5 Spectral fitting  

The frequency range covered by VLBI observations is very narrow (roughly, from 1 to 100GHz, with most of the observations done between 5 and 22GHz). The turnover frequency, $\nu_{\rm m}$, of the synchrotron spectrum often lies outside the range of observing frequencies (see the sketch in Fig. 6). Because of the limited frequency coverage, a straightforward application of the synchrotron spectral form to fitting may be ill-constrained (such as in the case B in Fig. 6). To provide an estimate of $\nu_{\rm m}$ in such cases, we first attempt to achieve the best fit of spectral data by polynomial functions, and then analyse local curvature of the obtained fit. From the measured curvature, an estimate (often only an upper limit) of $\nu_{\rm m}$ can be found. Details of spectral fitting are given in Sect. 5.4; application of the local curvature for determining the turnover frequency is discussed in Sects. 5.5-5.6.

 

Figure 6: Sketch illustrating the problems existing in determining the turnover frequency from VLBI data with insufficient frequency coverage. A homogeneous synchrotron source with isotropic pitch angle distribution is assumed. When the true turnover frequency lies outside of the range of observing frequencies, ad hoc information about low and high-frequency spectral shape is required, as well as the second order corrections based on the measured curvature of the observed part of the spectrum (see text for details)

5.1 Basic synchrotron spectrum  

For an extensive coverage of synchrotron emission and its properties, we refer to the works by Ginzburg & Syrovatskii (1969), Pacholczyk (1970), and Ternov & Mikhailin (1986). In our calculations, we consider synchrotron emission from a homogeneous plasma with isotropic pitch angle distribution and power law energy distribution $n(\gamma){\rm d}\gamma = n_{\gamma_{0}}\gamma^{\rm -s}{\rm d}\gamma$,for electron Lorentz factors $\gamma_{\rm L} < \gamma < \gamma_{\rm H}$.In this case, it would suffice to describe the emission within the range of frequencies $\nu_{\rm L} \ll \nu \ll \nu_{\rm H}$, where $\nu_{\rm L,H}$ are the low-frequency and high-frequency cutoffs given by:  

$$\nu_{\rm L,H} \approx \gamma^2_{\rm L,H}\frac{\Omega \rm _e}{\pi},$$ (6)

where $\rm \Omega_e$ is the electron gyro-frequency. Then, for a plasma with electron self-absorption, the spectral distribution of emission is (Pacholczyk 1970):  

$$I_\nu \propto \left( \frac{\nu}{\nu_1} \right)^{\alpha_{\rm ... ...1}{\nu} \right)^{\alpha_{\rm t}-\alpha_{\rm o}}\right]\right\},$$ (7)

where $\nu_1$ is the frequency at which the optical depth, $\tau_{\rm s}=1$, and $\alpha_{\rm t}$, $\alpha_{\rm o}$ are the spectral indices of the optically thick and optically thin parts of the spectrum (with spectral index defined by $S \propto \nu^\alpha$). It is clear from Eq. (7) that at frequencies $\nu \ll \nu_1$: $I_\nu \propto (\nu/\nu_1)^{\alpha_{\rm t}}$; and at frequencies $\nu \gg \nu_1$:$I_\nu \propto 1 - \exp [-(\nu_1/\nu)^{\alpha_{\rm t}-\alpha_{\rm o}}]$. For a plasma with a homogeneous synchrotron spectrum $\alpha_{\rm t}=2.5$. We will use the above $\alpha_{\rm t}$ and the description given by (7) in our calculations.

5.2 Fitting algorithm

The main steps of spectral fitting can be summarized as follows:

1) Make an approximate estimate of cutoffs in the spectrum, on the basis of all available spectral information. A fairly good guess can be attained by taking the average measured turnover frequency and optically thin spectral index in the compact source. Based on these values, calculate the high and low frequencies at which the corresponding flux density is at an arbitrarily low level (we use $S_{\rm cutoff} = 0.1$mJy). Add these spectral points to the measured data in each pixel, in order to ensure a negative curvature of the fitted curves, as required by the theoretical spectral form (7).

2) Fit the combined pixel spectra by polynomial functions, allowing for limited variations of the cutoff frequencies, and aiming at achieving the best fit to the measured data points. From the fits, determine the basic spectral parameters: the turnover frequency, turnover flux density, and integrated flux (with integration limited to the range of observing frequencies). Estimate the errors from Monte Carlo simulations, using the distribution of the $\chi^2$ parameters of the fits to the simulated datasets.

3. Calculate the local curvature of the fitted spectra within the range of observing frequencies. Compare the derived curvature with the values obtained from analytical or numerical calculations of the synchrotron spectrum. Derive the corrected value of the turnover frequency, by equating the fitted and the theoretical spectral curvatures.

4. Fit the data with the synchrotron spectral form described by (7), and using the corrected value of the turnover frequency.

The above procedure has been applied to spectral data obtained from modelling VLBI images of 3C345 by elliptical Gaussian components, and combining the models at different frequencies (Lobanov & Zensus 1998).

5.3 Spectral cutoffs

Because the transition between high-frequency ($\nu \gt \nu_1$) and low-frequency ($\nu < \nu_1$) spectral regimes determined by Eq. (7) is very sharp, the spectrum is determined by the synchrotron self-absorption at frequencies $\nu \ll \nu_1$, and by the electron energy distribution at $\nu \gg \nu_1$. Estimates of the typical turnover frequency and energy spectral index in the jet of 3C 345 based on our own calculations and on the results from Rabaça & Zensus (1994) give $\nu_1 \approx 10$GHz and $s\approx 2.6$.Using these values, we estimate the low-frequency, $\nu_{\rm L}\sim$ 1MHz, and high-frequency, $\nu_{\rm H}\sim$ 1000GHz, cutoffs in the spectrum, and use these values for the spectral fitting. However, the cutoffs cannot be well defined, and may change during the evolution of the jet emission. To account for this effect, we allow 15% variations of the cutoff frequencies so as to achieve the best fit to the data.

5.4 Spectral fits  

In order to calculate the spectral parameters of jet emission, we combine the component fluxes measured at frequencies $\nu_1,...,\nu_{\rm N}$, and add the cutoff information. The resulting spectral dataset ${S_{\rm s}, \sigma_{\rm s}^2}$ is represented by the flux densities,

$$\{S_{\rm s}\} = \{ S(\nu_{\rm L}),S(\nu_1),...,S(\nu_{\rm N}),S(\nu_{\rm H})\}\,,$$

and their respective variances,

$$\{\sigma^2_{\rm s}\} = \{ \sigma^2_{\rm S}(\nu_{\rm L}),\sig... ...ma^2_{\rm S}(\nu_{\rm N}), \sigma^2_{\rm S}(\nu_{\rm H}) \}\,,$$

where $S(\nu_{\rm L}) = S(\nu_{\rm H}) = 0.1$mJy is the cutoff flux density level.

By varying the spectral dataset, we then produce simulated spectral datasets  

$$S_{\rm sym} = {\cal V}(S_{\rm s})\vert _{\{\sigma^2_{\rm S}(\nu)\}}$$ (8)

assuming the Gaussian distribution of the errors in flux density measurements  

$$\phi (S) = \frac{1}{\sqrt{2\pi} \sigma_\nu} {\rm e}^{-1/2\left(\frac{S-S_\nu} {\sigma_\nu}\right)^2}$$ (9)

where $S_\nu$ and $\sigma_\nu$ are the measured flux density and its variance.

For each simulated spectral dataset, we apply the $M^{\rm th}$-order ($M\le N-1$) polynomial fitting by the basis functions $X_k(\nu)=\nu^k$, and minimize the $\chi^2$ parameter of the fit  

$$\chi^2 = \sum^N_{i=1}\frac{1}{\sigma^2_i}\left[ (S_{\rm sym})_i - \sum^{M}_{k=0} A_k X_k(\nu_i) \right]^2,$$ (10)

to obtain the least squares fit to the data, and derive the polynomial coefficients  

$$A_j = \sum^M_{k=0} [a]^{-1}_{jk} \beta_k, \quad \sigma^2(A_j) = [a]{-1}_{jj} \,.$$ (11)

Here a_j,k and $\beta_k$ are given by

$$a_{j,k} = \sum^N_{i=1}\frac{X_j(\nu_i)X_k(\nu_i)}{\sigma_i^2},$$

$$\beta_k = \sum^N_{i=1}\frac{(S_{\rm sym})_i X_k(\nu_i)}{\sigma_i^2} \, .$$

From the fit to the spectral dataset, we derive the basic parameters of the synchrotron spectrum: the integrated flux,  

$$S_{\rm int} = \int_{\nu_1}^{\nu_{\rm N}} \sum_{j=0}^{M}A_j\nu^j{\rm d}\nu \,,$$ (12)

the turnover frequency, $\nu_{\rm m}$, and the turnover flux density, $S_{\rm m}$: 

$$\frac{{\rm d}\sum_{j=0}^{M}A_j\nu^j}{{\rm d}\nu} = 0 \; \Rightarrow \; S_{\rm m}, \nu_{\rm m}$$ (13)

We then analyse the distribution of the $\chi^2$ parameters from the fits to all simulated datasets. From this analysis, standard deviations at the $3\sigma$ confidence level are calculated for $S_{\rm int}$, $S_{\rm m}$, and $\nu_{\rm m}$.

5.5 Curvature of the fits  

 

Figure 7: Theoretical curvature, $\kappa$, of the homogeneous synchrotron spectrum with spectral index $\alpha$. The $\xi$ axis denotes the ratio of the frequency at which the curvature is calculated to the turnover frequency

 

Figure 8: Curvature correction coefficients derived from simulating synchrotron spectra and subsequent fitting them by the polynomial functions described in Sect. 5.4

The local curvature of spectral fits is:  

$$\kappa = \frac{{\rm d}^2 S}{{\rm d}\nu^2} \left[1+\left(\frac{{\rm d}S}{{\rm d}\nu}\right)^2 \right]^{-1/3}\, .$$ (14)

The mathematical details of calculations are summarized in Appendix. If the spectral index, $\alpha_{\rm o}$, and the local curvature, $\kappa_{\rm o}$, of a polynomial fit are determined at a frequency $\nu_{\rm o}$, then the turnover frequency, $\nu_{\rm m}$, can be estimated from the adopted theoretical synchrotron spectrum. Using the derived $\alpha_{\rm o}$and $\kappa_{\rm o}$, we determine the ratio $\xi(\alpha_{\rm o}, \kappa_{\rm o}) = \nu_{\rm m}/\nu_{\rm o}$, from the adopted spectral form described by (7). The corresponding turnover frequency is then  

$$\nu_{\rm m} = \nu_{\rm o} \xi(\alpha_{\rm o}, \kappa_{\rm o})\, .$$ (15)

Figure 7 relates the curvature $\kappa$ of the homogeneous synchrotron spectrum described by (7) to the ratio $\xi(\alpha_{\rm o}, \kappa_{\rm o})$. For frequencies increasingly deviating from the turnover frequency (for which $\xi=1$), the curvature, $\kappa$, becomes progressively smaller, thereby limiting the ranges of applicability of the corrections described by (15). For data covering the frequencies from $\nu_1$ to $\nu_{\rm N}$, we expect the corrections to give reliable estimates for the turnover frequencies lying within the $0.05\nu_{1} < \nu_{\rm m} < 2\nu_{\rm N}$ range.

5.6 Numerical estimates of the curvature corrections  

An alternative method to estimate $\xi(\alpha_{\rm o}, \kappa_{\rm o})$ is to determine it numerically, by simulating the synchrotron spectrum with given turnover frequency and spectral index, and fitting it by the polynomial functions used in Sect. 5.4. We have performed such calculations, using the spectral form described in Sect. 5.1, and covering a range of turnover frequencies and spectral indices. The results are shown in Fig. 8 in which the ratio of fitted values to the theoretical values of the turnover frequency is plotted against the synchrotron spectral index. The contours show different values of $\xi$. One can see that the determined ratios do not depend strongly on the spectral index. Figure 8 can be used for the same purpose of correcting the fitted turnover frequencies obtained through the procedure described in Sect. 5.4.

 

Figure 9: Maps of 3C345 from the multi-frequency VLBA observation made on June 24, 1995. The restoring beam is 1.2 $\times$ 1.2mas. The contours are (1, 1.4, 2, 2.8, 4, ..., 51.2)$\times$14mJy