There are a number of ways to analyze these possibilities. Many of the
nonstandard models manifest themselves for neutrino production by leading
to a cooler solar core.
This may ultimately be due to changes in the metallicity, the opacities, or
the nuclear cross sections. The fluxes of high energy neutrinos are
especially sensitive to the core temperature, mainly because of the energy
dependence of the cross sections. We have carried out an analysis to see
whether the data can be described by a lower temperature [48] --
[50]. We start by assuming the Bahcall-Ulrich estimates [46]
of the temperature dependence of the 
and 
fluxes, namely 
and 
, where 
is the temperature
of the solar core in units of the SSM prediction, 
K. For
small deviations from the standard model 
is predicted to vary as
, i.e., the pp rate increases. This is
necessary to maintain 
constant, where 
is the solar luminosity. However, we will be using the power laws to
describe larger departures from the standard solar model than the usual
estimates, so instead we choose 
so that the
luminosity is constant. I.e., one knows that the basic energy
mechanism is the conversion
Assuming that the sun is quasi-static, the overall luminosity tells us the total energy generation rate, and this in turn puts a constraint on the total neutrino fluxes, namely
where the coefficients correct for the neutrino energies.
We will use this to constrain
. One then has
where R is the rate and 
is the
temperature, both relative to the standard solar model prediction. The
additional uncertainties in the formulas are from nuclear cross sections,
for the production rates within the sun and for the detector
cross sections, which have to be properly correlated between experiments.
``Small'' represents the contributions of minor flux components (pep and
CNO). The results are insensitive to how these are treated.
It is apparent that it is impossible to describe the data for any value of
. The best fit is nominally 
, but it has a
terrible 
of 
for 2 df, which is excluded at 99.96% cl. The
problem is that the different experiments require different temperatures,
namely 
, 
and 
for Kamiokande,
Homestake, and the combined gallium results, respectively.
The specific power laws, taken from Bahcall and Ulrich [46], were based
on the standard solar model, while here we are extrapolating to nonstandard
models. In fact one obtains equally strong conclusions for essentially any
temperature exponents, provided only that the 
neutrinos are more
temperature sensitive than the 
neutrinos. Since the 
neutrinos
are made first, this is certainly a reasonable assumption. One finds
similar conclusions even if the 
error is increased significantly.
In fact, the conclusion is strengthened if there is a smaller 
,
because the Kamiokande rate is then in better agreement with the
SSM, leaving less room to vary the temperature to account for the chlorine
experiment. Thus, it seems that the cool sun models are not an explanation
of the data.
One can generalize to more general fits, in which not only the
core temperature but the nuclear cross sections are allowed to vary. For
example, Dearborn, Shi, and Schramm [14]
have considered the case in which
, 
, and 
are all free parameters (
is
proportional to 
). Again,
they come to the conclusion that one cannot account for the data within
reasonable ranges for these parameters.
