tokens: and adds it to the envelope. We see that even with the large amount of energy that we add to the envelope as it expands to twice its original radius in a short time, the basic structure stays similar. For example, there is a density inversion in the outer part of the envelope and the convection speed stays below the sound speed. This implies that convection can carry even more energy than it carries now. Figure 3. Profiles of four quantities of our AGB stellar model in the outer layers of the envelope at t = 0 (the beginning of the plunge-in phase). The right edge of both panels is at the photosphere of the star. Upper panel: the temperature (black) and the density (magenta) in a logarithmic scale. Lower panel: the sound speed (blue) and the convective velocity (red). The orange, green and purple dotted vertical lines mark the zones of 90 per cent, 50 per cent and 10 per cent ionization of hydrogen, respectively. Figure 4. Like Fig. 3 but at from the beginning of the plunge-in phase. Figure 5. Like Fig. 3 but at the end of the plunge-in phase at . We next calculate the energy transport time from radius r to the surface of the star, R , by photon diffusion, as given in equation ( 1 ), and by convection, as given in equation ( 4 ). We present these energy transport times at the three evolutionary stages in Figs 6 – 8 , respectively. The convection velocity drops to very low values in the very outer part of the envelope, and so we calculate the convection time to the radius of 1 per cent hydrogen ionization fraction instead of the photosphere. This change is of no significance since in the outer region photon diffusion carries most of the energy. Figure 6. Energy transport times from radius r to the surface of the star at radius R for t = 0 when we start the energy injection. Since the convection velocity drops to very low values in the very outer part of the envelope, we calculate the convection time to the radius of 1 per cent hydrogen ionization fraction instead of the photosphere. This radius is also the right boundary of the graph. We present the photon diffusion time according to equation ( 1 ) by the red line, and the convection time according to equation ( 4 ) with the blue line. The diffusion time from R ion, 50 is . Note that the photon diffusion time becomes much shorter than the convection time in the outskirts of the envelope. Figure 7. Like Fig. 6 but at from the beginning of the plunge-in phase. At this time . Figure 8. Like Fig. 6 but at the end of the plunge-in phase at . At this time . From the figures we learn the following about the energy transport times in the inflated envelope. Convection can carry energy out from the entire hydrogen recombination zone very efficiently, as the convection transport time from 90 per cent hydrogen ionization fraction to the surface is shorter than the plunge-in time, in our case. The convection cells can move even faster, up to the sound speed that is larger than the convective speed of the
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