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Astron. Astrophys. 335, 691-702 (1998)

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6. Absolute and relative velocities: the role of diffusion

6.1. Absolute velocities

The profiles of the absolute velocities of the neutrals are relatively flat. As can be seen in the example of oxygen (Fig. 7, right panel), the gradients of the scaled velocity at the bottom and the top of the layer are small, while a noticeable change can only be found inbetween, in the region where the material gets ionized (go back to end of Sect. 4.2 for an explanation of what "scaled" means).

[FIGURE] Fig. 7. Relative mean velocity of oxygen to hydrogen (left and middle) and scaled absolute velocity of oxygen (right) in a hydrogen background following the numerical model discussed in Sect. 5.3 for four different background models.

This compares well with the (simpler) model of Marsch et al. (1995) who found the velocity of the neutrals to be constant (see their Figs. 3 and 4). This result is of great importance for the analytical model of Peter (1996) and the analytical studies in the present paper (see second part of Sect. 4.2).

As pointed out in Sect. 5.1 after Eq. (4) in the analytical model all the trace gases have higher velocities at the bottom than the background. This at first seemingly strange behaviour is revised somewhat in the numerical models.

In the numerical models the high-FIP elements are entering the layer with smaller velocities than the main gas (see Fig. 7, left and middle panel for the example of oxygen). But still, the low-FIPs are somewhat faster at the bottom than the background (not shown). How can this be understood?

Fist one has to note that in a one-dimensional diffusive model this behaviour has to be expected, because the observations tell that e.g. magnesium is enriched in relation to hydrogen (Sect. 2 and Fig. 1). The only way to do that in a model like this is to have a higher velocity of Mg than of hydrogen at the bottom, compare (9) below.

Marsch et al. (1995) discussed in their Sect. 3 a pure diffusion model. In a static situation an enrichment of e.g. Mg to hydrogen can be achieved by a "climbing up" of Mg-ions (see their Fig. 2): the net effect will be a (small) upward velocity of Mg, even though the main gas has zero velocity. The fractionation results from the different ability of the different species to climb up - some might even fall down. In a situation where the main gas is flowing up, this means that some species might be even (a bit) faster than the main gas, which explains the above problem. This is simply a result of the diffusion.

6.2. Relative velocities

Diffusion, i.e. the fact that the different elements have different velocities, plays a crucial role in the presented model. Fractionation as defined in (1) can also be expressed in terms of the velocities at the bottom of the ionization-diffusion layer by using the definition of the total particle flux of an element j, [FORMULA], as given by the total density [FORMULA] and the mean velocity [FORMULA] of the respective element.


where the indices [FORMULA] and [FORMULA] indicate the respective values to be taken at the top and the bottom of the layer.

Because of the conservation of flux (2) the total particle flux of an element at the bottom has to be the same as at the top, [FORMULA]. Additionally at the top of the layer the effective Coulomb-collisions lead to equal velocities, [FORMULA] of the different elements, which are ionized there. For this by definition the fractionation (8) is given by the quotient of the respective velocities (in the neutral phase) at the bottom,


Thus to calculate the fractionation one has to determine the diffusive equilibrium at the bottom of the ionization-diffusion layer.

One idea that might arise is the following: as the ionization layer is thin compared with the gravity scale height in the chromosphere, the medium has to be homogeneous at the bottom. In this case one assumes , that "far away", i.e. some ionization lengths from the ionization-diffusion layer, no diffusion can occur. Following (9) this would mean that fractionation is excluded.

But in a more general case, if homogeneity is not taken for granted and the diffusive equilibrium is calculated, it turns out that even at a distance of some ionization lengths a small but not-vanishing diffusion occurs! These (diffusion) velocities are small, but as the quotient of the velocities determines the fractionation, the latter one can be relatively large.

In Fig. 7 some aspects of the diffusion as following from the numerical models as discussed in Sect. 5.3 are shown for the example of oxygen in a hydrogen background. In the left panel it can be seen that the absolute value of the diffusion velocity ([FORMULA]) is small. At the bottom it is of the order of only 10 m/s. In the ionized phase a diffusive equilibrium is reached, were the diffusion velocity is finite (40 m/s) but small compared to the background speed, which can be of the order of some km/s. This finite diffusion velocity might be compared to the observed difference speeds in the solar wind of the order of some 10 km/s (Grünwald 1996).

But if the diffusion velocity is compared to the hydrogen velocity (middle panel, [FORMULA]) it turns out that in deed at the top the velocities are nearly equal, while at the bottom, i.e. in the neutral phase, the velocities can differ significantly, e.g. in the case of O and H up to 50%.

The conclusion from this must be that one cannot simply assume homogeneity at the bottom and thus rule out the fractionation. On the contrary, one has to calculate the diffusive equilibrium properly. No diffusion is a good assumption at the top in the ionized phase, where the very effective Coulomb-collisions are of importance (see Table 2 for a comparison of the different collisional rates). But at the bottom in the neutral phase the low efficiency of the neutral-neutral and neutral-ion collisions enables a small but not-vanishing diffusion, leading to a significant fractionation.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998