# Pulse Profiles, Spectra, and Polarization Characteristics of Nonthermal Emissions from the Crab-like Pulsars

**ABSTRACT** We discuss nonthermal emission mechanisms of the Crab-like pulsars with both a two-dimensional electrodynamic study and a three-dimensional model. We investigate the emission process in the outer gap accelerator. In the two-dimensional electrodynamic study, we solve the Poisson equation of the accelerating electric field in the outer gap and the equation of motion of the primary particles with the synchrotron and the curvature radiation processes and the pair-creation process. We show a solved gap structure that produces a gamma-ray spectrum consistent with EGRET observations. Based on the two-dimensional model, we construct a three-dimensional emission model to calculate the synchrotron and the inverse Compton processes of the secondary pairs produced outside the outer gap. We calculate the pulse profiles, the phase-resolved spectra, and the polarization characteristics in optical through gamma-ray bands for comparison with the observation of the Crab pulsar and PSR B0540-69. For the Crab pulsar, we find that the outer gap geometry extending from near the stellar surface to near the light cylinder produces a complex morphology change of the pulse profiles as a function of the photon energy. This predicted morphology change is quite similar to that of the observations. The calculated phase-resolved spectra are consistent with the data from the optical to the gamma-ray bands. We demonstrate that the 10%-20% of the polarization degree in the optical emissions from the Crab pulsar and the Vela pulsar is explained by the synchrotron emissions from the particle gyration motion. For PSR B0540-69, the observed pulse profile with a single broad pulse is reproduced for an emission region thicker and an inclination angle between the rotational axis and the magnetic axis smaller than the Crab pulsar.

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**ABSTRACT:**We review a proposed multicomponent model to explain the features of the pulsed emission and spectrum of the Crab Pulsar, on the light of the recent detection of pulsed emission above 25 GeV from the MAGIC atmospheric Cherenkov telescope. This model explains the evolution of the pulse shape and of the phase-resolved spectra, ranging from the optical/UV to the GeV energy band, on the assumption that the observed emission is due to several components, which have spectra modelled as log-parabolic laws. We show that the new MAGIC data are well consistent with the prevision of our model. Comment: 4 pages, 3 figures. Accepted for publication in Astronomy and AstrophysicsAstronomy and Astrophysics 03/2009; · 5.08 Impact Factor - SourceAvailable from: iopscience.iop.org[Show abstract] [Hide abstract]

**ABSTRACT:**We use a modified three-dimensional outer gap model to explain the features of the pulsed emission and spectra of the Crab pulsar from X-ray to above 25 GeV regimes. In such an outer gap model, the phase-averaged spectra below ~1 GeV are mainly produced through the synchrotron self-Compton mechanism, and the spectrum above ~1 GeV are due to the survival curvature photons. Our results show that (1) the observed phase-averaged spectrum from X-rays to γ-rays including the Fermi LAT and MAGIC data can be reproduced well, and (2) the basic properties of both the observed phase-dependent spectra of both X-ray and γ-ray up to >25 GeV can be interpreted in this model.The Astrophysical Journal 12/2009; 707(2):L169. · 6.73 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the phase-averaged spectra and luminosities of γ-ray emissions from young, isolated pulsars within a revised outer gap model. In the revised version of the outer gap, there are two possible cases for the outer gaps: the fractional size of the outer gap is estimated through the photon-photon pair process in the first case (Case I), and is limited by the critical field lines in the second case (Case II). The fractional size is described by Case I if the fractional size at the null charge surface in Case I is smaller than that in Case II, and vice versa. Such an outer gap can extend from the inner boundary, whose radial distance to the neutron star is less than that of the null charge surface to the light cylinder for a γ-ray pulsar with a given magnetic inclination. When the shape of the outer gap is determined, assuming that high-energy emission at an averaged radius of the field line in the center of the outer gap, with a Gaussian distribution of the parallel electric field along the gap height, represents typical emission, the phase-averaged γ-ray spectrum for a given pulsar can be estimated in the revised model with three model parameters. We apply the model to explain the phase-averaged spectra of the Vela (Case I) and Geminga (Case II) pulsars. We also use the model to fit the phase-averaged spectra of 54 young, isolated γ-ray pulsars, and then calculate the γ-ray luminosities and compare them with the observed data from Fermi-LAT.The Astrophysical Journal 02/2013; 765(2):124. · 6.73 Impact Factor

Page 1

arXiv:0707.3301v1 [astro-ph] 23 Jul 2007

Not to appear in Nonlearned J., 45.

PULSE PROFILES, SPECTRA AND POLARIZATION

CHARACTERISTICS OF NON-THERMAL EMISSIONS FROM

THE CRAB-LIKE PULSARS

J.Takata

Institute of Astronomy and Astrophysics, Academia Sinica; and Theoretical Institute for

Advanced Research in Astrophysics, Academia Sinica and National Tsing Hua University,

Taipei Taiwan

and

H.-K.Chang

Department of Physics and Institute of Astronomy, National Tsing Hua University,

Hsinchu, Taiwan

ABSTRACT

We discuss non-thermal emission mechanism of the Crab-like pulsars with

both a two-dimensional electrodynamical study and a three-dimensional model.

We investigate the emission process in the outer gap accelerator. In the two-

dimensional electrodynamical study, we solve the Poisson equation of the accel-

erating electric field in the outer gap and the equation of motion of the primary

particles with the synchrotron and the curvature radiation process and the pair-

creation process. We show a solved gap structure which produces a consistent

gamma-ray spectrum with EGRET observation. Based on the two-dimensional

model, we conduct a three-dimensional emission model to calculate the syn-

chrotron and the inverse-Compton processes of the secondary pairs produced

outside the outer gap. We calculate the pulse profiles, the phase-resolved spec-

tra and the polarization characteristics in optical to γ-ray bands to compare the

observation of the Crab pulsar and PSR B0540-69. For the Crab pulsar, we find

that the outer gap geometry extending from near the stellar surface to near the

light cylinder produces a complex morphology change of the pulse profiles as a

function of the photon energy. This predicted morphology change is quite similar

with that of the observations. The calculated phase-resolved spectra are consis-

tent with the data through optical to the γ-ray bands. We demonstrate that

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the 10∼20 % of the polarization degree in the optical emissions from the Crab

pulsar and the Vela pulsar are explained by the synchrotron emissions with the

particle gyration motion. For PSR B0540-69, the observed pulse profile with a

single broad pulse is reproduced with a thicker emission region and a smaller in-

clination angle between the rotational axis and the magnetic axis than the Crab

pulsar.

Subject headings: optical-X ray-gamma rays:theory-pulsars:Crab like -radiation

mechanisms:non-thermal

1.Introduction

The observed strong γ-ray emissions from the seven young pulsars (Thompson 2003)

show that electrons and positrons are accelerated up to ultra-relativistic regime in the pulsar

magnetosphere. The Crab pulsar (PSR B0531+21), which is one of the brightest and the

youngest γ-ray emitting pulsar, shows the non-thermal emission properties in optical to γ-

ray bands. The observed spectrum of the pulsed photons emitted from the Crab pulsar

extends continuously from optical to γ-ray bands with the spectral index αν, defined as

Iν∝ ν−αν, varying from αν∼ 0 in optical wavelengths, αν∼ 0.5 in X-ray bands, to αν∼ 2

in γ-ray bands. The pulse profile has two peaks in a single period, and the positions of

the pulse peaks across the wide energy range are approximately all in phase (Kuiper et

al. 2001). Interestingly, the pulse profile morphology changes significantly as a function of

the photon energy. The first peak (denoted Peak 1 in the following) dominates in optical

wavelengths. However, the second peak (Peak 2) becomes more and more pronounced for

increasing energies and eventually the Peak 2 emission dominates in soft γ-ray bands. Above

10 MeV photon energy, Peak 1 again dominates Peak 2. The electromagnetic spectrum of

the non-thermal emissions also changes with pulse phases. In the future, the phase-resolved

spectra above 10 MeV will be measured with a sensitivity better than that of the Energetic

Gamma-Ray Experiment Telescope on board the Compton Gamma-ray Observatory by, for

example, GLAST LAT. These observed detail properties for the pulse profiles and the phase-

resolved spectra will be useful to discriminate the proposed emission models.

In addition to the pulse profiles, Kanbach et al.(2005) measured the polarization char-

acteristics of the pulsed photons from the Crab pulsar in the optical wavelengths. The

observation revealed that the degree of the polarization at each pulse peak is lower than

10 % and a large swing of the position angle of the electric-vector of the radiation appears at

each pulse peak. The polarization measurements provide two additional observed properties,

namely, the degree and the position angle of the polarization. In the future, the polarization

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of X-ray and soft γ-ray emissions from the pulsar will probably be able to be measured

by ongoing projects such as PoGO (Kataoka et al. 2005) and NCT (Chang et al. 2007)

projects. Therefore, a theoretical study, particularly on the polarization characteristics, is

not only desired, but also timely for the understanding the non-thermal emission process in

the pulsar magnetospheres.

The polar cap accelerator (Ruderman and Sutherland 1975; Daugherty and Harding

1996) and the outer magnetospheric accelerator, the so called outer gap model (Cheng et

al. 1986a,b; Romani 1996), were proposed as the possible acceleration sites in the pulsar

magnetospheres. The traditional polar cap model assumes an acceleration region expanding

several stellar radii from the stellar surface around the magnetic pole. On the other hand,

the traditional outer gap assumes an acceleration region extending beyond the null surface

of the Goldreich-Julian charge density at the outer magnetosphere. The Goldreich-Julian

charge density is given by ρGJ∼ −Ω · B/2πc (Goldreich and Julian 1969) with Ω being the

rotational frequency of the star, B the magnetic field, and c the speed of light. Both models

assume the particle acceleration by an electric field parallel to the magnetic field line. In

the pulsar magnetosphere, the accelerating electric field arises in the region where the local

charge density differs from the Goldreich-Julian charge density.

The slot gap model (Muslimov & Harding 2004), which is an extended polar cap model,

predicts that the acceleration region extends up to near the light cylinder around the last-

open field lines because the pair-formation front, which screens the accelerating electric field,

occurs at higher altitude around there. Two-dimensional electrodynamical studies (Takata

et al. 2004, 2006; Hirotani 2006) suggested that the inner boundary of the outer gap locates

near (or at) the stellar surface because of the current through the outer gap. Although the

recent polar-slot gap and outer gap models both predict similar geometry of the acceleration

region, an important difference between the two models is the electric field configuration in

the accelerator. For the slot gap accelerator, the electric field is stronger nearer the stellar

surface and smaller at higher altitude. On the other hand, the outer gap model predicts a

stronger electric field beyond the null surface and a smaller one below the null surface due

to the screening effect of electron and positron pairs. This difference in the electric field

configuration, and the resultant difference in the acceleration and the emission structures

will appear as a difference in the predicted pulse profiles, the phase-resolved spectra and the

polarization characteristics, which can be examined by a three-dimensional model.

Within the framework of the traditional outer gap model, Romani & Yadigaroglu (1995)

considered a three-dimensional geometry and explained the general features of the observed

pulse profile such as two-peaks in a single period. Subsequently, Cheng et al. (2000, here-

after CRZ00) developed the three-dimensional outer gap model, in which the gap is sustained

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self-consistently by the pair-creation process between the primary photons emitted via the

curvature process and the surface X-ray photons. CRZ00 calculated the phase-resolved

spectrum in γ-ray region for the Crab pulsar with the synchrotron radiation and the inverse

Compton scattering of the electron and positron pairs produced outside the gap. Zhang &

Cheng (2002) reconsidered the CRZ00 model to examine the phase-resolved spectra in X-ray

region. However, it has been difficult to explain the detail features of the observed pulse

profiles and phase-resolved spectra with the traditional model. Furthermore, the previous

studies have not discussed the complex features of the observed pulse profiles from optical

to γ-ray bands. Recently, Takata et al. (2007) explained the observed polarization charac-

teristics in the optical wavelengths (Kanbach et al 2005) with the new outer gap geometry.

Jia et al. (2007) examined the phase-resolved spectra by taking account of the emissions

below null charge surface. However, these studies also did not consider the pulse profile,

the phase-resolved spectra and the polarization characteristics in optical to γ-ray bands,

simultaneously.

In this paper, we study the emission process of the Crab-like pulsars with the outer

gap accelerator model from both a two-dimensional electrodynamical model and a three-

dimensional emission model point of views. In first part (section 2) of this paper, we will

summarize the results of the two-dimensional electrodynamical study, in which the outer

gap structure for the Crab pulsar is solved with the Poisson equation, the particle motion,

the radiation process and the pair-creation process in meridional plane, following Takata

et al (2004, 2006) and Hirotani (2006). We will show a result which has a consistent GeV

spectrum with the observed phase-averaged spectrum of the Crab pulsar. In the second

part (sections 3 and 4), we will conduct a three-dimensional outer gap model based on the

results of the two-dimensional electrodynamical study. In the three-dimensional study, the

main purpose is to discuss the emission process of optical to γ-ray photons by examining the

morphology change of the pulse profile as a function of the photon energy and the phase-

resolved spectra for the Crab pulsar with the outer gap accelerator model. We will predict

the polarization characteristics through optical to γ-ray bands. We also apply the model

to a Crab-like pulsar, PSR B0540-69. The Crab pulsar and PSR B0540-69 are sometimes

called twin pulsars, because their pulsar parameters are very similar to each other. However,

the observed shapes of pulse profiles are very different to each other. This pair will give an

unique opportunity to examine the model capability.

Important differences between present and previous three-dimensional studies are as

follows. First, we take into account the emissions both below and beyond the null surface as

the electrodynamical study has predicted, while only the emissions beyond the null surface

were taken into account in CRZ00. Secondary we discuss the morphology change of the

pules profile by calculating local emissivity as a function of the photon energy, while the

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previous studies did not discuss the morphology change because they assumed a constant

emissivity when the pulse profiles were calculated. We deal the gyration motion of the pairs

because the gyration motion causes the depolarization for the synchrotron radiation. We

adopt the rotating dipole field in the observer frame, while the previous studies adopted it

in the co-rotating frame. Though these effects were considered in Takata et al. (2007), they

calculated only the synchrotron emission process and presented the phase-averaged spectrum

below MeV energy. In this paper, we extend the model spectrum up to γ-ray bands by

computing also the inverse Compton scattering. Finally, we calculate the collision angle of

the inverse Compton scattering between the pairs and the background synchrotron photons

by tracing the three-dimensional trajectory of the synchrotron photons, while the isotropic

distribution of the back ground photons was assumed in the previous studies (CRZ00). The

collision angle greatly affects to the emissivity of and the polarization characteristics of the

inverse Compton scattering. By including all these effects, we examine the pulse profiles,

the phase-resolved spectra and the polarization characteristics in optical to γ-ray bands,

simultaneously.

2. Results of Two-dimensional Electrodynamical Model

In this section, we summarize the results of the two-dimensional electrodynamical study

for the Crab pulsar. Following Takata et al (2004, 2006) and Hirotani (2006) we calculate

the spectrum of the synchrotron and curvature radiation processes of the primary particles

with the electric structure by solving the Poisson equation [∇2Φ = −4π(ρ − ρGJ)], the

equation of motion for the particles, the pair-creation process and the radiation process. As

discussed in Takata el al. (2004, 2006), the electric structure depends on the current and

the gap size, which are model parameters in their studies. In this section, we show a result,

which produces a consistent GeV spectrum with the observations. We ignore the effect of

the gravity which is not important for the dynamics of the outer gap accelerator. We adopt

static dipole field, while in the later section of the three-dimensional study, we apply the

rotating dipole field. The obtained electric structure with the static and the rotating dipole

field did not change very much, because the radial distances to the null charge points, that

is, to the gap position are similar to each other. For the pair-creation process in the gap, we

consider the thermal soft-photons coming from the stellar surface. We adopt kT = 170 eV

for the Crab pulsar (Yakovlev & Pethick 2004). The inclination angle is assumed as α = 50◦.

Thick solid line in Figure 1 shows the solved accelerating electric field along the field line

locating at 50 % of the trans-field thickness from the lower boundary (last-open field line).

Here, we assume 0.1Rlcof the gap thickness at the light cylinder and the outer boundary is

Page 6

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putted at near the light cylinder. We also assume that 5% of the Goldreich-Julian current,

0.05ΩB/2π, is injected at the outer boundary.

The position of the inner boundary is solved with the current, for which about 22 % of the

Goldreich-Julian current runs through the outer gap in the present case. From Figure 1, we

can see that the inner boundary (r ∼ 0.18Rlc) is inside of the null charge point (r ∼ 0.29Rlc).

As suggested by Takata et al. (2004), the inner boundary is located at the position, on which

jg+j2−j1∼ Bz/B is satisfies, where jgis non-dimensional current created in the gap, j1and

j2are non-dimensional current injected at the inner and the outer boundaries, respectively.

For example, for no injection currents, j1= j2= 0, the inner boundary is located at the

null charge surface, where Bz = 0, if no current is created inside of the gap (jg = 0) as

the vacuum case. On the other hands, if jg ∼ cosα, where α is the inclination angle, is

created, the inner boundary is located at the stellar surface on which Bz/B ∼ cosα is

satisfied around the magnetic pole. In the present case, the inner boundary is located at the

position of about 65 % of the radial distance to the null point with the current components

(jg, j1, j2) = (0.17, 0, 0.05).

Figure 2 shows the calculated synchrotron-curvature spectrum and compares with the

observed phase-averaged spectrum. Sold-line shows spectrum of the intrinsic radiation from

the outer gap, while the dashed-line represents the appearance spectrum after attenuation

of the photons via pair-creation process outside of the gap with the soft photon-field emitted

by the synchrotron process of the secondary pairs. We calculate the initial pitch angle of

the pairs from the propagating direction of the curvature photons and the magnetic field

direction at the pair-creation position. For obtaining the luminosity, we assume the gap

opening angle ∼ 250 degree in the azimuthal direction (see section 3.1).

As dashed-line shows the large amount of the curvature photons above 500 MeV are

converted into the secondary pairs outside of the gap via the pair creation process with

the X-ray photons from the secondary pairs. We find from Figure 2 that the shape of

the spectral energy distribution after absorption becomes relatively flat and explains the

observation above 100 MeV.

The inverse-Compton process of the primary particles in the gap is a possible mech-

anism for TeV emissions. However, the present model predicts the TeV flux for the Crab

pulsar is too low to detect the present Cherenkov telescopes. The soft-photons emitted by

secondary pairs above the gap may not be able to illuminate the gap due to the curvature

of the field lines, and only thermal photons from the stellar surface may be scattered by

the primary particles. In such a case, we found that the intrinsic flux on the Earth be-

comes ∼ 10−15erg/cm2s , which is much small compared with the sensitivity of the present

Cherenkov telescope. On the other hand, some soft-photons emitted by secondary pairs may

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illuminate the outer gap because of the effects of the pith angle. In such a case, we obtained

intrinsic TeV flux which is easily detected by present instruments. However, the optical

depth of the pair-creation for TeV photons is much larger than unity in the magnetosphere,

and the residual TeV photons are a very few, which is difficult to detect with the present

instruments.

From Figure 2, we can see that the synchrotron and curvature radiations of the primary

particles in the outer-gap does not explain the observed flux below 100 MeV. We consider that

the secondary pairs created outside gap produce below 100 MeV photons via the synchrotron

and the inverse-Compton process. Furthermore, the present two-dimensional model can

compare with only the phase-averaged spectrum. More detailed observation such like the

pulse profile, the phase-resolved spectra and the polarization require a three-dimensional

model. Following sections, therefore, we calculate the emission process of the secondary

pairs and conduct a three-dimensional model.

3.A Three-Dimensional Emission Model

In the following, we conduct a three-dimensional emission model. We anticipate that

the emission direction is coincide with the particle motion in the observer frame. In the

present paper, we adopt the rotating dipole field in the observer frame while it was assumed

in the co-rotating frame in the previous studies (Romani & Yadigaroglu 1995; Cheng et al

2000; Dyks et al 2004). As a results, the magnetic field configurations and resultant the

morphology of emission pattern in the observer frame are different between the present and

the previous studies, but the difference becomes to be important only near the light cylinder.

In the present study, furthermore, we discuss the model in the observer frame only, and we

do not introduce the co-rotating frame.

3.1.Electric field

We have to describe the accelerating electric field into the three-dimensional from. Based

on the result (Figure 1) of the two-dimensional electrodynamical study, we adopt the fol-

lowing three-dimensional form. First, we use the vacuum solution obtained by Cheng et al.

(1986a),

E||(r) =ΩB(r)f2(r)R2

lc

cs(r)

, (1)

beyond the null charge surface, where s(r) is the curvature radius of the magnetic field line

and f(r) is the fractional gap thickness. We can calculate the electric field at each point

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having a three-dimensional radial distance, r. Below null surface, we assume 50 % of the

strength of the electric field at the null point given by equation (1). The dashed-line in

Figure 1 shows the electric field strength of the approximation form in the meridional plane.

We can see that the typical strength of the electric field by the two-dimensional electro-

dynamical study (solid-line) in the meridional plane is in general described by the present

simple form (dashed-line). In fact, as long as the gap is geometrically thin in the trans-field

direction and the magnitude of the current is smaller than the Goldreich-Julian value, the

vacuum solution beyond the null charge surface approximately describes the typical strength

of the accelerating field in the meridional plane. Now, we assume that this simple form can

describe also the typical strength of the three-dimensional distribution of the accelerating

electric field.

At each point, the maximum Lorentz factor of the particles are determined by the

force balance between the acceleration by the electric and curvature radiation back re-

action, Γp(r) = [3s2(r)E||/2e]1/4, where Ω2 = Ω/100s−1.

high-energy photons as the curvature radiation process, whose typical energy is Ecurv(r) =

3hΓ3

The primary particle emits

p(r)c/4πs(r), and the local power of the curvature radiation is given by lcurv= eE||c.

It is important to estimate the polar cap opening angle of the active region of the outer

gap accelerator. When we consider an open-field line through the outer gap, the pair-creation

process between the primary curvature photons and the surface X-rays mainly occurs near

and below the null charge surface. Therefore, we may be able to relate the opening angle with

the pair-creation mean free path, which is estimated as l(r) ∼ [2s(r)f(r)Rlc] ∼ 2f1/2(Rlc/2)r,

at the null surface. In the present paper, we constrain the width of the polar cap angle of the

active gap by the condition that the mean-free path at the null charge point on the magnetic

field line becomes shorter than the light radius. This condition produces the width of the

polar cap angle of ∼ 250◦.

3.2.Distribution and motion of the secondary pairs

As we demonstrated in section 2, a significant amount of the curvature photons above

∼500 MeV convert into secondary pairs outside the gap via the photon-photon pair-creation

process with the soft-photons emitted by the synchrotron radiation of the secondary pairs.

From Figure 2, furthermore, we can read that the spectrum of the intrinsic emissions has the

photon index of about −1. Therefore, we may approximately describe the local curvature

spectrum with Fcurv ∼ lcurvjnGJ/EcurvEγ, where j represent the current in units of the

Goldreich-Julian value. Using the steady loss equation, d[˙Eedn/dEe]/dEe = Q(Ee), we

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obtain the distribution of the secondary pairs as

dne

dEe

∼

?lcurvjnGJln(Ecurv/2Ee)/˙EeEcurv

lcurvjnGJln(Ecurv/500 MeV)/˙EeEcurv

for 500 MeV < 2Ee< Ecurv

for 2Ee< 500 MeV

(2)

where˙Ee= 2e4B2(r)sin2θp(r)Γ2

of the secondary pairs, θp is the pitch angle, and Γe is the Lorentz factor of the pairs.

The local pitch angle will be expressed as sinθp(r) ∝

f(Rlc)(r/Rlc)3/2and s(r) ∼

sinθp(r) = (r/Rlc)1/2sinθp(Rlc) between the pitch angle of the local point and the light

cylinder along the field lines. Outside the gap, the pairs loose most of their energy via the

synchrotron process. Because the synchrotron loss rate˙Eeis proportional to square of the

particle energy, the power law index of the distribution given by equation (2) becomes p ∼ 2,

which produces a synchrotron spectrum with the spectral index of αν∼ 0.5.

In the observer frame, we may describe the particle motion outside the gap with

e/3m2

ec3is the energy loss rate of the synchrotron radiation

?2f(r)RL/s(r). Because f(r) =

√rRlc are satisfied for the dipole field, we may relate with

β = β0cosθpb + β0sinθpb⊥+ βcoeφ,(3)

where the first term and the second term in the right hand side represent, respectively,

the particle motion parallel to the magnetic field and gyration motion, and the third term

represents the co-rotational motion with the non-dimensional velocity βco= ̟/Rlc, where

̟ is the axial distance. The vector b is the unit vector along the field line and b⊥represents

the unit vector perpendicular to the magnetic field line, b⊥≡ ±(cosδφK + sinδφK × b),

where the sign + (or −) corresponds to gyration of the positrons (or electrons), K =

(b · ∇)b/|(b · ∇)b| is the unit vector of the curvature of the magnetic field line, and δφ

represents the phase of gyration around the magnetic field.

ultra-relativistic speed, we determine the value of the coefficient β0from the condition that

|β| = 1. We anticipate that the photons are emitted in the direction of the particle motion

of equation (3).

Because the pairs have an

We note that the synchrotron radiation after collecting of the photons is greatly depo-

larized due to the gyration motion of the pairs, although the intrinsic radiation is highly

polarized. Therefore, the observed small polarization degree ∼ 10 % at the optical bands

for not only the Crab pulsar, but also for the Vela pulsar (Mignami et al. 2007) are easily

reproduced by the synchrotron emission model (Takata et al. 2007).

We assume that the emission region of the secondary pairs extends just above the outer

gap with thickness of the mean free path of the pair-creation λ ∼ 107cm∼ 0.1Rlcfor the

Crab pulsar. Some secondary high-energy photons via the inverse Compton scattering may

convert into the tertiary pairs. The tertiary pairs will be produced above the emission region

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of the secondary pairs, and its initial Lorentz factor will be smaller than that of the secondary

pairs. We also take into account the effects of the emissions from the tertiary pairs.

3.3.Emission process of the secondary pairs and polarization

We consider that the synchrotron radiation and the inverse Compton scattering of

the secondary pairs are major emission mechanisms for the observed non-thermal radia-

tion through optical to γ-ray bands for the Crab-like pulsars. If we estimate the radiation

powers of the synchrotron radiation and the inverse Compton scattering, we obtain

Psyn

PIC

∼ 10

?

Uph

5 · 107erg/cm3

?−1?

B

106Gauss

?2?sinθp

0.1

?2

(4)

with Uph being the energy density of the synchrotron photons. The estimated value will

explain the observed flux ratio of 1 MeV and 100 MeV emissions of the Crab pulsar. We

calculate only the outward emissions, because the inward emissions are expected to be much

fainter than the outward emissions.

In the calculation, we firstly compute the volume emissivity of the synchrotron radia-

tion and its emitting direction for each radiating point (section 3.3.1). Then, we trace the

propagation of the synchrotron beam to simulate the scattering process by the pairs (sec-

tion 3.3.2). On each scattering point, we calculate the volume emissivity and the polarization

of the inverse Compton scattering for a specific viewing angle of the observer. We perform

this procedure for all calculation points to obtain the total radiation for the specific observer.

This procedure is equivalent with computing the radiation transfer,

dI(k1,ǫ1)

ds

= js(k1,ǫ1) + ji(k1,ǫ1),(5)

where I(k1,ǫ1) is the total intensity of the beam propagating in the direction of k1, ǫ1is the

energy of photons in units of the electron rest mass energy, js(k1,ǫ1) is the volume emissivity

of the synchrotron radiation, and ji(k1,ǫ1) represents the amount of the scattered photons

into the direction k1and to the energy ǫ1. We neglect the effects of the absorption, because

the synchrotron self-absorption is not important above optical photon energy, where we are

now interested in. Also, we ignore the effects of the scattering off from the direction k1of

the synchrotron photons, because the scattered photons are tiny amounts of total number

of the synchrotron photons.

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3.3.1.Synchrotron radiation

Assuming that all the synchrotron photons are radiated toward the particle motion

direction, k1= β, the volume emissivity of the synchrotron radiation is calculated from

js(ǫ1) ≡dIs

ds=ǫ1Fsyn(ǫ1)

δΩ

,(6)

where δΩ is the solid angle of the radiation, and Fsynis the photon spectrum described by

Fsyn(ǫ1) =31/2e3B(r)sinθp(r)

mc2hǫ1

? ?dne(r)

dEe

?

F(ǫ1/ǫsyn)dEe,(7)

where ǫsyn(r) = 3heΓ2

units of the electron rest mass energy, Γerepresents the Lorentz factor of the secondary pairs

and F(x) = x?∞

When we calculate the polarization of the synchrotron radiation, we anticipate that

direction of the electric vector of the electro-magnetic wave propagating toward the observer

is parallel to the projected direction of the acceleration of the particle on the sky, Eem∝

a−(k1·a)k1, (Blaskiewicz et al. 1991), where the acceleration vector a derived from equation

(3) is approximately written by a ∼ ±β0ωBsinθp(−sinδφK + cosδφK × b), where ωBis

the gyration frequency. We assume that the radiation at each point is linearly polarized

with degree of Πsyn = (p + 1)/(p + 7/2), where p is the power law index of the particle

distribution. Because the observed radiation is consist of the radiations from the different

particles with the different pitch angle, we assume that the circular polarization will cancel

out and become zero in the observed radiation. The Stokes parameters Qsynand Usynare,

respectively, calculated from dQsyn(k1,ǫ1)/ds = js(k1,ǫ1)cos2ηs(r) and dUsyn(k1,ǫ1)/ds =

js(k1,ǫ1)sin2ηs(r), where ηs(r) is the position angle defined by the angle between the electric

vector of the wave and the projected direction of the rotation axis on the sky, Ωp= Ω −

(k1· Ω)k1.

e(r)B(r)sinθp(r)/4πm2

ec3is the typical photon energy of the pairs in

xK5/3(y)dy with K5/3being the modified Bessel function of order 5/3.

3.3.2. Inverse Compton scattering

To simulate the scattering process, we trace the three-dimensional trajectory of the syn-

chrotron photons. When we trace the trajectory of the synchrotron photons, we define the

Cartesian coordinate such that z-axis is along the rotation axis and the x-axis is in the merid-

ional plane. By ignoring bending of the trajectory due to the gravity, the position of the pho-

tons after traveling distance δs is x(δs) = (x0+kx0δs)cos(δs/Rlc)+(y0+ky0δs)sin(δs/Rlc),

y(δs) = −(x0+kx0δs)sin(δs/Rlc)+(y0+ky0δs)cos(δs/Rlc) and z(δs) = z0+kz0δs, where the

Page 12

– 12 –

coordinates (x0, y0, z0) are the radiating point of the synchrotron photon, and (kx0, ky0, kz0)

represents the emission direction at (x0, y0, z0). The emission direction of the background

synchrotron radiation is calculated from equation 3, which takes account the aberration due

to the corotating motion.

Because the mean free path of a synchrotron photon of the scattering is much longer than

the light radius, one can consider that the scattering rate is constant along the path of the

synchrotron photons in the magnetosphere as the first order approximation. We determine

the scattering points at regular interval, which is much shorter than the gap size, along the

path of the synchrotron photons. In the calculation, we first compute the Stokes parameter

of the Compton process in the electron rest frame, and then we transform it to the observer

frame. In the following, the prime and ’non’-prime quantities represent the quantities in the

electron rest frame and the observer frame, respectively. A detail derivation of equations of

the Stokes parameters (8) and (10) are seen in Appendix A.

We denote the specific intensity of the synchrotron radiation propagating to the direction

k0in the observer frame with I0(k0,ǫ0), where ǫ0represents the energy of the background

photons in units of the electron rest mass energy. In the electron rest frame, the background

radiation becomes I′

Doppler factor, β is the velocity of the scattering particles in units of the speed of light,

and Γe = 1/?1 − β2. The polar angle θ0 is defined by the angle between the directions

of the particle motion and of the propagation of the background radiation, which becomes

θ0∼ 0.1 − 0.3 radian in numerically. For the particles with the Lorentz factor 103∼ 104,

optical to X-ray photons are mainly scattered. In this photon energy bands, the synchrotron

photons are distributed with a spectral index of αν∼ 0.5 because the cut-off energy of the

synchrotron spectrum is ∼ 1 MeV and because the particles are distributed with the index

p ∼ 2 (section 3.2). Because the synchrotron beam from each position is strongly collimated,

we approximately describe the background beam, in which the center of the beam is direct to

the polar angle θ0measured from the electron motion direction and the azimuthal direction

φ0, as I0(k0,ǫ)= C0ǫ−0.5δ(θ − θ0)δ(φ − φ0), where C0is evaluated from equation (6).

We are interested in the inverse Compton scattering with the background synchrotron

radiation, which is partially polarized with Πsyn∼ 70%. With unpolarized components of

the background radiation propagating to the direction k0, the volume emissivity juand the

Stoke parameter Quand Uuof the scattered radiation propagating to the direction k1are

calculated from (see appendix A)

0(k′

0,ǫ′

0) = D3

1I0(k0,ǫ0), where D1 = ǫ′

0/ǫ0 = Γ−1

e(1 + β cosθ′

0) is a

ju(k1,ǫ1) ≡ dIu(k1,ǫ1)/ds

dQu(k1,ǫ1)/ds

dUu(k1,ǫ1)/ds

= (1 − Πsyn)3σT

16πC0

?

dΓe

?dne

dΓe

?

Page 13

– 13 –

×

ǫ

′−a

0

Γ4+a

e

(1 − β cosθ1)2(1 + β cosθ′

0)a+1

?ǫ′

1

ǫ′

0

?2

?

q′

q′

ǫ′

ǫ′

ucos2ζ − u′

usin2ζ + u′

0

1+

ǫ′

ǫ′

1

0− sin2w′

s

?

,

usin2ζ,

ucos2ζ,

(8)

and

q′

u′

u

= sin2w′

= sin2w′

scos2η′,

ssin2η′,

u

(9)

where the Stokes parameters are measured from the rotation axis of the pulsar projected

on the sky, and ζ is defined by the angle between the directions of the rotation axis and

the particle motion projected on the sky (Figures 11). The polar angle θ1represents the

propagating direction of the scattered photons measured from the particle motion direction,

wsis the scattering angle defined by cosws= k0· k1and the azimuthal angle η is the angle

between the orthogonal direction to the scattering plane and the direction of the particle

motion projected on the sky.

For the polarized component of the background radiation, the volume emissivity and

the Stokes parameters are calculated from

jp(k1,ǫ1) ≡ dIp(k1,ǫ1)/ds

dQp(k1,ǫ1)/ds

dUp(k1,ǫ1)/ds

= Πsyn3σT

?ǫ′

16πC0

?

dΓe

?dne

dΓe

?

×

ǫ

′−a

0

Γ4+a

e

(1 − β cosθ1)2(1 + β cosθ′

0)a+1

1

ǫ′

0

?2

?

q′

q′

ǫ′

ǫ′

pcos2ζ − u,psin2ζ

psin2ζ + u′

0

1+

ǫ′

ǫ′

1

0− sin2w′

scos2λ′

p

?

,

pcos2ζ,

(10)

and

q′

u′

p

=

?sin2w′

?sin2w′

s− (1 + cos2w′

s− (1 + cos2w′

s)cos2λ′

s)cos2λ′

p

?cos2η′− 2cosw′

?sin2η′− 2cosw′

ssin2λ′

ssin2λ′

psin2η′,

pcos2eη′,

p

=

p

(11)

where λpis the angle between the polarization plane of the background radiation and the

plane of the scattering. By exploring the additive property of the Stokes parameters, the

total volume emissivity and Stokes parameters are, respectively, given by ji = ju+ jp,

dQi/ds = dQu/ds + dQp/ds and dUi/ds = dUu/ds + dUp/ds.

After collecting all photons from the possible points for each rotation phase Φ and a view-

ing angle ξ, the degree of the radiation and the position angle of the electric vector of the radi-

ation are, respectively, calculated from P(ξ,Φ,ǫ1) =

?Q2(ξ,Φ,ǫ1) + U2(ξ,Φ,ǫ1)/I(ξ,Φ,ǫ1)

Page 14

– 14 –

and χ(ξ,Φ,ǫ1) = 0.5atan[U(ξ,Φ,ǫ1)/Q(ξ,Φ,ǫ1)], where I(ξ,Φ,ǫ1), Q(ξ,Φ,ǫ1) and U(ξ,Φ,ǫ1)

are the Stokes parameters after collecting photons emitted via both synchrotron radiation

and the inverse Compton scattering. The position angle χ(ξ,Φ,ǫ1) is measured anticlockwise

from the axis of the rotation projected on the sky (Figure 11).

3.4.Model parameter

The inclination angle of the pulsars has been constrained by the polarization measure-

ments of the radio pulsed emissions. However, it has not been strongly constrained the

inclination angle for the Crab pulsar and PSR B0540-69. Therefore, we treat the inclina-

tion angle as a model parameter. The viewing angles ξ of the observer measured from the

rotational axis is also a model parameter. For this local model in the magnetosphere, the

current should be dealt as a model parameter and the position of the inner boundary de-

pends on the assumed current (Hirotani et al. 2003; Takata et al. 2004, 2006; Hirotani

2006). Instead of the current, however, the ratio of the radial distance to the inner boundary

and distance to the the null surface rnis parameterized and is assumed to be constant for

each field line, that is, rin(φ)/rn(φ)=constant, such that the inner boundary locates far away

from the stellar surface if the null surface locates far away. For example, from Figure 1, the

outer gap accelerator with the non-dimensional current j ∼ 0.22 has the inner boundary

at rin(φ)/rφ(φ) ∼ 0.65 in the meridional plane. The altitude of the emission region of the

secondary pairs is also model parameter, because the magnetic field will be modified by rota-

tional and plasma effects near the light cylinder and because the last-open field lines may be

different from the traditional magnetic file lines that are tangent to the light cylinder in the

vacuum case. To specify the upper surface of the outer gap, it is convenient to refer to the

footpoint of the magnetic surface on the star and to parameterize the fractional polar angle

af= θu/θlc, where θuand θlcare the polar angle of the footpoints of the magnetic surfaces

for the gap upper surface and the last-open field line in the vacuum case, respectively. We

constrain the boundary of the radial distance to the emission region to r = Rlc. In this

paper, we apply the model to the Crab pulsar in section 4.1 and to a Crab-like pulsar, PSR

0540-69, in section 4.2.

Page 15

– 15 –

4.Model results

4.1. The Crab pulsar

For the Crab pulsar, we adopt the inclination angle of α = 50◦, the viewing angle of

ξ = 100◦and the position of the inner boundary described by rin(φ)/rn(φ) = 0.67, which

were chosen in Takata et al. (2007). In this paper we chose the fractional angle of af= 1

to explain the phase-resolved spectra, that is, we assume the gap upper surface with the

magnetic field lines that are the conventional last-open field lines in vacuum. The opening

angle of the active outer gap in the azimuthal direction is set at δφ = 250◦, which as assume

in section 3.1.

Figure 3 is the photon mapping of the outwardly propagating photons, where the emis-

sion direction tangent to the local field lines, which is described by af= 1, were temporary

assumed. Figure 4 shows the variations of the typical radial distance to the emission points

of the photons measured by the observer with the viewing angle ξ = 100◦. The dashed-line

in Figure 4 represents the radial distance to the emission points that locate beyond the null

surface on the magnetic field lines coming from the north pole. And, the dotted-lines show

the distance to the points that locate below the null surface on the magnetic field from the

south pole. In the traditional study, only the emission beyond the null surface (dashed-line)

have been considered. We will see that the emission component below null surface is required

to explain the phase-resolved spectrum of Peak 1 (Figure 8). To calculate the phase-resolved

spectra, we define the phase intervals of Peak 1, Bridge, and Peak 2 as 0.06−0.16, 0.29−0.4,

and 0.49 − 0.6 (Figure 4).

4.1.1. Pulse profile and Polarization

Figures 5 and 6 show the predicted variations of the intensity (upper), the position angle

of the electric vector of the radiation (middle) and the degree of the polarization (lower) as a

function of the pulse phase from optical to γ-ray bands. To compare with the observed pulse

profiles in Kuiper et al. (2001), the results were calculated by integrating the photons within

the energy interval 1−10 eV, 0.1−2.4 keV, 20−100 keV, 100−315 keV, 0.75−10 MeV and

30−100 MeV. In the figure, we define the rotation phase Φ = 0 in abscissa axis as lying the

south pole, and the zero degree in the position angle of the bottom panel is corresponding

to the direction of the rotation axis projected on the sky.

In Figures 5 and 6, we see that the calculated pulse profile morphology changes signifi-

cantly as a function of the photon energy likewise the observational pulse profiles (see figure 5

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