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 210 G and i 0 3% difference Magnetic parts of Cross-Sections σ σ0 Δσ Δσ B dσν 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 dσν e σ dΩ e Ab f dΩ 12 f Magnetic Parts of Cross-Sections 0 S ,A S ,A 1 SS ,A cosi, 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 drdppuˆΔf ( p,r) Direction of Emitted Neutrino pθ P0 P1 cos KickKick Velocity Velocity 2 4 2 Kick Velocity E 4R 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 310 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.