The Phase-resolved High Energy Spectrum of
the Crab Pulsar
J.J. Jia a, Anisia P.S. Tang a, J. Takata b, H.K. Chang c,
K.S.Cheng a
aDepartment of Physics, The University of Hong Kong, Hong Kong, China,
hrspksc@hkucc.hku.hk
bASIAA/National Tsing Hua University - TIARA, PO Box 23-141, Taipei,
Taiwan
cDepartment of Physics and Institute of Astronomy, National Tsing Hua
University, Hsinchu 30013, Taiwan
Abstract
We present a modified outer gap model to study the phase-resolved spectra of the
Crab pulsar. A theoretical double peak profile of the light curve containing the
whole phase is shown to be consistent with the observed light curve of the Crab
pulsar by shifting the inner boundary of the outer gap inwardly to ∼ 10 stellar radii
above the neutron star surface. In this model, the radial distances of the photons
corresponding to different phases can be determined in the numerical calculation.
Also the local electrodynamics, such as the accelerating electric field, the curvature
radius of the magnetic field line and the soft photon energy, are sensitive to the
radial distances to the neutron star. Using a synchrotron self-Compton mechanism,
the phase-resolved spectra with the energy range from 100 eV to 3 GeV of the Crab
pulsar can also be explained.
Key words: neutron stars, pulsars, radiation mechanism, radiation processes
PACS: 97.60.Jd, 97.60.Gb, 95.30.Gv, 94.05.Dd
1 Introduction
There are eight pulsars have been detected in gamma-ray energy range (cf.
Thompson 2006 for a recent review) with period ranging from 0.033s to 0.237s
and age younger than million years old. Theoretically, it is suggested that
high-energy photons are produced by the radiation of charged particles that
are accelerated in the pulsar magnetosphere. There are two kinds of theoretical
Preprint submitted to Elsevier 18 November 2018
http://arxiv.org/abs/0704.1534v1
models: one is the polar gap model (e.g., Harding 1981, Daugherty & Hard-
ing 1996, for more detail review of polar cap model cf. Harding 2006), and
another is the outer gap model (e.g. Cheng, Ho, & Ruderman 1986a, 1986b;
Chiang & Romani 1994). Both models predict that electrons and positrons are
accelerated in a charge depletion region called a gap by the electric field along
the magnetic field lines and assume that charged particles lose their energies
via curvature radiation in both polar and outer gaps. The key differences are
polar gap are located near stellar surface and the outer gap are located near
the null charge surface, where are at least several tens stellar radii away the
star.
The continuous observations of powerful young pulsars, including the Crab,
the Vela and the Geminga, have collected large number of high energy photons,
which allow us to carry out much more detailed analysis. Fierro et al. (1998)
divided the whole phase into eight phase intervals, i.e. leading wing 1, peak 1,
trailing 1, bridge, leading wing 2, peak 2, trailing 2 and off-pulse. They showed
that the data in each of these phases can be roughly fitted with a simple power
law. However, the photon indices of these phases are very different, they range
from 1.6 to 2.6. Massaro et al. (2000) have shown that X-ray pulse profile is
energy dependent and the X-ray spectral index also depends on the phase of
the rotation.
The Crab pulsar has been extensively studied from the radio to the extremely
high energy ranges, and the phase of the double-pulse with separation of 144◦
is found to be consistent over all wavelengths. Recently Kuiper et al. (2001)
have combined the X-ray and gamma-ray data of the Crab pulsar, they showed
that the phase-dependent spectra exhibit a double-peak structure, i.e. one
very broad peak in soft gamma-rays and another broad peak in higher energy
gamma-rays. The position of these peaks depend on the phase. Although the
double-peak structure is a signature of synchrotron self-Compton mechanism,
it is impossible to fit the phase dependent spectrum by a simple particle energy
spectrum. Actually it is not surprised that the spectrum is phase dependent
because photons are emitted from different regions of the magnetosphere. The
local properties, e.g. electric field E(r), magnetic field B(r), particles density
and energy distribution are very much different for different regions. Therefore
these phase dependent data provide very important information for emission
region. Consequently, the phase-resolved properties provide very important
clues and constraints for the theoretical models. So far, the three-dimensional
outer gap model seems to be the most successful model in explaining both the
double-peak pulse profile and the phase-resolved spectra of the Crab pulsar
(e.g. Chiang & Romani 1992, 1994; Dyks & Rudak 2003; Cheng, Ruderman
& Zhang, 2000, hereafter CRZ). However, the leading-edge and trailing-edge
of the light curve cannot be given out, since the inner boundary of the outer
gap is located at the null charge surface in this model. Recently, the electro-
dynamics of the pulsar magnetosphere has been studied carefully by solving
the Poisson equation for electrostatic potential and the Boltzmann equations
for electrons/positrons (Hirotani & Shibata, 1999a,b,c; Takata et al. 2004,
2006; Hirotani 2005), and the inner boundary of the gap is shown to be located
near the stellar surface.
We will organize the paper as follows. We describe the modified outer gap
model in §2. In §3, we calculate the phase-resolved spectra and present the
fitting result of the spectra in different phase intervals. Finally, we discuss our
results and draw conclusions in §4.
2 A modified outer gap model
Originally proposed by Holloway (1973) that vacuum gaps may form in the
outer regions of pulsar’s magnetosphere, Cheng, Ho and Ruderman (1986a,
1986b; hereafter CHR) developed the idea of outer magnetosphere gaps and
explained the radiation mechanisms of the γ-rays from the Crab and Vela
pulsars. CHR argued that a global current flowing through the null surface
of a rapidly spinning neutron star would result in large regions of charge
depletion, which form the gaps in the magnetosphere. They assume the outer
gap should begin at the null charge surface and extend to the light cylinder.
In the gaps, a large electric field parallel to the magnetic field lines is induced
( ~E · ~B 6= 0), and it can accelerate the electrons or positrons to extremely
relativistic speed. Thus, those charges can emit high energy photons through
various mechanisms, and further produce copious e+e− pairs to sustain the
gaps and the currents.
Based on the CHR model, Chiang and Romani (1992, 1994) generated gamma
ray light curves for various magnetosphere geometries by assuming that gap-
type regions could be supported along all field lines which define the boundary
between the closed region and open field line region rather than just on the
bundle of field lines lying in the plane containing the rotation and magnetic
dipole axes. In their model, photons are generated and travel tangential to the
local magnetic field lines and there are beams in both the outward and inward
directions. They suggested that a single pole would produce a double-peak
emission profile when the line of sight crosses the enhanced regions of the γ-ray
beam, while the inner region of the beam results in the bridge emission between
these two pulses. The peak phase separation can be accommodated by choosing
a proper observer viewing angle. Because the location of emission of each point
in phase along a given line of sight can be mapped approximately in this model,
the outer gap is thus divided into small subzones. As the curvature radius,
photon densities and the local electrodynamics in different subzones are not
the same, the spectral variation of the high energy radiation in different phase
intervals varies. Later, Romani and Yadigaroglu (1995) developed the single
gap model by involving the effects of aberration, retarded potential and time of
flight across the magnetosphere. The light curve profiles in this modified model
is simply determined by only two parameters, which are magnetic inclination
angle α and the viewing angle ζ . They argued that the γ-ray emission can only
be observed from pulsars with large viewing angle (ζ ≥ 45◦), and we cannot
receive the γ photons but radio emissions from the aligned pulsars (α ≤ 35◦).
Furthermore, they showed the gap would grow larger as the pulsar slows down,
and more open field lines can occupy the outer gap, which means the older
pulsar are more efficient for producing GeV γ-ray photons (Yadigaroglu &
Romani, 1995).
However, the assumptions of the model proposed by Romani’s group are not
self-consistent. Why is there only a single pole and only outgoing current in
the magnetosphere? Cheng, Ruderman and Zhang (2000) proposed another
version of three dimensional outer gap model for high energy pulsars based
on the pioneering work of Romani, and made it more natural in physics. In
the CRZ model, two outer gaps and both outward and inward currents are
allowed (though it turns out that outgoing currents dominate the emitted ra-
diation intensities), and the azimuthal extension of the outer gap is restricted
on a bundle of fields instead of the whole lines. Like the previous work by
Yadigaroglu and Romani (1995), the CRZ model also contains the same two
parameters, but more self-consistent in gap geometry and radiation morphol-
ogy by using the pair production conditions. The electric field parallel to the
magnetic field lines is
ΩB(r)f 2(r)R2L
cs(r)
, (1)
where f(r) ∝ r3/2 and s(r) ∝ r1/2 are the fractional size of the outer gap and
the curvature radius at the distance r. The characteristic fractional size of the
outer gap evaluated at r ∼ rL, where rL is the light cylinder radius, can be
estimated by the condition of pair creation (Zhang & Cheng 1997; CRZ) and
is given by
f0 = 5.5P
26/21B
12 ∆Φ
1/7 , (2)
where ∆Φ is the azimuthal extension of the outer gap. CRZ estimates its value
by considering the local pair production condition and give ∆Φ ∼ 160◦ for the
Crab pular. It has been pointed out that if the inclination angle is small, f0
can be changed by a factor of several (Zhang et al. 2004). We want to remark
that equation (1) is the solution of vacuum solution, for regions near null
surface and the inward extension of the gap the electric field is shown to be
deviated from the vacuum solution (e.g Muslimov & Harding 2004; Hirotani
2006). Nevertheless for simplicity we shall assume the vacuum solution for the
entire gap.
In the numerical calculation, the outer gap should be divided into several layers
in space. The shape of each layer at the stellar surface is similar to that of the
polar cap, but smaller in size. Thus, for a thin gap, the calculation of only one
representative layer is enough; while for a thick one (e.g. Geminga), several
different layers should be added in the calculation (Zhang & Cheng, 2001). The
coordinate of the footprint of the last closed field lines on the stellar surface is
determined as (x0, y0, z0), then the coordinates values (x
0) of the inner
layers can be defined by x′0 = a1x0, y
0 = a1y0, and z
1− x′02 − y′02, where
a1 corresponds to the various layers in the open volume.
Inside the light cylinder, high energy photons will be emitted nearly tangent
to the magnetic field lines in the corotating frame because of the relativistic
1/γ beaming inherent in high energy processes unless |E×B| ∼ B2. Then the
propagation direction of each emitted photons by relativistic charged particles
can be expressed as (ζ ,Φ), where ζ is the polar angle from the rotation axis
and Φ is the phase of rotation of the star. Effects of the time of flight and
aberration are taken into account. A photon with velocity u = (ux, uy, uz)
along a magnetic field line with a relativistic addition of velocity along the
azimuthal angle gives an aberrated emission direction u′ = (u′x, u
z). The
time of flight gives a change of the phase of the rotation of the star. Combining
these two effects, and choosing Φ = 0 for radiation in the (x,z) plane from the
center of the star, ζ and Φ are given by cos ζ = u′z and Φ = −φu′ −~r · û′, where
φu′ is the azimuthal angle of û′ and ~r is the emitting location in units of RL.
In panel A of Fig. 1, the emission morphology in the (ζ , Φ) plane is shown.
For a given observer with a fixed viewing angle ζ , a double-pulsed structure is
observed because photons are clustered near two edges of the emission pattern
due to the relativistic effects (cf. panel B of Fig. 1).
In Fig. 1, we can see that this model can only produce radiation between two
peaks. However, the observed data of the Crab, Vela and Geminga indicate
that the leading wing 1 and the trailing wing 2 are quite strong, and even the
intensity in off-pulse cannot be ignored. Hirotani and his co-workers (Hirotani
& Shibata 2001; Hirotani, Harding & Shibata 2003) have pointed out that the
large current in the outer gap can change the boundary of the outer gap. They
solve the set of Maxwell and Boltzmann equations in pulsar magnetospheres
and demonstrate the existence of outer-gap accelerators, whose inner bound-
ary position depends the detail of the current flow and it is not necessarily
located at the null charge surface. For the gap current lower than 25% of the
Goldreich-Julian current, the inner boundary of the outer gap is very close to
the null surface (Hirotani 2005). On the other hand if the current is close to
the Goldreich-Julian current, the inner boundary can be as close as 10 stellar
radii. In Fig. 2, we show the light curve by assuming the inner boundary is
extended inward from the null charge surface to 10 stellar radii (cf. panel A
Fig. 1. Emission projection onto the (ζ,Φ) plane and pulse profile for the single pole
outer gap. The photons are emitted outwards from the outer gap. (a) The emission
projection (a1 = 0.9) and (b) the corresponding pulse profile (∆a1 = 0.03), for
Crab parameters α = 65◦ and ζ = 82◦.
of Fig. 2). In panel B of Fig. 2, the solid line represents emission trajectory of
outgoing radiation of one gap from the null surface to the light cylinder with
α = 50◦ and ζ = 75◦ and the dashed line represents the outgoing radiation
from another gap from the inner boundary to the null surface. In the presence
of the extended emission region from the near the stellar surface to the null
charge surface, leading wing 1, trailing wing 2 and the off-pulse components
can also be produced.
0 60 120 180 240 300 360
Phase
Fig. 2. Upper panel: the simulated pulse profile of the Crab pulsar; lower panel:
variation of radial distance with pulse phase for the Crab pulsar in units of RL,
where the bold line represents those in the outer magnetosphere, and the dashed
line represents those in the inner magnetosphere. The inclination angle is 50◦ and
the viewing angle is 75◦.
3 The phase-resolved spectra
3.1 radiation spectrum
The Crab pulsar has enough photons for its spectra to be analyzed, and the
phase-resolved spectra are useful for study of the local properties of the mag-
netosphere. Here, we summarize the calculation procedure of the radiation
spectrum given in CRZ, which is used to calculate the spectrum in different
phases.
The electric field of a thin outer gap (CHR) is given by E‖(r) =
ΩB(r)a2(r)
cs(r)
ΩB(r)f2(r)R2
cs(r)
, where a(r) is the thickness of the outer gap at position r, and
f(r) = a(r)/RL is the local fractional size of the outer gap. Assuming that
the magnetic flux subtended in the outer gap is constant in the steady state,
we get the local size factor f(r) ∼ f(RL)( rRL )
3/2, where f(RL) is estimated
by using the pair creation condition (cf. Zhang & Cheng 1997, CRZ). As the
equilibrium between the energy loss in radiation and gain from accelerating
electric field, the local Lorentz factor of the electrons/positrons in the outer
gap is γe(r) = (
eE‖(r)c)
For a volume element ∆V in the outer gap, the number of primary charged
particles can be roughly written as dN = nGJ∆A∆l, where nGJ =
is the
local Goldreich-Julian number density, B∆A is the magnetic flux through the
accelerator and ∆l is the path length along its magnetic field lines. (Here, We
would like to remark that this could overestimate the primary charge num-
ber density near the null surface, where the positronic charge density dom-
inates the Goldreich-Julian charge density. However, the observed radiation
comes from the wide range of magnetospheric region, an slight overestima-
tion of a small region should not cause a qualitative difference.) Thus, the
total number of the charged particles in the outer gap is N ∼ ΩΦ
RL, where
Φ ∼ f(RL)B(RL)R2L∆φ is the typical angular width of the magnetic flux
tube subtend in the outer gap. The primary e± pairs radiate curvature pho-
tons with a characteristic energy Ecur(r) =
~γ3e (r)
, and the power into
curvature radiation for dN e± pairs in a unit volume is dLcur
≈ lcurnGJ(r),
where lcur = eE‖c is the local power into the curvature radiation from a single
electron/positron. The spectrum of primary photons from a unit volume is
dV dEγ
≈ lcurnGJ
, Eγ ≤ Ecur. (3)
These primary curvature photons collide with the soft photons produced by
synchrotron radiation of the secondary e± pairs, and produce the secondary
e± pairs by photon-photon production. In a steady state, the distribution of
secondary electrons/positrons in a unit volume is
dV dEe
∫ d2Ṅ(E ′γ = 2E
dV dEγ
dE ′e ≈
lcurnGJ
), (4)
with Ėe the electron energy loss into synchrotron radiation, which is Ėe =
e4B2(r) sin2 β(r)
)2, where B(r) is the local magnetic field and β(r) the
local pitch angle, sin β(r) ∼ sin β(RL)( rRL )
1/2, sin β(RL) is the pitch angle at
the light cylinder. Therefore, the energy distribution of the secondary elec-
trons/positrons in volume ∆V (r) can be written as
dN(r)
dV dEe
∆V (r) ∼
lcurnGJ∆V (r)
). (5)
The corresponding photon spectrum of the synchrotron radiation is
Fsyn(Eγ , r) =
3e3B(r) sinβ
mec2h
dN(r)
F (x)dEe, (6)
where x = Eγ/Esyn, and Esyn(r) =
heB(r) sinβ(r)
is the typical photon
energy, and F (x) = x
x K5/3(y)dy, where K5/3(y) is the modified Bessel
function of order 5/3. Also, the spectrum of the inverse Compton scattered
photons in the volume ∆V (r) is
FICS(Eγ, r) =
dN(r)
d2NICS(r)
dEγdt
dEe, (7)
where
d2NICS(r)
dEγdt
nsyn(ǫ, r)F (%epsilon, Eγ , Ee)dǫ, and F (ǫ, Eγ , Ee) =
3σT c
4(Ee/mc2)2
[2q ln q+
(1 + 2q)(1− q) + (Γq)
2(1−q)
2(1+Γq)
], where Γ = 4ǫ(Ee/mec
2)/mec
2, q = E1/Γ(1− E1)
with E1 = Eγ/Ee and 1/4(Ee/mec
2) < q < 1. The number density of the
synchrotron photons with energy ǫ is nsyn(ǫ, r) =
Fsyn(ǫ)
cr2∆Ω
, where Fsyn is the
calculated synchrotron radiation flux, and ∆Ω is the usual beam solid angle.
Fig. 3 shows the observed data of the phase-resolved spectra from 100 eV
to 3 GeV of the Crab pulsar, and the theoretical fitting results calculated
by using the synchrotron self-Compton mechanism. The phase intervals are
defined by division given by Fierro (1998), and the amplitude of the spectrum
in each phase interval is proportional to the number of photons counted in it.
In this fitting, f(RL) = 0.21, and B = 3.0× 1012Gauss are used, which give a
consistent fitting of the phase-resolved spectra of the seven phase intervals. In
order to obtain a better fit, we treat the pitch angle (β) and the beam solid
angle (∆Ω) near the light cylinder as free parameters and vary from phase to
phase in the calculation. sin β(RL) = 0.06 and ∆Ω = 5.0 are chosen for trailing
wing 1, bridge and leading wing 2; sin β(RL) = 0.02,∆Ω = 1.0 for leading wing
1; sin β(RL) = 0.04,∆Ω = 3.5 for peak 1, sin β(RL) = 0.07,∆Ω = 3.0 for peak
2, and sin β(RL) = 0.03,∆Ω = 6.0 for trailing wing 2. Additionally, the phase-
averaged spectrum of the total pulse of the Crab pulsar is shown in Fig. 4,
where the parameters are chosen as sin β(RL) = 0.05 and ∆Ω = 5.0.
3.2 Analysis of the Phase-Resolved Spectra
The high energy spectra of the Crab pulsar is explained by using the syn-
chrotron self-Compton mechanism, which involves both the synchrotron radia-
tion and the Inverse Compton-Scattering (ICS) caused by the ultra-relativistic
electron/positron pairs created by the extremely high-energy curvature pho-
tons. The secondary e± pairs gyrate in the strong magnetic field and radiate
F(E) (MeV cm
Fig. 3. Phase resolved spectra of the Crab pulsar from 100 eV to 3 GeV in the 7
narrow pulse-phase intervals. Two spectra (for the TW1 and LW2) are displayed
twice. The curved line is calculated by the theoretical model, and the observed data
are taken from Kuiper et al. (2001).
synchrotron photons. While in the far regions of the magnetosphere where the
magnetic field decays rapidly, the relativistic pairs collide with the soft syn-
chrotron photons through the ICS process. Thus, the spectra of the radiation
contain two main components: one is the synchrotron radiation from the soft
X-ray to ∼10 MeV, and the other is the ICS component in the even higher
energy range. Usually, the synchrotron spectrum has stronger amplitude than
1x10
1x10

2
 F

 s
-
E (MeV)
Fig. 4. Phase-averaged spectrum of the Crab pulsar. The observed data are taken
from Kuiper et al. (2001).
that of ICS, and there is obvious turning frequency between these two com-
ponents, e.g. about 3MeV for peak 1. As we know, the power of synchrotron
radiation and ICS can be compared by the ratio of the local magnetic energy
density and the photon energy density, i.e.
∝ B(r)
ǫsyn(r)nsyn(ǫsyn, r)
, (8)
where ǫsyn(r) is the synchrotron photon energy in location r. In Fig. 3, the
spectra in trailing wing 1, bridge and leading wing 2 have broad synchrotron
spectra, which cover from 100 eV to ∼30 MeV. In Fig. 2, it is demonstrated
that the radiation of these three phase intervals are dominated by the photons
generated in the near surface region, where the magnetic field is so strong that
synchrotron radiation takes up the most emission. However, the radiation
of peak 1 and 2 are from the far regions near the light cylinder, where the
magnetic field decays rapidly (B ∝ r−3), thus, the ICS radiation becomes
more important above 3 MeV.
The peak of the synchrotron spectrum is determined by the characteristic
synchrotron photon energy. Since Esyn ∝ γ2eB sin β, where γe is the Lorentz
factor of the secondary pairs and β is the pitch angle of the electron/positron
to the magnetic field, the peak of the synchrotron spectrum can shift if the β
varies. Since the outward radiation direction covers a wider range than that of
the inward radiation, so the solid angle (∆Ω) is no longer the unity as assumed
in CRZ model. The solid angle can effect the amplitude of the ICS spectrum
because the number density of the synchrotron photons is proportional to 1
Therefore it is reasonable for us to choose β(RL) and ∆Ω as a set of parameters
in fitting the phase-resolved spectra of the Crab pulsar.
4 Conclusion and Discussion
We have tried to explain the high energy light curve and the phase-resolved
spectra in the energy range from 100 eV to 3 GeV of the Crab pulsar by mod-
ifying the three dimensional outer magnetosphere gap model. Compared to
the classical outer gap with the inner boundary at the null charge surface, the
modified model allows the outer gap to start at the region about several stel-
lar radii above the neutron star surface, and the ”inwardly-extended” part of
the outer gap contributes to the outer wings and off-pulse of the light curve.
Such modified outer gap geometry also plays a vital role in explaining the
optical polarization properties of the Crab pulsar (Takata et al. 2006). Two
adjustable parameters are used to simulate the light curve: one is the inclina-
tion angle of the magnetic axis to the rotational axis α, and the other is the
viewing angle also to the rotational axis ζ . As constrained by the phase sepa-
ration of the double peaks, we choose the values for these two parameters that
α = 50◦ and ζ = 75◦. So far, these two parameters have not been determined
from the observations. From radio observations, Rankin (1993) estimated that
α ≈ 84◦ and ζ is not known. Moffett and Hankins (1999) gave that α ≈ 56◦
and ζ = 117◦ by using the polarimetric observations at frequencies between
1.4 and 8.4 GHz. Of course, our values cannot be the true ones, and require
further observations to give strong restrictions of them.
In fitting the phase-resolved spectra of the Crab pulsar, our model performs
well from 100 eV to 1 GeV, but fails beyond 1 GeV. The inverse Compton
scattering spectrum of our results falls down quickly when the energy is over 1
GeV, but the observation data indicates that the spectrum still increases, es-
pecially in the first trailing wing, the bridge and the second leading wing phase
intervals. We have assumed that the curvature photons are all absorbed by
the magnetic field lines, however, some of these multi-GeV photons produced
near the light cylinder should be easily escaped from the photon-photon pair
creation process. In the spectrum fitting of peak 1, our result has a frequency
shift below 1 MeV, and we found that in order to well fit the spectrum we
should reduce the curvature photon energy by a quarter. The energy of the
curvature photon Ecur ∝ s−1(r), where s(r) is the local curvature radius. As
the high energy photons are produced in the far regions of the magnetosphere,
where s(r) maybe not follow the dipole form, we can change the photon energy
slightly.
Moreover, the stellar radius of a neutron star is usually treated as 106 cm when
calculating the strength of the surface magnetic field. However, the equation
of state inside the neutron star of the current theoretical models cannot give
a convincing value of the neutron star size. Thus, we can only determine the
magnetic moment, i.e. BpR
0, of the pulsar from the energy loss rate. Therefore,
we can rewrite the magnetic field of the Crab pulsar as B12R
6 = 3.8. In fitting
the phase-resolved spectra of the Crab pulsar, we choose Bp = 3×1012 G, not
the traditional value of 3.8× 1012 G, for it gives a better fitting on the lower
energy range below 10 keV.
Finally we want to emphasize that
This research is supported by a RGC grant of Hong Kong Government under
HKU 7015/05P.
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	Introduction
	A modified outer gap model
	The phase-resolved spectra
	radiation spectrum
	Analysis of the Phase-Resolved Spectra
	Conclusion and Discussion

  We present a modified outer gap model to study the phase-resolved spectra of
the Crab pulsar. A theoretical double peak profile of the light curve
containing the whole phase is shown to be consistent with the observed light
curve of the Crab pulsar by shifting the inner boundary of the outer gap
inwardly to $\sim 10$ stellar radii above the neutron star surface. In this
model, the radial distances of the photons corresponding to different phases
can be determined in the numerical calculation. Also the local electrodynamics,
such as the accelerating electric field, the curvature radius of the magnetic
field line and the soft photon energy, are sensitive to the radial distances to
the neutron star. Using a synchrotron self-Compton mechanism, the
phase-resolved spectra with the energy range from 100 eV to 3 GeV of the Crab
pulsar can also be explained.

Introduction
There are eight pulsars have been detected in gamma-ray energy range (cf.
Thompson 2006 for a recent review) with period ranging from 0.033s to 0.237s
and age younger than million years old. Theoretically, it is suggested that
high-energy photons are produced by the radiation of charged particles that
are accelerated in the pulsar magnetosphere. There are two kinds of theoretical
Preprint submitted to Elsevier 18 November 2018
http://arxiv.org/abs/0704.1534v1
models: one is the polar gap model (e.g., Harding 1981, Daugherty & Hard-
ing 1996, for more detail review of polar cap model cf. Harding 2006), and
another is the outer gap model (e.g. Cheng, Ho, & Ruderman 1986a, 1986b;
Chiang & Romani 1994). Both models predict that electrons and positrons are
accelerated in a charge depletion region called a gap by the electric field along
the magnetic field lines and assume that charged particles lose their energies
via curvature radiation in both polar and outer gaps. The key differences are
polar gap are located near stellar surface and the outer gap are located near
the null charge surface, where are at least several tens stellar radii away the
star.
The continuous observations of powerful young pulsars, including the Crab,
the Vela and the Geminga, have collected large number of high energy photons,
which allow us to carry out much more detailed analysis. Fierro et al. (1998)
divided the whole phase into eight phase intervals, i.e. leading wing 1, peak 1,
trailing 1, bridge, leading wing 2, peak 2, trailing 2 and off-pulse. They showed
that the data in each of these phases can be roughly fitted with a simple power
law. However, the photon indices of these phases are very different, they range
from 1.6 to 2.6. Massaro et al. (2000) have shown that X-ray pulse profile is
energy dependent and the X-ray spectral index also depends on the phase of
the rotation.
The Crab pulsar has been extensively studied from the radio to the extremely
high energy ranges, and the phase of the double-pulse with separation of 144◦
is found to be consistent over all wavelengths. Recently Kuiper et al. (2001)
have combined the X-ray and gamma-ray data of the Crab pulsar, they showed
that the phase-dependent spectra exhibit a double-peak structure, i.e. one
very broad peak in soft gamma-rays and another broad peak in higher energy
gamma-rays. The position of these peaks depend on the phase. Although the
double-peak structure is a signature of synchrotron self-Compton mechanism,
it is impossible to fit the phase dependent spectrum by a simple particle energy
spectrum. Actually it is not surprised that the spectrum is phase dependent
because photons are emitted from different regions of the magnetosphere. The
local properties, e.g. electric field E(r), magnetic field B(r), particles density
and energy distribution are very much different for different regions. Therefore
these phase dependent data provide very important information for emission
region. Consequently, the phase-resolved properties provide very important
clues and constraints for the theoretical models. So far, the three-dimensional
outer gap model seems to be the most successful model in explaining both the
double-peak pulse profile and the phase-resolved spectra of the Crab pulsar
(e.g. Chiang & Romani 1992, 1994; Dyks & Rudak 2003; Cheng, Ruderman
& Zhang, 2000, hereafter CRZ). However, the leading-edge and trailing-edge
of the light curve cannot be given out, since the inner boundary of the outer
gap is located at the null charge surface in this model. Recently, the electro-
dynamics of the pulsar magnetosphere has been studied carefully by solving
the Poisson equation for electrostatic potential and the Boltzmann equations
for electrons/positrons (Hirotani & Shibata, 1999a,b,c; Takata et al. 2004,
2006; Hirotani 2005), and the inner boundary of the gap is shown to be located
near the stellar surface.
We will organize the paper as follows. We describe the modified outer gap
model in §2. In §3, we calculate the phase-resolved spectra and present the
fitting result of the spectra in different phase intervals. Finally, we discuss our
results and draw conclusions in §4.
2 A modified outer gap model
Originally proposed by Holloway (1973) that vacuum gaps may form in the
outer regions of pulsar’s magnetosphere, Cheng, Ho and Ruderman (1986a,
1986b; hereafter CHR) developed the idea of outer magnetosphere gaps and
explained the radiation mechanisms of the γ-rays from the Crab and Vela
pulsars. CHR argued that a global current flowing through the null surface
of a rapidly spinning neutron star would result in large regions of charge
depletion, which form the gaps in the magnetosphere. They assume the outer
gap should begin at the null charge surface and extend to the light cylinder.
In the gaps, a large electric field parallel to the magnetic field lines is induced
( ~E · ~B 6= 0), and it can accelerate the electrons or positrons to extremely
relativistic speed. Thus, those charges can emit high energy photons through
various mechanisms, and further produce copious e+e− pairs to sustain the
gaps and the currents.
Based on the CHR model, Chiang and Romani (1992, 1994) generated gamma
ray light curves for various magnetosphere geometries by assuming that gap-
type regions could be supported along all field lines which define the boundary
between the closed region and open field line region rather than just on the
bundle of field lines lying in the plane containing the rotation and magnetic
dipole axes. In their model, photons are generated and travel tangential to the
local magnetic field lines and there are beams in both the outward and inward
directions. They suggested that a single pole would produce a double-peak
emission profile when the line of sight crosses the enhanced regions of the γ-ray
beam, while the inner region of the beam results in the bridge emission between
these two pulses. The peak phase separation can be accommodated by choosing
a proper observer viewing angle. Because the location of emission of each point
in phase along a given line of sight can be mapped approximately in this model,
the outer gap is thus divided into small subzones. As the curvature radius,
photon densities and the local electrodynamics in different subzones are not
the same, the spectral variation of the high energy radiation in different phase
intervals varies. Later, Romani and Yadigaroglu (1995) developed the single
gap model by involving the effects of aberration, retarded potential and time of
flight across the magnetosphere. The light curve profiles in this modified model
is simply determined by only two parameters, which are magnetic inclination
angle α and the viewing angle ζ . They argued that the γ-ray emission can only
be observed from pulsars with large viewing angle (ζ ≥ 45◦), and we cannot
receive the γ photons but radio emissions from the aligned pulsars (α ≤ 35◦).
Furthermore, they showed the gap would grow larger as the pulsar slows down,
and more open field lines can occupy the outer gap, which means the older
pulsar are more efficient for producing GeV γ-ray photons (Yadigaroglu &
Romani, 1995).
However, the assumptions of the model proposed by Romani’s group are not
self-consistent. Why is there only a single pole and only outgoing current in
the magnetosphere? Cheng, Ruderman and Zhang (2000) proposed another
version of three dimensional outer gap model for high energy pulsars based
on the pioneering work of Romani, and made it more natural in physics. In
the CRZ model, two outer gaps and both outward and inward currents are
allowed (though it turns out that outgoing currents dominate the emitted ra-
diation intensities), and the azimuthal extension of the outer gap is restricted
on a bundle of fields instead of the whole lines. Like the previous work by
Yadigaroglu and Romani (1995), the CRZ model also contains the same two
parameters, but more self-consistent in gap geometry and radiation morphol-
ogy by using the pair production conditions. The electric field parallel to the
magnetic field lines is
ΩB(r)f 2(r)R2L
cs(r)
, (1)
where f(r) ∝ r3/2 and s(r) ∝ r1/2 are the fractional size of the outer gap and
the curvature radius at the distance r. The characteristic fractional size of the
outer gap evaluated at r ∼ rL, where rL is the light cylinder radius, can be
estimated by the condition of pair creation (Zhang & Cheng 1997; CRZ) and
is given by
f0 = 5.5P
26/21B
12 ∆Φ
1/7 , (2)
where ∆Φ is the azimuthal extension of the outer gap. CRZ estimates its value
by considering the local pair production condition and give ∆Φ ∼ 160◦ for the
Crab pular. It has been pointed out that if the inclination angle is small, f0
can be changed by a factor of several (Zhang et al. 2004). We want to remark
that equation (1) is the solution of vacuum solution, for regions near null
surface and the inward extension of the gap the electric field is shown to be
deviated from the vacuum solution (e.g Muslimov & Harding 2004; Hirotani
2006). Nevertheless for simplicity we shall assume the vacuum solution for the
entire gap.
In the numerical calculation, the outer gap should be divided into several layers
in space. The shape of each layer at the stellar surface is similar to that of the
polar cap, but smaller in size. Thus, for a thin gap, the calculation of only one
representative layer is enough; while for a thick one (e.g. Geminga), several
different layers should be added in the calculation (Zhang & Cheng, 2001). The
coordinate of the footprint of the last closed field lines on the stellar surface is
determined as (x0, y0, z0), then the coordinates values (x
0) of the inner
layers can be defined by x′0 = a1x0, y
0 = a1y0, and z
1− x′02 − y′02, where
a1 corresponds to the various layers in the open volume.
Inside the light cylinder, high energy photons will be emitted nearly tangent
to the magnetic field lines in the corotating frame because of the relativistic
1/γ beaming inherent in high energy processes unless |E×B| ∼ B2. Then the
propagation direction of each emitted photons by relativistic charged particles
can be expressed as (ζ ,Φ), where ζ is the polar angle from the rotation axis
and Φ is the phase of rotation of the star. Effects of the time of flight and
aberration are taken into account. A photon with velocity u = (ux, uy, uz)
along a magnetic field line with a relativistic addition of velocity along the
azimuthal angle gives an aberrated emission direction u′ = (u′x, u
z). The
time of flight gives a change of the phase of the rotation of the star. Combining
these two effects, and choosing Φ = 0 for radiation in the (x,z) plane from the
center of the star, ζ and Φ are given by cos ζ = u′z and Φ = −φu′ −~r · û′, where
φu′ is the azimuthal angle of û′ and ~r is the emitting location in units of RL.
In panel A of Fig. 1, the emission morphology in the (ζ , Φ) plane is shown.
For a given observer with a fixed viewing angle ζ , a double-pulsed structure is
observed because photons are clustered near two edges of the emission pattern
due to the relativistic effects (cf. panel B of Fig. 1).
In Fig. 1, we can see that this model can only produce radiation between two
peaks. However, the observed data of the Crab, Vela and Geminga indicate
that the leading wing 1 and the trailing wing 2 are quite strong, and even the
intensity in off-pulse cannot be ignored. Hirotani and his co-workers (Hirotani
& Shibata 2001; Hirotani, Harding & Shibata 2003) have pointed out that the
large current in the outer gap can change the boundary of the outer gap. They
solve the set of Maxwell and Boltzmann equations in pulsar magnetospheres
and demonstrate the existence of outer-gap accelerators, whose inner bound-
ary position depends the detail of the current flow and it is not necessarily
located at the null charge surface. For the gap current lower than 25% of the
Goldreich-Julian current, the inner boundary of the outer gap is very close to
the null surface (Hirotani 2005). On the other hand if the current is close to
the Goldreich-Julian current, the inner boundary can be as close as 10 stellar
radii. In Fig. 2, we show the light curve by assuming the inner boundary is
extended inward from the null charge surface to 10 stellar radii (cf. panel A
Fig. 1. Emission projection onto the (ζ,Φ) plane and pulse profile for the single pole
outer gap. The photons are emitted outwards from the outer gap. (a) The emission
projection (a1 = 0.9) and (b) the corresponding pulse profile (∆a1 = 0.03), for
Crab parameters α = 65◦ and ζ = 82◦.
of Fig. 2). In panel B of Fig. 2, the solid line represents emission trajectory of
outgoing radiation of one gap from the null surface to the light cylinder with
α = 50◦ and ζ = 75◦ and the dashed line represents the outgoing radiation
from another gap from the inner boundary to the null surface. In the presence
of the extended emission region from the near the stellar surface to the null
charge surface, leading wing 1, trailing wing 2 and the off-pulse components
can also be produced.
0 60 120 180 240 300 360
Phase
Fig. 2. Upper panel: the simulated pulse profile of the Crab pulsar; lower panel:
variation of radial distance with pulse phase for the Crab pulsar in units of RL,
where the bold line represents those in the outer magnetosphere, and the dashed
line represents those in the inner magnetosphere. The inclination angle is 50◦ and
the viewing angle is 75◦.
3 The phase-resolved spectra
3.1 radiation spectrum
The Crab pulsar has enough photons for its spectra to be analyzed, and the
phase-resolved spectra are useful for study of the local properties of the mag-
netosphere. Here, we summarize the calculation procedure of the radiation
spectrum given in CRZ, which is used to calculate the spectrum in different
phases.
The electric field of a thin outer gap (CHR) is given by E‖(r) =
ΩB(r)a2(r)
cs(r)
ΩB(r)f2(r)R2
cs(r)
, where a(r) is the thickness of the outer gap at position r, and
f(r) = a(r)/RL is the local fractional size of the outer gap. Assuming that
the magnetic flux subtended in the outer gap is constant in the steady state,
we get the local size factor f(r) ∼ f(RL)( rRL )
3/2, where f(RL) is estimated
by using the pair creation condition (cf. Zhang & Cheng 1997, CRZ). As the
equilibrium between the energy loss in radiation and gain from accelerating
electric field, the local Lorentz factor of the electrons/positrons in the outer
gap is γe(r) = (
eE‖(r)c)
For a volume element ∆V in the outer gap, the number of primary charged
particles can be roughly written as dN = nGJ∆A∆l, where nGJ =
is the
local Goldreich-Julian number density, B∆A is the magnetic flux through the
accelerator and ∆l is the path length along its magnetic field lines. (Here, We
would like to remark that this could overestimate the primary charge num-
ber density near the null surface, where the positronic charge density dom-
inates the Goldreich-Julian charge density. However, the observed radiation
comes from the wide range of magnetospheric region, an slight overestima-
tion of a small region should not cause a qualitative difference.) Thus, the
total number of the charged particles in the outer gap is N ∼ ΩΦ
RL, where
Φ ∼ f(RL)B(RL)R2L∆φ is the typical angular width of the magnetic flux
tube subtend in the outer gap. The primary e± pairs radiate curvature pho-
tons with a characteristic energy Ecur(r) =
~γ3e (r)
, and the power into
curvature radiation for dN e± pairs in a unit volume is dLcur
≈ lcurnGJ(r),
where lcur = eE‖c is the local power into the curvature radiation from a single
electron/positron. The spectrum of primary photons from a unit volume is
dV dEγ
≈ lcurnGJ
, Eγ ≤ Ecur. (3)
These primary curvature photons collide with the soft photons produced by
synchrotron radiation of the secondary e± pairs, and produce the secondary
e± pairs by photon-photon production. In a steady state, the distribution of
secondary electrons/positrons in a unit volume is
dV dEe
∫ d2Ṅ(E ′γ = 2E
dV dEγ
dE ′e ≈
lcurnGJ
), (4)
with Ėe the electron energy loss into synchrotron radiation, which is Ėe =
e4B2(r) sin2 β(r)
)2, where B(r) is the local magnetic field and β(r) the
local pitch angle, sin β(r) ∼ sin β(RL)( rRL )
1/2, sin β(RL) is the pitch angle at
the light cylinder. Therefore, the energy distribution of the secondary elec-
trons/positrons in volume ∆V (r) can be written as
dN(r)
dV dEe
∆V (r) ∼
lcurnGJ∆V (r)
). (5)
The corresponding photon spectrum of the synchrotron radiation is
Fsyn(Eγ , r) =
3e3B(r) sinβ
mec2h
dN(r)
F (x)dEe, (6)
where x = Eγ/Esyn, and Esyn(r) =
heB(r) sinβ(r)
is the typical photon
energy, and F (x) = x
x K5/3(y)dy, where K5/3(y) is the modified Bessel
function of order 5/3. Also, the spectrum of the inverse Compton scattered
photons in the volume ∆V (r) is
FICS(Eγ, r) =
dN(r)
d2NICS(r)
dEγdt
dEe, (7)
where
d2NICS(r)
dEγdt
nsyn(ǫ, r)F (%epsilon, Eγ , Ee)dǫ, and F (ǫ, Eγ , Ee) =
3σT c
4(Ee/mc2)2
[2q ln q+
(1 + 2q)(1− q) + (Γq)
2(1−q)
2(1+Γq)
], where Γ = 4ǫ(Ee/mec
2)/mec
2, q = E1/Γ(1− E1)
with E1 = Eγ/Ee and 1/4(Ee/mec
2) < q < 1. The number density of the
synchrotron photons with energy ǫ is nsyn(ǫ, r) =
Fsyn(ǫ)
cr2∆Ω
, where Fsyn is the
calculated synchrotron radiation flux, and ∆Ω is the usual beam solid angle.
Fig. 3 shows the observed data of the phase-resolved spectra from 100 eV
to 3 GeV of the Crab pulsar, and the theoretical fitting results calculated
by using the synchrotron self-Compton mechanism. The phase intervals are
defined by division given by Fierro (1998), and the amplitude of the spectrum
in each phase interval is proportional to the number of photons counted in it.
In this fitting, f(RL) = 0.21, and B = 3.0× 1012Gauss are used, which give a
consistent fitting of the phase-resolved spectra of the seven phase intervals. In
order to obtain a better fit, we treat the pitch angle (β) and the beam solid
angle (∆Ω) near the light cylinder as free parameters and vary from phase to
phase in the calculation. sin β(RL) = 0.06 and ∆Ω = 5.0 are chosen for trailing
wing 1, bridge and leading wing 2; sin β(RL) = 0.02,∆Ω = 1.0 for leading wing
1; sin β(RL) = 0.04,∆Ω = 3.5 for peak 1, sin β(RL) = 0.07,∆Ω = 3.0 for peak
2, and sin β(RL) = 0.03,∆Ω = 6.0 for trailing wing 2. Additionally, the phase-
averaged spectrum of the total pulse of the Crab pulsar is shown in Fig. 4,
where the parameters are chosen as sin β(RL) = 0.05 and ∆Ω = 5.0.
3.2 Analysis of the Phase-Resolved Spectra
The high energy spectra of the Crab pulsar is explained by using the syn-
chrotron self-Compton mechanism, which involves both the synchrotron radia-
tion and the Inverse Compton-Scattering (ICS) caused by the ultra-relativistic
electron/positron pairs created by the extremely high-energy curvature pho-
tons. The secondary e± pairs gyrate in the strong magnetic field and radiate
F(E) (MeV cm
Fig. 3. Phase resolved spectra of the Crab pulsar from 100 eV to 3 GeV in the 7
narrow pulse-phase intervals. Two spectra (for the TW1 and LW2) are displayed
twice. The curved line is calculated by the theoretical model, and the observed data
are taken from Kuiper et al. (2001).
synchrotron photons. While in the far regions of the magnetosphere where the
magnetic field decays rapidly, the relativistic pairs collide with the soft syn-
chrotron photons through the ICS process. Thus, the spectra of the radiation
contain two main components: one is the synchrotron radiation from the soft
X-ray to ∼10 MeV, and the other is the ICS component in the even higher
energy range. Usually, the synchrotron spectrum has stronger amplitude than
1x10
1x10

2
 F

 s
-
E (MeV)
Fig. 4. Phase-averaged spectrum of the Crab pulsar. The observed data are taken
from Kuiper et al. (2001).
that of ICS, and there is obvious turning frequency between these two com-
ponents, e.g. about 3MeV for peak 1. As we know, the power of synchrotron
radiation and ICS can be compared by the ratio of the local magnetic energy
density and the photon energy density, i.e.
∝ B(r)
ǫsyn(r)nsyn(ǫsyn, r)
, (8)
where ǫsyn(r) is the synchrotron photon energy in location r. In Fig. 3, the
spectra in trailing wing 1, bridge and leading wing 2 have broad synchrotron
spectra, which cover from 100 eV to ∼30 MeV. In Fig. 2, it is demonstrated
that the radiation of these three phase intervals are dominated by the photons
generated in the near surface region, where the magnetic field is so strong that
synchrotron radiation takes up the most emission. However, the radiation
of peak 1 and 2 are from the far regions near the light cylinder, where the
magnetic field decays rapidly (B ∝ r−3), thus, the ICS radiation becomes
more important above 3 MeV.
The peak of the synchrotron spectrum is determined by the characteristic
synchrotron photon energy. Since Esyn ∝ γ2eB sin β, where γe is the Lorentz
factor of the secondary pairs and β is the pitch angle of the electron/positron
to the magnetic field, the peak of the synchrotron spectrum can shift if the β
varies. Since the outward radiation direction covers a wider range than that of
the inward radiation, so the solid angle (∆Ω) is no longer the unity as assumed
in CRZ model. The solid angle can effect the amplitude of the ICS spectrum
because the number density of the synchrotron photons is proportional to 1
Therefore it is reasonable for us to choose β(RL) and ∆Ω as a set of parameters
in fitting the phase-resolved spectra of the Crab pulsar.
4 Conclusion and Discussion
We have tried to explain the high energy light curve and the phase-resolved
spectra in the energy range from 100 eV to 3 GeV of the Crab pulsar by mod-
ifying the three dimensional outer magnetosphere gap model. Compared to
the classical outer gap with the inner boundary at the null charge surface, the
modified model allows the outer gap to start at the region about several stel-
lar radii above the neutron star surface, and the ”inwardly-extended” part of
the outer gap contributes to the outer wings and off-pulse of the light curve.
Such modified outer gap geometry also plays a vital role in explaining the
optical polarization properties of the Crab pulsar (Takata et al. 2006). Two
adjustable parameters are used to simulate the light curve: one is the inclina-
tion angle of the magnetic axis to the rotational axis α, and the other is the
viewing angle also to the rotational axis ζ . As constrained by the phase sepa-
ration of the double peaks, we choose the values for these two parameters that
α = 50◦ and ζ = 75◦. So far, these two parameters have not been determined
from the observations. From radio observations, Rankin (1993) estimated that
α ≈ 84◦ and ζ is not known. Moffett and Hankins (1999) gave that α ≈ 56◦
and ζ = 117◦ by using the polarimetric observations at frequencies between
1.4 and 8.4 GHz. Of course, our values cannot be the true ones, and require
further observations to give strong restrictions of them.
In fitting the phase-resolved spectra of the Crab pulsar, our model performs
well from 100 eV to 1 GeV, but fails beyond 1 GeV. The inverse Compton
scattering spectrum of our results falls down quickly when the energy is over 1
GeV, but the observation data indicates that the spectrum still increases, es-
pecially in the first trailing wing, the bridge and the second leading wing phase
intervals. We have assumed that the curvature photons are all absorbed by
the magnetic field lines, however, some of these multi-GeV photons produced
near the light cylinder should be easily escaped from the photon-photon pair
creation process. In the spectrum fitting of peak 1, our result has a frequency
shift below 1 MeV, and we found that in order to well fit the spectrum we
should reduce the curvature photon energy by a quarter. The energy of the
curvature photon Ecur ∝ s−1(r), where s(r) is the local curvature radius. As
the high energy photons are produced in the far regions of the magnetosphere,
where s(r) maybe not follow the dipole form, we can change the photon energy
slightly.
Moreover, the stellar radius of a neutron star is usually treated as 106 cm when
calculating the strength of the surface magnetic field. However, the equation
of state inside the neutron star of the current theoretical models cannot give
a convincing value of the neutron star size. Thus, we can only determine the
magnetic moment, i.e. BpR
0, of the pulsar from the energy loss rate. Therefore,
we can rewrite the magnetic field of the Crab pulsar as B12R
6 = 3.8. In fitting
the phase-resolved spectra of the Crab pulsar, we choose Bp = 3×1012 G, not
the traditional value of 3.8× 1012 G, for it gives a better fitting on the lower
energy range below 10 keV.
Finally we want to emphasize that
This research is supported by a RGC grant of Hong Kong Government under
HKU 7015/05P.
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	Introduction
	A modified outer gap model
	The phase-resolved spectra
	radiation spectrum
	Analysis of the Phase-Resolved Spectra
	Conclusion and Discussion
