2 Observing technique and calibration procedure

Due to elevation restrictions of the 100-m telescope, we were constrained to only include sources with a declination greater than $-15\hbox{$^\circ$}$. Also, sources with an angular extent greater than the half-power beam-width of the telescope, which is just under 3 min of arc, were excluded from the sample for the practical reason that more time-consuming 2-D raster scans of I, Q, and U around the sources could be avoided. Within the allotted observing time, we were able to obtain data for 154 sources, mostly radio galaxies, that were known to be significantly polarized.

The receiving system of the 100-m telescope employs a twin-horn system. The left-hand circular signal from the main horn is passed through a series of amplifiers while being mixed down to a frequency of 350 MHz. The right-hand circular signals from the main and reference horns are combined to produce sum and difference signals, which are also amplified and mixed down to a frequency of 350 MHz. The resultant three signals are then fed into a three-channel polarimeter having a 100-600 MHz bandpass, i.e. 500 MHz. The right-handed circular sum signal is then correlated with the left-handed circular signal to obtain the Stokes Q and U components, and the right-handed circular sum and difference signals are correlated to yield Stokes I. For gain stabilization, an intermittent linearly polarized noise calibration signal was introduced into the receiver system. This signal can then be extracted from the output of the polarimeter and used to normalize the Stokes parameters. The result is to further improve the stability of the receiver system already stabilized by the twin-horn method.

Each source was observed in an ON-OFF manner consisting of six "ON" source and six "OFF" source measurements, each with a duration of just less than $\sim\,30$ s. With a $T_\mathrm{sys}$ of 60 K, it was possible to attain a sensitivity of 5 mJy s^-1/2. Each observation was preceded by a gridded search in the sky to pin-point the exact location of the source. This was done to ensure that the sources were directly in the on-axis direction of the telescope. Under such conditions, we can assume that some linear relationship exists between the observed and true Stokes parameters. Since for our purposes circular polarization can be neglected, Stokes I, Q, and U are the components of the observed and true polarization vectors (i.e. $\mathbf{S}_\mathrm{OBS},\ \mathbf{S}_\mathrm{TRUE}$) and one can can write:

$$\mathbf{S}_\mathrm{OBS}=\mathbf{T}\ \mathbf{S}_\mathrm{TRUE}$$ (1)

where both $\mathbf{S}_\mathrm{OBS}$ and $\mathbf{S}_\mathrm{TRUE}$ are defined in the telescope's aperture plane, and $\mathbf{T}$ denotes the 3-by-3 polarization matrix of the whole receiving system, describing it's imperfections between the source and the polarimeter output. The first row of $\mathbf{T}$ determines the overall flux calibration and includes the cross-talk of polarization into the total power channel. The final two rows represent the effects of the instrumental polarization and the polarimeter ellipticity on the true polarization vector. A complete description of $\mathbf{T}$, along with schematics of the receiving system and polarimeter, can be found in Turlo et al. (1985).

For an altitude-azimuth mounted telescope, the true Q and U Stokes parameters in the sky rotate in the aperture plane with the parallactic angle. Crosstalk from the polarization channels to total power introduces a sinusoidal variation of Stokes I as a function of phase angle ($2\times$ parallactic angle + $2\times$ position angle). The length $\mathbf{I}_\mathrm{p}=\sqrt{Q^2+U^2}$ of the observed polarization vector varies in a more complicated fashion, since it is given by the distance with respect to the origin of a point moving on the circumference of an ellipse, being displaced by the instrumental polarization. These variations can be corrected for by determining the nine matrix elements of $\mathbf{T}$ from a set of at least 3 linearly independent input and output polarization vectors $\mathbf{S}_\mathrm{TRUE}$ and $\mathbf{S}_\mathrm{OBS}$. The collection of polarization vectors of a given source obtained at at least 3 different parallactic angles constitutes such a set of independent measurements. Since a set of redundant measurements is usually preferred, $\mathbf{T}$ is solved for by the least-squares method described by Thiel (1976). Once $\mathbf{T}$ has been found, the true polarization vector is given by

$$\mathbf{S}_\mathrm{TRUE}=\mathbf{T}^{-1}\ \mathbf{S}_\mathrm{OBS}$$ (2)

for all the program sources. We note that $\mathbf{S}_\mathrm{TRUE}$ is defined in the aperture plane and must be mapped back into the sky plane because of the variation of parallactic angle. The position angle ($\chi$) of the linear polarization can then be found in the usual manner from the second and third components of the polarization vector.

The matrix calibration technique includes an error analysis for thermal and systematic errors. The propagation of noise with respect to the Stokes parameters is described by Thiel (1976)

$$\Delta(\mathbf{S}_\mathrm{TRUE})^2= \left( \frac{\Delta\math... ...\mathbf{S}_\mathrm{OBS}} {\Delta\Vert\mathbf{T}\Vert} \right)^2$$ (3)

Forkert (1984) discussed the systematic errors, resulting from an imperfect correction of the polarimetric effects. Let $\mathbf{T}_\mathrm{true}$ and $\mathbf{T}_\mathrm{exp}$ denote the true and experimentally determined transformation matrices, respectively. An imperfect correction of the polarimetric effects is then described by a residual transformation matrix

$$\mathbf{R} = \mathbf{T}_\mathrm{exp}^{-1}\ \mathbf{T}_\mathrm{true} \ .$$ (4)

$\mathbf{R}$ should be very close to unity, but nevertheless, just like $\mathbf{T}$, gives rise to phase angle dependent variations in total power, polarized power $\mathbf{I}_\mathrm{p}$ and position angle $\chi$. The amplitudes of these residual variations are given by

$$\frac{\Delta \mathbf{I}} {\mathbf{I}}= \sigma_{\mathbf{I},1} + \sigma_{\mathbf{I},2} * m$$ (5)

$$\frac{\Delta \mathbf{I}_\mathrm{p}} {\mathbf{I}_\mathrm{p}} = \frac{\Delta m} {m} = \sigma_{m,1} + \sigma_{m,2} / m$$ (6)

$$\Delta \mathbf{\chi} = \sigma_{\mathbf{\chi},1} + \sigma_{\mathbf{\chi},2} / m$$ (7)

where the numerical constants (see below) are functions of the elements of the residual transformation matrix $\mathbf{R}$. Since, for a given source, the actual size and sign of the residual errors are dependent on phase angle and thus on the particular time of observation, they almost behave like an additional noise error, the standard deviation of which can conservatively be approximated by the above amplitudes. Thus, the overall error in Stokes I, $\mathbf{I}_\mathrm{p}$ and $\chi$ will be given by the squared sum of both noise and residual systematic contributions.

It is important to emphasise that the absolute errors in position angle will still be somewhat larger than those obtained from our error analysis, since they are subject to the accuracy of the adopted position angles of the calibrators, and to some low-level systematic contribution due to Faraday rotation of the polarization position angles. We estimate the latter to be at most $0\hbox{$.\!\!^\circ$}2$.

We also note that our implicit assumption of the applicability of Gaussian statistics for the calculation of the errors of the degree of polarization m and $\chi$ is valid only as a first-order approximation since a) the matrix elements may not be statistically independent under certain conditions and, b) the polarized flux is subject to Ricean statistics. The latter problem is important for sources with low signal-to-noise (i.e. $\leq 3$) in their polarized flux (cf. Wardle & Kronberg 1974, and references therein).

The reliability and repeatability of the above calibration scheme with the 100-m telescope has been tested by Turlo et al. (1985) and Forkert (1984). We applied the same calibration procedure as they did. Each observing session is divided into several sections corresponding to the availability of individual calibrators that had known and consistent parameters, and complete overlap in hour angle. In each section the elements of $\mathbf{T}$ are redetermined using the particular visible calibrator over the maximum possible range of parallactic angles. This approach is chosen, since aside from possible inconsistencies between calibrator parameters, there are variations in the matrix properties derived from different calibrators, which cannot be ascribed to thermal noise alone. Forkert (1984) demonstrated the presence of additional az-el dependent polarized components, these being most prominent at low elevations. Experience showed that the mixing of calibrators in order to determine $\mathbf{T}$ was inferior to the approach chosen. Another essential ingredient of the procedure is an elevation-dependent gain correction (peak-to-peak 3%).

Using 3C 286 as their primary calibrator, Turlo et al. find that the matrix elements of $\mathbf{T}$ within certain limits remain stable over periods of greater than four months at $\lambda$6 cm. Their estimates of the noise dependent scatter over a day for 3C 286 was 12.8 mJy, or 0.17% for $\mathbf{I}_\mathrm{p}$ and 0.09 degrees for $\chi$. For the same set of calibrators we used (see below), Forkert (1984) determined typical values for the constants describing the residual variations in amplitude (assuming 0 < m < 1):

For the data presented here, 3C 48 and 3C 286 were chosen as the calibrator sources; their total flux, m, and $\chi$ are listed in Table 1 where the parameters of 3C 48 were adjusted slightly to be consistent with 3C 286. A different matrix $\mathbf{T}$was computed for each observing session and used to reduce the data for that session. A log of the observing sessions is recorded in Table 2. We obtain a similar level of repeatability for the calibrators as Turlo et al. (1985) - see above, but we also noticed slight variations in the matrix elements amongst the set of matrices. We interpret these variations to be related to the extent of the parallactic angle coverage, since each observing session sampled different parallactic angle ranges and overlap in some cases is minimal. In some cases, a particular source was observed in more than one session. An example, 0010+04, observed in four different sessions, each reduced with a different matrix $\mathbf{T}$, is shown in Table 3. Since we found no reason to question the validity of any particular matrix, a straightforward unweighted average of the Stokes I, Q, and U values was used to produce the final results (Table 4).

  

Table 1: Calibrator polarization parameters

^a Baars et al. (1977).
^b Perley (1982); Ryle et al. (1975); Sastry et al. (1967).
^c Simard-Normandin et al. (1981).

  

Table 2: Record of observations at 6.3 cm