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

4. Model ingredients

4.1. Multi-fluid transport equations

For the description of the fractionation for every trace gas a set of transport equations (see e.g. Schunk 1975) has to be solved. Here the same formulation as in the background model of Peter & Marsch (1998) will be used. The equations of continuity and momentum for a species j for a one-dimensional case along the coordinate s parallel to the magnetic field and for a single temperature read

Here and are the particle density and the s-component of the velocity of the species j. The prime, ´, denotes the derivative with respect to s. The sources and sinks for the particle flux density are due to ionization and recombination with the respective rates . The sound speed is given by , with Boltzmann's constant , temperature T and atomic mass m. denotes the charge number of the species j, and g is the gravitational acceleration (pointing to the negative s-direction). The exchange of momentum between the species is due to ionization/recombination and elastic collisions with the respective rates and . The latter ones obey .

Only collisions of the respective trace gas (neutral and ionized components) with the background, i.e. hydrogen, are considered, while the interaction between the different trace gases is neglected: every trace gas is treated as a collection of test particles.

In the case of the trace gases, no energy equation has to be solved because the thermodynamics were solved together with the main gas (see Peter & Marsch 1998).

As in the case of the background model of Peter & Marsch (1998) the ionization rate is assumed to decay as where is the (mean) ionization cross section and the density of the neutrals. But one should keep in mind that the ionization rate for the minor elements is nearly constant, which is because of their low abundance.

That the radiation in the wavelength bands, which are relevant for the ionization of the minors, is nearly constant, can also be seen from the work of Vernazza et al. (1981), their Fig. 36. Their (static) semi-empirical radiative transport model includes also the radiation from the under-laying photosphere and lower chromosphere. Thus the present assumption that the radiation field and thus the ionization rate are more or less constant with altitude can at least be funded by their radiative transport model.

To solve the equations numerically, routines of the NAG library were used, as described in the background model of Peter & Marsch (1998).

4.2. Boundary conditions

For the numerical studies (in Sect. 5.3and 5.4)

The velocities of the neutral and ionized component are assumed to be equal at the top as well as at the bottom of the layer. This is motivated by the fact, that at the bottom the material is mostly neutral, at the top mostly ionized. Thus ionization and recombination cause the components to have the same speed.

At the bottom of the ionization-diffusion layer the abundance of an element and its degree of ionization (following the Saha equilibrium) are given. Finally the ionization rate at the top is set to the value as calculated by von Steiger & Geiss (1989), see their Table 2. These are reprinted in Table 1.

It should be noted, that no absolute value of the velocity or any diffusion velocity between two elements is presumed. The diffusion at the bottom of the ionization-diffusion layer results from the model (see Sect. 6.2).

For the analytical studies (in Sect. 5.1and 5.2)

For the analytical model, which follows Peter (1996), the following boundary conditions are chosen: all the material at the bottom is neutral, i.e. the density and the flux of the ions vanishes at the bottom. This is different from the boundary conditions for the numerical models as discussed above. In a way this is a stronger version of the above conditions, where the degree of ionization (non-zero above) is forced to be zero at the bottom.

Because the ionization rate is assumed to be constant in the analytical studies, there is no boundary condition needed for the ionization rate.

To solve for the fractionation Peter (1996) made an assumption, which can be considered as an additional boundary condition. This is one of the most critical points in that paper. It is assumed that there is no acceleration for the neutrals at the bottom, . This ad hoc assumption seems reasonable from a physical point of view - why should the neutrals be accelerated at the bottom?

But as every stationary model is determined largely by the given boundary conditions, this assumption has to be proven critically. This can be done with the numerical studies presented in Sect. 5.3, where no assumption was made on the value of the velocity or the acceleration at the bottom (compare above). As an example of a trace gas in the right panel of Fig. 7 the scaled absolute velocity of neutral oxygen as resulting from the numerical model is shown (see also Sect. 6.1).

The effect of the gravitational stratification leads to a velocity increasing with altitude (see Peter & Marsch 1998, their Sect. 5). To eliminate this, is scaled by the normalized total oxygen density, . The results as shown in Fig. 7 prove that the gradient of the velocity of the neutrals is in-deed small at the bottom, i.e. that the neutrals are flowing "smoothly" into the considered layer.

The numerical studies of the present paper, which are using much "weaker" boundary conditions, justify the critical assumption in the work of Peter (1996) and in the analytical studies in the present paper.

4.3. Atomic data

In Table 1 the used atomic data as collected from different references are shown. For every element the atomic mass , the photospheric abundance in a logarithmic scale , the first ionization potential FIP, the ionization time , the sum of radii of the colliding particles in the neutral phase (at K), the atomic polarizability , the recombination rate (at K) and the ionization diffusion speed as calculated from (6) for K and are given. The abundances are taken from Anders & Grevesse (1989) and are defined as , where is the total particle density of the respective trace gas and the one of hydrogen. , and are taken from von Steiger & Geiss (1989) and Marsch et al. (1995), from Landini & Monsignori Fossi (1990). Both, and depend weakly on temperature; see the above references for more details. The ionization rates are given by the inverse ionization times, .

In Table 2 the elastic collision frequencies for a trace gas in a hydrogen background are collected. The values are taken from von Steiger & Geiss (1989). denotes the charge number and A, and are as given in Table 1.

Table 2. Elastic collision frequencies of the minor species.

In both tables and are the temperature in K and the density in respectively.

Special care has to be taken in the case of oxygen: because O and H have nearly the same first ionization potential the charge exchange between both is very efficient. Following Arnaud & Rothenflug (1985) the ionization and recombination rate for oxygen due to this process is given at K by and . Both rates depend slightly on the temperature (see the original paper). This process is some orders of magnitude more efficient than the photoionization (compare Table 1).

© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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