AsymmetricAsymmetric Neutrino Neutrino Reaction Reaction and and Pulsar Pulsar Kick Kick inin Magnetized Magnetized Proto Proto--NeutronNeutron Stars Stars inin Fully Fully Relativistic Relativistic Framework Framework

TomoyukiTomoyuki MaruyamaMaruyama BRS,BRS, Nihon Nihon Univ. Univ. (Japan) (Japan) ToshitakaToshitaka KajinoKajino NaoNao,, Univ. Univ. of of Tokyo Tokyo (Japan)(Japan) NobutoshiNobutoshi YasutakeYasutake Univ.Univ. of of Tokyo Tokyo (Japan) (Japan) MyungMyung--kiki CheounCheoun SoongsSoongs Univ.Univ. (Korea)(Korea) ChungChung--YeolYeol RyuRyu SoongsSoongs Univ.Univ. (Korea) (Korea) ++ JunJun Hidaka Hidaka NaoNao (Japan)(Japan) G.J.G.J. Mathews Mathews Univ.Univ. of of Notre Notre Dome Dome (USA) (USA)

TM et al., Phys. ReV. D83, 081303(R) (2011) 1 §§1.1. IntroductionIntroduction 2 High Density Matter Study ⇒ Exotic Phases : Strange Matter, Ferromagnetism Meson Condensation, Quark matter They may exist inside Neutron Stars Observable Information ‥‥Neutrino Emissions S.Reddy, M.Prakash and J.M. Lattimer, P.R.D58 #013009 (1998) Influence from Hyperons Λ，∑

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

Magnetic Field → Large Asymmetry ？

P. Arras and D. Lai, P.R.D60, #043001 (1999) Neutrino Scattering & Absorption S. Ando, P.R.D68 #063002 (2003) in Surface Region Non-Relativistic

Our Works ⇒ Neutrino Reactions on High Density Matter wih Strong Mag. Fields PulsarPulsar Kick 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 http://chandra.harvard.edu/photo/ (almost Neutrino Emissions) 2004/casa/casa_xray.jpg 1% Asymmetry are sufficient to explain the Pulsar Kick

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

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

Estimating Kick Velocity 3 §§2.2. FormulationFormulation 4 Magnetic 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) §§22--11 Neutron Neutron--StarStar Matter Matter in in RMF RMF Approach Approach

RMF Lagrangian, N, 

Dirac Eq.

Scalar Field ⇒ Effective Mass EOS of Proto Neutron-Star-Matter 6

PM1-L1 * -3  2 BE  16 MeV, M N / M  0.7, K  200 MeV at   0.17 fm N 0 g ,  g , SU(3) 3 T.M, et al. PTP. 102, Charge Neutral （    ） & Lepton Fraction : Y = 0.4 p809 (1999) p e L §§22--22 Dirac Dirac Equation Equation under under Magnetic Magnetic Fields Fields

N B << εN (Chem. Pot) → B can be treated perturbatively B ~ 1017 G Landau Level can be ignored

Magnetic Part of Lagrangian

Dirac Eq.

Fermi Distribution

Deformed Mom.- Distribution negligibly small

Dirac Spinor Spin Vector The Cross-Section of Lepton-Baryon Scattering

Perturbative Treatment σ  σ0  Δσ Δσ  B

Non-Magnetic Part Magnetic Part Spin-indep. part

Spin-dep. Part 17  §3 Results ki   (neutrino chem.pot.) , B  210 G and i  0

3% difference Magnetic parts of Cross-Sections σ  σ0  Δσ Δσ  B dσν  ν  σ  dΩ e e Sc  i dΩ f

Scat.

Increasing  Integrating over the initial angle to Arctic Dir. Absorp.

Integrating over the final angle

dσν  e σ  dΩ e Ab  f dΩ 12 f Magnetic Parts of Cross-Sections

0  S ,A   S ,A 1 SS ,A cosi, f  §4 Estimating Pulsar Kick Velocities of Proto-Neutron Star

Neutrino Phase Space Distribution Function

f ( p, r)  f0 ( p, r)  Δf ( p, r) , f0 ( p, r)  1 1 exp( p  ) /T 

Equib. Part Non-Equib. Part

Neutrino Propagation ⇒ Boltzmann Eq. d d σ c f ( p,r)  c Δf ( p,r)  I  cb Δf ( p,r) , b  ab dx 0 dx coll  ν V

Neutrinos propagate on the strait line only absorption

z     1 z  Δf ( p,rT , z)  dx  f0(p,rT ,x) exp  d yb (y) , 0    x  Solution ⇒  x   c 

 d  z  r  pˆ, f0(p,rT ,z)  f0(p,rT ,z) z dz  NeutrinoNeutrino PropagationPropagation 1) Neutrinos propagate on the straight lines

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

Mean-Free Path

: σab/V Baryon density in Proto-Neutron Star M = 1.68Msolar

YL = 0.4

Calculating Neutrino Propagation above B = 0 1 Absorption Mean-Free Path A   A /V 

   0 1 S cos  B = 0 A A A i, f 30MeV SA: fitting function

50MeV

100MeV

200MeV

300MeV Angular Dependence of Emitted Neutrino  p   drdppuˆΔf ( p,r)

Direction of Emitted Neutrino  pθ  P0  P1 cos

KickKick Velocity Velocity 2 4 2 Kick Velocity E  4R P , p  P T 0 z 3 1

p P z  1  1.0 10 -2 , 6.0 10 -3 p, n ET 3P0  9.9 10 -3 , 2.6 10 -3 p, n, 

T = 20 MeV T = 30 MeV Assuming Neutrino Energy ~ 3×1053 erg D.Lai & Y.Z.Qian, Astrophys.J. 495 (1998) L103

53 M NS 1.68M solar [g], ET  310 p v  z  300, 180[km/s] p, n kick M  290, 77[km/s] p, n,  19 20 §§55 SummarySummary

 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 Estimating Pulsar Kick in Proto-Neutron-Star with Strong Magnetic Field

B = 2× 1017G ⇒ Perturbative Calculations

Vkick = 300 km/s ( p,n ) , 290 km/s (p,n,Λ） at T = 20 MeV = 180 km/s ( p,n ) , 77 km/s (p,n,Λ） at T = 30 MeV Antarctic Direction 400 km/s (Average of Observed Values）

Future Plans Fixed Temp. ⇒ Constant Entropy Exact Solution of Dirac Eq. Non-Perturbative Cal.

Low Density Contribution is large E < 50 MeV → Landau Level at least for Electron

E ≈  Lower T & Smaller B Spin-Spin interaction Poloidal Mag. Field ⇒ Toroidal Mag. Field 21 Neutrino Mean-Free-Path at Energy equal to Chem. Potential Magnetic Parts of Cross-Sections