THE EVOLUTION OF AmFm STARS, ABUNDANCE ANOMALIES AND TURBULENT TRANSPORT

Stellar evolution models of stars of 1.45 to 3.0M have been calculated including the atomic di usion of metals and radiative accelerations for all species in the OPAL opacities. As the abundances change, the opacities and radiative accelerations are continuously recalculated during evolution. These models develop iron peak convection zones centered at a temperature of approximately 200 000K. If one then assumes that there is su cient overshoot to homogenize the surface regions between the hydrogen, helium and iron peak convection zones, it is shown here that the surface abundances variations that are produced, without any arbitrary parameter, closely resemble the abundance anomalies of AmFm stars except that they are larger by a factor of about 3. Detailed evolutionary model calculations have been carried out varying the turbulence in the outer stellar regions in order to improve the agreement with the observed anomalies in AmFm stars. The outer mass mixed by turbulence has been varied as well as the density dependence of the turbulent di usion coe cient. It is shown that the anomalies depend on only one parameter characterizing turbulence, namely the depth of the zone mixed by turbulence. The calculated surface abundances are compared to observations of a number of recently observed AmFm stars. For Sirius A, 16 abundances (including 4 upper limits) are available for comparison. Of these, 12 are well reproduced by the model, while 3 are not so well reproduced and one is a very uncertain observation. 1CEntre de Recherche en Calcul Appliqu e (CERCA), 5160 boul. D ecarie, bureau 400, Montr eal, PQ, CANADA H3X 2H9 1 In cluster AmFm stars, the age and initial abundances are known. There is then less arbitrariness in the calculations but fewer chemical species have been observed than in Sirius. The available observations (Hyades, Pleiades and Praesepe stars are compared) agree reasonably well with the calculated models for the ve stars which are compared. The zone mixed by turbulence is deeper than the iron convection zone, reducing the abundance anomalies to values which are too small for iron peak convection zones to develop in many of the models. The origin of the mixing process then remains uncertain. There is considerable scatter in the observations between di erent observers so that it is premature to conclude that hydrodynamical processes other than turbulence are needed to explain the observations. We are not ruling out that this be the case but the observations do not appear to us good enough to establish it. Subject headings: di usion | stars:abundances | stars:evolution | stars:interiors To appear in the Astrophysical Journal, tentatively scheduled for Vol. 529, No. 1, January 20, 2000. 2 1. ASTROPHYSICAL CONTEXT The Am stars were rst recognized as a group by Titus & Morgan (1940) who noticed that the Ca K lines did not lead to the same classi cation as the hydrogen lines for some of the Hyades A stars. Since then, it has been noticed that most if not all slowly rotating A and F stars (Veq < 100 kms 1) exhibit abundance anomalies. The abundances of CNO are generally below solar while those of iron peak elements are generally above solar and the species in between can be either under or overabundant. Since abundance anomalies may a ect all slowly rotating stars, the cause of the anomaly must be general and related to some basic property. See Cayrel et al: (1991) for a review. In the wake of the success of nuclear astrophysics (Burbidge et al: 1957), attempts were made to explain the properties of the AmFm stars through nuclear reactions. None of those attempts were successful. It was generally tried to explain the magnetic ApBp stars at the same time, even though no explanation was given for the Te dependence of the observed anomalies. However, after it was suggested by Michaud (1970) that radiation driven di usion could explain the Ap stars and by Watson (1971) that the same process could lead to the AmFm stars, it has generally been accepted that atomic di usion was part of the explanation of the AmFm phenomenon. Watson noticed that, immediately below the H-convection zone, Ca is in the Ca iii (argon like) state in which it has a small radiative acceleration. He suggested that the separation occurs there, in which case Ca would be underabundant while iron peak elements would be overabundant, in accordance with observations. This requires the disappearance of the He convection zones which occurs after He has settled gravitationally. It has been shown to explain the absence of Scuti pulsators (Baglin 1972) among the AmFm stars. It is also compatible with the upper limit to the rotation velocity of AmFm stars (Michaud 1982; Charbonneau & Michaud 1988). It has also been shown that when radiative accelerations are included, atomic di usion can explain the envelope of the observed abundance anomalies (Michaud et al: 1976). However the expected anomalies are generally much larger than those observed in individual stars, suggesting that at least one competing hydrodynamical process is strong enough to reduce the e ects of atomic di usion. Currently the most favored competing process is mass loss (Michaud et al: 1983). Turbulence has been investigated (Vauclair et al: 1978) but it does not appear to reduce su ciently the overabundances of iron peak elements and rare earths if helium and calcium are to be underabundant. No calculation has ever successfully reproduced the detailed abundance anomalies of individual stars, including the small anomalies of some chemical species such as Mg. In this context, it appears appropriate to question some of the properties of the di usion model for AmFm stars. In this paper, we investigate an alternative to the suggestion of Watson (1971; see also Smith 1973) that the separation must occur immediately below the H-convection zone. Using the recently available atomic data (Rogers & Iglesias 1992b; Rogers & Iglesias 1992a; Iglesias & Rogers 1995; Iglesias & Rogers 1996), making it possible to calculate radiative accelerations throughout stellar models (Richer et al: 1998), detailed evolutionary models are calculated including the e ect of atomic and turbulent di usion for stars of 1.45 to 3.0M . In these calculations turbulent transport is parametrized using a turbulent di usion coe cient without explicit reference to any underlying turbulence model. The constraints that are determined here on turbulent transport may however be used in a subsequent step to constrain turbulence models such as those of Canuto (1998, 1999). This is however outside the scope of the present paper. Turbulence is varied in order to reproduce the observed abundances in AmFm stars in galactic clusters. Do atomic di usion coupled to turbulent transport explain the observed abundances in cluster AmFm stars at the age of the clusters? Can the abundance anomalies of individual stars be reproduced in detail with such a simple model? The evolutionary models calculated will be described in the next section for stars of 1.45 to 3.0M for di erent turbulences. All details concerning the simulation algorithm have already appeared elsewhere, except for the way turbulent transport is incorporated into the present models which is described in x 2; turbulence parameters as well as our model naming conventions are also de ned there. Global model properties and time dependent internal abundance and radiative force pro les are presented in x 2.1 and 2.2; x 2.2 is the key to under3 standing the resulting surface abundance anomalies. An attempt is made to summarize the results of a large number of simulations by analyzing the internal structure of two representative stellar models near the lower and upper limits of the AmFm mass range. The interdependence between abundance pro les, gravity, stellar luminosity and radiative force pro les is discussed. The determination of radiative accelerations throughout stellar envelopes leads to suggesting a new paradigm, di erent from that of Watson, to explain the AmFm stars. It is shown that the e ects of turbulence can be characterized by a single depth parameter, the mixed mass , which will be used throughout the paper. The e ect of di erent levels of turbulent transport on the evolving distributions of several elements is analyzed in detail. Alecian et al: (1993) have suggested that iron diffusion is important for stellar structure; this will be con rmed by new ndings in x 2.3; there we show that when turbulence is weak enough, an iron peak convection zone forms just below the usual He ii zone. Two examples show that these convection zones cause deeper mixed zones than is present in standard models with He di usion; this mixing can be simulated with appropriate turbulence pro les (see also Charpinet et al: 1997). Generic features of our results regarding surface abundances are presented in x 3: it is shown how surface abundances vary as a function of mixed mass, stellar mass, stellar age, mass at the turning point and initial metallicity. This section prepares the ground for the use of the models to t speci c stars and star clusters. A comparison to observed abundances in made in x 4. Some readers may prefer to read this section rst. Finally, the conclusion discusses the properties of AmFm stars that are explained by this model along with the work that is suggested for further confrontation with observations. 2. EVOLUTIONARY MODELS OF 1.45 TO 3.0M The evolutionary calculations take into detailed account the time dependent variations of 28 abundances induced by atomic di usion as well as their e ect on Rosseland opacity throughout stellar evolution. The opacity and radiative accelerations are calculated, by carrying out integrals over 104 frequency intervals, at each point of the model ( 1000{1500) at each time step ( 600{1000). The mixing length was calibrated using the Sun (Turcotte et al: 1998b). In most cases, the abundances given in Table 1 of Richer et al: (1998) are used as original abundances. For the Hyades, we used a metal mass fraction Z = 0:03, which is slightly higher than the value suggested by the observed Fe abundance in F and G stars (Cayrel et al: (1985); Boesgaard & Friel (1990)). The calculations were carried out as described in Turcotte et al: (1998b) and Turcotte et al: (1998a) with a few modi cations described in Richer et al: (1999) where the introduction of turbulent transport is further discussed. In the one dimensional calculations discussed here, turbulent transport is modeled as a di usion process by simply adding a pure di usion term DT@ lnXi @r to each element's di usion velocity, which tends to reduce its abundance gradient. Here DT is the so-called turbulent di usion coe cient, which is assumed to be the same for all elements; r is the radius and Xi is the mass fraction of element i at that point. Within each convection zone a mixing length approximation, DT '< v` >, has been used for DT. It always leads to large DTs and very homogeneous convection zone abundances. Turbulence has been assumed large enough to mix completely the regions between super cial convection zones. Below the deepest surface convection zone, the turbulent di usion coefcient has been assumed to obey a simple algebraic dependence on density given, in most calculations presented in this paper, by DT = !D(He)0 0 n (1) where n = 2, 3 or 4, and D(He)0 is the atomic di usion coe cient2 of He at density 0. The turbulent di usion coe cient is then equal to ! times the atomic di usion coe cient of He at the reference 2The values of D(He)0 actually used in this formula were always obtained | for programming convenience | from the simple analytical approximation D(He) = 3:3 10 15T 2:5=[4 ln(1 + 1:125 10 16T 3= )] (in cgs units) for He in trace amount in an ionized hydrogen plasma. These can di er signi cantly from the more accurate values used elsewhere in the code; in particular, the accurate values are used to de ne the mixing depths introduced in Table 1 and Figure 7. 4 density in the model of interest. The values used in calculations described below as well as our model naming conventions, are given in Table 1. [Table 1 goes about here.] A series of calculations was done by anchoring the turbulent di usion coe cient at a given temperature, T0, instead of at a given 0. This choice was motivated by the peculiar behavior of convection zones in some models to be discussed in x 2.3. Then 0 = (T0) (2) and Equation (2) is given by the stellar model. In words, in those calculations, 0 of Equation (1) is the density at which T = T0 in the evolutionary model. That density varies during evolution. Examples of turbulent di usion coe cients are shown in Figure 1 at three evolutionary time steps of a 2:5M model. All four turbulence pro les have ! = 1000. The dotted line corresponds to a coe cient with n = 2 anchored at 8 10 6 g cm 3, the dot dash line with n = 3 and the solid line is a turbulent di usion coe cient anchored at logT0 = 5:3 with n = 4. In the latter case, an iron convection zone has developed before 100My. Case (d) (top panel) only serves to show how much pro le (a) would be shifted if the anchoring temperature were increased from logT0 = 5:3 to 5.4; no evolutionary model was computed for this particular set of parameters, although one was computed with logT0 = 5:4 and ! = 106. The bottom of the convection zone appears as a vertical line in the gure. The bottom of the He ii convection zone may be located by the sudden changes in DT at3 log( M=M ) = 8 to 9. The extent of the mixing is further discussed below. 3In this paper, M always represents the mass of the spherical shell outside a certain radius. Fig. 1.| Some atomic and turbulent di usion coefcients used in 2.5M models. Two of these models (a and b) appear in Fig. 12 [curves (3) and (6)]. The pro les are shown at 10My, 100My, and 500My. The Fe atomic di usion coe cients shown are those used in the di usion calculations. For He, two pro les are shown in each panel; the top one is the accurate coefcient used in di usion calculations; the bottom one is a simple analytical estimate used only to compute DT from Eq. (1) (see footnote 2). Pro le (a) uses as density 0 (cf. Eq. 1) the value found at the point where log T = 5:3 ( 0 varies as the star evolves); the slope parameter n equals 4. Pro le (b) uses a xed 0 (here set to 8 10 6 g cm 3), and n = 2. Pro le (c) is identical to (b) except for the slope, n = 3. Pro le (d) shows the e ect of shifting the anchoring point of prole (a) to logT = 5:4. All four have ! = 1000. An iron convection zone appears spontaneously around 37My in case (a). 5 Fig. 2.| Evolution of Te , L, and log g for a number of models. More models were calculated than shown here. The 2.5 and 3.0M models were followed through the subgiant phase. 2.1. Global evolutionary model properties The Te , L, and log g are shown as a function of age in Figure 2 for a number of models calculated here. The chosen masses cover the Te interval of observed AmFm stars (7000 < Te < 10 000K). The 3:0M model drops below 10 000K for the nal 20{ 30% of its main{sequence life (when it is twice as luminous as at the beginning of its main{sequence life and so nearly 3 times more likely to be detected) while the 1:5M model is above 7000K at the beginning of its main{sequence life. On the scale shown, the inclusion of various degrees of turbulence does not modify global properties since the turbulence pro les used do not mix the central regions signi cantly (see Fig. 1): DT D(Fe) for log( M=M ) > 2. 2.2. Internal abundance variations and radiative accelerations The space variation of all chemical species is shown in Figure 3 for a model with M = 1:6M , (! = 100, n = 2) and in Figure 4 for two models with M = 3:0M , but with di erent turbulent di usion coe cient parametrizations, one [part (a)] with the same reference density (of Eq. 1) as the 1.6M model and the other [part (b)] with a reference density reduced by a factor of 2 and all with n = 2 and ! = 100. For the 3.0M models, only some of the abundances are shown. In the 1.6M model, evolution has been pursued long enough for the surface convection zone to get deeper. At 1 700My (dot-dash lines) the Fe and Ni abundances, for instance, have been considerably reduced from their peak value while there occurs dredge up of Cl from a small abundance peak that had accumulated at M=M 10 4. Compare the two models with the same turbulence parametrization Fig. 3.| Internal abundance pro les in an evolving 1.6M model with turbulence parameters 0 = 8 10 6g cm 3, ! = 100, and n = 2. Pro le ages (in My): 100 (long dash), 500 (dotted), 1000 (solid), 1500 (short dash), 1700 (dot-dash). but masses of 1.6 and 3.0M spanning the range of AmFm stars. In the 1.6M model, gravity is larger and the main{sequence life time is much longer. Since turbulence is assumed to have the same density dependence in both, the chemical species that are not supported below the mixed zone (e. g., He, C, N, O, Ca) are underabundant by a much larger factor in the 1.6 (of 1/20 for He) than in the 3.0M (of 1/2.5 for He) star. However the situation is more complex for those elements whose radiative acceleration is larger than gravity below the mixed zone. As the mass increases, so does Te and, consequently, radiative accelerations (see Figs. 5 and 6 for grad in the 1.6 and 3.0M models respectively). The radiative acceleration of Fe has a maximum in both stars at M=M ' 10 6 but the maximum is 3 times greater than g in the 1.6M and 10 times greater in the 3.0M model. The abundance anomalies for a given turbulence then have a tendency to be larger in the more massive star though the e ect is not by a large factor. For Fe, the overabundance is by a factor of 4 in the 1.6 and of 5 in the 3.0M 6 Fig. 4.| Internal abundance pro les in evolving 3.0M models with turbulence parameters ! = 100, n = 2; in the top panel (a), 0 = 8 10 6g cm 3 while in the bottom panel (b), 0 = 4 10 6g cm 3. Pro le ages (in My): 5 (long dash), 30 (dotted), 70 (solid), 260 (short dash). model. Consider the abundances of Fe and Ni in the 3.0M model with the smaller reference density (Fig. 4b) and so the smaller zone mixed by turbulence, Mm=M (the mass over which turbulence has a signi cant effect on the abundance gradients); at 100My of evolution, it extends down to log Mm=M ' 5:51 in this model. Above M=M ' 10 6, the abundances increase rapidly (within less than 70My) to a nearly constant value. An abundance gradient builds up where the turbulent di usion coe cient becomes small enough (it goes down as 2) and for Fe and Ni the maximum abundance is determined by the competition between gravity and grad in presence of turbulence and atomic di usion. The Fe and Ni abundance gradients are large over one decade in M=M (from log M=M 5:5 to 6:5). The mass of the zone where Fe and Ni are overabundant continues to get larger after the super cial abundance stops increasing. Even though the turbulent di usion coe cient deFig. 5.| Radiative accelerations (thick lines) and local gravity (thin lines) in a 1:6M model (1.60R100-2rho8) at age 100My (solid), 1Gy (dashed) and 1.7Gy (dotted). Vertical lines show the position of the bottom of the deepest surface convection zone. The bottom of the zone mixed by turbulence is at log Mm=M = 5:27 in this model, which corresponds to Fig. 3. creases smoothly toward the interior, the super cial mixed zone appears to have approximately the same frontier for all chemical species, for both under and overabundances. This turns out to be the case for all values of n used so that we will need, in practice, only one parameter, the value Mm=M where DT ' 2D(He); see Figure 7. From the insert in Figure 12, it follows that Mm=M is the mass over which turbulence has a signi cant e ect on the abundance gradients of all chemical species (except potassium; this seems related to grad(K) g changing sign close to the bottom of the mixed mass). This is not the same as the homogenized mass which is ten times smaller as one may see by looking at the abundance pro les of Fe or He. Within convection zones, the chemical species are homogenized. Below they are affected by the remaining turbulence. The mixed mass is given for all models in Table 1. For metals, the in7 Fig. 6.| Radiative accelerations (thick lines) and local gravity (thin lines) in a 3:0M model (3.00R100-2rho8) at age 70My (solid) and 311My (dotted). Vertical lines show the position of the bottom of the deepest surface convection zone. The mass mixed by turbulence extends approximately to log Mm=M = 5:08. crease in mass number, which multiplies (g grad) in the di usion equation, partially compensates for the decrease in the atomic di usion coe cient (/ 1=Z2) so that the mixed zone has approximately the same lower boundary for all atomic species. The grad for chemical species between Na and S are very nearly equal to gravity over the mass interval immediately below the homogenized zone (between the homogenized and the mixed mass). The abundances of these species are then especially sensitive to the accuracy of grad. It will turn out below that their calculated values are systematically slightly below the observed abundances. The most important characteristic of the grads for AmFm stars is probably that grad(Ca) < g in both models (Figs. 5 and 6) at M=M 10 6{10 5. This is true throughout main{sequence evolution. At that depth, Ca is in the Ne-like con guration and its Fig. 7.| Relationship between the surface abundances of two elements (one rising, one sinking with time) and the depth of the mixed zone [de ned here as the depth Mm=M at which DT = 2D(He)]. This depth was determined at a common age of 100My, for thirteen 2:50M models (large dots) with di erent turbulence pro les. Turbulence parameters for each model can be found in Table 1. Smooth lines represent predictions based on a simple exponential variation with time scale proportional to ( Mm=M )0:545, as proposed by Michaud (1977). Most models have n = 4, but the models with n = 2 and 3 are also well tted, showing that only the \mixed" mass is important. See also the insert in Fig. 12. grad is much reduced, while iron peak elements are supported. The grad were calculated for stars with masses in between (not shown) and a similar behavior is found. This suggests a di erent paradigm for the AmFm stars from that proposed by Watson (1971): calcium is generally underabundant while Fe peak elements are generally overabundant because the mixed region extends to a depth of Mm=M 10 6{10 5. This property is especially important since, as will be seen in x 2.3, an iron peak convection zone forms just above M=M 10 6 because of the drop in grad(Fe) and the grad of some other Fe peak elements occurring at M=M 10 7{10 6 (Figs. 5 and 6). Finally by looking at the Ti and Fe abundances 8 Fig. 8.| Di erence between radiative and adiabatic logarithmic temperature gradients in the 3.0M model with time independent DT( ) relationship. Turbulence parameters are: 0 = 8 10 6g cm 3, n = 2, and ! = 100 (M3R100-2rho8). Pro le ages are indicated in My. in Figure 5 one may note that they are substantially modi ed by di usion down to M=M 0:1%. This corresponds to 20% of the radius. The anomalies are not limited to the surface. 2.3. Iron peak convection zones and turbulence modeling In a forthcoming article (Richer & Michaud 1999) 1.5 and 2.5M models are used to analyze in detail how an Fe peak convection zone appears as the natural consequence of stellar evolution. Here are shown in Figures 8 and 9 the di erences between the radiative and adiabatic temperature gradients throughout 3.0 and 2.0M models respectively, calculated with xed amounts of turbulence. The zone mixed by turbulence includes the iron peak convection zones. Figure 8 shows, as a function of T within the 3.0M model (3.0R100-2), that there are four supercial convection zones. The two nearest the surface correspond to H and He i ionization, and are merged in cooler models (cf. Fig. 9); for that reason, they are usually viewed as a single convection zone. The next Fig. 9.| Same as Fig. 8 but for a 2.0M model with time varying turbulence: DT( ) is suddenly reduced by a factor 30 when the model reaches 300My. Turbulence parameters are: 0 = 8 10 6g cm 3, n = 3, and ! = 1000 initially (M2R1K-3var). one corresponds to He ii ionization and the fourth is the iron peak convection zone. All three major zones (H+He i, He ii, and iron peak) are present during most of the evolution. The convective regions are clearly linked to temperature and stay at the same T throughout main{sequence evolution. The Fe peak overabundance factors (a factor of 4 for Fe) are sufcient (but just barely su cient; a factor of 3 overabundance of Fe is not su cient, as other calculations, not shown, demonstrate) to cause convection. As evolution proceeds, the Fe abundance decreases slightly (see Fig. 11, solid line) because of the progressive mixing caused by the deepening iron convection zone which leads to mixing beyond that imposed by the assumed turbulence (Eq. 1). At 330My, the star is moving to the subgiant phase, the three convection zones have merged and extend much deeper than on the main{sequence. Figure 9 shows again the evolution of the temperature gradient di erences but in a 2.0M model with turbulence parameters xed only up to 300My (2.00R1000-3var). Turbulence below the hydrogen convection zone was then cut o (actually reduced to a very low level, to help maintain numerical stability) 9 and evolution allowed to proceed. Turbulence is weak enough for the He ii convection zone to disappear after some 200My so that there then only remains the H convection zone. There was no iron peak convection zone until turbulence was turned o : in presence of the originally assumed turbulence, Fe was overabundant by a factor of about 3 and this is not su cient to cause the iron peak convection zone in this model. After turbulence was turned o , the Fe abundance increased rapidly by a factor of 2 (see the next section) and a convection zone appeared. The chemical separation then occurred only below the iron peak convection zone. The evolution which followed would correspond rather closely to evolution with (2.00R50-3). The iron peak convection zone remained until the end of main{sequence evolution. The Fe convection zone xed the size of the mixed region: no additional mixing was assumed below. Whenever one tries to do evolution with a mixed region shallower than the Fe convection zone, the di usion of Fe peak elements imposes this convection zone. In both the 2.0 and 3.0M models, the Fe peak convection zone is centered at logT 5:3. That temperature was used in Equation (2) (logT0 = 5:3) for most models whose turbulence was linked to temperature. The imposed turbulence then mimics the e ect of the naturally occurring Fe peak convection zone with a weak (! = 100; n = 4) or strong (! = 106; n = 4) inward extension of turbulence. The models with ! = 100 and n = 4 have very nearly the same evolution of super cial abundances as models with an Fe convection zone and no extension. 3. SURFACE ABUNDANCE VARIATIONS, ABUNDANCE ANOMALIES AND TURBULENCE MODELS On Figures 10 and 11 are shown the time dependent variations of surface abundances of the 28 elements in four stellar models. The 2.0M (2.00R1K-3var) model (Fig. 10) is the same as used for Figure 9 in which turbulence was turned o after 300My; the (2.00R1K-3) model is identical except that turbulence was maintained at the same level throughout evolution. The 3:0M model (Fig. 11) was also used for Figure 8. The 2.5M (2.5T1M-4) model has a super cial mixed zone deeper than that of the 3.0M model. In the 2.5M star, the turbulence pro le is linked to temperature [Eq. (2)]. In both the 2.5 and 3.0M models one arrives, in the Fig. 10.| Surface abundance variations for two similar 2.0M stellar models with turbulence parameters 0 = 8 10 6g cm 3, ! = 1000, and n = 3. The only di erence between them is that in one model (2.00R1K-3var, solid line) turbulence was e ectively turned o after 300My (cf. Fig. 9). early evolution, at a plateau of over or under abundances that do not vary considerably during most of the main{sequence evolution. For chemical species that are not supported by radiative accelerations, the plateau occurs because the progressive increase, during evolution, of the mass above a point of a given T or of a given partially compensates the continuous ux of the various chemical species out of the mixed zone. This also applies to species that are supported by radiative accelerations except that, as will be discussed below, an equilibrium gradient sometimes develops for them. In the 2.0M models, one observes a similar evolution except for the imposed change in turbulence at 300My. The ensuing Fe peak overabun10 Fig. 11.| Surface abundance variations for two stellar models with turbulence. Solid line: M = 3:0M , 0 = 8 10 6g cm 3, ! = 100, and n = 2. Dotted line: M = 2:5M , logT0 = 5:3 [cf. Eq. (2)], ! = 106, and n = 4. dances are similar to those in the 3.0M model, while the underabundances of CNO and the lighter metals are more pronounced in the lower mass star. Iron reaches an overabundance by a factor of 8 once the depth of the turbulently mixed zone is xed by the depth of the iron peak convection zone with no extension of convective mixing, that is after 300My. This de nes the maximum anomalies which are possible, since they are limited by the appearance of the Fe peak convection zone. Si and S underabundances appear because the mixed zone shifts to a region where those elements are not supported by radiation pressure. When gravity gets below log g 3:0 (at the end of the evolution of the 3.0M star shown in Fig. 11), the convection zone becomes much deeper and the abundance anomalies disappear. This occurs as the star evolves through the subgiant phase. It de nes the end of the AmFm phenomenon. At the same epoch, the Li and Be abundances decrease rapidly. This is caused by the dredge up of material from deep enough in the star for Li and Be to have been burned. The boron abundance also decreases but by a smaller factor. The increase in the 3He and 13C abundances are due to the dredge up of nuclearly synthesized material. A similar e ect would appear in the 2.0 and 2.5M models if their evolution were continued a little longer. On Figure 12, are shown surface abundances versus Z, the atomic number (not to be confused with the stellar metallicity, which is also shown on the gure), in 100My old 2.5M models. The various curves correspond to di erent turbulence models (see Fig. 1 and Table 1). An evaluation of the depth of the mixed zone for each model is listed in Table 1. In the (5.30D100-4) model, the mixed zone extends only down to the bottom of the iron peak convection zone. In the (5.30D1M-4) model, it is 10 times deeper in mass. In the (5.40D1M-4) model, it is deeper by a further factor of 2.5 in mass. The deeper the mixing, the smaller the anomalies. An odd-even e ect is apparent for elements between Ne (Z = 10) and Ca (Z = 20). It is caused by saturation of the radiation eld which reduces grad most for the most abundant even Z elements (Mg, Si, S, . . . ). Since grad g is close to 0 for species with Z between 10 and 20, they are the most a ected by the odd even e ect, though it is also apparent for iron. The elements Na, Mg, and Al are least a ected by turbulence variations because gravity and grad are nearly equal for them. If, slightly below the bottom of the mixed zone in the models with least mixing, the sign of grad g changes, a super cial underabundance can be transformed into an overabundance as the depth of mixing is increased, as seen between S and Ca (for instance Cl). As the mixed mass is reduced, the overabundance of iron peak elements rst increases more rapidly than the underabundance of CNO but for the smaller mixed mass considered here (e.g., curves 1{ 3), it is the CNO abundances that are most sensitive. The global abundance patterns are not transformed by increasing/reducing turbulence, but the size of the anomalies is changed. On Figure 13 are shown abundance anomalies, in models of di erent masses but with a given density dependence of turbulence [labeled (b) on Fig. 1] and at an age of 300My. As the mass increases so does 11 Fig. 12.| Surface abundances in 2.5M models at 100My. Each curve corresponds to the use of a di erent turbulence pro le. They are labeled as follows (see Table 1 for parameters): (1) T5.30D100-4; (2) T5.30D200-4; (3) T5.30D1K-4; (4) T5.30D10K-4; (5) T5.30D10K-3; (6) R1K-2 (dotdash); (7) T5.25D1M-4 (dotted); (8) T5.30D1M-4; (9) T5.40D1M-4 (dashed). Some isotopes are included, slightly shifted horizontally. Horizontal bars on the right show the spread of corresponding surface metallicities [Z/H]. E ective temperatures range from 10300 to 10 400K. Inset: di erences between anomalies of two more models (10 & 11) and those of model (4); (10, dotted) R92-3 has the same Mm as (4), while (11, solid) R163-3 was adjusted to give a near perfect t for helium at 100My and has a slightly larger Mm. Te . As Te increases, one sees the e ect of the increased radiative ux on radiative accelerations. The overabundance of elements of the iron peak (and of P, Cl, and K) become larger. Between Al and Ti, some elements that are underabundant (Si, S, Ar) in the less massive stars become overabundant in the more massive stars. The same e ect may be seen as a function of Te in Figure 14 where the same turbulence parametrization was used for stars of di erent masses and the anomaFig. 13.| The same turbulence (R1K-2) was used to evolve a series of 16 models of various masses; here is a subset representative of AmFm stars: (1, dotted) 1.45M , (2, short dash) 1.6M , (3, solid) 1.9M , (4, long dash) 2.2M , and (5, dot dash) 3.0M . The surface abundances are shown at a common age of 300My. At that age, abundance anomalies have passed their peak in the 3.0M model (see Fig. 11). Lithium is shown relative to 12 + log(N(7Li)=N(H)) = 3:3, rather than to the solar value; otherwise, [N=H] has its usual meaning. In case (1), 7Li dips down to 1:3 . Horizontal bars on the right show the corresponding surface metallicities [Z=H]. lies plotted at three ages. For some chemical species, e. g., Fe, the anomaly is constant over a surprising Te range and for all ages. It is essentially determined by the turbulence model. For CNO and Ne, the underabundance is more a function of age. No suggestion is made here that the turbulence model chosen for this gure is optimal; choices that yield better ts to observations will be presented below. Figure 7 shows that this particular pro le is in fact one of the most deeply mixing ones that was tried, and is probably too deep. In Figures 3 and 4 of Hill (1995), there appears a correlation, in many A stars, between the Fe and 12 Fig. 14.| Te dependence of abundance anomalies, in models of di erent masses but a common turbulence model (R1K-2) and same initial composition. Crosses: at 100My; triangles: at 300My; open circles: at 670My. Note the di erent scales used in the top row and bottom right panel (metallicity). These gures may be compared directly to Fig. 7 of Burkhart & Coupry (1999), where observations of Li, Al, Si, S, Fe, and Ni are presented for clusters of ages ranging from 70 to 700My. Cr abundances and an anticorrelation between the Fe and C abundances. A similar correlation is present here in Figure 14 as well as in Figure 12. The size of the mixed zone is what determines the magnitude of the anomaly. We do not suggest this to explain Bootis stars, however. On Figure 15 are shown the surface abundances as a function of atomic number at ve di erent ages: 5, 20, 70, 151, and 355My. The anomaly patterns take about 70My to reach plateaus which they maintain throughout most of the evolution. As discussed in conjunction with Figure 12, the surface abundances vary little during the main{sequence except in its early phase so long as the turbulence parametrization is xed as a function of T or . The dredge up phase occurs at 520My and leads to the appearance on the surface of material from which Li and Be Fig. 15.| Evolution of surface abundances in a 2.50M model with turbulence parameters logT0 = 5:3, ! = 103, and n = 4. Ages in My are (also indicated on the right): 5 (stars), 20 (upright triangles), 70 (squares), 151 (circles), 355 (inverted triangles). Inset: late models in the dredge up phase; the only anomalies left are those due to burning of light trace elements. were burned (see the inset). This model developed an Fe peak convection zone before 70My and it coincides with the mixed zone. On Figure 16 are shown the abundances at a given Te and age but with models of di erent masses, all with the same turbulence parametrization. At that age, which coincides approximately with that of the Hyades, stars of mass 1.7 to 2.3M have a Te ' 8000K and are at the turn-o . The abundance of Fe peak elements is seen to be independent of mass. Since Te is the same for all, the radiative acceleration is the same at a given density where the turbulent diffusion coe cient is also the same. There follows that the abundance gradients which radiative acceleration can maintain are the same in all models. The underabundances are larger in the lower mass models, however, because gravity increases as the mass is reduced. This gure indicates the spread in abundance anomalies which can be expected in turn-o stars for 13