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Astron. Astrophys. 351, 347-358 (1999)

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1. Introduction

A number of well observed properties of the Sun cannot be explained in the framework of the Standard Solar Model (SSM). Their interpretation requires the introduction of a number of new physical processes in the description of the internal structure and evolution of the Sun, whose validity must be confirmed by the application to other stars.

Among these anomalous properties of the Sun we find the deficiency of the lithium abundance in the solar atmosphere and the deficiency of the solar neutrino flux, compared to the values predicted by the SSM. Given the fact that the nuclear reactions, responsible of lithium burning and neutrino production, take place at very different depths (near the bottom of the convective zone for the lithium, and in the central region for the neutrinos case), these controversial results provide information about the characteristics of the physical processes in these two very distant regions.

The lithium abundance in the solar atmosphere is about one hundred times smaller than the theoretical one and than the maximum observed in young clusters (Pleiades and [FORMULA]Per). The dependence on age, metallicity and mass of the lithium abundance has been well studied from the great number of observational data (see Rebolo 1990 for a review). Several mechanisms have been considered in order to produce chemical mixing in the stable region and then to explain these deviations from the standard model (Michaud & Charbonneau 1991). Those mechanisms, producing mixing by a transport process due to rotation, do not seem to provide very consistent results (Schatzman & Baglin 1991; Schatzman 1994; Montalbán 1995). Two diffusion processes linked to the internal waves have been considered by Press (1981). The first process, which has been studied by García López & Spruit (1991), assumes that the shear induced by random gravity waves generates turbulence. The effect is supposed to be limited to the region where the shear [FORMULA] is larger than the frequency of gravity waves. The consequence of that assumption is that the mixing process is limited to a region where the amplitude of gravity waves is sufficiently large. The second process is linked to the non-adiabatic propagation of internal waves in the stellar radiative region. The entropy losses during the oscillation allow to make an estimation of the mixing length and velocity, and provide a diffusion coefficient of the fourth order (Schatzman 1991, 1993). The transport process exists all over the region where gravity waves are propagating. Montalbán (1994), Schatzman & Montalbán (1995) and Schatzman (1996) introduced some improvements of the physical and mathematical treatment of the excitation and propagation of gravity waves, and they obtained a diffusion coefficient linked to the non-adiabatic propagation of internal waves. The application of this diffusion coefficient to predict light element abundances in the Sun and in solar type stars (Montalbán 1994; Montalbán & Schatzman 1996; Montalbán & Schatzman 1999) shown that this mixing mechanism can explain, in a consistent way, the features of lithium and beryllium abundances provided by the observational works. Therefore, it has been assumed that this mechanism is able to describe the transport process near the convective zone.

Recently, Frittz et al. (1998) have proposed another transport process associated to the gravity waves generated by convection at the boundary between turbulent and stratified regions in the Sun. As in Schatzman & Montalbán (1995) and Schatzman (1996), the internal waves could be excitated by the penetration of convective plumes in the radiative interior of the star. The propagation, dissipation and filtering of these waves in a sheared and stratified layer seem to impose a residual circulation within the solar interior that could account for the observed lithium depletion in the Sun. The ability to mix seems to be restricted to the shallow layers close to the so urce layer. The model predict no Be depletion at the age of the Sun. Given the fast decrease of the transport efficiency with the distance to the internal waves source, no effect is expected in the regions where the neutrinos are produced.

Solar neutrino production and some features derived by helioseismologycal data depend strongly on the physical conditions in the central region of the Sun. Consequently, the neutrino flux for three different energy ranges and helioseismology data provide information on the internal region of the Sun. Since 1967 there have been several experiments (Chlorine; Kamiokande; Gallium: GALLEX, SAGE) to measure the value of these fluxes and to test the validity of the SSM. Despite large uncertainties in the solar model and in the experimental determination, the theoretically predicted values have always been larger than the experimental one, suggesting that some aspects of the problem have been forgotten. The neutrino deficiency could be explained by the MSW effect (Mikheyev & Smirnov, 1986; Wolfenstein, 1978), but there remains the possibility of considering a Non-Standard Solar Model (NSSM), taking into account the physical process of macroscopic diffusion, which is ignored in SSM. This is the aim of the present paper.

There has been several attempts to explain the neutrino deficiency by a partially mixed core, either without specifying the mechanism responsible of that mixing (Ezer & Cameron, 1968; Shaviv & Baudet, 1968; Bahcall et al., 1968; Shaviv & Salpeter, 1968; Iben, 1968,1969), or introducing a turbulent diffusion process (Schatzman & Maeder 1981; Lebreton & Maeder 1987). With the transport of more fuel to the center, the nuclear energy production rate is larger. Due to the calibration constraint, the mixed core would experience a decrease of the central temperature which leads to a smaller production of high energy neutrinos. On the other hand, helioseismological data (J. Provost 1984) determine the sound velocity in the central regions and therefore, give some constraints over the internal structure of the Sun. The coherence of these data implies that the gradient of the mean molecular weight has to be larger than a given value. Consequently, the models with a large mixing process in the central region, invoked to solve the solar neutrino problem (Schatzman & Maeder 1981; Lebreton & Maeder 1987) appears to be ruled out, although, a small mixing could be present and would be compatible with the values of the sound speed curve. The neutrino production depends strongly on the central temperature ([FORMULA]) and a small variation of the chemical composition could imply a small diminution of temperature and a significant diminution of neutrino production.

On the other hand, in the models including microscopic processes, gravitational settling and thermal diffusion of 4He and 1H (Bahcall & Pinsonneault 1992), the computed neutrino fluxes are larger than in SSM. In fact, around the center, the abundance of 1H is depleted by the microscopic diffusion, while the abundance of 4He is increased. A direct consequence of this lowering of the available nuclear fuel, is a decrease of the thermonuclear energy production rate. The calibration process result is an increase of the central temperature and a concomitant increase of the neutrino flux. Therefore, the microscopic diffusion alone acts in a wrong direction; it is necessary to take into account the possible inhibition by other mixing processes.

Press (1981) noted that inward-propagating gravity waves could have an important effect over the neutrino production rate. In fact, these waves are focused as they approach the solar center, they therefore may become unstable and produce local mixing there, provided that they are generated with sufficiently large amplitude at the base of the convective zone and are not strongly damped along the way of propagation. Merryfield (1995) studied the likelihood that internal waves excitated by a classical convective zone, described by the mixing-length formalism, were able to generate turbulence in the center of the Sun. Gravity waves having wave numbers and frequencies given by time and space scales of convection are rapidly damped, and he concluded that turbulent mixing by convectively-driven internal waves is only possible if nonlinear interactions transform the emitted gravity waves into higher frequency waves that reach the core of the Sun.

Morel & Schatzman (1996) considered the effect on the center of the Sun of a non-turbulent macroscopic diffusion supposed to be induced by gravity waves. They used a phenomenological description based on the expression given by Schatzman & Montalbán (1995) (based on the second process noted by Press 1981). In the paper of Schatzman & Montalbán, a random field of internal waves is produced by inward-plumes of the convective zone (Rieutord & Zahn 1995) which are stopped at the bottom of the convective region. The transport process linked to internal waves is extended down to the central regions of the Sun, taking into account two dissipative effects: radiative damping, and work against the gradient of chemical composition generated during stellar evolution. The diffusion coefficient seems to have the necessary profile, respecting the helioseismological constraints and being able to reduce the solar neutrino flux. It presents first a fast decrease, due to radiative damping; when approaching the solar center, convergence of the waves produced a divergence of the diffusion coefficient, like [FORMULA]. Near the center, due to the gradient of the mean molecular weight, the diffusion process has to work against gravity. The total flux of mechanical energy decreases and the diffusion coefficient vanishes like [FORMULA] when r goes to zero. The result is that the diffusion coefficient presents a maximum at a distance [FORMULA] to the solar center of the order of [FORMULA]. This behavior corresponds to the one which, since the eighties, was considered as being able to explain simultaneously the light element abundances and the neutrino flux.

The theory of the diffusion process induced by stochastic gravity waves was still at that moment (and even now) in a preliminary state, and provides only indications of the magnitude and shape of the macroscopic diffusion coefficient curve. Morel & Schatzman (1996) presented the numerical experiment of a NSSM, with the macroscopic diffusion coefficient represented by a gaussian. They could neither reproduce the neutrinos flux nor the helioseismological data by changing only the parameters of the Gaussian representing the diffusion coefficient; it was necessary to introduce an asymmetric profile in the curve. They estimated that a diffusion coefficient with a maximum of the order of [FORMULA] at about [FORMULA] [FORMULA] and a half-width of the order of [FORMULA] was needed. Schatzman (1997) has shown, using the numerical results by Morel & Schatzman (1996),that it may be possible to chose the parameters in such way (maximum, [FORMULA]; center at [FORMULA] [FORMULA], and half-width of the order of [FORMULA]) that the frequency difference [FORMULA] is correct, and the neutrino flux production appreciably decreased. However, this phenomenological, time independent, diffusion coefficient is certainly an oversimplification, and it must be improved, including the complete description of the diffusion coefficient due to internal waves, the effect of microscopic diffusion, and the time dependent model of the macroscopic diffusion process. In fact, they considered a time independent diffusion coefficient whereas the position of the maximum near the center depends on the gradient of the mean molecular weight, and that gradient must change with the evolution of the star. On the other hand, the diffusion coefficient derived by Schatzman & Montalbán (1995), using Press' approximation, is valid for the outer regions and the magnitude and location of the maximum depend strongly on the value of the diffusion coefficient below [FORMULA] where the chemical gradient begins to appear. In fact, the diffusion coefficient which takes into account the dissipation of mechanical energy due to the work against [FORMULA] was written as:

[EQUATION]

where [FORMULA] is a function of the stellar structure and is close to zero every place except near the center of the star where [FORMULA] becomes important. Therefore, the correction term will be different to one only close to the stellar center, where it decreases rapidly. [FORMULA] is the diffusion coefficient without correction.

Another aspect of the effect of oscillatory motions on the diffusion process was considered by Knobloch & Merrifield (1992). They noted that the presence of an oscillatory motion can produce an increase of the microscopic diffusion, and that the increase in the case of gravity waves is very small. It is not clear how to compare the contribution of microscopic diffusion with recpect to the macroscopic diffusion studied here, as the oscillatory motion considered by Knobloch & Merrifield is not a random motion, whereas the macroscopic diffusion process we are studyding here is induced by macroscopic random motions. Remenber that these random properties are induced by the stochastic motion present in plumes, with characteristics which are defined at the boundary of the convective zone.

On the other hand, Gough (1992) studied the effect on the solar model of small amplitude temperature oscillations produced by the g-modes (gravity waves that penetrate down to the center where they are reflected). His estimations show that 7Be and 8B neutrino fluxes are decreasing functions of the oscillation amplitude. The contribution of non-adiabatic g-modes to the diffusion procces described here will be treated in a future paper. Here we only consider the contribution from the progressive gravity waves.

We will derive an expression for the diffusion coefficient associated to the non-adiabatic propagation of internal waves that is valid till the central regions of the Sun. In Sect. 2 using the complete equations for non-adiabatic oscillations given by Unno et al. (1989) we obtain, in the quasi-adiabatic approximation, the internal wave amplitude as a function of solar radius, and a new expression for the radiative damping effect, both valid up to the central regions of the Sun. In Sect. 3 we follow the procedure given in Schatzman (1996) and using Sect. 2 results, we derive the diffusion coefficient in the central regions. It is assumed that these waves are produced by turbulent flow in the convective inward-plumes which are braked in the overshooting region, at the bottom of the external convective zone. In Sect. 4 we analyse the dependence of diffusion coefficient at the central region of the Sun on turbulence description. Finally, in Sect. 5 we present the summary and conclusions from these computings and we suggest other aspects of the problem that should be considered in the future.

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999
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