2. On the convective boundaries in AGB stars
The criterion used in most stellar evolution calculations - and in all AGB models published to date - to delimit convective regions is that of Schwarzschild. According to that criterion, a layer is convective if , and radiative if . is the adiabatic temperature gradient, and the radiative temperature gradient given by
with T, L, , and m being, respectively, the temperature, luminosity, Rosseland mean opacity, and mass contained in the sphere interior to the radius r (other symbols having their usual meanings). The layer where defines the convection boundary, and is called the Schwarzschild layer .
Let us suppose for now that mixing of material inside a star can only occur through convection, and let us imagine the star as being composed of a finite number of shells located at increasing mass coordinates m. Each shell is characterized by a temperature , density , luminosity and chemical composition . All these quantities vary with time t. The Schwarzschild criterion tests locally the stability of each shell against convection by comparing derived from Eq. 1 with at the given shell independently of the values of those quantities in adjacent shells. Mixing of matter between two adjacent shells is allowed to operate only if both of them are found to be convective. Alternatively, the chemical composition of a radiative shell is not altered with time, according to the Schwarzschild criterion, unless and until its radiative gradient exceeds its adiabatic gradient. This could result from a change with time of T, and/or L (we neglect here changes due to nuclear burning). Applied in this strict definition, the local Schwarzschild criterion is called hereafter the strict Schwarzschild criterion . We note that the shell-representation of a star used in the above discussion corresponds to the numerical zoning performed during the computation of stellar models (by the finite difference method). Thus, stellar model calculations using the Schwarzschild criterion without any extra-mixing essentially use the strict Schwarzschild criterion. For this reason, the `strict Schwarzschild criterion' is equivalently called, hereafter, the Schwarzschild criterion without extra-mixing .
Convection, however, is essentially non-local. Convective bubbles reaching the convection boundary do penetrate, on a whatever small distance, into the radiative layers. The chemical composition of a radiative layer adjacent to a convective zone may thus, in reality, be modified by this convective penetration. For it to occur in models using the Schwarzschild criterion, however, an extra-mixing procedure must be specified in the code (whether purely numerically or according to a physical prescription). In most cases, this extra-mixing does not affect the stability of the radiative layers adjacent to the convective zone. A short discussion of these cases is presented in Sect. 2.1. In some cases, however, this extra-mixing does modify the stability of those radiative layers. Those are discussed in Sect. 2.2. They characterize, in particular, the dredge-up phase in AGB stars. A third situation may be encountered in model stars where both of the above cases are present simultaneously. This is discussed in Sect. 2.3.
2.1. Case a: Stable Schwarzschild boundary
The most common case is encountered when the convection boundary lies in a region of smooth profile, such as in chemically homogeneous regions as illustrated in Fig. 1a. This case characterizes, in particular, the bottom layers of the convective envelope in AGB stars during most of their pulse and interpulse phases.
It is easily seen that an extra-mixing of material from the convective envelope into the radiative layers does not alter the stability of those layers against convection. The Schwarzschild layer is thus said to be stable in the sense that an extra-mixing of matter from the convective zone into the underlying radiative layers does not, to first order, alter the location of the Schwarzschild layer.
Of course, the Schwarzschild layer does not necessarily locate correctly the boundary of the convective zone, since a penetration of the convective bubbles into the radiative layers (i.e. overshooting) may occur, the extent of which is still a question of debate. But such an overshooting does not, to first approximation, have any crucial consequence on the location of the Schwarzschild layer.
2.2. Case b: Unstable Schwarzschild boundary
During a 3DUP, the H-rich convective envelope penetrates into the H-depleted radiative layers, and gives rise to a discontinuous chemical profile (dotted line in Fig. 1b). As a result, a concomitant discontinuity develops in the profile (solid line in Fig. 1b; see also Iben 1976, Paczynski 1977, Frost & Lattanzio 1996). A similar situation occurs during the core helium burning phase of most stars when the He-depleted convective core grows into the overlying He-rich layers.
In models using the Schwarzschild criterion without extra-mixing, the border of the convective zone is automatically located at the position of the hydrogen abundance discontinuity. As a result, no alteration of any chemical abundance is expected to occur in the radiative layers. The convection boundary remains located at this position as long as the value of on the radiative side of the discontinuity is below . Evolutionary AGB model calculations performed without extra-mixing confirm that no penetration of the convective envelope occurs beyond that point of discontinuity (see Sect. 4). It should be stressed that the Schwarzschild layer is undefined in those models, a situation which is clearly unphysical.
In reality, some matter does travel across the discontinuity. This results from the finite velocity of the bubbles at the convection boundary, since is positive. Models should thus include some sort of extra-mixing. The consequences of such a penetration can readily be evaluated: mixing of matter from the envelope into the radiative regions enriches the latter region with hydrogen and renders them unstable against convection. This in turn leads to further penetration of the envelope. The situation is thus unstable , in the sense that an extra-mixing of matter from the convective zone into the underlying layers does alter the location of the convective boundary.
The challenge facing AGB modelers is to provide a reliable description of this envelope penetration with simple (i.e. computer feasible) convection prescription. The analysis presented in Sect. 5 actually reveals that the penetration of the envelope into the H-depleted layers is rather insensitive to the extra-mixing parameters such as its extent or mixing efficiency. That main conclusion to be obtained in Sect. 5 is most welcomed since not much - not to say nothing - is known on the extra-mixing characteristics in AGB stars. Sect. 5 also shows that the use of a diffusive algorithm for the mixing of chemical elements enables to avoid the development of a discontinuity in the models, and to properly define the Schwarzschild layer.
2.3. Case c: Existence of a minimum in the profile
In massive AGB stars, the profile can display a minimum close to the lower boundary of the envelope, as illustrated in Fig. 1c. A situation can occur where two convective zones develop, separated by a radiative region around that minimum. This case combines the features of both previous cases: the outer convective zone has a smooth profile as in case a , while the inner one presents a discontinuity as in case b .
In models using the Schwarzschild criterion with no extra-mixing, the inner convective zone does not penetrate into the H-depleted regions due to the discontinuity (see Sect. 2.2). The outer zone, on the other hand, is free to deepen until it reaches the inner convective zone. The two convective zones then merge together, and the situation resumes to case b .
When extra-mixing is allowed to operate, a specific scheme is expected to develop. Because of the small amount of mass involved in the inner convective zone (), any extra-mixing into the He-rich layers reduces the hydrogen content of that convective zone, and renders its outermost layers stable. Those outermost layers are then progressively excluded from the convective zone as they reach convective neutrality, and a semi-convective zone is expected to develop 1. Eventually, the main convective envelope penetrates into that semi-convective zone, and the case resumes to case b .
It is clear from the discussion in this section that models using the Schwarzschild criterion with no extra-mixing are inadequate to describe the 3DUP phenomenon. There is, however, a great deal of confusion in the literature on the third dredge-up predictions, and in particular on the role of extra-mixing. For the purpose of clarifying some of those issues, AGB models have been computed both with and without extra-mixing. Particular attention has been devoted to ensure that the code comply with the definitions given at the beginning of this section on the location of convection boundaries. The description and results of those calculations are presented in Sects. 3 to 5.
© European Southern Observatory (ESO) 1999
Online publication: March 18, 1999