- Email: [email protected]
Gaussian Modification of Neutrino Energy Losses in Electron Capture Processes of Nuclides 56Fe, 56Co, 56Ni and 56Mn in Stellar Interiors
CHINESE ASTRONOMY AND ASTROPHYSICS
ELSEVIER
Chinese Astronomy and Astrophysics 34 (2010) 386–393 ChineseAstronomy Astronomyand andAstrophysics Astrophysics3433 (2010) 386–393 Chinese (2010) 386–393
Gaussian Modification of Neutrino Energy Losses in Electron Capture Processes of Nuclides 56Fe, 56Co, 56Ni and 56Mn in Stellar Interiors LIU Jing-jing1
LUO Zhi-quan2
1
2
Institute of Science and Technology, Qiongzhou University, Sanya 572022 Institute of Theoretical Physics, China West Normal University, Nanchong 637002
Abstract By using the Gaussian modification method, the neutrino energy losses in the electron capture processes of nuclides 56 Fe, 56 Co, 56 Ni and 56 Mn are investigated. The results show that the energy loss rate of neutrinos is increased due to the Gaussian modification of the energy level distribution of the Gamow-Teller (GT) resonance transitions of nuclides. In the reactions dominated by the electron capture processes of the low-energy transitions, the Gaussian modification has a very small influence on the neutrino energy losses. When the high-energy G-T resonance transition is the main electron capture process, the influence on the neutrino energy losses will be greatly increased. For example, the correctional differences of nuclide 56 Fe are about 2 orders of magnitude when the density ρ7 = 100 (ρ7 is in units of 107 mol· cm−3 ) and the half width of Gaussian function Δ =14.3, 18.3, 22.3 Mev, and those of nuclide 56 Ni are about 60% and 40% when Δ =6.3, 18.3 Mev, respectively. Key words: stars: evolution — stars: interiors — neutrinos
† Supported by Special Foundation of Sanya Municipality for Higher Education, Natural Science Foundation of Hainan Province, Research Foundation of Education Department of Hainan Province for Higher Education and National Natural Science Foundation Received 2009–09–22 A translation of Acta Astron. Sin. Vol. 51, No. 2, pp. 144–150, 2010 [email protected]
0275-1062/09/$-see front matter © 2009 B.V. rights reserved. 0275-1062/10/$-see front matter © 2010 Elsevier All All rights reserved. c Elsevier 0275-1062/01/$-see front matter 2010B.V. Elsevier Science B. V. All rights reserved. doi:10.1016/j.chinastron.2010.10.007 doi:10.1016/j.chinastron.2010.10.007 PII:
LIU Jing-jing et al. / Chinese Astronomy and Astrophysics 34 (2010) 386–393
387
1. INTRODUCTION According to the theory of stellar evolution, the neutrino energy loss plays a very important role and even makes a major contribution in various kinds of the cooling mechanisms in stellar evolution, the models of X-ray bursts and the physical processes of the collapse in the cores of supernovae. The annihilation of electron pairs, photoneutrino process, plasma decay, nuclear bremsstrahlung process and other weak interaction processes, such as the electron capture and β decay, strongly influence the cooling rate and evolutionary timescale of stellar evolution. In the late evolutionary phase of massive stars, how the variation of this kind of cooling rate influences the structure and component of iron core in the mechanism of supernova outburst, which for a long time since lacks theoretical support, deserves our investigation. The electron capture process plays an important role in the process of supernova outburst. In the core collapse process of supernova, it is not only the main production base of neutrinos, but also plays an important role in influencing the neutronization process of the matter in cores, causing the core collapse of type II supernovae and the outburst of type Ia supernovae. The neutrinos produced in this process carry a great deal of energy and escape from celestial objects, making the stellar temperature rapidly decrease. Undoubtedly, the energy loss of neutrinos is of inestimable significance to the cooling of evolved stars. This highly interesting and challenging topic has been studied by many scholars. For example, Fuller et al.[1−3] made many pioneering works in this field. Liu et al.[4−8] , Luo et al.[9−10] and Brachwitz et al.[11] analyzed the relations of the energy distribution of the G-T transitions and the quenching effect with the electron capture, β decay, electron abundance of stars with masses in different ranges of solar mass. Langanke et al.[12] discussed in detail the G-T energy distributions of some isotopes with neutron-rich nuclei and the electron capture rates. Analyses indicate that in the environment of stellar interiors with both high temperature and high density, the Fermi energy of the electron gas is relatively high and may exceed 10 MeV. Therefore, in the electron capture process, the G-T resonance transition makes an important contribution, even plays a dominative role. This definitely influences the neutrino energy loss of the electron capture process. So it is very important and necessary to study the neutrino energy loss of the electron capture of the G-T resonance transition in the state of excitation. Based on the p − f shell model and according to the method adopted by Kar et al.[13] to treat the energy distribution of the G-T transitions, the Gaussian modifications are made to the energy states of the G-T resonance transitions in the state of excitation for typical nuclides 56 Fe, 56 Co, 56 Ni and 56 Mn in stellar interiors.
2. GAUSSIAN MODIFICATION OF NEUTRINO ENERGY LOSS IN ELECTRON CAPTURE PROCESS IN STELLAR INTERIORS It is assumed that after capturing an electron a nucleus k in stellar interior makes a transition from the initial state i to the final state f . When considering the contributions from all the possible final states which correspond to this initial state, the neutrino energy loss rate
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LIU Jing-jing et al. / Chinese Astronomy and Astrophysics 34 (2010) 386–393
λνk [14] of the electron capture process corresponding to the initial state i is λνk =
(2Ji + 1) e−Ei /KT G (Z, A, T )
i
λνif ,
(1)
f
where Ji and Ei represent the spin and the energy level of excitation state of the parent nucleus, respectively. G(Z, A, T ), K and T are the distribution function of nucleus, Boltzmann constant and temperature, respectively. The neutrino energy loss rate of the electron capture process from the initial state to all the possible final states is expressed as λνif . λνif 1 1 1 is equal to (flnt)2 fif and satisfies (f t) = (f t) . (f t)if is the comparative half-life F + (f t)GT if
if
if
if
of nuclide. It is related to the matrix elements of the G-T and Fermi transitions. Their relations are described as 1 GT
(f t)if
=
103.596 2
|M GT |if
,
1 F
(f t)if
=
103.79 2
|M F |if
.
fif = f (Qif ) is the factor of phase space and is given by the following expression f (Qif ) =
∞
3
εp (Qif + ε) F (z, ε) fe dε ,
(2) [4−5]
: (3)
ε0
where z is the nuclear charge number of nuclide, ε and p are the energy and momentum of electron, respectively. The threshold energy of electron capture is Qif = Q00 + Ei − Ef . Ei and Ef are the energies of the initial state and finial state of excitation, respectively. Q00 is the threshold energy of the transition from the parent nucleus to the child nucleus at the ground state. Q00 = Mp c2 − Md c2 , where Md and Mp are the masses of the parent nucleus and child nucleus, respectively. F (z, ε) is the factor of Coulomb correction. fe is the Fermi-Dirac distribution function of electrons. ε0 is determined by the following expression: Qif , (Qif < −1) . (4) ε0 = 1, (Qif ≥ −1) The chemical potential of the electron gas satisfies 1 ρ = 2 μe π NA λ3e
∞ (f−e − f+e )dp .
(5)
0
Here ρ is the mass density in units of g · cm−3 and μe is the average molecular weight of electrons. The Compton wavelength λe = h/me c. The distribution functions of electrons −1 −1 F −1 F +1 and f+e = 1 + exp ε+U , respecand positrons are f−e = 1 + exp ε−U KT KT tively. T , UF and NA represent the temperature, the chemical potential of the electron gas and Avogadro constant, respectively. The neutrino energy loss rate of the electron capture process can be further expressed as
ln 2 1 1 ν λk = λLJ = = λ0 + λGT . fif = ln 2fif + (6) (f t)if (f t)F (f t)GT if if
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λLJ and λ0 are the neutrino energy loss rate without Gaussian modification and the neutrino energy loss rate of the low-energy region around the ground state, respectively. λGT is the neutrino energy loss rate of the electron capture process of the G-T transition, which is far from the resonance region of the ground state. The contribution from the G-T resonance transitions is always an interesting and challenging problem. Aufderheide et al.[15] discussed the contribution from the resonance transitions, but not all the states of excitation were considered. According to References [13] and [16], we use a normalized Gaussian function, in which the central position is at the resonance point, as the distribution function [16] of the G-T resonance intensity 2 ε − E B GT GT LP = √ BGT . (7) exp − Δ2 πΔ LP and BGT are the modified and unmodified G-T resonance intensities, respectively. , BGT EGT and Δ are the energy of the state of excitation, the position of the resonance point of the G-T transition and the half width of Gaussian function, respectively. Therefore, the neutrino energy loss rate of the electron capture process is modified as
λGauss = λνif = if
= λGauss GT
ln 2 fif = λ0 + λGauss , GT (f t)if
∞ εGT
LP BGT dε
(8)
∞ ν fGT dε ,
(9)
Q
1/2 3 ν = ε ε2 − 1 (Q00 − ε + ε) F (z, ε ) fe (ε) . fGT
(10)
ν λGauss GT , GT and fGT are the modified neutrino energy loss rate of the electron capture process of the G-T resonance transition, the bottom energy of the high-energy resonance region and the modified factor of the phase space, respectively. Q is the least energy of the electron capture reaction and satisfies |Q00 − ε | , (Q00 − ε < −1) Q = . (11) 1, (Q00 − ε ≥ −1)
3. NUMERICAL CALCULATIONS AND ANALYSIS OF NEUTRINO ENERGY LOSS Figs.1 and 2 show, respectively, the functional relations of the unmodified neutrino energy loss rate λLJ and the modified neutrino energy loss rate λGauss with several possible half if widths of Gaussian function Δ= 2.3, 6.3, 10.3, 14.3, 18.3, 22.3 Mev with the density ρ7 for nuclides 56 Fe, 56 Co, 56 Ni and 56 Mn when T9 = 0.05 and 15 (T9 is in units of 109 K). As are not large for the cases of shown in the figures, the differences between λLJ and λGauss if high temperature and low density (e.g., T9 = 15, ρ7 = 20); on the other hand, the differences are quite large for the cases of high temperature. For the discussed possible half widths of
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LIU Jing-jing et al. / Chinese Astronomy and Astrophysics 34 (2010) 386–393
Gaussian function, the differences produced by the modifications of the neutrino energy losses are different. Fig.1 indicates the modification situation of the neutrino energy loss rates of the electron capture processes for several nuclides when T9 = 0.05. It can be seen from this figure that when Δ=2.3 MeV and ρ7 < 20, the Gaussian function has almost no influence on the modifications of the neutrino energy loss rates of the electron capture processes for nuclides 56 Fe and 56 Co. However, the differences caused by this kind of modifications will become larger with the increase of density. For example, for nuclides 56 Co and 56 Mn and under the condition of different half widths of Gaussian function, the differences of the modified and unmodified neutrino energy losses reach about one order of magnitude and they can reach 1.3 orders of magnitude for nuclide 56 Ni.
Fig. 1
Neutrino energy loss rate as a function of density ρ7 for nuclides
56 Fe, 56 Co, 56 Ni
and
56 Mn
when temperature T9 = 0.05
Fig.2 indicates the modification situation of the neutrino energy loss rates for nuclides Fe, 56 Co, 56 Ni and 56 Mn in the environment of relatively high temperature (T9 = 15). For the cases of high density with ρ7 = 100, this kind of modifications is very obvious. For example, the differences reach 2 orders of magnitude for nuclide 56 Fe when Δ=14.3, 18.3, 22.3 Mev. And the differences reach 60% and 40% for 56 Ni when Δ=6.3, 18.3 MeV, respectively. Comparing the results in these two figures, it can be seen that the G-T transition process of the electron capture process may not be dominant when Δ=2.3 MeV. This process is dominated by the low-energy transition, but the modifications of Gaussian function are mainly made to the high-energy transition part of the high-energy resonance region. There56
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fore, the effect produced by this kind of modifications is not very obvious. But for the cases with other half widths of Gaussian function, λLJ is always less than λGauss . We know that if the distribution of the electron gas with high temperature and high density must satisfy the Fermi-Dirac distribution. The G-T transition intensity of nuclide is distributed in the form of the centrosymmetric Gaussian function about the G-T resonance point. So the energies of the electrons taking part in the G-T resonance transitions in the high-energy range are not symmetric. The distribution of Gaussian function increases and includes more electrons to take part in the electron capture reactions. Therefore, the treatment of this kind of modifications obviously accelerates the progress of the electron capture process. It inevitably leads to great increases in the neutrino energy loss rates, so the modified neutrino energy loss rates are much larger than the unmodified ones.
Fig. 2
Neutrino energy loss rate as a function of density ρ7 for nuclides
56 Fe, 56 Co, 56 Ni
and
56 Mn
when temperature T9 = 15
From our numerical calculations, it can be seen that the lower the temperature and density, the less obvious the modifications of the neutrino energy loss rates of the electron capture processes. This is because when correcting the neutrino energy loss rates, the modifications of the neutrino energy losses by the electron capture processes of the G-T transitions in the high-energy region are mainly aimed at. Under the condition of low temperature, the average kinetic energy of electrons is relatively small, the Fermi energy of electrons is also relatively small for low density. The electrons are in the weakly degenerate state. The electron capture processes in the low-energy region make a comparatively large contribution to the neutrino energy losses, while the high-energy G-T transition processes may not dominate. So this kind of modifications is not obvious. On the contrary, in the environment of
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high temperature and high density, this kind of modifications will become necessary and the differences become larger. Since the higher the temperature, the larger the average energy of electrons, the density and the Fermi energy of the electron gas. The proportion occupied by the G-T transition increases and the contribution to the neutrino energy loss rate may dominate. Therefore, this makes the modifications of the G-T transitions of the electrons in the high-energy state relatively large in comparison with the energy loss rates for the cases without modifications.
4. CONCLUSIONS Based on the p − f shell model, we study the neutrino energy losses of the electron capture processes for nuclides 56 Fe, 56 Co, 56 Ni and 56 Mn by using the Gaussian modification method. The results indicate that the Gaussian modification of the energy level distribution of the excitation states of the resonance transitions of nuclides makes the neutrino energy loss rates comparatively large. In the reactions dominated by the electron capture processes of the low-energy transitions, the Gaussian modification has a very small influence on the neutrino energy losses, but the effect on the neutrino energy losses of the electron capture processes dominated by the high-energy G-T transitions will increase. For example, when Δ=2.3 MeV and ρ7 < 20, the Gaussian function has almost no influence on the modifications of the neutrino energy loss rates of the electron capture processes for nuclides 56 Fe and 56 Co. However, when ρ7 = 100, the differences of this kind of modifications are very obvious. The differences reach 2 orders of magnitude for nuclide 56 Fe when Δ=14.3, 18.3, 22.3 Mev. And when Δ=6.3, 18.3 MeV, the differences reach, respectively, 60% and 40% for nuclide 56 Ni. The cooling of cosmic objects, due to its strong influence on the progress of stellar evolution and evolutionary timescale, is always a challenging topic which attracts astrophysicists and some astronomers devoted to the research of stellar evolution. In the process of stellar evolution, the cooling and energy losses of celestial objects are inseparable. The energy conversion and loss always play an important role in the burning processes of the nuclear reactions in the interiors of celestial objects. In general, they appear in the form of electromagnetic radiation, gravitational waves and instable neutrino flows. For instance, the neutrino energy loss is the main cooling mechanism for white dwarfs, neutron stars and supernovae. The research on these cooling mechanisms will still be a long-term and arduous task. It will have an important and profound significance to better understand and study the final destinations, evolutionary paths and nuclear interaction model of cosmic objects, as well as the physical structure and state in the interiors of celestial objects in the environment of high temperature, high pressure and high density. Our conclusions may be helpful to the investigation of the late evolution of cosmic objects, the nucleosyntheses of heavy elements and the numerical calculations of stellar evolution. They may have an important significance to the further research on nuclear astrophysics and neutrino astrophysics. References 1
Fuller G. M., Fowler W. A., Newman M. J., ApJS, 1980, 42, 447
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Fuller G. M., Fowler W. A., Newman M. J., ApJ, 1982, 252, 715
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Fuller G. M., Fowler W. A., Newman M. J., ApJ, 1985, 293, 1
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Liu J. J., Luo Z. Q., Liu H. L., et al., IJMPA, 2007, 22, 3305
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Brachwitz F., Dean D. J., Hix W. R., et al., ApJ, 2000, 536, 934
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Kar K., Ray A., Sarkar S., ApJ, 1994, 434, 662
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393
ELSEVIER
Chinese Astronomy and Astrophysics 34 (2010) 386–393 ChineseAstronomy Astronomyand andAstrophysics Astrophysics3433 (2010) 386–393 Chinese (2010) 386–393
Gaussian Modification of Neutrino Energy Losses in Electron Capture Processes of Nuclides 56Fe, 56Co, 56Ni and 56Mn in Stellar Interiors LIU Jing-jing1
LUO Zhi-quan2
1
2
Institute of Science and Technology, Qiongzhou University, Sanya 572022 Institute of Theoretical Physics, China West Normal University, Nanchong 637002
Abstract By using the Gaussian modification method, the neutrino energy losses in the electron capture processes of nuclides 56 Fe, 56 Co, 56 Ni and 56 Mn are investigated. The results show that the energy loss rate of neutrinos is increased due to the Gaussian modification of the energy level distribution of the Gamow-Teller (GT) resonance transitions of nuclides. In the reactions dominated by the electron capture processes of the low-energy transitions, the Gaussian modification has a very small influence on the neutrino energy losses. When the high-energy G-T resonance transition is the main electron capture process, the influence on the neutrino energy losses will be greatly increased. For example, the correctional differences of nuclide 56 Fe are about 2 orders of magnitude when the density ρ7 = 100 (ρ7 is in units of 107 mol· cm−3 ) and the half width of Gaussian function Δ =14.3, 18.3, 22.3 Mev, and those of nuclide 56 Ni are about 60% and 40% when Δ =6.3, 18.3 Mev, respectively. Key words: stars: evolution — stars: interiors — neutrinos
† Supported by Special Foundation of Sanya Municipality for Higher Education, Natural Science Foundation of Hainan Province, Research Foundation of Education Department of Hainan Province for Higher Education and National Natural Science Foundation Received 2009–09–22 A translation of Acta Astron. Sin. Vol. 51, No. 2, pp. 144–150, 2010 [email protected]
0275-1062/09/$-see front matter © 2009 B.V. rights reserved. 0275-1062/10/$-see front matter © 2010 Elsevier All All rights reserved. c Elsevier 0275-1062/01/$-see front matter 2010B.V. Elsevier Science B. V. All rights reserved. doi:10.1016/j.chinastron.2010.10.007 doi:10.1016/j.chinastron.2010.10.007 PII:
LIU Jing-jing et al. / Chinese Astronomy and Astrophysics 34 (2010) 386–393
387
1. INTRODUCTION According to the theory of stellar evolution, the neutrino energy loss plays a very important role and even makes a major contribution in various kinds of the cooling mechanisms in stellar evolution, the models of X-ray bursts and the physical processes of the collapse in the cores of supernovae. The annihilation of electron pairs, photoneutrino process, plasma decay, nuclear bremsstrahlung process and other weak interaction processes, such as the electron capture and β decay, strongly influence the cooling rate and evolutionary timescale of stellar evolution. In the late evolutionary phase of massive stars, how the variation of this kind of cooling rate influences the structure and component of iron core in the mechanism of supernova outburst, which for a long time since lacks theoretical support, deserves our investigation. The electron capture process plays an important role in the process of supernova outburst. In the core collapse process of supernova, it is not only the main production base of neutrinos, but also plays an important role in influencing the neutronization process of the matter in cores, causing the core collapse of type II supernovae and the outburst of type Ia supernovae. The neutrinos produced in this process carry a great deal of energy and escape from celestial objects, making the stellar temperature rapidly decrease. Undoubtedly, the energy loss of neutrinos is of inestimable significance to the cooling of evolved stars. This highly interesting and challenging topic has been studied by many scholars. For example, Fuller et al.[1−3] made many pioneering works in this field. Liu et al.[4−8] , Luo et al.[9−10] and Brachwitz et al.[11] analyzed the relations of the energy distribution of the G-T transitions and the quenching effect with the electron capture, β decay, electron abundance of stars with masses in different ranges of solar mass. Langanke et al.[12] discussed in detail the G-T energy distributions of some isotopes with neutron-rich nuclei and the electron capture rates. Analyses indicate that in the environment of stellar interiors with both high temperature and high density, the Fermi energy of the electron gas is relatively high and may exceed 10 MeV. Therefore, in the electron capture process, the G-T resonance transition makes an important contribution, even plays a dominative role. This definitely influences the neutrino energy loss of the electron capture process. So it is very important and necessary to study the neutrino energy loss of the electron capture of the G-T resonance transition in the state of excitation. Based on the p − f shell model and according to the method adopted by Kar et al.[13] to treat the energy distribution of the G-T transitions, the Gaussian modifications are made to the energy states of the G-T resonance transitions in the state of excitation for typical nuclides 56 Fe, 56 Co, 56 Ni and 56 Mn in stellar interiors.
2. GAUSSIAN MODIFICATION OF NEUTRINO ENERGY LOSS IN ELECTRON CAPTURE PROCESS IN STELLAR INTERIORS It is assumed that after capturing an electron a nucleus k in stellar interior makes a transition from the initial state i to the final state f . When considering the contributions from all the possible final states which correspond to this initial state, the neutrino energy loss rate
388
LIU Jing-jing et al. / Chinese Astronomy and Astrophysics 34 (2010) 386–393
λνk [14] of the electron capture process corresponding to the initial state i is λνk =
(2Ji + 1) e−Ei /KT G (Z, A, T )
i
λνif ,
(1)
f
where Ji and Ei represent the spin and the energy level of excitation state of the parent nucleus, respectively. G(Z, A, T ), K and T are the distribution function of nucleus, Boltzmann constant and temperature, respectively. The neutrino energy loss rate of the electron capture process from the initial state to all the possible final states is expressed as λνif . λνif 1 1 1 is equal to (flnt)2 fif and satisfies (f t) = (f t) . (f t)if is the comparative half-life F + (f t)GT if
if
if
if
of nuclide. It is related to the matrix elements of the G-T and Fermi transitions. Their relations are described as 1 GT
(f t)if
=
103.596 2
|M GT |if
,
1 F
(f t)if
=
103.79 2
|M F |if
.
fif = f (Qif ) is the factor of phase space and is given by the following expression f (Qif ) =
∞
3
εp (Qif + ε) F (z, ε) fe dε ,
(2) [4−5]
: (3)
ε0
where z is the nuclear charge number of nuclide, ε and p are the energy and momentum of electron, respectively. The threshold energy of electron capture is Qif = Q00 + Ei − Ef . Ei and Ef are the energies of the initial state and finial state of excitation, respectively. Q00 is the threshold energy of the transition from the parent nucleus to the child nucleus at the ground state. Q00 = Mp c2 − Md c2 , where Md and Mp are the masses of the parent nucleus and child nucleus, respectively. F (z, ε) is the factor of Coulomb correction. fe is the Fermi-Dirac distribution function of electrons. ε0 is determined by the following expression: Qif , (Qif < −1) . (4) ε0 = 1, (Qif ≥ −1) The chemical potential of the electron gas satisfies 1 ρ = 2 μe π NA λ3e
∞ (f−e − f+e )dp .
(5)
0
Here ρ is the mass density in units of g · cm−3 and μe is the average molecular weight of electrons. The Compton wavelength λe = h/me c. The distribution functions of electrons −1 −1 F −1 F +1 and f+e = 1 + exp ε+U , respecand positrons are f−e = 1 + exp ε−U KT KT tively. T , UF and NA represent the temperature, the chemical potential of the electron gas and Avogadro constant, respectively. The neutrino energy loss rate of the electron capture process can be further expressed as
ln 2 1 1 ν λk = λLJ = = λ0 + λGT . fif = ln 2fif + (6) (f t)if (f t)F (f t)GT if if
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LIU Jing-jing et al. / Chinese Astronomy and Astrophysics 34 (2010) 386–393
λLJ and λ0 are the neutrino energy loss rate without Gaussian modification and the neutrino energy loss rate of the low-energy region around the ground state, respectively. λGT is the neutrino energy loss rate of the electron capture process of the G-T transition, which is far from the resonance region of the ground state. The contribution from the G-T resonance transitions is always an interesting and challenging problem. Aufderheide et al.[15] discussed the contribution from the resonance transitions, but not all the states of excitation were considered. According to References [13] and [16], we use a normalized Gaussian function, in which the central position is at the resonance point, as the distribution function [16] of the G-T resonance intensity 2 ε − E B GT GT LP = √ BGT . (7) exp − Δ2 πΔ LP and BGT are the modified and unmodified G-T resonance intensities, respectively. , BGT EGT and Δ are the energy of the state of excitation, the position of the resonance point of the G-T transition and the half width of Gaussian function, respectively. Therefore, the neutrino energy loss rate of the electron capture process is modified as
λGauss = λνif = if
= λGauss GT
ln 2 fif = λ0 + λGauss , GT (f t)if
∞ εGT
LP BGT dε
(8)
∞ ν fGT dε ,
(9)
Q
1/2 3 ν = ε ε2 − 1 (Q00 − ε + ε) F (z, ε ) fe (ε) . fGT
(10)
ν λGauss GT , GT and fGT are the modified neutrino energy loss rate of the electron capture process of the G-T resonance transition, the bottom energy of the high-energy resonance region and the modified factor of the phase space, respectively. Q is the least energy of the electron capture reaction and satisfies |Q00 − ε | , (Q00 − ε < −1) Q = . (11) 1, (Q00 − ε ≥ −1)
3. NUMERICAL CALCULATIONS AND ANALYSIS OF NEUTRINO ENERGY LOSS Figs.1 and 2 show, respectively, the functional relations of the unmodified neutrino energy loss rate λLJ and the modified neutrino energy loss rate λGauss with several possible half if widths of Gaussian function Δ= 2.3, 6.3, 10.3, 14.3, 18.3, 22.3 Mev with the density ρ7 for nuclides 56 Fe, 56 Co, 56 Ni and 56 Mn when T9 = 0.05 and 15 (T9 is in units of 109 K). As are not large for the cases of shown in the figures, the differences between λLJ and λGauss if high temperature and low density (e.g., T9 = 15, ρ7 = 20); on the other hand, the differences are quite large for the cases of high temperature. For the discussed possible half widths of
390
LIU Jing-jing et al. / Chinese Astronomy and Astrophysics 34 (2010) 386–393
Gaussian function, the differences produced by the modifications of the neutrino energy losses are different. Fig.1 indicates the modification situation of the neutrino energy loss rates of the electron capture processes for several nuclides when T9 = 0.05. It can be seen from this figure that when Δ=2.3 MeV and ρ7 < 20, the Gaussian function has almost no influence on the modifications of the neutrino energy loss rates of the electron capture processes for nuclides 56 Fe and 56 Co. However, the differences caused by this kind of modifications will become larger with the increase of density. For example, for nuclides 56 Co and 56 Mn and under the condition of different half widths of Gaussian function, the differences of the modified and unmodified neutrino energy losses reach about one order of magnitude and they can reach 1.3 orders of magnitude for nuclide 56 Ni.
Fig. 1
Neutrino energy loss rate as a function of density ρ7 for nuclides
56 Fe, 56 Co, 56 Ni
and
56 Mn
when temperature T9 = 0.05
Fig.2 indicates the modification situation of the neutrino energy loss rates for nuclides Fe, 56 Co, 56 Ni and 56 Mn in the environment of relatively high temperature (T9 = 15). For the cases of high density with ρ7 = 100, this kind of modifications is very obvious. For example, the differences reach 2 orders of magnitude for nuclide 56 Fe when Δ=14.3, 18.3, 22.3 Mev. And the differences reach 60% and 40% for 56 Ni when Δ=6.3, 18.3 MeV, respectively. Comparing the results in these two figures, it can be seen that the G-T transition process of the electron capture process may not be dominant when Δ=2.3 MeV. This process is dominated by the low-energy transition, but the modifications of Gaussian function are mainly made to the high-energy transition part of the high-energy resonance region. There56
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fore, the effect produced by this kind of modifications is not very obvious. But for the cases with other half widths of Gaussian function, λLJ is always less than λGauss . We know that if the distribution of the electron gas with high temperature and high density must satisfy the Fermi-Dirac distribution. The G-T transition intensity of nuclide is distributed in the form of the centrosymmetric Gaussian function about the G-T resonance point. So the energies of the electrons taking part in the G-T resonance transitions in the high-energy range are not symmetric. The distribution of Gaussian function increases and includes more electrons to take part in the electron capture reactions. Therefore, the treatment of this kind of modifications obviously accelerates the progress of the electron capture process. It inevitably leads to great increases in the neutrino energy loss rates, so the modified neutrino energy loss rates are much larger than the unmodified ones.
Fig. 2
Neutrino energy loss rate as a function of density ρ7 for nuclides
56 Fe, 56 Co, 56 Ni
and
56 Mn
when temperature T9 = 15
From our numerical calculations, it can be seen that the lower the temperature and density, the less obvious the modifications of the neutrino energy loss rates of the electron capture processes. This is because when correcting the neutrino energy loss rates, the modifications of the neutrino energy losses by the electron capture processes of the G-T transitions in the high-energy region are mainly aimed at. Under the condition of low temperature, the average kinetic energy of electrons is relatively small, the Fermi energy of electrons is also relatively small for low density. The electrons are in the weakly degenerate state. The electron capture processes in the low-energy region make a comparatively large contribution to the neutrino energy losses, while the high-energy G-T transition processes may not dominate. So this kind of modifications is not obvious. On the contrary, in the environment of
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high temperature and high density, this kind of modifications will become necessary and the differences become larger. Since the higher the temperature, the larger the average energy of electrons, the density and the Fermi energy of the electron gas. The proportion occupied by the G-T transition increases and the contribution to the neutrino energy loss rate may dominate. Therefore, this makes the modifications of the G-T transitions of the electrons in the high-energy state relatively large in comparison with the energy loss rates for the cases without modifications.
4. CONCLUSIONS Based on the p − f shell model, we study the neutrino energy losses of the electron capture processes for nuclides 56 Fe, 56 Co, 56 Ni and 56 Mn by using the Gaussian modification method. The results indicate that the Gaussian modification of the energy level distribution of the excitation states of the resonance transitions of nuclides makes the neutrino energy loss rates comparatively large. In the reactions dominated by the electron capture processes of the low-energy transitions, the Gaussian modification has a very small influence on the neutrino energy losses, but the effect on the neutrino energy losses of the electron capture processes dominated by the high-energy G-T transitions will increase. For example, when Δ=2.3 MeV and ρ7 < 20, the Gaussian function has almost no influence on the modifications of the neutrino energy loss rates of the electron capture processes for nuclides 56 Fe and 56 Co. However, when ρ7 = 100, the differences of this kind of modifications are very obvious. The differences reach 2 orders of magnitude for nuclide 56 Fe when Δ=14.3, 18.3, 22.3 Mev. And when Δ=6.3, 18.3 MeV, the differences reach, respectively, 60% and 40% for nuclide 56 Ni. The cooling of cosmic objects, due to its strong influence on the progress of stellar evolution and evolutionary timescale, is always a challenging topic which attracts astrophysicists and some astronomers devoted to the research of stellar evolution. In the process of stellar evolution, the cooling and energy losses of celestial objects are inseparable. The energy conversion and loss always play an important role in the burning processes of the nuclear reactions in the interiors of celestial objects. In general, they appear in the form of electromagnetic radiation, gravitational waves and instable neutrino flows. For instance, the neutrino energy loss is the main cooling mechanism for white dwarfs, neutron stars and supernovae. The research on these cooling mechanisms will still be a long-term and arduous task. It will have an important and profound significance to better understand and study the final destinations, evolutionary paths and nuclear interaction model of cosmic objects, as well as the physical structure and state in the interiors of celestial objects in the environment of high temperature, high pressure and high density. Our conclusions may be helpful to the investigation of the late evolution of cosmic objects, the nucleosyntheses of heavy elements and the numerical calculations of stellar evolution. They may have an important significance to the further research on nuclear astrophysics and neutrino astrophysics. References 1
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