2. Polarization of BL Lac objects
We follow the idea of the jet models ( Urry & Padovani 1990, Padovani & Urry 1990). The observed flux, , of a relativistic jet is related to its intrinsic flux, , by , where , the Doppler factor of the jet, is defined by , is the velocity in units of the speed of light, is the Lorentz factor, and is the viewing angle. The value of p depends on the shape of the emitted spectrum and the detailed physics of the jet ( Lind & Blandford 1985), is for a moving sphere and is for the case of a continuous jet, where is the spectral index. We consider a two-component model in which the total flux of a source, , is the sum of an unbeamed part and a jet flux . Assuming that the intrinsic flux of the jet is some fixed fraction f of the unbeamed flux, (Urry & Shafer 1984), we have . The direction of the magnetic field in the jet should in general, be random except for some of it along the direction of the jet. So if the flux is not totally polarized, and it is not unreasonable to assume that the jet flux consists of polarized and unpolarized parts and which are proportional to each other, namely , , where is a coefficient which determines the polarization of the emission in the jet, then the observed optical polarization can be expressed as
where intrinsic polarization is defined by
and is the Doppler factor in the optical band. It is clear that for . If is a constant for the same class of sources, then there should be a correlation between the Doppler factor and the observed polarization. From the expressions (1) and (2), two parameters, f and , must first be determined in order to give . In general, the condition of must be satisfied from equation (2).
2.2. Observed polarization and Doppler factor
The relevant data are listed in Table 1. Col. 1 gives the name of the source, Col. 2 the classification, Col. 3 the redshift, Col. 4 the maximum optical polarization, Col. 5 the references to Col. 4, Col. 6 the radio Doppler factor from the paper of Ghisellini (1993), Col. 7 the optical Doppler factor from the paper of Xie et al (1991). The object 0521-365, which is classified as an RBL in our paper, is classified as an XBL by other authors (Remero et al 1995). However, its optical spectral index of (Pian et al. 1994) is in the range of the spectral indices of RBLs, (Falomo et al. 1994). So, we believe that it should be classified as an RBL. For 0716+714, its violent optical variation of (Qian et al. 1995) is similar to that of typical RBLs ( 0851+202, 1215+285 (Cruz-Gonzales & Huchra 1984), its quasi-simultaneous radio, optical and X-ray data give , which is in the range of the optical spectral indices of RBLs (Falomo et al. 1994) and satisfies . So, although it has been observed in ROSAT all sky survey, we still classify it as an RBL. It should be pointed out that the radio Doppler factors given by Ghisellini (1993) are the lower limits of the Doppler factor , which are estimated from the SSC model with in the case of a spherical region of observed angular diameter. In the continuous jet model with , the Doppler factor, , can be estimated by (Ghisellini 1993, Urry & Padovani 1995), and is used ( Padovani & Urry 1992, Urry et al 1991). We consider two cases. First, we use the radio Doppler factors to deduce the optical Doppler factors by using (Fan et al 1993) if the optical Doppler factor is not available. The observed data for the spherical model and the continuous jet model are shown in Figs.1 and 2 (the open circles stand for RBLs and the filled circles for XBLs). Second, radio Doppler factors for some BL Lac objects listed in Table 1 have also been obtained by Schwartz & Ku (1982): for 0048-097, 28.0 for 0235+164, 3.7 for 0735+178, 3.6 for 0754+100, 35.0 for 0851+202, 1.6 for 1219+285, 32.0 for 1308+326 and 3.8 for 1538+149. for 0215+015 has been obtained by Kikuchi(1988). If we choose these alternative radio Doppler factors for these objects to estimate their corresponding optical Doppler factors and, for other objects, the Doppler factors in table 1 are used. Similar results as in Figs 1 and 2 can be obtained.
Table 1. BL Lac objects with known Doppler factors
We want to remark that it is reasonable to use the relation of . In fact, Ghisellini & Maraschi (1989) proposed that the bulk velocity of the plasma increases with increasing distance from the core and synchrotron X-rays are weakly beamed, while optical and radio emissions are more strongly beamed. This model seems to have got support from the results and (Padovani & Urry 1990, Urry et al 1991). Based on this accelerating model, we assume that the Doppler factor satisfies the expression . Therefore, the X-ray, optical and radio Doppler factors are correlated and any two of them will be known if the other one is known, since and (Fan et al 1993). When this relation is used, the corrected data of RBLs show much better multiwavelength correlations (Fan et al 1993) and they satisfy the same relation as that of XBLs (Fan & Xie 1996). This Doppler expression is also adapted to Seyfert galaxies (Xie et al 1995) and OVV/HPQs (Fan 1997), and has been confirmed (Fan et al 1996b) to be a good approximated expression from the superluminal motion (Vermeulen & Cohen 1994, Fan et al. 1996a).
2.3. Comparison with observations
In order to compare our relation with the observed data. Two parameters, f and , must first be determined. The parameter f is the ratio of the intrinsic luminosity of the jet to the unbeamed luminosity and its possible value is from 0.001 to 1.0 (Padovani & Urry, 1990, 1991; Urry et al. 1991; Urry & Padovani 1995). The parameter is chosen to be 0.6, which means polarization in the jet is about . We show comparisons of our results with the observed data for the spherical model and continuous jet model in Figs. 1 and 2 ( here ), where , 0.01, 0.1 and have been used, which correspond to , 0.38%, 3.4% respectively. On the other hand, it is obvious that the observed optical polarization is not obtained simultaneously with the Doppler factor. In order to reduce this effect, one can choose the maximum optical polarization and the largest optical Doppler factors to compare with the theoretical curves.
Our results in Figs. 1 and 2 show that the polarization increases with the increasing Doppler factor and tends to a constant as the Doppler factor increases. That means that the total flux will be dominated by the emission from the jet with high Lorentz factor and then the observed polarization should be determined by the polarization within the jet. Therefore, we should observe similar polarizations if the polarization in the jet is the same for a single class, which can explain the difference in polarizations between XBLs and RBLs since XBLs are weakly beamed (Padovani 1992; Perlman & Stocke 1993) while RBLs are strongly beamed. But it can be seen that there are some scattering points, which may result from (i) the radio Doppler factors are a lower limit, (ii) the maximum optical polarization and the Doppler factors are not obtained simultaneously, (iii) the polarization in the jet is not the same, especially for 1519- 273, and (iv) the maximum polarization has not been obtained for some objects because BL Lac objects do not spend much time at polarization as high as 30% (Jannuzi et al 1994). Comparing Figs. 1 and Fig. 2, it seems that the data points in Fig. 2 fits the theoretical curves better than those in Fig. 1, which supports the idea that the continuous jet model is a more realistic case.
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998