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Astron. Astrophys. 330, 79-89 (1998)

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4. Properties of ultracompact jets

To illustrate the implications of frequency dependence of VLBI core position, we calculate physical properties of ultracompact jets in 3C 345 and 5 other AGN with reported core position offsets. Table 3 summarizes the model parameters (panel A), and gives the measured and derived quantities (panel B). For all sources, we assume [FORMULA], [FORMULA], and use crude estimates [FORMULA], and [FORMULA] (since only the logarithms of these ratios affect the calculated luminosities). The value of [FORMULA] is determined in most cases, by setting [FORMULA] to its equipartition value given by (10). Synchrotron luminosity, [FORMULA], and maximum brightness temperature, [FORMULA], are calculated using the model of Blandford and Königl (1979). For the purpose of comparison between different sources, the offset measures, [FORMULA] are given, rather than the angular offsets cited in the original publications.


Table 3. Model and derived parameters of ultracompact jets


Table 3. (continued)

4.1. Cygnus A

We adopt the component identification made by Krichbaum et al. (1997), and use their position measurements to derive the offset of the core in Cygnus A between 22.2 and 43.2 GHz (observation epochs 1992.44 and 1992.40, respectively). Krichbaum et al. (1997) give Gaussian model fits describing the compact structure of the source. We use these models to calculate the shifts of the total of four features situated both in the jet (components J3, J4) and counter-jet (components C1, C2). The two closest jet-side components, J1 and J2, are blended together in the model at 22 GHz. The averaged shifts are: [FORMULA] as for J3 and J4, and [FORMULA] as for C1 and C2. The negative sign for the shift on the counter-jet side indicates that the component separations from the core are shorter at 43.2 GHz. This fact further supports the identification, as well as frequency dependence, of the core location in Cygnus A. We then estimate that the shift of the core position amounts to [FORMULA] as, and use this value for calculating the physical properties of the jet.

The estimated [FORMULA] G is consistent with the field strength ([FORMULA] G, for [FORMULA]) in a Poynting flux jet (Lovelace & Romanova 1996) driven by a [FORMULA] M [FORMULA] black hole. Carilli & Barthel (1996, CR96 hereafter) use the unresolved core flux density of Cygnus A at 43 GHz, and apply the minimum energy equation, to arrive at [FORMULA] G in a nuclear region of 0.15 mas (0.11 pc) in size. We have [FORMULA] pc, for the distance from the jet origin to the core observed at 43 GHz. With these values, [FORMULA] G.

The total jet power, [FORMULA], can be related to the luminosity of extended radio lobe, [FORMULA]: [FORMULA] (CR96), with [FORMULA] 0.01-0.1 (Leahy 1991) describing the efficiency of converting bulk kinetic energy into radio luminosity. In Cygnus A, the lobe radio luminosity [FORMULA] erg s-1 (CR96), which results in [FORMULA], for our derived [FORMULA].

The predicted location, [FORMULA], of the central engine in Cygnus A is offset by about 0.14 mas from the core. Comparison between the 22 GHz flux densities of two bright features (J2 on the jet side, and C1 on the counter-jet side) evenly separated from [FORMULA] gives [FORMULA] for the jet-to-counter-jet ratio (this figure should be viewed as an upper limit, if a foregorund absorbing medium is present). The calculated ratio results in [FORMULA] for the adopted [FORMULA]. Conversely, for [FORMULA] used in the model, the corresponding [FORMULA].

4.2. 3C 309.1

We use the position measurements of Aaron (1996 and priv. comm.), and adopt the jet geometry as modelled by Kus (1993). Only upper limits are available for the core offset at frequencies higher than 5 GHz, as the separation between the core and closest optically thin feature remains constant. The derived [FORMULA] decrease substantially towards higher frequencies. We take it as evidence for larger [FORMULA] due to increasing opacity at shorter distances from the jet origin. Between 1.5 and 2.3 GHz, [FORMULA] is likely to be close to unity, so we take the corresponding derived quantities as best estimates of physical conditions in the jet. The estimated [FORMULA] is larger than in the other objects listed in Table 3, possibly implying a stronger pressure confinement in 3C 309.1. The latter can be reconciled with the results from optical spectroscopy (Forbes et al. 1990) suggesting that 3C 309.1 may have a massive ([FORMULA] M [FORMULA] yr-1) cooling flow within a radius of 11.5 [FORMULA] kpc.

The variations of [FORMULA] necessary to account for the difference in measured [FORMULA] are compared in Fig. 5 with changes due to the pressure gradients described in section 2.3. Here we assign [FORMULA], treat upper limits of [FORMULA] as measured points, and use the same model parameters as in Fig. 2. The behavior of jet opacity displayed in Fig. 5 is in agreement with the measurements of Aaron et al. (1997), who report a moderately high rotation measure [FORMULA] with strong gradients across a nuclear region of [FORMULA] pc in size. It should be stressed however that the model shown in Fig 5 is used for purely illustrative purposes, and we do not attempt to make quantitative conclusions or rule out other explanations for the observed core offsets in 3C 309.1. For instance, external free-free absorption in the BLR clouds may result in similar changes of [FORMULA].

[FIGURE] Fig. 5. Opacity in the jet of 3C 309.1. Circles are the measured values of [FORMULA] in 3C 309.1 at different frequencies; solid line shows changes of [FORMULA] due to pressure gradients described in section 2.3. The model parameters are the same as in Fig. 2.

4.3. 3C 345

Properties of the ultracompact jet in 3C 345 have been discussed recently in several papers, based on X-ray (Unwin et al. 1994, 1997), VLBI (Zensus et al. 1995), and multifrequency observations of the source (Bregman et al. 1986; Webb et al. 1994; Stevens et al. 1996). We adopt a spectral index [FORMULA] (Unwin et al. 1994), and derive the geometric parameters for our model from the statistical properties of superluminal features embedded in the extended jet (Zensus et al. 1995; Lobanov 1996).

To obtain a plausible estimate of [FORMULA], one can assume that the jet carries the least kinetic power, so that [FORMULA] (the assumption is valid only if the observed speeds are those of bulk motions in the jet). Under this assumption, the variations of viewing angle are derived for the jet components in the immediate vicinity of the core (Lobanov 1996); and we estimate [FORMULA] and [FORMULA] for the Lorentz factor and viewing angle of the ultracompact jet in 3C 345. These values are close to the fitted values from the synchrotron self-Compton (SSC) models applied to 3C 345 (Unwin et al. 1994; Webb et al. 1994).

The jet opening angle, [FORMULA], is calculated from measuring the jet size within 1 mas distance from the core. The particle density [FORMULA] cm-3 is almost 20 times smaller than the value given in Unwin et al. (1994) for an SSC model of the VLBI core. Zensus et al. (1995) have indicated that the SSC value of [FORMULA] might be roughly 10 times smaller, to accommodate the magnetic field and particle density found in the jet superluminal features. The latter prediction is consistent with our estimate of [FORMULA] within a factor of 2.

The calculated offset measures in 3C 345 given in Table 3 decrease at lower frequencies. The corresponding [FORMULA] must decrease by about 20%, between 5 and 11 GHz (which is within the errors of the fit given in Fig. 3). Assuming that [FORMULA], the decrease in [FORMULA] corresponds to [FORMULA]. This implies that the optical depth in the jet also becomes smaller at higher frequencies - a situation that does not have a simple explanation in the frameworks of both synchrotron self-absorption and external free-free absorption. With this argument in mind, and also considering the magnitude of errors in the derived offset measures, it appears more likely that the decrease of [FORMULA] results from blending effects due to the limited resolution. With decreased angular resolution at lower frequencies, the blending between the core and a nearest jet feature (remaining unidentified in the VLBI maps) should become progressively stronger, resulting in a systematic trend in [FORMULA]. However, we cannot entirely exclude an explanation involving peculiar physical conditions in the source. We therefore take the average [FORMULA] as the reference value for 3C 345.

The corresponding average characteristic magnetic field, [FORMULA] G, is lower than most of the values obtained by Webb et al. (1994), but can be reconciled with the results from Unwin et al. (1994), considering the difference of the adopted particle densities.

The average total power of the jet is [FORMULA] erg s-1, and we find [FORMULA] erg s-1. With the derived magnetic field, the maximum brightness temperature is [FORMULA] K. The obtained value is consistent with the [FORMULA] K inverse Compton limit, similarly to the mean values of the maximum brightness temperature in a sample of superluminal sources (Vermeulen & Cohen 1994) . Arguments based on the hypothesis of equipartition between relativistic particle and magnetic energy density ([FORMULA]) predict a lower value of maximum brightness temperature [FORMULA] K, Readhead 1994). The relativistic particle energy density, [FORMULA], and the energy density in the magnetic field, [FORMULA]. This gives [FORMULA] for the ultracompact jet, and therefore the jet must lose its energy through X-ray emission due to inverse Compton scattering. Unwin et al. (1994) argue that physical conditions in the extended jet may as well deviate significantly from the equipartition. According to Unwin et al. (1994), [FORMULA] at [FORMULA] -100 pc, which can be a natural consequence of evolution and radiation losses in a plasma with original [FORMULA].

4.4. Magnetic field in the jet of 3C 345

Table 4 summarizes the derived positions and magnetic fields in the VLBI core of 3C 345 at different frequencies. According to the derived positions, the location of the core observed at 22 GHz should be about 4 pc away from the jet origin. The angular offsets are all smaller than the VLBI beam size at corresponding frequency; therefore, the central engine is likely to be blended with the VLBI core even if the opacity is low enough for the emission from the innermost regions of the jet to be observed. Under such circumstances, evidence for emission from the inner jet can only be found in the spectral index or turnover frequency maps (Lobanov et al. 1997).


Table 4. Derived parameters of the ultracompact jet

In Fig. 6, the derived magnetic field is compared with the values obtained from synchrotron emission calculations for the extended jet (Zensus et al. 1995), as well as with model predictions for the central engine in 3C 345. The theoretical values of magnetic field in the close vicinity of central engine are calculated from a model of thin, magnetically driven accretion disk (Field and Rogers 1993), and from a model of supermassive black hole surrounded by a strong magnetic field (Kardashev 1995) . The accretion disk model predicts [FORMULA] at [FORMULA], and [FORMULA] G at [FORMULA] ([FORMULA] is the Schwarzschild radius of the central black hole). The mass of central black hole can be related to [FORMULA], so that [FORMULA] M [FORMULA] ; and in 3C345: [FORMULA] M [FORMULA]. Kardashev (1995) suggests that a strong dipole magnetic field may exist in the vicinity of a supermassive black hole. The maximum field strength can be estimated from the equilibrium relations [FORMULA], where [FORMULA], [FORMULA], [FORMULA] are masses of black hole, its disk, and its magnetic field respectively. Then [FORMULA]. This gives for 3C 345: [FORMULA] G. For a dipole magnetic field, one obtains [FORMULA]), so that [FORMULA] G at 1 pc, slightly lower than our estimate in Table 3. To match the derived [FORMULA], the mass of central black hole in Kardashev's model must be [FORMULA].

[FIGURE] Fig. 6. Magnetic field distribution in 3C 345. Squares show the magnetic field in the compact jet derived from the frequency dependent shift of the core. Circles are the homogeneous synchrotron model estimates of magnetic field in the extended jet components (Zensus et al. 1995). Triangles show the characteristic magnetic field values from a model of magnetized accretion disk (Field and Rogers, 1993). Diamonds are the theoretical estimates from Kardashev's (1995) model for the dipole magnetic field around a supermassive rotating black hole.

4.5. 3C 395 and 4C 39.25

The core offsets are known from geodetic VLBI measurements between 2.3 and 8.4 GHz (Lara et al. 1996; Guirado et al. 1995). We use the models derived for the parsec-scale jets in these sources (Lara et al. 1994; Alberdi et al. 1993), and adopt the equipartition values of [FORMULA]. The resulting [FORMULA] are similar to the 2.3-5 GHz offset measure in 3C 309.1. Other derived parameters are also comparable to their counterparts in 3C 345 and 3C 309.1. The derived synchrotron luminosity of 4C 39.25 is quite high, compared to [FORMULA] (10-90 GHz) [FORMULA] erg s-1 given in Bloom & Marscher (1991), even taking into account the spectral limits.

Recent observations of 4C 39.25 at 22.2 and 43.2 GHz (Alberdi et al. 1997) indicate no detectable position shift in the source. From the measured [FORMULA] between 2.3 and 8.4 GHz, the expected shift between 22.2 and 43.2 GHz is less than 0.05 mas. The latter value is below the image resolution at both frequencies ([FORMULA] and [FORMULA] mas).

4.6. 1038+528 A

Marcaide & Shapiro (1984) reported a 0.7 mas shift between 2.3 and 8.4 GHz in 1038+528 A. Subsequently, Marcaide et al. (1985) derived [FORMULA], based on VLBI data at 1.4, 2.3, 8.4, and 10.6 GHz. Marcaide, Elósegui & Shapiro (1994) applied a relativistic shock model by Gomez et al. (1993) to explain the observed shift. Rioja & Porcas (1997) have made position measurements at 2.3, 8.4, and 15 GHz, and argued that at least a fraction of the observed offset can result from limited resolution and blending at lower frequencies. The [FORMULA] (2.3-8.4 GHz) calculated from the data of Rioja et al. (1997) is similar to the values obtained for 4C 39.25 and 3C 395 (although the projection effects should be kept in mind, in view of the different viewing angles in these sources). The upper limit given for the offset between 8.4 and 15 GHz corresponds to [FORMULA], making it rather difficult to be explained by synchrotron self-absorption. Even for free-free absorption, it would require very strong density gradients ([FORMULA]) in the absorbing medium, in order to reproduce the given limit on [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998