Polarization Evolution of Fast Radio Burst Sources in Binary Systems

In EM theory, it is customary to study polarization in the coordinate system of $\bm{k}$ along the $z$-axis. With $\bm{k}=\hat{\bm{x}}k_{x}+\hat{\bm{z}}k_{z}$, say that $E_{x^{\prime}}=\bm{E}\cdot(\hat{\bm{y}}\times\hat{\bm{k}})$ and $E_{y^{\prime}}=\bm{E}\cdot\hat{\bm{y}}$, where $\hat{\bm{k}}=\bm{k}/k$ is the unit wave vector. This introduces an alternative coordinate system $x^{\prime}y^{\prime}z^{\prime}$, in which $\bm{k}$ is along the $z^{\prime}$ axis while $\bm{B}$ is in the $x^{\prime}z^{\prime}$ plane, yielding $\hat{\bm{k}}\times\hat{\bm{B}}=-\sin\theta\,\hat{\bm{y}}$. It is worth emphasizing that even though the polarization is dependent on the chosen coordinate system, the dispersion relation remains Lorentz invariant in any coordinate system.

The coordinate transformations of the tensor satisfy $\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{\varepsilon}^{\prime}}}=\bm{Q}\cdot\stackrel{{\scriptstyle\leftrightarrow}}{{\bm{\varepsilon}}}\cdot\bm{Q}^{T}$, where $\bm{Q}$ is the rotation matrix, which in this context takes the form

Then we can write the wave equation in the $x^{\prime}y^{\prime}z^{\prime}$ coordinate system

The electric vector can be divided into transverse and longitudinal components. The longitudinal part corresponds to an electrostatic oscillating wave. We are mainly interested in weakly anisotropic medium (Melrose & McPhedran, 1991), which refers to the fact that EM waves could be split into two natural wave modes ($\mathit{anisotropic}$) when they propagate in the medium, and the difference in ray paths between these two components is not significant ($\mathit{weak}$, and $n^{2}\simeq 1$). Thus, we can ignore the effect of wave refraction. However, the refractive index difference between the two natural wave modes is still important, because it leads to the evolution of the relative phase between the two components. When these two components are recombined, the polarization state of the EM wave usually differs compared to that before incidence. Under the weak anisotropy approximation, the longitudinal part of the polarization is negligible (Melrose & Luo, 2004). Therefore, we only require to be concerned with the transverse part of the natural wave mode polarization, whose polarization ellipse can be completely described by the axial ratio

where $T=\pm 1$ corresponds to the opposite CP while $T=0,\infty$ corresponds to the mutually orthogonal LP. It is obtained by combining the first and third lines of Equation (9), which provides the correlation between the axial ratio $T$ of the polarization ellipse and the refractive index $n$, consequently substituting it into the dispersion relation (A24) will give

where $\mathcal{R}$ is defined as the polarization parameter and its detail expression is

where $X=\omega_{\rm p}^{2}/\omega^{2}$ and $Y=\omega_{\rm B}/\omega$ are two dimensionless parameters incorporating $\omega_{\rm p}$ and $\omega_{\rm B}$. It is very striking that it is the same form as the one required by the two approximations²²2By comparing the values of the two terms in the numerator of the dispersion relation, two approximate conditions can be given, which can simplify the dispersion relation. A detailed discussion is in Appendix A. we defined in Equations (A27) and (A28). This infers that $|\mathcal{R}|\gg 1$ corresponds to the quasi-transverse (QT) approximation while $|\mathcal{R}|\ll 1$ corresponds to the quasi-longitudinal (QL) approximation. Next, we will further investigate the connection between the polarization parameters and the ellipticity of the natural modes.

The explicit solution of the quadratic equation (11) satisfied by the axis ratio $T$ is

leading to $T_{+}T_{-}=-1$, which implies that the polarization ellipses of the two natural wave modes are orthogonal to each other. The LP degree is defined as $\Pi_{l}=(T_{\pm}^{2}-1)/(T_{\pm}^{2}+1)$ and the CP degree as $\Pi_{c}=2T_{\pm}/(T_{\pm}^{2}+1)$. More generally, the wave mode ellipticity angle $\chi_{\rm B}$ can be given by the axial ratio, that is $\tan\chi_{\rm B}=-T_{-}$ and $\cot\chi_{\rm B}=T_{+}$ or expressed directly in terms of the polarization parameter with $\tan\chi_{\rm B}=\mathcal{R}+{\rm sgn}(\mathcal{R})\sqrt{\mathcal{R}^{2}+1}$ (Broderick & Blandford, 2010; Melrose & McPhedran, 1991).

For the QT approximation (A27), which is equivalent to the one for $\mathcal{R}\gg 1$, we can immediately determine that $T_{\mathrm{O}}\simeq\infty$, $T_{\mathrm{X}}\simeq 0$, which means that the two natural wave modes are completely linearly polarized along the $x^{\prime}$ and $y^{\prime}$ axes, respectively. The subscripts are taken according to the customary definition that the polarization of the O-mode is in the plane of $\bm{B}$ and $\bm{k}$, while the polarization of the X-mode is perpendicular to this plane. It is worth noting that for a pure pair plasma, $\epsilon=0$, all propagation directions satisfy the QT approximation, which means that in this case, the two natural modes of the plasma are always orthogonally linearly polarized. Electrons and positrons contribute the same sign to the LP component but the opposite sign to the CP component, leading to CP originating only from the asymmetric distribution of electrons and positrons in the pair plasma. On the other hand, for the QL approximation (A28), $\mathcal{R}\ll 1$, one has $T_{\mathrm{R}}\simeq+1$ while the other has $T_{\mathrm{L}}\simeq-1$, which corresponds to two CP with opposite handedness.

The evolution of the polarization of an EM wave is determined entirely by its two natural wave modes decomposed in the medium, which can be described by the transfer equation for the Stokes parameters (Melrose & McPhedran, 1991)

here we assume that the polarized wave³³3Note that the EM wave described by Equation (23) is fully polarized. propagates along the $z$-direction. An alternative way to write Equation (23) is

where $\bm{\rho}=(\rho_{\rm Q},\rho_{\rm U},\rho_{\rm V})$ is the eigenvectors of the square matrix in Equation (23), which corresponds to one of the two natural modes, and $(Q,U,V)$ describes the polarization state of the EM wave.

To better understand the physical meaning of Equation (23), one usually introduces the concept of the Poincare sphere (Fig. 2). The Poincaré sphere represents a polarization state as a unit vector $\bm{P}$ determined by both latitude $2\chi$ and longitude $2\psi$, which characterize the ellipticity of the polarized wave and the relative phase of its LP component, respectively. The north ($V=1$) and south ($V=-1$) poles of the sphere represent the right and left CP, respectively, while the equator ($\chi=0$) corresponds to complete LP. The polarization of the natural wave modes are mutually orthogonal, as shown by Equation (13). They can be represented by two diametrically opposite points ($\pm 2\chi_{\rm B},\pm 2\psi_{\rm B}$ ) at two ends of the diameter axis through the center of the Poincaré sphere. This axis is called the modal axis. Together with Equation (24), the Poincaré sphere provides a geometric interpretation of the generalized Faraday rotation. It shows that the polarization point $P$ rotates around the modal axis at a constant latitude, and the speed of rotation $\rho$ is determined by the natural mode properties of the medium.

The parameters $\rho_{\rm Q}$, $\rho_{\rm U}$ and $\rho_{\rm V}$ are calculated from the dielectric tensor of the plasma (which is exactly equivalent to the natural wave modes we derived on the previous section), and their detailed expressions are

where $|\bm{\rho}|=\Delta k$ is the difference in wavenumber between the two natural modes, while $2\chi_{\rm B}$ is the latitude and $2\psi_{\rm B}$ is the longitude of the natural mode on the Poincaré sphere, and $\chi_{\rm B}$, $\psi_{\rm B}$ correspond to the ellipticity of the natural mode and the relative phase of the ray to the magnetic field, respectively.

For the ion-electron plasma (described by $\epsilon=1$) under the weak anisotropy approximation, following Equation (A24) we can derive the specific form of the wavenumber difference between the two natural modes

The polarization parameters $\mathcal{R}$ can be reduced to

For a given frequency, we can estimate the cyclotron resonance magnetic field $B_{\rm res}$

and the plasma resonance number density $n_{\rm res}$ in the plasma medium

The astrophysical environment we discuss hereafter is supposed to be away from these resonance regions, i.e., we adopt further approximations $B\ll B_{\rm res}$ and $n_{e}\ll n_{\rm res}$. With these plasma parameters, following equations (26) and (27), the approximation of $\Delta k$ can be treated as two cases (Melrose, 2010)

Except for unusual circumstances, at high frequencies ($Y\ll 1$) the second case occupies only a very narrow range around $\theta=\pi/2$, which we refer to as the QT region, where $\Delta k$ can be expected to be independent of $\theta$. Within the QT region, the natural mode is elliptically polarized, where it leads to the interconversion between CP and LP components, and we can define an FC rate to quantify this effect, which is

On the other hand, in the QL region, the natural mode is almost circularly polarized and the modal axis is almost along the vertical axis of the Poincaré sphere. In this scenario, it only changes the polarization angle (PA) $\psi$ of the incident EM wave, and similarly, we can define the Faraday rotation rate

Therefore, for a typical astrophysical environment, the radiative transfer usually performs in the QL approximation, i.e. $\rho_{\rm L}\ll\rho_{\rm V}$, with a difference of $Y/2$ between them.

Let us first consider a toy model in which the polarized radiation passes through the QL and QT regions in turn, where the mode axis is along the V-axis in the QL region and along the Q-axis in the QT region.

Given the initial polarization, the final polarization can be determined by

with

where the Faraday rotation angle

is a physical parameter to characterize the Faraday rotation effect, this implies that a sufficiently significant Faraday rotation can occur if the EM wave frequency is comparable to $\nu_{\rm FR}$ (i.e. $\theta_{\rm FR}\sim 1~{}{\rm rad}$), with

Based on this equation, we can derive the expression for the rotation measure (RM)

Similarly, the FC angle is

is a physical parameter that characterizes the Faraday conversion effect, and

We note that $\nu_{\rm FR}\gg\nu_{\rm FC}$, which suggests that Faraday rotation dominates in the usual magneto-ionic environment. Further, from equation (42), we can give an expression for the final CP

where the LP component is $L_{\rm i}=(Q_{\rm i}^{2}+U_{\rm i}^{2})^{1/2}$. We can therefore conclude that the change in $V_{\rm f}$ depends on whether the FC is significant as the wave passes through the QT region.

The PA is defined as $\psi=1/2\arctan(U_{\rm f}/Q_{\rm f})$. Substituting the specific expression for $Q_{\rm f}$ and $U_{\rm f}$, we notice that once the effect of the FC becomes significant, the PA no longer has a simple power-law relationship with frequency, i.e. we cannot measure the RM in the traditional sense (49), this is because the presence of changes in $V$ interferes the conversion between $Q$ and $U$. A common method for measuring RM is “QU fitting”, which models the Stokes Q and U quasi-periodic oscillations introduced by Faraday rotation (Mckinven et al., 2021; O’Sullivan et al., 2012).

For a more general discussion, the magnetic field could change direction gradually, which suggests that the modal axes are changing orientation continuously. Let us consider a region in whose center ($z=0$) the magnetic field along the line of sight suffers a sign reversal ($\theta=\pi/2$), which we call the rQT region, in which we can expand $\rho_{\rm V}$ into $\rho_{\rm V}=\rho_{\rm V}^{\prime}z$ and $\rho_{\rm L}$, $\rho_{\rm V}$ can be assumed to be constant. We consider that the magnetic field and the ray are always lying in the same plane (non-twisted magnetic field). Without loss of generality, we can choose a plane such that $\psi_{\rm B}=0$. Now we define $f\equiv\rho_{\rm V}/\rho_{\rm L}$ and treat $f=f(\tilde{z})$ as a function of the dimensionless variable $\tilde{z}=\rho_{\rm L}z$. The boundary of the rQT region $[-\tilde{z}_{0},\tilde{z}_{0}]$ is determined by the condition $f(\tilde{z})\gg 1$, which implies that the natural modes are nearly circular so that FC can be considered to be negligible at the boundary.

Therefore, the representation of a polarized wave passing through an rQT region on a Poincaré sphere is that the modal axis gradually moves away from one pole of the sphere on one side ($z<0$) of the rQT region, then crosses the equator of the sphere at the center ($z=0$) of the rQT region and approaches the other pole of the sphere on the other side ($z>0$) of the rQT region. The overall motion of the polarization state $\bm{P}$ is that it rotates at a constant latitude around this continuously evolving modal axis, which can only be determined by detailed calculations, but it is also necessary to carry out a qualitative analysis before doing so.

To proceed with the semi-quantitative calculation, we can define the modal coupling coefficient $C$ (Melrose & Robinson, 1994; Melrose et al., 1995). Physically, $C$ is the ratio of the velocity ($\left|\hat{\rho}^{\prime}\right|$) of the modal axis endpoint as it crosses the equator to the velocity ($\left|\bm{\rho}\right|$) of the rotation of the polarization point around the modal axis. The case of $C\ll 1$ is denoted as weak coupling, in which the rotation of the polarization point is dominant ($\left|\hat{\rho}^{\prime}\right|\ll|\bm{\rho}|$). As the modal axis turns from one pole of the sphere to the other, the polarization point rotates very fast so that the modal axis ”drags” the polarization point and flips over together, which leads to the CP component $V$ changing its direction of rotation across the rQT region, corresponding to the reversal of the $V$ sign. The case $C\gg 1$ is denoted as strong coupling, in which the modal axis rotates faster, the modes are tightly coupled at the center. As the modal axis reverses its direction, the polarization point does not move significantly due to the relatively slow rotation of the polarization point around the modal axis. Therefore, the polarization point remains near its original position and the final result is that the CP changes by a small fraction. The case of intermediate coupling ($C\sim 1$), however, is very elusive, but it can be expected that a small change from the plasma parameters will lead to a significant change in the final polarization.

Mathematically, a simple approximation of $C$ in the present context takes the form (Cohen, 1960; Melrose, 2010)

The evolution of the polarization of a polarized wave after crossing such an rQT region depends entirely on the value of $C$ at the center of the region, and it defines the transition frequency as

where $n_{0}$ and $B_{0}$ have the values taken from the center of the rQT region and $\rm L_{\rm\theta}$ is a characteristic length over which the magnetic field significantly changes direction, i.e., the size of the rQT region. For typical parameters, the transition frequency $\nu_{\rm T}$ is expected to be in the radio band, 100 MHz-10 GHz, and a strong FC is expected at EM wave frequencies below $\nu_{\rm T}$. The strong dependence of the coupling coefficient on frequency suggests that the transition from weak to strong coupling is very rapid as $\nu$ increases. It is noted that $\nu_{\rm T}$ depends only weakly on the size of the rQT region and the plasma density in the rQT region, but has a strong dependence on the magnetic field strength.

With the above definition and discussion, we can rewrite Equation (23) as

where

The final average value⁴⁴4The average value refers to the average over a specific frequency range. of the CP components $V$ can be given analytically by its initial value (Zheleznyakov & Zlotnik, 1964; Melrose & Robinson, 1994) ⁵⁵5Their work considers only the case of 100$\%$ CP, and we have redefined their derived equations based on the numerical results in the Fig. 3, as shown in Equation (57).

Combining equation (53), by this analytical formula we draw a plot of $V_{\rm f}$ versus frequency, as shown in Fig. 3. Furthermore, for a strong coupling case ($\nu_{\rm T}\ll\nu$), following Equation (52), one can perturbably determine the final CP as (Gruzinov & Levin, 2019)

where $\theta_{\rm FR}({z})\simeq\frac{\rho_{\rm V}^{\prime}}{2}({z}^{2}-{z}_{0}^{2})$ is the Faraday rotation angle at an arbitrary location within the rQT region, and $F(x)$ is the Fresnel class C integral, denoting the real part of the Fresnel integral, with $F(\infty)=1/2$. $L_{\rm i}$, $V_{\rm i}$ are the initial LP and CP, respectively. The above equation indicates that in the case of strong coupling, the final CP oscillates with frequency and, surprisingly, its amplitude decreases with increasing initial CP.⁶⁶6In particular, if the incident radiation is completely circularly polarized, i.e. $L_{\rm i}$= 0, the oscillations disappear entirely, which reduces to the result in (Zheleznyakov & Zlotnik, 1964). This is confirmed by our numerical results in Fig. 3. It is clear from the figure that $V_{\rm f}$ reverses sign at the low-frequency end and rapidly recovers towards $V_{\rm i}$ in the mid-frequency band, with a progressively smaller rate of oscillation, while at the high-frequency end, the oscillation behavior can be completely determined by Equation (58).

$\nu_{\rm CM}$ and $\nu_{\rm RM}$ determine the amplitude and frequency of the $V_{\rm f}$ oscillation respectively, and their expressions are

This suggests that large amplitudes ($\nu_{\rm CM}\simeq\nu$) of CP will typically induce very drastic oscillations of CP ($\nu_{\rm RM}\gg\nu$) for Gaussian-order magnetic field strength. Once the physical parameters at the center of the rQT region are given, we can adequately describe the polarization evolution of the EM wave.

In summary, Faraday rotation is due to the magneto-radio wave propagation effect that is observed as a rotation of the plane of LP, where the rotation angle $\psi$ is linearly proportional to the square of the wavelength, with the slope being the familiar RM (Equation 49). A common method of measuring RM is to fit the QU (Mckinven et al., 2021; O’Sullivan et al., 2012), which models Stokes quantities Q and U oscillations introduced by Faraday rotation, and the measurement of CP is a direct calculation of $V/I$. On the other hand, FC leads to interconversion between LP and CP, which is thought to be perhaps the origin of the CP observed by some radio sources. For FRBs, FC is thought to happen in special environments, such as (1) relativistic plasmas (Vedantham & Ravi, 2019); (2) pair plasmas (Lyutikov, 2022); (3) there is a magnetic field reversal along the line of sight (Gruzinov & Levin, 2019; Qu & Zhang, 2023; Li et al., 2023). FC has some unique observational features, such as the fact that CP and LP oscillate with frequency (Xu et al., 2022); CP profiles may be completely reversed after FC (Li et al., 2023). Lower (2021) and Kumar et al. (2022) also provided a systematic study about how to measure FC.

We define the mass of the companion star as $M_{\rm c}$, the radius as $R_{\rm c}$, and the mass loss rate as $\dot{M}$. We consider the companion star’s stellar wind as a radial, constant velocity outflow and attribute it to be the dominant contribution to the companion star’s mass loss (McKee & Ostriker, 2007), then the electron number density of the stellar wind at distance $r$ from the star is given by

where $\mu_{\rm m}$ is the mean molecular weight of the stellar wind material, which can be assumed to be 1.29 for a typical stellar outflow with a hydrogen abundance of 0.7, and the mass loss rate depends on the type of companion star, ranges from $10^{-14}$ to $10^{-5}M_{\odot}~{}{\rm yr}^{-1}$. The $n_{\mathrm{w},0}$ is the electron number density of the star surface

where the wind speed can be estimated as the escape velocity, i.e. $v_{\rm w}=(2GM_{\rm c}/R_{\rm c})^{1/2}$. The stellar wind causes a modification in the magnetic field structure of the companion star. If the stellar wind is strong enough, it will straighten the magnetic field line, which will lead to the original dipolar magnetic field becoming radial. We can therefore define the Alfvén radius, which corresponds to the radius where the magnetic field pressure is equal to the ram pressure of the stellar wind (Yang et al., 2023), i.e. $R_{\rm A}\simeq(B_{\rm c}^{2}R_{\rm c}^{6}/{2\dot{M}v_{\rm w}})^{1/4}$. Within the Alfvén radius the magnetic field is dipolar, i.e. $\bm{B}=B_{\rm c}R_{\rm c}^{3}(2\cos\theta\hat{\bm{r}}+\sin\theta\hat{\bm{\theta}})/2r^{3}$, and outside this the field is radial, i.e. $\bm{B}=B_{\rm c}R_{\rm c}^{2}\hat{\bm{r}}/r^{2}$. Here we ignore the toroidal field because the rQT region appears at the LOS only when the pulsar is at a specific position. Therefore, we consider that the magnetic field strength at a distance $r$ from the highly magnetized companion star’s center satisfies

In this work, we investigate the effect of companion stellar winds on the polarization properties of radio emissions in magnetized high-mass companion binary (HMCB) and low-mass companion binary (LMCB) systems involving an FRB source, respectively, whose parameter ranges are shown in Table 3.2. Before proceeding to the detailed calculations, it is useful to discuss qualitatively which type of binary systems are expected to cause a significant polarization evolution.

According to Kepler’s third law, we can derive the orbital radius

We refer to the catalog of Galactic X-ray binaries⁷⁷7Such binary systems are widely believed to contain neutron stars, but we don’t require the main star to have X-ray emission, which is why we named it in a different way. (Fortin et al., 2023; Avakyan et al., 2023), where the typical values of the binary parameters are those of maximum probability in the catalog. $M_{\rm c}$, $R_{\rm c}$, and $\dot{M}$ are dependent on the type of companion, and we first obtain typical values for $M_{\rm c}$, and then use the empirical relationships to derive typical values for $R_{\rm c}$ and $\dot{M}$ (Eker et al., 2018; Johnstone et al., 2015). The typical values⁸⁸8 With these typical binary parameters, the magnetized plasma satisfies $B\ll B_{\rm res}$ and $n_{e}\ll n_{\rm res}$, exactly the weakly anisotropic medium we discussed in Sec 2.1. for the two types of binary parameters are listed in brackets of Table 3.2. We note that there is a mass gap for the companion stars in the LMCB and HMCB, which is due to the fact that the parameter range was chosen with reference to the X-ray binary catalog.⁹⁹9 The lack of intermediate-mass companions in X-ray binary systems is mainly due to observational selection effect (Chaty, 2022). These systems are usually weak X-ray emission sources. In contrast, high-mass X-ray binaries can be directly accreted by the stellar wind and low-mass X-ray binaries can undergo an effective accretion process through Roche-lobe overflow (RLO) to produce sufficiently strong X-ray emission. In addition, when intermediate-mass X-ray binaries evolve towards RLO, this accretion phase only lasts for a short time and is thus not easy to detect.

For the LMCBs, we have $R_{\rm A}\gg R_{\rm orb}$, so that the companion field is dominated by a dipole field, and there exists an rQT region near the magnetic axis, as shown in Fig. 1(a). Then the characteristic transition frequency of the case can be estimated as

For the HMCB, one has $R_{\rm c}\ll R_{\rm A}\ll R_{\rm orb}$, so that the companion field is radially dominant, as shown in Fig. 1(b), and the characteristic transition frequency is

Here we assume that the size of the rQT region is proportional to the radius of its location ($L_{\theta}\propto R_{\rm orb}$), due to the self-similarity of the field. In both cases above, the dependence of the characteristic transition frequency on the radius is very dramatic. For the adopted parameters, we can see that the LMCBs have a weak FC, while the magnetized HMCBs have a strong FC. The characteristic Faraday transition frequencies $\nu_{\rm T}$ in the parameter space of orbital periods and mass loss rates are shown in Fig. 4. The FC favors occurring under conditions of higher mass loss rates and shorter periods (this corresponds to a smaller orbital radius).

In fact, strongly magnetized massive stars are usually rare. For example, less than 10$\%$ of O stars are highly magnetized (Wade & MiMeS Collaboration, 2015), this may be due to the absence of convection zones around massive stars. Thus, massive stars lack magnetic dynamo mechanisms near their surfaces to convert convective and rotational energy into magnetic energy. Any strong magnetic field at its surface is either a remnant of past magnetic activity or originates from the interior of the star, and long-lived magnetic fields generated by these mechanisms appear to be rare.

Next, we will solve the transfer equation (23) numerically by substituting the specific environmental parameters of the binary star system¹⁰¹⁰10Some of the code for this section can be available on https://github.com/xiaygyg/LHMXB. Let us now consider the binary system configuration illustrated in Fig. 7, where the $XYZ$ coordinate system is chosen so that the orbital plane lies in the $XY$ plane, the rotation axis $\bm{\Omega}$ of the companion star lies in the $XZ$ plane, and whose angles with the $Z$ axis and the magnetic axes are denoted $\theta_{\Omega}$ and $\alpha$ respectively. The unit vector in the direction of the line of sight (LOS) can be denoted as

For a typical LOS inclination of $\theta_{\rm L}=\pi/4$, we first plot the RM evolution with orbital phase $\phi_{\rm orb}$¹¹¹¹11The $\phi_{\rm orb}$ is the normalisation of the orbital period, $\phi_{\rm orb}=0.25$ for the pulsar is located in the superior conjunction. for the magnetized HMCBs and LMCBs, as shown in Fig. 5 and 6. For the magnetized HMCBs, we take the type parameters in Table 3.2. We find that for strongly magnetized HMCBs, the companion stellar wind can be expected to contribute RM values of up to $10^{5}~{}\mathrm{rad}~{}\mathrm{m}^{-2}$ and that the RM undergoes sign reversal easily, as shown in Fig. 5. However, the extremely large RM values were not found in some of the known pulsars with high-mass companions (Kaspi et al., 1994; Stairs et al., 2001; Lorimer et al., 2006; Lyne et al., 2015; Andersen et al., 2023). There are two reasons that high RM was not seen in these pulsars: (1) The high-mass companions associated with the six pulsars might have a lower surface magnetic field (e.g., $B\ll 1000~{}{\rm G}$), lower surface wind density ($n_{\rm w,0}\ll 10^{8}~{}{\rm cm^{-3}}$) or larger orbital period (e.g., $P_{\rm orb}\gg 100~{}{\rm day}$), leading to a smaller RM value. (2) An extremely large RM can cause the depolarization of linearly polarized component due to the limited frequency resolution or the multipath propagation, which makes the extremely large RM unobservable.

On the other hand, for the left panel of Fig. 5, the spin period of the companion star is shorter than the orbital period, and the evolution of RM is modulated mainly by the spin period; for the right panel, the orbital period is shorter, and the evolution of RM is modulated mainly by the orbital motion. We also investigated the influence of the angle $\theta_{\Omega}$ between the companion’s rotation axis and the orbital plane, and found that the different $\theta_{\Omega}$ simply introduced an asymmetry in the RM variations, which did not have much effect on the results. Further, one can expect to obtain very complex and diverse RM variations if we introduce eccentricity. Thus, we only consider the stellar wind contribution of the companion star, which is able to explain well the enriched and diverse RM variations (both in value and trend) in the recent FRB observations (Hilmarsson et al., 2021b; Mckinven et al., 2022; Xu et al., 2022; Anna-Thomas et al., 2023), and in particular to be advantaged in explaining significant RM reversal such as FRB 20190520B (Wang et al., 2022).

For LMCBs, the RM contributed by the companion star winds is very small, but their evolution trends are also diverse, as shown in Fig. 6. The substantial discrepancy in RM evolution between HMCB and LMCB is due to their different magnetic field geometry configurations.

Next, we investigate how the magnetic field of the companion star wind affects polarization. As shown in in Fig. 8, we note that after the polarized pulse passes through the companion’s magnetic field, even if the pulse is completely linearly polarized at incidence, a CP component will be induced, which comes from the LP conversion, and this conversion only occurs with a magnetic field perpendicular to the LOS ($B_{z}=0$). As we consider the emitted pulse with a certain frequency bandwidth (1-1.5 GHz, corresponding to the observed bandwidth of FAST), as shown in Fig. 9, we find that the final CP oscillates symmetrically around the point with the CP degree equal to zero (initial CP) and that the oscillation decreases in rate and amplitude as the frequency increases. The oscillating behavior of the polarization with frequency alters significantly if the pulsar is in a different orbital phase. The FC becomes weaker farther away from the superior conjunction.

In fact, for the HMCB, the CP oscillates with frequency at a very drastic rate due to its very strong plasma column density (the integration of the number density over the path of LOS). If the oscillations are so fast that the period of oscillation is smaller than the telescopic observed resolution, this will probably lead to the depolarization. Therefore, the FC in the HMCB is not favorable for producing significant CP. This may be the reason why CP in FRB 20190520B is very small and rare. On the other hand, according to Equation (61), for the FC to induce a significant CP, a $\sim$100G perpendicular magnetic field at the QT region center is required. This will allow the CP to have a large amplitude ($\nu_{\rm CM}$ $\sim\rm GHz$) while its oscillation rate ($\nu_{\rm RM}$ $\sim\rm GHz$) is not too fast.

Further, we can also obtain the evolution of polarization with orbital phase, as shown in Fig. 10. The polarization changes significantly only near the superior conjunction, the CP gradually becomes zero as the pulsar moves away from the superior conjunction, and then returns to its initial value at the normal phase.

Besides, we investigate the possibility that LMCBs can produce a strong FC by artificially altering the system’s LOS $\theta_{\rm L}$ and orbital radius $R_{\rm orb}$. The requirements for producing a strong FC are shown in the orange-shaded region in Fig. 11. Closer lines of sight to the orbital plane and smaller orbital radii can produce stronger FC, the binary system PSR B1744-24A (Li et al., 2023) is one such example.

Observations of three consecutive orbital periods of the binary system PSR B1744-24A in the 1.5 GHz and 2 GHz bands show that the CP profile of the radio pulses near the superior conjunction is completely opposite to that of the normal phase (Li et al., 2023), this provides strong evidence for FC. It emitted pulses that were also observed to appear irregular large RM variation, which is very similar to some FRBs. These observations suggest that the orbital motion of pulsar and the magnetic field of companion star play a decisive role. We applied our model¹²¹²12Here we have considered the magnetic axis perpendicular to the orbital plane, as the non-perpendicular case has only a minor effect on the results. to V and DM of PSR B1744-24A, as shown in Fig. 12, and give the residuals of the model with the data. The LP has a depolarization in many orbital phases, thus we neglected to overplot it. Notice that these results were obtained through numerical calculation via choosing appropriate fiducial parameters due to the model’s complexity.

The reason that RM is not measured near the superior conjunction may be that the strong FC makes the variation of the Stoke parameters $Q,U$ with frequency no longer oscillating quasi-periodically with each other, but very chaotic, and therefore RM cannot be measured.

Furthermore, we investigated how sensitive the model is to variations in the binary parameters in Fig. 13. The window of strong FC becomes narrower for higher observation frequencies $\nu$ and lower transition frequencies $\nu_{\rm T}$, owing to the coupling parameter $C\propto(\nu/\nu_{\rm T})^{4}$. The rQT region’s plasma parameters $n_{0}$ and $B_{0}$ are coupled in $\nu_{\rm T}$ (see Equation 54). The DM drops sharply near the superior conjunction, due to the more dense plasma here. The smaller $n_{\rm w,0}$ leads to a smaller DM.

However, it is noteworthy that there are other propagation effects that may arise as EM waves close to the companion, due to the large magnetic field there, such as synchrotron-cyclotron absorption (Qu & Zhang, 2023; Li et al., 2023), which modulates the CP profile finely and also changes the total polarization fraction, which is not achieved by both Faraday rotation and FC.

FRB 20201124A is the first repeating burst with significant CP. The CP is seen to oscillate periodically with frequency in some of these bursts (Xu et al., 2022), which seems to be consistent with the strong coupling case discussed in our model. In particular, in this observation of FAST (Xu et al., 2022), burst 779 has a smaller oscillating rate and amplitude compared to burst 926. This is consistent with our model that smaller oscillation amplitudes lead to smaller oscillation rates (Equation 61), as shown in Fig. 14. The large deviation of our model from LP data of burst 779 may be due to the presence of additional absorption effects (Xu et al., 2022; Li et al., 2023; Qu & Zhang, 2023) or depolarization. In particular, the fact that LP becomes less at lower frequencies is consistent with the picture of depolarization.