An EOS implementation for astrophyisical simulations

Transcription

1 Introduction Formalism Neutron Stars CCSN An EOS implementation for astrophyisical simulations A S Schneider 1, L F Roberts 2, C D Ott 1 1 TAPIR, Caltech, Pasadena, CA 2 NSCL, MSU, East Lansing, MI East Lansing, MI - April 2017

2 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;

3 Introduction Formalism Neutron Stars CCSN Nuclear EOS Physics Reports 621 (2016)

4 Introduction Formalism Neutron Stars CCSN Nuclear EOS Long-standing problem in nuclear physics Combines efforts from: heavy ion collision experiments; nuclear reaction experiments; computer simulations of astrophysical phenomena; computer simulations of dense matter; theoretical many-body calculations;

5 Nuclear EOS Nuclear forces are complicated combine different approaches! Hot-dense EOS available: 1 Lattimer and Swesty 2 H Shen et al 3 G Shen et al 4 Hempel et al 5 Steiner et al 6 Banik et al 7 Classification: 1 Relativistic vs Non-relativistic 2 Realistic potentials vs Effective potentials 3 SNA vs NSE vs reaction networks 4 Muons? Hyperons? Quarks?

6 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 matter constraints become available 3 Make the code open-source (soon)

7 The Lattimer & Swesty EOS The 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)

8 The Lattimer & Swesty EOS Free energy F(n,y,T) = F o + F h + F α + F e + F γ F o nucleons outside heavy nuclei (nucleon gas) F h 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 n ni : neutron density inside heavy nuclei n pi : proton density inside heavy nuclei n no : neutron density outside heavy nuclei n po : proton density outside heavy nuclei n α : alpha particle density

9 The Lattimer & Swesty EOS Heavy nuclei free energy: F h = F i + F S + F C + F T F i nucleons inside heavy nuclei F S surface free energy F C coulomb free energy F T translational free energy The EOS of each component: Nucleons local phenomenological Skyrme-type effective interaction Alpha particles hard spheres Electrons, positrons and photons background gas

10 The Lattimer & Swesty EOS Energy density of bulk nuclear matter with Skyrme-type interactions E B (n,y,t) = h2 2m τ n + h2 n 2m τ p + (a + 4by(1 y))n 2 p + (c i + 4d i y(1 y))n 1+δ i yn(m n m p ) i Nucleon effective mass m t h 2 2m t = h2 2m t + α 1 n t + α 2 n t where t = n t = p and vice versa, n n = (1 y)n and n p = yn Nucleon kinetic energy density τ t τ t = 1 2π 2 ( 2m t T h 2 ) 5 2 F3/2 (η t (n,y)),

11 The Lattimer & Swesty EOS a = t 0 4 (1 x 0), b = t 0 8 (2x 0 + 1), c i = t 3i 24 (1 x 3i), d i = t 3i 48 (2x 3i + 1), δ i = σ i + 1, α 1 = 1 8 [t 1(1 x 1 ) + 3t 2 (1 + x 2 )], α 2 = 1 8 [t 1(2 + x 1 ) + t 2 (2 + x 2 )]

12 Nuclear surface and Coulomb free energies F S = 3s(u) σ(y i,t) and F C = 4πα r 5 (y in i r) 2 c(u) s(u): surface shape function c(u): Coulomb shape function r: generalized nuclear size σ(y i,t): surface tension per unit area Nuclear virial Theorem: F S = 2F C r = 9σ [ ] s(u) 1/3 [ ] πα 1/3 where β = 9 (y i n i σ) 2/3 2β c(u) 15 F S + F C = β [ c(u)s(u) 2] 1/3 βd(u)

13 Nuclear surface and Coulomb free energies Lattimer & Swesty interpolate D(u) with D(u) = u(1 u) (1 u)d(u)1/3 + ud(1 u) 1/3 u 2 + (1 u) u 2 (1 u) 2 where D(u) = u1/ u 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

14 Introduction Formalism Neutron Stars CCSN Nuclear surface and Coulomb free energies Lim (2012)

15 Nuclear surface and Coulomb free energies Surface Tension σ(y,t) n ni n p n n n/n n pi R z (fm) n no n po

16 Nuclear surface and Coulomb free energies Set temperature T and proton fraction y i Solve equilibrium equations: P i = P o, µ ni = µ no, and µ pi = µ po, and y i = n pi n ni + n pi Set density to t = n,p n t (z) = n to + n ti n to 1 + exp((z z t )/a t )

17 Nuclear surface and Coulomb free energies Find z t and a t that minimizes + [ ] σ(y i,t) = F B (z) + E S (z) + P o µ no n n (z) µ po n p (z) dz where [q nn ( n n ) 2 + q np n n n p + q pn n p n n + q pp ( np ) 2 ] E S (z) = 1 2 and q nn = q pp = 3 16 [t 1(1 x 1 ) t 2 (1 + x 2 )], q np = q pn = 1 16 [3t 1(2 + x 1 ) t 2 (2 + x 2 )]

18 Solving the EOS Minimize F(n,y,T) wrt u, r, n ni, n pi, n no, n po, and n α where A 1 = P i B 1 P o P α = 0, A 2 = µ ni B 2 µ no = 0, A 3 = µ pi B 3 µ po = 0 B 1 = F u n i F, u n i B 2 = 1 [ yi F F u n i y i n i B 3 = 1 [ 1 yi u n i ], F + F y i n i ], with F = F S + F C + F T

19 Solving the EOS Constraints n = un i + (1 u)[4n α + n o (1 n α v α )], ny = un i y i + (1 u)[2n α + n o y o (1 n α v α )] µ α = 2(µ no + µ po ) + B α P o v α, r = 9σ [ ] s(u) 1/3, 2β c(u)

20 Solving the EOS log 10 [T (MeV)] log 10 [T (MeV)] (P/n) NRAPR y = 001 y = 010 y = 030 y = log 10 [n (fm 3 )] log 10 [n (fm 3 )] (MeV baryon 1 )

21 Solving the EOS

22 Nuclear Statistical Equilibrium Consider ensemble of nuclei at low densities Given an ensemble of nuclei i, solve for µ n and µ p such that µ i = m i + E c,i + Tlog = Z i µ p + (A i Z i )µ n that minimizes the free energy of the system [ ( ) ] n i 2π 3/2, g i m i T

23 Nuclear Statistical Equilibrium L&S EOS is obtained in the single nucleus approximation (SNA) Properties in the SNA can differ significantly from observed properties of nuclei No shell closure; No pairing; liquid drop model neglects many-body effects; Conversely, NSE breaks down close to nuclear saturation density n 0 016fm 3 ; Needs very large and very neutron rich nuclei at low y and/or high n n 0 ; No nuclear inversion (pasta phase)

24 Nuclear Statistical Equilibrium Use ad-hoc procedure to mix NSE and SNA free energies: Chose a(n) such that: F MIX = χ(n)f SNA + [1 χ(n)]f NSE χ(n) 0, if n n 0 χ(n) 1, if n n 0 /10 Corrections to thermodynamic quantities, eg P MIX = n 2 (F MIX/n) n = χ(n)p SNA + [1 χ(n)]p NSE + n 2 χ(n) T,y n (F SNA F NSE ) EOS is self consistent!

25 High density extension Skyrme parametrizations only constrained up to n 3n 0 NS maximum mass depend on EOS at n 10n 0 Most Skyrme EOS unable to reproduce NS maximum mass, M max 2M Add extra c i, d i, and δ i terms to Skyrme interactions: ε B (n,y,t) = h2 2m τ n + h2 n 2m τ p + (a + 4by(1 y))n 2 p + (c i + 4d i y(1 y))n 1+δ i yn(m n m p ) i Extra terms should: barely affect EOS for n 3n 0 ; increase M max 2M Problem: may imply in (c s /c) 1 for n (6 10)n 0

26 Introduction Formalism Neutron Stars CCSN Final EOS log 10 [n (fm 3 )] NSE EoS NSE + SNA SNA EoS High-ρ EoS

27 Introduction Formalism Neutron Stars CCSN Skyrme parametrizations Dutra et al [Phys Rev C (2012)] Analyzed over 240 Skyrme parametrizations available in the literature Only 11 fulfill all well established nuclear physics constraints! Not all 11 reproduce M max 2M! We produced hot-dense EOS tables for a few of these parametrizations

28 Skyrme parametrizations Average nuclear size Ā along s = 1k B baryon LS220 NRAPR SLy4 SkT1 SKRA LNS Skxs20 SQMC700 KDE0v1 NSE 23 NSE 837 NSE 3335 Ā y = 001 y = Ā 50 0 y = 030 y = log 10 [n(fm 3 )] log 10 [n(fm 3 )]

29 NS mass-radius relationship 20 PSR J Mgrav (M ) LS220 LS220 NRAPR SLy4 SkT1 SKRA LNS Skxs20 SQMC700 KDE0v1 Nättilä et al QMC+model A R (km)

30 NS mass-radius relationship 20 PSR J Mgrav (M ) SkT1 Modified SkT1: δ 2 = 4, c 2 = 0 δ 2 = 4, c 2 = d 2 δ 2 = 4, d 2 = 0 δ 2 = 5, c 2 = 0 δ 2 = 5, c 2 = d 2 δ 2 = 5, d 2 = 0 Nättilä et al QMC+model A R (km)

31 NS Structure 14M log 10 [ρ (g cm 3 )] LS220 NRAPR SLy4 SkT1 SKRA LNS Skxs20 SQMC700 KDE0v1 010 y R (km)

32 NS Structure M max 155 log 10 [ρ (g cm 3 )] y LS220 NRAPR SLy4 SkT1 SKRA LNS Skxs20 SQMC700 KDE0v R (km)

33 Spherically symmetric collapse of a 15M star ρc (10 14 g cm 3 ) LS220 NRAPR SLy4 SkT1 SkT1 SKRA LNS Skxs20 SQMC700 KDE0v1 Tc (MeV) t t bounce (s) 1

34 Spherically symmetric collapse of a 15M star Rs (km) SNA Ṁ300 (M s 1 ) Ās t t bounce (s) 1

35 Spherically symmetric collapse of a 40M star ρc (10 14 g cm 3 ) LS220 NRAPR SLy4 SkT1 SkT1 SKRA LNS Skxs20 SQMC700 KDE0v1 Tc (MeV) t t bounce (s) 1

36 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 stiffening of EOS for n 3n 0 Code converges for large region of parameter space Temperatures 10 4 MeV T MeV; Proton fractions 10 3 y 070; Densities fm 3 n 10fm 3 Successfully generated many new EOS tables to study CCSNe, NS mergers,

37 Introduction Formalism Neutron Stars CCSN Future Near future: publish results and make code open source; study EOS effects on CCSN and NS mergers EOS may affect 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;