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in a previous paper @xcite we have addressed the question of a possible energy - dependence of the solar - neutrino deficit coming to the conclusion that , at least for the time being , its existence must be regarded only as an attractive possibility and by no means as an established reality .
our analysis was based on the comparison between the existing experimental data @xcite and the theoretical predictions , assumed to be well represented by those of the `` reference model '' of ref .
@xcite .
the latter assumption is actually rather questionable .
there exist in fact several solar models capable of reproducing all experimentally known facts but giving rise to somewhat different neutrino flux predictions @xcite . as different models of the same sun can not be all simultaneously right , it follows that theoretical predictions can only be said to be known within indeterminations which are actually larger than those associated with the results of any single model .    in our previous analysis
@xcite we mentioned this effect but did not take it into account . in this paper , we try to quantify this additional theoretical uncertainty by looking at the spread of the various predictions .
the procedure adopted is described in section 2 . with modified theoretical predictions and their errors , the statistical analysis of ref .
@xcite is then repeated in section 3 .
section 4 summarises our conclusions .
an extensive review of solar models is presented in ref .
@xcite . we have restricted ourselves to `` standard '' models , i.e. to those capable of reproducing all solar observational results , including the helioseismology data .
the solar neutrino event rates for the gallium ( ga ) , clorine ( cl ) and kamiokande ( ka ) experiments predicted by the four considered models @xcite are reported in table 1 . within each model ,
the quoted errors reflect the uncertainties on the input parameters .
these errors are highly correlated .
however , it is only for the model of ref .
@xcite that the error correlation matrix is readily available @xcite .
consequently , we have taken the errors of this model to be typical of and to apply to all this type of calculations .
.solar neutrino event rates for the clorine ( cl ) , gallium ( ga ) and kamiokande ( ka ) experiments predicted by the four `` standard '' models . [ cols="<,^,^,^",options="header " , ]     following ref .
@xcite , the hypothesis of an energy - independent depletion of the solar @xmath0 flux due to oscillations is tested by studying the function    @xmath1    where :    * f is the common factor by which all calculated @xmath0 fluxes are reduced ; * the indices @xmath2 and @xmath3 run over the three ( ga , cl and ka ) experiments ; * @xmath4 are the experimental results ; * @xmath5 are the theoretical predictions
. they are obtained from the @xmath6 of table 2 through the relations @xmath7 where @xmath8 is the fraction of the kamiokande detection efficiency due to flavour - blind neutral currents ; * s is the covariance matrix obtained from that of table 2 by multiplying the elements @xmath9 with @xmath10 by @xmath11 , those with either @xmath2 or @xmath12 by @xmath13 $ ] and @xmath14 by @xmath15 ^ 2 $ ] and by adding in quadrature the experimental errors to the diagonal elements .
we have at first replaced all asymmetric errors by their average values . with this procedure ,
the @xmath16 analysis yields @xmath17 with @xmath18 corresponding to a confidence level ( c.l . ) of 1.7 @xmath19 .    for comparison , if the theoretical predictions and their covariance matrix are taken from the model of ref .
[ 6 ] alone , one obtains @xmath20 with @xmath21 corresponding to a c.l .
of 0.072 % .    to test the sensitivity of these results to the procedure adopted in treating the errors ,
we have introduced a modification . in all operations involving two quantities ( weighted averages of the two ga and ka results and evaluations of the @xmath16 terms )
we have used the positive ( negative ) error on the first and the negative ( positive ) error on the second if the first quantity was smaller ( larger ) than the second . in this case
the results are @xmath22 with @xmath23 corresponding to a c.l . of 2.2 % .
the results obtained by using the second procedure are shown in fig .
2 . in each plane
the two ellipses are the projections of the volumes in which the points representing respectively the experimental and calculated rates lie with 68.27 % probability .
although not unacceptably low , the confidence level for an energy - independent depletion of the solar @xmath0 flux due to oscillations obtained using the results of the model of ref .
[ 6 ] is admittedly rather marginal .    in the analysis presented in this paper
we have used the average values of the predictions of four models introducing also additional few - percent uncertainties on them due to their spread .
the confidence levels obtained are remarkably good .    in conclusion , for the time being
, oscillation solutions of the solar neutrino problem based on an energy modulation of the solar neutrino deficit must be considered as perhaps suggested but certainly not compelled by the data .
t. kirsten , in : proc .
xviith international conference on neutrino physics and astrophysics , helsinki , finland june 13 - 19 , 1996 edited by k. enqvist , k. huitu and j. maalampi , world scientific , singapore , 1997 , p. 3 .
the existing measurements of the solar neutrino flux are compared with the predictions of all models capable of reproducing the other solar observables . 
 these predictions are supplemented by the hypothesis of neutrino oscillations with mass differences large enough to render energy - independent the depletion of the solar @xmath0 flux . 
 it is concluded that the data are consistent with this hypothesis and that an energy - dependence of the solar neutrino deficit must be regarded as an attractive possibility but not as a compelling reality .    
 ub / fi 98 - 1 + april 1998    solar models and neutrino deficit + g. conforto , c. grimani , f. martelli and f. vetrano + universit degli studi , i-61029 urbino , italy + istituto nazionale di fisica nucleare , i-50125 firenze , italy