3 Linear coupling of waves in pulsar plasma

3.1 The region of quasi-longitudinal propagation

To study the problem of linear conversion of waves it is necessary to specify the location of the region of quasi-longitudinal propagation. Ray propagation in the magnetosphere is generally quasi-transverse. Given that a ray is emitted tangentially to the field line of a dipolar magnetic field and propagates along a straight line, the tilt of the wave vector to the ambient magnetic field increases monotonically along the trajectory due to magnetic line curvature. However, refraction of waves in pulsar plasma can be significant (Barnard & Arons 1986; Lyubarskii & Petrova 1998), so that both the wave vector direction and the ray trajectory can be considerably modified. Since the plasma number density decreases rapidly along the ray trajectory, refraction appears to cease at distances ${\sim} r_{\rm e}$; further on the rays follow a straight line and can pass through the region where they are almost aligned with the ambient magnetic field.

Given that the rays propagate in the plane of magnetic lines (the xz-plane), it is not difficult to find that after refraction the polar angle of the ray trajectory, $\chi$, is given by the expression:

$$\chi =\theta_{\rm f}-\frac{z_{\rm f}}{z}(\theta_{\rm f}-\chi_{\rm f}).$$ (13)

Here $\chi_{\rm f}$ is the polar angle at the distance $z_{\rm f}$, where refraction becomes inefficient and the propagation becomes straight-line, $\theta_{\rm f}$ the tilt of the wave vector to the magnetic axis at $z_{\rm f}$. Then

$$b_x=\frac{3}{2}\chi-\theta_{\rm f}=\frac{\theta_{\rm f}}{2}-\frac{3z_{\rm f}}{2z}(\theta_{\rm f}- \chi_{\rm f}).$$ (14)

One can see that at $z>z_{\rm f}$ the component b_x changes the sign on condition that

$$\theta_{\rm f}>\frac{3}{2}\chi_{\rm f}.$$ (15)

Refraction in the outer part of the open field line tube makes the wave vector deviate away from the magnetic axis, with the trajectory bending in the same direction. Numerical ray tracings show that the wave vector deviation is more prominent than the trajectory bending (cf., e.g., Fig. 1 in Lyubarskii & Petrova 1998). Provided that refraction in the outer part of the tube is efficient, the condition (18) is typically satisfied. The rays emitted in the inner part of the tube deviate toward the magnetic axis and can intersect it. As the ray comes into the opposite half of the tube, the wave vector almost immediately starts to deviate in the opposite direction, due to the oppositely directed plasma density gradient; the trajectory bends similarly, but much slower (see, e.g., Fig. 1c in Petrova 2000). So for the inner rays the inequality (18) can be satisfied as well.

Figure 1 shows the location of the region of quasi-longitudinal propagation, $z_{\rm c}$, for the rays with different final tilts to the magnetic axis (in the case of the central cut of the emission cone by the sight line, $\theta_{\rm f}$corresponds to pulse longitude) at two different frequencies. The data presented in this figure are obtained through numerical solution of ray equations in pulsar plasma (Eq. (9) in Petrova 2000). The distribution of the plasma number density across the open field line tube is taken in the form of two Gaussians centred on the characteristic field lines symmetrically with respect to the magnetic axis. Because of the symmetry of the problem only one half of the pulse profile is plotted.

Figure 1: Location of the region of quasi-longitudinal propagation, $z_{\rm c}$, for the rays with different final tilts to the magnetic axis; asterisks and points correspond to the lower and higher frequency, respectively; the frequency ratio is 1:11; $z_{\rm f}$ is fixed at $2r_{\rm e}$.

As is evident from Fig. 1, the regime of quasi-longitudinal propagation is characteristic of a number of rays (both inner and outer). Note that for both frequencies the same number of initially equidistant rays was considered. One can see that in Fig. 1 the rays at the higher frequency (points) are more numerous than those at the lower frequencies (asterisks). So at higher frequencies, where refraction is more efficient, the quasi-longitudinal regime holds for the larger part of the pulse. For most of the rays $z_{\rm c}\sim z_{\rm f}$, i.e. the region of interest lies not too high in the magnetosphere.

One more marked feature can be observed in Fig. 1: for the high-frequency inner rays the curve is double-valued. The point is that due to refraction two rays emitted at different angles to the magnetic axis can have the same final tilts, whereas their trajectories are different (for more details see, e.g., Petrova 2000), so that at a given pulse longitude one can observe the rays coming from different locations in the magnetosphere. Each of these rays passes only once through the region of quasi-longitudinal propagation (for the two rays at a fixed pulse longitude $z_{\rm c}$ has distinct values).

It should be mentioned that in the present consideration we concentrate on the qualitative features of ray behaviour on account of refraction which make possible the regime of quasi-longitudinal propagation. The quantitative results depend on the concrete form of the assumed plasma density distribution. For that reason we do not give the numerical values of ray frequencies in Fig. 1 (the frequency ratio is derived from the ratio of the emission altitudes on the basis of radius-to-frequency mapping, $\nu\propto r^{-3/2}$). Note that at a fixed frequency the observed profile widths and shapes are known to be quite distinct for different pulsars; this hints at an actual variety in the parameters of pulsar plasma.

Recall that above we have considered ray propagation in the plane of magnetic lines. In reality the waves can slightly bend from this plane either because of refraction in the plasma with a non-axisymmetric density distribution or because of the magnetosphere rotation. So in general b_y is not equal to zero, though far enough from $z_{\rm c}$, it is much less than b_x (for numerical estimates see Sect. 3.6).

3.2 Linear conversion in the limit $\mathsfsl{B_0\to \infty}$

Now let us turn to linear conversion of waves. For the sake of clarity we start from the case of an infinitely strong magnetic field. Then the set of Eqs. (12) is reduced to the form:

$$\frac{{\rm d}E_x}{{\rm d}z}+iRb_x(E_xb_x+E_yb_y)=0,$$

$$\frac{{\rm d}E_y}{{\rm d}z}+iRb_y(E_xb_x+E_yb_y)=0,$$ (16)

where

$$\begin{eqnarray*}R\equiv\sum_\alpha \frac{\omega_{{\rm p}\alpha}^2}{2\omega c\gamma_\alpha^3(1- \beta_{0\alpha}b_z)^2}\cdot \end{eqnarray*}$$

Given that $Rz\gg 1$ and at least one of the magnetic field direction cosines b_x, b_y is not too small, geometrical optics are valid and the solution of Eq. (13) can be searched for in the form:

$$E_{x,y}= (a_{0x,y}+\frac{1}{N}a_{1x,y}+\dots)\exp (iNF(z)),$$ (17)

with N being a large parameter of the order of Rz. Then it is easy to find that

$$E_x^{({\rm o})} \frac{b_x}{\sqrt{b_x^2+b_y^2}}\exp \left[-i\int R(b_x^2+b_y^2){\rm d}z\right],$$

$$E_y^{({\rm o})} \frac{b_y}{\sqrt{b_x^2+b_y^2}}\exp \left[-i\int R(b_x^2+b_y^2){\rm d}z\right],$$

$$E_x^{({\rm e})} \frac{b_y}{\sqrt{b_x^2+b_y^2}},$$

$$E_y^{({\rm e})} -\frac{b_x}{\sqrt{b_x^2+b_y^2}},$$ (18)

where the indices o and e correspond to the customarily-defined ordinary and extraordinary waves in the magnetized plasma.

In a narrow region close to $z_{\rm c}$ both b_x and b_y are so small that the approximation of geometrical optics is broken. On both sides out of this region the solution of Eq. (13) is the linear combination of the natural waves (Eq. (15)), with the coefficients at $z<z_{\rm c}$ and $z>z_{\rm c}$ being distinct. For example, if at $z<z_{\rm c}$ we have only the ordinary wave, at $z>z_{\rm c}$ both natural waves arise.

To find the intensity ratio of the outgoing waves it is necessary to obtain the exact solution of Eq. (13) in the region of conversion. Note that in this case we can use the quasi-homogeneous approximation: within the narrow region of interest, $\frac{\vert z-z_{\rm c}\vert}{z_{\rm c}}\ll 1$, one can present b_xin the form: $b_x=\theta (z-z_{\rm c})/z_{\rm c}$, where $\theta =\frac{{\rm d}b}{{\rm d}z/z_{\rm c}} \vert_{z=z_{\rm c}}\sim\theta_{\rm f},\chi_{\rm f}$, while all the other quantities have approximately the same values as at $z=z_{\rm c}$. Hence, for the region of conversion the set of Eq. (13) is written as

$$\frac{{\rm d}E_x}{{\rm d}u} iu^2E_x=-i\xi uE_y,$$

$$\frac{{\rm d}E_y}{{\rm d}u} i\xi^2E_y=-i\xi uE_x,$$ (19)

where $u\equiv (Rz_{\rm c}\theta^2)^{1/3}(z-z_{\rm c})/z_{\rm c}$, $\xi\equiv (Rz_{\rm c}\theta^2)^ {1/3}b_y/\theta$ and $R\equiv R(z_{\rm c})$. From Eq. (19) one can see that at $\xi\sim 1$ the wave amplitudes E_xand E_y evolve considerably as u changes from ${\sim} {-1}$ to ${\sim} 1$. Hence, the characteristic length of the region of conversion is $\Delta u\sim 1$, i.e. $\Delta z/z_{\rm c}\sim (Rz_{\rm c}\theta^2)^{-1/3}$. Recall that $Rz_{\rm c}\theta^2$ is the large parameter of the geometrical optics appriximation: $Rz_{\rm c}\theta^2\sim N\sim z_{\rm c}\omega/c$ (cf. Eq. (14)). On the one hand, $\Delta z$ is essentially small compared to $z_{\rm c}$. On the other hand, the region of conversion contains $N^{2/3}\gg 1$ wavelengths, so that it is large enough to allow significant conversion of the natural waves. In spite of considerable variation over $\Delta z$, the wave amplitudes change slowly on the scale of a wavelength, i.e. Eq. (6) is still valid.

Despite the considerable simplification provided by the quasi-homogeneous approximation, the set of Eq. (19) remains too complicated: its exact solution cannot be expressed via the known functions. Note that in contrast to the above situation, the problem of linear conversion in a non-relativistic, weakly magnetized plasma allows an exact analytical solution (it is expressed via the functions of a parabolic cylinder) and it is well studied in application to solar plasma (see, e.g., Zhelezniakov 1996 and references therein).

Let us assume that $\xi \ll1$. Then the solution of Eq. (19) can be found by means of successive approximations. Presenting the wave amplitudes in the form

$$\begin{eqnarray*}E_{x,y}=E_{0x,y}+\xi E_{1x,y}+\dots \end{eqnarray*}$$

and taking into account that the incident wave is ordinary (i.e., at $u\to -\infty$ E_y=0) one can obtain to the first order in $\xi$:

$$E_{0x}\,{=}\,C\exp (-iu^3/3),\ E_{1y}\,{=}\,-i\xi C\int_{-\infty}^u u\exp (-iu^3/3).$$ (20)

Setting $u\to\infty$ yields:

$$E_x=C,\quad E_y=-3^{1/6}\Gamma (2/3)C\xi,$$ (21)

where $\Gamma (2/3)$ is the gamma-function. To fit the solutions (15) and (21) together it is necessary to substitute $z=z_{\rm c}(1+(Rz_{\rm c}\theta^2)^{-1/3}u)$ in Eq. (15) and set $Rz_{\rm c}\theta^2 \to\infty$ at a fixed u. Keeping in mind that $\xi \ll1$, one can find that the ordinary wave is polarized along the x-axis, while the extraordinary one is polarized along the y-axis. So, as is obvious from Eq. (21), both natural waves are present in the outgoing radiation:

$$\begin{eqnarray*}{\vec E}=\alpha_1{\vec E}^{({\rm o})}+\alpha_2{\vec E}^{({\rm e})}. \end{eqnarray*}$$

It is of a particular interest to find the conversion coefficient, Q, i.e. the part of the energy of the primordial ordinary waves which is transmitted to the extraordinary mode:

$$\begin{eqnarray*}Q=\frac{\vert{\vec E}\cdot{\vec E}^{({\rm e})*}\vert^2}{\vert {\vec E}\vert^2}= \vert \alpha_2\vert^2. \end{eqnarray*}$$

(Note that the energy conservation, $\frac{{\rm d}}{{\rm d}z}(E_xE_x^*+E_yE_y^*)=0$, follows straightly from Eq. (19)). Apparently, in the case considered

$$\begin{eqnarray*}Q=E_y^2=3^{1/3}\Gamma^2(2/3)\xi^2. \end{eqnarray*}$$

Hence, as long as $\xi \ll1$, Q increases with $\xi$. Indeed, according to Eq. (19), as $\xi$ increases, the modes are more strongly coupled, in which case the conversion is more prominent. In the opposite limiting case, $\xi \gg 1$, the conversion is expected to be negligible, since the geometrical optics approximation is not violated at all.

The coefficient of conversion calculated numerically as a function of $\xi$is plotted in Fig. 2. One can see that the conversion is significant for . It should be pointed out that the physical meaning of the parameter $\xi$ is the phase difference of the natural modes acquired during their passing through the region of conversion. Indeed, the characteristic length of the region of conversion is $\Delta u\sim\xi^{-1}\sim (Rz_{\rm c}\theta^2) ^{-1/3}\theta/b_y$, i.e. $\Delta z/z_{\rm c}\sim (Rz_{\rm c}\theta^2)^{-1/3}\Delta u \sim (Rz_{\rm c}\theta^2)^{-2/3}\theta/b_y$. Taking into account that the difference of the refractive indices is $\Delta n=R\frac{c}{\omega}b_y^2$, we find: $\frac{\omega}{c}\Delta n\Delta z=(Rz_{\rm c}\theta^2)^{1/3}b_y/\theta \equiv\xi$.

Figure 2: The degree of mode conversion, Q, as a function of mode phase difference, $\xi$, in the limit of an infinitely strong magnetic field.

As is clear from Fig. 2, Q does not exceed 0.5. Hence, if the approximation of an infinitely strong magnetic field is the case, the extraordinary mode never dominates over the ordinary one, though the intensities can be comparable and, correspondingly, the outgoing radiation can be strongly depolarized.

3.3 Linear conversion in the limit $\mathsfsl{b_y=0}$

Let us turn to the opposite limiting case: b_y=0, whereas B₀ is finite and $G\neq 0$. Then the set of Eq. (12) is written as

$$\frac{{\rm d}a_x}{{\rm d}z}$$

$$\frac{{\rm d}a_y}{{\rm d}z}$$ (22)

where $E_{x,y}\equiv a_{x,y}\exp (i\frac{\omega}{c}\int B{\rm d}z)$ and $g\equiv G/A$. Given that $Rz\gg 1$ and b_x² is far enough from zero, geometrical optics holds and the solution of Eq. (22) takes the form:

$$a_x^{({\rm o})} \frac{i(b_x^2/2+\sqrt{b_x^4/4+g^2})}{\sqrt{g^2+[b_x^2/2+ \sqrt{b_x^4/4+g^2}]^2}}$$

$$\times\exp \left[-i\int R(b_x^2/2+\sqrt{b_x^4/4+g^2}){\rm d}z\right],$$

$$a_y^{({\rm o})} \frac{g}{\sqrt{g^2+[b_x^2/2+ \sqrt{b_x^4/4+g^2}]^2}}$$

$$\times\exp \left[-i\int R(b_x^2/2+\sqrt{b_x^4/4+g^2}){\rm d}z\right],$$

$$a_x^{({\rm e})} \frac{i(b_x^2/2-\sqrt{b_x^4/4+g^2})}{\sqrt{g^2+[b_x^2/2- \sqrt{b_x^4/4+g^2}]^2}}$$

$$\times\exp \left[-i\int R(b_x^2/2-\sqrt{b_x^4/4+g^2}){\rm d}z\right],$$

$$a_y^{({\rm e})} \frac{g}{\sqrt{g^2+[b_x^2/2- \sqrt{b_x^4/4+g^2}]^2}}$$

$$\times\exp \left[-i\int R(b_x^2/2-\sqrt{b_x^4/4+g^2}){\rm d}z\right].$$ (23)

At $g\to 0$ the electric field components of the natural waves in the above equation certainly coincide with those given by Eq. (15) at $b_y\to 0$.

In the region $\frac{\vert z-z_{\rm c}\vert}{z_{\rm c}}\ll 1$ Eq. (22) can be treated in the quasi-homogeneous approximation. So we obtain:

$$\frac{{\rm d}a_x}{{\rm d}u}+iu^2a_x=\eta a_y,$$

$$\frac{{\rm d}a_y}{{\rm d}u}=-\eta a_x,$$ (24)

where $\eta\equiv (Rz_{\rm c}/\theta)^{2/3}g$. The wave coupling described by Eq. (24) is qualitatively similar to that studied in the previous subsection. The parameter $\eta$ again can be interpreted as a phase difference between the modes which is gained while they are passing through the region of conversion. At $\eta =0$ the natural waves are independent. As $\eta$ increases, the wave conversion becomes more and more efficient. At $\eta \gg 1$ the waves are again independent. The numerically calculated dependence of the conversion coefficient on $\eta$ is presented in Fig. 3. One can see that the conversion is significant at and the peak value of Q slightly exceeds 0.5. So the outgoing radiation can be considerably depolarized, but the extraordinary mode cannot be noticeably stronger than the ordinary one.

Figure 3: The degree of mode conversion, Q, as a function of mode phase difference, $\xi$, in the limit by=0.

3.4 General case

The above results pose the question of whether the conversion is efficient at $\xi\sim\eta$, i.e. given that both b_y and g cannot be neglected. In the geometrical optics approximation the solutions of Eq. (12) are as follows:

$$\frac{{\rm d}F/{\rm d}z{+}Rb_y^2}{\sqrt{({\rm d}F/{\rm d}z{+}Rb_y^2)^2{+}Rb_x^2b_y^2+R^2g^2}}\exp (iF(z)),$$

$$\frac{iRg-Rb_xb_y}{\sqrt{({\rm d}F/{\rm d}z{+}Rb_y^2)^2{+}Rb_x^2b_y^2{+}R^2g^2}}\exp (iF(z)),$$ (25)

where $F(z)\equiv \int^zR[-(b_x^2+b_y^2)/2\pm\sqrt{(b_x^2+b_y^2)^2/4+g^2}]{\rm d}z$, the signs "+" and "-" refer to the extraordinary and ordinary waves, respectively.

The quasi-homogeneous problem is reduced to the form:

$$\frac{{\rm d}a_x}{{\rm d}u} iu^2a_x+i\xi ua_y-\eta a_y=0,$$

$$\frac{{\rm d}a_y}{{\rm d}u} i\xi ua_x+i\xi^2a_y+\eta a_x=0.$$ (26)

Figures 4a,b show the conversion coefficient obtained numerically on the basis of Eqs. (26) and (25) for various values of $\xi$ and $\eta$. As is evident from Fig. 4, the ordinary waves can almost completely be transformed into the extraordinary ones. Provided that and the extraordinary waves dominate in the outgoing radiation.

Figure 4: The degree of mode conversion for various $\xi$and $\eta$; a) Q versus $\xi$; $\eta =0.1,\, 0.3,\, 0.5$; b) Q versus $\eta$; $\xi =0.4,\, 0.6,\, 0.8$.

3.5 Polarization-limiting effect

After leaving the region of conversion the waves propagate independently until the geometrical optics approximation is violated. The plasma number density decreases along the trajectory and the waves tend to become the vacuum electromagnetic ones. As soon as the difference in the refractive indices becomes so small that the scale length for beatings between the natural waves becomes comparable to the scale length for change in the plasma parameters,

$$\frac{\omega}{c}r\Delta n\sim 1,$$ (27)

the natural modes are again coupled. At least, after the plasma density decreases considerably, the wave propagation is no longer affected by the medium and the polarization remains fixed. The mode coupling in the so-called polarization-limiting region leads to changes in the polarization state of the escaping waves. Namely, the waves acquire some circular polarization, with the position angle of the linear polarization being somewhat shifted.

It should be noted that the consequences of the mode coupling are essentially distinct from the linear conversion studied above. The problem is that in the region of conversion Eq. (27) is satisfied twice: at first $\Delta n$ decreases strongly because $b_x\to 0$ and then it increases up to its initial value. As $\Delta n\to 0$ the incident ordinary wave suffers the polarization changes similar to those resulting from the polarization-limiting effect. Later on, as $\Delta n$ starts increasing, this modified wave is split into two natural waves.

The polarization-limiting effect has been already studied in detail in application to the ordinary waves (Petrova & Lyubarskii 2000). It was found that a number of features of the observed polarization profiles can be attributed to this effect. Now we turn to it once more in order to compare the evolution of both orthogonal modes. Let us assume that the plasma number density decreases as z^-3 and $b_x={\rm const.}$, $b_y\propto z$. Although this model is somewhat abstract, it appears to be sufficient to clarify the qualitative picture of the polarization-limiting effect for both natural modes.

As was already discussed at the beginning of Sect. 2, far from the region of conversion the approximation of an infinitely strong magnetic field is appropriate. Then the set of Eqs. (12) can be presented in the form:

$$\frac{{\rm d}E_x}{{\rm d}w} is(w)wE_x-is(w)\mu E_y=0,$$

$$\frac{{\rm d}E_y}{{\rm d}w} is(w)\mu E_x-is(w)\mu^2/wE_y=0.$$ (28)

Here $w\equiv z_{\rm p}/z$, $z_{\rm p}$ is the polarization-limiting radius determined by the following relation:

$$\begin{eqnarray*}&&R(z_{\rm p})[b_x^2+b_y^2(z_{\rm p})]z_{\rm p}=1, \\ &&\mu\eq... ...)_{z=z_{\rm p}},\ s(w)\equiv\frac{1+\mu^2}{(1+\mu^2/w^2)^2}\cdot \end{eqnarray*}$$

From Eq. (28) one can see that the evolution of the wave polarization is completely determined by the parameter $\mu$. Recall that the waves are emitted in the plane of magnetic lines, i.e. initially b_y=0, and further along the trajectory the polarization is adjusted to the ${\vec k}\times {\vec b}$-plane. So $\mu \ll 1$ implies that in the regime of geometrical optics (i.e., before $z_{\rm p}$) the polarization plane rotates weakly. In the opposite case, $\mu \gg 1$, the electric field of a natural wave changes drastically. For example, the ordinary wave, which is initially polarized along the x-axis, enters the polarization-limiting region with the electric vector almost aligned with the y-axis. Hence, in the problem considered, the parameter $\mu$is a measure of rotation of the polarization plane of natural waves in the approximation of geometrical optics; at the same time, it completely determines wave mode coupling in the polarization-limiting region (cf. Eq. (28)) and can be interpreted as a phase difference between the modes which is acquired in the course of coupling.

Now let us turn to the numerical results on the final polarization state of the waves escaping from the plasma. The degree of circular polarization is characterized by the normalized Stokes parameter V:

$$\equiv \frac{i(E_x^*E_y-E_xE_y^*)}{E_xE_x^*+E_yE_y^*},$$ (29)

and the position angle, $\psi$, of the linear polarization is defined as follows:

$$\tan 2\psi\equiv\frac{E_yE_x^*+E_xE_y^*}{E_xE_x^*-E_yE_y^*}\cdot$$ (30)

The final values of V and $\psi$ calculated numerically from Eq. (28) as functions of $\mu$ are plotted in Fig. 5. In Fig. 5a the most important issue is that for two natural waves the resultant circular polarization has the opposite sense at any $\mu$. Furthermore, as is apparent from Fig. 5b, the position angle of both natural waves is shifted in exactly the same manner, so that the waves remain orthogonally polarized. Thus, given that the linear conversion is the case, the outgoing radiation is a mixture of the two types of natural waves and the waves with the dominant intensity determine both the position angle and the sense of circular polarization of pulsar radiation.

Figure 5: Consequences of the polarization-limiting effect; a) degree of the circular polarization versus mode phase difference, $\mu$; the upper curve corresponds to the extraordinary waves, the lower one refers to the ordinary waves; b) shift in the position angle of linear polarization versus $\mu$; the lower curve corresponds to the extraordinary waves, the upper one refers to the ordinary waves.

3.6 Numerical estimates

As is found in the previous subsections, the linear conversion is governed by the parameters $\xi$ and $\eta$, while the polarization-limiting effect is determined by $\mu$. Now we proceed to the numerical estimates of these parameters in application to typical pulsar conditions.

Although R is the sum over the particle species, it is clear that either one of the addends contributes chiefly or both contributions are nearly equal. So for our estimates one can write: $Rz_{\rm c}\sim\frac{\omega z_{\rm c}}{c}\frac{\omega_{\rm p}^2\gamma}{\omega^2}$; here it is taken into account that in the region of conversion $b_x,b_y\ll 1/\gamma$and $\omega_{\rm H}\gg \omega^\prime$. The waves are supposed to originate at the characteristic plasma frequency, i.e. at the emission origin $\omega\gamma \theta_{\rm e}^2\sim\omega_{\rm pe}\sqrt{\gamma}$, and further along the trajectory the plasma number density, N, decreases as z^-3, so that $\omega_{\rm p} \propto\sqrt{N}\propto z^{-3/2}$. Assuming that $\omega =f\omega_{\rm pe}\sqrt {\gamma}$ with $f\equiv (\theta_{\rm e}^2\gamma)^{-1}\sim 1$ one can obtain:

$$Rz_{\rm c}\theta^2=2.7\times 10^{4}f_{-0.5}^{-2}\theta_{-1}^2\nu_9z_8(5z_{\rm e}/z_{\rm c})^3,$$ (31)

where $f_{-0.5}\equiv\frac{f}{0.3}$, $\theta_{-1}\equiv\frac{\theta}{0.1}$, $\nu_9\equiv\frac{\nu}{10^9\,{\rm Hz}}$, $z_8\equiv\frac{z_{\rm c}}{10^8\, {\rm cm}}\cdot$ The minimum value of b_yis determined by the rotational effect and can be written as

$$\begin{eqnarray*}b_y\approx\frac{z_{\rm c}}{r_{\rm L}}\sin\alpha , \end{eqnarray*}$$

where $r_{\rm L}\equiv 5\times 10^9 P$ cm is the light cylinder radius, P the pulsar period, $\alpha$ the angle between the magnetic and rotational axes of the pulsar. Then

$$\xi\equiv (Rz_{\rm c}\theta^2)^{1/3}b_y/\theta =f_{-0.5}^{-2/3}\theta_{-1}^{-1/3} \nu_9^{1/3}z_8^{1/3}(5z_{\rm e}/z_{\rm c})\zeta_{-1.5},$$ (32)

where $\zeta_{-1.5}\equiv\frac{b_y/\theta}{3\times 10^{-2}}\cdot$

Note that in contrast to R the expression for G contains the difference of the contributions of electrons and positrons. Therefore one can write:

$$\begin{eqnarray*}g\sim\frac{\varepsilon\omega}{\gamma\omega_{\rm H}}, \end{eqnarray*}$$

where $\varepsilon\equiv\Delta v/c$ is the normalized difference in the velocities of the two particle species. The value of $\varepsilon$ is determined by the multiplicity factor of the secondary plasma, $\kappa$: $\varepsilon\sim\kappa^{-1}$ (Cheng & Ruderman 1977) or it may be somewhat less (Buschauer & Benford 1977). Unfortunately, $\kappa$ will be known exactly only after the self-consistent theory of the electron-positron cascade is developed. Here we take $\kappa =10^2$. In our model the conversion occurs far from the radius of cyclotron resonance, $r_{\rm H}$, for which the condition $\omega\gamma\theta^2/2=\omega_{\rm H}$ is satisfied. Let us take that $z_{\rm c}/r_{\rm H}\sim 0.1$ and correspondingly, $\omega\gamma\theta^2/\omega_{\rm H}\sim 10^{-3}$, since $\omega_{\rm H}\propto B_0\propto z^{-3}$. Hence, using Eq. (31) we find:

$$\eta \,{\equiv}\, (Rz_{\rm c}\theta^2)^{2/3}g/\theta^2{=}0.1\vare... ...2}f_{-0.5}^{-4/3} \theta_{-1}^{-8/3}z_8^{2/3}\nu_9^{2/3}(5z_{\rm e}/z_{\rm c}).$$ (33)

In the right-hand sides of Eqs. (32) and (33) all the quantities are normalized by their characteristic values. Although in reality they can be somewhat different, the conditions for efficient conversion, and , are expected to be easily satisfied. The explicit frequency dependence of $\xi$ and $\eta$ is not too strong; in addition, it is necessary to keep in mind that all the other quantities (except $\varepsilon$) can also depend on $\nu$ implicitly. One can expect that significant conversion can occur over a broad enough frequency range.

The result of the polarization-limiting effect is determined by the parameter

$$\begin{eqnarray*}\mu =\frac{z_{\rm p}}{r_{\rm L}\theta}\sin\alpha . \end{eqnarray*}$$

Unfortunately, to date there is no reliable estimate of the polarization-limiting radius. If $z_{\rm p}/r_{\rm L}\sim 0.1$ and $\theta \sim 0.1$, then $\mu\sim 1$. As can be seen from Fig. 5, in the region $\mu\sim 1$ even relatively small variations of $\mu$ can lead to a significant change in the polarization of the outgoing waves. In particular, this may account for the observed randomization of the position angle.