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Astron. Astrophys. 348, 993-999 (1999)

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3. Dynamic models of the solar core

3.1. Static core

Obtaining a Ga neutrino-temperature is [FORMULA], this leads to a pp luminosity of the Sun around [FORMULA]. The remaining part of the solar luminosity should be produced by the hot bubbles, [FORMULA]. The runaway nuclear reactions proceeding in the bubbles (and possibly in the microinstabilities) should also produce neutrinos, and this additional neutrino-production, [FORMULA] should generate the surplus terms in the chlorine and water Cherenkov detectors as well. At present, I was not able to determine directly, which reactions would proceed in the bubbles, and so it is not possible to determine directly the accompanying neutrino production as well. Nevertheless, it is plausible that at high temperatures [FORMULA] (Grandpierre, 1996), such nuclear reactions occur as at nova-explosions or other types of stellar explosions. Admittedly, these could be rapid hydrogen-burning reactions, explosive CNO cycle, and also nuclear reactions producing heat but not neutrinos, like e.g. the explosive triple-alpha reaction (Audouze et al. 1973, Dearborn et al. 1978). At present, I note that the calculated bubble luminosity ([FORMULA]) may be easily consistent with the calculated non-pp,CNO neutrino fluxes [FORMULA], [FORMULA], and [FORMULA], [FORMULA].

The above results are in complete agreement with the conclusion of Hata et al. (1994), namely: "We conclude that at least one of our original assumptions are wrong, either (1) Some mechanism other than the pp and the CNO chains generates the solar luminosity, or the Sun is not in quasi-static equilibrium, (2) The neutrino energy spectrum is distorted by some mechanism such as the MSW effect; (3) Either the Kamiokande or Homestake result is grossly wrong." These conclusions are concretised here to the following statements: (1) a runaway energy source is present in the solar core, and the Sun is not in a thermodynamic equilibrium, (2) this runaway source distorts the standard neutrino energy spectrum, and perhaps the MSW effect also contributes to the spectrum distortion (3) The Homestake, Gallex and SuperKamiokande results contains a term arising from the non-pp,CNO source, which has the largest contribution to the SuperKamiokande, less to the Homestake, and the smallest to the Gallex.

The helioseismic measurements are regarded as being in very good agreement with the SSM. However, the interpretation of these measurements depends on the inversion process, which uses the SSM as its basis. Moreover, the different helioseismic measurements at present are contradicting below [FORMULA] (Corbard et al., 1998).

We can pay attention to the fact that the energy produced in the solar core do not necessarily pours into thermal energy, as other, non-thermal forms of energy may also be produced, like e.g. energy of magnetic fields. The production of magnetic fields can significantly compensate the change in the sound speed related to the lower temperature, as the presence of magnetic fields may accelerate the propagation of sound waves with the inclusion of magnetosonic and Alfven magnetohydrodynamical waves.

The continuously present microinstabilities should produce a temperature distribution with a double character, as part of ions may posses higher energies. Their densities may be much lower than the respective ions closer to the standard thermodynamic equilibrium, and so they may affect and compensate the sound speed in a subtle way. Recent calculations of the non-maxwellian character of the energy distribution of particles in the solar core (Degl'Innocenti et al., 1998) indicate that the non-maxwellian character leads to lowering the SSM neutrino fluxes and, at the same time, produces higher central temperatures. This effect may also compensate for the lowering of the sound speed by lowering of central temperature.

At the same time, an approach specially developed using helioseismic data input instead of the luminosity constraint, the seismic solar model indicates a most likely solar luminosity around [FORMULA](Shibahashi & Takata, 1996, Figs. 7-10), which leads to a seismological temperature lower than its SSM counterpart, [FORMULA]. On the other hand, as Bludman et al. (1993) pointed out, the production of high energy [FORMULA] neutrinos and intermediate energy [FORMULA] neutrinos depends very sensitively on the solar temperature in the innermost [FORMULA] of the Sun's radius.

Accepting the average value of [FORMULA] [FORMULA], this value gives [FORMULA]. With [FORMULA], the derived gallium-temperature will be [FORMULA]. With [FORMULA] (Cleveland et al. 1998), [FORMULA]. The result that [FORMULA] can arise from the circumstance that the gallium detectors are less sensitive to intermediate and high-energy neutrinos than the chlorine one, which detects less runaway neutrino than the SuperKamiokande. Therefore, if thermonuclear runaways produce intermediate- and/or high-energy neutrino flux in the Sun, it results a relatively smaller contribution in the gallium detectors than in the chlorine one. Moreover, the SuperKamiokande can detect also runaway muon and tau neutrinos besides the high-energy electron neutrinos, therefore they can contribute with an extra term which would give account why the Kamiokande observes a larger neutrino flux than the Homestake. Therefore, the deduced three temperatures actually indicate that the solar core is actually cooler than the standard one by an amount around [FORMULA]. Therefore, the beryllium neutrino flux in the dynamical solar model is estimated as [FORMULA] of its SSM expected value. The luminosity of the SSM-like core is around [FORMULA] of [FORMULA], therefore the bubble luminosity has to be [FORMULA]. The boron neutrino flux of the SSM like core will be [FORMULA] of the SSM value. Therefore, the bubbles has to produce the remaining [FORMULA] of the high energy neutrinos observed by the SuperKamiokande. This requirement may be easily satisfied and it may be consistent with the result obtained that the bubble luminosity is [FORMULA] of the solar luminosity, too.

The dynamic solar model predicts a beryllium neutrino flux [FORMULA] of the SSM value, corresponding to a temperature of [FORMULA]. This estimation offers a prediction for the Borexino neutrino detector [FORMULA]. Regarding the SNO detector, I can assume that the neutral currents are produced by the electron neutrinos of the SSM-like core plus all kinds of neutrinos produced by the hot bubbles. Therefore, the prediction of the DSM is [FORMULA]. These predictions differ significantly from the MSW SSM-values. Therefore, the future observations may definitively decide which model describes better the actual Sun, the SSM-based MSW effect or the dynamic solar model. In the interpretation of the future measurements it will be important also to take into account the possible dependence of the neutrino fluxes on the solar cycle.

3.2. Around activity maximum

Similarly, we can apply the equations given to derive the temperatures as seen by the different neutrino detectors in relation to the phases of solar activity. Around solar activity maximum the Kamiokande reported no significant deviancies from the averaged neutrino flux, therefore I can take [FORMULA] which leads to [FORMULA] again. With the data of Cleveland et al. (1998), neutrino fluxes were measured in two solar activity maximum period, in 1980 the result was [FORMULA] and in 1989 around [FORMULA], which compares to the reported average value of [FORMULA]. Since the average absolute flux is [FORMULA], this refers to an expected flux of [FORMULA]. These values lead to [FORMULA]. Also, the Gallex collaboration did not report about activity related changes in their observed neutrino data, therefore [FORMULA] can be used. Solving the neutrino flux equations for an assumed SSM-like solar core, the resulting temperatures will be [FORMULA] and [FORMULA].

The obtained results, [FORMULA], [FORMULA], [FORMULA], are consistent with the assumption that in the solar maximum the Gallex and the Homestake detect only the neutrinos from the SSM-like solar core, which has a temperature around [FORMULA] lower than in the SSM, or that in the solar maximum the neutrinos produced by the hot bubbles contribute mostly to the SuperKamiokande data. This results may be regarded as well fitting to the main point of the paper, namely that the neutrino flux produced by the hot bubbles produce muon and tau neutrinos, axions and anti-neutrinos, to which only the SuperKamiokande is sensitive.

3.3. Around solar activity minimum

Using a value [FORMULA] for the minimum of the solar activity, the Kamiokande temperature will be [FORMULA]. With the data presented in Cleveland et al. (1998), in the periods around solar minimum the Homestake measured [FORMULA] around 1977, [FORMULA] around 1987, and [FORMULA] around 1997. These values average to [FORMULA], suggesting an [FORMULA] SNU. With this [FORMULA] the neutrino flux equations leads to [FORMULA]. Now the Gallex results marginally indicates larger than average counts around 1995-1997, as reported by the Gallex-IV measurements of [FORMULA]. This value leads to a [FORMULA], i.e. an anti-correlation with the solar cycle. For a temperature of [FORMULA] the [FORMULA] would be [FORMULA].

The results obtained above suggest that around solar minimum all the neutrino detector data are consistent with a uniform temperature [FORMULA]. In this case the results would suggest that all the neutrino detectors observe only the SSM-like solar core and not the neutrinos arising from the hot bubbles of the thermonuclear runaways. In the dynamic solar model there is a quick and direct contact between the solar surface and the solar core. In the dynamic solar model the transit time scale of the hot blobs from he solar core to the surface is estimated to be around one day (Grandpierre, 1996), therefore, the absence of the surface sunspots may indicate the simultaneous absence (or negligible role) of runaways in the core. Therefore, the result that in solar minimum no bubble neutrino flux are observed in each of the neutrino detectors, is consistent with the fact that in solar minimum there are no (or very few) sunspots observed at the solar surface.

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© European Southern Observatory (ESO) 1999

Online publication: August 13, 199