## 6. Third dredge-up laws## 6.1. The dredge-up processThe important conclusion that the dredge-up rate is insensitive to
the extra-mixing parameters can be understood from the following
considerations. The AGB star is basically formed by an
-degenerate core of
0.10-0.15 surrounded by an
extended H-rich envelope. As dredge-up proceeds, the outermost
H-depleted layers of the core become part of the envelope and expand
(which translates into a negative gravothermal energy production in
those layers, see Fig. 8c). This results in an increase of the
potential energy of those layers engulfed in the convective envelope,
which must be supplied by the luminosity provided by the inner layers.
Let us estimate that dredge-up rate from simplified arguments. The potential energy of a mass located at the H-He discontinuity is given by where is the mass of the H-depleted core and its radius (i.e. during dredge-up). The energy necessary to lift the dredged-up material into the envelope is provided by the luminosity at the core edge, . The thermal time-scale for the dredge-up is then All quantities in this section are expressed in solar units and years. The dredge-up rate can then be estimated by Typical values for the star are (Fig. 7a), (Fig. 8d) and (Fig. 4b). Eq. 13 then gives /yr. This estimate of the dredge-up rate, while giving only an order of magnitude, convincingly supports the rate of /yr obtained in the full evolutionary models (Sect. 5). It is also consistent with the results obtained by Iben (1976), who estimated the dredge-up rate to be about /yr in a star with , and with those of Paczynski (1977), who found /yr in a star with . The question of why the dredge-up rate is independent of the extra-mixing parameters can further be understood by comparing the time-scale of the thermal readjustment of the envelope with the typical time-scale of convective bubbles to cross the H-He transition zone. The velocity of the convective bubbles as they approach the H-He discontinuity is of the order of /yr, which translates in terms of mass to about /yr (from Fig. 8d). This rate is much higher than the dredge-up rate of /yr established above. The deepening of the envelope into the H-depleted layers is thus primarily fixed by the thermal relaxation time-scale of the envelope, rather than by the speed of the convective bubbles penetrating into the H-depleted layers. ## 6.2. Dredge-up characteristicsEq. 13 reveals a formal dependence of the dredge-up rate on , and . Those three stellar parameters are not independent of each other.
Refsdal & Weigert (1970) and Kippenhahn (1981), for example, show
from homology considerations that most red giant star properties are
functions of the H-depleted core mass and radius. As the burning shell
advances, the structure of the intershell layers evolves, to first
approximation, like homologous transformations. Let us consider two
times The dredge-up rate at time
can then easily be evaluated from
that at time The homology transformations thus suggest a In standard AGB calculations (i.e. without dredge-up),
is a linear function of
(Eq. 8 for our
star). Eq. 14 then recovers the
classical linear - When the feedback of dredge-up on the structural evolution is taken
into account, the core radius is no more a linear relation of the core
mass, and the linear - The implications of the linear - relation on the core mass evolution are analyzed in Sect. 7.1. © European Southern Observatory (ESO) 1999 Online publication: March 18, 1999 |