A&A Supplement series, Vol. 123, June II 1997, 449-454

Received May 31; accepted October 11, 1996

**F. Ciacio -
S. Degl'Innocenti
- B. Ricci**

*Send offprint request: *S. Degl'Innocenti, Dipartimento di Fisica Università
scilla@vaxfe.fe.infn.it

Dipartimento di Fisica dell'Università di Ferrara,
via Paradiso 12, I-44100 Ferrara, Italy

Max-Planck Institut for Astrophysics, K. Schwarzschild Str. 1,
D-85740 Garching bei München, Germany

Istituto Nazionale di Fisica Nucleare,
Sezione di Ferrara, via Paradiso 12, I-44100 Ferrara, Italy

We present an updated version of our standard solar model (SSM) where helium and heavy elements diffusion is included and the improved OPAL equation of state (Rogers 1994; Rogers et al. 1996) is used. In such a way the EOS is consistent with the adopted opacity tables, from the same Livermore group, an occurrence which should further enhance the reliability of the model. The results for the physical characteristics and the neutrino production of our SSM are discussed and compared with previous works on the matter.

**keywords:** the Sun: evolution -- the Sun: general -- the Sun:
particle emission

In the last decades evolutionary computations of Standard Solar Models (SSM) played a fundamental role in understanding the inner solar structure and, in particular, in approaching the well known problem of solar neutrinos. Bahcall & Pinsonneault (1995; hereafter BP95) have already shown that to reach the agreement with observational evidence given by helioseismology one needs, in addition to the best available input physics, an accurate treatment of element diffusion all along the solar structure. The recent availability of an improved equation of state (EOS), as given by the OPAL group (Rogers 1994; Rogers et al. 1996) obviously suggests to investigate the influence of such an improvement on SSM. In this paper we discuss this scenario, presenting an updated SSM resulting from a recent version of FRANEC (Frascati Raphson Newton Evolutionary Code), where helium and heavy elements diffusion is included and the OPAL equation of state is used. Note that in such a way the EOS is consistent with the adopted opacity tables, from the same Livermore group (Rogers & Iglesias 1995), an occurrence which should further enhance the reliability of the model. In addition, updated values of the relevant nuclear cross sections are used and more refined values of the solar constant and age (BP95) are adopted.

Before entering into the argument, let us recall that
in recent years helioseismology has added
important pieces of
information on the solar structure, producing severe
tests for standard solar model calculations.
According to Christensen-Dalsgaard et al. (1993)
one can accurately determine the depth of the solar
convective
zone and the speed of sound at its bottom:

Richard et al. (1996; hereafter RCVD96) recently confirmed the
value of , finding
. In addition several determinations of the
helium photospheric
abundances have been derived from inversion of helioseismological data,
with results in the range (see Castellani et al. 1996 and refs.
therein):

The much smaller errors often quoted,
reflect the observational frequency errors only. The results actually
depend on the method of inversion and on the starting physical inputs
(e.g. the EOS), see RCVD96.

Note that, since in building a SSM one is dealing with three free parameters
(mixing length, original He content and original *Z*/*X*), with these three
additional constraints (, and ) no free
parameter is left for SSM builders.

After discussing the effect of the physical inputs, we compare our SSM with other recent solar model calculations, all including diffusion of helium and heavy elements, finding an excellent agreement. We present neutrino fluxes and the expected signals in ongoing experiments. A detailed presentation of our new model (profiles of density, temperature, chemical composition...) is available on World Wide Web at the address http://dns.unife.it/Fiorentini/index.htm and a more extensive discussion will be published elsewhere.

As for the computations, FRANEC has been described in previous papers (see e.g. Chieffi & Straniero 1989; Castellani et al. 1992). Recent determinations of the the solar luminosity () (BP95) and of the solar age () (BP95) are used. The present ratio of the solar metallicity to solar hydrogen abundance by mass corresponds to the most recent value by Grevesse & Noels (1993): . Plasma screening is treated according to the Graboske et al. (1973) calculation.

Following the standard procedure (see e.g. Bahcall & Ulrich 1988),
for each set of assumed physical inputs the initial *Y*, *Z* and the
mixing length parameter were varied until the radius,
luminosity and *Z*/*X* at the solar age matched the observed values
within a tenth of percent or better.
We considered the following steps:

a) As a starting model we used the Straniero (1988) equation of state, the version of the OPAL opacity tables available in 1993 (Rogers & Iglesias 1992; Iglesias et al. 1992) for the Grevesse & Noels (1993) solar metallicity ratio, combined with the molecular opacities by Alexander & Fergusson (1994); diffusion was ignored. This model is useful for a comparison with BP95 (without diffusion) which uses the same chemical composition.

b) Next, we introduced the OPAL equation of state. With respect to other commonly used EOS, this one avoids an ad hoc treatment of the pressure ionization and it provides a systematic expansion in the Coulomb coupling parameter that includes various quantum effects generally not included in other computations (see Rogers 1994; Rogers et al. 1996 for more details).

c) Next, OPAL 1993 tables were substituted with the latest OPAL opacity tables (Rogers & Iglesias 1995), again for Grevesse & Noels (1993) solar metallicity ratio. With respect to Rogers & Iglesias (1992) the new OPAL tables include the effects on the opacity of seven additional elements and some minor physics changes; moreover the temperature grid has been made denser.

d) Furthermore, we included the diffusion of helium and heavy
elements. The diffusion coefficients have been calculated using the
subroutine developed by Thoul (see Thoul et al. 1994).
The variations of the abundance of H, He, C, N, O and Fe are followed
all along the solar structure;
all these elements are treated as fully ionized.
According to Thoul et al. (1994)
all other elements are assumed
to diffuse at the same rate as the fully-ionized iron.
To account for the effect of heavy element diffusion
on the opacity coefficients
we interpolated (by a cubic spline interpolation)
between
opacity tables with different total metallicity
(*Z*=0.01, 0.02, 0.03, 0.04).

e) As a final point, we investigated the effect of updating the nuclear
cross sections for and reactions,
following a recent reanalysis of all available data (see Castellani et al.
1996, and Table 6 (click here)). For , a polynomial fit
gives (energy in MeV, *S* in MeV barn):

Alternatively, by using an exponential parametrization, as frequently
adopted in the literature, one obtains an equally good fit to
experimental data:

At the energies of interest, , the second expression
yields an *S* factor larger by about 5%, a value which is
indicative of the uncertainty on
at these low energies.
In the calculations, we used Eq. (4 (click here)).

The resulting solar models are summarized in Table 1 (click here), which deserves the following comments:

**Table 1:** Comparison among solar models obtained with different versions
of the FRANEC code. The labels (a) to (e) correspond to the models
defined in the text. Our best Standard Solar Model is (e).
The last column shows the helioseismological results as discussed in the text.
Here: S88 = Straniero (1988)

(): the introduction of the new OPAL EOS reduces appreciably the initial helium abundance. The Straniero (1988) EOS understimates the Coulomb effects neglecting the contribution due to the electrons, which are considered as completely degenerate, whereas the OPAL EOS includes corrections for Coulomb forces which are correctly treated (see Rogers 1994 for more details). The models with Straniero EOS have a higher central pressure and a higher central temperature, and correspondingly a higher initial helium abundance. The effect of an understimated Coulomb correction was discussed in Turck-Chièze & Lopes (1993) and in Dzitko et al. (1995). Note that the helium abundance is higher than the helioseismological determination by more than 2. Moreover, the transition between radiative and convective regions is not correctly predicted by the model, the convective region being definitely too shallow.

(): the updating of the radiative opacity coefficients has minor effects. The surface helium abundance is now within 2 from the helioseismological determination, but the convective zone is again too shallow.

(): this step shows the effects of diffusion. Helium and heavy elements sink relative to hydrogen in the radiative interior of the star because of the combined effect of gravitational settling and of thermal diffusion. This increases the molecular weight in the core and thus the central temperature is raised. The surface abundances of hydrogen, helium and heavy elements are appreciably affected by diffusion: the initial value is reduced to the present photospheric value , within 2 from the helioseismological results. More important, the predicted depth of the convective zone and the sound speed are now in good agreement with helioseismological values. With respect to models without diffusion, in the external regions the present helium fraction is reduced while the metal fraction stays at the observed photospheric value. Thus the opacity increases and convection starts deeper in the Sun. As an example of the diffusion process we show in Fig. 1 (click here) the time dependence of the He profile and in Fig. 2 (click here) the present H profile, calculated with and without diffusion.

(): The modifications of our Solar Model arising from the new values of the nuclear cross sections are negligible with respect to the other improvements just presented.

**Table 2:** Neutrino fluxes and signals obtained with different versions
of the FRANEC code. The labels (a) to (e) correspond to the models
defined in the text. Our best prediction is (e)

**Figure 1:** Time dependence of the He profile of our model e). The He
profile at is that of our "best'' standard solar model

**Figure 2:** H profile for a solar model without diffusion (our model c)
and with diffusion (our model e)

Our "best'' Standard Solar Model, model (e), appears in good agreement with recent calculations (Table 3 (click here)) by several authors, all including microscopic diffusion and using (slightly) different physical and chemical inputs, summarized in Table 4 (click here). Our ingredients are very close to those of BP95 and one notes a substantial similarity with that model. We only have a slightly lower central temperature, possibly an effect of the different EOS. Among the different models, diffusion looks less efficient in RVCD96, as a consequence of the inclusion of rotational induced mixing.

**Table 3:** Comparison among several SSMs, all including diffusion.
The correspondence between acronyms and references is as
follows: Cox et al. (1989),
Proffit
(1994), Dar & Shaviv (1996),
Richard et al. (1996).
BP95 indicates here the "best model with diffusion'' of
Bahcall & Pinsonneault
(1995).
FRANEC96 indicates our best model with diffusion, model (e).
The star indicates values of calculated by us
assuming fully ionized gas EOS

**Table 4:** Physical inputs of the SSMs in Table 3 (click here):
C&S70 = Cox & Steward (1970), MHD =
Mihalas et al.
(1988), CEFF = EOS of Eggleton et al. (1973)
with the Coulombian correction added (see
Christensen-Dalsgaard &
Dappen 1992), EFF =
Eggleton et al. (1973), BP92 =
Bahcall & Pinsonneault (1992), G&N93 =
Grevesse & Noels
(1993), G91 =
Grevesse (1991), F75 = Fowler et
al. (1975), C&F88 = Caughlan et al. (1988)

The predicted neutrino fluxes and signals from our SSM are summarized in Table 2 (click here). All in all, the results are rather stable with respect to the changements we have introduced as long as diffusion is neglected. The new equation of state () and the improved opacity () yield a somehow smaller (and also and CNO) neutrino flux as a consequence of the smaller central temperature, see also Turck-Chièze & Lopes (1993) and Dzitko et al. (1995) where a similar result is obtained. On the other hand, due to the higher central temperature, model (d) has significantly higher and CNO neutrino fluxes. It is essentially the increase of the boron flux which enhances the predicted Chlorine signal. The slight changement in the nuclear cross section weakly affects neutrino fluxes and signals: the and fluxes are reduced by about 5%, as a consequence of the correspondingly smaller value of . Should we use the polinomial parametrization of Eq. (3) one would get a further 5% decrease. As a final remark, we want to stress that again our results are in good agreement with other recent calculations including diffusion (see Table 5 (click here)).

**Table 5:** Comparison among the neutrino fluxes and signals of the SSMs in
Table 3 (click here)

Regarding the comparison with BP95, the small difference in the neutrino production reflects well the small difference in the central temperature, already remarked, and in the adopted values of (see Table 6 (click here)).

**Table 6:** Astrophysical *S*-factors
[MeV barn] and their derivatives with respect to energies *S*' [barn]
for our "best" solar model, for the "best SSM with diffusion'' of
Bahcall &
Pinsonneault (1995) (BP95) and for our models

*Acknowledgements*

We warmly thank V. Castellani and G. Fiorentini for their advice. F. Ciacio and B. Ricci are grateful to CNR for a fellowship. S. Degl'Innocenti and B. Ricci acknowledge the support of the European Union, through the Human Capital & Mobility Program. This work was also supported by MURST (Ministero dell' Università e della Ricerca Scientifica).

web@ed-phys.fr