Theory and Simulations ofParticle Accelerationin Collisionless Non-relativistic Shocks
Takanobu Amano(University of Tokyo)
In collaboration with:
M. Hoshino, Y. Matsumoto, T. SaitoShocks in the Universe

TSURUTANI AND RODRIGUEZ: OVERVIEW UPSTREAM WAVES AND PARTICLES

Tsurutani&Rodriguez’81

4317

604

SN1006

Cygnus A

RYU ET AL.

Vol. 593

CME

Ryu+’03

Fig. 4.—Three-dimensional shock surfaces in a volume of ð25 h!1 Mpc)3 around the same complex as in Fig. 3. The color bar shows the values of MachParticle Acceleration

X-ray image by
Chandra [Bamba+2003]

In-situ obs. by Geotail[Shimada+1999]

“Standard Model”• Diffusive Shock Acceleration (DSA)– established in late 70’s [e.g., Bell ’78]
– predicts a universal power-law; N E-2– simple; comparison with observations isrelatively easy

M. Scholer

Questions
1. What fraction of the total energy
is converted into accelerated
particles ?
2. What is the maximum particle
energy achievable ?
Key Issues− Injection− Nonlinear Feedback− Particle Transport

Energetics
Energy conversion
efficiency ~ 10% ?
(needed for SNRs)
Not bad at CME-driven
shocks. But, themaximum energy ismuch smaller.
Mewaldt+(2006)

Positive Feedback
DSA is an intrinsically efficient process !
softerBerezhko & Ellision (1999)

Figure 3.1: A profile of the background plasma flow u(x) and the spatial distribution function of CRs
f (x, p) with the different momentum p (from Berezhko and Ellison [1999]). A thick solid line showsu(x) and three thin solid lines show f (x, p) at the low (p < mc), middle (p = mc) and high (p > mc)
energy.

harderAmato & Blasi (2005)

Recall that the standard DSA theory predicts the spectrum of the form:f (p) / p

q

q=

3r

where the background plasma flow is smoothly decelerated in the upstream precursor region:u=

8<u(x),:(x < 0)

r

1

(3.6)

(r is the shock compression ratio)Possible Negative Feedback ?
• Injection
If the injection occurs predominantly at thesubshock, the reduction of Mach number inthe precursor may lower the injection rate.

• Turbulent Heating
Turbulence driven by streaming CRs in the
precursor becomes so strong, so that one
expects turbulent dissipation may reduce theoverall efficiency.

Injection

The Injection Problem
The seed population must have sufficiently large
energy so that they• easily traverse the shock: v >> Vshock• scatter by waves for isotropization
cyclotron resonance condition!

kvk = ⌦/for! ⌧ ⌦/

krg ⇠ 1

relatively easy for ions, but
serious difficulty for electrons

Protons

Russell & Hoppe (1983)

Protons

Sugiyama (2011)

Electrons
Only relativistic electrons can satisfy the resonancecondition with low-frequency MHD waves!

kvk = ⌦/

Possible solutions to the electron injection problem:² Generation of high-frequency (whistler) wavesor² Pre-acceleration to > 100keV

Generation of whistlers
Q : How to generate whistlers ?
A : Consider mirrorly reflected energetic electrons.
Critical Mach number
resonance with whistlers

MHD regime(Alfven waves)

Amano & Hoshino (2010, PRL)In-situ Observations
Oka+2006 argued that the electron acceleration
efficiency at the bow shock is regulated by awhistler critical Mach number Mcritw
This by chance corresponds to the critical Mach
number of ours (within a numerical factor ~1)
Critical Mach number
used by Oka+2006

Our definition ofcritical Mach number

Oka+2006

Note: α ~ βe ~ 1

First Principles Approach
To understand possible pre-acceleration mechanisms:• Shock internal structure• Kinetic instabilities• Plasma waves
must be considered. This involves extremely complicatednonlinear physics. Fully kinetic Particle-In-Cell simulation is
the only option to investigate the mechanisms.
Caveat:Any (!) simulations employ artificial parameters such as• ion-to-electron mass ratio: mi/me• plasma-to-cyclotron frequency ratio: ωpe/Ωce vA
Unfortunately, plasma instabilities are sometimes sensitive
to these parameters.

Shock Surfing Acceleration
Plausible mechanism at very high Mach number shocks:• 1D: McClements+(2001), Hoshino&Shimada(2002), Amano&Hoshino(2007)• 2D : Amano&Hoshino(2009), Matsumoto+(2012)
Amano&Hoshino(2009)

Hoshino&Shimada (2002)

Spontaneous Reconnection

Upstream Flow Energy↓
Generation of B-field↓
Magnetic Reconnection↓
Particle Acceleration
Ma=45, mi/me=225
Matsutmoto, Amano, Kato, Hoshino (Science, 2015)

Magnetic Field Amplification

Extremely fast decay of X-ray hot spots
Uchiyama+(2007, Nature)

Magnetic field amplification by CR
streaming instability
Lucek&Bell(2000), Bell(2004)

Non-adiabatic Heating
Wave kinetic equation (for shear Alfven wave)

@
@t

✓

2

B
4⇡

◆

+

@
@x

✓

2

B
4⇡

◆✓

3
u
2

vA

◆

=u

@
@x

✓

2

B
8⇡

◆

+ vA

@
Pc
@x

L

is coupled with the CR diffusion-convection and hydrodynamic equations.

Work done by CR pressure gradient (wave generation)
Dissipation of wave energy leading to entropy production
⇢

1

@
u
1 @x

✓

Pg
⇢

◆

=L

@Pc
Assumption of a specific form of dissipation: L = ↵vA@x[e.g., McKenzie & Voelk, 1982]

0.00

0.02

0.04

0.06

ν
Figure 3.8: Precursor compression ratio R = u0 /u1 as a function of the normalized injection parametern = (4p /3)(mc2 /r0 u20 )p40 fsub (p0 ). The format is the same with Figure 3.7.

Semi-analytical Solution
1.0

100

Overall Efficiency

80

0.6

standard NLDSA

0.4

test particle

standard NLDSA

60
M1

pc,1/ρ0 u20

0.8

Subshock Mach No.

0.2
0.0
0.00

40
20

w/ dissipation
0.02
0.04
0.06
ν
injection parameterw/ dissipation
0.00
0.02
0.04
ν
injection parameter

0.06

Figure 3.7: Downstream CR pressure pc as a function of the normalized injection
parametern =Figure 3.9:
Mach numberof the subshock M1 as a function of the normalized injection parameter n =2 )p4 f (p ). The format is the same with Figure 3.7.
(4p /3)(mc2 /r0 u20 )p40 fsub (p0 ) with M0 = 100. The solid lines show the results(4ofva = 02 /andp /3)(mcr0 udotted
0 0 sub 0line shows test-particle limit. The blue diamonds show the results of va,0 /u0 = 1.1 ⇥ 10 2 (b0 = 1),
ah = 0 and x = 10. The red squares show the results of va,0 /u0 = 1.1 ⇥ 10 2 (b0 = 1), ah = 1.
75

Non-adiabatic heating substantially reduces the subshock
Mach number ! The acceleration efficiency is degradedfrom the standard NLDSA solution, but yet resides above
the test-particle limit.

T. Saito (Ph.D thesis)

Conclusions• Particle acceleration efficiency of 10-20% (interms of energy conversion rate) seems to bepossible.
• Conventional understanding is that nonlinearity
enhances the efficiency, in an essentiallyunlimited manner.
• There must be something that would suppress
otherwise the unlimited acceleration.
• The injection and turbulence have yet remained
the key issues in the shock acceleration theory.