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Astron. Astrophys. 342, 704-708 (1999)

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3. Discussion

The internal structure of the expanding models presented in the preceding section consists of:

  1. a completely degenerate isothermal inner core of radius [FORMULA] (in fact its white dwarf radius) containing almost all of [FORMULA];

  2. a non-degenerate isothermal outer core, located still below the interface, extending to radii [FORMULA] very much larger than [FORMULA], but containing very little mass;

  3. an interface between the He-core and the H-envelope, where the shell source is located; and

  4. an extended envelope above the interface.

There is, most importantly, the intermediate layer 2 which reduces the strong gravitational pull of the compact core upon the envelope by elevating the core-envelope interface.

Formally, layer 2 is needed if the temperature at the interface, [FORMULA], i.e. the temperature of the shell source, is prescribed. For a model consisting only of parts 1 and 4 only the quantities M and [FORMULA] may be prescribed, because of the universally fixed factor K in Eq. (7). Physically layer 2 is produced by the thermostatic action of the shell source. If it would not exist, the temperature [FORMULA] would have to be much higher than [FORMULA] to balance the pressure exerted by the envelope. Then the shell source would produce much more energy than could be transported by the envelope and the surplus energy would be used to lift [FORMULA] against gravity - until [FORMULA] is restored. We are not able to follow these processes within the present framework. However, it is very suggestive that exact numerical simulations yield models of the same structure as obtained here (see Sect. 3).

This intermediate layer can only serve its purpose if, for decreasing energy w in the outward direction, the thermal energy u does not decrease too much as well. This requirement is well satisfied, because no energy is released from the core and hence the layer is isothermal. This layer can even extend into the shell source region, where the temperature is kept constant by hydrogen burning. This behaviour is not realized in the present model, where the shell source region is nearly a "sheet", but is found in more realistic simulations. The formation of the intermediate isothermal layer may be the reason for an increasing polytropic index in stars evolving off the main sequence, as pointed out by Eggleton and Faulkner (1981).

Since the intermediate layer is below the shell source and since it must be non-degenerate, the shell source itself must always be located in the non-degenerate regime. Furthermore, the intermediate layer is an extension of the degenerate central part of the core; hence this part must be an almost complete polytrope [FORMULA] (i.e. a white dwarf inside the star). Both properties are well known from numerical simulations.

The hydrostatic arguments outlined so far may be visualized by the specific energies w and u as functions of the fractional mass [FORMULA] - see Fig. 2 - and for [FORMULA] and [FORMULA]. According to the virial theorem, the areas below these curves must be equal. The run of u is fixed by [FORMULA] at [FORMULA], which is very small compared to the contribution of the degenerate core. Hence, almost all of the internal energy of the whole star is concentrated in the core. Accordingly, the specific gravitational energy must also attain very small values at [FORMULA], otherwise the virial theorem would not be satisfied. Thus, a drop of w at [FORMULA] by more than a factor 10 becomes necessary. This drop is due to the intermediate layer, which, in this representation, appears almost like a discontinuity in radius. Fig. 2 also shows why the naive argument for the behaviour of R given in the introduction must fail. The main contribution to the energies are made by the core - and in fact it does show decreasing [FORMULA] for increasing [FORMULA], whereas R is completely decoupled from the energy balance. Because of the weakly bound envelope, R must be very large - regardless of the detailed structure of the envelope, which only fixes the exact value of the large radii.

[FIGURE] Fig. 2. Specific internal and gravitational energy [FORMULA] and [FORMULA] as functions of fractional mass [FORMULA] for the [FORMULA]-model with [FORMULA].

These specific energies w and u as functions of [FORMULA] may now be compared with those obtained from numerical simulations. For this purpose Stix (1997) ran a solar model from the main sequence to a state where, after [FORMULA]y, a He-core of [FORMULA] has developed - as in the model discussed in the preceding paragraph. The functions w and u taken from the numerical results are shown in Fig. 3. There is good agreement between Figs. 2 and 3, especially the behaviour of the specific gravitational energy. Detailed inspection reveals that the drop of w at [FORMULA] does not exclusively occur below the shell source but extends into it; the drop is also slightly smaller than in our model. This behaviour shows that the shell source itself is also contributing to the separation between core and envelope - a result possible only if the shell source is resolved (see Fig. 3). Again, the envelope is unimportant as far as energies are concerned. Taking its detailed structure into account, a total radius [FORMULA]cm is obtained, compared to [FORMULA]cm for an envelope with constant density in our model.

[FIGURE] Fig. 3. Specific internal and gravitational energy [FORMULA] and [FORMULA] as functions of fractional mass [FORMULA] for the sun with [FORMULA] as obtained by numerical simulation (Stix, 1997).

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999