Astron. Astrophys. 355, 1168-1180 (2000)

3. Observational consequences of propagation effects

3.1. Wave mode coupling introduced by refraction

According to the standard models of electron-positron cascade, close to the magnetic axis as well as beyond the edge of the open field line tube the plasma density is negligible. So we assume that the plasma is distributed in the tube in such a way that the region near the magnetic axis constrained by characteristic open magnetic lines is free from the plasma. As the distance from the neutron star increases, the open field lines diverge and the plasma flow widens. In addition, the plasma density distribution is expected to be nonaxisymmetric (Arons & Scharlemann 1979). In agreement with the above considerations, we choose the following distribution of the plasma number density:

Recall that is radius of the emission point, which is assumed to be well within the light cylinder. Taking into account that

and recalling the definitions (4) one can rewrite the set of Eqs.  (7) in the following form:

Here

the quantities r, , , N are normalized by their initial values and the terms of the order of are neglected. Provided that the rays suffer strong deviation towards the magnetic axis and ultimately enter the region containing no plasma. The polarization-limiting region, where refraction is already insignificant, but the influence of the medium on wave propagation is still essential, should lie near the inner edge of the plasma flow at distances from the neutron star. Hence, neglecting magnetosphere rotation is really a good approximation in the case considered.

We assume that initially radiation is emitted along the magnetic field. Solution of Eqs. (17) yields direction of the outgoing wave, and . Then we solve Eqs. (13) and find the final polarization of this wave. The final circular polarization of outgoing waves can be characterized by the normalized Stokes parameter V:

It is easy to show (e.g., see Fig. 10.4 in Manchester & Taylor 1977) that the rays detectable by an observer should satisfy the following relation:

where is the wave vector tilt to the rotation axis. The pulse phase, , is then given by the expression:

Since the rays deviate on account of refraction, one cannot know in advance what initial conditions correspond to the rays detected by an observer. So to find the pulse profile we traced the wave vector evolution for the rays emitted all over the open field line tube and then chose those satisfying Eq. (19) to calculate the final polarization. The profiles of circular polarization obtained through numerical solution of Eqs. (17) and (13) are presented in Fig. 3. It is clear that the wave mode coupling introduced by refraction in the plasma with the number density Eq. (16) can produce significant circular polarization. The polarization profiles appear to be antisymmetric, with the sense of circular polarization being changed at the pulse centre. In agreement with the fourth equation of the set (7), it is the azimuthal density gradient that causes the wave vector deviation away from the initial field line plane, the direction of the deviation being opposite to the gradient. Then the change in the sign of the azimuthal density gradient leads to the corresponding change in the sign of the -plane turn, so that the resulting circular polarization reverses the sense. With the density distribution Eq. (16), the sense reversal occurs at the pulse centre. Note that only the extremums projected near the pulse centre can cause the observable sense reversal. The point is that refraction is essential only for the rays close enough to the magnetic axis, . Note that although the rays deviate intensely given , the wave mode coupling introduced by refraction can be efficient at the weaker condition, as well, so that the width of the antisymmetrical region of the profile appears to be compatible with that observed. The polarization profiles shown in Fig. 3 are similar to those observed for a number of pulsars (see, e.g., Han et. al. 1998 and references therein). The sense reversal is really observed only near the pulse centre.

Fig. 3. Profiles of circular polarization resulting from the wave mode coupling introduced by refraction in the plasma with nonaxisymmetric density distribution: a , b , c ; , , , , ,

One more issue worth discussion is the frequency evolution of such profiles. Let us suppose that the waves are emitted at frequencies of the order of the characteristic plasma frequency, . Because the plasma flow widens with the distance, , each frequency originates at a corresponding radius, . Open field lines of the dipolar magnetic field diverge with the distance so that for a chosen magnetic line . As obvious from Eq. (17), the efficiency of refraction is determined by the factor . Hence, with the distance from the neutron star refraction becomes less efficient and, correspondingly, wave mode coupling ceases. Thus, the lower the frequency the smaller amounts of circular polarization are achievable. This trend is demonstrated in Fig. 3, where the curves are obtained for the case, when the polar angles of the boundary magnetic line are related to each other as , that is, the frequency ratio is . Such frequency evolution of the antisymmetric polarization profiles is compatible with the observed one (see, e.g., the frequency sets of V-profiles for PSR 1451-68 and PSR 1857-26 in Rankin 1983).

Another striking observational fact is that at sufficiently low frequencies these antisymmetrical profiles can turn into symmetrical ones; then the sense of circular polarization becomes the same all over the pulse, save that the nulling of V and small amounts of the reverse polarization can be met in the wings of the pulse. So the circular polarization profile of PSR 1859+03 given by Radhakrishnan & Rankin (1990) as an example of the antisymmetric species ( MHz) appears to be symmetrical at the frequencies 631 MHz (McCulloch et. al. 1978) and 430 MHz (Rankin & Benson 1981). Similarly, the V-profile of PSR 1917+00, which is antisymmetric at MHz (Rankin et. al. 1989), becomes symmetrical at MHz (Rankin & Benson 1981). This testifies for the fact that at low frequencies the wave mode coupling introduced by refraction is dominated by that introduced by magnetosphere rotation.

3.2. Wave mode coupling introduced by magnetosphere rotation

For the sake of simplicity let us consider the axisymmetric distribution of the plasma density. Then refraction does not contribute to the turn of the -plane along the trajectory and wave mode coupling occurs only on account of magnetosphere rotation. Let

that is the plasma is confined between the two open field lines and the density decreases towards both boundaries. The set of equations describing refraction can be easily obtained from Eq. (17) putting and replacing by for the region of outward density decrease. Numerical solution of Eqs. (17) and (13) then yields almost symmetrical profile of circular polarization shown in Fig. 4. Apparently, the wave mode coupling introduced by the magnetosphere rotation can result in significant amounts of circular polarization, so that the highest observed , for PSR 1702-19 (Biggs et. al. 1988), can be explained by the effect. The sense of the circular polarization remains the same throughout the pulse. The polarization profile in Fig. 4 exhibits a slight departure from the symmetrical shape. The matter is that rotation of the open field line tube breaks the symmetry in wave propagation at negative and positive azimuths.

Fig. 4. The profile of circular polarization resulting from the wave mode coupling introduced by magnetosphere rotation; , , , , , , , ,

As can be seen from Eq. (13), the sign of V should depend on the signs of and . According to Eq. (14), the component is purely determined by the magnetosphere rotation, the sign being the same all over the pulse. Now we are to examine the behaviour of the component . As clear from Eq. (14), if magnetosphere rotation is sufficiently slow and the ray emitted along the open field line is not affected by refraction, that is, , remains positive in each point of the trajectory. Given that refraction causes ray deviation towards the magnetic axis, , is all the more positive. Strong enough outward deviation of the ray, however, can provide negative values of the component in the vicinity of the starting point of straight-line propagation (), while at larger distances, , becomes positive because of the magnetic line curvature. Hence, if refraction is strong and polarization-limiting radius lies not too far from the emission region, , the waves deviating away from the magnetic axis should gain the circular polarization of the sense opposite to that resulting from the wave mode coupling in case of wave deviation towards the axis. Note that if refraction outwards the axis is slightly weaker or is located slightly further, changes sign at , so that the wave has enough time to start acquiring the circular polarization of another sense. Obviously, this can lead to an essential decrease in the emergent circular polarization. The above considerations are illustrated in Fig. 5. Take note of the sense reversal and depolarization at the wings of the profiles. It should be pointed out that both the peculiarities are confirmed by the observations. Indeed, the lack of circular polarization in conal components is a well-known observational fact (see, e.g., Rankin 1983). The V-profiles, in which the circular polarization changes sense at the periphery are also quite abundant, especially in the high-frequency reviews (see, e.g., Rankin et. al. 1989).

Fig. 5. The profiles of circular polarization given the efficient outward refraction at the outer edges of the open field line tube: a , , , , , , , , , b the same parameters, save that , ,

Although the theory of polarization-limiting effect because magnetosphere rotation is worked out sufficiently minutely (Cheng & Ruderman 1979; Stinebring 1982; Barnard 1986; Lyubarskii & Petrova 1999), so far there is no reliable estimate of the polarization-limiting radius. The point is that turns out to be essentially dependent on a number of parameters, which can vary within some orders of magnitude. In addition, the value of depends crucially upon the angle the wave vector makes with the magnetic field, which is in turn determined by the structure of the magnetosphere. Unfortunately, up to date there is no self-consistent quantitative description of the rotating magnetosphere containing the plasma. So one cannot firmly specify the locus of the polarization-limiting radius in the magnetosphere and we treated it as a free parameter. However, it is that conditions the final value of the circular polarization in the case considered. As is near the emission region, , rotation effect is weak and the emergent circular polarization is small. Polarization-limiting effect taking place at large distances, , also results in a weak circular polarization, since in the outer magnetosphere the turn of the -plane along the trajectory ceases. The variety of the maximum values of circular polarization observed in the symmetrical V-profiles testifies for the variety of conditions in pulsar magnetospheres. Due to the lack of a reliable expression for , it is not possible to say anything about the evolution of the symmetrical V-profiles with the frequency. Note that in reality both trends of the frequency evolution are observed (see, e.g., Manchester, Hamilton, McCulloch 1980).

3.3. The swing of position angle of linear polarization with allowance for wave mode coupling

The polarization of normal waves originating in the magnetospheric plasma is certainly connected with the local orientation of the magnetic field. So the position angle of linear polarization should vary as the sight line traverses the open field line tube. Given that the propagation effects are ignored, the position angle swing through the pulse is known to be S-shaped (e.g., Manchester & Taylor 1977). Wave propagation in the vicinity of the emission region can be treated in terms of geometrical optics. The wave polarization then follows the local orientation of the -plane. Barnard (1986) was the first to investigate the influence of this effect on the observed position angle swing. It was found that given the total swing through the pulse should be essentially diminished. Note that polarization transfer in the polarization-limiting region, where geometrical optics approximation is violated, is also expected to result in position angle variation. Indeed, at distances wave polarization has not enough time to follow the -plane turn and the waves suffer birefringence. Then the polarization becomes elliptical, with the polarization ellipse turning along the trajectory. So the polarization-limiting effect influences the position angle swing as well as produces the circular polarization in the outgoing radiation. It should be pointed out that irregular position angle swing and large amounts of circular polarization are really observed together, both being the attributes of the so-called core emission (e.g., Rankin 1990). In the present subsection we are to study the position angle swing across the pulse taking into account wave propagation in the polarization-limiting region.

If the position angle, µ, were determined by the orientation of the magnetic field, it would be presented as follows (Barnard & Arons 1986; Barnard 1986):

This can be easily reduced to the form (see also Manchester & Taylor 1977):

Generally the position angle should be determined by the direction of the major axis of polarization ellipse, , beyond the polarization-limiting region. Then, taking into account that the waves considered are transverse ones, i.e. , one can rewrite Eq. (22) as

Following Landau & Lifshits (1988), one can express the components of the vector via the wave amplitudes and . The wave electric field can be presented in the form:

where , , and are the principal axes of the polarization ellipse, so that . Proceeding from the equality:

where the complex conjugation is marked by asterisk, it is easy to find the inclination of the major axis to the x-axis ():

Taking into account that

we obtain finally:

Using in Eq. (27) the limiting values of the wave amplitudes found through the numerical solution of Eqs. (17) and (13) yields the curves of position angle swing presented in Fig. 6. The swing without allowance for the polarization-limiting effect is shown by the dashed lines. The swing in Fig. 6a is obtained at the same parameters as the V-profile in Fig. 3b. One can see that under such condition the polarization-limiting effect does not alter essentially the character of the swing. Position angle variation remains quite smooth, the total swing through the pulse being almost the same. The value of the total swing is close to implying that the sight line traverses the open field line tube close enough to the centre. Obviously, at a fixed the smaller the pulse width the larger is the normalized impact parameter . As clear from Fig. 6a, in case of non-central sight-line trajectories the polarization-limiting effect something diminishes the total position angle swing. Position angle variation shown in Fig. 6a is similar to that observed for some pulsars with the antisymmetric V-profiles (e.g., PSR 1508+55, PSR 2111+46, and PSR 0942-13 (Lyne & Manchester 1988)).

Fig. 6. The swing of position angle of linear polarization with allowance for wave propagation in the polarization-limiting region: a , b ; the rest parameters are the same as on Fig. 1b, c the parameters are the same as on Fig. 4

If one considers another parameter, the influence of polarization-limiting effect on the position angle swing can be more significant (see Fig. 6b). One can see that the transition in the pulse centre becomes more distinct and the total swing exceeds . The latter is really the case in some pulsars. The abrupt change of position angle near the pulse centre corresponding to the sense reversal in V-profile is observed, e.g., for PSR 1451-68 (McCulloch et. al. 1978).

Fig. 6c shows the position angle variation in case of symmetrical V-profile. The essential departure from the S-shape corresponds to the peak of V-profile. It should be mentioned that as a rule the position angle swing observed in the conal components of triple profiles is smooth and can be regarded as the S-shaped, whereas the swing in the central components can be quite irregular, usually with abrupt transitions, which are not necessarily orthogonal (see, e.g., McCulloch et. al. 1978, Rankin et. al. 1989). So in the whole position angle swing presented in Fig. 6c also agrees with the observations.

3.4. Interpretation of total-intensity profiles in terms of refraction

The hollow-cone model (Radhakrishnan & Cooke 1969) is commonly accepted as a basis for geometrical explanation of pulsar total-intensity profiles. From the physical point of view, pulsar radio emission is associated with the magnetospheric plasma, which is believed to be distributed within the open field line tube, with the density close to the magnetic axis being negligible. Given the radius-to-frequency mapping is the case the widening of double profiles with the wavelength as well as the increase of separation between the components are naturally explained by the open field line tube widening with the distance from the neutron star. However, as first established by Backer (1976), in addition to the basic double structure the observed profiles often contain the component near the pulse centre, whereas some profiles exhibit two pairs of almost symmetrical components near the central component. So the hollow-cone model called for modification.

Thorough analysis of the wide observational data led Rankin (1983) to the conclusion that the emission from the central part of the open field line tube (core emission) differs from that generated at the tube edges (conal emission) in physical properties, namely in polarization and spectral behaviour. Later on Lyne & Manchester (1988) argued for a smooth variation of these properties with the distance from the magnetic axis, without respect to the concrete components of total-intensity profiles. In the previous subsections we demonstrated that the propagation effects in pulsar magnetosphere can account for both considerable circular polarization and irregular position angle swing close to the pulse centre, whereas the emission at the wings should be characterized by small amounts of circular polarization and quite orderly swing. This is well compatible with the observations.

Now we turn to the spectral properties of pulsar total-intensity profiles. As found out by Rankin (1983), core components have a steeper spectrum than the conal ones, so that at high frequencies the conal emission becomes more prominent. The frequency evolution of triple profiles was well simulated theoretically in terms of purely geometrical effects (Sieber 1997). The author proceeded from the assumption that pulsar emission beam contained a gaussian beam centered on the magnetic axis as well as shifted gaussian beams corresponding to conal emission. Our aim is to suggest the physical basis for such geometrical considerations. We are to show that refraction in the magnetospheric plasma can provide the separation of pulse components. Note that the beam separation on account of refraction should be frequency-dependent.

Consider refraction in the plasma with the following density distribution:

Let the emission be generated in the region filled with the plasma and radius-to-frequency mapping be the case. For simplicity we assume that the intensity distribution is uniform. Since the plasma number density decreases towards the both boundaries of the plasma flow, the transverse density gradient should change the sign. Then the rays emitted near the inner edge of the plasma flow should deviate towards the magnetic axis, while those emitted near the outer tube edge should tend away from the axis. Thus, refraction really leads to the angular separation of uniformly emitted radiation. At high frequencies refraction is more efficient (the tube is narrower and transverse density gradient is larger) and the component separation should become more distinct.

Using Eq. (29) in Eq. (7) one can find the total-intensity profiles shown in Fig. 7. They are obtained for the set of tube widths, that is for the set of frequencies related to each other as . Note that the profiles in Fig. 7 are calculated at the assumption of the uniform intensity distribution throughout the emission region, while in reality it is not so. It is apparent that refraction in the hollow-cone beam causes the triplicity of the observed profiles, with the component separation becoming more distinct at higher frequencies. Note that at proper conditions refraction can split the hollow-cone beam into the five components (see Fig. 7f). The inner conal components are then formed by the rays emitted at . The trajectories of such rays intersect the critical field line, so that the rays get finally into the region with the oppositely directed density gradient and suffer a slight deviation towards the outer tube edge. It should be mentioned that the five-component profiles are the most typical among the observed multiple profiles. Given that the inner boundary of the plasma flow lies far enough from the magnetic axis and the sight-line trajectory across the pulsar beam is sufficiently central, the central component can turn into a pair of components. So the recently discovered six-component profiles of PSR 0329+54 (Kuzmin & Izvekova 1996) and PSR 1237+25 (Kuzmin et al. 1997) can also be explained in the spirit of the present discussion. Thus the whole variety in the morphology of pulsar total-intensity profiles can be interpreted in terms of the common hollow-cone model with allowance for refraction in the magnetospheric plasma.

Fig. 7. The separation of components in total-intensity profiles due to refraction: a , b , c , d , e , f ; , , , ,

© European Southern Observatory (ESO) 2000

Online publication: March 21, 2000
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