Sub-Nuclear Matter in Neutron Stars and Sub-Nuclear Matter in Neutron Stars and SupernovaeSupernovae

Nuclear Pasta and Complex FluidsNuclear Pasta and Complex Fluids

W.G.NewtonW.G.Newton11, J.R.Stone, J.R.Stone1,21,2

11University of Oxford, UKUniversity of Oxford, UK22Physics Division, ORNL, Oak Ridge, TN, USAPhysics Division, ORNL, Oak Ridge, TN, USA

OutlineOutline

Overview of NS, SN MatterOverview of NS, SN Matter Anatomy of Supernovae (SNe) and Neutron Stars (NSs)Anatomy of Supernovae (SNe) and Neutron Stars (NSs) Superfluidity in NSsSuperfluidity in NSs

The Transition to Uniform MatterThe Transition to Uniform Matter Astrophysical consequencesAstrophysical consequences Frustration and Complex FluidsFrustration and Complex Fluids Nuclear PastaNuclear Pasta

Self-consistent modelsSelf-consistent models QMDQMD Hartree-FockHartree-Fock

A new Hartree-Fock study of nuclear pastaA new Hartree-Fock study of nuclear pasta Computational MethodComputational Method Preliminary ResultsPreliminary Results

ConclusionsConclusions

AnatomyAnatomy of Core Collapse of Core Collapse SNSN

Collapse proceeds until Collapse proceeds until core reaches few times core reaches few times nuclear saturation nuclear saturation density (density (≈2.4×10≈2.4×101414g cmg cm-3-3 or 0.16 baryons fmor 0.16 baryons fm-3-3))

Neutrinos initially Neutrinos initially trapped above densities trapped above densities of of ((≈10≈101212g cmg cm-3-3), ), temperatures reach up temperatures reach up to ≈100MeV and the to ≈100MeV and the proton fraction is roughly proton fraction is roughly constant at ≈0.3constant at ≈0.3

Anatomy of a Neutron StarAnatomy of a Neutron Star

<10<101111 g cm g cm-3-3 Nuclear physics relatively well known (heavy nuclei) Nuclear physics relatively well known (heavy nuclei) >4×10>4×101111 g cm g cm-3-3 (neutron drip): nuclear models begin to diverge (neutron drip): nuclear models begin to diverge >5×10>5×101414 g cm g cm-3-3 physics is extremely uncertain (Hyperons? Meson physics is extremely uncertain (Hyperons? Meson

condensates? Quarks? When does description in terms of condensates? Quarks? When does description in terms of nucleonic degrees of freedom become unphysical?nucleonic degrees of freedom become unphysical?

Superfluids in Neutron StarsSuperfluids in Neutron Stars

At temperatures below a critical temperature of At temperatures below a critical temperature of ≈≈ 10 1099K - K - 10101010K, neutrons in the inner crust and core are expected to K, neutrons in the inner crust and core are expected to become superfluid (and, in the core, protons become superfluid (and, in the core, protons superconducting).superconducting).

Superfluids have zero viscosity, and so cannot support Superfluids have zero viscosity, and so cannot support bulk rotation.bulk rotation.

If a fluid, rotating with period P(s), is cooled below the If a fluid, rotating with period P(s), is cooled below the critical temperature, it arranges itself into quantized critical temperature, it arranges itself into quantized vortices of spin, density 10vortices of spin, density 104 4 /P cm/P cm-2-2

Superfluid VorticesSuperfluid Vortices

Quantized vortices in a sodium gas cooled into a Bose-Einstein Quantized vortices in a sodium gas cooled into a Bose-Einstein condensate and set into rotation (Onofrio et al, Phys Rev Lett 85, 2228, condensate and set into rotation (Onofrio et al, Phys Rev Lett 85, 2228, 2001)2001)

Transition to Uniform MatterTransition to Uniform Matter

The density regime 1013 < ρ < 2×1014 g/cm3 is important

It marks the transition from the outer crystalline crust of a NS, or the gas of nuclei, neutrons and leptons in a core collapse, to the liquid, homogeneous phase above nuclear saturation density

The Transition to Uniform Matter: The Transition to Uniform Matter: Astrophysical ConsequencesAstrophysical Consequences

Neutrino opacities and emission Neutrino opacities and emission mechanismsmechanisms during core collapse during core collapse neutron star coolingneutron star cooling

Pulsar GlitchesPulsar Glitches star-quakes star-quakes superfluid vortex dynamicssuperfluid vortex dynamics

Pinned vortices? Pinned vortices? Change in crustal composition and Change in crustal composition and

reheating during accretionreheating during accretion NS OscillationNS Oscillation

GWsGWs

Frustration and ComplexityFrustration and Complexity If a system contains energetically favourable (attractive) and If a system contains energetically favourable (attractive) and

unfavourable (repulsive) interactions operating over the same unfavourable (repulsive) interactions operating over the same range, matter will be range, matter will be frustrated.frustrated.

Prototypical frustrated Prototypical frustrated system: Ising anti-system: Ising anti-ferromagnet on triangular ferromagnet on triangular lattice.lattice.

- Impossible to minimize Impossible to minimize energy with respect to all energy with respect to all interactions simultaneouslyinteractions simultaneously

- Large number of low energy Large number of low energy configurations resultconfigurations result

• At densities just below nuclear saturation (1013 – 1014 g cm-

3) the distances between Coulomb repelling nuclei becomes comparable with the range of the attractive nuclear interaction that binds nuclei. Complex structures thus develop – nuclear pasta.

Nuclear PastaNuclear Pasta

• Competition between surface tension and Coulomb repulsion of closely spaced heavy nuclei results in a series of shape transitions from the inner crust to the core (Ravenhall et al Phys. Rev. Lett. 50, 2066, 1983 and Hashimoto et al, Progress of Th. Physics, 71, 2, 320, 1984).• The basic sequence is(a) spherical (meatball/gnocchi) → (b) rod (spaghetti) → (c) slab (lasagna) → (d) tube (penne) → (e) bubble (swiss cheese?) → uniform matter

Nuclear Pasta vs Complex Nuclear Pasta vs Complex FluidsFluids

•A wide range of mechanical properties are exhibited (liquid crystal, sponge, rubber…)•Pethick, C.J. and Potekhin, A.Y. – Liquid Crystals in the Mantles of Neutron Stars – Phys. Lett. B, 427, 7, 1998

Self-consistent Modeling: QMDSelf-consistent Modeling: QMD

0.1ρ0 0.175ρ0 0.35ρ0 0.5ρ0 0.55ρ0

Quantum Molecular Dynamics (QMD): semi-Quantum Molecular Dynamics (QMD): semi-classical dynamical simulations with nucleonic classical dynamical simulations with nucleonic degrees of freedom (Watanabe and Sonoda, nucl-degrees of freedom (Watanabe and Sonoda, nucl-th/0512020).th/0512020).

Pasta shapes emerge without pre-conditioning.Pasta shapes emerge without pre-conditioning. Pasta formation from compression and cooling Pasta formation from compression and cooling demonstrated.demonstrated.

Self-consistent Modeling: Mean Self-consistent Modeling: Mean fieldfield

Magierski and Heenen Magierski and Heenen PRC65 045804 (2001): 3D PRC65 045804 (2001): 3D HF calculation of nuclear HF calculation of nuclear shapes at bottom of shapes at bottom of neutron star crust at zero Tneutron star crust at zero T

When examined self-When examined self-consistently in three consistently in three dimensions, many more dimensions, many more configurations emergeconfigurations emerge

- has effect of smoothing - has effect of smoothing EoSEoS

An important new An important new phenomenon emerges: the phenomenon emerges: the fermionic Casimir effectfermionic Casimir effect. . Scattering of unbound Scattering of unbound nucleons off nuclear nucleons off nuclear structures leads to an structures leads to an effective interaction effective interaction between those structures of between those structures of order the energy difference order the energy difference between configurationsbetween configurations

Computational Method: Skyrme HFComputational Method: Skyrme HF Choose phenomenological nuclear interaction (Skyrme)Choose phenomenological nuclear interaction (Skyrme) Assume one can identify (local) unit rectangular cells of matter at Assume one can identify (local) unit rectangular cells of matter at

a given density and temperature, calculate one unit cell a given density and temperature, calculate one unit cell containing A nucleons (A up to 5000)containing A nucleons (A up to 5000)

Hartree-Fock approximation: system can be represented by a Hartree-Fock approximation: system can be represented by a single Slater determinant.single Slater determinant.

Minimize energy w.r.t. single particle wavefunctions: SchrMinimize energy w.r.t. single particle wavefunctions: Schrödinger ödinger equation for A nucleons → A equation for A nucleons → A SchrSchrödinger equations (A up to 5000)ödinger equations (A up to 5000)

Periodic boundary conditions Periodic boundary conditions φφ(x,y,z) = (x,y,z) = φφ(x+L,y+L,z+L) (More (x+L,y+L,z+L) (More generally Bloch boundary conditions generally Bloch boundary conditions φφ(x,y,z) = e(x,y,z) = eiikrkr φφ(x+L,y+L,z+L))(x+L,y+L,z+L))

Impose parity conservation in the three dimensions: tri-axial Impose parity conservation in the three dimensions: tri-axial shapes allowed, but not asymmetric ones.shapes allowed, but not asymmetric ones. Solution only in one octant of cellSolution only in one octant of cell

Additional free parameters: A, (proton fraction yAdditional free parameters: A, (proton fraction ypp), proton and ), proton and neutron quadrupole moments Qneutron quadrupole moments Qp,20p,20, Q, Qp,22p,22

Unconstrained calculation at Unconstrained calculation at 8 densities between 0.01fm8 densities between 0.01fm--

33 and 0.12fm and 0.12fm-3-3, T=0MeV, , T=0MeV, yypp=0.03: Self-consistent =0.03: Self-consistent

dissolution of nuclear dissolution of nuclear structurestructure

IntegratedIntegrated Densities at n Densities at nbb = 0.0195fm = 0.0195fm-3-3

Integrated Densities at nIntegrated Densities at nbb = 0.0312fm = 0.0312fm-3-3

Integrated Densities at nIntegrated Densities at nbb = 0.0390fm = 0.0390fm-3-3

Integrated Densities at nIntegrated Densities at nbb = 0.0507fm = 0.0507fm-3-3

Integrated Densities at nIntegrated Densities at nbb = 0.0585fm = 0.0585fm-3-3

Integrated Densities at nIntegrated Densities at nbb = 0.0702fm = 0.0702fm-3-3

Integrated Densities at nIntegrated Densities at nbb = 0.0780fm = 0.0780fm-3-3

Integrated Densities at nIntegrated Densities at nbb = 0.0976fm = 0.0976fm-3-3

Minimization with respect to Minimization with respect to AA

T = 2.5MeV, nb=0.04fm-3

• Minimization with respect to quadrupole moments is obtained in a similar way

T=5MeVT=5MeVnnbb=0.12fm=0.12fm-3-3

Boundary Conditions Boundary Conditions and Shell Effectsand Shell Effects

• Pasta phase superimposesartificial and real oscillations,and real minima, on the curve

Conclusions and FutureConclusions and Future The properties of matter in the density The properties of matter in the density

region region 1013 < ρ < 2×1014 g/cm3 are an are an important ingredient in NS and SN models important ingredient in NS and SN models

Thorough microphysical description of Thorough microphysical description of transition to uniform matter – the nuclear transition to uniform matter – the nuclear pasta phases – is underwaypasta phases – is underway Generalize boundary conditions to the Bloch Generalize boundary conditions to the Bloch

form: form: φφ(x,y,z) = e(x,y,z) = eiikrkr φφ(x+L,y+L,z+L)(x+L,y+L,z+L) Calculate entrainment coefficientCalculate entrainment coefficient Examine response of matter to perturbationExamine response of matter to perturbation

neutrino interactionsneutrino interactions mechanical propertiesmechanical properties

Investigate effects of BCS pairingInvestigate effects of BCS pairing The Future(?)The Future(?)

Hydrodynamical modeling of pasta phasesHydrodynamical modeling of pasta phases Mesoscopic structuresMesoscopic structures