2. Basic equations and the SSM-like approach
The basic equations of the neutrino fluxes in the standard solar models are the followings (see e.g. Heeger & Robertson, 1996):
with a notation similar to that of Heeger & Robertson (1996): the subscripts i = 1, 7 and 8 refer to , and B reactions. The -s are the observed neutrino fluxes at the different neutrino detectors, in dimensionless units, j = K, C, G to the SuperKamiokande, chlorine, and gallium detectors. The observed averaged values are (Fukuda et al., 1998), (Cleveland et al., 1988) and (Cleveland et al., 1998). are measured in . Similar equations are presented by Castellani et al. (1994), Calabresu et al. (1996), and Dar & Shaviv (1998) with slightly different parameter values. Using these three detector-equations to determine the individual neutrino fluxes , I derived that
Now let us see how these equations may serve to solve the solar neutrino problems. There are three solar neutrino problems distinguished by Bahcall (1994, 1996, 1997): the first is related to the lower than expected neutrino fluxes, the second to the problem of missing beryllium neutrinos as relative to the boron neutrinos, and the third to the gallium detector data which do not allow a positive flux for the beryllium neutrinos in the frame of the standard solar model. It is possible to find a solution to all of the three neutrino problems if we are able to find positive values for all of the neutrino fluxes in the above presented equations. I point out, that the condition of this requirement can be formulated with the following inequality:
If we require a physical , with the numerical values of the detector sensitivity coefficients, this constraint will take the following form:
In the obtained solutions the total neutrino flux is compatible with the observed solar luminosity , but the reactions involved in the SSM (the pp and CNO chains) do not produce the total solar luminosity. The detector rate inequalities (7) or (9) can be fulfilled only if we separate a term from , which represents the contribution of non-pp,CNO neutrinos to the SuperKamiokande measurements (Fukuda et al. 1998) (and, possibly, also allow the existence of and ). The presence of a non-electron neutrino term in the SuperKamiokande is interpreted until now as indication to neutrino oscillations. Nevertheless, thermal runaways are indicated to be present in the solar core that may produce high-energy electron neutrinos, as well as muon and tau neutrinos, since is predicted for the hot bubbles (Grandpierre, 1996). Moreover, the explosive reactions have to produce high-energy axions to which also only the SuperKamiokande is sensitive (Raffelt, 1997, Engel et al., 1990). Also, the SuperKamiokande may detect electron anti-neutrinos arising from the hot bubbles. This indication suggests a possibility to interpret the neutrino data with standard neutrinos as well.
To determine the terms I introduced the "a priori" knowledge on the pp,CNO chains, namely, their temperature dependence. This is a necessary step to subtract more detailed information from the neutrino detector data. In this way one can derive the temperature in the solar core as seen by the different type of neutrino detectors. I note that finding the temperature of the solar core as deduced from the observed neutrino fluxes does not involve the introduction of any solar model dependency, since the neutrino fluxes of the SSM pp,CNO reactions depend on temperature only through nuclear physics. Instead, it points out the still remaining solar model dependencies of the previous SSM calculations and allowing other types of chains, it removes a hypothetical limitation, and accepting the presence of explosive chains as well, it probably presents a better approach to the actual Sun.
The calculations of the previous section suggested to complete the SuperKamiokande neutrino-equation with a new term
where T is the dimensionless temperature . The one-parameter allowance describes a quiet solar core with a temperature distribution similar to the SSM, therefore it leads to an SSM-like solution of the standard neutrino flux equations (see Grandpierre, 1999). An essential point in my calculations is that I have to use the temperature dependence proper in the case when the luminosity is not constrained by the SSM luminosity constraint, because another type of energy source is also present. The SSM luminosity constraint and the resulting composition and density readjustments, together with the radial extension of the different sources of neutrinos, modify this temperature dependence. The largest effect arises in the temperature dependence of the pp flux: for the SSM luminosity constraint (see the results of the Monte-Carlo simulations of Bahcall & Ulrich, 1988), but without the SSM luminosity constraint. Inserting the temperature-dependence of the individual neutrino fluxes for the case when the solar luminosity is not constrained by the usual assumption behind the SSM (Turck-Chieze & Lopes, 1993) into the chlorine-equation, we got the temperature dependent chlorine neutrino-equation
Similarly, the temperature-dependent gallium-equation will take the form:
Now let us first determine the solutions of these equations in the case . The obtained solutions will be relevant to the SSM-like solar core. Now we know that the Sun can have only one central temperature T. Therefore, the smallest -s will be the closer to the actual T of the SSM-like solar core, and the larger -s will indicate the terms arising from the runaways. In this way, it is possible to determine the desired quantities .
From the observed values, it is easy to obtain , and . With these values, the chlorine neutrino-temperature from (11) , the gallium neutrino-temperature is from (12) and the SuperKamiokande neutrino-temperature is from (10) . The neutrino flux equations are highly sensitive to the value of the temperature. Assuming that the actual Sun follows a standard solar model but with a different central temperature, the above result shows that the different neutrino detectors see different temperatures. This result suggest that the different neutrino detectors show sensitivities different from the one expected from the standard solar model, i.e. some reactions produce neutrinos which is not taken into account into the standard solar model, and/or that they are sensitive to different types of non-pp,CNO runaway reactions. Let us explore the consequences of this conjecture.
© European Southern Observatory (ESO) 1999
Online publication: August 13, 199