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

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 #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

Birth of Proto-neutron Star 5 5

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 http://chandra.harvard.edu/photo/ 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

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 #013009 (1998)

§2-1 RMF Approach for Neutron-Star Mattter

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

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

§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.

Dirac Spinor Spin Vector 3 negligibly small Dirac Spinor Spin Vector 1414

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

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

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

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

Spin-dependent Part

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

Final electron contribution in ne → e-

§3 Results of Cross-Sections No magnetic field

Differential Cross-Sections in Magnetic Field 3% difference

Initial Angle-Dependence

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

Magnetic Parts of Cross-Sections

Neutrino Mean-Free Paths scattering absorption

Contribution of Each Element in Scattering Part

Contribution of Each Element in Absorption Part

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 1.7 % Absorption 2.2 % at ρB=3ρ0 and T = 20 MeV 30

§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　　 http://chandra.harvard.edu/photo/ 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

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 ⇒

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

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

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

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

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

§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

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

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

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

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

§5 Summary EOS of Neutron-Star-Matter with p, n, Λ in RMF Approach 43 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 1.7 % Absorption 2.2 % at ρB=3ρ0 and T = 20 MeV 　⇒　Convection, Pulsar-Kick

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

EOS in Iso-Entropical Model

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

Dirac Spinor Spin Vector negligibly small Dirac Spinor Spin Vector

Spin-indep. part Spin-dep. Part

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

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

§3 Results 3% difference

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

Magnetic Parts of Cross-Sections

Magnetic Parts of Cross-Sections