Michael Kachelrieß NTNU, Trondheim · Introduction Outline Outline of the talk 1 Introduction GMF...

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Anisotropic Diffusion and the Cosmic Ray Anisotropy

Michael Kachelrieß

NTNU, Trondheim

with G.Giacinti, A.Nerenov, V.Savchenko, D.Semikoz

Introduction Outline

Outline of the talk

1 Introduction

◮ GMF & calculation of CR trajectories

◮ Consequences of anisotropic diffusion: single sources

2 Imprint of a single source on anisotropy

3 Conclusions

Michael Kachelrieß (NTNU Trondheim) Anistropy of a Single Cosmic Ray Source ECRS, 9. July ’18 2 / 13

Introduction GMF and the trajectory approach

Propagation in turbulent magnetic fields:

Galactic magnetic field: regular + turbulent component

turbulent: fluctuations on scales Lmin ∼AU to Lmax ∼ (10− 150) pc






CRs scatter mainly on field fluctuations B(k) with kRL ∼ 1.




Galactic magnetic field: regular + turbulent componentturbulent: fluctuations on scales Lmin ∼AU to Lmax ∼ (10− 150) pc


all fluctuations between Lmax and ∼ RL/10 have to be included⇒ makes trajectory approach computationally very expansive







⇒ often diffusion as effective theory








slope of power spectrum P(k) ∝ k−α determines energy dependenceof diffusion coefficient for Breg = 0 as D(E) ∝ Eβ as β = 2− α:

Kolmogorov α = 5/3 ⇔ β = 1/3Kraichnan α = 3/2 ⇔ β = 1/2










injection spectrum dN/dE ∝ E−δ modified to dN/dE ∝ E−δ−β










injection spectrum dN/dE ∝ E−δ modified to dN/dE ∝ E−δ−β

anisotropy δi = −3Dij∇i ln(n) ∝ Eβ



Our approach:

use model for Galactic magnetic field: Jansson-Farrar, Psirkhov etal.,. . .

calculate trajectories x(t) via F L = qv ×B.



Our approach:

use model for Galactic magnetic field: Jansson-Farrar, Psirkhov etal.,. . .

calculate trajectories x(t) via F L = qv ×B.

as preparation, let’s calculate diffusion tensor in pure, isotropicturbulent magnetic field



Eigenvalues of Dij = 〈xixj〉/(2t) E = 1015 eV, Brms = 4µG

1e+28

1e+29

0.1 1 10 100

D(1

PeV

) (c

m2 /s

)

Time (kyr)

d3d2d1D

[Giacinti, MK, Semikoz (’12) ]




1e+28

1e+29

0.1 1 10 100

D(1

PeV

) (c

m2 /s

)

Time (kyr)

d3d2d1D

-10 -5 0 5 10

-10

-5

0

5

10

0.01 0.1 1





1e+28

1e+29

0.1 1 10 100

D(1

PeV

) (c

m2 /s

)

Time (kyr)

d3d2d1D


asymptotic value is ∼ 50 smaller than standard value



Is isotropic diffusion possible? [Giacinti, MK, D. Semikoz ’17 ]

for isotropic diffusion: for α = 5/3

D =cL0

3

[

(RL/L0)2−α + (RL/L0)

2]

∝ B−1/3



Is isotropic diffusion possible? [Giacinti, MK, D. Semikoz ’17 ]

for isotropic diffusion: for α = 5/3

D =cL0

3

[

(RL/L0)2−α + (RL/L0)

2]

∝ B−1/3

10

100

103

104

10-5 10-4 10-3 0.01 0.1 1 10

l c/p

c

Brms/µG

KraichnanKolmogorov



Anisotropic diffusion – 2 options:

anisotropic turbulence

dominance of regular field, Brms ≪ B0 ⇒ D‖ ≫ D⊥






10 27

10 28

10 29

10 30

10 31

10 32

10 33

10 15 10 16 10 17

D(E

) [c

m2 /s

]

E [eV]

B/C,θ=90o

η = 0.1η = 0.5

η = 1

η = 2η = 4η = ∞

[Giacinti, MK, Semikoz ’17 ]






⇒ anisotropic CR propagation

⇒ relative importance of single sources is changed



Consequences of anisotropic propagation:

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 8.45

8.5

8.55

8.6

8.65

8.7

8.75

8.8

-0.1-0.08-0.06-0.04-0.02

0 0.02 0.04 0.06 0.08

2000 yr1000 yr

400 yrObserver

Source



Consequences of anisotropic propagation:

109

1010

1011

1012

1013

1014

1015

109 1010 1011 1012 1013 1014 1015 1016 1017 1018

E2.

5 F(E

) [e

V1.

5 /cm

2 /s/s

r]

E [eV]

PAMELA 2011CREAM-2011KASCADE 2013KASCADE GrandeVela 12 pcVela 25 pcVela face onGeminga face on

⇒ local sources contribute only, if d⊥ is small


Single source Diffusion

Anisotropy of a single source

number density is Gaussian with σ2 = 2DT

δ =3D

c

∇n

n=

3R

2T= 5× 10−4 R

200pc

2Myr

T




if only turbulent field:diffusion = random walk = free quantum particle


δ =3D

c

∇n

n=

3R

2T= 5× 10−4 R

200pc

2Myr

T






δ =3D

c

∇n

n=

3R

2T= 5× 10−4 R

200pc

2Myr

T

2 options:

◮ old, nearby source dominating flux

◮ young, nearby source (dominating?) flux, suppression






δ =3D

c

∇n

n=

3R

2T= 5× 10−4 R

200pc

2Myr

T

2 options:

◮ old, nearby source dominating flux

◮ young, nearby source (dominating?) flux, suppression

what happens for general fields?


Single source Trajectory approach

Anisotropy of a single source: only turbulent field

0.01

0.1

1000 10000 100000

Anis

otr

opy ×

(R

/ 5

0 p

c)-1

Years

B0 = 0 G

kpc

kpc

3/2 × 50 pc / ct

0.1

1

[Savchenko, MK, Semikoz ’15 ]



Anisotropy of a single source: plus regular

0.01

0.1

1000 10000 100000

Anis

otr

opy ×

(R

/ 5

0 p

c)-1

Years

B0 = 1 µG

kpc

kpc

3/2 × 50 pc / ct

0.1

1




Anisotropy of a single source:

0.01

0.1

1000 10000 100000

An

iso

tro

py

× (

R /

50

pc)

-1

Years

B0 = 1 µG

kpc

kpc

3/2 × 50 pc / ct

0.1

1

regular field changes n(x), but keeps it Gaussian

∇n and Dii are not misaligned

⇒ no change in δ, no suppression



Dipole anisotropy

10-5

10-4

10-3

10-2

1011 1012 1013 1014 1015 1016 1017 1018

100 TeV

1 PeV

δ

E [eV]

all dataIceCube 2013

fit 1 SNGalactic

totalGalactic




Dipole anisotropy

10-5

10-4

10-3

10-2

1011 1012 1013 1014 1015 1016 1017 1018

100 TeV

1 PeV

δ

E [eV]

all dataIceCube 2013

fit 1 SNGalactic

totalGalactic


suggests low-energy cutoff ⇒ source is off-setsame cutoff responsible for breaks in spectra


Conclusions

Conclusions

1 Single source: anisotropy

◮ dipole formula δ = 3R/2T holds universally in quasi-gaussian regime

◮ plateau of δ and phase flip point to dominance of single sources

◮ see no suppression due to misalignement

2 Source with T ∼ 2− 3Myr and R ∼ 100 pc:

◮ consistent explanation of p̄ and e+ fluxes, breaks and B/C

◮ consistent with 60Fe

3 local geometry of GMF is important: Local Bubble and Loop I


Michael Kachelrieß NTNU, Trondheim · Introduction Outline Outline of the talk 1 Introduction GMF...

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