Towards a self-consistent theoryof pulsar magnetospheric
HE/VHE emissions
HIROTANI, Kouichi
ASIAA/TIARA, Taiwan
Nagoya University
July 26, 2013
Crab nebula: Composite image of X-ray [blue] and optical [red]§1 γ -ray Pulsar Observations
After 2008, LAT aboard Fermi has detected
more than 117 pulsars above 100 MeV.

Large Area
TelescopeFermi/LAT point sources (>100 MeV)
Geminga

Fermi γ -ray
space telescope

Crab
Vela

2nd LAT catalog (Abdo+ 2013)§1 γ -ray Pulsar Observations
Ground-based, Imaging Air Cherenkov Telescopes (IACTs)
found pulsed emission above 25 GeV from the Crab pulsar.
MAGIC (25-416 GeV)

VERITAS ( > 120 GeV)

Aleksić+ (2011a,b)

Aliu+ (2011, Science 334, 69)
OFF

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Pulsed broad-band spectraνFν

the particle
accelerator)(Thompson, EGRET spectra)

Crab
B1059-58

103
yrs

Vela
B1706-44

NS ageHigh-energy (~ GeV)
photons are emitted mainly
via curvature process by
ultra-relativistic, primarye-’s/e+’s.
(created in

B1951+32
Geminga
B1055-52eV hν 100 MeV

105
yrs

Pulsed broad-band spectra
CrabHigh-energy (> 100MeV)
photons are emitted mainly
via curvature process by
ultra-relativistic e±’s.

B1059-58

What is the curvature process?

B1706-44

103
yrs

NS ageVela

νFνConsider relativistic charges moving along curved B. They emit
(Thompson,
EGRET
spectra)curvatureradiation,
provided
P||» P┴B1951+32. A charge e with Lorentz factor
γ emits the following synchrotron radiation,
cf. synchrotron case
105
3 3 c
3Geminga
3 cyrsωc = γcharacteristic energy: ω = γ
curv

2
rg
2
Rc
B1055-52222
23e 4  c 
3e 2 4  c 2 
Psynch = 3 γ  radiation power: Pcurv = 3 γ  r hν 100 MeV
c
2
2c
R
eV
 c
 g

Pulsed broad-band spectra
CrabHigh-energy (> 100MeV)
photons are emitted mainly
via curvature process by
ultra-relativistic, primary e±’s.

B1059-58

However, > 20 GeV,ICS by secondary &tertiary e±’s contributes.

For pulsar VHE emissions,
Klein-Nishina effect
B1706-44
becomes important,because ε*i » mec2.

NS ageVela

B1951+32

Fig. Two
Lorentz frames
when a photon
is up-scatteredby a relativisitic
e -.

103
yrs

Geminga
B1055-52

§2 Pulsar Emission Models

Let us begin with considering how andwhere such incoherent, high-energy
photons are emitted from pulsars.

§2 Pulsar Emission Models
High-energy emissions are realized when therotational energy of the NS is electrodynamically extracted and partly dissipated in
the magnetosphere.
(e.g., unipolar inductor)

Magnetic and rotation
axes are generally
misaligned.
Pulsars:
rapidly rotating, highly
magnetized NS

§2 Pulsar Emission Models
Pulsar emission takes
place at …
 Polar gap(r <30 km), near NS surface

 Outer gap, or slot gap(r ~103 km), near the lightcylinder(outside the null surface)Crab Wind region
We neglect the emission
from the wind region,
because they are not pulsed.

ChandraHST§2 Pulsar Emission Models
Pulsar emission takes
place at …
 Polar gap(r <30 km), near NS surface

 Outer gap, or slot gap(r ~103 km), near the lightcylinder(outside the null surface)

E|| arises in a limited volume near
the PC surface due to heavy
screening by pair discharge.
E|| arises in a greater volume in
the higher altitudes due to lessefficient pair production.

§2 Pulsar Emission Models
Early 80’s, the polar-cap (PC) model was proposed.
(Daugherty & Harding ApJ 252, 337, 1982)
A single PC beam can produce a variety of pulse profiles.
Rotationaxis
Openmagnetic
field lines

Radio/
Polar cap
beam

Slot
gap
Gamma
rays
Outer
gap

§2 Pulsar Emission Models
Early 80’s, the polar-cap (PC) model was proposed.
(Daugherty & Harding ApJ 252, 337, 1982)
A single PC beam can produce a variety of pulse profiles.
Rotation
However, the emission solid
angle (∆Ω «1 ster) was too
axis
small to reproduce
the wide-separated double peaks.
Openmagnetic
field lines

Radio/
Polar cap
beam

Slot
gap

Wide-separated
double peaks
Gamma
rays

Outer
gap

§2 Pulsar Emission Models
Early 80’s, the polar-cap (PC) model was proposed.
(Daugherty & Harding ApJ 252, 337, 1982)
A single PC beam can produce a variety of pulse profiles.
However, the emission solid angle (∆Ω «1 ster) was too
small to reproduce the wide-separated double peaks.
Thus, a high-altitude emission drew attention.

§2 Pulsar Emission Models
To contrive a higher-altitude emission model, the polar
gap was extended into higher altitudes (in fact, by hand).
Muslimov & Harding (2004a, ApJ 606, 1143;
2004b, ApJ 617, 471)
Dyks, Harding & Rudak (2004, ApJ 606, 1125)
Harding+ (2008, ApJ ApJ 680, 1378)

They explained, e.g., the widely separated double peaks.
widely separated double peaks§2 Pulsar Emission Models
To contrive a higher-altitude emission model, the polar
gap was extended into higher altitudes (in fact, by hand).
Muslimov & Harding (2004a, ApJ 606, 1143;
2004b, ApJ 617, 471)
Dyks, Harding & Rudak (2004, ApJ 606, 1125)
Harding+ (2008, ApJ ApJ 680, 1378)

They explained, e.g., the widely separated double peaks.
However, unfortunately, the higher-altitude SG model
contains two fatal electro-dynamical inconsistencies.

§2 Pulsar Emission Models
Problem 1: insufficient luminosity
This numerical conclusion is confirmed also analytically(KH’08), showing that a SG can produce only anegligible γ-ray flux.

For details, please also
refer to my talk last year
at CTA-Japan meeting.

KH (2008) ApJ 688, L25

§2 Pulsar Emission Models
Problem #2: unphysical assumption of ρGJ /B
In the SG model, they assume ρGJ /B distribution that
contradicts with the Maxwell equation.

For details, pleasealso refer to mytalk last year at
CTA-Japan
meeting.

Unfortunately, thisassumption is unphysical.

§2 Pulsar Emission Models
Problem #2: unphysical assumption of ρGJ /B
In fact, to solve the insufficient flux problem (prob. #1),
a geometrically thick version of the higher-altitude SG
model, the pair-starved PC (PSPC) model, was proposed(Venter+ 2009, ApJ 707, 800). However, the PSPC model
adopts the same ρGJ/B distribution, which means that thesame difficulty applies.
Higher-latitude SG model & PSPC model contradict
with Maxwell eq. That is, higher-altitude extensionof the polar-cap model failed.

§2 Pulsar Emission Models
As an alternative possibilityof high-altitude emission
model, the outer gap model
was proposed.
Cheng, Ho, Ruderman(1986, ApJ 300, 500)
So far, there have been found noserious electro-dynamical
problems in the OG model (unlike
SG or PSPC model).
Thus, let us concentrate on the
OG model in what follows.

§2 Pulsar Emission Models
Indeed, the sub-TeV components from the Crab pulsar
shows that pulsed γ-rays are emitted from the outermagnetosphere (γB→ee).
Crab (P1+P2)(a) Crab Pulsar, P1+P2



We thus consider theouter-gap model

E2 dN/dE [ TeV cm 2 s 1 ]10-10(Cheng+ 86, ApJ 300,500)

in this talk.

Aleksic + (2012)
A&A 540, A69

10-11
10-12
10-13

1010

(P1+P2)MAGIC Stereo, this work
thisMwork(P1+P2)MAGIC Stereo, this work
thisE work
(P1+P2)E MAGIC Mono (Aleksić et al. 2011)
MAGIC-mono(P1+P2)
M Fermi-LAT (Abdo et al. 2010)
Fermi-LAT(P1+P2)E Fermi-LAT (Aleksić et al. 2011)
Fermi-LAT(P1+P2)
V VERITAS (Aliu et al. 2011)
VERITAS
Nebula, MAGIC Stereo, this work
Nebula, MAGIC Stereo, ICRC 2011
Nebula, Fermi-LAT (Abdo et al. 2010)
tot. pulsed, OG+pairs (this work)

syst.

1
2
1010
10
100
Energy [ GeV ]§2 Pulsar Emission Models
Various attempts have been made on recent OG model:
3-D geometrical model→ phase-resolved spectra (Cheng + ’00; Tang + ’08)
→ atlas of light curves for PC, OG, SG models(Watters + ’08)

2-D self-consistent solution(Takata + ’06; KH ’06)

3-D self-consistent solution→ phase-resolved spectra, absolute luminosity
if we give only P, dP/dt, α, kT (+ζ ) (this talk)In this talk, I’ll present the most recent results obtainedin my 3-D version of self-consistent OG calculations.

§2 Pulsar Emission Models
3-D self-consistent OG model
Death line of normal and millisecond PSRs on (P,P) plane(Wang & KH 2011, ApJ 736, 127)

Spectral hardening of trailing light-curve peak
(KH 2011, ApJ 733, L49)

Evolution of γ-ray luminosity of rotation-powered PSRs
(KH 2013, ApJ 766, 98)

Crab pulsars HE-VHE pulsed emission

Today’s talk I.
Today’s talk II.

(Alkesic + 2011, ApJ 742, 43; Alkesic + 2012, AA 540, A69)

Comment on Lorentz invariance violation tests.

§3 Modern Outer-gap Model: Formalism
Self-sustained pair-production cascade in a rotating
NS magnetosphere:e±’s are accelerated by E||
Relativistic e+/e- emit γ-rays via
synchro-curvature, and IC processes
γ-rays collide with soft photons/B tomaterialize as pairs in the accelerator

§3 Modern OG Model: Formalism
Poisson equation for electrostatic potential ψ :
2
2
2
∂
ψ
∂
ψ
∂
ψ
2
− ∇ ψ = − 2 − 2 − 2 = 4π ( ρ − ρGJ ) ,
∂x
∂y
∂z
where

∂ψ
ΩiB
E ≡ −, ρ GJ ≡ −
,
∂x
2π c
∞

π

1

0

ρ (x) ≡ e ∫ d γ ∫ d χ [ N + (x, γ , χ ) − N − (x, γ , χ )]+ρion (x),x = ( x, y , z ) .

N+/N-: distrib. func. of e+/eγ : Lorentz factor of e+/eχ : pitch angle of e+/e-§3 Modern OG Model: Formalism
Assuming ∂t+Ω∂φ =0 , we solve the e±’s Boltzmann eqs.
  v   ∂N ±Iν∂N± 
+ v ⋅∇N ± +  eE  + × B  ⋅  = S IC + S SC + ∫ αν dν ∫ dω
∂t
c
hν

 ∂p
together with the radiative transfer equation,dIν= −αν Iν + jνdl
N±: positronic/electronic spatial # density,
E||: mangnetic-field-aligned electric field,
SIC: ICS re-distribution function, dω: solid angle element,Iν: specific intensity,l : path length along the rayαν: absorption coefficient, jν: emission coefficient§3 Modern OG Model: Formalism
Boundary Conditions
To solve the elliptic-type differential eq. (Poisson eq.),
we impose

Ψ=0

at inner, lower, upper BDs∂Ψ= 0 at outer BD∂x§3 Modern OG Model: Formalism
At the inner BD
To solve thehyperbolic-type PDE(e± Boltzmann eq.),
we imposeinN + ( x , z, γ ) = 0
To solve the ODE(radiative transfer eq.),
we imposeinIν ( x , z ,θγ ) = 0, where 0<θ γ < π / 2 (outgoing)
That is, no e±/γ-ray injection across the BD.

§3 Modern OG Model: Formalism

At the outer BD

N − ( x out , z , Γ) = 0
Iν ( x out , z , θγ ) = 0, where π / 2<θ γ < π (in-going)
That is, no e±/γ-ray injection across the BD.

§4 Gamma-ray vs. Spin-down Luminosities
0.5
L
∝
L
First, we demonstrate the observed γ.
spin

Lγ ∝ Lspin 0.5

2nd LAT catalog(Abdo + 2013)§4 Gamma-ray vs. Spin-down Luminosities
To begin with, let us analytically examine the condition
for an OG to be self-sustained. An OG emits the energy
flux (KH 2008, ApJ 688, L25)
2 4

(ν Fν ) peak ≈ 0.0450hm

3

µ Ω 1
c

3

d

2

,

at distance d by curvature process, where hm denotes
dimensionless OG trans-B thickness, µ the dipole moment.
OG luminosity can be, therefore, evaluated as22Lγ ≈ 2.36(ν Fν ) peak × 4π d f Ω ≈ 1.23 f Ω hm
Thus, hm controls the luminosity evolution.

3

µ Ω
c

3
i

∝E

4

.

§4 Gamma-ray vs. Spin-down Luminosities
To examine hm, consider the condition of self-sustained OG.
An inward e- emits Nγin~104 synchro-curvature photons,
Nγinτ in ~10 of which materialize as pairs.
Each returned, outward e+ emits Nγout~105 curvature photons,
Nγoutτ out ~0.1 of which materialize as pairs.

§4 Gamma-ray vs. Spin-down Luminosities
To examine hm, consider the condition of self-sustained OG.
An inward e- emits Nγin~104 synchro-curvature photons,
Nγinτ in ~10 of which materialize as pairs.
Each returned, outward e+ emits Nγout~105 curvature photons,
Nγoutτ out ~0.1 of which materialize as pairs.
That is, gap trans-B-field thickness hm is automaticallyregulated so that Nγinτ in Nγoutτ out =1 is satisfied.

§4 Gamma-ray vs. Spin-down Luminosities
inin and N
outconditionout are expressed
To examine,considertheof self-sustained
Step
1: BothhNττin terms OG.
ofmγγinτ in N outτ out =1 gives
P,µ,α
,T,
and
h
.
Thus,
N
γ
γ
An inward e- emits Nγinm~104 synchro-curvature
photons,
hm = hm (P,µ,α,T).
in
inNγ τ ~10 of which materialize as pairs.

Each returned, outward e+ emits Nγout~105 curvature photons,
Nγoutτ out ~0.1 of which materialize as pairs.
That is, gap trans-B-field thickness hm is automaticallyregulated so that Nγinτ in Nγoutτ out =1 is satisfied.

§4 Gamma-ray vs. Spin-down Luminosities
Step 1: express Nγinτ in and Nγoutτ out with P,µ,α,T, hm.
OG model predicts

µ
2
E ≈
h
.
3 m
2ϖ LC
Particles (e±’s) saturate at Lorentz factor,
1/4

 3ρ

E  ,
γ =
 2e

2
c

emitting curvature photons with characteristic energy,
33 γhν c = c .
2 ρc

§4 Gamma-ray vs. Spin-down Luminosities
Step 1: express Nγinτ in and Nγoutτ out with P,µ,α,T, hm.
An inward e- or an outward e+ emitsin( Nγ ) = eEµ l2 /2hν c , ( Nγ )out= eEl1 / hν c

E ≈
h .
3 m
photons while
the distance l2 or l1.
2ϖrunning
LC
1/4

 3ρ

E  ,
γ =
 2e

2
c

3

3 γ
hν c = c .
2 ρc

§4 Gamma-ray vs. Spin-down Luminosities
Step 1: express Nγinτ in and Nγoutτ out with P,µ,α,T, hm.
An inward e- or an outward e+ emitsin( Nγ ) = eEµ l2 /2hν c , ( N γ )out= eEl1 / hν c

E ≈
h .
3 m
photons while
the distance l2 or l1.
2ϖrunning
LC

Such photons materialize as pairs with probability
1/4
2
in 3ρ
outτ=l= l1 F1σ 1 / c
c2 F2σ 2 / c, τE  ,γ =
2e
where F1, F2 denotesthe X-ray flux and σ1, σ2 the pairproduction cross section.
Quantities l1, l2, F1, F2, σ1, σ2 can be expressed by
P,µ,α,T, and hm, if we specify the B field configuration.

§4 Gamma-ray vs. Spin-down Luminosities
Step 1: Both Nγinτ in and Nγoutτ out are expressed in terms of
P,µ,α,T, and hm. Thus, Nγinτ in Nγoutτ out =1 gives
hm = hm (P,µ,α,T).
Step 2: Specifying the spin-down law, P=P(t,α), and thecooling curve, T=T(t), we can solve hm = hm (t, α).
iiStep 3: Independently, P=P(t,α) gives E = E (t , α ).
Step 4: Therefore, we can relate
3

and

i

Lγ = Lγ (t , α ) ∝ hm Eii

E = E (t , α )with intermediate parameter, pulsar age, t.

§4 Gamma-ray vs. Spin-down Luminosities
Step 2: Give spin-down law and NS cooling curve.
Assume dipole-radiation formula,2 µ 2Ω4−I Ω Ω =33 ci→ P = P(t , α )Adopt the minimum cooling scenario (i.e., without any
direct-Urca, rapid cooling processes).

→ T = T (t )§4 Gamma-ray vs. Spin-down Luminosities
Step 2: Cooling curves in the minimum cooling scenario:(contaminated by H, He, C, O)

Page +, ApJS
155, 623 (2004)

§4 Gamma-ray vs. Spin-down Luminosities
Step 2: Now we can solve hm = hm (t).

§4 Gamma-ray vs. Spin-down Luminosities
Step 1: Both Nγinτ in and Nγoutτ out are expressed in terms of
P,µ,α,T, and hm. Thus, Nγinτ in Nγoutτ out =1 gives
hm = hm (P,µ,α,T).
Step 2: Specifying the spin-down law, P=P(t,α), and thecooling curve, T=T(t), we can solve hm = hm (t, α).
iiStep 3: On the other hand, P=P(t,α) gives E = E (t , α ) .
Step 4: Therefore, we can relate
3

and

i

Lγ = Lγ (t , α ) ∝ hm Eii

E = E (t , α )with intermediate parameter, pulsar age, t.

§4 Gamma-ray vs. Spin-down Luminosities
ii

Step 3: We can immediately solve E = E (t , α ) by the
spin-down law.
2•i

E = − I Ω Ω = C (α )

µ Ω
c

4

3

Ω = Ω(t , α )ii

E = E (t , α )2 2
C = sin α for magnetic-dipole spin-down
32≈ 1+sin α for force-free spin-down§4 Gamma-ray vs. Spin-down Luminosities
Step 1: Both Nγinτ in and Nγoutτ out are expressed in terms of
P,µ,α,T, and hm. Thus, Nγinτ in Nγoutτ out =1 gives
hm = hm (P,µ,α,T).
Step 2: Specifying the spin-down law, P=P(t,α), and thecooling curve, T=T(t), we can solve hm = hm (t, α).
iiStep 3: On the other hand, P=P(t,α) gives E = E (t , α ) .
Step 4: Therefore, we can relate
3

and

i

Lγ = Lγ (t , α ) ∝ hm Eii

E = E (t , α )with intermediate parameter, pulsar age, t.

§4 Gamma-ray vs. Spin-down Luminosities
3

i

i

i

Step 4: Use hm=hm (t) to relate Lγ ∝ hm E with E = E (t ).

Heavy element envelope

Light element envelope§4 Gamma-ray vs. Spin-down Luminosities
Numerical solution is consistent with the analytical one.
Lγ ~Lspin 0-0.4
if Lspin>1036
erg/s
However,Lγ declines
rapidly
if Lspin < 1035.5
erg/s

KH (2013) ApJ 766, 98

§4 Gamma-ray vs. Spin-down Luminosities
To convert the observed flux Fγ into luminosity,
Lγ= 4π fΩ Fγ d 2, it is (conventionally) assumed fΩ=1.
However, we underestimate Lγ if fΩ>1.

View from equator

View from rotation axis

§4 Gamma-ray vs. Spin-down Luminosities
To convert the observed flux Fγ into luminosity,
Lγ= 4π fΩ Fγ d 2, it is (conventionally) assumed fΩ=1.
For example, we obtain fΩ>3 with probability ~50%.

~50%

View from equator

View from rotation axis

§4 Gamma-ray vs. Spin-down LuminositiesfΩ » 1

KH (2013) ApJ 766, 98

§5 Application to the Crab pulsar
We can apply the same numerical scheme to the Crabpulsar.
Today, we assume■ magnetic inclination angle α = 60o,
■ cooling NS surface temperature kT=100eV,(consistent with the cooling curve of aheavy-element envelope)§5 Application to the Crab pulsar
3-D OG distribution: trans-B thickness, D┴.,
projected on the last-open B field line surface.

§5 Application to the Crab pulsar
Distribution of acceleration E field, E||.

Max(E||) areprojected on
the last-open B
surface.

§5 Application to the Crab pulsar
Sky map of OG emission
Secondary/tertiarysynchrotron emission
Primary
curvature &Max(E||) aresecondary/tertiary
projected
onthelast-open
B
SSC
emissionsurface.
Secondary/tertiary
SSC emission

§5 Application to the Crab pulsar
If we cut the sky map at a specific viewing angle,
we obtain the pulse profile.
One NS rotation> 51 GeV> 51 GeV

Max(E||) areprojected on
the last-open B
surface.
> 51 GeV> 51 GeV§5 Application to the Crab pulsar
From X-ray observations (of the Crab nebula),
ζ~120o is suggested.

Max(E||) areprojected on
the last-open B
surface.

Introduce artificial meridional straightening andtoroidal bending of B field (due to current):

Bθ = (1 − c1ϖ / ϖ LC ) Bθ ,vacBφ = (1 + c2ϖ / ϖ LC ) Bφ ,vac§5 Application to the Crab pulsar
Peak separation increases if B field is toroidally
bent and meridionally straightened moderately.

Max(E||) areprojected on
the last-open B
surface.

§5 Application to the Crab pulsar
Peak separation increases if B field is toroidally
bent and meridionally straightened moderately.
We can in principle discriminate B field structurenear the LC, which has been highly unknown.
Max(E||) areprojected on
the last-open B
surface.

§5 Application to the Crab pulsar
Schematic picture of cascading pairs and their emissions:§6 Lorentz invariance violation tests
Finally, let us consider the Lorentz invariance violation
tests using pulsars.
Quantum gravity can be tested e.g., by measuring anenergy-dependent dispersion relation of mass-less particles.
Ex.) Photons would propagate at the speed E v( E )= 1± 
c
 EQG n

For n=1, for instance, two photons with different energies
E1 and E2 will arrive with time difference,

L E2 − E1
∆t =
c EQG

§6 Lorentz invariance violation tests

L E2 − E1
∆t =
c EQG

or

L E2 − E1
EQG =
c ∆t

If two photons are emitted at the same place (i.e., same L),
we can derive EQG (or set a lower bound of EQG) from ∆t.
Shot-time events (small ∆t) with large photon-energy
separation (greater E2-E1) are ideal.

§6 Lorentz invariance violation tests
They applied this method torotation-powered pulsars andexamined ∆t at different
Pair production
location
photon energies, assuming that
GeV and TeV photons are
emitted from the same location Max(E||) arein the magnetosphere.
projected on
See e.g.,

Pair production
the last-openBlocationsurface.

Zitzer +, ICRC2013, Rio de Janeiro
Otte, ICRC, Beijing
McCann +, 4th Fermi sympo, Monteley
Pair production
location§6 Lorentz invariance violation tests
They applied this method torotation-powered pulsars andexamined ∆t at different
Pair production
location
photon energies, assuming that
GeV and TeV photons are
emitted from the same location Max(E||) arein the magnetosphere.
projected on
However, higher energy
photons are emitted fromhigher altitudes by different
emission mechanisms.

Pair production
the last-openBlocationsurface.

Pair production
location§6 Lorentz invariance violation tests
Indeed, curvature
photons and SSC ones
appear close in phase.
However, if they
appear in different
phases, it may merelymean that they are
emitted from different
locations in themagnetosphere.

Max(E||) areprojected on
the last-open B
surface.

§6 Lorentz invariance violation tests
For example, is B field
is moderately bent
toroidally in the
counter-rotation
direction, HE and VHE
pulses will arrive at
different phase.
This has nothing to dowith the LIV.

Bθ = (1 − c1ϖ / ϖ LC ) Bθ ,vacBφ = (1 + c2ϖ / ϖ LC ) Bφ ,vac

Max(E||) areprojected on
the last-open B
surface.

Summary
High-energy pulsar observations have made rapid
progress in recent five years by the advent of Fermi/LAT.
Development of pulsar emission theory is highly required.
Now we can predict the HE emissions from pulsar outermagnetospheres, by solving the set of Maxwell (divE=4πρ)and Boltzmann eqs., if we specify P, dP/dt, αincl, kTNS.
The solution coincidentally corresponds to a quantitativeextension of classical outer gap model. However, we nolonger have to assume the gap geometry, E||, e± distributionfunctions.
i 0 − 0.4
i
36.5
-1
L
∝
E
γ-ray luminosity evolves as γ
when E > 10 erg s ,which is consistent with Fermi/LAT observations.
 Crab pulsar’s phase-resolved spectrum can be explainedby the current outer-magnetospheric accelerator theory.

Thank you.