The Astrophysical Journal, 608:470–479, 2004 June 10 # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
NEUTRINO EFFECTS BEFORE, DURING, AND AFTER THE FREEZEOUT OF THE r-PROCESS M. Terasawa,1 K. Langanke,2 T. Kajino,1 G. J. Mathews,3 and E. Kolbe4 Received 2003 March 30; accepted 2004 February 20
ABSTRACT We study the effects of neutrino interactions before, during, and after the operation of the r-process in the context of a relativistic hydrodynamic neutrino-energized wind model that includes a Boltzmann solver for the dominant neutrino transport. We also employ newly available charged- and neutral-current interaction rates. Studies are made in a model with a short dynamical timescale that gives a fair reproduction of the solar system r-process abundance curve and hence could be a fair approximation to the true supernova environment. We confirm that charged- and neutral-current interactions can have specific unique effects on the final abundances. Early on, charged-current interactions determine the electron fraction, while later on, neutrino-induced neutron emission can continue to provide a slight neutron exposure even after the freezeout of the r-process. We propose two new potentially observable effects that derive from the use of a more realistic hydrodynamic model and /or new neutrino interaction rates. One is an enhanced odd-even effect in the final abundances, and the other is an enhancement of light A ¼ 68 76 nuclei in the final abundances. These effects are consistent with some recent observations of r-process abundances in metal-poor halo stars and might help to identify the neutrino fluxes present near the freezeout of the r-process. Subject headings: neutrinos — nuclear reactions, nucleosynthesis, abundances
1. INTRODUCTION
can completely dominate the environment just outside a newly born neutron star, their effects must be included in the nucleosynthesis calculations. Even though the site and conditions for the r-process are still unknown, core-collapse supernovae remain as a likely site. It is important, therefore, to search for abundance signatures that could identify uniquely that the r-process had occurred in a neutrino-energized environment. One goal of this paper has been to further investigate whether such a signature might exist. To make this investigation we will consider one possibly realistic set of conditions in a neutrinoenergized environment that produces a solar-like r-process abundance curve. We select parameters for a relativistic hydrodynamic model (see Sumiyoshi et al. 2000; Terasawa et al. 2002) that includes a Boltzmann solver for neutrino transport. In addition, we utilize new and much more detailed calculations of the charged- and neutral-current interaction rates in the present work than have been previously employed. Our intent is not to clam that this particular model is the site for the r-process, but merely to use it as a plausible schematic a framework in which we might search for new possible signatures of neutrino interactions. However small and difficult to measure such potential signatures may be, it is an important to carefully identify what possibilities might exist.
The rapid capture of neutrons by heavy nuclei (the r-process) is responsible for roughly half of the abundance of nuclei heavier than iron. However, the astrophysical site for this nucleosynthesis process is still a mystery that remains a major focus of nuclear astrophysics. Nevertheless, a popular current model (Woosley et al. 1994) involves the flow of neutrinoheated material into the high-entropy bubble above a nascent proto–neutron star in a core-collapse supernova. The r-process occurs in the region between the surface of the neutron star and the outward moving shock wave (Meyer et al. 1992). In this region the entropy is so high that the nuclear statistical equilibrium favors abundant free neutrons and protons rather than heavy nuclei. This is, therefore, an ideal r-process site that satisfies the requirement from observations that the yields be metallicity-independent (Sneden et al. 1996, 1998, 2000; Westin et al. 2000; Johnson & Bolte 2001; Cayrel et al. 2001; Honda et al. 2003, 2004). Using this model, Woosley et al. (1994) obtained an excellent fit to the solar r-process abundance pattern. However, the required high entropy in their simulation has not been duplicated by other numerical and semianalytic calculations (e.g., Witti et al. 1994; Takahashi et al. 1994; Qian & Woosley 1996; Otsuki et al. 2000; Thompson et al. 2001). It has also been pointed out (Terasawa et al. 2001) that the light-element reaction network used in those calculations may be too limited. Of particular interest for the present paper, however, is that they did not consider all possible neutrino-nucleus interactions (see Meyer 1995), although they were included (Meyer et al. 1992) to help to smooth the final abundance pattern. Since neutrinos
1.1. Background Neutrino-nucleus interaction processes during the r-process have been considered by a number of authors in the context of somewhat more schematic models than that of the present work (e.g., Meyer 1995; Fuller & Meyer 1995; McLaughlin et al. 1996; McLaughlin & Fuller 1996; 1997; Qian et al. 1997; Meyer et al. 1998). The present study differs from the previous investigations in that (1) we utilize a fully relativistic hydrodynamic model with a relativistic Boltzmann solver for the dominant neutrino transport and (2) we utilize new and more detailed charged and neutron-current interaction rates. In that sense, the present work is complementary to the important conclusions of those earlier studies. In particular, those studies
1 National Astronomical Observatory, Osawa, Mitaka, Tokyo 181-8588, Japan. 2 Institute for Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark. 3 Center for Astrophysics, University of Notre Dame, Notre Dame, IN 46556. 4 National Cooperative for the Disposal of Radioactive Waste, CH-5430 Wettingen, Switzerland.
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NEUTRINO EFFECTS AND r-PROCESS FREEZEOUT have shown (e.g., Meyer 1995) that neutrino interactions can hinder the r-process by decreasing the neutron-to-seed abundance ratio. This places important constraints on the supernova explosion model. However, a number of authors (e.g., Cardall & Fuller 1997; Qian & Woosley 1996; Hoffman et al. 1997; Otsuki et al. 2000; Sumiyoshi et al. 2000; Meyer 2002) have all shown that a short dynamic timescale model plus general relativistic effects can lead to successful r-process nucleosynthesis. This is because the temperature and density decrease very rapidly. Hence, charged particle reactions to make seed nuclei do not proceed efficiently, and only a small number of seed nuclei are produced. Moreover, since electronneutrino interactions on neutrons (converting neutrons into protons) is diminished, the neutron density remains high (Terasawa 2002; Langanke & Martinez-Pinedo 2003, Fig. 30) and a robust r-process follows. Thus, in spite of its problems, the neutrino-heated bubble remains a plausible and likely model for the r-process. Nevertheless, since this environment is dominated by neutrino interactions, it is important to carefully identify all possible influences of these neutrino interactions on the final individual elemental abundances. Only in this way can one hope to find a unique signature to identify the presence of neutrino interactions during the r-process. Such a signature could be used, for example, to finally unambiguously identify the supernova bubble as the r-process site. Among such possible signatures it was also shown by Qian et al. (1997) that even after freezeout of the r-process abundances, neutral-current and chargedcurrent postprocessing can cause a spreading of the abundance peaks and damping of the most pronounced features (peaks and valleys). They pointed out in particular that two mass regions, A ¼ 124 126 and 183–187, are particularly sensitive to this postprocessing effect. In this paper we attempt a complementary and in some respects more detailed exploration of the specific effects that the neutrino interactions might have on the final produced abundances. The present study is somewhat more detailed in that we use new (and partially unpublished) neutrino-induced reaction rates for nuclei heavier than -particles (see Langanke & Kolbe 2001, 2002; E. Kolbe & K. Langanke 2003, private communication; Kolbe et al. 2003). These rates involve a detailed calculation of nuclear states rather than a schematic treatment of the GT resonances as in previous studies. Also, rather than using a simple steady state or exponential expansion timescale (e.g., Qian et al. 1997; Meyer et al. 1998), we analyze detailed effects of both charged-current and neutral-current interactions in the context of a neutrino-energized relativistic hydrodynamic wind model with a Boltzmann solver for the dominant neutrino transport. We use a numerical hydrodynamic calculation based on the supernova core-collapse model of Yamada (1997) and Yamada et al. (1999). Although still schematic, this model is based on more detailed physics than the previous studies and is therefore arguably more realistic. We find, as in previous studies, that the effects of charged-current interactions in particular are identifiable in the final r-process abundance distribution even after freezeout and decay back to the line of stability. In addition to effects previously noted (e.g., Qian et al. 1997; Meyer et al. 1998), we find previously unreported effects that can be traced to the use of a hydrodynamic model and /or the use of new more detailed neutrino interaction rates. One is that the maximum sensitivity to neutrino reactions occurs in the region of light nuclei, A ¼ 68 76. Another is an enhanced odd-even effect in the final abundance distribution. This may be a new abun-
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dance signature that might be used to characterize the neutrino fluence during the r-process freezeout. 2. CALCULATIONS OF THE r-PROCESS IN NEUTRINO-DRIVEN WINDS As previously noted, a number of studies have been devoted to understanding the mechanism of supernova explosions and the r-process in neutrino-driven winds. Among these, some have explored the neutrino process before, during, and after the r-process. Our goal in this paper is to better quantify particular effects on the final abundance pattern due to charged- and neutral-current neutrino interactions on nuclei, especially after the freezeout of the r-process. For this purpose, we will adopt a relativistic hydrodynamic neutrinodriven wind model with a Boltzmann solver for the dominant neutrino transport effects. It has been shown (e.g., Otsuki et al. 2000; Sumiyoshi et al. 2000; Terasawa et al. 2002; Meyer 2002) that conditions with a short dynamical timescale in such a model give a good fit to the solar r-process abundance distribution. This model should in some ways be more realistic than previous schematic steady state or exponential expansion models of the wind. Here we give a brief summary of the detailed physics of this wind model and the nucleosynthesis calculation. 2.1. Supernova Model As in Terasawa et al. (2002) and Sumiyoshi et al. (2000), we employ a numerical supernova model based on implicit general-relativistic and spherically symmetric Lagrangian hydrodynamics (Yamada 1997; Yamada et al. 1999). This model has been modified to simulate the evolution of the bubble environment above the proto–neutron star. Relativity is incorporated using a spherically symmetric metric of the form ds2 ¼ e2ðt; mÞ c2 dt 2 e2kðt; mÞ
2 G dm2 c2
r 2 (t; m)(d2 þ sin2 d2 ):
ð1Þ
With this metric choice, the hydrodynamic equations of motion become @ 1 @ 4r2 F ¼ ; (4r2 U ) @t @m r ˜ @U @p q m ¼ 4r2 þ e 2 4r( p þ p ); @t h @m 4r2 r 1 @ 4r2 F @" 2 ¼ p (4r U ) p e Q; @t @m r 1 @ h @ 1 2 2 (4r pU ) 2 e U ¼ @m @t 2 h @ 1 ˜ þ 2 me @t r hU @r2 1 2 2e ( p þ p ) Uq @t p þ 4rF Q; Z @Ye f d3p ¼ mu e ; @t k coll: p0 e
@ ( 43 r3 )=@m ¼ ;
ð2Þ ð3Þ ð4Þ
ð5Þ ð6Þ ð7Þ
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where 1=b is the volume per unit baryon mass and is the U e ð@r=@t Þ Dt r is the radial fluid velocity. general relativistic gamma factor, and h is the specific enthalpy. The main adaptation of this model for the present purposes is to treat the proto–neutron star as a fixed inner boundary except for material above the neutrinosphere, which is heated in the wind. Since we consider very short expansion timescales, it is sufficient to treat the neutrino luminosity at the neutrinosphere as constant (see Yamada 1997; Yamada et al. 1999; Sumiyoshi et al. 2000 for further details).
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where the average energy, hEi i, Li and Ri are given as model parameters as described below. The number density of neutrinos, ni , at radius, r, is given by 1x hEi i2 fi0 42 1 x Li ; ¼ 2 hEi iR2i
ni ¼
e þ n $ e þ p; ¯e þ p $ eþ þ n;
i þ e $ i þ e ; i þ e þ $ i þ e þ ; ¯i þ e $ ¯i þ e ; ¯i þ eþ $ ¯i þ eþ ;
þ
i þ ¯i $ e þ e ;
ð8Þ ð9Þ ð10Þ ð11Þ ð12Þ ð13Þ ð14Þ
where the index i of i stands for neutrino flavors (i ¼ e; ; ). The first two reactions involving neutrino absorption on nucleons were evaluated (e.g., Yamada et al. 1999) using energy integrals containing the distribution functions of neutrinos and electrons from the Boltzmann solver. The Pauli blocking effects for electrons and positrons and the neutronproton mass difference are thus properly taken into account. The evolution of the electron fraction, Ye, was solved together with hydrodynamics using the collision terms for the neutrino absorption reactions of the Boltzmann solver (Yamada 1997; Yamada et al. 1999). For the remaining reactions above, however, it was sufficient to evaluate the rates as in Qian & Woosley (1996), whereby the neutrino spectra are approximated with simple distribution functions at each Lagrangian mesh point. For the parts of the calculation for which the Boltzmann solver was not employed, neutrino spectra were treated in a manner consistent with the usual assumptions (Qian & Woosley 1996) in neutrino-heated wind calculations. That is, the neutrino distribution functions, fi (Ri ), at the neutrinosphere were approximated as emerging isotropically at an average energy, fi (Ri ) ¼ fi0 (Ei hEi i):
ð15Þ
The coefficient, fi0 , is determined once the neutrino luminosity, Li , and neutrinosphere radius, Ri , are specified by fi0 ¼
2Li hEi i3 R2i
;
ð16Þ
ð18Þ
where x is defined as
2.2. Neutrino Heating/Cooling Neutrino heating and cooling processes within the bubble were calculated as described in Sumiyoshi et al. (2000), using a complete relativistic Boltzmann equation solver (Yamada et al. 1999) for the dominant reactions. The following neutrino reactions were included as sources of heating and cooling in the wind:
ð17Þ
R2 x ¼ 1 2i r
!1=2 :
ð19Þ
The solid angle subtended by the neutrinosphere is included as in Qian & Woosley (1996). Using the number density defined above, the neutrino distribution at a radius, r, is just fi (r) ¼ fir (Ei hEi i);
ð20Þ
where fir is given by fir ¼
2(1 x)Li hEi i3 R2i
:
ð21Þ
Further details on the model can be found in Sumiyoshi et al. (2000). 2.3. Model Parameters For the present investigation, we have adopted a numerical model (similar to those described in Sumiyoshi et al. 2000) for which the neutron-star gravitational mass is 1.4 M and the neutrinosphere radius is fixed at 10 km for all species. As for other parameters, the neutrino luminosity at the neutrinosphere is set to L;i ¼ 1051 ergs s1 for each species (i ¼ e, , , and their antineutrinos) and average neutrino energies of hEi i ¼ 10; 20; and 30 MeV are adopted for electron, antielectron, and / neutrinos, respectively. A constant pressure of Pout ¼ 1020 dyn cm2 is maintained at the outer boundary. This leads to a dynamical timescale of dyn 23 ms and an entropy per baryon of S=k 200. Because of this lower boundary pressure, the asymptotic temperature is also low, T9 0:4. This model was chosen because it gives a reasonable reproduction of the r-process abundance distribution between the second and third peaks for a single trajectory, even though the entropy is lower than the successful model with a high entropy (S=k 400) in Woosley et al. (1994) and the dynamical timescale is longer than the successful models with a short dynamical timescale ( dyn 10 ms) in Otsuki et al. (2000), Sumiyoshi et al. (2000), and Thompson et al. (2001). In view of the differences between our adopted model and some of the other models in the literature, we should clarify the underlying physics of this model. We emphasize, however, that this model is chosen for convenience in that it gives a nearly solar abundance distribution in a single trajectory so that an easy comparison with and without neutrino interactions can be made. Our point is not to claim this as a definitive but rather as an illustrative model. Nevertheless, this is a plausible model in that it actually produces a reasonable
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heavy-element abundance distribution, and it is no more extreme than a variety of models (many with less input physics) that have been proposed in the literature (see Meyer 2002). Regarding the conditions calculated by other groups, we note that our model indeed reproduces the results of Thompson et al. (2001) and Otsuki et al. (2000) when the code is matched to similar conditions. This was clearly demonstrated in Sumiyoshi et al. (2000). (See, e.g., Table 2 and model c01 in that paper. This model gives S=k ¼ 131 and dyn ¼ 50 ms for M ¼ 1:4 M, R ¼ 10 km, L ¼ 1051 ergs s1. The comparable model from Thompson et al. [2001] gives S=k ¼ 138, dyn ¼ 50 ms.) We did not use this trajectory, however, because it does not give a good r-process abundance distribution. We chose instead a model that gives a shorter dynamical time, a higher entropy, and a lower asymptotic temperature and therefore a good r-process abundance distribution (Terasawa et al. 2002). An explanation of how this trajectory could arise physically is therefore warranted. Our specification of the boundary condition on the model in terms of the outer boundary pressure is a potentially confusing point, which we now clarify. There is an important distinction between the hydrodynamic model described here and the steady state wind models of Thompson et al. (2001), Qian & Woosley (1996), Otsuki et al. (2000), and so on. Our hydrodynamic model is based on Lagrangian mass coordinates. A boundary condition must be imposed on the outer mass point. For this model we have specified the outer boundary by imposing equality of pressure (and other thermodynamic variables) once the pressure diminishes to a particular value. That is, we specify a pressure at which the last two zones are in pressure equilibrium. This is different from the steady state models (e.g., Otsuki et al. 2000; Thompson et al. 2001) in which an outer fixed point is used to constrain the trajectory. For example, in Otsuki et al. (2000) an asymptotic temperature of 0.1 MeV at R ¼ 104 km was assumed. This fixes the total energy in the wind. The model of Thompson et al. (2001) fixes the outer boundary by choosing the radius of the sonic point (wind speed equal sound speed). In Sumiyoshi et al. (2000) it was shown that the trajectories of the steady state model of Otsuki et al. (2000) and the hydrodynamic model used here are equivalent (except for a minor equation-of-state effect) when the outer boundary condition (outer pressure ¼ 1022 dyn cm2) is chosen to match the Otsuki et al. (2000) conditions at the fixed point. In essence, the outer boundary condition (taken here as a boundary pressure) fixes the asymptotic radius at which the velocity slows and begins to coast outward. Thus, choosing the boundary pressure to be 1020 dyn cm2 is equivalent to choosing a very large radius at which the radial velocity begins to slow and coast. In the wind model of Sumiyoshi et al. (2000) mentioned above (i.e., boundary pressure ¼ 1022 dyn cm2), material is beginning to slow as it crosses the radius at which the temperature drops below 0.5 MeV. (We define the timescale dyn as the e-folding time for the temperature to decrease once it falls below 0.5 MeV so the -process can begin.) In our adopted (low outer boundary pressure) model, material is not yet slowing down when it crosses T ¼ 0:5 MeV. Hence, the timescale for the temperature to drop by a factor of 1/e is shorter. In our models with a low asymptotic temperature, -reactions to produce seed nuclei are slow relative to the expansion rate at the relevant time. Hence, the neutron-to-seed ratio becomes high and heavy elements can be synthesized. Also, since the wind speed is ˙ Thus, greater, the baryon density is reduced for the same M.
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TABLE 1 Different r-Process Interactions Case
cc -N
n-emission
nc N
1....................... 2....................... 3....................... 4....................... 5....................... 6.......................
No Only n,p All nuclei All nuclei All nuclei All nuclei
No No No All nuclei All nuclei All nuclei
No No No No All nuclei Only
Notes.—Different r-process scenarios are studied in this work. The various cases differ in the treatment of neutrino-nucleus reactions. The abbreviations have the following meanings: cc N = charged-current reactions on nuclei; n-emission = neutrino-induced neutron emission by charged-current reactions; nc N = neutralcurrent reactions on nuclei (this case always includes neutrinoinduced neutron emission).
the photon-to-baryon ratio is larger and the entropy per baryon is slightly larger in the wind. Our model also produces slightly higher entropy than models with the same luminosity as the Sumiyoshi et al. (2000) model because we choose a slightly a deeper mass cut below the surface for which to treat in the hydrodynamic simulations. Since our models contained a slightly larger amount of matter between the neutrinosphere and the surface than in the models of Sumiyoshi et al. (2000), there is less energy per baryon deposited at the surface and fewer baryons are ejected for the same neutrino luminosity. This leads to a slightly higher entropy per baryon in the wind. 2.4. Nucleosynthesis Network For the calculations of r-process nucleosynthesis, we employ the reaction network of Terasawa et al. (2001), which includes over 3000 species from the stability line to the neutron drip line. In the present study we have added chargedand neutral-current neutrino-nucleus interactions, as described in the next section. For our purpose, we have considered six cases with and without charged- and /or neutral-current interactions and also with and without the effects of neutrino-induced neutron emission. These are summarized in Table 1. The last case in Table 1 includes all charged-current interactions but with only -particles allowed to experience neutrino-induced neutron emission and neutral-current interactions. This is to compare with previous work (Meyer 1995) in which it has been shown that neutral-current interactions and neutrino spallation of neutrons and protons from -particles can be important. Here we reinvestigate the importance of this spallation process to the final abundances when a short dynamical timescale is employed as opposed to a somewhat longer exponential timescale. 3. NEUTRINO-NUCLEUS CROSS SECTIONS The charged-current reaction rates used in this study are based on calculations of Langanke & Kolbe (2001, 2002), Kolbe et al. (2003) and E. Kolbe & K. Langanke (2003, private communication). We have included the charged-current reactions, e.g., ! XN 1 þ e ; e þZ XN Zþ1
ð22Þ
along with the analogous reactions for and neutrinos. (Reactions of ¯e with heavy nuclei were found to be negligible
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in this study and therefore were not included in the results discussed here.) We also included neutral-current reactions: XN þ 0 : þZ XN ! Z
ð23Þ
As in Qian et al. (1997), it is convenient to write the neutrino reaction rate per nucleus for each neutrino species as L MeV k 4:97 hE i 1051 ergs s1 2 100 km h i ; ð24Þ s1 ; r 1041 cm2 where L and hE i are the luminosity and average energy, respectively, of the neutrino species responsible for the reaction and h i is the corresponding cross section averaged over the neutrino spectrum. The spectrum-averaged neutrino reaction cross section is XZ
f (E ) f (E )dE ; ð25Þ h i ¼ f
where the sum extends over all possible final nuclear states f. For our purposes a good approximation (e.g., Woosley & Weaver 1995; Woosley et al. 1990) is to replace the neutrino spectrum derived from the Boltzmann solver with the analytic expression f (E ) by f (E ) ¼
1 E2 ; F2 ()T 3 exp ½(E =T ) þ 1
ð26Þ
where T and are parameters fitted to numerical spectra and F2 () normalizes the spectrum to unit flux. The neutrino spectra from Yamada et al. (1999) are well fitted with (T ; ) ¼ (4; 0) for e and (8, 0) for ; neutrinos and their antiparticles. Our rates differ from Qian et al. (1997) in that rather than a simple phenomenological Gaussian or singlestate approximation to the Gamow-Teller or Fermi strength functions, we utilize a formalism based on the prescription of Walecka (1975). This approach involves an evaluation of the various multipole operators for the charge longitudinal and the electric and magnetic transverse parts of the weak nuclear current. These can be written (Walecka 1975) as one-body operators in the nuclear Hilbert space. In the extreme relativistic limit (final lepton energy El 3 lepton mass ml c2 ) the neutrino (antineutrino) differential cross section for excitation of a discrete nuclear state is given by d i!f (GF Vud )2 pl El cos2 ð=2Þ F(Z 1; l ) ¼ (2 Ji þ 1) d l ;¯ " # 1 1 X X ;
JCL þ
JT J ¼0
ð27Þ
J ¼1
where !˜ LJ (q)jjJi > j2 ; q ! q2 J 2
T ¼ 2 þ tan 2 2q
˜ J (q) þ
JCL ¼ j < Jf jjM
ð28Þ
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h i ; j Jf jjJ˜Jmag (q)jjJi j2 þ j Jf jjJ˜Jel (q)jjJi j2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2
tan þ tan2 2 2 q2 h i ; 2Re Jf jjJ˜Jmag (q)jjJi Jf jjJ˜Jel (q)jjJi :
ð29Þ
Here denotes the angle between incoming and outgoing q ) (q ¼ j~ qj) is the four-momentum leptons, while q ¼ (!;~ transfer. The minus or plus sign before the second half of equation (29) refers to the neutrino or antineutrino cross ˜ J ; L˜ J ; J˜ el and J˜ mag are section, respectively. The quantities M J J multipole operators for the charge, the longitudinal, and the transverse electric and magnetic parts of the four-current, respectively. The cross section thus involves the reduced matrix elements of these operators between the initial state Ji and the final state Jf . For low-energy electrons and positrons the Fermi function F(Z; El ) in equation (27) accounts for the Coulomb interaction between the final charged lepton and the residual nucleus in the charged-current processes. This Coulomb correction is determined from a numerical solution of the Dirac equation for an extended nuclear charge: F(Z; El ) ¼ F0 (Z; El )L0 ; with 2 2ð 1Þ ( þ iy) y F0 (Z; El ) ¼ 4(2pl R) (2 þ 1) e ;
ð30Þ
where Z is the atomic number of the residual nucleus in the final state, El is the total lepton energy (in units of ml c2 ), pl is the lepton momentum (in units of ml c), R is the nuclear radius (in units of f=ml c), and and y are given by ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 (Z )2
and
y ¼ Z
El ; pl
ð31Þ
where is the fine-structure constant. The numerical factor L0 in equation (30) is of order unity. It accounts for finite-charge distribution and screening corrections. At higher energies and for muons at essentially all energies, the Fermi function (though valid for s-wave leptons) is a poor approximation for the Coulomb effect since higher partial waves also contribute for pl R 1. In that case the Coulomb effects are included in the ‘‘effective momentum approximation,’’ whereby the outgoing lepton momentum pl is replaced by an effective momentum: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m2 ; EeA ¼ El VC (0); ð32Þ peA ¼ EeA l where VC (0) ¼ 3e2 Z=2R is the Coulomb potential at the origin. In Kolbe et al. (2003) the Coulomb effect is taken into account not only by using the effective momentum, but also by replacing the phase space factor pl El by peA EeA. In practice a smooth interpolation between these two regimes is employed. The total cross sections to be used in equation (25) is finally obtained from the differential cross sections (eq. [27]), by summing (or integrating) over all possible final nuclear states and by numerical integration over the angles. The cross section then involves the reduced matrix elements of these operators between initial and final nuclear states of good angular momentum and parity. As a nuclear model for the
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evaluation of these matrix elements, the random phase approximation with a proton /neutron formalism was adopted. That is, proton and neutron degrees of freedom are distinguished for the particle and hole states (Rowe 1968; Kolbe et al. 1994, 2000). For example, the charged-current reaction in this model changes a neutron in the parent nucleus into a proton hole state in the daughter. Our parent ground states are described by the lowest independent particle model state. Partial occupancy was assigned to the last shell if this state is not completely occupied. The same shell is included among the hole states but is appropriately partially blocked (Rowe 1968). The partial occupation formalism necessarily assumes that the parent ground state is spherical and has a spin and parity of 0+. This assumption is, of course, incorrect for odd-A and most odd-odd nuclei. Nevertheless this shortcoming is not expected to significantly affect the since data and shell model calculations, where available, indicate no differences in the gross structures of the multipole responses between even-even, odd-A, and odd-odd nuclei (although for the latter two cases the multipole strength is distributed over three different angular momenta in the daughter nucleus [see Caurier et al. 1999 and references therein]). The particle and hole states were determined from a WoodsSaxon potential with standard parameters. The depth of the potential adjusted to reproduce the proton and neutron separation energies in the parent nucleus. As a residual interaction the Landau-Migdal force given in Rinker & Speth (1978) and Plumlet et al. (1997) was used. This parameterization was chosen to reproduce simultaneously the energies of the isobaric analog states (IAS) in 48Ca and 208Pb. However, for the charged-current reactions studied here we usually have to shift the hole energies slightly to reproduce the position of the IAS state. For the nuclear binding energies we adopted the experimental masses where known. If these are not available, we used the mass compilation of Duflo & Zuker (1995, 1999). In our calculations, we considered all multipole transitions with k 3 and both parities. From shell model calculations it is well established that the Gamow-Teller strength requires an additional quenching factor (Brown & Wildenthal 1985; Kajino et al. 1988), which we take from Martinez-Pinedo et al. (1996) as (0.7)2. Hence, the Ikeda sum rule (Fujita & Ikeda 1965) is also modified by the same factor. For the other multipole operators, there exists no firm indication of a need for such additional quenching factors. For the q-dependence of the nuclear form factors we use the standard dipole form. We use the appropriate multipole operators for finite momentum transfer (Walecka 1975). Thus, only in the limit q ! 0 do our 0+ and 1+ operators reduce to the Fermi and Gamow-Teller operators, respectively. Finally, the relevant neutral- and charged-current cross sections for reactions induced by supernova neutrinos are obtained by folding the RPA-based cross sections calculated as a function of neutrino energy in the initial channel, with the appropriate supernova neutrino spectrum. Most of the multipole transition strength for the neutrinoinduced reactions studied here resides above the particle (i.e., neutron) emission threshold. Hence, the excited daughter state will decay by the emission of one or several neutrons. We calculate the partial cross sections for the emission of k neutrons as described in Hektor et al. (2000). In our neutralcurrent calculations we allow k 5. Because of the rather large Q-values encountered for the heavy neutron-rich nuclei, we consider k 8 neutrons emitted in charged-current reac-
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tions. Again we use the compilation of Duflo and Zuker (1995; 1999) to derive the relevant neutron thresholds for neutron-rich nuclei. In our r-process network calculations, to be discussed below, we do not consider the emission of k neutrons as individual channels but describe neutrino-induced neutron emission ‘‘on average,’’ i.e., we determine for every reaction the average number hki of neutrons emitted and assume that the reaction on the nucleus (Z; N ), where Z and N are the charge and neutron numbers of the parent nucleus, leads to the final nucleus (Z; N hki) for neutral-current reactions and to (Z þ 1; N hki þ 1) for charged-current reactions. We do, however, include all neutron emission channels during the decay back to the line of stability. We have calculated neutral- and charged-current neutrinoinduced cross sections for about 1500 nuclei relevant to the r-process. Our cross section data base includes all nuclei with neutron numbers up to N ¼ 135, encompassing at least those nuclei with neutron separation energies less than 4 MeV. These cross sections have been partially published in (Langanke & Kolbe 2001; 2002; E. Kolbe & K. Langanke 2003, private communication). These compilations include the total and partial cross sections for nuclei with N 41. For the present study we have extended the data base to include additional light nuclei with N < 41, which can be important (Terasawa et al. 2001) when the r-process occurs in a rapidly expanding neutrino-wind, as assumed.5 Our neutrino-nucleus compilation is completed by the well known rates (e.g., Bruenn 1985) for -reactions on free protons and neutrons. Regarding the +4He cross sections, in our model, the 4He ground state is described by a closed-shell configuration. Hence, Gamow-Teller transitions do not contribute and the cross section is dominated by first-forbidden transitions. However, it is well known that the 4He ground state has a small (by a few percent) admixture in which the four nucleons are coupled to S ¼ 2 and L ¼ 2 (e.g., Barnes et al. 1987; Assenbaum & Langanke 1987; Arriaga et al. 1991). This admixture allows for Gamow-Teller transitions, which might affect the +4He cross sections. Woosley et al. (1990) included this possibility in their calculation of the cross section, which was based on a 2 f! shell model. Their cross sections, however, are rather similar to our RPA results. This might be caused by the fact that the Gamow-Teller strength in 4He resides at excitation energies that are too high to affect supernova neutrino cross sections. Indeed, a recent large-scale shell model calculation (G. Martinez-Pinedo 2002, private communication) places the strength at excitation energies above 30 MeV. 4. RESULTS AND DISCUSSION Figures 1 and 2 show a comparison of model calculations with and without various neutrino interaction effects included (see Table 1). In Figure 1 we consider charged-current interactions with and without neutrino-induced neutron emission. In Figure 2 neutral-current interactions are also included. One can get an idea of the magnitude of the effects apparent in these figures from a simple estimate of the neutrino interaction probability per nucleus, P , P k t; 5
ð33Þ
The extended compilation is available upon request from the authors, E. Kolbe & K. Langanke;
[email protected],
[email protected].
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Fig. 1.—Comparison of final r-process abundance yields for various calculations with and without charged-current interactions. The dashed, longdashed, solid, and dotted lines show cases 1, 2, 3, and 4, respectively. Note that cases 2 and 3 are very similar to each other. See Table 1 for an explanation of the different cases.
Fig. 2.—Comparison of final r-process abundance yields for various calculations with and without neutral-current interactions. The dotted, dashed, and solid lines represent the calculated results in cases 4, 5, and 6, respectively. Note that the dashed line (case 5) is very similar to the solid line (case 6). See Table 1 for an explanation of the different cases.
where k is the neutrino interaction rate per nucleus (eq. [24]) and t is the timescale over which the material is exposed. For illustration, consider a neutrino luminosity of 1051 ergs s1 per species and an average energy of hEi i 10 MeV at a distance of 3000 km during freezeout that occurs over a timescale of t 1 s. Then for a typical charged-current cross section of 1040 cm2, the number of interactions per nucleus is of order 102 to 103. Hence, the effects of neutrino interactions in Figures 1 and 2 can be subtle except in a few cases, which we now highlight.
tron star surface the two contributions from neutrino interactions on free neutrons and protons are almost equal. As the temperature drops, however, neutrons and protons begin to assemble into -particles. The resultant constituents are then mainly free neutrons and alphas (because of the neutron richness of the r-process environment). Even after the neutrinoproton interactions cease because of the lack of protons, neutrino-neutron interactions continue to convert neutrons into protons. These protons quickly lock up into alphas and cannot reconvert to neutrons. As matter flows farther from the proto–neutron star, the neutrino flux becomes weak and the neutrino interaction rate diminishes. In this paper, we adopt an expansion model with a relatively short dynamical timescale, (i.e., 20 ms). Since the temperature drops so rapidly in our hydrodynamic expansion model, the -process does not efficiently work to produce seed nuclei. Instead, the seed nuclei are mainly synthesized by neutron captures in a region far removed from the neutron star (Terasawa et al. 2001). Hence, the most important neutrino-nucleus interactions are those involving free neutrons and protons. These interactions, also, mainly determine the electron fraction and therefore the number of neutrons per seed nuclei in the r-process. Furthermore, we note the dominant effect from neutrino-induced neutron emission by a comparison between case 3 and case 4 in Figure 1. It is generally known that -delayed neutron emission smooths out the final abundance pattern after freezeout (Kodama & Takahashi 1976; Meyer et al. 1992; Qian et al. 1997). From this fact one might expect that neutrinoinduced neutron emission may have the same smoothing effect. On the contrary, however, we find in our hydrodynamic model that there is a larger scatter in the abundance yields in case 4 when neutrino-induced neutron emission is included than for case 3, in which it is excluded. To better illustrate this effect we show the ratio of abundances in case 4 to case 3 in Figure 3. We observe effects near A ¼ 125 and 185 previously noted in Qian et al. (1997), but at a much reduced level. There are, however, two additional effects to emerge from our
4.1. Effects of Charged-Current Interactions Considering Figure 1 first, the dotted line labeled case 4 includes charged-current interactions and neutrino-induced neutron emission on all nuclei. In the solid line labeled case 3, neutrino-induced neutron emission has been removed. The long-dashed line labeled case 2 is for charged-current interactions on free neutrons and protons only. For comparison, all neutrino-nucleus interactions are suppressed in the results labeled case 1 (dashed line). From this figure, it is readily apparent that elements beyond the third peak are abundantly synthesized when neutrino-nucleus interactions are suppressed (case 1), while for cases 2, 3, and 4 a much lower abundance of heavy elements is synthesized. This highlights the importance of charged-current interactions on the final abundance pattern even, which has been previously pointed out (e.g., Meyer 1995; Qian et al. 1997; Meyer et al. 1998). The observed equality between cases 2 and 3 implies that the dominant effect from charged-current interactions is just that due to neutrino interactions with free neutrons and protons. This justifies our limitation of the full Boltzmann solver to describe only heating and cooling from neutrino absorption reactions on neutrons and protons. The reason these reactions dominate simply derives from the fact that, in the hightemperature region near the surface of the proto–neutron star, the nuclear statistical equilibrium initially favors a gas of free neutrons and protons (see Meyer et al. 1998). Near the neu-
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Fig. 4.—Comparison of neutron density as a function of time for models with (dotted line) and without (solid line) neutrino-induced neutron emission.
Fig. 3.—Ratio of abundances in case 4 to in case 3 as a function of atomic mass number.
short-timescale model. For one, the largest change of abundances (nearly a factor of 2) occurs in the range A ¼ 68 76. The other is an enhanced odd-even effect. The magnitude of these effects may be smaller than the uncertainty due to the use of nuclear mass models. Nevertheless, the observed universality of r-process abundances (e.g., Cowan & Sneden 2002 and references therein) significantly constrains the uncertainty due to the nuclear mass model (see Terasawa 2003), as discussed below. Hence, these abundance changes by neutrino-induced neutron emission may be significant when a comparison with observed abundances is made. Regarding the observations of r-process abundances, heavy-element abundances have been detected in a number of metal-poor stars. These observations have shown that excellent agreement with the solar system r-process abundance pattern is obtained for elements with 56 Z 76 (Cowan & Sneden 2002). More recently, Honda et al. (2003, 2004) have also observed several stars using the Subaru / HDS. Although they obtained a reasonable agreement with the previous observations, they also found several interesting differences in the details. Some stars agree with the solar r-process abundances within the observational error bar. Others, however, clearly exhibit some differences with solar abundances for light nuclei (10 < Z < 30), and even in the region between 56 Z 69. Especially, odd-Z nuclei tend to have a smaller abundance than the solar abundance (when normalized to the solar abundance of Eu with odd-Z). This is of particular interest in the present context, since we deduce that at least some of these light elements may be the ones most affected by neutrino interactions. As for numerical calculations of the r-process, studies of nucleosynthesis yield a variety of physical conditions for which the r-process environments have been made (Terasawa 2002, 2003; Wanajo et al. 2002; Schatz et al. 2002; Otsuki et al. 2003a, 2003b). It generally appears that an almost perfect universality can be theoretically realized as long as the third r-process peak is reasonably produced, regardless of how high the peak is. Thus, the small changes of abundances in our present calculations are large enough to break the
universality, although they are too small to explain the observational gaps. Terasawa (2002, 2003) have also shown that there is an unique decay path for the r-process and that the universality can be broken by changes in the -decay path due to -delayed neutron emission after the freezeout of the r-process. Therefore, an underproduction of odd nuclei, as suggested by the observations, may be caused by changes in the decay path due to neutrino-induced neutron emission after the freezeout of the r-process. One would hope to constrain the neutrino emission model in supernovae by a detailed study of such effects on the final abundances. Therefore, we speculate that if precise observations of r-process abundance patterns can be obtained for many stars, it may be possible to unfold the relationship between the neutrino luminosity and the progenitor star. As described before, a change of the -decay path causes changes of abundances. The reason for this change can be traced to the availability of neutrons even after freezeout. Figure 4 displays neutron densities as a function of time. The solid and dotted lines correspond to these in cases 3 and 4, respectively. At early times (t 1 s) these two lines are nearly identical. These gradually show appreciable departure at late times as the dotted line (case 4) rises above the solid line (case 3). This increase of neutron density is caused by neutrinoinduced neutron emission. These late-time neutron captures alter the path of decay chains, since some nuclei more readily capture neutrons than others. The final abundance pattern therefore reflects differences in the neutron capture cross sections of nuclei near the stability line. When an odd-A nucleus captures a neutron, the resultant nucleus has an even mass number. The cross section is determined by the level density of the resultant nuclei. Even-A nuclei typically have a lower density of states. Therefore, odd-A nuclei have a larger neutron capture cross section. This causes an enhancement of the odd-even effect in the final abundances. In other words, when a strong neutrino flux continues even after freezeout, the neutron density can remain sufficiently high that an enhanced odd-even effect in the final abundance can emerge. In the present calculations, we have adopted a neutrino luminosity of L ¼ 1051 ergs s1. With this choice, the maximum difference in abundances between the cases with and without neutrino induced neutron emission is about factor of 1.9. However, if we adopted a larger neutrino luminosity of 1052 ergs s1, the maximum gap also increases to over 1 order of magnitude. This is because
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the increased luminosity leads to a similar increase in the neutron density. In order to explain the gaps in the abundance patterns observed in Honda et al. (2003, 2004), a neutrino luminosity greater than 1052 ergs s1 is probably needed. However, if the neutrino luminosity is too great, heavy elements are not synthesized. These facts combined lead us to conclude that these observations suggest that the r-process may be occurring in an environment with a neutrino flux about 1052 ergs s1 and that neutrino flux decreases slowly. This flux is consistent with the gravitational binding energy of the neutron star. This conclusion regarding the neutrino flux is also similar to that of Yoshida et al. (2004), who considered neutrino emission models that realize both the light (7Li and 11B) and r-process elements. Finally, we also mention the effect of neutrino-induced neutron emission on nuclear cosmochronometry. Several studies (Meyer & Truran 2000; Goriely & Arnould 2001; Terasawa 2002, 2003; Wanajo et al. 2002; Otsuki et al. 2003a, 2003b) indicate that the Th-Eu chronometer may be uncertain because Th and Eu are positioned on either side of the thirrd peak, and the initial Th-Eu ratio can vary widely. Both Th and U have an even proton number, and they differ in atomic number by only 2. Therefore, we expect from our calculations that both nuclei will be affected similarly by neutrino-processes. Hence, conclusions regarding the Th-U ratio may not change even when the effects of neutrino-process are large. 4.2. Effects of Neutral-Current Interactions Figure 2 shows the effects of adding in the neutral-current interactions. For neutrino-alpha interactions we consider only 0 the 4He(, n)3He and 4He(, 0 p)3H reactions, even though there are other possible spallation reactions on -particles. The neutron and proton branching ratios are taken from Woosley et al. (1990). The solid line (case 6) is the result of including only neutrino-alpha reactions. The dashed line labeled case 5 shows the effects of neutral-current reactions with all nuclei. These two lines are almost identical. This means that neutralcurrent interactions with nuclei heavier than alphas have little effect on the final abundance pattern. This is in contrast to the charged-current reactions where neutrino interactions with heavy nuclei remarkably affect the final abundance pattern. Neutral-current interactions on heavier nuclei are not important because the cross sections are much smaller than those of the charged-current interactions and the number of neutrons emitted by neutral-current interactions is smaller (Langanke & Kolbe 2001, 2002). Note that even though the effects of neutral-current interactions become larger as the neutrino flux increases, the charged-current interactions still dominate over the neutral-current interactions in determining the final abundance pattern. For comparison, we also plot in Figure 2 the results of case 4, with all charged-current interactions being included but not neutral-current interactions. Previously, Meyer (1995) has shown that neutral-current interactions with alphas can have a significant influence on the r-process yields. He found that elements heavier than the second peak can be greatly reduced even in the very high-entropy environment S=k 400 of Woosley et al. (1994). This effect is mainly caused by neutrinoinduced proton spallation from -particles. However, in the present calculations, there is little effect on the final abundances from neutral-current interactions with alphas. This is because
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the dynamical timescale of our model is much shorter than that of Woosley et al. (1994). Hence, interactions with alphas are suppressed in our model. 5. SUMMARY We have studied the effects of neutrino-interactions on r-process nucleosynthesis in the context of a relativistic hydrodynamic neutrino-driven wind model with a Boltzmann neutrino transport solver and new neutrino interaction rates. We have calculated the full dynamic r-process nucleosynthesis network for material above neutron star mass of 1.4 M. For these studies a particular model (Terasawa et al. 2002) with model a short expansion timescale (20 ms), a moderately high entropy (S=k 200), and a low asymptotic temperature (T9 ¼ 0:4) was chosen because it gives a reasonable r-process abundance curve for a single trajectory and so may be representative of the true physical environment. In this hydrodynamic model, we find (as in other studies) that the most important neutrino interactions are those involving free nucleons at an early phase of the expansion near the surface of the proto–neutron star. These reactions determine the initial electron fraction and the end point of synthesized elements by the r-process. Later, after -synthesis, these interactions unilaterally change neutrons into protons that quickly lock up into alphas. Therefore, the amount of heavy elements produced diminishes as these neutrino interactions increase. We have also found two interesting and (to our knowledge previously unreported) effects from our hydrodynamic simulation. One is that neutrino-induced neutron emission, especially by charged-current interactions, may most strongly affect the lighter r-process nuclei with A ¼ 68 76. The other is that late-time neutron captures from neutrino-induced neutron emission can cause an enhanced odd-even effect in the abundances. These new effects may help to explain some recently reported differences between observational abundance patterns in metal-poor stars (Honda et al. 2003, 2004) and solar system abundances. When neutrino-induced neutron emission is important, nuclei with smaller neutron-capture cross sections become more abundant. This is because there is a late-time flux of neutrons generated by neutrino-induced neutron emission. This causes neutron captures in addition to
-decay even after the freezeout of the nominal r-process. Differences in capture cross sections leads to an enhanced production of some light nuclei and an enhanced even-odd effect in the final abundances. The effect observed here is larger than in previous studies because of the new neutrino interaction rates, which take into account the detailed differences in nuclear level densities. These calculated effects are at least suggestive of some of the features recently observed in metal-poor stars. We speculate, therefore, that detailed studies of this effect might allow one to constrain the neutrino spectrum in the neutrino-driven winds and deduce a relationship between the neutrino spectra and progenitor stars. For this, however, more studies of supernova explosions are necessary, along with more observations of the heavy-element abundance patterns in metal-poor stars. The effects of neutral-current interactions are not as significant. However, we have found that they can also increase the amount of light nuclei with A 80 and reduce the heavy A 200 nuclei. This tendency may shed light on the observed depletions of some lighter nuclei relative to solar abundances (Sneden et al. 2000). We have also studied the effects of
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neutron interactions on the Th and U cosmochronometers and find that these chronometers are largely unaffected by the neutrino interactions. One of the authors (M. T.) wishes to acknowledge the fellowship of the Japan Society for Promotion of Science (JSPS).
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This work has been supported in part by Grants-in-Aid for Scientific Research (12047233, 13640313, 14540271) and for Specially Promoted Research (13002001) of the Ministry of Education, Science, Sports, and Culture of Japan. Our work was partially supported by the Danish research Council. Work at the University of Notre Dame supported by the US Department of Energy under Nuclear Theory grant DE-FG02-95-ER40934.
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