Astron. Astrophys. 321, 305-310 (1997)

2. Method, observations, and data reduction

We observed with a standard radioastronomy receiver of wide frequency bandpass: a linearly polarized antenna, a receiver followed by a detector, time constant, and a recording system. No polarimeter and no dedispersion system were used, and no calibration (except for the ionosphere) was necessary. We recorded individual pulsar pulses, for our purpose these pulses should have a strong linearly polarized component. As these pulses drift downwards in frequency through the bandpass following the delay in arrival time (due to the cold interstellar plasma dispersion law) they will experience a rotation of the plane of polarization (Faraday effect) (Fig. 1). It is very common to have a pulsar pulse with a variable linearly polarized component, and where the (intrinsic) sky angle of the direction of the linear polarization is variable through the pulse (e.g. Rankin & Benson 1981, Stinebring et al. 1984, Rankin et al. 1989). Nevertheless, the signal component within a pulse which has the maximum linearly polarized flux has a well defined polarization direction in many pulsars, and we will take this to be the dominant linear polarization angle. The intrinsic sky angle of polarization at source is taken to be constant over the receiver bandpass at any given pulse longitude, including that of the dominant linear polarization. After propagation through the interstellar medium the direction of this dominant linearly polarized electric field vector rotates as a function of frequency. If observed with an antenna-receiver system that measures full intensity (e.g. two cross-polarized antennas with a receiver each, both detected outputs summed together) followed by our backend and recording system we will simply see a dispersed pulse (Fig. 1b). If instead we use a single linearly-polarized antenna-receiver system that receives a signal with a linearly-polarized component whose electric vector rotates as a function of frequency (Fig. 1c), then the voltage induced in the linearly-polarized antenna is the dot product of the electric vector and a unit vector in the direction of the antenna polarization. As a function of time within the dispersed pulse this will result in a sine wave: the electric vector either aligns itself with the linearly-polarized antenna at certain frequencies, where we will have the maximum of the sine, or is perpendicular to it at others, where we will have the minimum of the sine, or, for in-between frequencies, we will have the sine projection values. All other pulse components (circularly polarized, non-polarized, etc.) will also be received by this linearly-polarized antenna-receiver system, and will be the non-polarized component of the pulse, as shown in Fig. 1.

Fig. 1. Cartoon representation of the output of the pulsar pulse when observed with three different receiver systems. a Single pulse as observed with a narrowband filter (not dispersed), or at source (intrinsic). b Dispersed pulse at the output of our wideband receiver system, if we observe intensity (non-polarized system). Amount of linear polarization contribution is indicated. c Same as b but observed with a linearly polarized antenna. Alignment of the dominant linearly polarized electric vector and the antenna dipole is indicated at different frequencies (times) as the dispersed pulse descends in frequency within the passband.

The time delay (sec) of the arrival of a pulse at the frequency f (MHz) from the arrival at due to interstellar medium dispersion is

where and DM is the dispersion measure, pc cm^-3. The angle of rotation (rad) of the plane of linear polarization at the frequency f from the intrinsic sky angle due to Faraday rotation is

where the rotation measure RM () is

where , R is the distance pulsar - observer, in µGauss, and if the magnetic field is pointed towards the observer. Consider two frequencies: and , between which the electric field vector (or the plane of the dominant polarization) executes one half of a full turn () (that is, sinusoid maximum to maximum, one full period)

Between and there is a dispersion delay

Hence, for one turn of the polarization plane there is a delay that we obtain from Eq. (2) and (3) (Smirnova 1991) (after replacing constants)

Note that depends exclusively on here, and will be constant through the bandpass. is a constant in the temporal domain, that is is the same at all receiver frequencies. does not depend on the sign of RM or , hence it tells us their absolute value, however, it is possible to determine indirectly the sign for new measurements (see below). The detected output of our wideband receiver shows the pulsar pulse as wide pulses, smeared by dispersion over the bandpass (Fig. 1a and b). With a linearly-polarized antenna-receiver system within these widened pulses we will see the sine modulation of period as the dominant linearly-polarized component of the pulsar pulse changes its angle continuously with respect to the direction of polarization of the antenna (Fig. 1c). That remains constant throughout the BW is a good assumption, since we select , where is the center receiver frequency. We emphasize that, having eliminated the dependence in the measured parameter , the primary parameter measured with our method is the magnetic field (Eq. (4)), which is of physical interest, and has errors due only to our measurement of and reduction procedures. The RM is a derived parameter in our experiment, and includes the errors of the DM measurements done separately.

We chose the observation frequency and the bandpass so that we have a dispersion delay over the bandpass as large as possible, so the number of rotations of the electric vector would be as large as possible. But it must be , where is the pulsar period, so that consecutive pulses do not overlap. BW should also be consistent with the receiver (and interference!) constraints. For a proper fitting we need a minimum of two or three sine cycles (better more), that is BW should be larger than or , depending in part on the signal-to-noise ratio. If these conditions are not met a different receiver frequency may solve the problem.

For observation we have selected mostly pulsars of a tabulated average flux mJy at 400MHz, however, because of signal variability some pulsars weaker than this limit produced good results, and some pulsars stronger than this limit produced poor or no results. If polarization characteristics were known high linear polarization pulsars were preferred.

Observations were carried out in August - September, 1992, at Arecibo. Strong interference at the preferred 130MHz band precluded observations. At 318MHz seven pulsars turned out to be too weak, and two others (PSR 1845-01 and PSR 1900 06) had no sinusoidal modulation superimposed, presumably because of a low linear polarization component. Positive results were obtained for seven pulsars, all at 318MHz. Table 1 lists them, the bandwidth BW used, their dispersion smearing and period .

Table 1. Pulsars observed at 318 MHz and their parameters: , pulsar period; BW, observational bandwidth; , dispersion delay over the observational bandwidth; N, number of points of running average; n, number of 15-minute observations used in the processing.

The detected output of the receiver (integrated by a 2 msec time constant) was sampled with an A/D converter with a period of of 0.5 msec, and continuous 15-minute recordings on tape were done, n recordings per pulsar (Table 1). Then the time average of the dispersion-smeared pulse profile was obtained by folding the data with a period , where 512 bins were distributed. No pulse selection was done. Since the beginning of the integration had an arbitrary pulse phase we normally rotated cyclically the integrated pulse so that the beginning of the pulse corresponded to the low order bins. A running average smoothing of N consecutive points (Table 1) was done to improve the signal-to-noise ratio. To obtain the period of the sine a least-squares fitting of the sum of a sine and a second degree polynomial was done, and the sine frequency, phase and amplitude, the three polynomial coefficients, and their errors were obtained from the fit. Using higher degree polynomials did not improve the fit. Of the fitted parameters only the frequency of the sine is of interest. As the modulated part of the dispersed pulse was less than only the part of the pulse period with sinusoidal modulation was used in the fitting procedure.

Different parts of the pulse may have linear polarization components that have different polarization sky angles. Each will produce sines of the same frequency but different phase. Because the sum of sines of constant frequency (which depends only on ) but arbitrary phase is also a sine of the same frequency, we do not have to separate the contributions from different pulse longitudes. In many pulsars the polarization angle swings over the whole pulse longitudes range (e.g. Rankin & Benson 1981) is of the order of or less, the resulting sine will be of larger amplitude than the original sines. Fitting to the resulting sine produces the same result (frequency) as fitting to separate pulse component sines. Nevertheless, in most cases one component at a specific pulse longitude predominated.

The signal inside the receiver goes through a square-law detector, however, the reasoning developed above is valid for the voltage induced by the electric field in the antenna, not the power. The voltage sinusoid that we obtain in this experiment after going through the square-law detector becomes the sum of a sine and double frequency cosine. The contribution of the double-frequency cosine term is small, and it becomes significant only if the pulse linear polarization component approaches 100%, not common in pulsars. In any case double-frequency distortion adds little to the error in the fitting procedure, where we are primarily interested in the frequency of the sine.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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