4 Discussion

We have found that in the open field line tube of a pulsar the ordinary waves can be partially converted into extraordinary ones. For the reasonable parameters of the magnetospheric plasma, the conversion can be significant. As a result of this effect two natural modes of the plasma arise. In the light of our model of the OPM-phenomenon the standard definitions of both disjoint and superimposed modes should be somewhat reconsidered. Originally a single mode is emitted. It is the conversion that causes OPMs, and none of the modes observed is the mode produced directly by the emission mechanism. In the region of conversion the primordial mode turns into two modes, the ordinary and extraordinary ones, which can be easily recognized as the superimposed OPMs. On the other hand, the OPM-transitions are due to the switching between significant and insignificant conversion, and only one of these states is the case at a fixed instant and at a fixed longitude; therefore the radiation resulting from significant and insignificant conversion can be presented as disjoint modes. Then at any instant the disjoint mode is the sum of two superimposed modes, so that its partial polarization is explained naturally.

The conversion takes place in the region where the wave vector is almost aligned with the ambient magnetic field. The latter can be the case for the rays which have been refracted outwards from the magnetic axis and further propagate along straight lines in the magnetic field with the diverging field lines. As was shown in Petrova & Lyubarskii (2000), these rays form the conal components of the total intensity profile. The regime of quasi-longitudinal propagation is most easily realizable for the outermost rays. Note that for many pulsars the orthogonal transitions are indeed particularly apparent at the outer edges of the conal components (e.g., Rankin 1983). In this respect it is interesting to mention the mode-separated double profiles of PSR 0525+21 calculated in the assumption of superposed OPMs (McKinnon & Stinebring 2000). In this case the profile of the secondary mode has very steep inner edges and nearly zero intensity in the central part. So one can conclude that the secondary mode exists only at the outer edges of the profile, at longitudes beyond some threshold value, where the conversion becomes significant.

The quasi-longitudinal propagation and the consequent conversion are peculiar not only for the periphery of the profile. This also happens for the inner rays which deviate toward the magnetic axis and intersect it with the subsequent turn in the opposite direction (cf. Fig. 1). The prominent orthogonal transitions can really be seen in the central parts of the profiles (e.g., Gil et al. 1992). For example, in the inner conal component of PSR 0329+54, only the secondary (disjoint) mode is present (Gil et al. 1992), indicating that the conversion is particularly significant.

Thus the relationship between the observed loci of the most prominent OPM-transitions and the components of the total-intensity profiles can be explained in our theory. Moreover, the geometry of magnetospheric refraction suggests that the regime of quasi-longitudinal propagation can be in principle met for the rays at any pulse longitude. So the conversion can occur throughout the pulse, with the efficiency at various longitudes being substantially different. This is compatible with the general picture of the observed OPM-phenomenon (e.g., Stinebring et al. 1984).

As a result of the conversion, the OPMs emerge with a certain intensity ratio (which is governed by the parameters $\xi$ and $\eta$) and the position angle of the outgoing radiation is that of the dominant mode. So the question arises: what is the nature of the switching between the OPMs. First of all, the switching mechanism can be based on the alternation between the significant conversion and the absence of conversion, in which case refraction is too weak to provide the quasi-longitudinal propagation of rays. In addition, even if the ray trajectory contains the segment of quasi-longitudinal propagation, the efficiency of the conversion (i.e. the intensity ratio of the outgoing waves) can vary considerably on account of change in the conditions inside the region of conversion (i.e., because of changes in the values of $\xi$ and $\eta$).

Note that the difference in the energy and shape of the single pulses of a pulsar hints at significant temporal variations of the plasma density distribution within the open field line tube. Firstly, the intensity variations can arise because the emission mechanism is associated with the plasma; secondly, changes in the profile shape can be attributed to refraction (which depends on the plasma density distribution). Being related to the variations of the plasma density distribution as well, the OPM-switching should be correlated with the variations in the intensity and shape of the single pulses. The results on the correlation of the observed OPMs with the intensity of the pulse imply that the conversion occurs only if the intensity is above some threshold value (Cordes et al. 1978; Suleymanova & Pugachev 2000). However, it is difficult to comment on this conclusion in the absence of a notion about the emission mechanism.

The connection between the OPMs and the variations in the profile shape has an observational support. The conal components of one of the two mode-separated profiles (obtained with the assumption of disjoint OPMs) in some pulsars are shifted out of the profile centre (e.g., Gupta et al. 2000). So the enhancement of refraction is simultaneously responsible for both the component shifting and significant conversion. In a number of cases the asymmetry in the distribution of OPMs across the profile can be related to the asymmetry in the profile shape (Gil 1987; Gil et al. 1991, 1992), both being the consequences of refraction in the asymmetric plasma density distribution. The most apparent results can be obtained for the pulsars with the mode-changing profiles, in which case one can expect significant variations of the plasma density. For example, the normal and abnormal modes of the intensity profile of PSR 1604-00 appear to correspond to the primary and secondary polarization modes, respectively (Rankin 1988).

The observations of the OPMs at different frequencies show slight shifts in the dominance of OPMs at a given longitude and changes in the longitude range over which they are visible (Stinebring et al. 1984). The first feature can be attributed to variation of the parameters $\xi$ and $\eta$ with the frequency and consequent variation of the efficiency of conversion, whereas the second one can be explained by changes in the orientation of the rays (including those which pass through the region of quasi-longitudinal propagation) on account of the frequency dependence of refraction.

At high enough frequencies refraction is so strong that the regime of quasi-longitudinal propagation holds for most of the rays and, correspondingly, the conversion becomes efficient for most of the pulse. This is believed to result in considerable depolarization of pulsar radiation. Most of the pulsars really show significant depolarization toward high frequencies (e.g., Morris et al. 1981). As a general rule, at low frequencies the degree of linear polarization varies slightly, but beyond some critical frequency, $\nu_{\rm c}$, it begins to rapidly decrease with the frequency. Within the scope of the above considerations, $\nu_{\rm c}$ can be interpreted as the frequency at which refraction is significant for most of the open field line tube. Note that $\nu_{\rm c}$is usually close to the critical frequency in the frequency dependence of pulse width (e.g., McKinnon 1997; Suleymanova & Pugachev 1998). The latter frequency has the same interpretation: it is the frequency at which refraction becomes efficient for most of the rays (for more details see Lyubarskii & Petrova 1998). Moreover, the index of depolarization appears to be correlated with the width of the core component: less core widths signify stronger depolarization (Xilouris et al. 1995). This correlation is also explained naturally if one takes into account that refraction acts to decrease the width of the core component (the rays deviate toward the magnetic axis).

Toward very low frequencies (${\sim}100$ MHz) the degree of linear polarization increases strongly (Suleymanova & Pugachev 2000) indicating that most of the rays do not suffer conversion. Indeed, at low frequencies refraction is too weak to provide the regime of quasi-longitudinal propagation over a wide range of pulse longitudes. At the same time, in those parts of the profile where both OPMs still exist the secondary polarization mode can become more prominent compared to the primary one (Suleymanova & Pugachev 2000). One can suppose that in this situation the values of $\xi$ and $\eta$ are particularly favorable for the efficient conversion in those scarce regions where the quasi-longitudinal propagation still occurs (though refraction is weak).

Although at low frequencies refraction generally tends to weaken, the quantity of this weakening is determined by the concrete conditions in the plasma of a given pulsar. Therefore at a fixed frequency the efficiency of refraction (and the consequent effects) can differ essentially for different pulsars. A diversity of physical conditions in pulsar plasma reveals itself, e.g., in the fact that the critical frequency for depolarization along with the depolarization index vary from pulsar to pulsar. While studying the consequences of the propagation effects it should be kept in mind that low frequencies are not those beyond some numerical value, but those at which refraction is sufficiently weak.

As was found in Sect. 3.5, in the polarization-limiting region the ordinary and extraordinary waves acquire the circular polarization of opposite signs. Therefore the sign of the resultant circular polarization of the outgoing radiation is determined by the dominant mode. Hence, the OPM-transitions should be accompanied by change in the sense of circular polarization. This is in complete agreement with the observations (e.g., Cordes et al. 1978). In addition, the polarization-limiting effect leads to the shift in position angle of linear polarization. The value of the shift is determined by the parameter $\mu$ (Eq. (34)) and it is equal for both modes. Given that $\mu$ is subjected to temporal variations (e.g., $\theta$ is changed due to variations in the strength of refraction or $z_{\rm p}$ is altered because of change in the plasma parameters), the distribution of OPMs in position angle should be broadened. This is just what is observed (e.g., Backer & Rankin 1980; Stinebring et al. 1984; McKinnon & Stinebring 1998). In some cases the modes are clearly non-orthogonal (Stinebring et al. 1984; Gil et al. 1991, 1992). The non-orthogonality of the modes follows naturally if one supposes that the variations of the conditions in the region of conversion are correlated with those in the polarization-limiting region. Then substantial change in the efficiency of conversion and the consequent switching between the modes are accompanied by considerable variation of the shift in position angle on account of the polarization-limiting effect. Note that the shift in position angle of each mode is connected with the amount of circular polarization, both being the consequences of the polarization-limiting effect. Unfortunately, for the observational data this relationship has not been investigated yet.

Note that any technique of separating the OPMs in the observed radiation can give reasonable results only if the polarization-limiting effect is allowed for (or if it is negligible). Examples of such reasonable results for the model of superposed modes can be found in McKinnon & Stinebring (2000), where the pulsars with small amounts of circular polarization (i.e. weak polarization-limiting effect) are considered.