Stochastic Particle Acceleration via
Turbulence in Various High-Energy
Astrophysical Phenomena
Katsuaki Asano（ICRR）

Non-thermal Emission
Acero+ 2010

Supernova Remnant SN1006

Electron energy distribution:
Single power-law with the index of 2.1,
cut off at 10 TeV and B=30μG.
20𝜂𝜂𝜂𝜂
6𝜋𝜋𝑚𝑚𝑒𝑒2 𝑐𝑐 3𝑡𝑡acc =& 𝑡𝑡cool =2𝜎𝜎𝑇𝑇 𝐵𝐵2 𝐸𝐸
3𝑒𝑒𝑒𝑒𝑒𝑒𝛽𝛽SNR

Bohm Limit (𝜂𝜂 = 1) & 𝑣𝑣SNR =1000km→ 𝐸𝐸max = 8TeV
sConsistent with the diffusive shock acceleration (Fermi I)+Bohm Limit!
All non-thermal phenomena are due to shocks?

Troubles in Blazar Emission
1. Index harder than 2 for the electron
injection spectrum.

2. Lower maximum energy
Mrk 421：
B=38mG
If 𝜂𝜂 = 1, even for 𝛽𝛽sh = 0.1,𝐸𝐸max = 7 TeV.
But actual Max. Energy 50 GeV

The Bohm factor should be ∼ 104

Inoue & Takahara 2002

Kino, Takahara, Kusunose 2002

3. Too sharp break in the electronspectrum.
Compared to the cooling break,the difference in the indicesseems large.

Electron Index in Fermi BL Lacs
Yan+

Fermi Bubbles
Emission from thin shell

Surface brightness

Limb Brightening

Uniform Surface brightness

Uniform Emissivity

Center Brightening
indicates thick shell >> Electron cooling timescale× shock speed (90pc for TeV e)

Hadronic?
Shock velocity (Crocker+ 2015)

M82 starburst galaxy (𝐿𝐿 ∼ 3 × 1042 erg/s)

Mach number a few
1040 erg s −1 → 3 × 1055 erg@108 yr

Hadronic model requires>1057 erg.
（CRs in the Galactic Disk: ～6×1054 erg）

If the past GC activity depositedan energy>1057 erg as the hadronicmodel implies, the structure may
be more prominent as M82 shows.
In a sense, the Fermi bubble is shabby
or faint.

Weak shock + Thick Shell + Leptonic → continuous electron acceleration via turbulenceAlternative ModelContinuous Acceleration by Scattering withTurbulence. (2nd Order Fermi Acceleration)
See e.g. Stawarz & Petrosian 2008
Energy gain per scattering∆E4≡ξ ≅ β2
E3
Alfvenic Wave (transverse/incompressible)
𝑘𝑘 −1

𝛿𝛿𝛿𝛿

𝐵𝐵𝐵𝐵 2
pitch angle diffusion → mean free path 𝑙𝑙 ∼ 𝑘𝑘𝑘𝑘𝐵𝐵2 𝑘𝑘 𝑟𝑟L , 𝑘𝑘 ∼ 1/𝑟𝑟L
resonance condition
22
̅< Δ𝐸𝐸 > 𝜉𝜉𝐸𝐸
∝ 𝐸𝐸 𝑞𝑞𝛿𝛿𝐵𝐵2 𝑘𝑘 ∝ 𝑘𝑘 −𝑞𝑞 → 𝐷𝐷𝐸𝐸𝐸𝐸 =∼Δ𝑡𝑡𝑙𝑙/𝑐𝑐

Compressible wave
Acoustic Wave (longitudinal/compressible)
Mirror ForceΔ𝑝𝑝 𝑝𝑝⊥ 𝑣𝑣⊥
𝑝𝑝⊥ 𝑣𝑣⊥
𝑘𝑘𝑘𝑘𝑘𝑘(𝑘𝑘)∼𝛻𝛻𝐵𝐵 ∼2𝐵𝐵2𝐵𝐵Δ𝑡𝑡

𝑐𝑐 2 < Δ𝑝𝑝2 > 𝐸𝐸 2 𝑐𝑐 213 𝑘𝑘𝑘𝑘 2 𝛿𝛿𝐵𝐵 2 (𝑘𝑘)𝐷𝐷𝐸𝐸𝐸𝐸 ∼∼∫𝑑𝑑∥
8𝐵𝐵2Δ𝑡𝑡𝑘𝑘∥ 𝑣𝑣ph

𝐷𝐷𝐸𝐸𝐸𝐸 ∼ 𝐸𝐸

For fast wave with a typical eddy size 𝐿𝐿22𝑣𝑣
𝛿𝛿𝐵𝐵
ph𝐹𝐹
2𝑐𝑐𝑐𝑐 𝐵𝐵2𝑘𝑘max

�𝑘𝑘min𝑑𝑑 𝐿𝐿𝐿𝐿

𝑣𝑣ph 2 𝑐𝑐∝ 𝐸𝐸 2𝑐𝑐𝐿𝐿𝑡𝑡acc ∝ 𝐸𝐸 0𝐵𝐵2 ∼ 𝛿𝛿𝐵𝐵𝐹𝐹2(Hard sphere)

𝐿𝐿𝐿𝐿 1−𝑞𝑞 ∼ 𝐸𝐸 2Ptuskin 1988; Cho & Lazarian 2006;
Yan & Lazarian 2008; Lynn et al. 2014

Focker-Planck Equation
𝐷𝐷𝐸𝐸𝐸𝐸 = 𝐾𝐾𝐸𝐸 𝑞𝑞

Kolmogorov+Alfvenic q=5/3, Compressible q=2 (hard sphere)𝜕𝜕𝑁𝑁e (𝜀𝜀, 𝑡𝑡)
𝜕𝜕𝜕𝜕𝑁𝑁e 𝐸𝐸, 𝑡𝑡=𝐷𝐷𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝐸𝐸Diffusion
𝑁𝑁 𝜀𝜀𝜕𝜕 2𝐷𝐷𝐸𝐸𝐸𝐸− 𝐸𝐸̇ cool 𝑁𝑁e 𝐸𝐸, 𝑡𝑡
𝐸𝐸𝜕𝜕𝜕𝜕
Acceleration Cooling−2−1 or − or 23∝ 𝜀𝜀Nocooling, continuous injection, time evolution
2εe Ne(εe)εε0

t=t0-1Injection

harder than the shock case 𝑁𝑁 𝜀𝜀 ∝ 𝜀𝜀 −2No injection, balance with cooling, steady𝜀𝜀 2 𝑁𝑁(𝜀𝜀)13t0+ 𝑁𝑁̇ e,inj (𝐸𝐸, 𝑡𝑡)

Hard Sphere
Kolmogorov
Bohm

t=t0
2t0

0.01

10-1t0

q=2

t=t0

ε

10-1/3t0-2/30.0001

10-2/3t0q=5/3

𝜀𝜀 210-2t0

1e-06

q=1

107

108

109

1010

1011

1012
εe [eV]

1e-08
0.01

0.1

110

𝜀𝜀

Fermi Bubble
Sasaki, Asano & Terasawa 2015
𝑞𝑞 = 2 with escape effect from the disturbed region
Evolution of Electron spectrum

Elapsed time since the shock passed

Decaying turbulence
Escaped & Cooled Component

Spatial diffusion coefficient𝐸𝐸tot= 2.6 × 1053 erg

Growing escape efficiency

L: Region size～180pc

Slow process so the temporal evolution is essential.

Fermi bubble
Gamma-ray spectrum

Surface brightness

The spectrum and surface brightness are reproduced by the phenomenologicalstochastic acceleration model.

Turbulence in AGN jets
Spine-Sheath structure
Hardee+ 2007 MHD simulation

Kelvin Helmholtz Instability
Mizuno+ 2007

Double layer in M87 jetExtreme Hard Blazar １ES 1101-232
Lγ = 2.6 ×10 43 erg s -1

Asano+ 2014
Electron spectrum
32-2 ε' n'(ε') [erg/cm ]1ES 1101-232

10εf(ε) [erg/cm /s]
210-10

Photon spectrum
1ES 1101-232

2.0-4

3.0

10

1.5

5.0

10-11

10.0
15.0

10-6

20.0

10-12

10-8
10-10

R/R0=1.01

1078101091010101112

1010-13 -3
101310ε' [eV]

100103106

109

Kolmogorov value q=5/3

Expanding jet → No steady state → Temporal evolution is essential

1012 ε [eV]

Mrk421
𝑞𝑞 = 2

Low maximum energy and curved electron spectrum are naturally reproducedwith temporal evolution effects.

1023e [erg/cm ]-2 ε'e n'(ε')

Mrk 421

10Tramacere+ 2009-9εf(ε) [erg/cm2/s]

Abdo+ 2011

10-4

2.0

5.0

1.5

10-10

10.0

3.0

15.0
20.0

10-6

10-11

10-8

10-12
R/R0=1.01

10-13

10-10107108

Γ = 15, 𝐵𝐵0109

1010= 0.16G, 𝑊𝑊 ′ =1011

1012

1013 10ε'e [eV]-1410-51001051010

ε [eV]

𝑅𝑅0
𝑊𝑊 ′′
16
= 10 cm, Δ𝑇𝑇𝑖𝑖𝑖𝑖𝑖𝑖 =, 𝐾𝐾 = 3.7 × 10−6 s −1 , 𝑁𝑁̇ = 9.8 × 1046 s−1
Γ𝑐𝑐See also Kakuwa+ 2015

FSRQ 3C 279

10-8εf(ε) [erg/cm2/s](b)10Asano & Hayashida 2015
Hayashida+2012
10Steady Stateεf(ε) [erg/cm2/s](b)

-9

10-9

SSC Component constrains
the emission radius

10-10-82Γ′′𝑇𝑇UV= 10ΓeV, 𝑈𝑈UV=8erg cm−315

Flare State

3.5h
6.2h

11h
20h
2h
35h

10-10

10-11

10-11

10-12

10-12

10-13 -61010-3𝑞𝑞 = 2100

103

106

109
ε [eV]

10-13 -61010-3100103106109
ε [eV]

3C 279 FlareLightcurve

Evolution of energy density
2Photon Flux [Phs/cm /s] (>0.1GeV)
1.6(×10-5)

Energy Density [erg/cm3]
101

U'UV

1.4
U'e

100

1.2
110-1

0.8

10-2

0.6

U'γ

10-3

U'B

0.4
10-4

0.2
0-3

036912 15 18 21 24 27 30
tobs[hours]

123456 7 8 9 10R/R0

Weak magnetic field:
Acoustic waves rather than Reconnection

Gamma Ray Burst• Power-law injection
with a minimum energy
leads to an electron
spectrum below.
• The index for cooledelectrons becomes 2,
then photons index
should be 1.5.

𝛼𝛼 = −1.5𝛼𝛼 = −1𝛼𝛼 = 1∝ γ −2n(γ )∝ γ − ( p +1)

Softer than the thermal spectrum,
but harder than the synchrotron spectrum.
See Daigne, Bosnjak, Dubus 2011

γcγmγ

Balance between Acceleration and Cooling
Magnetic Field is so strong that balance with acceleration.
Analytic Steady Solution for Electrons

1

q=1

0.01

0.0001

q=2

Hard Sphere
Kolmogorov
Bohm∝ 𝐸𝐸𝑒𝑒2

1e-06

1e-08
0.01

0.1

1

10

See Lefa+ 2011

Synchrotron Photon
χ2 Fq(χ)
101
0.1
0.01
0.001

Hard Sphere
Kolmogorov
Kraichnan
Bohm

0.0001
0.001

0.01

0.1

110

𝜀𝜀0 =100 1000
χ=ε/ε03𝜋𝜋𝜋𝜋𝜋𝜋𝜋 2𝛾𝛾𝑚𝑚𝑒𝑒 𝑐𝑐 pk

Consistent with Band function
for 2 orders of mag.

Numerical Simulation in Asano & Terasawa 2009

Decaying TurbulenceZrake 2014

Temporal Evolution of the Spectrum
Electron Number: Increase with injection

Electron peak energy:
Decay as the efficiency ofacceleration/cooling drops.
The temporal evolutions of two combinations determine the final photon spectrum

High-energy photon index

NoteBand-like Spectra
Asano & Tersawa 2015

q=5/3, a=21000000

εp=0.13, α=-1, β=-2.5
b=010000b=0.5
b=1100

a=2.05, b=1.4
b=1.51b=20.01
0.001

0.01

0.1

1101001000

The required indices and MHD simulations seem to be reasonable.

Ultra High-Energy Cosmic Ray from GRBs
-1010

Based on GRB rate & Luminosity functionin Wanderman & Piran 2010.

ε J(ε) [erg/cm /s/sr]
22Assuming E-2-spectrum,

AugerCRs

Lp

LB
= 10,
= 0.1
LγLγ

10-11
IC40+59

10-12

In this shock acceleration
model, only the highest
energy region is explained.

Neutrino

10-13
GZK-Neutrino

10-14 14101015

1016

1017

Asano & Meszaros 2014

1018

1019

1020
ε [eV]

Summary• The stochastic acceleration is a slow process, but enoughto explain the Fermi bubble.
• In relativistic jets (AGN, GRB…), the acceleration
timescale can be comparable to the shock acceleration.
• The curved spectra in blazars are reproduced by thestochastic acceleration model with the temporal evolution
effect.
• The problem in the GRB spectra is solved by the balance
between the acceleration and cooling with decaying
turbulence.
• The hard spectrum in this GRB model is favorable toagree with the UHECR flux.