Astron. Astrophys. 324, 725-734 (1997)

3. The results of the Kappa exospheric model

In this section, we present the plasma densities, bulk speed and temperatures obtained at 1 AU with our new exospheric model. In order to explore the influence of a change of the index , we have used four different values for the electron index: , which is the lowest permissible value; , and finally , which coincides almost to a Maxwellian electron VDF.

3.1. The solar wind at 1 AU: model calculations

The electrical potential (a), density n (b) and electron (c) and proton (d) average temperatures and obtained at 1 AU with our kinetic model are displayed in Figs. 4 a,b,c,d, as a function of the electron temperature at the exobase. The average temperatures are defined by , and derived from the pressure tensor components parallel and perpendicular to the magnetic field direction. As indicated in Table 1, the choice of a value for determines uniquely the other boundary conditions (, , ) at the exobase. The five successive points on the curves shown in Fig. 4 correspond to the five sets of exobase condition given in Table 1. Each curve corresponds to a different value of .

Fig. 4. The electrical potential (a), density n (b) and electron (c) and proton (d) average temperatures and obtained at 1 AU with the present kinetic model, as functions of the electron temperature at the exobase and for the four values , , and .

3.2. Comparison with solar wind observations at 1 AU

The horizontal dotted lines in Fig. 4 correspond to the ranges within which the four physical quantities are usually observed at 1 AU. The density and proton temperature are usually observed to range between 1.0 and 30 cm^-3 and between 10⁴ and 2 10⁵ K respectively (see Feldman et al. 1978). Taking into account the ranges within which the core/halo electron densities and temperatures were observed by Feldman et al. (1975), the average total electron temperature range roughly between 5 10⁴ and 1.2 10⁶ K at 1 AU. The interplanetary electrical potential difference between the exobase and 1 AU can be determined assuming that the core/halo parametrization reflects the existence of two electron populations predicted by some previous exospheric theories of the solar wind (Jockers 1970; Schulz & Eviatar 1972; Perkins 1973). With this assumption, the core/halo breakpoint energy could be associated to the difference of electrical potential energy (Feldman et al. 1975; McComas et al. 1992). These authors found that the electrical potential at 1 AU, deduced from the breakpoint energy, ranges between 20 and 130 Volts.

It can be seen in Fig. 4 that the values of the density and temperatures observed at 1 AU and the electrical potential difference are well reproduced in our model with values of ranging between 2 and 6.

3.3. The high speed solar wind and its origins

The aim of this section is to compare the results on the solar wind expansion speed obtained with our kinetic approach to those of the classical hydrodynamic one.

3.3.1. The expansion speed at 1 AU

The expansion bulk speed at 1 AU obtained in our model is displayed in Fig. 5 as a function of and for three values of the index: , and . Note first a general tendency for the bulk speed V to increase with the temperatures at the exobase, and . Indeed, the average velocity of the escaping particles, which is the solar wind bulk speed, is an increasing function of the temperature and the width of the VDFs at the exobase: like in planetary atmospheres, when the width of a VDF is enhanced at the exobase, the number of particles with a velocity greater than the escaping velocity is equally enhanced, and thus the average velocity of those particles is also increased.

Fig. 5. The expansion bulk speed obtained at 1 AU with the present model, as a function of and for the three values , and : One can note that the high speed solar wind streams (600-800 km/s) can be explained if the electron VDFs in the corona have high velocity tails,,i.e., small values of (2-3).

But, as shown in Fig. 5, increasing the spread of the VDFs at the exobase is not the only way of increasing the solar wind bulk speed at 1 AU. The same result can be obtained for the electron VDF by decreasing . Indeed, we have shown that the average bulk velocity of the escaping particles increases when decreases; in other words:

This is the main result of the present study. It confirms a similar conclusion by Scudder (1992b), who analyzed the implication of his "velocity filtration " mechanism on the asymptotic form of Parker's (1963) isothermal solution for the bulk speed. Therefore, there is less need to refer to any additional (ad-hoc) heating mechanism in the outer corona, in order to increase the solar wind bulk speed, as it is often postulated in hydrodynamic model calculations.

In our model, the only physical input ingredients are the coronal density and temperatures at the exobase. Among the various hydrodynamic models of the solar wind, the one which is most comparable to our exospheric model, in terms of sophistication, is Parker's (1963) thermally driven solar wind model. This latter model does not include pressure anisotropies of the coronal fluid and no magnetic effects (which is equivalent to a radial magnetic field or a non rotating Sun); as our exospheric model, Parker's model is based on the assumption of a unique energy source: heat entering at the base of the corona which is converted into expansion bulk energy like in a de-Laval nozzle. Note also that Parker (1963) showed rather generally that the temperature achieved at the critical point determines the asymptotic wind speed in a way almost totally independant of the form of the energy equation. In Fig. 5 the solar wind bulk speed at 1 AU obtained from Parker's solutions (heavy dashed line) compare our exospheric approach. Parker's solutions have been computed for a spherical coronal expansion and a heliospheric temperature distribution of , which corresponds to an average gradient obtained from spacecraft observations. It is interesting to point out that Parker's bulk speed at 1 AU is almost identical to that for the kinetic solutions with Maxwellian VDFs or for . On the other hand, one can see in Fig. 5 that if one diminishes the value of below 6, then the Kappa exospheric model yields bulk speeds larger than the hydrodynamic one. Unlike the thermally driven hydrodynamic model, our exospheric model for the solar wind yields high bulk speed winds (600 - 800 km/s), provided the value of is small enough (2 - 3). Therefore our simple kinetic model of the solar wind leads to high velocities comparable to those obtained in hydrodynamic models when additional MHD wave energy deposition is artificially added in the outer corona. Similar conclusions are reached by Scudder (1992b). Our work is therefore reinforcing his pioneering work.

3.3.2. The expansion speed/coronal temperature anticorrelation

In the second half of the 70's, the Skylab observations indicated that the coronal holes are the sources of the fast solar wind streams (Neupert & Pizzo 1974; Nolte et al. 1976; Krieger et al. 1973). The coronal holes plasma has a lower temperature than the equatorial active regions of the corona: 10⁶ K for the coronal holes and 2 10⁶ for the equatorial active regions (Withbroe & Noyes 1977). According to Parker's hydrodynamic model, the asymptotic solar wind bulk velocity decreases when the coronal temperature is lowered. The anticorrelation observed between the solar wind bulk speed and the coronal temperature seemingly has remained a puzzling feature for many years.

Our exospheric Kappa model offers an explanation for this anticorrelation. The three profiles of the bulk speed at 1 AU displayed in Fig. 5 have been obtained for constant values of . But, of course, the index may change with the temperature at the exobase. Indeed, as it is shown Table 1, the Coulomb collision times are very different for the case A ( 1520 seconds) and for the case E ( 160 seconds), assuming the radial electron density distribution remains the same in both cases. The difference is also important regarding the m.f.p. of the particles in the case A (3.4 ) and in the case E (0.7 ). Considering that non-Maxwellian features develop preferentially in cold and less dense coronal hole region, where the Coulomb collision time is large, one may infer that the value of for the low temperature model (A) can be smaller than that of the hotter models (B, C, D & E). Therefore, it is likely that in the case A the tail of the VDF at the exobase is more populated than in the case E: the value of for the case A should therefore be smaller than for the case E. Based on this hypothesis, Fig. 5 offers a possible explanation for the anticorrelation between the bulk speed observed at 1 AU and the coronal temperatures & . Indeed, from Fig. 5 and Table 2 it can be seen that in case A, corresponding to a low coronal temperature ( K) but with a highly non-Maxwellian electron VDF (), one obtains a higher speed solar wind at 1 AU, than for instance in the case C corresponding to a higher coronal temperature ( K) and a nearly Maxwellian electron VDF (). Fairfield & Scudder (1985) came to the same conclusion more than ten years ago. Our quantitative calculation reinforces their contention.

Table 2. The 1 AU bulk speed/coronal temperature anticorrelation

3.3.3. Conclusion

The results presented in this paper indicate that high speed solar wind streams can be explained provided the electron VDFs in coronal holes have enhanced high velocity tails, in contrast to the VDFs in the equatorial regions, which would be closer to that of a Maxwellian VDF due to the larger collision frequency there. It is possible also that the coronal magnetic field topology, whose effects have been neglected in the first series of exospheric models, plays a role by modifying the electron density and consequently the frequency of Coulomb collisions. In coronal holes the lines are "open", while in equatorial active regions there are closed loops where the particles can accumulate and collide more frequently. The consequence is that in the equatorial regions the Coulomb collision time is indeed smaller compared to that in the polar regions where magnetic field lines are more stretched out into interplanetary space. Therefore, the plasma should be closer to the Maxwellian equilibrium near the equator than in coronal holes. If the overall dipolar magnetic field topology of the Sun is the cause of the latitudinal density distribution, with higher densities at the equator than over the polar regions or in coronal holes, our assumption about the variation of the index versus heliographic latitude is then fully justified.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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