PARTICLE ACCELERATOR IN PULSAR MAGNETOSPHERES: SUPER-GOLDREICH-JULIAN CURRENTWITH ION EMISSION FROM THE NEUTRON STAR SURFACE

Kouichi Hirotani1

ASIAA/National Tsing Hua University-TIARA, P.O. Box 23-141, Taipei, Taiwan; hirotani@tiara.sinica.edu.tw

Received 2006 April 2; accepted 2006 August 1

ABSTRACT

We investigate the self-consistent electrodynamic structure of a particle accelerator in the Crab pulsar magneto-sphere on the two-dimensional poloidal plane, solving the Poisson equation for the electrostatic potential togetherwith the Boltzmann equations for electrons, positrons, and �-rays. If the transfield thickness of the gap is thin, thecreated current density becomes sub-Goldreich-Julian, giving the traditional outer-gap solution but with negligible�-ray luminosity. As the thickness increases, the created current increases to become super-Goldreich-Julian, giving anew gap solution with substantially screened acceleration electric field in the inner part. In this case, the gap extendstoward the neutron star with a small-amplitude positive acceleration field, extracting ions from the stellar surface as aspace-charge-limited flow. The acceleration field is highly unscreened in the outer magnetosphere, resulting in a �-rayspectral shape that is consistent with the observations.

Subject headinggs: gamma rays: observations — gamma rays: theory — magnetic fields — methods: numerical —pulsars: individual (Crab)

Online material: color figures

1. INTRODUCTION

The Energetic Gamma-Ray Experiment Telescope (EGRET)aboard theCompton Gamma-Ray Observatory has detected pulsedsignals from at least six rotation-powered pulsars (e.g., for the Crabpulsar; Nolan et al. 1993; Fierro et al. 1998). Since interpreting�-rays should be less ambiguous compared with reprocessed, non-thermal X-rays, the �-ray pulsations observed from these objectsare particularly important as a direct signature of basic nonther-mal processes in pulsar magnetospheres, and potentially shouldhelp to discriminate among different emission models.

The pulsarmagnetosphere can be divided into two zones (Fig. 1):the closed zone filled with a dense plasma corotating with the star,and the open zone in which plasma flows along the open field linesto escape through the light cylinder. The last open field lines formthe border of the open magnetic field line bundle. In all the pulsaremission models, particle acceleration and the resulting photonemissions take place within this open zone.

On the spinning neutron star surface, an electromotive force,EMF � �2B�r

3� /c

2 � 1016:5 V, is exerted from themagnetic poleto the rim of the polar cap. In this paper, we assume that both thespin and magnetic axes reside in the same hemisphere; that is,6 = m > 0, where6 represents the rotation vector and m the stel-lar magnetic moment vector. This strong EMF causes the mag-netospheric currents that flow outward in the lower latitudes andinward near the magnetic axis (Fig. 2, left). The return current isformed at large distances where Poynting flux is converted intokinetic energy of particles or dissipated (Shibata 1997).

Attempts to model the particle accelerator have traditionallyconcentrated on two scenarios: polar-cap models with emissionaltitudes of�104 cm to several neutron star radii over a pulsar po-lar cap surface (Harding et al. 1978; Daugherty & Harding 1982,1996; Dermer & Sturner 1994; Sturner et al. 1995), and outer-gapmodels with acceleration occurring in the open zone located nearthe light cylinder (Cheng et al. 1986a, 1986b, hereafter CHR86a,

CHR86b; Chiang & Romani 1992, 1994; Romani & Yadigaroglu1995). Both models predict that electrons and positrons are accel-erated in a charge depletion region, a potential gap, by the electricfield along the magnetic field lines to radiate high-energy �-raysvia the curvature and inverse Compton (IC) processes.

In the outer magnetosphere picture of Romani (1996), he es-timated the evolution of high-energy flux efficiencies and beamingfractions to discuss the detection statistics, by considering howpair creation on thermal surface flux can limit the accelerationzones. Subsequently, Cheng et al. (2000, hereafter CRZ00) devel-oped a three-dimensional outer magnetospheric gap model, self-consistently limiting the gap size by pair creation from collisionsof thermal photons from the polar cap that is heated by the bom-bardment of gap-accelerated charged particles. The outer gap mod-els of these two groups have been successful in explaining theobserved light curves, particularly in reproducing the wide separa-tion of the two peaks commonly observed from �-ray pulsars(Kanbach 1999; Thompson 2001), without invoking a very smallinclination angle. In these outer gapmodels, they consider that thegap extends from the null surface, where the Goldreich-Julian (GJ)charge density vanishes, to the light cylinder, beyond which thevelocity of a corotating plasmawould exceed the velocity of light,adopting the vacuum solution of the Poisson equation for the elec-trostatic potential (CHR86a).

However, it was analytically demonstrated by Hirotani et al.(2003, hereafter HHS03) that an active gap, which must be non-vacuum, possesses a qualitatively different properties from thevacuum solution discussed in traditional outer-gap models. For ex-ample, the gap inner boundary shifts toward the star as the createdcurrent increases and at last touch the star if the created currentexceeds the GJ value at the surface. Therefore, to understand theparticle accelerator, which extends from the stellar surface to theouter magnetosphere, we have to merge the outer-gap and polar-cap models, which have been separately considered so far.

In traditional polar-cap models, the energetics and pair cascadespectrum have had success in reproducing the observations. How-ever, the predicted beam size of radiation emitted near the stellarsurface is too small to produce the wide pulse profiles that are

A

1 Postal address: TIARA,Department of Physics,National TsingHuaUniversity,101, Section 2, Kuang Fu Road, Hsinchu 300, Taiwan.

The Astrophysical Journal, 652:1475Y1493, 2006 December 1

# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.

1475

observed from high-energy pulsars. Seeking the possibility of awide hollow cone of high-energy radiation due to the flaring offield lines, Arons (1983) first examined the particle accelera-tion at the high altitudes along the last open field line. This typeof accelerator, or the slot gap, forms because the pair formationfront (PFF), which screens the accelerating electric field, Ek, inawidth comparable to the neutron star radius, occurs at increasinglyhigher altitude as the magnetic colatitude approaches the edgeof the open field region (Arons & Scharlemann 1979). Muslimov&Harding (2003, hereafter MH03) extended this argument by in-cluding two new features: acceleration due to spacetime dragging,and the additional decrease of Ek at the edge of the gap due to thenarrowness of the slot gap. Moreover, Muslimov & Harding(2004a, 2004b, hereafter MH04a, MH04b) matched the high-altitude slot gap solution for Ek to the solution obtained at loweraltitudes (MH03), and found that the residual Ek is small andconstant, but still large enough at all altitudes to maintain theenergies of electrons, which are extracted from the star, above5 TeV.

It is noteworthy that the polar-slot gap model proposed byMH04a and MH04b is an extension of the polar-cap model intothe outer magnetosphere, assuming that the plasma flowing in thegap consists of only one sign of charges. This assumption is self-consistently satisfied in their model, because pair creation in theextended slot gap occurs at a reduced rate and the pair cascade dueto inward-migrating particles does not take place. In the polar-slotgap model, the completely charge-separated, space-charge-limitedflow (SCLF) leads to a negative Ek for6 = m > 0. However, weshould note here that the electric current induced by the negativeEk (Fig. 2, right) contradictswith the global current patterns (Fig. 2,left), which is derived by the EMF exerted on the spinning neutron-star surface, if the gap is located near the last-open field line. (Notethat the return current sheet is not assumed on the last-open fieldline in the slot gap model.)On these grounds, we are motivated by the need to contrive an

accelerator model that predicts a consistent current direction withthe global requirement. To this aim, it is straightforward to extendrecent outer-gap models, which predict opposite Ek to polar-cap models, into the inner magnetosphere. Extending the one-dimensional analysis along the field lines in several outer-gapmodels (Hirotani & Shibata 1999a, 1999b, 1999c; HHS03), Takataet al. (2004, hereafter TSH04) and Takata et al. (2006, hereafterTSHC06) solved the Poisson equation for the electrostatic po-tential on the two-dimensional poloidal plane, and revealed thatthe gap inner boundary is located inside of the null surface owingto the pair creation within the gap, assuming that the particle mo-tion immediately saturates in the balance between electric andradiation-reaction forces.In the present paper, we extend TSH04 and TSHC06 by solv-

ing the particle energy distribution explicitly, and by consideringa super-GJ current solution with ion emission from the neutronstar surface. In x 2,we formulate the basic equations and boundaryconditions. We then apply it to the Crab pulsar in x 3 and comparethe solution with MH04 in x 4.

2. GAP ELECTRODYNAMICS

In this section, we formulate the basic equations to describethe particle accelerator, extending the method first proposed byBeskin et al. (1992) for black hole magnetospheres.

Fig. 1.—Schematic figure (side view) of the two representative acceleratormodels. The small filled circle represents the neutron star.

Fig. 2.—Schematic picture of electric current in the pulsar magnetosphere. Left: Global electric current due to the EMF exerted on the spinning neutron star surface when6 = m > 0. Right: Current (downward arrows) derived in the inner-slot gap (shaded region). [See the electronic edition of the Journal for a color version of this figure.]

HIROTANI1476 Vol. 652

2.1. Background Geometry

Around a rotating neutron star with angular frequency �,mass M, and moment of inertia I, the background spacetimegeometry is given by (Lense & Thirring 1918)

ds2 ¼ gtt dt2þ 2gt’ dt d’þ grr dr

2 þ g�� d�2 þ g’’ d’

2; ð1Þ

where

gtt � 1� rg

r

� �c2; gt’ � ac

rg

rsin2�; ð2Þ

grr � � 1� rg

r

� ��1

; g�� � �r2; g’’ � �r2 sin2�; ð3Þ

rg � 2GM /c2 indicates the Schwarzschild radius, and a �I�/(Mc) parameterizes the stellar angular momentum; secondand higher order terms in the expansion of a/rg are neglected.At radial coordinate r, the inertial frame is dragged at angularfrequency

! � �gt’g’’

¼ I

Mr2�

rg

r�

r�

r

� �3

� ¼ 0:15�I45r�36 ; ð4Þ

where r� represents the stellar radius, I45 � I /1045 ergs cm2,and r6 � r/10 km.

2.2. Poisson Equation for Electrostatic Potential

The first kind of equation we have to consider is the Poissonequation for the electrostatic potential, which is given by Gauss’slaw as

9�Ft� ¼ 1ffiffiffiffiffiffi�g

p @�

ffiffiffiffiffiffi�gp

�2w

g�� �g’’Ft� þ gt’F’�

� �� �¼ 4�

c2�;

ð5Þ

where9 denotes the covariant derivative, the Greek indices runover t, r, �, and ’, �gð Þ1/2¼ grrg���

2w

� �1/2¼ cr2 sin �, and

�2w � g2t’ � gttg’’ ¼ c2 1� rg

r

� �r2 sin2�: ð6Þ

If there is an ion emission from the stellar surface into the mag-netosphere, the total real charge density � is given by

� ¼ �e þ �ion; ð7Þ

where �e denotes the sum of positronic and electronic chargedensities, while �ion does the ionic one. The six independent com-ponents of the field-strength tensor give the electromagnetic fieldobserved by a distant static observer (not by the zero-angular-momentum observer) such that (Camenzind 1986a, 1986b)

Er ¼ Frt; E� ¼ F�t; E’ ¼ F’t; ð8Þ

Br ¼gtt þ gt’�ffiffiffiffiffiffi�gp F�’; B� ¼

gtt þ gt’�ffiffiffiffiffiffi�gp F’r; B’ ¼ � �2

wffiffiffiffiffiffi�gp Fr�;

ð9Þ

where F�� � A�;� � A�;� and A�;� denotes the vector potentialA� partially differentiated with respect to x� . The strength ofthe poloidal component of the magnetic field is defined as

Bp � c2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�grr Brð Þ2�g�� B�ð Þ2

qgtt þ gt’�

: ð10Þ

Assuming that the electromagnetic fields are unchanged in thecorotating frame, we can introduce the noncorotational potential� such that

F� t þ �F�’ ¼ �@��(r; �; ’� �t); ð11Þ

where � ¼ t; r; �; ’. If FAt þ �FA’ ¼ 0 holds for A ¼ r, �, theangular frequency � of a magnetic field is conserved along thefield line. On the neutron-star surface, we impose F�t þ �F�’ ¼0 (perfect conductor) to find that the surface is equipotential; thatis, @�� ¼ @t�þ �@’� ¼ 0 holds. However, in a particle ac-celeration region, FAt þ �FA’ deviates from 0 and the magneticfield does not rigidly rotate (even though the deviation fromuniform rotation is small when the potential drop in the gap ismuch less than the EMF exerted on the spinning neutron starsurface). The deviation is expressed in terms of �, which givesthe strength of the acceleration electric field that is measured by adistant static observer as

Ek �B

B= E ¼ Bi

B(Fit þ �Fi’) ¼

B

B= (�:�); ð12Þ

where the Latin index i runs over spatial coordinates r, �, ’,and an identity BrFr’ þ B�F�’ ¼ 0 is used.

Substituting equation (11) into equation (5), we obtain thePoisson equation for the noncorotational potential,

� c2ffiffiffiffiffiffi�gp @�

ffiffiffiffiffiffi�gp

�2w

g��g’’@��

� ¼ 4� �� �GJð Þ; ð13Þ

where the general relativistic GJ charge density is defined as

�GJ �c2

4�ffiffiffiffiffiffi�g

p @�

ffiffiffiffiffiffi�gp

�2w

g��g’’(�� !)F’�

� �: ð14Þ

Using g rr ¼ 1/grr, g�� ¼ 1/g��, g

r� ¼ g�r ¼ 0, g tt ¼ �g’’ /�2w,

g’t ¼ gt’ /�2w, and g’’ ¼ �gtt /�

2w, taking the limit r3 rg, and

noting that @r½r (�g��)1/2B�� � @�½(�grr)

1/2Br� gives the toroidalcomponent of : < B, we find that equation (14) reduces to theordinary, special relativistic expression of the GJ charge density(Goldreich & Julian 1969; Mestel 1971).

Instead of (r, �, ’), we adopt in this paper the magnetic co-ordinates (s, ��, ’�) such that s denotes the distance along a mag-netic field line, and �� and ’� represent the magnetic colatitudeand the magnetic azimuth of the point where the field line inter-sects the stellar surface, respectively. Defining that �� ¼ 0 cor-responds to the magnetic axis and ’� ¼ 0 to the plane on whichboth the rotation and the magnetic axes reside, we can computespherical coordinate variables as

r s; ��; ’�ð Þ ¼ r� þZ s

0

Br s0; �; ’� �tð ÞB s0; �; ’� �tð Þ ds0; ð15Þ

� s; ��; ’�ð Þ ¼ � 0; ��; ’�ð Þ þZ s

0

B� s0; �; ’� �tð ÞB s0; �; ’� �tð Þ ds0;

ð16Þ

’ s; ��; ’�ð Þ � �t ¼ ’� þZ s

0

B’ s0; �; ’� �tð ÞB s0; �; ’� �tð Þ ds0; ð17Þ

where �(0; ��; ’�) satisfies sin � 0; ��; ’�ð Þ cos ’� sin � inc þcos � 0; ��; ’�ð Þ cos � inc ¼ cos ��; � inc represents the angle be-tween the rotation and magnetic axes. We can numerically com-pute the transformation matrix @xi/@x j 0 and its inverse @xi

0/@x j

PARTICLE ACCELERATOR IN PULSAR MAGNETOSPHERES 1477No. 2, 2006

from equations (15)Y(17), where x1 ¼ r, x2 ¼ �, x3 ¼ ’, x10 ¼ s,

x20 ¼ ��, and x3

0 ¼ ’�. Substituting

@

@xi¼ @s

@xi@

@sþ @��

@ xi@

@��þ @’�

@ xi@

@’�ð18Þ

into equation (13), and utilizing @/@t ¼ ��@/@’, we obtain thefollowing form of Poisson equation, which can be applied toarbitrary magnetic field configurations:

�c2g’’�2w

(gss@ 2s þg����@ 2

��þg’�’�@ 2

’�þ2g s��@s@��þ2g��’�@��@’�

þ2g’�s@’�@s)�� As@sþA��@�� þA’�@’�

� �� ¼ 4�(�� �GJ);

ð19Þ

where (see Appendix for explicit expressions)

g i0j 0 ¼ g��

@ xi0

@ x�@ x j 0

@ x�

¼ g rr@ xi

0

@r

� �;’

@ x j 0

@r

� �;’

þg��@ xi

0

@�

� ’; r

@ x j 0

@�

� ’; r

� k0

�2w

@ xi0

@’

� r; �

@ x j 0

@’

� r; �

ð20Þ

and

Ai 0 � c2ffiffiffiffiffiffi�gp

(@r

g’’�2w

ffiffiffiffiffiffi�gp

g rr@ xi

0

@r

� �;’

" #

þ @�g’’�2w

ffiffiffiffiffiffi�gp

g��@ xi

0

@�

� ’;r

" #)

�c2g’’�2w

k0

�2w

@ 2xi0

@’2

� r;�

: ð21Þ

The light surface, which is a generalization of the light cylinder,is obtained by setting k0 � gtt þ 2gt’�þ g’’�

2 to be zero (e.g.,Znajek 1977; Takahashi et al. 1990). It follows from equation (12)that the acceleration electric field is given by Ek ¼ �(@�/@s)��;’�

.Let us briefly consider equation (19) near the polar cap sur-

face of a nearly aligned rotator. Since s � r � r�, �T1, andB’B

’TB2, we obtain (Scharlemann et al. 1978, hereafterSAF78; Muslimov & Tsygan 1992, hereafter MT92)

� 1

r2@

@rr2

@�

@r

� � 1

r2(1� rg=r)

@��@�

� 2

’; r

;1

��

@

@����

@�

@��

� þ 1

�2�

@ 2�

@’2�

� �¼ 4�(�� �GJ): ð22Þ

Noting that the solid angle element in the metric of magneticcoordinates is given by (to the lowest order in �2)

g���� d�2� þ g’�’� d’

2� ¼ r2�

B(0; ��; ’�)

B(s; ��; ’�)d�2� þ sin2��d’

2�

� �;

ð23Þ

we find that the factor

g���� ¼ 1

r2@��@�

� 2

’;r

¼ 1

r2�

B(s; ��; ’�)

B(0; ��; ’�)ð24Þ

expresses the effect of magnetic field expansion in equation (22).In the same manner, in the general equation (19), magnetic fieldexpansion effect is essentially contained in g���� , g��’� , and g’�’� ,or equivalently, in the coefficients of the second-order transfieldderivatives. In what follows, we assume that the azimuthal dimen-sion is large compared with the meridional dimension and neglect’� dependences.

2.3. Particle Boltzmann Equations

The second kind of equations we have to consider is theBoltzmann equations for particles. At time t, position r, and mo-mentum p, the distribution function Nþ of positrons (or N� ofelectrons) obeys the following Boltzmann equation:

@N�

@tþ v = :N� þ qEþ v

c< B

� �=@N�

@p¼ S�(t; r; p); ð25Þ

where v � p/(me�);me refers to the restmass of the electron, q thecharge on the particle, and � � 1/ 1� ( vj j/c)2

�1/2is the Lorentz

factor. In a pulsar magnetosphere, the collision term Sþ (or S�)consists of the terms representing the appearing and disappearingrates of positrons (or electrons) at r and p per unit time per unitphase-space volume due to pair creation, pair annihilation, and theenergy transfer due to IC scatterings and synchrocurvature process.Imposing a stationary condition

@

@tþ �

@

@�¼ 0; ð26Þ

utilizing: = B ¼ 0, and introducing dimensionless particle den-sities per unit magnetic flux tube such that n� ¼ N�/(�B/2�ce),we can reduce the particle Boltzmann equations as

c cos @n�@s

þ dp

dt

@n�@p

þ d

dt

@n�@

¼ S�; ð27Þ

where the upper and lower signs correspond to the positrons(with charge q ¼ þe) and electrons (q ¼ �e), respectively; p �j pj and

dp

dt� qEk cos � PSC

c; ð28Þ

d

dt� �

qEk sin

pþ c

@ ln B1=2� �

@ssin ; ð29Þ

ds

dt¼ c cos : ð30Þ

( Introduction of the radiation-reaction force, PSC/c, is discussedin the next paragraph.) For outward- (or inward-) migrating par-ticles, cos > 0 (or cos < 0). Sincewe consider relativistic par-ticles, we obtain � ¼ p/(mec). The second term on the right-handside of equation (29) shows that the particle’s pitch angle evolvesdue to the variation of B (e.g., x 12.6 of Jackson 1962, p. 591). Forexample, withoutEk, inward-migrating particles would be reflectedby the magnetic mirrors. Using n�, we can express �e as

�e ¼�B

2�c

ZZnþ(s;��;’�;�;)� n�(s; ��; ’�; �; )½ � d� d:

ð31Þ

HIROTANI1478 Vol. 652

The radiation-reaction force due to synchrocurvature radiationis given by (Cheng & Zhang 1996; Zhang & Cheng 1997)

PSC

c¼ e2�4Q2

12rc1þ 7

r2c Q22

� ; ð32Þ

where

rc �c2

rB þ �cð Þ c cos =�cð Þ2þrB!2B

; ð33Þ

Q22 � 1

rB

r2B þ �crB � 3�2c

�3c

cos4þ 3

�ccos2þ 1

rBsin4

� ;

ð34Þ

rB � �mec2 sin

eB; !B � eB

�mec; ð35Þ

and �c is the curvature radius of the magnetic field line. In thelimit of ! 0 (or �c ! 1), equation (32) becomes the expres-sion of pure curvature (or pure synchrotron) emission.

Let us briefly discuss the inclusion of the radiation-reactionforce, PSC/c, in equation (28). Except for the vicinity of the star,the magnetic field is much less than the critical value (Bcr � 4:41 ;1013 G) so that quantum effects can be neglected in synchrotronradiation. Thus,we regard the radiation-reaction force,which is con-tinuous, as an external force acting on a particle. Near the star, if�(B/Bcr) sin > 0:1 holds, the energy loss rate decreases fromthe classical formula (Erber 1966). If �(B/Bcr) sin > 1 holdsvery close to the star, the particle motion perpendicular to thefield is quantized and the emission is described by the transitionsbetween Landau states; thus, equations (28) and (32) break down.In this case, we artificially put ¼ 10�20, which guarantees purecurvature radiation after the particles have fallen onto the ground-state Landau level, avoiding discussions of the detailed quan-tum effects in the strong-B region. This treatment will not affectthe main conclusions of this paper, because the gap electrody-namics is governed by the pair creation taking place not veryclose to the star.

Collision terms are expressed as

S�(s; ��; ’�; �; ) ¼

�Z 1

�1

d�c

ZE�<�

dE��IC(E�; �; �c)n�(s; ��; ’�; �; )

þZ 1

�1

d�c

Z�i>�

d�i eIC(�i; �; �c)n�(s; ��; ’�; �i; )

þZ 1

�1

d�c

ZdE�

;@��(E�;�; �c)

@�þ @�B(E�; �; �c)

@�

� �B�

Bg�(r; E�; k)

� ;

ð36Þ

where �c refers to the cosine of the collision angle between theparticles and the soft photons for inverse Compton scatterings(ICS), between the �-rays and the soft photons for two-photonpair creation, and between the �-rays and the local magnetic fieldlines for one-photon pair creation. The function g represents the�-ray distribution function divided by �B�/(2�ce) at energy E�,momentum k, and position r, where B� denotes the polar-cap mag-netic field strength. Here k should be understood to represent the

photon propagation direction, because E� and k are related withthe dispersion relation (see x 2.4). Since pair annihilation is neg-ligible, we do not include this effect in equation (36).

If we multiply d� on both sides of equation (36), the first (orthe second) term on the right-hand side represents the rate of par-ticles disappearing from (or appearing into) the energy intervalmec

2� andmec2(�þ d�) due to IC scatterings; the third termdoes

the rate of two-photon and one-photon pair creation processes.The IC redistribution function �IC(E�; �; �c) represents the

probability that a particle with Lorentz factor � upscatters pho-tons into energies between E� and E� þ dE� per unit time whenthe collision angle is cos�1�c. On the other hand,

eIC(�i; �; �c)

describes the probability that a particle changes Lorentz factorfrom �i to � in a scattering. Thus, energy conservation gives

eIC �i; �f ; �c

� �¼ �IC �i � �f

� �mec

2; �i; �c

�: ð37Þ

The quantity �IC is defined by the soft photon flux dFs/dEs andthe Klein-Nishina cross section �KN as follows (HHS03):

�IC(E�; �; �c) ¼ (1� ��c)

;

Z Emax

Emin

dEs

dFs

dEs

Z bi

bi�1

dE�

dE 0�

dE�

Z 1

�1

d�0�

d�0KN E�; �; �c

� �dE 0

� d�0�

;

ð38Þ

where � � (1� 1/�2)1/2 is virtually unity, �� is the solid angleof upscattered photon, the prime denotes the quantities in the elec-tron (or positron) rest frame, and E� ¼ (bi�1 þ bi)/2. In the restframe of a particle, a scattering always takes place well above theresonance energy. Thus, theKlein-Nishina cross section can be ap-plied to the present problem. The soft photon flux per unit photonenergy Es (in units of s

�1 cm�2) is written as dFs/dEs and is givenby the surface blackbody emission with redshift corrections ateach distance from the star.

The differential pair-creation redistribution function is givenby

@��@�

(E�; �; �c) ¼ 1� �cð ÞZ 1

Eth

dEs

dFs

dEs

d�p(E�; �; �c)

d�;

ð39Þ

where the pair-creation threshold energy is defined by

Eth �2

1� �c

1

E�; ð40Þ

and the differential cross section is given by

d�p

d�¼ 3

8�T

1� � 2CM

E�

;1þ � 2

CM 2� �2CM

� �1� � 2

CM�2CM

�2�4

CM 1� �2CM

� �21� � 2

CM�2CM

� �2" #

; ð41Þ

�T refers to the Thomson cross section, and the center-of-massquantities are defined as

�CM � � 2�mec2 � E�

�CME�; � 2

CM � 1� 2 mec2ð Þ2

1� �cð ÞEsE�: ð42Þ

Since a convenient form of @�B/@� is not given in the liter-ature, we simply assume that all the particles are created at the

PARTICLE ACCELERATOR IN PULSAR MAGNETOSPHERES 1479No. 2, 2006

energy �mec2 ¼ E� /2 for magnetic pair creation. This treatment

does not affect the conclusions in the present paper.Let us briefly mention the electric current per magnetic flux

tube. With projected velocities, c cos , along the field lines, elec-tric current density in units of �B/(2�) is given by

jgap(s; ��) ¼ je(s; ��)þ jion(��); ð43Þ

where

je �Z Z

n� þ nþð Þ cos dp d; ð44Þ

jion denotes the current density carried by the ions emitted fromthe stellar surface. Since dp/dt and d/dt in equation (27) dependon momentum variables p and , je and hence jgap are not con-served along the field line in an exact sense.

Nevertheless, jgap is kept virtually constant for s. This is be-cause most of the particles have relativistic velocities projectedalong themagnetic field lines at each point. For example, considera situation inwhich a pair is created inwardly at position s ¼ s1. Inthis case, the positron will return after migrating a certain distance(say, s, which is positive). In s1 � s < s < s1, the positron doesnot contribute for the electric current, because both the inward andthe outward current cancel each other in a stationary situation weare dealing with, provided that the projected velocity along thefield line is relativistic. Only when the positronic trajectory on (s,p cos ) space becomes asymmetricwith respect to the p cos ¼0 axis, owing to the synchrotron radiation, which is important ifjcos jT1 (see Fig. 14 of Hirotani & Shibata 1999a), does thereturning positron have a nonvanishing contribution for thecurrent density at s � s1 � s. In s > s1, positronic pitch angle issmall enough to give a spatially constant contribution to the cur-rent density (per magnetic flux tube). For electrons, it always hasan inward relativistic projected velocity and hence gives a spa-tially constant contribution to the current density. In practice, thecontribution of the returning particles with an asymmetric trajec-tory around � 90

�on the current density, can be neglected when

we discuss the jgap. Thus, we can regard jgap as virtually conservedeven though dp/dt and d/dt have p- and -dependences.

2.4. �-Ray Boltzmann Equations

The third kind of equationswehave to consider is theBoltzmannequation for �-rays. In general, the distribution function g ofthe �-rays with momentum k obeys the following Boltzmannequation:

@g

@tþ c

k

kj j= :g(t; r; k) ¼ S�(t; r; k); ð45Þ

where jkj2 � �kiki; S� is given by

S� ¼�Z 1

�1

d�c

Z 1

1

d�@p r; �; �cð Þ

@�g r; E�;

k�

kr;k’

kr

�

þZ 1

�1

d�c

Z 1

1

d��IC E�; �; �c

� � B

B�n� s; ��; ’�; �; ð Þ

þZ �

0

d

Z 1

1

d�SC E�; �; � � B

B�n� s; ��; ’�; �; ð Þ;

ð46Þ

where SC is the synchrocurvature radiation rate (in units of s�1)

into the energy interval between E� and E� þ dE� by a particle

migrating with Lorentz factor �, and is the pitch angle of theparticles. An explicit expression of SC is given by Cheng &Zhang (1996).Imposing the stationary condition (26), or equivalently, as-

suming that g depends on ’ and t as g ¼ g r; �; ’� �t; kð Þ, weobtain

ck’

jkj � �

� @g

@’þ c

k r

kj j@g

@rþ c

k�

kj j@g

@�

¼ S� r; �; ’; c kj j; k r; k�; k’� �

; ð47Þ

where ’ ¼ ’� �t. To compute ki, we have to solve the photonpropagation in the curved spacetime. Since the wavelength ismuch shorter than the typical system scales, geometrical opticsgives the evolution of momentum and position of a photon by theHamilton-Jacobi equations,

dkr

dk¼ � @kt

@r;

dk�

dk¼ � @kt

@�; ð48Þ

dr

dk¼ @kt

@kr;

d�

dk¼ @kt

@k�; ð49Þ

where the parameter k is defined so that c dk represents the dis-tance (i.e., line element) along the ray path. The photon energyat infinity kt and the azimuthal wavenumber �k’ are conservedalong the photon path in a stationary and axisymmetric spacetime(e.g., in the spacetime described by eqs. [1]Y[3]). TheHamiltoniankt can be expressed in terms of kr, k�, k’, r, and � from the dis-persion relation k�k� ¼ 0, which is a quadratic equation of k�(� ¼ t; r; �; ’). Thus, we have to solve the set of four ordinarydifferential equations (48) and (49) for the four quantities kr, k�,r, and � along the ray. Initial conditions at the emitting point aregiven by ki/jkj ¼ �Bi/jBj, where i ¼ r, �, ’; the upper (or lower)sign is chosen for the �-rays emitted by an outward- (or inward-)migrating particle. When a photon is emitted with energy Elocal

by the particle whose angular velocity is ’, it is related to kt and�k’ by the redshift relation, Elocal ¼ (dt/d�)(kt þ k’’), wheredt/d� is solved from (dt/d�)2(gtt þ 2gt’’þ g’’’

2) ¼ 1. To ex-press the energy dependence of g, we regard g as a function ofkt ¼ E� (i.e., observed photon energy).In this paper, in accordance with the two-dimensional analysis

of equations (19) and (27), we neglect the ’-dependence of g, byignoring the first term in the left-hand side of equation (47). In ad-dition, we neglect the aberration of photons and simply assumethat the �-rays do not have angular momenta and put k’ ¼ 0. Theaberration effects are important when we discuss how the outward-directed �-rays will be observed. However, they can be correctlytaken into account only when we compute the propagation of emit-ted photons in the three-dimensional magnetosphere. Moreover,they are not essential when we investigate the electrodynamics,because the pair creation is governed by the specific intensity ofinward-directed �-rays, which are mainly emitted in a relativelyinner region of the magnetosphere. Thus, it seems reasonable toadopt k’ ¼ 0 when we investigate the two-dimensional gapelectrodynamics.We linearly divide the longitudinal distance into 400 grids from

s ¼ 0 (i.e., stellar surface) to s ¼ 1:4$LC, and the meridionalcoordinate into 16 field lines from �� ¼ �max

� (i.e., the last-openfield line) to �� ¼ �min

� (i.e., gap upper boundary), and consideronly the ’� ¼ 0 plane (i.e., the field lines threading the stellarsurface on the plane formed by the rotation andmagnetic axes). Tosolve the particle Boltzmann equations (27), we adopt the cubic

HIROTANI1480 Vol. 652

interpolated propagation (CIP) schemewith the fractional step tech-nique to shift the profile of the distribution functions n� in thedirection of the velocity vector in the two-dimensionalmomentumspace (e.g.,Nakamura&Yabe 1999). To solve the�-rayBoltzmannequation (47), on the other hand, we do not have to compute theadvection of g in the momentum space, because only the spatialderivative terms remain after integrating over �-ray energy bins,which are logarithmically divided from �1 ¼ 0:511 MeV to �29 ¼28:7 TeV into 29 bins. The �-ray propagation directions, k�/k r, aredivided linearly into 180 bins every ��� ¼ 2�. Since the specificintensity in the ith energy bin at height �� ¼ � k

� ¼ �max� � (k/16)

(�max� � �min� ) is given by

gi; k; l(s) ¼c

������

Z bi

bi�1

g s; � k� ; E�; k �=k r

� �l

h idE�; ð50Þ

the observed �-ray energy flux at distance d is calculated as

Fi;l ¼�y

Pk �zkgi; k; ld 2

; ð51Þ

where�y denotes the azimuthal dimension of the gap at longitudi-nal distance s (¼ $LC in this paper), �zk denotes the meridionalthickness between two field lines with �� ¼ ��k and ��kþ1, andi ¼ 1, 2, 3, : : : , 28, k ¼ 1, 2, 3, : : : , 15, and l ¼ 1, 2, 3, : : : , 180.To compute the phase-averaged spectrum, we set the azimuthalwidth of the �-ray propagation direction, ��� , to be � radians.

Equation (46) describes the �-ray absorption and creation ratewithin the gap. However, to compute observable fluxes, we alsohave to consider the synchrotron emission by the secondary, ter-tiary, and higher-generation pairs that are created outside of thegap. If an electron or positron is created with energy �0mec

2 andpitch angle 0, it radiates the following number of �-rays (inunits of �B�/2�ce) in energies between bi�1 and bi:

dgidn

¼ 2�ce

�B�

Z 1

0

dt

Z bi

bi�1

1

E�

dW

dt dE�dE�; ð52Þ

where

dW

dt dE�¼

ffiffiffi3

pe3B sin 0

hmec2F

E�

Ec

� ; ð53Þ

mec2 d�

dt¼ � 2

3

e4B2 sin20

m2e c

3�2; ð54Þ

F(x) � x

Z 1

x

K5=3(�) d�; ð55Þ

K5/3 is the modified Bessel function of 5/3 order, and Ec �(3h/4�)(eB�2 sin i)/(mec) is the synchrotron critical energy atLorentz factor �. Substituting equations (53) and (54) into equa-tion (52), we obtain

dgidn

¼ 2�ce

�B�

3ffiffiffi3

pm2

e c3

2heB sin 0

Z �0

1

d�

�2

Z bi=Ec

bi�1=Ec

dy

Z 1

y

K5=3(�) d�:

ð56Þ

Note that we assume that particle pitch angle is fixed at ¼ 0,because ultrarelativistic particles emit radiation mostly in the in-stantaneous velocity direction, preventing pitch-angle evolution.Once particles lose sufficient energy, they preferentially lose per-

pendicular momentum; nevertheless, such less energetic particleshardly emit synchrotron photons above MeV energies. On thesegrounds, to incorporate the synchrotron radiation of higher gen-eration pairs created outside of the gap, we add

R11

(dn/d�0)(dgi/dn) d�0 to compute the emission of �-rays in the energyinterval [bi�1, bi] in the right-hand side of equation (45), wheredn/d�0 denotes the particles created between position s and sþ dsin the Lorentz factor interval ½�0; �0 þ d�0�.

2.5. Boundary Conditions

In order to solve the set of partial differential equations (19),(27), and (47) for �, n�, and g, we must impose appropriateboundary conditions. We assume that the gap lower boundary,�� ¼ �max

� , coincides with the last open field line, which is de-fined by the condition that sin �(�grr)

1/2Brþcos �(�g��)1/2B� ¼ 0

is satisfied at the light cylinder on the surface ’� ¼ 0. Moreover,we assume that the upper boundary coincides with a specific mag-netic field line and parameterize this field line with �� ¼ �min� . Ingeneral, �min� is a function of ’�; however,we consider only’� ¼ 0in this paper. Determining the upper boundary from physical con-sideration is a subtle issue, which is beyond the scope of the presentpaper. Therefore, we treat �min� as a free parameter. We measure thetransfield thickness of the gap with

hm � �max� � �min��max�

: ð57Þ

If hm ¼ 1:0, it means that the gap exists along all the open fieldlines. On the other hand, if hmT1, the gap becomes transverselythin and �� derivatives dominate in equation (19). To describe thetransfield structure, we introduce the fractional height as

h � �max� � ���max�

: ð58Þ

Thus, the lower and upper boundaries are given by h ¼ 0 and hm,respectively.

The inner boundary is assumed to be located at the neutronstar surface. For the outer boundary, we solve the Poisson equa-tion to a large enough distance, s ¼ 1:4$LC, which is locatedoutside of the light cylinder. This mathematical outer boundary isintroduced only for convenience in order that the Ek distributioninside of the light cylinder may not be influenced by the artifi-cially chosen outer boundary positionwhenwe solve the Poissonequation. Since the structure of the outermost part of the mag-netosphere is highly unknown,we artificially setEk ¼ 0 if the dis-tance from the rotation axis, $, becomes greater than 0:90$LC.Under this artificially suppressed Ek distribution in$ > 0:90$LC,we solve the Boltzmann equations for outward-migrating parti-cles and �-rays in the range 0< s < 1:4$LC. For inward-migratingparticles and �-rays, we solve only in the range$ < 0:9$LC. Theposition of the mathematical outer boundary (1:4$LC in this case)little affects the results by virtue of the artificial boundary condition,Ek ¼ 0 for $ > 0:9$LC. On the other hand, the artificial outerboundary condition, Ek ¼ 0 for $ > 0:9$LC, affects the calcula-tion of outward-directed �-rays to some degree; nevertheless, it littleaffects the electrodynamics in the inner part of the gap (s< 0:5$LC),which is governed by the absorption of inward-directed �-rays.

First, to solve the elliptic-type equation (19), we impose� ¼ 0on the lower, upper, and inner boundaries. At the mathematicalouter boundary (s ¼ 1:4$LC), we impose @�/@s ¼ 0. Generallyspeaking, the solution ofEk ¼ �(@�/@s)s!0 under these boundary

PARTICLE ACCELERATOR IN PULSAR MAGNETOSPHERES 1481No. 2, 2006

conditions does not vanish at the stellar surface. Let us considerhow to cancel this remaining electric field.

For a super-GJ current density in the sense that �e � �GJ < 0holds at the stellar surface, equation (19) gives a positive electricfield near the star. In this case, we assume that ions are emittedfrom the stellar surface so that the additional positive charge inthe thin nonrelativistic region may bring Ek to zero (for the pos-sibility of free ejection of ions due to a low work function, seeJones 1985; Neuhauser et al. 1986, 1987). The column density inthe nonrelativistic region becomes (SAF78)

�NR ¼ 1

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic�B�

q=mjion

s; ð59Þ

where q/m represents the charge-to-mass ratio of the ions andjion gives the ionic current density in units of �B�/(2�). Equat-ing 4��NR to�(@�/@s)s!0 calculated from relativistic positrons,electrons, and ions, we obtain the ion injection rate jion that can-cels Ek at the stellar surface.

For a sub-GJ current density in the sense that �e � �GJ > 0holds at the stellar surface, � increases outward near the starto peak around s ¼ 0:02$LCY0:10$LC, depending on � inc and�e(s ¼ 0), then decreases to become negative in the outer mag-netosphere. That is, �(@�/@s)s!0 < 0 holds in the inner regionof the gap. In this case, we assume that electrons are emitted fromthe stellar surface and fill out the region where � > 0; thus, weartificially put� ¼ 0 if � > 0 appears. Even though a nonvanish-ing, positive Ek remains at the inner boundary, which is locatedaway from the stellar surface, we neglect such details. This is be-cause the gap with a sub-GJ current density is found to be inac-tive and hence less important, as is demonstrated in x 3.

Second, to solve the hyperbolic-type equations (27) and (47),we assume that neither positrons nor �-rays are injected acrossthe inner boundary; thus, we impose

nþ sin; ��; �; � �

¼ 0; g sin; ��; E�; ��� �

¼ 0 ð60Þ

for arbitrary ��, �, 0 < < �/2, E� , and cos (�� � �B) > 0,where �B designates the outward magnetic field direction. In thesame manner, at the outer boundary, we impose

n� sout; ��; �; ð Þ ¼ 0; g sout; ��; E�; ��� �

¼ 0 ð61Þ

for arbitrary ��, �, �/2 < < �, E� , and cos (�� � �B) < 0.

3. APPLICATION TO THE CRAB PULSAR

Since the formulation described in the foregoing section isgeneric, we specify some of the quantities in x 3.1 before turningto a closer examination in xx 3.2Y3.8.

3.1. Assumptions on Magnetic Field and Soft Photon Field

First, let us specify the magnetic field. Near the star, we adoptthe static (unperturbed by rotation and currents) dipole solutionobtained in the Schwarzschild spacetime (e.g., MT92, and ref-erences therein). That is, in equation (9), we evaluateF�’ andF’r

as

F�’ffiffiffiffiffiffi�gp ¼ f (r)

2�

r3cos�; ð62Þ

F’rffiffiffiffiffiffi�gp ¼ � �

r2d

dr

f (r)

r

� �sin�; ð63Þ

where � is the angle measured from the magnetic axis and

f (r) ¼ �31

2

r

rgþ r

rg

� 2

þ r

rg

� 3

ln 1� rg

r

� " #: ð64Þ

At high altitudes (but within the light cylinder), the open fieldlines deviate from the static dipole to be swept back in the oppositedirection of the rotation and bent toward the rotational equator.There are two important mechanisms that cause the deviation:charge flow along the open field lines, and retardation of an in-clined, rotating dipole. Both of them appear as the first order cor-rection in $/$LC expansion to the static dipole. To study theformer correction,Muslimov&Harding (2005) employed the space-charge-limited longitudinal current solved by MT92, and theyderived an analytic solution of the correction. However, if we dis-card the space-charge-limited flow of emitted electrons and con-sider copious pair creation in the gap, we have to derive a moregeneral correction formula that is applicable for arbitrary longi-tudinal current distribution. To follow up this general issue fur-ther would involve us in other factors than the electrodynamicsof the accelerator and would take us beyond the scope of thispaper. Thus, we consider only the latter correction and adopt theinclined, vacuummagnetic field solution obtained by CRZ00 (theireqs. [B2]Y[B4]).Second, we consider how the toroidal current density, J ’, af-

fects �GJ near the light cylinder. In the outer magnetosphere, gen-eral relativistic effects are negligible; thus, equation (14) becomes

�GJ ¼ �6 = B

2�cþ $

$LC

J ’

c: ð65Þ

Since J ’ is of the order of (�B/2�)($/$LC), the second termappears as a positive correction that is proportional to ($/$LC)

2

and will become comparable to the first term if $/$LC � 1.Thus, to incorporate this special relativistic correction, we adopt

�GJ ¼ �6 = B

2�c1þ �

$

$LC

� 2" #

; ð66Þ

where the constant � is of the order of unity. For example,CHR86a adopted � ¼ 1. Even though a larger value of � is pref-erable to reproduce a harder curvature spectrum above 5 GeVand a larger secondary synchrotron flux around 100 MeV, weadopt a conservative value � ¼ 0:5 in the present paper.Third, we have to specify the differential soft photon flux,

dFs/dEs, which appears in equations (38) and (39). As the pos-sible soft photon fields illuminating the gap, we can consider thefollowing three components in general:

1. Photospheric emission from the hot surface of a coolingneutron star. For simplicity, we approximate this component witha blackbody spectrum with a single temperature, kTs. We assumethat this component is uniformly emitted from the whole neutronstar surface.2. Thermal soft X-ray emission from the neutron star’s polar

cap heated by the bombardment of relativistic particles stream-ing toward the star from the magnetosphere. Since we consider ayoung pulsar in this paper, this component is negligible com-pared to the first component.3. Nonthermal, power-law emission from charged relativistic

particles created outside of the gap in the magnetosphere. Theemitted radiation can be observed from the optical to the �-rayband. Since the nonthermal emission will be beamed away from

HIROTANI1482 Vol. 652

the gap, we assume that this component does not illuminate thegap. The major conclusions in this paper will not be affected bythis assumption, except that the pair creation would increase tosuppress the potential drop and hence the �-ray luminosity if thiscomponent illuminates the gap.

We apply the scheme to the Crab pulsar, adopting four freeparameters, � inc, �, kTs, and hm. Other quantities such as gapgeometry on the poloidal plane, exerted Ek and potential drop,particle density, and energy distribution, as well as the �-ray fluxand spectrum, are uniquely determined if we specify these fourparameters.We consider transversely thin and thick cases in x 3.2and xx 3.3Y3.8, respectively.

3.2. Sub-GJ Current Solution: Traditional Outer-Gap Model

To begin with, let us consider the magnetic inclination � inc ¼70

�, which is more or less close to the value (65

�) suggested by

a three-dimensional analysis in the traditional outer gap model(CRZ00). We adopt kTs ¼ 100 eVas the surface blackbody tem-perature, which is consistent with the observational upper limit,kTs < 180 eV (Tennant et al. 2001). In xx 3.2Y3.4, we adopt � ¼4:0 ; 1030 G cm3 as the magnetic dipole moment, which givesB� ¼ 1:46 ; 1013 G (eq. [10]), assuming r� ¼ 106 cm andM ¼1:4 M. If we evaluate � from the spin-down luminosity E ¼2�4�2/3c3, we obtain � ¼ 3:8 ; 1030 G cm3 for E ¼ 4:46 ;1038 ergs s�1 (e.g., Becker & Trumper 1997). The dependenceof the solution on kTs, �, and � inc is discussed in xx 3.4Y3.6.

Examine a sub-GJ current solution, which is defined by je <�GJ/(�B/2�c)j js¼0 � (1� !/�)Bz�/B�, where Bz� refers to thesurface magnetic field component projected along the rotationaxis; the right-hand side is evaluated at s ¼ 0 and �� ¼ �max

� . Tothis aim, we consider a transversely thin gap, hm ¼ 0:047. Thesolution becomes similar to the vacuum one obtained in the tra-ditional outer-gap model (CHR86a), as the left panel of Figure 3indicates. In this figure, we present Ek(s; h) at discrete height h of2hm/16, 5hm/16, 8hm/16, 11hm/16, and 14hm/16 with dashed, dot-ted, solid, dashYtriple-dotted, and dash-dotted curves, respectively;they are depicted in the right panel with a larger hm (¼ 0:200) forillustration purposes. For one thing, for such a transversely thin,nearly vacuum gap, the inner boundary is located slightly inside ofthe null surface. What is more, Ek maximizes at the central height,

h ¼ hm/2, and remains roughly constant in the entire region of thegap. The solvedEk distributes almost symmetricallywith respect tothe central height; for example, the dashed and dash-dotted curvesnearly overlap each other. Similar solutions are obtained for athinner gap, hm < 0:047.

3.3. Super-GJ Current Solution: Hybrid Gap Structure

Next, let us consider a thicker gap, hm ¼ 0:048. In this case, jebecomes comparable to or greater than j�GJ/(�B/2�c)js¼0 in theupper half region h > hm/2 of the gap, causing the solution todeviate from the vacuum one. In the left panel of Figure 4, wepresent Ek(s; h) at five discrete heights h in the same way asFigure 3. It follows that Ek is screened by the discharge of createdpairs in the innermost region (s < 0:3$LC) in the higher latitudes(h > hm/2). For example, Ek at h ¼ 7hm/8 (dash-dotted curve)deviates from the unscreened solution at the lower latitudes h ¼hm/8 (dashed curve). On the other hand, along the lower fieldlines (h < hm/2), Ek(s) is roughly constant as in the traditionalouter-gap model.

Let us further consider a thicker gap, hm ¼ 0:060. The rightpanel of Figure 4 shows thatEk(s; h) is substantially screened com-pared to themarginally super-GJ case, hm ¼ 0:048 (left panel). Be-cause the gap transfield thickness virtually shrinks in s < 0:5$LC

due to screening in the higher latitudes, Ek in the lower latitudesalso decreases compared to smaller hm cases, as the dashed linesin the left and right panels indicate.

To understand the screening mechanism, it is helpful to ex-amine the Poisson equation (22), which is a good approximationin s < 0:4$LC. In the transversely thin limit, it becomes

� 1

(1� rg=r)

1

r2�

B(r)

B�

@ 2�

@�2�� 4�(�� �GJ)

� 2�B(r)

c

�

�B=2�c� �GJ

�B=2�c

� :

ð67Þ

Since we are interested in the second-order �� derivatives, thisequation is valid not only for a nearly aligned rotator, but also foran oblique rotator. We could directly check it from the general

Fig. 3.—Left: Field-aligned electric field of a sub-GJ current solution at five discrete heights (see right panel) for� inc ¼ 70� and hm ¼ 0:047. The abscissa indicates thedistance along the field line from the star in the unit of the light-cylinder radius. The null surface position at the height h ¼ hm/2 is indicated by the down arrow. Right:Magnetic field lines on the poloidal plane in which both the rotational and magnetic axes reside. Instead of hm ¼ 0:047, hm ¼ 0:200 is adopted for clarity. The thick solidcurves denote the lower and upper boundaries, while the thin dashed, dotted, solid, dashYtriple-dotted, and dash-dotted ones give the same h/hm values as those in the leftpanel; they are h/hm ¼ 2/16, 5/16, 8/16, 11/16, and 14/16, respectively. [See the electronic edition of the Journal for a color version of this figure.]

PARTICLE ACCELERATOR IN PULSAR MAGNETOSPHERES 1483No. 2, 2006

equation (13). Factoring out the magnetic field expansion factor,B(r)/B�, from the two sides, we obtain

� � 1� rg

r

� �B�c

�

�B=2�c� �GJ

�B=2�c

� ; r2� �� � �min

�� �

�max� � ��

� �: ð68Þ

We thus find that Ek � �@�/@s is approximately proportionalto�@(�/B� �GJ/B)/@s. It is, therefore, important to examine thetwo-dimensional distribution of �/B and �GJ/B to understandEk(s; h) behavior.

In Figure 5, we present �/(�B/2�c), �e/(�B/2�c), and�GJ/(�B/2�c), as the solid, dash-dotted, and dashed curves, atnine discrete magnetic latitudes, ranging from h ¼ (4/16)hm,(5/16)hm, : : :, to (12/16)hm for the same parameters as the rightpanel of Figure 4. If there is a cold-field ion emission from thestar, the total charge density (solid curve) deviates from the createdcharge density (dash-dotted curve). It follows that the current is sub-GJ for h (5/16)hm ¼ 0:0187 and super-GJ for h � (6/16)hm ¼0:0225. Along the field lines with super-GJ current, �e � �GJbecomes negative close to the star. This inevitably leads to a posi-tive Ek, which extracts ions from the stellar surface. In this paper,we assume that the extracted ions consist of protons; nevertheless,the conclusions are little affected by the composition of the ex-tracted ions.

In the outer region, �/B levels off in s > 0:5$LC for h > hm/2.Since �GJ/B becomes approximately a linear function of s, Ekremains nearly constant in s > 0:5$LC, in the same manner as intraditional outer-gapmodel. In the inner region, on the other hand,inward-directed �-rays propagate into the convex side due to thefield line curvature, increasing particle density exponentially withh. As a result, the lower part (i.e., smallerh region) becomes nearlyvacuum.For example, ath ¼ hm/4 ¼ 0:015 (Fig. 5, top left panel),the positive �e � �GJ leads to a negative Ek at the stellar surface,inducing no ion emission. Even though the created current is sub-GJ at h ¼ 0:0187, ions are extracted from the surface. This is be-cause the negative�eA � ���GJ in the higher latitudes h� 0:0225cancels the relatively small positive �eA along h ¼ 0:0187 to in-duce a positive Ek at the stellar surface. Such a two-dimensionaleffect in the Poisson equation is also important in the higher alti-tudes (0:1$LC < s < 0:3$LC) along the higher-latitude field lines(h � 0:0225). Outside of the null surface, s > 0:09$LC, there is a

negative �eA in the sub-GJ current region (h 0:015). This neg-ative �eA works to prevent Ek from vanishing in the higher lat-itudes, where pair creation is copious. However, the created pairsdischarge until Ek vanishes, resulting in a larger gradient of � thanthat of�GJ in the intermediate latitudes in 0:0225 h 0:0262. Inthe upper half region (0:03 h < hm ¼ 0:06), @�/@s does nothave to be greater than @�GJ/@s in order to screen Ek.In short, the gap has a hybrid structure: The lower latitudes

(with small h) are nearly vacuum, having sub-GJ current densi-ties, and the inner boundary is located slightly inside of the nullsurface, because the � > 0 region will be filled with the electronsemitted from the stellar surface. The higher latitudes, on the otherhand, are nonvacuum, having super-GJ current densities, and theinner boundary is located at the stellar surface, extracting ions atthe rate such that their nonrelativistic column density at the stellarsurface cancels the strong Ek induced by the negative �� �GJ ofrelativistic electrons, positrons, and ions. The created pairs dis-charge such that Ek virtually vanishes in the region where paircreation is copious. Thus, in the intermediate latitudes betweenthe sub-GJ and super-GJ regions, @�/@s > �GJ/@s holds.Even though the innermost region of the gap is inactive,

general relativistic effect (spacetime dragging effect, in this case)is important to determine the ion emission rate from the stellar sur-face. For example, at h ¼ hm/2 for hm ¼ 0:060 (i.e., central panelin Fig. 5), jion is 69% greater than what would be obtained in theNewtonian limit, �GJ ¼ �6 = B/2�c. This is because the reducedj�GJj near the star (about 15% less than the Newtonian value)enhances the positiveEk, which has to be canceled by a greater ionemission (compared to the Newtonian value). The current, jion, isadjusted so that j�eAjmay balance with the transfield derivative of� near the star. The resulting j�eAj becomes small compared toj�GJj, in the same manner as in traditional polar-cap models, whichhas a negativeEk with electron emission from the star. Although thenonrelativistic ions have a large positive charge density very closeto the star (within 10 cm from the surface), it cannot be resolved inFigure 5. Note that the present calculation is performed from thestellar surface to the outer magnetosphere and does not contain aregion with Ek < 0. It follows that an accelerator having Ek < 0(e.g., a polar-cap or a polar-slot-gap accelerator) cannot existalong the magnetic field lines that have an super-GJ current den-sity created by the mechanism described in the present paper.It is worth examining how Ek changes with varying hm. In

Figure 6, we present Ek(s; h) at the central height h ¼ hm/2. In

Fig. 4.—Same as Fig. 3: Ek(s; h) of two super-GJ current solutions, but for hm ¼ 0:048 (left) and 0.060 (right) at five discrete h values. [See the electronic edition of theJournal for a color version of this figure.]

HIROTANI1484

Fig. 5.—Total (solid curves), created (dash-dotted curves), and Goldreich-Julian (dashed curves) charge densities in �B(s; h)/(2�c) units, for � inc ¼ 70� andhm ¼ 0:060 at nine transfield heights, h. If there is an ion emission from the stellar surface, the total charge density deviates from the created one. [See the electronic editionof the Journal for a color version of this figure.]

Fig. 6.—Field-aligned electric field at h ¼ hm/2 as a function of s/$LC for � ¼ 4:0 ; 1030 G cm3, kTs ¼ 100 eV, and � inc ¼ 70�. Left: The dotted, solid, dashed, anddash-dotted curves correspond to h ¼ 0:047, 0.048, 0.060, and 0.100, respectively. Right: The dash-dotted, dashYtriple-dotted, solid, and dashed curves correspond toh ¼ 0:100, 0.160, 0.200, and 0.240, respectively. [See the electronic edition of the Journal for a color version of this figure.]

the left panel, the dotted, solid, dashed, and dash-dotted curvescorrespond to hm ¼ 0:047, 0.048, 0.060, and 0.100, respectively,while in the right panel, the dash-dotted, dashYtriple-dotted, so-lid, and dashed curves correspond to 0.100, 0.160, 0.200, and0.240, respectively. It follows that the inner part of the gap be-comes substantially screened by the discharge of created pairs ashm increases. It also follows that themaximumof Ek increaseswithincreasinghm forhm < 0:2, because the two-dimensional screeningeffect due to the zero-potential walls becomes less important fora larger hm. To solve particle and �-ray Boltzmann equations, weartificially put Ek ¼ 0 in $ > 0:9$LC, or equivalently in s >1:1$LC for � inc ¼ 70�, as mentioned in x 2.5.

The created current density, je, is presented in Figure 7, as afunction of h. The thin dashed line represents j�GJ/(�B/2�c)js¼0;if je appears below (or above) this line, the created current is sub-(or super-) GJ along the field line specified by h. The open andfilled circles denote the lowest and highest latitudes that are usedin the computation. For hm ¼ 0:047, the solution (dotted curve)is sub-GJ along all the field lines; thus, screening due to the dis-charge is negligible as the dotted curve in the left panel of Fig-ure 6 shows. As hm increases, the solution becomes super-GJfrom the higher latitudes, as indicated by the solid (hm ¼ 0:048),

dashed (hm ¼ 0:060), dash-dotted (hm ¼ 0:100), and dashYtriple-dotted (hm ¼ 0:160) curves in Figure 7. As a result, screeningbecomes significant as hm increases, as Figure 6 shows. Thisscreening of Ek has a negative feedback effect in the sense that jeis regulated below unity. Even though it is not resolved in Fig-ure 6, in the lower latitudes, je grows across the gap height expo-nentially, as CHR86a suggested. For example, je ¼ 3:0 ; 10�10,2:1 ; 10�9, 2:1 ; 10�8, 1:0 ; 10�7, 4:3 ; 10�7, 1:0 ; 10�6, 4:6 ;10�5, and 1:3 ; 10�1 at h/hm ¼ 1/16, 2/16, 3/16, : : :, 8/16, re-spectively, for hm ¼ 0:048 (solid curve). This is because the paircreation rate at height h is proportional to the number of �-raysthat are emitted by charges on all field lines below h.Let us now turn to the emitted �-rays. Figure 8 shows the

phase-averaged �-ray spectrum calculated for three differenthm values. Open circles denote the pulsed fluxes detected byCOMPTEL (below30MeV;Ulmer et al. 1995;Kuiper et al. 2001)andEGRET (above 30MeV;Nolan et al. 1993; Fierro et al. 1998),while the open square does the upper limit obtained by CELESTE(de Naurois et al. 2002). It follows that the sub-GJ (i.e., traditionalouter-gap) solution with hm ¼ 0:047 predicts too little �-ray fluxcomparedwith the observations. (Note that in traditional outer-gapmodels, particle number density is assumed to be the GJ value,while Ek is given by the vacuum solution of the Poisson equation,which is inconsistent.) The maximum flux, which appears aroundGeVenergies, does not become greater than 1011 JyHz for any sub-GJ solutions, whatever values of � inc, �, and kTs we may choose.Thus, we can rule out the possibility of a sub-GJ solution for theCrab pulsar.As hm increases, the increased Ek results in a harder curvature

spectrum, as the solutions corresponding to hm ¼ 0:100 and 0.200indicate. As discussed in x 4.1, the problem of insufficient �-rayfluxesmay be solved ifwe consider a three-dimensional gap struc-ture. However, the secondary synchrotron flux emitted outsideof the gap is too small to explain the flat spectral shape below100MeV. The �-ray spectrum is nearly unchanged for 0:2 hm 0:3. For hm > 0:3, the �-ray flux tends to decrease, because theEk(s) peaks outside of the artificial outer boundary, r sin � ¼0:9$LC, which corresponds to s ¼ 1:1$LC for � inc ¼ 70

�. For

hm > 0:4, the gap virtually vanishes because of the discharge ofcopiously created pairs; in other words, the gap is located out-side of r sin � ¼ 0:9$LC. On these grounds, we cannot reproducethe observed flat spectral shape if we consider kTs ¼ 100 eV, � ¼4:0 ; 1030 G cm3, and � inc ¼ 70�, no matter what value of hmis adopted. Therefore, in the next three subsections, we examine

Fig. 7.—Created current density je (in units of �B/2�) as a function of thetransfield thickness h for � ¼ 4:0 ; 1030 G cm3, kTs ¼ 100 eV, and � inc ¼ 70�.The dotted, solid, dashed, dash-dotted, and dashYtriple-dotted curves corre-spond to h ¼ 0:047, 0.048, 0.060, 0.100, and 0.160, respectively. [See the elec-tronic edition of the Journal for a color version of this figure.]

Fig. 8.—Calculated phase-averaged spectra of the pulsed, outward-directed �-rays for� inc ¼ 70�, kTs ¼ 100 eV, and � ¼ 4:0 ; 1030 G cm3, with three different gapthickness, hm. The flux is averaged over the meridional emission angles between 44� and 58� (solid curve), 58� and 72� (dashed curve), 72� and 86� (dash-dotted curve),86� and 100� (dotted curve), and 100� and 114� (dashYtriple-dotted curve), from the magnetic axis on the plane in which both the rotational and magnetic axes reside.[See the electronic edition of the Journal for a color version of this figure.]

HIROTANI1486 Vol. 652

how the solution changes if we adopt different values of kTs, �,and � inc.

3.4. Dependence on Surface Temperature

In the same manner as in x 3.3, we calculate Ek(s; h ¼ hm/2)for kTs ¼ 150 eVand find that their distribution is similar to thekTs ¼ 100 eVcase (i.e., Fig. 6). For example, sub/super-GJ currentsolutions are discriminated by the condition whether hm is greaterthan 0.048 or not, and the maximum of Ek is 7:2 ; 108 V m�1 for0:20 < hm < 0:24. Other quantities, such as the particle and �-raydistribution functions, are also similar. Therefore,we can concludethat the solution is little subject to change for the variation of kTs,even though the photon-photon pair production rate increaseswith increasing kTs. This is due to the negative feedback effect,which is discussed in x 4.2.

3.5. Dependence on Magnetic Moment

Let us examine how the solution depends on the magneticmoment, �. In Figure 9, we present Ek(s; h ¼ hm/2) for sevendiscrete hm values with � ¼ 6:0 ; 1030 G cm3; in the left panel,the dotted and solid curves correspond to h ¼ 0:039 and 0.041,respectively, instead of h ¼ 0:047 and 0.048 as in Figure 6. It fol-lows that the exerted Ek is greater than the case of � ¼ 4:0 ;1030 G cm3 (Fig. 6), because �GJ increases 1.5 times. The neg-ative feedback effect cannot cancel the increase of �, unlike theincrease of kTs, because the right-hand side of equation (19) is

more directly affected by the variation of� through �GJ than by thevariation of kTs through pair creation.

We also present the �-ray spectrum for three different values of� in Figure 10. It follows that both the peak energy and the flux ofcurvature �-rays (around GeVenergies) increase with increasing�. This is because �GJ, and hence Ek, increases with �. It alsofollows that the secondary synchrotron flux (below 100 MeV)increases with �, because the magnetic field strength increases inthe magnetosphere. We find that a larger magnetic dipole mo-ment, � � 6 ; 1030 G cm3, is preferable to explain the observedpulsed flux from the Crab pulsar.

If we adopt � ¼8 ;1030 G cm3, which is about twice as largeas the dipole deduced value, the spectral shape becomes moreconsistent compared with smaller � cases. We should note herethat the moment of inertia, I, has to be large in this case. Forexample, if we assume a pure magnetic dipole radiation, the spin-down luminosity becomes LSD ¼ (2 sin2�inc/3)(�

2�4/c3)¼ 1:7 ;1039 ergs s�1 for � inc ¼ 70�. Equating �E ¼ �I�� with thisLSD, we obtain I ¼ 3:9 ; 1045 g cm2, which is consistent withthe limit (I > 3:04 ; 1045 g cm2) derived from the considerationof energetics of the Crab nebula (Bejger & Haensel 2002). How-ever, solving the time-dependent equations of force-free electro-dynamics, Spitkovsky (2006) derived LSD � (1þ sin2� inc) ;(�2�4/c3), which gives �5:6 ; 1039 ergs s�1 for � inc ¼ 70�.This large spin-down luminosity results in I � 1:2 ; 1046 g cm2,which is too large even compared with those obtained for a stiff

Fig. 9.—Same as Fig. 6, but with � ¼ 6:0 ; 1030 G cm3; in the left panel, the dotted and solid curves correspond to h ¼ 0:039 and 0.041, respectively. For the twoother curves in the left panel and the four curves in the right panel, h takes the same values as in Fig. 6. [See the electronic edition of the Journal for a color version of thisfigure.]

Fig. 10.—Same as Fig. 8 for � inc ¼ 70�, kTs ¼ 100 eV, and hm ¼ 0:200, but with a different dipole moment, �. [See the electronic edition of the Journal for a colorversion of this figure.]

PARTICLE ACCELERATOR IN PULSAR MAGNETOSPHERES 1487No. 2, 2006

equation of state (Serot 1979a, 1979b; Pandharipande & Smith1975). Thus, it implies either that � should be less than 8 ;1030 G cm3 or that the deduced magnetospheric current is toolarge when the relationship LSD � (1þ sin2� inc)(�

2�4/c3) is de-rived (for a discussion of the magnetospheric current determina-tion, see x 4 of Hirotani 2006).

3.6. Dependence on Magnetic Inclination

Let us examine how the solution depends on themagnetic incli-nation angle, � inc. Figure 11 shows the �-ray spectra for � ¼ 6 ;1030 G cm3, kTs ¼ 100 eV, and hm ¼ 0:200, with three differentinclination, � inc ¼ 50

�, 60

�, and 80

�, in the same manner as in

Figure 8. The flux is averaged over themeridional emission anglesbetween 44� and 58� (solid line), 58� and 72� (dashed line), 72�

and 86�(dash-dotted line), 86

�and 100

�(dotted line), and 100

�

and 114� (dashYtriple-dotted line) from the magnetic axis on theplane in which both the rotational and magnetic axes reside. Wefind that the �-ray flux reaches a peak of�2 ; 1013 Jy Hz around2 GeVand that this peak does not strongly depend on� inc. This isbecause the pair creation efficiency, which governs the gap elec-trodynamics, crucially depends on the distance from the star, whichhas a small dependence on� inc. However, we also find that the fluxtends to be emitted into larger meridional angles from the magneticaxis (i.e., from outer regions) for smaller � inc. The reasons arefourfold:

1. �GJ decreases with decreasing � inc at a fixed s; as a result,the null surface appears at larger s for smaller � inc.

2. �GJ(s) ¼ ��GJ(0) is realized at larger s for smaller � inc.3. In the region where �GJ < ��GJ(0) holds, Ek is substan-

tially screened by the discharge of created pairs. (Compare Fig. 5and the right panel of Fig. 4.)

4. The unscreened Ek tends to appear at larger s for smaller� inc, resulting in a �-ray emission which concentrate in largermeridional angles.

We can alternatively interpret the explanation above as follows:

a) j�GJ(0)j increases with decreasing � inc.b) The created current density, je, which is greater than

cj�GJ(0)j for a super-GJ solution, increases with decreasing � inc.For example, we obtain je ¼ 0:62, 0.67, 0.74, and 0.77 for� inc ¼80�, 70�, 60�, and 50�, respectively, at h ¼ hm/2 ¼ 0:100 with� ¼ 6 ; 1030 G cm3 and kTs ¼ 100 eV.

c) A larger je results in a larger injection of the discharged pos-itrons and the emitted ions into the strong-Ek region from thestellar side.

d ) A larger injection of positive charges from the stellar sideshifts the gap outward by the mechanism discussed in x 2 ofHirotani & Shibata (2001a).

For a smaller inclination, � inc 40�, the gap is located in$ > 0:9$LC; that is, no super-GJ solution exists. Therefore, theobserved �-ray flux cannot be explained by the present theory, ifthe magnetic inclination is constrained to be less than 40� bysome other methods.

3.7. Particle Distribution Functions

Let us examine how the particle distribution function evolvesat different positions. Figure 12 represents the evolution of posi-tronic distribution function from s ¼ 0:85$LC (dashed curve),0:90$LC (dash-dotted curve), 0:95$LC (dashYtriple-dotted curve),1:00$LC (solid curve), to 1:30$LC (dotted curve). It showsthat the positrons are injected into the strong-Ek region (s >0:85$LC) with energies 104 < � < 3 ; 106 because of the ac-celeration by the small-amplitude, residual Ek in s < 0:85$LC

(Fig. 9, right panel, solid curve). The positrons are acceleratedoutward to attain � � 4 ; 107 at s � 1:0$LC. They are subse-quently decelerated by curvature cooling in$> 0:9$LC (or equiv-alently s > 1:1$LC for � inc ¼ 70

�), where we artificially put

Fig. 11.—Same as Fig. 8 for kTs ¼ 100 eV, � ¼ 6:0 ; 1030 G cm3, and hm ¼ 0:200, but with different magnetic inclination angle, � inc. [See the electronic editionof the Journal for a color version of this figure.]

Fig. 12.—Energy spectrum of positrons at s ¼ 0:85$LC (dashed curve),0:90$LC (dash-dotted curve), 0:95$LC (dashYtriple-dotted curve), 1:00$LC

(solid curve), and 1:30$LC (dotted curve), for� inc ¼ 70�, � ¼ 6:0 ; 1030 G cm3,kTs ¼ 100 eV, and hm ¼ 0:20. [See the electronic edition of the Journal for acolor version of this figure.]

HIROTANI1488 Vol. 652

Ek ¼ 0, to escape outward with� � 107 at s � 1:3$LC. There is asmall population of the positrons that have upscattered surfaceX-rays to possess smaller energies than the curvature-limited pos-itrons. For example, within the gap (s < 1:1$LC), the dash-dotted,dashYtriple-dotted, and solid curves have the broad, lower-energycomponent, which connects with the curvature-limited peak com-ponent. However, in s > 1:1$LC, we artificially put Ek ¼ 0; as aresult, the positrons that have lost energies by ICS cannot be re-accelerated, forming a separate component in� < 4 ; 106 from thecurvature-limited peak component, as the dotted curve shows. Theupscattered photons obtain several TeVenergies; however, they aretotally absorbed by the strong magnetospheric infrared radiationfield. Therefore,we depict only the photon energies below100GeVin Figures 8, 10, and 11.

Since most of the pairs are created inward, positrons return tomigrate outward by the small-amplitude Ek in s < 0:85$LC (for� ¼ 6:0 ; 1030 G cm3 and � inc ¼ 70�), losing significant trans-verse momenta via synchrotron process to fall onto the ground-state Landau level in the strong-B region. Thus, their emission inthe strong-Ek region is given by a pure-curvature formula.

Next, we consider the distribution function of electrons. In Fig-ure 13 we present their evolution along the field line h ¼ hm/2 ¼0:100 from s ¼ 0:90$LC (dashYtriple-dotted curve), 0:40$LC

(dotted curve), 0:20$LC (dash-dotted curve), 0:08$LC (dashedcurve), to 0 (solid curve). Left panel shows the energy spectrum.Electrons created in s < 0:85$LC cannot be accelerated by Ekefficiently; thus, their energy spectrumbecomes broad, as the dotted,dash-dotted, and dashed curves indicate (i.e., particle Lorentzfactors do not concentrate at the curvature-limited terminal value).From s ¼ 0:08$LC to 0, electrons are decelerated via synchro-curvature radiation and nonresonant IC scatterings. Finally,they hit the stellar surface with � < 3 ; 105. Assuming thatthe azimuthal gap width is � radians, we obtain LPC ¼ 2:6 ;1031 ergs s�1 as the heated polar-cap luminosity. Thus, theX-ray emission due to the bombardment is negligible, com-pared with the total soft X-ray luminosity (0.1Y2.4 keV) of 7:6 ;1034 ergs s�1(��X/sr) (e.g., Becker & Trumper 1997), where��X refers to the emission solid angle.

Let us see the pitch-angle evolution of the electrons, which ispresented in the right panel of Figure 13. In the outer part of thegap, the electrons are created by the collisions between the outward-directed �-rays and the surfaceX-rays. Thus, created electrons haveoutward momenta initially, then return by the positive Ek, losing

their perpendicular momentum substantially via synchrotron radi-ation. Thus, at s ¼ 0:90$LC, the dashYtriple-dotted curve showsthat their pitch angles, , are less than 8 ; 10�5. However, mostof the pairs are created by the inward-directed �-rays in s <0:6$LC; thus, electrons have initial inward momenta to migrateinward by the small-amplitude, residual Ek. Since such inward-created electrons do not change their migration direction, theirpitch angles are greater than those of the outward-created ones,as the dotted curve demonstrates. As the electronsmigrate inward,they lose perpendicular momenta via synchrocurvature radiationin the strong B-field, reducing their pitch angles, as the dotted,dash-dotted, and dashed curves indicate.

We should point out that a pure curvature formula cannot beapplied to the electrons. For example, at s ¼ 0:4$LC, the dottedcurve in Figure 13 demonstrates that electrons have 104 <� < 107 and 10�8 < sin < 10�3:5. Noting that we have B ¼3:0 ; 107 G at s ¼ 0:4$LC, we find that newly created electrons,which have lower energies (� < 106) and larger pitch angles(sin > 10�5), emit synchrocurvature radiation, rather than pure-curvature radiation, as Figure 14 shows. At s ¼ 0:2$LC, we haveB ¼ 3:6 ; 108 G; thus, the dash-dotted curve shows that the pure-curvature formula is totally inapplicable as we consider a smallerdistance from the star. The only exception is the innermost region(s < 5r� ¼ 0:031$LC and B > 5 ; 1010 G), where electrons suf-fer substantial de-excitation via synchrotron radiation, falling at lastonto the ground-state Landau level. In this region, electrons emit viapure curvature radiation.We thus artificially assume sin ¼ 10�20,which guarantees pure curvature emission, and do not depict thesolid curve in the right panel of Figure 13. Since pair creation andthe resulting screening of Ek is governed by the inward-directed�-ray flux and spectrum, it is essential to adopt the correct ra-diation formula by computing the pitch-angle evolution of inward-migrating particles.

3.8. Formation of Magnetically Dominated Wind

Let us finally consider the magnetic dominance within thelight cylinder. First, we introduce the magnetization parameter,

�� B2

4��mec

2

ZZZNþþN�ð Þd 3pþ�ionmionc

2

ZZZNiond

3p

� ��1

¼ e(B=2)$LC

je�mec2 þ jion�ionmionc2; ð69Þ

Fig. 13.—Distribution function of electrons at s ¼ 0:9$LC (dashYtriple-dotted curves), 0:4$LC (dotted curves), 0:2$LC (dash-dotted curves), 0:08$LC (dashedcurves), and 0 (solid curves), for � inc ¼ 70�, � ¼ 6:0 ; 1030 G cm3, kTs ¼ 100 eV, and hm ¼ 0:20. Left panel shows the Lorentz factor dependence, while the right oneshows the pitch-angle dependence. [See the electronic edition of the Journal for a color version of this figure.]

PARTICLE ACCELERATOR IN PULSAR MAGNETOSPHERES 1489No. 2, 2006

where Nion and �ion refer to the distribution function and the av-eraged Lorentz factor of the ions, respectively. Evaluating equa-tion (69) at the light cylinder, and noting �ionmionc

2 ¼ je��gapj,where je��gapj � (B/4)h2

m$LC refers to the potential drop in thegap (see eq. [68]), we obtain

�LC � 1

h2m

2

je�mec2= e��gap

�� ��þ jion: ð70Þ

Recalling jion < je < 1, �mec2 < je��gapj, and hm < 1, we can

conclude that the magnetic energy flux is always greater than theparticle kinetic energy flux at the light cylinder, regardless of thespecies of the accelerated particles, of the sign of Ek, or of the gapposition (i.e., whether inner or outer magnetosphere). Only whenthe gap is formed along most of the open field lines (i.e., hm � 1),�LC can be of the order of unity. Since the second factor in equa-tion (70) does not change significantly for different values ofparameters except for hm, �LC solely depends on hm. Substitutingjion � 0:3, je � 0:6, �mec

2/je��gapj � 0:25 in (3/16)hm h (13/16)hm for � inc ¼ 70

�, � ¼ 6 ; 1030 G cm3, and hm ¼ 0:20,

we obtain �LC � 110.In short, along the open field lines threading the gap, Poynting

flux dominates particle kinetic energy flux by a factor of 100 atthe light cylinder. Positrons escape from the gap with � � 107,while ions escape with �ion � 104.

4. SUMMARY AND DISCUSSION

To conclude, we investigated the self-consistent electrodynamicstructure of a particle accelerator in the Crab pulsar magnetosphereon the two-dimensional surface that contains the magnetic fieldlines threading the stellar surface on the plane in which both therotation and magnetic axes reside. We regard the transfield thick-ness, hm, of the gap as a free parameter, instead of trying to con-strain it. For a small hm, the created current density, je, becomessub-Goldreich-Julian (sub-GJ), giving the traditional outer-gapsolution but with negligible �-ray flux. However, as hm increases,je increases to become super-GJ, giving a new gap solution withsubstantially screened acceleration electric field, Ek, in the innerpart. In this case, the gap extends toward the neutron star with asmall-amplitude positive Ek, which extracts ions from the stellar

surface. It is essential to examine the pitch-angle evolution of thecreated particles, because the inward-migrating particles emit�-rays, which govern the gap electrodynamics through pair cre-ation, via a synchrocurvature process rather than a pure-curvatureone. The resulting spectral shape of the outward-directed �-rays isconsistent with the existing observations; however, their predictedfluxes appear insufficient. The pulsar wind at the light cylinder ismagnetically dominated: along the field lines threading the gap,the magnetization parameter, �LC, is about 10

2.

4.1. How to Obtain Sufficient �-Ray Flux

The obtained �-ray fluxes in the present work are all below theobserved values. Nevertheless, this problemmay be solved if weextend the current analysis into a three-dimensional configurationspace. As mentioned in x 2.4, we assume k’ ¼ 0 and neglect theaberration of photon emission directions. However, in a realisticthree-dimensional pulsarmagnetosphere, �-rayswill have angularmomenta andmay be emitted in a limited solid angle as suggestedby the caustic model (e.g., Dyks&Rudak 2003; Dyks et al. 2004),which incorporates the effect of aberration of photons and that oftime-of-flight delays. In particular, in the trailing peak of a highlyinclined rotator, photons emitted at different altitudes s will bebeamed in a narrow solid angle to be piled up at the same phase ofa pulse (Morini 1983; Romani&Yadigaroglu 1995), resulting in a�-ray flux that is an order of magnitude greater than the presentvalues. Thus, the insufficient �-ray fluxes do not suggest the in-applicability of the present method. It is noteworthy that the me-ridional propagation angles of the emitted photons (e.g., differentcurves in Figs. 8, 10, and 11) can be readily translated into theemissivity distribution in the gap as a function of s. Therefore, inthis work, we do not sum up the �-ray fluxes emitted into dif-ferent meridional angles taking account of the aberration of light.

4.2. Stability of the Gap

Let us discuss the electrodynamic stability of the gap, by con-sidering whether an initial perturbation of some quantity tends tobe canceled or not. In the present paper, we consider that the softphoton field is given and unchanged when gap quantities vary.Thus, let us first consider the case in which the soft photon field isfixed. Imagine that the number of created pairs is decreased as aninitial perturbation. It leads to an increase of the potential dropdue to less efficient screening by the discharged pairs, and henceto an increase of particle energies. Then the particles emit synchro-curvature radiation efficiently, resulting in an increase of the cre-ated pairs, which tends to cancel the initial decrease of createdpairs.Let us next consider the case in which the soft photon field

also changes. Imagine again that the created pairs are decreasedas an initial perturbation. It leads to an increase of particle en-ergies in the same manner as discussed just above. The increasedparticle energies increase not only the number and density of syn-chrocurvature �-rays, but also the surface blackbody emission fromheated polar caps and the secondary magnetospheric X-rays. Eventhough neither the heated polar-cap emission nor the magneto-spheric emission is taken into account as the soft photon field il-luminating the gap in this paper, they all work, in general, toincrease the pair creation within the gap, which cancels the initialdecrease of created pairs more strongly than for the case of thefixed soft photon field.Because of such negative feedback effects, solutions exist in a

wide parameter space. For example, the created current density isalmost unchanged for a wide range of hm (e.g., compare the dash-dotted and dashYtriple-dotted curves in Fig. 7). On these grounds,

Fig. 14.—Pure-curvature (solid curve) vs. synchrocurvature radiation-reactionforces. For the latter, we adopt the pitch angles ¼ 10�5, 10�6, and 10�7 rad forthe dashed, dash-dotted, and dotted curves, respectively, and B ¼ 3 ; 107 and109 G for the thin and thick curves, respectively. Curvature radius is assumed to be0:4$LC ¼ 6:37 ; 107 cm. [See the electronic edition of the Journal for a colorversion of this figure.]

HIROTANI1490 Vol. 652

although the perturbation equations are not solved under appro-priate boundary conditions for the perturbed quantities, we conjec-ture that the particle accelerator is electrodynamically stable,irrespectivewhether theX-ray field illuminating the gap is thermalor nonthermal origin.

4.3. Local versus Global Currents

Let us briefly look at the relationship between the locally de-termined current density jgap (eq. [43]) and the globally requiredone, jglobal. It is possible that jglobal is constrained independentlyfrom the gap electrodynamics by the dissipation at large distances(Shibata 1997), which provides the electric load in the currentcircuit, or by the condition that the magnetic flux function shouldbe continuous across the light cylinder, as discussed in the recentforce-free argument of the transfield equation (Contopoulos et al.1999; Goodwin et al. 2004; Gruzinov 2005; Spitkovsky 2006).In either case, jglobal will be more or less close to unity (i.e., typ-ical GJ value). On the other hand, as demonstrated in x 3.3, forsuper-GJ cases, jgap is automatically regulated around 0.9 for awide parameter range. Thus, the discrepancy between jgap andjglobal is small provided that hm is large enough to maintain thecreated current density at a super-GJ value. The small imbalancejglobal � jgap may have to be compensated by a current injectionacross the outer boundary (if the gap terminates inside of the lightcylinder, charged particles could be injected from the outer bound-ary), or by an additional ionic emission from the stellar surface (ifthe imbalance leads to an additional residual Ek at the surface). Inany case, the injected current is small compared with jgap; thus, itwill not change the electrodynamics significantly, even though thegap active regionmay be shifted to some degree along themagneticfield lines, as demonstrated by Hirotani & Shibata (2001a, 2001b,2002), TSH04, and TSHC06.

4.4. Created Pairs in the Inner Magnetosphere

Let us devote a little more space to examining the particle fluxalong the open field lines that do not thread the gap (i.e., hm <h < 1). Since Ek vanishes, the created, secondary pairs emit syn-chrotron photons, which are capable of cascading into tertiary andhigher generation pairs by �-� or �-B collisions. Examining thecascade, we can calculate the rate of pair creation, which takesplace mainly in the inner magnetosphere. Denoting that the paircreation rate is �w(h)(�B/2�e) per unit area per second, we find�w ¼ 2:2 ; 104, 2:1 ; 104, 1:9 ; 104, 1:8 ; 104, 1:6 ; 104, 1:1 ;104, 1:0 ; 104, 0:99 ; 104, 0:95 ; 104, 0:96 ; 104, and 1:0 ; 104

at h ¼ 0:20125, 0.2025, 0.20375, 0.205, 0.305, 0.405, 0.505,0.605, 0.705, 0.805, and 0.905, respectively, for � inc ¼ 70�, � ¼6:0 ; 1030 G cm3, and hm ¼ 0:200. Thus, the averaged creationrate becomes �w ¼ 1:4 ; 104(�B/2�e) pairs per unit area persecond, giving Npair ¼ 3:8 ; 1038 s�1 as the pair creation rate inthe entire magnetosphere. It should be noted that this Npair ap-pears less than the constraints that arise from consideration ofmagnetic dissipation in the wind zone (Kirk & Lyubarsky 2003,who derived 1040 s�1), and of Crab Nebula’s radio synchrotronemission (Arons 2004, who derived 1040.5 s�1).

Due to strong synchrotron radiation, these inwardly createdparticles quickly lose energy to fall onto the lowest Landau level,preserving longitudinal momentum�mec per particle. These non-relativistic particles possess momentum flux of 2Npairmec/(�R

2PC)

� 1:0 ; 1012 dyn cm�2, whereRPC � (r3� /$LC)1/2 denotes the po-

lar cap radius. On the other hand, the surface X-ray field has the up-ward momentum flux of 6:0 ; 108(kT /100 eV)4 dyn cm�2.Thus, the created pairs will not be pushed back by resonant scat-terings. They simply fall onto the stellar surface with nonrela-tivistic velocities. The luminosity of the e�-eþ annihilation line

is about 3 ; 1026 ergs s�1, which is negligible (e.g., comparedwith the �-ray luminosity �1034.5 ergs s�1). On these grounds,for the Crab pulsar, we must conclude that the present work failsto explain the injection rate of the wind particles, in the samewayas in other outer-gap models.

4.5. Comparison with Polar-Slot Gap Model

It is worth comparing the present results with the polar-slot-gap model proposed by MH03, MH04a, and MH04b, who ob-tained a quite different solution (e.g., negativeEk in the gap) solvingessentially the same equations under analogous boundary con-ditions for the same pulsar as in the present work. The only dif-ference is the transfield thickness of the gap. Estimating thetransfield thickness to be�lSG � hmr� r/$LCð Þ1/2, which is a fewhundred times thinner than the present work, they extended thesolution (near the polar cap surface) that was obtained by MT92into the higher altitudes (toward the light cylinder). Because ofthis very small�lSG, emitted �-rays do not efficientlymaterializewithin the gap; as a result, the created and returned positrons fromthe higher altitudes do not break down the original assumption ofthe completely charge-separated SCLF near the stellar surface.

To avoid the reversal of Ek in the gap (from negative near thestar to positive in the outer magnetosphere), or equivalently, toavoid the reversal of the sign of the effective charge density,�eA ¼ �� �GJ, along the field line, MH04a andMH04b assumedthat �eA/B nearly vanishes and remains constant above a certainaltitude, s ¼ sc, where sc is estimated to be within a few neutronstar radii. Because of this assumption, Ek is suppressed at a verysmall value and the pair creation becomes negligible in the entiregap. In another word, the enhanced screening is caused not onlyby the proximity of two conducting boundaries, but also by theassumption of @(�eA/B)/@s ¼ 0 within the gap (see eq. [68]). Tojustify this �/B distribution, MH04a and MH04b proposed anidea that � should grow by the cross-field motion of charges dueto the toroidal forces, and that �eA/B is a small constant so thatc�eA/B may not exceed the flux of the emitted charges from thestar, which ensures the equipotentiality of the slot-gap boundaries(see x 2.2 of MH04a for details).

The cross-field motion becomes important if particles gainangular momenta as theymigrate outward to pick up energies thatare a nonnegligible fraction of the difference of the cross-fieldpotential between the two conducting boundaries. Denoting thefraction as �, we obtain �mec

2’�($/c)2 ¼ �eB�lSG (Mestel et al.1985; eq. [12] of MH04a). If we substitute their estimate�lSG �r�/20, we obtain � � 0:33(’/�)�7B

�16 r�3

6 ($/$LC)2, where �7 ¼

�/107, B6 ¼ B�/106 G, and r6 ¼ r�/10 km; therefore, the cross-

field motion becomes important in the outer magnetospherewithin their slot-gap model. A larger value of �lSG is incom-patible with the constancy of �eA/B due to the cross-field motionin the higher altitudes.

As for the equipotentiality of the boundaries, it seems rea-sonable to suppose that cj�eAj/B < cj��j/B� should be held at anyaltitudes in the gap, as MH04a suggested, where �� denotes thereal charge density in the vicinity of the stellar surface. However,the assumption that �eA/B is a small positive constant may be toostrong, because it is only a sufficient condition of cj�eAj/B <cj��j/B�.

In the present paper, on the other hand, we assume that themagnetic fluxes threading the gap are unchanged, consideringthat charges freely move along the field lines on the upper (andlower) boundaries. As a result, the gap becomesmuch thicker thaninMH04a andMH04b; namely,�lSG � 0:5hm$LC, which gives� < 10�3. Therefore, we can neglect the cross-field motion andjustify the constancy of �/B in the outer region of the gap, where

PARTICLE ACCELERATOR IN PULSAR MAGNETOSPHERES 1491No. 2, 2006

pair creation is negligible. In the inner magnetosphere, �eA/Bbecomes approximately a negative constant, owing to the dis-charge of the copiously created pairs. Because of this negativity of�eA/B, a positive Ek is exerted. For a super-GJ solution, we obtainjeþ jion � 0:9 > �eA/(�B/2�), which guarantees the equi-potentiality of the boundaries. For a sub-GJ solution, a problemmay occur regarding the equipotentiality; nevertheless, we are notinterested in this kind of solutions.

In short, whether the gap solution becomes MH04a-like (witha negative Ek as an outward extension of the polar-cap model)or like the model in the present paper (with a positive Ek as an in-ward extension of the outer-gap model) entirely depends on thetransfield thickness and on the resulting �eA/B variation along thefield lines. If �lSG � r�/10 holds in the outer magnetosphere,�eA/B could be a small positive constant by the cross-field mo-tion of charges (without pair creation); in this case, the current isslightly sub-GJ with electron emission from the neutron star sur-face, asMH04a,MH04b suggested. On the other hand, if�lSG >$LC/40 holds in the outer magnetosphere, �eA/B takes a smallnegative value by the discharge of the created pairs (see Fig. 5); in

this case, the current is super-GJ with ion emission from the sur-face, as demonstrated in the present paper. Since no studies haveever successfully constrained the gap transfield thickness, there isroom for further investigation on this issue.

The author is grateful to J. G. Kirk, B. Rudak, S. Shibata, K. S.Cheng, A. K. Harding, J. Arons, R. Taam, H. K. Chang, andJ. Takata for helpful suggestions. Some important parts of thisworkwere prepared while the author studied at Max-Planck-Institut furKernphysik, Heidelberg. This work is supported by the Theoret-ical Institute for Advanced Research in Astrophysics (TIARA)operated under Academia Sinica and the National Science CouncilExcellence Projects program in Taiwan administered through grantNSC 94-2752-M-007-001. Also, this work is partly supported byKBN through grant 2P03D.004.24 to B. Rudak, which enabled theauthor to use the MEDUSA cluster at CAMK Torun. The finalcomputations were carried out with the aid of the Blade Tank serv-ers at TIARA, Taipei.

APPENDIX

Explicit expressions of equations (20) and (21) are as follows:

g ss ¼ grr@s

@r

� 2

�;’

þ g��@s

@�

� 2

’;r

� k0

�2w

@s

@’

� 2

r;�

; ðA1Þ

g���� ¼ g rr@��@r

� 2

�;’

þ g��@��@�

� 2

’;r

� k0

�2w

@��@’

� 2

r;�

; ðA2Þ

g’�’� ¼ g rr@’�

@r

� 2

�;’

þ g��@’�

@�

� 2

’;r

� k0

�2w

@’�

@’

� 2

r;�

; ðA3Þ

gs�� ¼ grr@s

@r

� �;’

@��@r

� �;’

þ g��@s

@�

� ’;r

@��@�

� ’;r

� k0

�2w

@s

@’

� r;�

@ ��@’

� r;�

; ðA4Þ

g��’� ¼ grr@��@r

� �;’

@’�@r

� �;’

þ g��@��@�

� ’;r

@’�@�

� ’;r

� k0

�2w

@��@’

� r;�

@’�@’

� r;�

; ðA5Þ

g’�s ¼ grr@’�

@r

� �;’

@s

@r

� �;’

þ g��@’�

@�

� ’;r

@s

@�

� ’;r

� k0

�2w

@’�

@’

� r;�

@s

@’

� r;�

; ðA6Þ

and

As � c2ffiffiffiffiffiffi�gp @r

g’’�2w

ffiffiffiffiffiffi�gp

g rr@s

@r

� �;’

" #þ @�

g’’�2w

ffiffiffiffiffiffi�gp

g��@s

@�

� ’;r

" #( )�

c2g’’�2w

k0

�2w

@ 2s

@’2

� r;�

; ðA7Þ

A�� � c2ffiffiffiffiffiffi�gp @r

g’’�2w

ffiffiffiffiffiffi�gp

grr@��@r

� �;’

" #þ @�

g’’�2w

ffiffiffiffiffiffi�gp

g��@��@�

� ’;r

" #( )�

c2g’’�2w

k0

�2w

@ 2��@’2

� r;�

; ðA8Þ

A’� � c2ffiffiffiffiffiffi�gp @r

g’’�2w

ffiffiffiffiffiffi�gp

g rr@’�@r

� �;’

" #þ @�

g’’�2w

ffiffiffiffiffiffi�gp

g��@’�@�

� ’;r

" #( )�

c2g’’�2w

k0

�2w

@ 2’�@’2

� r;�

: ðA9Þ

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