Supercomputing for exploring
electron accelerationsat astrophysical shock waves
Yosuke Matsumoto
Division of Plasma Astrophysics
ICEHAP

Collaborators:
T. Amano & M. Hoshino (Univ. of Tokyo)
T. N. Kato (NAOJ)
ICEHAP seminar

Outline
Introduction
Collision-less shockFermi acceleration (Why e- acceleration?)
Magnetic reconnection

Methods
Basic eqs.
Ab initio particle-in-cell simulation
Shock experiments on supercomputer systems

Results
e- shock surfing acceleration (YM+ '13 PRL)
Stochastic e- acceleration (YM+ '15 Science)

SummaryJi & Zweibel '15 ScienceIntroductionShock, Shock, Shock! (radio, X, g)
SNR shocks

radio(Reynolds+)

X-ray
Chandra

g-ray
HESSn
IceCube?

Shock, Shock, Shock! (in-situ)
Planetary shocks

Bow shock @ Saturn

Interplanetary shock @ 1AU

Russel & Greenstadt '79

30 years later...

Masters+ '11,'13

High-energy electrons at strong shock waves
upstream

downstream

hard-X
SN1006soft-X
Bamba+ '03

Saturn's
bow shock

Masters+ '13

Why electron? (99% of CR are nuclei/protons... )Observationally evident, theoretically puzzling.
(contrary to protons, viz. leptonic vs. hadronic for TeV g)

HESS collaboration '10

Fermi acceleration (Fermi, '49)
Lab. frame

co-moving framep'out, E'outMv (G)qp'in, E'inq'mLorentz transformationpin
Ein
Eout = G(E'out-Vp'incos(q'))=G2Ein(1+V/c2-2vVcos(q))

p'out cos(q') = -p'in cos(q')
E'in = G(Ein-Vpincos(q))
E'out = E'in

1st & 2nd order Fermi accels. (v~c, G~1)
2
E out −E i n Δ EVV==−2 cos θ+Ei nEcccos(q)q = 0 → over taking : energy loss
q = p → head-on : energy gain
2nd ordercos(q) ~ -1 ~ +1(V/c)2 term remains
1st order if the head-on collisions
dominate<cos(q)> ~ -1
the fist term (V/c) is the leading order()Diffusive shock acceleration (1st order Fermi)

M. Scholer

V1-V2 > 0 (div(V)<0) → head-on
1st order Fermi accelerationg = r+2/r-1, where r is the compression
ratio. g=2 for strong shocks (r=4)

Theoretical issues
Injection
Shock scale L~ali >> rge
Thermal electrons arestrongly magnetizedge > ~ 10 can be injected
Pre-accelerations forelectrons are necessary
Bell, 1974

Scattering bodies
Alfven waves – cannot scatter thermal
electrons
magnetic clouds – found by YM+ '15 Sciencemagnetic field amplification (x ~100)

Magnetic reconnection
Solar flare

AuroraTopology change of anti-parallel
magnetic field lines
Conversion of magnetic energy toplasma kinetic energy
Fundamental process in planetary
and astrophysical phenomena

Methods

Vlasov equation∂fsqsv+ v⋅∇ f s +E+ ×B ⋅∇ u f s =0
∂tmsc()

configuration (3D) + velocity space (3D) = 6Ddistribution function
in velocity space

uyux

uy

particle-in-cellsimulationux

Maxwell equation∂B=−c ∇×E∂t∂E=c ∇ ×B−4 π J∂twith∇⋅B=0∇⋅E=4 π ρeImplicit FDTD method()

4π( I−( θ c Δ t ) ∇ ) δ B=θ(c Δ) ∇ B + c ∇ ×J t +Δ t /2 −c Δ t ∇×Et
q : implicitness factor
solved within ~10 interations by the conjugate gradient method2222t

particle-in-cell (PIC) simulation
Particle pushd xp up=γpdtd up qup=E+×Bdt
mc γpupJ=∑ p q p γp()
E, B, J

Maxwell equation∂B=−c ∇ ×E∂t
∂E=c ∇×B−4 π J∂tCharacteristic scales in PIC simulations
Dh ~ Debyle length lD:

12(1λ D [ m]=7.4 T [eV ]n[ cm−3 ]12)

Dt ~ electron plasma frequency wpe-1:
11
1−32
ω [ sec]=10−39 n [cm ]−1pe()

Proton-to-Electron mass ratio M/m:

M/m ~ O(10) (↔ 1836)
parsec and 103-6 yrs in astrophysics!

Characteristic scales of SNR shocks
Shock speed
Vsh = 1000 – 10000 km/s
non-relativistic shocks
Magnetic field (upstream)a few mG : Alfven speed VA ~ 10 km/s (n~0.1 /cc)
(Alfven) Mach number M > 100 !
Dynamic ranges
shock scale : MHD (L >> rgi >> rge)
Ion to Electron mass ratio M/m=1836
relativistic electrons : v~c (>>Vsh > VA)

VA

Vshc

Shock creation - Injection methodyshock front propagates toward right →boundaryB
downstream
upstreamV

Moving injector(Riquelme & Spitkovsky '11)

2D PIC simulations
Perpendicular shocks (B ⊥ n)
Non-relativistic shock speeds (Vsh/c ~ 0.3)
High Mach numbers (Vsh/Cs and Vsh/VA) ~ 40
M/m: O(100)xShock experiments on supercomputer systems

Optimized on K computer at AICS, RIKENMPI+OpenMP hybrid parallelization
Scalable up to 105 cores

Collision-less shock simulations

Super-critical (MA> ~3) shock structures
key: Reflected ions

Leroy '82; Wu+ '84

Physics in high MA shocks
k ⊥ B0elecron pre-acceleration

k // B0
magnetic field turbulenceElectron shock surfing acceleration (eSSA)
1D PIC simulation

Hoshino & Shimada '02

Linear unstable conditionTrapping condition
Shimada & Hoshino '00
McClements+ '01

Matsumoto+ '12

eSSA in multi dimensions ?
M/m=25, MA~15

multi-dimensinonal effects
Amano & Hoshino '09

2Dw/oB

1D

mass ratio dependence(M/m=25, 100, 400)

2DwB
Ohira & Takahara '07*periodic system modelingfoot region

Riquelme & Spitkovsky '11

2D PIC simulations of perpendicular shocks

M A ∝(M / m)

12

Trapping condition

Linear unstable condition

eSSA

Matsukiyo & YM

Matsumoto+ '12

e- acceleration at M/m=225, MA~45 shock

downstream|Ees|/B0:accelerated:thermalupstream

Matsumoto+ '13 PRL

Electron downstream energy spectra
young SNR shocks

Matsumoto+ '12, '13

Physics in high MA shocks
k ⊥ B0elecron pre-acceleration

k // B0
magnetic field turbulenceIn-plane B field caseMA~5

Ion-scale ripples along the shock
surface
Ion cycrotron-instability
Ion-beam Weibel instability
Origin of ion-scale magnetic field
turbulence
Burgess, '06

MA~100

Kato & Takabe, '11

M/m=225, MA~42 shock (in-plane B case)downstream

log10(Ne/N0)

upstreamreconnection

magnetic cloudYM+ '15 ScienceCurrent sheet formation via ion Weibel instability
cf. Kato & Takabe '08, '10

shock front

upstream ion

Stochastic e- acceleration

blue/red: Bz, gray: in-plane B field lines, circle: electron orbit

Stochastic e- acceleration (contd.)
Vx

NE>E0/NBz

Time evolution of energy distribution

Condition for turbulent reconnection
ion Weibel Biw >> upstream B01-2％ of beam energy (0.5r0V02) can be converted to themagnetic energy Biw/8p

EB/EKEx10-2Biw ∼0.1 √ 4 π ρ 0 V ≫B 020M A ≫10

Kato & Takabe '08

Electron accelerations in perp. shocks
Trapping condition

Linear unstable condition

M A ∝(M / m)

12young SNR shocks

eSSAMA >> 10Whistler wave

Summary
High-energy electrons are observationallyevident, but puzzle in theory
High performance computing helps toreveal generation of cosmic rays at
astrophysical shock waves
On-going 3D PIC simulations arepromising!
Hopefully, hadron/lepton accelerations tocollaborate with IceCube