MOJAVE. XIX. Brightness Temperatures and Intrinsic Properties of Blazar Jets

Homan et al. (2006) used the 25% point in the brightness temperature distribution of a given source as its characteristic brightness temperature; however, that choice was driven by the desire to avoid too many lower limits in a relatively small set of brightness temperature measurements available at the time. Our new data set is far larger, both in terms of numbers of epochs on individual sources and for the number of sources in our sample as whole. Consequently we now simply use the median brightness temperature of a given source as its characteristic brightness temperature. Only those jets that have a median T_b value, not a limit, are used in the analysis. Limits are ambiguous in the statistical comparison and do not allow robust estimates of the relativistic properties. Fortunately only twelve of the 321 sources with viable observed speeds have median brightness temperature limits, and none of them are part of the MOJAVE 1.5 Jy QC flux-density-limited subsample.

In addition to summarizing the brightness temperature properties of each AGN jet, Table 3 also includes a summary of the distribution of apparent speed of features reported in MOJAVEXVIII. For characterizing the speed distribution of a given source, we only consider features with significant motions, ≥ 3σ, in the approaching jet and discard those features identified as "inward" moving in MOJAVEXVIII. For each source, Table 3 reports the number of measured speeds, N_s , which meet these criteria and lists the maximum apparent speed, median apparent speed, and speed of the feature that was closest to the VLBI core in its first measured epoch. Unlike MOJAVEXVIII, which required at least five robust features to identify a median speed, here we report a fastest, median, and closest speed for every jet with at least one motion meeting the criteria described above.

In our previous papers we have taken the fastest observed speed in a given jet as the most representative of the underlying flow (e.g., Lister et al. 2009, 2019); however, the range of speeds in a source with many moving features can span a factor of a few, often including some very slow features. Jets with at least five features meeting our criteria have a median speed that is, on average, about 60% of the magnitude of their maximum observed speed. Because the features we observe may be propagating shocks (e.g., Marscher & Gear 1985; Hughes et al. 1989), they may travel at a different speed than the flow itself and the best observed speed to use in representing the flow remains an open question. To address this issue we directly compare three different choices for characterizing the observed speed of a jet to the median observed brightness temperature of the jet cores for those sources with several moving features, N_s ≥ 5.

Figure 5 compares median brightness temperature of the core with the fastest observed speed, ${\beta }_{\max }$, the median speed, β_med, and the speed of the feature closest to the core, β_close. The same 83 jets with at least five moving features are shown in each panel; the only difference is the speed used to represent each jet on the y-axis. The strongest correlation with median T_b is for the fastest apparent speed (see panel (a)) with a Spearman ρ = 0.63, while the median and closest features have ρ = 0.58 and ρ = 0.36 respectively. It is important to note that even with ideal measurements, we do not expect a perfect correlation between the observed brightness temperature and apparent speed. At the "critical" angle that maximizes apparent superluminal motion with $\cos \theta =\beta$,

$$\begin{eqnarray}&&{\beta }_{\mathrm{obs}}=\beta \delta =\beta {T}_{{\rm{b}},\mathrm{obs}}/{T}_{{\rm{b}},\mathrm{int}},\end{eqnarray} \tag{ 3 }$$

which would indeed suggest a strong correlation given that β is typically very nearly unity for powerful AGN jets; however, some jets may lie at smaller or larger angles than the critical angle and consequently have larger or smaller Doppler factors respectively. Indeed we will see this effect below when we look at the full data set; however, this subset of 83 jets includes only those that have at least five moving features meeting the criteria outlined above. Jets where we can identify and follow several moving features may be more likely to be near the critical angle where we are viewing the jet structures from the side in the comoving frame, and the strong correlation seen in panel (a) is consistent with that expectation. In our view, the fastest observed speed, ${\beta }_{\max }$, is the best speed to use in comparing to core brightness temperatures across the sample, and we use ${\beta }_{\max }$ in the analysis that follows. In Section 3.2.2, we revisit this question in the light of possible jet acceleration and consider the effects on our results if the median speed is used instead.

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In a complete, flux-density-limited sample, jets are more likely to be observed at a smaller angle to the line of sight than the critical angle due to Doppler beaming selection (e.g., Cohen et al. 2007). Lister & Marscher (1997) found that a typical beamed jet in a flux-density-limited sample like the MOJAVE 1.5 Jy QC sample has an angle to the line of sight about one-half of the critical angle, and Homan et al. (2006) used a simulation of a flux-density-limited sample to estimate that about 75% of the jets should lie inside the critical angle with a Doppler beaming factor:

$$\begin{eqnarray}&&\delta \gt \sqrt{1+{\beta }_{\mathrm{obs}}^{2}}\simeq {\beta }_{\mathrm{obs}}.\end{eqnarray} \tag{ 4 }$$

To update this estimate, we created 1000 Monte Carlo simulations of a 174-source, flux-density-limited sample based on the parameters estimated by Lister et al. (2019), and we find that 69% of the simulated jets lie within the critical angle. While the full results of the Monte Carlo simulation reported in that paper are based on the luminosities and apparent speeds of the MOJAVE 1.5 Jy QC quasars at that time, in this work we only use the fraction of simulated jets within the critical angle to allow us to estimate the typical median intrinsic brightness temperature, T_b,int, of our sample as a whole.

Following Homan et al. (2006) we start by assuming that every source in our sample has the same median intrinsic brightness temperature, and therefore that any differences in observed median brightness temperatures between sources are due to their Doppler beaming factor. With this assumption we can calculate the expected observed median brightness temperature for jets at the critical angle: ${T}_{{\rm{b}},\mathrm{obs}}=\sqrt{1+{\beta }_{\mathrm{obs}}^{2}}{T}_{{\rm{b}},\mathrm{int}}$. Jets with larger observed median brightness temperatures are therefore more highly beamed and located inside the critical angle. We vary T_b,int until 69% of our sample lie inside the critical angle.

There are 178 sources in the MOJAVE 1.5 Jy QC sample with both observed median brightness temperatures and observed speeds, 149 of which are quasars. Using the whole 1.5 Jy QC sample, we find the best estimate for the median intrinsic brightness temperature to be T_b,int = 10^10.609 K, and restricting the sample to only quasars does not change this value appreciably. We estimate the uncertainty in this value in two ways: (1) by creating 10,000 samples of 178 sources by randomly drawing with replacement from the data itself to include the effects of a limited sample size; and (2) by changing our fraction of sources within the critical angle by ±5% and repeating this estimate using 64% and 74% of sources within the critical angle. Including these uncertainties, our best estimate for the typical median intrinsic brightness temperature of the sample is T_b,int = 10^{10.609±0.067} K = 4.1(±0.6) × 10¹⁰ K.

Figure 6 shows plots of maximum observed jet speeds versus observed median brightness temperature for both our entire sample (panel (a)) and for the MOJAVE 1.5 Jy QC sample (panel (b)). The superimposed lines use our estimated value for the intrinsic median brightness temperature. The first curve is a red-orange line through the center of the plot, which shows where jets with intrinsic brightness temperature = 10^10.609 K would fall if viewed at the critical angle, $\cos \theta =\beta$. The second curve is a blue "envelope," which shows where jets with the same intrinsic brightness temperature and a Lorentz factor of 50 would fall if seen at the full range of angles to the line of sight. If all of the jets in our sample have this same median intrinsic brightness temperature, jets with Lorentz factors <50 should fall below the blue curve, and jets viewed inside the critical angle should fall to the right of the red-orange curve.

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For each source in our sample, we use the assumption that they all have the same intrinsic median brightness temperature found above, T_b,int = 10^{10.609±0.067} K, to estimate their Doppler factor from their median observed brightness temperature, δ = T_b,obs/T_b,int. We then use their maximum observed speeds, ${\beta }_{\max }$, to find their Lorentz Factor, Γ, angle to the line of sight, θ, and angle to the line of sight in the source fluid frame, θ_src, as follows, e.g., Jorstad et al. (2017):

$$\begin{eqnarray}&&{\rm{\Gamma }}=({\beta }_{\max }^{2}+{\delta }^{2}+1)/2\delta ;\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\theta =\arctan \displaystyle \frac{2{\beta }_{\max }}{{\beta }_{\max }^{2}+{\delta }^{2}-1};\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\theta }_{\mathrm{src}}=\arccos \displaystyle \frac{\cos \theta -\beta }{1-\beta \cos \theta }.\end{eqnarray} \tag{ 7 }$$

These values are listed in Table 4, with distributions of δ, Γ, and θ shown in Figure 7.

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Table 4. Doppler Factors and Derived Properties

Source	Tb,med	${\beta }_{\max }$	δ	Γ	θ	θsrc
	(log10 K)				(deg)	(deg)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
0003+380	11.550	4.61 ± 0.36	8.7	5.6	5.5	56.1
0003−066	11.079	7.08 ± 0.21	3.0	10.1	13.8	135.4
0006+061	>11.021	...	>2.6	...	...	...
0007+106	11.729	1.58 ± 0.29	13.2	6.7	1.0	13.7
0010+405	>11.425	6.92 ± 0.64	>6.5	...	...	...
0011+189	>11.207	4.54 ± 0.46	>4.0	...	...	...
0012+610	>10.747 a	...	>1.4	...	...	...
0014+813	11.223	9.47 ± 0.91	4.1	13.1	10.2	133.5
0015−054	11.246	...	4.3	...	...	...
0016+731	11.902	7.64 ± 0.32	19.6	11.3	2.0	42.6
...

Notes. The complete version of this table appears in the online journal. Table of source properties deduced from the brightness temperature versus speed analysis. All 447 sources are included in this table, but only 309 sources have both measured apparent speeds and non-limit-brightness temperatures, making them suitable for the full analysis as described in Section 2.2. Columns are as follows: (1) source name in B1950.0 coordinates; (2) median peak Gaussian brightness temperature; (3) fastest apparent speed; (4) Doppler factor, assuming T_b,int = 10^10.609 K as found in Section 2.2; (5) Lorentz factor derived from δ and β_max; (6) angle to the line of sight derived from δ and β_max; and (7) angle to the line of sight in the comoving jet frame.

^a Lower limit value (z = 0) only on account of unknown source redshift.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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It is interesting to compare Doppler factors we estimated from the median core brightness temperature to the values obtained by different methods. Doppler factors have been estimated for a large number of sources by flare modeling using the data of the single-dish monitoring programs at the OVRO 40 m radio telescope at 15 GHz (Liodakis et al. 2018), at the Metsähovi Radio Observatory at 22 and 37 GHz (Hovatta et al. 2009), and at the Effelsberg 100 m and IRAM 30 m telescopes within the F-GAMMA project at the frequencies from 2.64 to 86 GHz (Liodakis et al. 2017). Jorstad et al. (2017) estimated Doppler factors by another method, using the flux-density decay timescale of VLBI superluminal components at 43 GHz. Figure 8 shows the comparison of these values with our results. There is a statistically significant correlation between our Doppler factors and those obtained from the single-dish monitoring programs (panels (a)–(c)): p-values determined by the Kendall partial (given redshift) correlation test, accounting also for lower limits, are no more than 10⁻³.

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The most significant correlation, p ≈ 10⁻¹², is with the OVRO values (Figure 8(a), upper panel). These values also have the smallest median offset, about 10%, from our estimates (Figure 8(a), lower panel). The Doppler factors presented here and in the OVRO results are estimated by two very different methods, in different states of the sources with quite different corresponding estimates for T_b,int in those states. As described in Section 3.3, our typical intrinsic core brightness temperature for the median state is at or below the equipartition value while the flaring state intrinsic core brightness temperature from Liodakis et al. (2018) is only 2 times smaller than the inverse Compton limit (Kellermann & Pauliny-Toth 1969; Readhead 1994). The fact that the resulting Doppler factors are in such a good agreement lends confidence to both methods, although we note that the two approaches are not totally independent as Liodakis et al. (2018) used population modeling of an earlier set of MOJAVE kinematics to help constrain their value of T_b,int in the flaring state.

The values from Hovatta et al. (2009) and Liodakis et al. (2017) also correlate with ours, but are, on average, about two times smaller (Figures 8(b) and (c)). In both of these works, the authors used as intrinsic brightness temperature its equipartition value T_eq = 5 × 10¹⁰ K (Readhead 1994). Re-scaling their Doppler factors to the higher T_b,int = 2.8 × 10¹¹ K value used by Liodakis et al. (2018) would decrease them by about a factor of two, increasing their difference from our estimates. Liodakis et al. (2018) discuss several possible reasons for this disagreement between the otherwise similar variability approaches, including possibly insufficient cadence of the earlier observations. Our Doppler factors and those from Jorstad et al. (2017) are poorly correlated, regardless of which Doppler factor values for individual jet components from Jorstad et al. (2017) are used to represent each source: the maximum, the median, or the average value. For Figure 8(d), the maximum values are used. The Doppler factors estimated by Jorstad et al. (2017) may simply have a larger scatter if the assumption that the observed flux density decay timescale of jet components equals their light-crossing time divided by the Doppler factor is not always satisfied.

Figure 3 showed histograms of the median observed brightness temperatures for our sample as a whole (panel (a)) and the MOJAVE 1.5 Jy QC subsample (panel (b)), and beneath these panels we showed box plots illustrating the range of median brightness temperature values for different optical classes. Quasars (n_ws = 271, n_m15 = 158) ¹⁵ , BL Lacs (n_ws = 136, n_m15 =37), and galaxies (n_ws = 23, n_m15 = 6) appear to differ in their median brightness temperatures. Because some of our median brightness temperatures are lower limits, we use a pair-wise log-rank test from the Numerical Python "lifelines" distribution (Davidson-Pilon et al. 2020) to account for this censored data. We find that galaxies are very unlikely to be drawn from the same distribution as quasars (p_ws < 0.001, p_m15 < 0.001) or BL Lacs (p_ws < 0.001, p_m15 < 0.001). BL Lacs appear to differ from quasars for our whole sample (p_ws = 0.028) but we detect no difference in the flux-density-limited MOJAVE 1.5 Jy QC subsample (p_m15 = 0.93).

The BL Lacs in our flux-density-limited, MOJAVE 1.5 Jy QC sample are strongly dominated by sources with a spectral energy distribution characterized by a low synchrotron peak (LSP). In Figure 9, we compare the median brightness temperatures of LSP BL Lacs (n = 75) to those with intermediate or high-synchrotron-peaked (HSPs), ISP (n = 35) and HSP (n = 26), which are better represented in our whole, heterogeneous sample. HSP BL Lacs have distinctly lower median brightness temperatures when compared to ISP or LSP BL Lacs as confirmed by a log-rank test with p < 0.001 for both comparisons; however, we detect no difference between the median brightness temperature distributions of ISP and LSP BL Lac classes (p = 0.14). Figure 10 shows a plot of SED peak frequency in the galaxy rest frame versus median brightness temperature. BL Lac objects in particular show a strong negative correlation between SED peak frequency and median brightness temperature.

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If the median observed brightness temperature is a good proxy for the Doppler beaming factor, these results mean that radio galaxies are less beamed than BL Lacs and quasars as one would expect from unification arguments (e.g., Urry & Padovani 1995); however, we do not detect a difference between BL Lacs and quasars in the flux-density-limited MOJAVE 1.5 Jy QC sample. The apparent difference between these two classes in our larger, heterogeneous sample is likely due to differences within the BL Lac optical class itself. The differences in median brightness temperature between HSP and LSP sources suggest that HSP BL Lacs are less beamed than those whose SEDs peak at lower frequencies, consistent with earlier findings (e.g., Nieppola et al. 2008; Lister et al. 2011).

In Figure 11 we plot γ-ray luminosity versus median brightness temperature for 291 Fermi/LAT-detected AGN. The luminosity values are computed from the Fermi/LAT 10 yr point-source catalog (Ajello et al. 2020) using their 0.1–100 GeV energy flux and power-law spectral index following the approach given by Lister et al. (2011), Equation (3). To allow computation of their luminosity and to avoid issues related to galactic foreground subtraction, only sources with known redshifts and with a galactic latitude ∣b∣ > 10° are included in this plot. The histogram at the bottom of the plot shows the 60 sources meeting the same criteria that do not have Fermi/LAT detections in the 10 yr point-source catalog.

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We see a strong, positive correlation between γ-ray luminosity and median observed brightness temperature. Figure 11 includes lower limits on the median brightness temperature of only 13/291 of our LAT-detected AGN, and we measure a significant Spearman rank correlation for the remaining 278 sources of ρ = 0.54 (p < 0.001). However, we must be cautious in interpreting this correlation, as selection effects must be considered as well as common factors that affect both T_b and the γ-ray luminosity.

The observed brightness temperature in the frame of the host galaxy depends only weakly on redshift, see Equation (1), and even if we divide out the factor of (1 + z), the correlation remains significant (ρ = 0.32, p < 0.001). Another possible confounding factor is that many sources in our sample are selected on the basis of their radio flux density as part of the flux-density-limited MOJAVE 1.5 Jy QC subsample, and sources at large distances are likely to be highly beamed to meet this criterion, creating a natural correlation between Doppler factor and luminosity distance. In this same group of 278 sources we find a correlation of ρ = 0.44 (p < 0.001) between median brightness temperature and luminosity distance squared, ${D}_{{\rm{L}}}^{2}$. If we divide the γ-ray luminosity by ${D}_{{\rm{L}}}^{2}$, the correlation with median brightness temperature still remains significant with ρ = 0.33 (p < 0.001).

We can test the relationship between median brightness temperature and γ-ray emission further by comparing these results to those of Kovalev et al. (2009) and Lister et al. (2011) who found that γ-ray detected jets in earlier LAT catalogs had higher brightness temperatures than non-detected jets. Here we use a log-rank test to compare the distributions of median brightness temperature of the detected γ-ray sources (n = 291) to the non-detected sources (n = 60) in Figure 11, and we find the two groups are very unlikely to be drawn from the same distribution (p < 0.001) with the detected sources having distinctly larger median brightness temperatures on average. The Log-Rank test correctly accounts for the lower limits on some of our brightness temperature values, and by simply comparing the detected versus non-detected distributions we are not biased by a possible luminosity distance correlation with median brightness temperature through our flux-density-limited radio sample.

Taken together these results imply a common Doppler boosting of both the γ-ray emission and the brightness temperature of the radio core and will be discussed further in Section 3.2.1.

As described in Section 2.1, we characterize the brightness temperature variability of each jet by using a fractional measure of the variability between the 25% and 75% points in the brightness temperature distribution over time, see Equation (2). Figure 4 showed histograms of this brightness temperature variability index for our whole sample (panel (a)) and the MOJAVE 1.5 Jy QC subsample (panel (b)). Box plots below each histogram showed the distribution of variability index for different optical classes. Across the whole sample, quasars (n = 269) ¹⁶ appear to have higher variability and a log-rank test confirms that their distribution differs significantly from both BL Lacs (n = 132, p = 0.006) and radio galaxies (n = 22, p = 0.011), although we detect no difference between BL Lacs and radio galaxies when compared to each other (p = 0.36). For the MOJAVE 1.5 Jy QC flux-density-limited sample, we are unable to detect any difference in variability index distributions between quasars (n = 158), BL Lacs (n = 37), and radio galaxies (n = 6) with p ≥ 0.48 for each paired comparison.

The right panel of Figure 9 showed box plots of the brightness temperature variability index of ISP (n = 71), LSP (n = 35), and HSP (n = 26) BL Lacs in our sample as a whole, and paired log-rank tests show that HSP and LSP BL Lacs differ significantly from each other (p = 0.004); however, we do not detect differences from ISP BL Lacs for either of them (p = 0.18 versus LSP and p = 0.21 versus HSP).

Figures 12, 13, and 14 show scatter plots of viewing angle versus Lorentz factor for our entire heterogeneous sample, the MOJAVE 1.5 Jy QC flux-density-limited sample, and BL Lacs divided by SED class respectively. Each of these scatter plots is accompanied by two sets of box plots, which show the distributions of these quantities as a function of optical or SED class. Note that these figures and the following discussion are complementary to the brightness temperature plots and discussion in Section 3.1.1 as we are taking brightness temperature to be directly proportional to the Doppler factor.

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Figure 12 for our whole, heterogeneous sample has 233 quasars, 56 BL Lacs, 17 radio galaxies, and 3 narrow-line Seyfert I galaxies. Quasars have larger Lorentz factors and smaller viewing angles than both BL Lacs and galaxies as confirmed by Anderson–Darling tests that show the probability they are drawn from the same distribution is p < 0.001 in each case. If we restrict the comparison to just LSP quasars (n = 227) and BL Lacs (n = 27), the Lorentz factor difference still holds (p = 0.010), but we no longer detect a viewing angle difference (p = 0.22), consistent with the findings of Liodakis et al. (2018).

The MOJAVE 1.5 Jy QC flux-density-limited sample has 149 quasars, 23 BL Lacs, and just 6 radio galaxies in Figure 13. For Lorentz factor, we find all three distributions differ from one another (p = 0.004 for quasars versus BL Lacs, p < 0.001 for quasars versus galaxies, p = 0.011 for BL Lacs versus galaxies), with quasars having the largest Lorentz factors and galaxies the smallest in the sequence. For viewing angles, we can detect no difference between the classes with our Anderson–Darling tests, although we note the number of galaxies is quite small (n = 6) and includes M87, which may have had its viewing angle underestimated as described in Section 3.2.

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We note that the Lorentz factor differences between quasars and radio galaxies described above are driven by our flux-density-limited selection criteria where only nearby radio galaxies have sufficient flux-density without the need for large Doppler beaming factors to make it into our sample.

Finally, we look at BL Lacs as a function of SED class in Figure 14, which has 27 LSP, 12 ISP, and 17 HSP BL Lacs. We cannot detect a difference in either Lorentz factor or viewing angle between LSP and ISP BL Lacs with p > 0.25 for both quantities; however, HSP BL Lacs have smaller Lorentz factors and larger viewing angles than both LSPs (p < 0.001 for both quantities) and ISPs (p = 0.002 for Lorentz factor and p = 0.001 for viewing angle). When combined with our finding in Section 3.1.1 that HSP BL Lacs have lower brightness temperatures, and therefore lower Doppler factors, than the other classes, we get the consistent picture in Figure 15, which shows all three quantities as a function of SED peak frequency. HSP BL Lacs appear distinct from ISP and LSP BL Lacs with lower Doppler and Lorentz factors and larger viewing angles. This is consistent with the analysis of Piner & Edwards (2018) who estimate a maximum Lorentz factor of about 4 for this class on the basis observed motions.

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In Section 3.1.1, we investigated a correlation between γ-ray Luminosity and median brightness temperature, most likely due to a common Doppler boosting of the radio cores and the γ-ray emission. Figure 16 examines this question further by plotting γ-ray luminosity against each of the intrinsic quantities estimated by our analysis. The strongest correlation is clearly with the Doppler factor, and the somewhat weaker correlations with Lorentz factor and viewing angle are likely a consequence of their necessary role in producing highly Doppler boosted emission. This is consistent with the finding of Savolainen et al. (2010) that LAT γ-ray detected blazars differ significantly in their Doppler factor distribution from non-LAT detected blazars. We do not see a strong trend with the angle to the line of sight in the comoving emission frame, θ_src, in contradiction to the results of Savolainen et al. (2010) from a smaller sample, but consistent with the findings of Liodakis et al. (2018), who do not detect a difference in source-frame viewing angle distribution between LAT detected and non-detected sources.

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In Section 2.2 we examined three possible choices for representing the apparent jet speed in this analysis, and we chose to use the fastest apparent speed as it correlated most strongly with median brightness temperature and was the least likely to be contaminated by slowly moving, "quasistationary," features in the jets. An additional complicating factor is that jets are still becoming organized on these length scales and show evidence for acceleration and collimation (e.g., Komissarov et al. 2007; Homan et al. 2015; Chatterjee et al. 2019; Kovalev et al. 2020c), and it is possible that choosing the fastest apparent speed may better characterize the jet downstream from the core, rather than the core region itself where the brightness temperature measurements are made. When we looked at the speed of the feature that was closest to the jet core in its first epoch, we found it correlated much more poorly with apparent brightness temperature, likely due to contributions from quasistationary shocks near the jet origin (e.g., Lister et al. 2009; Jorstad et al. 2017); however, the median jet speed correlated almost as well with core brightness temperature as the fastest speed and might have made a reasonable alternative for this analysis.

If we had chosen to represent jets by their median apparent speed rather than their fastest apparent speed, very few of our results would change. We would conclude the intrinsic brightness temperature was about 40% larger, T_b,int = 10^10.762 K, and would find correspondingly lower Doppler factors for each source. Those lower Doppler values combined with their median speeds would lead to smaller estimated Lorentz factors and larger estimated viewing angles for most sources by a similar factor. However, despite these changes to δ, Γ, and θ, the relationships between these quantities and optical class, SED class, and γ-ray luminosity all remain the same without any appreciable change to the significant statistical relationships and trends discussed in our analysis above using the fastest speed.