VHE Pulsed Emissions from
Rotation-Powered Pulsar
HIROTANI, Kouichi
ASIAA/TIARA-NTHU, Taiwan
CTA Workshop
Kashiwa, Japan
September 25, 2012
Crab nebula: Composite image of X-ray [blue] and optical [red]§1 g -ray Pulsar Observations
After 2008, LAT aboard Fermi has detected
more than 121 pulsars above 100 MeV.
Large Area
TelescopeFermi/LAT point sources(>100 MeV)

Fermi g -ray
space telescope

Vela

GemingaCrabr

1st LAT catalog (Abdo+ 2009)§1 g -ray Pulsar Observations
Recent IACTs found pulsed emission in 25-400 GeV fromthe Crab pulsar.
VERITAS ( > 120 GeV)
Aliu+ (2011, Science 334, 69)

MAGIC (25-416 GeV)
Aleksić+ (2011a,b)

Broad-band spectra(pulsed)CrabAbove 20 GeV, ICS bysecondary/tertiary pairs
contributes.

B1059-58
Vela
B1706-44

NS agenFn

High-energy (> 100MeV)
photons are emitted mainly
via curvature process by
ultra-relativistic (~10 TeV)
e’s accelerated in pulsar
magnetosphere.

103
yrs

B1951+32
Geminga
B1055-52eV hn 100 MeV

105
yrs

Broad-band spectra(pulsed)CrabAbove 20 GeV, ICS bysecondary/tertiary pairs
contributes.

B1059-58
Vela
B1706-44

NS agenFn

High-energy (> 100MeV)
photons are emitted mainly
via curvature process by
ultra-relativistic (~10 TeV)
e’s accelerated in pulsar
magnetosphere.

103
yrs

B1951+32
Geminga-e

B1055-52eV hn 100 MeV

105
yrs

High-energy (> 100MeV)
photons are emitted mainly
via curvature process by
ultra-relativistic (~10 TeV)
e’s accelerated in pulsar
magnetosphere.
Some of the primary
g-rays are absorbed in
the NS magnetosphereand reprocessed in lower
energies via
synchrotron process.

Crab
B1059-58

103
yrs

Vela
B1706-44

NS ageBroad-band spectra(pulsed)B1951+32
Geminga
B1055-52eV hn 100 MeV

105
yrs

§2 Rotating NS Magnetosphere
The observed high-energy emissions are
realized when the rotational energy of the NS
is electro-dynamically extracted and partly
dissipated in its magnetosphere.
(e.g., unipolar inductor)

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

§2 Rotating NS Magnetosphere
Observer anglezInclination angleaPossible sites of particle
acceleration


Ideal MHD condition
holds in most ofmagnetosphere,

E B  0.


In some limited regions,deficient charge supply
leads to E B  0.

In charge deficit region,E|| is solved from the
Poisson eq.,

 E||  4 (   GJ ).
Credit: A. Harding§2 Rotating NS Magnetosphere
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.

A great deal of effort has been made; however, one has toinvoke a very small inclination, a, and viewing angles, z,
to reproduce the widely separated pulse peaks.
In addition, localization of gap altitudes (« r*) prohibits
enough Lg (< 0.3Lspin) as observed. (Lradio~10-5Lspin is OK.)Thus, a high-altitude emission drew attention.

§3 Higher-altitude Pulsar Emission Models
Big breakthrough: Muslimov & Tsygan (1992, MN 255, 61)
found that a polar gap can extend into the higher altitudes
by virtue of the frame-dragging effect.

§3 Higher-altitude Pulsar Emission Models
Extending the original idea of Arons (1983, ApJ 266, 215),
Muslimov & Harding (2003, ApJ 588, 430) proposed thelower-altitude slot-gap (SG) model.
Slot gap = a pair-free space formed between the last-open
field lines and the pair-formation front (PFF).

§3 Higher-altitude Pulsar Emission Models
However, a lower-altitude SG is still limited within a few r*.
Thus, the same difficulty (DW «1 ster) still remains.

Energetics is solved!!!

§3 Higher-altitude Pulsar Emission Models
To contrive a higher-altitude emission model,
Muslimov & Harding (2004a, ApJ 606, 1143; 2004b, ApJ 617,
471) and Dyks, Harding & Rudak (2004, ApJ 606, 1125)
extended the lower-altitude slot-gap solution into higher
altitudes (in fact, by hand).
They explained, e.g., the widely separated double peaks.
widely separated double peaks§3 Higher-altitude Pulsar Emission Models
Assuming that the gap extends from the NS surface to thelight cylinder with constant emissivity, Dyks & Rudak(2003, ApJ 598, 1201) demonstrated the formation of double
peaks, which arise from the crossing of two caustics
associated with different poles.
caustics

§3 Higher-altitude Pulsar Emission Models
Assuming that the gap extends from the NS surface to thelight cylinder with constant emissivity, Dyks & Rudak(2003, ApJ 598, 1201) demonstrated the formation of double
peaks.
In this higher-altitude slot-gap model, most observers
catch emission from both (N/S) poles.
caustics

trailing-side emission
leading-side emission

main peaks
inter-peak & off-pulse§3 Higher-altitude Pulsar Emission Models
Dyks, Harding, Rudak (2004, ApJ 606, 1125) showed that
Crab pulsar’s optical polarization characteristics can bequalitatively reproduced by their SG model.

caustics

§3 Higher-altitude Pulsar Emission Models
Dyks, Harding, Rudak (2004, ApJ 606, 1125) showed that
Crab pulsar’s optical polarization characteristics can bequalitatively reproduced by their SG model.
Harding+ (2008, ApJ ApJ 680, 1378) examined Crab’s g-ray
emission and found that the pulse profiles and the phaseresolved spectrum are qualitatively reproduced.
caustics

§3 Higher-altitude Pulsar Emission Models
Dyks, Harding, Rudak (2004, ApJ 606, 1125) showed that
Crab pulsar’s optical polarization characteristics can bequalitatively reproduced by their SG model.
Harding+ (2008, ApJ ApJ 680, 1378) examined Crab’s g-ray
emission and found that the pulse profiles and the phaseresolved spectrum are qualitatively reproduced.
caustics

However, unfortunately, the higher-altitude SG model
contains two fatal electro-dynamical inconsistencies.

§4 Difficulties in SG Model: Insufficient Luminosity
KH (2008) ApJ 688, L25
Problem 1: insufficient luminosity
Adopting the same parameter as Harding+(2008), one
obtains too small g-ray flux from a higher-altitude SG.

Fig. Phaseaveraged SG
spectrum for
four discrete
viewing angles,
90o, 100o, 110o,and 120o.

90o, 100o,
110o, 120o§4 Difficulties in SG Model: Insufficient Luminosity
KH (2008) ApJ 688, L25
Problem 1: insufficient luminosity
Adopting the same parameter as Harding+(2008), one
obtains too small g-ray flux from a higher-altitude SG.

Analytically predicted g-ray flux of the Crab pulsar:(n Fn )peak  0.0450 f k
3

mW 1
2

4

c

3



d

2

,

k 1.

 E/ d 2 : spin-down flux
f : fractional gap width ( f « 1 denotes a thin gap)
The difference between OG and SG models appears
through f, k, and assumed m (magnetic moment).

§4 Difficulties in SG Model: Insufficient Luminosity
Problem 1: insufficient luminosity

KH (2008) ApJ 688, L25(n Fn )peak  0.0450 f 3k

m 2 W4 1
c3

d2

Apply this general result to the Crab pulsar (W=190 rad s-1).
(I) For OG model ( f ~0.14, k ~0.3, m =41030 G cm3),(nFn)peak ~ 410-4 MeV s-1 cm-2 ~ EGRET flux.
(II) For SG model ( f ~0.04, k ~0.2), even with a large m,(nFn)peak ~ 310-5 (m /8  1030)2 MeV s-1 cm-2
< 0.1 EGRET flux.

§4 Difficulties in SG Model: Insufficient Luminosity
Problem 1: insufficient luminosity
In short, both analytical and numerical results show thata SGthiscangeneralproducea negligible
g-ray
flux.
Apply
resultto the Crabpulsar
(W=190 rad s-1).
(I) For OG model ( f ~0.14, k ~0.3, m =41030 G cm3),(nFn)peak ~ 410-4 MeV s-1 cm-2 ~ EGRET flux.
(II) For SG model ( f ~0.04, k ~0.2), even with a large m,(nFn)peak ~ 310-5 (m /8  1030)2 MeV s-1 cm-2
< 0.1 EGRET flux.
OG model remains as the only possible g-ray pulsar model.
KH (2008) ApJ 688, L25

§5 Difficulties in SG Model: Unphysical GJ Charge
Problem 2: unphysical assumption of GJ charge
density (per B flux tube)
Without pair creation, electron density per B will be
constant along the field line. However, it results in areversal of E|| due to the sign change of  -GJ.
E||<0

E||>0

§5 Difficulties in SG Model: Unphysical GJ Charge
Problem 2: unphysical assumption of GJ /B
To prevent a sign reversal of E||, they assumed that GJ /B
tends to a constant in the higher altitudes

§5 Difficulties in SG Model: Unphysical GJ Charge
Problem 2: unphysical assumption of GJ /B
To prevent a sign reversal of E||, they assumed that GJ /B
tends to a constant in the higher altitudes.

However, Maxwell eq. uniquely gives  g mnc2
GJ =
 m  2 g g  W    Fn  .
4  g   wUnfortunately, thisassumption is unphysical.

§5 Difficulties in SG Model: Unphysical GJ Charge
Problem 2: unphysical assumption of GJ /B
Since the pair-starved PC (PSPC) model adopts the sameGJ distribution, this difficulty applies not only to thehigher-altitude SG model, but also to the PSPC model(Venter+ 2009, ApJ 707, 800).

Unfortunately, thisassumption is unphysical.

§5 Difficulties in SG Model: Unphysical GJ Charge
Problem 2: unphysical assumption of GJ /B
Since the pair-starved PC (PSPC) model adopts the sameGJ distribution, this difficulty applies not only to thehigher-altitude SG model, but also to the PSPC model(Venter+ 2009, ApJ 707, 800).
In short, higher-latitude SG model & PSPC model
contain serious electro-dynamical problem that
contradicts with Maxwell eq.
Unfortunately, thisassumption is unphysical.

§6 Classic Outer-gap (OG) Models
As an alternative
possibility of high-altitude
emission model, the outergap model was proposed.
Cheng, Ho, Ruderman(1986, ApJ 300, 500)
So far, there have been foundno serious electrodynamical
problems in the OG model(unlike SG or PSPC model).
Thus, let us concentrate on the
OG model in what follows.

§6 Classic OG Models
Mid 80’s, the outer-gap (OG) model was proposed.
(Cheng, Ho, Ruderman ApJ 300, 500, 1986)
Emission altitude > 100 rNS

hollow cone emission(DW > 1 ster)Mid 90s’, OG model was further developed by including
special relativistic effects.
(Romani ApJ 470, 469)
Explains wide-separated double peaks.
Outer-gap model
became promising.

one NS rotation§6 Classic OG 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, a, kT (+z ) (this talk)In this talk, I’ll present the most recent results obtainedin my 3-D version of self-consistent OG calculations.

§7 Modern Outer-gap Model
Self-sustained pair-production cascade in a rotating
NS magnetosphere: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

§7 Modern Outer-gap Model
The Poisson equation for
the electrostatic potential ψ
is given by    4 (    GJ ) ,
2

where

ΩB
E 
,  GJ  
,
x
2 c


  e  dp  N  (x, p)  N  (x, p) +ion .
30

N+/N-: distrib. func. of e+/ep : momentum of e+/e-§7 Modern Outer-gap Model
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 
t
c
hn

 p

together with the radiative transfer equation,dIn an In  jndlN: positronic/electronic spatial # density,
E||: mangnetic-field-aligned electric field,
SIC: ICS re-distribution function, d: solid angle element,
In: specific intensity,l : path length along the ray
an: absorption coefficient, jn: emission coefficient§7 Modern Outer-gap Model
Specify the three parameters: (period P is known)
 magnetic inclination (e.g., ainc=45o, 75o), magnetic dipole moment of NS (e.g., m =41030G cm3)
 neutron-star surface temperature (e.g., kTNS=50 eV)
Solve Poisson eq. + Boltzmann eqs + radiative transf. eq.
I first solved (in 6-D phase space) 3-D gap geometry, acceleration electric field distribution, E||, particle density and energy spectrum,
 photon specific intensity (→predicts g-ray properties),by specifying these three parameters, assuming B-fieldstructure by vacuum rotating dipole solution (Cheng + ’00).

§8 ICS spectrum of the Crab pulsar
Let us apply this numerical method to the Crab pulsar.

Maxwell &Boltzmann eqs.,

OG 3-D geometry,
E|| distribution,e+/e- distribution functions,photon specific intensity

Apply this method to the Crab pulsar, assuming
m =3.8×1030 G cm3, a =60o , kT= 100 eV.

§8 ICS spectrum of the Crab pulsar
3-D distribution of the particle accelerator (i.e., highenergy emission zone) solved from the Poisson eq.:
last-open
B field lines
NS

e-/e+
accelerator§8 HE/VHE Pulsation from the Crab Pulsar
3-D geometry: Trans-field gap thickness is self-regulated
by pair production.
Crab, a =60o
Fractional gap thickness projection on the last-open B line surfaceD┴ is reduced by createdpairs in lower altitudes.

null surface§8 HE/VHE Pulsation from the Crab Pulsar
E|| is also self-regulated by pair production.

(→ Curvature photon energy changes little for various pulsars.)

f*: azimuth around B axis [rad]Crab, a =60o
E|| is efficiently screened by createdpairs in lower altitude, leading side.

Copious pair
production
Large E||
remains in thehigher altitudes

s: distance along B field lines / light-cylinder radius§8 HE/VHE Pulsation from the Crab Pulsar
E|| is also self-regulated by pair production.

(→ Curvature photon energy changes little for various pulsars.)

f*: azimuth around B axis [rad]Crab, a =60o

P1

leading side
later
pulse
phase

bridgeP2

trailing side

s: distance along B field lines / light-cylinder radius§8 HE/VHE Pulsation from the Crab Pulsar
Crab 60o
Using Ell, computeemissivity at each position.

latitude

Intensity distribution shows
caustic pattern in the sky
map.

-azimuth + phase lag

inwardSky map:
north OG
only

outwardpulse phase [deg]one NS rotation

§8 HE/VHE Pulsation from the Crab Pulsar
Crab 60o

inwardUsing Ell, computeemissivity at each position.

latitude

Intensity distribution shows
caustic pattern in the skymap.

Sky map:
north OG
only

outwardnorth+south
pulse
phase [deg]one NS rotation-azimuth + phase lag
Consider photons emitted
from OGs connected toboth poles.

outward,north-pole OG

inward,south-pole OG§8 HE/VHE Pulsation from the Crab Pulsar
Crab 60osky map:north+south

Cut the sky map at
a viewing angle z
to obtain a pulse
profile.
With energydependent sky map,
we obtain pulse
profiles at different
energies.

> 90 MeV> 3 GeV> 9 GeVinwardemission§8 HE/VHE Pulsation from the Crab Pulsar

P1

P2

higher energies

If we look at thedetails, however,
the energydependent pulse
profile does not
reproduce theFermi and MAGIC
observations.

§8 HE/VHE Pulsation from the Crab Pulsar
Some details …

The peak widthappears to beroughly constantfrom 0.1 to 10 GeV,
while observed
peak width
sharpens in higher
energies.

Crab 60o> 90 MeV> 3 GeV> 9 GeV§8 HE/VHE Pulsation from the Crab Pulsar
Phase-averaged spectrum
Crab, a =60osolid: primary (bef. absorption)dashed: secondary (bef. absorption)
dotted: tertiary (bef. absorption)
green: all (after absorption)
1

Magnetospheric IR-X photons
are up-scattered by the gapaccelerated e+’s v > 0.7 v LC.

1

§8 HE/VHE Pulsation from the Crab Pulsar
Phase-averaged spectrum
Crab, a =60o12Primary ICS component
is totally absorbed by
gg→ee andreprocessed as
secondary component.

§8 HE/VHE Pulsation from the Crab Pulsar
Phase-averaged spectrum
Crab, a =60o

2

The secondary SSC
component is absorbed again
to be reprocessed as thetertiary synchrotron/SSC
components.
That is, the tertiary SSC
component explains the Crab’s
pulsation in 30-400 GeV.
(see also Lyutikov+ 2011)34

§8 HE/VHE Pulsation from the Crab Pulsar
Phase-resolved spectrum (in LAT-defined phase bins):
Crab, a =60o§8 HE/VHE Pulsation from the Crab Pulsar
Phase-resolved spectrum (in LAT-defined phase bins):
Crab, a =60o

Since LS is efficiently illuminated bymagnetospheric IR photons, the tertiarycomponent is substantially absorbed(although it is emitted near the LC).

§8 HE/VHE Pulsation from the Crab Pulsar
Viewing angle dependence of phase-averaged spectrum:
Crab, a =60o§8 HE/VHE Pulsation from the Crab Pulsar
Schematic picture of cascading pairs and their emissions:§8 HE/VHE Pulsation from the Crab Pulsar
Schematic picture of cascading pairs and their emissions:
Unfortunately, the IC flux is vulnerable to B geometry
near LC.

§8 HE/VHE Pulsation from the Crab Pulsar
Schematic picture of cascading pairs and their emissions:
Unfortunately, the IC flux is vulnerable to B geometry
near LC.
Incorporation of correct B geometry near LC is crucial.

§9 Theoretical derivation of Lg ~ Lspin0.5
Let us finally derive the observed relationship,
Lg  Lspin 0.5 , both analytically and numerically.

LAT 1st catalog(Abdo+, 2011)§9 Theoretical derivation of Lg ~ Lspin0.5
To specify the gg pair production rate, which governs
gap evolution, we adopt the minimum coolingscenario.
light element (H, He, C, O)
envelopheavy element(e.g., Fe)envelop
Minimal coolingscenario(Page + 2004)

§9 Theoretical derivation of Lg ~ Lspin0.5
Gap closure condition, (Ngt)in (Ngt)out= 1, gives
analytical predictions of Lg evolution.

§9 Theoretical derivation of Lg ~ Lspin0.5
Also derive Lg evolution numerically, consideringboth cooling NS emission & heated PC emission.

§9 Theoretical derivation of Lg ~ Lspin0.5
Numerical solution is consistent with analytical one.
Greater Lg thananalytical ones.
Becauseemission fromhigher altitudes(LS:P1 & TS:P2)
also contribute,in addition tolower altitudes(bridge).

§9 Theoretical derivation of Lg ~ Lspin0.5
Numerical solution is consistent with analytical one.
 declines at
Lspin<1035 erg/s↓
Lg declines
sharply.
For light element
envelop, Lg
peaks around
30000 yrs, aspredicted byanalytical
calculation.

§9 Theoretical derivation of Lg ~ Lspin0.5
Numerical solution is consistent with analytical one.
A realistic NS will
have envelop
composition
between the two
extreme cases,
light and heavy
element envelops.
Thus, the actual
Lg will distribute
between the redand blue curves.

SummaryWe can now solve pulsed high-energy emissions from theset of Maxwell (divE=4) and Boltzmann eqs., if we
specify P, dP/dt, aincl, kTNS. We no longer have to assume
the gap geometry, E||, e distribution functions.
By SSC of secondary/tertiary pairs, Crab pulsar’s total
and phase-resolved spectrum shows a power-law-like shape.
Observed relationship, Lg  Lspin 0.5 , can now be derived
both analytically and numerically under the OG model.