Kinetic Theory for Supernova Explosions

Transcription

1 Proc. Int. Workshop on Hot and Dense Matter in Relativistic Heavy Ion Collisions (2004) Budapest 2004 International Workshop Budapest, Hungary March 24 27, 2004 Kinetic Theory for Supernova Explosions Wolfgang Bauer National Superconducting Cyclotron Laboratory, and Department of Physics and Astronomy, Michigan State University East Lansing, MI , USA URL: bauer/ Abstract. Coupled kinetic equations for the time evolution of baryons and neutrinos are introduced in order to present a new solution approach for the problem of type II supernova core collapse and explosion. A test-particle method is introduced in order to cast these kinetic equations into numerically tractable form. 1. Introduction The modeling of type II supernova core collapse and subsequent explosion events is one of the most challenging tasks in the field of computational physics today. Through decades of work, the delayed shock mechanism suggested by [1] is now the accepted paradigm for core collapse supernovae. The success of this explosion mechanism (or lack of success!) crucially depends on the cross sections for neutrino capture on and scattering off nucleons and nuclei, the neutrino production rates, as well as convection behind the stalled shock. Depending on these ingredients simulations of the delayed shock mechanism yielded successful explosions [2,3] while they failed in other recent cases [4 7]. Several two dimensional simulations have been carried out in the last ten years in which especially the impact of convection between the neutrino sphere and the stalled shock has been studied [2, 3, 8, 9]. It is widely believed today that convection in this region helps the success of the explosion by making a larger fraction of the energy converted to neutrinos available. [10] recently extended their simulations to three dimensions and found that their results remain qualitatively the same as in two dimensions. Why the simulation of supernovae is such a complicated task is sketched in Fig. 1. Here we show a representative iron core in a gravitational potential. Obviously, before collapse this system is gravitationally self-bound. A reduction of the Fermi pressure of the electron gas due to neutrino capture leads to the onset of collapse. However, this alone is not enough to create the explosion. For the explosion to be successful, much of the baryonic matter needs to remain in a highly compact state, for example a neutron star. This frees up energy to eject some of the matter. However, we need to also reserve enough energy in this accounting to provide for the copious production of photons and neutrinos, several foes (1 foe = erg), plus the binding energy it takes to transform the iron nuclei onto ISBN c 2004 EP Systema, Budapest

2 66 W. Bauer Fig. 1. Schematic representation of the energy considerations in supernova explosions or lack thereof. heavier or lighter nuclei. Thus a rather delicate redistribution of the energy between the baryons is necessary. It has be argued that a reason for the inability of many of the most realistic recent simulations to produce an explosion lies in a partly incorrect treatment of the underlying microphysics [11]. However, it is also conceivable that phenomena that can only occur in three dimensions play a key role in the explosion. Moreover, the impact of angular momentum on the core collapse and the explosion mechanism has been studied relatively little. Some studies came to the result that rotation does not have dramatic effects on the explosion mechanism [3,9,12]. Other effects may include an incorrect, non-self-consistent, or missing treatment of the magnetic fields generated and their influence on the motion of matter. What all present simulations have in common is the utilization of hydrodynamical simulations for the baryon matter. It is exceedingly complicated to couple these hydro simulations to the Boltzmann transport for neutrinos. This has not been achieved yet in a full three-dimensional simulations. The purpose of this present note is to rethink this approach in a fundamental way. 2. Kinetic Theory To introduce our kinetic theory considerations, we first refer to the many body wave function of the complete iron core of the star. From this we will construct the n-body density operator and a recursion relation for it in terms of a hierarchy. A physically motivated truncation of this hierarchy will provide an equation that we can solve in the test-particle

3 Kinetic Theory 67 approximations, which we will introduce in the following section. We will also see the relationship to the commonly used hydrodynamic formalism, as well as the relative ease with which the coupled transport problem including the neutrino transport can then be addressed. Our treatment here will follow closely the formalism for a transport theory for mixed species we have developed for relativistic heavy ion collisions [13]. The n-body baryon density operator ˆρ n is defined as ˆρ n (1,2, n;1,2, n ) = ψ (1 )ψ (2 ) ψ (n )ψ(n) ψ(1), (1) with n = x n = (t, r n,m n ), n = x n = (t, r n,m n ), (2) where m n is the spin isospin quantum number and we have used that t = t. For this density matrix, we can derive the recursion relation ˆρ n = 1 ˆN n Tr 1 (n+1) ˆρ n+1 = Tr (n+1) ˆρ n+1 ˆN n. (3) So far this is a completely general formalism that applies to any many-body system. A choice of an effective Hamiltonian fixed the equation of motion for the density matrix, i ˆρ n t = [Ĥ h (n), ˆρ n ]+Tr (n+1) [ ˆV (n + 1), ˆρ n+1 ]. (4) Here Ĥ h (n) contains the one-body gravitational and electron gas mean field potential and two-body baryon baryon, baryon photon and baryon neutrino interactions, and is thus specific to the problem of supernova collapse and explosion. A similar equation can be written down for the neutrino density matrix. This equation of motion, together with the above density matrix recursion relations, leads to an equation of motion for the one-body density matrix as a function of the two-body density matrix, and an equation of motion of the two-body density matrix as a function of the three-body density matrix. This hierarchy of equations of motion needs to be truncated in order to obtain closed sets of equation that can be solved numerically. Truncation to lowest order yields the TDHF equations. However, this approximation is totally insufficient for the present problem, because the dynamics is dominated by two-body collisions. Thus we need to employ a truncation on the next level. This derivation is analogous to the one performed in [13] for the coupled system of nucleons, pions, and delta resonances. We can now introduce the Wigner transformation for the one-body baryon and neutrino density matrices, the phase space distribution functions for baryons and neutrinos, fˆ b (r, p,t) = ρ(r + s 2,m;r s 2,m ) e i p s ds, (5) fˆ ν (r, k,t) = ρ ν (r + s 2,m;r s 2,m ) e i p s ds. (6) A rather lengthy derivation leads to the equations of motion for these phase space distribution functions: f b (xp) + i t Eb (p) x i f b (xp) µ E b (p) x i U µ(x) i p f b(xp)+ M b E b (p) x i U s i p f b(xp) = Ibb b (xp)+ I bν b (xp) (7)

4 68 W. Bauer for the particular state b of the baryon. For any neutrino species we have a simpler equation of motion that only contains the streaming term and the collision term, but no mean field contributions, f ν (xk) + k x t E ν (k) f ν(xk) = Ibν ν (xk). (8) The baryon baryon collision term, Ibb b, is given by Ibb b (xp) = π Mb (2π) 9 dp 1 dp 2 dp M α 1 Mα 2 Mα 3 3 Eb E α 1 Eα 2 Eα (9) 3 α 1 α 2 α 3,m b s δ(e b (p)+ E α 1 (p 1 ) E α 2 (p 2 ) E α 3 (p 3 ))δ( p+ p 1 p 2 p 3 ) M bb [ f α2 (xp 2 ) f α3 (xp 3 ) f α1 (xp 1 ) f b (xp) f α2 (xp 2 ) f α3 (xp 3 ) f α1 (xp 1 ) f b (xp)]. In principle, the baryon baryon matrix elements M bb need to be evaluated as a function of relative energy, spin and isospin. However, in the hydrodynamical limit this is not needed, as infinitely short mean free path implies collision number that are just sufficiently high to guarantee local thermalization, a limit that can be readily implemented at the end of our derivation. Here we use f b (xk) 1 f b (xk). The appearance of these factors f b (xk) are a consequence of the Pauli Exclusion Principle for the final scattering states. Fig. 2. Physical processes represented by a) the elastic baryon baryon collision integral, b) the neutrino gain term and c) the neutrino loss term. The physical processes represented by the gain and loss terms of the baryon neutrino collision integral, Ibν b, are shown in Fig. 2. Explicitly, The gain term is given by Igain b (xp) = 1 16(2π) 5 να m b s Ibν b (xp) = I gain b (xp) I loss b (xp). (10) M b M α E b (p)e α (p ) M νb (11) [ f ν (xk) f α (xp ) f b (xp)δ(e b (p)+ E ν(k) E α (p ))δ( p k p) + f ν (xk) f α (xp ) f b (xp)δ(e b (p) E ν(k) E α (p ))δ( p + k p)] dp dk,

5 Kinetic Theory 69 while the loss term is I b loss (xp) = 1 16(2π) 5 να m b s M b M α E b (p)e α (p ) M bν (12) [ f ν (xk) f α (xp ) f b (xp)δ(e b (p)+ E ν(k) E α (p ))δ( p k p) + f ν (xk) f α (xp ) f b (xp)δ(e b (p) E ν(k) E α (p ))δ( p + k p)] dp dk, where f ν (xk) = 1 f ν (xk) and the subscript ν has been used to specify the flavor of the neutrino. The matrix elements for neutrino creation and absorption, M bν and M νb, need to be calculated in detail, because the mean free paths of neutrinos are not small, even in dense matter. 3. Test Particles, Numerical Realization So far, it is not easy to see how Eqs. (7) (12) lend themselves to a numerical solution of the supernova time evolution problem. However, one can use an idea that has been very fruitfully employed in heavy ion physics, the introduction of test particles [14]. To do this, we represent the phase space distribution function by a sum over delta functions, f (xp) = i δ(x x i )δ(p p i ). (13) The initial coordinates of these delta-function point particles (= test particles) have to be determined by some physical input, for example by sampling a rotating iron core [15]. In heavy ion physics, this approach has been followed and met with incredible success [16 22] when compared to experimental observables. Among other finding, it has been shown in particular that the hydrodynamic limit and shock wave formation are contained within this theory [19]. This is important for the supernova physics with the need to correctly describe the shock wave, because it implies that the formalism sketched here is applicable to this problem. Using the test-particle ansatz, we can derive first order differential equations for the centroid coordinates of these test particles as d dt p j = U EoS,e (r j )+ F G, j +C(p j )+C ν (p j ), (14) d dt r p j j =, m 2 + p 2 j (15) j = 1,..., N, (16) where C(p j ) and C ν (p j ) symbolize the effects of two-body collisions with other baryons and neutrinos, respectively, on the test-particle momenta. N is the number of test particles used and should be at least 10 6, but 10 8 or even larger is also technically feasible with present-day computers. The equations of motion for the neutrino test particles are identical to those in Eqs. (14) (16), except that only the last term on the right hand-side of Eq. (14) does not vanish

6 70 W. Bauer for the neutrino propagation we neglect the effect of gravity on the neutrinos, and we assume that there is not collective mean field potential of significance that acts on the neutrinos. This does not, by the way, preclude the inclusion of neutrino oscillations in matter, as this is represented by an operation in flavor space, which is not included as an additional index in this short-hand notation. The inclusion of neutrino transport on equal footing in the same formalism as the baryon transport is perhaps the biggest advantage of the present approach. The CPU time requirements to include a 3d Boltzmann transport for neutrinos increase the total requirements only by a factor of less than 2. For the nuclear equation of state (EoS) and the associated mean field potential, U EoS,e, one can insert any of the currently available models. Using the boundaries established in heavy ion collisions on parameters like the compressibility and the isospin dependence of the nuclear EoS, one is then able to find cross-disciplinary interaction between these two important areas of research. Finally, for the gravitational force, we utilize the Newtonian monopole approximation, Gm 2 # { } i {1,..., N} : r i < r j F G, j = r j. (17) With an efficient quicksort-type algorithm, the CPU time requirements for the self-consistent calculation of gravity only scales as N log N. For a further discussion of some of the numerical details, the reader is referred to [23, 24]. r 3 i 4. Conclusions Hopefully, I have convinced the reader that the present approach, based on kinetic theory, is a promising alternative to the commonly employed baryon hydrodynamics + neutrino Boltzmann transport solution methods for the supernova explosion problem. Acknowledgments This research was supported by the U.S. National Science Foundation under grant PHY and an Alexander-von-Humboldt Foundation Distinguished Senior U.S. Scientist Award. References 1. J.R. Wilson, Numerical Astrophysics, Jones and Bartlett, Boston, M. Herant, W. Benz, W.R. Hix, C.L. Fryer and S.A. Colgate, Astrophys. J. 435 (1994) C.L. Fryer and A. Heger, Astrophys. J. 541 (2000) R. Buras, M. Rampp, H.-Th. Janka and K. Kifonidis, Phys. Rev. Lett. 90 (2003) H.-Th. Janka, R. Buras and M. Rampp, Nucl. Phys. A 718 (2003) 269c.

7 Kinetic Theory A. Mezzacappa, M. Liebendörfer, O.E.B. Messer, W.R. Hix, F.-K. Thielemann and S.W. Bruenn, Phys. Rev. Lett. 86 (2001) T.A. Thompson, A. Burrows and P.A. Pinto, Astrophys. J. 592 (2003) M. Herant, W. Benz and S. Colgate, Astrophys. J. 395 (1992) S. Yamada and K. Sato, Astrophys. J. 434 (1994) C.L. Fryer and M.S. Warren, Astrophys. J. 574 (2002) L K. Langanke, G. Martínez-Pinedo, J.M. Sampaio, D.J. Dean, W.R. Hix, O.E.B.Messer, A. Mezzacappa, M. Liebendörfer, H.-Th. Janka and M. Rampp, Phys. Rev. Lett. 90 (2003) K. Kotake, S. Yamada, K. Sato and T.M. Shimizu, Nucl. Phys. A 718 (2003) 629c. 13. S.J. Wang, B.-A. Li, W. Bauer and J. Randrup, Annals of Physics 209 (1991) C.-Y. Wong, Phys. Rev. C 25 (1982) A. Heger, N. Langer and S.E. Woosley, Astrophys. J. 528 (2000) G.F. Bertsch et al., Phys. Rev. C 29 (1984) H. Kruse et al., Phys. Rev. Lett. 54 (1985) W. Bauer, G.F. Bertsch, W. Gassing and U. Mosel, Phys. Rev. C 34 (1986) H. Stöcker and W. Greiner, Phys. Rep. 137 (1986) G.F. Bertsch and S. Das Gupta, Phys. Rep. 160 (1988) P. Schuck et al., Prog. Part. Nucl. Phys. 22 (1989) W.G. Gong, W. Bauer, C.K. Gelbke and S. Pratt, Phys. Rev. C 43 (1991) T. Bollenbach and W. Bauer, in Exotic Clustering, eds. S. Costa, A. Insolia and C. Tùve, American Institute of Physics Conference Proceedings, Melville, New York, 2002, Vol. 644 (2002) W. Bauer, 19th Winter Workshop on Nuclear Dynamics, Breckenridge, Colorado, 2003, Acta Phys. Hung. A 21 (2004) 371.

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