Astron. Astrophys. 360, 952-968 (2000)

4. Overshoot at the bottom of the convective envelope

Overshoot at the bottom of the convective envelope has an effects on the modeling of the TDUP and on the formation of a pocket after the end of the dredge-up phase.

4.1. The third dredge-up

It has been a matter of discussion whether convective overshoot at the bottom of the convective envelope is a prerequisite for dredge-up Mowlavi 1999. This question is closely linked to the general problem of determining the convective boundary of mixing within the framework of a local theory of convection together with its numerical implementation. As reported by previous authors (e.g. Paczynski 1977), our models without overshoot develop a discontinuity in when the bottom of the envelope convection zone reaches the He-rich buffer layer. This discontinuity can prohibit the TDUP under certain conditions. In our models of AGB stars with not so massive cores the flux on the stable side of the convective boundary is not large enough to lift above .

In Fig. 1 the situation in the dredge-up region is shown for several thermal pulses computed with different assumptions on overshoot. After an early TP of the 3 sequence computed without any overshoot (panel a) the bottom of the envelope convection does not approach the H-free core close enough for any dredge-up. For more advanced TP (panel b) the convection clearly approaches the H-free core for a period of about but due to the effect of the abundance discontinuity shown in Fig. 3 the lack of overshoot prohibits the TDUP. This is evident from panel (c) which shows the same situation with overshoot switched on below the bottom of the envelope convection. In this case a considerable dredge-up of material is possible. Panel (d) shows that overshoot applied also to the preceding PDCZ increases the DUP efficiency even further, which is discussed in detail in Sect. 5. Note that if overshoot only at the base of the convective envelope had been applied for the 5th TP shown in Fig. 1, this would have had no effect on DUP because the stable layer between the bottom boundary of convection and the H-free core is too large compared to the tiny extent of the overshoot layer. Fig. 3 demonstrates the situation of panel (b) of Fig. 1 as a set of profiles. The bottom of the convective envelope recoils at the He-rich region which encloses the core, despite the fact that contact between the convective envelope and the H-free core is established over a few hundred years. In Fig. 3 the time after the eighth TP of the 3 sequence with overshoot has been recalculated without overshoot at the bottom of the envelope convection. A profile at the beginning of the TDUP of the corresponding original sequence with overshoot is shown in Fig. 2. Close inspection and comparison with Fig. 3 shows that of He-rich material have already been dredged-up. The additional mixing removes the sharp abundance discontinuity. This in turn dissolves the sharp discontinuity in the radiative gradient which sensitively depends on the opacity. In the envelope, hydrogen has a mass fraction of and a higher opacity than helium. If this material is mixed into the stable region the radiative gradient will be lifted and dredge-up can occur.

Fig. 1a-d. Evolution of mass coordinate of convective boundary (, solid line) and H-free core (, dashed line). The mass scale has been set to zero at the mass coordinate of the H-free core at the time of the He-flash (TP5: ; TP14: ). The time scale has bee set to zero at the peak of the He-flash luminosity (TP5: ; TP14: ). The spike (solid line) at shown in a , b and d is the top of the pulse-driven convection zone of the He-flash. The upper solid line shows the bottom boundary of the envelope convection zone. Panel a : 5th TP of the 3 sequence without overshoot; panel b : 14th TP of the same sequence; panel c : the same TP as in b started after the PDCZ has vanished and with overshoot at the bottom of the envelope convection zone; panel d : again the 14th TP recalculated from before the onset of the He-flash, with overshoot at the PDCZ and during the following TDUP episode.

Fig. 2. Temperature gradients, opacity and abundances for the mass fraction in the AGB star during the TDUP () after the eighth TP with overshoot. after peak helium luminosity. Compare Fig. 3: there the bottom of the convective envelope does not reach into the He-rich buffer zone at , whereas in the case shown here the envelope has started to engulf the He-rich buffer zone.

Fig. 3. A time series of the region of the former hydrogen-burning shell after the eighth TP of the 3 sequence. The dredge-up phase after the thermal pulse of the overshoot sequence has been recalculated without overshoot and as a result dredge-up is prevented (compare Fig. 1) and Fig. 2. Abundances in mass fractions of (short-dashed), (dot-dashed) and hydrogen (dashed) refer to the left scale while the adiabatic (, long-dashed) and radiative (, solid line) temperature gradient refer to the right scale. For each panel the abscissa ranges from the bottom of the envelope at the right to the top of the intershell region at the left. In the middle of each panel a region of almost pure has been formed by hydrogen-burning during the previous interpulse phase. The lower boundary of the He-rich region is formed by the He-flash convection zone, which has at homogenized the intershell layers below . The four snapshots show how the bottom of the envelope convection zone advances downward by mass during the phase when TDUP would occur with overshoot. The order is from top to bottom and left to right and the ages are given with respect to the flash peak of the previous pulse. Note how the discontinuity of is related to the abundance discontinuity which starts to build up (). The discontinuity grows further () and finally when the region resumes contraction the bottom of the envelope convection retreats without being able to dredge-up material from the H-exhausted regions.

This principle is not dependent on the efficiency of overshoot within a factor of maybe two (Fig. 13). At the lower edge of the AGB envelope the convective velocities are of the order of while very efficient mixing at this convective boundary is still achieved by convective velocities down to . An instantaneously mixed region can be identified in which the downward directed flows decelerate from the initial velocity at the edge to the much lower velocity where the chemical abundance is not homogenized. Here, the turbulent velocity is so slow that mixing becomes inefficient. If the overshoot efficiency is small the region of homogenized composition may be very thin. This will still suffice to allow TDUP if the resolution is chosen such that overshoot is numerically able to smear out the abundance discontinuity (see below). Fig. 2 gives an impression of the masswise size of the overshoot region which can be barely identified in the abundance profiles on this scale. If the time steps are too large with a small overshoot, the smoothing effect on the abundance discontinuity is not efficient and DUP is prevented as is the case in the examples shown in Fig. 1b and Fig. 3.

In our calculations without overshoot and are first compared at each mass grid zone which defines stable and unstable grids. If a discontinuity has already built up, grids are included at the discontinuity in order to still resolve the abundance gradient. No mixing is allowed from an unstable grid to a stable grid. This approach closely realizes the MLT which predicts that no convective bubble can leave the convectively unstable region. However, one can argue equally well that the average of the gradients between neighboring grids should be compared. This criterion would allow material in stable grids to be mixed with material from the neighboring grid if the latter is unstable. Under certain conditions dredge-up may occur with the second treatment which would not occur with the first treatment.

During the dredge-up the upper part of the intershell region is engulfed by the convective envelope at the rate where is the mass coordinate of the convective boundary. Then, the extent (in Lagrangian coordinates) of the stable layer which is practically homogenized by extra mixing, depends on the overshoot efficiency and the rate . is larger for a larger efficiency. Moreover, will be larger if is smaller and vice versa because element mixing is treated time dependently. With the efficiency and we find . In order to resolve the overshoot region the mesh size at and around the convective boundary should not exceed .

The time step is limited by the extent of the homogenized stable layer. Current stellar evolution calculations typically separate the solution of the structural equations and the solution of the mixing equations (often a diffusion-like equation) in order to save computational resources. This separated solution is justified if the two processes described by the two sets of equations operate on different time scales. If the numerical time step is of the order then the mass coordinate of the envelope convection bottom after this time step would be below the previously homogenized layer: . In that case the separated solution of the structure equation and the equations of abundance change is no longer valid because the abundances at cannot be approximated by the abundances . Only if this approximation is valid can the separate computation of structure and abundance change give correct results. This is only the case if .

The actual spatial resolution is controlled by a combination of criteria. Throughout the star we insert mass shells if any of the conditions , , or is fulfilled between two mass shells. After the He-flash, when the dredge-up is expected we increase the resolution in a region of about around the mass coordinate at the bottom of the envelope convection zone by requiring . This prescription for the spatial resolution ensures the above mentioned mesh size constraints in an adaptive way. Any further increase of resolution does not affect the amount of dredge-up obtained. Adapting the mesh may cause additional numerical diffusion. In the situation of the advancing convective boundary associated with the TDUP this does not prevent the discontinuity (Fig. 3) if no minimum grid size is enforced. However, numerical diffusion may easily distort the abundance profiles in the pocket (Fig. 4) when the grid density after the end of the TDUP is first reduced and then increases afterwards when H-burning starts.

Fig. 4. Development and destruction of the pocket after the eighth TP of the 3 sequence. The panels show the abundance profiles of H, and other isotopes in a mass range that contains the contact region of envelope and core at the end of the TDUP. The top panel shows a model after the end of the dredge-up episode (about after the He-flash). The second panel shows a model about later when hydrogen burning has set in again. The third panel shows a model at the very end of the interpulse phase when has already been destroyed by reaction . The last panel shows the profiles at the onset of the next thermal pulse. The solid lines in the top and bottom panel are the diffusion coefficient (right scale) of these models. In the top panel the diffusion coefficient results from the overshoot of the envelope convection while in the bottom panel it is the diffusion coefficient of the upper overshoot zone of the He-flash convection zone during the ninth pulse.

4.2. The formation of the pocket and of an adjacent pocket

During the TDUP the hydrogen rich and convectively unstable envelope has contact with the radiative carbon rich intershell region. If the boundary of the envelope convection zone is treated according to the exponential diffusive overshoot method hydrogen diffuses into the carbon rich layer below. When the bottom of the envelope convection zone has reached the deepest position by mass, a region forms were protons from the envelope and from the intershell region coexist (Fig. 4, top panel). Note that the whole mass range shown in Fig. 4 is part of the overshoot region of the convective envelope. The solid line in the top panel shows the exponentially declining diffusion coefficient at the end of the dredge-up episode. In the right half of the panel D is large enough to cause a homogeneous element mixture. The profiles have developed cumulatively over a period of during which the mass coordinate of the base of the envelope convection remains almost constant. In the left part the hydrogen abundance decreases as increases. Thus, the H/ number ratio decreases continuously from in the envelope to zero in the intershell region.

When the temperature increases during the succeeding evolution proton captures transform and H according to the continuously decreasing H/ ratio into a pocket and a prominent pocket side by side. The forms in the deeper layers where the H-abundance is below . Here the newly formed cannot be further processed into because all protons are already consumed. The abundance reaches a maximum mass fraction of 0.09 at . Above this mass coordinate the H/ ratio is larger and after the formation of some there are still protons available which transform into . The ratio changes continuously in this region and at any position it is almost entirely determined by the initially present H/ ratio. Moreover, a pocket is a direct consequence of the continuous variation of the H and abundances from the core to the envelope. Therefore, if the efficiency of mixing is decreasing smoothly in the overshoot region a substantial pocket will inevitably form. The maximum abundance is given by the amount of in the intershell (see Sect. 5.2). In a narrow region (albeit larger than the region occupied by the pocket) becomes the most abundant isotope at a mass fraction of 0.45. The bend of the profile, e.g. at originates from the changing abundance ratios of , H and which are relevant for the production of .

Note that at the time when proton captures start to form the and pockets the radiative region above the burning region is well established. The isotopes made in this region cannot reach the surface at this time. Instead they are processed under radiative conditions Straniero et al. 1995 and the products of this processing will be engulfed by the next He-flash convection zone. Thus, the production of after dredge-up with overshoot does not decrease the observed ratio in giant stars as suspected by Wallerstein & Knapp Wallerstein & Knapp (1998).

During the whole interpulse period the region displayed in Fig. 4 is heating and contracting (Fig. 6). When the temperature has reached about , is destroyed (Fig. 5) by and neutrons are released. Towards the end of the interpulse period (third panel Fig. 4) most of the is burnt by capture. The remaining small fraction is engulfed by the outwards reaching He-flash convection zone of the ninth pulse. This is displayed in the bottom panel of Fig. 4. The straight solid line shows the diffusion coefficient in the overshoot region at the top of this convection zone. Although the absolute value of D is larger than in the top panel, the impact on the abundance appears to be smaller. This effect is caused by the smaller velocity of the convective boundary of the envelope during the dredge-up compared to that of the boundary of the He-flash convection zone.

Fig. 5. The maximum abundance of the pocket after the fifth and eighth pulse of the sequence displayed over one pulse cycle respectively. The zero time is set to the flash peak respectively.

Fig. 6. The temperature and density at the location of the maximum abundance for the eighth TP corresponding to Fig. 5.

Fig. 5 shows that after more advanced pulses (like the eighth in this case), the formed at the hydrogen - carbon interface does indeed burn almost completely under radiative conditions Straniero et al. 1995. The which formed after earlier pulses (like the fifth in this case) burns only partly under radiative conditions. After early thermal pulses a certain fraction of the will be mixed into the pulse-driven convective zone of the succeeding TP where it is processed under convective conditions in the intershell. In Fig. 6 the density and temperature at the position of the pocket are given for the case of the eighth TP. This shows the conditions under which the neutron capture operates.

With our overshoot description and assumption about the efficiency () the layer in which the pocket forms is very thin (see Fig. 4). The total mass of contained in the pocket comprises only . Estimates of the s-process element distribution from the neutron source demand that the pocket contains in order to model the main s-process component Straniero et al. 1995. However, the numbers are not comparable at face value because the result of Straniero et al. is based on models without overshoot. Their hydrogen profile in the carbon rich region is not the same as our overshoot hydrogen profile. Also, the contribution and modification of the s-process element distribution from the neutron source reaction during the high-temperature phase of the flash should be different in our models (see Sect. 5.1 and 6). Finally, the additional pocket which forms due to the overshoot model must be considered. On one hand is known to be an important neutron poison because of its large cross section. On the other hand a large fraction of the neutrons lost by this reaction may be reproduced by . Moreover the additional amount of to be ingested into the next He-flash convection zone will also be converted into additional . All those processes will affect the s-process nucleosynthesis, which cannot be reduced to the mere amount of produced.

Preliminary tests with different overshoot efficiencies have shown that the shape of the abundance profiles in the region of the pocket is scaled with respect to the mass coordinate but conserved with respect to the abundance ratios. Intervals with certain mass ratios of, e.g. and hydrogen are just larger with a larger f. This means that a dedicated study of the s-process nucleosynthesis with a pocket according to our overshoot description should find that within a certain range of efficiencies the f-value does only determine the total amount of s-process elements in the dredged-up material but not its distribution. It should be investigated whether the functional form of an exponential velocity decay can reproduce a s-process distribution in compliance with the solar main component.

Apart from overshoot the process of rotationally induced mixing is a promising mechanism for the origin of the pocket Langer et al. 1999. It is surprising that the overall amount of found in the pocket due to rotation is almost identical to the amount found here with overshoot (a few ).

© European Southern Observatory (ESO) 2000

Online publication: August 23, 2000
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