Abstract We study the nucleosyntheses of the light elements 7 Li and 11 B and the rprocess elements in Type II supernovae in the lights of the progress in neutrino spectrum modeling and the studies of galactic chemical evolution. We investigate the influence of total energy, E ν , and decay time, τ ν of supernova neutrinos on these two nucleosynthesis processes. Common models of the neutrino spectra are adopted in order to look for unified neutrino spectrum modeling for both processes. We adopt the model of the supernova explosion of a 16.2 M ⊙ star, which corresponds to SN 1987A, and study the nucleosynthesis of the light elements by the postprocessing. We find that the ejected masses of 7 Li and 11 B are roughly proportional to the total energy of supernova neutrinos and weakly dependent on the decay time of the neutrino luminosity. As for the r-process nucleosynthesis, we adopt the same models of neutrino spectra in the neutrino-driven wind models of a neutron star with the mass of 1.4M ⊙ . We find that the r-process nucleosynthesis is affected through the peak neutrino luminosity. The observed r-process abundance pattern is better reproduced in low total neutrino energy and long decay time to attain the low peak neutrino luminosity. We also discuss an unresolved problem of the overproduction of 11 B in the galactic chemical evolution of the light elements. We first identify that the ejected mass of 11 B is a factor of 2.5 to 5.5 overproduced in Type II supernovae when one adopts the standard neutrino parameters similar to those in the previous studies, i.e. E ν = 3.0 × 10 53 erg, τ ν = 3 s, and the neutrino temperature T νµ,τ = Tν µ,τ = 8.0 MeV/k. We have to assume E ν ≤ 1.2 × 10 53 erg in order to avoid the overproduction of 11 B, which is too small to accept in comparison with 3.0 × 10 53 erg deduced from the observation of SN1987A. We here propose to reduce the temperature of ν µ,τ and their anti-particles to T νµ,τ = Tν µ,τ = 6.0 MeV/k. This modification of the neutrino temperature is shown to resolve the overproduction problem of 11 B while still keeping a successful r-process abundance pattern. Subject headings: Galaxy: evolution -neutrinos -nuclear reactions, nucleosynthesis, abundances -stars: abundances -supernovae: general Introduction During supernova (SN) explosions, a huge amount of neutrinos is emitted from a protoneutron star and blows off surface materials of the proto-neutron star. The neutrinos interact with nuclei in the supernova ejecta, and the neutrino emission is strong enough to change the compositions despite their small cross sections for neutrino-nucleus interactions. The neutrino-induced reactions affect mainly two kinds of proposed nucleosynthetic processes which occur during supernova explosions: One is "the synthesis of light elements such as Li and B" through the ν-process in the He-layer, and the other is "the r-process" in the neutrino-driven winds above the surface of the neutron star. Introduction The production of the light elements through the ν-process during SN explosions was first suggested by Domogatsky, Eramzhyan, & Nadyozhin (1977) . Woosley et al. (1990) precisely evaluated the roles of the ν-process and showed that a large amount of 7 Li and 11 B is produced during supernova explosions. Woosley & Weaver (1995) tabulated the abundances of the elements including the light elements with grids of stellar masses and metallicities. Their results have been adopted to the studies on the galactic chemical evolution (GCE) (e.g., Fields et al. 2000; Ramaty et al. 2000b; Ryan et al. 2001 ). Olive et al. (1994) pointed out, however, that there remains a serious problem of overproduction of 11 B from the SN ν-process in the GCE models of the light elements. Here, the overproduction means that the predicted 11 B abundance in theoretical calculations is overabundant compared with the observed one when we adopt theoretical yields of Woosley & Weaver (1995) without any renormalization. Studies on the GCE have shown that the light elements are mostly produced from the galactic cosmic-ray (GCR) interactions with interstellar medium (ISM). The GCR model was improved by taking account of the primary acceleration of heavy elements from SN ejecta in addition to the secondary acceleration of the engulfing ISM (Ramaty et al. 1997; Yoshii, Kajino, & Ryan 1997; Vangioni-Flam et al. 1998; Suzuki, Yoshii, & Kajino 1999) . This explains naturally linear metal-dependence of the GCE of the amounts of Be and B, i.e., [BeB/H] ∝ [Fe/H], but one still needs another contribution to 7 Li and 11 B (Olive et al. 1994; Vangioni-Flam et al. 1996; Fields & Olive 1999; Romano et al. 1999 ). The production of 11 B in SN ν-process is identified to be the most important process to explain very precise data of meteoritic 11 B/ 10 B abundance ratio. Several authors studied the GCE models of the light elements by taking account of the contribution of SN ν-process and showed that the amount of 11 B is too large by a factor of two (Fields et al. 2000) to five (Ramaty, Lingenfelter, & Kozolvsky 2000a) , while the other light elements 6 Li, 7 Li, 9 Be, and 10 B are well reproduced in appropriate amount. Introduction Mass loss of the outer envelope in the presupernova evolutionary phase would decrease the efficiency of 11 B production in the ν-process during the SN explosion. Wolf-Rayet stars in fact exhibit strong activities in their stellar atmosphere such as convection and mass loss. They originate from stars as massive as 40 M ⊙ (e.g., Abbott & Conti 1987; Meynet et al. 2001) . Since the SNe which we discuss in the present article have main sequence mass of 13 M ⊙ ∼ 30 M ⊙ , there is no need to take account of such a mass loss effect. Introduction The production of 7 Li and 11 B during the SN explosion may depend on the SN models. It also depends on the details of total neutrino luminosity and its time variation. Moreover, yet unclarified neutrino temperature should strongly affect the ν-spallation cross sections of 4 He that provide seed elements for the production of 7 Li and 11 B (Woosley & Weaver 1995) . Since the neutrino spectra would thereby affect the final abundance of 7 Li and 11 B (Fields et al. 2000; Yoshida, Emori, & Nakazawa 2000) , we should investigate the dependence of these elemental abundances on the neutrino spectra in order to solve the overproduction problem. Introduction As for the SN r-process nucleosynthesis in the neutrino-driven winds, the r-process occurs in "hot bubble" region between the surface of a neutron star and the outward shock wave, in which the density is relatively low and the temperature and entropy are high. Although Woosley et al. (1994) showed that the r-process occurs successfully in the hot bubble region with very high entropy, 400k, where k denotes Boltzmann constant, their nucleosynthesis calculation did not include neutrino-nucleus interactions after the winds are driven by neutrino heating. It was subsequently pointed out that the supernova neutrinos convert neutrons into protons during the nucleosynthesis and that the r-process has difficulty in producing 3-rd peak elements due to the neutron deficiency even in high entropy hotbubble (Fuller & Meyer 1995; Meyer 1995) . It was also reported that independent simulations of neutrino-driven winds have difficulty in producing the required high entropy condition . Thus, the neutrinodriven winds were suspected as a site of the r-process. Introduction Recently, neutrino-driven wind models have revived as promising sites of the r-process nucleosynthesis by taking a moderately high entropy, ∼ 200k, and very short expansion time scale, ∼ 10 ms Sumiyoshi et al. 2000) . They assumed massive (∼ 2.0M ⊙ ) and compact (10 km) neutron star models in order to obtain such conditions. It is known, however, that a typical mass of neutron star is about 1.4M ⊙ and that a radius is about 10 km. Terasawa et al. (2002) have recently shown a possibility that successful r-process abundance pattern emerges from the neutron star model with a typical mass of 1.4M ⊙ and a radius of 10 km by setting suitable outer boundary condition of slightly low asymptotic temperature of the neutrino-driven winds. In all these simulations, they set mean energies of neutrinos to be about 10, 20, and 30 MeV for ν e ,ν e , and ν i (i = µ, τ , and their antineutrinos), respectively, to match with those adopted in the previous theoretical study (Qian & Woosley 1996) . Introduction In the light of successful r-process nucleosynthesis in the neutrino-driven winds, it is of current interest and importance to study how to solve the overproduction problem of the light elements in the ν-process in the same supernova. Since neutrinos are very weakly interacting particles, their energy spectrum would not change in the ejecta unless neutrino oscillation is considered. Nevertheless, there was no attention to the fact that the neutrino spectrum, which is used for the light element synthesis during SN explosions, should be identical to the spectrum adopted to the neutrino-driven wind models. Introduction In the present article we set common neutrino luminosity which decreases with time in the application to both light element synthesis in the He-layer and r-process synthesis in neutrino-driven winds. We investigate the sensitivity of light element synthesis in SN ejecta to the neutrino luminosity with two parameters of the decay time, τ ν , and the total energy of the supernova neutrinos, E ν . At the same time, we simulate the neutrino-driven wind models with the same neutrino luminosity parameters and calculate the r-process abundance pattern in those models. We thus discuss the consistency between the light element production and the abundance distribution of the r-process elements and try to solve the overproduction problem of the light elements in the GCE. Introduction In addition to the ambiguity of the neutrino luminosity, neutrino temperature is still a controversial problem. Although a lot of numerical studies of neutrino transfer have been done by several groups (e.g., Burrows et al. 2000; Janka, Buras, & Rampp 2001; Keil, Raffelt, & Janka 2002) , neutrino spectrum has not been uniquely determined. Hence, it is also important to investigate the sensitivity of the light element production and the r-process abundance pattern to the temperature of e-, µ-, and τ -neutrino families using a common neutrino spectra. Recent theoretical studies on the explosive nucleosynthesis in supernovae (e.g. Rauscher et al. 2002) have shown smaller ejected mass of 11 B in a good direction to solve the overproduction problem by assuming the temperature of µ-and τ -type neutrinos and their antineutrinos slightly lower than the temperature adopted in Woosley & Weaver (1995) . We therefore explore many different neutrino luminosity with different neutrino temperature in order to look for successful ejected mass of 11 B and an appropriate r-process abundance pattern. This result would in turn strongly constrain the spectra of supernova neutrinos. Calculations The Neutrino Spectra In order to investigate the relation between the ejected mass of the light elements and the r-process elements, we set a common model for neutrino spectra based on Woosley et al. (1990) and Woosley & Weaver (1995) . The neutrino luminosity, L ν i (ν i = ν e , ν µ , ν τ , and their antineutrinos), is the same for all species and exponentially decreases with a decay time, τ ν , The Neutrino Spectra where E ν is the total neutrino energy, r is the radius, c is the speed of light, Θ(x) is the step function defined by Θ(x) = 1 for x > 0 and 0 otherwise. The total neutrino energy and the decay time of the neutrino luminosity are parameters. In the above two papers, they fixed τ ν = 3 s and E ν = 3 × 10 53 erg. Here, we adopt more wide range of these parameters. We set the decay time of the neutrino luminosity in the range between 1 s and 3 s. The total neutrino energy, E ν , is evaluated approximately as six times the total energy emitted asν e 's. The determined total energy ofν e 's from SN 1987A (Hirata et al. 1987; Bionta et al. 1987) leads to the total neutrino energy ranging 1.4 × 10 53 erg E ν 6.1 × 10 53 erg (Suzuki 1994) . The Neutrino Spectra Since error bars are very large, depending on different methods of the maximum likelihood analysis of the observed data, we vary the total neutrino energy, E ν , in the range between 1.0 × 10 53 erg and 6.0 × 10 53 erg in the present study. The Neutrino Spectra When we evaluate the reaction rates of the ν-process and thermal evolution of the neutrino-driven winds, we further need to know the temperature of all species of neutrinos. The Neutrino Spectra We set the temperature of µ-and τ -type neutrinos and their antineutrinos, T ν i , as follows: The Neutrino Spectra This set is taken from Woosley et al. (1990) and Woosley & Weaver (1995) . The temperature corresponds to the mean energy of 25 MeV. The temperature of electron-type neutrinos and their antineutrinos are set to be The Neutrino Spectra and The Neutrino Spectra respectively. The corresponding mean energy of the neutrinos are 10 MeV and 16 MeV. Chemical potentials of all species of neutrinos are assumed to be zero. The Neutrino Spectra Let us remark that the temperatures of ν e andν e are different from those in Woosley & Weaver (1995) , in which they assumed the same temperature 4 MeV/k for both ν e andν e . We note that T νe is not equal to Tν e in the previous studies of the r-process nucleosynthesis and the supernova neutrinos (e.g., Woosley et al. 1994; Mezzacappa et al. 2001; Janka et al. 2001) . We also note that the temperature of µ-and τ -type neutrinos and their antiparticles lower than 8 MeV/k has been reported in recent Monte Carlo study of SN neutrino spectra formation (Keil et al. 2002) . Even the temperature lower than in equation (2), i.e., T ν i = 6 MeV/k (i = µ, τ , and their antiparticles), is also used in the later discussions as well as 8 MeV/k. Supernova Explosion Model for Light Element Nucleosynthesis We study the light element synthesis by the postprocessing nucleosynthesis calculation of a supernova explosion. The presupernova model is 14E1 model (Shigeyama & Nomoto 1990) which corresponds to a model for SN 1987A. This model is constructed from the precollapse 6 M ⊙ helium star (Nomoto & Hashimoto 1988 ) and 10.2 M ⊙ H-rich envelope. Chemical composition of the presupernova model is taken from 14E1 model. In the H-rich envelope, the mass fractions of 1 H and 4 He are set to be X = 0.565 and Y = 0.43, respectively. For the postprocessing calculation, the abundance distribution of the CNO-elements is assumed to be the equilibrium values of the CNO-cycle, and the abundances of heavier elements are assumed to be one-third of those of the solar-system abundances, i.e., Z = 0.005. Supernova Explosion Model for Light Element Nucleosynthesis In order to carry out the postprocessing calculation of the light element synthesis in the supernova, we have to know time evolutions of the temperature, density, and radius during the supernova explosion. In the present study we evaluate the propagation of a shock wave during the supernova explosion using a spherically symmetric Lagrangian PPM (piecewise parabolic method) code (Colella & Woodward 1984; Shigeyama et al. 1992 ) which includes a small nuclear reaction network containing thirteen α-nuclei. The explosion energy is set to be 1 × 10 51 erg and the location of the mass cut is assumed to be 1.61 M ⊙ . Supernova Explosion Model for Light Element Nucleosynthesis A nuclear reaction network for the postprocessing calculation of the light element synthesis consists of 291 nuclear species up to Ge (Table 1) . Reaction rates in this network are adopted from NACRE (Angulo et al. 1999) , Caughlan & Fowler (1988) , Rauscher & Thielemann (2000) , Bao et al. (2000) , Chart of the nuclides (Horiguchi et al. 1996 ), Fuller, Fowler, & Newman (1982 , and Oda et al. (1994) . Reaction rates of the ν-process are adopted from Hoffman & Woosley (1992) . Neutrino-Driven Wind Models for the r-Process Nucleosynthesis We adopt models of the neutrino-driven winds for the r-process nucleosynthesis in the core-collapse supernova explosion. When the proto-neutron star is formed in the supernova explosion, a plenty of neutrinos is emitted and heats surface materials to form the hot bubble at high entropy. This hot bubble, which blows off from the proto-neutron star, is called the neutrino-driven winds. The thermodynamic conditions in the hot bubble region strongly depend on the neutrino properties. We use the same neutrino spectra as we described in section 2.1 for numerical simulations of the neutrino-driven winds and the r-process nucleosynthesis. Instead of following the time evolution of the neutrino luminosity, we run a set of different simulations of the neutrino-driven winds with constant and different neutrino luminosities, and superpose the calculated results of nucleosynthesis as employed by Wanajo et al. (2001 Wanajo et al. ( , 2002 . This is justified because the decay time of the neutrino luminosity with τ ν = 1 ∼ 3 s is slow enough compared with expansion time scale of the neutrino-driven winds and nuclear reaction time scale of α-and r-processes Terasawa et al. 2001) . We refer the time t = t end when the luminosity decays to as low as L ν i = 3.5 × 10 51 erg s −1 . We choose three representative times at 0, t end /2, and t end for each model of neutrino spectra and superpose three calculated results of hydrodynamical simulations and r-process nucleosynthesis (see Table 2 ). Neutrino-Driven Wind Models for the r-Process Nucleosynthesis To follow the hydrodynamical evolution of neutrino-driven winds, we employ an implicit numerical code for the general-relativistic and spherically symmetric hydrodynamics (Yamada 1997; Sumiyoshi et al. 2000) including the heating and cooling processes due to neutrinos (Qian & Woosley 1996) . As an initial condition, we put thin surface material upon a neutron star with the typical mass of 1.4M ⊙ and radius of 10 km. We obtain the structure of this matter by solving the Oppenheimer-Volkoff equation. As an outer boundary condi-tion, we put a constant pressure, P out , next to the outermost grid point of Lagrangian mesh. The value of P out is taken to be 10 20 dyn cm −2 for all simulations, since the low asymptotic temperature due to the low outer pressure is favorable for the r-process nucleosynthesis (Terasawa et al. 2002) . Neutrino-Driven Wind Models for the r-Process Nucleosynthesis We start the network calculations of the nucleosynthesis at the time when the temperature drops to T 9 = 9.0 (in units of 10 9 K) and follow the time evolution of abundances. The reaction network covers over 3000 species of nuclei from the β-stability line to the neutron drip line including light neutron-rich unstable nuclei (Terasawa et al. 2001) . It includes only the charged current neutrino-nucleus interactions for all nuclei. Since the neutral current interactions have little influence on the final composition of material when a time scale of explosions is very short (Terasawa et al. 2003) as in our present model, we did not include these reactions in the current studies of the r-process. Neutrino-Driven Wind Models for the r-Process Nucleosynthesis In order to estimate the total ejected mass of each isotope, M eject,i , for a given neutrino spectra, we sum up three results of nucleosynthesis ensembles with different neutrino luminosities in the following trapezoid-formula, Neutrino-Driven Wind Models for the r-Process Nucleosynthesis where t end is the time when the luminosity becomes 3.5 × 10 51 erg s −1 , as defined before, M 0,i ,Ṁ half,i , andṀ end,i are the mass ejection rate of isotope, i, obtained from the calculation for three neutrino luminosities L ν i = L ν i ,0 , L ν i ,half , and L ν i ,end at t = 0, t end /2, and t end , respectively. Results Abundances of 7 Li and 11 B In this subsection we examine the influence of the neutrino spectra on the ejected masses of 7 Li and 11 B. We first present the calculated mass fraction distribution of the light elements and explain main production processes of 7 Li and 11 B in the case of E ν = 3 × 10 53 erg and τ ν = 3 s, which is the same parameter set as that adopted in Woosley & Weaver (1995) . Figure 1 shows the mass fractions of the light elements as a function of the mass coordinate, M r . We see that 7 Li and 11 B are abundantly produced in the He/C-layer in the range of 4.6M ⊙ M r 5.8M ⊙ and 4.2M ⊙ M r 5.0M ⊙ ,respectively. In addition, 10 B is produced in the regions below the He/N-layer but less abundantly. A small amount of 6 Li and 9 Be is also produced in the O/C-layer and the outer part in the He/C-layer. Abundances of 7 Li and 11 B Let us first discuss the main production process of 7 Li in the 7 Li production region (4.6M ⊙ M r 5.8M ⊙ ). At first, neutrinos emitted from the collapsed core break up 4 He through 4 He(ν, ν ′ p) 3 H and 4 He(ν, ν ′ n) 3 He. The produced 3 H and 3 He capture 4 He to produce 7 Li through 3 H(α, γ) 7 Li and 3 He(α, γ) 7 Be(e − , ν e ) 7 Li. The main production process of 11 B in the 11 B production region (4.2M ⊙ M r 5.0M ⊙ ) needs another step in addition to the 7 Li production: The produced 7 Li leads to 11 B through 7 Li(α, γ) 11 B. Note that the α-capture reaction from 7 Be, i.e., 7 Be(α, γ) 11 C(e + ν e ) 11 B also proceeds but only small amount of 11 B is produced through this reaction sequence. Abundances of 7 Li and 11 B In both sides of inner and outer mass coordinates surrounding the 7 Li and 11 B production regions mentioned above, the mass fractions of 7 Li and 11 B are smaller. In the region below the He/C-layer, the mass fraction of 7 Li is much smaller than that in the He/C-layer because the mass fraction of 4 He, which is the seed nuclei of 7 Li, is very small. The mass fraction of 11 B is also smaller. In this region 11 B is produced through 12 C(ν, ν ′ p) 11 B and 12 C(ν, ν ′ n) 11 C(e + ν e ) 11 B. In the O/C-layer, a large fraction of 11 B is also produced through the ν-process of 12 C(ν, ν ′ p) 11 B. However, the total mass of 11 B in the O/C-layer is much smaller than that in the He/C-layer because the total mass of the O/C-layer is only 0.1 M ⊙ . In the range of M r 4.2M ⊙ in the He/C-layer, the temperature becomes so high upon the shock arrival that the produced 7 Li and 11 B capture 4 He to produce 11 B and 14 N through 7 Li(α, γ) 11 B and 11 B(α, n) 14 N, respectively. In the H-rich envelope, the maximum temperature does not become high enough to proceed 3 H(α, γ) 7 Li and 3 He(α, γ) 7 Be(e − , ν e ) 7 Li so that 7 Li is not produced. In this region, 11 B is produced through 12 C(ν, ν ′ p) 11 B but the produced mass fraction is extremely small because of the small mass fraction of 12 C. Now, we describe the relation of the total ejected masses of 7 Li and 11 B to the parameters of the neutrino luminosity, i.e., the total neutrino energy, E ν , and the decay time of the neutrino luminosity, τ ν . Figures 2a and 2b show the ejected masses of 7 Li and 11 B as a function of the total energy, E ν . We find two clear features of the parameter dependence. One is that the total ejected mass is almost proportional to the total energy, E ν , for a given decay time, τ ν . The other is that the ejected mass for a given E ν is insensitive to the decay time, τ ν . In Table 2 we list the sets of the two parameters adopted for the r-process calculations in the ranges of the total energy of 1 × 10 53 erg ≤ E ν ≤ 3 × 10 53 erg and the the decay time of 1 s ≤ τ ν ≤ 3 s. In the given range of the total energy, the ejected masses of 7 Li and 11 B change within a factor of 2.7 and a factor of 2.9. The variation of the ejected masses of 7 Li and 11 B due to the decay time are within 10 % for any E ν values. Hence, we can conclude that, in the light element nucleosynthesis during supernova explosions, the ejected masses of 7 Li and 11 B are almost proportional to the total energy and the variation is within a factor of three, but insensitive to the decay time. Abundances of 7 Li and 11 B Simple relation between the ejected mass and the total neutrino energy arises from the following specific properties of the reaction processes to produce 7 Li and 11 B. In the He/Clayer, 7 Li and 11 B are mainly produced through the three reaction chains: 4 He(ν, ν ′ p) 3 H(α, γ) 7 Li(α, γ) 11 B, 4 He(ν, ν ′ n) 3 He(α, γ) 7 Be(e − , ν e ) 7 Li, and 12 C(ν, ν ′ p) 11 B. At higher temperature 11 B(α, n) 14 N reaction also occurs. All these reaction sequences are triggered by the neutrino spallations, and followed by α-capture reactions. Total neutrino energy is proportional to the neutrino number flux. The spallation reaction rates for 4 He(ν, ν ′ p) 3 H, 4 He(ν, ν ′ n) 3 He, and 12 C(ν, ν ′ p) 11 B are also proportional to the total neutrino energy. The amounts of seed nuclei of 7 Li and 11 B, namely 4 He and 12 C, are determined solely by the composition of the presupernova and independent of the neutrino spectra. Although the mass of 4 He and 12 C in the He-layer changes depending on presupernova models, the ejected masses of 7 Li and 11 B are almost proportional to the total neutrino energy alone because the production regions of these nuclei are limited within the middle of the He-layer. Abundances of 7 Li and 11 B We turn to the insensitivity of the ejected masses of 7 Li and 11 B to the decay time of the neutrino luminosity as evidenced in Figure 2 . In the 7 Li and 11 B production region, their mass fractions do not strongly depend on the decay time of the neutrino luminosity. This is because all neutrinos promptly pass through the region before the shock arrival, and 7 Li and 11 B are not effectively processed after the shock passes by. In the bottom of the He/C-layer (M r 4.2M ⊙ ), where both 7 Li and 11 B are scarcely produced, the shock wave arrives earlier (about 10 s) and further α-capture reactions on the produced 7 Li and 11 B proceed in several seconds after the shock arrival. These α-capture processes do not depend on τ ν . However, there is a slight difference, about 10 %, between the two cases of τ ν = 1 s and 3 s. This arises from the competition between the shock arrival and the time scale of the production processes of 7 Li and 11 B through the ν-process which depends on τ ν . In the case of τ ν =3 s, a fractional part of 7 Li and 11 B is still being produced through the ν-process even after the shock arrival. In the case of τ ν =1 s, however, the ν-process ends up quickly before the shock arrival, so that the produced 7 Li and 11 B capture 4 He when the shock arrives. This difference leads to a small decrease of the amounts of 7 Li and 11 B in the case of τ ν =1 s compared with the case of τ ν = 3 s. Abundances of 7 Li and 11 B Finally, we compare our result with the ejected masses of 7 Li and 11 B in S20A model of Woosley & Weaver (1995) . In the case of E ν = 3 × 10 53 erg and τ ν =3 s, the obtained masses of 7 Li and 11 B in our calculation are 7.46 × 10 −7 M ⊙ and 1.92 × 10 −6 M ⊙ , respectively. The masses of 7 Li and 11 B in the S20A model are 6.69 × 10 −7 M ⊙ and 1.85 × 10 −6 M ⊙ , respectively (see solid horizontal line in Figure 2 ). Our result is in reasonable agreement with Woosley & Weaver (1995) though the calculated masses of 7 Li and 11 B are 12% and 4% larger than the corresponding masses in the S20A model. Therefore, the difference due to the temperatures of ν e andν e does not affect the ejected masses of 7 Li and 11 B largely. This agreement confirms that the overproduction problem of 11 B in the context of the GCE of the light elements still remains independently of the specific SN model. We will discuss how to solve this problem in section 4. Abundances of the r-Process Elements The final total isotopic distribution of the ejected mass of the r-process elements is presented as a function of mass number in Figure 3 for the three sets of neutrino spectrum listed in Table 2 . Dashed line refers to the case with low total neutrino luminosity, E ν = 1.0 × 10 53 erg, and long decay time, τ ν = 3.0 s (LL model). Solid line corresponds to the case with E ν = 1.0 × 10 53 erg but short decay time τ ν = 1.0 s (LS model). Dotted line is the case with high total neutrino energy, E ν = 3.0 × 10 53 erg, and long decay time, τ ν = 3.0 s (HL model). We can observe in this figure that the 3-rd peak elements are synthesized in all three models, though the height of the peak and the ejected mass of these elements largely differ from each other. Abundances of the r-Process Elements Let us first compare the result of the LL model with that of the LS model in order to investigate the dependence of final abundance distribution on the decay time. Figure 3 displays that the ejected mass in the LS model is larger than that in the LL model. Since the value of E ν is the same between these two models, the difference of τ ν brings about directly the different value of neutrino luminosity (see eq. [1] and Table 2 ). In the LS model with short decay time scale, τ ν = 1.0 s, the peak luminosity, L ν i ,0 , is higher than the peak luminosity in the LL model with the long decay time scale, τ ν = 3.0 s, because of a common total neutrino energy, E ν = 1.0 × 10 53 erg. It is known that the mass ejection rate increases faster than the neutrino luminosity (Woosley et al. 1994) . The increase rate of the total ejected mass becomes higher than the increase rate of the peak luminosity. Therefore, the total ejected mass becomes larger as the decay time becomes shorter. Abundances of the r-Process Elements Secondly, we find an interesting fact that the abundance ratio of the 3-rd to the 2-nd peak elements in the LL model is larger than that in the LS model. The reason is as follows. From previous studies of neutrino-driven winds, favorable conditions for a successful r-process have been identified: They are higher entropy (s/k), shorter dynamical time scale (τ dyn ), lower electron fraction (Y e ) (e.g. Meyer & Brown 1997) , and lower asymptotic temperature (T out ) (Terasawa et al. 2002) . In our present model calculations, Y e and T out are almost the same because we employ a common neutrino temperature and the same outer boundary conditions. It was also found that the entropy and the dynamical time scale become larger as the neutrino luminosity is lower (Qian & Woosley 1996; Otsuki et al. 2000; Sumiyoshi et al. 2000) . The values of s/k and τ dyn change depending on τ ν through the change of the neutrino luminosity. Even combining the above theoretical findings, one cannot simply explain the difference of the 3-rd/2-nd peak abundance ratios between the LS model and the LL model because resultant effects from the change in s/k and τ dyn counteract the production efficiency of the r-process elements. From our numerical calculations, we find that the gain for the r-process nucleosynthesis due to the increase of entropy is quantitatively larger than the loss due to the increase of dynamical time scale. This means that more abundant 3-rd peak elements relative to 2-nd peak elements are synthesized as the luminosity becomes lower. In both the LS and LL models, efficiency of producing the 3-rd peak elements is higher for L ν i = L ν i ,end than for L ν i = L ν i ,0 and L ν i ,half . In addition, as we discussed in the previous paragraph, the peak luminosity in the LL model is lower than that in the LS model (see Table 2 ). For these reasons, the efficiency of producing the 3-rd peak elements in the LL model is most prominent, leading to larger abundance ratio of the 3-rd to the 2-nd peak elements in the LL model than that in the LS model. Abundances of the r-Process Elements We, next, consider the dependence on the total neutrino energy, E ν . We compare the results of the LL and HL models in Figure 3 . Since these two models have the different total energies (see Table 2 ) with a common decay time, the total ejected mass is larger and the 3-rd to 2-nd peak ratio is smaller in the HL model than those in the LL model for the reasons discussed in the previous paragraphs. Abundances of the r-Process Elements Finally, we compare the HL model with the LS model. We obtain the result that the pattern of isotopic abundance distribution is exactly the same in these two models, though the ejected mass in the HL model is larger. Note that the values of L ν i ,0 , L ν i ,half , and L ν i ,end are exactly the same between the two models by the definition of L ν i in equation (1). As a result, we find that the key quantity is the neutrino luminosity for determining the pattern of the r-process abundance distribution. Abundances of the r-Process Elements In summary, the lower peak luminosity and the longer decay time are preferable in order to obtain the successful r-process abundance pattern. In Figure 3 , we also display the comparison of our calculated results with the solar r-process abundance pattern in arbitrarily units (Käppeler et al. 1994) . We can conclude that the LL model is the best among our adopted three models in order to reproduce the solar r-process abundance pattern. Discussion In this section, we discuss the constraint on neutrino spectra from the contribution of the two nucleosynthesis processes to the galactic chemical evolution. We then propose new neutrino spectrum modeling in order to solve the overproduction problem of 11 B. We also discuss some implication of our predicted masses of 7 Li and 11 B from the SN ν-process in the different sources of the light elements including the primordial nucleosynthesis. Galactic Chemical Evolution of 11 B Let us first discuss the overproduction problem of 11 B. Recently, the studies on the GCE of the light elements have shown that 7 Li and 11 B originate from supernova explosions as well as galactic cosmic ray (GCR) interactions with interstellar medium (ISM) (e.g., Olive et al. 1994 ). The contribution from supernova explosions is evaluated so that the predicted 11 B/ 10 B ratio at the solar metallicity agrees with the meteoritic ratio. The evaluated contribution of 11 B from supernovae is smaller than that predicted from the supernova explosion models in Woosley & Weaver (1995) . We refer their models as WW95 hereafter in this section. Several authors have investigated the GCE of the light elements and have introduced a reduction factor, f ν , as the ratio of the amount of 11 B determined in the GCE model to that evaluated in WW95 (e.g., Vangioni-Flam et al. 1996; Fields & Olive 1999) . Namely, the amount of 11 B is the same as that in WW95 for f ν = 1 and is less than that for f ν < 1. Fields et al. (2000) , Ramaty et al. (2000a) , Ramaty et al. (2000b) , and Alibés, Labay, & Canal (2002) have evaluated the factor f ν to be 0.40, 0.18, 0.28, and 0.29, respectively. The scatter in f ν is mainly caused by the different treatment of GCRs: Assumed chemical composition and energy spectra of the GCRs. Since the factor f ν still depends on the treatment of GCRs and has not been precisely determined, we set acceptable range of the reduction factor to be 0.18 ≤ f ν ≤ 0.40, Galactic Chemical Evolution of 11 B in this paper. The ejected masses of 11 B with the largest and smallest values are shown in Figure 2b as denoted by f ν = 0.40 and f ν = 0.18 together with that of WW95. Figure 2b shows that the calculated ejected mass of 11 B in our standard neutrino spectra described in section 2.1 meets with the above range of f ν , (see eq. [6]), only when the total neutrino energy is as low as E ν 1.2 × 10 53 erg. This energy is much lower than 3 × 10 53 erg, which is the value used in WW95 in accordance with the value constrained from the observation of SN 1987A. In this energy the ejected mass of 7 Li is also smaller than that in WW95 as shown in Figure 2a . r-Process Nucleosynthesis Constraint We turn now to the constraint from the r-process nucleosynthesis. The total ejected mass of heavy r-process elements, M eject , is between 9.0 × 10 −6 M ⊙ (for the LL model) and r-Process Nucleosynthesis Constraint 1.1 × 10 −4 M ⊙ (for the HL model). This range is consistent with the GCE of the r-process elements: Assuming that the Type II supernova rate is on the order of 10 −2 y −1 over the entire history of the Galaxy evolution, current mass of the r-process elements in the Galaxy is estimated to be 9 × 10 2 M ⊙ (for the LL model) and 1.1 × 10 4 M ⊙ (for the HL model). Since the total baryonic mass of the Galaxy is ∼ 10 11 M ⊙ , our models lead to the present mass fraction in r-process elements of order ∼ 10 −8 (for the LL model) and ∼ 10 −7 (for the HL model). These values are in reasonable agreement with the observed solar mass fraction of ∼ 10 −7 . We recall furthermore the discussion in section 3.2 that the 3-rd to 2-nd peak ratio is sensitive to the neutrino luminosity and that the LL model in Table 2 (E ν = 1.0 × 10 53 erg and τ ν = 3.0 s) is most favorable for explaining the observed r-process abundance pattern. r-Process Nucleosynthesis Constraint Thus, summarizing the constraints from the two nucleosynthesis processes, the neutrino luminosity with E ν = 1.0 × 10 53 erg can resolve the overproduction problem of 11 B and the r-process abundance pattern as far as the decay τ ν is longer than or equal to 3 s. New Neutrino Spectrum Model The neutrino spectrum model suggested above, however, encounters a potential conflict between the total neutrino energy E ν and the gravitational mass of the neutron star which is formed in Type II supernova explosion. Lattimer & Yahil (1989) suggested an approximate relation between the gravitational binding energy E BE and the neutron star mass M NS . Since it is known that almost 99 % of the binding energy is released as supernova neutrinos, E BE is equal to E ν to a very good approximation. Their suggested relation is thereby expressed as New Neutrino Spectrum Model This formulation is shown to be a reasonable approximation in the theoretical studies of several non-relativistic potential models and field theoretical models (Prakash et al. 1997; Lattimer & Prakash 2001) . By using this formulation, the total neutrino energy for the neutron star mass of 1.4 M ⊙ turns out to be 2.4 × 10 53 erg E ν 3.5 × 10 53 erg, New Neutrino Spectrum Model within ±20 % error bars. This range is displayed by two vertical lines in Figures 2a and 2b . Although we summarized above that the most suitable total neutrino energy is E ν = 1.0×10 53 erg from the constraints on the two nucleosynthesis processes, it is inconsistent with equation (8) which is based on the neutron star formation conjecture in Type II supernova explosion. New Neutrino Spectrum Model In order to solve this inconsistency, we modify the temperature of µ-and τ -type neutri-nos (ν µ and ν τ ) and their antineutrinos, which we set to be 8 MeV/k as in WW95. Rauscher et al. (2002) have tried to reduce the amount of 11 B by decreasing these neutrino temperature from 8 MeV/k to 6 MeV/k. The value of these neutrino temperature has also been suggested in recent studies (e.g., Myra & Burrows 1990; Keil et al. 2002) . We therefore investigate the ejected masses of 7 Li and 11 B by adopting the same lower temperature for ν µ , ν τ , and their antineutrinos of 6 MeV/k. New Neutrino Spectrum Model We show the calculated result in the case of τ ν = 3 s by a dash-dotted line in Figure 2 . The ejected masses of 7 Li and 11 B decrease drastically. This is because smaller amounts of seed nuclei, such as 3 H and 3 He for the production of 7 Li and 11 B, are provided from the neutrino spallation of 4 He due to smaller cross sections of neutral-current interactions at the lower neutrino temperature. The ejected mass of 11 B now turns out to be in the proper range as required from the GCE model analyses when the total neutrino energy E ν is between 1.6 × 10 53 erg and 3.5 × 10 53 erg (see Figure 2 ). Combining this restriction with neutron star mass constraint in equation (8), we finally obtain the range of mass of 11 B between 5.1 × 10 −7 M ⊙ and 7.3 × 10 −7 M ⊙ which corresponds to the reduction factor between 0.27 and 0.40. Hence, we can conclude that the ejected mass of 11 B required from the GCE models is successfully reproduced with the appropriate total neutrino energy of equation (8) which is carried away in neutrinos from supernova when one adopts the neutrino temperature of 6 MeV/k. When we reduce the neutrino temperature from 8 MeV/k to 6 MeV/k, the dynamical time scale of expansion of the neutrino-driven winds becomes slightly longer. A long time scale of explosion hinders the synthesis of the r-process heavy elements. However, this effect by reducing neutrino temperature does not drastically change the r-process abundance pattern as for 11 B. More details will be reported in the forthcoming paper. Implication of Predicted 7 Li and 11 B We can predict how much amount of 7 Li is produced in the SN ν-process in our best model to explain the GCE of 11 B. From Figures 2a and 2b , we summarize that the ejected masses of 7 Li and 11 B are Implication of Predicted 7 Li and 11 B and Implication of Predicted 7 Li and 11 B The predicted mass of 7 Li would help understand the still controversial origin of 7 Li. This nucleus has at least five different production sources: The primordial nucleosynthesis, the GCR spallation and α + α fusion reactions which also produce 6 Li, 9 Be, and 10,11 B, the AGB star nucleosynthesis (Forestini & Charbonnel 1997) , the nova nucleosynthesis (José & Hernanz 1998) , and the SN nucleosynthesis via the ν-process as we discussed in this article. Implication of Predicted 7 Li and 11 B Since the three sources of the GCR, AGB, and nova nucleosyntheses have been studied theoretically, we now discuss an implication of our result in the primordial nucleosynthesis. Implication of Predicted 7 Li and 11 B Identification of the primordial abundance of 7 Li depends on the flatness of "lithium plateau" in Population II stars at low metallicity region [Fe/H] ≤ -1 where the SN ν-process and the GCR α + α fusion reaction still make appreciable contributions in addition to the primordial abundance. If one applies our predicted mass of 7 Li to the GCE model of Fields & Olive (1999) , we would more precisely determine the depleted abundance level of the primordial 7 Li from the observed "lithium plateau" (Pinsonneault, Deliyannis, & Demarque 1992; Kurucz 1995) . Thus inferred depletion factor of the primordial abundance of 7 Li would make an impact on the concordance of cosmological baryon-density parameter Ω b between the standard Big-Bang nucleosynthesis model prediction (Burles, Nollett, & Turner 2001; Kirkman et al. 2003) and the WMAP data of the cosmic microwave background anisotropy (Spergel et al. 2003) . The depletion factor would also be compared directly to the value determined in a new method which is to use the observed isotopic abundance ratios of ISM 7 Li/ 6 Li which were detected by the use of SUBARU/High-Dispersion-Spectrograph (Kajino et al. 2000b; Kawanomoto 2002; Kawanomoto et al. 2003) . Implication of Predicted 7 Li and 11 B The predicted masses of 7 Li and 11 B would also be valuable to test an alternative cosmological theory of the inhomogeneous Big-Bang nucleosynthesis models. These models predict enhanced primordial 9 Be and 11 B (Kajino & Boyd 1990; Orito et al. 1997 ) which originate from unstable nuclear reactions in proton and neutron segregated inhomogeneous baryon-number density distribution (Kajino, Mathews, & Fuller 1990; Kajino, Otsuki, & Orito 2002) . Applying our predicted mass of 11 B to the GCE would more precisely describe the linear metal-dependence of the 11 B abundance, i.e. [B/H] ∝ [Fe/H], because the SN ν-process for the production of 11 B is a primary process. One can thereby expect that the primordial "plateau" in 11 B will be found in very metal-deficient halo stars in the region [Fe/H] ≤ -3, as expected similarly to 9 Be (Kajino et al. 2000a) , if the inhomogeneous Big-Bang nucleosynthesis operated in the early Universe. Detailed studies of the GCE of the light elements are highly desirable with the use of our predicted masses of 7 Li and 11 B. Summary We investigated the influence of the neutrino spectra, i.e., the total neutrino energy and the decay time of the neutrino luminosity, on the light element synthesis. We also investigated the r-process nucleosynthesis in neutrino-driven winds with the same neutrino spectra. We summarize our findings as follows. Summary First, the ejected masses of 7 Li and 11 B are roughly proportional to the total neutrino energy. The difference due to the decay time of the neutrino luminosity is small, i.e., within 10 %. Therefore, in order to obtain the ejected masses of the light elements precisely, it is important to determine the total neutrino energy rather than the decay time of the neutrino luminosity. Summary Secondly, results of the r-process nucleosynthesis depend on the decay time and the peak luminosity of neutrinos. Although the r-process is sensitive to the total energy of neutrinos, only the ejected mass is affected strongly. In
{'paper_id': '11676791', 'title': 'Nucleosynthesis of Light Elements and Heavy r-Process Elements through the ν-Process in Supernova Explosion', 's2_url': 'https://api.semanticscholar.org/CorpusID:11676791', 'source': 'S2ORC'}