The immediate post main sequence state of stars is one of the best known phases of stellar evolution, observationally as well as theoretically. After using up their central hydrogen content by nuclear burning, stars produce their luminosity in a thin shell around a continually growing helium core. At this stage, they leave the main sequence of the Hertzsprung-Russell diagram and evolve towards higher luminosities and larger radii.
Which physical effects are responsible for this expansion? Simple physical reasoning based on the virial theorem would rather suggest decreasing radii as a consequence of increasing internal energy: due to the outward burning shell source a growing part of the star is at temperatures of K or higher. Results of numerical simulations show that this reasoning is wrong. What causes the stars to expand? There have been several suggestions in the literature:
One of the earliest explanations is due to Hoeppner and Weigert (1973). Experimenting with their stellar evolution code, they found large expansion after artificially placing a strong point mass in the center of a main sequence star. This result was further investigated by Weiss (1983). Thus, a strong gravitational field is needed for expansion.
Eggleton and Faulkner (1981) compared the structure of expanding stars with polytropes of index , which, at finite masses, have infinite radii. Indeed they found an increasing local polytropic index in those stars. Yahil and van der Horn (1985) investigated composite models and suggested a nearly isothermal structure just below the shell source as a cause for expanding envelopes - thus corroborating the reasoning of Eggleton and Faulkner (1981).
Applegate (1988) inspected the behaviour of the shell source: If the rate of energy generation exceeds a critical amount determined by the envelope the latter starts to expand. Thus the temperature of hydrogen burning is brought into play.
Finally, Whitworth (1989) investigated the overall problem: he considered series expansions around the main sequence state using parametrized properties of stellar matter. From his elegant but very involved formalism he concluded that it must be a complicated interaction between core and envelope which leads to expansion.
Is there really no simple physical answer? In what follows we study - very much in the line of Eggleton and Faulkner (1981) and Yahil and van der Horn (1985) - simple hydrostatic configurations with masses . The cores of these stars start to contract after using up their nuclear energy sources and soon reach a state of electron degeneracy. Then, the contraction stops and the cores become isothermal - due both to the lack of gravitational energy release and high thermal conductivity of degenerate electrons.
In constructing a composite configuration consisting of a fully degenerate isothermal core and a polytropic envelope with the hydrogen burning shell source at the interface, the following remarkable difficulty arises. The specific gravitational energy at the interface due to a polytropic core of and is
The specific gravitational energy in excess of the specific internal energy strongly limits the mass of the envelope, which may be estimated as follows. The pressure at the interface exerted by an envelope of mass is approximately
According to the equation of state, this pressure is determined by the thermal conditions at the interface, and , and hence the following inequality must be satisfied:
Substituting the specific energies from Eqs. (1) and (2), we see that an envelope of only can be in hydrostatic balance with the compact core - a result well-known from the theory of white dwarfs.
This difficulty can be avoided either by increasing u or by decreasing w (and hence increasing ). Since is fixed by the conditions of hydrogen burning, the only possibility to increase to the values needed for our composite model - - 10, say - is to lift the interface to appreciably larger distances from the center. The envelope would then be considerably less bound to the core and it would expand.
How can a star manage to lift its core-envelope interface? To find an answer, the following composite hydrostatic model is considered: an isothermal core, in which the transition from non to complete degeneracy is taken exactly into account; and a hydrogen envelope of polytropic structure fitted to the core. Such a model is uniquely determined by prescribing the total mass, M, the (hydrogen burning) temperature at the interface, , and the core mass, . By continuously increasing at fixed M, an "evolutionary sequence" of the model star is obtained. This model is presented in Sect. 2 and its solutions are discussed in Sect. 3. Here it is especially helpful to consider the specific energies w and u as functions of the fractional mass for one of the expanding models; the functions w and u are interpreted in the light of the virial theorem and they are compared with those of an evolved solar model obtained by numerical simulation.
© European Southern Observatory (ESO) 1999
Online publication: February 23, 1999