Dynamical Simulations of Supernovae Collapse and Nuclear Collisions via the Test Particle Method - Similarities and Differences

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1 Proc. 19th Winter Workshop on Nuclear Dynamics (2003) th Winter Workshop on Nuclear Dynamics Breckenrige, Colorao, USA February 8 15, 2003 Dynamical Simulations of Supernovae Collapse an Nuclear Collisions via the Test Particle Metho - Similarities an Differences Wolfgang Bauer 1,a 1 Department of Physics an Astronomy an National Superconucting Cyclotron Laboratory Michigan State University East Lansing, MI , USA Abstract. Test particle methos have been applie very successfully to the numerical simulation of heavy ion reactions at intermeiate an high beam energies. Here we will show that the same techniques can be use successfully to simulate the ynamics of the collapse of type II supernovae precursors. We will focus special attention on the effects of collective angular momentum on the resulting supernova ynamics. Keywors: Heavy ion collision, test particle simulation, particle spectra, flow, interferometry, supernova, type II, precursor, phase space ynamics, angular momentum. PACS: i, Lx, z, q, x, Bw 1. Nuclear Dynamics Wong [1] showe in 1982 that it is possible to approximate the quantum nuclear many boy problem solution on the mean fiel level (TDHF) via a semiclassical approach base on test particles formulation. In this metho, one follows the initially (completely or partially) occupie cells in the 6-imensional phase space as a function of time. At about the same time, Cugnon an collaborators [2, 3] implemente ieas of intra-nuclear cascaes, i.e. an approximation without a nuclear mean fiel, in which two-boy collisions exclusively etermine the nuclear ynamics. Bertsch an collaborators [4 7] an then several other groups aroun the Worl [8 15] combine both ieas to prouce a solution of the nuclear transport equation

2 2 W. Bauer Fig. 1. Time evolution of a typical heavy ion collision at intermeiate energy. Shown here is the baryon ensity in the reaction plane for a 60 AMeV Au + Au collision at an impact parameter of 5 fm. The 4 frames shown were taken at times 0, 25, 50, an 100 fm/c. [19]. (Boltzmann-Uehling-Uhlenbeck, BUU, also VUU, BN,...), t f( r, p, t) + p m r f( r, p, t) r U p f( r, p, t) (1) g = 2π 3 m 2 3 q 1 3 q 2 3 q 2 ( ) 1 δ 2m (p2 + q2 2 q2 1 q2 2 ) δ 3 ( p + q 2 q 1 q 2 ) σ Ω { ( ) ( ) f( r, q 1, t) f( r, q 2, t) 1 f( r, p, t) 1 f( r, q 2, t) ( ) ( f( r, p, t) f( r, q 2, t) 1 f( r, q 1, t) 1 f( r, q 2, t)) }. in the test particle approximation. In this approximation the solution of the above integro-ifferential equation can be reuce to the simultaneous solution of N cou-

3 Dynamics of Supernovae 3 ple first-orer ifferential equations in time: t p i = U( r i ) + q i q j ( r i r j ) 2 + C( p i), j i t r i = p i, (2) m 2 i + p 2 i i = 1,..., (A t + A p )N, where A t an A p are the target an projectile masses, respectively, an N is the number of test particles per nucleon (usually taken > 100 to reuce artificially generate numerical fluctuations). The term C( p i ) represents the solution of the collision integral via an intranuclear cascae for the test particles. The test particle collisions respect the Pauli exclusion principle ue to the presence of the factors (1 f), which are numerically implemente via a Monte Carlo rejection metho. In our particular numerical realization the values of f( r, p, t) are store in a six-imensional lattice so that the computation of the factors (1 f( r, p, t)) only requires the call of 2 6 lattice elements for a six-imensional interpolation [18]. Fig. 1 shows a typical time evolution of a heavy ion collision that results from this approach. We have grown confient that this simulation captures the essentials of the nuclear ynamics, because BUU-type approaches have been increibly successful in reproucing all kins of experimental observables, such as the emission spectra of protons an neutrons, the coalescence of small fragments, the nuclear collective flow, an prouction of seconary particles (photons, pions, kaons,...), an even Hanbury-Brown-Twiss type interferometry [16, 17]. 2. Type II Supernovae We currently istinguish two types of supernova events. In type I supernovae, a white warf excees its Charasekhar Mass ( 1.4 M ) ue to accretion an collapses. We want to focus here on type II supernova events associate with the violent eath of stars at the en of their thermonuclear fuel cycle. These are powere by the gravitational energy release uring a star s late stage iron core collapse, cause by instabilities ue to electron capture (p + e n + ν e ) or photoisintegration ( 56 Fe + γ 14 4 He + 4n, 4 He + γ 2p + 2n). The first process ominates for precursor mass ranges between 10 an 20 solar masses ZAMS (= zero age main sequence; mass of the star at the beginning of its evolution), an the secon for masses in the range between 20 an 40 solar masses ZAMS. The numerical stuy of supernova explosions has been a mainstay of astrophysics for the last few ecaes, relying in hyroynamical coes that run on the largest available computer systems. It has turne out that these hyro simulations are of increible complexity, ue to the rapily changing length scales uring the collapse phase, ue to the changing viscosity as a function of time, ue to the relatively long evolution that one has to simulate (only a few millisecons in real time,

4 4 W. Bauer but a very large number of time steps in the computer), an ue to the coupling to the neutrino Boltzmann transport. In particular the moeling of the neutrinos has proven to be ifficult, an the outcome of the simulation (explosion or stalling of the shock) has turne out to be very sensitive on the etails of this part of the calculation. 3. Test Particle Approach for Supernova Dynamics Can the test particle approach iscusse in section 1 be utilize for the ynamics of core collapse simulations of type II supernovae? The answer is yes. We can re-cast the hyroynamical time evolution equations for the iron core matter in terms of test particle equations. The first question, however, is what a test particle is to represent in physical space. It we utilize a total number of N tp test particles in our simulations, an we consier a typical iron core mass of orer λm, then the mass of each test particle, m tp, is given by λm /N tp. Using 10 7 test particles, for example, we obtain typical test particle masses of 1/10 of that of Earth. The resulting test particle time evolution equations are: t p j = U EoS,e ( r j ) + F G,j ( r 1,..., r Ntp ) + C( p j ) + C ν ( p j ) t r p j j = m 2 tp + p2 j j = 1,..., N tp, (3) where F G,j enotes the force on particle j ue to gravity an F EOS,j the force ue to the equation of state. Gravity is moele using the Newtonian monopole approximation: { } m 2 tp # i {1,..., N tp } : r i < r j F G,j = G r j 3 r j. (4) This approximation is obviously only appropriate as long as the eviations from spherical symmetry are sufficiently small. As a realistic EOS, F EOS,j, for core collapse conitions we use a combination of the nuclear EOS by Lattimer & Swesty [20] an the Helmholtz EOS by Timmes (which is an EOS for the electron/positron gas) [21]. The former is use for ρ g/cm 3, the latter for ρ < g/cm 3 where the nuclear contribution to the pressure is negligible. The Term C( p j ) represents again the effects of the test-particle cascae, i.e. two-boy scattering events between test particles. During the initial phase of the collapse the velocities scale with the raial istance, an the effects of this two-boy scattering term is small, however, in the late phases, an in particular uring the shock formation, it becomes ominant.

5 Dynamics of Supernovae 5 The term C ν ( p j ) represent the effects of the scattering of neutrinos off the baryonic test particles. The corresponing time evolution equations for the neutrinos have a source term originating from two-particle collisions an form couple equations with the above baryon time evolution equations. This part of the ynamics is not implemente as of yet an will be work to be accomplishe in the near future. However, we anticipate that the formalism is in complete analogy to the implementation of couple transport equations for baryons, resonances, an mesons in relativistic heavy ion collisions [14]. 4. First Results In an initial implementation of our ieas, we have concentrate on investigating the effects of total angular momentum, i.e. the rotation of the precursor star, on the collapse ynamics [22, 23]. Stellar evolution calculations for a rotating progenitor one by Heger [24] inicate that it is a very goo approximation to assume that the inner core (initially) rotates like a rigi boy. Therefore we use a constant initial angular velocity ω 0. N tp = 10 6 test particles, the gri parameters N r = 110, N cos θ = 100, an a backgroun ensity ρ min = g/cm 3 were applie in all runs of the series. Fig. 2 shows the ensity profiles of three ifferent runs with ifferent values of the initial angular velocity (70, 160, an 220 s 1, from left to right). This correspons to a ratio of rotational to gravitational energy of 1.3%, 6.8%, an 13%, respectively. While the bounce times are relatively insensitive to this parameter, the maximum central ensity reache is strongly epenent on it. We reach ρ max = 2.35ρ 0 for the low angular momentum run, an values of 0.56ρ 0 an 0.21ρ 0 for the high angular momentum runs. 5. Conclusions Test particle methos have been applie very successfully to the numerical simulation of heavy ion reactions at intermeiate an high beam energies. Here we have shown that the same techniques can be use successfully to simulate the ynamics of the collapse of type II supernovae precursors. While neutrino transport is not fully implemente into our program, it seems clear that approaches base on test particle methos are excellent caniates to provie a fully 6-imensional phase space simulation of the supernovae collapse an subsequent explosion, incorporating in particular the effects of collective angular momentum. Acknowlegements This research was supporte by NSF grants PHY an INT , as well as by an A. v. Humbolt U.S. Distinguishe Senior Scientist awar.

6 6 W. Bauer Fig. 2. Mass ensity in a slice in the x-z-plane at (a) onset of simulation, (b) after 2 ms, (c) 3 ms, () core bounce, (e) shortly after core bounce. All plots have the same raius scale ( 120 km) inicate by the black line in the top left. From left to right, the initial angular velocities use in the simulation runs were 70, 160, an 220 s 1.

7 Dynamics of Supernovae 7 Notes a. bauer@pa.msu.eu Home Page: bauer/ References 1. C.Y. Wong, Phys. Rev. C 25 (1982) J. Cugnon, T. Mizutani, an J. Vanermeulen, Nucl. Phys. A 352 (1981) J. Cugnon, D. Kinet, an J. Vanermeulen, Nucl. Phys. A 379 (1982), G.F. Bertsch, H. Kruse, an S. Das Gupta, Phys. Rev. C 29 (1984) W. Bauer, G.F. Bertsch, W. Cassing, U. an Mosel, Phys. Rev. C 34 (1986) G.F. Bertsch an S. Das Gupta, Phys. Rep. 160 (1988) P. Danielewicz an G.F. Bertsch, Nucl. Phys. A 533 (1991) H. Kruse et al., Phys. Rev. Lett. 54 (1985) H. Stöcker an W. Greiner, Phys. Rep. 137 (1986) J. Cugnon an M.C. Lemaire, Nucl. Phys. A 489 (1988) P Schuck et al., Prog. Part. Nucl. Phys. 22 (1989) W. Cassing an U. Mosel, Prog. Part. Nucl. Phys. 25 (1990) B.A. Li an W. Bauer, Phys. Lett. B254 (1991) S.J. Wang et al., Ann. Phys. (N.Y.) 209 (1991) A. Ono et al., Prog. of Theor. Phys. 87 (1992) W. Bauer, C.K. Gelbke, an S. Pratt, Annu. Rev. Nucl. Part. Sci. 42 (1992) W. Bauer, Prog. in Part. an Nucl. Phys. 30 (1993) W. Bauer, Phys. Rev. Lett. 61 (1988) A color version of this figure is on the back cover of the proceeings of the 18th Winter Workshop on Nuclear Dynamics. 20. J. Lattimer, an F. Swesty, Nucl. Phys A, 535 (1991) 331; Equation of state version 2.7 (ls eos v2.7), F. Timmes, an F. Swesty, Astrophys. J. Suppl. S. 126 (2000) 501; fxt/eos.shtml. 22. T. Bollenbach, M.S. Thesis, Michigan State University (2002). 23. T. Bollenbach an W. Bauer, in Exotic Clustering, eite by S. Costa, A. Insolia, an C. Tùve, American Institute of Physics Conference Proceeings, Volume 644 (Melville, New York, 2002), p A. Heger, N. Langer, an S. Woosley, Astrophys. J., 528 (2000) 368.