Updating standard solar models

Astronomy and Astrophysics Supplement Series, Jul 2018

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.

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Updating standard solar models

Astron. Astrophys. Suppl. Ser. Updating standard solar models F. Ciacio 0 2 3 S. Degl'Innocenti 0 1 2 B. Ricci 0 2 0 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara , via Paradiso 12, I-44100 Ferrara , Italy 1 Max-Planck Institut for Astrophysics, K. Schwarzschild Str. 1, D-85740 Garching bei Mu ̈nchen , Germany 2 Send o print requests to: S. Degl'Innocenti , Dipartimento di Fisica Universita 3 Dipartimento di Fisica dell'Universita 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. 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 di usion 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 di usion 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 re ned 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: Rb=R = 0:710 0:716 cb = 0:221 0:225 (Mm/s) Richard et al. (1996 ; hereafter RCVD96) recently conrmed the value of Rb, nding Rb=R = 0:7137. 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) : Yphoto = 0:233 0:268: 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 (Rb, cb and Yphoto) no free parameter is left for SSM builders. After discussing the e ect of the physical inputs, we compare our SSM with other recent solar model calculations, all including di usion of helium and heavy elements, nding an excellent agreement. We present neutrino fluxes and the expected signals in ongoing experiments. A detailed presentation of our new model (pro les 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. (1) (2) As for the computations, FRANEC has been described in previous papers (see e.g. Chie & Straniero 1989; Castellani et al. 1992) . Recent determinations of the the solar luminosity (L = 3:844 1033 erg=s) (BP95) and of the solar age (t = 4:57 109 yr) (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): (Z=X)photo = 0:0245. 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) ; di usion was ignored. This model is useful for a comparison with BP95 (without di usion) 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 e ects 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 e ects 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 di usion of helium and heavy elements. The di usion coe cients 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 di use at the same rate as the fully-ionized iron. To account for the e ect of heavy element di usion on the opacity coe cients we interpolated (by a cubic spline interpolation) between opacity tables with di erent total metallicity (Z = 0:01, 0.02, 0.03, 0.04). e) As a nal point, we investigated the e ect of updating the nuclear cross sections for 3He +3 He and 3He +4 He reactions, following a recent reanalysis of all available data (see Castellani et al. 1996, and Table 6) . For S(0)34, a polynomial t gives (energy in MeV, S in MeV barn): S34(E) = (4:8 − 2:9 E + 0:9 E2) 10−4: (3) Alternatively, by using an exponential parametrization, as frequently adopted in the literature, one obtains an equally good t to experimental data: S34(E) = 5:1 10−4 exp(−0:83 E + 0:25 E2): (4) At the energies of interest, E0 20 KeV, the second expression yields an S factor larger by about 5%, a value which is indicative of the uncertainty on S34 at these low energies. In the calculations, we used Eq. (4). The resulting solar models are summarized in Table 1, which deserves the following comments: (a ! b): the introduction of the new OPAL EOS reduces appreciably the initial helium abundance. The Straniero (1988) EOS understimates the Coulomb e ects 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 e ect of an understimated Coulomb correction was discussed in Turck-Chieze & 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 de nitely too shallow. (b ! c): the updating of the radiative opacity coe cients has minor e ects. The surface helium abundance is now within 2 from the helioseismological determination, but the convective zone is again too shallow. (c ! d): this step shows the e ects of di usion. Helium and heavy elements sink relative to hydrogen in the radiative interior of the star because of the combined e ect of gravitational settling and of thermal di usion. 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 di usion: the initial value Yin = 0:269 is reduced to the present photospheric value Yphoto = 0:238, 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 di usion, 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 di usion process we show in Fig. 1 the time dependence of the He pro le and in Fig. 2 the present H pro le, calculated with and without di usion. (d ! e): The modi cations of our Solar Model arising from the new values of the nuclear cross sections are negligible with respect to the other improvements just presented. Our \best" Standard Solar Model, model (e), appears in good agreement with recent calculations (Table 3) by several authors, all including microscopic di usion and us0.3 0.2 (d) ing (slightly) di erent physical and chemical inputs, summarized in Table 4. 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 e ect of the di erent EOS. Among the di erent models, di usion looks less e cient in RVCD96, as a consequence of the inclusion of rotational induced mixing. The predicted neutrino fluxes and signals from our SSM are summarized in Table 2. All in all, the results are rather stable with respect to the changements we have introduced as long as di usion is neglected. The new equation of state (a ! b) and the improved opacity (b ! c) yield a somehow smaller 8B (and also 7Be and CNO) neutrino flux as a consequence of the smaller central temperature, see also Turck-Chieze & 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 signi cantly higher 8B 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 a ects neutrino fluxes and signals: the 7Be and 8B fluxes are reduced by about 5%, as a consequence of the correspondingly smaller value of S34. Should we use the polinomial parametrization of Eq. (3) one would get a further 5% decrease. As a nal remark, we want to stress that again our results are in good agreement with other recent calculations including di usion (see Table 5). 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F. Ciacio, S. Degl'Innocenti, B. Ricci. Updating standard solar models, Astronomy and Astrophysics Supplement Series, 449-454, DOI: 10.1051/aas:1997168