Do pulsars rotate clockwise or counterclockwise?
Renaud Gueroult, Yuan Shi, Jean-Marcel Rax, Nathaniel J. Fisch
aa r X i v : . [ a s t r o - ph . I M ] M a r Do pulsars rotate clockwise or counterclockwise?
Renaud Gueroult
LAPLACE, Universit´e de Toulouse, CNRS, INPT, UPS, 31062 Toulouse, France
Yuan Shi
Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
Jean-Marcel Rax
Universit´e de Paris XI - Ecole Polytechnique, LOA-ENSTA-CNRS, 91128 Palaiseau, France
Nathaniel J. Fisch
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08540, USA (Dated: March 5, 2019)Pulsars are rotating neutron stars which emit lighthouse-like beams. Owing to their unique prop-erties, pulsars are a unique astrophysical tool to test general relativity, inform on matter at extremedensities, and probe galactic magnetic fields. Understanding pulsars physics and emission mech-anisms is critical to these applications. Here we uncover that mechanical-optical rotation in thepulsars’ magnetosphere affects polarisation in a way which is indiscernible from Faraday rotation inthe interstellar medium for typical GHz observations frequency, but which can be distinguished inthe sub-GHz band. Besides being essential to correct for possible systematic errors in interstellarmagnetic field estimates, our novel interpretation of pulsar polarimetry data offers a unique means todetermine whether pulsars rotate clockwise or counterclockwise, providing new constraints on mag-netospheric physics and possible emission mechanisms. Combined with the ongoing development ofsub-GHz observation capabilities, our finding promises new discoveries, such as the spatial distri-butions of clockwise rotating or counterclockwise rotating pulsars, which could exhibit potentiallyinteresting, but presently invisible, correlations or features.
Pulsars are strongly magnetized rotating neutron stars.Because of rotation, pulsars emit two intense radiationbeams [1]. For a distant observer, emission appears as apulse each time the beam sweeps across his line-of-sight.Owing to their unique properties, pulsars have played,and continue to play, a critical role in the developmentof astronomy and astrophysics. For instance, pulsars’extreme density makes them one-of-a-kind tools to testboth the equation of state of superdense matter [2] andthe theory of general relativity in the strong field limit [3–6], while their unparalleled emission stability could allowdetecting nanohertz gravitational waves [7]. Millisecondpulsars also enabled the first detection of an extra-solarplanetary system [8].Pulsars’ highly polarised emission and compactnessalso make them unmatched sources to probe the mag-netic fields through Faraday rotation [9], and pulsars havebeen instrumental in mapping magnetic field propertiesin the interstellar medium (ISM) of the Milky Way [10–12]. These studies generally rely on the assumption thatpolarisation rotation ∆ φ results only from the Faradayeffect experienced in the non-moving magnetised plasmabetween the source and the observer. For wave frequency ω much greater than the plasma frequency ω pe , such asradio-waves in the ISM (see Table I), one can then showthat ∆ φ F = RM λ , with λ the vacuum wavelength. In-formation on the magnetic field orientation and strengthalong the line of sight is then derived from the propor-tionality coefficient RM , called the rotation measure. However, pulsars are surrounded by a magnetosphere.Although pulsar magnetospheric physics, and with itthe mechanism responsible for pulsars’ emission, remainspoorly understood [13, 14], it is widely accepted thatthe magnetosphere is populated by relativistic electron-positron pairs, and that it, or at least its inner region,co-rotates with the neutron star. The analysis of pul-sar’s signal should hence in principle not only account forpropagation in the interstellar medium between the pul-sar and the observer (between points Q and R in Fig. 1),but also for propagation in the rotating magnetosphere(between points P and Q in Fig. 1). In particular, pulsarpolarimetry ought to consider both the the well knownFaraday rotation induced by intervening plasma screensand the possible polarisation rotation in the magneto-sphere [15]. TABLE I. Typical plasma parameters in the interstellarmedium and in pulsars’ magnetosphere. Rotation measure RM are typically observed at ω ∼ ω pe and ω ce arethe plasma and electron cyclotron frequency, respectively. B [T] n [m − ] Ω [s − ] ω pe /ω ω ce /ω Interstellar medium 10 − − − − Pulsar magnetosphere 10
10 10 FIG. 1. Illustration of the different contributions to pulsars’emission polarisation rotation. Polarisation rotation is typi-cally assumed to stem from Faraday rotation between Q and R . But wave polarisation also contains information on themagnetosphere properties between points P and Q , and inparticular on the magnetosphere rotation Ω. RESULTS
The effect of the rotating magnetosphere on the po-larisation rotation can be both significant and revealing.First of all, any deduction of the intervening magneticfield between the pulsar and the observer through Fara-day rotation will have to be corrected for the additionalpolarisation rotation. Second of all, significant new infor-mation can be obtained about the pulsar. For example,absent accounting for the effect of the rotating magne-tosphere, the observation of a pulsar from a single dis-tant point will uncover the pulsar rotational frequency,but not whether it is rotating clockwise or counterclock-wise. However, as we show here, because the wavelengthdependency of the polarisation rotation due to rotatingmagnetosphere differs from that due to Faraday rotation,these rotation directions can in fact be disambiguatedeven when observed from a single distant point. This isimportant because the sign of Ω · B constrains possiblemagnetosphere compositions (electron-positron only oralso proton) and particle acceleration mechanisms, andin turn, possible radio emission mechanisms [13]. Polarisation in rotating gyrotropic media.
To seethis, consider for simplicity the propagation of a wave along the axis of an aligned rotator, that is to say whenthe obliquity α = 0 in Fig. 1 (i.e., the rotation and mag-netic axes are aligned). For perfect alignment of the axes( α = 0 ), the radiation pulsing vanishes, but the key ef-fects are retained. This case is also important since thebeam axis tends to align with the rotational axis ( α → φ = ∆ φ F + ∆ φ M (Ω) . (1)The first term ∆ φ F is the classical Faraday rotationwhich occurs in a stationary gyrotropic medium. Thesecond term ∆ φ M (Ω) stems from the medium’s rotationat frequency Ω, and is referred to as mechanical-opticalrotation (MOR) [17, 18]. Information on Ω is thereforeimprinted in wave polarisation. This result is expectedto hold when the wave propagates along the magneticfield ( k k B ), even if the mechanical and magnetic axesare only nearly aligned. Rotation can thus in principlebe retrieved from ∆ φ M in pulsars’ pulsating signal. Mechanical-optical rotation in e − p magneto-sphere. In a rotating magnetised plasma, the combinedeffects of Faraday rotation and MOR makes eliciting theeffect of mechanical rotation difficult. Yet, it happensthat Faraday rotation cancels in the particular case ofa cold electron-positron ( e − p ) plasma symmetrical indensity ( n e = n p = n ). We thus take advantage of thiscoincidence to shed light onto how polarisation may beaffected as a wave propagates in the rotating magneto-sphere.In an e − p plasma rotating at Ω >
0, we show (seeMethods) that the left-handed circularly polarised (LCP)wave only propagates above a cut-off frequency ω lc whichdepends on Ω and the e − p plasma density n in the mag-netosphere. Above this cut-off frequency, both LCP andRCP waves propagate, and the difference in wave index∆ n = n l − n r introduced by mechanical rotation leads toMOR. Importantly, ω lc ∼ s − for plasma parameterstypical of pulsar magnetospheres (see Table I), which ison the lower end of the frequency range used for pulsarpolarimetry (typically GHz) [19]. This implies that MORshould be present in a large fraction of pulsar polarisationdata.For frequencies at least a few times ω lc , we see (seeFig. 2) that ∆ n ( ω ) ∝ ω − , and therefore ∆ φ M ∝ ω − .It follows that MOR in the rotating e − p magnetosphereis a priori indiscernible from Faraday rotation in the in-tervening interstellar medium since both contributionsare proportional to λ . When fitting observations usingthe relation ∆ φ = RM λ , the rotation measure RM inthis frequency range not only portrays Faraday rotationbut also any possible MOR. Attributing RM to the effectof magnetic fields in the interstellar medium alone, as isoften done in pulsar polarimetry, thus risks systematicerrors. Quantitatively, the magnetic field strength along FIG. 2. Mechanical contribution to polarisation rotation ina rotating symmetrical e − p plasma. δ M = ∆ φ M /l is themechanical-optical rotation (MOR) per unit length. Wellabove the cut-off ω lc , MOR scales like ω − , similarly toFaraday rotation. In a limited frequency band above ω lc ([ ω/ω lc − ≪ B = 10 T, n = 10 m − and Ω = 10 s − , ω lc ∼
185 MHz. the line of sight will be respectively over- and under-estimated for negative and positive rotation of the sourcepulsar since Faraday rotation and MOR are in oppositedirections for Ω · B > Low-frequency wavelength-scaling deviation.
However, we uncover here that the peculiar behaviourof MOR near the LCP cut-off may retire this apparentambiguity. As illustrated in Fig. 2, mechanical-opticalrotation features a different wavelength scaling for ν = ω − ω lc ≪ ω lc . In this frequency band, ∆ n increasesas p ν/ω lc (∆ n < > φ PQ ( ω ) between point P and Q in Fig. 1 thenscales like ω √ ω − ω lc . In contrast, for the same wavefrequency, ∆ φ QR ( ω ) due to Faraday rotation is verywell approximated by the classical ω − law since ω lc is over 1000 times larger than ω pe in the interstellarmedium (see Table I). Combined with the jump inpolarisation angle which is predicted to take place at thecut-off frequency, these different frequency scalings offera conceptual means to separate MOR in the rotating e − p magnetosphere from Faraday rotation in theinterstellar medium. MOR could then provide insightsinto the pulsar magnetosphere dynamics. In particular,a negative deviation from the λ high-frequency fit abovethe cut-off will indicate a counter-clockwise rotation ofthe pulsar magnetosphere, while a positive deviationwill indicate clockwise rotation. DISCUSSION
While this symmetrical e − p plasma magnetospheremodel is overly simplified, we show that accounting forrelativistic-quantum effects and density asymmetry inthe magnetosphere does not qualitatively modify this pic-ture (see Methods). It suggests that while the detailedfrequency dependence ∆ φ ( ω ) near the cut-off changes asthe magnetosphere model is refined, the existence of a fre-quency cut-off and the associated non λ behaviour nearthis cut-off are robust features of mechanical rotation.Our finding that MOR and Faraday rotation show dif-ferent frequency dependences near the cut-off is partic-ularly relevant in light of the recent observations thatpolarisation rotation in certain pulsars does not fol-low the classical λ law [20], even at higher frequency f ≥
730 MHz. If these observed deviations were at-tributable to MOR, it stands to reason that observationsat lower frequency (closer to the predicted ω lc ) will re-veal a greater fraction of pulsars displaying a non λ relation. The ongoing development of observation ca-pabilities at radio frequencies below 250 MHz for grav-itational waves detection with pulsar timing arrays [7]should enable confirming this conjecture by measuringthe frequency dependence of pulsars’ polarisation closerto the predicted cut-off frequency [21, 22]. If success-ful, this would provide unique means to advance our un-derstanding of the magnetospheric structure and pulsarradio emission mechanism. Finally, if observed devia-tions were indeed traceable to MOR, polarisation dataobtained at shorter wavelengths where ∆ φ ∝ λ holdstrue may have to be revisited to correct magnetic fieldestimates in light of the mechanical rotation contribution. METHODS
Polarisation rotation in gyrotropic media.
Con-sider a typical magneto-optic medium described by thesusceptibility tensor ¯ χ (¯ ω ) = ¯ χ ⊥ − j ¯ χ × j ¯ χ × ¯ χ ⊥
00 0 ¯ χ k . (2)The magneto-optical activity translates into right-handed circularly polarised (RCP) and a left-handed cir-cularly polarised (LCP) eigenmodes propagating along ˆz ,with indices ¯ n r = (1 + ¯ χ ⊥ + ¯ χ × ) / and ¯ n l = (1 + ¯ χ ⊥ − ¯ χ × ) / , respectively. Here left- and right-handed wavesare defined from the point of view of the source in thedirection of propagation of the wave.The difference in wave index n r and n l of RCP andLCP waves associated with the non zero off-diagonalterm ¯ χ × leads to a rotation of the plane of polarisationof a linearly polarised wave. After propagating over adistance l , the polarisation has been rotated by∆ φ ( ω ) = [ n l ( ω ) − n r ( ω )] ωl c . (3)The polarization rotation per unit length, also known asthe specific rotary power, is δ ( ω ) = ∆ φ ( ω ) /l .A magnetised plasma can be considered as ananisotropic dielectric. Writing the background magneticfield B = B ˆz and assuming a cold and collisionlessplasma, the components of the susceptibility tensor inthe plasma rest frame are [23]¯ χ ⊥ ( ω ) = X α ω pα ω cα − ω (4a)¯ χ × ( ω ) = X α ǫ α ω cα ω ω pα ω − ω cα (4b)¯ χ k ( ω ) = − X α ω pα ω , (4c)where ω cα = | q α | B /m α and ω pα = [ n α e / ( m α ǫ )] / are the cyclotron frequency and plasma frequency ofspecies α , respectively, and ǫ α = q α / | q α | .Typically, plasma parameters in the Faraday screen inbetween the pulsar and the observer are such that ω cα ≪ ω and ω pα ≪ ω for the GHz wave of radio-telescopemeasurements (see Table I). In this limit, 1 ≫ | ¯ χ ⊥ | ≫| ¯ χ × | , ¯ χ ⊥ < χ × <
0, so that n l ( ω ) ≥ n r ( ω ) and,from Eq. (3), ∆ φ >
0. Quantitatively, n l ( ω ) − n r ( ω ) ∼ ω ce ω pe ω , (5)which yields the classical scaling ∆ φ ∝ λ . Parallel propagation in rotating gyrotropic me-dia.
Let us now assume that the medium defined byEq. (2) is rotating with Ω = Ω ˆz , and that the dielectricproperties in the medium’s rest frame are not modifiedby rotation, i. e. χ ′ = ¯ χ . Here p ′ refers to the laboratoryframe variable p in the gyrotropic medium’s rest frame.In the rotating frame, the constitutive relations write B ′ = µ H ′ (6a) D ′ = ǫ [ I + ¯ χ ( ω ′ )] E ′ . (6b)Using Lorentz transformation from the dielec-tric rest frame rotating at instantaneous velocity v = T ( − Ω y, Ω x,
0) to laboratory frame (see, e. g. ,Ref. [24]), we get the constitutive relations in the labframe B = µ H − v c × ¯ χ ( ω ′ ) · E (7a) D = ǫ ε · E + ¯ χ ( ω ′ ) · (cid:16) v c × H (cid:17) . (7b)The second term in Eqs. (7a) and (7b) represent, to firstorder in v/c , the effect of rotation. This set of consti-tutive relations Eq. (7) is complemented by Maxwell’s equations ∇ · B = 0 (8a) ∇ · D = 0 (8b) ∇ × E = − ∂ B ∂t (8c) ∇ × H = ∂ D ∂t . (8d)Using Eq. (8d) into the curl of Eq. (7a), and pluggingin Eq. (7b), one gets c ∇ × B = 1 c ∂∂t [( I + ¯ χ ( ω ′ )) · E ] + ∂∂t [ ¯ χ ( ω ′ ) · ( β × µ H )] − ∇ × ( β × ¯ χ ( ω ′ ) · E ) , (9)with β = v /c . To first order in β , B can be substitutedto µ H in the second term on the right hand side. Fol-lowing Player [18], we consider the particular case of awave propagating along the rotation axis, i. e. k = k ˆz .Eq. (8a) and Eq. (8b) require respectively that B and D are transverse. Eq. (7a) and Eq. (7b) then imply that H and E have longitudinal amplitudes of order β . Tofirst order in β , the operator ∇ can thus be replaced by ˆz ∂/∂z when it operates on field quantities [18]. Underthese assumptions, and after some algebra, the last termin Eq. (9) can be rewritten ∇ × [ β × ¯ χ ( ω ′ ) · E ] = Q [ ¯ χ ( ω ′ ) · E ] (10)where we have defined the operatorQ = Ω c [Q · ∇ × + Q · + ˆe z × ] (11)withQ = x y − x − y and Q = − y xy − x ∂∂z . (12)Further derivation shows that the product of the last twoterms of the operator A in Eq. (11) with ¯ χ ( ω ′ ) · E dependsonly on ∂E z /∂z , which is negligible to first order in β as aresult of Eq. (8b) and Eq. (7b). Using the vector identityEq. (28), and noting that [ ¯ χ ( ω ′ ) · ∇ ] × E = ¯ χ k ∇ × E ,Eq. (10) then writes to first order in β ∇ × [ β × ¯ χ ( ω ′ ) · E ] = Ω c Q ¯ χ † · ( ∇ × E ) (13)with ¯ χ † = ¯ χ ⊥ − j ¯ χ × j ¯ χ × ¯ χ ⊥
00 0 2 ¯ χ ⊥ − ¯ χ k . (14)Using Eq. (8c) in Eq. (13), plugging it into Eq. (9), andtaking the curl, we get c ∇ × ∇ × B = 1 c ∂∂t ( ∇ × [ I + ¯ χ ( ω ′ )] · E )+ 1 c ∂∂t [ ∇ × ¯ χ ( ω ′ ) · ( β × B )]+ Ω c ∇ × Q ¯ χ † ( ω ′ ) ∂ B ∂t . (15)Using once more the vector identity Eq. (28), and ([ I + ¯ χ ( ω ′ )] · ∇ ) × E = (1 + ¯ χ k ) ∇ × E , the first term in thebracket on the right hand side of Eq. (15) reads ∇ × [ I + ¯ χ ( ω ′ )] · E = (cid:2) I + ¯ χ † ( ω ′ ) (cid:3) · ∇ × E . (16)Finally, plugging Eq. (8c) in Eq. (16), a wave equationfor B is obtained, ∇ × ∇ × B = − c (cid:2) I + ¯ χ † ( ω ′ ) (cid:3) · ∂ B ∂t + 1 c ∂∂t ∇ × ¯ χ ( ω ′ ) · ( β × B )+ Ω c ∇ × Q ¯ χ † ( ω ′ ) ∂ B ∂t . (17)Writing B = T ( B x , B y ,
0) exp[ j ( kz − ωt )] and introduc-ing the wave index n = kc/ω , Eq. (17) leads to (cid:18) χ ⊥ − n − jχ × jχ × χ ⊥ − n (cid:19) (cid:18) B x B y (cid:19) = (cid:18) (cid:19) (18)with χ ⊥ = ¯ χ ⊥ − Ω ω ¯ χ × (19a) χ × = ¯ χ × − Ω ω (cid:0) ¯ χ k + ¯ χ ⊥ (cid:1) . (19b)In deriving Eq. (18), terms in ∂ β /∂t and ∂ β /∂t havebeen neglected since they are respectively of order β and β . Mechanical contribution to polarisation rotation.
From Eq. (18), we see that the wave indexes for RCP( B y = jB x ) and LCP ( B y = − jB x ) waves are modifiedby rotation and now write n r ( ω ) = 1 + χ ⊥ ( ω ′ ) + χ × ( ω ′ )= 1 + ¯ χ ⊥ ( ω ′ ) + ¯ χ × ( ω ′ ) − Ω ω (cid:2) ¯ χ × ( ω ′ ) + ¯ χ k ( ω ′ ) + ¯ χ ⊥ ( ω ′ ) (cid:3) , (20a)and n l ( ω ) = 1 + χ ⊥ ( ω ′ ) − χ × ( ω ′ )= 1 + ¯ χ ⊥ ( ω ′ ) − ¯ χ × ( ω ′ ) − Ω ω (cid:2) ¯ χ × ( ω ′ ) − ¯ χ k ( ω ′ ) − ¯ χ ⊥ ( ω ′ ) (cid:3) . (20b)Owing to Doppler shift, ω ′ = ω − Ω for the RCP, and ω ′ = ω + Ω for the LCP.Just like polarisation rotation in a stationary gy-rotropic medium arose from ¯ χ × = 0, Eq. (20) shows thatpolarisation rotation in a rotating gyrotropic mediumstems from χ × = 0. However Eq. (19b) indicates that po-larisation rotation can now stem either from anisotropyof the medium ( ¯ χ × = 0) or from mechanical rotation(Ω = 0), or a combination of the two effects.In the limit of an isotropic dielectric, ¯ χ ⊥ = ¯ χ k = ǫ r − ǫ r the dielectric relative permittivity, and ¯ χ × = 0. Polarisation rotation hence results only from mechanicalrotation. Assuming slow rotation (Ω ≪ ω ), Eqs. (20)rewrite n l/r ( ω ) ∼ p ǫ r ( ω ′ ) ± "p ǫ r ( ω ′ ) − p ǫ r ( ω ′ ) Ω ω . (21)Taylor expanding the refractive index difference ∆ n = n l − n r , one recovers from Eq. (3) the result∆ φ = ∆ nωlc = (cid:0) n g − n − (cid:1) Ω lc (22)first obtained by Player [18] and later generalised byGoette [25] to account for wave optical angular momen-tum. Here n g = n + ωdn/dω is the group index and n = ǫ r . Polarisation rotation in a rotating symmetricalelectron-positron magnetosphere.
For a symmet-rical and cold electron-positron ( e − p ) plasma, n = n e = n p , ǫ e = − ǫ p = 1 and m = m p = m e . The non-diagonalterm ¯ χ × of the susceptibility tensor in Eq. (4b) hencealso cancels. Electrons and positrons interact symmet-rically with RCP and LCP waves, respectively, and nopolarisation rotation is found in the absence of rotation(no Faraday rotation). Polarisation rotation is in thiscase a purely mechanical effect, as it is the case for anisotropic dielectric [18].For the typical pulsar magnetosphere parameters givenin Table I and GHz radio waves typically used by radio-telescopes, the ordering ω c ≫ ω p ≫ ω ≫ Ω holds.In these conditions, | ¯ χ k | ≫ ≫ ¯ χ ⊥ . Since ¯ χ k < ω lc ∼ (cid:0) ω p Ω (cid:1) / (23)below which the LCP does not propagate assumingΩ >
0. Note that reversing the pulsar sense of rota-tion (Ω <
0) simply changes the LCP cut-off into aRCP cut-off at the same frequency. For the parametersgiven here, ω lc ∼ s − , which is on the lower end ofradio-telescope observations. For Ω > n r ( ω ) ≥ n l ( ω )above ω lc , and Eq. (3) shows that ∆ φ M <
0. Conversely,∆ φ M > <
0. Depending on the pulsar senseof rotation, mechanical-optical rotation in the rotating e − p magnetosphere can hence add to or subtract frompolarisation rotation associated with the magneto-opticaleffect in the low-density Faraday screen between the pul-sar and the observer. Since the situation is symmetrical,we consider the case Ω > ω lc ≪ ω ≪ (cid:0) ω c Ω (cid:1) / , 1 ≫ | ¯ χ k | Ω /ω ≫ ¯ χ ⊥ and n l ( ω ) − n r ( ω ) ∼ Ω ω ¯ χ k ∼ − ω p Ω ω . (24)From Eq. (3), polarisation rotation ∆ φ is hence propor-tional to ω − , similarly to Faraday rotation in a station-ary magnetised plasma for wave frequencies much largerthan the plasma frequency ω pe .Interestingly, different behaviour is found near the cut-off. Taylor expanding the left and right wave indexes, onefinds, to lowest order in ν = ω − ω lc , n l ( ω ) − n r ( ω ) = −√ r νω lc + O (cid:18) νω lc (cid:19) . (25)In this frequency band, polarisation rotation ∆ φ hencescales like ω √ ω − ω lc .Finally, a jump in polarisation angle ∆ φ lc should beobserved when passing ω = ω lc for Ω >
0. Below thecut-off frequency, only the RCP wave propagates, and thechange in polarisation angle after propagating a distance l will hence be ∆ φ = − n r ωl/c . The polarisation angleshould hence jump by∆ φ lc = n r ( ω lc ) ω lc l c ∼ (cid:18) ω p Ω √ (cid:19) / lc (26)when ω increases past ω lc . Effects of magnetosphere model refinement.
Whilethe symmetrical e − p plasma model used so far conve-niently highlights the role of mechanical rotation, it failsto account for two features which are typical of pulsar’smagnetosphere.First, magnetospheres are generally assumed to havenon-zero space charge, so that n e = n p . The densityasymmetry leads to non-zero non-diagonal susceptibil-ity ¯ χ × . This makes polarisation rotation more compli-cated with now both Faraday rotation and MOR tak-ing place in the magnetosphere. If the charge density isequal to the Goldreich-Julian value N GJ [26], the rela-tion η (1 − f ) = 1 holds true with f = n p / ( n e + n p )the positron fraction and η = ( n e + n p ) /N GJ ≥
1. Themultiplicity factor η is generally assumed to be large(10 − ), so that f is close to 0 .
5. To illustratethe effect of density asymmetry, we choose n p = n and n e = (1 − f ) /f n with f = 0 .
49. This corresponds to aspace charge larger than the Goldreich-Julian value forthe pulsar parameters given in Table I and used in thesymmetrical model for which η = 285 so that f ∼ . > λ scaling persists. The observed upshift in cut-offfrequency stems from the increase in ω pe .Second, the relativistic-quantum effects associatedwith the extremely strong magnetic fields found in pul-sars should also be considered [27]. The plasma suscepti-bility tensor Eq. (4) is then replaced by its QED form [28] ¯ χ ⊥ = − X α m α ω pα m α ω N α ( ω, n ) D α ( ω, n ) , (27a)¯ χ × = − X α ǫ α ω cα ω pα ω m α D α ( ω, n ) , (27b)¯ χ k = − X α m α ω pα m α ω κ ω (1 − n ) − m α κ ω (1 − n ) − m α , (27c)with N α ( ω, n ) = κ ω (1 − n ) − κ (1 − n ) m α ω cα − m α , D α ( ω, n ) = [ κω (1 − n ) − m α ω cα /ω ] − m α . Here κ = ¯ h/c and m α = p m α + eB ¯ h/c the shiftedground-state mass of the charged particle. Compared tothe classical model, the components of the susceptibilitytensor now depend on the wave vector k , but implicitexpressions can be found for the wave refractive indexes n r and n l . The numerical solution for our default pulsarparameters is shown in Fig. 3. This result shows that thedeviation from the λ scaling near the cut-off frequencypersists even when QED effects are taken into consider-ation. FIG. 3. Comparison of polarisation rotation per unit lengthpredictions δ = ∆ φ/l obtained for different e − p magneto-sphere models. While QED corrections and e − p densityasymmetry do affect polarisation rotation near the cut-off, allthree cases are found to deviate from the ω − scaling. Thesymmetrical case is the baseline computed for B = 10 T, n = 10 m − and Ω = 10 s − . The non-symmetrical iscomputed for n p = n and n e = (1 − f ) /fn with f = 0 . symmetrical QED corrected non-symmetrical Vector identity.
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Y. Shi’s work was performed un-der the auspices of the U.S. Department of Energy atLawrence Livermore National Laboratory under Con-tract DE-AC52-07NA27344 and was supported by theLawrence Fellowship through LLNL-LDRD Program un-der Project No. 19-ERD-038. NJF was supported, inpart, by NNSA Grant No. DE-NA0002948.
Competing Interests
The authors declare that theyhave no competing financial interests.