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An EOS implementation for astrophyisical simulations A. S. Schneider - - PowerPoint PPT Presentation

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. . . . . . . . . . . . Introduction . Formalism Neutron Stars CCSN An EOS implementation for astrophyisical simulations A. S. Schneider 1 , L. F. Roberts 2 , C. D. Otu 1 1 TAPIR, Caltech, Pasadena, CA 2 NSCL, MSU, East Lansing, MI

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Introduction Formalism Neutron Stars CCSN

An EOS implementation for astrophyisical simulations

  • A. S. Schneider1,
  • L. F. Roberts2,
  • C. D. Otu1

1TAPIR, Caltech, Pasadena, CA 2NSCL, MSU, East Lansing, MI

East Lansing, MI - April 2017

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Introduction Formalism Neutron Stars CCSN

Nuclear EOS

Equation of state (EOS): ⇒ ε(n,y,T), P(n,y,T), s(n,y,T), … Astrophysical relevance

Core-collapse supernovae; NS structure and evolution; Merger of compact stars; r-process nucleosynthesis; …

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Introduction Formalism Neutron Stars CCSN

Nuclear EOS

Physics Reports 621 (2016) 127–164

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Introduction Formalism Neutron Stars CCSN

Nuclear EOS

Long-standing problem in nuclear physics. Combines efgorts from: heavy ion collision experiments; nuclear reaction experiments; computer simulations of astrophysical phenomena; computer simulations of dense matuer; theoretical many-body calculations; …

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Introduction Formalism Neutron Stars CCSN

Nuclear EOS

Nuclear forces are complicated ⇒ combine difgerent approaches! Hot-dense EOS available:

1

Latuimer and Swesty

2

  • H. Shen et al.

3

  • G. Shen et al.

4

Hempel et al.

5

Steiner et al.

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Banik et al.

7

… Classifjcation:

1

Relativistic vs Non-relativistic

2

Realistic potentials vs Efgective potentials

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SNA vs NSE vs reaction networks

4

Muons? Hyperons? Qvarks?

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Introduction Formalism Neutron Stars CCSN

Nuclear EOS

Goals:

1

Write code to construct EOS tables for astrophysical simulations.

2

Easy to update EOS as new nuclear matuer constraints become available.

3

Make the code open-source. (soon)

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Introduction Formalism Neutron Stars CCSN

Tie Latuimer & Swesty EOS

Tie Lattimer & Swesty EOS [Nucl. Phys. A 535, 331 (1991)] Most used EOS for simulations of CCSNe and NS mergers. Non-relativistic compressible liquid-drop description of nuclei. Contains

1

Nucleons;

2

alpha particles;

3

electrons and positrons;

4

photons. Nucleons may cluster to form nuclei. LS EOS use the single nucleus approximation (SNA).

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Introduction Formalism Neutron Stars CCSN

Tie Latuimer & Swesty EOS

Free energy F(n,y,T) = Fo +Fh +Fα +Fe +Fγ Fo ≡ nucleons outside heavy nuclei (nucleon gas) Fh ≡ nucleons clustered into heavy nuclei In this work, F depends on seven variables: u: volume fraction occupied by heavy nuclei r: generalized size of heavy nuclei nni: neutron density inside heavy nuclei npi: proton density inside heavy nuclei nno: neutron density outside heavy nuclei npo: proton density outside heavy nuclei nα: alpha particle density

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Introduction Formalism Neutron Stars CCSN

Tie Latuimer & Swesty EOS

Heavy nuclei free energy: Fh = Fi +FS +FC +FT Fi ≡ nucleons inside heavy nuclei FS ≡ surface free energy FC ≡ coulomb free energy FT ≡ translational free energy Tie EOS of each component: Nucleons ⇒ local phenomenological Skyrme-type efgective interaction. Alpha particles ⇒ hard spheres. Electrons, positrons and photons ⇒ background gas.

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Introduction Formalism Neutron Stars CCSN

Tie Latuimer & Swesty EOS

Energy density of bulk nuclear matuer with Skyrme-type interactions EB(n,y,T) = ¯ h2 2m∗

n

τn + ¯ h2 2m∗

p

τp +(a+4by(1−y))n2 +∑

i

(ci +4diy(1−y))n1+δi −yn(mn −mp). Nucleon efgective mass m∗

t

¯ h2 2m∗

t

= ¯ h2 2mt +α1nt +α2n−t. where t = n ⇒ −t = p and vice versa, nn = (1−y)n and np = yn. Nucleon kinetic energy density τt τt = 1 2π2 (2m∗

t T

¯ h2 ) 5

2

F3/2(ηt(n,y)),

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Introduction Formalism Neutron Stars CCSN

Tie Latuimer & Swesty EOS

a = t0 4 (1−x0), b = t0 8 (2x0 +1), ci = t3i 24(1−x3i), di = t3i 48(2x3i +1), δi = σi +1, α1 = 1 8 [t1(1−x1)+3t2(1+x2)] , α2 = 1 8 [t1(2+x1)+t2(2+x2)] .

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Introduction Formalism Neutron Stars CCSN

Nuclear surface and Coulomb free energies

FS = 3s(u) r σ(yi,T) and FC = 4πα 5 (yinir)2c(u). s(u): surface shape function c(u): Coulomb shape function r: generalized nuclear size σ(yi,T): surface tension per unit area Nuclear virial Tieorem: FS = 2FC r = 9σ 2β [s(u) c(u) ]1/3 where β = 9 [πα 15 ]1/3 (yiniσ)2/3 FS +FC = β [ c(u)s(u)2]1/3 ≡ βD(u).

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Introduction Formalism Neutron Stars CCSN

Nuclear surface and Coulomb free energies

Latuimer & Swesty interpolate D(u) with D(u) = u(1−u)(1−u)D(u)1/3 +uD(1−u)1/3 u2 +(1−u)2 +0.6u2(1−u)2 where D(u) = 1− 3

2u1/3 + 1 2u

as u → 0 reproduces free energy of spherical nuclei as u → 1 reproduces free energy of “bubble nuclei” intermediate u: reproduces free energy of pasta phases:

cylinders; slabs; cylindrical holes.

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Introduction Formalism Neutron Stars CCSN

Nuclear surface and Coulomb free energies

Lim (2012)

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Introduction Formalism Neutron Stars CCSN

Nuclear surface and Coulomb free energies

Surface Tension σ(y,T). 0.0 0.2 0.4 0.6 0.8

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1 2 3 4 nni nno npi npo ∆R n/n0 z (fm) np nn 0.0 0.2 0.4 0.6 0.8

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1 2 3 4

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Introduction Formalism Neutron Stars CCSN

Nuclear surface and Coulomb free energies

Set temperature T and proton fraction yi. Solve equilibrium equations: Pi = Po , µni = µno , and µpi = µpo , and yi = npi nni +npi Set density to nt(z) = nto + nti −nto 1+exp((z−zt)/at) t = n,p.

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Introduction Formalism Neutron Stars CCSN

Nuclear surface and Coulomb free energies

Find zt and at that minimizes σ(yi,T) =

∫ +∞

−∞

[ FB(z)+ES(z)+Po − µnonn(z)− µponp(z) ] dz. where ES(z) = 1 2 [ qnn (∇nn)2 +qnp∇nn ·∇np +qpn∇np ·∇nn +qpp ( ∇np )2 ] and qnn = qpp = 3 16 [t1(1−x1)−t2(1+x2)] , qnp = qpn = 1 16 [3t1(2+x1)−t2(2+x2)] .

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Introduction Formalism Neutron Stars CCSN

Solving the EOS

Minimize F(n,y,T) w.r.t. u, r, nni, npi, nno, npo, and nα. A1 = Pi −B1 −Po −Pα = 0, A2 = µni −B2 − µno = 0, A3 = µpi −B3 − µpo = 0. where B1 = ∂F ∂u − ni u F ∂ni , B2 = 1 u [yi ni ∂F ∂yi − ∂F ∂ni ] , B3 = −1 u [1−yi ni ∂F ∂yi + ∂F ∂ni ] , with F = FS +FC +FT.

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Introduction Formalism Neutron Stars CCSN

Solving the EOS

Constraints n = uni +(1−u)[4nα +no(1−nαvα)], ny = uniyi +(1−u)[2nα +noyo(1−nαvα)]. µα = 2(µno + µpo)+Bα −Povα , r = 9σ 2β [s(u) c(u) ]1/3 ,

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Introduction Formalism Neutron Stars CCSN

Solving the EOS

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log10[T (MeV)] y = 0.01

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1 2 10−2 10−1 100 101 102 103 (P/n)NRAPR (MeV baryon−1) y = 0.10 10−2 10−1 100 101 102 103 (P/n)NRAPR log10[T (MeV)] log10[n (fm−3)] y = 0.30

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10−2 10−1 100 101 102 103 (P/n)NRAPR log10[n (fm−3)] y = 0.50

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10−2 10−1 100 101 102 103 (P/n)NRAPR

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Introduction Formalism Neutron Stars CCSN

Solving the EOS

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Introduction Formalism Neutron Stars CCSN

Nuclear Statistical Equilibrium

Consider ensemble of nuclei at low densities. Given an ensemble of nuclei i, solve for µn and µp such that µi = mi +Ec,i +Tlog [ ni gi ( 2π miT )3/2] , = Ziµp +(Ai −Zi)µn that minimizes the free energy of the system.

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Introduction Formalism Neutron Stars CCSN

Nuclear Statistical Equilibrium

L&S EOS is obtained in the single nucleus approximation (SNA). Properties in the SNA can difger signifjcantly from observed properties of nuclei. No shell closure; No pairing; liquid drop model neglects many-body efgects; … Conversely, NSE breaks down close to nuclear saturation density n0 ≃ 0.16fm−3; Needs very large and very neutron rich nuclei at low y and/or high n ∼ n0; No nuclear inversion (pasta phase).

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Introduction Formalism Neutron Stars CCSN

Nuclear Statistical Equilibrium

Use ad-hoc procedure to mix NSE and SNA free energies: FMIX = χ(n)FSNA +[1− χ(n)]FNSE. Chose a(n) such that: χ(n) →0, if n ≪ n0 χ(n) →1, if n ≲ n0/10 Corrections to thermodynamic quantities, e.g.

PMIX = n2 ∂(FMIX/n) ∂n

  • T,y

= χ(n)PSNA +[1− χ(n)]PNSE +n2 ∂χ(n) ∂n (FSNA −FNSE).

EOS is self consistent!

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Introduction Formalism Neutron Stars CCSN

High density extension

Skyrme parametrizations only constrained up to n ≲ 3n0. NS maximum mass depend on EOS at n ∼ 10n0. Most Skyrme EOS unable to reproduce NS maximum mass, Mmax ∼ 2M⊙. Add extra ci, di, and δi terms to Skyrme interactions: εB(n,y,T) = ¯ h2 2m∗

n

τn + ¯ h2 2m∗

p

τp +(a+4by(1−y))n2 +∑

i

(ci +4diy(1−y))n1+δi −yn(mn −mp). Extra terms should: barely afgect EOS for n ≲ 3n0; increase Mmax ∼ 2M⊙. Problem: may imply in (cs/c) ≳ 1 for n ∼ (6−10)n0.

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Introduction Formalism Neutron Stars CCSN

Final EOS

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1 log10[n (fm−3)]

NSE EoS NSE + SNA SNA EoS High-ρ EoS

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Introduction Formalism Neutron Stars CCSN

Skyrme parametrizations

Dutra et al. [Phys. Rev. C 85 035201 (2012)] Analyzed over 240 Skyrme parametrizations available in the literature. Only 11 fulfjll all well established nuclear physics constraints! Not all 11 reproduce Mmax ∼ 2M⊙! We produced hot-dense EOS tables for a few of these parametrizations.

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Introduction Formalism Neutron Stars CCSN

Skyrme parametrizations

Average nuclear size ¯ A along s = 1kB baryon−1

50 100 150 50 100 150

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¯ A LS220 NRAPR SLy4 SkT1 SKRA LNS Skxs20 SQMC700 KDE0v1 NSE 23 NSE 837 NSE 3335 50 100 150

y = 0.01 y = 0.10

¯ A log10[n(fm−3)] 50 100 150

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y = 0.30

log10[n(fm−3)]

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y = 0.50

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Introduction Formalism Neutron Stars CCSN

NS mass-radius relationship

0.0 0.5 1.0 1.5 2.0 6 7 8 9 10 11 12 13 14 15 16 Mgrav (M⊙) R (km)

LS220∗ LS220 NRAPR SLy4 SkT1 SKRA LNS Skxs20 SQMC700 KDE0v1 PSR J0348+0432 N¨ attil¨ a et al. QMC+model A

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Introduction Formalism Neutron Stars CCSN

NS mass-radius relationship

0.0 0.5 1.0 1.5 2.0 6 7 8 9 10 11 12 13 14 15 16 Mgrav (M⊙) R (km) SkT1 Modified SkT1: δ2 = 4, c2 = 0 δ2 = 4, c2 = d2 δ2 = 4, d2 = 0 δ2 = 5, c2 = 0 δ2 = 5, c2 = d2 δ2 = 5, d2 = 0

PSR J0348+0432 N¨ attil¨ a et al. QMC+model A

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Introduction Formalism Neutron Stars CCSN

NS Structure

1.4M⊙

13.5 14.0 14.5 15.0 0.00 0.05 0.10 2 4 6 8 10 12 log10[ρ (g cm−3)]

LS220 NRAPR SLy4 SkT1 SKRA LNS Skxs20 SQMC700 KDE0v1

13.5 14.0 14.5 15.0 y R (km) 0.00 0.05 0.10 2 4 6 8 10 12

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Introduction Formalism Neutron Stars CCSN

NS Structure

Mmax

13.5 14.0 14.5 15.0 15.5 0.00 0.05 0.10 0.15 0.20 2 4 6 8 10 log10[ρ (g cm−3)]

LS220 NRAPR SLy4 SkT1 SKRA LNS Skxs20 SQMC700 KDE0v1

13.5 14.0 14.5 15.0 15.5 y R (km) 0.00 0.05 0.10 0.15 0.20 2 4 6 8 10

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Introduction Formalism Neutron Stars CCSN

Spherically symmetric collapse of a 15M⊙ star

3 4 5 6 7 8 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ρc (1014g cm−3)

LS220 NRAPR SLy4 SkT1 SkT1∗ SKRA LNS Skxs20 SQMC700 KDE0v1

3 4 5 6 7 8 Tc (MeV) t − tbounce (s) 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 1.2

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Introduction Formalism Neutron Stars CCSN

Spherically symmetric collapse of a 15M⊙ star

60 80 100 120 140 1 2 3 52 54 56 58 0.00 0.05 0.10 0.15 0.20 Rs (km) SNA 23 82 206 837 3335 60 80 100 120 140 ˙ M300 (M⊙ s−1) 1 2 3 ¯ As t − tbounce (s) 52 54 56 58 0.00 0.05 0.10 0.15 0.20

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Introduction Formalism Neutron Stars CCSN

Spherically symmetric collapse of a 40M⊙ star

4 8 12 16 20 20 40 60 80 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ρc (1014g cm−3)

LS220 NRAPR SLy4 SkT1 SkT1∗ SKRA LNS Skxs20 SQMC700 KDE0v1

4 8 12 16 20 Tc (MeV) t − tbounce (s) 20 40 60 80 0.0 0.2 0.4 0.6 0.8 1.0 1.2

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Introduction Formalism Neutron Stars CCSN

Summary

Generalize L&S formalism to obtain hot dense EOS for most Skyrme parametrizations. Improved calculation of surface properties. Added smooth ad-hoc transition from SNA to NSE EOS. Extended formalism to allow stifgening of EOS for n ≳ 3n0. Code converges for large region of parameter space.

Temperatures 10−4MeV ≲ T ≲ 102.5MeV; Proton fractions 10−3 ≲ y ≲ 0.70; Densities 10−13fm−3 ≲ n ≲ 10fm−3.

Successfully generated many new EOS tables to study CCSNe, NS mergers, …

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Introduction Formalism Neutron Stars CCSN

Future

Near future: publish results and make code open source; study EOS efgects on CCSN and NS mergers ⇒ EOS may afgect neutrino and GW emissions; perform 2D and 3D simulations; add an improvement treatment of neutron skins to the EOS; add reaction network treatment for low temperatures/densities; …