Astron. Astrophys. 322, 846-856 (1997)

6. Discussion

We noted earlier that at 2.25 GHz we only obtained right-hand circular signals. This can lead to uncertainties in the analysis since the emission of many pulsars exhibits a significant circularly polarised component, so that the location of the measured profile midpoint might differ in comparison to a total power profile. Moreover, the propagation time of a signal through the ionized interstellar medium depends also on the sense of the wave polarisation. The difference in arrival times between left-hand and right-hand circularly polarised signals is given by

whereas is the Rotation Measure of the source in units of rad m^-2 (e.g. Phillips 1991b). The value of is a measure of the Faraday rotation occurring to the pulsar signals along the line-of-sight. Measured values are as large as rad m^-2 (Taylor et al. 1993). However, even for such large s the time difference at our lowest frequency, viz. at 1.41 GHz, is only 21 ns and therefore far beyond the accuracy of our measurements. The effect of possible profile changes on the measured TOAs can be estimated by comparing TOA offsets between profiles obtained from LHC and RHC data at 1.41 GHz. For our sample of sources, the amount of circular polarization at 1.41 GHz is less than 15% (Gould 1994), decreasing at higher frequencies (Xilouris et al. 1996). Therefore, TOA offsets determined between LHC and RHC profiles at 1.41 GHz obviously represent safe upper limits for the uncertainty which is introduced if we use RHC signals rather than total power signals at 2.25 GHz. As the result of such an analysis, we find that the TOA-offsets are largest for B0329+54 (i.e. 670 s) but always smaller than one sampling interval. Our results are therefore not affected by the characteristics of the received signals at 2.25 GHz.

Another effect which could influence the measurements of our TOAs as a function of the emission height, and thus frequency, is that due to curved spacetime near the neutron star. This effect might become important if the emission is (as we will see) created close to the pulsar surface. Gonthier & Harding (1994) examined the importance of such general relativistic corrections and found that, because of spacetime curvature, a photon is observed to have an extra delay in its travel time by as much as 80 s for an emission height directly at the pulsar surface. Depending on the direction in which the photon is emitted this time delay might slightly increase by another 10 s. Even for low emission altitudes the extra time delay is therefore smaller than 90 s and obviously negligible compared to our sampling time and thus our measurement accuracy when observing slowly rotating pulsars. In fact, the largest uncertainties in our results are certainly determined by the general limited S/N at high frequencies. However, the applied technique to determine the profile midpoint and the quality of the timing models meant that we only had to adjust the pulse phase. This ensures sufficient accuracy necessary to draw the following conclusions.

Since the observed change in profile width is generally negligible (see Figs. 1 & 2 and Xilouris et al. 1996), the hollow cone model implies that the emission comes from well inside the light cylinder, i.e.  . Otherwise we would expect significant changes over small frequency intervals. Therefore, we can safely ignore the magnetic field sweep back in the timing analysis. The expected difference in the TOAs of pulse profiles measured at different frequencies is, then, solely determined by the retardation and aberration effect. If the dispersion delay has been removed from the data, we obtain

Knowing the magnetic inclination, , we can use the upper limits derived from the data to constrain the size of the emission region, , i.e.

For many sources, magnetic inclination angles, , have been determined by various authors (e.g. Lyne & Manchester 1988; Rankin 1990; BCW; Rankin 1993a, b). Depending on the applied technique the obtained values may differ, although in the cases discussed here the published values for generally agree among various authors. In Table 2 we list those angles (and their associated reference) which we adopted for our analysis. The resulting maximum size of the emission region for radiation observed at frequencies between 2.25 GHz (or 1.41 GHz, respectively) and 32 GHz are presented in column 5 of the same table. Nevertheless, the true value of might differ from the one adopted, so that we additionally computed the height of the emission region for the case of an aligned geometry, upper limit (column 6). Apparently, the emission seems to be created within a very compact region and, in fact, our data seem also to be consistent with the assumption that all high frequency emission originates from virtually the same place.

Table 2. Origin of high frequency radio emission. We quote the pulse period (column 2) and the magnetic inclination, (column 3) used for the calculations. The upper limit for the time delay between TOAs measured in a range from 2.25 GHz to 32.00 GHz, (column 4), leads to the maximum size of the corresponding emission region, , quoted in column 5 in units of stellar radii, km. Since the actual value of could be different from the value quoted in column 2, we calculated also the maximum height of the emission region assuming aligned rotators, i.e.  =0 (column 6). Assuming a dipolar magnetic field, Xilouris et al. (1996) derived absolute emission heights from profile widths, , listed in column 7. The size of the emission region derived for frequencies between (generally) 0.1 GHz and 32.0 GHz by Xilouris et al. (1996), , is given in column 8 also in units of . Upper limits for the RFM index obtained from Eq. (17) are listed in column 9.

So far, the obtained results constrain only the size of the emission region. In contrast, a geometrical method using a comparison between measured pulse widths and geometrical predictions from dipolar models can yield absolute emission heights. Such absolute values are, however, model dependent and necessarily underestimated for two reasons. First, for an application of the method we have to assume filled emission beams, i.e. the profile boundaries have to be determined by the last open field lines, which might not hold true for some pulsars (cf. Lyne & Manchester 1988). Second, only the application of a full profile width, viz measured at a very low intensity level (i.e. %), could yield true values (cf. Gil & Kijak 1993). However, profile widths are generally measured at a 10%-level, thus, leading only to lower limits for emission altitudes (see discussion by Xilouris et al. 1996).

The derived lower limits for the absolute emission heights are nevertheless useful in order to investigate quantitatively the index of a RFM, , introduced in Eq. (1). Using lower limits for emission altitudes of 32 GHz radiation, , we can yield an upper limit for from the simple relation

whereas is the maximum time delay between emission at the frequency measured in GHz and 32 GHz. Applying emission altitudes, , presented by Xilouris et al. (1996) and also listed in Table 2, we derive upper limits, , quoted in column 9 of Table 2. The actual value of probably depends on pulsar dependent parameters like period or period derivative, but if we average the listed values, we can obtain a "mean upper limit" of . Such value agrees well with the result of BCW (), PW92 () and Kijak & Gil 1996 (). Kramer et al. (1994) and Hoensbroech & Xilouris (1997) investigated in particular the high frequency behaviour of a RFM (i.e. for GHz), and found a typically lower value of and , respectively. Therefore, RFM might be less significant at high frequencies, which is supported by our results (see Figs. 1 & 2) and those of Xilouris et al. (1996), i.e. the pulse shapes and profile widths seem to stop changing above a few GHz, indicating that the emission originates from the same magnetospheric region.

It is of interest to investigate how our derived upper limits for compare with theoretical models. Certainly, our upper limit for is at variance with the classical Ruderman & Sutherland (1975) model, which predicts a value of . In the theory of Beskin et al. (1988, 1993), is expected to be 0.33 which seems to be only just consistent with our upper limit. Luo (1993) expects a similar value of . In all these models it is, however, assumed that the Lorentz factors of the emitting particles are frequency independent. Any frequency dependence of the Lorentz factors would affect the scaling. Barnard & Arons (1986) suggest that all radiation is created in a narrow range of radius, and that actually refraction effects in the pulsar magnetosphere broadens low frequency pulses, i.e. they basically propose 0. Although this model is therefore consistent with the results shown here, it may have some problems to explain the observed high frequency polarisation properties of some pulsars, e.g. of B1929+10 (cf. Xilouris et al. 1996).

We emphasize that a comparison of the observed data to predictions of various models is in general difficult or even not possible. This is because most of the models are derived to explain only low frequency data and, thus, try to account for a relation similar to Eq. (1), i.e. . However, including data obtained at the highest radio frequencies, the observations can be apparently better described by a relation like

where a constant offset is present (see also Thorsett 1991). Such a relation fits the observations of Xilouris et al. (1996) very well, suggesting that after a well behaving RFM at low frequencies, the emission altitude saturates above a few GHz to a constant value close to the surface. This lower limit, , can be then identified with the emission altitude derived for 32 GHz emission, , listed in Table 2. A fit of Eq. (1) to data which seem to be better described by Eq. (18) will necessarily lead to inconsistent results.

The result that the emission seems to originate from a small magnetospheric region is, however, unaffected by the actual scaling law. In fact, all major methods applied to derive the size of emission region, i.e. the geometrical method, timing analysis or the polarimetric method introduced by BCW, yield similar results. Emission altitudes derived by BCW using both the polarimetric and geometrical approach for radiation between 0.43 GHz and 1.40 GHz are found within 200 from the star, while in most cases the upper bound is only 10-30 . Phillips (1992) analyzed timing and pulse width data between 47 MHz and 4.80 GHz. He found that the emission of that frequency range originates within a region of 20 in height. Moreover, for PSR B1133+16 in particular, Phillips locates the 4.80 GHz emission in a distance of less than 14 from the star, while Cordes (1978) finds for the 1.40 GHz emission of B1133+16 an altitude of less than 63 . BCW studied, in addition to B1133+16, two other pulsars of our sample, B0540+23 and B1929+10. Their results are also consistent with our emission altitudes (Table 1) since they determine for B0540+23, for B1133+16 and for B1929+10, respectively, which also agrees well with the results of Hoensbroech & Xilouris (1997).

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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