P H Y S I C A L R E V I E W L E T T E R S week ending15 AUGUST 2003VOLUME 91, NUMBER 7

Polarized X-Ray Emission from Magnetized Neutron Stars:Signature of Strong-Field Vacuum Polarization

Dong Lai and Wynn C. G. HoCenter for Radiophysics and Space Research, Department of Astronomy, Cornell University, Ithaca, New York 14853, USA

(Received 27 March 2003; published 13 August 2003)

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In the atmospheric plasma of a strongly magnetized neutron star, vacuum polarization can induce aMikheyev-Smirnov-Wolfenstein type resonance across which an x-ray photon may (depending on itsenergy) convert from one mode into the other, with significant changes in opacities and polarizations.We show that this vacuum resonance effect gives rise to a unique energy-dependent polarizationsignature in the surface emission from neutron stars. The detection of polarized x rays from neutronstars can provide a direct probe of strong-field quantum electrodynamics and constrain the neutron starmagnetic field and geometry.

DOI: 10.1103/PhysRevLett.91.071101 PACS numbers: 97.60.Jd, 95.30.Gv, 95.85.Nv, 98.70.Qy

namics (QED) that in a strong magnetic field the vacuumbecomes birefringent [9–13]. Acting by itself, the bi- where ui � �EBi=E�2 and H� � jdz=d ln�j is the density

Surface emission from isolated neutron stars (radiopulsars and radio quiet neutron stars, including magne-tars [1]) provides a useful probe of the neutron star (NS)interior physics, surface magnetic fields, and composition.The advent of x-ray telescopes in recent years has madethe detection and detailed study of NS surface emission areality [2]. On the other hand, x-ray emission from accret-ing x-ray pulsars has long yielded information on thedynamics of accretion and radiative mechanisms in thestrong gravity, strong magnetic field regime [3].

In the magnetized plasma that characterizes NS atmos-pheres, x-ray photons propagate in two normal modes: theordinary mode (O mode) is mostly polarized parallel tothe k-B plane, while the extraordinary mode (X mode) ismostly polarized perpendicular to the k-B plane, wherek is the photon wave vector and B is the external mag-netic field [3]. This description of normal modes appliesunder typical conditions, when the photon energy E ismuch less than the electron cyclotron energy EBe ��heB=�mec� � 11:6B12 keV [where B12 � B=�1012 G�], Eis not too close to the ion cyclotron energy EBi �6:3B12�Z=A� eV (where Z and A are the charge numberand mass number of the ion), the plasma density is not tooclose to the vacuum resonance (see below), and �kB (theangle between k and B) is not close to zero. Under theseconditions, the X mode opacity (due to scattering andabsorption) is greatly suppressed compared to the Omode opacity, �X � �E=EBe�

2�O [3,4]. As a result, the Xmode photons escape from deeper, hotter layers of the NSatmosphere than the O mode photons, and the emergentradiation is linearly polarized to a high degree (as high as100%) [5–7]. Measurements of x-ray polarization, par-ticularly when phase resolved and measured in differentenergy bands, could provide unique constraints on themagnetic field strength and geometry and the compact-ness of the NS [6–8].

It has long been predicted from quantum electrody-

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refringence from vacuum polarization is significant(with the index of refraction differing from unity bymore than 10%) only for B * 300BQ, where BQ �m2ec3=�e �h� � 4:414� 1013 G is the critical QED fieldstrength. However, when combined with the birefringencedue to the magnetized plasma, vacuum polarization cangreatly affect radiative transfer at much smaller fieldstrengths. A ‘‘vacuum resonance’’ arises when the con-tributions from the plasma and vacuum polarization to thedielectric tensor ‘‘compensate’’ each other [14–17]. For aphoton of energy E, the vacuum resonance occurs at thedensity �V ’ 9:64� 10�5Y�1

e B212E21f

�2 g cm�3, whereYe is the electron fraction, E1 � E=�1 keV�, and f �f�B� is a slowly varying function of B and is of orderunity (f � 1 for B � BQ and f ! �B=5BQ�

1=2 for B BQ; see Refs. [17,18]). For � > �V (where the plasmaeffect dominates the dielectric tensor) and � < �V (wherevacuum polarization dominates), the photon modes (forE � EBe, E � EBi and �kB � 0) are almost linearly po-larized; near � � �V , however, the normal modes becomecircularly polarized as a result of the ‘‘cancellation’’ ofthe plasma and vacuum effects — both effects tend tomake the mode linearly polarized, but in mutually or-thogonal directions. When a photon propagates in aninhomogeneous medium, its polarization state willevolve adiabatically (i.e., following the K� or K� curvein Fig. 1) if the density variation is sufficiently gentle.Thus, a X mode (O mode) photon will be converted intothe O mode (X mode) as it traverses the vacuum reso-nance, with its polarization ellipse rotated by 90� (Fig. 1).This resonant mode conversion is analogous to theMikheyev-Smirnov-Wolfenstein neutrino oscillation thattakes place in the Sun [19,20]. For this conversion to beeffective, the adiabatic condition must be satisfied [17,21](cf. [16])

E * Ead � 2:52�f tan�kBj1�uij�2=3

�1 cm

H�

�1=3keV; (1)

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VacuumResonance

O−modePhotosphere

X−modePhotosphere

O−mode

X−modeX−mode

X−mode X−mode

O−modeO−mode

O−mode

E = 1 keV E = 5 keV

FIG. 2 (color online). A schematic diagram illustrating howvacuum polarization affects the polarization state of the emer-gent radiation from a magnetized NS atmosphere. This diagramapplies to the ‘‘normal’’ field regime [B & 7� 1013 G; seeEq. (3)] in which the vacuum resonance lies outside the photo-spheres of the two photon modes. The photosphere is definedwhere the optical depth (measured from outside) is 2=3 and iswhere the photon decouples from the matter. At low energies(such as E & 1 keV), the photon evolves nonadiabaticallyacross the vacuum resonance (for �kB not too close to 0), andthus the emergent radiation is dominated by the X mode. Athigh energies (E * 4 keV), the photon evolves adiabatically,with its plane of polarization rotating by 90� across the vacuumresonance, and thus the emergent radiation is dominated by theO mode. The plane of linear polarization at low energies istherefore perpendicular to that at high energies.

FIG. 1 (color online). The polarization ellipticity of the pho-ton mode as a function of density near the vacuum resonance.The two curves correspond to the two different modes. In thisexample, the parameters are B � 1013 G, E � 5 keV, Ye � 1,and �kB � 45�. The ellipticity of a mode is specified by theratio K � �iEx=Ey, where Ex (Ey) is the photon’s electric fieldcomponent along (perpendicular to) the k-B plane. The Omode is characterized by jKj 1, and the X mode jKj � 1.

P H Y S I C A L R E V I E W L E T T E R S week ending15 AUGUST 2003VOLUME 91, NUMBER 7

scale height (evaluated at � � �V) along the ray. For anionized hydrogen atmosphere, H� ’ 2kT=�mpg cos�� �1:65T6=�g14 cos�� cm, where T � 106T6 K is the tem-perature, g � 1014g14 cm s

�2 is the gravitational accel-eration, and � is the angle between the ray and the surfacenormal. In general, the mode conversion probability isgiven by [17,21]

Pcon � 1� exp���"=2��E=Ead�3�: (2)

The probability for a nonadiabatic ‘‘jump’’ is �1� Pcon�.Because the two photon modes have vastly different

opacities, the vacuum resonance can significantly affectthe transfer of photons in NS atmospheres. When thevacuum polarization effect is neglected, the decouplingdensities of the O mode and X mode photons (i.e., thedensities of their respective photospheres, where the op-tical depth measured from outside is 2=3) are approxi-mately given by (for hydrogen plasma and �kB not tooclose to 0) �O ’ 0:42T�1=4

6 E3=21 G�1=2 g cm�3 and �X ’

486T�1=46 E1=21 B14G

�1=2 g cm�3, where G � 1� e�E=kT

[17]. There are two different magnetic field regimes:For normal magnetic fields,

B< Bl ’ 6:6� 1013T�1=86 E�1=4

1 G�1=4 G; (3)

the vacuum resonance lies outside both photospheres(�V < �O; �X); for the magnetar field regime, B > Bl,the vacuum resonance lies between these two photo-spheres, i.e., �O < �V < �X (the condition �V < �X issatisfied for all field strengths and relevant energies andtemperatures). These two field regimes yield qualitativelydifferent x-ray polarization signals.

Consider the normal field strengths, 1012 G & B & Bl,which apply to most NSs (see Fig. 2). In this regime, theatmosphere structure and total spectrum can be calcu-

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lated without including vacuum polarization [to be moreaccurate, we require �B=Bl�

4 � 1 for this to be valid]. Forconcreteness, we consider emission from a hot spot (mag-netic polar cap) on the NS; the magnetic field at the hotspot (with size much smaller than the stellar radius) isperpendicular to the stellar surface. Let the specific in-tensities of the O mode and X mode emerging from theirrespective photospheres (which lie below the vacuumresonance) be I�0�O and I�0�X , which we calculate using ourH atmosphere models developed previously [18,22]. Fora given B and Teff , both I�0�O and I�0�X depend on E and �kBat emission (the hot spot). As the radiation crosses thevacuum resonance, the intensities of the O mode and Xmode become IO � �1� Pcon�I

�0�O � PconI

�0�X and IX �

�1� Pcon�I�0�X � PconI

�0�O . In calculating Pcon, we use the

temperature profile of the atmosphere model to determinethe density scale height at the vacuum resonance. We notethat in principle, circular polarization can be producedwhen a photon crosses the vacuum resonance [21], but thenet circular polarization is expected to be zero whenphotons from a finite-sized polar cap are taken intoaccount.

To determine the observed polarization, we mustconsider propagation of polarized radiation in the NS

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P H Y S I C A L R E V I E W L E T T E R S week ending15 AUGUST 2003VOLUME 91, NUMBER 7

magnetosphere, whose dielectric property in the x-rayband is dominated by vacuum polarization [8]. As aphoton (of a given mode) propagates from the polar capthrough the magnetosphere, its polarization state evolvesadiabatically following the varying magnetic field it ex-periences, up to the ‘‘polarization-limiting radius’’ rpl,beyond which the polarization state is frozen. We setup a fixed coordinate system XYZ, where the Z axis isalong the line of sight and the X axis lies in the planespanned by the Z axis and � (the spin angular velocityvector). The direction of the magnetic field B follow-ing the photon trajectory in this coordinate system isspecified by the polar angle �B�s� and the azimuthalangle *B�s� (where s is the affine parameter along theray). The X; Y components of the electric field of thephoton mode are �EX; EY� � �cos*B; sin*B� for the Omode and �� sin*B; cos*B� for the X mode. The adia-batic condition requires jnX � nOj * 2� �hc=E�jd*B=dsj(where nX; nO are the indices of refraction of the twomodes); this condition breaks down at r � rpl, which ismuch greater than the stellar radius for all parame-

FIG. 3 (color online). The phase evolution of the observedspectral flux (upper panel, in arbitrary units), the linear po-larization FQ=FI (middle panel), and the polarization degreePL � �F2Q � P2U�

1=2=FI (lower panel) produced by the mag-netic polar cap of a rotating NS. The model parameters are themagnetic field B � 1013 G, the effective temperature of thepolar cap Teff � 5� 106 K, the angle between the spin axis andthe line of sight 0 � 30�, and the angle between the magneticaxis and spin axis / � 70�. The different curves are fordifferent photon energies as labeled. In the upper panel, theflux at 5 keV has been multiplied by 10 relative to the othercurves. The definition of the Stokes parameter is such thatFQ=FI � 1 corresponds to linear polarization in the planespanned by the line-of-sight vector and the star’s spin axis(the zero phase corresponds to the polar cap in the same plane).Note that the polarization plane of high-energy (E * 4 keV)photons is perpendicular to that of low-energy photons (E &

1 keV). Also note that for E around Ead, the polarization degreePL can be zero at certain phases such that Pcon � 1=2.

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ter regimes of interest in this paper. The observedStokes parameters (normalized to the total inten-sity I) are given by Q=I � �2Pcon � 1�pe cos2*B�rpl�and U=I � �2Pcon � 1�pe sin2*B�rpl�, where pe �

�I�0�X � I�0�O �=�I�0�X � I�0�O � is the ‘‘intrinsic’’ polarizationfraction at emission, and *B�rpl� is the azimuthal angleof the magnetic field at r � rpl [for rpl � c= or for spinfrequency � 70 Hz, *B�rpl� ’ "�*B�R�]. We calculatethe observed spectral fluxes FI; FQ; FU (associated withthe intensities I; Q;U) using the standard procedure, in-cluding the effect of general relativity [23].

Figure 3 shows the total flux and polarization ‘‘lightcurves’’ generated from the hot polar cap of a rotating NS.As the star rotates, the plane of linear polarization ro-tates. The light curves obviously depend on the geometry(specified by the angles / and 0). Note that FQ for low-energy (E & 1 keV) photons is opposite to that for high-energy photons (E * 4 keV), which implies that theplanes of linear polarization at low and high energiesare orthogonal. This feature also manifests in the phase-averaged linear polarization (see Fig. 4). This is a uniquesignature of photon mode conversion induced by vacuumpolarization.

In the magnetar field regime, B > Bl, the vacuumresonance lies between the photospheres of the twomodes. At low energies (e.g., E & 1 keV), no mode con-version occurs at the vacuum resonance, and vacuumpolarization makes the X mode photosphere lie above(i.e., at lower density) its ‘‘original’’ location (i.e., whenvacuum polarization is turned off) because the photonopacity exhibits a large spike at the vacuum resonance[17]. At high energies (e.g., E * 4 keV), the effective Xmode photosphere lies very near the vacuum resonance.For both low and high energies, the emergent radiationis dominated by the X mode (for �kB not too close to 0).

FIG. 4 (color online). The phase-averaged linear polarizationas a function of photon energy. The model parameters are B �1013 G and Teff � 5� 10

6 K, and the three curves correspondto three different sets of angles (0; /) as labeled. The averagedhFUi � 0. The dashed curves depict the results when thevacuum polarization effect is turned off.

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FIG. 5 (color online). Same as Fig. 3, except for B � 5�1014 G and Teff � 5� 10

6 K. The different curves are fordifferent photon energies (1 keV dashed, 2 keV long-dashed,5 keVsolid lines). The thicker curves are for 0 � 30�, / � 70�,and the thin curves for 0 � 60�, / � 40�. In contrast to Fig. 3and 4, in the magnetar field regime, the planes of linear polar-ization at different photon energies coincide with each other.

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Therefore the planes of linear polarization at differ-ent energies coincide and evolve ‘‘in phase’’ as the starrotates (see Fig. 5). These polarization signals are quali-tatively different from the normal field regime. Note thatin the magnetar field regime, vacuum polarization sig-nificantly affects the total spectral flux from the atmo-sphere [17,18,21]: (i) Vacuum polarization makes theeffective decoupling density of X mode photons (whichcarry the bulk of the thermal energy) smaller, therebydepleting the high-energy tail of the spectrum and mak-ing the spectrum closer to blackbody; (ii) vacuum polar-ization suppresses the proton cyclotron line and otherspectral line features [24] in the spectra by making thedecoupling depths inside and outside the line similar. Aswe discussed in previous papers [18,24], the absence oflines in the observed thermal spectra of several magnetarcandidates [25] may be considered as a proof of thevacuum polarization effect at work in these systems.Measurement of x-ray polarization would provide anindependent probe of the magnetic fields of these objects.

Finally, we note that although the specific resultsshown in Figs. 3–5 refer to emission from a hot polarcap on the NS, we expect the vacuum polarization sig-nature (e.g., that the planes of linear polarization at E &

1 keV and at E * 4 keV are orthogonal for B &

7� 1013 G) to be present in more complicated models(e.g., when several hot spots or the whole stellar surfacecontribute to the x-ray emission). This is because the

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polarization-limiting radius (due to vacuum polarizationin the magnetosphere) lies far away from the star (seeabove), where rays originating from different patches ofthe star experience the same dipole field [8]. Our resultstherefore demonstrate the unique potential of x-ray polar-imetry [26] in probing the physics under extreme condi-tions (strong gravity and magnetic fields) and the natureof various forms of NSs.

This work was supported by NASA Grant No. NAG 5-12034 and NSF Grant No. AST 0307252.

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