Gravitational settling has become part of the new standard solar model. It also appears that atomic di usion plays a major role in Bp, Ap, Am, and Fm stars where.
CONSISTENT SOLAR EVOLUTION MODEL INCLUDING DIFFUSION AND RADIATIVE ACCELERATION EFFECTS Turcotte1 , S., Teoretisk Astrofysik Center, Aarhus Universitet, DK-8000, Aarhus C, Denmark Richer2 , J., Michaud2 , G., Departement de physique, Universite de Montreal, Montreal, Canada, H3C 3J7 Iglesias, C. A. and Rogers, F. J. Lawrence Livermore National Laboratory, PO Box 808, Livermore, CA 94550
ABSTRACT The solar evolution has been calculated including all the e ects of the di usion of helium and heavy elements. Monochromatic opacities are used to calculate radiative accelerations and Rosseland opacities at each evolution time step taking into account the local abundance changes of all important (21) chemical elements. The OPAL monochromatic data are used for the opacities and the radiative accelerations. The Opacity Project data are needed to calculate how chemical species and electrons share the momentum absorbed from the radiation ux. A detailed evaluation of the impact of atomic di usion on solar models is presented. On some elements thermal di usion adds approximately 50 % to the gravitational settling velocity. While gravitational settling had been included in previous solar models, this is the rst time that the impact of radiative accelerations is considered. Radiative accelerations can be up to 40 % of gravity below the solar convection zone and thus a ect signi cantly chemical element di usion contrary to current belief. Up to the solar age, the abundances of most metals change by 7.5 % if complete ionization is assumed, but by 8.5 to 10 % if detailed ionization of each species is taken into account. If radiative accelerations are included, intermediate values are obtained. Di usion leads to a change of up to 8 % in the Rosseland opacities if one compares to those of the original mixture. Most of this e ect can be taken into account by using tables with several values of Z . If one isolates the e ects of radiative accelerations, the abundance changes they cause alter the Rosseland opacity by up to 0.5 %; the density is a ected by up to 0.2 %; the sound speed is a ected by at most 0.06 %. The inclusion of radiative accelerations 1 Departement de physique, Universite de Montreal, Montreal, Canada, H3C 3J7 2 CEntre
H3X 2H9
de Recherche en Calcul Applique (CERCA), 5160 boul. Decarie, bureau 400, Montreal, PQ, CANADA
{2{ leads to a reduction of 3 % of neutrino uxes measured with 37 Cl detectors and 1 % with 71 Ga detectors. The partial transformation of C and O into N by nuclear reactions in the core causes a 1 % change in the opacities which cannot be modeled by a change in Z alone. The evolution is allowed to proceed to 1010 yr in order to determine the impact at the end of the main sequence life of solar type stars. It is found that immediately below the convection zone, the radiative acceleration on some iron peak elements is within a few percents of gravity. The abundance anomalies reach 18 % for He in the convection zone but are kept within 12 % and 15 % for most because of grad . They would have reached 18 % in the absence of grad . Subject headings: stars: structure | stars: evolution | Sun: interior | di usion
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1. Introduction In standard stellar evolution it has generally been assumed, following Eddington's (1926) arguments, that the e ects of atomic di usion were small enough to be neglected. However it has recently been shown that He gravitational settling has a measurable impact on the solar pulsation spectrum. Indeed, now that very accurate helioseismological data are available, numerous recent models, (Bahcall & Pinsonneault 1992; Bahcall & Pinsonneault 1995; Bahcall et al: 1997; Basu et al: 1996; Guzik & Cox 1992; Prott 1994; Richard et al: 1996), show that the radius of the convection zone as well as pulsation frequencies can be reconciled with observations only if the surface abundance of helium has decreased over time. The He abundance pro le must also have changed. The only mechanism yet suggested to explain this is gravitational settling. If helium sinks in the gravitational well, other massive elements are bound to do the same. Guenther & Demarque (1997) have shown that this further improves the agreement. Gravitational settling has become part of the new standard solar model. It also appears that atomic di usion plays a major role in Bp, Ap, Am, and Fm stars where radiative acceleration leads to the formation of abundance anomalies (Michaud 1970). Atomic di usion has been suggested to play a role in explaining the Li abundance gap that appears at Te ' 6700 K in the Hyades and other clusters. Atomic di usion might then a ect most slowly rotating stars. None of the stellar evolutionary models that have been calculated up to now are fully self-consistent. A basic physical process, such as atomic di usion, may only be neglected if one introduces a competing process that wipes out its e ects. The e ects of all competing processes must be calculated. This generally has not been done. Even when di usion was included its e ect was never fully taken into account. In particular the Rosseland opacities depend on the relative composition of all elements. Iron is the main contributor to the Rosseland opacity over a signi cant fraction of the stellar envelope according to OPAL (Iglesias et al: 1992). The relative abundances of the elements change during evolution if atomic di usion is present. A self consistent solution to the stellar evolution problem then requires recalculating the Rosseland opacity from monochromatic opacities of each important element, at the same time as the abundances evolve. Radiative accelerations result from selective absorption by the various chemical species present of part of the net outgoing momentum ux carried by photons. Usually this absorption produces a net outward force on each absorbing species, which can often exceed the particle weight (see Gonzalez et al: 1995b; Richer et al: 1998). If radiative accelerations are present there appear both over and underabundances, mostly as a result of the imbalance between radiative forces and gravity. The models that have included atomic di usion have neglected the e ect of radiative acceleration on di usion and the local overabundances they can create within the star. For all species contributing signi cantly to Rosseland mean opacities, the radiative accelerations have to be recalculated continuously as their abundances change and a ect the local radiation spectrum. In most evolutionary stellar models calculated up to now, the Rosseland mean is interpolated
{4{ in a series of tables parametrized in terms of temperature, of a function of density (or an equivalent parameter; the parameter R =(T=106 )3 is used in the OPAL tables for example) and of the relative abundance of hydrogen, helium, and metals. The relative abundance of each metal with respect to the total mass fraction of metals is assumed constant and uniform for all opacity tables and thus throughout the star. The absolute abundance of each individual metal is merely scaled by metallicity (Z ). This is an ad hoc hypothesis which is not justi ed in most stars if atomic di usion is taken into account. Calculating the contribution of an element to the Rosseland averaged opacity involves the same atomic data as the calculation of its radiative acceleration and ideally both should be calculated with the same data for consistency. Attacking this problem has been made possible by the availability of large atomic data banks such as OP (Seaton et al: 1994) and OPAL (Iglesias & Rogers 1996) and by the development of faster computers. The mean opacity tables generally distributed up to now are not sucient since one needs the monochromatic opacities (the absorption spectra ) for all important species. This paper is the rst involving self-consistent detailed calculations of radiative accelerations and of the impact of radiatively driven di usion on stellar structure. It is important to determine the size of the e ect and the stellar models where the impact is largest. In this paper, our approach will be described and applied to the Sun. Although the e ects are expected to be relatively small for the Sun, it is important to verify to what extent the di usion of individual metals may be neglected. At what level of accuracy of the reproduction of the pulsation spectrum should one expect to have to treat the detailed di usion of individual metals, including the e ects of radiative accelerations? Is it ever larger than other sources of uncertainty? These questions can only be answered by doing the di usion calculation in detail. The paper is structured as follows. The use of the monochromatic data in the evolution code is described in x 2 while the radiative accelerations and opacities from TOPbase (called MONO11) are described in the Appendix and those from OPAL by Richer et al: (1998). The physics included in the evolution code is detailed in x 3. The di erences with other codes are mostly limited to the more elaborate treatment of di usion required to take full advantage of the monochromatic data. The introduction of physics requiring free parameters is avoided as much as possible. Solar models are discussed in detail. Rosseland mean opacity tables are used rst (x 4), in order to compare our models to those in the literature. Then solar models calculated using the monochromatic opacities and radiative accelerations are investigated (x 5). All evolutionary models were calculated with OPAL opacities, initially the 1992 tables, then the 1996 tables including the contributions of 21 elements. Some of the solar models were computed using radiative accelerations calculated with atomic data from TOPbase; these are compared with results obtained using OPAL data for the radiative accelerations. Since the OP data were rst available to us, some of the comparisons between models calculated entirely with opacity tables and models calculated with monochromatic data were done using the OP data. They were not repeated with OPAL data when it was evident that the result would not change. The e ect of the monochromatic data on helioseismological results
{5{ are discussed. We seek to determine at which level of accuracy the use of the monochromatic data is required. A brief description of a complete solar mass model calculated with OPAL monochromatic data up to an age of 1010 yr is presented. It is followed by a discussion of our main results and of the suggestion for including atomic di usion self consistently in solar models calculations. A table appearing near the end of this paper (Table 7) summarizes the principal features of the models presented here.
2. Opacities The di usion approximation for radiative transfer is adequate throughout stellar interiors and it is therefore appropriate to use the Rosseland averaged opacity in the structure equations. To compute the Rosseland opacity while taking into account the variation of the abundance of every element at every point in the star one needs to have a detailed spectrum for each element. Once the individual spectra, multiplied by the local concentration of each element, have been added the usual simple integration over the total spectrum, 1 = Z 1 dB d Z dB d ; (1)
R
dT
dT
yields the Rosseland opacity (R ). This is time consuming because the spectra must be added and the integral has to be evaluated at each mesh point and at every time step. Two series of models were calculated. All have Rosseland opacities equal to OPAL93 or OPAL96 at solar abundances so as to isolate the di erential e ects of di usion and radiative accelerations. In one series of models (MONO11) all variations caused by di usion were calculated with the atomic data stored at Strasbourg by the OP project. In a second series of models, the monochromatic opacities of OPAL were used. By OPAL93, we mean OPAL's previous library of Rosseland opacities (Rogers & Iglesias 1992a, referred to here as OPAL92) with extrapolations by Prott (1994); by OPAL96 we mean the more recent Rosseland opacity tables (Iglesias & Rogers 1996). We complement these opacities at low temperatures (T < 12 000 K) with the Kurucz93 (Prott 1994) opacity tables.
2.1. The MONO11 data The MONO11 spectra, opacities and radiative accelerations were computed from the atomic data of the OP team publicly accessible in the TOPbase database at Strasbourg as part of the Opacity Project. Aside from H and He, the elements included are C, N, O, Ne, Mg, Si, S, Ar, and Fe. They are the most abundant metals in a solar mix and are the most important for stellar
{6{ opacities. The solar mixture used for the MONO11 data is that of Grevesse & Noels (1993) scaled so as to give a total metal mass fraction of 0.02 (see Table 1).
[Table 1 goes about here.] The TOPbase data have been used previously by Gonzalez et al: (1995a), Gonzalez et al: (1995b), LeBlanc & Michaud (1993), and Seaton (1997), to calculate radiative accelerations. The spectra for heavy elements are constructed as described in Gonzalez et al: (1995b, hereafter referred to as GLAM). The spectra span a range from 0 to 20 in u h=kT and are divided into 4000 intervals of equal width. The spectra for each element contain every transition available in TOPbase for all ionization states as well as the photoionization continuum opacity using the cross sections also provided by TOPbase. The total cross section of each line is spread uniformly over the width of the interval which contains the line center whatever the line width (except for H, for which multi-interval line pro les are used). The spectra are computed on a grid of 50 temperatures [the same as for the Rogers & Iglesias (1992a) tables] and 21 Re values, where Re is a density parameter related to the electron number density Ne by Re Ne =T 3 (in cgs units), as in Gonzalez et al: (1995b). The data for Fe do not include ne structure and have been shown to be missing many lines and important continuum contributions (Iglesias et al: 1992; Iglesias & Rogers 1995). The e ect of using more complete Fe data will be seen in the comparison with OPAL. The e ect is however so large at some temperatures and densities that it was necessary to correct the spectra calculated with TOPbase in order for the Rosseland opacities to be consistent with OPAL93 or OPAL96 at solar abundances. The procedure is described in the Appendix. This correction procedure is approximate and was found to lead to inconsistencies at solar center in some models. This will be further discussed in x 5.1 below. The improved data used by Seaton (1997) were not available to us for this work; however even this larger database [including the so-called PLUS DATA (Seaton et al: 1994)] still leads to solar opacities signi cantly lower than the OPAL results (Iglesias & Rogers 1995), and the corresponding spectra would presumably still require ad hoc corrections, in our approach3 . Using the TOPbase data permits introducing physical processes in the calculation of radiative accelerations that are important (Gonzalez et al: 1995b) and which cannot be calculated directly with OPAL spectra. They are introduced in the OPAL radiative accelerations as described in Richer et al: (1998). 3 Iglesias
& Rogers (1995) have shown that the PLUS DATA is missing important photoabsorption cross sections for all elements, not only Fe.
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2.2. The OPAL data The OPAL data (Iglesias & Rogers 1996) are de ned on a mesh of 75 temperatures, 25 R ( =T63 , in cgs units), for a mixture of 21 elements at 104 frequency points, for a total of a little less than 4 108 opacity values. The spectra do not represent average absorption over a frequency interval but are calculated precisely at the grid frequencies. The opacities at frequency points are exact (in as much as OPAL physics is exact) but the frequency sampling is not sucient to resolve most lines at low to medium temperatures (see Richer et al: 1998). This can have a signi cant impact on radiative force calculations. The chemical composition is given in Table 1. The relative abundances of metals from that table are used as initial values for the Sun. However note that the OPAL spectra were computed for H and He mass fractions of 0:35 and 0:63, respectively, which corresponds to the current solar center composition. Where composition di ers from that used for the OPAL spectra, the principle of corresponding states (Rogers & Iglesias 1992b) is invoked to justify interpolation in log T and log Ne (or log Re , but not in log R), to build spectra at the new composition (with the same Ne but a di erent ). Note that the OPAL orthogonal (T; R) grid becomes irregular in terms of (T; Ne ). Concretely, the procedure amounts to computing all the necessary integrals on a small subset of the OPAL (Ti ; Rj ) grid, not necessarily rectangular, and then to weigh the results based on the desired Ne rst, for each Ti , then on T . The spectrum for an individual element includes the bound-bound and bound-free transitions, the free-free absorption as well as the electron scattering caused by the electrons provided by the ionization of the element. The total monochromatic opacity is simply the sum of the 21 individual monochromatic spectra properly weighted by concentrations. The amount of OPAL data is much larger than that of MONO11 data because of the larger number of elements and of the ner frequency mesh.
2.3. Interpolations and smoothing Obtaining the Rosseland mean opacity for a speci c point in a stellar model involves calculating it on the closest grid points in the Re vs T plane and then interpolating it to the point of the model. It is important for numerical stability that the interpolation lead to smooth gradients of the opacity and of other parameters dependent on opacity. For models constructed with mean opacity tables, we use the interpolation routines supplied with the tables. These procedures include a preliminary smoothing of the original table values based on an algorithm written by Seaton (1993), and designed to provide such improved di erentiability. For models calculated with monochromatic opacities, one cannot construct an interpolated spectrum at the model's temperature because of the temperature dependence of the frequency mesh. However since computing the Rosseland mean is by far the most time consuming operation
{8{ in our models, it is important to use the most e ective method. We found that the best approach was to limit the order of interpolation and to smooth the resulting opacity pro les. For both MONO11 and OPAL monochromatic data, the interpolation in the log Re or log R dimension is always linear. The quantities log Re and log R do not vary much in most of the stellar interior; they do vary more in regions close to the photosphere, but these regions are typically in convection zones where the opacity is not relevant for stellar structure. Interpolation in log T is based on lagrangian polynomials. Examples of interpolation with MONO11 in a one solar mass model are shown in boxes (a) through (e) of Figure 1. Interpolation is linear in (a), quadratic in (b), and cubic in (c). Non uniform interpolation is used in the remaining cases to minimize discontinuities associated with a shift in the interpolation range: cubic about half-way between grid points and quadratic around grid points in (d), and, in (e), linear between two temperature grid points but quadratic near a grid point. Higher order interpolations, uniformly cubic as in (c) for example, give smoother results but even in this case there remain di erences between the opacities calculated from the monochromatic data and those interpolated from the tables of the order of a few percents. On grid points the ratios would be 1.0 (by virtue of the corrections introduced in Eq. (A6)). The di erences come entirely from di erences in the interpolation procedures. The largest features are well correlated with regions where the models cross from one interval in the OPAL93 log R grid to an adjacent interval. Some features are correlated to similar crossovers in the MONO11 log Re grid. The interpolation errors are of the order of a few percents which is similar to the precision of the interpolation procedure of the OPAL tables. In the evolutionary models, interpolation in MONO11 is done as in (e) whereas interpolation in the monochromatic OPAL data is done as in (a) because of the length of each calculation.
[Figure 1 goes about here.] Part (f) of the gure shows the importance of the equation of state in the comparison between opacities. The OPAL tables require log R as input whereas MONO11 data requires log Re as input. When using MONO11 we take care to use the same equation of state between log R and log Re as used to compute monochromatic data. Otherwise, one introduces spurious variations as shown in part (f) of the gure where were used both the Eggleton et al: (1973) EOS used in the evolution model (see x 3.2) and a simple MHD EOS (Gonzalez et al: 1995b; see Hummer & Mihalas 1986, for more on the MHD formalism) which takes into account neither degeneracy nor the contribution of metals to the electronic density but does include many excitation states for each ionization state of H and He. The opacity di erences resulting from the use of these two di erent electronic densities can be as high as 10 %. After calculating the opacity pro le throughout the model, it is necessary to smooth it. This is done locally by least-square tting a parabola on a limited number of points (15) centered on each mesh point, in succession. The value for the central opacity given by the t is substituted
{9{ for the original central value. This smoothing method, somewhat analogous to Savitsky{Golay ltering (Press et al: 1986) is very fast. Small structure in the opacity pro le is preserved by this ltering, but interpolation slope discontinuities are erased almost completely. Figure 2 illustrates this smoothing action in the vicinity of the convection zone inner boundary, in the current Sun. The dashed line shows the Rosseland opacity pro le computed from OPAL monochromatic data; the solid line is the smoothed pro le. The dot-dash line gives the corresponding relative change in the opacity due to smoothing. These changes never exceed 1 % throughout the model, and most corrections are less than 0:4 %.
[Figure 2 goes about here.] 2.4. Radiative forces The calculations of radiative accelerations, grad , using OPAL data were carried out as described in x 2 and 3 of Richer et al: (1998). The calculations described there were carried out at the same time as the evolutionary calculations. For the MONO11 calculations, the radiative accelerations were described in GLAM, with some additions described in the present Appendix. The radiative forces are subject to the same interpolation and smoothing procedures as the opacities.
3. The models The code evolved from the one described in Prott (1994) and Prott & Michaud (1991). The part which computes stellar structure and evolution has not changed much from the original Vandenberg (1985) code (see also Kippenhahn et al: 1967). The standard Lagrangian structure equations are solved using a Henyey scheme assuming spherical symmetry.
3.1. Surface boundary condition The surface boundary condition is applied using a triangulation method as described in Kippenhahn et al: (1967). The integration of the photospheric pressure is performed using a fourth-order Runge{Kutta method. For solar models the usual Krishna-Swami relation was used (Krishna-Swamy 1966). Some solar models were calculated with the grey atmosphere T ( ) relation: one needs to determine for this boundary condition since it will be used to calculate evolutionary models for other stellar masses (see Turcotte et al: 1998). Tabulated model atmosphere, e. g., from Kurucz, are often preferred to a grey atmosphere in non-solar models but an analytical t is more appropriate for our simulations because it can be used for any chemical composition unlike the models which are currently available only for a xed metal composition.
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3.2. Equation of state The equation of state is that of Eggleton et al: (1973) including the Coulomb correction, known as the CEFF equation of state (Christensen-Dalsgaard & Dappen 1992). The main advantage of this EOS is that all thermodynamical derivatives are analytical. The main disadvantage is that the handling of ionization and the Coulomb correction is only approximate. The problem is that one needs to know the ionization equilibrium in order to compute the Coulomb correction and vice versa. One has to iterate or alternatively to make some assumptions for the ionization. In the current version of the EOS, the Coulomb correction is calculated with the assumption that H and He are fully ionized. This leads to an overestimate of the correction at lower temperatures. We have encountered problems with this formulation when core-hydrogen is exhausted but the EOS behaves correctly for main-sequence models. The OPAL EOS gives better agreement with pulsation spectra (Bahcall et al: 1997) and its use would have been preferable but it was not available for arbitrary compositions. Since the main aim of our study is to determine the di erential e ects caused by abundance changes due to di usion and radiative accelerations, it is important to use an equation of state, such as the CEFF, where the di erential e ects of abundance changes are included.
3.3. Energy generation The nuclear energy generation routine of Bahcall & Pinsonneault (1992) was used; in particular, neutrino cross sections are from Bahcall (1989). The 17 ON cycle is treated as a correction to the CN cycle. The only isotopes explicitly involved in nuclear reactions are 1 H, 3 He, 4 He, 12 C, 13 C, 14 N, and 16 O. Other reactions in the PP chains and the CN cycle are fast reactions that are assumed in equilibrium. The triple- reaction is not included. The total energy generated in the star also takes into account the gravitational contribution and the net loss of energy due to neutrinos. The abundance variations due to nuclear reactions are treated simultaneously to atomic di usion. Nucleosynthesis adds sink and source terms to the di usion equations for H, He, and CNO (see Eq. (2) below).
3.4. Convection Delimitation of the convective zones is based on the Schwarzschild criterion. The energy
ux in the convection zones is computed with the Bohm-Vitense mixing length formalism (Bohm-Vitense 1992). Mixing is supposed to be instantaneous in the convection zones and homogenizes the abundances therein. When there are two convection zones near the surface, such as in A-type stars, they are assumed linked by overshoot for the purpose of di usion (Latour et al: 1981). Elsewhere, there is no overshoot of turbulent motions in radiative regions. Turbulent
{ 11 { pressure is ignored.
3.5. Spatial and temporal discretization As one computes the evolution of stars of various masses, including the di usion of many elements simultaneously, one needs to adapt the mesh to varying conditions in the star. The mesh needs to be re ned where the structure variables vary rapidly, for example in the core and around the boundary of convective zones. It also needs to be suciently dense where an abundance gradient forms in order to avoid numerical instabilities. Abundance gradients created by di usion can be very steep and thus require a ne mesh. The mesh thus has to satisfy the con icting demands of structure and di usion. Each dependent structure variable and all abundances are given equal weight in our models. A ne mesh is required near the boundaries of convection zones since abundance gradients tend to form there and solving the di usion equations is dicult. The basic rezoning process is similar to that described by Richer & Michaud (1993). There can be a maximum of 1500 mesh points and a typical main-sequence model has between 600 and 1000 mesh points. The size of time steps is mainly determined by the rate of variation in time of the dependent structure variables, of the mass of the convection zones, and of the e ective temperature. Abundance variations (with the exception of H and He) have been given little in uence on the evolutionary time step since it was divided into a number of sub-time steps for the di usion equations. The smallest evolutionary time step is 104 yr, the maximum is 300 106 yr.
3.6. Abundance variations and di usion We follow in detail the abundances of all elements listed in Table 1, the isotopes 3 He, 13 C plus 6 Li, 7 Li, 9 Be, 10 B, 11 B totaling 18 elements for the MONO11 runs and 28 for the OPAL runs. The di erential equations for the abundances in spherical symmetry are of the form
dXi = ? @ 4r2 X w ? X j X + X j ; i i create burn i dt @Mr j j
(2)
where wi is the net average (over ionization states | see below) velocity of element i in the center-of-mass frame of reference, and the 's are the rates for nuclear reactions involving element i. The abundances of all elements included in the opacity and the nucleosynthesis are updated at every iteration over the star's structure. At each iteration, the di usion equations, excluding nuclear terms, are rst solved for each metal, and then the non-linear system of equations for the di usion of H and He and all the nuclear terms from the PP-chains and the CNO cycle (see
{ 12 { Prott & Michaud 1991 for details) are solved. The abundances of other trace elements (Li, Be, . . . ) are computed only once the nal structure for a given evolution time-step is converged. The di usion equations for metals (all included species except H and He) are solved using a nite element formalism as described by Turcotte & Charbonneau (1993) (see also Charbonneau & Michaud 1991). The di usion equations for metals are solved throughout the star including the convection zones. Mixing in the convective zones is enforced by introducing a large turbulent di usion coecient, typically of the order of 105 times the atomic di usion coecient. The abundances of H and He and the correction to CNO from nuclear reactions are solved iteratively until the maximum relative variation in successive iterations is smaller than one part in ten thousand4 . The di usion coecients, velocities and nuclear rates are taken as constant over the evolution time-step. All quantities in convective zones are averaged over the mass of the convective zone and are assigned to one mesh point at the boundary of the solution domain. They are connected to the radiative regions by the boundary conditions. The di usion coecients (including thermal di usion coecients) and velocities are determined by solving Burgers' ow equations for ionized gases (Burgers 1969, chapter 3, eqs. 18.4{6) for all di using elements as in Prott & Michaud (1991). The collision integrals from Paquette et al: (1986) are used. The equations are solved for an average ionization state for all elements except H and He since the radiative forces are determined for an average ionization state by Richer et al: (1998). The average charge for each metal is obtained directly from the OPAL tables, interpolating exactly as for opacities, or by using the MHD formalism, depending on the model, as summarized in Table 7. Hydrogen is assumed to be fully ionized in di usion calculations, which is justi ed since regions where hydrogen is partially ionized are strongly convective. Helium ionization is an output of the CEFF equation of state; for di usion, He i is added to He ii as only He ii and He iii are considered5. These approximations would not be valid in stars with no super cial convection zone or with only a very thin one, because the very large di usion coecients of H i and He i in cooler envelope layers can compensate for their low concentration and let them carry an important part of the element di usive ux. Radiative forces on H and He are set to zero; this is a good approximation for the solar interior. Some di usion velocity comparisons will be made in x 5.3 with the approximate analytical ts that have been proposed for solar work by Thoul et al: (1994) in replacement for the full solution of Burgers' equations. is sucient to require convergence to the 10?4 level since convergence is quadratic, and nal errors are then smaller than that by several orders of magnitude almost everywhere in the model. Since each layer only has a very small e ect on global solar properties such as its luminosity or radius, it is still possible to have these parameters converge to a better precision. 5 Like H i, He i is abundant only in the solar atmosphere and convection zone, so adding it to He ii (to insure that concentrations add up to one) introduces a totally insigni cant error in di usion calculations. 4 It
{ 13 {
3.7. Computational ow The equations of structure, of di usion, and for nuclear reactions are solved successively in a loop until the relative corrections to the dependent structure variables for two successive loops (radius, temperature, luminosity, and pressure) are smaller than 10?4 (see Footnote 5). The surface boundary conditions are veri ed and the spatial mesh is optimized. If needed the structure and abundances are converged again. The evolution of the star is followed from an initial homogeneous, fully convective pre-main-sequence model.
4. Solar models based on Rosseland opacity tables In this section solar models computed with the evolutionary code described here but without the monochromatic data (i. e., using the OPAL93 and OPAL96 Rosseland opacity tables) are compared to some of the more recent models available in the literature as well as to observational data. These models are needed in order to determine the di erential e ects of using radiative accelerations and monochromatic opacities. The Sun may be used as a laboratory to test a stellar evolutionary code because of the wealth of information on its structure from helioseismology as well as because of the number of independent solar models available in the literature. Though the radius and luminosity of the Sun are tted constraints and thus cannot be used as tests, the depth of the convection zone and the surface helium abundance are well constrained by helioseismology and so are independent tests of the models. The pro le of the square of the isothermal sound speed (u = P=) is also reproduced with an accuracy better than 0.5 % by existing models.
4.1. Parameters of the models Solar models have two main free parameters, the mixing-length parameter ( ) and the initial helium abundance (Yo ). The usual procedure (see e. g., Bahcall & Ulrich 1988; Prott & Michaud 1991) of computing a number of models while varying and Yo until the observed solar radius R and luminosity L are reproduced within a predetermined range was followed. Iterations were made until R and L were reproduced with a precision of the order of 10?5 . As a general rule, Yo depends mostly on the interior structure and nuclear rates while depends more on surface conditions such as the treatment of the atmosphere and convection. For the models discussed here R = 6:9599 1010 cm, L = 3:86 1033 erg/s and a solar age of 4.60 Gyr were adopted. The initial metallicity is chosen such that the surface value at the solar age of Z=X = 0:0245 be approximately reproduced. Values of L varying from 3.844 to 3:9 1033 erg/s and of the solar age from 4.55 to 4.63 Gyr have been used by di erent authors. The acceptable uncertainties in inputs is small due to the high precision helioseismological data. E ects of varying constraints such as age, R and L were previously analyzed by Prott & Michaud (1991) in the context of determining precise di erential e ects of di usion on solar models.
{ 14 { Our models are evolved from the pre-main sequence while some of the comparison models such as Bahcall & Pinsonneault's (1995, hereafter BP95), are evolved from a zero-age main sequence star. As Prott (1994) pointed out, one needs to take into account the time spent on the pre-main sequence in order to compare models of similar ages. Our model of 4.6 Gyr is of approximately the same age as BP95's 4.57 Gyr model. An upper limit to possible di erences was obtained by converging one of our models to an age of 4.57 Gyr. As would be expected the surface abundance variations are smaller in the younger model with di usion, the core is a little cooler and less dense. The overall di erences are well under one percent. The largest single di erence is 1.3 % for . Models using a luminosity of 3:844 1033 erg/s preferred by BP95 were also calculated (to 4.6 Gyr). The surface parameters are virtually unchanged while the core is again cooler and less dense than in the canonical model. In this case, the biggest change is a 2.3 % reduction of the predicted neutrino ux detected by Cl detectors. Initial metal abundances have some impact on solar opacities and neutrino uxes. Models calculated by Turck-Chieze & Lopes (1993) and Richard et al: (1996, hereafter RVCD96) show that for a variation of approximately 5 % in the initial metallicity, all the global model parameters vary by less than one percent, except for the neutrino uxes which are highly sensitive to the core temperature and which increase by as much as 6 %. When di usion is included, our nal metal abundances are constrained to be approximately within 1 % of the adopted metallicity at the solar age. Grevesse & Noels (1993) and Grevesse et al: (1992) nd that measured photospheric abundances agree, within measurement errors (which for most elements are of the order of 10 %), with measurements in meteors which are understood to be primordial abundances. In that sense, their quoted Z=X of 0.0245 can be taken as either the primordial or current value. However Dar & Shaviv (1996) argued that since, for the volatile elements, it is the current photospheric abundance which is used in the tables of Grevesse & Noels (1993) and Grevesse et al: (1992), it is the solar age abundance of the volatiles which should be compared to the values in the tables. This correction has not been introduced here. As will be seen below, di usion causes a reduction of Z of about 9 % but reduction of the abundance of individual elements between 7.5 and 10 % . Since those di erences are smaller than the current accuracy of photospheric abundance determinations we will not attempt to force the convergence of each abundance. It is then not meaningful to force convergence of solar Z=X to better than 1 %. For our most accurate comparisons, we will instead use exactly the same primordial Z=X for the two models being compared. The e ect of any remaining di erence in Z=X on the solar age model may then best be evaluated using the dependence on Z=X that Turck-Chieze & Lopes and Richard et al: obtained.
{ 15 {
4.2. Models without di usion Solar models without di usion are helpful to validate the code by comparing with models in the literature and are needed to determine the di erential e ects of di usion. Our results are compared with non di usion models computed by BP95 and RVCD96 in Table 2. The model calculated using the OPAL93 opacities (model A) has a slightly lower core temperature than the BP95's. Checking our opacities with those published in Bahcall & Pinsonneault (1992) as well as those listed in Iglesias & Rogers (1991), we nd that our opacities are systematically smaller in conditions typical of the solar center (as was previously noticed by Prott 1994). Taking into account these opacity di erences, the non di usion models are compatible with the other models.
[Table 2 goes about here.] The di erences amongst neutrino capture rates predicted by our models and those given by other key studies (see Tables 2, 3, and 6) are due in part to variations in the nuclear cross sections used in the models. A quantitative comparison of some of these rates is deferred until x 6.
4.3. Models with di usion The importance of thermal di usion may be seen from Figure 3 (top part).
[Figure 3 goes about here.] In this gure, w(A) includes the contribution of both gravitational settling and thermal di usion while wT (A) includes only the contribution of thermal di usion. For C, O, and Fe the thermal di usion contribution equals between 25 and 50 % that of gravitational settling. Thermal di usion clearly should not be neglected. The e ect of thermal di usion is sometimes calculated by using the simpli ed approximation suggested by Michaud & Prott (1992, hereafter MP92). Figure 4 shows the results of three di erent
[Figure 4 goes about here.] calculations for the net (or total) di usion velocities of H, He, O, and Fe in the envelope of a 30 Myr Sun. This young age was chosen so that the contributions of concentration gradients to di usion velocities would be negligible; radiative forces were also turned o . What is left is the e ect of gravity, the electric eld, thermal di usion, and dragging forces between di erent species. In all three calculations, averaged metallic ions were used (as explained in x 3.6), to limit the size of the system of velocity equations to 1 or 2 (depending on the method) equation(s) per atomic species. The thick lines include all e ects; in particular, friction terms between all di erent pairs
{ 16 { of particles were taken into account simultaneously. Symbols represent velocities as calculated by the evolution code. To speed up calculations, the evolution code computes di usion velocities in two steps, separating trace elements from the main plasma constituents. The latter are treated in a fully consistent manner (as far as dragging terms and thermal di usion is concerned), neglecting only collision terms with metals. Then metals velocity equations are solved using the already calculated H, He and electron velocities, and neglecting only collision terms among di erent metallic species. This two{step solution underestimates the hydrogen velocity by about 5 %. The third calculation (thin lines) was made using the MP92 analytic approximations, in which metals and the main constituents are again treated separately, but in a simpler manner. The gure shows that all three methods are in reasonable agreement, the largest di erence occurring with the MP92 approximations for He and Fe. This re ects in part the crudeness of the MP92 thermal di usion correction; it is estimated to be accurate to only about 20 % (see the bottom part of Figure 3). This gure also shows that to a rst approximation, di usion velocities in the Sun increase roughly linearly with radius (see also Fig. 13). Models including the di usion of all elements are compared in Tables 3 and 4 (model H is the subject of x 5). Predicted abundance variations, shown in Table 4, are the most immediate e ect of di usion. Both BP95 and RVCD96 are used as comparison. Model C has relative abundance variations similar to those of BP95. RVCD96 nd variations in the super cial metal abundances smaller by a factor of two. This is surprising insofar as they use a formalism similar to that used in model C to calculate their di usion velocities. Their result can however be understood if they did not include thermal di usion. The He, C and O abundance variations in the center are similar for all models since they are dominated by nuclear processes. However for Z , there is a contribution both from the transformation of C into N and from the settling of metals. BP95 used the numerical algorithm of Thoul et al: (1994) for di usion but assumed that all metals di use like fully ionized iron; this overestimates collision integrals for Fe ions because their charges are in fact closer to +21 on average, near the center, and even lower in the envelope. As a result, their metals di use more slowly than ours near the solar center and they obtain a smaller Z variation than we do (5.3 % instead of 6.6 %). The analytical ts of Thoul et al: would have resulted in larger di usion velocities (at least for oxygen and iron | see x 5.3) and using them would have resulted in a larger Z variation than we obtain near the center.
[Table 3 goes about here.] Comparing the more global properties of the models (see Table 3) one notices an overall agreement for all models with similar physics. The radius of the convection zone of all models are consistent with the observational data (0:713 0:003 R , Christensen-Dalsgaard et al: 1993), although it is somewhat high for model D. The only di erence between models C and D are the opacities. The ratio of these opacities (Figure 5) is nearly one with the exception of a 5 % dip near
{ 17 { the bottom of the convection zone and a 10 % increase near the surface. The opacity di erence at the base of the convection zone, shown by Iglesias & Rogers (1996) to be an e ect of the improvement of the equation of state used to calculate the opacities, would explain its relative shallowness.
[Figure 5 goes about here.] [Table 4 goes about here.] 4.4. Comparison to helioseismology Models derived from helioseismological data (so called seismic models) give an observed value for the function u = P= which is closely related to the sound velocity. One can compare the structure of a theoretical model to a seismic model like, for instance, that of Dziembowski et al: (1994, hereafter DG94)6 . Predicted and observed isothermal sound velocities (actually P=) have been shown to agree to within 0.5 % between 0.2 and 0.65 R , and around 1 % up to 0.9 R . The comparisons are less accurate close to the core because of uncertainties in the seismic models and close to the atmosphere as a result of approximations in the modelization of convection and atmospheres in stellar models. It has been veri ed that using Basu et al:'s (1996) more accurate data as a reference, instead of DG94, has an impact on the general behavior of the comparisons but has none on the following analysis. Results for standard models are shown in Figure 6. Models including di usion are substantially closer to the seismic model than models neglecting di usion. This is in agreement with numerous other studies, notably BP95's whose best model is also shown in the gure (their non-di usion model is not shown but would be almost identical to model A).
[Figure 6 goes about here.] The shape of the curves inside 0.2 R is a feature of the DG94 seismic model and can also be seen in FRANEC96 results (Ciacio et al: 1997; Degl'Innocenti et al: 1997). The apparent divergence of the models near the surface is an artifact of comparing models having slightly di erent radii and density pro les on a common normalized radius scale. If one compares the models on a common lagrangian mesh the curves remain approximately at in these outer regions. As was mentioned for the boundary of the convection zone, replacing the OPAL93 tables by the OPAL96 tables leads to a slight degradation of the comparison with the seismic model in the region where the correction in the OPAL EOS has an impact on the opacity. When comparing 6 The model was taken from the SOHO website and is credited to Dziembowski, W. A. and Goode, P. R., 1994
{ 18 { their models with their versions of OPAL92 and OPAL96 opacities, a similar behavior has been obtained by FRANEC96 as well as Christensen-Dalsgaard (1997) but, surprisingly, the opposite trend has been observed by both Bahcall et al: (1997) and Morel et al: (1997). No evident explanation was found when reviewing the papers as all the cited works use the latest OPAL equation of state and treat di usion in similar fashion to BP95. They also essentially agree on primordial abundances, and other macroscopic solar parameters. This indicates that previously insigni cant modeling di erences may have an impact at this level of precision.
4.5. Solar models with Eddington's grey atmosphere As we will use the Eddington's grey atmosphere T ( ) relation rather than the KrishnaSwamy's relation for other models we have converged solar models with the Eddington's T ( ) relations to calibrate the for those models. The resulting values for Y0 and are listed in Table 5 for later use. Results are shown for both the EFF and CEFF equations of state.
[Table 5 goes about here.] 5. Impact of the monochromatic data All converged solar models with di usion including the monochromatic data are summarized in Table 6.
[Table 6 goes about here.] Model F based on the MONO11 opacities is compared to model C in Figure 7. In Figure 8, models G and H
[Figure 7 goes about here.] [Figure 8 goes about here.] using the OPAL monochromatic data are compared to model D. Also shown is another model (D') similar to model D but based on additional approximations for the di usion of metals, that is, full ionization and settling of all elements more massive than oxygen at a rate identical to that of iron. All models are interpolated on DG94's mesh. These plots illustrate the impact of di erential settling on opacities and the isothermal sound speed pro le as well as the density and temperature structure of the models. The e ect of including radiative accelerations and of assuming full ionization can only be seen in Figure 8.
{ 19 {
5.1. The opacities The ratio between opacities calculated from the MONO11 and OPAL93 data as well as the OPAL monochromatic and OPAL96 data will be close to unity whenever the chemical mix remains close to the initial mix. Corrections to the opacity brought on by settling and by the use of the monochromatic data is shown in the bottom left boxes of Figures 7 and 8. In Figure 7, the relative di erence of MONO11 and OPAL93 opacities is shown to be nearly zero, within the interpolation errors, in the radiative zone with the exception of the nuclear burning core where it reaches 7 %. The di erence increases proportionally to hydrogen depletion, down to 80 % in an eventual pure helium core. This was traced to an underestimate in MONO11 of the iron contribution compounded with uncertainties in both the hydrogen and helium free-free absorption opacities. This e ect is NOT real. For conditions typical of the solar core, the opacity is largely dominated by H, He, and Fe (as previously shown by Iglesias & Rogers (1991), and illustrated in Fig. 10 below). The opacity is especially sensitive to a change in the relative abundances of H and He if Fe is signi cantly underestimated, as in the MONO11 data. To correct this problem the iron contribution would have to be increased by a factor of 4 or 5 over its value in MONO11 (factor of 20 over the raw data, i. e., with a Fe of 20 and other of one in Equation (A6)). The relative opacity di erences between models using the OPAL monochromatic data and the OPAL96 Rosseland opacity tables are dominated by di erences in the interpolation scheme used in each case. Di erences rise to as much as 4 % in the convection zone where opacity does not a ect the structure. As models D and G di er only in the opacities whereas model H includes radiative forces one would expect model G to be more similar to model D than to model H, which is not the case. As a matter of fact, for radii upwards of 0:5 R the 1 % opacity di erence between models D and H (or G) is large enough to have a visible impact on u, and T . On the other hand, relative di erences between models using the same interpolations scheme are more reliable. Figure 8 shows that neglecting radiative forces but using monochromatic opacities has an e ect of the order of 0.5 % on opacities at the boundary of the convection zone which translates into a 0.05 % percent di erence in u just beneath the convection zone. The impact on density and temperature at the boundary of the convection zone is of the same size. When one assumes that metals are fully ionized and settle at the same rate as iron, one introduces a 0.1 % di erence in u. Radiative forces and di usion velocities are discussed further in x 4.3, 5.2, and 5.3. The solar core is a region where relative abundances change drastically over the Sun's history and the present models are the rst that can give us an accurate picture of opacities there. Consider Figure 9 where, in a solar model calculated with di usion and radiative accelerations (model H),
[Figure 9 goes about here.]
{ 20 { the opacities calculated for mixtures used in Rosseland opacity tables are compared to the Rosseland opacities calculated with the actual abundance values. The short-dashed line gives the ratio when the actual local Y and Z values in the Sun are used for the numerator. The changes in the mixture giving Z are seen to lead to errors of at most 0.8 % in the opacity. This only occurs close to the center and is mainly due to the transformation of C into N by nuclear reactions. Bahcall & Ulrich (1988) noticed the e ect of this composition change and took it into account by using opacity correction tables derived from analytical ts to Los Alamos opacity tables of various metal compositions. With more recent opacity tables, Prott (1994) took a di erent approach by introducing a simple correction formula [his Eq. (3)] which is used here to produce another opacity estimate (long-dashed line). The correction is too large and does not improve the accuracy of the opacity determination7. Figure 10 (bottom part) shows H, He, and Fe are the main contributors to the opacity near the center.
[Figure 10 goes about here.] Prott's formula gives a good estimate of the factor required to scale the opacities of metals that di use at approximately the same rate as Fe (O follows Fe well enough); Z scaling can work if these are the only metals contributing signi cantly to the Rosseland mean. Carbon and nitrogen invalidate this approximation: their abundances do not follow that of other metals (N increases by a factor of 5, while C drops near zero), and their initial or nal contribution to R is not negligible, at this level of accuracy. Prott's correction cancels out these special contributions; to avoid this one would need to take into account supplementary data about the contribution of C and N to R , possibly in the form of opacity derivative tables. To verify these armations, we added to Figure 9 another curve (dot-dashed ) where all metals except C and N are scaled according to Prott's prescription, while X (C)=Z and X (N)=Z are unchanged. Separately, C or N changes a ect the opacity by more than 1 %, but these e ects cancel out almost completely in this last calculation. The remaining error is due mostly to oxygen.
5.2. The radiative accelerations Radiative accelerations for some of the more important metals are shown in Figure 11 for MONO11 data and Figure 12 for OPAL data.
[Figure 11 goes about here.] [Figure 12 goes about here.] 7 Adjusting X
and/or to keep + + = 1 during this correction process makes little di erence to the nal result, which is still overcorrected by roughly the same amount. Y
X
Y
Z
{ 21 { The radiative acceleration from the MONO11 data is signi cantly smaller than gravity for all elements other than iron. For iron it is about 1=5 of gravity very close to the convection zone which is large enough to a ect signi cantly surface iron depletion. The OPAL grad for iron-peak elements not available in the MONO11 data can be as much as 40 % of gravity at the base of the convection zone. The OPAL grad (Fe) below the convection zone is 30 % larger than the one calculated with MONO11. A comparison of the radiative accelerations of CNO calculated using OPAL and MONO11 is shown on Figure 7 of Richer et al: 1998. Note that the grad from OPAL spectra are not corrected for momentum redistribution to electrons (see Richer et al: 1998). This does not modify the radiative acceleration immediately below the convection zone but it a ects them close to the solar center. The radiative accelerations in 1.0 M , 1010 yr old model are also shown for three elements. They are larger than in the Sun by a factor of about 2 for corresponding elements. This is mainly caused by a density increase in these models compared to solar age models. The radiative acceleration becomes nearly equal to gravity but di usion is still downwards because of the contribution of thermal di usion that was mentioned above. The e ect of radiative accelerations on the structure of stellar models may be estimated by evaluating the Rosseland opacity change that they cause, through abundance variations, immediately below the convection zone. From the comparison of the abundances in the di erent models one concludes that the radiative acceleration on Fe causes a 1.6 % change in its surface abundance. Since Fe contributes 20 % of the opacity (see Fig. 10 above, or Fig. 4b of Iglesias & Rogers (1991)), the radiative acceleration on Fe causes a 0.3 % change of the total opacity. One obtains similarly from the change caused by the combination of all other elements, another change of 0.3 % for a total opacity change of 0.6 % caused by the abundance change due to radiative acceleration. As may be seen from Figure 9 (dotted line, scale on right hand side) the total opacity change caused by di usion is of order 8 %, while not taking into account the changes in the relative abundances of metals brought about by radiative forces causes a 0.1 % variation of the Rosseland opacity. If one uses tabulated radiative accelerations such as those of GLAM, one may evaluate accurately enough the abundance changes caused by radiative accelerations. The main error made would be in evaluating the Rosseland opacity from tables; as we just said, this is a 0.1 % error.
5.3. The di usion velocities Comparing di usion velocities for O and Fe (in Figure 13), we see that the di usion velocity of iron at the base of the convection zone with radiative forces (OPAL) is smaller than without radiative forces by 18 %. The oxygen di usion velocity is reduced by 4 %. Di usion velocities calculated assuming complete ionization are also shown. They underestimate the di usion velocity of iron by 34 % compared to calculations ignoring radiative forces and by 20 % compared to calculations including radiative accelerations.
{ 22 {
[Figure 13 goes about here.] For comparison, the analytic ts of Thoul et al: (1994) have also been used to compute O and Fe di usion velocities in the same model. They are shown as dot-dashed curves in Figure 13. They deviate most from our results near the bottom of the convection zone. Near the center, where the ts should work best, their velocities are still larger than ours by 20{30 % if radiative forces are ignored, and by nearly 60 % if their w(Fe) is compared to our calculation with radiative forces. A good part of the discrepancy could be due their use of xed e ective Coulomb cuto s evaluated at the solar center (they assumed a xed charge of +21 for Fe ions) while we used collision integral data (Paquette et al: 1986) calculated using screened Coulomb potentials (see also Michaud 1991).
5.4. The abundances The surface abundance variations caused by di usion in three di erent models are shown in Figure 14.
[Figure 14 goes about here.] The three upper curves are at the solar age while the lower curve is after 1010 yr. The uppermost curve (squares , model D'), having the smallest e ect of gravitational settling, was obtained by carrying out calculations in a way that closely resembles BP95's best model, that is, all elements are assumed to be fully ionized below the convection zone and all elements more massive than O are assumed to di use at the same rate as fully ionized iron; BP95 made all metals di use like fully ionized iron. The triangles (model G) were calculated using the ionization of OPAL96 for all elements but without radiative accelerations. The di erences with the upper curve show the e ect of partial ionization. The circles (model H) were calculated including the e ect of radiative accelerations. The di erences with the triangles show the e ect of radiative accelerations, which are clearly largest for Fe peak elements. Di usion causes the abundances to decrease by some 10 %. This is currently at the limit of the uncertainties of the abundance determinations of Grevesse & Noels (1993). Verifying observationally the 2 % e ect of radiative accelerations is however much more dicult. After 1010 yr, the e ects of di usion range from 12 % for Ti to 18 % for He. This includes a reduction of the gravitational settling by about 2{4 % caused by radiative accelerations. A 14 % reduction in Fe peak abundances is signi cant and may be related to some of the discrepancies measured by Hipparcos between the measured isochrones of old clusters and those calculated for their measured super cial metal abundances (Cayrel 1997). Metallic abundance variations in the core are shown in Figure 15. These are the variations in the central ball, of radius 0:01 R . As before, the rightmost symbols (above the letter Z ) represent changes in the total metallicity. The lightest elements have been left out of the gure
{ 23 { in order to focus more on di usion e ects. In the center, most of the C, N, and O abundance variations (and about half of the Z variation) are caused by nuclear reactions. As in Figure 14 one can see that radiative forces slow down metal settling, resulting in di usion velocities and abundances variations that are accidentally close to those obtained by following the BP95 prescription.
[Figure 15 goes about here.] Internal abundance pro les are shown in Figure 16
[Figure 16 goes about here.] both at the solar age and at 1010 yr. Since gravitational settling always dominates for all elements, the pro les are similar; however one notes the larger light element settling compared to iron-peak element, close to the convection zone. All elements are overabundant in the center because of gravitational settling, except for those destroyed by nuclear reactions. In the central regions, the Ne and Mg settling for instance, is similar to the settling of Ti and Cr. Below the convection zone the settling of Ti and Cr is slowed by their radiative acceleration (see Fig. 12) and is smaller than that of Ne and Mg. At 1010 yr, the convection zone has started to get deeper and dredge up is about to begin.
6. Discussion A fully consistent stellar evolution model including di usion was presented. Two sets of monochromatic data were used in order to accurately gauge the impact of the di usion of the more abundant heavy elements on the structure of the Sun. The opacities calculated from the MONO11 data su er from the incomplete iron data available in TOPbase. The use of the OPAL monochromatic opacities for the purpose of computing the Rosseland opacity is required, at least until the improved OP data are made available (however see Footnote 3). A summary of the models presented here is shown in Table 7. It was shown that the fully consistent models in that set yield results compatible with those of published models. The e ect of the changes in the relative abundances of metals due to di usion have been evaluated in detail and allow us to specify the accuracy limit which is possible without using monochromatic data. The procedure by which one separates the increase in metallicity in the core due to the CNO cycle from the total metallicity variation leads to a 0.7 % error in the opacity. Including radiative accelerations from tables such as those of Seaton (1997) or GLAM allows to calculate their main e ect on the abundances. They are high enough for iron-peak elements that they should be taken into account if one wishes to calculate abundance variations of those elements. There would remain a 0.1 % variation in the Rosseland opacity due to relative abundance changes which cannot
{ 24 { be calculated without detailed monochromatic tables. In more massive stars, radiative forces will be much larger because of the higher luminosity and atomic di usion will be more important in the outer envelope. Partial ionization and thermal di usion have large e ects and should be included if grad are included. The models maximize the impact of element di usion since no mixing in radiative zones is added. Turbulence is at the moment the only known mechanism which can explain the surface lithium depletion. Mixing and overshoot in the radiative zone are a major source of uncertainty for di usion in stars. In the Sun, Monteiro et al: (1993) have estimated that the adiabatic extension of convection in the underlying radiative region can at most be of the order of 0.006 R . This small overshoot cannot resolve the lithium problem. One can expect some mixing to be caused by turbulent di usion from di erential rotation (see RVCD96 or Chaboyer et al: 1995) or possibly by gravity waves (Schatzman 1996). Turbulence is outside the scope of this paper. One may use the results obtained above for model G (calculated without grad s) and model H (calculated with grad s) to obtain precise estimates of the e ect of neglecting grad s on various quantities. For instance, how is the neutrino ux a ected by neglecting grad s in solar models converged to given super cial metal abundances? Our predictions for solar neutrino uxes are compatible with those of BP95 if one takes into account that our predicted neutrino uxes were not calculated with the most recent nuclear data (see Table 3). It is however not possible to use the neutrino uxes of models G and H to evaluate the e ect of grad s on neutrino uxes since both models were calculated with the same original abundances of all metals. Instead, we now combine models G or H from Table 6 and BP95's models 8 (with He di usion) and 9 (with He and metal di usion) from their Tables iii and iv to evaluate, in two di erent ways, the e ect of grad s on solar neutrinos. In the rst evaluation, we assume that in solar models converged to the same current luminosity, radius, and surface metallicity [as given by the (Z=X (H))surf surface abundance ratio], di ering only by a di erent metal di usion, changes in other model properties are, to rst order, linear functions of the di erence Zsurf = Zsurf ? Z0 . Here Z0 is the starting value of Z throughout the pre-main{sequence Sun and Zsurf its present day surface value. BP95 nd that the di erence (Zsurf) = ?0:00202 between their models 8 and 9 is accompanied by neutrino detection rate increases of 15 % and 5 % for 37 Cl and 71 Ga detectors respectively. After rescaling our Zsurfs very slightly for models G and H, to make them correspond exactly to the nal (Z=X (H))surf adopted by BP95, the Zsurf due to metal di usion are found to be very close to BP95's (as in Figure 14). The Zsurf is increased by 0.00008 when radiative forces are included. From this di erential e ect, one can estimate that the neutrino detection rate increases due to metal di usion, as reported by BP95, are overestimated by 8/202 or about 4 %, because BP95 ignored radiative forces. In other words, the 15 % and 5 % increases they quoted should be 14.4 % and 4.8 %, respectively. However, in the second evaluation, noting that metal di usion a ects neutrinos mostly through the solar center opacity, we are not concerned with changes in Z , but rather in X (Fe),
{ 25 { because the contribution of iron to the central opacity exceeds by far that of any other metal (cf. the lower panel in Fig. 10). While grad s have little e ect on the abundance of CNO in the Sun, iron is signi cantly supported by its grad : about one fth of its surface abundance drop is cancelled by grad (Fe) (see Fig. 14). For that reason, we use the present observed value of (X (Fe)=X (H))surf to constrain solar models, instead of (Z=X (H))surf . Figure 14 shows that, for iron, the surface abundance drop, (X (Fe)surf ? X (Fe)0 ), is 20 % larger when radiative forces are ignored. For models with and without radiative forces to converge to the same nal value of (X (Fe)=X (H))surf , the former would require an initial value of X (Fe)0 lower by 2 %. Assuming that neutrino variations due to metal di usion calculated by BP95 are indeed proportional to variations of X (Fe) in the center of the Sun, one nds that the e ects of grad s are 5 times larger than if (Z=X (H))surf is used for calibration (as in the rst estimate above); the BP95 corrections should then be reduced from +15 % to +12 % and from +5 % to +4 % for 37 Cl and 71 Ga detectors respectively. The actual e ect of grad s on neutrino uxes is probably close to the last estimate; this could be veri ed with more simulations. In these, the solar models would however need to be converged not to the same (Z=X (H))surf but to the same surface value for each individual metal, for the second estimate given above to be improved signi cantly. The e ect of grad s on solar neutrino
uxes is however rather small and the above estimate is currently sucient for most purposes. Finally, the di erences between our model H and the solar structure as obtained from helioseismology are small (compare for instance the radius of the convection zone obtained from helioseismology to those of models G and H) and the code can be con dently applied to stars of higher masses where di usion of metals is expected to be important (for an application to F stars, see Turcotte et al: 1998). This research was partially supported at the Universite de Montreal by NSERC. We thank Charles Prott for his code including the OPAL93 and Kurucz93 opacity tables, and for his comments on the paper. We also thank John N. Bahcall, J rgen Christensen-Dalsgaard, Francis LeBlanc, Gerard Massacrier, and Sylvaine Turck-Chieze for helpful discussions. Work by FJR and CAI performed under the auspices of the Department of Energy by Lawrence Livermore National Laboratory under contract W-7405-Eng-48. ST thanks the Space Telescope Science Institute where part of this research was conducted as part of the summer student program.
A. Computing opacities from OP data This Appendix describes some of the physics and approximations that went into the preparation of our MONO11 monochromatic opacity tables from TOPbase data. The pro les of hydrogen lines were parametrized according to Clausset et al: (1994). Opacities
{ 26 { due to H were computed by Artru (1994). Hydrogen lines were treated separately from the other elements because the lines are typically very wide and span too many frequency intervals to include them in only one. Individually, they are also better known than those of metals. Helium opacities were also computed by Artru (1994), but were not spread over multiple intervals; they were treated like metal opacities. The H spectra are de ned over the same frequency mesh and the same (T; Re ) grid as the other elements. In addition to the process described above one needs to take into account electron scattering, free-free absorption by ions, stimulated emission and the e ect of missing elements and missing iron lines. All spectra are cut o at the plasma frequency up given approximately by ! p 2 q 1 m ec 3 4 4 Ne a0 ' 3 T ; in cgs units ; (A1) up = kT 6 where a0 is the Bohr radius, the ne structure constant, T6 = T (K)=106 , and other symbols have their usual meaning. | Free-free absorption by H and He ions can be a major source of opacity at high temperatures (such as in stellar cores) when few lines are present. It is complicated by plasma and partial degeneracy e ects in conditions such as prevail in the solar core. The following approximation was used for the free-free cross section for ions of charge Zion ,
!7=2
g (u) u?3 ; (A2) [1 ? (ucuto =u)2 ] in which g is the free-free Gaunt factor. We used Lang's (1980) formula for g , which in the high 3 2 32 p a50 8 (u) = NeZion 3 6
temperature limit takes the form
p
mec2 kT
constant 3 g (u) = ln : u
(A3)
The constant was adjusted (a value of 11 was used) so as to reproduce reasonably well OPAL's high temperature free-free opacity for He (g is set to zero when the argument of the logarithm becomes smaller than one). | The absorption spectrum for an element is proportional its abundance; hence the spectrum of an element must be multiplied by the ratio of the model's element abundance to that for which the spectrum has been calculated, i. e., X (Ai )model=X (Ai )solar , this scaling taking place at xed electron density. | The stimulated emission correction factor (1 ? e?u ) is applied to absorption processes. | Electron scattering is added to the total monochromatic opacity. The scattering cross section is a simple constant if plasma, degeneracy, and relativistic e ects are neglected. Assuming complete ionization, the electron scattering opacity is approximated by 1 (A4) e 0:75 X (H) + X (He) 2 mH T ;
{ 27 { where T is the Thomson cross section. The factor 0.75 was added to the standard scattering term as a crude correction, to bring our calculations in better agreement with the work of Boercker (1987); Boercker showed that, in conditions where electronic scattering is signi cant, such as in stellar cores, the plasma e ects reduce the cross section by roughly 25{27 %. Iglesias & Rogers (1991) estimated that including electronic scattering without taking into account plasma e ects would lead to an overestimate of the Rosseland opacity of approximately 7 % in the solar core. In stellar envelopes, electronic scattering is small compared to atomic absorption processes. | In order to compensate for missing data in the OP database, especially Fe data, a correction procedure is applied at every (T; Re ) grid point, for solar abundances. A correction factor is rst de ned by (OPAL93 : X ) : (A5) = R(MONO11 :X ) R
If is less than one, this factor is applied as is to all spectra including electronic scattering. If is greater than one, which is generally the case, the iron spectrum is corrected by a factor Fe so as to give the net correction on R we seek. If the Fe thus determined ranges between one and six, no other correction is applied. If it is greater than six, Fe is reduced to six, in order to avoid excessive increase of the iron spectrum, and an additional correction other is found which, applied to all other spectra and to the electronic scattering term, leads to the desired value. The other factor is greater than one in only a small part of the (T; Re ) grid. With such a correction there is a risk of overestimating/underestimating the importance of an element in some conditions when the abundance mix becomes strongly non-solar. By correcting the iron opacity separately one should obtain the main correction. The limitations of this approach are discussed in x 5.1. Assembling all corrections and additions to the monochromatic spectra read from the tables (i (u)), the total (MONO11) monochromatic opacity which is used in Equation (1) is
1 0 X X ( A ) i model ? u i (u) + ;i (u) (1 ? e ) X (A ) other A + eother total (u) = @ i solar i6=Fe + (u) + (u) (1 ? e?u ) X (Fe)model : Fe
;Fe
X (Fe)solar
Fe
(A6)
{ 28 {
Table 1: Abundances used for monochromatic spectra Elementa Mass fraction for MONO11b Mass fraction for OPALc H 0.70 0.70 He 0.28 0.28 ? 3 C 3.4710 3.46610?3 ? 3 N 1.0610 1.06310?3 O 9.6510?3 9.64510?3 Ne 1.9210?3 1.97310?3 Na | 3.99710?5 ? 4 Mg 7.4610 7.51510?4 Al | 6.47610?5 Si 8.0510?4 8.10410?4 P | 7.09910?6 ? 4 S 4.2010 4.22810?4 Cl | 9.11710?6 Ar 1.2910?4 1.07610?4 K | 4.19210?6 Ca | 7.46910?5 Ti | 4.21510?6 Cr | 2.00910?5 Mn | 1.09710?5 Fe 1.5510?3 1.43610?3 Ni | 8.91810?5
1 10?5 and 13 C= 1 % of 12 C+13 C(=C) in both cases. b To represent missing elements, a mass fraction of 2.510?4 was allocated to a dummy, non-di using, non-absorbing element. c The listed abundances are used as starting values for all models in Turcotte et al (1998). In the present paper only the relative metal abundances are relevant since starting and are adjusted.
a 3 He=
:
Y
Z
{ 29 {
Table 2: Solar models without di usion Parameters
A Opacities OPAL93 L=L 1.00000 R=R 1.00000 1.83540 Yo 0.26652 Zo 0.01754 (Z=X )o 0.02450 Yc 0.61936 Zc 0.01800 RZC=R 0.734 TZC (106 K) 2.040 6 Tc (10 K) 15.47 ? 3 c (g cm ) 150.7 Cl (SNU) 6.5 Ga (SNU) 123 age (Gyr) 4.60 33 ? 1 L (10 erg s ) 3.86
Models B BP95 OPAL96 OPAL92 1.00004 | 1.00004 | 1.86959 1.96 0.26678 0.26793 0.01753 0.01743 0.02450 0.02450 0.62175 0.62126 0.01799 0.01743 0.731 0.726 2.045 2.085 15.50 15.56 150.8 152.4 6.6 7.0 124 126 4.60 4.57 3.86 3.844
RVCD96 OPAL92 1.00011 1.00011 1.652 0.2782 | | 0.6346 0.0195 0.725 2.100 15.56 150.7 7.4 127 4.60 3.851
{ 30 {
Table 3: Solar models with di usion Parameters
C OPAL93 L=L 0.99994 R=R 1.00003 2.03561 Yo 0.27532 Zo 0.02005 (Z=X )o 0.02845 Ys 0.24473 Zs 0.01798 (Z=X )s 0.02439 Yc 0.64312 Zc 0.02137 RZC =R 0.713 6 TZC (10 K) 2.200 6 Tc (10 K) 15.75 c (g cm?3 ) 155.0 Cl (SNU) 8.53 Ga (SNU) 134 age (Gyr) 4.6 L (1033 erg s?1 ) 3.86 Opacities
D OPAL96 1.00002 0.99995 2.01737 0.27636 0.02002 0.02846 0.24771 0.01826 0.02487 0.64339 0.02117 0.717 2.168 15.77 154.4 8.74 135 4.6 3.86
Models FRANEC96 RVDG96 BP95 OPAL96 OPAL92 OPAL92 | 1.00011 | | 1.00011 | | 1.789 2.09 0.269 0.2798 0.27753 0.0198 0.01976 0.02 0.0266 0.02821 0.02847 0.238 0.2513 0.24695 0.0182 0.01891 0.01798 0.0245 0.02591 0.02446 | 0.6464 0.64564 | 0.021 0.02106 0.716 0.714 0.714 2.17 2.178 2.204 15.69 15.70 15.843 151.8 154.2 156.2 7.4 8.6 9.3 128 133 137 4.57 4.6 4.57 3.844 3.851 3.844
{ 31 {
Table 4: Abundance variations at the solar age in models including di usion Model model C model H BP95 RVDG96
4 He
?11:1 ?10:4 ?11:0 ?10:2
surface (in %) 16 O Z ?10:0 ?10:1 ?10:3 ?8:6 ?8:4 ?8:5 ?10:2 ?10:0 ?10:1 ?6:1 ?5:4 ?4:3 12 C
center (in %) 12 C 16 O 133:0 ?99:3 ?2:0 131:7 ?99:3 ?3:0 132:6 ?99:4 ?1:5 131:0 | | 4 He
Z 6:6 5:7 5:3 6:3
{ 32 {
Table 5: Parameters for solar models with di usion, with a grey atmosphere boundary condition. Opacities Yo Equation of state OPAL93 0.27554 1.69164 CEFF OPAL96 0.27636 1.68510 CEFF OPAL96 0.28467 1.68742 EFF
{ 33 {
Table 6: Solar models using monochromatic data Parameters
Models
E F G H Opacities MONO11 MONO11 OPAL mono OPAL mono Di usion no yes yes yes grad no no no yes L=L 0.99995 1.00001 0.99998 1.00002 R=R 1.005 0.99999 0.99997 1.00004 1.87322 2.11684 2.09424 2.09635 Yo 0.26740 0.28471 0.27769 0.27769 Zo 0.01752 0.01982 0.01999 0.01999 (Z=X )o 0.02450 0.02845 0.02846 0.02846 Ys 0.26740 0.25234 0.24871 0.24875 Zs 0.01752 0.01770 0.01821 0.01829 (Z=X )s 0.02450 0.02425 0.02485 0.02495 Yc 0.617 0.649 0.6463 0.6464 Zc 0.01798 0.02115 0.02113 0.02113 RZC=R 0.736 0.715 0.71775 0.71756 6 TZC (10 K) 2.027 2.169 2.1574 2.1597 Tc (106 K) 15.38 15.69 15.801 15.801 ? 3 c (g cm ) 151 158 154.80 154.85 Cl (SNU) 6.18 8.50 8.85 8.86 Ga (SNU) 122 135 136 136
{ 34 {
Table 7: Features of the models presented in this paper Models
opacities
A B C
OPAL93 OPAL96 OPAL93
D D'a E F G H
a Model
di usion
none none all elements independent OPAL96 all elements independent OPAL96 all metals A > 16 settle as Fe+26 MONO11 none MONO11 all elements independent OPAL96 all elements monochromatic independent OPAL96 all elements monochromatic independent
ionization radiative of metals forces N/A N/A N/A N/A OPAL EOS no OPAL EOS
no
fully ionized N/A MHD EOS
no N/A no
OPAL EOS
no
OPAL EOS
yes
D' was an attempt to imitate BP95's best model. It corresponds to the dotted abundance pro les in Figs. 14 and 15. It di ers slightly from the BP95 model in that C, N, and O di use independently; all other metals di use like fully ionized Fe.
{ 35 {
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This preprint was prepared with the AAS LATEX macros v4.0.
{ 37 {
Figure captions
Fig. 1.| Ratio of the Rosseland mean opacity from MONO11 spectra (including all corrections shown in Eq. (A6)) over the value obtained from the tables used in a solar-like model. For the denominator interpolation is always done with the routines supplied with the OPAL93 tables. If there were no interpolation problems, all ratios would be identically one. The results shown are for interpolations that are: a) uniformly linear, b) quadratic, c) cubic, d) quadratic between grid points, cubic at or around grid points, e) quadratic between grid points, linear at or around grid points, and f) dashed line same as e), solid line is for the same interpolation but using the EFF Ne as input. Fig. 2.| E ect of our Savitsky{Golay-like ltering procedure on Rosseland opacities, in the vicinity of the convection zone inner boundary, in the current Sun. The dashed line shows the original opacity pro le (with R in cm2 /g) computed from OPAL monochromatic data. The solid line is the smoothed pro le. The dot-dash line gives the corresponding relative change in the opacity due to smoothing. Fig. 3.| Thermal di usion velocities of a few elements in a young (30 Myr) solar model. The top part of the gure shows the fraction of the net di usion velocity that is due to thermal di usion, for ve elements, as given by a fully self-consistent one{step solution of Burgers' ionized gases ow equations for the full mixture. The calculation is limited to the radiative part of the star. The bottom part shows a comparison of analytic approximations proposed by Michaud & Prott (1992) with the more accurate results of the top gure. Helium is not shown there, because in the MP92 approximation this quantity is never needed.
{ 38 { Fig. 4.| Di usion velocities (note the signs ) for a few elements according to three di erent approximations. The calculation was made in the radiative part of a young, nearly homogeneous (30 Myr) solar model, to exclude the e ects of concentration gradients. Thick lines (solid , dashed , dotted , or dash-dotted ) show the fully consistent simultaneous solutions, including drag terms and energy exchange terms betweem all types of particles. Thin lines represent the Michaud & Prott (1992) solutions and symbols are the values calculated by the stellar evolution code (see text). For oxygen, the thick and thin lines coincide almost perfectly. Note: 100 pm/s = 0.00453 R /Gyr. Fig. 5.| Ratio of the OPAL96 and OPAL93 Rosseland opacities in the solar interior for a nondi usion model (dashed line) and a full di usion model (solid line). The models were computed using the OPAL96 opacities. The dotted line marks the limit of the OPAL96 di usion model convection zone. Fig. 6.| Comparison of computed models and the observed solar model of DG94. u=u = (umodel ? uobserved )=uobserved , where u = P=. The models are: OPAL93 with di usion (model C, solid line), OPAL93 without di usion (model A, long dash ), OPAL96 with di usion (model D, short dash ), and the BP95 model (dotted ). The dotted line marks the limit of the convection zone for the OPAL93 model with di usion. Fig. 7.| Comparison of solar models calculated with monochromatic opacities (MONO11, model F) and with tabulated Rosseland opacities (OPAL93, model C). Model C is the reference. Fig. 8.| Comparisons of di erent OPAL96 models. In each case the reference model is model H (cf. Table 6), which uses OPAL monochromatic data for opacities and radiative forces, includes partial ionization and di usion with radiative forces for all elements. This is compared to various models with di usion but without radiative forces. These di er only in the way Rosseland opacities are obtained, and ionization of metals is handled. Solid : detailed opacity calculation for arbitrary composition, detailed ionization (model G). Dashed : (standard) Rosseland opacity tables, detailed ionization (model D). Dotted : (standard) Rosseland opacity tables, fully ionized metals, emulating BP95 except for CNO which di use at their own speed (model D'). Fig. 9.| E ect of composition variations on Rosseland opacities in a solar model including di usion and radiative forces (model H). Rosseland opacity estimates are compared to the actual Rosseland opacity for the true local composition. Mean opacities calculated from spectra with metals in standard OPAL proportions (those of Table 1) are used in the numerator for the long-dashed line, with X , Y , and Z the same as in the model. For the short-dashed line X and Y are the same as in the model, while Z is adjusted to compensate for CNO nuclear changes as suggested by Prott (1994); in this case, Z is equal to Prott's Z . The dot-dashed line shows the same calculation but with the C=Z and N=Z ratios maintained equal to their values in the model. (Z is used to scale the other metals). For reference, the dotted line (scale on right hand side) shows the opacity for the initial (homogeneous) model composition.
{ 39 { Fig. 10.| Contribution of each element to the Rosseland opacity at the bottom of the solar convection zone (top panel), and at the solar center (bottom panel). Opacities are compared with (21 ) and without (20 ) the element of atomic number Z , without renormalizing the 20-element mix abundances; the local model composition is used for the 21-element mix. Fig. 11.| Radiative forces on the most abundant metals, normalized to gravity, in model H. Fig. 12.| Radiative forces on some iron-peak elements, in model H at 4.6 Gyr, and in the same model at 10 Gyr. A rough fion correction was applied to Fe only. Curves end at the bottom of the convection zone, on the right. R (10 Gyr)=R (4.6 Gyr) = 1:41 . Fig. 13.| Di usion velocities of Fe and O with radiative forces (solid ), without radiative forces (dashed ), and assuming complete ionization and no radiative forces (dotted ), in model H. The calculation is done with full collisional coupling between all elements. For comparison, the analytic ts of Thoul et al: (1994) are also shown (dot-dashed ), for the same model. The curves end on the right at the base of the convection zone. Fig. 14.| Surface abundance variations for all elements. Open symbols show abundances after 4.6 Gyr, lled symbols after 10.0 Gyr. Circles : di usion with radiative accelerations from the OPAL data and detailed ionization. Triangles : di usion without radiative accelerations but with detailed ionization. Squares : di usion similar to BP95; all elements more massive that O settle at the same rate as fully ionized iron and there is no radiative acceleration. Isolated symbols on the right show the corresponding Z ( 1 ? X (H) ? X (He)) variation. Note that in BP95 CNO di use as fully ionized iron and have the same surface abundance variation as other metals. Fig. 15.| Abundance variations at the solar center. The meaning of symbols is as in Fig. 14, except that the 10 Gyr curve has been omitted, as well as elements lighter than oxygen, to allow focusing on small di erences among metals that can be related to the treatment of di usion (see also Fig. 16). Nuclear e ects dominate over di usion e ects for CNO. Fig. 16.| Internal abundance pro le changes in the Sun (model H) at 4.6 Gyr (solid line) and 10 Gyr (dashed ). For 3 He, the logarithm has been divided by 20 to t the scale.
Figures
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
100 P
κ20/κ21 in %
Na Al He
C
N
Mg
S
ClAr K Ti Ca
CrMn
Ni
Ne
H
80
Si
Fe
60
T=2.177E+06
O
100
κ20/κ21 in %
C
N O
Na Al P Cl K Ti Ne Mg Si S Ar Ca
CrMn
Ni
80 H Fe
T=1.577E+07
He
60
0
10
20
Z(element removed)
Figure 10
30
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
