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

. . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . East Lansing, MI - April 2017

. . . . . . . . . . . . Introduction . Formalism Neutron Stars CCSN Nuclear EOS Equation of state (EOS): Astrophysical relevance Core-collapse supernovae; NS structure and evolution; Merger of compact stars; r -process nucleosynthesis; . . . . . . . . . . . . . . . . . . . . . . . . . . . … ⇒ ε ( n , y , T ) , P ( n , y , T ) , s ( n , y , T ) , …

. . . . . . . . . . . . . . . . Introduction Formalism Neutron Stars CCSN Nuclear EOS . . . . . . . . . . . . . . . . . . . . . . . . Physics Reports 621 (2016) 127–164

. . . . . . . . . . . . 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; . . . . . . . . . . . . . . . . . . . . . . . . . . . …

. 1 4 G. Shen et al. 3 H. Shen et al. 2 Latuimer and Swesty Hot-dense EOS available: 5 Nuclear EOS CCSN Neutron Stars Formalism Introduction . Hempel et al. Steiner et al. . Realistic potentials vs Muons? Hyperons? 4 networks SNA vs NSE vs reaction 3 Efgective potentials 2 6 Non-relativistic Relativistic vs 1 Classifjcation: … 7 Banik et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qvarks? Nuclear forces are complicated ⇒ combine difgerent approaches!

. 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)

. Tie Latuimer & Swesty EOS . . . . . . . . Introduction Formalism Neutron Stars CCSN 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).

. . . . . . . . . . . . Introduction . Formalism Neutron Stars CCSN Tie Latuimer & Swesty EOS 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 n α : alpha particle density

. . . . . . . . . . . . . . . Introduction Formalism Neutron Stars CCSN Tie Latuimer & Swesty EOS Heavy nuclei free energy: Tie EOS of each component: interaction. . . . . . . . . . . . . . . . . . . . . . . . . . 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 Nucleons ⇒ local phenomenological Skyrme-type efgective Alpha particles ⇒ hard spheres. Electrons, positrons and photons ⇒ background gas.

. Energy density of bulk nuclear matuer with Skyrme-type interactions . . . . . . . . Formalism Neutron Stars CCSN Tie Latuimer & Swesty EOS h 2 . n h 2 p i t h 2 t h 2 2 m t 1 h 2 2 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E B ( n , y , T ) = ¯ τ n + ¯ τ p +( a + 4 by ( 1 − y )) n 2 2 m ∗ 2 m ∗ ( c i + 4 d i y ( 1 − y )) n 1 + δ i − yn ( m n − m p ) . + ∑ Nucleon efgective mass m ∗ ¯ = ¯ + α 1 n t + α 2 n − t . 2 m ∗ where t = n ⇒ − t = p and vice versa, n n = ( 1 − y ) n and n p = yn . Nucleon kinetic energy density τ t ) 5 ( 2 m ∗ τ t = F 3 / 2 ( η t ( n , y )) , t T 2 π 2 ¯

. . . . . . . . . . . . . . Introduction Formalism Neutron Stars CCSN Tie Latuimer & Swesty EOS t 0 t 0 t 3 i t 3 i 1 . . . . . . . . . . . . . . . . . . . . . . . . . . a = 4 ( 1 − x 0 ) , b = 8 ( 2 x 0 + 1 ) , c i = 24 ( 1 − x 3 i ) , d i = 48 ( 2 x 3 i + 1 ) , δ i = σ i + 1 , α 1 = 1 8 [ t 1 ( 1 − x 1 )+ 3 t 2 ( 1 + x 2 )] , α 2 = 8 [ t 1 ( 2 + x 1 )+ t 2 ( 2 + x 2 )] .

. . . . . . . . . . . . . . Introduction Formalism Neutron Stars CCSN Nuclear surface and Coulomb free energies r and r : generalized nuclear size where 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 4 πα F S = 3 s ( u ) σ ( y i , T ) F C = 5 ( y i n i r ) 2 c ( u ) . s ( u ) : surface shape function c ( u ) : Coulomb shape function σ ( y i , T ) : surface tension per unit area Nuclear virial Tieorem: F S = 2 F C ] 1 / 3 ] 1 / 3 [ πα r = 9 σ [ s ( u ) ( y i n i σ ) 2 / 3 β = 9 2 β c ( u ) c ( u ) s ( u ) 2 ] 1 / 3 ≡ β D ( u ) . [ F S + F C = β

. . . . . . . . . . . . . . Introduction Formalism Neutron Stars CCSN Nuclear surface and Coulomb free energies where 2 u intermediate u : reproduces free energy of pasta phases: cylinders; slabs; . . . . . . . . . . . . . . . . . . . . . . . . . . cylindrical holes. Latuimer & 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 ) 2 + 0 . 6 u 2 ( 1 − u ) 2 2 u 1 / 3 + 1 D ( u ) = 1 − 3 as u → 0 reproduces free energy of spherical nuclei as u → 1 reproduces free energy of “bubble nuclei”

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

. . . . . . . . . . . . . . . . . Formalism Neutron Stars CCSN Nuclear surface and Coulomb free energies . Introduction . . . . . . . . . . . . . . . . . . . . . . Surface Tension σ ( y , T ) . 0.8 0.8 n p n ni n n 0.6 0.6 n/n 0 0.4 0.4 n pi ∆ R 0.2 0.2 n no n po 0.0 0.0 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 z (fm)

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

. . . . . . . . . . . . Introduction . Formalism Neutron Stars CCSN Nuclear surface and Coulomb free energies where 1 2 and 3 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . − ∞ [ ] q nn ( ∇ n n ) 2 + q np ∇ n n · ∇ n p + q pn ∇ n p · ∇ n n + q pp ( ) 2 ∇ n p E S ( z ) = q nn = q pp = 16 [ t 1 ( 1 − x 1 ) − t 2 ( 1 + x 2 )] , q np = q pn = 16 [ 3 t 1 ( 2 + x 1 ) − t 2 ( 2 + x 2 )] .

. . . . . . . . . . . . Introduction . Formalism Neutron Stars CCSN Solving the EOS where u u n i u n i . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimize F ( n , y , T ) w.r.t. u , r , n ni , n pi , n no , n po , and n α . 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 F , ∂ u − n i ∂ n i [ y i ∂ F − ∂ F ] B 2 = 1 , ∂ y i ∂ n i [ 1 − y i ] ∂ F + ∂ F B 3 = − 1 , ∂ y i ∂ n i with F = F S + F C + F T .

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