Multiwavelength Polarization of Rotation-powered Pulsars

We use a simplified version of the radiation model presented by Harding & Kalapotharakos (2015, HK15), who simulated SR, CR, and synchrotron self-Compton emission (SSC) from a population of primary electrons and electron–positron pairs moving away from the neutron star polar cap. The particles originate at points on the neutron star surface that fill concentric rings near the edge of the polar cap. The primary electrons are injected with a low Lorentz factor (γ = 2000) and accelerated by a constant electric field component parallel to the magnetic field, ${E}_{\parallel }$, from the neutron star surface up to maximum radius ${r}_{\max }=2.0\,{R}_{\mathrm{LC}}$, where ${R}_{\mathrm{LC}}=c/{\rm{\Omega }}$ is the LC radius, and their energies are limited by the CR reaction, while the pairs are injected with the energy distribution of the polar cap cascade spectrum computed for the Crab pulsar (shown in Figure 1 of HK15) and assumed not to be accelerated. Since the details of this model are described in HK15, we will note here only the modifications we have made for the polarization calculation of this paper. First, we simulate here only SR from a spectrum of pairs and CR or SR from primary electrons, ignoring SSC since it appears at very high energies $\gt 1\,\mathrm{GeV}$. Second, the treatment of SR is greatly simplified. In HK15, the particles gained pitch angles through cyclotron resonant absorption of radio photons and suffered SR losses, which is time consuming to compute. Here, we simply assume that the pairs, with a range of Lorentz factors γ, have zero momentum perpendicular to the magnetic field, ${p}_{\perp }$, from the start of their trajectories up to a radius ${r}_{\min }$ and maintain a constant ${p}_{\perp }$ above a radius ${r}_{\min }$ to maximum radius ${r}_{\max }$. We explore different constant pitch angles $\psi ={p}_{\perp }/p$. In this way, we can explore how the polarization characteristics change for radiation at different altitudes. Third, we modulate the particle flux in each ring by a Gaussian of width ${\rm{\Delta }}r=0.05$ centered at ${r}_{0}=0.9$ in units of polar cap radius, which is the same for primaries and pairs, while HK15 assumed uniform particle flux between inner and outer rings at ${r}_{\mathrm{in}}$ and ${r}_{\mathrm{out}}$, respectively, that had different values for primaries and pairs. Use of a Gaussian profile produces smoother light curves and phase-resolved polarization. As in HK15, the photon flux from the injected test particles are normalized to the Goldreich–Julian flux, ${\dot{n}}_{\mathrm{GJ}}={n}_{\mathrm{GJ}}{{cA}}_{\mathrm{inj}}$, where ${n}_{\mathrm{GJ}}={B}_{0}{\rm{\Omega }}/2\pi {ec}$ is the Goldreich–Julian number density at the neutron star surface and ${A}_{\mathrm{inj}}$ is the area on the polar cap over which the injection occurs.

Computation of particle trajectories and radiation is done in the inertial observers frame (IOF) since the force-free magnetic field structure has been determined through numerical calculations in the IOF (see Kalapotharakos & Contopoulos 2009; Kalapotharakos et al. 2012). As described in HK15, we extend the numerical solution to a neutron star surface of $0.01{R}_{\mathrm{LC}}$ by joining it to the analytic Deutsch (1955) retarded vacuum dipole solution below $0.2{R}_{\mathrm{LC}}$, using a ramp function between 0.2 and $0.4{R}_{\mathrm{LC}}$. At each location above $0.2{R}_{\mathrm{LC}}$, we interpolate the magnetic field components from stored numerical table values. We assume all photons are emitted along the direction of the particle velocity and the Stokes Parameters for the emission (see Section 2.2) at steps along the particles trajectories are accumulated in sky maps for 24 energy intervals.

All of the parameters for these calculations are set for the Crab pulsar, with period ${P}_{\mathrm{rot}}=0.033\,{\rm{s}}$ and period derivative ${\dot{P}}_{\mathrm{rot}}=4.22\times {10}^{-13}\,{\rm{s}}\,{{\rm{s}}}^{-1}$, and the pair multiplicity is set to ${M}_{+}={10}^{5}$. The results are most applicable to the Crab and Crab-like pulsars such as PSR B0540-69 and PSR B1509-58 that show higher levels of optical and non-thermal X-ray emission and thus are more likely to have detectable phase-averaged and phase-resolved polarization measurements.

To determine the polarization of the emitted radiation, we calculate the Stokes Parameters at each step along the particle trajectories for each spectral interval,

$$\begin{eqnarray}I(\omega ) & = & {N}_{\mathrm{CR}}(\omega )+{N}_{\mathrm{SR}}(\omega )\\ Q(\omega ) & = & {N}_{\mathrm{CR}}(\omega )\cos (2{\chi }_{\mathrm{CR}}){P}_{\mathrm{CR}}(\omega )\\ & & +{N}_{\mathrm{SR}}(\omega )\cos (2{\chi }_{\mathrm{SR}}){P}_{\mathrm{SR}}(\omega )\\ U(\omega ) & = & {N}_{\mathrm{CR}}(\omega )\sin (2{\chi }_{\mathrm{CR}}){P}_{\mathrm{CR}}(\omega )\\ & & +{N}_{\mathrm{SR}}(\omega )\sin (2{\chi }_{\mathrm{SR}}){P}_{\mathrm{SR}}(\omega ),\end{eqnarray} \tag{ 1 }$$

where ${N}_{\mathrm{CR}}(\omega )$ and ${N}_{\mathrm{SR}}(\omega )$ are the spectral flux, ${\chi }_{\mathrm{CR}}$ and ${\chi }_{\mathrm{SR}}$ are the PAs, and ${P}_{\mathrm{CR}}(\omega )$ and ${P}_{\mathrm{SR}}(\omega )$ are the intrinsic polarization degree of CR and SR at photon energy ω. To compute the degree of intrinsic polarization of both SR and CR, we use the expression (Westfold 1959)

$$\begin{eqnarray}&&P(\omega )=\displaystyle \frac{{K}_{2/3}(\omega /{\omega }_{c})}{{\int }_{\omega /{\omega }_{c}}^{\infty }{K}_{5/3}(x){dx}},\end{eqnarray} \tag{ 2 }$$

where ${K}_{2/3}(x)$ and ${K}_{5/3}(x)$ are modified Bessel functions and ${\omega }_{c}$ is the critical energy for CR

$$\begin{eqnarray}&&{\omega }_{\mathrm{CR}}=\displaystyle \frac{3}{2}\displaystyle \frac{c\,{\gamma }^{3}}{{\rho }_{c}}\end{eqnarray} \tag{ 3 }$$

and for SR

$$\begin{eqnarray}&&{\omega }_{\mathrm{SR}}=\displaystyle \frac{3}{2}{\gamma }^{2}\,B^{\prime} \,\sin \psi ,\end{eqnarray} \tag{ 4 }$$

where γ is the particle Lorentz factor, ${\rho }_{c}$ the particle trajectory radius of curvature, $B^{\prime}$ the local magnetic field strength in units of the critical field, ${B}_{\mathrm{cr}}=4.4\times {10}^{13}$ G, and ψ is the particle pitch angle. Asymptotic values of $P(\omega )$ are 0.5 for $\omega \ll {\omega }_{c}$ and 1.0 for $\omega \gg {\omega }_{c}$. At $\omega ={\omega }_{c}$, $P\simeq 0.75$ (Westfold 1959).

The direction of polarization for each process is determined by the orientation of the photon electric vector, which is parallel to the direction of particle acceleration (Blaskiewicz et al. 1991; Hirschmann & Arons 2001). The direction of the particle acceleration in the IOF is the derivative of the velocity (Takata et al. 2007),

$$\begin{eqnarray}&&{\boldsymbol{\beta }}={{\boldsymbol{\beta }}}_{d}+f(\cos \psi \hat{{\boldsymbol{b}}}+\sin \psi \cos {\varphi }_{g}\hat{{\boldsymbol{n}}}+\sin \psi \sin {\varphi }_{g}\hat{{\boldsymbol{k}}}),\end{eqnarray} \tag{ 5 }$$

where ${{\boldsymbol{\beta }}}_{d}$ is the particle drift velocity component

$$\begin{eqnarray}&&{{\boldsymbol{\beta }}}_{d}=\displaystyle \frac{{\boldsymbol{E}}\times {\boldsymbol{B}}}{{B}^{2}},\end{eqnarray} \tag{ 6 }$$

ψ is the particle pitch angle, and ${\varphi }_{g}$ is the gyro phase. By requiring $\beta =1$ and outward motion, the scalar quantity f can be determined at each point along the trajectory. The unit vector $\hat{{\boldsymbol{b}}}={\boldsymbol{B}}/B$ is the direction of the local magnetic field. The unit vectors $\hat{{\boldsymbol{n}}}$ and $\hat{{\boldsymbol{k}}}$ are orthogonal to $\hat{{\boldsymbol{b}}}$ and are oriented such that $\hat{{\boldsymbol{n}}}=(\hat{{\boldsymbol{b}}}\cdot {\rm{\nabla }})\hat{{\boldsymbol{b}}}/| (\hat{{\boldsymbol{b}}}\cdot {\rm{\nabla }})\hat{{\boldsymbol{b}}}|$ is in the plane of the field line curvature and $\hat{{\boldsymbol{k}}}=\hat{{\boldsymbol{n}}}\times \hat{{\boldsymbol{b}}}$. In the case of CR, $\psi =0$ and we determine the particle acceleration direction, and thus the electric vector ${\hat{{\boldsymbol{e}}}}_{\mathrm{CR}}$, through a numerical derivative of the particle velocity, which is the change in direction of the trajectory at two successive positions. In the case of SR, one would need to resolve the particle gyro motion with ${\varphi }_{g}={\varphi }_{0}+{{\rm{\Omega }}}_{B}t$ in order to numerically compute the velocity derivative, where ${{\rm{\Omega }}}_{B}={eB}/\gamma {mc}$ is the gyro frequency. Since it is impractical to resolve the gyro motion in the high pulsar magnetic field, we can use the analytic derivative of the particle velocity in Equation (5) (Takata et al. 2007),

$$\begin{eqnarray}&&{{\boldsymbol{a}}}_{\mathrm{SR}}=f{{\rm{\Omega }}}_{B}\sin \psi (\cos {\varphi }_{g}\hat{{\boldsymbol{k}}}-\sin {\varphi }_{g}\hat{{\boldsymbol{n}}}),\end{eqnarray} \tag{ 7 }$$

which is a very good approximation if ${{\rm{\Omega }}}_{B}\gg {\rm{\Omega }}$. The SR electric vector direction is then ${\hat{{\boldsymbol{e}}}}_{\mathrm{SR}}={{\boldsymbol{a}}}_{\mathrm{SR}}/| {{\boldsymbol{a}}}_{\mathrm{SR}}|$. In the particle gyro motion, the phase ${\varphi }_{g}$ rotates around the magnetic field direction, changing orientation over $2\pi$ each gyro period. This affects both the photon emission direction, which is along the particle velocity (Equation (5)), which broadens the caustics in the sky map and the direction of the radiation electric vector causing depolarization of the SR. To capture this in our calculation, we average the emission over a range of ${\varphi }_{g}$ values, evenly spaced between 0 and $2\pi$ at each point in the trajectory.

We define the PA as the angle in the counterclockwise direction between the electric vector $\hat{{\boldsymbol{e}}}^{\prime}$ and the projected pulsar spin axis ${\boldsymbol{\Omega }}^{\prime}$ on the plane of the sky:

$$\begin{eqnarray}\hat{{\boldsymbol{e}}}{^{\prime} }_{\mathrm{CR}} & = & {\hat{{\boldsymbol{e}}}}_{\mathrm{CR}}-(\hat{{\boldsymbol{\eta }}}\cdot {\hat{{\boldsymbol{e}}}}_{\mathrm{CR}})\hat{{\boldsymbol{\eta }}}\\ \hat{{\boldsymbol{e}}}{^{\prime} }_{\mathrm{SR}} & = & {\hat{{\boldsymbol{e}}}}_{\mathrm{SR}}-(\hat{{\boldsymbol{\eta }}}\cdot {\hat{{\boldsymbol{e}}}}_{\mathrm{SR}})\hat{{\boldsymbol{\eta }}}\\ {\boldsymbol{\Omega }}^{\prime} & = & {\boldsymbol{\Omega }}-(\hat{{\boldsymbol{\eta }}}\cdot {\boldsymbol{\Omega }})\hat{{\boldsymbol{\eta }}},\end{eqnarray} \tag{ 8 }$$

where $\hat{{\boldsymbol{\eta }}}$ is the direction of the photon emission, assumed to be the particle velocity direction. The PA for CR and SR is then

$$\begin{eqnarray}&&\chi (\omega )=\mathrm{atan}\left(\displaystyle \frac{{\hat{e}}_{y}^{\prime }}{{\hat{e}}_{x}^{\prime }}\right).\end{eqnarray} \tag{ 9 }$$

Stokes Parameters ${I}_{i,j,k}$, ${U}_{i,j,k}$, and ${V}_{i,j,k}$ from all particle trajectories are accumulated for each energy ${\omega }_{i}$, observer angle ${\zeta }_{j}$, and phase ${\phi }_{k}$ with respect to ${\boldsymbol{\Omega }}$ to form energy-dependent sky maps $(\omega ,\zeta ,\phi )$. We can then make sky maps of flux,

$$\begin{eqnarray}&&F(\omega ,\zeta ,\phi )={I}_{i,j,k},\end{eqnarray} \tag{ 10 }$$

polarization fraction,

$$\begin{eqnarray}&&P(\omega ,\zeta ,\phi )=\displaystyle \frac{{[{Q}_{i,j,k}^{2}+{U}_{i,j,k}^{2}]}^{1/2}}{{I}_{i,j,k}},\end{eqnarray} \tag{ 11 }$$

and PA,

$$\begin{eqnarray}&&\mathrm{PA}(\omega ,\zeta ,\phi )=0.5\,\mathrm{atan}\left(\displaystyle \frac{{U}_{i,j,k}}{{Q}_{i,j,k}}\right).\end{eqnarray} \tag{ 12 }$$

For phase-resolved polarization, cuts across the sky maps at constant ζ for a given energy range ${\omega }_{n}$ to ${\omega }_{m}$ give the flux, polarization fraction, and PA as a function of phase,

$$\begin{eqnarray}{F}_{\zeta }(\phi ) & = & \displaystyle \sum _{i=n}^{m}F({\omega }_{i},\zeta ,\phi )\\ {P}_{\zeta }(\phi ) & = & \displaystyle \sum _{i=n}^{m}P({\omega }_{i},\zeta ,\phi )\\ {\mathrm{PA}}_{\zeta }(\phi ) & = & \displaystyle \sum _{i=n}^{m}{PA}({\omega }_{i},\zeta ,\phi ).\end{eqnarray} \tag{ 13 }$$

We also compute the phase-averaged spectral flux,

$$\begin{eqnarray}&&\langle {F}_{\zeta }(\omega )\rangle =\displaystyle \sum _{k}F(\omega ,\zeta ,{\phi }_{k}){\rm{\Delta }}\phi /2\pi ,\end{eqnarray} \tag{ 14 }$$

polarization fraction

$$\begin{eqnarray}&&\langle {P}_{\zeta }(\omega )\rangle =\displaystyle \frac{{\displaystyle \sum }_{k}P(\omega ,\zeta ,{\phi }_{k})F(\omega ,\zeta ,{\phi }_{k})}{{\displaystyle \sum }_{k}F(\omega ,\zeta ,{\phi }_{k})},\end{eqnarray} \tag{ 15 }$$

and PA

$$\begin{eqnarray}&&\langle {\mathrm{PA}}_{\zeta }(\omega )\rangle =\displaystyle \frac{{\displaystyle \sum }_{k}{PA}(\omega ,\zeta ,{\phi }_{k})F(\omega ,\zeta ,{\phi }_{k})}{{\displaystyle \sum }_{k}F(\omega ,\zeta ,{\phi }_{k})},\end{eqnarray} \tag{ 16 }$$

where the $\langle {P}_{\zeta }\rangle$ and $\langle {\mathrm{PA}}_{\zeta }\rangle$ have been weighted with the flux at each phase.

We present the phase-resolved polarization results for four different energy ranges, in optical (1–10 eV), soft X-ray (2–10 keV), hard X-ray (0.1–10 MeV), and γ-ray (0.1–100 GeV). In Figure 1, we show sky maps of the intensity and polarization degree for the soft X-ray and γ-ray energy bands for a pulsar inclination angle of $\alpha =60^\circ$ and emission radius range $r=0.7\mbox{--}1.3\,{R}_{\mathrm{LC}}$, derived using Equations (10) and (11). In the soft X-ray band, the radiation is SR from pairs, shown for two different pitch angles ($\psi =0.01$ and 0.1), while in the γ-ray band, the radiation is primarily CR from primary electrons. The intensity pattern (left panels) shows bright caustics at locations where photons from a range of radii arrive at an observer at the same phase. The caustic formation is somewhat different at radii inside and outside the LC. Inside the LC, the particle trajectories in a force-free magnetosphere are parallel to curved, mostly poloidal magnetic field lines in the corotating frame. In the IOF, aberration and time-of-flight produce phase delays that cancel the phase differences of emission at different radii along the trailing field lines (Morini 1983; Dyks et al. 2004), causing the emission to bunch along a preferred direction and phase toward the observer. Outside the LC, the poloidal field lines straighten and the toroidal field becomes dominant. The particle trajectories (defined always in the IOF) become radial and when emission occurs near the current sheet, as it does when particle trajectories lie near the last open field line or separatrix, the caustics trace the current sheet as it moves up and down in latitude with phase (Bai & Spitkovsky 2010). At $r=0.7\mbox{--}1.3\,{R}_{\mathrm{LC}}$, the caustics are formed partly inside and partly outside the LC. At each observer angle ζ, peaks in the light curve arise when the light of sight (horizontal lines) cuts once or twice across the caustic. The range of gyro phase of SR produces different particle trajectories that broaden the soft X-ray caustics to a greater degree for larger pitch angle (Takata et al. 2007). As is apparent, the light-curve peaks are in phase for different energy ranges. The right panels show sky maps of polarization degree for the soft X-ray and γ-ray energy ranges and display patterns of depolarization. For emission inside the LC, the dip in polarization is due to the superposition of different field directions from a large range of radii, whereas for emission outside the LC near the current sheet, the depolarization comes from superposition of emission from field lines of opposite polarity on either side of the sheet. The polarization dips should therefore come nearly in phase with the light-curve peaks. For SR in soft X-rays, there are also large regions of depolarization following the peaks due to the range of gyro phases for which the photon electric vector points in different directions.

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Figure 2 shows the light curves, PA, and polarization fraction for $\alpha =60^\circ$ and $\zeta =70^\circ$ for different energy bands, emission radii, $r=0.7\mbox{--}1.3\,{R}_{\mathrm{LC}}$ and $r=1.3\mbox{--}2.0\,{R}_{\mathrm{LC}}$, and two different pitch angles. The light curve for this geometry shows two Crab-like peaks at the same phases at all energy bands. The polarization of SR in the optical, soft X-ray, and hard X-ray bands are nearly the same, with swings of PA in the same direction in both peaks and depolarization at each peak. Due to the range of different gyro phases, the depolarization is quite strong, with large regions of phase where the polarization is below 10%. The polarization of CR in the γ-ray band is quite different, with swings of PA at the peaks but much higher levels of polarization greater than 40% even at the peaks. In our calculation, the projection of the pulsar rotation axis on the plane of the sky is 0° (see Equation (9)). For $r=1.3\mbox{--}2.0\,{R}_{\mathrm{LC}}$, the PA swings are larger and the polarization dips in the γ-ray band are deeper and wider, as the PA changes by nearly 180° due to the change in field line polarity as the line of sight crosses the current sheet. One interesting feature is that the PA in the bridge region for optical and X-ray bands is 90° different from the γ-ray bridge PA for the $r=0.7\mbox{--}1.3\,{R}_{\mathrm{LC}}$ case but similar to the γ-ray bridge PA for $r=1.3\mbox{--}2.0\,{R}_{\mathrm{LC}}$. This is due to the change in direction of the magnetic field as it transitions from mostly poloidal inside the LC to mostly toroidal outside the LC, so that the SR electric vector flips by nearly 90°. On the other hand, the particle acceleration vector stays at the same orientation, so that the CR electric vector does not change from inside to outside the LC. The PA swings for SR are also larger for the higher pitch angle case and the polarization fraction is somewhat higher between the peaks for emission at smaller radii. Note that there is a symmetry in the PA such that the curves for ζ and $180^\circ -\zeta$ are flipped vertically about 0°. That means that the PA swings in opposite directions for 110° and 70°.

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Since the polarization characteristics seem to be most sensitive to emission radius, in Figure 3 we show the light curves, PA, and polarization fraction for smaller emission ranges inside, $r=0.7\mbox{--}1.0\,{R}_{\mathrm{LC}}$, and just outside the LC, $r=1.0\mbox{--}1.3\,{R}_{\mathrm{LC}}$, as well as further outside the LC, $r=1.3\mbox{--}2.0\,{R}_{\mathrm{LC}}$, for pitch angle $\psi =0.01$. Since, as noted earlier, there are minimal variations in the SR polarization between optical and hard X-ray bands, we show only the optical and γ-ray bands here. There are indeed dramatic differences in polarization behavior with radius. In general, the PA swings become larger and there is more depolarization as the emission radius increases. For emission inside the LC, the PA swings for SR are smallest and the polarization degree is high (50%) outside the peaks with sudden dips to around 5% on either side of the peaks. For emission just outside the LC, the regions of depolarization are wider and the polarization fraction is lower between the peaks. For emission well outside the LC, where the current sheet is well formed, the PA swings and depolarization are maximum.

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To explore the polarization dependence on inclination angle, Figure 4 shows the light curves, PA, and polarization fraction at $\zeta =70^\circ$ for the optical and γ-ray energy bands for $\alpha =45^\circ ,60^\circ$, and 75°, for emission radii, $r=0.7\mbox{--}1.3\,{R}_{\mathrm{LC}}$ and $r=1.3\mbox{--}2.0\,{R}_{\mathrm{LC}}$. The polarization behavior is more strongly dependent on emission radius than on inclination angle, with the swings in polarization being larger for the larger emission radii at all three inclination angles. One noticeable trend is that at $r=0.7\mbox{--}1.3\,{R}_{\mathrm{LC}}$, the PA swings for CR in the γ-ray band are larger with increasing inclination angle.

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The phase-averaged flux, polarization degree, and PA as a function of energy are calculated from Equations (14), (15), and (16), respectively. Figure 5 shows the combined phase-averaged SR and CR spectral energy distributions (SEDs), PA, and polarization percent for three different inclination angles, $\alpha =45^\circ ,60^\circ$, and 75°, three different viewing angles, $\zeta =40^\circ ,60^\circ$, and 70°, and two emission radius ranges, $r=0.7\mbox{--}1.3\,{R}_{\mathrm{LC}}$ (top panels) and $r=1.3\mbox{--}2.0\,{R}_{\mathrm{LC}}$ (bottom panels). The SR peak in the SED at lower energy is broader than the CR peak at higher energy since it is emission from a broad range of pair energies (see HK15), while the CR is from primary particles. The ranges of angles are chosen primarily to model the Crab pulsar, whose viewing angle has been constrained to 63° ± 1.3° by modeling the Chandra image of the pulsar wind nebula (Ng & Romani 2008). The inclination angle is unconstrained by flux observations but, as we will discuss, could potentially be constrained by multiwavelength polarization measurement.

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For the smaller emission radii, there is little variation with inclination angle. The general trend is for the polarization to be quite low, $\lt 10$%, at energies below the SR to CR transition around 100 MeV and to rise sharply at higher energy to about 45% and then increase to 60% beyond the CR SED peak. As discussed earlier, the low polarization degree for SR is caused both by depolarization in the caustic peaks and by the combined emission from a range of different gyro phases that have different electric vector directions. The CR maintains the intrinsic degree of polarization outside the peaks since the acceleration and electric vector always points in the direction of the particle trajectory curvature. The PA for SR is nearly aligned with the projection of the spin axis on the plane of the sky (0°) but at the transition to CR, the PA increases suddenly for the lower altitude emission range. There is a decrease in PA with increasing energy, most strongly for $\alpha =75^\circ$. There is little variation with viewing angle except at $\alpha =45^\circ$, where the PA and degree of SR polarization are smaller for $\zeta =40^\circ$.

For the larger emission radii outside the LC, the PA for SR is again nearly aligned with the rotation axis projection, but is constant with energy except for $\zeta =40^\circ$ at $\alpha =60^\circ$ and 75° where there is a decrease with increasing energy. Above the SR–CR transition, there are significant variations with inclination and viewing angle with the PA showing both increases and decreases. The SR polarization percent is higher than for the smaller emission radii, averaging around 18% and varying significantly with viewing angle, with the lowest values of around 10% for $\zeta =40^\circ$ and higher values around 20% for $\zeta =70^\circ$. In contrast, the CR polarization degree is lower than for the smaller emission radii, except for $\zeta =40^\circ$ at $\alpha =60^\circ$ and 75°. Figure 6 shows the phase-averaged polarization for pitch angle $\psi =0.1$, $\alpha =60^\circ$, and the two ranges of emission radii, illustrating that there is very small variation with pitch angle.

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There are generally larger changes in PA with energy for SR emission starting inside the LC where the magnetic field is poloidal and the magnetic field and electric vector directions are changing with distance from the neutron star. For all α, the PA of SR decreases with energy for most ζ until the jump around 10–100 MeV, after which the CR PA increases with energy. Outside the LC, the toroidal field direction is roughly constant with distance from the neutron star so that there is less change of PA with energy for both SR and CR. It would thus be possible to constrain the emission radius and whether the emission mechanism at γ-ray energies is SR or CR by measuring the polarization at different energies. Detecting a change of PA as a function of energy below 10 MeV would indicate that the pair SR originates at least partly inside the LC. Detecting a large increase in polarization degree between hard X-ray and γ-ray bands and a jump in change PA would suggest that the γ-ray emission is CR. To test this hypothesis, we simulated the phase-averaged broadband polarization for the case where both pairs and primary particles emit SR by artificially suppressing primary CR and giving the primary particles pitch angles of ${p}_{\perp }=\,5\times {10}^{-5}\gamma$ for emission radii $r=0.7\mbox{--}1.3\,{R}_{\mathrm{LC}}$. The result, shown in Figure 7, is that the PA changes by only a small amount over the entire energy range, with the exception of a small dip at the transition between the two SED components, and the polarization degree remains low at all energies. Increasing the radial extent of the emission increases the effects of depolarization of SR and the jumps of polarization degree at the SR/CR transition. For minimum emission radii below $0.5\,{R}_{\mathrm{LC}}$, the phase-averaged polarization degree drops below 5%, which could not agree as well with the pulsar observations discussed below.

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The phase-averaged polarization of the Crab pulsar and nebula has been measured in optical and X-ray bands and the results depend on the field of view (FOV). Measurements with HST (Moran et al. 2013a, 2013b) for an FOV of $1\buildrel{\,\prime}\over{.} 7\times 1\buildrel{\,\prime}\over{.} 7$ centered on the pulsar gives a PA = 155° and P = 19%, and for a smaller FOV of $2^{\prime\prime} \times 2^{\prime\prime}$ that includes the pulsar and inner knot gives PA = 105° and P = 5.2%. Measurement of the inner nebula by INTEGRAL (Dean et al. 2008) for an FOV of $25^{\prime\prime} \times 33^{\prime\prime}$, that includes the pulsar, inner knot, X-ray jet, and wisps, gives PA = 123° ± 11° and P = 46 ± 10%. The PA of the inner nebula is consistent with the projected direction of the pulsar rotation axis of 124°, while the PA = 105° of the pulsar may be aligned with the pulsar proper motion vector, PA = 110° ± 2° ± 9° (where both measurement and reference frame uncertainties are given; Kaplan et al. 2008). Slowikowska et al. (2009) obtained a phase-averaged polarization percent of only 5.5% after subtracting an assumed constant nebular component of 33%. Forot et al. (2008) obtained a much higher phase-averaged pulsar polarization fraction of ${0.47}_{-0.13}^{+0.19}$ at soft γ-ray energies with INTEGRAL/IBIS (200 keV–5 MeV), where the bridge and off-peak flux of the pulsar is much higher. To compare with our results, the measured sky projection of the pulsar spin axis of 124° corresponds to a PA of 0° in our models. Our results for the SR phase-averaged PA in the optical band is a few degrees larger than 0°, which is roughly consistent with observations. The measured low degree of polarization of the pulsar emission at optical energies is more consistent with our result of $\lt 10$% polarization for the lower range of emission radii starting inside the LC in our model. Our result of ∼20% polarization for the emission range $r=1.3\mbox{--}2.0\,{R}_{\mathrm{LC}}$ indicates that the pulsed emission does not come from only far outside the LC. Measuring a change in PA with energy above the optical band could further discriminate between different inclination angles and emission radii.