Nuclear “Pasta” in Compact Stars

Hidetaka Sonoda

University of Tokyo Theoretical Astrophysics Group

Collaborators (G. Watanabe, K. Sato, K. Yasuoka, T. Ebisuzaki)

Content

• Introduction

• Quantum Molecular Dynamics (QMD)

• Pasta Phases at zero and finite temperatures

• Neutrino opacity of Pasta phases

• Summary

Supernovae and Nuclear “Pasta”

Core-collapse Supernova Explosion

Neutrino transport in supernova coresEOS of dense matter

No successful simulation with realistic settings

Nonspherical nuclei --- “Pasta” phases

Possible key element

“Bounce” triggered by nuclear repulsive force

Scenario

Just before bounce (just before nuclear matter phase)

Neutron Stars and Nuclear “Pasta”Neutron Stars

Pasta phases in the deep inside inner crust

Core

Outer Crust

Inner crust

Pasta Phases

10 km1 km

Solid of heavy nuclei

Liquid of nuclear matter(quark matter, hyperons)

Transition region from nuclei to nuclear matter

What is Nuclear “Pasta” ?

(K.Oyamatsu, Nucl.Phys.A561,431(1993))

Nonspherical nuclei in dense matter ～ 1014g/cc

Sphere→ Rod → Slab → Rod-like Bubbles → Spherical Bubbles →Uniform Nuclear Matter

MeatballSpaghettiLasagnaAnti-spaghettiCheese→”Pasta” Phases

(Ravenhall et al. 1983,Hashimoto et al.1984)

Phase Diagram of Pasta Phases

Motivation

How pasta phases appear in collapsing cores ?And in cooling neutron stars?How transition from sphere to uniform matter ?

Pasta phases are dynamically formed as equilibrium-state of hot dense matter in supernovae ? as ground-state in neutron stars ?

Why QMD ?

Quantum Molecular Dynamics (QMD) gives us a picture for

How nuclei are deformed into uniform nuclear matter

No assumptions on nuclear shapes.Nuclear system is treated in degrees of freedom of nucleons.Thermal fluctuations are included.

QMD is suitable to answer the above question

Quantum Molecular Dynamics

Model Hamiltonian 1（ Chikazumi et al　 Phys.Rev.C 63 024602(2001))

Pauli Potential Nuclear Force Coulomb EnergyKinetic Energy

Nucleons obey Equation of Motion of QMD

Saturation properties of symmetric nuclear matterBinding energy and rms radius of stable nuclei

Hamiltonian is constructed to reproduce …

Model Hamiltonian 2（ Maruyama et al　 Phys.Rev.C 57 655(1998))

Simulation settings

2048 or 10976 nucleons in simulation boxPeriodic boundary conditionProton fraction x=0.3

Simulation Settings

Ground state is obtained by cooling of hot matter

Equilibrium state at finite temperature is obtained by Nose-Hoover thermostat for MD pot.

Pasta at zero temperature

Sphere Rod Slab

Spherical Bubbles

0.100ρ 00.200ρ 0 0.393ρ 0

0.575ρ 000.490ρRod-like Bubbles

Red : ProtonsBlue: Neutrons

ρ =0.168 fm-3

（ Nuclear density ）

0

Cooling of hot nuclear matter (~10 MeV) below 0.1 MeV

Sponge-like StructureBetween rod and slab, slab and rod-like bubblesMultiply connected “Sponge-like” structure appears

10976 nucleons at 0.3ρ0

Between rod and slab

10976 nucleons at 0.45ρ0

Between slab and rod bubbles

These intermediate phases at least meta-stable

Phase diagram at zero temperature

Model 1

Model 2

SP

SP

C

C

S SH

SHS

CH

CH

SP: sphereC: cylinder

S: slabCH: cylindrical hole

SH: spherical hole

Uniform

Uniform

(C,S)

SP&C coexist. (S,CH) CH,SH coexist.

(C,S)

SP&C 共存 (S,CH)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ρ/ρ ０

ρ/ρ ０

(a)

(b)

（密度）

（密度）

( , ): intermediateSphere→Rod → Slab → Rod-like bubbles → Spherical bubbles →Uniform matter

Pasta at finite temperatures

0.393ρ0 (Slab nuclei at zero temperature)

Evaporated Neutrons Connected Slab

T= 2 MeVT= 1 MeVT= 0 MeV

Slab Nuclei

Increasing dripped neutronsDiffusive nuclear surface

Pasta at finite temperature

Cannot identifynuclear surface

Phase separationdisappears

Rodlike Bubble-like structure

T=3MeV T=5MeV T=6MeV

Phase transition, Melting surface, Dripped protons,Disappearance of phase separation

Phase diagram at finite temperatures

SP

1

2

3

45

6

7

8

910

T (MeV)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80ρ/ρ0

SP ： SphereC ： CylinderS ： SlabCH ： C bubbleSH ： S bubble( , ) ： Intermediate

Phase separation line (T=6 ~ 10 MeV)

Surface line(T=4 ~ 6 MeV)

Thermal fluctuation increases volume fraction of nucleiAbove T= 4 ~ 6 MeV, cannot identify surfaceAt T= 6 ~ 10 MeV, Liquid-gas phase separation

SH

(C,S)

CS CH

(S,CH)

Phase separation

Summary of Phase Diagram

• Performed simulation of nuclear matter at sub-nuclear densities with QMD

• Pasta Phases are obtained by QMD Ground-state by cooling hot matter Equilibrium-state of hot matter• How structure of nuclear matter change in the

density-temperature plane is examined

Neutrino Opacity of Pasta Phases

Motivation

Neutrino transport --- a key element for success of supernovae

How pasta phases change neutrino transportin collapsing cores ?

Neutrinos are trapped in collapsing phaseLepton fraction affects EOS

Cross section of neutrino-Pasta

Cross section of neutrino-nucleon system coherent scattering

Neutrino-neutron cross section

Amplification factor(Static structure factor)

Total transport cross section

→Amplification factor by structure

Method

1. Comparison cases with and without pasta phases using BBP liquid drop model

2. Show the results obtained by QMD as realistic model

Prediction by Liquid Drop Model

Energy of neutrino (MeV)

Red: with PastaBlack: without Pasta

Peak at 30~40 MeVPeak monotonically decreasesBelow 25 MeV incoherent

T=0 MeV・ YL=0.3

Amplification factor

Existence of Pasta phases increases peak energy, and decreases opacity at lower energy

QMD resultsYe=0.3, ρ=0.0660fm-3 (Slab at T=0)

・ Peak is lowered by increasing temp.

・ Transition from slab to rod-like bubbles dramatically changes peak energy and peak height

T= 1 MeV T= 3 MeV

Phase transitions can largely change neutrinoopacity with low energy (~25-30 MeV)

Summary of neutrino opacity

• Pasta phases decrease neutrino opacity at low energy

• Phase transitions at finite temperatures complicate neutrino opacity

Summary

• Pasta phases appear with QMD simulation

• How nuclei are deformed into uniform nuclear matter has been examined

• Pasta phases decrease neutrino opacity at low energy side

• Phase transition at finite temperature complicate neutrino opacity