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Astron. Astrophys. 341, 560-566 (1999)

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6. Timescales

Is is likely that a star may come into equilibrium? This clearly depends on the timescales over which the relevant processes act. Here the important timescales are (i) the diffusion time (time taken for the ions to diffuse round that star), (ii) the main-sequence lifetime of the star, (iii) the lifetime of each layer (if mass-loss is present).

6.1. Diffusion, stellar and mass-loss timescales

For an asymmetry to occur in the elemental distribution along an equipotential, the metals must diffuse a significant distance from their initial latitude to the equator. This therefore defines a drift timescale [FORMULA], dependent on latitude [FORMULA], which is that over which a latitudinal asymmetry in the metals will be generated:

[EQUATION]

The star only has a finite time in which to allow metals diffuse around its equipotentials. The main sequence lifetimes for these models are [FORMULA]yr and [FORMULA]yr for the 2[FORMULA] and 5[FORMULA] star respectively.

The mass-loss timescale within the envelope depends on the mass exterior to that point ([FORMULA]) and the mass loss rate [FORMULA]. If the mass-loss rate [FORMULA] is a function of azimuthal position [FORMULA] for rotating stars (for line-driven winds see Friend & Abbott 1982 and for radiation-driven dusty winds see Dorfi & Höfner 1996), then the mass-loss timescale also varies with [FORMULA]. However, as a first approach [FORMULA] is taken to be independent of [FORMULA]:

[EQUATION]

6.2. Where can the asymmetry exist?

In order for any significant metallic asymmetry to be generated, the diffusion timescale must be less than the mass-loss timescale. This ensures that the diffusion process has enough time to act before that layer of the star is lost in an outflow. The diffusion timescale must also be shorter than the typical main-sequence timescale else no significant asymmetry will build up over the lifetime of the star. Therefore, chemical abundance variations may be generated in regions for which the two inequalities

[EQUATION]

are satisfied.

The timescales are displayed in Fig. 4. The lines are the diffusion timescale around the star for [FORMULA], [FORMULA], [FORMULA] and [FORMULA] with the lower line corresponding to [FORMULA] decreasing upwards. The solid lines are for solar iron abundance, and the dotted lines are for [FORMULA] solar abundance. The oblique dashed lines are the mass loss timescale, corresponding to mass-loss rates of [FORMULA] [FORMULA] (lower), [FORMULA] [FORMULA], [FORMULA] [FORMULA] (upper). Fig. 4 should be interpreted as follows: the region to the right and lower than the mass-loss timescale lines may, if the diffusion velocity is larger than the diffusive velocity ([FORMULA] the meridional velocity [FORMULA] see Sect. 2) develop metallic abundance gradients around an equipotential. Therefore, the 5[FORMULA] model may have enough time to develop a significant abundance gradient across most of the meridian in the outer regions is the mass-loss rate is [FORMULA] [FORMULA].

[FIGURE] Fig. 4. The timescales in the star: the diffusion timescale around the star for [FORMULA], [FORMULA], [FORMULA] and [FORMULA] with the lower line corresponding to [FORMULA] decreasing upwards. The solid (dashed) lines correspond to solar ([FORMULA] solar) abundance of iron. The dotted lines are the mass-loss timescales corresponding to mass-loss rates of [FORMULA] [FORMULA] (lower ), [FORMULA] [FORMULA], [FORMULA] [FORMULA] (upper ).

Fig. 4 suggests that during the main sequence lives of the [FORMULA]solar abundance 2[FORMULA] and both 5[FORMULA] stars, the drift velocity due to the anisotropic flux in the star [FORMULA] is large enough that a latitudinal abundance gradient may be generated. However, this statement must be confined to stars which are rotating at significant fractions of break-up, and only applies to bands near to the equator.

Although this effect may produce metallic drift, it seems from Fig. 4 unlikely that the whole of an equipotential s will be involved. It appears that only regions close to the equator will be effected and consequently the metallic distribution will not come into equilibrium during the star's main-sequence lifetime. The distribution of metals is therefore difficult to calculate - indeed the dynamical modelling of the distribution is out of the scope of this paper and is flagged for further study.

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© European Southern Observatory (ESO) 1999

Online publication: December 4, 1998
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