Stars as Laboratories for Fundamental Physics - MPP Theory Group

12 Chapter 1
One of the main difficulties at calculating the opacity is that heavier
elements, notably iron, are only partially ionized for typical conditions.
Resonant transitions of electrons between different bound states are
very important, an effect which causes stellar models to be rather sensitive
to the amount of “metals” (elements heavier than helium). The
construction of an opacity table is a major effort as it requires including
huge numbers of electronic energy levels. Widely used were the Los
Alamos and the Livermore Laboratory opacity tables.
Recently, the Livermore tables were systematically overhauled (Iglesias,
Rogers, and Wilson 1990; Iglesias and Rogers 1991a,b), resulting
in the new OPAL tables which since have become the standard in stellar
evolution calculations. The main differences to the previous tables
are at moderate temperatures so that no substantial changes in the
deep interior of stellar structures have occurred. However, envelope
phenomena are affected, notably convection near the surface and stellar
pulsations. A number of previous discrepancies between theory and
observations in this area have now disappeared.
If the equation of state, the energy generation rate, and the opacity
are known one can construct a stellar model for an assumed composition
profile by solving the stellar structure equations with suitable
boundary conditions. (It is not entirely trivial to define surface boundary
conditions because the star, strictly speaking, extends to infinity.
A crude approach is to take T = 0 and p = 0 at the photosphere.)
It may turn out, however, that in some locations this procedure yields
a temperature gradient which is so steep that the material becomes
unstable to convection—it “boils.”
An adequate treatment of convection is one of the main problems
of stellar evolution theory. A simplification occurs because convection
is extremely effective at transporting energy and so the temperature
gradient will adjust itself to a value very close to the “adiabatic gradient”
which marks the onset of the instability. At this almost fixed
temperature gradient a nearly arbitrary energy flux can be carried by
the medium. This approximation tends to be justified for regions in the
deep interior of stars while the “superadiabatic convection” found near
the surface requires a substantial refinement. One usually applies the
“mixing length theory” which contains one free parameter, the ratio between
the convective mixing length and the pressure scale height. This
parameter is empirically fixed by adjusting the radius of a calculated
solar model to the observed value.
Main-sequence stars like our Sun with M < ∼ M ⊙ have a radiative
interior with a convective surface which penetrates deeper with decreas-