaberge, M. 2005, A&A, 432, 401, doi: 10.1051/0004-6361:20041404
  * Giovannini et al. (2018) Giovannini, G., Savolainen, T., Orienti, M., et al. 2018, Nature Astronomy, 2, 472, doi: 10.1038/s41550-018-0431-2
  * Giroletti et al. (2004) Giroletti, M., Giovannini, G., Feretti, L., et al. 2004, ApJ, 600, 127, doi: 10.1086/379663
  * González-Morán et al. (2021) González-Morán, A. L., Chávez, R., Terlevich, E., & et al. 2021, MNRAS, 505, 1441, doi: 10.1093/mnras/stab1385
  * H. E. S. S. Collaboration et al. (2021) H. E. S. S. Collaboration, Abdalla, H., Adam, R., et al. 2021, A&A, 648, A23, doi: 10.1051/0004-6361/202038949
  * Hallum et al. (2022) Hallum, M. K., Jorstad, S. G., Larionov, V. M., et al. 2022, arXiv e-prints, arXiv:2202.00061.  https://arxiv.org/abs/2202.00061
  * Hartman et al. (1999) Hartman, R. C., Bertsch, D. L., Bloom, S. D., et al. 1999, ApJS, 123, 79, doi: 10.1086/313231
  * Homan et al. (2009) Homan, D. C., Kadler, M., Kellermann, K. I., et al. 2009, ApJ, 706, 1253, doi: 10.1088/0004-637X/706/2/1253
  * Homan et al. (2015) Homan, D. C., Lister, M. L., Kovalev, Y. Y., et al. 2015, ApJ, 798, 134, doi: 10.1088/0004-637X/798/2/134
  * Homan et al. (2002) Homan, D. C., Ojha, R., Wardle, J. F. C., et al. 2002, ApJ, 568, 99, doi: 10.1086/338701
  * Homan et al. (2006) Homan, D. C., Kovalev, Y. Y., Lister, M. L., et al. 2006, ApJ, 642, L115, doi: 10.1086/504715
  * IceCube Collaboration et al. (2018a) IceCube Collaboration, Aartsen, M. G., Ackermann, M., et al. 2018a, Science, 361, 147, doi: 10.1126/science.aat2890
  * IceCube Collaboration et al. (2018b) —. 2018b, Science, 361, eaat1378, doi: 10.1126/science.aat1378
  * Isler et al. (2015) Isler, J. C., Urry, C. M., Bailyn, C., et al. 2015, ApJ, 804, 7, doi: 10.1088/0004-637X/804/1/7
  * Isobe et al. (1986) Isobe, T., Feigelson, E. D., & Nelson, P. I. 1986, ApJ, 306, 490, doi: 10.1086/164359
  * Jekel & Venter (2019) Jekel, C. F., & Venter, G. 2019, pwlf: A Python Library for Fitting 1D Continuous Piecewise Linear Functions.  https://github.com/cjekel/piecewise_linear_fit_py
  * Jorstad & Marscher (2016) Jorstad, S., & Marscher, A. 2016, Galaxies, 4, doi: 10.3390/galaxies4040047
  * Jorstad et al. (2004) Jorstad, S. G., Marscher, A. P., Lister, M. L., et al. 2004, AJ, 127, 3115, doi: 10.1086/420996
  * Jorstad et al. (2001a) Jorstad, S. G., Marscher, A. P., Mattox, J. R., et al. 2001a, ApJ, 556, 738, doi: 10.1086/321605
  * Jorstad et al. (2001b) —. 2001b, ApJS, 134, 181, doi: 10.1086/320858
  * Jorstad et al. (2005) Jorstad, S. G., Marscher, A. P., Lister, M. L., et al. 2005, AJ, 130, 1418, doi: 10.1086/444593
  * Jorstad et al. (2013) Jorstad, S. G., Marscher, A. P., Smith, P. S., et al. 2013, ApJ, 773, 147, doi: 10.1088/0004-637X/773/2/147
  * Jorstad et al. (2017) Jorstad, S. G., Marscher, A. P., Morozova, D. A., et al. 2017, ApJ, 846, 98, doi: 10.3847/1538-4357/aa8407
  * Joshi et al. (2014) Joshi, M., Marscher, A. P., & Böttcher, M. 2014, ApJ, 785, 132, doi: 10.1088/0004-637X/785/2/132
  * Kaplan & Meier (1958) Kaplan, E. L., & Meier, P. 1958, Journal of the American Statistical Association, 53, 457.  http://www.jstor.org/stable/2281868
  * Keivani et al. (2018) Keivani, A., Murase, K., Petropoulou, M., et al. 2018, ApJ, 864, 84, doi: 10.3847/1538-4357/aad59a
  * Kellermann et al. (1998) Kellermann, K. I., Vermeulen, R. C., Zensus, J. A., & Cohen, M. H. 1998, AJ, 115, 1295, doi: 10.1086/300308
  * Kellermann et al. (2004) Kellermann, K. I., Lister, M. L., Homan, D. C., et al. 2004, ApJ, 609, 539, doi: 10.1086/421289
  * Komissarov (2001) Komissarov, S. S. 2001, MNRAS, 326, L41, doi: 10.1046/j.1365-8711.2001.04863.x
  * Komissarov (2005) —. 2005, MNRAS, 359, 801, doi: 10.1111/j.1365-2966.2005.08974.x
  * Königl (1981) Königl, A. 1981, ApJ, 243, 700, doi: 10.1086/158638
  * Kovalev et al. (2020) Kovalev, Y. Y., Pushkarev, A. B., Nokhrina, E. E., et al. 2020, MNRAS, 495, 3576, doi: 10.1093/mnras/staa1121
  * Kovalev et al. (2009) Kovalev, Y. Y., Aller, H. D., Aller, M. F., et al. 2009, ApJ, 696, L17, doi: 10.1088/0004-637X/696/1/L17
  * Kovalev et al. (2016) Kovalev, Y. Y., Kardashev, N. S., Kellermann, K. I., et al. 2016, ApJ, 820, L9, doi: 10.3847/2041-8205/820/1/L9
  * Krawczynski et al. (2004) Krawczynski, H., Hughes, S. B., Horan, D., et al. 2004, ApJ, 601, 151, doi: 10.1086/380393
  * Lähteenmäki & Valtaoja (1999) Lähteenmäki, A., & Valtaoja, E. 1999, ApJ, 521, 493, doi: 10.1086/307587
  * Lähteenmäki & Valtaoja (2003) —. 2003, ApJ, 590, 95, doi: 10.1086/374883
  * Larionov et al. (2013) Larionov, V. M., Jorstad, S. G., Marscher, A. P., et al. 2013, ApJ, 768, 40, doi: 10.1088/0004-637X/768/1/40
  * Larionov et al. (2020) —. 2020, MNRAS, 492, 3829, doi: 10.1093/mnras/staa082
  * LaValley et al. (1992) LaValley, M., Isobe, T., & Feigelson, E. 1992, in Astronomical Society of the Pacific Conference Series, Vol. 25, Astronomical Data Analysis Software and Systems I, ed. D. M. Worrall, C. Biemesderfer, & J. Barnes, 245
  * León-Tavares et al. (2015) León-Tavares, J., Chavushyan, V., Lobanov, A., Valtaoja, E., & Arshakian, T. G. 2015, in Extragalactic Jets from Every Angle, ed. F. Massaro, C. C. Cheung, E. Lopez, & A. Siemiginowska, Vol. 313, 43–47, doi: 10.1017/S1743921315001842
  * León-Tavares et al. (2011) León-Tavares, J., Valtaoja, E., Tornikoski, M., Lähteenmäki, A., & Nieppola, E. 2011, A&A, 532, A146, doi: 10.1051/0004-6361/201116664
  * León-Tavares et al. (2013) León-Tavares, J., Chavushyan, V., Patiño-Álvarez, V., et al. 2013, ApJ, 763, L36, doi: 10.1088/2041-8205/763/2/L36
  * Li et al. (2020) Li, X., An, T., Mohan, P., & Giroletti, M. 2020, ApJ, 896, 63, doi: 10.3847/1538-4357/ab8f9f
  * Liodakis et al. (2018) Liodakis, I., Hovatta, T., Huppenkothen, D., et al. 2018, ApJ, 866, 137, doi: 10.3847/1538-4357/aae2b7
  * Liodakis & Pavlidou (2015) Liodakis, I., & Pavlidou, V. 2015, MNRAS, 454, 1767, doi: 10.1093/mnras/stv2028
  * Liodakis et al. (2017a) Liodakis, I., Pavlidou, V., Hovatta, T., et al. 2017a, MNRAS, 467, 4565, doi: 10.1093/mnras/stx432
  * Liodakis et al. (2017b) Liodakis, I., Zezas, A., Angelakis, E., Hovatta, T., & Pavlidou, V. 2017b, A&A, 602, A104, doi: 10.1051/0004-6361/201629902
  * Liodakis et al. (2020) Liodakis, I., Blinov, D., Jorstad, S. G., et al. 2020, ApJ, 902, 61, doi: 10.3847/1538-4357/abb1b8
  * Liska et al. (2018) Liska, M., Hesp, C., Tchekhovskoy, A., et al. 2018, MNRAS, 474, L81, doi: 10.1093/mnrasl/slx174
  * Lister (2001) Lister, M. L. 2001, ApJ, 561, 676, doi: 10.1086/323528
  * Lister et al. (2018) Lister, M. L., Aller, M. F., Aller, H. D., et al. 2018, ApJS, 234, 12, doi: 10.3847/1538-4365/aa9c44
  * Lister & Marscher (1997) Lister, M. L., & Marscher, A. P. 1997, ApJ, 476, 572, doi: 10.1086/303629
  * Lister et al. (2009) Lister, M. L., Cohen, M. H., Homan, D. C., et al. 2009, AJ, 138, 1874, doi: 10.1088/0004-6256/138/6/1874
  * Lister et al. (2013) Lister, M. L., Aller, M. F., Aller, H. D., et al. 2013, AJ, 146, 120, doi: 10.1088/0004-6256/146/5/120
  * Lister et al. (2016) —. 2016, AJ, 152, 12, doi: 10.3847/0004-6256/152/1/12
  * Lister et al. (2019) Lister, M. L., Homan, D. C., Hovatta, T., et al. 2019, ApJ, 874, 43, doi: 10.3847/1538-4357/ab08ee
  * Lobanov & Roland (2005) Lobanov, A. P., & Roland, J. 2005, A&A, 431, 831, doi: 10.1051/0004-6361:20041831
  * Longair (2011) Longair, M. S. 2011, High Energy Astrophysics
  * Lyubarsky (2009) Lyubarsky, Y. 2009, ApJ, 698, 1570, doi: 10.1088/0004-637X/698/2/1570
  * MacDonald et al. (2017) MacDonald, N. R., Jorstad, S. G., & Marscher, A. P. 2017, ApJ, 850, 87, doi: 10.3847/1538-4357/aa92c8
  * MacDonald et al. (2015) MacDonald, N. R., Marscher, A. P., Jorstad, S. G., & Joshi, M. 2015, ApJ, 804, 111, doi: 10.1088/0004-637X/804/2/111
  * MAGIC Collaboration et al. (2021) MAGIC Collaboration, Acciari, V. A., Ansoldi, S., et al. 2021, A&A, 647, A163, doi: 10.1051/0004-6361/202039687
  * Mann & Whitney (1947) Mann, H. B., & Whitney, D. R. 1947, The Annals of Mathematical Statistics, 18, 50 , doi: 10.1214/aoms/1177730491
  * Marscher (1987) Marscher, A. P. 1987, in Superluminal Radio Sources, ed. J. A. Zensus & T. J. Pearson, 280–300
  * Marscher (1990) Marscher, A. P. 1990, in Parsec-scale radio jets, ed. J. A. Zensus & T. J. Pearson, 236
  * Marscher & Gear (1985) Marscher, A. P., & Gear, W. K. 1985, ApJ, 298, 114, doi: 10.1086/163592
  * Marscher et al. (2002) Marscher, A. P., Jorstad, S. G., Gómez, J.-L., et al. 2002, Nature, 417, 625, doi: 10.1038/nature00772
  * Marscher et al. (2010) Marscher, A. P., Jorstad, S. G., Larionov, V. M., et al. 2010, ApJ, 710, L126, doi: 10.1088/2041-8205/710/2/L126
  * Marti-Vidal et al. (2013) Marti-Vidal, I., Marcaide, J. M., Alberdi, A., & Brunthaler, A. 2013, arXiv e-prints, arXiv:1301.4782.  https://arxiv.org/abs/1301.4782
  * Max-Moerbeck et al. (2014) Max-Moerbeck, W., Hovatta, T., Richards, J. L., et al. 2014, MNRAS, 445, 428, doi: 10.1093/mnras/stu1749
  * McHardy et al. (1990) McHardy, I. M., Marscher, A. P., Gear, W. K., et al. 1990, MNRAS, 246, 305
  * Meier (2012) Meier, D. L. 2012, Black Hole Astrophysics: The Engine Paradigm
  * Mertens et al. (2016) Mertens, F., Lobanov, A. P., Walker, R. C., & Hardee, P. E. 2016, A&A, 595, A54, doi: 10.1051/0004-6361/201628829
  * Mignone et al. (2010) Mignone, A., Rossi, P., Bodo, G., Ferrari, A., & Massaglia, S. 2010, MNRAS, 402, 7, doi: 10.1111/j.1365-2966.2009.15642.x
  * Mimica et al. (2015) Mimica, P., Giannios, D., Metzger, B. D., & Aloy, M. A. 2015, MNRAS, 450, 2824, doi: 10.1093/mnras/stv825
  * Moll et al. (2008) Moll, R., Spruit, H. C., & Obergaulinger, M. 2008, A&A, 492, 621, doi: 10.1051/0004-6361:200810523
  * Morozova et al. (2014) Morozova, D. A., Larionov, V. M., Troitsky, I. S., et al. 2014, AJ, 148, 42, doi: 10.1088/0004-6256/148/3/42
  * Morris et al. (2019) Morris, T., Jarvis, C., Cragg, W., et al. 2019, BioMedical Journal Open, 9, doi: 10.1136/bmjopen-2019-030215
  * Murase et al. (2018) Murase, K., Oikonomou, F., & Petropoulou, M. 2018, ApJ, 865, 124, doi: 10.3847/1538-4357/aada00
  * Nagai et al. (2014) Nagai, H., Haga, T., Giovannini, G., et al. 2014, ApJ, 785, 53, doi: 10.1088/0004-637X/785/1/53
  * Nakamura et al. (2001) Nakamura, M., Uchida, Y., & Hirose, S. 2001, New A, 6, 61, doi: 10.1016/S1384-1076(01)00041-0
  * Nalewajko et al. (2014) Nalewajko, K., Begelman, M. C., & Sikora, M. 2014, ApJ, 789, 161, doi: 10.1088/0004-637X/789/2/161
  * Nieppola et al. (2011) Nieppola, E., Tornikoski, M., Valtaoja, E., et al. 2011, A&A, 535, A69, doi: 10.1051/0004-6361/201116818
  * O’Sullivan & Gabuzda (2009) O’Sullivan, S. P., & Gabuzda, D. C. 2009, MNRAS, 400, 26, doi: 10.1111/j.1365-2966.2009.15428.x
  * Pacholczyk (1970) Pacholczyk, A. G. 1970, Radio astrophysics. Nonthermal processes in galactic and extragalactic sources
  * Pelletier & Roland (1989) Pelletier, G., & Roland, J. 1989, A&A, 224, 24
  * Perucho et al. (2012) Perucho, M., Kovalev, Y. Y., Lobanov, A. P., Hardee, P. E., & Agudo, I. 2012, ApJ, 749, 55, doi: 10.1088/0004-637X/749/1/55
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., & et al. 2020, A&A, 641, 6, doi: 10.1051/0004-6361/201833910
  * Pu et al. (2015) Pu, H.-Y., Nakamura, M., Hirotani, K., et al. 2015, ApJ, 801, 56, doi: 10.1088/0004-637X/801/1/56
  * Pushkarev et al. (2005) Pushkarev, A. B., Gabuzda, D. C., Vetukhnovskaya, Y. N., & Yakimov, V. E. 2005, MNRAS, 356, 859, doi: 10.1111/j.1365-2966.2004.08535.x
  * Pushkarev et al. (2009) Pushkarev, A. B., Kovalev, Y. Y., Lister, M. L., & Savolainen, T. 2009, A&A, 507, L33, doi: 10.1051/0004-6361/200913422
  * Pushkarev et al. (2017) —. 2017, MNRAS, 468, 4992, doi: 10.1093/mnras/stx854
  * Qian et al. (2019) Qian, S. J., Britzen, S., Krichbaum, T. P., & Witzel, A. 2019, A&A, 621, A11, doi: 10.1051/0004-6361/201833508
  * Raiteri et al. (2017) Raiteri, C. M., Villata, M., Acosta-Pulido, J. A., et al. 2017, Nature, 552, 374, doi: 10.1038/nature24623
  * Raiteri et al. (2021) Raiteri, C. M., Villata, M., Larionov, V. M., et al. 2021, MNRAS, 504, 5629, doi: 10.1093/mnras/stab1268
  * Readhead (1994) Readhead, A. C. S. 1994, ApJ, 426, 51, doi: 10.1086/174038
  * Richards et al. (2011) Richards, J. L., Max-Moerbeck, W., Pavlidou, V., et al. 2011, ApJS, 194, 29, doi: 10.1088/0067-0049/194/2/29
  * Riess et al. (2016) Riess, A. G., Macri, L. M., Hoffmann, S. L., et al. 2016, ApJ, 826, 56, doi: 10.3847/0004-637X/826/1/56
  * Rodrigues et al. (2018) Rodrigues, X., Fedynitch, A., Gao, S., Boncioli, D., & Winter, W. 2018, ApJ, 854, 54, doi: 10.3847/1538-4357/aaa7ee
  * Savolainen et al. (2002) Savolainen, T., Wiik, K., Valtaoja, E., Jorstad, S. G., & Marscher, A. P. 2002, A&A, 394, 851, doi: 10.1051/0004-6361:20021236
  * Savolainen et al. (2006) Savolainen, T., Wiik, K., Valtaoja, E., & Tornikoski, M. 2006, A&A, 446, 71, doi: 10.1051/0004-6361:20053753
  * Schmitt (1985) Schmitt, J. H. M. M. 1985, ApJ, 293, 178, doi: 10.1086/163224
  * Seabold & Perktold (2010) Seabold, S., & Perktold, J. 2010, in 9th Python in Science Conference
  * Shepherd (1997) Shepherd, M. C. 1997, Astronomical Society of the Pacific Conference Series, Vol. 125, Difmap: an Interactive Program for Synthesis Imaging, ed. G. Hunt & H. Payne, 77
  * Sobacchi et al. (2021) Sobacchi, E., Nättilä, J., & Sironi, L. 2021, MNRAS, 503, 688, doi: 10.1093/mnras/stab562
  * Sol et al. (1989) Sol, H., Pelletier, G., & Asseo, E. 1989, MNRAS, 237, 411, doi: 10.1093/mnras/237.2.411
  * Stirling et al. (2003) Stirling, A. M., Cawthorne, T. V., Stevens, J. A., et al. 2003, MNRAS, 341, 405, doi: 10.1046/j.1365-8711.2003.06448.x
  * Tavecchio & Ghisellini (2008) Tavecchio, F., & Ghisellini, G. 2008, MNRAS, 386, 945, doi: 10.1111/j.1365-2966.2008.13072.x
  * Tavecchio et al. (2010) Tavecchio, F., Ghisellini, G., Bonnoli, G., & Ghirlanda, G. 2010, MNRAS, 405, L94, doi: 10.1111/j.1745-3933.2010.00867.x
  * Tchekhovskoy et al. (2010) Tchekhovskoy, A., Narayan, R., & McKinney, J. C. 2010, ApJ, 711, 50, doi: 10.1088/0004-637X/711/1/50
  * Terasranta & Valtaoja (1994) Terasranta, H., & Valtaoja, E. 1994, A&A, 283, 51
  * Urry & Padovani (1995) Urry, C. M., & Padovani, P. 1995, PASP, 107, 803, doi: 10.1086/133630
  * Valtaoja & Terasranta (1995) Valtaoja, E., & Terasranta, H. 1995, A&A, 297, L13
  * van Moorsel et al. (1996) van Moorsel, G., Kemball, A., & Greisen, E. 1996, Astronomical Society of the Pacific Conference Series, Vol. 101, AIPS Developments in the Nineties, ed. G. H. Jacoby & J. Barnes, 37
  * Vermeulen & Cohen (1994) Vermeulen, R. C., & Cohen, M. H. 1994, ApJ, 430, 467, doi: 10.1086/174424
  * Villata & Raiteri (1999) Villata, M., & Raiteri, C. M. 1999, A&A, 347, 30
  * Vlahakis & Königl (2003) Vlahakis, N., & Königl, A. 2003, ApJ, 596, 1080, doi: 10.1086/378226
  * Wardle et al. (1984) Wardle, J. F. C., Moore, R. L., & Angel, J. R. P. 1984, ApJ, 279, 93, doi: 10.1086/161868
  * Weaver et al. (2019) Weaver, Z. R., Balonek, T. J., Jorstad, S. G., et al. 2019, ApJ, 875, 15, doi: 10.3847/1538-4357/ab0e7c
  * Weaver et al. (2020) Weaver, Z. R., Williamson, K. E., Jorstad, S. G., et al. 2020, ApJ, 900, 137, doi: 10.3847/1538-4357/aba693
  * Wehrle et al. (2012) Wehrle, A. E., Marscher, A. P., Jorstad, S. G., et al. 2012, ApJ, 758, 72, doi: 10.1088/0004-637X/758/2/72
  * Williamson et al. (2014) Williamson, K. E., Jorstad, S. G., Marscher, A. P., et al. 2014, ApJ, 789, 135, doi: 10.1088/0004-637X/789/2/135
  * Xie et al. (2012) Xie, W., Lei, W.-H., Zou, Y.-C., et al. 2012, Research in Astronomy and Astrophysics, 12, 817, doi: 10.1088/1674-4527/12/7/010
  * Yang et al. (2021) Yang, S., Yan, D., Zhang, P., Dai, B., & Zhang, L. 2021, ApJ, 907, 105, doi: 10.3847/1538-4357/abcbff
  * Zhang et al. (2020) Zhang, L., Chen, S., Xiao, H., Cai, J., & Fan, J. 2020, ApJ, 897, 10, doi: 10.3847/1538-4357/ab9180


* [148] O. Tsai, _The STAR Forward Upgrade – An Overview_ , talk given at the XXVIII International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS21), Stony Brook (NY), USA, April 2021, STAR upgrade webpage.
  * [149] A. Bazilevsky, _Cold QCD physics program with sPHENIX_ , talk given at the XXVIII International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS21), Stony Brook (NY), USA, April 2021, and sPHENIX webpage.
  * [150] F. Close, _Particle Physics - A Very Short Introduction_ , Oxford University Press (2004), ISBN 9780192804341.
  * [151] J. Trefil, _From Atoms to Quarks - An Introduction to the Strange World of Particle Physics_ , Charles Scribner’s Sons (1980), ISBN 978-0684164847.
  * [152] M. Thomson, _Modern particle physics_ , Cambridge University Press (2013), ISBN 9781107034266.
  * [153] P.A. Zyla et al. [Particle Data Group], _Review of Particle Physics_ , PTEP 2020, no.8, 083C01 (2020), 10.1093/ptep/ptaa104.
  * [154] A. Bacchetta, U. D’Alesio, M. Diehl, C. A. Miller, _Single-spin asymmetries: The Trento conventions_ , Phys. Rev. D 70, 117504(2004), 10.1103/PhysRevD.70.117504.