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Supernova electron capture rates
N
ELSEVIER
NUCLEAR PHYSICS A Nuclear Physics A654 (1999) 904c-907c
www.elsevier.nl/locate/npe
Supernova Electron Capture Rates G. Martinez-Pinedo a and K. Langanke a ~Institute of Physics and Astronomy and Center for Theoretical Astrophysics, University of •rhus, DK-8000 Arhus, Denmark We have calculated the Gamow-Teller strength distributions for the ground states and low lying states of several nuclei that play an important role in the precollapse evolution of supernova. The calculations reproduce the experimental GT distributions nicely. The GT distribution are used to calculate electron capture rates for typical presupernova conditions. The computed rates are noticeably smaller than the presently adopted rates. The possible implications for the supernova evolution are discussed. 1. I n t r o d u c t i o n The core of a massive star becomes dynamically unstable when it exhausts its nuclear fuel. If the core mass exceeds the appropriate Chandrasekhar mass, electron degeneracy pressure cannot longer stabilize the center and it collapses. As pointed out by Bethe et aI. [1,2] the collapse is very sensitive to the entropy and to the number of leptons per baryon, Y~. In the early stage of the collapse Y~ is reduced as electrons are captured by Fe peak nuclei. Knowing the importance of the electron capture process, Fuller et aI. (usually called FFN) have systematically estimated the rates for nuclei in the mass range A = 45 - 60 putting special emphasis on the importance of capture to the GamowTeller (GT) giant resonance [3]. The GT contribution to the rate has been parametrized by FFN on the basis of the independent particle model. To complete the FFN rate estimate, the GT contribution has been supplemented by a contribution simulating lowlying transitions. Recently the FFN rates have been updated and extended to heavier nuclei by Aufderheide et al. [4]. These authors also considered the wellknown quenching of the Gamow-Teller strength by reducing the independent particle estimate for the GT resonance contribution by a common factor of two. After experimental (n, p) data clearly indicated that the Gamow-Teller strength is not only quenched (usually by more than a factor 2 compared to the independent particle model), but also fragmented over several states at modest excitation energies in the daughter nucleus [5-9], the need for an improved theoretical description has soon been realized [10-12]. These studies have been performed within the conventional shell model diagonalization approach, however, in strongly restricted model spaces and with residual interactions, which turned out to neither reproduce the quenching nor the position of the GT strength sufficiently well. These model studies therefore had only a limited value, as they required experimental input informations, and they had no predictive power. Fortunately shell model diagonalizations techniques have made significant progress in the last 0375-9474/99/$ see front matter © 1999 ElsevierScienceB.V. All rights reserved.
G. Martinez-Pinedo, K. Langanke /Nuclear Physics A654 (1999) 904c-907c
905c
couple of years and now allow for basically complete p f shell diagonalizations for low-lying states in the A = 56 mass range [13]. 2. Shell M o d e l C a l c u l a t i o n s The Gamow-Teller strength distribution needed for the calculation of the electron capture rates have been computed by shell model diagonalizations in the p f-shell using the code A N T O I N E [14]. The enormous dimensions involved, however, forced us to truncate the model space. The truncations are done limiting the number of particles, n, t h a t are allowed to move out of the f~/2 orbit to the rest (P3/2, f5/2,Pl/2) of the shell. The value for n is limited by the dimensions of the calculations, they range from 1,053,302 for 5~Co with n = 4 to 20,886,472 for 6°Co with n = 9, but it is large enough to guarantee virtually converged results. Given a choice n = np for a parent state, to ensure respect of the Ikeda sum rule, the truncation level of the dauther states must be taken to be nd =np + 1. As effective interaction we use an improved version of the KB3 interaction [15] which corrects the defects of the KB3 interaction near the f7/2 shell closure and for the upper part of the p f-shell [16]. The Gamow-Teller distributions are obtained through W h i t e h e a d ' s prescription [17] as described in [18]. As electron capture takes place at finite t e m p e r a t u r e (T ~-, 300 keV) we have included in our rate calculations also the capture from t h e r m a l l y excited states in the parent nucleus at excitation energies below ] MeV. Any calculation t h a t aims to compute week interaction rates should be able to reproduce the experimentally measured G T distributions. Figure 1 compares the computed and experimental distributions [5-8], where we have quenched our results by a factor (0.74) 2 [19]. For a meaningful comparison the discrete calculated G T distribution has been folded with Gaussians simulating the experimental resolution and then transformed into a histogram with 1 MeV bins. The agreement is quite satisfactory. In the figure the F F N estimation of the position of the G T resonance is indicated by an arrow. There are systematic differences in the location of the main G T resonance strength compared to the p a r a m e t r i z a t i o n of F F N . In even-even nuclei the G T strength resides at lower excitation energies in the daughter than assumed by F F N , while in odd-A nuclei the G T strength is centered at higher excitation energies. These findings agree with the Shell Model Monte Carlo results of reference [20]. For odd-odd nuclei there is no experimental information, but our calculations show that the G T resonance is at higher energies than the F F N estimate. 3. E l e c t r o n C a p t u r e
Rates
Once the G T distributions are known the electron capture rate can be calculated as outlined in [3,4]. The computed rates for some selected nuclei are shown in figure 2 as a function of t e m p e r a t u r e (T9 measures the t e m p e r a t u r e in 109 K) and at those densities at which the individual nuclei are relevant [4] for the electron capture process. The present shell model rates are compared to the F F N rates at the same densities. A d d i t i o n a l l y the figure indicates the rate of ref. [4] taken from their Tables 15-17. For even-even nuclei both rates agree; but while the F F N the rates are dominated by transitions to low-lying states we find t h a t nearly 50% of the capture rate is due to the strong transition to the G T resonance, which is located nearly 1 MeV lower in excitation energy than predicted
G. Martinez-Pinedo, K. Langanke/Nuclear Physics A654 (1999) 904c-907c
906c
by the independent particle model. For odd-odd nuclei and odd-A nuclei our rates are in general between a factor 10 and 100 lower than F F N as our calculations place the resonance at higher excitation energies.
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Figure 1. Experimental GT+ strength for several p f-shell nuclei compared to our shell model calculations. The arrow shows the position of the F F N estimate of the G T resonance.
Figure 2. Electron capture rates as a function of temperature at those densities at which the individual nuclei are relevant for the capture process. The F F N [3] and Aufderheide et al. [4] rates are also shown.
4. C o n c l u s i o n We do believe that the nuclei studied here reflect a typical, rather than an exceptional sample. Accepting this point of view one is lead to the conclusion that the current compilations of electron capture rates are based on a parametrization which places the G T centroid for odd-odd and odd-A parent nuclei at 1oo low excitation energies. Consequently the electron capture rates on these nuclei, as recommended in [3] and [4], are too large. Then, one expects that the total electron capture rate relevant for the presupernova collapse at densities P7 _~ 1000 is smaller than currently believed, as this is dominated by capture on odd-odd and odd-A nuclei [4]. As a consequence of a slower electron capture rate, the core radiates less energy away by neutrino emission, keeping the core on
G. Martinez-Pinedo, K. Langanke/Nuclear Physics A654 (1999) 904c-907c
907c
a trajectory with higher temperature and entropy. However, drawing conclusions about possible effects which lower electron capture rates might have on the collapse mechanism, in particular on the size of the homologous core are premature and prohibited at this stage. First, one has to compile a complete set of shell model based capture rates for all relevant nuclei. Secondly, during the collapse electron capture has to compete with ~-decay and preliminary results indicate that the shell model roughly confirms the total FFN rates. If true, electron capture and ~-decay rates balance during the stellar collapse and might lead to a cooling of the star without changing its Y~ value. This possibility has already been suggested in Ref. [21] on the basis of a few experimental GT distributions. REFERENCES
1. H.A. Bethe, G. E. Brown, J. Applegate, and J. M. Lattimer, Nucl. Phys. A 324 (1979) 487. 2. H.A. Bethe, Rev. Mod. Phys. 62 (1990) 801. 3. G.M. Fuller, W.A. Fowler, and M.J. Newman, Astrophys. J. Suppl. Ser. 42 (1980) 447; 48 (1982) 279; Astrophys. J. 252 (1982) 715; Astrophys. J. 293 (1985) 1. 4. M.B. Aufderheide, I. Fushiki, S.E. Woosley, and D.H. Hartmann, Astrophys. J. Suppl. Ser. 91 (1994) 389. 5. A.L. Williams et al., Phys. Rev. C 51 (1995) 1144. 6. W.P. Afford et al., Nucl. Phys. A 514, (1990) 49 7. S. E1-Kateb et al., Phys. Rev. C 49 (1994) 3129. 8. T. RSnnquist et al., Nucl. Phys. A 563 (1993) 225. 9. M.C. Vetterli et al., Phys. Rev. C 40 (1989) 559. 10. M.B. Aufderheide, Nucl. Phys. A 526 (1991) 161. 11. M.B. Aufderheide, S.D. Bloom, D.A. Ressler, and G.J. Mathews, Phys. Rev. C 4"/ (1993) 2961. 12. M.B. Aufderheide, S.D. Bloom, D.A. Ressler, and Mathews G.J., Phys. Rev. C 48 (1993) 1677. 13. E. Caurier, G. Martinez-Pinedo, F. Nowacki, and A. Poves, LANL archive nuc1th/9809068, submitted to Phys. Rev. C. 14. E. Caurier, computer code ANTOINE, CRN, Strasbourg (1989), unpublished 15. A. Poves and A. P. Zuker, Phys. Rep. 70 (1981) 235. 16. E. Caurier and A. Poves, private communication. 17. R.R. Whitehead, in Moment Methods in Many Fermion Systems, edited by B.J. Dalton et al. (Plenum, New York, 1980). 18. E. Caurier, A. Poves, and A.P. Zuker, Phys. Rev. Lett. 74 (1995) 1517. 19. G. Martfnez-Pinedo, A. Poves, E. Caurier, A. P. Zuker, Phys. Rev. C 53 (1996) R2602. 20. D.J. Dean, K. Langanke, L. Chatterjee, P.B. Radha, and M.R. Strayer, Phys. Rev. C 58 (1998) 536. 21. M.B. Aufderheide, I. Fushiki, G.M. Fuller, and T.A. Weaver, Astrophys. J. 424 (1994) 257.
ELSEVIER
NUCLEAR PHYSICS A Nuclear Physics A654 (1999) 904c-907c
www.elsevier.nl/locate/npe
Supernova Electron Capture Rates G. Martinez-Pinedo a and K. Langanke a ~Institute of Physics and Astronomy and Center for Theoretical Astrophysics, University of •rhus, DK-8000 Arhus, Denmark We have calculated the Gamow-Teller strength distributions for the ground states and low lying states of several nuclei that play an important role in the precollapse evolution of supernova. The calculations reproduce the experimental GT distributions nicely. The GT distribution are used to calculate electron capture rates for typical presupernova conditions. The computed rates are noticeably smaller than the presently adopted rates. The possible implications for the supernova evolution are discussed. 1. I n t r o d u c t i o n The core of a massive star becomes dynamically unstable when it exhausts its nuclear fuel. If the core mass exceeds the appropriate Chandrasekhar mass, electron degeneracy pressure cannot longer stabilize the center and it collapses. As pointed out by Bethe et aI. [1,2] the collapse is very sensitive to the entropy and to the number of leptons per baryon, Y~. In the early stage of the collapse Y~ is reduced as electrons are captured by Fe peak nuclei. Knowing the importance of the electron capture process, Fuller et aI. (usually called FFN) have systematically estimated the rates for nuclei in the mass range A = 45 - 60 putting special emphasis on the importance of capture to the GamowTeller (GT) giant resonance [3]. The GT contribution to the rate has been parametrized by FFN on the basis of the independent particle model. To complete the FFN rate estimate, the GT contribution has been supplemented by a contribution simulating lowlying transitions. Recently the FFN rates have been updated and extended to heavier nuclei by Aufderheide et al. [4]. These authors also considered the wellknown quenching of the Gamow-Teller strength by reducing the independent particle estimate for the GT resonance contribution by a common factor of two. After experimental (n, p) data clearly indicated that the Gamow-Teller strength is not only quenched (usually by more than a factor 2 compared to the independent particle model), but also fragmented over several states at modest excitation energies in the daughter nucleus [5-9], the need for an improved theoretical description has soon been realized [10-12]. These studies have been performed within the conventional shell model diagonalization approach, however, in strongly restricted model spaces and with residual interactions, which turned out to neither reproduce the quenching nor the position of the GT strength sufficiently well. These model studies therefore had only a limited value, as they required experimental input informations, and they had no predictive power. Fortunately shell model diagonalizations techniques have made significant progress in the last 0375-9474/99/$ see front matter © 1999 ElsevierScienceB.V. All rights reserved.
G. Martinez-Pinedo, K. Langanke /Nuclear Physics A654 (1999) 904c-907c
905c
couple of years and now allow for basically complete p f shell diagonalizations for low-lying states in the A = 56 mass range [13]. 2. Shell M o d e l C a l c u l a t i o n s The Gamow-Teller strength distribution needed for the calculation of the electron capture rates have been computed by shell model diagonalizations in the p f-shell using the code A N T O I N E [14]. The enormous dimensions involved, however, forced us to truncate the model space. The truncations are done limiting the number of particles, n, t h a t are allowed to move out of the f~/2 orbit to the rest (P3/2, f5/2,Pl/2) of the shell. The value for n is limited by the dimensions of the calculations, they range from 1,053,302 for 5~Co with n = 4 to 20,886,472 for 6°Co with n = 9, but it is large enough to guarantee virtually converged results. Given a choice n = np for a parent state, to ensure respect of the Ikeda sum rule, the truncation level of the dauther states must be taken to be nd =np + 1. As effective interaction we use an improved version of the KB3 interaction [15] which corrects the defects of the KB3 interaction near the f7/2 shell closure and for the upper part of the p f-shell [16]. The Gamow-Teller distributions are obtained through W h i t e h e a d ' s prescription [17] as described in [18]. As electron capture takes place at finite t e m p e r a t u r e (T ~-, 300 keV) we have included in our rate calculations also the capture from t h e r m a l l y excited states in the parent nucleus at excitation energies below ] MeV. Any calculation t h a t aims to compute week interaction rates should be able to reproduce the experimentally measured G T distributions. Figure 1 compares the computed and experimental distributions [5-8], where we have quenched our results by a factor (0.74) 2 [19]. For a meaningful comparison the discrete calculated G T distribution has been folded with Gaussians simulating the experimental resolution and then transformed into a histogram with 1 MeV bins. The agreement is quite satisfactory. In the figure the F F N estimation of the position of the G T resonance is indicated by an arrow. There are systematic differences in the location of the main G T resonance strength compared to the p a r a m e t r i z a t i o n of F F N . In even-even nuclei the G T strength resides at lower excitation energies in the daughter than assumed by F F N , while in odd-A nuclei the G T strength is centered at higher excitation energies. These findings agree with the Shell Model Monte Carlo results of reference [20]. For odd-odd nuclei there is no experimental information, but our calculations show that the G T resonance is at higher energies than the F F N estimate. 3. E l e c t r o n C a p t u r e
Rates
Once the G T distributions are known the electron capture rate can be calculated as outlined in [3,4]. The computed rates for some selected nuclei are shown in figure 2 as a function of t e m p e r a t u r e (T9 measures the t e m p e r a t u r e in 109 K) and at those densities at which the individual nuclei are relevant [4] for the electron capture process. The present shell model rates are compared to the F F N rates at the same densities. A d d i t i o n a l l y the figure indicates the rate of ref. [4] taken from their Tables 15-17. For even-even nuclei both rates agree; but while the F F N the rates are dominated by transitions to low-lying states we find t h a t nearly 50% of the capture rate is due to the strong transition to the G T resonance, which is located nearly 1 MeV lower in excitation energy than predicted
G. Martinez-Pinedo, K. Langanke/Nuclear Physics A654 (1999) 904c-907c
906c
by the independent particle model. For odd-odd nuclei and odd-A nuclei our rates are in general between a factor 10 and 100 lower than F F N as our calculations place the resonance at higher excitation energies.
0,jl 1.0
o~ II
54
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,
.
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,
•
.
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10o
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.
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,
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51
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: •
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10 2
0
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7.0
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10.0
"
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7.0
10.0
T9
E (MeV)
Figure 1. Experimental GT+ strength for several p f-shell nuclei compared to our shell model calculations. The arrow shows the position of the F F N estimate of the G T resonance.
Figure 2. Electron capture rates as a function of temperature at those densities at which the individual nuclei are relevant for the capture process. The F F N [3] and Aufderheide et al. [4] rates are also shown.
4. C o n c l u s i o n We do believe that the nuclei studied here reflect a typical, rather than an exceptional sample. Accepting this point of view one is lead to the conclusion that the current compilations of electron capture rates are based on a parametrization which places the G T centroid for odd-odd and odd-A parent nuclei at 1oo low excitation energies. Consequently the electron capture rates on these nuclei, as recommended in [3] and [4], are too large. Then, one expects that the total electron capture rate relevant for the presupernova collapse at densities P7 _~ 1000 is smaller than currently believed, as this is dominated by capture on odd-odd and odd-A nuclei [4]. As a consequence of a slower electron capture rate, the core radiates less energy away by neutrino emission, keeping the core on
G. Martinez-Pinedo, K. Langanke/Nuclear Physics A654 (1999) 904c-907c
907c
a trajectory with higher temperature and entropy. However, drawing conclusions about possible effects which lower electron capture rates might have on the collapse mechanism, in particular on the size of the homologous core are premature and prohibited at this stage. First, one has to compile a complete set of shell model based capture rates for all relevant nuclei. Secondly, during the collapse electron capture has to compete with ~-decay and preliminary results indicate that the shell model roughly confirms the total FFN rates. If true, electron capture and ~-decay rates balance during the stellar collapse and might lead to a cooling of the star without changing its Y~ value. This possibility has already been suggested in Ref. [21] on the basis of a few experimental GT distributions. REFERENCES
1. H.A. Bethe, G. E. Brown, J. Applegate, and J. M. Lattimer, Nucl. Phys. A 324 (1979) 487. 2. H.A. Bethe, Rev. Mod. Phys. 62 (1990) 801. 3. G.M. Fuller, W.A. Fowler, and M.J. Newman, Astrophys. J. Suppl. Ser. 42 (1980) 447; 48 (1982) 279; Astrophys. J. 252 (1982) 715; Astrophys. J. 293 (1985) 1. 4. M.B. Aufderheide, I. Fushiki, S.E. Woosley, and D.H. Hartmann, Astrophys. J. Suppl. Ser. 91 (1994) 389. 5. A.L. Williams et al., Phys. Rev. C 51 (1995) 1144. 6. W.P. Afford et al., Nucl. Phys. A 514, (1990) 49 7. S. E1-Kateb et al., Phys. Rev. C 49 (1994) 3129. 8. T. RSnnquist et al., Nucl. Phys. A 563 (1993) 225. 9. M.C. Vetterli et al., Phys. Rev. C 40 (1989) 559. 10. M.B. Aufderheide, Nucl. Phys. A 526 (1991) 161. 11. M.B. Aufderheide, S.D. Bloom, D.A. Ressler, and G.J. Mathews, Phys. Rev. C 4"/ (1993) 2961. 12. M.B. Aufderheide, S.D. Bloom, D.A. Ressler, and Mathews G.J., Phys. Rev. C 48 (1993) 1677. 13. E. Caurier, G. Martinez-Pinedo, F. Nowacki, and A. Poves, LANL archive nuc1th/9809068, submitted to Phys. Rev. C. 14. E. Caurier, computer code ANTOINE, CRN, Strasbourg (1989), unpublished 15. A. Poves and A. P. Zuker, Phys. Rep. 70 (1981) 235. 16. E. Caurier and A. Poves, private communication. 17. R.R. Whitehead, in Moment Methods in Many Fermion Systems, edited by B.J. Dalton et al. (Plenum, New York, 1980). 18. E. Caurier, A. Poves, and A.P. Zuker, Phys. Rev. Lett. 74 (1995) 1517. 19. G. Martfnez-Pinedo, A. Poves, E. Caurier, A. P. Zuker, Phys. Rev. C 53 (1996) R2602. 20. D.J. Dean, K. Langanke, L. Chatterjee, P.B. Radha, and M.R. Strayer, Phys. Rev. C 58 (1998) 536. 21. M.B. Aufderheide, I. Fushiki, G.M. Fuller, and T.A. Weaver, Astrophys. J. 424 (1994) 257.