Astron. Astrophys. 327, 392-403 (1997)

4. Summary and conclusions

The model presented in the paper is the first one that simultaneously takes into account three populations of particles: galactic and anomalous cosmic rays as well as pick-up ions, interacting with the magnetized solar wind plasma. The interaction of the solar wind ions with the LISM neutrals is considered in a self-consistent way. When comparing the present model with the classical, two-fluid approach (solar wind + LISM), the main qualitative differences are:

- slowing down of the flow, as the result of additional forces acting in the system (the upwind wind velocity in the outer heliosphere becomes strongly dependent on the heliocentric distance) - changes in the shock compression ratio. The interaction with LISM neutrals, i.e. the mass loading of the solar wind by the pickup ions, significantly decreases the wind Mach number and the shock becomes weaker; the compression ratio decreases. Taking into account the production of the superthermal population at the shock leads, on the other hand, to large compression ratios; the shock generating the superthermal population is stronger than the normal gasdynamical one. The net effect depends on the relative strength of these mechanisms. - changes in the shock radius. The upwind gradient forces, , as well as the loss of momentum connected with acceleration of the pickup ions, slow down the upwind flow, which leads to the decrease of ; the shock moves inward. On the other hand, the sink of the gas energy flux at the shock and the negative gradient of P lead to the increase of ; the shock moves outward from the Sun. Because these effects are nonlinear, the net result of these mechanisms is difficult to predict without numerical calculations.

Quantitatively, the above effects can be summarized as follows:

- A. The solar wind termination shock.
The efficiency of ACR production at the shock is in the range 5-10%, if the (ecliptic) shock is at 80-85 AU (i.e. the distance corresponding to the correct values of and . It can, however, be greater than 15% for distant shocks ( AU), with the drawback of a too large (). The shock compression ratios lie in the range: 1.6-3.5, i.e. in most cases are much smaller than 4, which means that the mass loading effect prevails over the ACR production. Introduction of ACRs into the model moves the shock out, while the GCRs move it in. The result of the interaction with LISM neutrals is rather to move the shock in. We want to point out that this result is difficult to predict without numerical calculations: one could expect, that the increased upwind pressure will result in the shock being more distant from the Sun, but the decrease in the shock compression ratio is so important, that, in fact, the shock radius is smaller in the presence of pickup ions. Our results agree with the previous findings obtained in simpler approaches (Ko et al. 1988., Ziemkiewicz 1994) that for one obtains AU in the ecliptic plane. - B. The outer heliosphere.
The set of outer heliospheric configurations has been presented for both ecliptic and polar solar winds, showing the spectrum of the possible profiles of the wind velocity and cosmic rays pressure. In the presence of the anomalous component cosmic rays the solar wind is significantly decelerated: to 0.5-0.7 of its value at the Earth's orbit, i.e. from 400 km/s at 1 AU to 200-300 km/s at 80 AU (ecliptic plane). The approximate value of the ACR pressure at the solar wind termination shock is in the range .

It has already been pointed out several times, that the model presented here is a steady-state one. The heliosphere should, however, be treated as a very unstationary medium; because of the numerous effects, the conditions of the flow are strongly time dependent. Here the problem of the response of the system on the nonlinear, time dependent changes arises; the stationary models are valid only if the relaxation time is short. In the recent literature one can find several models studying the termination shock interactions with different types of time dependent disturbances: from shocks to the very long Alfvén waves (Donohue & Zank 1993, Ziemkiewicz 1995). Donohue & Zank estimated the reformation time after the interaction with an interplanetary shock as about 6 months. With an average termination shock speed of about 100 km/s (Ziemkiewicz, 1995) this corresponds to termination shock oscillations with an amplitude of the order of 10 AU. During the motion of the termination shock its parameters, e.g. the compression ratio, will change. One can consider our steady-state results as corresponding to an average shock radius; on the other hand one can also roughly assess from them what will be the shock structure at different stages of interaction.

Another apparent disadvantage of our approach is the assumption of spherical symmetry. Therefore, our model describes quite well the upwind region and these parts of the downwind region which are either close to the shock or to the stagnation line. Recently, several groups (e.g. Pauls & Zank, 1996, Zank et al. 1996b) published their multi-dimensional models of the solar wind - LISM interaction, which can reproduce the real shape of the heliopause. These models, however, do not take into account the full interaction with the cosmic rays. Incidentally, their results concerning the termination shock radius are very similar to our values both for the ecliptic and polar cases. For example, the Pauls & Zank's (1996, Table 3) shock radii are 87.5 AU (ecliptic) and 120 AU (polar). In that way, quite surprisingly, our simple approach finds its justification in calculations that are more sophisticated from this point of view.

© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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