爆発前の星風中を伝播する超新星残骸における宇宙線の加速と逃走
Shoma Kamijima，Yutaka Ohira
The University of Tokyo

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

Origin of Galactic Cosmic Rays (GCRs)
Ohira, Yamazaki, Terasawa 2012

Knee (~1015.5 eV)

Galactic Cosmic Rays

Energy of Cosmic Rays (eV)

SN1006 (Radio, Optical, X-ray)

Diffusive Shock Acceleration (DSA)𝑑𝑁∝ 𝐸 !"𝑑𝐸𝑢# /𝑢$ +2
𝑠== 2 (for strong shock)𝑢# /𝑢$ − 1

Axford 1977, Krymsky 1977,Blandford & Ostriker 1978,Bell 1978

Which shock accelerates GCRs, parallel shock or perpendicular shock ?
rapid acceleration

What types of supernovae accelerate GCRs up to PeV ?

Escape & Environment around SNRs
Cosmic-ray escape can determine the maximum energy of cosmic rays
and the spectral index of escaped cosmic rays.

(Ptuskin & Zirakashvili 2003, 2005, Ohira et al. 2010, Ohira & Ioka 2011)

The gyration makes crucial role to realize rapid acceleration in a perpendicular shock.
à We cannot use the diffusion approximation that is assumed in previous work.
(Takamoto & Kirk 2015, Kamijima, Ohira, Yamazaki 2020)

The environment around SNRs and the shock geometry are important.

Type Ia

core collapse

Environment: Interstellar medium (ISM)

Environment: ISM, stellar wind

ISM magnetic field

spherical𝐸!"#,%&' ~10
TeV ≪ 3 PeV(w/o upstream BShockfield amplification)
Kamijima & Ohira 2021

Parker-spiralmagnetic field

sphericalshockAligned Rotator & Oblique Rotator
Aligned Rotator (𝛼 = 0)

Oblique Rotator (𝛼 ≠ 0)
rotation axis
wavy current sheet
magnetic axis
𝛼current sheetprogenitor

Neuman & Kopp 1970

Straus et al. 2012

The current sheet becomes the wavy structure if the rotation axis andthe magnetic axis are misaligned.
4Motivation
Previous work about cosmic-ray escape assumed the diffusion approximation.
(The rapid acceleration in perpendicular shocks is induced by a gyration.)
We found that typical type Ia SNRs cannot accelerate CRs to PeV
without the upstream magnetic amplification(Kamijima & Ohira 2021).
à How about SNRs in the circumstellar medium?
We investigate the escape process and the maximum energy limited byescape from a SNR in the circumstellar medium.
In this study,・We exactly solve the gyration in the upstream region.
・We consider the shock geometry, the upstream magnetic field
configuration, and the current sheet structure.
5

Our Simplified Model・plane shock approximation
・stationary wavy structure

upstream𝐵#𝑢!"planeshock downstream𝑧gyration
Bohm
diffusion

・The same wavy structure infinitelycontinue in y direction.
・turbulent magnetic field in thedownstream region

𝐵' = const.

𝑦 0

𝜆current
sheet・magnetic field in the wind(only uniform component)

𝐵% = 𝐵& = 0𝑥𝛼

𝐵#6

Perpendicular Shock Acceleration Model
𝑩𝟐

10uniform
B field𝒖𝟏

gyration
residence timeΔ𝑡! = 𝜋Ω%!
#,!
aceleration time

turbulent
B field𝒖𝟐

Bohm diffusion
4𝜅"Δ𝑡" =𝑢" 𝑣𝑡&'' = Δ𝑡! + Δ𝑡" 𝑝/Δ𝑝

10Drury 1983

10102

∝ "#%!"

1

0

10

-3

-2

10𝑢 /𝑣

ush/c !"

10-1

104

∆𝑝 𝑢!
=𝑝𝑣

Our model can realize the rapid accelerationand the canonical spectrum, 𝑑𝑁/𝑑𝑝 ∝ 𝑝%" ,simultaneously.

𝐵! = 10#𝐵"

103

10𝑢!" ,- ,- 16 𝐵$ ,- 𝑢!" ,$ ,𝑡)*+ = 𝜋
Ω%,- +Ω%,𝑣
3 𝐵𝑣∝ "#$!"

4

𝐵! = 10!𝐵"

0𝑩𝟏(Kamijima, Ohira, Yamazaki, 2020)
𝐵! = 10𝐵"
tacc,‖,Bohm,B1

5Ω%&'()**-1
tacc,⊥ [Ω𝑡g,100p ]

shock

106

10
2p dN/dp
𝑝$ 𝑑𝑁/𝑑𝑝

Upstream

downstream𝑑𝑁∝ 𝑝,$𝑑𝑝3

102

10Δ𝑡! > Δ𝑡"
Δ𝑡! < Δ𝑡"

1

100

100

101

𝑝/𝑝#

p/p0

102

7

Simulation Setup: Oblique Rotator
・test particle simulation (upstream)+ Monte-Carlo (downstream)・shock velocity 𝑢!" = 0.1𝑐 (const.)
・downstream flow velocity 𝑢$ = (3/4)𝑢!"shockupstream
Solve E.o.M.
𝑑𝑢 𝑞 𝑢=×𝐵𝑑𝑡 𝑚 𝛾𝑐Monte-Carlo𝑢!"

・Downstream particles are isotropically scatteredin the local downstream fluid rest frame.

downstream

𝐵#3𝑧𝑢$ = 𝑢!"4・upstream B field 𝐵- = 1µG(only uniform component)・polarity: pole à equator・downstream B field 𝐵$ = 100𝐵-equator・impulsive injection at the initial time
(isotropic velocity distribution, 𝐸# = 1 TeV)
・the wavelength of the wavy current sheet
structure: 𝜆 = 100 𝑟%,#・the angle between the rotation axis and themagnetic axis: 𝛼 = 𝜋/6・particle splitting method𝑥!",#𝜆𝑥

𝛼𝑦 0

𝐵#@upstream rest frame

8Oblique Rotator (Pole à Equator)𝑢!" /𝑐 = 10,- , 𝛼 = 𝜋/6,𝐵$ /𝐵- = 10$ , 𝜆/𝑟%,# = 10$

𝐵#

Escapeupdown

equator𝐵#

Particles in the wavy structure move along the current sheet.
Particles escape to the far upstream region along the equatorial plane.
9Energy Scalesnapshotshockupdownwavy current sheet

equatorThere are two characteristicenergy scales.

10Maximum Energy① Particles injected far from the equator:
The maximum energy is given by the potential drop.
shock𝑢./downstream
𝐵# (𝑧012 − 𝑥./,3 tan 𝛼) upstream
à 𝐸()*,,- = 𝑒𝑢!"
𝑐② Particles injected near the equator:

𝐵#The maximum energy is given by
the half wavelength of the wavy
structure (𝒓𝐠 = 𝝀/𝟐).
𝜆à 𝐸()*,4/$ = 𝑒𝐵#2

à

𝜆 𝑢./ !#𝑧012,>/ =+ 𝑥./,3 tan 𝛼
2 𝑐①𝑧()*,+,②equator

The transition between ① and ② isgiven by 𝑬𝐦𝐚𝐱,𝐏𝐃 = 𝑬𝐦𝐚𝐱,𝝀/𝟐 .

𝑧𝑧()*𝑥!",#𝜆𝑥𝛼𝑦 0𝑥!",# tan 𝛼𝐵#11

Oblique Rotator (Pole à Equator)
half wavelength ofwavy current sheet(𝑧 ≤ 𝑧123,4" )(𝑧 ≥ 𝑧123,4" )potential drop

12

Oblique Rotator (Pole à Equator)𝑢&(𝜋𝐸!"#,)* = 𝑒
𝐵+− 𝛼 − 𝜃,-. 𝑅&(𝑐2injectedparticle

𝜆𝐸!"#,//1 = 𝑒𝐵+2
𝜆Escape𝑅∗ Ω∗
𝐵' = 𝐵∗
𝑉@01A

Particles movetowards the equator.

𝜃012

𝛼𝑅!"

𝑅∗
sin 𝜃𝑟Escapeshock𝑢&(13Estimation of Maximum Energy𝑅∗ Ω∗𝐵5 ~𝐵∗
𝑉7128

・Emax given by potential drop

𝑅∗𝑅!"𝑢&(𝜋𝜋𝑢&( 𝑅∗ Ω∗𝐸!"#,)* = 𝑒
𝐵+− 𝛼 𝑅&( =−𝛼𝑒𝐵∗ 𝑅∗𝑐22𝑐 𝑉3,-4
Emax does not depend on the shock radius.
If the shock velocity is constant (e.g. free expansion phase), Emax becomes
constant during this phase.
Red Supergiant𝑢1,!"𝐸-&.,/0 ~7.5×10 eV
0.01𝑐

𝑉2()3 %!
Ω∗10 km/s
5×10%5 s %!

𝑇'(4 = 2𝜋/Ω∗ ~36 ± 8 yr àΩ∗ ~5×10,9 s ,WR star

𝑢1,!7𝐸-&.,/0 ~3.2×10 eV
0.01𝑐𝐵∗
10Gfor Betelgeuse%!
𝑉2()3Ω∗1000 km/s2×10%8 s %!

𝑇'(4 = 2𝜋/Ω∗ ~10 hours àΩ∗ ~2×10,: s ,-for WR46

𝑅∗
100𝑅⨀

"

(Kervella et al. 2018)

𝐵∗
1kG𝑅∗
𝑅⨀"

(Hubrig et al. 2020)14

Estimation of Maximum Energy
・Emax given by 𝒓𝐠 = 𝝀/𝟐

𝜆 1𝑅∗ Ω∗ 𝑅∗
𝑅∗
𝐸()*,:/" = 𝑒𝐵' = 𝑒 𝐵∗
𝑉@01A 𝑇BC> = 𝜋𝑒𝐵∗𝑅∗
2 2
𝑉@01A 𝑅./𝑅./
𝑉'(4𝐵5 ~𝐵∗
𝑉7128

𝑅∗
𝑅!"𝑉'(4 = 𝑅∗ Ω∗2𝜋
𝜆 = 𝑉7128 𝑇'(4 = 𝑉7128
Ω∗Emax depend on the shock radius.
Even if the shock velocity is constant, Emax decreases as the shock expands.

𝐵∗
67
Red Supergiant 𝐸!"#,5/1 ~10 eV
10G𝑅∗
100𝑅⨀

𝐵∗
eV1kG

1WR star

67𝐸!"#,5/1 ~10𝑅∗𝑅⨀

1𝑅&(6000𝑅⨀

𝑅&(20𝑅⨀

96

96

We confirmed that 𝐸-&.,:/" can be larger than 𝐸-&.,/0 in more realistic simulation.
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Summary & Future Work
SummaryPole à Equator

・Particles injected inside the wavy current sheet structure move along
the current sheet (meandering motion).
・Particles move towards the equator while being accelerated.
・Particles escape to the far upstream region while moving along the equator.
two types of maximum energy𝜋𝑢&(① potential drop: 𝐸!"# =−𝛼 𝑒𝐵+ 𝑅&(2𝑐𝜆② the half wavelength of the wavy structure: 𝐸!"# = 𝑒𝐵+
2Future WorkWe investigate escape from a spherical shock in the Parker-spiralmagnetic field.

16