2 General equations

Let us consider the polarization evolution along the wave trajectory in the magnetospheric plasma. Such a problem has already been studied in application to polarization-limiting effect (Lyubarskii & Petrova 1999; Petrova & Lyubarskii 2000). In that case the plasma was supposed to be embedded in an infinitely strong magnetic field. Indeed, well within the magnetosphere the magnetic field strength is so high that the critical angle for the circularly polarized natural modes is too small. So in general the wave propagation is quasi-transverse with respect to the magnetic field, i.e. the non-coupled natural waves are linearly polarized. Then for the polarization evolution there is no essential distinction between whether the magnetic field strength is regarded as finite or infinite.

In the region of quasi-longitudinal propagation, polarization of the natural modes can differ essentially becoming almost circular. Although this region is rather narrow, it is of a particular interest for the present consideration, since it is the region where the conversion of waves is expected to occur. So it is reasonable to take into account both the finiteness of the magnetic field strength and the difference in the distribution functions of the plasma constituents.

Note that the region of conversion lies at distances ${\sim} r_{\rm e}$ (see below), in which case we can ignore the magnetosphere rotation. As was found earlier (Petrova & Lyubarskii 2000), the corrections allowing for the rotational aberration are not significant even in the polarization-limiting region, though it is situated much higher.

The wave fields ${\vec E}$ and ${\vec B}$ are described by the Maxwell's equations:

$$\nabla\times {\vec B}=-\frac{i\omega}{c}{\vec E}+\frac{4\pi}{c}\sum_\alpha {\vec j}_{\alpha},$$ (1)

$$\nabla\times {\vec E}=\frac{i\omega}{c}{\vec B},$$ (2)

$$-i\omega q_{\alpha}n_{\alpha}+{\rm div}{\vec j}_{\alpha}=0.$$ (3)

Here ${\vec j}_{\alpha}$ is the linearized current density for each of the particle species $\alpha$ ( $\alpha =1,2$):

$$\begin{eqnarray*}{\vec j}_{\alpha}\equiv q_{\alpha}[n_{\alpha}{\vec v}_{0\alpha}+n_{0\alpha} {\vec v}_{\alpha}], \end{eqnarray*}$$

$q_{\alpha}=\pm e$ the particle charge, ${\vec v}_{0\alpha}$ and $n_{0\alpha}$the particle velocities and number densities unperturbed by the wave, ${\vec v} _\alpha$ and $n_{\alpha}$ the small perturbations of these quantities. In our consideration the plasma is assumed to be cold. Although for the real pulsars it is not so, the results are believed to be qualitatively the same. Indeed, despite the particles of each species streaming along the magnetic lines with a large spread in Lorentz-factors, $\gamma\sim 10{-}10^3$ (e.g., Arons 1981), only those with the momenta close to some characteristic value contribute essentially to ${\vec j}_{\alpha}$ (for more details see, e.g., review by Lyubarskii 1995). Since electrons and positrons move along the curved magnetic lines in the rotating magnetosphere, their velocities should be somewhat different (e.g., Cheng & Ruderman 1977). This follows directly from combining the continuity equation with the requirement to supply the Goldreich-Julian current density in each point of the magnetosphere.

The equation of the particle motion in the external magnetic field of the strength ${\vec B}_0$ is written as follows:

$$\frac{{\rm d}{\vec p}_{\alpha}}{{\rm d}t}=q_\alpha\left({\vec E}+... ...alpha \times {\vec B}_0}{c}+\frac{{\vec v}_{0\alpha}\times {\vec B}}{c}\right),$$ (4)

where ${\vec p}_\alpha$ is the linearized particle momentum caused by the wave, $\frac{\rm d}{{\rm d}t}\equiv -i\omega +{\vec v}_0\cdot\nabla$ is the total derivative. Note that for the ultrarelativistic particles streaming along the magnetic lines

$$\begin{eqnarray*}\frac{{\rm d}{\vec p}_\Vert}{{\rm d}t}&=&m\gamma^3\frac{{\rm d}... ...perp}{{\rm d}t}&=&m\gamma\frac{{\rm d}{\vec v}_\perp}{{\rm d}t}, \end{eqnarray*}$$

where ${\vec p}_\Vert$ and ${\vec p}_\perp$ are the momentum components parallel and perpendicular to the external magnetic field, ${\vec v}_\Vert$ and ${\vec v}_\perp$ the corresponding components of the particle velocity. The set of Eqs. (1)-(4) yields a complete description of the wave fields and the plasma particle motion in these fields.

The tracing of the polarization evolution of the waves starts from the region where refraction is already inefficient (i.e., $1-n\ll 1$, where n is the refractive index of the ordinary superluminous waves; the extraordinary waves are not subject to refraction at all). At the same time, in this region the difference between the refractive indices of the waves still should be high enough to prevent the mode coupling: $\frac{\omega}{c}r\Delta n\gg 1$. Then the geometrical optics approximation is valid and the wave behaviour is well known: the polarization plane follows the orientation of the ambient magnetic field.

Seeing that the waves propagate along straight lines, one can choose a three-dimensional Cartesian coordinate system with the z-axis along the wave trajectory. So all the quantities in Eqs. (1)-(4) depend only on the z-coordinate. Moreover, further simplification is possible. The wave modes propagating in opposite directions do not interact with each other given that the refractive indices are far from zero (i.e. in the absence of reflections). Hence, we shall treat only the waves propagating in the positive direction of the z-axis. Since the refractive indices of the waves considered are very close to unity, the spatial dependence of the wave electric field can be presented as follows:

$$E_{x,y,z}=\widetilde{E}_{x,y,z}(z)\exp\Big(i\frac{\omega}{c}z\Big),$$ (5)

where $\widetilde{E}_{x,y,z}$ are the slowly varying wave amplitudes:

$$\frac{{\rm d}\widetilde{E}_{x,y,z}}{{\rm d}z}\ll \frac{\widetilde{E}_{x,y,z}\omega}{c}\cdot$$ (6)

All the remaining perturbed quantities can certainly be presented in the same form. Hereafter the tildes will be omitted for the sake of convenience. The scale length for change in the unperturbed medium parameters is also assumed to be much larger than the wavelength.

Keeping in mind the above considerations and using Eq. (2) one can write the component equations for the equation of motion (4) as follows:

$$-i\omega m\gamma_\alpha^3(1-\beta_{0\alpha} b_z)\left [v_{x\alpha}b_x+v_{y\alpha}b_y+ v_{z\alpha}b_z\right ]=q_\alpha(E_xb_x+E_yb_y+E_zb_z),$$

$$-i\omega m\gamma_\alpha (1-\beta_{0\alpha} b_z)\left [v_{y\alpha}b_x- v_{x\alpha}b_y\right ]=q_\alpha (E_yb_x-E_xb_y)(1-\beta_{0\alpha} b_z)$$

$$+q_\alpha\frac{B_0}{c} \left [v_{z\alpha}(b_x^2+b_y^2)-v_{x\alpha... ...z)\left [v_{z\alpha}(b_x^2+b_y^2)- v_{x\alpha}b_xb_z-v_{y\alpha}b_yb_z\right ]=$$

$$q_\alpha (E_xb_x+E_yb_y) (b_z-\beta_{0\alpha})-q_\alpha E_z(b_x^2+b_y^2)+q_\alpha\frac{B_0}{c} \left [v_{y\alpha}b_x-v_{x\alpha}b_y\right ],$$ (7)

where b_x,y,z are the Cartesian components of the unit vector of the external magnetic field, ${\vec B}_0=B_0{\vec b}$, $\beta_{0\alpha}$ is the unperturbed particle velocity in units of c, and ${\vec v}_{0\alpha}\Vert {\vec b}$. From Eq. (7) one can find the perturbations of the particle velocity:

$$v_{x\alpha} \frac{q_\alpha^2B_0\left [E_yb_z{-}\beta_{0\alpha}E_xb_xb_y{-} \b... ...y(b_y^2{+}b_z^2){-}E_zb_y\right ]}{m^2c(\omega_{\rm H}^2{-}\omega^ {\prime 2})}$$ =

$$\frac{iq_\alpha\omega^\prime}{m(\omega_{\rm H}^2-\omega^{\prime^2... ...pha b_x[E_xb_x+E_yb_y+E_zb_z]}{m\gamma_\alpha^3\omega (1- \beta_{0\alpha}b_z)},$$ +

$$v_{y\alpha} \frac{q_\alpha^2B_0\left [E_x(\beta_{0\alpha} (b_x^2{+}b_z^2){-}b... ...\alpha}E_yb_xb_y{+}b_xE_z\right ]}{m^2c(\omega_{\rm H}^2{-}\omega^ {\prime 2})}$$ =

$$\frac{iq_\alpha\omega^\prime}{m(\omega_{\rm H}^2-\omega^{\prime^2... ...pha b_x[E_xb_x+E_yb_y+E_zb_z]}{m\gamma_\alpha^3\omega (1- \beta_{0\alpha}b_z)},$$ +

$$v_{z\alpha} \frac{iq_\alpha b_z[E_xb_x+E_yb_y+E_zb_z]}{m\gamma_\alpha^3\omega... ...yb_x-E_xb_y)(1-\beta_{0\alpha}b_z)} {m^2c(\omega_{\rm H}^2-\omega^{\prime 2})},$$ = (8)

where $\omega_{\rm H}\equiv\frac{eB_0}{mc}$ is the gyrofrequency, $\omega^\prime \equiv\gamma_\alpha\omega (1-\beta_{0\alpha}b_z)$ the frequency in the particle rest frame. From the continuity Eq. (3) one easily obtains:

$$n_\alpha =\frac{n_{0\alpha} v_{z\alpha}/c}{1-\beta_{0\alpha}b_z}\cdot$$ (9)

Combining Eqs. (1), (2) and (9) yields:

$$\frac{{\rm d}E_x}{{\rm d}z}{+}\frac{2\pi}{c}\sum_\alpha \frac{q_\... ...a}b_z}[v_{x\alpha}(1{-}\beta_{0\alpha}b_z){-}v_{z\alpha} \beta_{0\alpha}b_x]=0,$$ (10)

$$\frac{{\rm d}E_y}{{\rm d}z}{+}\frac{2\pi}{c}\sum_\alpha \frac{q_\... ...a}b_z}[v_{y\alpha}(1{-}\beta_{0\alpha}b_z){+}v_{z\alpha}\beta_{0\alpha} b_y]=0,$$

$$\frac{4\pi i}{\omega}\sum_\alpha \frac{q_\alpha n_{0\alpha}v_{z\alpha}}{1- \beta_{0\alpha}b_z}=0.$$

From Eqs. (8) and (10) one finds:

$$\sum_\alpha \frac{\omega_{\rm p\alpha}^2b_z}{\gamma_\alpha\omega^{\prime 2}} (E_xb_x+E_yb_y+E_zb_z)$$

$${+}\sum_\alpha \frac{\omega_{\rm p\alpha}^2\gamma_\alpha}{\omega_... ...mega^{\prime 2}}[(E_xb_x{+}E_yb_y)(b_z{-}\beta_{0\alpha}){-}E_z(b_x^2{+}b_y^2)]$$

$$+\sum_\alpha \frac{i(q_\alpha /e)\omega_{\rm H}\omega_{\rm p\alph... ...ime (\omega_{\rm H}^2-\omega^{\prime 2})}(E_yb_x-E_xb_y)(1-\beta_{0\alpha}b_z),$$ (11)

where $\omega_{\rm p\alpha}\equiv\sqrt{\frac{4\pi e^2n_{0\alpha}}{m}}$ is the plasma frequency for each of the particle species. Taking into account that in the emission region $\frac{\omega_{\rm p}^2\gamma}{\omega^{\prime 2}}\sim 1$ and $\frac{\omega^\prime}{\omega_{\rm H}}\ll 1$, whereas further on $\omega_{\rm p}$and $\omega_{\rm H}$ decrease with the altitude, one can see that far enough from the cyclotron resonance region (where $\omega^\prime\sim\omega_{\rm H}$) $E_z\ll E_x,E_y$. The cyclotron resonance radius is believed to lie even higher than the polarization-limiting region, so that in our consideration the waves are almost transverse. Then using Eqs. (8) and (10) we obtain finally:

$$\frac{{\rm d}E_x}{{\rm d}z} \frac{i\omega}{2c}[Ab_x(E_xb_x+E_yb_y)-BE_x+iGE_y]=0,$$

$$\frac{{\rm d}E_y}{{\rm d}z} \frac{i\omega}{2c}[Ab_y(E_xb_x+E_yb_y)-BE_y-iGE_x]=0,$$ (12)

where

$$\begin{eqnarray*}A&\equiv&\sum_\alpha \frac{\omega_{\rm p\alpha}^2}{\gamma_\alph... ...\beta_{0\alpha}-b_z)} {\omega_{\rm H}^2-\omega^{\prime 2}}\cdot \end{eqnarray*}$$

The set of Eqs. (12) describes the polarization evolution of the waves in an ultrarelativistic highly magnetized plasma. Setting $\omega_{\rm H}\to\infty$ immediately turns the above equations into Eq. (13) in Petrova & Lyubarskii (2000), which corresponds to the case of an infinitely strong magnetic field. Neglecting the z-dependence of the unperturbed plasma parameters in Eq. (12), one comes to the homogeneous problem. Then the solution can be found in the form:

$$\begin{eqnarray*}E_{x,y,z}(z)\propto\exp \left(-\frac{i\omega}{c}(1-n)z\right). \end{eqnarray*}$$

Setting the determinant of the system equal to zero, one can obtain the customary expressions for the refractive indices of the ordinary and extraordinary waves in the homogeneous highly magnetized ultrarelativistic plasma (cf., e.g., Melrose 1979, given that $b_x=\theta$, b_y=0).