辐射在脉冲星磁层中的传播效应 国家天文台 王陈 2013 年脉冲星天文学讲习班

Motivation – Circular polarization Lyne & Manchester (1988) P.A. Single sign & Sign reversalSingle sign & Sign reversal Weak and strongWeak and strong Intrinsic emission mechanism Propagation effect Origin

Motivation - Orthogonal mode emission P.A. Stinebring et al. (1984) V I %L

辐射高度 ~ a few - 100’s R NS. 辐射高度 ~ a few - 100’s R NS. 初始线偏振 初始线偏振 O 模： E // k-B plane X 模： E ⊥ k-B plane 磁层 B * =10 8 G – GB * =10 8 G – G 充满相对论流动的 ee + 等离 子体（开放磁力线区域） 充满相对论流动的 ee + 等离 子体（开放磁力线区域） 沿磁层流动 N/N GJ ~ 10s – 1000s 沿磁层流动 N/N GJ ~ 10s – 1000s γ ~ 10s – 1000s γ ~ 10s – 1000s 最终偏振 状态？ Ω μ k B 传播效应 波模耦合波模耦合 回旋吸收回旋吸收 准切点效应准切点效应 脉冲星磁层中 辐射的传播效应 法拉第效应法拉第效应

Outlines Previous studiesPrevious studies –Dispersion relation and natural wave modes in pulsar magnetosphere. –Some propagation effects Our worksOur works –On some special propagation effects Vacuum resonance (Wang, Lai & Han 2007)Vacuum resonance (Wang, Lai & Han 2007) Quasi-tangential effect (Wang & Lai 2009)Quasi-tangential effect (Wang & Lai 2009) Wave mode coupling effect (Wang, Lai & Han 2010)Wave mode coupling effect (Wang, Lai & Han 2010) Intrinsic Faraday Rotation effect in pulsar magnetosphereIntrinsic Faraday Rotation effect in pulsar magnetosphere (Wang, Han & Lai 2011) (Wang, Han & Lai 2011) –Numerical simulations on Polarization profile changes due to all the propagation effects. (Wang, Lai & Han 2010) Conclusion

Previous studies Dispersion relation and natural wave modes in pulsar magnetosphere. (Melrose & Stoneham 1977, Arons & Barnard 1986, von Hoensbroaech et al. 1998, Lyubaskii 1998, Melrose et al. 1999)Dispersion relation and natural wave modes in pulsar magnetosphere. (Melrose & Stoneham 1977, Arons & Barnard 1986, von Hoensbroaech et al. 1998, Lyubaskii 1998, Melrose et al. 1999) Special Propagation effectsSpecial Propagation effects –Adiabatic Walking (Cheng & Ruderman 1979) –Wave mode coupling (or limiting-polarization effect) (Cheng & Ruderman 1979, Petrova 2000, 2006) –Circularization (Cheng & Ruderman 1979) –Refractive effect of O-mode (Melrose 1979, Allen & Melrose 1982, Barnard & Arons 1986, Lyubarskii & Petrova 1998, Weltevrede et al. 2003, Fussell & Luo 2004) –Cyclotron absorption (Luo & Melrose 2001, 2006, Fussell et al. 2003)

B=∞ limit (Tsytovitch & Kaplan 1972; Arons & Barnard 1986) –Cyclotron frequency >> wave frequency –Don’t consider QED effect in dielectric tensor. Two natural wave modes Two natural wave modes –ordinary mode, or O-mode –extraordinary mode, or E-mode Polarized in k-B plane Polarized perpendicular to k-B plane Dispersion relation and natural wave modes in pulsar magnetosphere k B k B O-mode E-mode In the real case, the two modes are elliptically polarized.

Dispersion relation and natural wave modes in pulsar magnetosphere Dispersion relation of O-mode In the plasma rest frameIn the lab frame B=∞ limit

Propagation effects ： Adiabatic walking The polarization direction follows B ⊥ field in adiabatic conditionThe polarization direction follows B ⊥ field in adiabatic condition Make the final PA different from the initial emissionMake the final PA different from the initial emission B⊥B⊥B⊥B⊥ E φBφB φ PA 垂直磁场 方向 电场 方向 Adiabatic Non- Adiabatic Adiabatic Condition

Refractive effect of O-mode Only happens when wave frequency is close to proper plasma frequency, near the emission height. n is not so close to unity.Only happens when wave frequency is close to proper plasma frequency, near the emission height. n is not so close to unity. Refraction direction depends on the density gradient.Refraction direction depends on the density gradient. Can cause the separation of natural waves, => OPM phenomenon. ( Melrose 1979, Allen & Melrose 1982)Can cause the separation of natural waves, => OPM phenomenon. ( Melrose 1979, Allen & Melrose 1982) Outward density decrease causes ray deviation away from magnetic axis, which may widen the emission beam width. ( Lyubarskii & Petrova 1998)Outward density decrease causes ray deviation away from magnetic axis, which may widen the emission beam width. ( Lyubarskii & Petrova 1998) The refraction induced wave mode coupling may cause CP with sign reversal (Petrova & Lyubarskii 2000)The refraction induced wave mode coupling may cause CP with sign reversal (Petrova & Lyubarskii 2000) (Melrose 1979, Allen & Melrose 1982, Barnard & Arons 1986, Lyubarskii & Petrova 1998, Weltevrede et al. 2003, Fussell & Luo 2004)

ω′= eB/mc. r = r cr RCP absorbed by electrons RCP absorbed by electrons LCP absorbed by positrons LCP absorbed by positrons Optical depth with γ >>1Optical depth with γ >>1 circular polarization can be generated by the asymmetric cyclotron absorption of electrons and positrons. B + E = scattered e Cyclotron Resonance/Absorption B p (Luo & Melrose 2001, 2006, Fussell et al. 2003)

对称的正负电子回旋吸收 不对称的正负电子回旋吸收

A small summary to previous studies Dispersion relation and natural wave modes in some simple approximations was analytically derived.Dispersion relation and natural wave modes in some simple approximations was analytically derived. A few kinds of propagation effects were studied qualitatively (early years) and using numerical calculations (recent works).A few kinds of propagation effects were studied qualitatively (early years) and using numerical calculations (recent works). Almost none of the previous studies has calculated the final polarization profiles with all of these propagation effects included in a self-consistent way within a single theoretical frameworkAlmost none of the previous studies has calculated the final polarization profiles with all of these propagation effects included in a self-consistent way within a single theoretical framework

Our works Wave and modes amplitude evolution equationWave and modes amplitude evolution equation On some special propagation effectsOn some special propagation effects –Vacuum resonance (Wang, Lai & Han 2007) –Wave mode coupling (Wang, Lai & Han 2010) –Quasi-tangential effect (Wang & Lai 2009) –Intrinsic Faraday Rotation in magnetosphere (Wang, Han & Lai 2011) (Wang, Han & Lai 2011) Numerical calculations on Polarization profile changes considering all the propagation effects. (Wang, Lai & Han 2010)Numerical calculations on Polarization profile changes considering all the propagation effects. (Wang, Lai & Han 2010)

Wave evolution equation => numerical ray integrations is necessary. Wave evolution equationWave evolution equation Determined by dielectric tensor Some propagation effects have not analytic solutions. Different effects are coupled and not easy to be separated. Plasma properties Wave frequency Magnetic field

Mode amplitude evolution equation E+E+ E-E- Two elliptically polarized modes in the lab frame The ellipticity of the modes, 0 o or 90 o : linearly polarized; 45 o : circularly polarized The orientation of the modes, dominated by B field Mode amplitude evolution equation x y Y X B⊥B⊥ Adiabatic Condition i i

Our works on propagation effects Vacuum resonance (Wang & Lai 2007, MNRAS)Vacuum resonance (Wang & Lai 2007, MNRAS) Wave mode coupling (Wang, Lai & Han 2010, MNRAS)Wave mode coupling (Wang, Lai & Han 2010, MNRAS) Quasi-tangential effect (Wang & Lai 2009, MNRAS)Quasi-tangential effect (Wang & Lai 2009, MNRAS) Intrinsic Faraday effect in pulsar magnetosphereIntrinsic Faraday Rotation effect in pulsar magnetosphere (Wang, Han & Lai 2011, MNRAS) (Wang, Han & Lai 2011, MNRAS)

Vacuum resonance (the competition between plasma and QED effect) 1. Correction to dielectric tensor 2. Inverse permeability tensor X-ray band, Vacuum polarization (or QED effect) dominates the dispersion relationX-ray band, Vacuum polarization (or QED effect) dominates the dispersion relation Radio band, plasma effect dominatesRadio band, plasma effect dominates Where is the boundary? What happens in the boundary regime QED dominatedplasma dominated Vacuum resonance The two effects “cancled” with each other

Wave modes with QED effect “Avoid mode crossing” occurs, the two modes (O-mode & E-mode) coupled.“Avoid mode crossing” occurs, the two modes (O-mode & E-mode) coupled.  define two new modes:  define two new modes: “ ＋ ” mode “ ＋ ” mode “ － ” mode. “ － ” mode. O-mode X-mode

QED dominated plasma dominated Vacuum Resonance O-mode X-mode Mode conversion due to vacuum resonance Helicity unchanged

Results on Vacuum resonance If mode evolution across vacuum resonance is adiabatic, mode conversion (from O to X-mode or reversal) occurs. Location in dipole B fieldLocation in dipole B field vacuum resonance can occur for sufficiently high frequencies and strong surface magnetic fields. => high-frequency radio emission from the transient magnetar AXP XTE J1810−197 (Camilo et al. 2006, 2007; Kramer et al. 2007). optical radiation emitted from the NS surface or near vicinity may experience the vacuum resonance

Our works on propagation effects Vacuum resonance (Wang & Lai 2007, MNRAS)Vacuum resonance (Wang & Lai 2007, MNRAS) Wave mode coupling (Wang, Lai & Han 2010, MNRAS)Wave mode coupling (Wang, Lai & Han 2010, MNRAS) Quasi-tangential effect (Wang & Lai 2009, MNRAS)Quasi-tangential effect (Wang & Lai 2009, MNRAS) Intrinsic Faraday effect in pulsar magnetosphereIntrinsic Faraday Rotation effect in pulsar magnetosphere (Wang, Han & Lai 2011, MNRAS) (Wang, Han & Lai 2011, MNRAS)

Wave Mode Coupling The evolution of two linear eigenmodes from adiabatic to non- adiabatic.The evolution of two linear eigenmodes from adiabatic to non- adiabatic. r pl - polarization limiting radius, defined byr pl - polarization limiting radius, defined by –r << r pl, adiabatic mode evolution –r >> r pl, non-adiabatic mode evolution Before WMC, PA follows the B field line planeBefore WMC, PA follows the B field line plane After WMC, the polarization states are frozen After WMC, the polarization states are frozen Circular polarization generated because of mode coupling.Circular polarization generated because of mode coupling.

Single Photon evolution along the ray CP generated by wave mode coupling Cyclotron absorption

An Interesting Application for WMC Conal-double pulsars, PA increase  V < 0 PA decrease  V > 0 Can be explained easily by wave mode coupling effect CP generated by Wave mode coupling:

Our works on propagation effects Vacuum resonance (Wang & Lai 2007, MNRAS)Vacuum resonance (Wang & Lai 2007, MNRAS) Wave mode coupling (Wang, Lai & Han 2010, MNRAS)Wave mode coupling (Wang, Lai & Han 2010, MNRAS) Quasi-tangential effect (Wang & Lai 2009, MNRAS)Quasi-tangential effect (Wang & Lai 2009, MNRAS) Intrinsic Faraday effect in pulsar magnetosphereIntrinsic Faraday Rotation effect in pulsar magnetosphere (Wang, Han & Lai 2011, MNRAS) (Wang, Han & Lai 2011, MNRAS)

Quasi-Tangential Effect Tangential and Quasi-Tangential pointTangential and Quasi-Tangential point k is in the magnetic field line plane, there is a tangential point where θ kB =0 and B ⊥ change 180 o suddenly 。 k B BtBt k B BtBt Wang & Lai 2009, MNRAS k is not in the magnetic field line plane, there is a quasi-tangential point where θ kB reaches its minimum value (not 0) and B ⊥ change 180 o continuely. B⊥B⊥ B⊥B⊥

Ω μ k B Quasi-tangential point (a few R NS ) Hot spot X-ray emissoin

Sketch Map of the model evolution across QT point

Polarization State after QT effect for the X-ray emission from polar cap region （ Initially 100% O-mode LP ） Wang & Lai 2009, MNRAS WtWt

Polarization intensity (Q) changes due to Quasi-tangential effect （ total intensity does not change) Wang & Lai 2009, MNRAS

The phase evolution of the modification of linear polarization by the QT effect, B_surf = 10^13GB_surf = 10^14G Conclusion for the QT effect: the QT effect will have at most modest effect on the observed X-ray polarization signals from magnetized NSs. => linear polarization is weakened, LP profiles will be modified.

Our works on propagation effects Vacuum resonance (Wang & Lai 2007, MNRAS)Vacuum resonance (Wang & Lai 2007, MNRAS) Wave mode coupling (Wang, Lai & Han 2010, MNRAS)Wave mode coupling (Wang, Lai & Han 2010, MNRAS) Quasi-tangential effect (Wang & Lai 2009, MNRAS)Quasi-tangential effect (Wang & Lai 2009, MNRAS) Intrinsic Faraday Rotation effect in pulsar magnetosphereIntrinsic Faraday Rotation effect in pulsar magnetosphere (Wang, Han & Lai 2011, MNRAS) (Wang, Han & Lai 2011, MNRAS)

Intrinsic Faraday Rotation in pulsar magnetosphere Faraday rotation effect : two natural circular polarized modes have different phase velocities. FR of Pulsars in ISM (non-relativistic electrons, B~uG) is used to measure interstellar B field. RM = RM_ISM + RM_PSR Pulsar Magnetosphere – –Strong B field – –relativistic streaming plasma Δk =Δnω/c – Natural modes are linear polarized in inner magnetosphere and circularly polarized in outer magnetosphere –Δk no longer prop. to λ^2

Pair plasma case ， Ne ~ Np, Np–Ne = N GJ Pair plasma case, where FR effect is negligible LP CP FR effect negligible Pulsar parameters ： α=35 ， β=5 ， γ=100 ， η=100 ， Np-Ne=N GJ ， Bs=1e12G ， P=1s ， r_em=50Rs ， Ψi=0

Pure electrons case ， N = Ne = 1000 N GJ LP CP FR effect significant Pulsar parameters ： α=35 ， β=5 ， γ=100 ， η=1000 ， N=Ne ， Bs=1e12G ， P=1s ， r_em=50Rs ， Ψi=0 Pure electrons case, where FR effect is significant

k μ aligned μ k k μ inversely aligned k μ

Phased resolved RM profile Pulsar parameters ： α=35 ， γ=100 ， η=1000 ， Bs=5e12G ， P=1s ， r_em=50Rs

For symmetric pair plasma case (e.g. Goldreich-Julian model), intrinsic Faraday rotation in pulsar magnetosphere is negligible Only for the assumed highly asymmetric plasma (e.g., a electrons-ions streams with Ne >> N GJ ), FR maybe significant. FR angle is proportional to λ^~0.5, not 2 The intrinsic RM for mainpulse and interpulse should be opposite sign. Which may be checked in precise RM observations. Results on Intrinsic Faraday rotation effect

Our works Wave and modes amplitude evolution equationWave and modes amplitude evolution equation On some special propagation effectsOn some special propagation effects –Vacuum resonance (Wang, Lai & Han 2007) –Wave mode coupling (Wang, Lai & Han 2010) –Quasi-tangential effect (Wang & Lai 2009) –Intrinsic Faraday Rotation in magnetosphere (Wang, Han & Lai 2011) (Wang, Han & Lai 2011) Numerical calculations on Polarization profile changes considering all the propagation effects. (Wang, Lai & Han 2010)Numerical calculations on Polarization profile changes considering all the propagation effects. (Wang, Lai & Han 2010)

Numerical calculations on Polarization profile changes considering all the propagation effects. (Wang, Lai & Han 2010) Wave evolution equationWave evolution equationAssumptions: Photon emitted along the tangential direction of local B fieldPhoton emitted along the tangential direction of local B field Initially 100% linearly polarized, (generally O-mode).Initially 100% linearly polarized, (generally O-mode). Single gamma (cold plasma)Single gamma (cold plasma) Uniformly distributed plasma in the open field line region.Uniformly distributed plasma in the open field line region. all emissions are from the same heightall emissions are from the same height

Solid line (red): Dashed line (blue): Dotted line (green):

回旋吸收各相位不同 波模耦合产生足够强的圆偏振 偏振位置角曲线移动很大 正交模式现象 ? 经过磁层传播效应后的偏振轮廓 （一个例子，回旋吸收和波模耦合主导）

2D polarization profiles

Possible examples in observation: orthogonal modes

总结作用波段总强度线偏振圆偏振备注 绝热行走 Radio ～ X-ray 无有无影响偏振位置角 波模耦合 无有有产生圆偏振 回旋吸收 有有有吸收辐射 O 模折射 Radio无有有 分离 2 种波模 准切点效应 Radio ～ X-ray 无有无影响偏振谱 圆化效应Radio无有有产生圆偏振 真空极化效应 无有无可能导致正交模式 法拉第效应Radio无有无 产生本征 RM 传播效应对辐射尤其偏振有很大影响

很多脉冲星偏振观测现象可以由传播效应来解 释。 很多脉冲星偏振观测现象可以由传播效应来解 释。 – 锥双峰脉冲星中圆偏振的起源 – 部分正交模式现象 需要更详细模型下的理论计算 需要更详细模型下的理论计算 需要更多的与观测的结合 需要更多的与观测的结合总结

谢谢！

谢谢

观测事实

Ω μ k B Hot spot 辐射区 刚出大气层时热斑各 处偏振不一致 绝热行走使 各处偏振一致

中子星整个表面各处 准切点效应对 X-ray 线偏振度的影响 QT 效应影响区域

PA evolution along the way for different parameters

Dielectric tensor in pulsar magnetosphere Basic parameters: density. Using the Goldreich–Julian number density as a fiducial value Lorentz factor. here we consider cold plasma case with single cyclotron frequency and plasma frequency. and define

Dielectric tensor for an streaming electron-positron plasma Particle (electron and position) motion equationParticle (electron and position) motion equation Current densityCurrent density Dielectric tensorDielectric tensor Conductivity tensor

Dielectric tensor caused by plasma effect Wave modes in magnetosphere: dielectric tensor with B k θ z′z′ x′x′ Coordinate system x′y′z′

Dielectric tensor corrections due to Vacuum Polarization (QED effect) 1. Correction to dielectric tensor 2. Inverse permeability tensor For and

Subtract the influence to PA from other propagation effects Rotation Measure for the assumed pure electrons case RM defined by Pulsar parameters ： α=35 ， β=5 ， γ=100 ， η=1000 ， Bs=1e12G ， P=1s ， r_em=50Rs ， Ψi=0