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Astron. Astrophys. 335, 303-308 (1998)

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3. Results

The minor (species 2) ions we take to be [FORMULA] (J) or [FORMULA] (J) (in the latter case the major species is the proton-alpha mixture with u, n and T). As explained in Appendix A, the temperature profiles are obtained from those derived in the one-ion model of McKenzie et al. (1997) by simple re-scaling. For the ion temperatures this means assuming T, T for both parallel and perpendicular cases (T is the plasma temperature and m the effective ion mass used in the one-ion model). K represents additional heating of the heavy ion component; in our calculations we use K, 1.2, 1.4, 1.6, 1.8 and 2.0. The resulting profiles (with K for T curves) are presented in Fig. 1. The limits of the calculation region are 1 [FORMULA] and 1 A.U. The general agreement of the resulting proton flow speed with observations is essentially assured by the input temperature profile for the protons.

[FIGURE] Fig. 1. Temperature profiles T, T, T (solid lines), T (dashed line) and T (dotted line). For the ions both perpendicular (upper lines) and parallel (lower lines) temperatures are shown.

Fig. 2 shows the flow speeds corresponding to the critical solution for [FORMULA] case at K. On the same plot the combinations of the corresponding thermal and wave motion speeds w responsible for broadening of the spectral lines along the line of sight are shown to compare with the UVCS/SOHO data (Cranmer et al. 1997, Kohl et al. 1997). The observed w start to grow from [FORMULA] [FORMULA] where w km/s, w km/s and keep increasing up to the outer limit [FORMULA] [FORMULA] of the observation region, reaching 250 km/s for hydrogen and 600 km/s for [FORMULA]. The [FORMULA] flow speed (obtained from Doppler dimming) also starts increasing at [FORMULA] [FORMULA]. The different behaviour of w and the flow speeds in our model follows from the assumed form of the heating source (Eq. (A1)), which is concentrated within [FORMULA] [FORMULA] from the coronal base. Note that the value of T as inferred from the UVCS/SOHO data can be as high as 90 at [FORMULA] [FORMULA].

[FIGURE] Fig. 2. The flow speeds u (solid line) and u (dashed line) for the ([FORMULA]) system at K. Also shown is the combination of the ion thermal (v) and wave (v) speeds w for [FORMULA] (dashed-dotted line) and [FORMULA] (dotted line).

The relative flow speeds u are illustrated in Fig. 3. Note that at given K the [FORMULA] ions are lagging after the [FORMULA]. This is a consequence of higher value of Z of the latter. The Alfven speed is exceeded only for the cases of high excess heating (K=1.8, 2.0) at large distance (50 and 100 [FORMULA], respectively).

[FIGURE] Fig. 3. The relative speed u for ([FORMULA]) (solid line) and ([FORMULA]) (dotted). The values of the extra heating factor K are (from below) 1.0, 1.2, 1.4, 1.6, 1.8, 2.0. The dashed line is the Alfven speed.

The associated changes in density ratios are shown in Fig. 4. Because the heavy ions thermal speeds are high, we cannot expect the mechanism of proton flux regulation (Leer et al., 1992) to apply in this case. Neither is there a significant rise of the alpha density in inner corona (Burgi, 1992).

[FIGURE] Fig. 4. The density ratio n for K (solid line), K (dashed line) and K (dotted line).

Fig. 5 illustrates the effects of the wave force and Coulomb friction terms. The wave force contribution to acceleration is seen to be significant. We note that despite the dissipation due to the second term in Q (Eq. (A1)) the relative amplitude of the Alfven waves is above 1 as far as 100 [FORMULA]. The maximum value is above 2.0 at 10 [FORMULA].

[FIGURE] Fig. 5. The proton and alpha (upper line) flow speeds at K compared to the cases with no wave force (dashed lines) or no Coulomb friction (dotted lines).

Figs. 6 and 7 show the (u) plane projection of the critical lines and the critical solution (for clarity only the part u km/s is shown). The slow mode critical curve has two branches, only one of which being crossed by the critical solution. When the minor ion temperature is increased the crossing point moves to another branch. The two critical (crossing) points are close to each other at [FORMULA]. See Appendix B for a description of the method of finding the critical solution.


[FIGURE] Fig. 6. The (u) plane projection of the critical lines (solid lines) and the critical solution trajectory (dashed line) for the ([FORMULA]) system at K. The solution crosses one of the branches of the slow mode (S) critical line before crossing the critical line (F) corresponding to the fast mode.

[FIGURE] Fig. 7. The [FORMULA] plane projection of the critical lines (solid lines) and the critical solution trajectory (dashed line) for the ([FORMULA]) system at K. The solution crossing point on the slow mode critical line moved to another branch.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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