Wave Modes in the Magnetospheres of Pulsars and Magnetars

We study the wave propagation modes in the relativistic streaming pair plasma of the magnetospheres of pulsars and magnetars, focusing on the effect of vacuum polarization. We show that the combined plasma and vacuum polarization effects give rise to a vacuum resonance, where ``avoided mode crossing'' occurs between the extraordinary mode and the (superluminous) ordinary mode. When a photon propagates from the vacuum-polarization-dominated region at small radii to the plasma-dominated region at large radii, its polarization state may undergo significant change across the vacuum resonance. We map out the parameter regimes (e.g., field strength, plasma density and Lorentz factor) under which the vacuum resonance occurs and examine how wave propagation is affected by the resonance. Some possible applications of our results are discussed, including high-frequency radio emission from pulsars and possibly magnetars, and optical/IR emission from neutron star surfaces and inner magnetospheres.


INTRODUCTION
The magnetospheres of pulsars and magnetars consist of relativistic electron-positron pair plasmas, plus possibly a small amount of ions. These plasmas can affect the radiation produced in the inner region of the magnetosphere or the stellar surface. Understanding the property of wave propagation in pulsar/magnetar magnetospheres is important for the interpretation of various observations of these objects.
Radio emission from pulsars (at least "normal"-non millisecond -pulsars) likely originates from close to the stellar surface, within a few percent of light cylinder radius (e.g., Blaskiewicz et al. 1991, Kramer et al. 1997. A number of studies have been devoted to the propagation effect of radio waves in pulsar magnetospheres (e.g., Cheng & Ruderman 1979; Barnard 1986;Lyubarskii & Petrova 1998;Melrose & Luo 2004;Petrova 2006). Some of the observed polarization properties of pulsar emission, such as orthogonal modes (in which the polarization position angle exhibits a sudden ∼ 90 • jumps; e.g. Stinebring et al. 1984aStinebring et al. , 1984b and circular polarization (e.g., Radhakrishnan & Rankin 1990;Han et al. 1998;You & Han 2006) may be explained by the propagation effect. In addition, optical/IR radiation may be produced in the inner magnetosphere or surface of magnetized neutron stars. For example, while for most radio pulsars the optical and near IR flux is thought to be dominated by magnetospheric emission, several middle-aged pulsars (PSR B0656+14, PSR B0950+08 and Geminga) also exhibit a surface optical component (e.g., Mignani, de Luca & Caraveo 2004;Kargaltsev et al. 2005, Mignani et al. 2006. The optical emission detected in several thermally emitting, isolated neutron stars mostly likely has a surface origin (e.g., Kaplan et al. 2003;Haberl et al. 2004;Haberl 2005;van KerKwijk & Kaplan 2006). Finally, the optical/IR emission detected from a number of magnetars may originate from a hot corona near the stellar surface (Beloborodov & Thompson 2006).
Wave modes in pulsar magnetospheres have been studied in a number of papers under different assumptions about the plasma composition and the velocity distribution of electron-position pairs (e.g., Melrose & Stoneham 1977;Lyutikov 1998;Melrose et al. 1999;Asseo & Riazuelo 2000). In this paper, we reinvestigate the property of wave propagation in the magnetospheres of pulsars and magnetars, focusing on the competition between the plasma effect and the effect of vacuum polarization. It is well known that in the strong magnetic field typically found on a neutron star, the electromagnetic dispersion relation is dominated by vacuum polarization (a prediction of quantum electrodynamics; e.g., Heisenberg & Euler 1936;Adler 1971; see Schubert 2000 for extensive bibliography) at high photon frequencies (e.g. X-rays) and by the plasma effect at sufficiently low frequencies (e.g., radio waves). But where is the "boundary" at which the two effects are "equal" and what are the mode proportion in the "boundary" regime? These are the questions we are trying to address in this paper. We show that the combined plasma and vacuum polarization effects give rise to a vacuum resonance: For a given plasma parameters and external magnetic field, there exists a special photon frequency at which the plasma effect and vacuum polarization effects "cancel" each other. A more physical way to describe the resonance is as follows: Consider a photon propagating in the inhomogeneous pulsar/magnetar magnetosphere with varying plasma density (and/or distribution function) and magnetic field. For certain parameter regimes of the photon frequency, plasma density and magnetic field strength -to be determined in the following sections, the photon may traverse from the vacuum-polarization-dominated region to the plasma-dominated region or vice versa. This transition point (location) is the vacuum resonance. When the photon crosses this resonance, its polarization state may undergo significant change. The goal of our paper is to map out the parameter regimes under which the vacuum resonance may occur and to elucidate how wave propagation may be affected by the resonance.
Vacuum resonance in cold, non-streaming plasmas have been studied before (e.g., Gnedin et al. 1978;Mészáros & Veutura 1979;Lai & Ho 2002, 2003a. In the atmospheres of highly magnetized neutron stars, the resonance can significantly affect the surface emission spectrum and polarization (Ho & Lai 2003;Lai & Ho 2003a, b;van Adelsberg & Lai 2006). We note that while some previous papers on wave modes in pulsar magnetospheres (e.g. ) did include the vacuum polarization contributions to the dielectric tensor, the vacuum resonance phenomenon was neglected because it is unimportant at the low frequencies and magnetic fields they considered.
Our paper is organized as follows. In §2, we give the expression for the dielectric tensor of a relativistic pair plasma charactering the magnetosphere of pulsars/magnetars, including the contribution due to vacuum polarization. In §3, we derive the general expression for wave modes in the combined "plasma + vacuum" medium, and show that the vacuum resonance arises for a wide range of magnetosphere parameters. In §4 we study the evolution of wave mode across the vacuum resonance. In most of this paper, we consider cold, streaming plasma with a single Lorentz factor γ. We examine the effect of a more general γ distribution in §5 and the case of opposite plasma streams in §6. In §7 we discuss possible applications of our results.

DIELECTRIC TENSOR FOR AN STREAMING ELECTRON-POSITRON PLASMA
We consider an electron-positron plasma in the magnetosphere of a neutron star (NS). Let N−, N+ be the number densities of electrons and positrons, respectively, N = N− + N+ the total density, and f = N+/N the positron fraction. The corotation region of the magnetosphere is usually assumed to have Goldreich-Julian charge density where Ω is the angular velocity of the star. This is not necessarily the case in the open-field line region. In this paper we shall use the Goldreich-Julian number density as a fiducial value: where B12 = B/(10 12 G), P1 is the spin period in units of 1 s. The actual particle density N is larger than NGJ by a factor of η, i.e. N = ηNGJ, with η 1. If the charge density is equal to the Goldreich-Julian value, then η(1 − 2f ) = 1, but we will not entirely restrict ourselves to this constraint. Although the plasma is expected to be relativistic, it is useful to define the (nonrelativistic) cyclotron frequencies and plasma frequencies of electron and position: Because of the very short cyclotron/synchrotron decay time of elections and positions (≈ 3 × 10 −16 B −2 12 γ sec) , all the particles in the magnetosphere quickly lose their transverse momenta and stay in the lowest Landau level. Thus the magnetosphere pair plasma can be considered as one-dimensional, with the particles streaming along the field line. The Lorentz factor γ of the streaming motion is uncertain. In the polar-cap region of a pulsar, primary particles may be accelerated to very high energy (γ ∼ 10 6 − 10 7 ) by a field-aligned electric field. The bulk of the plasma produced in an electromagnetic cascade may have lower energies, γ ∼ 10 2 − 10 4 , with multiplicity factor η ∼ 10 2 − 10 5 (e.g., Dangherty & Harding 1982;Hibschman & Arons 2001). Kunzl et al. (1998) argued that too high a density of secondary particles in the magnetosphere is in contradiction to the observed low-frequency emission from radio pulsars, implying η < ∼ 100. The physical parameters for the plasma in the closed-field-line region of a pulsar are also not well constrained. It was suggested that a pair plasma density larger than NGJ may be present, maintained by conversion of γ-rays from the pulsar's polar-cap and/or out-gap accelerators (see Wang et al. 1998;Ruderman 2003).
For magnetars, recent theoretical work suggests that a corona consisting mainly of relativistic pairs with γ ∼ 10 3 (and a wide spread in γ) may be generated by crustal magnetic field twisting/shearing due to starquakes (Thompson et al. 2002;Beloborodov & Thompson 2006). The plasma density is of order N ∼ |∇ × B|/(4πe) ∼ B/(4πer) (for a twist angle of order unity), implying η = N/NGJ ∼ c/(2Ωr) ≃ 2 × 10 3 (R/r) (where R is the stellar radius). There are roughly equal amount of electrons and positrons, f ≃ 1/2, with the electrons and positrons streams in opposite directions.

Cold, Streaming Pair Plasma
We first consider a cold electron-positron plasma with all the electron streaming with velocity V −,0 and positron all with V +,0 which is also along the magnetic filed B0. The dielectric tensor for such a plasma was derived by Melrose & Stoneham (1977) based on Lorentz transformation (see also Melrose 1973). Here we outline a derivation based on classical magneto-ionic theory.

Correction due to Vacuum Polarization
Vacuum polarization contributes a correction to the dielectric tensor: where I is the unit tensor andB = B/B is the unit vector along B (here we use B,B to denote B0,B0 for simple notations). The magnetic permeability tensor µ also deviates from unity because of vacuum polarization, with the inverse permeability given by In the low frequency limithω ≪ mec 2 , general expressions for the vacuum polarization coefficients a, q, and m are given in Adler (1971) and Heyl & Hernquist (1997). For B ≪ BQ = m 2 e c 3 /(eh) = 4.414 × 10 13 G, they are given by (2.35) and αF = e 2 /hc = 1/137 is the fine structure constant. For B ≫ BQ, simple expressions for a, q, m are given in Ho & Lai (2003) (see also Potekhin et al. 2004 for general fitting formulae). When |∆ǫ , the plasma and vacuum contributions to the dielectric tensor can be added linearly, i.e., ǫ = ǫ (p) + ∆ǫ (v) . In the frame withB alongẑ ′ , with S ′ = S +â, P ′ = P +â + q andâ = a − 1,

WAVE MODES IN A COLD STREAMING PLASMA
Here we consider the case of a pair plasma all streaming with the same velocity β along the field line. The effect of finite spread in γ will be studied in §5, and the case of opposite streams will be considered in §6.

Equations for the Wave Modes
Using the electric displacement D = ǫ · E and equation (2.33) in the Maxwell equations, we obtain the equation for plane waves with E ∝ e i(k·r−ωt) (henceforth we use E to denote δE, and use B to denote B0) where n = ck/ω is the refractive index andk = k/k. In the coordinate system xyz with k along the z-axis and B in the x-z plane, such thatk ×B = − sin θBŷ, the components of dielectric tensor are given by [compared to eq. (2.36)] ǫxx = S ′ cos 2 θB − 2A sin θB cos θB + P ′ sin 2 θB, ǫyy = S ′ , ǫzz = S ′ sin 2 θB + 2A sin θB cos θB + P ′ cos 2 θB, The z-component of equation (3.37) gives Reinserting this back into equation (3.37) yields where r = 1 +r ≡ 1 + (m/a) sin 2 θB and The above expressions are valid for the general dielectric tensor (Eq. 2.36). We now consider the B = ∞ limit introduced by e.g., Tsytovitch & Kaplan (1972) and . In this regime, the magnetic field is sufficiently large so that the cyclotron frequency are large compared to the Lorentz-shifted wave frequency. At the same time, B is not really infinity so that the wave propagation is dominated by plasma effect and we neglect the QED correction in the dielectric tensor. The approximate elements of the dielectric tensor are (where k = k cos θB, k ⊥ = k sin θB) for the ordinary mode (O-mode) and n 2 = 1 (3.51) for the extraordinary mode (X-mode). The polarization of the O-mode is given by   Thus, the X-mode is a transverse wave with the electric field vector in the k × B direction, while the O-mode is polarized in the plane spanned by k and B. Figures 1 -3 depict the refractive indices n = ck/ω of different modes as a function of (ωp/ω)γ −3/2 for γ = 1, 1.1 and 10 3 , respectively, all with θB = 30 • . The O-modes have two branches: the superluminous branch (w > ck or n < 1) and the subluminous branch (w < ck or n > 1), the latter corresponds to plasma oscillations. At low densities, (ωp/ω)γ −3/2 ≪ 1, the superluminous O-mode becomes transverse vacuum electromagnetic wave which can escape from the magnetosphere.
In the very low density region, (ωp/ω)γ −3/2 ≪ 1, the QED effect may not be neglected compared to the plasma effect. The "competition" between the vacuum polarization effect and the plasma effect gives rise to a vacuum resonance, at which the superluminous O-mode and the X-mode may be coupled with each other. In the remainder of this paper, we will focus on this vacuum resonance phenomenon.  Figure 2 except for γ = 10 3 . In the insert, the x-axis gives 10 5 (ωp/ω)γ −3/2 and the y-axis 10 8 (n − 1).

Vacuum Resonance
For |βp| ≫ 1, the two modes are (almost) linearly polarized: the mode with |K| ≃ 2|βp| ≫ 1 is polarized in thek-B plane, and is usually called ordinary mode (O-mode); the mode with |K| ≃ 1/(2|βp|) ≪ 1 is polarized perpendicular to thek-B plane, and is called extraordinary mode (X-mode). From equation (3.62), we see that for a general θB which is not too close to 0 o or 180 o , and for almost all values of B, N , ν, γ's, the inequality |βp| ≫ 1 is satisfied either when the condition fη f12 is satisfied, or when is satisfied. The exception occurs when fη + q + m = 0 (3.71) or βV = 1 − q + m vγ −3 (1 − β cos θB) −2 = 0. (3.72) This defines the "vacuum resonance". For given ν, γ and B, the resonance occurs at the density is equal to unity for b = B/BQ ≪ 1 and is at most of order a few for B < ∼ 10 15 G (see Fig. 1 of Ho & Lai 2003). It's obvious that NV ∝ γ 3 for γ ≫ 1 and θB not too close to 0 • (see Figure 4). For γ = 1, equation (3.73) agrees with the result of Lai & Ho (2002). The dependence of the resonance density NV on γ and θB can be easily understood from the cold (non-streaming) plasma limit and Lorentz transform. The wave freqency in the plasma rest frame is ω ′ = γ(ω − βk ) ≃ γω(1 − β cos θB). Note that in this frame, the external magnetic field and the dielectric tensor due to vacuum polarization are unchanged. The rest frame plasma density at the vacuum resonance is N ′ V ∝ B 2 ω ′2 F (b), and the corresponding "lab frame" density is NV = γN ′ V . We can rewrite the resonance density in terms of the Goldreich-Julian density The physical meaning of the resonance is clear: For given ν, γ and B, the dielectric property of the medium is dominated by the plasma effect when N ≫ NV , while it is dominated by vacuum polarization when N ≪ NV ; at N = NV , the plasma effect and vacuum polarization compensate each other, and the wave modes become exactly circular polarized. There are other ways to view the vacuum resonance. For example, at given B, γ and density N (or η), we can define the vacuum resonance frequency: Thus, wave modes with ν ≪ νV are determined by the plasma effect, while those with ν ≫ νV are determined by the vacuum polarization effect. The characteristic width of the resonance region can be estimated by considering |βp| = 1 as defining the edge of the resonance. Since βV = 1 − NV /N , we find that the densities at the edges of the resonance are NV ± ∆N , with where cos θB − ζ sin θB ≃ (cos θB − β)/(1 − β cos θ). Figures 4 and 5 show the mode properties near vacuum resonance for different values of γ and f . It's obvious that the resonance density and width scale with γ 3 (for γ ≫ 1), and the resonance density doesn't change with f , while the resonance region becomes narrow when f is close to 0.5. satisfied. If these "linear polarization" conditions are not satisfied, the modes will be approximately circular polarized even away from the resonance, and no dramatic change in the mode properties takes place around the vacuum resonance (see Fig. 6).

MODE EVOLUTION ACROSS THE VACUUM RESONANCE
Consider a photon (or electromagnetic wave) of a given frequency ν and polarization state propagating in the NS magnetosphere. The magnetosphere is inhomogeneous because of variations in B, N and possibly γ. How does the polarization of the photon evolve, particularly as the photon traverses the vacuum resonance region (e.g. from the plasma-dominated region to the vacuum-dominated region)? Clearly, if the variations of the magnetosphere parameters (B, N , etc.) are sufficiently gentle, the polarization state of the photon will evolve adiabatically, i.e. a photon in a definite wave mode will stay in that mode. Then Figure 4 and 5 show that across the vacuum resonance, the photon polarization ellipse will rotate by 90 o , with the mode helicity unchanged.
To quantify the mode evolution, it is convenient to introduce the "mixing" angle θm via tan θm = 1/K+, so that tan 2θm = β −1 p , (4.78) where we have used |r − 1| ≪ 1. The transverse eigenvectors of the modes are E+T = (i cos θm, sin θm) and E−T = (−i sin θm, cos θm). Clearly, at the resonance, θm = 45 • , the X-mode and O-mode are maximally "mixed". A general polarized electromagnetic wave with frequency ω traveling in the z-direction can be written as a superposition of the two modes: Note that both A± and E± depend on z. Substituting equation (4.79) into the wave equation where ′ stands for d/dz, ∆k = k+ − k−. In deriving equation (4.81), we have assumed that E±(z) and A±(z)exp −i z k±dz vary on a length scale much larger than the photon wavelength, and we have used k+ ≃ k− and |k ′ ± /k±| ≪ |k±|. Clearly, when |θ ′ m | ≪ |∆k/2|, or the polarization vector will evolve adiabatically (e.g., a photon in the plus-mode will remain in the plus-mode). Using equations (3.62) and (4.78), we find . (4.83) The difference in refractive indices of the two modes is (4.84) Thus equation (4.82) becomes (1 − β cos θB) 4 cos θB − ζ sin θB sin θB 2 ≫ 1, (4.85) where H ≡ |β0/β ′ p | specifies the length scale of variation of βp along the ray. Equation (4.85) gives the general condition for adiabatic mode evolution along the photon path.

THE INFLUENCE OF VELOCITY DISTRIBUTION
The results of Sections 3 and 4 are for cold streaming plasma with a single Lorentz factor γ. GHz.
(5.91) Figure 7 shows the polarization ellipticities and the refractive indices of the two wave modes in plasmas with different ∆γ's. Thus, a broad distribution of γ's does not qualitatively affect the vacuum resonance behavior.

DISCUSSION
In previous sections, we have studied the property of wave propagation in the magnetospheres of pulsars or magnetars for various plasma parameters. We have focused the vacuum resonance phenomenon, arising from the combined effects of plasma and vacuum polarization. The possible occurrence of the vacuum resonance and the related wave property depends on the plasma parameters, magnetic field and the wave frequency. The key equations are (assuming single-stream plasma): Note that at the vacuum resonance, the waves are always circular polarized. Figures 9 and 10 summarize these conditions for two different sets of parameters. In both figures, we see that when the vacuum resonance induces significant "avoided mode crossing" (cf. Figs. 4 and 5), i.e. when the resonance lies above the "Linear I" or "Linear II" line, wave evolution across the vacuum resonance is nonadiabatic. In general, this can be understood as follows. We define the cross frequency of the "Linear I" line and "Linear II" line (as well as "Vacuum Resonance" line) as vcross. We find For typical parameters(e.g., f = 0 − 0.5, η = 10 2 − 10 3 , P ∼1s, θ = 45 • , H6 = 1), νcross is less than the adiabatic frequency. Adiabatic mode evolution across the vacuum resonance with appreciable mode crossing is possible for larger η and H6. We now discuss possible implications of our results for various radiation processes in pulsars and magnetars. We assume a dipole magnetic field, with  .87)] means that the adiabatic condition is satisfied to the right of the line when the wave evolves across the resonance. The line labeled "ω ′ ≪ ωc" [Eq. (3.59)] means that the Doppler-shifted frequency is much less than the electron cyclotron frequency above the line, and the line labeled "weak dispersion" [Eq. (3.60)] means that the medium is weakly dispersive (i.e., index of refraction close to unity) below the line; Our analytical expression for the wave modes and vacuum resonance are valid in this parameter regime (below the "weak dispersion" line and above the "ω ′ ≪ ωc" line). Above the "Linear I" line [Eq. Since the dispersion due to vacuum polarization is of order q + m ∝ F (b)B 2 ∝ r −6 , while the plasma effect is measured by ∼ vγ −3 ∝ N γ −3 ∝ ηBγ −3 ∝ ηγ −3 r −3 , if ηγ −3 does not vary rapidly, we find that for a given photon frequency ν, the wave dispersion is dominated by the vacuum effect for r < ∼ rV and by the plasma effect for r > ∼ rV . First consider the radio emission from the open field line region of a pulsar. The emission angle relative the local magnetic field line is θB ∼ 1/γ, so that 1 − β cos θB ≃ γ −2 . This would imply rV /R * ≪ 1, even for B * ∼ 10 15 G and for high frequencies (e.g. ν = 20 GHz). Along the ray trajectory, the angle θB increases. In the small angle approximation (θB ≪ 1), we have θB ≈ 3 4 rem R * θ0 1 − rem r , (7.109) where rem is the radius of the emission point, and θ0 is the polar angle at the stellar surface of the emission field line. Thus θB increases from 0 (at r = rem) to (3/4)(rem/R * )θ0. As an example, for rem = 2R * and θ0 ∼ R * /RLC ≃ 0.0145P This means that for radio emission along open, dipole field lines, plasma effects always dominate the property of wave propagation and vacuum resonance will not occur. Radio emission may also come from the large-curvature magnetic field structure (e.g., field lines with curvature radius ∼ R * ). In this case, even if θB ≪ 1 at emission, it will become significantly large (∼ 45 • ) after the wave propagates a short distance of order R * . Thus, according to Eq. (7.108), vacuum resonance can occur for sufficiently high frequencies and strong surface magnetic fields. This could be the case with the high-frequency radio emission from the transient AXP XTE J1810-197 (Camilo et al. 2006).
Finally, optical/IR radiation emitted from the neutron star surface or near vicinity may experience the vacuum resonance while propagating through the magnetosphere. The polarization of such radiation may probe the physical conditions of the magnetosphere.