Asymmetric Neutrino Emission fromAsymmetric Neutrino Emission fromStrongly Magnetized Neutron StarStrongly Magnetized Neutron Star

（強磁性中性子星からの非対称ニュートリノ放出）（強磁性中性子星からの非対称ニュートリノ放出）

Asymmetric Neutrino Emission fromAsymmetric Neutrino Emission fromStrongly Magnetized Neutron StarStrongly Magnetized Neutron Star

（強磁性中性子星からの非対称ニュートリノ放出）（強磁性中性子星からの非対称ニュートリノ放出）

1

Collaborators

ToshitakaToshitaka KAJINOKAJINO 　 　 　 　 Nao, Univ. of Tokyo Nao, Univ. of Tokyo

(Japan)(Japan)

NobutoshiNobutoshi YASUTAKEYASUTAKE INFN, Catania INFN, Catania (Italy)(Italy)

Myung-kiMyung-ki CHEOUNCHEOUN Soongs Univ. (Korea)

Chung-YeolChung-Yeol RYU RYU Soongs Univ. (Korea)

Jun HIDAKA Nao (Japan)

G.J. MATHEWS Univ. of Notre Dome 　　 (USA)

＋

　　 Takami KURODA Nao (Japan)

Collaborators

ToshitakaToshitaka KAJINOKAJINO 　 　 　 　 Nao, Univ. of Tokyo Nao, Univ. of Tokyo

(Japan)(Japan)

NobutoshiNobutoshi YASUTAKEYASUTAKE INFN, Catania INFN, Catania (Italy)(Italy)

Myung-kiMyung-ki CHEOUNCHEOUN Soongs Univ. (Korea)

Chung-YeolChung-Yeol RYU RYU Soongs Univ. (Korea)

Jun HIDAKA Nao (Japan)

G.J. MATHEWS Univ. of Notre Dome 　　 (USA)

＋

　　 Takami KURODA Nao (Japan)

TomoyukiTomoyuki MARUYAMA MARUYAMA BRS, Nihon Univ. (Japan)BRS, Nihon Univ. (Japan)TomoyukiTomoyuki MARUYAMA MARUYAMA BRS, Nihon Univ. (Japan)BRS, Nihon Univ. (Japan)

Structure of Proto-Neutron Stars : Strange Matter,

Ferromagnetism

　　 Observable Information ‥‥Neutrino Emissions

S.Reddy, M.Prakash and J.M. Lattimer, P.R.D58 #013009 (1998) 　　　　　　 Influence from Hyperons Λ ，∑

Ferromagnetism 　→　 Large Asymmetry ？

　　 P. Arras and D. Lai, P.R.D60, #043001 (1999)

S. Ando, P.R.D68 #063002 (2003)

Magnetar 1015G in surface 1017-19G inside (?)

　 Our Works ⇒ Neutrino Reactions on High Density Matter with with Strong Mag. Fields

in the Relativistic Mean-Field (RMF) Approach TM et al., PRD83, 081303(R) (2011)

Neutrino Scattering& Absorption

in Surface Region

4

§1. §1. IntroductionIntroduction§1. §1. IntroductionIntroduction

5

Birth of Proto-neutron StarBirth of Proto-neutron Star

core collapse

HHe

C+OSi

Fe

trapping core bounce

NSNS

shock propagation in coreshock in envelopeSN explosion

5

8

Pulsar KickPulsar KickPulsar KickPulsar Kick

Asymmetry of Supernova Explosion kick and translate Pulsar with

Kick Velocity: Average … 400km/s ， Highest … 1500km/s 　　

Explosion Energy ~ 1053 erg

(almost Neutrino Emissions)

　 1% Asymmetry are sufficient to explain the Pulsar Kick

http://chandra.harvard.edu/photo/2004/casa/casa_xray.jpg

CasA

A.G.Lyne, D.R.Lomier, Nature 369, 127 (94)

Present Work Neutrino Scattering and Absorption ‥ In Hot and Dense Neutron-Star-Matter

Estimating Kick Velocity and Spin of NS

9

§2. §2. FormulationFormulation§2. §2. FormulationFormulationMagnetic Field :

Baryon Lepton B & L – Mag.

Weak Interaction

e + B → e + B : scattering

e + B → e- + B’ : absorption 　 S.Reddy, M.Prakash and J.M. Lattimer, P.R.D58 #013009 (1998)

1. Proto-Nuetron-Star (PNS) Matter without Mag. Field

2. Baryon Wave Function under Mag. Field in Perturbative Way

3. Cross-Sections for reactions

§2-1§2-1 RMF Approach for Neutron-Star MattterRMF Approach for Neutron-Star Mattter

EOS of EOS of PM1-1PM1-1 -3

0* fm 0.17 at MeV 200 ,7.0/ MeV, 16 KMMBE NN

3/2

gg

gg

T.M, H. Shin, H. Fujii, & T. Tatsumi,

Prog. Theo. Phys. Vol.102, P809

EOS of Proto Neutron-Star-MatterEOS of Proto Neutron-Star-Matter

Charge Neuttral （ ） & Lepton Fraction : Yp =

0.4 ep

§2-2 Dirac Equation under Magnetic Fields§2-2 Dirac Equation under Magnetic Fields

　 N B << εF (Chem. Pot) → B can be treated perturbatively　　　　 　 Landau Level can be ignored 　

Dirac Eq.

~ 1017 G

Magnetic Part

of Lagrangian

Spin VectorDirac Spinor

negligibly small

3

14

　 Fermi

Distribution

Momentum Distr. of Major Part

is deformed as oblate

Relativistic Effects of Magnetic Contributions

1)Momentum Dependent of Spin Vector

2)Deformation of Fermi Distribution

15

ElectronElectron

Landau Level

When B → 0 , the wave function → plane wave

§2-3 The Cross-Section of ν-B

0 BΔσΔσσσ Perturbative

Treatment

Non-Magnetic Part Magnetic Part

　 Fermi Distribution

Deformed Distribution

17

The Cross-Section of Lepton-Baryon The Cross-Section of Lepton-Baryon ScatteringScattering

with

Spin-independent part

Spin-dependent Spin-dependent PartPart

Electron ContributionElectron Contribution

Single Particle Energy

Sum of Landau Level

Expectation Value of Quantity A

Perturbation

Spin vector

Final electron contribution in Final electron contribution in ee →→ ee--

§3 Results of Cross-§3 Results of Cross-SectionsSections

No magnetic field

pot.) chem.-( ik

3% difference

0 andG102 , pot.) chem. (neutrino i17 Bki

Differential Cross-Sections in Magnetic Field Differential Cross-Sections in Magnetic Field

Initial Angle-DependenceInitial Angle-Dependence

0 andG102 , pot.) chem. (neutrino i17 Bki

25

f

eeiSc dΩ

ννdσdΩσ

f

efAb dΩ

eνdσdΩσ

Increasing in Dir. parallel to B

Scat.

Absorp.

0 BΔσΔσσσ

Integrating over the initial angle

Integrating over the final angle

Magnetic parts of Cross-SectionsMagnetic parts of Cross-Sections

0 andG102 , pot.) chem. (neutrino i17 Bki

Magnetic Parts of Cross-SectionsMagnetic Parts of Cross-Sections

cos1 ,,0

,, fiASASAS S

Neutrino Mean-Free PathsNeutrino Mean-Free Paths

scattering absorption

)180()0(2

1

VV

Contribution of Each Element in Contribution of Each Element in Scattering PartScattering Part

Contribution of Each Element in Contribution of Each Element in Absorption PartAbsorption Part

30

EOS of Neutron-Star-Matter with p, n Λ in RMF AppraochEOS of Neutron-Star-Matter with p, n Λ in RMF Appraoch

Cross-Sections of Neutrino Scattering and Absorption Cross-Sections of Neutrino Scattering and Absorption

under Strong Magnetic Field, calculated in Perturbative Wayunder Strong Magnetic Field, calculated in Perturbative Way

Neutrinos are More Scattered and Less Absorbed Neutrinos are More Scattered and Less Absorbed

in Direction in Direction Parallel to Magnetic FieldParallel to Magnetic Field

　　　⇒　　　　⇒　 More Neutrinos are Emitted in Arctic AreaMore Neutrinos are Emitted in Arctic Area　　　　

　　　　　　　　　　Scattering 1.7 % Scattering 1.7 %

Absorption 2.2 % at Absorption 2.2 % at ρρB==33ρρ0 and T = 20 MeV

31

Asymmetry of Supernova Explosion kick and translate Pulsar with

Kick Velocity: Average … 400km/s ， Highest … 1500km/s 　　

http://chandra.harvard.edu/photo/2004/casa/casa_xray.jpg

CasA

A.G.Lyne, D.R.Lomier, Nature 369, 127 (94)

Estimating Kick Velocity of PNS with T = 20 MeV and B B = 2× 10= 2× 1017GG

PoloidalPoloidal Magnetic Field Magnetic Field 2‐32‐3%% Asymmetry Asymmetry inin AbsorptionAbsorption

§4 Estimating Pulsar Kick §4 Estimating Pulsar Kick VelocitiesVelocities

of Proto-Neutron Star of Proto-Neutron Star

Explosion Energy ~ 1053 erg

(almost Neutrino Emissions)

　 1% Asymmetry is sufficient to explain the Pulsar Kick

D.Lai & Y.Z.Qian, Astrophys.J. 495 (1998) L103

Neutrino Transportation Neutrino Transportation

, ),(),( ),(),( 00 V

σbΔfcbIΔfcfcfc ab

νcoll

rprpr

rpr

rpr

Neutrino Propagation ⇒ Boltzmann Eq. 　

Neutrino Phase Space Distribution Function

,z)(p,rfdz

d,z)(p,rf

zpz

yybd,x)(p,rfx

dxzΔf

TT

z

x

z

TTc

00

0 0

,ˆ

, )(exp ),,( 1

r

rp

/)(exp11),( , ),( ),(),( 00 TfΔfff prprprprp

Equil. Part Non-Equil. Part

Solution ⇒

Neutrinos Propagate on Strait Line only absorption

Other Pieces are equilibrium

Mean-Free Mean-Free PathsPaths

Scattering

Absorption 21in

10

0ρρB

AS

Magnetic Parts

cos1 ,0

fiAAA S

SA: fitting function

0,, / ASAS V

Baryon density in Proto-Neutron StarBaryon density in Proto-Neutron Star

Calculating Neutrino Propagation above B = 0

M = 1.68Msolar

YL = 0.4

Baryon density in Proto-Neutron Star

T = 20 MeV

M = 1.68 Msolar

YL = 0.4

Calculating

Neutrino Propagation above B = 0

\θ

Neutrino Neutrino PropagatioPropagatio

nn 1) Neutrinos propagate on

the straight lines

2) Neutrino are created and absorbed at all positions on the lines

Mean-Free Path

: σab/V

Angular Dep. of Emitted Neutrinos in Uniform Poloidal Mag. Field

[erg] 103

[g], 68.1

53

T

solarNS

E

MM

　　　　　　　　　　　

p, n

p, n,

　　　

T = 20 MeV

[km/s] 600 M

pv zkick

Observable Average … 400km/s ，

cos 10 PPp n

[G] 102 17B

2

2

0

1

1094.1

1076.1 3

-

-

T

z

PP

E

p

38

Stability of Magnetic Field in Compact ObjectsStability of Magnetic Field in Compact Objects

((Braithwaite & Spruit 2004Braithwaite & Spruit 2004))

Toroidal Magnetic Toroidal Magnetic Field is stable !!Field is stable !!

§5 Angular Deceleration in Toroidal §5 Angular Deceleration in Toroidal Magnetic FieldMagnetic Field

Mag. FieldParallel

to Baryonic Flow

Assym. of -Emit.must decelerate

PNS Spin

No poloidal Magnetic Field at the No poloidal Magnetic Field at the beginningbeginning

by T. Kuroda

Single Toroidal

Finite Polaidal Magnetic Field at the beginning

by 黒田Anti-paralleled Double Toroidal

Toroidal Magnetic

Field

φφe

rz

rzzG

rRr

rRrrG

ezGrGBzr

L

T

TTT

LTTT

)0,cos,sin(ˆ

/exp1

/exp)(

/)(exp1

/)(exp16)(

ˆ)()(),(

2

20

0

0

B

z = 0

T = 20MeV

r = 0.5 (km)

R0 = 8 (km) Mag.‐A

R0 = 5 (km) Mag.‐B

42

Angular DecelerationAngular DecelerationAngular DecelerationAngular Deceleration

zLLS

z pΔfdpddcdt

dLN

prnrnr ),,(

Neutrino Luminosity

(dET/dt) ~ 3×1052 erg/s

/

/11

dtdE

dtcdL

dt

dE

Idt

dL

I T

Z

NS

Z

NS

T

dt

d

68.1 solarNS MM Period P = 10s

Mag

Distr.Bary.

B [G]

at B = 0 (cm) ( emis.) ( rad.)

Mag-A p,n 1.0×1016 46.7 3.02×10-2 4.82×10-21.0×10-8

p,n, 1.6×1016 70.6 5.19×10-2 8.26×10-22.6×10-8

Mag-B p,n 2.6×1013 0.157 1.01×10-4 1.61×10-46.6×10-14

p,n, 4.2×1013 0.259 1.90×10-4 3.03×10-4 1.7×10-13

/

/

dtdE

dtcdL

T

Z PPPP

In Early Stage ( ～ 10 s) Asymmetric Emission must affect PNS Spin

More Significantly than Magnetic Dipole -Radiation

Magnetic Dipole Rad. [G] 102.3 39 PPB

EOS of Neutron-Star-Matter with p, n, Λ in RMF ApproachEOS of Neutron-Star-Matter with p, n, Λ in RMF Approach

Exactly Solving Dirac Eq. with Magnetic Field in Perturbative WayExactly Solving Dirac Eq. with Magnetic Field in Perturbative Way

Cross-Sections of Neutrino Scattering and Absorption Cross-Sections of Neutrino Scattering and Absorption

under Strong Magnetic Field, calculated in Perturbative Wayunder Strong Magnetic Field, calculated in Perturbative Way

Neutrinos are More Scattered and Less Absorbed Neutrinos are More Scattered and Less Absorbed

　　　　　　 　　　　　　　　　　　　　　　　　　　　 　　　　　　　　　　　　　　　　　　　　 in Direction in Direction Parallel to Parallel to

Magnetic FieldMagnetic Field

　　　⇒　　　　⇒　 More Neutrinos are Emitted in More Neutrinos are Emitted in 　　 ArcticArctic 　 　 AreaArea　　　　

　　　　　　　　　　Scattering 1.7 % Scattering 1.7 %

Absorption 2.2 % at Absorption 2.2 % at ρρB==33ρρ0 and T = 20 MeV

⇒ Convection, Pulsar-Kick

43

§5§5 SummarySummary§5§5 SummarySummary

44

Pulsar Kick in Poroidal Mag. Field B = 2× 1017G 　⇒ Perturbative Cal.

　　 　　 vvkickkick = 580 km/s ( p,n ) , 610 km/s (p,n, = 580 km/s ( p,n ) , 610 km/s (p,n,ΛΛ ） ） at at T T = 20 MeV= 20 MeV

　 　　 　 　　 = 230 km/s ( p,n ) , 270 km/s (p,n,= 230 km/s ( p,n ) , 270 km/s (p,n,ΛΛ ） ） at at TT = 30 MeV = 30 MeV 　 　　　 　　

400 km/s400 km/s (Average of Observed Values (Average of Observed Values ）　　　　）　　　　

Spin Deceleration Spin Deceleration in in Toroidal Magnetic Field Magnetic Field

Asymmetric Asymmetric --Emission plays an important role in PNS Spin in Early Time Emission plays an important role in PNS Spin in Early Time

Future Plans Future Plans 　　　　 -Scattering-Scattering

Fixed Temp. Fixed Temp. ⇒⇒ Constant Entropy Constant Entropy

Exact Solution of Dirac Eq. in Non-Perturbative Cal.Exact Solution of Dirac Eq. in Non-Perturbative Cal.

　 →　 　 →　 Landau LevelLandau Level at least for Electron at least for Electron

Antarctic Dir.

Mag

Distr.

(P = 10 s)

p,n p,n,

Mag-A 4.8×10-3 5.2×10-3

Mag-B 6.1×10-3 7.2×10-3

Dpl. Rad.

9.8×10-11

][s 1-PP

Reestimatingwhen B = 1015G

in the surface with = 0

EOS in Iso-Entropical ModelEOS in Iso-Entropical Model

§2-1 Neutron-Star Matter in RMF §2-1 Neutron-Star Matter in RMF ApproachApproach

RMF Lagrangian, N,

Dirac Eq.

Scalar Field ⇒ Effective Mass

Spin VectorDirac Spinor

negligibly small

Spin-indep. part

Spin-dep. Part

\θ

Neutrino Neutrino PropagatioPropagatio

nn 1) Neutrinos propagate on

the straight lines

2) Neutrino are created and absorbed at all positions on the lines

Mean-Free Path

: σab/V

Absorption Mean-Free PathAbsorption Mean-Free Path

cos1 ,0

fiAAA S

1/ VAA

B = 0

30MeV

50MeV

100MeV

200MeV

300MeV

SA: fitting function

0 andG102 , pot.) chem. (neutrino i17 Bki§3 Results§3 Results

3% difference

Neutrino Mean-Free-Path at Energy equal to Neutrino Mean-Free-Path at Energy equal to Chem. PotentialChem. Potential

Magnetic Parts of Cross-SectionsMagnetic Parts of Cross-Sections

Magnetic Parts of Cross-SectionsMagnetic Parts of Cross-Sections

cos1 ,,0

,, fiASASAS S