electron-positron flows around magnetars andrei m. beloborodov physics department and columbia...

8/13/2019 ELECTRON-POSITRON FLOWS AROUND MAGNETARS Andrei M. Beloborodov Physics Department and Columbia Astr

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arXiv:1209.4

063v2

[astro-ph.H

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Draft version September 10, 2013Preprint typeset using LATEX style emulateapj v. 5/2/11

ELECTRON-POSITRON FLOWS AROUND MAGNETARS

Andrei M. BeloborodovPhysics Department and Columbia Astrophysics Laboratory, Columbia University, 538 West 120th Street New York, NY 10027;

[email protected]

Draft version September 10, 2013

ABSTRACT

The twisted magnetospheres of magnetars must sustain a persistent flow of electron-positron plasma.The flow dynamics is controlled by the radiation field around the hot neutron star. The problem ofplasma motion in the self-consistent radiation field is solved using the method of virtual beams.The plasma and radiation exchange momentum via resonant scattering and self-organize into theradiatively locked outflow with a well-defined, decreasing Lorentz factor. There is an extended zonearound the magnetar where the plasma flow is ultra-relativistic; its Lorentz factor is self-regulatedso that it can marginally scatter thermal photons. The flow becomes slow and opaque in an outerequatorial zone, where the decelerated plasma accumulates and annihilates; this region serves as areflector for the thermal photons emitted by the neutron star. The e flow carries electric current,which is sustained by a moderate induced electric field. The electric field maintains a separationbetween the electron and positron velocities, against the will of the radiation field. The two-streaminstability is then inevitable, and the induced turbulence can generate low-frequency emission. In

particular, radio emission may escape around the magnetic dipole axis of the star. Most of the flowenergy is converted to hard X-ray emission, which is examined in the accompanying paper.Subject headings: plasmas stars: magnetic fields, neutron X-rays

1. INTRODUCTION

The observed activity of magnetars is believed to becaused by their surface motions, which are driven bystrong internal stresses. The magnetosphere is anchoredin the neutron star and twisted by the surface motions,resembling the behavior of the solar corona. As a resultit becomes non-potential, B= 0, and threaded byelectric currents (Thompson et al. 2002; Beloborodov& Thompson 2007). The currents flow along B, i.e.,

the twisted magnetosphere remains nearly force-free,j B= 0. Numerical models of dynamic twisted magne-tospheres of magnetars (Parfrey et al. 2012, 2013) showhow the twist creates spindown anomalies and initiatesgiant flares when the magnetosphere is overtwisted andloses equilibrium.

Free energy stored in the magnetospheric twist is grad-ually dissipated through the continual electron-positrondischarge that sustains the electric current. As a result,the magnetosphere tends to untwist on the character-istic ohmic timescale of a few years. Electrodynamics ofuntwisting is quite peculiar: whenever the crustal mo-tions stop or slow down, the electric currents tend tobe quickly removed from the magnetospheric field lineswith small apex radii R

max (Beloborodov 2009). Cur-

rents have longest lifetimes on field lines with Rmax R(where R 10 km is the star radius) and form an ex-tended j-bundle (Figure 1). The j-bundle has a sharpboundary, which gradually shrinks toward the magneticdipole axis. In particular, the footprint of the j-bundleon the neutron star surface, which may be observed asa hot spot, shrinks with time. Shrinking hot spots wereindeed reported in transient magnetars whose magne-tospheres were temporarily activated and then graduallyrelaxed back to the quiescent state (see Figure 5 in Be-loborodov 2011a and refs. therein).

The j-bundle must be filled with relativistic plasma

that carries the electric current j= (4/c) B. Theplasma is continually created by e discharge near thestar and must expand along the extended field lines. Theplasma emits persistent nonthermal emission, convert-ing the dissipated twist energy to radiation. In the ac-companying paper (Beloborodov 2013), we calculate thespectrum of produced radiation and show that it formsthe observed hard X-ray component with a peak around1 MeV. The present paper studies in more detail the dy-

namics of the e

plasma.Previously, semi-transparent plasma flows aroundmagnetars were invoked to explain the deviation of theobserved 1-10 keV emission from a thermal spectrum(Thompson et al. 2002). It was usually assumed thatthe magnetar corona is filled with positive and negativecharges that are counter-streaming with mildly relativis-tic speeds. The counter-streaming picture was motivatedby the fact that electric current j must flow along thetwisted magnetic field lines. The coronal plasma mustbe nearly neutral; it can easily carry the required cur-rent if the opposite charges with densitiesn+ = n flowin the opposite directions, so that j = e(v+n+ vn)wherev+v < 0. The velocitiesvare free parameters inthis phenomenological model, which may be adjusted so

that resonant scattering of thermal X-rays in the coronareproduces the 1-10 keV part of the magnetar spectrum(e.g. Fernandez & Thompson 2007; Nobili et al. 2008;Rea et al. 2008).

The counter-streaming model has, however, a prob-lem. Note that the thermal radiation of the magne-tar (h 1 keV) is resonantly scattered at large radiir 10R 100 km where B 1011 1012 G.1 In thisregion, the plasma is strongly pushed by radiation awayfrom the star and the counter-streaming model needs an

1 This assumes that photons are scattered by electrons, not ions.
http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2http://arxiv.org/abs/1209.4063v2


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Fig. 1. Snapshot of a slowly untwisting magnetosphere. In this example, a global twist with a uniform amplitude = 0.2 wasimplanted into the dipole magnetosphere at t = 0, and the snapshot shows the magnetosphere at t 1 yr. Details of the calculationsare described in Beloborodov (2009). The plane of the figure is the p oloidal cross section of the magnetosphere. The black curves are the

poloidal magnetic field lines. The magnetosphere is symmetric about the vertical axis and the equatorial plane; therefore, the figure onlyshows one quarter of the poloidal cross section. The neutron star is shown by the black circle (radius R 10 km). Left panel: Currentdensityj normalized to BR/c. The region from which currents have been pulled into the star (the potential cavity with j = 0) is shownin white. The boundary between the cavity and the j-bundle (magenta curve) expands with time, i.e. the j-bundle shrinks toward thevertical axis. Right panel: Twist amplitude at the same time. The twist amplitude is defined for each closed field line as the azimuthaldisplacement of its footpoint in the southern hemisphere relative to the footpoint in the northern hemisphere.

electric field E that forces charges of the right sign tomove toward the star against the radiative drag. At thesame time, the electric field acts on the opposite chargesand accelerates them away from the star, cooperatingwith the radiative push. In the presence ofe plasma(which is inevitably created near magnetars), the out-ward acceleration generates relativistic particles, and noself-consistent solution exists for the mildly-relativistic

counter-streaming model.In this paper (and the accompanying paper Be-loborodov (2013)), we develop a different picture ofplasma circulation in the magnetar corona. It is schemat-ically shown in Figure 2. The outer corona is inevitablyfilled with e pair plasma of a high density n, which islarger than j/ecby the multiplicity factorM 1; inthis respect it resembles the flow along the open field linesof a rapidly rotating, strongly magnetized neutron star(e.g. Hibschman & Arons 2001; Thompson 2008a; Medin& Lai 2010). Pairs are created in the adiabatic zoneB >1013 G where the flow energy is reprocessed into par-ticles with Lorentz factors 20 (Beloborodov 2013);their multiplicity M 102 is basically set by energy con-servation and mainly controlled by the discharge voltage.

Both electrons and positrons outflow from the magnetar,and radiation pressure forces the particles to accumulatein the equatorial plane of the magnetic dipole, wherethey annihilate. The required current j= (c/4) Bis sustained in the outflow by a moderate electric fieldE. This field is self-consistently generated to main-tain a small difference between the velocities of the charges, (v+ v)/v M1 1, so that the condi-tione(n+v+ nv) =j is satisfied with n+ n andv+v > 0. In the simplest, two-fluid model (Section 3)the velocities v tend to be locked by the balance oftwo forces, electric and radiative.

The coronal outflow significantly changes the radiationit interacts with via scattering. The problem of out-flow dynamics can be formulated as a problem of self-consistent radiative transfer where particles and photonsexchange energy and momentum as they flow away fromthe neutron star. This problem is solved in this paperusing a specially designed numerical method.

The paper is organized as follows. Section 2 discusses

the creation ofe

pairs and their circulation in the innerand outer magnetosphere. Section 3 presents the modelof a radiatively locked outflow in its simplest version us-ing a two-fluid description and assuming an optically thinmagnetosphere. Section 4 discusses the two-stream in-stability in the e flow and the origin of low-frequencyemission from magnetars. Then, in Sections 5 and 6we formulate and solve the full problem where an out-flow with a broad momentum distribution function andsignificant optical depth interacts with the neutron-starradiation. The numerical method and results are de-scribed in Section 6. Our conclusions are summarized inSection 7.

2. CREATION AND CIRCULATION OF E PAIRS

The creation ofe pairs by an accelerated particle isa two-step process: the particle generates a high-energyphoton (via resonance scattering) and then the photonconverts to an e pair in the strong magnetic field. Anaccelerated electron (or positron) can resonantly scatterphotons of energy once it reaches the Lorentz factorrequired by the resonance condition(1cos )= B,whereis the angle between the photon and the electronvelocity and B = eB/mec. When is not small, theresonance condition gives

sc 103B14

1 keV

1. (1)


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The scattered photons are boosted in energy by the fac-tor of 2sc. Such high-energy photons quickly convertto e pairs in the strong magnetic field, creating moreparticles near the star. A similar process ofe creationoperates in the polar-cap discharge of ordinary pulsars,but in a different mode. In ordinary pulsars, the high-energy photons convert to e with a significant delay.The scattered photon initially moves nearly parallel to B

and converts to e only when it propagates a sufficientdistance where its angle with respect to B increasesso that the threshold condition for conversion is satisfied.This delay leads to the large unscreened voltage in pulsarmodels.

In contrast, the magnetic field of magnetars is so strongthat pair creation can occur immediately following reso-nant scattering (Beloborodov & Thompson 2007). Theenergy of a resonantly scattered photon is related to itsemission angle by

E() = EB

sin2

1

cos2 +

m2ec4

E2Bsin2

1/2,

(2)

whereEB = (2B/BQ+1)1/2mec2 is the energy of the firstexcited Landau level andBQ= m2ec

3/e 4.4 1013 G.The scattered photon may immediately be above thethreshold for conversion, E > Ethr = 2mec2/ sin , ifB > 4BQ. Therefore, e discharge in magnetars canscreen E more efficiently than in ordinary pulsars andbuffer the voltage growth once the Lorentz factors of ac-celerated particles reach sc given in Equation (1).

2.1. Pair creation on field lines with apexesRmax 1

M ~ 1

+

+

+

Fig. 2. Schematic picture of plasma circulation in the mag-netosphere with surface B 1015 G. Two regions are indicated.(1) Inner corona. Here e have a moderate multiplicity M 1.The particles do not stop in the equatorial plane. The electricfield E ensures that electrons and positrons circulate in the op-posite directions along the magnetic field lines, maintaining theelectric current demanded by B. The particles are lost as

they reach the footpoints of the field line and continually replen-ished by pair creation. (2) Outer corona extended field lineswith Rmax R. Electrons and positrons are created by the dis-charge near the star and some of them flow outward to the regionof weaker B . Here resonant scattering enhances the pair multiplic-ity, M 1, and decelerates the outflow. The e particles stopat the apexes of magnetic field lines (blue region in the equato-rial plane), accumulate, and annihilate there. The number fluxesof electrons and positrons toward the annihilation region differ bya small fraction M1, so that the outflow carries the requiredelectric current j = (c/4) B. Electrodynamics of the twistdissipation implies that the inner corona is less likely to be ac-tive, as the electric currents are erased by the expanding cavity(Figure 1); the observed activity tends to concentrate on extendedfield lines that form the outer corona.

The average plasma density in the circuit n is close to

the minimum density nmin = j/ec, as required by thecriticality conditionM =n/nmin 1.The global-discharge picture applies only to field lines

with Rmax


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cross section for resonant scattering res = 22rec/res

increases asB1 (herere = e2/mec2 is the classical elec-tron radius). Practically all photons scattered by theoutflowing particles in the region B > 1013 G convertto e; detailed Monte-Carlo simulations of this processare presented in the accompanying paper (Beloborodov2013). In essence, the particles outflowing from the dis-charge zone lose energy to photon scattering, and this en-ergy is transformed to new generations ofe. As a result,thee multiplicity of the outflow increases fromM 1to M 100. This implies that there is no charge starva-tion in the outer corona there are plenty of charges toconduct the current demanded by the twisted magneticfield.

Pair creation sharply ends near the surface of B 1013 G; outside this surface the resonantly scattered pho-tons are not absorbed. The steady relativistic outflowwithout pair creation maintainsM = const along themagnetic field lines. This follows from conservation ofmagnetic flux, charge, and particle number, which given/B = const, j/B = const, and n/j = const along thefield line.

2.3. Global circulation of pair plasma

The picture of plasma circulation in the magnetar mag-netosphere is summarized in Figure 2. The inner corona(field lines with Rmax of a few neutron-star radii) isfilled with the ultra-relativistic counter-streaming e ande+. In this region, resonant scattering is marginally effi-cient and a global discharge operates as described in Sec-tion 2.1, with multiplicityM 1. Outside this region,pair multiplicity is much higher and the electric current isorganized with bothe ande+ outflowing from the star.The exact location of the boundary between the two re-gions depends on the strength of the magnetic field ofthe star.

In the outer corona, the opposite flows in the nothernand southern hemispheres meet in the equatorial plane ofthe magnetic dipole and stop there. Two effects preventtheir inter-penetration. (1) Radiative drag is strong inthe outer corona and pushes both nothern and southernflows toward the equatorial plane (see Section 3 below).(2) When the two opposite flows try to penetrate eachother, a two-stream instability develops. As a result,a strong Langmuir turbulence is generated, which in-hibits the penetration. This effect is particularly impor-tant in the transition region between the outer and innercorona. For these field lines, resonant scattering is effi-cient enough to generateM > 1 but not strong enoughto stop the pair plasma in the equatorial plane. Then thecolliding northern and southern flows are stopped by thetwo-stream instability. This behavior contrasts with theinner corona where the induced electric field enforces thecounter-streaming ofe and e+, i.e. the opposite flowswithM 1 are forced to penetrate each other despitethe two-stream instability.

The density ofe pairs accumulated in the outer equa-torial region (shown in blue in Fig. 2) is regulated by theannihilation balance. In a steady state, the annihilationrate is given by Nann 2M(I/e), whereIis the electriccurrent through the annihilation region. The correspond-ing annihilation luminosity is Lann = 2mec

2M I/e.

3. RADIATIVELY LOCKED CORONAL FLOW

Dynamics of thee flow in the outer corona (r R) isinfluenced by resonant scattering, which exerts a strongforce Fon the particles along the magnetic field lines. Fvanishes only if the particle has the saturation momen-tumpsuch that the radiation flux measured in the restframe of the particle is perpendicular to B. In the sim-plest case of a weakly twisted dipole magnetosphere ex-posed to central radiationp is given by (Appendix B),2

p(r, ) = 2cos

sin , (3)

where momentum is in units ofmec. The radiative forcealways pushes the particle towardp = p. The strengthof this effect may be measured by the dimensionless dragcoefficient,

D rFp mec2

. (4)

Momentum p is a strong attractor in the sense thatdeviations p p generateD 1 in the outer corona(see Appendix A).

Even an extremely strong radiative drag does notimply that e+ and e acquire exactly equal velocities+ = = = p(1 + p2)

1/2. Such a single-fluidflow would be unable to carry the required electric cur-rent j, and E must be induced to ensure a sufficientvelocity separation+ . Below we describe the two-fluid model with +=.

3.1. Two-fluid model: basic equations

Consider an e flow in the region where no new pairsare produced. In a steady state, the fluxes of electronsand positrons are conserved (n v) = 0. This implies j = 0 where j =env are the contributionsof electrons and positrons to the net electric current j=j

+

+ j

. Since e move along the magnetic field lines,j = Bwhereare scalar functions. From j =0 and B= 0 one gets B = 0. Therefore,

j+B

=const, j

B =const, (5)

are constant along the field line. The net current j =j+ +j is fixed by the conditionj = (c/4)| B|. Themultiplicity of pairs is defined by

M = j++ |j|j

. (6)

Here + corresponds to positrons, which carry currentj+ > 0 and corresponds to electrons, which carryj < 0; we assume a positive net current j = j++ jfor definiteness. A counter-streaming model (v+v < 0)would haveM = 1. A charge-separated outflow (n =0) would also haveM = 1. We study here pair-richoutflows withM >1.

The outflowing e plasma must be nearly neutral,3

n+ n, (7)2 This expression is valid in the region where 1Br/B >(R/r)2.

In this region, stellar radiation may be approximated as a centralflow of photons, neglecting the angular size of the star R/r.

3 Here we neglect rotation of the neutron star and its magne-tosphere, which is a good approximation everywhere except theopen field-line bundle that connects the star to the light cylin-


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otherwise a huge electric field would be generated thatwould restore neutrality. Using this condition, one findsthaten+v+ env = j is satisfied if

1 +

= 2

M + 1 . (8)

A deviation from this condition implies a mismatch be-tween the conduction currentj

++j

and (c/4)

B,

which would induce a growing electric field according toMaxwell equation E/t = c B 4j. An electricfield E must be established in the outflow to sustainthe condition (8) against the radiative drag that tends toequalize and + at . This moderate electric fieldmust be self-consistently generated by a small deviationfrom neutrality,n= n+ n n.

The two-fluid dynamics of the outflow is governed bytwo equations,

mec2 d

dl = F() eE, (9)

wherel is length measured along the magnetic field line.In the region of strong drag,

|D| 1, the left-hand side

is small compared with F; then the radiative and electricforces on the right-hand side nearly balance each other,F(+) eE andF() eE. This implies,

F(+) F() (|D| 1). (10)Equations (8) and (10) describe the radiatively lockedtwo-fluid current with pair multiplicity M. The solution to these equations exists ifM > 1. Note that < < + in the radiatively-locked state.

Let us now relax the assumption |D| 1. Thenthe inertial term mec2d/dl must be retained in Equa-tions (9), i.e. we now deal with differential equationsfor . Since and + are not independent theyare related by condition (8) it is sufficient to solve

one dynamic equation, e.g. for+ (and use the dynamicequation for to exclude E). Straightforward algebragives,

mec2 d+

dl =

F(+) + F()1 +d/d+

, (11)

dd+

=

M 1M + 1

2 +

3. (12)

3.2. Sample numerical model

Suppose that e plasma is injected near the star witha given multiplicity, e.g. M = 50, and a given highLorentz factor, e.g. + = 100. The corresponding is

determined by Equation (8). Suppose that the plasmais illuminated by the blackbody radiation of the star oftemperature kT = 0.5 keV and neglect radiation fromthe magnetosphere itself. This approximation is valid

der. In a more exact model, the Gauss law in the co-rotatingframe E = 4( + v) includes the effective vacuum chargedensity v = B/2c where is the angular velocity of thestar (Goldreich & Julian 1969). Then the neutrality condition be-comes e(n+ n) +v = 0. Magnetars rotate slowly (typical 1 rad/s) and hence a small fraction of their magnetic flux is

open, typically v is prone to the two-stream instability (e.g. Krall & Trivelpiece 1973). Thegrowth rate of the instability is obtained from the dis-persion relation for Langmuir modes with frequency and wavevector k , which can be derived by consideringperturbations of the two-fluid system and using con-


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Fig. 3. Lorentz factors + (left panel) and (right panel) in the two-fluid model of the e flow. In this example, the electric currentis carried by the e outflow of a fixed multiplicity M = 50. The plasma is injected at radius r = 2R and outflows along the magneticfield lines (white curves). The flow is illuminated by the star with temperature kT = 0.5 keV (magenta circle at the origin), and theradiation exerts the forces F() on the positron (+) and electron () fluids. The Lorentz factors + and change as the flow entersthe drag-dominated region |D| 1. The region|D| > 3 is shown by the thick black curve and shadowed in black. D < 0 for positrons(+ > ) and D > 0 for electrons ( < ). The radiative drag stops the plasma in the equatorial plane outside 8R. A nearly dipolemagnetic field (weakly twisted) with Bpole= 10

15 G is assumed in this example. R is the neutron-star radius.

tinuity, Euler, and Poisson equations,

1 2+

3+( kv+)2

2

3( kv)2 = 0, (14)

where2 = 4ne2/me. The plasma is nearly neutral,

n+ = n, and hence + = . Equation(14) has a so-lution(k) with a positive imaginary part that describesan unstable mode. The solution simplifies in the follow-ing two limits:(1) + , which is valid whenM 2(see Equation 8). Then the flow is convenient to viewin its center-of-momentum frame that moves with (+ + )/2. In this frame, the two fluids with densi-ties n =

1n move in the opposite directions withnon-relativistic velocities v = v, so the dispersion re-lation (14) simplifies. It gives the most unstable mode

k= (

3/2) /v with a growth rate = /2.(2) +/ 1. Then the contribution of positrons tothe dispersion relation is small compared to that of theelectrons. The growth rate of the instability is given by 1/2 1+ p (e.g. Lyubarsky & Petrova 2000).

This estimate gives the characteristic length-scale ofthe instabilityc/, which is much shorter than the elec-tron free path to resonant scattering, sc. Hence the ra-diation drag and the induced electric fieldeE = F(p)are unable to lock the positive and negative chargesat the momenta p+ and p calculated in the two-fluidmodel. The instability will generate plasma oscillationsthat should broaden the momentum distribution so thatparticles fill the region p< p < p+.

The generated plasma oscillations may be expected to

introduce an anomalous resistitivity. The fluctuating Ein the oscillations creates a stochastic force that tendsto reduce the free-path of a charged particle. A simplestestimate suggests that this effect could be very strong.Suppose a substantial fraction of the energy density of

the flowmec

2

nis given to plasma oscillations. Then thecharacteristic electric field is E (8mec2n)1/2; it ismuch stronger thanEin the radiatively locked two-fluidmodel, however it is irregular and quickly changes sign.Suppose that the stochastic electric force exerted on theparticle randomly changes sign on a timescale t 1p ,where p is the plasma frequency. The stochastic Egives the momentum kicks P eE1p and causesdiffusion of particles in the momentum space with thediffusion coefficient

Dp (P)2

t (eE)

2

p. (15)

Diffusion in momentum spacep2(t)

Dptimplies a small

free-path of the particle, ()2m2ec3p/(eE)2, muchsmaller than the mean free path to resonant scattering.Thus, a large anomalous resistivity could, in principle,be possible, and then a large longitudinal voltage wouldbe generated to maintain the electric current.

4.2. Numerical experiment

To explore the role of the two-stream instability andanomalous resistivity, we designed the following numer-ical experiment. Keeping in mind that particles aroundmagnetars can flow only along the magnetic field lines,consider the simple one-dimensional problem. Suppose


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An important implication of the two-stream instabil-ity is the excitation of a strong plasma turbulence thatcan generate coherent low-frequency radiation. The two-stream instability is often considered in pulsar modelsas a mechanism feeding radio emission from the openfield-line bundle (e.g. Sturrock 1971; Cheng & Rud-erman 1977). A related model invoking radiative dragwas considered by Lyubarsky & Petrova (2000).4 In the

e flows around magnetars, the two stream instabilityis naturally driven by the strong electric current in thesystem that tends to lock itself in the two-fluid configura-tion as described in Section 3. The instability is continu-ally pumped by the radiative drag in the dense radiationfield of the magnetar. The generated low-frequency ra-diation has the best chance to escape near the magneticaxis, where the plasma density is lowest and its Lorentzfactor is highest. Note that the two-stream instabilityoperates on the closed (twisted) field lines, which carrysignificant magnetic flux. This allows the low-frequencyemission to be unusually bright, even though the volt-age e 109 1010 V is small by the ordinary-pulsarstandards.

Radio pulsations have been detected and studied in de-tail in two magnetars XTE 1810-197 and 1E 1547.0-5408(Camilo et al. 2006; 2007). The estimated radio luminos-ityLr 1030 erg s1 requires a sufficiently high ohmicdissipation rate, Ie > Lr, which cannot be generatedin the open field-line bundle unless e exceedsLr/ILC10111r Lr,30 V. Here r is the efficiency of radio emis-sion, ILC is the electric current circulating in the openbundle, ILC c/R2LC, and RLC =cP /2 1010 cm isthe light cylinder for a magnetar rotating with the pe-riod P of a few seconds. As we argued above, voltageis likely near the threshold ofe discharge, which givese 1011 V.

Therefore, we conclude that the electric current asso-ciated with observed radio emission is large, I

ILC,

and should flow in the closed (twisted) magnetosphere,giving a bright and relatively broad radio beam. Thisis consistent with the unusually broad radio pulses ofmagnetars, much broader than the typical pulse of ordi-nary pulsars with similar periods (Camilo et al. 2006;2007). Note also that the plasma density in the twistedclosed magnetosphere reaches much higher values thanin the open field-line bundle, and the plasma frequencymay approach the infrared band. This may explain theobserved hard radio spectra.

One could consider the possibility that the radio lumi-nosity of magnetars is generated by enhanced dissipationin the open field-line bundle and its immediate vicin-ity. Thompson (2008b) suggested that the diffusion of

magnetic twist in the closed magnetosphere initiates astrong Alfvenic turbulence near the light cylinder, witha high dissipation rate. This picture assumes that themagnetic twist tends to spread due to ohmic diffusion, asobserved in normal laboratory plasma with a finite resis-

4 They suggested that a broad momentum distribution of rel-ativistic particles in the open field-line bundle can evolve into atwo-hump distribution as a result of resonant scattering losses, asthe lower energy particles lose their energy faster than the moreenergetic ones. In contrast, the two-fluid flow described in Sec-tion 3 is shaped by the induced E so that it sustains the electriccurrentj ; without E radiative drag would equalize the velocitiesof all particles at v.

tivity. However, later work (Beloborodov 2009; 2011a)showed that the twists in neutron-star magnetospheresevolve differently: the twist is erased inside out ratherthan spreads diffusively. The twist evolution /t iscontrolled by voltage induced along the magnetic fieldlines, e, so that/t de/d , where

is the mag-

netic flux function labeling the field lines ( = 0 o nthe magnetic dipole axis). An increased voltage near the

axis would imply a large negative de/d , which impliesa large negative/t, i.e. rapid untwisting. Thus, thehigh-e twist near the open field-line bundle cannot besustained. The ohmic effects in the closed magnetospherecan pump the twist near the open bundle only if eis re-ducedtoward the magnetic dipole axis, i.e. de/d > 0.Then the twist pumping continues until the expandingcavity j = 0 reaches the open bundle; a snapshot ofthis evolution is shown in Figure 1. The pumping ofleads to outbursts and spindown anomalies (Parfrey etal. 2012, 2013).

Our scenario for the low-frequency emission from mag-netars may be summarized as follows. The emission isgenerated by the two-stream instability on the twisted

closed field lines with the apex radii Rmax such thatR Rmax RLC. These field lines carry a large elec-tric currentI c/R2max ILC and a modest voltagee 109 1010 V. The high plasma density and thebroad beam of radiation expected on these field lines ex-plain the unusual radio pulsations of magnetars.

5. DYNAMICS OF OUTFLOW WITH A BROADMOMENTUM DISTRIBUTION

5.1. Waterbag model

The plasma instability discussed in Section 4 is gener-ated by the gradient of the distribution function dfe/dp,and the feedback of the excited plasma waves tends tomake the distribution flatter. The numerical simulation

in Section4illustrates how two interpenetrating cold flu-ids ofe+ ande withfe(p) = (1/2)[(pp)+ (pp+)]quickly evolve into a state with a broad and smoothfe(p). Below we design a simple modification of the two-fluid model that takes this effect into account.

The simplest model has a top-hat distribution func-tion. In plasma physics, this approximation is oftencalled waterbag model. Like the two-fluid model, theoutflow is described by two parametersp(or ). How-ever, now instead of two delta-functions we require thee distribution to be flat between p andp+,

fe(p) =

(p+ p)1 p< p < p+

0 p < p or p > p+(16)

This distribution includes both electrons and positrons.The plasma must be nearly neutral, n+ = n, and thetotal density of particlesn = n++n is given by

n=N

c=

Mjec

, (17)

where N =Mj/e is the particle number flux and isthe average velocity,

=

(p) fe(p) dp=

+ p+ p . (18)


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9

Fig. 7. Waterbag distribution of minimum width.

Similar to the two-fluid model,p+and pare not inde-pendent, as the outflow must carry currentj with a givenmultiplicity M; this would be impossible if, for instance,

p+ = p. The minimum possible width of the distribu-tion function (i.e. minimum p+p) is achieved if allnegative charges are slower than all positive discharges,as shown in Figure 7. We will adopt this idealized mo-mentum distribution in our numerical simulations. Thisis a rather crude approximation to more realistic distri-butions ofe+ ande, which overlap in momentum space(cf. Figure 6).5 Nevertheless, it is useful as it allows oneto explore all basic features of the e outflow using aconcrete relation between p+ and p. This relation is

determined by the outflow multiplicityM. It is easy toshow that the generalization of the two-fluid Equation (8)toe flows with any distribution function is given by

1 +

= 2

M + 1 , (19)

where + and are the average velocities of the posi-tive and negative charges, respectively. For the waterbagdistribution shown in Figure 7 this condition gives

+

= (p) + (p) = 1

2

M + 1 , p=p+p+

2 , (20)

where(p) = (1 + p2)1/2. Equation (20) determines the

relation betweenp+ andp (for a given M); it is shownin Figure 8.

Taking into account the relation between p+ and p,the outflow has essentially one degree of freedom besides

5 The minimum-width waterbag model may be particularlycrude near the star where the outflow just begins to experiencesignificant radiative drag. The faster positive charges will experi-ence drag first while the slower negative charges still move freely.As p+ decreases, the minimum-width model would require an in-crease in p. In reality, p may not react to the reduced p+ untilparticles withp = palso begin to experience the drag force (whichis positive as p < p). After this point, the outflow may not befar from the minimum-width waterbag state.

Fig. 8. Relation between p+ and p for the waterbag modelshown in Figure 7. Each curve describes an outflow of a givenmultiplicity M, which is indicated next to the curve; p = 0 isindicated by the dotted line.

the multiplicityM. One can chose, e.g.,p+ as an inde-pendent variable, or any convenient combination of p+andp,(p+, p), that is independent fromM(p+, p).Below we will choose variable and formulate the dy-namic equation for; it will describe how the interactionwith radiation governs the outflow dynamics along themagnetic field lines.

5.2. Momentum equation

Since the outflow is bound to move along the mag-

netic field lines, we will need the projection of momentumequation onto B. It is convenient to write this equationin a covariant form that is valid in any curved coordinatesystemxi (e.g. spherical coordinates). In a steady state,the momentum equation reads

BiB

kTik = dPdtdV

, (21)

wheredP/dtdV is the rate of momentum exchange withradiation (per unit volume),k is the covariant deriva-tive (indices i, k run from 1 to 3), and Tik are the com-ponents of the plasma stress tensor,

Tik =mec2 uiuk

dn

=n mec2 B

iBk

B

2 p2

fe(p) dp.

(22)Hereui =p Bi/Bare the spatial components of the four-velocity vector of a particle with dimensionless momen-tump, dn = nfe(p)dp is the number density of particleswith momenta (p, p+dp), and dn/ is the correspond-ing density in the frame moving with (p) = (1 +p2)1/2.Then the left-hand side of Equation(21) takes the form,

BiB

kTik =BiB

k

n mec2 B

iBk

B2 p

= mec2 Bkk

nB

p

, (23)


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10

where we used Bik(BiBk/B) = 0 (which follows fromkBk = 0). The directional derivative Bkk equalsB d/dl where l is length measured along the magneticfield line. Equation (23) can be further simplified usingn/BM = const, which follows from the relation

n= M jec

, (24)

andj /B = const(cf. Equation5). This gives

BiB

kTik =mec2 n Md

dl

M

p

. (25)

WhenM = const (i.e. no new pairs are created),Mcancels out. Finally, the substitution of Equation (25)to Equation (21) gives the momentum equation in thefollowing form,

d

dl =

F mec2

, p

, (26)

where F=n1dP/dtdVis the average force exerted byradiation per particle. If

F= 0 then remains constantalong the field lines, which impliesp = const the mo-

mentum distribution remains unchanged along the flow.For the waterbag model, is given by Equation (18).

The quantity pcan also be expressed in terms ofp,using the indefinite integral

p dp=

p d=

p

2 1

4ln

1 +

1

+ const. (27)

This gives

= 1

+

1

2(p++ p) 1

4ln

(1 ++)(1 )(1 +)(1 +)

.

(28)

In our numerical simulations, we use as the variablethat describes the dynamical state of the outflow, andsolve the differential Equation (26) for . The values of

p that correspond to a given andM are found fromEquations (20) and(28).

The force Fexperienced by an outflow with a given isdetermined by the local radiation field, which is describedby the intensities in the two polarization modes I(, n)and I(, n) (Appendix A). In general, the force actingon a plasma with a distribution function fe(p) is givenby Equation (A20). Force F is easily calculated in thesimple case of an optically thin magnetosphere exposedto the central blackbody radiation (Appendix B). Thenthe right-hand side of Equation (26) is a known function

of and it is straightforward to numerically solve thisequation; the results are similar to the two-fluid modeldescribed in Section 4. The optically thin approximationis, however, invalid for active magnetars. The radiationis not central; insteadI andI must be calculated self-consistently, together with the outflow dynamics. Thisrequires radiative transfer simulations.

Note that we assume here that p > 0, so that allparticles move away from the star until they reach theequatorial plane and disappear (annihilate) there. Thisapproximation is reasonable for the flow along the ex-tended field lines with apex radii Rmax > 10R. Nearthe apexes, i.e. near the equatorial plane = /2, the

radiative drag enforces p p 1, creating a denselayer of slow particles where they annihilate (Fig. 2). Amore general model of the flow could allow the parti-cles to cross the equatorial plane and enter the oppositehemisphere. By symmetry, this would be equivalent tothe reflection boundary condition, i.e. the mirror imageof the outflow approaching the equatorial plane wouldemerge from the equatorial plane. Then the distribution

functionfe(p) must extend to negativep. This modifica-tion would be required for the plasma flow on field lineswith smallRmax, where radiative drag is less efficient andthe plasma can cross the equatorial plane with a large p.In this paper, we focus on the field lines withRmax > 10,where this does not happen. The field lines with smallRmax are assumed to form a cavity with j = 0 and anegligible plasma density (Figure 1).

6. SELF-CONSISTENT RADIATIVE TRANSFER

The problem of radiative transfer in a relativisticallymovinge plasma whose velocity is controlled by the ra-diation field is not unique to magnetars. A similar situ-ation may occur in accretion disk outflows (Beloborodov

1998) and gamma-ray bursts (Beloborodov 2011b). Thestrong radiative drag (measured by the coefficient D 1,Equation[4]) was previously shown to simplify the prob-lem, as it forces the plasma to keep the saturation mo-mentum p such that the net radiation flux vanishesin the plasma rest frame. This equilibrium transferhas one additional integral compared with the classicalChandrasekhar-Sobolev transfer problem for a mediumat rest. The transfer in magnetar magnetospheres has,however, two special features that complicate the prob-lem. First, the electric current and plasma instabili-ties imply additional (electric) forces that broaden themomentum distribution around p, as discussed in Sec-tions 4 and 5. Therefore, the equilibrium condition

p= p is not satisfied even whereD 1. Second, opac-ity is dominated by resonant scattering, whose rate issensitive to the particle momentum. Below we developa method to solve the self-consistent transfer for magne-tars.

6.1. Monte-Carlo technique and the virtual beammethod

Radiative transfer in a magnetosphere filled withplasma with given parameters can be calculated usingthe standard Monte-Carlo technique (e.g. Fernandez &Thompson 2007; Nobili, Turolla & Zane 2008). Black-body photons are injected into the magnetosphere at theneutron-star surface and their trajectories are followed

until they escape the magnetosphere. We implement thismethod using the scattering opacity given in Appendix Aand keeping track of the photon polarization, which canswitch in the scattering events.

Calculation of transfer in an outflow with self-consistent dynamics is a more ambitious goal, as theplasma parameters are not known in advance. A nat-ural approach is iterative. One can start with a trialoutflow, calculate radiative transfer to find the radiationintensity that would correspond to this outflow, and thenre-calculate the outflow dynamics in the obtained radi-ation field. These iterations can be repeated until theyconverge. The problem is simplified for the waterbag


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plasma model (Section 5) as the outflow is described byone dynamic variable . As the first trial initiating theiterations one can take the outflow solution (r, ) forthe optically thin magnetosphere exposed to the centralblackbody radiation.

One then encounters the following difficulty. For thecalculation of next iteration of outflow dynamics, oneneeds to know I(r, , n) and I(r, , n) everywhere in

the magnetosphere. In axisymmetric magnetospheres,the radiation intensity is a function of five variables: twofor location (radius r and polar angle ), one for spec-trum (), and two for angular distribution (unit vector nis described by two angles). To determine intensities Iand I with sufficient accuracy, one has to introduce afive-dimensional grid and accumulate large photon statis-tics for each grid cell during the Monte-Carlo simulationof radiative transfer. A grid of sizeNfor each of the fivevariables hasN5 cells. Accumulation of photon statisticsin each cell requires the calculation of a huge number ofMonte-Carlo realizations of the photon trajectory in themagnetosphere. This is expensive and gives poor accu-racy.

There is, however, a more efficient method that doesnot require the knowledge of intensities I,(r, , n).They are not, in fact, needed for the calculation ofoutflow dynamics. What enters the momentum Equa-tion (26) is the average force F(r, ), which may be tab-ulated in the two-dimensional space of r, . This forcecan be calculated directly during the Monte-Carlo sim-ulation of radiative transfer if we find a way to evaluatethe contribution of each simulated photon to F(r, ) asit propagates through the magnetosphere.

This can be achieved if we imagine that the photon isreplaced by a beam of radiation. As we follow the photonalong its trajectory, we can calculate the force that wouldbe applied by the imaginary beam to any given outflowthat can be imagined in the magnetosphere. In the wa-

terbag model, the outflow is described by one dynamicalparameter, and so the force applied by the beam canbe tabulated on a grid of.

Once we know how to calculate the force created byeach photon trajectory in our Monte-Carlo simulation,we should be able to average it over all simulated pho-tons and thus accurately evaluate F(r,,). This givesthe force that would be applied by our radiation fieldto an outflow with any given at any point r, . Atthe next iteration step F(r,,) is used to obtain thenew outflow solution (r, ) by integrating the momen-tum equation d/dl = W(l, ), where W = F/mec2(Equation26). Then the Monte-Carlo simulation can berepeated to calculate radiative transfer in the new out-

flow and find F(r,,) for the new radiation field. Thesesteps can be repeated until the outflow solution (r, )

converges, i.e. remains practically unchanged by new it-erations.

The concrete implementation of this strategy is as fol-lows. Let Lth be the thermal luminosity of the star andN=Lth/2.7kTbe the number of photons emitted by thestar per unit time. In our Monte-Carlo simulation we fol-lowKMC 107 random photon trajectories. This can bethought of as dividing N into KMC random monochro-matic beams. Each beam has a random start at thestar surface (the photon energy is drawn from the Planck

distribution) and follows one random realization of thephoton trajectory, which can involve multiple scatteringevents in the magnetosphere. The photon number fluxin each monochromatic beam is Nb = N/KMC, and theenergy flux in the beam is

Eb = Nb . (29)

Note that Nb = const along the beam (i.e. along thephoton trajectory in the Monte-Carlo simulation) whilethe photon energychanges after each scattering in themagnetosphere. The collection ofKMC beams representthe state of the radiation field around the star.

As we follow each realization of the photon trajectory,in parallel we calculate the force applied by the virtualbeam to the plasma. We can imagine that the beam hasa small cross section A (it will cancel in the final result)

and flux density F = Eb/A. Equation (A19) gives ageneral expression for the force exerted by radiation ona plasma with a given distribution function fe(p). Fora monochromatic beam of frequency propagating atanglewith respect to the magnetic field, this expressiongives the following momentum deposition rate,

dP

dtdV = 22renF

B2

[1fe(p1) 2fe(p2)] , sin B(30)

(dP/dtdV= 0 if sin > B), where

p1,2(, ) = B

sin2

cos

1

2

2Bsin2

, (31)

and the factor depends on the beam polarization,

=

1, 1

2

2Bsin2 , (32)

The rate of momentum deposition by the beam (i.e. theexerted force) per unit length along its trajectory is givenby

dP

dtds=

dP

dtdV A. (33)

We need to tabulate the force on a spatial grid, whichis used to calculate the outflow dynamics at the nextiteration. Therefore, we need to evaluate the net forceapplied to a given spatial cell. The number of particlesin the cell is nVc where Vc is the cell volume, and theforce exerted by the beamper particlein the cell is givenby

d

Fb

ds = 1

nVc

dP

dtds = 22 re

Vc

B Eb2 [1fe(p1) 2fe(p2)] .

(34)In the transfer calculation, we track the photon trajec-tory using small stepss, much smaller than the cells ofthe spatial grid. To obtain the force Fb(r,,) appliedby the beam in a given cell we integrate Equation (34)along the photon path where it crosses the cell. Notethat a finer spatial grid implies a larger d Fb/dsV1c ;however, it also implies that the cell is less frequentlyvisited by photons in our Monte-Carlo simulation, andthe photons spend a shorter time in the cell; therefore,the final result does not depend on the grid.


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12

In an axisymmetric magnetosphere, the spatial cells(i, j) are tori of volumeVi,j = 2r

2i sin jr . The net

force applied per particle in a given cell (i, j) is obtainedby summing up the contributions Fb(ri, j , ) from allsimulated beams,

F(ri, j, ) =

N

KMC

KMC

k=1

2re

Vi,j

cell(i,j)

B

[1fe(p1) 2fe(p2)] ds. (35)

Note that the beam may cross a given cell multiple timesin an opaque magnetosphere, and all crossings contributeto the path integral in Equation (35). The e distri-bution function fe(p) is determined by the parameter (assuming a given pair multiplicityM, see Section 5).

As a test, one can apply Equation (35) to the simplestcase of a transparent outflow exposed to the central ther-mal radiation. Then the expected Fcan be directly cal-culated using Equation (B17) in Appendix B. We analyt-ically verified that in this case Equation (35) is reduced

to Equation (B17). We also tested our numerical code;it reproduced the analytical result.

6.2. Results

The obtained self-consistent solution for the outflowdynamics is shown in Figure 9. We showp+ rather thanour dynamical parameter, becausep+ is closely relatedto the radiation emitted by the outflow. We find thatonly particles with the highest momenta p p+ reso-nantly scatter thermal photons in the relativistic zone

p+ > 1; the remaining, dominant part of the momen-tum distribution does not participate in scattering. TheLorentz factor of the scattering particles is

sc (1 +p2+)1/2. (36)The solutionp+(r, ) shown in Figure 9 was calculated

assuming that the star has radius R = 10 km, a uni-form surface temperature kT = 0.3 keV, and a moder-ately twisted dipole magnetic field with Bpole = 1015 Gand = 0.3; the multiplicity of the e flow was fixedatM = 200. The flow is injected at r = 2R with

p+(2R) = 100. The choice of the boundary conditionis not important as we are interested in the flow be-havior outside 5R, where the scattered photons avoidconversion to e pairs and can escape, i.e. where theobserved hard X-rays are produced (Beloborodov 2013).For any reasonable boundary value p+(2R), the flow re-

laxes to the same solution p+(r, ) outside a few stellarradii. Remarkably, our simulations gave practically thesame solution for a broad range of parameters ,M, Tthat is relevant for magnetars. This demonstrates a ro-bust self-regulation mechanism; this important feature isexplained below (see also the accompanying paper).

There are two distinct zones in the outflow:I. Non-relativistic zonep+ 1, which is near the equa-

torial plane (red zone in Figure 9). This zone has a hugeoptical depth and scatters essentially all thermal photonsthat impinge on it from the star (10% of the thermalluminosity Lth). We will call this zone the equatorialreflector.

Fig. 9. Self-consistent outflow solution. Color shows the pa-rameter p+ of the waterbag distribution function. Observed hardX-rays (discussed in the accompanying paper Beloborodov (2013)),are produced by particles with p p+. The simulation assumedthat the active j-bundle occupies the field lines with apex radiiRmax> 10R (Section 1).

II. Relativistic zonep+ > 1 (blue to green in Figure 9).This zone is transparent for essentially all thermal pho-tons flowing from the star, except for rare photons in thefar Wien tail of the Planck spectrum, which may be ne-

glected. Basically, the outflow in this zone does not seethe thermal radiation flowing from the star. It mainlyinteracts with (and is decelerated by) the quasi-thermalphotons that flow from the equatorial equator.

It is easy to see why the outflow in the relativisticzone + 1 is regulated so that it interacts with atiny fraction of the thermal photons around the magne-tar. Scattering on average boosts the energy of a thermalphoton by a factor comparable to 2, and hence theenergy lost by the outflow per scattering is 2. Ifwe imagine that each thermal photon is scattered oncewith an average blueshift of2, the generated hard X-rayluminosity2Lth would exceed the kinetic power of theoutflow L, which is impossible. The scattering rate mustbe kept low, just sufficient for the self-consistent grad-ual deceleration of the outflow. The outflow visionof targets for scattering is controlled by the resonancecondition(1 cos ) = B, where is the angle be-tween the photon direction and the particle velocity .The scattering rate is greatly reduced ifis so low thatthe resonance condition is satisfied only for rare photonsin the Wien tail of the thermal spectrum. The situa-tion may be compared with the regulation of the nuclearburning rate in the sun. The self-consistent temperatureis sufficiently low so that fusion reactions occur only inthe far tail of the Maxwell distribution; as a result onlyrare particles participate in burning, and these particles


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are concentrated in a narrow interval of (large) momenta,a phenomenon known as the Gamow peak. Similarly, ourself-consistent outflow moves sufficiently slow so that res-onant scattering is only enabled between rare particleswith the highest pp+ and rare thermal photons withthe highest energies in the particle rest frame. Theselucky photons have the largest (gained after reflectionfrom the equatorial reflector) and large

2.7kT.

The inward direction (cos < 0) of the reflected photonsgives them particularly large blueshift in the outflow restframe and hence reduces the energy requirement in thelab frame. This makes the reflected photons the domi-nant targets for scattering even though their density ismuch smaller than the density of photons flowing directlyfrom the star. The reflected photons have a diluted quasi-thermal spectrum of temperature T.

In essence, we observe in our simulations that the out-flow moves fast enough to resonantly interact with re-flected photons of energy (710)kT (the low-density exponential tail of the thermal spectrum), andslow enough to not interact with the main peak of the re-flected thermal spectrum 3kT. This condition, to-gether with the resonance condition(1cos )= Band cos 0.5, determines that the scattering plasmamoves with Lorentz factor

sc mec2

10kT

B

BQ, (37)

as long as sc 1. In this regime, the number of tar-get photons visible to an outflowing particle has a strongexponential sensitivity to the particle momentum p. Es-sentially all scattering must be done by particles withthe highest momenta in the distribution function, i.e.with p p+ for the waterbag distribution, as indeedobserved in our numerical simulation. Therefore,sc isassociated withp+. Equation (37) serves as a simple andreasonably accurate approximation to the exact numer-ical results shown in Figure 9. Its applicability is notlimited to the specific simulation with its Bpole, T, ,M the approximation works well for other magnetarparameters, because of the robust self-regulation effectdescribed above. This fact is further discussed and illus-trated in Figure 2 in Beloborodov (2013).

Besides Equation (37), the transfer problem is char-acterized by the position of the equatorial reflector. Itis described by a simple formula, which can be used toscale the results shown in Figure 9 to models with otherparameters. The formula is based on the following fact(see Appendix B): if a given active magnetic loop extendsto the region where B R1 where

R1 80

1033 G cm3

1/3 kT

1 keV

1/3km. (38)

Here = R3Bpole/2 is the magnetic dipole moment ofthe star. Equation (38) describes the position of the inneredge of the non-relativistic (red) zone; e.g. the model inFigure 9 has R1 10R.

It is instructive to compare the result of the full trans-fer calculation in Figure 9 with the simplest, opticallythin two-fluid model in Figure 3. The relativistic zone

p+ > 1 remains practically transparent to thermal radi-ation. The key difference is the presence of the opaqueequatorial reflector. The reflector weakly affects thespectrum of thermal photons supplied by the star, how-ever it significantly changes their angular distributionin the magnetosphere. As a result, the radiation exertsa stronger drag on the outflow and p+ decreases fasteralong the magnetic field lines. In Figure 3, +

p+

1

remains huge near the magnetic axis the central ra-diation is unable to decelerate the plasma because thephotons flow from behind and have small angles withrespect to the plasma velocity. In Figure 9, the equatorialreflector supplies photons with large , which efficientlydecelerate the outflow, according to Equation (37).

The upscattered photons of energy E 2Et arebeamed along the relativistic outflow. Therefore, theybecome unable to decelerate the plasma, even thoughthey can scatter multiple times before escaping. Theoutflow significantly loses energy when it scatters a pho-ton propagating at a large angle with respect to theoutflow velocity; only in this case the scattering booststhe photon energy by the factor of 2sc 1. Afterthe scattering, the photon angle is reduced to

1sc ,and its subsequent scatterings have a small effect on the

outflow dynamics. The beamed radiation initially movestogether with the plasma and then escapes. Our transfersimulations include all scattering events, however practi-cally the same p+(r, ) would be obtained if only singlescattering were allowed in the relativistic zone p+ 1.

7. CONCLUSIONS

This paper examined the behavior of the relativis-tic plasma created by e discharge around magnetars.Plasma circulation in the magnetosphere is schematicallyshown in Figure 2. We focused on large magnetic loops,which must be heavily loaded with e pairs. The plasma

momentum is controlled by the radiation field aroundthe star, which interacts with e+ and e via resonantscattering. We developed a method to calculate radia-tive transfer in the self-consistently moving plasma andobtained the solution for the e flow (Figure 9). The so-lution is a strong attractor the behavior of the plasmaoutside a few stellar radii does not depend on how theplasma is injected near the star and its initial Lorentzfactor 0, as long as 0 30 (0 > 103 is expected,which corresponds to the discharge voltage e 109 V).Thee flow shows the following features:

(1) The relativistic flow scatters radiation with a welldefined Lorentz factor sc given in Equation (37); scdecreases proportionally to B along the magnetic field

lines.(2) The relativistic flow remains transparent to thermalphotons emitted by the star until it decelerates to non-relativistic momenta p


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to the region ofB 1013 G are immediately absorbed (Beloborodov 2013. Interesting radiativetransfer occurs outside this region, where B BQ. In particular, the observed hard X-rays of energy up to severalMeV are radiated where B BQ. Electron recoil is small for resonant scattering in such relatively weak fields, andthe scattering cross section is particularly simple.

A.1. Scattering cross sectionIn classical language, the electro-magnetic wave (photon) with frequency B =eB/mec resonates with the Larmor

rotation of electron. Then the wave strongly accelerates the charged particle and generates scattered radiation. Thecorresponding cross section is largest for waves with the right-hand circular polarization e = e that matches theelectron Larmor rotation. Here e= 2

1/2(ex iey) and {ex, ey, ez} is a Cartesian basis with the z -axis anti-parallelto B. For a wave with an arbitrary polarization vector e, only the projection ofeon eis responsible for the resonance,and the cross section is reduced by the factor |e e|2, where e is the complex conjugate ofe. In quantum language,the resonance occurs because the photon energy matches the energy B needed for the electron transition from theground Landau state to the first excited state. The resonance has a finite width. It equals the natural width of thecylcotron line , which corresponds to the lifetime of the excited electron to spontaneous transition back to the groundLandau state.

For an electron at rest, the differential cross section for photon scattering into solid angle d is given by (Canutoet al. 1971; Ventura 1979),

d

d =r2

e

2

( B)2 + (/2)2|e

e|

2

|e

e|

2, (A1)

wherere =e2/mec

2, is the photon frequency, e and e are the polarization vectors of the photon before and afterscattering. Equation (A1) retains only the resonance peak of the cross section and neglects the non-resonant part.Positron cross section is described by the same equation except that e is replaced by e+ = 2

1/2(ex + iey) (positronsgyrate in the opposite sense). The cross section can also be derived in the framework of quantum electrodynamics(Herold 1979; Daugherty & Harding 1986). For B mec2 (which corresponds to B BQ) the result is reduced toEquation (A1).

The resonance line is very narrow, /B = (4/3)(B/BQ)1, where = e2/c= 1/137 (Daugherty & Ventura1978; Herold et al. 1982), and the resonance factor [( B)2 + (/2)2]1 is well approximated by the delta-function21( B). Then the cross section may be written as

d

d = 2

d

d = 32rec ( B) |e e|2 |e e|2, (A2)

where +/ correspond to scattering by positron/electron, = cos , and is the angle of the scattered photon withrespect to the magnetic field B. The distribution of scattered photons is axially symmetric about B; this fact hasbeen used in Equation (A2).

Two polarization states (eigen modes) exist for the photon. They are controlled by the dielectric tensor of themagnetosphere. In the considered regionr


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where is the photon angle with respect to B. The two modes have slightly different propagation speeds c/Nand therefore they adiabatically track, i.e. the photon propagating through the curved magnetic field preserves itspolarization state. The adiabaticity condition reads klB(N N) 1 wherelB r is the characteristic scale of thespatial variation ofB(see e.g. Fernandez & Davis 2011 for a detailed discussion). This condition is satisfied for X-raysin the considered region of the magnetosphere where scattering occurs. Thus, in our transfer problem, the photon canswitch its polarization state only in a scattering event.

As the photon can be in either polarization state, calculation of radiative transfer involves four scattering processes

,

,

, and

. The corresponding cross sections are given by Equation (A2) with

|e

e

|2 =

|e e|2 = 1/2,|e e|2 =2/2, and|e e|2 =2/2. Note that the cross sections of electron and positron areequal, as|e e|2 = |e e+|2 for any linear polarization e.

Equation (A2) describes the cross section of electron (or positron) at rest. In our transfer problem, the particles aremoving along Band the above equations should be used in the rest frame of the particle. Note that the polarizationstates and B are invariant under Lorentz boosts along B. Consider a photon with energy and propagation anglewith respect to B, in the orpolarization state. We are interested in its scattering by an electron (or positron)that moves with velocity c along the magnetic field line. The total cross section in the lab frame may be obtainedby integrating the differential cross section in the electron rest frame and then multiplying the result by 1 cos ,

tot = 22 rec ( B) (1 ), (A4)

where= (1 ) , (A5)

is the photon frequency in the electron rest frame and = cos . The factor depends on the photon polarization

and is given by

1, 2,

, =

1 , (A6)

where = cos and is the photon angle with respect to B in the electron rest frame. The cross section tot issummed over the final polarization states.

The outcome of the scattering may be either or photon propagating at angle in the electron frame. Theprobability distribution for = cos is given by

P() = 1

tot

d

d =

3

8. (A7)

where = 1 or 2, depending on the final polarization state. The integral

P() d equals 3/4 for thestate and1/4 for the state. Thus, 3/4 of scattering events produce photons with uniform distribution P() = const andthe remaining 1/4 of scattering events produce photons withP(

) 2

. The distribution of the final photons overangle and polarization does not depend on the initial state of the photon before scattering.The standard description of resonant scattering summarized above assumes the transition between the ground and

first excited Landau levels, which has the largest cross section. Transitions to higher levels are neglected in our transfercalculations.

A.2. Opacity

Optical depth of a relativistic plasma to resonant scattering was discussed previously in detail (e.g. Fernandez &Thompson 2007 and refs. therein). Here we write down relevant equations and introduce notation that is used belowin the discussion of radiative drag. Consider a photon of energy that propagates through e plasma with densityn= n++n. The plasma particles move along B with momentum distribution fe(p) that is normalized to unity

fe(p) dp= 1. (A8)

The optical depthdseen by the photon as it propagates distance ds is given byd

ds =n

totfe(p) dp. (A9)

Assuming that the magnetosphere has a small optical depth to non-resonant scattering (Appendix C), we include onlyresonant scattering and use Equation (A4) for tot. Integration over p may be carried out using the identity,

( B) =k

(p pk)|d/dp| , (A10)

whered

dp =

d

dp(p) = ( ), (A11)


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and pk are all possible solutions of equation (p) = B. The delta-functions (p pk) express the fact that thephoton is scattered by electrons or positrons with momenta pk for which the resonance condition is met. The relation sin = sin together with the resonance condition = B determines sin = (/B)sin and leaves two

possibilities for = cos,

=

1 2

2Bsin2

1/2. (A12)

Angle exists (i.e. the resonance is in principle possible) for photons that satisfy the condition sin B. ThenEquation (A12) defines two electron velocities1,2, which may be found from the Doppler transformation of the photonangle, = ( )/(1 ). It yields,

= 1 , 1,2 =

||1 || . (A13)

The corresponding Lorentz factors = (1 2)1/2 and dimensionless momenta p = are

1,2 = 1 ||sin sin

, p1,2 = ||sin sin

. (A14)

Substitution of Equations(A4) and (A10) to Equation (A9) and integration over p gives

d

ds = 2

2

re

c

|| n [fe(p1) +fe(p2)] . (A15)Here= 1 for photons and = 2 forphotons (EquationA6).

A.3. Radiative drag force

Consider now a radiation field with intensity I(, k) in a given polarization state,or. As a result of scattering,radiation exerts a force on the e plasma. We are interested in the component of this force along the magnetic field.The force applied to unit volume of the plasma is given by

dP

dtdV =

d

d

dp

I(, k)

n fe(p) totP. (A16)

Here

d is the solid-angle integration over photon directions n = k/k, and P is the average momentum (perscattering) passed to an electron or positron with Lorentz factor by a photon (, k). In the electron rest frame,

Pequals the photon momentum along B, as its average momentum after scattering vanishes (resonant scatteringis symmetric in the electron frame when B mec2). Thus, P = /c and the Lorentz transformation of thefour-momentum vector to the lab frame gives

P =B

c , (A17)

where we used the condition = B since we consider only resonant scattering.Radiation is described by two intensities I(, k) and I(, k) in the two polarization states. They scatter with

cross sections tot that differ by the factor of 2 (see Equations (A4) and (A6)). Substituting Equation (A4) toEquation (A16) and taking the sum over polarizations, one finds the net force exerted by I and I,

dP

dtdV =

d

d

dp

(I+ 2I)

n fe(p) 2

2rec ( B)(1 ) P. (A18)

Integration overdp similar to that in Section A.2 gives

dP

dtdV =

d

B sin 0

d22reB2

I+

2I

n [1fe(p1) 2fe(p2)] , (A19)

wherep1,2(, ) and 1,2 are given by Equations (A14) and (A12). The upper limit in the integral over takes intoaccount that only photons with Bsin may be resonantly scattered (Section A.2). Equation (A19) is usefulbecause it shows the contribution of each photon (, k) (in theorpolarization state) to the drag force. It can beused even if the radiation intensity is not known in advance and needs to be found self-consistently with the plasmadynamics (Section 6).

An alternative way of simplifying Equation (A18) is to first integrate over , which gives

dP

dtdV =

d

dp22re(I+

2I) n fe(p) ( ), (A20)


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17

where I and I are evaluated at = 1(1 )1B. Equation (A20) is convenient to use when the radiation

intensity is known.Note that Equations (A19) and (A20) are valid only where B BQ. Near the star, where the field is stronger, the

drag force is modified (Baring et al. 2011; Beloborodov 2013). In this paper, we do not need the strong-field corrections,as radiative transfer occurs in the region ofB BQ (where the scattered photons avoid the quick conversion to epairs); use of the full relativistic cross section would be an unnecessary complication.

B. OPTICALLY THIN OUTFLOW

Consider a magnetosphere that is optically thin to resonant scattering, so that intensity I is dominated by theunscattered radiation from the star. We will assume that the neutron star emits approximately blackbody radiationwith the polarization (the photons dominate the surface radiation because they have a larger free path below thesurface, e.g. Silantev & Iakovlev 1980). Then

I = 3

83c2[exp(/kT) 1] , I= 0, (B1)

and we deal with the outflow dynamics in the known radiation field. This case was studied in detail in previous work(e.g. Sturner 1995).

B.1. Scattering rate for one particle

Consider an electron (or positron) located atr, and moving outward with Lorentz factor = (1 2)1/2 along amagnetic field line. The number of photons scattered by the electron per unit time is

Nsc =

d

d

I(, n)

tot =

22rec

I(res[n],n)

resd, (B2)

where we substitutedtot(eq.A4). The resonant frequency depends on the photon direction, res = 1(1n)1B

where n= k/k is the unit vector corresponding tod.At large radii r R all photons at a given location r have approximately the same direction n r. The angle

between the stellar photons and the particle velocity, , is given by = cos Br/B (assuming > R/r). Thenintegration overd in Equation (B2) is reduced to multiplication by (R/r)2, the solid angle subtended by thestar when viewed from radiusr. This gives,

Nsc =23rec

x2

I(res)

resres =

B(1 ) , (B3)

wherex = r/R. It is instructive to write Nsc in the following form,

Nsc = 2 c4x2

g(y)y

, (B4)

where = /mec, = kT/mec2 103 and g(y) is the dimensionless Planck function evaluated at the frequencyres= 1(1 )1B,

g(y) = y3

ey 1 , y= res

kT =

b

(1 ) , (B5)

whereb = B/BQ.

B.2. Drag force exerted on one particle

The drag force applied by the central blackbody radiation to the electron isF = NscP where P is given byEquation (A17). This yields,

F() = 2

4x2mec2

re 3

g(y) ( ), (B6)where = . ForceF vanishes if= ; in this case the radiation flux measured in the rest frame of the particleis perpendicular to B and cannot accelerate or decelerate it. In a weakly twisted magnetosphere, the magnetic fieldin the outer corona is approximately dipole. Then the saturation velocity at a point r, (spherical coordinates)depends only on and is given by

= =Br

B =

2cos

(1 + 3 cos2 )1/2, p =

2cos

sin . (B7)

The radiative force always pushes the particle towardp = p. This effect may be measured by the drag coefficient,

D rc

1

p

dp

dt. (B8)


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Consider an electron (or positron) with momentum p p = . A small deviation p p causes dragD p p,which may be written as

D= D

1 pp

, (B9)

where

D= 2

4

R

re

3g(y)

x 2 4 10

4

x

g(y)

2 kT

0.5 keV3

. (B10)

Herey =b/ corresponds to photons that are resonantly scattered by the electron with pp. The momentump is a strong attractor ifD1. The value ofD is sensitive to y. In particular, in the equatorial plane, we have= 1 and

y 1.6 104

Bpole1015 G

kT

0.5 keV

1 rR

3, (= /2), (B11)

whereBpole is the dipole field at the magnetic pole. The conditionD > 1 corresponds to y 10R.

B.3. Optical depth in the single-fluid approximation

The single-fluid flow has a distribution function fe(p) = (p p) where p(r, ) is the flow momentum. Then any

photon of energy may only be scattered on the infinitesimally thin resonance surface defined by (1 )= B,where = cos describes the photon angle relative to the flow velocity. The optical depth of the resonant surfacemay be obtained from Equation (A9), which gives

d

ds = 22rec n (1 )( B), (B12)

where = (1 ). One can use the identity,

( B) =k

(s sk) dds( B)

. (B13)The locationsk on the photon trajectory is where the photon crosses the resonant surface. Performing the integrationovers along the photon trajectory, one finds the optical depth for one crossing of the resonant surface,

=22rec n (1 )

dds ( B)

. (B14)

If we specialize to the case of central photons emitted by the neutron star with the polarization,6 then= 1 and= Br/B. For a moderately twisted dipole magnetosphere, the electric current density is given by j cB/4Rmax(Beloborodov 2009), and

n= M jev

MB4eRmax

. (B15)

Note that n is small on field lines with a large Rmax, which implies a low optical depth near the axis; this fact wasalso emphasized by Thompson et al. (2002) who used a self-similar twist model.

The expression for the optical depth becomes particularly simple if the outflow has the equilibrium momentump= p. Then d/ds= 0, i.e. remains constant along the radial ray through the outflow. This fact can be derivedby noting that the Doppler factor (1 ) = 1 is a function of only it does not depend on s = r for theapproximately dipole magnetosphere. It is also easy to see that dB/ds= 3B/r, and we find

=

12 M

sin4

cos (1 + 3 cos2 )1/2. (B16)

The single-fluid model with p = p may approximate the outflow only sufficiently close to the equatorial plane where1 > M1. Nevertheless, the approximate Equation (B16) shows a general feature: the optical depth seen by thecentral photons is dramatically increased toward the equatorial plane ( sec ) and dramatically reduced toward theaxis ( sin4 ). As a result, a distant observer can see the unscattered radiation from the neutron-star surface whenthe line of sight is within a moderate angle < /4 from the polar axis. This feature becomes even more pronouncedin the full radiative transfer problem where the relativistic outflow is decelerated by the reflected radiation from theouter corona. Then scattering of the central radiation is negligible in the entire relativistic zone of the outflow.

6 The neutron-star radiation is dominated by the polarization(Silantev & Iakovlev 1980). In addition, for the outflow with theequilibrium momentum p considered below, the scattering of

photons (even if they were included) would be suppressed. In theoutflow rest frame, the photons move perpendicular to B, and theresonant cross section for the polarization mode vanishes.


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B.4. Drag force exerted on a plasma with a broad distribution function

Equation (B6) describes the drag force exerted by the central thermal radiation on a particle with a given momentump = . One can also consider a collection of particles with a momentum distribution fe(p) and derive the averageforce per particle F=n1(dP/dV dt). Equation (A20) gives

F= re4c2

R

r

23B

( ) fe(p)2(1 )3(exp y 1)dp. (B17)

where y is given in Equation (B5). The same result is obtained by averaging the forceF given by Equation (B6),F= F(p)fe(p) dp.

Equation (B17) simplifies when the plasma is described by the waterbag distribution function fe (Section 5.1); itleads to a straighforward calculation of the flow dynamics in the central radiation field. We use this simple outflowmodel as the first trial to initiate the iterations that converge to the solution shown in Figure 9. In the final solution,the drag exerted by the central radiation turns out negligible in the relativistic zone; instead, the outflow decelerationis controlled by the radiation streaming from the equatorial reflector, as discussed in Section 6. Then the force Fderived in this section may be of interest only in the non-relativistic zone.

C. NON-RESONANT SCATTERING

Non-resonant scattering is not limited by the resonance condition, and hence many more photons can participatein scattering, although with a smaller cross section. Below we discuss the effect of non-resonant scattering on thedynamics ofe flow around magnetars.

Non-resonant scattering occurs mainly with photons in the Wien peak of the thermal radiation flowing directly fromthe neutron star, which dominates the photon density around the star. Relativistic particles see the thermal photons(of typical energyE 3kT) blueshifted as E= (1 )Ewhere = cos describes the photon direction relative tothe particle velocity in the lab frame. Sufficiently far from the star (whereR2/r2 < 1 Br/B) the radiation can beapproximated as a narrow radial beam; then = Br/B. In general, is a function of the particle position r, , inthe magnetosphere. For an approximately dipole field, = 2 cos (1 + 3 cos2 )1/2 is a function of the polar angle only.

Magnetic field strongly affects the non-resonant scattering cross section if E < B = bmec2. If the electronis relativistic, the target photons are aberrated in the electron rest frame, cos = = ( )/(1 ). In thelimit 1, even photons with the polarization have electric fields almost perpendicular to B, which makes theirscattering inefficient. For photons with E B, the non-resonant scattering cross section is given by (e.g. Canutoet al. 1971)

T E

B

2

+sin2

2

,

T E

B

2

, E

B . (C1)

We assume E mec2 and neglect Klein-Nishina corrections. Most of the radiation emitted by the neutron star hasthe polarization.

The energy loss of the electron due to scattering is given by

Ee =

d

dE(1 ) (E)I(n, E)

E

E E , (C2)

wheren is the unit vector describing the photon direction in solid angled, andE =Eis the mean expectation forthe photon energy after scattering. This gives,

Ee =

d

dE(1 ) 2(1 ) 1 I(n, E) (E) dE. (C3)

In the simplest case of Thomson scattering of isotropic radiation, averaging over random gives the standard result

Ee =(4/3)T c U 22, where U is the energy density of radiation. In our case, < T, and the radiation field isnot isotropic; far from the star it is better approximated as a central beam.

Using Equation (C3) one can show that non-resonant scattering makes a small contribution to the radiative dragcompared with resonant scattering, and hence its inclusion in the calculation does not significantly change the outflowsolution shown in Figure 9. Consider first the non-relativistic zonep+ < 1. An upper bound on the non-resonant Eeis obtained if we substitute into Equation (C3) (E) = T and = 0. This gives,

Ee= Tc p2 U. (C4)

The drag coefficient due to non-resonant scattering is defined similar to Equation (B8). Using U Lth/4r2c anddp/dt= Ee/mec

2, one obtains

D TLth4 r mec3

0.2 Lth,35 r16 1. (C5)


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In the relativistic zone p+ > 1, the upper bound given by Equation (C5) increases proportionally to and becomesuseless, because it does not take into account the strong reduction of the scattering cross section below T. In thiszone, the outflow adjusts so that it can resonantly scatter photons with E 7kT and 0.5 (photons flowing fromthe equatorial reflector). This implies that the main targets for non-resonant scattering (photons flowing from the star

with E 3kT and > 0) have energies well below the resonance energy, E (0.1 0.2)B. Then the scatteringcross section is strongly reduced below T according to Equation (C1). When this reduction is taken into account,one obtainsD

electron-positron flows around magnetars andrei m. beloborodov physics department and columbia...

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