1. Neutron stars swollen under strong magnetic fields Chung-Yeol Ryu Soongsil University, Seoul, Korea Vela pulsar

2. Outline 1. Motivations - Equation of state from Heavy ion collisions and neutron stars - Magnetar (neutron star with strong B fields) 2. Model for neutron star in strong magnetic fields - Baryons (QHD) - Kaons and kaon condensation 3. Results 4. Summaries

3. Motivation I -Heavy ion collisions and neutron stars

4. Dense matter - Heavy ion collisions and neutron star Neutron star

5. Constraints - Heavy ion collisions and neutron stars T. Klahn et al., Phys. Rev. C 74, 035802 (2006) : np Heavy Ion Collisions Neutron stars

6. The masses of neutron stars

7. The structure of neutron star

8. Theoretical calculations 1) No strangeness : 2-3 M ๏ 2) Strangeness : 1-2 M ๏ - hyperons(Λ, Σ, Ξ) - kaon condensation - quark matter(u, d, s) The masses of neutron stars are very large. Are all exotic phases (hyperons, quark matter, kaon condensation) ruled out ?

9. Motivation II -Magnetar

10. The histroy of soft gamma repeater (Magnetar)

11. The observed magnetars and candidates C.Y.Cardall, M. Prakash, J.M.Lattimer, Astrophys. Jl. 554, 322 (2001) Surface magnetic field of neutron star : B ~ 10 14 – 10 15 G Interior magnetic field from scalar virial theorem : B ~ 10 18 G ~ 10 5 B e c where B e c is the electron critical field (4.414 x 10 13 G) We need to investigate neutron star in strong magnetic fields with strangeness.

12. Models - Magnetic fields - Baryons (QHD) - Kaons

13. The magnetic fields in neutron star Magnetic flux conservation : Ф M = R 2 Scalar virial theorem for non-rotating star : T + W + 3П + M = 0 Where T : Kinetic energy, W : Gravitational potential, П : Internal energy, M : magnetic energy. A core of superonvae with B = 10 4 G and R = R ⊙ A neutron star with B = 10 14 G for R = 10 km – surface magnetic field B ∼ 2 x 10 8 (M/M ⊙ )(R ⊙ /R) 2 ∼ 10 18 G : the interior of the star

14. Landau quantization under strong magnetic fields Lorentz force In quantum mechanics, the orbits of charged particles are quantized and then charged particles can be confined in strong magnetic fields.  Landau quantization

15. σ-ω-ρ model in nuclear matter N N Long range attraction (σ meson) + Short range repulsion (ω meson) + Isospin force : ρ meson Other mesons are neglected !! pion : (-) parity, other mesons : small effects, simplicity H H attraction (σ* meson) + repulsion (φ meson)

16. QHD Lagrangian in magnetic field Hadronic phase in magnetic fields - Quantum hadrodynamics (QHD) Baryon octet, leptons and five meson fields

17. Energies for fermions in strong magnetic fields Energy spectra for charged, neutral baryons and leptons Here ν = n + ½ - sgn(q) s/2 = 0, 1, 2 … enumerates the Landau levels of charged fermions where s = +1 (↑) and s=-1 (↓).

18. Kaon fields in magnetic fields Kaons under magnetic fields where covariant derivative is And the effective mass of a kaon is

19. The energy of an antikaon Antikaon condensation under magnetic fields S-wave condensation : ) )

20. Equation of state in magnetic fields The energy density : The pressure : where + ε K

21. The conditions in neutron star 1)Baryon number conservation : 2)Charge neutrality : 3)chemical equilibrium (Λ, Σ, Ξ) μ n - μ p = μ K

22. TOV equation Macroscopic part – General relativity Einstein field equation : Static and spherical symmetric neutron star (Schwarzschild metric) Static perfect fluid Diag T μν = (ε, p, p, p) TOV equation : equation of state (energy density, pressure)

23. Density dependent magnetic fields Magnetic field from surface to interior in the star B 0 = B 0 * x B e c where B e c = 4.414 x 10 13 G (electron critical magnetic field) Thus, B 0 * is a free parameter.

24. 3. Results

25. Baryon octet (npH)

26. Populations of particles (npH)

29. Equation of state (npH)

30. Mass-radius relation (npH)

31. Baryon octet + kaon condnesation (npHK - )

32. Populations of particles (npHK)

35. Equation of state (npHK)

36. Mass-radius relation (npHK) In preparation

37. 4. Summaries I 1.The large masses (M > 2 M solar) of neutron stars in observations : Are all exotic phases like hyperons and kaon condensation ruled out ? 2.But we can explain them with exotic phases by considering very strong magnetic fields B ~ 10 18 G. 3.We assume very strong magnetic fields due to scalar virial theorem. 4.Strong magnetic fields cause the Landau quantization of charged particles. 5.Hyperons and kaon condensation with strong magnetic fields can explain around 2 M solar.

38. Summaries II 6.If strong magnetic fields are possible in the center of a neutron star, proto-neutron stars may have strong magnetic fields which can cause pulsar kick through the emission of neutrinos shown in previous talk by Maruyama san.

39. Thank you for your attention !