Astron. Astrophys. 336, 455-478 (1998)

Appendix A: a new method for finding rotation measures

The `classical' method for finding Rotation Measures is to determine the position angle of the polarized emission in each individual channel, and to plot these as a function of the square of the wavelength. The RM is then the slope of the line that fits these datapoints best. This method has proved itself extensively, but it provides little insight in the spatial distribution of the Rotation Measure when studying extended structures.

For this purpose, a new routine has been developed in the NFRA NEWSTAR reduction and analysis package, called Rotation Measure Mapping. This routine combines the maps of the Stokes Q and U parameter of each separate frequency channel in the following way. Using an `a priori' guess of the RM, the Q and U vectors in each channel are rotated `backwards' over the angle that they would have been rotated over by the assumed RM, relative to the frequency of the highest frequency channel. Then, the Q (and U) maps of all channels are vector summed. Only if the chosen RM is close to the real RM, the rotated Q (and U) vectors in each channel will be the same, and the length of the summed vectors will be maximal. Using these vector summed Q and U vectors, which we will refer to as and , respectively, we can calculate the summed polarized intensity , which will also be at its maximum when the correct RM is chosen. Using a range of possible RMs, the RM for which the summed polarized intensity is maximal, must be closest to the real value.

The main advantage of the NEWSTAR method is that maps of summed polarized intensity can be made for a wide range of RMs. These can be displayed in a movie form, so that for extended sources, like WNB 0313+683, one can look for systematic changes in the RM over the source. But it can also be used for a more strict analysis of the data.

In the classical method, the spectral index and depolarization of the emission, when using widely separated frequencies, will affect the error in the measured polarization angle, and thus the accuracy of the result. In what manner exactly strongly depends on the nature of the source observed. The intrinsic degree of polarization for synchrotron radiation with a powerlaw spectrum is not frequency dependent. Therefore, in radio sources with negative spectral slopes, e.g. , the drop in total intensity at a higher frequency can, at least partly, be counteracted by the lower amount of depolarization at that frequency.

In order to test the accuracy and reliability of the NEWSTAR method, we have performed computer simulations, in which we analysed the same artificial input data set with the NEWSTAR and the classical method. As input we provided 500 random Q and U intensities, which we assumed to have a frequency of 1400 MHz (cf. the NVSS survey). For each datapoint, we then determined the polarization angle at 1400 MHz and calculated the polarization angles at the other frequencies, using a fixed Rotation Measure of 10 rad m^-2. From this, the Q and U intensities at each frequency have been determined. Then noise was added using a Gaussian noise distribution with a standard deviation equal to the rms noise of our real Q and U data.

We find that both methods recover the input RM very well, as long as the polarized intensity , where is the noise in the averaged polarized intensity map corrected for Ricean bias (see Fig. A1). For polarized intensities , the input RM is not reliably recovered. Especially note the two `sidelobes' in the RM distribution, which are separated by  rad m^-2 from the input RM. They are the result of the rad ambiguity of the determination of the polarization angle in the high frequency (NVSS) data. Due to the added noise, chosing a value which is 180^o higher or lower sometimes gives a better fit through the datapoints. We find that of the datapoints in our simulations which have polarized intensities between 8 and lie in these two sidelobes.

We have used this method to correct for the observed position angles of the E-field in the 1400-MHz NVSS observations. The result is shown in Fig. A2. This should be compared with the orientation of the magnetic field that is observed at 10.45 GHz with the Effelsberg telescope (i.e. the vectors plotted in Fig. 4a rotated by ). The  rad m^-2 ambiguity causes the small wiggle in the B-field vectors in the southern lobe of WNB 0313+683.

Fig. 15. Plots showing the result of the simulations of the NEWSTAR method and the `classical' method for finding Rotation Measures (see text for details). Each plot is a realisation of 500 points with random signal-to-noise values. We used a RM of 10 rad m-2 as input. Both methods give very similar results.

Fig. 16. Contour plot of the WNB 0313+683 at 1400 MHz from the NVSS survey (see also Fig. 1a and b). The orientation of the vectors represent the orientation of the `true' B-field after rotating the observed E-field with the amount of rotation expected from the RM-analysis described in this Appendix. The position angles are almost identical to those inferred from the high-frequency Effelsberg observations. The apparent `wiggle' is caused by the  rad m-2 uncertainty that is described in the text.

© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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