Neutron stars swollen under strong magnetic fields

Chung-Yeol RyuSoongsil University, Seoul, Korea

Vela pulsar

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

Motivation I- Heavy ion collisions

and neutron stars

Dense matter- Heavy ion collisions and neutron star

Neutron star

Constraints- Heavy ion collisions and neutron stars

T. Klahn et al., Phys. Rev. C 74, 035802 (2006) : np

Heavy Ion Collisions Neutron stars

The masses of neutron stars

The structure of neutron star

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 mat-ter, kaon condensa-

tion) ruled out ?

Motivation II- Magnetar

The histroy of soft gamma repeater (Mag-netar)

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 ~ 105 Be

c where Be

c is the electron critical field (4.414 x 10 13 G)

We need to investigate neutron star in strong magnetic fields with

strangeness.

Models- Magnetic fields

- Baryons (QHD)

- Kaons

The magnetic fields in neutron star

Magnetic flux conservation : ФM = < B > R2

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 = 104 G and R = R⊙

A neutron star with B = 1014 G for R = 10 km – surface magnetic field

B ∼ 2 x 108 (M/M⊙)(R⊙ /R)2

∼ 1018 G : the interior of the star

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

σ-ω-ρ model in nuclear matter

N N

Long range attraction (σ meson)+

Short range repulsion (ω me-son)

+ Isospin force : ρ meson

Other mesons are neglected !! pion : (-) parity, other mesons : small effects, simplic-ity

H

H

attraction (σ* meson) + repulsion (φ me-son)

QHD Lagrangian in magnetic field

Hadronic phase in magnetic fields- Quantum hadrodynamics (QHD)

Baryon octet, leptons and five meson fields

Energies for fermionsin strong magnetic fields

• Energy spectra for charged, neutral baryons and lep-tons

Here ν = n + ½ - sgn(q) s/2 = 0, 1, 2 … enumerates the Landau levels of charged fermions where s = +1 (↑) and s=-1 (↓).

Kaon fields in magnetic fields

Kaons under magnetic fields

where covariant derivative is

And the effective mass of a kaon is

The energy of an antikaon

Antikaon condensation under magnetic fields

S-wave condensation : ) )

Equation of state in magnetic fields

The energy density :

The pressure :

where

+ εK

The conditions in neutron star

1) Baryon number conservation :

2) Charge neutrality :

3) chemical equilibrium (Λ, Σ, Ξ)

μ n - μ p = μK

TOV equation

• Macroscopic part – General relativity

• Einstein field equa-tion :

Static and spherical symmetric neutron star (Schwarzschild metric)

Static perfect fluidDiag Tμν = (ε, p, p, p)

• TOV equation :

• equation of state (energy density, pressure)

Density dependent magnetic fields

Magnetic field from surface to interior in the star

B0 = B0* x Be

c

where

Bec = 4.414 x 10 13

G(electron critical magnetic field)

Thus, B0* is a free

parameter.

3. Results

Baryon octet (npH)

Populations of particles (npH)

Populations of particles (npH)

Populations of particles (npH)

Equation of state (npH)

Mass-radius relation (npH)

Baryon octet +

kaon condnesation (npHK-)

Populations of particles (npHK)

Populations of particles (npHK)

Populations of particles (npHK)

Equation of state (npHK)

Mass-radius relation (npHK)

In preparation

4. Summaries I1. The large masses (M > 2 M solar) of neutron stars in observa-

tions : Are all exotic phases like hyperons and kaon conden-sation ruled out ?

2. But we can explain them with exotic phases by considering very strong magnetic fields B ~ 1018 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.

Summaries II

6. If strong magnetic fields are possible in the center of a neu-tron star ,

proto-neutron stars may have strong magnetic fields which can cause pulsar kick through the emission of neu-trinos shown in pre-vious talk by Maruyama san.

Thank you for your attention !