M. S. Kalinin and M. B. Krainev
P. N. Lebedev Physical Institute, Moscow, Russia
The data obtained by a network of spacecraft (IMP 8, Pioneer 10, and Voyager 2) during more than 2 decades and covering large spatial scales (up to r = 65 AU) allow comparative analysis of the GCR intensity behavior during two successive 11-yr solar cycles. In spite of a limited amount of data, this analysis [Webber and Lockwood, 1997] has revealed a rather unexpected intensity behavior in the distant heliosphere. If we consider only successive solar minima - the GCR intensity measured by IMP 8 ( r = 1 AU) in 1996 was approximately equal to that in 1976 - an unexpected very weak increase in the GCR intensity with radial distance is revealed in the distant heliosphere ( r 40 AU) in 1995-1996 as compared with 1987. For the 130-230 MeV proton channel, the ratio between intensities in 1996 and 1987 was J96/J87 1.23 at the Earth's orbit, 0.44 at r 42 AU, and as low as 0.29 at r 64 AU.
These data, supplemented by Ulysses data on global latitudinal GCR gradients [Heber et al., 1996] and spectra measured by IMP 8 at the Earth's orbit, have been described by numerical solutions of the transport equation with standard boundary conditions [Potgieter, 1997]. It has been shown that in order to satisfactorily describe the spatial GCR distribution in the heliosphere at two successive intensity maxima (solar minima) periods, the main parameter of the model - radial diffusion coefficient - should be taken to be 5-6 times greater for the intensity maximum in 1987 (when the radial projection of the interplanetary magnetic field in the Northern Hemisphere was negative, A = -1 ) than for the intensity maximum in 1996 (when the projection was positive, A = 1 ). Another characteristic of the spatial distribution - latitudinal intensity gradients at 40-60 AU - proved to be too small for the period with A = -1 in this case.
The goal of this work was to show, based on numerical solution of the transport equation for protons, that experimental data on the radial intensity behavior can be satisfactorily described if we introduce additional modulation beyond the outer heliosphere boundary due to the modulating action of the electrostatic potential that changes its sign at the TSMF polarity reversal.
The spatial distribution of proton intensity was calculated by numerically solving the two-dimensional (with respect to spatial variables r (radial distance) and q (polar angle)) axisymmetric stationary diffusion equation including drift [Toptygin, 1983]
where U is the particle density in the phase space ( r,p) related to the intensity J as J = p2U; Kij is the total diffusion tensor (DT) whose nonsymmetric part describing drifts corresponds to coefficients KT KTf(q,a) for the tilt angle of the current sheet of a = 5o modified according to Potgieter and Moraal  and Jokipii and Kota ; and Vi is the solar wind (SW) velocity having only radial component depending on both spatial coordinates. The actual dependence of the SW velocity on coordinates was qualitatively consistent with the Ulysses data and was approximated as
where r0 is the photosphere radius. At q< 60o, the SW velocity was latitude-independent and corresponded to the solution of (2) for q = 60o.
The symmetric part of the DT describing diffusion was taken to be fully anisotropic; and the components normal to the magnetic field were assumed to be proportional to the field-aligned component K r = a1K, K q = a2K. Fixed coefficients a1 and a2 were chosen so as to fit the radial intensity behavior and to ensure the ratio between the intensities at the pole and helioequator for radial distances r 1 AU limited by 1.4-1.5 [Heber et al., 1996]. The field-aligned diffusion coefficient is given by
Here, the dimensional coefficient K0 measured in terms of 6 1020 cm2 s-1 was adjusted in calculations; b = n /c, where n is the particle velocity and c is the speed of light; and function f1(R), where R is rigidity, was chosen so as to fit experimental data. In practice, a simple dependence
The antisymmetric coefficient KT describing drift was taken in the standard form KT = K0(a) f(q, a) b R/3B, where B is the value of the interplanetary magnetic field (IMF); and coefficient K0(a), typically equal to unity, was varied to control the effect of drift. Function f2(r) = 1 +rn, where r is expressed in AU.
Equation (1) was solved with the standard initial condition U( r,pm) = UH(pm), where UH(p) is the unmodulated particle density in the Galaxy, pm = 100 GeV s-1, and conditions at the outer boundary r = rM of the modulation region are U(rM, q, p) =UH(p).
The effect of external modulation beyond the modulation region was provided by the modulating action of the electrostatic potential (EP), which was described by the expressions obtained earlier by Kalinin and Krainev [1992, 1997] (see also Krainev  and Jokipii and Levy ), but shifted by the value a which was chosen so as to fit experimental data
where A = 1 indicates the TSMF polarity, rs is the radius of the magnetic field source surface, w is the angular speed of rotation of the Sun, Bs is the radial component of the magnetic field at the source surface, and function f(q) describes the latitudinal dependence according to Kalinin and Krainev . In calculations, coefficient (Br02w/c) equal to 0.25 GV, typical of the heliosphere was used. The particle density was recalculated for the outer boundary of the modulation region by using the relation U(rM, q, p) = UH(p + Dp), where rM = 100 AU is the modulation region radius (Liouville theorem). In this expression, p and p' = p + Dp are related by the energy integral
Here, q is the particle charge.
As a result, a radial intensity profile of the 200-MeV protons at the helioequator was obtained and then compared with the data of the 130-230 MeV proton channel at the spacecraft obtained by Webber and Lockwood .
It is clear from the magnitudes of the parameters given in the caption to Figure 1 that even in the case of a two-fold decrease in drift effects, coefficient K r, responsible for radial gradients in the distant heliosphere, must be chosen low enough to provide the required large intensity drop within the interval of distances 20 AU from the outer boundary of the modulation region.
The major conclusion inferred from the above consideration is that the transport equation is not able to describe the experimental spatial GCR distribution within the heliosphere at successive solar minima periods in terms of the standard approach involving the same set of kinetic coefficients. To adequately describe measurements, at least the diffusion coefficients responsible for the radial distribution of the GCR intensities must considerably differ in these sets (by a factor of 5-6).
Thus, the obtained results confirm the conclusions made by Potgieter , who considered an analogous problem in terms of another model dependence of the diffusion coefficients on the rigidity. Note also that our results are not the consequence of the diffusion coefficient dependence on the radial distance in the form K (1+r) that we have chosen. For instance, the K 1/B dependence (where B is the IMF value) often used gives the same qualitative picture. The difference is in insignificant changes in the transport equation parameters.
The results given in the previous section suggest that other modulating factors can exist. By considering them, it is possible to avoid inconsistencies arising in description of the measured spatial GCR distribution. It is quite natural that these factors must be associated with the Sun - the only cause and source of modulation - and, in addition, they must exhibit a "correct" dependence on the sign of the 22-yr solar cycle phase (that is, on the sign of A ) and on heliolatitude. In a correct dependence, the effects of these factors fit the experimentally observed GCR behavior.
If we extend the radial intensity dependence to the outer heliosphere boundary using the same gradients as in the distant heliosphere, it becomes evident that two successive solar minima can have different intensity levels at the outer heliosphere boundary. In this case, the ratio between unmodulated intensities for the energy interval considered must be 3-4. However, the attempt to describe the radial intensity behavior at two successive solar minima by varying only the value of the intensity spectrum at the modulation region boundary, without changing its shape (that is, with unvaried energy dependence), and by using the same set of transport equation coefficients for different signs of A is not justified, on the one hand, and does not give the desired result, on the other hand.
Kalinin and Krainev [1992, 1997] derived the expression for the heliospheric electrostatic potential (EP) averaged over the azimuthal variable relative to infinity on the basis of the Parker spiral IMF structure and high electric conductivity of the SW plasma. Since the EP induced in this heliosphere model is completely included in the transport equation, which is solved for the inner regions of the heliosphere, the action of such induction field beyond the heliosphere can be reduced, in the first approximation, to the effect of its EP on the intensity spectrum at the heliosphere boundary. In this case, the dependence of EP on global sign-defining multiplier A describing polarity of the 11-yr cycle and on latitude provides the necessary and correct dependence on it of the modulating effect beyond the heliosphere. The EP value necessary to provide the required modulation level at the modulation region boundary for different A is determined by the intensity difference in the distant heliosphere. It depends also on how completely physical mechanisms affecting the true unmodulated intensity spectrum at large distances beyond the heliosphere are taken into account. If we ignore energy losses of particles during their travel to the outer heliosphere boundary from the local interstellar medium and assume the efficient action of nondissipative scattering, the particle density along their trajectories in the phase space must be preserved, according to the Liouville theorem. If, in determining the phase trajectories, we restrict ourselves to the EP effect and ignore the IMF influence, all trajectories will belong to the constant-energy surfaces with uniform filling. In this case, the spectrum at the modulation region boundary can be found from the expression: U( rM, p) = UH(p') (see the expression in the above discussion). Note that this approach can be regarded only as the first approximation to the actual picture, but it is useful owing to its simplicity [Jokipii and Levy, 1979].
The qualitative picture of the modulating EP effect beyond the heliosphere in terms of the approach described above is sufficiently clear and follows from (6). It is not reduced to a mere change in the spectral amplitude at the heliosphere boundary. For the period with A = -1, the EP effect at low latitudes leads to a considerable shift of the initial unmodulated intensity spectrum toward higher kinetic energies (that is, the spectrum goes higher and simultaneously is cut from the left).
Thus, the intensity spectrum at the heliosphere boundary for any above-threshold energy determined by the EP amplitude (~125 MeV) at A = 1 is lower in near-equatorial regions and higher in the polar regions as compared with that at A = -1, thereby providing the necessary sign of the effect. In addition, this picture of the influence on the unmodulated intensity spectrum should lead to increasing latitudinal gradient in the distant heliosphere, thus providing a better fit to measurements.
The obtained results show that
1. It is impossible to fit the measurements of spatial GCR distribution in the heliosphere at successive solar minima in terms of the standard approach to the solution of the transport equation involving the use of the same set of diffusion coefficients. There is a question: whether the difference in transport equation parameters corresponds to the difference in actual physical conditions of the GCR particle propagation in the heliosphere at successive solar minima periods is distinguished, according to modern ideas, only by the TSMF sign. The question still remains unresolved.
2. Since the required difference in diffusion coefficients is associated with a low intensity level in the distant heliosphere at the solar minimum when the TSMF sign was positive and the drift mechanism had to be effective, the question about the actual contribution of drifts into modulation arises. The contribution of drifts derived from the standard first-order orbit theory is likely to be overestimated, and the fit to measurements can be achieved only by assuming the contribution of this mechanism to be half its standard value. Since a much lower radial diffusion coefficient in the distant heliosphere is also required to fit measurements during this period, this result can be interpreted as the effect of diffusion on the drift efficiency (a low diffusion coefficient corresponding to intense scattering leads to a weaker drift mechanism).
3. One of the ways to avoid inconsistencies accompanying the standard approach is to assume that the radius of the Sun's action on charged GCR particles extends beyond the boundaries of the modulation region, which is typically thought of as heliosphere sizes. A direct physical mechanism in this case is the EP effect on the GCR intensity spectrum beyond the modulation region. According to the results given in section 3, the EP having the amplitude of 250-300 MV and depending on the TSMF sign is needed to fit measurements. This potential differs from the EP typically associated with the heliosphere by a shift by a constant value, Br02w/2c 125 MV; the latitudinal dependence remains unvaried. This EP value can be associated with the positively charged heliosphere when the total charge within any radius is not zero. This possibility cannot be excluded, though there are no arguments in favor of this hypothesis as well. Another possible way to increase the EP amplitude can be taken into account: the latitudinal dependence of the magnetic field at the source surface, which is usually neglected for simplicity [Krainev, 1981]. Consideration of both assumptions is beyond the scope of this paper. In conclusion, a conceptually similar approach, but in terms of a more realistic model, was successfully employed by Jokipii and Kota  to describe the radial behavior of the anomalous CR component.
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