ON PARTIAL MIXING ZONES IN HORIZONTAL-BRANCH STELLAR CORES

We present a semianalytical treatment describing some aspects of the c ore evolution in horizontal-branch (HB) stars, In particular, we deriv e and discuss a criterion for the existence of a partial mixing zone ( i.e., a region with a composition gradient) in a general stellar model , using a method similar to that of Naur & Osterbrock (1953). The resu lt we derive implies that, if temperature gradient in the layers above the convective core is required to the equal to (at most) the adiabat ic gradient, a zone with a composition gradient must result, however t he material mixes. Further, the point where partial mixing is initiate d is approximately fixed in mass throughout HB evolution; in the early stages of HB evolution the outer extent of the convective core lies w ell within this point but reaches it later in the evolution, initiatin g a partially mixed zone. We have used our result to compute stellar m odels according to the ''canonical semiconvection prescription,'' whic h assumes that the region above the fully convective core will have a composition such that del = del(r) = del(ad). We then consider the con sequences of this assumption for behavior of the core close to central helium exhaustion. We find that with the degree of mixing that is imp lied by the constraint on the temperature gradient, the opacity in the core tends to a limit late in the evolution (Y(c) approximately 0.1), and then declines. The central luminosity has similar behavior as the fuel supply shrinks. Numerical methods that depend precisely on chang es in central opacity with evolution will inevitably become ill-behave d as the opacity itself becomes less sensitive to that evolution. As a result of these decreases in central opacity and flux, the convective region has a strong tendency to shrink, rendering the partially mixed zone radiative. The controversial ''breathing pulse'' phenomenon can operate only if a massive mixing event is able to trigger an expansion in the core and reverse the evolutionary direction. We discuss this p ossibility and note also that the formulae we derive show a direct rel ationship between the thermal energy generated in the core and the abi lity to produce breathing pulses; looking for solutions in thermal equ ilibrium is both predicted and found to suppress them altogether. We c ontrast the possibilities for this late stage of evolution with the '' shell flash'' phenomenon of the AGB. Assuming that the mean evolutiona ry trend is followed, our program is able to follow the HB evolution t hrough central helium exhaustion to the lower asymptotic giant branch (AGB) with ease. The sequences we have calculated give values of R2 = N(AGB)/N(HB) in the range 0.14 to 0.17, the small variation being depe ndent on the total mass of the model sequence.