4.4.3. Doppler and Lorentz factors
Ghisellini et al. (1993) have derived a lower limit of the Doppler factor for a sample of 19 BLLs, 13 BL/HPQs and 24 HPQs from the condition that the synchrotron self Compton flux should not exceed the observed flux at high frequencies. They found that the distribution for BLLs and HPQs are significantly different, with BLLs having lower values of . 19/24 HPQs have > 5, while only 3/19 BLLs have such a high value of (PKS0048-09, PKS0735+17 and PKS1519-27); 8/13 BL/HPQs have > 5.
The equipartition Doppler factors are derived assuming that the sources are at or near equipartition of energy between radiating particles and magnetic field (). Güijosa & Daly (1996) have computed the equipartition Doppler factor for the objects in the Ghisellini et al. sample. They found that 5/19 BLLs have > 2, while 10/13 BL/HPQs and 18/23 HPQs have such a large value of .
Comparisons between the inverse-Compton and equipartition Doppler factors shows a high correlation between the two estimates, suggesting that they are both reliable (; ).
Fig. 7 shows the distributions of the Doppler factors for BLLs, HPQs and BL/HPQs as derived by the two methods. Again, we find that weak broad line objects are more similar to HPQs than to objects with no detected broad lines.
The apparent transverse velocity of a moving feature is a = sin / (1 - cos). The maximum transverse velocity for a given value of is a = (which is larger than 1 when > 2-0.5) or a ~ when ~ 1.  have shown that the typical VLBI component apparent speed a in BLLs are systematically lower than those in HPQs. To get this result, they used measurements of 12 BLLs. Five of these objects have been reclassified as BL/HPQs; the difference between the two distributions becomes even more striking, the median values of a being 4 and 9 for the BLLs and the HPQs respectively. The BLL Mark421 has no superluminal motion which could arise if the source is aligned almost exactly (within 0.4°) with the line of sight ().
Figure 7. Distributions of blazar Doppler factors determined in two different ways by Ghisellini et al. (1993) and by Güijosa & Daly (1996), plotted separately for the HPQs, BL/HPQs and BLLs.
Several authors have estimated the mean value of R, the ratio of the compact- to-extended radio luminosity, for the RBLs; they found that it is much larger than for FR Is (; ; Maraschi & Rovetti 1994; ); the estimates of the ratio of the mean R values for BLLs to that for FR Is range from 200 to 4000; if this is taken as the mean value of R/R0, it indicates values of in the range 3-6 if p = 2. According to , the distribution of R for BLLs and HPQs are undistinguishable; however, as the mean value of R is larger for FR Is than for FR IIs sources, it follows that R0 is on average larger for BLLs than for HPQs, implying a smaller average value for .
Again, we find that BL/HPQs have Doppler and Lorentz factors more similar to HPQs than to BLLs.
A weak unresolved optical nuclear source has been detected with the HST in the great majority (85%) of a complete sample of 33 FR I galaxies, the luminosities of the optical and radio cores being strongly correlated. The detection of a compact optical core in most FR Is indicates that we have a direct view of the innermost regions and that the obscuring torus, if any, must be very thin; the correlation between the luminosities of the optical and radio cores suggests that the optical core is emitted by the jet itself rather than by a weak accretion disk ().
The luminosities of the optical cores found in FR I radio galaxies are 2 102 to 3 105 times weaker than the core luminosity of BLLs selected for having similar host galaxy magnitude and extended radio luminosity, suggesting that BLLs are optically dominated by the beamed radiation from a relativistic jet with Lorentz factor ~ 5-10 (); this range of values is significantly larger than the range obtained from radio data!