X-Ray image of Crab Nebula Pulsar Credit: NASA/CXC/ASU/J. Hester et al.) Liliana Caballero With Pf. Horowitz Molecular Dynamics Simulations of Non-uniform.

X-Ray image of Crab Nebula Pulsar Credit: NASA/CXC/ASU/J. Hester et al.)

Liliana CaballeroWith Pf. Horowitz

Molecular Dynamics Molecular Dynamics Simulations of Non-uniform Simulations of Non-uniform Dense Matter and Neutrino Dense Matter and Neutrino Interactions in SupernovaeInteractions in Supernovae

Indiana University

NNPSS June/13/2005

OutlineOutline

Introduction: Supernova and Neutron Introduction: Supernova and Neutron StarsStars

Ion ResponseIon ResponseNeutrino-Nucleus scattering Neutrino-Nucleus scattering ModelModelSimulation ResultsSimulation Results

Future WorkFuture Work

Supernova NeutrinosSupernova Neutrinos

Collapse of a Massive StarCollapse of a Massive Star

Gravitational Force vs. Nuclear FusionGravitational Force vs. Nuclear Fusion Iron Core -> energy barrier->collapseIron Core -> energy barrier->collapseMost of the energy is lost by neutrino emission: p+e -> n+Most of the energy is lost by neutrino emission: p+e -> n+ννeeDensity increment->Strong Force & Fermi Degeneracy->Outward Density increment->Strong Force & Fermi Degeneracy->Outward

PressurePressureNeutron StarNeutron Star

SN1987A

Image taken from Dany Pierre Page, Neutron Star (Theory) Group at UNAM

Mass= 1.4 Sun MassRadius=10 km

~10~101212 g/cm g/cm33

Medium= Plasma Electrons and NucleiMedium= Plasma Electrons and NucleiNeutrino Trapping in SupernovaNeutrino Trapping in Supernova

Neutrino-Nucleus elastic scatteringNeutrino-Nucleus elastic scattering Ion, Electron Screening have important effectIon, Electron Screening have important effect

Ion ResponseIon Response

Ion ResponseIon Response

Linear Response -> Neutrino-Nucleus Elastic Linear Response -> Neutrino-Nucleus Elastic Cross Section:Cross Section:

Correlation Function g(r)Correlation Function g(r) Static Structure Factor S(q)Static Structure Factor S(q) Dynamic Structure Factor S(q,w)Dynamic Structure Factor S(q,w)

FreedEd

dqS

dEd

d

),(

Correlation Function g(r)Correlation Function g(r)

Probability of finding another ion a distance r Probability of finding another ion a distance r from a given ion.from a given ion.

r

Static Structure FactorStatic Structure Factor

)exp(1)(1)( 3 rqirgrdqS

2

0000 )(ˆ)(ˆ)(ˆ1

)( qqqN

qS

N

iirqiq

1

)exp()(Charge Density

Ground State 0

ModelModel

Classical Approximation Classical Approximation Ions interact via screened Coulomb potentialIons interact via screened Coulomb potential

e

ij

ijij

r

r

eZrV

exp4

)(22

e electron screening

rij distance between Ions

c

e

ekFe

4

,2

ijB r

Monte Carlo SimulationMonte Carlo SimulationT= 1 MevT= 1 MevN ionsN ions 5656FeFe

ρρ =1x10 =1x1012 12 g/cmg/cm33

Periodic Boundary ConditionsPeriodic Boundary Conditions

rrii at equilibrium, i=1,N at equilibrium, i=1,N

jijiij xxLxxMinx ,

1

2

1

2

L

Correlation FunctionCorrelation FunctionN=1000 λe=10 fm L=382 fm

r (fm)

0 100 200 300 400

g (r

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Static Structure FactorStatic Structure FactorN=1000 λe=10 fm L=382 fm

q (fm-1)

0.0 0.2 0.4 0.6 0.8 1.0

S(q

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Dynamic ResponseDynamic Response

Dynamic Structure FactorDynamic Structure Factor

)0,(),()0,(),(1

),( qtqqtqN

tqS

dttqStiqS

),()exp(2

1),(

1

0

1111

),(),(1

)0,(),(t

dttqttqt

qtq

N

ii trqitq

1

))(exp(),(

Molecular Dynamics AlgorithmMolecular Dynamics Algorithm

Verlet AlgorithmVerlet Algorithm

tttattvttv

ttrVm

tta

ttatvttv

ttattvtrttr

)(2

1)2/()(

)(1

)(

)(2

1)()2/(

)(2

1)()()( 2

Molecular Dynamic SimulationMolecular Dynamic Simulation

T= 1 MeVT= 1 MeVN ionsN ions 5656FeFe

ρρ =1x10 =1x1012 12 g/ccg/ccPeriodic Boundary ConditionPeriodic Boundary Condition

rrii(t), i=1,N(t), i=1,N

Dynamic Structure FactorDynamic Structure FactorN=500 λe=10 fm L=304 fm

(fm-1)

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012

S(q

, )

0

200

400

600

800

1000

q=2/L

q=221/2/L

q=231/2/L

q=241/2/L

q (fm-1)q (fm-1) *10*10-4 -4 (fm-1)(fm-1) ΩΩpp*10*10-4-4(fm-1)(fm-1)

qq00=2=2ππ/L/L 3.33.3 4.24.2

qq00* 2* 21/21/2 4.54.5 5.75.7

qq00 *3 *31/21/2 5.15.1 6.96.9

qq00 *4 *41/21/2 66 7.87.8

22

222 4

e

ip q

q

MZc

PeaksPeaks

Future WorkFuture Work

Ion-Ion Screening model break down at Ion-Ion Screening model break down at large densitylarge density

N=100000 nucleons MDGRAPE N=100000 nucleons MDGRAPE Dynamical Response PastaDynamical Response Pasta