4. Discussion of basic assumptions
The analysis presented, in particular the concept of gravity darkening, is largely based on two assumptions: (i) Energy transport is due to and described by radiation diffusion, convection is disregarded. (ii) The angular velocity is constant over cylinders centered about the axis of rotation (, "pseudo-barotrope").
Assumption (i) limits the validity of our discussion to the region below the "photosurface". Provided the second assumption is valid, a proportionality between the total energy flux and the gradient of the effective potential similar to Eq. (4) still holds, if convection contributes to the energy transport. However, then the diffusion coefficient is not necessarily constant over a surface of constant effective potential, which leads to a modification of von Zeipel's law of gravity darkening. Its precise form will depend on the details of the description of convection (in the presence of rotation), for which a reliable theory is not available. On the other hand, to derive the Eddington limit and the condition for critical rotation the energy transport equation is used only at the "photosurface", where in general convection is not present. Thus the results of Sect. 2 remain unaffected, even if convection is reponsible for energy transport in deeper layers of the star.
Rather than the first restriction the condition of a pseudo-barotrope is crucial being necessary for the existence of the effective potential (defined as the sum of gravitational and centrifugal potential), which many further arguments are based on. On the other hand, consideration of pseudo-barotropes may be justified as a dependence on z of would lead to thermal instability (Fricke 1968, Goldreich & Schubert 1967). We note that the general derivation of the Eddington limit (Eq. 13) remains valid, even if .
The Eddington limit for rotating pseudo-barotropes depends on the quantity f which has been estimated to vary between 1 for zero rotation and for critical rotation. This estimate is based on evaluating the definition (7) for rigid rotation and assuming - similar to the approximation (5) - the gravitational potential to be spherically symmetric, i.e., we adopt a multipole expansion for the potential and keep only the monopole term. The latter is justified, if the mass contained in the stellar envelope is small compared to that of the stellar core (e.g., for evolved massive stars) and if the ratio of centrifugal and gravitational forces decreases from the surface to the center (e.g., for rigid rotation): Then the gravitational potential is determined by a spherical stellar core.
© European Southern Observatory (ESO) 1998
Online publication: September 30, 1998