IN SOLAR-TYPE STARS 1. INTRODUCTION: THE LIGHT-ELEMENT ABUNDANCES

Transcription

1 THE ASTROPHYSICAL JOURNAL, 511:466È480, 1999 January 20 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. ANGULAR MOMENTUM TRANSPORT IN MAGNETIZED STELLAR RADIATIVE ZONES. III. THE SOLAR LIGHT-ELEMENT ABUNDANCES G. BARNES,1 P. CHARBONNEAU, AND K. B. MACGREGOR High Altitude Observatory, National Center for Atmospheric Research P.O. Box 3000, Boulder, CO ; barnesg=hao.ucar.edu, paulchar=hao.ucar.edu, mac=hao.ucar.edu Received 1997 October 9; accepted 1998 August 27 ABSTRACT We calculate the depletion of the trace elements lithium and beryllium within a solar-mass star during the course of its evolution from the zero-age main sequence to the age of the present-day Sun. In the radiative layers beneath the convection zone, we assume that these elements are transported by the turbulent Ñuid motions that result from instability of the shear Ñow associated with internal di erential rotation. This turbulent mixing is modeled as a di usion process, using a di usion coefficient that is taken to be proportional to the gradient of the angular velocity distribution inside the star. We study the evolution of the light-element abundances produced by rotational mixing for models in which internal angular momentum redistribution takes place either by hydrodynamic or by hydromagnetic means. Since models based on these alternative mechanisms for angular-momentum transport predict similar surface rotation rates late in the evolution, we explore the extent to which light-element abundances make it possible to distinguish between them. In the case of an internally magnetized star, our computations indicate that both the details of the surface abundance evolution and the magnitude of the depletion at solar age can depend sensitively on the assumed strength and conðguration of the poloidal magnetic Ðeld inside the star. For a conðguration with no direct magnetic coupling between the radiative and convective portions of the stellar interior, the depletion of lithium calibrated to the solar lithium depletion at the solar age is similar at all ages to the lithium depletion of a model in which angular-momentum transport occurs solely by hydrodynamical processes. However, the two models can be distinguished on the basis of their respective beryllium depletions, with the depletion of the magnetic model being signiðcantly smaller than that of the nonmagnetic model. Subject headings: Sun: abundances È Sun: interior È Sun: magnetic Ðelds È Sun: rotation 1. INTRODUCTION: THE LIGHT-ELEMENT ABUNDANCES IN SOLAR-TYPE STARS It has long been recognized that the light elements lithium and beryllium can be used to constrain the amount of mixing that takes place in stellar radiative zones. Because these elements undergo nuclear reactions at relatively low temperatures (D2.7 ] 106 K and 3.5 ] 106 K, respectively), they can survive only in the convection zone and outermost layers of the radiative core in main-sequence stars of solar mass. Thus, observations of the surface abundances of these elements provide information about mixing in the outer portion of the stellar radiative interior. If di erential rotation in this region gives rise to instabilities and hence to turbulent mixing, then these observations may also provide information on the internal rotation of solar-type stars. The Ðrst detection of lithium in the solar photosphere (outside of sunspots) was carried out by Greenstein & Richardson (1951), who estimated its abundance (in comparison to other metallic species) to be a factor of D100 smaller than on Earth, in approximate agreement with the abundance inferred earlier from sunspot spectra (Russell 1929). Interestingly, these authors also pointed out that the very fact that lithium is seen at all in the Sun implies rather stringent upper limits on the efficiency of any mixing mechanism operating below the convective envelope. Goldreich & Schubert (1967) appear to have been the Ðrst to suggest that lithium is depleted as a result of turbulent mixing, driven by di erential rotation in the radiative portion of the SunÏs interior. According to their scenario, 1 Also Department of Physics, Cornell University, Ithaca, NY the respective time histories of the solar angular momentum and light-element abundances are linked, by virtue of the fact that the nonuniform rotation responsible for the latter is produced by the wind-related torque that causes the former. Observational support for this notion was presented shortly thereafter by Conti (1968) and further extended by Skumanich (1972). Endal & Sophia (1978, 1981) developed a particular approach to modeling the coupled evolutions of rotation and chemical abundances. Their work was subsequently elaborated upon (for solar-type stars) by Pinsonneault et al. (1989), Pinsonneault, Kawaler, & Demarque (1990), and Pinsoneault, Deliyannis, & Demarque (1991) and more recently by Chaboyer, Demarque, & Pinsonneault (1995a, 1995b). Whichever physical process is presumed to give rise to the turbulent motions responsible for compositional mixing in the upper part of the solar radiative core, the associated lithium depletion cannot yet be computed from Ðrst principles. Faced with the formidable difficulties of calculating transport properties of stratiðed, turbulent Ñows in stellar radiative interiors, early attempts to quantify the efficiency of rotational mixing utilized methods developed to treat similar problems in geophysical Ñuid dynamics. In particular, the functional forms of the di usivity terms in the transport equations were retained, but enhanced values ( eddy ÏÏ or turbulent ÏÏ di usivities) were adopted for the transport coefficients themselves. In his initial work on the problem of the solar lithium depletion, Schatzman (1969) used this approach and found that eddy di usivities several orders of magnitude larger than the ordinary microscopic di usion coefficient were required. Attempts to extend this picture to stars of di erent masses and/or ages showed, not sur-

2 SOLAR LIGHT-ELEMENT ABUNDANCES 467 prisingly, that the assumption of constant eddy di usivities was an oversimpliðcation. Considerable e ort has since gone into constructing more detailed expressions for turbulent di usivities, in the hope of reproducing the observed variations of the lithium abundance with time and stellar mass on the main sequence. Many such models have come and gone, and it would not be particularly useful to review here the massive literature that has arisen from these various attempts. The interested reader is referred to Schatzman & Baglin (1991) and Zahn (1992) as suitable starting points for bibliographic searches. A characteristic of most of these models is the rather large number of free parameters required to deðne the mixing coefficients and the practical impossibility of reliably estimating their magnitudes from Ðrst principles. Because of this, the approach generally adopted has been to turn the problem around and use observables such as the lithium abundance to constrain the free parameters of the mixing models. Of course, the key to this approach is (at the very least) to secure as many observables as free parameters, which in practice implies determining the abundances of other species in addition to lithium. For reasons similar to those discussed above in the case of lithium, the light elements beryllium (Be) and boron (B) are also quite sensitive to proton capture under stellar interior conditions. They are much more difficult to observe, however, as their more readily accessible spectral lines lie close to Be or beyond B the atmospheric cuto, in a rather crowded region of the ultraviolet spectrum. Moreover, non-lte e ects contribute to the formation of these lines, so that the derived abundances are more sensitive to atmospheric modeling assumptions than is the case for lithium. The identiðcation of beryllium lines in the solar spectrum was already well established in the early decades of the twentieth century (see Russell 1929), but the Ðrst reliable determination of its abundance is probably that of Greenstein & Tandberg-Hanssen (1954), who found that beryllium is markedly less depleted than lithium in the solar atmosphere, being underabundant (with respect to terrestrial and meteoritic abundances) by a factor of 2È4 at most. Boron was Ðrst detected in the Sun at a much later date (Kohl, Parkinson, & Withbroe 1977). Current abundance determinations indicate that both beryllium and boron are depleted by at most a factor of 2 (see, e.g., Anders & Grevesse 1989) and may not be depleted at all (Balachandran & Bell 1998). Figure 1 shows the expected proðles of light-element abundances in a present-day solar model in which only nuclear reactions have altered the abundances. The aforementioned extreme sensitivity of the light elements to proton capture reactions translates into steplike abundance proðles, with the abundances going from undepleted to zero over a very narrow temperature interval. Moving downward from the base of the convective envelope, lithium vanishes below T B 2.7 ] 106 K, beryllium below 3.5 ] 106 K, and boron below 5 ] 106 K. These temperatures occur at depths of r/r B 0.65, 0.54, and 0.4, respectively. Clearly, a near-normal _ beryllium abundance and a lithium depletion by a factor of D200 (see, e.g., Anders & Grevesse 1989) impose a limit on the spatial extent of the region below the solar convective envelope in which mixing acts efficiently. From the bottom of the convection zone, mixing must be efficient down a distance of at least 0.1 R, but not more _ than 0.15È0.2 R. Alternatively, the mixing region must _ include material within at least 1 scale height of the base of the envelope, but less than 2. This represents an extremely stringent constraint on any global mixing mechanism operating in the upper portion of the solar radiative interior; the timescale over which mixing occurs must increase rapidly with depth. Throughout this paper, we ignore the e ects of large-scale Ñows and chemical separation due to gravitational settling on the abundance of light elements. Instead, we assume that their transport inside the star is dominated by turbulent Ñows that result from the occurrence of any of the hydrodynamical instabilities to which nonuniform internal rotation is susceptible. We treat chemical mixing by such turbulence as a di usion process, so that the mass fraction of a given light element, X, obeys a conservation equation of the E form o LX E Lt \ $ Æ (od$x E ) [ os E X E, (1) FIG. 1.ÈExpected variations with depth of the abundances of lithium, beryllium and boron in a present-day solar model, in the absence of any kind of mixing in the radiative interior. All abundances are normalized to their value on the ZAMS. The extreme temperature sensitivity of the (p, a) nuclear reactions responsible for the depletion of the light elements leads to steplike abundance proðles. The base of the convective envelope is located at r/r ^ 0.74, where the temperature is T ^ 2 ] 106 K. The dotted tick marks _ along the lower x-axis are separated by one density scale height, working downward from the base of the convective envelope. where S (in s~1) describes the rate at which nuclei of the species in E question are destroyed by proton capture. The reaction rates for the computations reported on in this paper are taken from Fowler, Caughlan, & Zimmerman (1975) using the treatment of electron screening from Graboske et al. (1973). The density and other background thermodynamic variables are taken from a model for the interior of the present-day Sun calculated by R. Gilliland. This model arrives on the zero-age main sequence (ZAMS) at an age of approximately 4 ] 107 yr, which is where we begin our calculations with a uniform initial abundance. We neglect variations in solar structural properties during the course of evolution on the main sequence; the error made as a result of this omission is likely to be less than that stemming from uncertainties in the speciðcation of the di usion coefficient. Because convection is very efficient at mixing chemical species, we adopt a value for the di usion coefficient D in the convection zone that is large enough to ensure

3 468 BARNES, CHARBONNEAU, & MACGREGOR Vol. 511 that the concentration of any element therein is nearly uniform. In the radiative core, D is generally expressed as the product of a length scale and a speed characterizing the turbulent Ñow Ðeld associated with an instability. If this estimate is less than the microscopic di usivity, the di usion coefficient is set equal to the microscopic value (see, e.g., Spitzer 1962): D \ 2.2 ] 10~15 T 5@2 cm2 s~1 (2) 12 o ln " where " \ 1.0 ] 10~5(T /o1@2). Using the physical conditions at the base of the convection zone in the solar interior model described above, equation (2) yields D \ 27.2 cm2 s~1. 12 As a preface to the computations described in subsequent sections, it is instructive to estimate the magnitude of the present-day solar lithium depletion that could be produced by a microscopic di usion coefficient that is constant in space and time. Given the strong time dependence of the observed lithium surface abundances during main-sequence evolution, it is likely that the di usion coefficient is also a function of space and time. However, by treating it as constant, we can estimate the order of magnitude by which D 12 must be enhanced in order to account for the observed solar depletion. Let X(i) and X(f) denote, respectively, the initial Li Li and Ðnal values of the lithium mass fraction. If t (D5 ] 109 yr) is the age of the Sun and l (D0.09 R ) is the depth below _ the convection zone at which all lithium is destroyed, equation (1) can be approximated as X(f)[X(i) X(f)]X(i) Li Li D [D Li Li, (3) t 12 2l2 where we have assumed that no lithium is destroyed at depths less than l. Microscopic processes would thus be expected to produce a fractional depletion of lithium of order X(f) Li X(i) D 1 [ (D 12 t/2l2) B 0.92, (4) 1 ] (D t/2l2) Li 12 which is much less than the observed depletion, X(f)/X(i) B Li Li 5 ] 10~3. In order to produce the current solar lithium abundance, a di usion coefficient of order B D D A2l2 1 [ (X(f)/X(i)) Li Li t 1 ] (X(f)/X(i)) B 6 ] 102 cm2 s~1 B 20D 12 Li Li (5) is required. Hence, a relatively modest enhancement of the microscopic di usivity is sufficient to account for the measured depletion of lithium in the Sun. Assuming l D 0.2 R and performing a similar analysis for the element beryllium, _ we Ðnd a depletion of magnitude X(f)/X(i) B 0.98 at the age of present-day Sun. As above, the transport Be Be implied by the observed solar beryllium depletion (B5 ] 10~1) requires a di usion coefficient of magnitude D B 8 ] 102 cm2 s~1 D 30D. On the basis of this rudimentary examination of the abundances 12 of lithium and beryllium in the Sun, we conclude that the physical process(es) responsible for the transport of these elements inward from the solar surface layers must give rise to an e ective di usion coefficient D whose magnitude lies within the fairly narrow range 6 ] 102 [ D [ 8 ] 102 cm2 s~1. In this paper, we address the following question regarding the abundances of lithium, beryllium, and boron in a 1 M star: in what ways do the depletions of these light elements _ with time depend on the nature of the mechanism(s) responsible for the redistribution of angular momentum inside the star? This question is motivated by the following considerations regarding the angularmomentum histories of cool, dwarf stars. A number of investigators have constructed detailed models for the main sequence and preèmain-sequence rotational evolution of solar-type stars. In all such models, angular momentum is removed from a star by the magnetized wind that emanates from the hot, coronal layers of its atmosphere. In the case of low-mass dwarf stars, it is generally assumed that the torque associated with this wind acts only on the outer, convective portion of the stellar interior. Hence, magnetic braking instantaneously decelerates the rotation of just the surface layers of the star, thereby causing a shearing stress to be applied to the material in the underlying radiative interior. It is in response to the nonuniform rotation that arises from this shear that angular momentum is redistributed internally, from the inner core to the outer envelope. The models alluded to earlier utilize di erent physical mechanisms to e ect the angular-momentum transport noted above. For example, if the instantaneous internal angular velocity distribution is susceptible to the occurrence of hydrodynamic instabilities, then the turbulence produced by them might provide for both the transport of angular momentum and the mixing of light metallic species, from the stellar surface layers down to depths where the thermodynamic conditions are suitable for destructive nuclear processes to take place. In the models constructed by Endal & Sophia (1978, 1981), Pinsonneault et al. (1989, 1990, 1991), Chaboyer et al. (1995a, 1995b), Krishnamurthi et al. (1997), it is assumed that turbulent Ñows of this type dominate the transport of angular momentum and light elements in the solar interior. Alternatively, Charbonneau & MacGregor (1993; hereafter Paper II) have investigated how the interaction between internal di erential rotation and a large-scale poloidal magnetic Ðeld can redistribute angular momentum inside a star. In their models, the locally nonuniform rotation produced by the solar wind torque leads to the generation of a toroidal magnetic Ðeld component. Associated with the sheared internal magnetic Ðeld is an azimuthally directed Lorentz force, whose action causes the stellar interior to evolve over time toward a state in which the angular velocity of rotation is constant along poloidal Ðeld lines. There are a number of other alternative mechanisms for transporting angular momentum and mixing in solar-mass stars, including meridional circulation and gravity waves (see, e.g., Schatzman 1996). The Eddington-Sweet time for the Sun is t D 1010 yr () /))2, which means that merid- ional circulation ES is unlikely _ to be important on the main sequence because of the rapidity with which solar-mass stars spin down, quickly leading to timescales longer than the main sequence lifetime of the Sun. On the preèmain sequence, when the rotation rate is signiðcantly higher, meridional circulation may well be important, but since we begin our models on the main sequence, we choose to ignore it in our calculations. Downward-propagating gravity waves generated in the outer convective envelope also o er a means of transporting angular momentum. The di erential absorption of these waves can lead to angular-

4 No. 1, 1999 SOLAR LIGHT-ELEMENT ABUNDANCES 469 momentum redistribution (see, e.g., Zahn, Talon, & Matias 1997; Kumar & Quataert 997). The associated momentum Ñux also generates a secondary circulation that may contribute to light-element transport and lead to changes in the local angular velocity that a ects the propagation characteristics of the waves themselves (see Garcia Lopez & Spruit 1991; Fritts, Vadas, & Andreassen 1998). We have not attempted to model the e ects of gravity waves here, but it should be noted that any mechanism that can transport angular momentum without signiðcant chemical mixing has the potential to produce light-element abundances similar to models in which the transport of angular momentum is dominated by a magnetic Ðeld. To further complicate matters, the downward propagation of gravity waves from the base of the convection zone can be greatly modiðed by the presence of any extant magnetic Ðeld in the upper portion of the radiative core (Barnes. MacGregor, & Charbonneau 1998). The presence of such a magnetic Ðeld is also likely to a ect any large-scale circulatory Ñow therein in a signiðcant manner (see, e.g., Gough & McIntyre 1999). Modeling the e ects of all these transport processes simultaneously is an MHD problem of enormous complexity. At this juncture, the use of parametric models for angularmomentum transport, coupled with detailed studies of each transport mechanism in isolation, represents perhaps the most practical way to disentangle the various physical processes at play in the solar radiative core and hopefully to establish their relative contributions to mass and angularmomentum transport. In subsequent sections of this paper, we conduct a detailed comparison of the computed main-sequence evolutions of light-element abundances within magnetic and nonmagnetic solar models. Our objective in this regard is to determine the extent to which model-dependent di erences in elemental depletions as functions of age might be used as diagnostics of angular-momentum transport mechanisms in the Sun and solar-type stars. In 2, we describe a onedimensional, hydrodynamical model in which the so-called secular shear instability (see, e.g., Zahn 1993) is assumed to be responsible for the turbulent di usion of angular momentum and chemical composition. In 3, we consider a magnetohydrodynamical (MHD) model in which internal angular-momentum redistribution is controlled by the magnetic mechanism investigated in Paper II, but chemical mixing results from turbulence due to unstable shear Ñow. Among other things, in this section we extend the description of turbulent di usion from one to two dimensions and examine the e ects of changing the strength and geometry of the prescribed poloidal Ðeld inside the Sun. Finally, in 4 we compare the computed time histories of the lithium and beryllium abundances from both models with solar and cluster abundance measurements, in an e ort to discern any observationally signiðcant di erences between magnetic and nonmagnetic models. 2. A HYDRODYNAMICAL MODEL For comparison with the results of later MHD calculations, we have developed a simpliðed hydrodynamical model, similar to the ones used by Pinsonneault et al. (1989, 1990, 1991) and Chaboyer et al. (1995a, 1995b), among others. In accord with the Ðndings of these authors, we assume that the transport of angular momentum and chemical species in the stellar interior is dominated by the turbulent mixing that results from the occurrence of the so-called secular shear instability (Zahn 1974; Endal & Sophia 1978; Pinsonneault et al. 1989) As noted by Zahn (1993), the principal consequence of this shear instability is the formation of vortices having a variety of sizes from an otherwise laminar Ñow. The minimum vortex size is determined by the requirement that the Reynolds number of the shear Ñow exceed a critical value (D103) for the instability to take place. The maximum vortex size is approximately established by the following argument. In the stably strati- Ðed solar radiative interior, buoyancy would preclude any radial mixing by adiabatic perturbations to the existing shear Ñow. However, heat exchange by radiative di usion will limit the e ectiveness of buoyancy as a restoring force by eliminating the temperature and density contrast between a displaced Ñuid parcel and the background in which it is immersed (Townsend 1958; Zahn 1974, 1993). Hence, the largest vortex is the one for which the overturning time is about equal to the time required for cooling by radiative di usion. In this way, the size of the largest eddies depends on the magnitude of the thermal di usivity in the Ñuid (see Zahn 1993). We anticipate that these largescale vortices will provide most of the angular-momentum transport and chemical mixing. As noted in the preceding section, we suppose that mixing due to turbulence generated by shear instabilities can be described as a di usion process. Chaboyer & Zahn (1992) have argued that mixing should be very efficient in the horizontal (i.e., meridional) direction, so we assume spherical symmetry and treat just radial di usion. We must therefore solve the coupled, nonlinear equations describing the radial redistribution of angular momentum and chemical composition, namely and or4 L) Lt \ L Lr or2 LX E Lt \ L Lr C or4d()) L) LrD ] or4sj ) ; (6) C LX or2fd()) E D ] or2se X, (7) Lr E where D is the di usion coefficient and S (in s~1) describes J the loss of angular momentum through the action of the torque applied to the star by a magnetized wind. As in Paper II, we calculate S by using the wind model of Weber & Davis (1967) to J obtain exact solutions of the MHD equations describing the Ñow of a polytropic wind in the equatorial plane of the star. The instantaneous angularmomentum loss rate is evaluated from these solutions, under the assumption that the coronal magnetic Ðeld strength is proportional to the angular velocity of the stellar surface layers. There remains then one parameter in the Weber-Davis model that is not directly given by observation, namely the assumed polytropic index; this quantity is set by requiring the model to reproduce the observed mass Ñux of the present-day Sun. For numerical reasons explained in Paper II, angular momentum is removed from the entire convection zone, rather than just at the surface. In equation (7), f is an adjustable parameter that accounts for the fact that the turbulent transport of angular momentum and chemical species may take place with di erent efficiencies (see, Pinsonneault et al. 1989). In the convection zone, the di usion coefficient is given the value D \ 106 cm2 s~1, in an e ort to simulate the efficient compositional mixing that arises from convective Ñuid motions. Through-

5 470 BARNES, CHARBONNEAU, & MACGREGOR Vol. 511 out the radiative interior, we use an expression for the di usion coefficient that is based on the analysis performed by Zahn 1993, D \ 2c 27G K d ln T dr [ 2 3 d ln o dr K~1 r4 Ad). (8) iom drb2 In equation (8), i (in cm2 g~1) is the opacity and m(r) is the mass contained within a spherical shell of radius r. The expression given for D in equation (8) was evaluated numerically using opacity tables kindly provided by D. VandenBerg, assuming a stellar composition X \ , Y \ 0.27, and Z \ Although we have used this expression for the di usion coefficient throughout the radiative interior, we note that in the innermost portion of the radiative core, mixing associated with the secular shear instability is not likely to be the dominant transport process. However, because the trace elements with which we are concerned are essentially absent from material interior to r B 0.5 R (i.e., having been completely destroyed by nuclear reactions _ at larger radii) and since there is relatively little angular momentum below this depth, our results should not be signiðcantly a ected. We begin our calculations on the zero-age main sequence, assuming that the star rotates as a solid body at a rate ) \ 50 ). Likewise, we ignore any light-element 0 _ depletion on the preèmain sequence. While it would be interesting to begin our calculations on the preèmain sequence, our code does not allow for the structural changes in a solar-mass star as it contracts to the main sequence. We do not believe starting on the ZAMS makes a large di erence in the depletion of lithium as the models of Chaboyer et al. (1995a) deplete lithium by only a factor of 2È3 byan age of approximately 4 ] 107 yr. Adopting the efficiency parameter value f \ in order to reproduce the observed present-day solar lithium depletion, we obtain the solutions displayed in Figure 2. The similarities between these results and those of Chaboyer et al. (1995a) are remarkable, given the simpliðed nature of the present model and the fact that we have treated wind-related angularmomentum loss di erently. Perhaps the most interesting aspect of the rotational evolution as mediated by hydrodynamical processes is that signiðcant di erential rotation is present in the central regions of the star, even at times Z4 ] 109 yr.2 In fact, at the solar age the radial gradient in angular velocity is still so large as to be inconsistent with inferences from helioseismology regarding the internal solar rotation. Such measurements yield no evidence for a signiðcant increase in the rate of rotation, from just below the convection zone down to r B 0.2 R (Tomczyk, Schou, & Thompson 1995; Charbonneau et _ al. 1998). The angular 2 Spiegel & Zahn (1992) have argued that a sufficiently large horizontal turbulence could limit the depth to which the latitudinal di erential rotation of the convective envelope could penetrate into the radiative core. The radial di erential rotation set up at the core-envelope interface by the spin-down of the envelope is a di erent matter altogether and cannot be obliterated by horizontal turbulence. The transport of angular momentum associated with the large-scale secondary circulation set up by the spindown, on the other hand, can be altered by strong horizontal turbulence. In the spirit of comparison with the one-dimensional calculations of Chaboyer et al. (1995a, 1995b), we have chosen to neglect such e ects. FIG. 2.ÈAngular velocity and light-element abundance evolution in the hydrodynamical model. The efficiency of the transport of chemical species is a factor f \ smaller than that of angular momentum so as to reproduce the solar lithium depletion. (a) Large gradients in the angular velocity, extending all the way to the center of the star, persist until at least the solar age. (c) Lithium abundance as a function of depth at the same times as in (a). (b and d) Surface values as a function of time for the angular velocity and light-element abundances, respectively, with the tick marks at the bottom showing the times displayed in (a) and (c).

6 No. 1, 1999 SOLAR LIGHT-ELEMENT ABUNDANCES 471 velocity gradient depicted in Figure 2 develops in the following way. Initially, the wind removes angular momentum from the nearly rigidly rotating convection zone at a rate that is faster than microscopic viscosity can resupply it from the underlying radiative material. Consequently, a shear layer forms at the base of the convective envelope, within which the radial gradient in ) grows until the turbulent di usion (see, e.g., eq. [8]) of angular momentum from below has a magnitude that is approximately equal to that of the solar wind torque. The shear layer subsequently thickens as the lower boundary of the region in which (d)/dr) D 0 propagates inward, ultimately spanning the entire radiative interior. At later times, the angular velocity gradient adjusts its magnitude in order to maintain a near balance with the decreasing rate of angular-momentum loss. That the angular velocity of rotation increases inward throughout the radiative portion of the solar interior at all times has two consequences for the abundances of light elements. First, because the nonzero radial gradient in ) persists to the solar age (and presumably beyond), the depletion of light elements takes place at a rate such that L log (X )/L log (t) is nearly constant during main sequence E evolution. Second, because at later times d)/dr D 0 at all depths in the radiative interior, trace elements that are less easily burned than lithium (i.e., beryllium) can sustain signiðcant depletion. 3. LITHIUM DEPLETION IN A MAGNETOHYDRODYNAMICAL MODEL In the model in 2, it was assumed that the redistribution of angular momentum was a purely hydrodynamical process. However, it was shown in Paper II that is possible to construct a model that is consistent with both the evolution of the main-sequence surface rotation rate with time and with helioseismic determination of the SunÏs internal di erential rotation in which the redistribution of angular momentum in a solar-mass star is dominated by magnetic stresses associated with the shearing of a large-scale poloidal magnetic Ðeld. We will now try to estimate what such a model predicts for the depletion of light elements T he Choice of the Di usion Coefficient The instabilities believed to be responsible for the transport of light elements (and possibly angular momentum) have, for the solar interior, at best been studied in the linear regime. Since it is the nonlinear behavior of these instabilities that should dominate the transport, the di usion coefficients based on the linear regime are likely to be only crude approximations to the true behavior. We therefore adopt an anisotropic di usivity with components that are proportional to the gradient of the angular speed K L) K D \ D, Dk K \ D L) K 0, (9) r 0 Lr r Lk where D is a constant. This can be justiðed by arguing that transport 0 of light elements should be most efficient for the largest scale eddies. As before, an estimate for the di usivity can be constructed by taking the product of a characteristic speed with a lengthscale. In this case, the speed would be the di erence in speed between the base of the convection zone and the depth at which lithium burns. This is a thin layer, and so to lowest order the di erence in speed can be approximated by the derivative of the angular speed times the depth of this region. When particular instabilities are considered, the estimated di usivity generally depends on the radial derivative of the angular speed to some low power (e.g., 2 for the secular shear instability as treated by Zahn 1993), so this seems like a reasonable approach to the problem since we are more concerned here with qualitative di erences between models, not the precise factor by which lithium is depleted. For the case where the transport of angular momentum is assumed to be dominated by magnetic processes, the internal angular-velocity proðle is inherently two dimensional, so we must solve a two-dimensional di usion equation for the light-element abundance B L ] D or2 LX E Lt \ L Lr A or2dr LX E Lr Lk C o(1 [ k2)dk LX E Lk ] or2s X, (10) nuc E with k \ cos h and the prescription for the di usion tensor given in equation (9), where D is chosen to produce a lithium depletion of 5 ] 10~3 by 0 an age of 4 Gyr. In the convective envelope, a nearly uniform abundance was enforced by letting D \ D \ 5 ] 106 cm2 s~1. Equation r k (10) is solved numerically using a modiðed version of the Ðnite element based two-dimensional particle transport code discussed in Charbonneau & Michaud (1991) T he Magnetic Model We consider the two poloidal Ðeld conðgurations that were labeled D1 and D3 in Paper II. The poloidal Ðeld is given by B P (r, h) \ B 0 f B C 1 r2 sin h L'(r, h) eü Lh r [ 1 r sin h L'(r, h) eü Lr hd, (11) where f is a normalization factor chosen so that o B o \ B B P 0 midway between the surface and the center along the polar axis and '(r, h) \ q r2(r [ r) sin2 h r ¹ R r 2 2 (12) s 0 r [ R. 2 This produces a dipole-like Ðeld contained within a spherical shell of radius r \ R with a strength determined by B. The D1 conðguration has 2 R \ 0.9 R, meaning that the 0 Ðeld connects the radiative 2 core to the _ convection zone while the D3 conðguration has R \ R \ 0.74 R so that the Ðeld is contained entirely within 2 the cz core (see Fig. _ 3). As standard ÏÏ models, we consider each conðguration with Reynolds numbers R \ R \ 105 [deðned, as in Paper II, as R \ R u /l l and m R \ R u /g, where u \ B /(4no l )1@2 _ is A the poloidal m Alfve n _ speed A and o is A the density 0 CE at the base of the convection zone], and a CE poloidal Ðeld strength of B \ 1 G. In each case, D has been adjusted so that lithium 0 is depleted by a factor of 0 D200 by the age of 4 Gyr. As can be seen in Figure 4, these two cases produce markedly di erent surface abundances at early ages before slowly converging (by construction) to the present-day solar lithium depletion. For the D1 conðguration, this is a twodimensional process, as can be seen by examining the internal distribution of lithium at various times (Fig. 5). After an

7 472 BARNES, CHARBONNEAU, & MACGREGOR Vol. 511 FIG. 3.ÈPoloidal magnetic Ðeld conðgurations in a meridional [r,h] plane, with the symmetry axis oriented vertically and coincident with the left edge. The dashed line is the base of the convection zone. Note the magnetic dead zone ÏÏ for the D1 case, indicated by the darker contour, where the Ðeld lines are contained completely within the core. The D3 conðguration has all the Ðeld lines contained within the core, so there is no direct magnetic coupling between the core and the convective envelope. initial period of phase mixing (t [ 107 yr for B \ 1 G, see Paper II), there are no signiðcant gradients in 0 the angular velocity at high and middle latitudes because the Ðeld lines in these regions are all tied into the convective envelope in which solid-body rotation is enforced and angular momentum is transported very efficiently along Ðeld lines, so the di usion coefficient is essentially the unenhanced microscopic di usivity. However, near the equator, there is a magnetic dead zone ÏÏ in which the magnetic Ðeld lines do not penetrate the convection zone. At the boundary of this region, larger gradients develop in the angular speed giving rise in turn to large di usion coefficients. That is, angular momentum is transported efficiently only along Ðeld lines, and none of the Ðeld lines in this region connect to the convection zone, so large gradients in the angular velocity can develop across the Ðeld lines. Initially, this allows the lithium in the convection zone to be transported from the poles toward the equator, and then down through the dead zone to depths at which it can be destroyed by nuclear reactions. This rapidly depletes the lithium in the convection zone by a factor of order 103 but leaves at high latitudes a large reservoir of lithium between the base of the convection zone and the depth at which lithium burns. However, the rate at which angular momentum is lost from the surface decreases with time, so the magnitude of the gradient in the dead zone also decreases with time, and lithium is no longer transported as rapidly from the convection zone down to depths at which it can be destroyed. There still exists a large gradient in the lithium abundance just below the base of the convection zone at middle and high latitudes, which allows the microscopic di usivity to transport lithium from the reservoir into the convection zone. The rate at which this happens eventually exceeds the (decreasing) rate at which lithium is being transported from the convection zone to the core, leading to an increase in surface abundance. This gradual increase continues for at least 1 Gyr, until the reservoir has been depleted primarily just by microscopic di usivity. Finally, the surface abundance starts to decrease once more, although this decrease is fairly slow as the di usion coefficient is only slightly enhanced relative to the microscopic value, even in the dead zone. This is because, by then, di u-

8 No. 1, 1999 SOLAR LIGHT-ELEMENT ABUNDANCES 473 FIG. 4.ÈSurface abundance (solid lines) and surface rotation rate (dotted lines) as a function of time for the two standard magnetic models. The initial rapid depletion in the D1 case is due to the large gradients in the magnetic dead zone, while the increase at around 1 Gyr is due to the convection zone being replenished from the reservoir of light elements at high latitudes just below the convection zone. The D3 conðguration behaves essentially as a onedimensional system and so shows a monotonic decrease in abundance. While the light-element abundances have dramatically di erent surface-abundance evolutions, the surface rotation rate for the two conðgurations is quite similar. sive coupling across Ðeld lines has reinstated nearèsolidbody rotation throughout the radiative interior. By contrast, the D3 conðguration behaves essentially one dimensionally. Since the poloidal Ðeld does not cross the interface between the radiative core and the convection zone, a large gradient in the angular speed develops for r [ R giving rise to signiðcant radial di usion at all lati- cz tudes (see eq. [9]). This causes rapid depletion of lithium at early times. Then, as the star again approaches solid-body rotation, the rate of depletion decreases until, by the solar age, it is due almost entirely to the microscopic di usivity. Below the boundary between the core and the convection zone, at the greater depths at which beryllium survives, the magnetic Ðeld in the D3 conðguration is able to enforce nearly uniform rotation, so that there is very little depletion of beryllium. The D1 conðguration, on the other hand, rapidly depletes large amounts of beryllium for the same reason that it depletes lithium, then starts to replenish the convection zone from a deeper reservoir, so that by the solar age, the abundance of beryllium in the convection zone is still increasing, although its surface abundance is still much lower than the D3 conðguration. In Figure 6 we show the e ects of changing the poloidal Ðeld strength. For the D1 conðguration, decreasing the poloidal Ðeld strength leads to a much greater amount of lithium depletion, almost independent of the value adopted for the viscosity. This can be understood as follows. The weaker the poloidal Ðeld is, the more it must be sheared to produce a magnetic stress large enough to equilibrate the applied torque (see Paper II, 3.5). Weakening the Ðeld strength results in larger gradients in the angular velocity, which persist for longer times. Since, in this conðguration, the poloidal Ðeld connects the convection zone to the depth at which lithium is destroyed, this larger gradient in turn leads to a greater enhancement of the di usion coefficient for longer times, and thus greater depletion. In the D3 conðguration, the edge of the region containing the poloidal Ðeld is in the layer through which lithium is transported, so the Ðeld strength is less important than the viscosity, which is the only mechanism available for coupling Ðeld lines and thus is responsible for the magnitude of the gradient in the angular velocity in this region. This makes the D3 conðguration quite sensitive to the value of the microscopic viscosity. Small viscosities allow large gradients to develop at the base of the convection zone, which in turn give rise to large di usivities and hence greater depletion. This is an artifact of the model as presumably these smaller viscosities would tend to grow as the gradient increased. That is, the enhancement of the di usivity over a truly microscopic value is proportional to (some power of) the gradient of the angular speed, and so large gradients would not develop and the di usivity would not be as large as predicted. In order to understand the essential features of the evolution of the surface abundance of lithium in the D3 conðguration, we constructed a simple analytic model. To a good approximation, only the following conditions are needed to account for the behavior of the D3 conðguration. Nuclear reactions must e ectively destroy all lithium below a certain depth but have no appreciable e ect above that depth. The distance below the convection zone at which this happens must be sufficiently small that a plane-parallel approx-

9 474 BARNES, CHARBONNEAU, & MACGREGOR Vol. 511 FIG. 5.ÈLithium abundance shown on a gray scale in an upper meridional quadrant with the symmetry axis along the left edge and the equatorial plane along the bottom of each panel for both the D1 conðguration and D3 conðguration standard models. At early times, the D1 conðguration clearly shows the lithium reservoir between r ^ 0.65 R and r ^ R, and h \ 0 and h ^ 60 where the magnetic Ðeld imposes nearèsolid-body rotation, and so prevents much turbulent mixing. By approximately _ 1 Gyr, the lithium cz in the reservoir has started to di use out into the convection zone and is almost as depleted as in the convection zone by solar age. The D3 conðguration has a Ðeld that is nowhere connected to the convection zone, so large gradients develop all along the base of the convection zone, and the depletion is nearly one dimensional. imation can be made and the abundances must be e ectively one dimensional. The di usion coefficient must consist of a small microscopic value enhanced by a few orders of magnitude initially, but dropping to the microscopic value after a few hundred million years. It need not depend on depth, and the only important part of the dependence on the (gradient of) the rotation rate is that it decreases with time. The details of the model and how it compares to the D3 case can be found in the Appendix. The point we wish to make is that it is comparatively easy to construct a simple model that produces a surface abundance that evolves in a manner that is similar to what is predicted by more elaborate formulations T he Possible Importance of Magnetohydrodynamical Instabilities We have so far assumed that, even though the redistribution of angular momentum is due largely to magnetic processes, the transport of light elements is due to purely hydrodynamical instabilities. However, a toroidal Ðeld of the magnitudes produced in the model of Paper II would almost certainly be susceptible to magnetohydrodynamic instabilities. How efficiently these instabilities would mix light elements is uncertain, but one can argue that the abundance should still obey a di usion equation if, as suggested by Mestel, Taylor, & Moss (1987), the e ect of the instabilities is to limit the growth of the toroidal Ðeld. For a constant magnetic di usivity, under axisymmetry and ignoring viscosity and any meridional circulation, the / components of the induction and momentum equations are LB Õ Lt ] g$2b Õ \ (r sin h)b Æ ($)) ; (13) P L) Lt \ 1 4nr2o sin h B P Æ [$(B r sin h)]. (14) Õ Now assume that the e ect of the magnetic di usivity is to limit the growth of the toroidal Ðeld after some time. That is, at some point, the growth of the instability is halted by dissipative processes that operate on a timescale q, so that the second term on the left-hand side of equation (13) may be replaced by simply B /q. Substituting for this term and Õ

10 No. 1, 1999 SOLAR LIGHT-ELEMENT ABUNDANCES 475 FIG. 6.ÈLithium depletion as a function of poloidal Ðeld strength. (a and b) show the amount of lithium depletion at an age of 4 Gyr as a function of the poloidal Ðeld strength for the D1 and D3 conðgurations respectively. (c and d) Evolution of the surface abundance for Ðeld strengths of B \ 0.1 G, 1.0 G, and 10.0 G (crosses [ ] ÏÏ], bulls eyes [ x ÏÏ], and diamonds [ ) ÏÏ], respectively). For each Ðeld conðguration, the case with B \ 1.0 G and 0 l \ g \ 5.4 ] 105 cm2 s~1 (the standard model) has been calibrated to the solar lithium depletion (bulls eyes [ x ÏÏ]), then the same value for the 0 constant D has been used to generate the other plots. For the D1 conðguration, D \ 3.0 cm3 while for the D3 conðguration the larger gradients produced by the 0 lack of magnetic coupling of the core and the envelope require the much 0 smaller value of D \ 2.6 ] 10~2 cm3. The magnitude of the poloidal Ðeld in the D1 case determines both the size and duration of the gradient in the angular velocity below the 0 convection zone. Weaker poloidal Ðelds allow larger gradients to develop and to persist for longer times, allowing much more lithium to be transported to where it can be destroyed. Since the D3 conðguration has no magnetic coupling between the convective envelope and the core, the gradient just below the convection zone depends mainly on the viscosity. Smaller viscosities allow larger gradients and hence greater lithium depletion. This means that for the D1 conðguration, the amount of lithium dependence is extremely sensitive to the Ðeld strength, while in the D3 conðguration, the Ðeld strength is less critical than the viscosity, which determines the viscous boundary layer just below the base of the convection zone. di erentiating with respect to time gives A L2B Õ Lt2 ] 1 LB Õ q Lt \ (r sin h)b P Æ $ L) Lt \ (r sin h)b P Æ A $ G 1 4nr2o sin h B P B ] C $(B Õ r sin h) DHB, (15) where the expression for L)/Lt from equation (14) has been used. For times much longer than the timescale of the instability, one would expect the term involving the Ðrst time derivative to dominate, so C L(B r sin h) Õ ^ (r sin h)2$ Æ qb P 2 Lt 4nr2o sin h $(B r sin h)d Õ for t? q, (16) which is a di usion equation for B r sin h with a di usion coefficient Õ D \ qb P 2 (4no sin h). (17) One might hope, then, that the light-element abundances might also obey a similar di usion equation, possibly with the same di usion coefficient. In the preceding sections, we solved such a di usion equation for light-element abundances, but with a di erent prescription for the di usion coefficient. Based on those results, let us suppose that the magnitude of the magnetic Ðeld-based di usion coefficient must be of the same order of magnitude as the hydrodynamical coefficients in order to reproduce the observed lithium and beryllium depletion qb2 P D 104 cm2 s~1. (18) 4no cz This would imply that the dissipation must occur on a timescale of the order q D 4no cz 104 cm2 s~1 D 2 ] 104 s. (19) B2 P There are a number of instabilities that may be at work in the Sun and could lead to dissipation on such a timescale, such as the one described by Balbus & Hawley (1991, 1994). We have not been able to estimate a dissipation time for this instability; however, one may recall that it is characterized by a growth rate of the order of the rotation rate; in a global

11 476 BARNES, CHARBONNEAU, & MACGREGOR Vol. 511 steady state it would appear natural to equate the dissipation time to the growth rate, so a corresponding dissipation time on the order of 104 s seems plausible. A similar instability, possibly related to the one described by Balbus & Hawley, has recently been discussed by Gilman & Fox (1997). These authors have shown that the solar tachocline is unstable, in a two-dimensional sense, in the presence of a wide range of di erential rotation and toroidal Ðeld values. The growth rate of this instability is on the order of a few months to a few years and hence is also plausible for having a dissipation time on the order of 104 s. However, it is not clear that it can be responsible for the radial transport of trace elements since it is inherently two dimensional. 4. DISCUSSION AND CONCLUSIONS For stars of a given mass, there are two basic sets of observations that constrain the abundance evolution of light elements: the depletion as a function of age and the spread in abundances at a given age. In addition to the Sun (Anders & Grevesse 1989; Libbrecht & Morrow 1991), extensive measurements of both lithium and rotation rates are available for a number of clusters, including a Persei at an age of 50 Myr (Balachandran, Lambert, & Stau er 1988, 1996;Prosser 1994), the Pleiades at 70 Myr (Soderblom et al. 1993b), the Ursa Major Group at 300 Myr (Soderblom et al. 1993a), the Hyades at 700 Myr (Thorburn et al. 1993; Boesgaard & Budge 1988), NGC 7789 at 1.6 Gyr (Pilachowski 1986), NGC 752 at 1.7 Gyr (Hobbs & Pilachowski 1986a), and M 67 at 5.0 Gyr (Hobbs & Pilachowski 1986b). Stars for which measurements of both lithium and rotation rate are available (with e ective temperatures between 5500 and 6000 K) are plotted in Figure 7. It should be noted that these clusters span a small but signiðcant range of metallicities, which introduces yet another variable FIG. 7.ÈObserved v sin i and depletion of lithium for a number of clusters along with the depletion as a function of age for the models considered. Circles are actual measurements of lithium, while triangles denote upper limits. Likewise, open symbols are actual measurements of v sin i, while Ðlled symbols are upper limits. The sizes of the symbols are determined by the magnitude of v sin i. Only stars with e ective temperatures in the range 5500È6000 K are plotted. The Sun is represented by a bulls eye ( x ÏÏ). log N(Li) is deðned as log N(Li) \ log (7Li/H) ] 12. A small (5%) random scatter in age has been introduced to make it easier to distinguish individual measurements. With the exception of the D1 magnetic case, the models are quite similar and qualitatively consistent with the observations. into the picture. Nevertheless, the trend of decreasing lithium with time is clear. It is also clear that there is a small (a factor of [10) but signiðcant spread in abundance at a given age whose magnitude does not appear to change much with time. It might increase slightly for older stars (t Z 1 Gyr); however, there are relatively few observations of older stars, so this is far from certain. Superimposed on the plot of the observations is the depletion of lithium as a function of time for the various mixing models considered in this paper. Except for the D1 magnetic case, the curves are qualitatively similar, although in the D3 conðguration lithium is depleted more rapidly at earlier times and more slowly at later times, than in the hydrodynamical model. In the D1 conðguration, lithium is depleted very rapidly at early epochs (t [ 0.1 Gyr) and levels o to an approximately constant abundance at later times, comparing rather poorly to observations. Although all models have been adjusted to reproduce the observed solar depletion, the rate of depletion at the solar age di ers markedly between the hydrodynamical model and the magnetic models. By an age of a few Gyr, the magnetic models have all returned to nearèsolid-body rotation, resulting in little, if any, enhancement to the microscopic di usivity. By contrast, the hydrodynamical model retains a rapidly rotating core, so there remains considerable enhancement of the di usivity, which persists considerably past the solar age. This has a potentially observable consequence: although all the models are calibrated so as to reproduce the solar lithium depletion by the solar age, in the magnetic cases, lithium is not depleted much further beyond the solar age, while in the hydrodynamical model the depletion continues for a considerable time. Given the uncertainties in mixing coefficients and the presence of adjustable parameters in the model, lithium abundances up to the solar age do not provide a clear cut distinction between, say, the hydrodynamical and D3 magnetic conðguration models. In conjunction with lithium abundance determinations in older solar-type stars, however, such a distinction may be possible. Another observationally signiðcant di erence between the hydrodynamic and magnetic models is the variation of the di usion coefficient with depth, which leads to signiðcant di erences in beryllium abundances. Figure 8a shows the evolution of Be abundances predicted by the hydrodynamic model of 2 and by the D3 magnetic model of 3, after adjusting both f and D to recover the present day solar Li abundance. The depth 0 dependence of the corresponding di usion coefficients is shown in Figures 8b, 8c, and 8d for a few representative times. At early times, the di usion coefficient for the magnetic case (solid line) between r ^ 0.54 R and r \ R is larger than in the hydrodynamic model. _ However, it cz decreases more rapidly with time, so that by t ^ 0.1 Gyr both models yield comparable values of D below the convection zone. Beyond D1 Gyr, for the magnetic model, D is more than an order of magnitude smaller than in the hydrodynamical models. This reñects the very di erent angular velocity proðles associated with the internal redistribution of angular momentum by magnetic stresses versus enhanced viscous transport. In the former case angular velocity gradients are largest below the convection zone and decrease rapidly with depth, while in the latter case gradients in ) persist to much greater depths and to much later times (see Fig. 2a). Once calibrated to reproduce the observed solar Li depletion, the