4 Data reduction and calibration

Figure 2: Example of fitting the simulated beam of RATAN-600 (lines) to the source response (dots) for 4C 39.25 from Fig. 1

Data reduction has been done using a YURZUF software package, which had been specially designed for the automatic reduction of the broad band spectra monitoring observations (Kovalev 1998). Fitting a beam which is simulated at the source elevation allows us to compute the amplitude of a source response at each frequency. A Singular Value Decomposition subroutine from Forsythe et al. (1977) was applied. Routine functions, designed by V.R. Amirkhanyan, were also included via an additional interface as a subroutine to the YURZUF software to simulate the main antenna beam together with the secondary lobe. Before the reduction, the quality of such a fitting has been checked and a simulation of the beam has been optimized by tuning control parameters using the sample of 30-50sources which are strong and compact at all frequencies, and distributed on different elevations (see an example of fitting in Fig. 2).

   

Table 4: Parameters of calibration sources: flux density, Jy, and correction factors g_ext and g_pol(from top to bottom)

Source	$\lambda$, cm
	1.4	2.7	3.9	7.7	13	31
0134+32	1.216	2.431	3.540	6.765	10.79	21.90
( $h=79\hbox{$.\!\!^\circ$ }3$)	1.016	1.004	1.002	1.000	1.000	1.000
	1.032	1.046	1.039	0.959	1.004	1.002
0237-23	0.700	1.470	2.200	4.030	5.590	6.610
( $h=23\hbox{$.\!\!^\circ$ }0$)	1.000	1.000	1.000	1.000	1.000	1.000
	0.979	0.979	0.979	1.025	0.992	0.983
0518+16	1.135	2.008	2.700	4.413	6.221	10.28
( $h=62\hbox{$.\!\!^\circ$ }8$)	1.000	1.000	1.000	1.000	1.000	1.000
	0.954	0.956	0.923	1.104	0.930	0.964
0624-05	1.400	2.764	4.156	8.112	12.80	24.10
( $h=40\hbox{$.\!\!^\circ$ }3$)	1.034	1.009	1.004	1.001	1.000	1.000
	1.027	1.020	1.021	1.025	0.942	1.017
1328+30	2.563	4.244	5.529	8.576	11.50	17.20
( $h=76\hbox{$.\!\!^\circ$ }7$)	1.016	1.004	1.002	1.000	1.000	1.000
	1.053	0.948	0.957	1.049	0.954	0.954
2037+42	...	...	...	17.40	12.10	5.000
( $h=88\hbox{$.\!\!^\circ$ }5$)	...	...	...	1.099	1.077	1.013
	...	...	...	1.000	1.000	1.000
2105+42	5.330	5.940	6.100	5.050	2.850	...
( $h=88\hbox{$.\!\!^\circ$ }4$)	1.297	1.078	1.034	1.008	1.003	...
	0.992	0.992	0.999	1.000	1.000	...

Figure 3: The flux density calibration factor Fcalversus elevation h at all wavelengths. Solid and dashed lines at 1.4, 2.7 and 3.9 cm represent Fcal for each horn separately. All data (crosses and points with errors), shown for 6-7 calibrators, were averaged during the observational set. Seven calibrators are shown at 7.7 and 13 cm, but the data for two calibrators at $h=88\hbox{$.\!\!^\circ$ }4$ and $h=88\hbox{$.\!\!^\circ$ }5$ are plotted in the same spot

The following seven flux density calibrators were applied to obtain the calibration curve in the scale of Baars et al. (1977): 0134+32, 0237-23, 0518+16, 0624-05, 1328+30, 2037+42 (for calibration at 7.7, 13 and 31 cm only), 2105+42 (excluding 31 cm calibration). They were recommended by Baars et al. (1977), excluding 0237-23 which is the traditional RATAN-600 flux density calibrator at low elevations. Measurements of some calibrators were corrected, where necessary, on angular size and linear polarization, following the data, summarized in Ott et al. (1994) and Tabara & Inoue (1980) respectively. Response to an extended calibrator was simulated as a two-dimensional convolution of the beam and brightness distribution in the published model of a calibrator. The best fit to the observed response was found by optimization of the angular size of an extended calibrator at each frequency. The correction factor due to an angular extension g_ext was calculated numerically by integrating over the solid angle of the optimized brightness distribution and the convolution of the distribution with the beam. Following Ott et al. (1994), we applied Gaussian profiles of the brightness distribution over right ascension and declination for 0134+32, 0625-05, 1328+30, 2037+42 and the elliptical disk model for 2105+42, additionally making an axial ratio to be equal to the measured one in Masson (1989). The correction factor due to linear polarization g_pol of the calibrators was calculated in the standard way (Kuzmin & Salomonovich 1964; Kraus 1966) as $g_\mathrm{pol}=1/[1+p\cos (2\varphi)]$, where p is the linear polarization degree and $\varphi$ is the angle between polarization planes of a source and the antenna. The corrected amplitude of the response to a calibrator is calculated as the observed one, multiplied by the factors g_ext and g_pol.

The flux densities of the calibrators, in Janskies, the factors g_ext and g_pol are summarized in Table 4 for each source (at the elevation h) at each wavelength from top to bottom respectively. The flux densities were calculated from polynomial approximations (Taylor 1999) of the VLA measurements (relative to the spectrum of 3C 295) for 0134+32, 0518+16, 1328+30; from spline and polynomial approximations of the data by Ott et al. (1994) for 0624-05, 2037+42, 2105+42 (in relation to the spectra of 3C 295 and 3C 286), and from the polynomial approximation (Kühr et al. 1979) of the spectrum for 0237-23. For 0134+32 and 1328+30 at 31 cm, we give preference to the flux densities extrapolated from the approximations of Ott et al. (1994). Taking into account all available data, we also used two following extrapolated values: 0.70 Jy for 0237-23 and 1.4 Jy for 0624-05 at the wavelength of 1.4 cm. With these extrapolations, we have obtained reasonable results.

Amplitude measurement of a source in flux density units has been done in relation to an amplitude of a flux density calibrator by comparing both with the amplitude of a stable signal from a noise generator, using standard methods. In our observations the elevations of seven flux density calibrators are fixed. Because of this fact we computed a regression curve to obtain dependence of the flux density calibration factor F_cal on the elevation for each horn on each frequency (Fig. 3). F_cal is equal to the amplitude of the noise generator signal, calibrated in flux density units. In fact, the obtained calibration curves F_cal(h) show the dependence of the mean measured flux density for a source on the elevation h, if its antenna temperature is equal to that of the noise generator signal T_ns (Kovalev 1998): F_cal(h)=2kT_ns/A^aa_eff(h), where $A^\mathrm{aa}_\mathrm{eff}(h)= A_\mathrm{eff}(h)\, q^\mathrm{ab}(h)\, q^\mathrm{atm}(h)$, A_eff(h) is the effective area of the antenna in the focus, q^ab(h) - the factor of aberration (due to transversal shifts of a feedhorn from the focus), q^atm(h) - the atmosphere attenuation factor, k - the Boltsman's constant. We did not make any additional atmospheric correction during the set (the altitude of RATAN-600 site is 970 m above see-level).

The total relative rms error of each individual flux density measurement $\sigma/F$ is estimated from the following relation (Kovalev 1998):

$$\left(\frac{\sigma}{F}\right)_{\nu}^2 = \left(\frac{\sigma_\... ... \left(\sigma^\mathrm{r}_\mathrm{{scale}}\right)_{\nu}^2 \, ,$$

where the first term inside the brackets on the right hand is the relative error of the amplitude A_s of a source (after fitting the simulated response to the observed one), the second - the relative error of the amplitude A_ns of a response to the noise generator signal, the third - the relative error of our flux density calibration F_cal, averaged on the set, and the last - the relative error $\sigma^{r}_{scale}$ of the absolute flux density scale. Usually, the last term is excluded from presented errors, but we show it in the relation to emphasize its importance, because different calibrators may be used in various works.

The total error $\sigma$ is calculated, excluding only the $\sigma^{r}_{scale}$ error, which is estimated by Baars et al. (1977), Ott et al. (1994) and Taylor (1999) as about 10% at 1.4 cm and 3-5% at other wavelengths. It is better to increase $\sigma^{r}_{scale}$ to 10-15% at 1.4 cm for the sources with declinations less than $-5^\circ$ because of the above mentioned extrapolation of the flux density values for 0624-05 and 0237-23. Errors $(\sigma_\mathrm{cal}/F_\mathrm{cal})$ of the calibration depend on elevation and are formally less than 2.6, 0.7, 1.4, 1.1, 0.7 and 0.9% at 1.4, 2.7, 3.9, 7.7, 13 and 31 cm respectively (the errors are averaged here over two horns at 1.4-3.9 cm, Fig. 3).

Mean values are always calculated as the mean weighted values, if several measurements have been made, with corrections by the Student's factor to increase the reliability to the standard value 0.683 for one sigma error. The dispersion of frequent measurements of calibrators (and, consequently, $\sigma_{cal}$ and $\sigma$) as well as calculated errors of mean spectra measurements represent also random instrumental instabilities and a variability due to atmosphere conditions during the set.

Systematic errors caused by various reasons including calibration are known to be often the main real errors. We have compared our results with published observations of other authors to check the residual systematic errors, using several tens of strong objects distributed on elevations with constant or slightly variable broad band spectra. The agreement is found to be quite good within the total accuracy of the data.