Pulsar outer-gap model:
phase-resolved spectrum of
the Crab pulsar
HIROTANI, Kouichi
ASIAA/TIARA, Taiwan
CTA Workshop
ICRR
October 3, 2014
Crab nebula: Composite image of X-ray [blue] and optical [red]§1 g -ray Pulsar Observations
Large Area
Telescope

After 2008, LAT aboard Fermi has detected
more than 117 pulsars above 100 MeV.
Fermi/LAT point sources (>100 MeV)
Geminga

Fermi g -ray
space telescope
CrabVela
LAT radio-loud PSRs
LAT radio-quiet PSRs

2nd LAT catalog (Abdo+ 2013)

LAT MSPs

Pulsed broad-band spectra of young pulsars

Fermi/LAT(sensitive in 20 MeV – 300 GeV)nFnparticleaccelerator)

Crab
B1059-58

103
yrs

Vela
B1706-44

NS ageHigh-energy (~ GeV)
photons are emitted mainly
via curvature process by
ultra-relativistic, primary(created ine-’s/e+’s.

B1951+32
Geminga
B1055-52eV hn 100 MeV

105
yrs

Pulsed broad-band spectra of young pulsars
Crab103yrs

However, > 20 GeV, Inverse-Compton scatterings(ICS) by the cascaded e’s contribute.
(created outside the accelerator)

VERITASSensitive in 50 GeV – 50 TeV

Fig. TwoLorentz frames
when a photon
is up-scatteredby a relativisitice -.

NS ageHigh-energy (>100 MeV)
photons are emitted mainly
via curvature process byultra-relativistic e’s.

MAGIC
Sensitive in 25 GeV – 30 TeV
Geminga
B1055-52

§2 Pulsar Emission Models

Let consider how and where suchincoherent, high-energy photons
are emitted from pulsars.

§2 Pulsar Emission Models
If copious charges are (somehow) supplied, they realize aforce-free magnetosphere, E·B=0, and corotate with themagnetosphere under the corotational electric field,𝑬⊥ ≡ −𝑐 −1 (𝜴 × 𝒓) × 𝑩.

Charges corotateby 𝑬⊥ × 𝑩 drift,𝒗j ≡ 𝜴 × 𝒓.

§2 Pulsar Emission Models
If copious charges are (somehow) supplied, they realize aforce-free magnetosphere, E·B=0, and corotate with themagnetosphere under the corotational electric field,𝑬⊥ ≡ −𝑐 −1 (𝜴 × 𝒓) × 𝑩.

Decoupling E into E^ and Enon-corotate, we obtain from theMaxwell eq.
𝛻 · 𝑬⊥ + 𝑬non−corotate = 4𝜋𝜌,that is,𝛻 · 𝑬non−corotate = 4𝜋(𝜌 − 𝜌GJ),
where 𝜌GJ≡ 𝛻 · 𝑬⊥ ~ − 𝜴 · 𝑩.
If r deviates from rGJ in some region,
E|| =𝑬non−corotate · 𝑩/B arises around that region.

§2 Pulsar Emission Models
If copious charges are (somehow) supplied, they realize aforce-free magnetosphere, E·B=0, and corotate with themagnetosphere under the corotational electric field,𝑬⊥ ≡ −𝑐 −1 (𝜴 × 𝒓) × 𝑩.

Thus, the problem reduces to …

Decouple
E intoandE^ andeq. gives
non-corotate“WherehowEdoes
the. Maxwellcharge deficit
𝑬non−corotate = 4𝜋𝜌,(|𝜌| 𝛻< ·|𝜌𝑬appear?”
GJ⊥|)+
that is,This vacuum gap (E||≠0) should also account for𝛻 · 𝑬ofnon−corotate4𝜋(𝜌− 𝜌GJ),
the supplycharges =thatrealizes
themagnetosphereoutside of it.
where force-free𝜌GJ≡ 𝛻 · 𝑬⊥ ~ − 𝜴 · 𝑩.

If r deviates from rGJ in some region,
E|| =𝑬non−corotate · 𝑩/B arises around that region.

§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.

§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 (DW «1 ster) was too
small to reproduce the wide-separated double peaks.

Wide-separated
double peaks§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 (DW «1 ster) was too
small to reproduce the wide-separated double peaks.
In addition, localization of gap altitudes («r*) lead to too
small Lg («0.3Lspin), although Lradio~10-5Lspin is OK.
Moreover, the detection of VHE (> 20GeV) pulsed
emission from the Crab pulsar, which should avoid strong
magnetic absorption, clearly rules out PC emissions.
Thus, a high-altitude emission drew attention.

§2 Pulsar Emission Models
Higher-altitude emission models concentrate on …
 slot-gap (SG) model (Muslimov & Harding 2003, 2004)
 pair- starved polar-cap (PSPC) model (Venter + 2009) outer gap (OG) model (Cheng + 1986; Romani 1996)
 striped-wind synchrotron (SWS) model (Petri 2013)
 wind-inverse-Compton (WIC) model (Aharonian + 2012)
SG, PSPC models: e- are extracted as in PC model
OG model: e±’s created by g-g coll. and accelerated by E||SWS model: HE pulsed photons emitted from current sheet

WIC model: VHE pulsed photons emitted via ICS byultra-relativistic e±’s accelerated at r< 50 RLC

§2 Pulsar Emission Models
SG model, classic OG models:
have very thin meridional thickness (w ≪ 1),
reproduce only 10-1~10-3 Lg(KH 2008 ApJ 688, L25)

Therefore, the PSPC model (w≤1.0) was proposed.
However, the PSPC model contradicts with div(B)=4pr, inthe same way as the SG model.
(KH 2011, High Energy Emission fromPulsars and Their Systems, p. 117–37)

Thus, as long as the emissions inside LC are concerned, themodern OG model (w>0.1), survives as the only model thatquantitatively describes the pulsed HE/VHE emissions.

However, in all the models above, B configuration is not
solved consistently with the magnetospheric currents.

§2 Pulsar Emission Models
How about the emissions outside the light cylinder?
In SWS or WIC model, B configuration is consistently
solved with magnetospheric electric currents, whereas
particle creation & radiation are artificially set up.
In the SWS model, plasma collective effects (e.g., waveparticle interactions) are considered as a heating mechanismof plasmas in the current sheet.
(Chkheidze + 2013)
In the WIC model, the physical mechanism that converts thePoynting energy into the plasmas’ kinetic energy has notbeen solved by MHD or PIC (particle-in-cell) simulations.

§2 Pulsar Emission Models
The B structure can be solved e.g., by the PIC simulation.
This approach is valid for coherent pulsar radio emissions.
(1) A bunch of electrons move in phase in PC region.
The PIC code is most suited to solve plasma collective
effects from the first principles. (Timokhin & Arons 2013)(2) Spatial size < coherent scale (<50 cm at 600 MHz)The microscopic cell size in the PIC code favors
such localized phenomena (e.g., strong shocks)

However, such exact treatment are unnecessary to study
incoherent high-energy (> 0.001 eV) emissions, because(1) plasma collective effects are negligible as n >nplasma,(2) spatial size >1000 km for typical young pulsars.
The macroscopic PIC cell size disfavors such
non-localized phenomena.

§2 Pulsar Emission Models
It is, therefore, possible to investigate incoherent pulsar
HE/VHE emissions by solving the set of(1) e± Boltzmann equations,(2) radiative transfer equation, and(3) the Poisson equation for the electro-static potential(i.e., one of the Maxwell equations),
without taking account of plasma collective effects in theBoltzmann equations.

Instead of solving the B field configuration near the lightcylinder, we parameterize how the vacuum dipole B field is
deformed into monopole-like, and compare the prediction
with the g-ray observations.

§2 Pulsar Emission Models
As a model of high-altitude
emissions, we investigate theouter gap scenario.
Cheng, Ho, Ruderman(1986, ApJ 300, 500)

Emission altitude~ light cylinderhollow cone emission(DW > 1 ster)
OG model was further
developed by including
special relativistic effects.
Romani (1996, ApJ 470, 469)

§2 Pulsar Emission Models
As a model of high-altitude
emissions, we investigate theouter gap scenario.
Cheng, Ho, Ruderman(1986, ApJ 300, 500)

Emission altitude~ light cylinderhollow cone emission(DW > 1 ster)

one NS rotation

OGmodel wasfurther wideSuccessfully
explained
developedby including
separated doublepeaks.
special relativistic effects.
OGRomanimodel(1996,becameApJpromising.
470, 469)

§3 Modern Outer-gap Model: Formalism
I quantify the classic OG model by simultaneously
solving the pair-production cascade in a rotating NSmagnetosphere:e’s are accelerated by E||Relativistic e+/e- emit g-rays via
synchro-curvature, and IC processes
g-rays collide with soft photons/B tomaterialize as pairs in the accelerator

§3 Modern OG Model: Formalism
Poisson equation for electrostatic potential ψ :222
2     2  2  2  4p ( r  r GJ ) ,xyzwhereΩBE , r GJ  ,x2p cp10r (x)  e  d g  d   N  (x, g ,  )  N  ( x, g ,  ) +rion ( x),x  ( x, y , z ) .

N+/N-: distrib. func. of e+/eg : Lorentz factor of e+/e : pitch angle of e+/e-§3 Modern OG Model: Formalism
Assuming t+Wf =0 , we solve the e’s Boltzmann eqs.
 N Inv
N
 v N    eE   B   S IC  S SC   an dn  d w
tchn p

together with the radiative transfer equation,dIn an In  jndl
N: positronic/electronic spatial # density,
E||: mangnetic-field-aligned electric field,
SIC: ICS re-distribution function, dw: solid angle element,
In: specific intensity,l : path length along the ray
an: absorption coefficient, jn: emission coefficient§3 Application to the Crab pulsar
This numerical scheme can be applied to arbitrary
pulsars. Today, we apply it to the Crab pulsar.
Recent force-free, MHD, and PIC simulations suggestthat B field approaches monopole-like near andbeyond the light cylinder.

Thus, we consider
vacuum, rotating dipole B+ b * split-monopole B (Michael’74)b=0: pure dipoleb=1: Bdipole=Bmonopole @ LC§3 Application to the Crab pulsar
First, let us see why the double-peak light curve is formed.
Examine the vacuum (i.e., non-screened) solution of E||.
E|| for vacuum, rotating dipole B field (b=0 case)
Leading side

Null surface
Forms P1
Max(E||) is projectedon last-open B surface.

Forms P2
Trailing side§3 Application to the Crab pulsar
E|| is governed by the rGJ distribution, which issolely determined by B geometry.

j*=0o45o-45o 90o-90o

Puredipoleb=0.

KH 2014,MNRAS 442, L43

§3 Application to the Crab pulsar
E|| is governed by the rGJ distribution, which issolely determined by B geometry.

j*=0o45o-45o 90o-90oLarge gradient of rGJ
in the LS & TS.

Pure
dipoleb=0.

Large E|| toform P1, P2.
KH 2014,MNRAS 442, L43

§5 Application to the Crab pulsar
E|| is heavily screened by the produced pairs. Nevertheless,the essential features of P1/P2 formation is unchanged.
3-D gap solution (non-vacuum)Max(E||) areprojected on
the last-open Bsurface.

§4 Results: pulse profiles
The resultant g-ray light curves changes as a function of
the observer’s viewing angles:
b=0 (pure rotating vacuum dipole)

z=

z=

P2

B inclination:a=65o
0.1-30 GeV>30 GeV (x 10)

P1

z=z=z

One NS rotationObs.

§4 Results: pulse profiles
b=0 (pure rotating vacuum dipole)
0.1-30 GeV>30 GeV (x 10)

P1

b=0.5 (dipole + weak monopole)P2

a=65o
b=1.0 (+ moderate monopole)b=1.0 (+ strong monopole)§4 Results: pulse profiles
b=0 (pure rotating vacuum dipole)b=0.5 (dipole + weak monopole)

From P1/P2 behavior,a moderate superpositionof monopole is preferable.
That is, the true solution
will be found between theb=1.0 (+ moderatemonopole)pure dipole
and theforcefree solutions.

b=1.0 (+ strong monopole)

P1/P2 increases as B
approaches monopole.
P1/P2 decreases withincreasing photon energy.

§4 Results: pulse profiles
b=0 (pure rotating vacuum dipole)
0.1-30 GeV>30 GeV (x 10)

P1

b=0.5 (dipole + weak monopole)

P2

X-ray obs. of
PWN suggests

a=65o
b=1.0 (+ moderate monopole)b=1.0 (+ strong monopole)
0.1-30 GeV>30 GeV (x 10)

§4 Results: pulse profiles
b=0 (pure rotating vacuum dipole)
0.1-30 GeV>30 GeV (x 10)

P1

b=0.5 (dipole + weak monopole)P2

Cf. observed peak separation ~ 140o
120o

a=65o

With typical a~60o-65o, it is difficult toreproduce the observed peak separation.

b=1.0 (+ moderate monopole)b=1.0 (+ strong monopole)

Note: no observational constraint on a .

Greater a (e.g., 80o) is necessary.
(on-going work)§8 Results: Application to the Crab pulsar
Total and phase-resolved spectrum for b=0, a =60o, z =95o§8 Results: Application to the Crab pulsar
Total and phase-resolved spectrum for b=0, a =60o, z =100o§8 Results: Application to the Crab pulsar
Total and phase-resolved spectrum for b=0, a =60o, z =105oSummaryWe can predict the HE/VHE emissions from pulsar outermagnetospheres, by solving the set of Maxwell (divE=4pr)and e± Boltzmann eqs., radiative transfer eq., if we specify
P, dP/dt, aincl, kTNS.
The solution corresponds to a quantitative extension ofclassic outer gap model. We no longer have to assume thegap geometry, E||, e distribution functions.
Moderate B deformation (b~0.5) near LC is preferable toreproduce P1/P2 ratio and relatively large peak separation.
Bridge emission reduces due to strong screening.
Forz=120o (as inferred from X-ray torus obs.), Crabpulsar’s g-ray peak separation becomes < 120o for a <65o,whereas it should be ~140o. (a=70o-80o cases are on-going.)
If z=100o, a=60o with b=0 (dipole) gives an acceptable fit.