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Astron. Astrophys. 342, L57-L61 (1999)

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2. Analysis and results

Presently we have data for three pulsars, where a clear change from linear to circular polarization towards high frequencies is observed (see Fig. 5 in von Hoensbroech et al. 1998b). It is obvious from Fig. 1 that the three objects have very different rotational periods P and period time derivatives [FORMULA]. Hence they do not form an isolated group with respect to their rotational parameters. Since the change from linear to circular polarization is superposed by general depolarization effects, which affect both types of polarization likewise, the degrees of polarization cannot be compared directly with the theory. Therefore we choose the ratio


between the degrees of linear ([FORMULA]) and the circular ([FORMULA]) polarization as an depolarization independent parameter. R and its statistical error [FORMULA] can easily be calculated from the data.

[FIGURE] Fig. 1. Position of the three pulsars in the [FORMULA]-diagram for which the change from linear to circular polarization has been observed. They do not form a coherent group within the pulsar sample. One of them is very young, one is rather average and the third one is a relatively old object.

The polarization data we used for this analysis were accessed through the online EPN-database (http://www.mpifr-bonn.mpg.de/pulsar/data/ ). The calibrated data were extracted in the EPN-format (Lorimer et al. 1998). The degrees of polarization were calculated using the same routine for all profiles. All relevant parameters and references are listed in Table 1.


Table 1. Polarization data and references for all three pulsars. Note: [FORMULA] Reference-code: [1] Gould & Lyne (1997), [2] von Hoensbroech et al. (1998a), [3] Qiao et al. (1995), [4] unpublished Effelsberg data.

The theoretical functional dependence of R can be derived from Eq. (25) in (von Hoensbroech et al. 1998b). Applying a Taylor-expansion in inverse frequency [FORMULA] to the first order yields


Here [FORMULA] is the local electron gyro frequency at the PLR, [FORMULA] the angle between the propagating wave and the direction of the local magnetic field at the PLR and [FORMULA] the streaming velocity of the plasma. Higher order terms can be neglected as long as the wave frequency is different from [FORMULA].

Obviously, a [FORMULA]-frequency dependence of R is required by the theory. A comparison of the observed frequency dependence of R with the predicted one is therefore a strong test for the theory. Figs. 2-4 show a comparison of the data points and the theoretical curve (dashed line) for R.

[FIGURE] Fig. 2. PSR B0144+59, [FORMULA] versus frequency. The theoretical change of R (dashed line) is compared to measured data. The horizontal dotted line corresponds to [FORMULA], i.e. linear and circular polarization are of equal strength (here [FORMULA] at [FORMULA] GHz). Parameters for theoretical line: [FORMULA] of [FORMULA].

[FIGURE] Fig. 3. PSR B1737-30, see caption of Fig. 2 and text for details (here [FORMULA] at [FORMULA] GHz). See text about the two "outriders'. Parameters for theoretical line: [FORMULA] of [FORMULA].

[FIGURE] Fig. 4. PSR B1913+10, see caption of Fig. 2 and text for details (here [FORMULA] at [FORMULA] GHz). Parameters for theoretical line: [FORMULA]

There are a couple of parameters which indirectly enter Eq. (2) as scaling factors. [FORMULA] is proportional to the local magnetic field strength [FORMULA]. This value again depends on basic pulsar parameters such as the period, its time derivative and, if known, on the inclination angle between the rotation- and the magnetic dipole axis. Furthermore the angle [FORMULA] between the propagating wave and the local magnetic field depends on the chosen field line and the assumed emission height. For the background plasma Lorentz factor we made the assumption [FORMULA]. Finally we chose the PLR at 20% of [FORMULA]. As the values of the PLR and the emission height (2% of [FORMULA] are given as fractions of [FORMULA], their absolute values depend on the period. The combination of [FORMULA] and PLR was chosen without restriction of generality as various other combinations yield the same result (see Fig. 5).

[FIGURE] Fig. 5. Possible combinations of PLR and [FORMULA] which yield consistent results for the prefactor in Eq. (2) which fit to the observed data of each pulsar. A variation of this factor results into a parallel displacement of the theoretical curves in Figs. 2-4. The solid line below corresponds to the assumed emission height at 2% of the [FORMULA] as the minimum PLR.

However, apart from the known intrinsic pulsar parameters P and [FORMULA], the same set of parameters was used for all three pulsars. Please note that these parameters only enter Eq. (2) as scaling factors, yielding a parallel displacement of the function. Hence they determine the frequency range where the transition from linear to circular polarization takes place, but not the functional dependence [FORMULA].

2.1. PSR B0144+59

This pulsar is the first one in which we found the effect of increasing circular polarization to high frequencies. Its rotational parameters are [FORMULA] ms and [FORMULA], yielding to a weak surface magnetic field of `only' [FORMULA] T, an average value for [FORMULA] W and the characteristic age [FORMULA] yrs. Reasonable data in full polarization was available between 610 MHz and 4.85 GHz.

Fig. 2 shows the measured values and the theoretical curve for the change of R with frequency.

2.2. PSR B1737-30

PSR B1737-30 has a spin period of [FORMULA] ms and a period derivative of [FORMULA]. The resulting spin down energy loss [FORMULA] places it amongst the top 10% of the pulsar sample. The very low characteristic age [FORMULA] yrs and the very high surface magnetic field [FORMULA] T make this one an extreme object. Note that in terms of the surface magnetic field, this object is at the opposite end of the "normal" pulsar sample compared to PSR B0144+59.

The change of R with frequency is shown in Fig. 3. Although the two low frequency points do not fit the theoretical function, the frequency change of R smoothly follows this function at higher frequencies. Note that the two outrider profiles are significantly affected by interstellar scattering which could also alter the polarization state when emission from different pulse phases with different polarization states are superposed (see also Sect. 2.4).

2.3. PSR B1913+10

With a spin period [FORMULA] ms and its temporal derivative [FORMULA] this is an average pulsar. This is also reflected by its parameters [FORMULA] W, [FORMULA] T and [FORMULA] yrs.

Fig. 4 shows the measured values for R and the theoretical function. As for PSR B1737-30 the lower frequency points are too small, but at higher frequencies the theoretical curve is matched perfectly. As for the previous pulsar, the two outrider profiles are significantly scattered.

2.4. Outriders

The systematic deviation of the low frequency points is certainly a draw back of these observations. However they can be understood through the following argument: von Hoensbroech et al. (1998a) have shown that the polarization properties in general are much less systematic at low radio frequencies compared to higher ones. This indicates that the polarization of pulsars undergo some sort of randomization at low radio frequencies. This can be caused either through intrinsic variations - e.g. non constant PLR at low frequencies - or through additional propagation effects in the highly magnetized medium close to the pulsar, which mainly affect low radio frequencies.

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999