6 Turnover frequency mapping  

On the basis of the method described above, we have developed a code for mapping the turnover frequency distribution from multi-frequency VLBA data. Fitting is performed in every valid pixel of the image; the validation is based on clipping the pixels with low flux density or low SNR. Flux density errors are estimated from the noise level and flux density gradients in the total intensity maps. For each pixel, we average the values of pixels within selected bin, and add, in quadratures, the averaging standard deviations to the estimated noise level. This results in slightly increased errors for pixels in the areas with steep flux density gradients, providing more conservative error estimates. The use of the gradients for error estimation is optional, and can be turned off by setting the bin size to 1.

The output of the mapping procedure can be the turnover frequency distribution, turnover flux density distribution, integrated flux distribution, or total intensity map at a given frequency. The last option allows us to predict, from the fitted spectral shape, the expected source structure at any frequency within the range of observing frequencies of the maps used for the spectral fitting. This can also be used for testing the quality of the spectral fit, by comparing the predicted and observed images at the same frequency (given that the observed image was not used for producing the above spectral fit).

 

Figure 10: Turnover frequency distribution in the extended jet of 3C345. The central region is saturated, for better representation of the turnover frequency variations in the jet. The contours are drawn at 0.1, 0.15, 0.2, 0.3, 0.7, 1, 2, 5, 8, 10, 12, and 15GHz. All values below 5 GHz should be regarded as upper limits

6.1 Mapping the turnover frequency distribution in 3C345

The blazar 3C345 (z=0.594, Hewitt & Burbidge 1993) is a strongly variable core-jet type source with a compact core responsible for most of the source radio emission, and a curved, parsec-scale jet (Zensus et al. 1995) containing enhanced emission regions (bright components) travelling along curved trajectories, with speeds of up to 20c (Zensus et al. 1995). Synchrotron spectra of the core and the nearest bright components are often peaked around 10GHz (Lobanov & Zensus 1998), and show a remarkable evolution. The emission from the core and the components is believed to be produced by condensations of highly-relativistic electron-positron plasma injected in the jet, and losing their energy first through the inverse-Compton mechanism (Kellermann & Paulini-Toth 1969; Unwin et al. 1997), and later on due to the synchrotron emission from adiabatically expanding relativistic shocks (Wardle et al. 1994; Zensus et al. 1995).

The turnover frequency procedure was applied to the multi-frequency VLBA observation of 3C345 made on June 24, 1995. The source was observed at 5, 8.4, 15.4, and 22.2GHz. At each frequency, there was roughly one 5 minutes scan made every 20 minutes. After the correlation, the data were fringe-fitted and mapped in AIPS and DIFMAP (Shepherd 1993). The data were tapered at 150 M$\lambda$, and the maps were produced with a circular restoring beam of 1.2mas in diameter. The core shift with respect to the reference frequency (22.2GHz) was applied to the data at 5, 8.4, and 15.4GHz. The magnitude of the core shift for the data at 15.4GHz was determined from the fit $r_{\rm core} \propto \nu^{-1.04\pm0.16}$, whereas for the data at 5 and 8.4GHz the measured values were used (Lobanov 1998). The resulting maps are shown in Fig. 9; the main characteristics of the maps are given in Table 3. Marked in the maps are the source core "D'' and jet component "C7'' which dominated the source emission at the epoch of observation.

 

Table 3: Parameters of the VLBA maps  

Notes: 1 - observing frequency; 2 - total CLEAN flux; 3 - peak flux density; 4 - minimum flux density; 5 - measured noise; 6 - major axis, minor axis, and position angle of the tapered beam; 7 - uv-range of the data; 8 - half-power taper size.

The turnover frequency map produced from the VLBA maps shown in Fig. 9 is presented in Fig. 10. Figure 11 shows a map at 11GHz obtained from the spectral fit to the combined VLBA data at 4 frequencies. One can see that the main features in the predicted 11GHz image are consistent with the structures seen in the original VLBA maps.

 

Figure 11: Map of 3C345 at 11GHz obtained from the spectral fit. The restoring beam and contour levels are the same as in Fig. 10. Spectral profiles in Fig. 13 are taken along the horizontal line crossing the nuclear region of the source

6.2 Nuclear region

In Fig. 10, there are two regions of higher turnover frequency in the nucleus of 3C345 oriented nearly transversely to the direction of the jet. These regions match the locations of the core and C7 fairly well. The increased turnover frequency may indicate that the emission is coming from a shocked plasma. The transverse extension is then consistent with strong shocks that are likely to be oriented almost perpendicularly to the jet direction. Figure 12 shows spectral profiles made along the horizontal line crossing the center of the core (horizontal line in Fig. 11. The core and C7 are both visible in the turnover frequency profile. The turnover flux distribution is very smooth and peaks almost precisely at the center of the core. From the turnover frequency and turnover flux distributions, we can derive the profile of magnetic field in the central region using the relation (Cawthorne 1991)  

$$B(r) = C_0 \nu_{\rm m}^5 r^4 S_{\rm m}^{-2},$$ (16)

where C₀ is the proportionality coefficient. C₀ can be determined empirically from the estimates of the absolute position ($r_{\rm core}\approx 5$pc) and magnetic field ($B_{\rm core} \approx 0.3$G) of the core at 22.2GHz (Lobanov 1998):

$$C_0 = B_{\rm core} S_{\rm m,core}^2 \nu_{\rm m,core}^{-5} \approx 1.2\ 10^{-5}\,,$$ (17)

for the measured $S_{\rm m,core} = 5.6$Jy and $\nu_{\rm m,core} = 15.1$GHz. Equation (16) expresses the magnetic field strength due to the compression that the plasma has undergone during shock formation. Therefore, the magnetic field also depends on the strength of the underlying magnetic field in the location of the jet where the shock is formed. We postulate that the underlying magnetic field $B_{\rm amb}\propto r^{-m}$, and consider the cases, with m=1 and m=2. The jet is assumed to have a constant opening angle $\phi=2.4^{\circ}$ (Lobanov 1998). For the magnetic field in an arbitrary pixel p, formula 16 yields  

$$B_{\rm p} = C_0 \nu_{\rm m,p}^5 S_{\rm m,p}^{-2} (r_{\rm p}/r_{\rm core})^{4-m}\,\,\,\,[G].$$ (18)

In this formula, $\nu_{\rm m}$ is measured in GHz, $S_{\rm m}$ is in Jy, and r is in parsecs. The resulting magnetic field profiles are plotted in Fig. 12. The magnetic field rises sharply, close to the outer edge of C7. This can signify the amount of plasma compression in the shock. The increased magnetic field on the opposite side (particularly visible in the $B\propto r^{-2}$ profile) may reflect a larger electron plasma density near the jet origin. In the relativistic jets, the case $B\propto r^{-1}$ is expected to be more likely. A somewhat high value of the magnetic field in C7 ($B_{\rm C7}^{\rm max} \approx 8.5$G) in this case may also be caused by possible errors in the estimates of the core magnetic field. However, the derived shape of the magnetic field profile is consistent with C7 being a strong shock embedded in the jet of 3C345.

 

Figure 12: Profiles of turnover frequency, $\nu_{\rm m}$, turnover flux, $S_{\rm m}$, and magnetic field, B(r), along the line $\Delta\delta=-3$mas crossing the center of the core (horizontal line in Fig. 12). The underlying magnetic field decreases along the jet as $r^{\rm -m}$

6.3 Extended jet

Almost everywhere in the extended jet shown in Fig. 10, the turnover frequency is lower than 5GHz, posing a problem for both the spectral fitting and assessing the results from the fits--we therefore resort to regarding all values of $\nu_{\rm m}\le 5$GHz as upper limits. Apparently, there are no strong shocks dominating the extended jet of 3C345, or their turnover points may have evolved rapidly due to strong adiabatic cooling. Because the derived turnover frequencies are too low, we cannot make quantitative statements about the physical conditions in the extended jet. Observations at lower frequencies (1.6, 1.4, 0.6, 0.3GHz) are required for a better understanding of the turnover frequency changes in these regions. With the available data, we can only make general comments about the gradients observed in the turnover frequency map. The bright patterns elongated along the jet ridge line may indicate the presence of an ultra-relativistic channel inside the jet (e.g. Sol et al. 1989). The extended patterns seen in the jet at oblique angles to the ridge line resemble the patterns of Kelvin-Helmholtz instabilities (see Hardee et al. 1995, for the results from 3D simulations of the KH-instability driven jets). As has been noted above, the turnover frequency is exceptionally sensitive to the variations of plasma speed and density. Therefore, the observed patterns may reflect the velocity gradients and/or density gradients existing in the jet perturbed by the Kelvin-Helmholtz instability. However, the low frequency data are needed for making a better substantiated conclusion about the observed gradients.