2. Critical rotation and Eddington luminosity
General assumptions and approximations are largely identical with those adopted by Langer (1997, 1998). In particular, we restrict our analysis to rigidly rotating configurations. Following Langer, the -limit is derived by considering the momentum balance in the equatorial plane at the stellar surface. Ignoring the contribution of gas pressure and describing the radiative acceleration within the diffusion approximation we are left with:
In Eq. (1) which is almost identical with Eq. (5) of Langer (1997) denote the gravitational constant, the speed of light and the opacity, respectively. and are mass, radius, angular velocity and the radial component of the energy flux of the star.
being the Eddington factor. Langer emphasizes that the critical rotation velocity (2) vanishes for .
Crucial in the derivation sketched is the assumption of a spherically symmetric radiation field by which the effect of gravity darkening (von Zeipel 1924) is discarded. The latter means that in any pseudo-barotrope, in particular in the rigidly rotating models considered, the energy flux on the stellar surface is proportional to the gradient of the effective potential (see, e.g., Tassoul 1978, Sect. 7.2). Thus - rather than by - is given by:
denote (cylindrical) radial coordinate, density, temperature and radiation pressure constant and the gradient of the effective potential is in the equatorial plane of the stellar envelope to first approximation given by:
Moreover, in a pseudo-barotropic star the meridional circulation - developing due to von Zeipel's paradox - transports no net energy over a level surface defined by constant (Schwarzschild 1958, Roxburgh et al. 1965, Tassoul 1978). Therefore a luminosity L constant in the envelope may be defined as the integral of the energy flux over a surface determined by constant which implies the relation:
rather than with the critical rotation speed (2) ("-limit") postulated by Langer. Thus the existence of an "-limit" has to be considered as an artifact based on disregarding gravity darkening. In a correct treatment the Eddington factor has no effect on the condition for critical rotation.
where f varies between 1 (zero rotation) and (critical rotation) obviously corresponds to the Eddington limit for rotating pseudo-barotropes.
In principle, Eq. (12) may also be interpreted as condition for a "second" critical rotation rate depending on the Eddington factor. Thus, the only basis for the definition of an "-limit" would be the solution of Eq. (12) for the Eddington-limit of rotating configurations. An analysis using the condition for critical rotation modified by radiative forces without taking into account gravity darkening is, however, not selfconsistent and therefore not admissible.
Rather than using Eq. (12) for the definition of an -limit - which would then be conceptionally different from that derived by Langer - we adopt the more conventional point of view and interpret (12) as the dependence on rotation of the critical Eddington factor. Compared to the case without rotation it can be reduced by up to 40 per cent for critical rotation. However, depending on the rotation law, and super-Eddington luminosities are also possible. Such cases have been considered in connection with accretion tori by Abramowicz et al. (1980). This paper also contains a generally valid derivation of the Eddington limit of rotating objects which yields for the critical Eddington factor (where a flux-weighted average over the stellar surface is to be used for the opacity):
The r.h.s. of Eq. (13) can be shown to be identical with the factor f occurring in (12). We emphasize that - contrary to the -limit (2) - both these criteria for the critical Eddington factor rely on globally defined quantities. In particular, the dependence on rotation involves an integral of a function of the angular velocity over the entire configuration. While the interpretation of (13) and (12) as conditions for the critical Eddington factor with given angular velocity is straightforward their global nature in general excludes the opposite procedure: Solving, e.g., (13) for a critical rotation rate with given Eddington factor, i.e., inverting the integral involving the rotation rate, is ambiguous. This can be done only by adopting further restrictive assumptions on the rotation law thus supporting the point of view that (13) and (12) have to be understood as critical conditions for the Eddington factor with given rotation rate but not vice versa.
Eq. (13) clearly demonstrates that the critical Eddington factor sensitively depends on the rotation law in a global rather than a local way. For suitably chosen the critical Eddington factor may even exceed unity.
The discussion presented demonstrates that in any case (also for zero rotation rate) the Eddington factor is to be interpreted in a global way. Considering as a local quantity does not imply any limit for the object: Should exceed unity in some region of the envelope this only means (at least for a nonrotating star) that a positive gas pressure gradient is necessary to guarantee equilibrium. As a consequence, the density gradient is also positive there and the region is convectively unstable. In an attempt to rephrase earlier comments on the connection between convection, density inversions and super Eddington luminosities by Glatzel & Kiriakidis (1993) this point was discussed by Langer (1997), however, by erroneously inverting the implication. Langer argues that strongly nonadiabatic convection may lead to density inversions which imply a positive gas pressure gradient. (The latter is equivalent to .) This line of reasoning is not correct: Even if convection is sufficiently inefficient to provide a density inversion this does not necessarily imply gas pressure inversion and thus -locally- super Eddington luminosities. E.g., common Cepheid models having small Eddington factors exhibit density inversions. The inverse implication is valid: If locally prevails, a gas pressure inversion (implying a density inversion) is necessary for equilibrium, which leads to negative entropy gradients indicating convective instability. If convection is already taken into account in the model, this means that convective transport of energy is inefficient and corresponds to a strongly nonadiabatic environment. For a thorough discussion of this issue we refer the reader to Glatzel & Kiriakidis (1993).
Throughout any of Langer's papers mentioned both the Eddington limit and the condition for critical rotation (also the -limit) are referred to as "stability" limits. For clarification, we would like to point out that none of the limits and conditions considered here are based on a stability analysis. Rather they are derived by considering equilibria and correspond to a limit for the existence of an equilibrium solution.
© European Southern Observatory (ESO) 1998
Online publication: September 30, 1998