3. Radiation forces
3.1. Flux around the star
Assuming that the potential of the outer parts of a rotating star may be represented using the Roche approximation, the radius of equipotential surfaces r is dependent on the angle to the pole , and is found by the solution of
where is the angular velocity, is the radius in the polar direction, and M is the mass contained within the equipotential surface. This has been solved for geometrically by Collins & Harrington (1966). Deep within the star the Roche approximation will be incorrect, although for the present discussion, it is adequate. Indeed the region in which it is possible to generate latitudinal metallic abundance anisotropies is found, a posteriori , to be the outer parts of the envelope and so , and Eq. 5 will be a good representation.
The flux of the star f is proportional to the local gravity (von Zeipel 1924):
Along the surface of any equipotential the local flux f is higher at the pole than the equator (gravity darkening). This difference between pole and equator is the root cause of the latitudinal variation of the metallic abundance. It is noted that this gravity darkening actually causes the meridional circulation, else energy would not be conserved in the star. However, the change in flux is not destroyed by the motions.
3.2. Radiation force on a trace element
The force on ions due to radiation is dependent on the gradient of the flux passing through that point, i.e. . To facilitate the calculation of the force along an equipotential the ratio of the forces in the radial and latitudinal directions is now calculated. In the outer parts of the star, the luminosity is a constant, and hence the flux varies with distance r from the centre of the star as . As the radiative force is proportional to , then the ratio of the latitudinal to radial components is
where s is the distance along an equipotential. Again, note this is only an approximation as the unit vectors and are not orthogonal. This ratio is shown in Fig. 1 for different rotation rates. As can be seen it is typically , and therefore, perhaps surprisingly, the radiative force around an equipotential is typically similar to the radial radiative acceleration.
Michaud et al. (1976) see also Alecian & Artru (1990) have discussed the radiative force due to lines on heavy elements. Their expressions, along with Eq. 7, lead to the latitudinal force on on a trace element (with a negligible gradient in the concentration of the ions) of
where and are the temperature and effective temperature in units of K respectively, r is the radius and is the stellar radius. The atomic parameters are enveloped in , and is the atomic mass of the element i.
With the inclusion of a non-zero concentration of ions the driving lines become saturated, and the line force is reduced from Eq. 8:
(Alecian 1985). The reference concentration is
where is the Rosseland mean opacity, is the number density of electrons and is an atomic parameter containing line broadening effects.
© European Southern Observatory (ESO) 1999
Online publication: December 4, 1998