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: {\it Fermi}-LAT has detected pulsed gamma-ray emissions with high confidences from more than 40 millisecond pulsars (MSPs). Here we study the phase-averaged gamma-ray properties of MSPs by using revised version of a self-consistent outer gap model. In this model, a strong multipole magnetic field near the stellar surface for a MSP is assumed and such a field will be close to the surface magnetic fields ($\sim 10^{11}- 10^{12}$ G) of young ulsars; the outer gap of a MSP is controlled by photon-photon pair production process, where the effects of magnetic inclination angle ($\alpha$) and magnetic geometry have been taken into account, therefore the fractional size of the outer gap is a function of not only pulsar's period and magnetic field strength but also magnetic inclination angle and radial distance to the neutron star, the inner boundary of the outer gap can be estimated by the pair production process of the gamma-ray photons which are produced by the back-flowing particles through the null charge surface; inside the outer gap, a Gaussian distribution of the parallel electric field along the trans-field thickness is assumed, and the gamma-ray emission is represented by the emission from the average radial distance along the central field lines of the outer gap. Using this model, the phase-averaged gamma-ray spectra are calculated and compared with the observed spectra of 37 MSPs given by the second {\it Fermi}-LAT catalog of gamma-ray pulsars, our results show that the {\it Fermi}-LAT results can be well explained by this model. The thermal X-ray emission properties from MSPs are also investigated.11/2013; - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: Since the launch of the Fermi telescope more than five years ago, many new gamma-ray pulsars have been discovered with intriguing properties challenging our current understanding of pulsar physics. Observation of the Crab pulsar furnish today a broad band analysis of the pulsed spectrum with phase-resolved variability allowing to refine existing model to explain pulse shape, spectra and polarization properties. The latter gives inside into the geometry of the emitting region as well as on the structure of the magnetic field. Based on an exact analytical solution of the striped wind with finite current sheet thickness, we analyze in detail the phase-resolved polarization variability emanating from the synchrotron radiation. We assume that the main contribution to the wind emissivity comes from a thin transition layer where the dominant toroidal magnetic field reverses its polarity, the so-called current sheet. The resulting radiation is mostly linearly polarized. In the off-pulse region, the electric vector lies in the direction of the projection onto the plane of the sky of the rotation axis of the pulsar. This property is unique to the wind model and in good agreement with the Crab data. Other properties such as a reduced degree of polarization and a characteristic sweep of the polarization angle within the pulses are also reproduced. These properties are qualitatively unaffected by variations of the wind Lorentz factor, the lepton injection power law index, the contrast in hot and cold particle, the obliquity of the pulsar and the inclination of the line of sight.Monthly Notices of the Royal Astronomical Society 08/2013; 434(3). · 5.23 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. · 6.28 Impact Factor
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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
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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
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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)
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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
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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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