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Astron. Astrophys. 342, 474-479 (1999)

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2. Observations and data reduction

The pulsar PSR B1133+16 was observed with the 100-m Effelsberg radiotelescope of MPIfR. Observations were made at the center frequency 1.41 GHz with a bandwidth of 40 MHz, using a tunable HEMT-receiver with a system temperature of about 28 K. The two circular polarizations are separated in the receiver and amplified. The signals are then fed into an adding polarimeter, a passive device with four output channels which allows further online signal processing. Using an online dedispersion device, a four unit [FORMULA] kHz filterbank, the pulse-smearing caused by the dispersion due to the interstellar medium is removed. The output of each channel is then detected and converted into a digital signal by a fast A/D converter. After a time delay according to the dispersion measure, the outputs of all channels are added and then recorded by the backend. After a careful calibration, Stokes parameters are obtained from the four recorded output channels. The system description and the calibration procedure are given by von Hoensbroech & Xilouris (1997).

We recorded three sets of single pulse data (each set containing about 500 individual pulses) with time resolutions of 1160, 500 and 150 µs on three different observing sessions during September 1996. PSR B1133+16 is strong at 1.41 GHz, and has a small dispersion measure (DM=4.8 cm-3 pc), which makes it a good source for studying orthogonal polarization modes with high time resolution. The average polarization parameters: intensity I, linear L and circular V polarization are plotted as a function of pulse phase in Fig. 1a. The continuous line curve indicates I while the broken and dotted ones represent L and [FORMULA] respectively, in arbitrary units. We do not find any significant difference between this average profile and the other average profiles obtained from 500 µs and 150 µs time resolutions data. The polarization properties of the average profile, a well-resolved double lobe, show notable differences between the leading component and the rest of the profile. The leading component is only [FORMULA]% polarized while the saddle and trailing components are [FORMULA]% polarized (Manchester, Taylor & Huguenin 1975; Cordes & Hankins 1977).

[FIGURE] Fig. 1a-d. The average polarization profile of PSR B1133+16 in arbitrary flux units (a.u) and polarization angle [FORMULA] gray-scale maps. The panels b , c and d represent the polarization angle gray-scale maps obtained from individual pulses with time resolutions of 1160, 500, and 150 µs, respectively. A marker of time duration 2 ms is given at the upper-left corner of panel a . The average polarization angle curves and the RVM fits are superposed. The shade is a number in the range 0 (white) and 1 (black) obtained by linear interpolation between the background and foreground levels.

2.1. Polarization angle swing of orthogonal modes

The gray-scale maps in Fig. 1b-d show the frequency of occurrence of orthogonal polarization modes with respect to the pulse phase for three different time resolutions. We used the pgplot routines developed by Pearson (1989) for making gray-scale maps. The fortran subroutine pggray draws gray-scale map of an array in the [FORMULA]-[FORMULA] plane, by determining the shade of each point from the corresponding array value. The shade is a number in the range from 0 to 1 obtained by linear interpolation between the background level (white) and the foreground level (black). The white regions in the maps are with shade = 0 and darkest regions are with shade = 1.

This technique has become a powerful tool in analyzing the pulsar polarization properties (e.g. Stinebring et al. 1984a,b). The panels b-d in Fig. 1 represent the polarization angle gray-scale maps at the time resolutions of 1160, 500 and 150 µs, respectively. To compare the three time resolutions we have given a marker of time duration 2 ms at the upper-left corner of Fig. 1a. The darkest shades represent the most probable regions of occurrence. At each pulse phase bin, the full range of polarization angle was divided into 200 intervals. In this way, the gray-scale maps were made from all those phase bins where the linear polarization L is above [FORMULA] level. Here [FORMULA] is the rms of L in the off pulse region. All those phase bins, where the condition [FORMULA] was not met, were excluded as they lead to spurious polarization quantities.

The curves represented by points, associated with error bars, indicate the integrated polarization angle swing. They follow closely the mode which is strong at any pulse phase. The RVM was proposed to explain the polarization angle swing in integrated pulse profiles (e.g. Manchester and Taylor 1977):


where the parameters are [FORMULA] the rotation phase, [FORMULA] the angle between rotation and magnetic axes, and [FORMULA] the impact parameter, i.e., the angle between the magnetic axis and the line-of-sight.

We fitted the function [FORMULA] to the observed polarization angle using two different numerical algorithms. First, the robust Simplex Algorithm (Nelder & Mead 1965) was applied to get close to the minimum of [FORMULA] and the minimum was then optimized with the Levenberg-Marquardt Algorithm (Marquardt 1963). The goodness of the fit was tested using the classical [FORMULA]-test and Kolmogorov-Smirnov test (e.g. Press et al. 1992). We found with [FORMULA] and [FORMULA] the RVM can be fitted to all three data sets. Using these values [FORMULA] and [FORMULA], the RVM was fitted with the polarization angle swing of orthogonal polarization modes, as indicated by the continuous line curves in Figs. 1b-d.

In Fig. 1b, orthogonal polarization modes appear separated only under the leading component, while under the trailing component one mode has become too weak. In the case of incoherent addition of radiation fields, Stokes parameters become additive (Gangadhara 1997). Therefore, it is the stronger mode, which determines the polarization angle in the case of low resolution observations. In Fig. 1c, orthogonal polarization modes are not clearly seen under the trailing component, as they are not yet fully separated, while in Fig. 1d they are separated in both components.

The gray-scale maps and the integrated polarization angle swings clearly indicate that the discrepancies between the RVM and the polarization angle swing in average pulses arises due to the presence of orthogonal polarization modes. The non-orthogonal (random) radiation, which does not fit with the RVM model, exists in all our data sets. The polarization angle closely follows the stronger mode in any phase bin.

To investigate the effect of time resolution on orthogonal polarization modes, we consider the pulse phase between [FORMULA] where both modes are clearly active. Nearly [FORMULA] polarization angle swing over this interval of pulse phase was removed by applying rotation in the opposite direction. Let i=1, 2, 3... N be the intervals of polarization angle with each one having the width 180/N. If [FORMULA] is the number of pulses (frequency) which appear in the [FORMULA] interval, then the total number of pulses, which appear in all the N intervals, is given by


Therefore, the probability of occurrence (relative mode frequency) of pulses in the [FORMULA] interval is given by


such that the total probability is unity.

Using N=200, we computed [FORMULA] separately for each data set, and plotted as function of [FORMULA] in Fig. 2. The dotted line represents the pulses with 1160 µs resolution, while the dashed and continuous line curves represent the pulses with resolutions of 500 and 150 µs, respectively. Note that we have applied a position angle rotation of [FORMULA] while plotting Fig. 2. This rotation was just to make the two humps clearly visible. It can be clearly seen that the polarization angle preferentially occurs around the two angles separated by about [FORMULA] At low resolutions the primary mode, which is at [FORMULA] is observed more often than the secondary mode, which is at [FORMULA] At the higher resolution, orthogonal polarization modes are better resolved and the probability of observing the secondary mode becomes comparable to that of primary mode. Hence the probability of detection of orthogonal polarization modes is affected by the data resolution.

[FIGURE] Fig. 2. The probability of occurrence of pulsar radiation at different polarization angles. The dotted, broken and continuous line curves represent the pulses with time resolutions of 1160, 500, and 150 µs, respectively. To display the two humps clearly position angle rotation of [FORMULA] has been applied.

2.2. Depolarization

To investigate the role of time resolution on the polarization properties of pulsar radiation, we have analyzed single pulses with different time resolution. We used [FORMULA] threshold on I and [FORMULA] and considered the full range of pulse phase [FORMULA] The gray-scale maps which represent the correlation between the fractional linear [FORMULA] and circular [FORMULA] polarization and [FORMULA] are shown in Figs. 3 and 4.

[FIGURE] Fig. 3a-c. Grey-scale maps representing the correlation between [FORMULA] and [FORMULA] in the individual pulses at resolutions of 1160, 500, and 150 µs. The shade is a number in the range 0 (white) and 1 (black) obtained by linear interpolation between the background and the foreground levels.

[FIGURE] Fig. 4a-c. Grey-scale maps representing the correlation between [FORMULA] and I in the individual pulses at resolutions of 1160, 500, and 150 µs.

When the resolution is high, both linear and circular polarization are also high as indicated by Figs. 3c and 4c. This is because the short time scale structures such as micropulses are often highly polarized, which leads to high polarization when observed with a higher resolution.

Figs. 3a-c indicate the correlation between the linear polarization and circular polarization. It is evident that the linear polarization becomes higher when circular polarization is at minimum. This is in agreement with the predictions of curvature radiation (Gil et al. 1993; Gangadhara 1997). Figs. 4a-c indicate the behaviour of circular polarization with respect to the intensity at different time resolutions. They indicate that at lower intensity and higher resolution, circular polarization is higher.

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© European Southern Observatory (ESO) 1999

Online publication: February 22, 1999