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

5. Results

5.1. Velocity-dependent fractionation: slow and fast wind

By simplifying Eqs. (2) and (3) Peter (1996) found a simple formula describing the velocity-dependent fractionation. To achieve this, gravity and recombination were neglected (compare the ionization rate with in Table 1). Furthermore he assumed a subsonic flow and the interaction between the neutral and the ionized component of the trace gas to be small compared to the effects of the collisions with the main gas.

Using the boundary conditions as discussed in the second part of Sect. 4.2, one can solve for the total flux of a trace gas t,

where is the total trace gas density at the bottom. denotes the (mean) main gas velocity and is the ionization-diffusion speed as defined below in (6).

In an up-streaming situation () only the "+"-sign gives a physical solution, because otherwise the main and the trace gas would flow in opposite directions. For the same reason in a down-streaming situation () only the "-"-sign is allowed. Within the above approximations it follows from (4) that all the trace gases are faster than the main gas, . As the interest is here only on the trace gases and not on the absolute fractionation, this will be discussed in more detail in Sect. 6.1.

At the top of the layer the material is fully ionized: due to the effective Coulomb-collisions the species have equal velocities there. For this the fractionation (1) can be written as the quotient of the (constant) total fluxes and the total densities at the bottom of two trace gases, , see Peter (1996). It is easy to show that in an up- as well as in a down-streaming situation the above considerations lead to a fractionation given by

The most important factor entering this formula is the ionization-diffusion speed of the respective elements, defined as

where is the (reduced) collision frequency between the neutrals of the species j and neutral hydrogen. Generally it is defined as . By definition is always positive. The speeds are listed for a number of elements in Table 1.

In the case of small main gas velocities the fractionation following (5) is simply given by the quotient of the respective ionization diffusion speeds, as already calculated by Marsch et al. (1995).

As a rough estimate, the photon flux in the UV can be assumed to be constant, which results in the quotient of the ionization rates, , to be given by the quotient of the respective ionization cross sections, . As can be seen from Table 2 the quotient of the collision frequencies depends only weak on the masses of the respective elements and is mainly given by , where can be interpreted as the cross section for the elastic collisions, .

Thus the fractionation in the case of small velocities can be written as

This clearly shows that the processes of photoionization () and diffusion () are most relevant and that the fractionation is basically given by atomic parameters, namely the cross sections for photoionization and elastic collisions. It does not depend on the assumed geometry of the considered region (see also Sect. 3).

But as can be seen from (5) the result of Peter (1996) goes much further: it also includes a dependence on the main gas velocity in the chromosphere and can thus reproduce the fractionation not only in the slow wind, but also in the fast wind. As it was not done by Peter (1996) this velocity dependence will be discussed in more detail and especially compared to measurements in the following.

In Fig. 3 the resulting fractionation as following from (5) and the measurements in the solar wind as shown in Fig. 1 are plotted versus the first ionization potential for the elements listed in Table 1. For chromospheric velocities of 100 m/s and 400 m/s at a density of the theoretical results are matching the measurements in the slow and the fast wind quite well. Please note that here a slightly higher and more realistic density than in Peter (1996) was used. Therefore these results are comparable to the numeric studies in Sect. 5.3.

Fig. 3. Calculated fractionation for two different main gas velocities in the chromosphere matching the measurements in the slow and fast wind.

In the case of the fractionation Mg/O the velocity-dependence is shown in more detail in Fig. 4 for three different chromospheric densities. This shows that the fractionation is vanishing for higher velocities, in which case the transit time through the ionization-diffusion layer is to short for any process to establish a fractionation (see end of Sect. 3). It also shows that the dependence on density is only weak and that sometimes no fractionation can be expected in high speed events (compare observations of Bochsler et al. 1996)

Fig. 4. Velocity-dependence of the fractionation of Mg/O. The three curves show the theoretical dependence of for different densities (upper and right axis). The diamonds show the measured data points from Fig. 5, one for every day of the epoch (lower and left axis).

To prove this model result it is of interest whether the predicted velocity-dependence can also be found in the measurements in the solar wind. For this in Fig. 5 the relative abundance of Mg/O and the solar wind speed, represented by the -particle velocity, are shown for one epoch as measured by SWICS/Ulysses (Geiss et al. 1995). Now the relation between the speed and the abundance as derived from these measurements is over-plotted on the theoretical predictions in Fig. 4. According to Anders & Grevesse (1989) the relative abundance in the photosphere is (see Table 1). If this value is measured in the solar wind the fractionation is .

Fig. 5. Solar wind speed and the Mg/O abundance for one epoch as measured by SWICS/Ulysses (from Geiss et al. 1995).

For the curve of the theoretically predicted velocity-dependence of the fractionation is matching the observed data points quite well! Thus, assuming that the velocity structure in the chromosphere can be more or less directly mapped to the solar wind speed, the change of the fractionation with the solar wind speed becomes understandable.

5.2. Fractionation of the heavy noble gases

As outlined in Sect. 2.3 and Fig. 1 the fractionation of the noble gases (in the slow wind) does not fit into the two plateau structure of the other elements, but forms its own line. In Fig. 3 the theoretical results also for the noble gases as following from (5) are shown.

The three noble gases Xe, Kr and Ar are perfectly on a line, with Xe and Kr clearly enhanced to the high-FIP plateau. But the enhancement is much weaker than indicated by the observations shown in Fig. 1. Nevertheless this discrepancy should not be taken too seriously. The lunar samples from which the noble gas abundances were derived are quite old (Sect. 2.3). As the older ones (some years) show a stronger fractionation by 30% than the younger ones ( years) (see Wieler & Baur 1995), one might speculate that the actual fractionation of the noble gases is much weaker than indicated in Fig. 1 and might match the model results in Fig. 3. In every case the qualitative behaviour can be understood by the model.

The special behaviour of the noble gases is reasoned by their atomic properties, mainly their different ionization times. This is analogous to their chemical properties, which are different from the other high-FIPs, too.

5.3. Numerical models: absolute fractionation and hydrogen

To study the variation of the absolute abundances of the elements, e.g. the fractionation of hydrogen to oxygen, a more sophisticated numerical model has to be applied: the density of the trace gases has to be calculated in the background of the main gas. For this purpose (2) and (3) are solved numerically in a given hydrogen-proton background (see the first part of Sect. 4.2 for the boundary conditions).

In the present paper the main gas models of Peter & Marsch (1998) are serving as a background. They have calculated a variety of different hydrogen models which can be characterized by the typical mean velocity (taken at the maximum of the proton density) in the ionization-diffusion layer (see their Sect. 5.3). In their Fig. 3 they show the models with a density at the bottom of 7, 8 and with corresponding typical velocities of 890, 350 and 20 m/s. In the present study main gas models of Peter & Marsch (1998) with typical velocities of 367, 233, 155, 89 and 62 m/s respectively are used (see Fig. 6).

Fig. 6. Absolute fractionation as a function of altitude in the numerical models (Sect. 5.3and 5.4). The results are plotted for different background models characterized by the typical background velocity.

For the main gas models with (typical) velocities of m/s and m/s the fractionation of hydrogen to oxygen turns out to be and respectively at the top of the layer (see Fig. 6, left panel): using main gas models with the typical velocities in the range of those velocities giving the fractionation for the slow and fast wind in the analytical model (Sect. 5.1, Fig. 3), the numerical study gives the measured fractionation of hydrogen to oxygen in the slow and fast wind. (compare Sect. 2.2 and Fig. 3).

At these respective velocities in the numerical models as well the fractionation of the low- to high-FIPs for the slow and fast wind is reproduced; see the example of Mg to O (4 and 2). There is also no fractionation within the high-FIP plateau (N/O); see Fig. 6, middle (N/O) and right (Mg/O) panel, compare with Fig. 3.

In summary these numerical calculations confirm the results of the analytical model, but they can also explain the fractionation of hydrogen to oxygen within a consistent picture. One should note especially that the same (typical) main gas velocities in the numerical calculations and in the analytical model of Sect. 5.1 lead to the same fractionation. Thus all the effects included additionally in the numerical model, mainly gravitation and recombination, do not play an important role for the fractionation.

5.4. Polar plumes

In the analytical study the fractionation of magnesium can only reach a maximum value of about 6 (see Fig. 4). In contrast to this in the numerical model a much stronger fractionation can be found: for very low mean background velocities of only 50 m/s the fractionation Mg/O can reach a factor of 10 (see right panel of Fig. 6).

This corresponds well to the observations in polar plumes, e.g. by Widing & Feldman (1992) or Sheeley (1996), who found strong Mg/O fractionations of up to 10 or even 15. For this they conclude that the plumes cannot contribute a significant fraction to the solar wind mass flux, because of the different abundances. Thus the plumes have to be quasi-static, which means that the velocities in the chromosphere of the plumes are indeed very low.

Thus their results for the plumes, namely low velocity and strong fractionation, can be understood (also quantitatively) with the presented fractionation model.

© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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