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

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2. The model

As already outlined, we consider a polytropic envelope. Since the exact value of the polytropic index is of no great importance for the qualitative behaviour of the solutions, [FORMULA] was chosen for simplicity; an envelope of constant density can be treated analytically. Its structure is determined by

[EQUATION]

[EQUATION]

The procedure to solve theses equations is straight-forward and will not be repeated here. Only the approximate solution for extendend envelopes should be considered more closely. In this case, [FORMULA] in Eq. (3) may be replaced by [FORMULA] because in extended envelopes the variation of the gravity acceleration [FORMULA] is by far more due to the variation of r by orders of magnitude than to the moderate variation of [FORMULA]. Thus

[EQUATION]

is obtained, where the constant of integration is determined at the interface. Applying Eq. (5) to the outer boundary [FORMULA], [FORMULA] and expressing [FORMULA] by [FORMULA] and [FORMULA], we obtain

[EQUATION]

Eq. (6) relates the specific energies w and u at the interface - considered in the introduction - to the total radius R. Indeed, for [FORMULA], only a small radius and hence a small mass of the envelope is possible. As w approaches u, the core-envelope interface is raised, the binding energy of the envelope goes to zero, and there are no restrictions on R and [FORMULA] anymore.

The structure of the isothermal core, allowing for non-relativistic electron degeneracy, is again determined by Eqs. (3) and (4) and the corresponding equation of state

[EQUATION]

[EQUATION]

where [FORMULA],and [FORMULA] are the numbers of electrons, and nuclei per unit atomic weight respectively, and [FORMULA] and [FORMULA] are the Fermi-Dirac integrals. For [FORMULA], the equations of state for complete and for non-degeneracy are recovered. The former is

[EQUATION]

with negligible contribution of the nuclei. This is a polytropic relation for [FORMULA] and a fixed polytropic constant K; with which w in Eq. (1) has been estimated.

The numerical solutions for [FORMULA] consist of two fairly distinct parts: a completely degenerate central part and a non-degenerate peripheral region on top of it with almost no transition in between. The core and envelope are fitted to each other at the interface [FORMULA]. There, the shell source is located, assumed to be burning at [FORMULA]K. Hence, the chemical discontinuity and jump in density due to a He-core and a H-envelope

[EQUATION]

has been properly taken into account.

Solutions were obtained for three different total masses [FORMULA], 1.0, [FORMULA]. All three sequences start at [FORMULA] ("main sequence state") and eventually reach a phase of rapidly increasing radii - see Fig. 1, where radius R is plotted against [FORMULA].

[FIGURE] Fig. 1. Radius [FORMULA] as a function of [FORMULA] for models of three different masses [FORMULA], [FORMULA], [FORMULA]. Solid curves refer to expanding models; dotted and dashed parts belong to phases preceeding the formation of degenerate cores in Schoenberg-Chandrasekhar scenario.

Two different evolutionary phases are clearly discernible in Fig. 1. For small mass fractions of the core, there is only a moderate increase of the radius. This phase is terminated when the Schoenberg-Chandrasekhar limiting mass is reached. Evolution with increasing mass fraction only becomes possible after the formation of a degenerate core. The ensueing evolutionary phase is characterized by a dramatic increase of radius. It is this second phase which is relevant to our problem.

Obviously there is no quasistatic transition between the two phases. This is the consequence of an isothermal non-degenerate core (Schoenberg-Chandrasekhar scenario) during the first phase. According to numerical simulations, evolution towards shell source models proceeds along a somewhat different (strictly quasistatic) way. After some rearrangements on a thermal timescale (responsible for the Hertzsprung gap), the stars become structurally very similar to those of the above second phase (see Sect. 3 for comparison with a detailed numerical simulation).

Thus, sequences of expanding composite stellar models are obtained on an essentially hydrostatic basis for properly chosen equations of state. What they are teaching us physically will be discussed in the next section.

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
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