Multi-wavelength Pulsed Emission
from Fermi Pulsars: Vela & Crab
——Annular Gap Model
Du Y. J., Qiao G. J., Han, J. L., Lee K. J. & Xu R. X. 2010, MNRAS, 406, 2671-2677
Du Y. J., Han, J. L., Qiao G. J. & Chou C. K. 2011, ApJ, 731, 2 (Vela)
Du Y. J. 2011, ApJ, to be submitted (Crab)
Reporter:
YuanJie Du (NAOC)
Supervisor：JinLin Han (NAOC)
GuoJun Qiao (PKU)
RenXin Xu (PKU)
Outline
Overview
Background
Vela Fermi observations
Crab multi-wavelength
observations
Other models
Annular Gap Model
Our work —— Vela （☻）
Our work —— Crab
Summary
Background — Pulsed Emission
 Before 2008, only 7 γ-ray
pulsars were discovered. 3
candidates
 After Fermi launching,
about 100 γ-ray pulsars
were discovered so far
 γ-ray selected pulsars,
 radio selected pulsars
 millisecond pulsars.
 Golden age for pulsar high
energy emission studies:
opportunity and challenge
Pulsar Emission
Background——Pulsar Emission Physics
Emission Physical Picture
Problems：Emission Region and Emission Mechanism
Yong Pulsar
MSP
J0437-4715
J0218+4232
Background— Fermi telescope
Fermi ： LAT and GBM
 LAT: a pair conversion telescope
 Large effective area: ~ 8000 cm2
 Large view of field: ~ 2.4 sr
 Wide energy band:
0.02GeV to 300 GeV
 High angular resolution:
0.6o for 1 GeV; 0.1o for 10 GeV
 GBM: for gamma-ray burst
(not discussed here)
LAT data reduction for pulse profile
Data collect (based on radio timing
solution)
Data selection (Diffuse, 0.1－300GeV, 2o,
zenith angle<105o)
Energy selection   max[1.6 - 3 log10(E GeV ), 1.3]
Timing each photon → phase information
(Tempo2 with Fermi plug-in)
Light curves plotting
P1
Du et al. 2011, ApJ, 731, 2
The Vela Pulsar
• Two sharp peaks, P1 is
narrower than P2
• P1/P2 varies with
energy
• A third peak (P3) in
the bridge, the location
and intensity shift with
energy
P3
P2
GeV phase-averaged spectrum for Vela
Hyper-exponential power law cut-off
Multi-wavelength observed light curves for Crab
Two peaks
Bridge
Phase-aligned
Observations of phase-averaged spectrum
and phase-resolved spectra for Crab
Kuiper et al. 2001, A&A, 378, 918
Fermi data
Observation hints
• Vela: Two sharp γ-ray peaks with a large
separation 0.42 —— high emission height !
• Vela: P1/P2 and P3 varying with energy ——
single pole or two pole?
• Vela & Crab: The “radio lag” problem needs to
be solved self-consistently —— Outer Gap
model?
• Modelings of multi-waveband light curves and
phase-resolved spectra are needed for the Crab
pulsar —— The Annular Gap model
Outline
Background
Other models
Annular Gap Model
Summary
Our work —— Vela （☻）
Our work —— Crab
High Energy Emission Models
 Outer Gap (Cheng)
 Polar Cap (Harding)
 Two-pole caustic (Dyks)
 Slot Gap (Harding)
 Annular Gap (Qiao)
 Two separatrix layer
Polar Cap Model for Vela
Daugherty & Harding 1996
 Emission height:
2-3 Rns
 Nearly aligned rotator:
αvery small
Outer Gap Model for Vela
Romani & Yadigaroglu 1995
• Both radio and γshown
• γemission is from single
pole, whereas radio comes
from the other polar cap.
Comments from Lommen et al. 2007
“Our results imply a connection
between the radio and X-ray
emission mechanisms for Vela
that is not consistent with outer
gap model … It is not clear how a
correlation could exist between
the radio and high energy regimes
in these models”.
Caustics in water
Two-pole caustic model for Vela
Dyks & Rudak 2003
Static dipole field
rmax< 0.95RLC
A REVISIT OF THE TWO-POLE CAUSTIC MODEL
Fang & Zhang 2010, ApJ
Retarded dipole field
Yu, Fang & Jiang 2009
（Wang, Takata, Cheng 2011, arXiv: 1102.4474）
Two-layer outer gap model
α=57◦，ζ=80◦
Annular Gap Model —— Our work
 Vela (Fermi Gamma-ray studies)
 Concepts & Methods
 Light curve
 Phase-averaged spectrum
 Phase-resolved spectra
 Crab (Multi-wavelength studies)
 Light curve
 Phase-averaged spectrum
 Phase-resolved spectra
Annular Gap + Core Gap
 The annular gap radius is
much larger for pulsars with
short spin periods, and can be
a excellent accelerator for
pulsar γ-ray emission.
Ruderman & Sutherland 1975
 The open field line region is
divided into core gap and
annular gap regions by the
critical field line.
Zhang, Qiao & Han 1997, 491, 891
Primary particles and pairs
 Primary particles are accelerated
to ultra-relativistic energy with γ~
107 by the induced acceleration
electric field.
 Three modes of pairs: CR, thermal
ICS and resonant ICS.
 Thermal ICS induced pairs usually
have larger Lorentz factors up to
γth~ 105 .
 Radio emssion comes from pairs.
High energy emission comes from
primary particles.
Electric field
Pair production
Light curve modeling thread
rN (0)
Re (α,ψ)
Central
Emission height
α,ψ
rN (ψ)
θnull (α,ψ)
rs(ψ)  λκ rN(ψ)  (1  λ)κ rN(0)
Gaussian distribution
I ( , ) 
Io
2
Aberration
retardation
View angle
I(φ,ζ)
256 bins of φandζ
(l  l 0) 2
exp[ 
]
2
2
Project onto
the sky
I(φ0,ζ0)
Light Curve Modeling Steps (Vela)
 Dividing polar cap
 Projected intensity
 Emissiom direction
 Emission phase
 Light curve
Step Ⅰ: Dividing polar cap
Dividing polar cap
Projected intensity
Emissiom direction
Emission phase






Magnetic inclination angle α: 70 deg
Viewing angle ζ: 64 deg
Critical field line θN (ψ)
Last open field line θP (ψ)
Footpoint in each open field line
40 rings for both core and annular gap
Light curve plotting
Torus
ζ
fitting
(Ng & Romani 2008)
PA fitting
α
β
(Johnston 2005)
Core Gap
Annular Gap
Step Ⅱ：Projected intensity
Dividing polar cap
Projected intensity
Emissiom direction
Emission phase
Light curve plotting
Two types of Gaaussian emission intensities
are assumed, i.e.,
• A Gaussian distribution on a field line
(parameter: κ,λ,σarc_AG,σarc_CG, ratio, I1, I2 , ICG)
• Another Gaussian distribution between field
lines with same magnetic azimuthal (ψ)
(parameter: σpeak_AG,σpeak_CG)
• Model parameters are different between core
and annular gap.
Step Ⅲ: Emission direction
Dividing polar cap
Emission direction of each emission spot nB in the magnetic frame
Projected intensity
Matrix Tα
Emissiom direction
Emission phase
Light curve plotting
nspin in the spin frame
Aberration
nobserver = {nx, ny, nz} in the observer frame
0  arctan(ny /n x )
 0  arccos(nz / n x 2  n y 2  n z 2 )
Step Ⅳ: Emission phase
Dividing polar cap
Projected intensity
Emissiom direction
Emission phase
Light curve plotting
• “Retardation effect” is needed for the final
photon emission phase φ.
• A phase shift Δφret retardation is because
of the photon flight time at a certain
emission height.
This leads to photon generated at higher
height comes to the Earth earlier.
• Finally, φ= φ0 -Δφret
ret  r cos(  ) / RLC
Wang et al. 2006
Du et al. 2011, ApJ, 731, 2
Step Ⅴ: Light curve plotting
Dividing polar cap
Projected intensity
Emissiom direction
Emission phase
Light curve plotting
• Observations (red solid
lines in 256 bins) versus
similations (thick black
solid lines in 128 bins)
in the framework of
Annular + Core gap
model.
• P1 and P2 originate
from the annular gap
region.
• P3 and bridge emission
come from the core gap
region
Du et al. 2011, ApJ, 731, 2
Radio lag
• A radio lag ~0.13 is
shown.
• Radio emission
originates from high
altitude and narrow
regions in the
annular gap.
• Single-pole annular
and core gap model
is favored for Vela.
Du et al. 2011, ApJ, 731, 2
Phase-averaged spectrum for Vela
 3 Components
（P1、P2 and P3）
 Emission position：
P1: 0.62RLC, ψ=-110°
P2: 0.75RLC, ψ=131°
P3: 0.28RLC, ψ=-104°
 3 free parameters：
γmin、 γmax 、Ω
 GeV emission is
generated from Synchrocurvature radiation from
primary particles and
synchrotron radiation
from secondaries (pairs)
Du et al. 2011, ApJ, 731, 2
Phase-resolved spectra
P1
P2
• P1 and P2：located
in AG region，
larger pitch angle,
synchrotron is
important for
< 1 GeV band
• P3：located in CG
region， pitch angle,
CR dominated
P3 low-energy
P3 high-energy
Du et al. 2011, ApJ, 731, 2
Dependencies of flux and emission height
Emission intensities are not uniform along an open field line. They
are likely to have a gaussian distribution near the peak position.
Annular Gap Model —— Our work
 Vela (Fermi Gamma-ray studies)
 Concepts & Methods
 Light curve
 Phase-averaged spectrum
 Phase-resolved spectra
 Crab (Multi-wavelength studies)
 Light curve
 Phase-averaged spectrum
 Phase-resolved spectra
AG multi-wavelength results for the
Crab pulsar
• Light curves
• Phase-averaged spectrum
• Phase-resolved spectra
Outer Gap for Crab
Tang et al. 2008, ApJ, 676, 562
Harding et al. 2008, ApJ, 680, 1378
Slot Gap for Crab
Li & Zhang 2010, ApJ, 725, 2225
Outer Gap for Crab
Phase-averaged spectrum
Photon sky-map
Du et al. 2010, MNRAS, 406, 2671
Brief Introduction to the AG work
• This is a fast work mainly focusing on the light
curve simulations for millisecond and young
pulsars.
• Concepts and methods are announced in details
for our annular gap model, although the results
are rough.
年轻脉冲星
Du et al. 2010, MNRAS, 406, 2671
毫秒脉冲星
Du et al. 2010, MNRAS, 406, 2671
Outline
Background
Annular Gap Model
Summary
Summary
• Under our self-consistent annular gap model, multiwavelength light curves and phase-averaged and phaseresolved spectra for the Vela pulsar and the Crab pulsar
are well reproduced with comparison of the
observations.
• The features (spectra and light curves) of P3 for Vela
are well described by our model.
• Our model explains the radio lag problem for Vela and
Crab, and they are different.
• Existence of both annular and core gaps could be
verified by the Vela pulsar and the Crab pulsar.
谢谢各位老师和同学！
Crab
Vela
补充材料
• MSP J0437-4715
• Crab
MSP J0437-4715
Both the simulated radio and
γ-ray
light
curves
for
millisecond pulsar J04374715. The radio lag problem
can be well solved by our
model.
One part of Poster for 38th
Cospar conference in Bremen
Crab脉冲轮廓
Fermi 观测到的毫秒脉冲星
E‖计算结果
• 粒子数守恒和磁
通量守恒
• 泊松方程：
环间隙加速电场(E‖)产生
• 产生的原因：在光速圆柱半径附近，磁流管中的带电
粒子由于不能随中子星共转而逃逸出磁层。为了保持
整个系统的电中性，中子星必须补偿所逃逸的带电粒
子，当这些粒子流动时，由于偏离了当地的GJ电荷密
度，故平行与磁力线方向的电场因此而产生。
• 电流回路闭合: 环区和核区输出电荷符号相反的带电
粒子。
E‖计算
微磁流管（beamlet）
逃逸高度：r1 = RLC
逃逸粒子的电荷密度:
b (r1 )  GJ ( RLC )
Ω
RLC
确定辐射高度
 P1：高度分布范
围小
 P3 + bridge：
高度分布范围广
 P2：高度分布范
围广，但由于光
性差效应和辐射
高度差效应被压
缩
引言——脉冲星
奇妙的脉冲星：
① 集四大力为一身的天然实验室
② 宇宙中的灯塔
两次获得诺贝尔物理学奖：
Faster Reaction!
Ⅰ 1974年，Hewish，第一颗脉冲星的发现
Ⅱ 1993年，Hulse & Talor, PSR J1913+16
引言——脉冲星
• 约2000颗脉冲星类
致密星
• 种类多：
正常脉冲星
毫秒脉冲星
AXP、SGR磁星
DTN、CCO、RRAT
• Fermi发现将近70
颗伽玛射线脉冲
星，其中包括29颗
毫秒脉冲星