c Pleiades Publishing, Inc., 2018. ISSN 1063-7737, Astronomy Letters, 2018, Vol. 44, No. 5, pp. 309–314.  c I.V. Panov, S.I. Glazyrin, F.K. R¨ Original Russian Text  opke, S.I. Blinnikov, 2018, published in Pis’ma v Astronomicheskii Zhurnal, 2018, Vol. 44, No. 5, pp. 331–337.

Nucleosynthesis during a Thermonuclear Supernova Explosion 4, 5 ¨ I. V. Panov1, 2 , S. I. Glazyrin2, 3* , F. K. Ropke , and S. I. Blinnikov2, 6, 7 1

“Kurchatov Institute” National Research Center, pl. Kurchatova 1, Moscow, 123182 Russia 2 Institute for Theoretical and Experimental Physics, ul. Bol’shaya Cheremushkinskaya 25, Moscow, 117259 Russia 3 Dukhov All-Russian Research Institute of Automatics, Sushchevskaya ul. 22, Moscow, 127055 Russia 4 Zentrum fur ¨ Astronomie der Universitat ¨ Heidelberg, Heidelberg, Germany 5 Heidelberg Institute for Theoretical Studies, Heidelberg, Germany 6 KAVLI Institute for the Physics and Mathematics of the Universe, Tokyo, Japan 7 Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia Received October 23, 2017

Abstract—Supernovae are such bright objects that they can be observed even at high redshifts. Some types of such events, for example, type Ia (thermonuclear), have peculiarities of the light curve, which allows them to be used for cosmological applications. The light curve is determined by the details of the explosion dynamics and nucleosynthesis: in particular, it depends on the amount of iron-peak elements produced during the explosion. We discuss the burning processes in such objects and the peculiarities of turbulence simulations in them, which is needed for a proper hydrodynamic description of the explosion process. A direct nucleosynthesis calculation is performed for the temperature and density profiles derived in the available 3D hydrodynamic explosion simulations. We show that in the supernova progenitor model considered the calculated abundances of elements from carbon to iron-peak elements are in good agreement both with the observations and with the calculations of other authors. At the same time, no r-elements are produced even at the maximum neutron excess for this model (Ye ∼ 0.47) due to the slow evolution of the density and temperature. DOI: 10.1134/S1063773718050031 Keywords: nuclear astrophysics, nucleosynthesis; supernovae and supernova remnants; nuclear reactions, beta decay.

INTRODUCTION The discovery of an accelerated expansion of the Universe in 1998, in which thermonuclear (type Ia) supernovae (SNe) played a major role (Riess et al. 1998; Perlmutter et al. 1999), changed dramatically our view of the structure of the Universe and showed the significance of SNe for its study. Thermonuclear SNe have an important property that allows them to be used as “standardizable candles.” Such explosions are not universal: their light curves are different, but there exists a correlation between individual characteristics of these curves. The peak luminosity correlates with the initial decline rate of the light curve (Pskovskii 1977; Phillips 1993) and, therefore, for each event its absolute luminosity can be calculated. The method of determining cosmological distances is *

E-mail: [email protected]

based on this property. By applying it, we can study mysterious substances, dark matter and dark energy. This correlation is an empirical law and was deduced on the basis of observations from the nearest SNe Ia. A theoretical or numerical validation of this correlation is the goal of most thermonuclear SN studies. The shape of the light curve depends strongly on chemical composition, in particular, on the mass of the iron-peak elements produced during explosive nucleosynthesis. Therefore, proper nucleosynthesis calculations reconciled with hydrodynamics are also needed for a full-scale description of the observed characteristics of these objects. This will allow one to find not only the amount (mass) of new elements but also their spatial distribution. There exist several scenarios under which an SN Ia explosion is possible. A detailed overview of the scenarios and models is presented in Hillebrandt and Niemeyer (2000). One of them is an explosion 309

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in a binary system with degenerate (white dwarf) and nondegenerate stars. Mass transfer onto the white dwarf occurs in such a system and its mass increases. When the mass of the white dwarf reaches its maximum (Chandrasekhar) limit for such stars, the central region of the star heats up and the nuclear reaction rate begins to rapidly rise. As a result of the emerging explosive nucleosynthesis initiated by the combustion wave, in the bulk of the star its chemical composition changes completely. The details of the burning regime (initially deflagration, subsequently passing to detonation) play a major role for the explosion itself, but they are not considered in this paper. Such burning leads to complete destruction of the original white dwarf: ≈2.2 × 1051 erg is released as the entire star with a mass of ∼1.4 M composed of 0.512 C + 0.516 O is burnt out to the ironpeak elements (56 Ni). Subtracting the gravitational binding energy (≈(5−6) × 1050 erg) and taking into account the fact that not all material is burnt out to Ni (but intermediate elements like Mg, Si, S, and Ca are partially produced), we obtain an observed explosion energy of ∼1.3 × 1051 erg. In turn, the energy being released in the chain of decays 56 Ni → 56 Co →56 Fe explains the main characteristics of the light curve. A detailed analysis of the observed light curves and spectra allows the remnant density and composition to be determined. This confirms the complete destruction of the star and the mass of the produced iron-peak elements, ∼0.5−0.6 M , and a smaller amount of intermediate elements Si, S, Ar, Ca, etc. As calculations show, the amount of iron-peak elements is very sensitive to the density and temperature during their synthesis (Travaglio et al. 2004). Full-scale simulations of such objects require describing both the dynamics of an exploding star and the nuclear transformations occurring in it. Simultaneous simulations of nucleosynthesis with SN Ia explosion hydrodynamics are a very complex problem primarily because of the huge requirements imposed on the computational resources. At present, the simulations are divided into two physical processes: the hydrodynamic part, in which nucleosynthesis is taken into account in a simplified form, and then fullfledged simulations of nuclear transformations are performed in accordance with the temperature, density, and other profiles calculated in the SN progenitor model. One came to this approach after decades of SN Ia nucleosynthesis simulations and calculations in spherically symmetric models (Nomoto et al. 1984; Iwamoto et al. 1999; Brachwitz et al. 2000; Imshennik and Zabrodina 1999). Present-day multidimensional calculations are more resource-consuming

even without any detailed nuclear transformations and, therefore, the duration of the simulated physical time is short, being a few seconds (Reinecke ¨ et al. 1999, 2002; Travaglio et al. 2004; Ropke 2005), enough for nucleosynthesis to proceed. The method that was first implemented for SNe Ia by Travaglio et al. (2004), although it was also applied previously for other problems, is used in most of the calculations to preserve the data from the hydrodynamic stage and, subsequently, to perform detailed nucleosynthesis. Tracer particles that are passively advected with the material are added to the calculations (this is done, because the main approach in the calculations is Eulerian, when the medium flows through a reference frame at rest specified by a grid of cells): dxα = v(xα , t). (1) dt Here, v(xα ) is the flow velocity of the medium at the position of a Lagrangian particle xα . The evolution of the parameters (density, temperature) is written along the trajectories of these particles. This approach has been actively used until now (see, e.g., Seitenzahl et al. 2013; Townsley et al. 2016). In this paper we also apply it. THE HYDRODYNAMIC EXPLOSION MODEL The dynamics of SN Ia explosions is determined by the propagation of the thermonuclear burning flame through the star. The flame itself is a very thin surface separating the reactants (unburnt material) and nuclear reaction products (burnt material). The emerging turbulence in the star affects significantly the flame dynamics. Instabilities in the flame itself are the source of this turbulence, demanding careful selfconsistent simulations. In turn, describing the turbulence requires using various models. Such models exist and are actively used in calculations (see, e.g., Schmidt et al. 2006a, 2006b (based large eddy simulations, LES); Woosley 2009; Glazyrin 2014). One promising model describing the propagation of a turbulent flame was suggested by Glazyrin (2014). This model belongs to the class of Reynolds-averaged ones (Reynolds-averaged Navier–Stokes, RANS), which has some advantages. It requires less computational resources than do the LES models and allows calculations in lower dimensions to be performed: the properties of 3D turbulence are properly reproduced even in 1D calculations. This is especially useful for the problems of determining the influence of multidimensional effects on the SN Ia dynamics and nucleosynthesis. These properties of the model are achieved due to an excessive averaging of the turbulent pulsations and, as a result, the model ASTRONOMY LETTERS

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describes poorly the events that deviate strongly from the average behavior of the system. Just as other turbulence models, it contains a certain set of empirical constants (these constants are selected based on a comparison with the results of instability development). The model equations include, first, an ordinary system of hydrodynamic equations (the equations for the density ρ, velocity v, and specific energy E): ∂t ρ + ∂i (ρvi ) = 0,

(2)

∂t (ρvi ) + ∂j (ρvi vj + pδij ) = −∂j Rij ,

(3)

∂t E + ∂i (vi (E + p))

(4)

= −G2 + ρ + ∂i (pai −

QTi ) +

S,

to which the turbulent contributions Rij , G2 , ρ, ai , and QT , each having a certain physical meaning, were added (for details, see Glazyrin 2014). Second, explicit equations for the specific turbulent energy k: ∂t (ρk) + ∂i (ρkvi ) = G1 + G2 − ρ + ∂i (ρck D∂i k),

(5)

and an analogous equation for the turbulent energy dissipation  were added to Eqs. (2)–(4). In this case, the mass fraction of elements Xα evolves as ∂t (ρXα ) + ∂i (ρXα vi ) = ∂i (ρD∂i Xα ) + Rα , (6) where the mass fraction is advected as a passive tracer (advection term), is spread due to turbulent diffusion (diffusion term ∝ generalized diffusion coefficient D dependent on turbulence intensity), and changes through nuclear reactions (term R). The energy release from the reactions is taken into account in Eq. (4) by adding the term S. In “ideal” calculations it is necessary to take into account the full nuclear reaction network, but since the computational resources are limited, simplified schemes are used in hydrodynamic calculations. The variants of two schemes are commonly used: (1) only some of the most important reactions are considered; (2) it is assumed that the material is always in nuclear statistical equilibrium (NSE) and an energy source is specified on the burning surface. We emphasize that the turbulent diffusion of the components Xα should be taken into account (the term with D), because the turbulence in the star is quite strong. As in other papers, such a model does not purport to resolve the burning front. Therefore, we will also use the method of passive tracer particles that are advected with the medium. The dynamics of passive particles is usually described by Eq. (1) in this method. Nevertheless, when turbulent flows are calculated, the velocity of the medium is the velocity averaged over the turbulent pulsations and, hence, the ASTRONOMY LETTERS

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contribution from the turbulent diffusion is neglected when solving Eq. (1). In LES the error from such an approximation can be small and is related only to the subgrid turbulence flows larger than the grid cell size are resolved explicitly (nevertheless, this approximation requires an additional analysis). In the case of a RANS turbulence model, this diffusion should be taken into account explicitly, which is described by adding a Brownian process to Eq. (1). The dynamics of the burning front itself is tracked by the level-set method, which has long been used for flame calculations and was tested for SNe Ia in Reinecke et al. (1999). In that case, the velocity of the flame front relative to the medium should be specified explicitly (it is not calculated self-consistently) by an analytical expression defining the burning velocity as a function of the turbulence intensity at a given point. Several such analytical expressions have been suggested and are used in the calculations of various authors. This model allows the propagation of a turbulent flame in a star to be simulated (Glazyrin 2014) and the numerical implementation is currently being upgraded to include nucleosynthesis calculations. Therefore, we performed the nucleosynthesis calculations presented below based on the well-tested hy¨ drodynamic model presented in Ropke (2005), but according to our nucleosynthesis scheme. NUCLEOSYNTHESIS At the initial stage of explosive nucleosynthesis in the SN Ia explosion scenario described above, new elements are produced mainly during the thermonuclear burning of carbon and oxygen through reactions with charged particles. As the temperature and density drop and as the nucleosynthesis rate decreases, after the completion of the α-process (Woosley and Hoffman 1992), the production of new nuclei can pass, in the presence of a sufficient number of free neutrons, into a dynamic r-process (Panov and Chechetkin 2002). As in many previous papers (Nomoto et al. 1984; Mazzali et al. 2007; and references therein), when simulating the SN Ia explosion process, we assumed the composition of the white dwarf to be close to Y (12 C) ≈ Y (16 O) ≈ 0.5. A detailed analysis of the observed light curves for SNe Ia and a comparison with the synthetic spectra obtained from explosion simulations and available observations lead to the conclusion that the initial model composition (YC = 0.5, YO = 0.475, and YNe = 0.025) used by us is close to the observed one, while the change in nucleosynthesis conditions depends more strongly on ¨ the mass accretion rate (Ropke et al. 2007; Mazzali et al. 2007).

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To properly describe the nucleosynthesis in the inner ejecta, including that under the action of neutrons, it is important to know Ye = Z/A. This quantity changes in weak-interaction reactions under explosive burning during the capture of electrons and positrons, beta decay, and the capture of neutrinos and antineutrinos. At a high temperature and a large density (Nomoto et al. 1984), when the photodissociation processes prevail, the material is composed predominantly of neutrons and protons and the following reactions dominate: νe + n  p + e− and ν˜e + p  n + e+ . Therefore, we supplemented the code used (Nadyozhin et al. 1998) with weak-interaction reactions (Langanke and Martinez-Pinedo 2000). These processes are especially important at high temperatures (T > 5 × 109 K) and densities (ρ > 108 g cm−3 ), which, in particular, leads to a change in Ye . For the conditions in the ejecta material under consideration the quantity Ye on a time scale comparable to the hydrodynamic one changes mainly due to the weak interaction before freeze-out and increases due to the beta decay after freeze-out of the reactions with protons and alpha particles, at a neutron excess needed for the r-process. Since the nucleosynthesis results based on a 3D model agree with the observations in released energy and nickel mass much better than do the calculations based on a 2D one (Contardo et al. 2000), in this paper the explosive nucleosynthesis calculations were performed along the trajectories obtained from the hydrodynamic 3D SN progenitor model c3-3d¨ 256-10s (Kozma 2005; Ropke 2005). The simulations of nucleosynthesis during an SN Ia explosion were performed along a series of trajectories with known temperature and density profiles by taking into account not only all reactions with neutrons, protons, alpha particles, and photons but also the weak interaction. The α- and r-processes are usually simulated separately, within the framework of various mathematical models and codes, which imposes certain constraints on the physical models in the domain of solutionjoining parameters. We used a nucleosynthesis model (Nadyozhin et al. 1998) applicable to a consideration of the pattern of evolution of the chemical composition from the SN progenitor stage at high densities and temperatures to the end of nucleosynthesis in the ejecta, where either explosive nucleosynthesis (at Ye ∼ 0.5) or even a weak r-process (at a low electron-to-baryon ratio Ye , less than 0.46) is realized. This model realizes the full nuclear reaction network by taking into account not only all reactions with neutrons, protons, alpha particles, and photons but also the

carbon, oxygen, silicon burning processes and weak interactions. The region of nuclei under consideration included all isotopes of the elements from helium to ruthenium (Z = 44), and we took into account all pair reactions between 1072 isotopes as well as the betadecay and weak-interaction reactions disregarded in our previous paper (Panov et al. 2016), where the possibility of the synthesis of elements heavier than iron on trajectories with T9 < 5 was considered. RESULTS As the first approximation of nucleosynthesis in the described scenario, we used 10 trajectories of the c3-3d-256 model (Kozma et al. 2005) obtained in hy¨ drodynamic SN progenitor simulations (Ropke 2005) and reflecting the regions with different maximum densities and temperatures. The trajectories calculated in the c3-3d-256-10s model represent the hydrodynamic evolution of medium parameters with a duration of 10 s. This time is enough for the reactions to freeze out for all trajectories. Note two important (from the viewpoint of nucleosynthesis) processes that occur along the trajectories: (1) the change in the neutron excess due to weak interactions (the interactions of electrons and positrons with nucleons and nuclei), especially at high temperatures (T9 > 5) and densities (ρ > 109 g cm−3 ); (2) the burning of light and intermediate nuclei to produce a large amount of iron-peak elements. The figures reflecting our results present both the integrated (over all of the trajectories considered) abundance of elements and the relative yield of elements normalized to the observed values in the Solar System. For example, Fig. 1 presents the relative abundances of chemical elements along each of the 10 trajectories (Fig. 1a) and the integrated (over all of the trajectories considered) abundance of isotopes with a mass number A (Fig. 1b). Figure 2 shows the range of change in the neutron excess of the medium during the first 300 ms (Fig. 2a), when the role of weak reactions can be either minor, the upper two lines (T9 < 5.5), or great (at high temperatures and densities)—the lower four lines. Figure 2b shows the abundances of chemical elements at the end of nucleosynthesis normalized to the observed values in the Solar System. Note that the results of our calculations presented in Fig. 2b are in good agreement with the results from Travaglio et al. (2004) and Maeda et al. (2010), although the SN progenitor models and nuclear data for nucleosynthesis were different. Note that the results normalized to the observational data are given in ASTRONOMY LETTERS

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¨ Fig. 1. (a) The abundances of elements along each of the 10 trajectories (Ropke et al. 2007) for a C/O core with Ye0 = 0.498. (b) The integrated (over all trajectories) abundance of elements.

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Al

B 10

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Fig. 2. (а) Dependence of the neutron excess during the first 300 ms on weak reactions. The curves reflect the range of Ye variations on trajectories with different maximum temperatures and densities. For the upper two curves the maximum temperature is T9 = 2.4 (dotted line) or 5.3 (dash-dotted line). The lower four curves indicate the pattern of decrease in Ye at T9 = 7 with rising density (from top to bottom). (b) The relative yields of elements normalized to the observed values in the Solar System.

Travaglio et al. (2004) in solar units, while in our figures they are given in relative units and were obtained only for the selected set of tracer particle trajectories. CONCLUSIONS The results obtained here, i.e., the abundances of iron-peak elements, including iron and nickel, needed to calculate the light curves, generally show good agreement both with other calculations and with the observational data. An analysis of the results shows that virtually no heavy elements beyond the iron peak are synthesized when the characteristics of the medium evolve slowly (when compared with the rate of change of the parameters in another type of ASTRONOMY LETTERS

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supernovae, core-collapse ones). Our calculations of the abundances of iron-peak elements allow us to ascertain whether both predicted nuclear data, such as the cross sections and decay rates, and observational data are reliable. In comparison with the calculations of other authors, note some overestimation of the amount of the isotope 54 Fe. The presented calculations reflect the results of nucleosynthesis along several trajectories with different dynamics, which differ by the maximum values of the peak temperature and are not a representative sample of ejecta material. In addition, such a discrepancy can be caused by local inaccuracies of the model and the nuclear data (for example, the reaction

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rate) appearing in the nucleosynthesis equations and the observational data from Lodders (2010) used for the normalization. The results obtained will be used to determine more accurately the abundances of chemical elements based on calculations along a large number of trajectories, to consider the nucleosynthesis in terms of a new model (Glazyrin 2014), and to increase the accuracy of light-curve calculations. After a further elaboration of the hydrodynamic model described above and the addition of the same scheme of nucleosynthesis calculations as that used in this paper, we will be able to determine the influence of the hydrodynamic description on the chemical composition of the elements produced during an explosion, which is planned to be reported in our next paper. ACKNOWLEDGMENTS We are grateful to C. Travaglio for the useful discussions of SNe Ia explosion models and the details of their presentation to optimize the nucleosynthesis calculations. This work was supported by the Russian Science Foundation (project no. 16-12-10161). REFERENCES 1. F. Brachwitz, D. J. Dean, W. R. Hix, K. Iwamoto, K. Langanke, G. Mart ´ınez-Pinedo, K. Nomoto, M. R. Strayer, et al., Astrophys. J. 536, 934 (2000). 2. G. Contardo, B. Leibundgut, and W. D. Vacca, Astron. Astrophys. 359, 876 (2000). 3. S. I. Glazyrin, Astrophys. Space Sci. 350, 683 (2014). 4. W. Hillebrandt and J. C. Niemeyer, Ann. Rev. Astron. Astrophys. 38, 191 (2000). 5. V. S. Imshennik and E. A. Zabrodina, Astron. Lett. 25, 93 (1999). 6. K. Iwamoto, F. Brachwitz, K. I. Nomoto, N. Kishimoto, H. Umeda, W. R. Hix, and F.-K. Thielemann, Astrophys. J. Suppl. Ser. 125, 439 (1999). 7. C. Kozma, C. Fransson, W. Hillebrandt, C. Travaglio, J. Sollerman, M. Reinecke, F. K. Ropke, and J. Spyromilio, Astron. Astrophys. 437, 983 (2005). 8. K. Langanke and G. Mart ´ınez-Pinedo, Nucl. Phys. A 673, 481 (2000). 9. K. Lodders, Principles and Perspectives in Cosmochemistry, Lecture Notes of the Kodai School on “Synthesis of Elements in Stars,” Ed. by A. Goswami and B. Eswar Reddy (Springer, Berlin, Heidelberg, 2010), p. 379.

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Translated by V. Astakhov

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