1. Asymmetric Neutrino Emission from Strongly Magnetized Neutron Star （強磁性中性子星からの非対称ニュートリノ放出）
Tomoyuki MARUYAMA BRS, Nihon Univ (Japan)
Collaborators
　Toshitaka KAJINO　 　 Nao, Univ. of Tokyo 　(Japan)
　Nobutoshi YASUTAKE INFN, Catania 　(Italy)
　Myung-ki CHEOUN Soongs Univ 　(Korea)
　Chung-Yeol RYU Soongs Univ 　(Korea)
　Jun HIDAKA Nao (Japan)
　G.J. MATHEWS Univ. of Notre Dome 　　 (USA) ＋
　　Takami KURODA Nao (Japan) 1

4. in the Relativistic Mean-Field (RMF) Approach
§1. Introduction
Structure of Proto-Neutron Stars　:　Strange Matter, Ferromagnetism
　　Observable Information ‥‥Neutrino Emissions
S.Reddy, M.Prakash and J.M. Lattimer, P.R.D58 # (1998)
　　　　　　Influence from Hyperons Λ，∑
Ferromagnetism　→　Large Asymmetry ？
　　P. Arras and D. Lai, P.R.D60, # (1999)
S. Ando, P.R.D68 # (2003)
Magnetar 1015G in surface G 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, (R) (2011)
Neutrino Scattering
& Absorption
in Surface Region 4

5. Birth of Proto-neutron Star5 5

8. In Hot and Dense Neutron-Star-Matter
Pulsar Kick
CasA
A.G.Lyne, D.R.Lomier, Nature 369, 127 (94)
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
2004/casa/casa_xray.jpg
Present Work ‥ Neutrino Scattering and Absorption
In Hot and Dense Neutron-Star-Matter
Estimating Kick Velocity and Spin of NS 8

9. S.Reddy, M.Prakash and J.M. Lattimer, P.R.D58 #013009 (1998)
§2. Formulation
Magnetic Field :
Lepton
B & L – Mag.
Baryon
Proto-Nuetron-Star (PNS) Matter without Mag. Field
Baryon Wave Function under Mag. Field in Perturbative Way
Cross-Sections for n reactions
Weak Interaction
e + B → e + B : scattering
e + B → e- + B’ : absorption 　
S.Reddy, M.Prakash and J.M. Lattimer, P.R.D58 # (1998)

10. §2-1 RMF Approach for Neutron-Star Mattter

11. T.M, H. Shin, H. Fujii, & T. Tatsumi,
EOS of PM1-1
T.M, H. Shin, H. Fujii, & T. Tatsumi,
Prog. Theo. Phys. Vol.102, P809

12. EOS of Proto Neutron-Star-Matter
Charge Neuttral （ ） & Lepton Fraction : Yp = 0.4

13. §2-2 Dirac Equation under Magnetic Fields
　N B << εF (Chem. Pot) → B can be treated perturbatively　　
　　 　 Landau Level can be ignored　
~　1017 G
Magnetic Part
of Lagrangian
Dirac Eq.

14. Dirac Spinor Spin Vector3
negligibly small
Dirac Spinor
Spin Vector
1414

15. Momentum Distr. of Major Part
　 Fermi
Distribution
Momentum Distr. of Major Part
is deformed as oblate
Relativistic Effects of Magnetic Contributions
Momentum Dependent of Spin Vector
Deformation of Fermi Distribution 15

16. When B → 0 , the wave function → plane wave
Electron
Landau Level
When B → 0 , the wave function → plane wave

17. §2-3 The Cross-Section of ν-B
　 Fermi
Distribution
Deformed Distribution
Perturbative
Treatment
Non-Magnetic Part
Magnetic Part 17

18. The Cross-Section of Lepton-Baryon Scattering
with
Spin-independent part

19. Spin-dependent Part

20. Electron Contribution
Single Particle Energy
Sum of Landau Level
Expectation Value
of Quantity A
Perturbation
Spin vector

21. Final electron contribution in ne → e-

22. §3 Results of Cross-Sections
No magnetic field

23. Differential Cross-Sections in Magnetic Field
3% difference

24. Initial Angle-Dependence

25. Magnetic parts of Cross-Sections
Scat.
Increasing 
in Dir. parallel to B
Integrating over the initial angle
Absorp.
Integrating over the final angle 25

26. Magnetic Parts of Cross-Sections

27. Neutrino Mean-Free Paths
scattering
absorption

28. Contribution of Each Element in Scattering Part

29. Contribution of Each Element in Absorption Part

30. EOS of Neutron-Star-Matter with p, n Λ in RMF Appraoch
Cross-Sections of Neutrino Scattering and Absorption
under Strong Magnetic Field, calculated in Perturbative Way
Neutrinos are More Scattered and Less Absorbed
in Direction Parallel to Magnetic Field
　　　⇒　More Neutrinos are Emitted in Arctic Area
Scattering %
Absorption % at ρB=3ρ0 and T = 20 MeV 30

31. §4 Estimating Pulsar Kick Velocities of Proto-Neutron Star
CasA
A.G.Lyne, D.R.Lomier, Nature 369, 127 (94)
Asymmetry of Supernova Explosion
kick and translate Pulsar with
Kick Velocity: Average … 400km/s，
Highest … 1500km/s　　
2004/casa/casa_xray.jpg
　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
Estimating Kick Velocity of PNS with T = 20 MeV and B = 2× 1017G
Poloidal Magnetic Field 2‐3% Asymmetry in Absorption 31

32. Neutrino Transportation
Neutrino Phase Space Distribution Function
Equil. Part
Non-Equil. Part
Other Pieces are equilibrium
Neutrino Propagation ⇒　Boltzmann Eq.　
Neutrinos Propagate on Strait Line
only absorption
Solution ⇒

33. Mean-Free Paths Magnetic Parts SA: fitting function Scattering
Absorption

34. Baryon density in Proto-Neutron Star
M = 1.68Msolar
YL = 0.4
Calculating Neutrino Propagation above rB = r0

35. Baryon density in Proto-Neutron Star
T = 20 MeV
M = 1.68 Msolar
YL = 0.4
Calculating
Neutrino Propagation
above rB = r0

36. θ Neutrino Propagation Neutrinos propagate on the straight lines
2) Neutrino are created and absorbed at all positions on the lines
Mean-Free Path
: σab/V \ θ

37. Angular Dep. of Emitted Neutrinos in Uniform Poloidal Mag. Field
　　　　　　　　　　　 p, n
p, n, L
T = 20 MeV
Observable Average
… 400km/s，

38. §5 Angular Deceleration in Toroidal Magnetic Field
Stability of Magnetic Field in Compact Objects
(Braithwaite & Spruit 2004)
Toroidal Magnetic Field is stable !!
Mag. Field
Parallel
to Baryonic Flow
Assym. of n-Emit.
must decelerate
PNS Spin 38

39. No poloidal Magnetic Field at the beginning
Single Toroidal
by T. Kuroda

40. Finite Polaidal Magnetic Field at the beginning
Anti-paralleled Double Toroidal
by 黒田

41. Toroidal Magnetic Field
T = 20MeV
Dr = 0.5 (km)
R0 = 8 (km) Mag.‐A
R0 = 5 (km) Mag.‐B
z = 0

42. Angular Deceleration B [G] Neutrino Luminosity
(dET/dt)n ~　3×1052 erg/s
Period P = 10s
Magnetic
Dipole Rad.
Mag Distr.
Bary.
B [G]
at rB = r0
(cm)
(n emis.)
(g rad.)
Mag-A
p,n
1.0×1016
46.7
3.02×10-2
4.82×10-2
1.0×10-8
p,n,L
1.6×1016
70.6
5.19×10-2
8.26×10-2
2.6×10-8
Mag-B
2.6×1013
0.157
1.01×10-4
1.61×10-4
6.6×10-14
4.2×1013
0.259
1.90×10-4
3.03×10-4
1.7×10-13
In Early Stage (～ 10 s) n Asymmetric Emission must affect PNS Spin
More Significantly than Magnetic Dipole g-Radiation 42

43. §5 Summary EOS of Neutron-Star-Matter with p, n, Λ in RMF Approach43
EOS of Neutron-Star-Matter with p, n, Λ in RMF Approach
Exactly Solving Dirac Eq. with Magnetic Field in Perturbative Way
Cross-Sections of Neutrino Scattering and Absorption
under Strong Magnetic Field, calculated in Perturbative Way
Neutrinos are More Scattered and Less Absorbed
　　　　　　　　　　　　　　　　　　　　 in Direction Parallel to Magnetic Field
　　　⇒　More Neutrinos are Emitted in 　Arctic　 Area
Scattering %
Absorption % at ρB=3ρ0 and T = 20 MeV
　⇒　Convection, Pulsar-Kick

44. Pulsar Kick in Poroidal Mag. Field B = 2× 1017G ⇒ Perturbative Cal.
　　 vkick = 580 km/s ( p,n ) , 610 km/s (p,n,Λ） at T = 20 MeV
　 　　 = 230 km/s ( p,n ) , 270 km/s (p,n,Λ） at T = 30 MeV 　 　　
400 km/s (Average of Observed Values）　　　　
Spin Deceleration
　in Toroidal Magnetic Field
Asymmetric n-Emission plays an important role in PNS Spin in Early Time
Future Plans 　　n-Scattering
Fixed Temp. ⇒　Constant Entropy
Exact Solution of Dirac Eq. in Non-Perturbative Cal.
　 →　 Landau Level at least for Electron
Antarctic Dir.
Mag Distr.
(P = 10 s)
p,n
p,n,L
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
Reestimating
when B = 1015G
in the surface with r = r0 44

45. EOS in Iso-Entropical Model

46. §2-1 Neutron-Star Matter in RMF Approach
RMF Lagrangian, N, L, s, , r
Dirac Eq.
Scalar Field ⇒ Effective Mass

47. Dirac Spinor Spin Vector
negligibly small
Dirac Spinor
Spin Vector

48. Spin-indep. part
Spin-dep. Part

49. θ Neutrino Propagation Neutrinos propagate on the straight lines
2) Neutrino are created and absorbed at all positions on the lines
Mean-Free Path
: σab/V \ θ

50. Absorption Mean-Free Path
30MeV
SA: fitting function
50MeV
100MeV
200MeV
300MeV

51. §3 Results
3% difference

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

53. Magnetic Parts of Cross-Sections

54. Magnetic Parts of Cross-Sections