Solar Nuclear Energy Generation and the Chlorine Solar Neutrino Experiment

The study of solar neutrinos may provide important insights into the physics of the central region of the Sun. Four solar neutrino experiments have confirmed the solar neutrino problem but do not clearly indicate whether solar physics, nuclear physics, or neutrino physics have to be improved to solve it. Nonlinear relations among the different neutrino fluxes are imposed by two coupled systems of differential equations governing the internal structure and time evolution of the Sun. We assume that the results of the four neutrino experiments are correct and are concerned not with the discrepancy between the average rate and the predicted rate, but with a possible time dependence of the argon production rate as revealed in the Homestake experiment over a time period of 20 years. Based on the subtlety of the solar neutrino problem we review here qualitatively the physical laws employed for understanding the internal solar structure and conjecture that the interlink between specific nuclear reactions of the PPIII-branch of the proton-proton chain may allow the high-energy solar neutrino flux to vary over time. 1 Solar Neutrino Detection: Results The Sun is supposed to be a simple main-sequence star that generates its energy through the proton-proton chain and to a much less extend through the CNO cycle, thereby producing a copious flux of neutrinos. The Sun is considered to be a simple star because very basic physical laws can be employed to describe the evolution and the internal structure of the Sun, namely Newton’s laws of gravity and motion, the first two laws of thermodynamics, Einstein’s law of equivalence of mass and energy, Boyle’s law and Charles’ law of perfect gases, and Heisenberg’s uncertainty principle. To probe the simplicity and consistency of theoretically developed standard solar models, over the past two and a half decades four experiments to detect the solar neutrino flux have been established (Bahcall and Pinsonneault 1992). (1) The chlorine experiment in the Homestake gold mine (USA), in operation since 1967, observed an average Ar production rate by solar neutrinos with energies Eν > 0.8MeV of (2.2± 0.2)SNU for runs 18-109 over the time period 1970.8 to 1990.0 (Davis 1993). (2) The neutrino-electron scattering experiment in the Kamiokande mine (Japan), in operation since 1986 for two time periods of 1040 days (Kamiokande II: Jan87-Apr90, 450 days with Eν > 9.3 MeV and 590 days with Eν > 7.5 MeV) and 220 days (Kamiokande III: Dec90-Dec91, Eν > 7.5 MeV), detected solar neutrinos with event rates < Φ(B)obs >= [0.47 ± 0.05(stat) ± 0.06(syst)]Φcalc( B) of that predicted by standard solar models (Nakamura 1993). For the chlorine experiment the standard solar model with the best input parameters predicts an event rate of (8.0 ± 3.0) SNU (Bahcall and Pinsonneault 1992). The experimental results referred to in (1) and (2) above revealed that the measured solar neutrino fluxes are significantly below those predicted by standard solar models, known as the “solar neutrino problem”. (3) The gallium experiment under a mountain in the North Caucasus at Baksan (Russia) began to detect solar neutrinos with energies Eν > 0.23 MeV in 1988 and reported a Ge production rate of (85 −32[stat]± 20[syst]) SNU (Anosov et al. 1993). (4) The European gallium experiment in the Gran Sasso tunnel (Italy), in operation since 1991, measured a capture rate of solar neutrinos by Ga 1SNU (solar neutrino unit)=10 captures per atom per second of (79± 10[stat]± 6[syst]) SNU in 30 runs between May 1991 (GALLEX I: May91-March92) and October 1993 (GALLEX II: Aug92-Oct93)(Anselmann et al. 1994). The detected solar neutrino fluxes for the two gallium experiments almost agree with each other but are in conflict with standard solar models which predict a capture rate of (132 −17)SNU (Bahcall and Pinsonneault 1992). The result from the neutrino-electron scattering experiment (Kamiokande) constrains only the high-energy B neutrino flux, while the chlorine and gallium experiments (Homestake, Baksan, Gran Sasso) constrain both the Be and B solar neutrino fluxes. The results of the four neutrino experiments, despite of extensive experimental and theoretical efforts to reveal the origin of the discrepancies, do not indicate clearly whether solar physics, nuclear physics, or neutrino physics have to be improved to settle the account of the solar neutrino problem. The net reaction for the proton-proton chain and the CNO cycle is the conversion of hydrogen into helium, 4p −→ α+ 2νe + 2e +Q,Q = 26.7MeV, where two neutrinos are produced in the Sun per 26.7 MeV release of nuclear energy. This reaction allows an estimate of the copious solar neutrino flux on Earth, assuming that the solar nuclear energy generation equals the Sun’s luminosity and does not vary over time periods short in comparison to the nuclear time scale, Φν⊙ = 2L⊙ Q− 2Eν 1 4π(AU) ≈ 6.5× 1010ν⊙cm−2s−1, where L⊙ = 3.86× 10ergs is the Suns’s luminosity, AU = 1.5× 10cm is its average distance from Earth, and Eν ≈ 0.26 MeV is the average energy of the produced neutrinos. The net reaction assumes that baryon number, charge flavour, and energy are conserved quantities. In the following it will be assumed that the reported results of the four neutrino experiments are correct. 2 Fundamental Physical Constants and Solar Structure: Differential Equations There are two distinct types of basic equations in physics: dynamical equations exhibiting the reversible (Newtonian) time and transport equations reflecting the irreversible (Boltzmann-Gibbsian) time. The Sun contains a large number of hydrogen atoms, and evolves as a main-sequence star in nuclear time scale, an extremely long time in comparison to thermal diffusion time (Helmholtz-Kelvin time scale) and to its fundamental pulsation mode. Accordingly, the time evolution of the Sun is managed by the change of its chemical composition governed by a system of coupled nonlinear kinetic equations which can be solved separately from the system of coupled partial differential equations of solar structure, boundary conditions, and the constraint that the model luminosity at the present epoch must be equal to the observed solar luminosity. The four basic differential equations required to calculate the internal structure of the Sun are equations which represent respectively the distribution of mass within the Sun, the balance of gravity and pressure giving hydrostatic equilibrium (involving Newton’s gravitational constant G), the outward flow of energy driven by the temperature gradient inside the Sun (involving the velocity of light c and Stefan’s constant a), and the equation of nuclear energy generation within the Sun which continually replenishes that radiated away (involving Planck’s quantum of action h̄). In order to solve these differential equations, it is necessary to specify the equation of state (involving Boltzmann’s constant k), the opacity of the solar material and the nuclear energy generation rates. Boundary conditions have to be satisfied at the surface and at the centre of the Sun. From a more general point of view with regard to the calculation of the internal structure of the Sun it is only necessary to make an assumption about the distribution of the energy sources within the Sun, not necessarily knowing what produces the energy. Due to this fact, calculations of the internal structure of solar-type stars made already considerable progress before the production of energy by nuclear synthesis was understood in any detail (Eddington 1928, cp. Chandrasekhar 1984). The differential equations for the internal structure of the Sun contain fundamental constants of physics that make it possible to gain qualitative insight into the solutions of the system of differential equations by simple dimensional analysis. In quantum field theory the Sommerfeld finestructure constant αel plays the role of a dimensionless coupling constant for the Coulomb force: α el = h̄c e ≈ 137. Equivalent to this quantity one can define the gravitational fine-structure constant αg, α g = h̄c Gm2p ≈ 10, where mp is the mass of the proton. This quantity measures the smallness of the gravitational force between two protons, similar to αel which measures the smallness of the Coulomb force between two electrons. In terms of stability of matter, α el gives the maximum positive charge of the central nucleus that will allow a stable electron-orbit around it. It can be further shown that the combination of the fundamental constants in α el and α −1 g are playing an important role in the evolution and structure of the Sun as expected from the differential equations (Salpeter 1966). Electromagnetic and gravitational interactions are long range interactions, where the unit of electromagnetic interaction is e and the unit of gravitational interaction is mp(mp = 1837me). The fourth fundamental constant of physics, Boltzmann’s k, is an exceptional case since temperature can be defined in terms of energy (Gamow 1970). However, if the equation of state for solar material followed exclusively the perfect gas law, one would not have any preferred units for density and temperature. This leads necessarily to the consideration of radiation pressure (involving Planck’s constant h̄) and electron degeneracy (involving h̄ and the electron mass me) for the internal structure of the Sun. 3 The Sun is Massive Because Gravity is Weak: Virial Theorem The Sun is a system of N nucleons with a mean separation of the order of magnitude d, M⊙ ≈ Nmp, R⊙ ≈ Nd, where M⊙ and R⊙ denote the mass and radius of the Sun, respectively. Observationally the Sun ought to be considered close to hydrostatic equilibrium while from the theoretical point of view the Sun is considered to be in complete hydrostatic equilibrium. The virial theorem implies that the gravitational binding energy of the Sun must be of the order of its internal energy,