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

3. Results

We consider several models defined by combinations of input parameters controlling the solutions of Eqs. (26)-(30), They belong to three different groups. First of all, one can choose between the equatorial and the polar solar wind. In the former case the initial values of hydrodynamic parameters at 1 AU are (Schwenn, 1990):

while those of the polar wind are (McComas et al., 1995):

The pressures above correspond to temperatures of K and K, respectively. The pressures of the cosmic rays components are the same in both models:

The pressure of galactic cosmic rays is close to the value of , typical for the inner heliosphere (Jokipii, 1990).
The next set of parameters determines the pick-up ion source terms (Eqs. (5)-(7)). At the solar minimum the densities of neutral components of LISM and their ionization rates at 1 AU are (Lee, 1995):

The above values correspond to the lower limit of the referenced values. The recent Ulysses data (Witte et al. 1993) suggest that the density values may even be more than 2 times larger. Also at solar maximum the ionisation rate can significantly increase. Therefore, we will use also the following values:

The last group of parameters consists of the diffusion coefficients for the galactic and anomalous cosmic rays, . Unfortunately, it is not possible within the framework of the present model to calculate self-consistently the diffusive coupling between the ambient medium (plasma and magnetic field) and the cosmic rays. Moreover, since the diffusion coefficients averaged over the spectrum of energies of each kind of cosmic rays are not well known, must in fact be treated as free parameters of the model. It is well known that the diffusion coefficient is a function of the turbulence level and of the magnetic field B itself; the diffusion in the direction perpendicular to the field is generally less efficient than the diffusion along the field (). As the upwind field decreases with r, we assume that the upwind diffusion coefficients increase:

In the ecliptic plane, where the field is azimuthal, the cross-field diffusion is appropriate; in the polar heliosphere the parallel diffusion coefficient should be taken into account. In the post-shock region both the magnetic field () and the turbulence level increase, but, as the functional dependence of K on the field and on the number of scattering centers is not fully known, the value of the downwind diffusion coefficient is difficult to specify. In order not to multiply the amount of numerical results, in this paper we assume that the downwind diffusion coefficients, , are constant and equal to the upwind values at the shock. According to Potgieter et al. (1987), the diffusion coefficient is proportional to the thermal velocity of diffusing particles, so that , corresponding to particles with energy 10 MeV, is assumed to be 10 times smaller than , which describes particles in the GeV energy range. We follow this prescription in calculating the equatorial models, and represent by linear functions with moderate (, ) and steep (, ) gradients. The values of and are chosen in that way that the average value of in the upwind region (i.e. over 100 AU), is in both cases similar and amounts to . In the polar wind, the average value must be increased to avoid singularities in the solutions. To simplify the considerations we take the constant value of ), which is probably the upper limit of possible coefficients, and tune the value of to obtain the physically meaningful solutions.

All models are summarized in Table 1.

Table 1. Models: input parameters and main results

Two other quantities, important in evaluating the obtained solutions, are the pressure of galactic cosmic rays far away from the Sun ("at infinity") and the pressure of the local interstellar medium. According to Lee & Axford (1988), the LISM pressure should be in the range , the upper value being more likely. These values are uncertain mainly due to our very limited knowledge about soft (MeV) galactic cosmic rays in the LISM. In consequence, one can argue for either subsonic or supersonic LISM (Zank et al, 1996b). For the galactic rays pressure, we assume (Axford, 1985).

We have found that the correct value of the LISM pressure at the heliopause is obtained for only a narrow interval of , which, however, changes for different . Values of smaller than the lower limit of the mentioned interval result in an increase of the downwind ACR pressure with distance, which corresponds to the source at infinity, not at the shock. The large , on the other hand, forces a steep negative gradient of the ACR pressure at the shock, which, in consequence, leads to unphysical, negative values of the ACR pressure in the downwind region (Fig. 1).

3.1. Equatorial models

The results of four equatorial models are showed in Figs. 2-5. Three quantities: velocity, galactic and anomalous cosmic rays pressure are plotted here as functions of the heliocentric distance for the assumed shock positions at 65, 80, and 95 AU. The presented solutions correspond to the largest possible still yielding a non-negative ACR pressure in the downwind region.

Fig. 2. Velocity v and cosmic rays pressures and as functions of a distance for model 1. Three different positions of the shock are indicated by dotted lines; from each one of them the downstream intergration starts and results in a different profile of the considered quantities. The production efficiency is chosen for each shock as a maximum possible value from the range (0,0.15) still yielding a nonnegative ACR pressure downstream of the shock. It is equal 0.0, 0.01, and 0.06 for the shock at 65, 80, and 95 AU, respectvely.

Fig. 3. The same plot as in Fig. 2 for model 2. The values are 0.0, 0.04, and .

Fig. 4. The same plot as in Fig. 2 for model 3. The values are 0.0, 0.01, and 0.05.

Fig. 5. The same plot as in Fig. 2 for model 4. The values are 0.0, 0.04, and .

Velocity. As the result of mass loading the upwind velocity decreases with distance, first linearly, then with an increasing gradient. Clearly, the models 3 and 4, with a smaller pick-up ion production also show a smaller velocity decrease. The steeper gradient of the diffusion coefficients (models 2 and 4) results in a very efficient slowing down of the flow; velocities at the most distant shock position (95 AU) are quite low, about 200 km/s. The velocity profiles for AU show a strong precursor structure; in each case one observes a thick region of a large negative velocity gradient, that can be interpreted as a precursor of weak shock. In the inner heliosphere the dominant terms in Eq. (24) determining the velocity profile are: the gas pressure gradient and the momentum exchange with neutrals, while near the shock the cosmic rays pressure and the momentum loss to the neutrals dominate.

GCR pressure. The galactic cosmic rays pressure increases monotonically, with a faster rate in the upwind, and more slowly in the downwind region. The obtained GCR pressure is, for all models, in the range , in good agreement with the canonical value of - . The more distant the shock, the larger the GCR pressure at the heliopause (i.e., practically, "at infinity").

ACR pressure. The anomalous cosmic rays pressure shows the largest variations among different models. Also, for a given model, the ACR pressure at the shock is very sensitive to the shock position. This follows from a strong dependence of on the velocity; as was explained earlier, v varies significantly near the shock. One can say also that the ACR diffusion is most efficient at places where the density of scattering centers (solar wind) varies most, i.e. in a heavily mass loaded wind, close to the shock. The form of in the downwind region is, to a large extent, determined by ; this parameter appears in the conservation equation (21) and influences the post-shock value of - the intial condition for the downwind integration. One can see (Table 1), that the shock at 65 AU allows for a very small () regardless of the model. For AU one gets , the upper limit obtained for models with a steep gradient of . Finally, large values () are still acceptable for the most distant shocks (at 95 AU).

Gas pressure and magnetic pressure. Gas pressure dominates the overall pressure near the Sun. It falls down rapidly with the distance, following the density variation. At the shock, P shows a step-like increase and then decreases slowly towards the heliopause (Fig. 6). This pattern is very similar for all models considered, and the quantitative differences are not large. At the heliopause P amounts to and is smaller than . The magnetic pressure at 200 AU is larger than P by a factor ranging from 2 ( AU, models 3 and 4) up to 15 ( AU, models 1 and 2). The sum of both pressures, which should balance at the heliopause, vary between , depending on the model and . In general, it decreases with the shock distance and is larger for a smaller pick-up ion production rate. In Fig. 6 we also show a profile of the polytropic pressure variation, that starts from the same value as P at 1 AU. Even if the difference between the polytropic and nonpolytropic pressure at the shock is somehow exaggerated here (the density profile in the polytropic model would be different from the value of obtained in our model), one still may come to the conclusion that quantitative results obtained from the polytropic models must be treated with great caution.

Fig. 6. Gas pressure P and magnetic pressure as functions of distance for model 1. The shock is set at 80 AU. The polytropic pressure (dotted line) is shown for comparison with P obtained from a nonpolytropic model.

Density. The source term in Eq. (26) increases the solar wind density above the value that would be obtained for a mass conserving flow. For an efficient mass loading (models 1 and 2) the increase amounts to 10% at 80 AU and 12% at 95 AU. The models with low source-rate of pick-up ions show an increase of about 2-3%.

Compression ratio. For a single-fluid, classical shock with one obtains . As mentioned earlier, two opposite effects included in our equations can modify this canonical value: production of ACRs leads to an increase of q, the mass loading of the wind with pick-up ions results in q smaller than 4. In Fig. 7 we present the compression ratio as a function of obtained for four equatorial models and two values of : 0.05 and 0.15 (Note, that only for AU () and for AU () some of our models give meaningful solutions for in the downwind region). For , the mass loading effect dominates the ACR production and the resulting q is smaller than 4. It can be as small as 2.7-2.8 for a distant shock, small (), and the model with the efficient pick-up ion source.

Fig. 7. Shock compression ratios q as functions of the shock distance for the ecliptic solar wind models: 1 (), 2(), 3(), 4(), and for two values of .

3.2. Polar models

Similarly as for the equatorial models, we consider two different production rates of pick-up ions. The main difference with respect to the equatorial case is a change of the cosmic rays diffusion coefficients. An attempt to use the same values as for the equatorial models was unsuccessful: either the obtained was too large, or, what was even worse, the solutions of (26)-(30) appeared to be singular in the upwind region. Therefore, we had to change and . An increase of to about ensures that the obtained is approximately correct. In order to avoid singularities in the solution, must be larger than . The above numbers correspond to the upper limits of diffusion coefficients considered in the literature. In what follows, we will present the results obtained for a constant and varying from to (Figs. 8-11). The ACR pressure close to the shock is extremly sensitive to the value.
Since the expected distance to the heliopause should be larger in the polar than in the equatorial plane (in the direction towards the stagnation point), we will now consider in the range 80-120 AU.

Fig. 8. The same plot as in Fig. 2 for model 11. The values are 0.0, 0.01, and 0.04.

Fig. 9. The same plot as in Fig. 2 for model 12. The values are 0.0, 0.07, and .

Fig. 10. The same plot as in Fig. 2 for model 13. The values are 0.0, 0.01, and 0.10.

Fig. 11. The same plot as in Fig. 2 for model 14. The values are 0.0, 0.04, and .

From the physical point of view, the two important differences of the polar flow as compared with the equatorial one are: (i) the wind is faster in the upwind region, and (ii) the magnetic pressure in the downwind region vanishes. The magnetic pressure stabilizes the flow; in the equatorial models the upwind solutions are always regular, while the polar models almost always fail before reaching the heliopause: either the velocity becomes negative or the denominator in (27) goes to zero. One could argue that behind the shock the magnetic field is no longer parallel to the shock and, therefore, the magnetic pressure is not negligible there. However, we decided not to introduce the magnetic pressure and follow the downwind solutions as far as possible. At the point where the calculations are terminated, the solar wind and the GCR pressures are practically constant and can be extrapolated to the heliopause in order to be compared with and .

Velocity. The polar models show a relatively larger decrease of velocity on the way from 1 AU to the shock than the equatorial models, but this effect results from the larger distance to the shock, not the larger velocity gradient. Again, the models with a less efficient pick-up ion source show a smaller velocity decrease. The impressive slowing down of the solar wind at 100-120 AU in models 12 and 14 follows from a very steep ACR pressure gradient.

GCR pressure. The galactic cosmic ray pressure shows similar profiles as in the equatorial cases, it is, however, larger at the heliopause () because more distant shocks provide an extra room for its increase. The value of can be easily decreased by a different choice of initial conditions or .

ACR pressure. The anomalous cosmic ray pressure is very sensitive to the diffusion coefficient. Its moderate change, from to , yields a dramatic increase of the ACR pressure at the shock (approximately a factor of 4). Such an effect has not been observed for equatorial models, but could not be excluded - we have not tried to explore the whole parameter space.

Gas pressure and density. The upwind variations of the pressure are similar in form to the equatorial profiles. The magnetic pressure is zero, therefore downstream of the shock only the gas pressure contributes to the pressure balance at the heliopause. Its value at the heliopause (or at the point where calculations are stopped) decreases with the distance of the shock; for shocks at 80-100 AU, P is equal to , which is too large as compared with the expected LISM pressure. From this point of view only the shocks at 110-120 AU are acceptable - they yield pressures of . For models with large anomalous cosmic ray production at the shock (12 and 14), the upstream density shows an interesting feature - it decreses, as one could expect, to a distance of 100 AU but then starts to increase, forming a kind of wall in front of a distant (very weak) shock ( AU). This increase is, of course, correlated with a very steep velocity decrease in this region. The mass loading effect amounts to 11-18% at AU for the efficient ionization (models 11, 12), but is 3-4 times smaller for models 13 and 14.

Compression ratio. Perhaps the most interesting issue is an appearance of very weak shocks for models 12 and 14. The value of q can be as small as 1.6-1.8 (Fig. 12). In fact, such structures are very close to a shock-free transition from one flow to another.

Fig. 12. Shock compression ratios q as functions of the shock distance for the polar solar wind models: 11 (), 12(), 13(), 14(), and for two values of .

© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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