Nadyozhin, D

Yudin, A

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PoS(NIC-IX)197Equation of state and neutrino signal fromcollapsing stellar coresA. Yudin   and D. NadyozhinA.I. Alikhanov Institute for Theoretical and Experimental PhysicsE-mail: yudin@itep.ru, nadezhin@itep.ruDuring the infall stage of collapse up to nuclear densities, the properties of stellar matter are welldescribed by the Nuclear Statistical Equilibrium (NSE) approach . Even with this simplificationthere still exists a variety of effects in the Equation Of State (EOS) to be taken into account.We start from the problem of nuclear partition functions then continue with a nuclear networkcompleteness question, closely connected with a neutron-rich nuclei problem. Separate studiesare devoted to the problem of Coulomb interaction in multi-component systems. Our final goal isto explore these effects and to estimate their impact on the structure of core-collapse supernovaeand the properties of the neutrino signal they produced.International Symposium on Nuclear Astrophysics - Nuclei in the Cosmos - IX25-30 June 2006CERNSpeaker.cCopyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(NIC-IX)197Equation of state and neutrino signal from collapsing stellar cores A. Yudin1. IntroductionUnder the condition of NSE, the properties of EOS are determined by only three parameters:density ρ , temperature T , and neutron-to-proton ratio θ . In addition to these explicit parameters,there exist a number of “degrees of freedom" which are connected to the microphysics involved.For example, one has to specify the set of nuclei (nuclear network) to be explicitly included inthe calculations. Thus, there arises the problem of nuclear network completeness. One must besure that the nuclear network specified accurately reproduce both chemical and thermodynamicproperties of matter. After some numerical experiments we settled on the network that includesfree neutrons and protons, the set of α-nuclides from 4He through 40Ca, and isotopes of iron-peakelements from Sc through Kr (in total 137 nuclides). If necessary, we supplement this basic networkwith a group of the neutron-rich nuclei.2. Nuclear partition functionsThe correct evaluation of nuclear partition functions is a long-standing problem. Our approach 1  is similar to that of Rauscher et al. 2  except the high-temperature domain. At high tempera-tures, our approach satisfies to a natural requirement that the average excitation energy of a nucleusEex  should not exceed it’s binding energy QA  Z. Hence, for the high-temperature domain we usethe relationEex  QA  Zχ  T  . In principle, one can use an arbitrary form for the monotonouslyincreasing with temperature function χ  T  that satisfies the condition 0 	 χ  T 	 1. Then, byintegrating the relationEex  T d lnωd lnT one can obtain the partition function ω . The results areshown in Fig. 1. The left panel shows the partition function of 56Fe as a function of temperature.Red line represents our approach, blue line corresponds to the calculations of Rauscher et al., andblack one shows the well-known Bethe’s formula. Right panel shows the corresponding averageexcitation energies. One can see that for the approaches of Bethe and Rauscher et al.Eex  exceedsthe binding energy already at T9  100 indicating that the corresponding partition functions areinapplicable for such a high temperature. Our method allows to estimate the partition functions ina reasonable way even at very high temperatures. This is especially important for the case of theneutron-rich nuclei when the binding energy is often the only parameter known.And what is the effect of the nuclear excitations on the thermodynamic properties of matter?One result is shown in Fig. 2 where the level lines of of the adiabatic index are plotted. One cansee that calculation with an allowance for the nuclear excitations (solid lines) results in a wideningof upper part of the instability ravine — the domain where adiabatic index γ  4  3 (the ravine edgeis shown by a thick red lines).3. Neutron-rich nucleiThe importance of the neutron-rich nuclei for the advanced stages of the collapse have beenconfirmed by many authors. These nuclei are significant not only for elucidating the chemicalcomposition and thermodynamical properties described by EOS but also for a correct descriptionof the kinetics of nuclear weak-interactions that control the temporal behavior of θ during thecollapse. Figure 3 shows how such important thermodynamic quantities as dimensionless entropy2PoS(NIC-IX)197Equation of state and neutrino signal from collapsing stellar cores A. YudinFigure 1: Average excitation energy and partition function for 56Fe.per nucleon and adiabatic index vary with θ . Dashed lines represents the calculation carried outwith the standard set of nuclei mentioned above. Solid red lines correspond to the calculation withseven neutron-rich isotopes of Se (83   89Se) being added to the standard set. Until θ does not exceedits individual value 1.44 for 83Se the addition of 83   89Se virtually makes no difference. However,for θ  1  5 the difference becomes significant and can even destabilize those regions which werestable against the collapse (γ  4  3).4. Coulomb InteractionTo account for the Coulomb interaction under the conditions of NSE is a rather complicatedproblem because it influences not only the thermodynamic quantities but also can significantlychange the condition of chemical equilibrium and thereby the nuclide abundances Yi. The free en-ergy formalism is the most adequate for this issue. The free energy of the system with interactionis represented as a sum of free energy without interaction (ideal part) and extra term, determinedby the Coulomb interaction: F  ρ  T Yi    Fid  ρ  T Yi  	 F  ρ  T Yi   . The new set of equi-librium concentrations can be found by minimization of F  ρ  T Yi   with respect to Yi at constantT , ρ , and θ .First of all, one must specify a form of the interaction-free part  F and then address to theproblem of Coulomb interaction in a multi-component system. The most known and carefullystudied is the case of one-component system where the interaction term has the form  FOCS nkT f  Γ  , where f is a universal function of the only one parameter ΓZe  2kT 4pi3 n 1  3, n being3PoS(NIC-IX)197Equation of state and neutrino signal from collapsing stellar cores A. YudinFigure 2: Level lines of adiabatic index γ with (solid lines) and without (dashed lines) nuclear excitation.the number of particles per unit volume and Ze their individual charge. There exist two well-known limits: the limit of weak interaction (the Debye limit) and that of strong interaction calledlinear mixing rule. In the latter case the interaction part of free energy can be represented as FLMR  kT ∑ni f  Γi  , Γi  Z5  3i Γe, where Γe is the electron interaction factor. Thus, the linearmixing rule is defined as a linear sum over the component concentrations and uses the same one-component function f  Γ  for all the component involved. The validity of this rule in the domain ofstrong interaction have been confirmed by many authors. But for astrophysical implementation weneed an approach that would be applicable in the whole domain of the interaction.The simplest way to reach such an aim is to use the average-ion model. It based on the one-component model with all the quantities involved being averaged over chemical components, i.e. FAIM  kT  ∑ni  f Γ ,Γ ∑niΓi  ∑ni. Using this approach one can cover both strong andweak-interaction limits. However, such an approach always fails to adequately describe interme-diate interaction regimeΓ 1 causing nonphysical behavior of equilibrium concentrations andthermodynamic quantities (there appear oscillations and even breaks). Thus we came to the conclu-sion that various forms of average-ion approach is inapplicable to describe the Coulomb-interactioneffects under the conditions of NSE.The other way is to start from the linear-mixing rule and to try to expand its applicability toweak interaction. But doing so in the spirit of average-ion model, i.e. modifying it by some addi-tional parameters averaged over chemical components, one comes to the same difficulties. Aftersome logical attempts and numerical experiments we have found a special modification technique4PoS(NIC-IX)197Equation of state and neutrino signal from collapsing stellar cores A. YudinFigure 3: Influence of neutron-rich nuclei on the entropy (left panel) and adiabatic index (right panel).which is free from the problems listed above and predicts smooth behavior for all the essentialquantities 3  .Why is this intermediate interaction region so important? Figure 4 gives the answer. It showsthe moment after the bounce when the shock wave just has begun to propagate outwards fromstellar center. We have calculated the average interaction factorΓfor three cases: EOS withoutnuclear excitation (black dashed line), with excitation (red solid line) and with Coulomb interac-tion and excitation (blue solid line). The central region and that above the shock wave belong tostrong interaction domain Γ  1, the region passed by the shock belongs to the weak interaction do-main. The transition between them occurs somewhere at the shock front and one needs the correctdescription of this important region.5. Neutrino signalThe spectra and light curves of neutrino radiation are the most important issues of any collapsecalculation. Here we present our first results of the calculations for a 2M   stellar core collapse.These calculations were done without an allowance for Coulomb interaction. Figure 5 representsthe calculated neutrino light curves for the electron neutrino νe. The first luminosity maximum oc-curs at the point 2 approximately 1.25ms before the bounce (point 1) and is due to the temperatureincrease during the infall. The second peak 1  38ms after bounce is caused by a rapid expansionof shock front emitting surface. The average emitted neutrino energy that was smoothly growingbefore the bounce begins rapidly to rise after it up to 15 MeV. This effect is caused by changes5PoS(NIC-IX)197Equation of state and neutrino signal from collapsing stellar cores A. YudinFigure 4: Coulomb interaction factor as a function of mass coordinate.in the structure of collapsing core in the vicinity of moving shock wave: behind the shock thereappears a layer filled with free neutrons and protons. After that both the neutrino luminosity andthe mean individual νe energy begin to decrease. As for the electron antineutrino  νe, these valuescontinue to increase smoothly. However, the contribution of  νe to the total neutrino flux is small.The total, emitted in form νe and  νe energy at 15ms after bounce reaches  1052erg.Figure 6 shows the spectra of νe (red solid line) and  νe (magenta line) at the moment of bounce(0 ms) and 5 ms later. The calculation neglecting the effect of the neutrino-electron scattering(NES) are shown by a black dashed line.At the moment of bounce, NES noticeably shifts spectrum peak to lower energies as comparedto the non-NES calculations. At 5 ms after bounce this effect is less pronounced, the spectrum getsstiffer and broader.The final goal of our study is to explore how the properties of the neutrino signal dependon different physical effects especially such as the Coulomb interaction, partition functions, andproperties of the neutron-rich nuclei. This work is underway.Acknowledgement. We thank the NIC-9 Organizing Committee for the invitation to attendthe Conference. The work was supported by the grants: RFBR No. 04-02-16793 and SNSFNo. IB7320-110996.6PoS(NIC-IX)197Equation of state and neutrino signal from collapsing stellar cores A. YudinFigure 5: The electron neutrino light curves in different energy groups. The total (energy-integrated) neu-trino luminosity is shown by purple solid triangles, (left panel, upper curve). The right panel shows thenear-peak region   0  005  t  0  015 in more detail (t  0 corresponds to the moment of bounce).7PoS(NIC-IX)197Equation of state and neutrino signal from collapsing stellar cores A. YudinFigure 6: Electron neutrino and antineutrino spectra.References[1] D.K. Nadyozhin and A.V. Yudin, Equation of State under Nuclear Statistical Equilibrium Conditions,Pis’ma Astron. Zh. 30, 697 (2004) [Astron. Lett. 30, 634 (2004)].[2] T. Rauscher, F.K. Thielemann, and K.L. Kratz, Nuclear Level Density and the Determination ofThermonuclear Rates for Astrophysics, Phys.Rev. C 56, 163 (1997).[3] D.K. Nadyozhin and A.V. Yudin, The Influence of Coulomb Interaction on the Equation of Stateunder Nuclear Statistical Equilibrium Conditions, Pis’ma Astron. Zh., 31, 4 (2005) [Astron. Lett. 31,4 (2005)].8

Equation of state and neutrino signal from collapsing stellar cores

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