垂直衝撃波での加速時間とエネルギースペクトル上島 翔真 (青山学院大学)
大平 豊 (東京大学)
山崎 了 (青山学院大学)

Energy Fluxes of Cosmic Rays
E2dF/dE [eV/(m2 s sr)]

Cosmic Rays (CRs)ref : Ohira, Yamazaki and Terasawa (2012)

Galactic Origin(Galactic Comic Rays (GCRs))

Knee (~1015.5 eV)Standard Model for GCRs
Accelerator: Supernova Remnants
Mechanism: Diffusive Shock Acceleration

Energy of Cosmic Rays (eV)Diffusive Shock Acceleration (DSA)
UpDownu2

u1

CRs

Assumption
・existence of shock・existence of scatter
dN/0r+223∝ps==2r−1
dp(compression ratio r = 4)
It does not depends on kinds of particles.

Shockscatter(turbulence)@ shock rest frame

momentum gain (one cycle)
Δp 4(u' − u) )=p3v

Acceleration in Supernova Remnants
Supernova Remnants (type-Ia)
The size of
The coherent length
supernova remnants <of ISM(~ several pc)(< 100 pc)
Main acceleration region

Perpendicularshock

We focus on

Parallel shock
High injection rate?
Slow acceleration?

Magnetic
Parallel shock
field

Perpendicular shockvs.

Low injection rate?
Rapid acceleration!!!

SNRshockAcceleration Rate of a Perpendicular ShockAcceleration Rate [/parallel shock acc. rate]u The acceleration rate of a perp. shock is much larger than theacceleration rate of a parallel shock.
u Particles are efficiently accelerated in the case that themagnetic fluctuation is weak.

Assumption
The motion of particles is diffusive.
up shock downB

ref : Jokipii (1987)

θηB=100B

ηB: Bohm factor

ηB=10parallelB=)
short time to accelerate
η: ~δB ) ?@3×(1+ηB2)
90,θ perpendicular

B CRslong time to accelerate(shock rest frame)

Energy Spectrum of a Perp. Shock Acceleration
The energy spectrum of a perpendicular shock acceleration becomessorter than that of the standard DSA prediction in the case thatthe magnetic fluctuation is weak in downstream region.
ref: Takamoto & Kirk (2015)

Observations and simulations show thatthe magnetic field turbulence is amplified in the downstream region.
ref: observation : Bamba et al.(2003), Ohira and Yamazaki(2017)
simulation : Ohira (2016), Caprioli and Spitkovski(2013), Inoue et al.(2009)
Giacalone and Jokipii(2007)

In this study, we assume
the strong magnetic field amplification in the downstream region andthe random walk in the downstream region.

Motivation
Magnetic turbulence in this study
rapid acceleration à upstream : weak fluctuation (δB/B0<1)
observations and simulations in a downstream region
à downstream : strong fluctuation (δB/B0≥1)

u Acceleration time at a perpendicular shocku Energy spectrum of accelerated particles at a perp. shockClassical Model and Our Model
Magnetic turbulence in an upstream region
rapid acceleration à upstream : weak fluctuation (δB/B0<1)
Under the condition that the upstream magnetic fluctuation is weak,
The particle motion is gyration, not diffusion.

B

Classical Modelup
CRs down

Our Model
up
Bdown
CRs

perpendicular
diffusion
shock

Bohm
gyration diffusion
shock

Acceleration TimeClassical Model
up
down
CRs
B

Our Model
Bup

down
CRs

perpendicular
diffusionshock

the one cycle timeΔt =Acceleration timep
TEFF ==
Δp/Δt

Bohm
gyrationdiffusion
shock

4D:NBO,)u) c
4DQ,' 4DQ,)+u' cu) cπΩ2'K,' +Acceleration mechanism : DSA
Δp u3B=pc
Assumption: downstream
Bohm diffusion & B amplification

λmfp,2 ~ rg,2gyration + Bohm
perpendicular diffusion

gyrationu3B 2' 2' 16 B) 2' u3B 2) 2'π
ΩK,' +
ΩK,'+ Bohmc
3 B'c4DQ,' 16DQ,)+
perpendicular diffusion))
u3B
u3B

Dependences
Acceleration Time of our modelp
u3B 2' 2' 16 B) 2' u3B 2) 2'TEFF ==πΩK,' +
ΩK,'
Δp/Δtc
3 B'cu dependence of the magnetic field amplification in downstream region
u dependence of the shock velocity
1st term (residence time in an upstream region) is dominant.
(large B2/B1 or fast shock velocity)à Tacc ∝ ush-1
2nd term (residence time in a downstream region) is dominant.
(small B2/B1 or slow shock velocity)à Tacc ∝ ush-2
c.f. If the particle motion in the upstream region is also diffusion (classical model),à Tacc ∝ ush-2 for all cases

4DQ,' 16DQ,)TEFF = ) +u3Bu)3B

Setup : No Magnetic Fluctuation Case
forward shock velocity (constant)ush/c = 0.001, 0.00316, 0.01, 0.0316, 0.1, 0.316(up rest frame)
shockupAssumption
Bohm diffusion in a downstream region
B = B0 ush(isotropic scattering in downstream rest frame) z
B2/B1= 1, 10, 100, 300, 1000

down3ush/4

scattering time : tscat ∝ p
impulsive injection @t=0 γ0=15, isotropic
The magnetic field in the upstream region
consists of only uniform magnetic field.
à The weakest fluctuation limit

B= = 0, 0, B=B0 : const.

xsolve E.O.M.
dpv= q ×B=dtcMonte-Carlo

Spectrum of accelerated particles
104ush/c = 0.001
B2/B1=100

p2dN/dp

103

102101standard DSA dN:∝ p2)
prediction
dpThis is because the particle motionin the downstream region is isotropic scattering.

10010-1

100

101

102
p/p0103

Acceleration Time Tacc vs. B field Amplification B2/B1Tacc [rg,0/c]

107

u3B 2' 2' 16 B) 2' u3B 2) 2'TEFF = π
ΩK,' +ΩK,'c
3 B'cstrong B2 à short Tacc
(Because particles in downstream region
are scattered in a short time.)

106

105

104

ush=0.01cp/p0=100
This is the limit of the residence timein the upstream region (half gyro period)
100

101

102
B2/B1103

Acceleration Time Tacc vs. Shock Velocity ush/c
108u3B 2' 2' 16 B) 2' u3B 2) 2'TEFF = π
ΩK,' +
ΩK,'c
3 B'cTacc [rg,0/c]

107: B2/B1=10: B2/B1=100: B2/B1=1000
p/p0=100

106
105
104
103
102

10-310-2ush/c

10-1

100

Acceleration Time Tacc vs. Shock Velocity ush/c
108u3B 2' 2' 16 B) 2' u3B 2) 2'TEFF = π
ΩK,' +
ΩK,'c
3 B'cTacc [rg,0/c]

107: B2/B1=10: B2/B1=100: B2/B1=1000
p/p0=100∝ ush-2

106
105∝ ush-1

104
103
102

10-3

10-2
ush/c

10-1

100

Summary
・We investigated a particle acceleration by a perpendicular shock.
・Weak and strong magnetic field turbulences are assumed
in the upstream and downstream region, respectively.
Our Model
Gyration in the upstream region + Bohm diffusion in the downstream region
Energy spectrum of accelerated particles
・The energy spectrum at perp. shock become E-2 in the case thatthe magnetic fluctuation is sufficiently strong in downstream region.
Acceleration time・The shock velocity dependence of the acceleration time changes.
When the upstream residence time is dominant, Tacc ∝ ush-1
When the downstream residence time is dominant, Tacc ∝ ush-2