Modulation of anomalous protons: Effects of ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, A01106, doi:10.1029/2005JA011066, 2006

Modulation of anomalous protons: Effects of different solar wind speed profiles in the heliosheath U. W. Langner1 Institut fu¨r Theoretische Physik IV, Ruhr-Universita¨t Bochum, Bochum, Germany

M. S. Potgieter Unit for Space Physics, School of Physics, North-West University, Potchefstroom, South Africa

H. Fichtner and T. Borrmann Institut fu¨r Theoretische Physik IV, Ruhr-Universita¨t Bochum, Bochum, Germany Received 11 February 2005; revised 20 October 2005; accepted 3 November 2005; published 21 January 2006.

[1] Two termination shock acceleration modulation models are used to study the

modulation of anomalous protons, in particular the effects of different scenarios for global solar wind speed (V) variations in the heliosheath. The first numerical model simulates a symmetric heliosphere and the second simulates an asymmetric heliosphere with respect to the Sun. The modulation differences between these models are illustrated and discussed. The geometry of the heliosphere in the latter model is deduced from a time-dependent three-dimensional hydrodynamic model of the heliosphere which provides the different scenarios for the V-profiles in the heliosheath. The modulation models include the solar wind termination shock, global drifts, adiabatic energy changes, diffusion, convection, and a heliosheath. The anomalous protons are kinetically described using the Parker transport equation. A solar wind speed decreasing stronger than the generally assumed V / 1/r2 dependence, with r the radial distance from the Sun, is studied as well as an extreme scenario with V / r2. The stronger decrease produces a compressive flow in the heliosheath which results in additional acceleration of anomalous protons in the heliosheath. The solutions are shown for solar minimum and moderate maximum modulation conditions for both heliospheric magnetic field polarity cycles. Significant modulation differences are found to occur between these different scenarios for V in the heliosheath. If the stronger than V / 1/r2 scenarios in the heliosheath are real, the anomalous intensities should increase beyond the TS, which should be measurable by the Voyager 1 spacecraft in the near future. Citation: Langner, U. W., M. S. Potgieter, H. Fichtner, and T. Borrmann (2006), Modulation of anomalous protons: Effects of different solar wind speed profiles in the heliosheath, J. Geophys. Res., 111, A01106, doi:10.1029/2005JA011066.

1. Introduction [2] The two Voyager spacecraft have been measuring the spectra of anomalous protons for several years in the outer heliosphere [e.g., Webber and Lockwood, 2004]. These particles seem absent at the Earth’s orbit [Christian et al., 1988, 1995; Cummings and Stone, 1998]. The origin of anomalous protons is generally assumed to be interstellar neutral hydrogen atoms that are ionized by charge exchange with the solar wind protons relatively close to the Sun. These ionized particles are then ‘‘picked up’’ by the solar 1 Also at Unit for Space Physics, School of Physics, North-West University, Potchefstroom, South Africa.

Copyright 2006 by the American Geophysical Union. 0148-0227/06/2005JA011066

wind and because they have significantly higher random velocities than the solar wind protons, they are more easily accelerated to energies at which diffusive shock acceleration by the termination shock (TS) becomes effective [Pesses et al., 1981]. Since their discovery [Garcia-Munoz et al., 1973a, 1973b] they also have been the subject of numerous theoretical papers, studying their acceleration and propagation in the heliosphere with numerical models [e.g., Jokipii, 1986; Jokipii, 1990; Potgieter and Moraal, 1988; Steenberg and Moraal, 1996; Sreenivasan and Fichtner, 2001; Langner and Potgieter, 2004b]. These particles are also the subject of study in this paper. [3] The interest in the large-scale structure and geometry of the heliosphere and the transport of cosmic rays within this configuration is not limited to an understanding of anomalous protons in the outer heliosphere but has strongly grown since the approach of the TS by the Voyager 1

A01106

1 of 14

A01106

LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH

spacecraft [e.g., Webber et al., 2001; Krimigis et al., 2003; McDonald et al., 2003; Stone and Cummings, 2003]. For recent reviews on the structural shape of the heliosphere, see Zank [1999], Fichtner [2001] and Florinski et al. [2004a]. In this work a two-dimensional (2-D) TS modulation model is used to study the acceleration of the anomalous protons at the TS in an asymmetrically bounded heliosphere [see Langner and Potgieter, 2005; Langner et al., 2005] and their subsequent propagation into the heliosheath and toward the Sun. This model was further extended [Langner et al., 2005] to include different scenarios for the global solar wind speed (V) in the heliosheath for galactic protons, since it became apparent that the change of V in the heliosheath may be different than the previously assumed V / 1/r2 dependence [e.g., Baranov and Malama, 1993; Pauls et al., 1995; le Roux and Fichtner, 1997; Fahr et al., 2000; Florinski et al., 2004b]. [4] The effect of different scenarios for V in the heliosheath on anomalous protons is now studied in this work with a 2-D TS model, using as input the different solar wind speed profiles and the shape of the heliosphere from a threedimensional (3-D) hydrodynamic model developed by Borrmann [2005] [see also Borrmann and Fichtner, 2005]. This is done because a reasonable treatment is possible only if a hydrodynamic approach is used in accordance with the particle propagation approach, thus establishing a simplified ‘‘hybrid model.’’ These approaches must be used together because a hydrodynamic approach relies on a momentum-averaged diffusion coefficient, resulting in the problem being overdetermined [Florinski et al., 2003]. A fluid approach also does not yield particle spectra that can be compared to available observations to validate assumptions, whereas the particle approach, on the other hand, cannot predict the geometry of the heliosphere, simply relying in this regard on informed assumptions. Therefore a time-dependent 3-D hydrodynamic model of the dynamics of the heliosphere is used to provide the different scenarios for the solar wind speed profiles in the heliosheath, for the heliospheric nose and tail regions. So far, no comprehensive hybrid model exists that effectively combines these effects in a self-consistent treatment of the transport of CRs [Fichtner, 2005], although first steps were made by Ferreira et al. [2004] and Ferreira and Scherer [2004] for galactic electrons and by Scherer and Ferreira [2005] for protons. The solutions resulting from the combination of the results of these models are therefore a step closer to a more realistic self-consistent modulation model, which provides reasonable scenarios for the modulation of anomalous protons in the heliosphere, which, to our knowledge, has not been done before. [5] The TS model is first used to study the modulation differences caused in a symmetrical heliosphere by the different profiles of V in the heliosheath which are (1) V decrease as V / 1/r2, (2) V decrease as V / 1/r8, and (3) V increase as V / r2. Although some of these scenarios are unphysical for a symmetrical heliosphere, they were used for an empirical study in order to quantify the effects of the different V profiles. This study is then repeated for an asymmetrical bound heliosphere for two scenarios: (1) a scenario where V decrease as V / 1/r2 at all latitudes in the heliosheath and (2) a scenario where V change over latitude from a decrease of V / 1/r8 in the heliospheric nose region

A01106

to an increase of V / r2 in the tail. This change over latitude of the radial component of V and its strong decrease in speed in the heliospheric nose region are a combination of flow stagnation, charge exchange, and the structure of the heliosheath due to the subsonic solar wind flow there. In particular, the role of the r V in the modulation of anomalous protons will be investigated as the global profile of V changes in the heliosheath. The solutions are shown for solar minimum and moderate maximum conditions for both polarity cycles of the heliospheric magnetic field. The heliopause is assumed to separate the solar and interstellar plasmas and is considered the outer modulation boundary with the heliosheath defined as the region between the heliopause and the TS. For this work, the intensities of anomalous protons are forced to approach zero at the heliopause for numerical reasons and as a first approximation, although the anomalous particles could, in reality, diffuse into the interstellar medium. There is a reasonable consensus that the TS should be in the vicinity of (90 ± 5) AU [e.g., Stone and Cummings, 2001] in the heliospheric nose direction in which the heliosphere is moving (the TS was recently crossed at 94 AU by the Voyager 1 spacecraft), although over a solar cycle the TS may move outward and inward [e.g., Steinolfson, 1994; Karmesin et al., 1995; Baranov and Zaitsev, 1998; Whang et al., 2004]. The position of the heliopause is less certain; 30 – 50 AU beyond the TS in the nose direction, while in the tail direction the heliosphere is most probably an open structure, causing a considerable asymmetry with respect to the Sun [e.g., Scherer and Fahr, 2003; Zank and Mu¨ller, 2003; Ferreira et al., 2004; Scherer and Ferreira, 2005]. The positions computed with the hydrodynamic model could not be used directly in the asymmetric TS model because of computer power constraints. However, the solutions presented here do establish a reasonable presentation of modulation in an asymmetric heliosphere and give a good representation of what modulation effects are to be expected if these changes in V would occur in the heliosheath.

2. Numerical Models 2.1. Kinetic Model [6] The modulation model is based on the numerical solution of the time-dependent cosmic ray transport equation [Parker, 1965]: @f 1 @f ¼ ðV þ hND iÞ rf þ r ðK S rf Þ þ ðr V Þ @t 3 @ ln p ð1Þ þ Jsource ;

where f(r, p, t) represents the omnidirectional cosmic ray distribution function, p is the particle momentum, r is position, and t is time, with V(r, q) = V(r, q)er the solar wind velocity. Terms on the right-hand side represent convection, gradient and curvature drifts, diffusion, adiabatic energy changes caused by the r V, and a source function, respectively. In this work the role of the r V in equation (1) will be investigated as the global profile of V changes in the heliosheath. The tensor KS consists of a parallel diffusion coefficient (kk) and of perpendicular diffusion coefficients (k?). The averaged guiding center drift velocity for a nearly isotropic cosmic ray distribution is given by

2 of 14

A01106


hnDi = r (kT eB), with eB = B/Bm, where Bm is the magnitude of the modified background heliospheric magnetic field (HMF) as described below; here kT (sometimes indicated as kA) is the diffusion coefficient specified by the antisymmetric elements of the generalized diffusion tensor K, which describes gradient and curvature drifts in the large-scale HMF. The function Jsource represents in this work a source of anomalous protons, which are injected at the TS position as a delta function at an energy of 2.0 MeV. Although this injection energy seems high, it was shown previously that solutions of this particular model is independent of this energy, as long as it is well below the spectral ‘‘cutoff’’ energy for anomalous protons [Langner, 2004]. [7] The TS model combines diffusive shock acceleration, drifts, and a heliosheath, neglecting any azimuthal dependence. This two-dimensional TS modeling approach was described originally by Jokipii [1986] for a symmetrical heliosphere and has been applied numerous times, recently by, e.g., Ferreira et al. [2001], Potgieter and Ferreira [2002], Langner et al. [2003], and Langner and Potgieter [2004a, 2004b]. The HMF was assumed to have a basic Parkerian geometry in the equatorial plane but was modified in the polar regions similar to the approach of Jokipii and Ko´ta [1989]. 2.2. Diffusion Tensor [8] The basic expressions for the diffusion coefficients kk, k?, and kT were given by Burger et al. [2000] for a steady-state model, except for minor changes caused by the introduction of the TS in this model as described by Langner et al. [2003]. Of importance is that perpendicular diffusion is assumed to enhance toward the poles [e.g., Potgieter, 1996]. These diffusion coefficients were used previously with success for a symmetrical model for various species of particles and are therefore not changed for this work. They are optimal for modeling without solar maximum effects, e.g., global merged interaction regions. For a complete description of these diffusion coefficients and details of the model, see Langner et al. [2003], Langner [2004], and Langner and Potgieter [2004a]. For a description of the asymmetric boundary configuration of this model, see also Langner and Potgieter [2005].

where the primed variables are the transformed coefficients. The exact formulations of the coefficients in equation (3) as well as the difference formulae were given in detail by Haasbroek [1997] [see also Haasbroek and Potgieter, 1997]. The numerical grid was chosen to range from q = 0 to 360. The equatorial plane of the heliosphere is then at q = 90 (nose direction) and at q = 270 (tail direction), respectively. For the asymmetrical model the different positions of the heliopause (range of u in equation (2)) are then assumed to be at rHP = 120 AU in the equatorial region in the nose direction of the heliosphere, rHP = 140 AU at the poles, and rHP = 180 AU in the equatorial region in the tail direction of the heliosphere, respectively. The TS positions are given in section 2.5. 2.4. Hydrodynamical Model [10] The flow fields in the heliosheath were computed with a hydrodynamical model. This model [Borrmann, 2005; Borrmann and Fichtner, 2005] is a 3-D two-fluid model with the solar wind and the interstellar plasma forming one quasi-neutral electron-proton fluid and with the interstellar neutral gas, which is treated as a pure atomic hydrogen fluid. With the additional often-made assumption that magnetic fields do not have a direct dynamical influence [see, e.g., Fahr et al., 2000; Zank and Mu¨ller, 2003], the basic equations of this two-fluid model take the following form for the rest frame of the Sun: @t ri ¼ divðri ui Þ þ Qri

@t ui ¼

ð2Þ

f(r, q, p) is then transformed to g(u, v, w) with x and y variables that can be changed in order to give the desired locations of the TS and heliopause so that equation (1) in polar coordinates becomes @g @2g @2g @g @g @g @2g ¼ a0 2 þ b0 2 þ c0 þ d0 þ e0 þ j0 ; ð3Þ @t @u @v @u @v @ ln w @u@v

1 grad pi ðui gradÞ ui þ Qm;i ui Qri = ri ri

@t Ei ¼ divððEi þ pi Þ ui Þ þ Qe;i þ ui Qm;i u2i Qri = 2

ð4Þ

ð5Þ

ð6Þ

with the mass density ri, velocity ui, and the total energy density Ei = 12 riu2i + ei, where ei denotes the internal energy. The source terms Qri, Qm,i, and Qe,i contain the mutual interactions between all species and read explicitly (at the example of species i 6¼ j):

2.3. Numerical Representation of an Asymmetrical Heliosphere [9] For the first part of the study, the locations of the heliopause and the TS in the symmetrical model are assumed to be at r HP = 120 AU and r s = 90 AU, respectively. For the asymmetrical model, equation (1) is derived by using the coordinate transformation: u ¼ rð x y sin qÞ; v ¼ q; w ¼ p;

A01106

Qr;i ¼ nij rj nji ri

ð7Þ

Qm;i ¼ nij rj uj nji ri ui

ð8Þ

Qe;i ¼ nij ej nji ei þ

2 1 i n r uj ui ; 2 j j

ð9Þ

where the nij are the ionization frequencies describing the processes of charge exchange, photoionization, and electron-impact ionization. For their explicit values, see Rucinski et al. [1996]. Most important is the charge exchange for protons and hydrogen, which is modeled according to Holzer [1972], who gave an approximation valid for the supersonic region of the solar wind. While this expression has been used for the heliosheath as well, a refined treatment should employ the result obtained by

3 of 14

A01106


A01106

Figure 1. Three different solar wind speed (V) profiles (top panels) as a function of radial distance from 1 to 120 AU in the equatorial plane. In the middle panels, the corresponding divergence of the solar wind speed is shown. The right panels show the same as the left panels but enlarged for the heliosheath region from 85 AU to 120 AU. The bottom panels show V as used in the kinetic asymmetric TS model as fitted to the solutions for the solar wind speed profiles as obtained with the threedimensional (3-D) HD model in the heliospheric nose and tail regions in the heliosheath. Note that the TS is in the bottom panels (for the 3-D HD model) in a different location than for the top and middle panels (for the 2-D asymmetric TS model).

Holzer and Banks [1969], who derived an expression for the case that the thermal speeds are higher than the streaming speeds. Terms related to the gravitational potential are omitted because the inner boundary of our computational domain is a sphere with radius 10 AU and at this and greater heliocentric distances gravitational effects are negligible. [11] For the present study, only the HD solution for the solar wind plasma in the heliosheath between the solar wind termination shock and the heliopause is of interest. Such a solution was presented as a contour plot of the density by Langner et al. [2005] and, e.g., McComas et al. [2004, Figure 6], which shows a typical result of a computation of the interaction of a nonspherically symmetric solar wind with latitude-independent mass flow, a speed varying from 400 km s1 in the equatorial plane to 800 km s1 above the solar poles, a number density of 8.3 cm3 in the equatorial plane, and a temperature of 5 104 K at the Earth orbit. The

interstellar medium moves with 26 km s1 relative to the Sun and has a number density of 0.1 cm3 and a temperature of 8000 K. [12] The velocity profiles in the upwind and the downwind direction are given in the lower left and right panels of Figure 1, respectively (open circles). These profiles show the expected behavior: in the upwind direction a strong decrease of the velocity is followed by a ramp-like structure towards the heliopause (compare, e.g., with the results shown in Figure 2 of Pauls et al. [1995]), and in the downwind direction there is a moderate increase of the velocity in the direction of the heliotail. This, at a first glance, counterintuitive result is, however, well-known [see, e.g., Zank et al., 1996; Pauls and Zank, 1997; Florinski et al., 2003]: it is due to a charge-exchange-induced narrowing of the heliotail that as a consequence of the continuity of the mass flow, results in a local acceleration. While we cannot

4 of 14

A01106


A01106

Figure 2. Solutions of a symmetric model of the heliosphere in the nose region (q = 90) for anomalous protons during solar minimum conditions (a = 10) in the A > 0 (top panels) and A < 0 (bottom panels) polarity cycles for different solar wind speed scenarios in the heliosheath. (left) Energy spectra at radial distances of 1 AU, 60 AU, and the TS position (90 AU in these cases). (right) Differential intensities as a function of radial distance at energies of 16 MeV and 200 MeV, respectively. The TS position is indicated in the right panels with a vertical line. Black lines represent the reference solutions with V / 1/r2 in the heliosheath, thick gray (red in online version) lines for V / 1/r8, and dark gray (blue in online version) lines for V / r2. exclude the possibility that the narrowing of the tail is somewhat overestimated in our model due to the heliotail boundary conditions used, note, however, that the acceleration effects remain small. 2.5. Implementation of the Fluid-Model Results Into the Kinetic Model [13] The intensities of anomalous protons are assumed to be zero in the interstellar medium for both models. The position of the TS in the asymmetrical model, where the anomalous protons are injected at all latitudes, is assumed at rs = 90 AU in the equatorial nose region, rs = 95 AU at the poles, and rs = 100 AU in the equatorial tail region of the heliosphere, respectively. The compression ratio, s = 3.2, and a shock precursor scale length of L = 1.2 AU [see also Langner et al., 2003] are used. This means that up to the shock, the solar wind speed V decreases by 0.5 s starting at L, then abruptly as a step function to the downstream value, in total V/s. Beyond the TS, V changes as 1/r2, 1/r8, and r2, respectively, up to rHP for the symmetrical model. For the asymmetrical model, scenarios were investigated where V / 1/r2 in the heliosheath at all latitudes and then where V / 1/r8 in the heliosheath in the heliospheric nose region changing to V / r2 in the tail region. It must be noted that the modulation model assumes spherical geometry and a purely radial solar wind velocity. This means that some flow divergence is already built in. Using velocity profiles from the hydrodynamic model in this case leads to a significant over- or underestimation (depending on

whether velocity is decreasing faster or slower than 1/r2) of the actual r V because of the extra r2 expansion factor. In particular, in a 3-D HD model flow slowdown toward the stagnation point near the nose is accompanied by an overexpansion (the flow is spreading into the wings of the heliosheath) and the actual divergence is larger than in the spherical model with V / 1/r8. Conversely, the tail flow is essentially one-dimensional and its acceleration as r2 would lead to an overestimation of r V by the spherical model. Basically, the heliosheath flow is nearly incompressible (and hence has zero divergence) all by itself and the effects that cause deviations from this are mostly related to charge exchange with neutral hydrogen. This implies that the difference between the extreme cases, which is already small, will be further reduced. [14] Concerning the global latitudinal dependence of the solar wind speed, it is assumed that V changes from 400 km s1 in the equatorial plane (q = 90, q = 270) to 800 km s1 in the polar regions for solar minimum conditions [Phillips et al., 1995] and that V has no clear latitudinal dependence for moderate solar maximum conditions as was observed by Ulysses [McComas et al., 2000]. (Note that q is counted from 0 to 360 in order include the downwind heliosphere for q 180.) The factor of 2.0 increase for solar minimum conditions happens in the heliosphere for 60 q, q 300, and for 240 q 120. For moderate solar maximum conditions it is assumed to be on average 400 km s1 in the whole heliosphere. A modified version of the current

5 of 14

A01106


A01106

Figure 3. Similar to Figure 2 but for moderate solar maximum modulation conditions (a = 75).

sheet model of Hattingh and Burger [1995], which emulates the waviness of the current sheet in two spatial dimensions, was used [see also Langner, 2004]. The current sheet ‘‘tilt angles,’’ a, as calculated by J. T. Hoeksema of Wilcox Solar Observatory (available at http://wso.stanford. edu), were assumed to represent moderate solar maximum modulation conditions with a = 75 and solar minimum conditions with a = 10 during A > 0 (e.g., 1990 – 2001) and A < 0 (e.g., 1980 – 1990) polarity cycles. This is accompanied by a change in V with changing solar activity and an increase in the values of perpendicular diffusion where the latter implies decreasing drift with increasing solar activity [see also Ferreira et al., 2004]. [15] The r V determines the energy losses and gains of charged particles in the heliosphere, e.g., r V < 0 implies further acceleration of particles in the heliosheath, although clearly not to the same extent as at the TS where its value becomes significantly negative; r V > 0 in the heliosheath implies adiabatic deceleration of particles, the same as normally occurring inside the TS where it gets larger with decreasing radial distances to produce the characteristic E1 spectral slopes for protons at Earth. For an incompressible fluid approach, r V = 0 with V / 1/r2 so that neither energy losses nor gains for charged particles can then play a role in the heliosheath, which is the simplest assumption as used before [Langner and Potgieter, 2004a, 2005].

3. Modeling Results and Discussion [16] In Figure 1 the different V profiles and the resulting r V in the equatorial plane as a function of radial distance are shown. The bottom panels of Figure 1 also show the fits for V as used in the kinetic asymmetric TS model to the solutions for the solar wind speed profiles as obtained with the 3-D HD model in the heliospheric nose and tail

regions in the heliosheath. From these fits it is clear that the increase in the solar wind speed (as discussed in the previous section) will probably occur for distances in the heliosheath for the heliospheric tail region >350 AU, after an initial strong decrease. The strong decrease of V in the nose region (V / 1/r8) is necessitated by the fact that the solar wind speed must become zero when the heliopause is reached. While principally in agreement with (part of) the numerically computed velocity profiles, the power law fits shown in the top panels of Figure 1, which were used in the asymmetric TS model, must therefore be considered extreme cases. [17] The first results illustrate the effect of the different V-profiles used in the heliosheath region on the modulation of anomalous protons for a symmetrical model of the heliosphere (with the Sun in the center) by showing in Figures 2 and 3 differential intensities for the solutions in the nose region (q = 90), for solar minimum (a = 10) and moderate solar maximum modulation conditions (a = 75) for both polarity cycles. The energy spectra are shown at radial distances of 1 AU, 60 AU, and the TS position (rs = 90 AU in these cases) and the differential intensities as a function of radial distance at energies of 16 MeV and 200 MeV, respectively. These solutions represent what happens in the nose region of the heliosphere. Although these scenarios for the solar wind profiles are strictly speaking unrealistic for a symmetrical heliosphere, it is shown here for comparative purposes only. The dominant feature in the spectra for anomalous protons is the wellreported ‘‘acceleration cutoff’’ in energy [Potgieter and Moraal, 1988; Moraal et al., 1999; Potgieter and Langner, 2004, 2005]. This ‘‘cutoff’’ clearly moves significantly to higher energies when V is assumed to decrease stronger than 1/r2 in the heliosheath. The opposite happens when V increases in the heliosheath. This shift in the ‘‘cutoff’’ in the

6 of 14

A01106


A01106

Figure 4. Solutions of the asymmetric model for anomalous protons in the heliospheric nose ((top) q = 90), for the Voyager 1 trajectory latitude ((middle) q = 55), and for the heliospheric tail ((bottom) q = 270) regions. This is done for solar minimum conditions (a = 10) in the A > 0 polarity cycle for different solar wind speed scenarios in the heliosheath. (left) Energy spectra at radial distances of 1 AU, 60 AU, and at the TS position (90 AU in the nose region, 95 AU at the poles, and 100 AU in the tail region). (right) Differential intensities as a function of radial distance at energies of 16 MeV and 200 MeV, respectively. The TS position is indicated in the right panels with a vertical line. Black lines are for solutions where V / 1/r2 in the heliosheath for all latitudes, gray (red in online version) lines are solutions where V changes from V / 1/r8 in the nose region of the heliosheath to V / r2 in the tail region of the heliosheath. spectra is probably caused by the values of the diffusion coefficient just outside the TS. For V / 1/r8 krr just outside the TS will be smaller than for the V / 1/r2 case, therefore reducing the rate of escape for higher-energy particles, which lead to the shift in the ‘‘cutoff’’ to higher energies. The opposite will happen for the V / r2 case. The acceleration process, as measured by the energy where the ‘‘cutoff’’ occur, is even more effective for the A < 0 cycle because shock-drift is in the same direction as perpendicular diffusion in the theta direction, therefore also reducing the escape rate for the particles in this cycle. When V / 1/r8 in the heliosheath, additional acceleration of the anomalous protons occurs in the heliosheath, evident from the increasing intensities as a function of radial distance in this region. When V / r2 in the heliosheath, additional deceleration (adiabatic cooling) of particles occur in this

region, resulting in the decreasing intensities as a function of radial distance in the heliosheath. It must be noted that the sharp decrease in intensities in front of the heliopause is caused by the assumed boundary conditions which sets the solutions to zero at the heliopause for all the cases because it is assumed that there are no sources or sinks of anomalous protons at the heliopause, which might be an oversimplification. The effects of the different solar wind profiles in the heliosheath on the spectra are therefore only a product of the conditions in the heliosheath region and just downstream of the TS. These factors have a considerable effect on the anomalous proton spectra inside the TS; they differ for the A > 0 cycle at solar minimum at Earth for the V / 1/r2 and V / r2 scenarios in the heliosheath by more than a factor of 10 at 1 GeV and more than a factor of 100 for the V / 1/r2 and V / 1/r8 scenarios. These differences are

7 of 14

A01106


A01106

Figure 5. Similar to Figure 4 but for the A < 0 polarity cycle at solar minimum modulation conditions (a = 10). enhanced for the A < 0 cycle. The differences which occur inside the TS with decreasing radial distances are a product of the global modulation effects. The different V-profiles obviously cause differences in the intensities at the TS at different latitudes, which in turn have an effect on the global modulation, as shown in Figures 2 and 3. The differences are also caused by the harder spectrum at the TS for the V / 1/r8 scenario, caused by the shift in the ‘‘cutoff’’ to higher energies, which in turn led to a supplement for the lowenergy population through adiabatic cooling of the highenergy particles. If the stronger than V / 1/r2 scenario in the heliosheath would be real, the anomalous intensities should increase beyond the TS, which should then be measurable by the Voyager 1 spacecraft in the near future. [18] For moderate solar maximum modulation shown in Figure 3 the ‘‘cutoff’’ in the spectra moves to even higher energies, as noted before by Potgieter and Langner [2003] which again should be a measurable feature in order to distinguish between these scenarios and to establish which one is closer to reality. [19] In Figures 4 to 9 the differences in modulation are illustrated that occur for the asymmetrical TS model of the heliosphere when V / 1/r2 in the heliosheath at all latitudes, compared to when V changes from V / 1/r8 in the helio-

spheric nose region (q = 90) to V / r2 in the tail region (q = 270) over latitude in the heliosheath. As mentioned, the latter V-profile was calculated with the hydrodynamic model and is therefore not arbitrary. It must, however, be noted that this is one of a series of possible profiles computed for V and that these profiles can be highly dynamic and timedependent. The solar wind profiles used for the heliosheath in the asymmetric TS model are therefore only used here to show the maximum effects of a variable solar wind speed. [20] Solutions are shown in Figures 4 to 7 for the heliospheric nose (q = 90), the Voyager 1 trajectory latitude (q = 55), and the tail (q = 270) regions of the heliosphere for solar minimum conditions (a = 10) in the A > 0 cycle (Figure 4), for solar minimum conditions in the A < 0 cycle (Figure 5), for moderate solar maximum conditions (a = 75) in the A > 0 cycle (Figure 6), and for moderate solar maximum conditions in the A < 0 cycle (Figure 7). This is done for the two different V scenarios for the asymmetrical model in the heliosheath. Energy spectra are shown at radial distances of 1 AU, 60 AU, and the TS position (90 AU in the nose region, 95 AU at the polar regions, and 100 AU in the tail region) and the differential intensities as a function of radial distance at energies of 16 MeV and 200 MeV, respectively.

8 of 14

A01106


A01106

Figure 6. Similar to Figure 4 but for the A > 0 polarity cycle at moderate solar maximum conditions (a = 75). [21] Comparing Figures 4 to 7, several modulation aspects can be noted that will not be discussed again here in detail, e.g., the difference between A > 0 and A < 0 cycles, etc. The emphasis is rather on the effects of the different V-scenarios in the heliosheath. For a discussion of other modulation effects relevant to the anomalous component, see Langner and Potgieter [2004a, 2004b]. It follows from these figures that the modulation differences caused by the different V-scenarios are the largest for the A < 0 polarity cycle. The intensities are consistently less for the scenario where V / 1/r2 in the heliosheath for all latitudes than for V / 1/r8 ! r2 in the heliosheath. The differences are for most cases less in the tail region than in the nose region. The intensities for the V-scenarios differ markedly in the heliosheath as a function of radial distance for 16 and 200 MeV particles, especially in the nose regions of the heliosphere. Since the anomalous component dominates at these energies, observations from the Voyager 1 spacecraft should give clear indications which of the presented scenarios are indeed more reasonable. [22] The corresponding anomalous proton intensities at 16 MeV and 200 MeV are shown as contour plots in the meridional plane in Figure 8, with V / 1/r2 for all polar angles in the heliosheath and in Figure 9 with V / 1/r8 ! r2

in the heliosheath. This is done again for the A > 0 and A < 0 cycles and for solar minimum and maximum. The intensities in the heliosheath at these energies differ markedly for the two scenarios. Interesting features of the asymmetric model follow from this comparison. For the V / 1/r2 scenario for both magnetic cycles during solar minimum, the 16 MeV intensities in the heliosheath are moderately latitude-dependent, with the highest intensities in the equatorial region, which get less latitude-dependent for moderate maximum solar activity. Apart from the heliosheath being wider in the tail region, not much modulation difference occurs between the tail and nose. This scenario predicts that there should be an abundance of these anomalous particles in most of the heliosheath at almost all latitudes. At 200 MeV the latitude dependence gets significantly stronger, especially with a = 10 during the A < 0 cycle. With a = 75 the intensity distribution in the heliosheath becomes more uniform in latitude despite a change in polarity. For the second scenario, on the other hand, this picture of the heliosheath changes to one where the intensities are much less in the tail region and differently distributed with the highest intensity clearly in the nose region of the heliosphere in all modulation situations. The spectra at the TS position are also altered the most in the

9 of 14

A01106


A01106

Figure 7. Similar to Figure 4 but for the A < 0 polarity cycle at moderate solar maximum conditions (a = 75). heliospheric nose region as shown in Figures 4 to 7. For this scenario the clear abundance of anomalous particles may occur only in the nose direction of the heliosphere, fortunately also in the directions the two Voyager spacecraft are moving. The asymmetrical modulation features are significantly enhanced by this approach.

4. Summary and Conclusions [23] A two-dimensional asymmetric TS model is used to study the effects on the modulation of anomalous protons of different solar wind speed profiles and consequently different profiles for the r V in the heliosheath. This is done for both polarity cycles and with modulation changes from solar minimum to moderate solar maximum conditions. First, several V scenarios in the heliosheath were studied with a symmetrical heliosphere, with the

Sun placed in the center. Although some of the V scenarios are unphysical for a symmetrical heliosphere, it were done as an empirical study in order to quantify the effects of the different V profiles. Second, two different scenarios for V in the heliosheath were studied with an asymmetrical model of the heliosphere. For the first scenario for V it was assumed that the solar wind is incompressible in this region, resulting in V / 1/r2 in this region. For the second scenario a three-dimensional timedependent hydrodynamic model was used to calculate possible spatial dependencies for V in the heliosheath. An approximation of this calculation, where V is changing from V / 1/r8 in the heliospheric nose region to V / r2 in the heliospheric tail region over latitude, was used as input for the asymmetric TS model. [24] The following were found for the solutions of the symmetrical model: (1) The cutoff in energy spectra move

Figure 8. Contour intensities for anomalous protons at (top) 16 MeV and (bottom) 200 MeV for an asymmetrical heliosphere and a scenario where V / 1/r2 in the heliosheath for all polar angles, for solar minimum and moderate solar maximum conditions, and the (left) A > 0 and (right) A < 0 polarity cycles. Note that the legends of the contours correspond to the exponent of the base 10 on a logarithmic scale and that the scaling of the legends differ for the 16 MeV and 200 MeV plots. The TS and the heliopause are indicated in the plots by black lines. 10 of 14

A01106


Figure 8 11 of 14

A01106

A01106


Figure 9. Similar to Figure 8, but for a scenario where V changes from V / 1/r8 in the heliosheath nose to V / r2 in the heliosheath tail. 12 of 14

A01106

A01106


significantly to higher energies when V is assumed to decrease stronger than 1/r2 in the heliosheath. (2) The opposite occurs when V is assumed to increase in the heliosheath. (3) When V / 1/r8 in the heliosheath, additional acceleration of the anomalous protons takes place in the heliosheath region, resulting in altered accelerated spectra at the TS. (4) When V / r2 in the heliosheath, additional deceleration of particles occur in this region, resulting in decreasing intensities in the heliosheath as a function of radial distance. (5) These factors cause the spectra at Earth for the V / 1/r2 and V / r2 scenarios to differ by more than a factor of 10 for the A > 0 cycle at solar minimum conditions at 1 GeV and by more than a factor of 100 for the V / 1/r2 and V / 1/r8 scenarios. [25] For the solutions of the asymmetrical model, with a scenario where the solar wind speed profile in the heliosheath was kept constant at V / 1/r2 for all polar angles and a scenario where it changes from V / 1/r8 in the heliospheric nose region to V / r2 in the heliospheric tail region, it was found that (1) modulation differences between the two V-scenarios are the largest for the A < 0 polarity cycle. (2) For V / 1/r2 at all latitudes the intensities are less than for V / 1/r8 ! r2 in the heliosheath. These differences are also less in the tail region of the heliosphere than in the nose region. (3) The intensities as a function of radial distance for 16 and 200 MeV particles differ markedly in the heliosheath for the two V-scenarios, especially in the nose region. (4) The mild dependence of the 16 MeV intensities in the heliosheath as a function of polar angle for the V / 1/r2 scenario for solar minimum and moderate maximum conditions shifted to one where the intensities are significantly higher in the nose region only for the V / 1/r8 ! r2 scenario. (5) The highest intensities for the V / 1/r2 scenario occurred in the equatorial regions with little difference between the nose and tail regions, despite the heliosheath being much wider in the tail region. [26] It is concluded that the Voyager 1 spacecraft is in an advantageous position so that future measurements of anomalous protons in the heliosheath should provide us with a modulation picture of which solar wind speed profile is most realistic in this region; the predicted modulation differences are large enough to be observable. From the asymmetrical model it was found that the effectiveness of particle acceleration at the TS, as seen by the higher intensities at the TS, can be altered by changing the V-profile in the heliosheath over latitude, especially in the A < 0 cycle, but not significantly. Similarly, this can also happen for the symmetrical model with a V profile that decreases stronger than the previously assumed 1/r2 dependence because particles then are accelerated to significantly higher energies at the TS and in the heliosheath. The V / 1/r2 scenario for all latitudes predicts that there should be an abundance of anomalous particles in the heliosheath at almost all latitudes. For the scenario where V / 1/r8 ! r2, thus a strongly varying V with latitude in the heliosheath from the nose to the tail, the abundance of anomalous particles may occur only in the nose direction of the heliosphere. [27] This modeling work, which includes an asymmetrical heliospheric structure and a change over latitude for the radial solar wind speed profile in the heliosheath, gives insight into the modulation of anomalous protons that may

A01106

occur at the TS and beyond in the heliosheath, which is presently been probed by the Voyager 1 spacecraft. [28] Acknowledgments. We thank the SA National Research Foundation (NRF) and the Deutsche Forschungs Gemeinschaft (DFG) for partial financial support under the bilateral DFG/NRF agreement, GUN 2049412, and DFG SCHL201/14-1/14-2/14-3. UWL also wishes to thank the NRF for partial financial support during his postdoctoral research in Germany and the Claude Leon Foundation for financial support during his postdoctoral research in South Africa. [29] Shadia Rifai Habbal thanks Eric R. Christian and another referee for their assistance in evaluating this paper.

References Baranov, V. B., and Y. G. Malama (1993), Model of the solar wind interaction with the local interstellar medium: Numerical solution of self-consistent problem, J. Geophys. Res., 98(A9), 15,157. Baranov, V. B., and N. A. Zaitsev (1998), On the problem of the heliospheric interface response to the cycles of the solar activity, Geophys. Res. Lett., 25(21), 4051, doi:10.1029/1998GL900044. Borrmann, T. (2005), Ein hydrodynamisches 3D-mehrkomponente modell der Heliosphaere und ihrer wechselwirkung mit kosmicher strahlung, Ph.D. thesis, Ruhr-Univ. Bochum, Bochum, Germany. Borrmann, T., and H. Fichtner (2005), On the dynamics of the heliosphere on intermediate and long time-scales, Adv. Space. Res., 35(12), 2091, doi:10.1016/j.asr.2005.08.039. Burger, R. A., M. S. Potgieter, and B. Heber (2000), Rigidity dependence of cosmic ray proton latitudinal gradients measured by the Ulysses spacecraft: Implications for the diffusion tensor, J. Geophys. Res., 105(A12), 27,447. Christian, E. R., A. C. Cummings, and E. C. Stone (1988), Evidence for anomalous cosmic-ray hydrogen, Astrophys. J. Lett., 334, L77. Christian, E. R., A. C. Cummings, and E. C. Stone (1995), Observations of anomalous cosmic-ray hydrogen from the Voyager spacecraft, Astrophys. J. Lett., 446, L105, doi:10.1086/187941. Cummings, A. C., and E. C. Stone (1998), Anomalous cosmic rays and solar modulation, Space Sci. Rev., 83(1/2), 51. Fahr, H. J., T. Kausch, and H. Scherer (2000), A 5-fluid hydrodynamic approach to model the solar system-interstellar medium interaction, Astron. Astrophys., 357, 268. Ferreira, S. E. S., and K. Scherer (2004), Modulation of cosmic-ray electrons in the outer heliosphere, Astrophys. J., 616(2), 1215. Ferreira, S. E. S., M. S. Potgieter, R. A. Burger, B. Heber, H. Fichtner, and C. Lopate (2001), Modulation of Jovian and galactic electrons in the heliosphere: 2. Radial transport of a few MeV electrons, J. Geophys. Res., 106(A12), 29,313. Ferreira, S. E. S., M. S. Potgieter, and K. Scherer (2004), Modulation of cosmic-ray electrons in a nonspherical and irregular heliosphere, Astrophys. J., 607(2), 1014, doi:10.1086/383566. Fichtner, H. (2001), Anomalous cosmic rays: Messengers from the outer heliosphere, Space Sci. Rev., 95(3/4), 639. Fichtner, H. (2005), Cosmic rays in the heliosphere: Progress in the modelling during the past ten years, Adv. Space Res., 35(4), 512, doi:10.1016/j.asr.2005.01.012. Florinski, V., G. P. Zank, and N. V. Pogolerov (2003), Galactic cosmic ray transport in the global heliosphere, J. Geophys. Res., 108(A6), 1228, doi:10.1029/2002JA009695. Florinski, V., N. V. Pogorelov, and G. P. Zank (2004a), Physics of the Outer Heliosphere: Third International IGPP Conference, Riverside, California, 8 – 13 February 2004, Am. Inst. of Phys., Melville, N.Y. Florinski, V., et al. (2004b), Do anomalous cosmic rays modify the termination shock?, Astrophys. J., 610(2), 1169, doi:10.1086/421901. Garcia-Munoz, G. M., G. M. Mason, and J. A. Simpson (1973a), A new test for solar modulation theory: The 1972 May-July low-energy galactic cosmic-ray proton and helium spectra, Astrophys. J., 182, L81. Garcia-Munoz, G. M., G. M. Mason, and J. A. Simpson (1973b), The anomalous 1972 low energy galactic cosmic ray proton and helium spectra, Proc. 13th Int. Conf. Cosmic Rays, 2, 1304. Haasbroek, L. J. (1997), The transport and acceleration of charged particles in the heliosphere, Ph.D. thesis, Potchefstroom Univ., Potchefstroom, South Africa. Haasbroek, L. J., and M. S. Potgieter (1997), Cosmic ray modulation in a non-spherical heliosphere during solar minimum conditions, Adv. Space Res., 19(6), 921. Hattingh, M., and R. A. Burger (1995), A new simulated wavy neutral sheet drift model, Adv. Space Res., 16(9), 213. Holzer, T. E. (1972), Interaction of the solar wind with the neutral component of the interstellar gas, J. Geophys. Res., 77, 5407.

13 of 14

A01106


Holzer, T. E., and P. M. Banks (1969), Accidentally resonant charge exchange and ion momentum transfer, Planet Space Sci., 17, 1074. Jokipii, J. R. (1986), Particle acceleration at a termination shock: 1. Application to the solar wind and the anomalous component, J. Geophys. Res., 91, 2929. Jokipii, J. R. (1990), The anomalous component of cosmic rays, in Physics of the Outer Heliosphere, edited by S. Grzedzielski and D. E. Page, p. 169, Elsevier, New York. Jokipii, J. R., and J. Ko´ta (1989), The polar heliospheric magnetic field, Geophys. Res. Lett., 16, 1. Karmesin, S. R., P. C. Liewer, and J. U. Brackbill (1995), Motion of the termination shock in response to an 11 year variation in the solar wind, Geophys. Res. Lett., 22(9), 1153. Krimigis, S. M., et al. (2003), Voyager 1 exited the solar wind at a distance of 85 AU from the Sun, Nature, 426(6962), 45. Langner, U. W. (2004), Effects of termination shock acceleration on cosmic rays in the heliosphere, Ph.D. thesis, Potchefstroom Univ., Potchefstroom, South Africa. Langner, U. W., and M. S. Potgieter (2004a), Solar wind termination shock and heliosheath effects on the modulation of protons and anti-protons, J. Geophys. Res., 109, A01103, doi:10.1029/2003JA010158. Langner, U. W., and M. S. Potgieter (2004b), Effects of the solar wind termination shock and heliosheath on the heliospheric modulation of galactic and anomalous Helium, Ann. Geophys., 22, 3063. Langner, U. W., and M. S. Potgieter (2005), Modulation of galactic protons in an asymmetrical heliosphere, Astrophys. J., 630(2), 1114, doi:10.1086/ 431547. Langner, U. W., M. S. Potgieter, and W. R. Webber (2003), Modulation of cosmic ray protons in the heliosheath, J. Geophys. Res., 108(A10), 8039, doi:10.1029/2003JA009934. Langner, U. W., M. S. Potgieter, T. Borrmann, and H. Fichtner (2005), Effects of diffferent solar wind speed profiles in the heliosheath on the modulation of cosmic ray protons, Astrophys. J., in press. le Roux, J. A., and H. Fichtner (1997), A self-consistent determination of the heliospheric termination shock structure in the presence of pickup, anomalous, and galactic cosmic ray protons, J. Geophys. Res., 102(A8), 17,365. McComas, D. J., et al. (2000), Solar wind observations over Ulysses’ first full polar orbit, J. Geophys. Res., 105(A5), 10,419. McComas, D. J., et al. (2004), The interstellar boundary explorer (IBEX), in Physics of the Outer Heliosphere, edited by?, AIP Conf. Proc., 719, 162. McDonald, F. B., et al. (2003), Enhancements of energetic particles near the heliospheric termination shock, Nature, 426(6962), 48. Moraal, H., C. D. Steenber, and G. P. Zank (1999), Simulations of galactic and anomalous cosmic ray transport in the heliosphere, Adv. Space Res., 23(3), 425. Parker, E. N. (1965), The passage of energetic charged particles through interplanetary space, Planet. Space Sci., 13, 9. Pauls, H. L., and G. P. Zank (1997), Interaction of a nonuniform solar wind with the local interstellar medium: 2. A two-fluid model., J. Geophys. Res., 102, 19,779. Pauls, H. L., G. P. Zank, and L. L. Williams (1995), Interaction of the solar wind with the local interstellar medium, J. Geophys. Res., 100(A11), 21,595. Pesses, M. E., D. Eichler, and J. R. Jokipii (1981), Cosmic ray drift, shock wave acceleration, and the anomalous component of cosmic rays, Astrophys. J., 246, L85. Phillips, J. L., et al. (1995), Ulysses solar wind plasma observations from pole to pole, Geophys. Res. Lett., 22(23), 3301. Potgieter, M. S. (1996), Heliospheric modulation of galactic electrons: Consequences of new calculations for the mean free path of electrons between 1 MeV and 10 GeV, J. Geophys. Res., 101(A11), 24,411. Potgieter, M. S., and S. E. S. Ferreira (2002), Effects of the solar wind termination shock on the modulation of Jovian and galactic electrons in

A01106

the heliosphere, J. Geophys. Res., 107(A7), 1089, doi:10.1029/ 2001JA009040. Potgieter, M. S., and U. W. Langner (2003), Modulation of anomalous protons with increasing solar activity, Adv. Space Res., 32(4), 687, doi:10.1016/S0273-1177(03)00359-4. Potgieter, M. S., and U. W. Langner (2004), Modulation and acceleration of anomalous protons in the outer heliosphere, Adv. Space Res., 34(1), 132, doi:10.1016/j.asr.2003.02.062. Potgieter, M. S., and U. W. Langner (2005), Modulation of cosmic rays: Perpendicular diffusion and drifts in a heliosphere with a solar wind termination shock, Adv. Space Res., 35(4), 554, doi:10.1016/ j.asr.2005.01.008. Potgieter, M. S., and H. Moraal (1988), Acceleration of cosmic rays in the solar wind termination shock. I - A steady state technique in a spherically symmetric model, Astrophys. J., 330, 445, doi:10.1086/166482. Rucinski, D., A. C. Cummings, G. Gloeckler, A. J. Lazarus, E. Mbius, and M. Witte (1996), Ionization processes in the heliosphere - Rates and methods of their determination, Space Sci. Rev., 78, 73. Scherer, K., and H. J. Fahr (2003), Solar cycle induced variations of the outer heliospheric structures, Geophys. Res. Lett., 30(2), 1045, doi:10.1029/2002GL016073. Scherer, K., and S. E. S. Ferreira (2005), A heliospheric hybrid model: Hydrodynamic plasma flow and kinetic cosmic ray transport, ASTRA, 1, 17. Sreenivasan, S. R., and H. Fichtner (2001), ACR modulation inside a non-spherical modulation boundary, in The Outer Heliosphere: The Next Frontiers, COSPAR Colloq. Ser., vol. 11, edited by K. Scherer et al., p. 207, Elsevier, New York. Steenberg, C. D., and H. Moraal (1996), An acceleration/modulation model for anomalous cosmic-ray hydrogen in the heliosphere, Astrophys. J., 463, 776, doi:10.1086/177289. Steinolfson, R. S. (1994), Termination shock response to large-scale solar wind fluctuations, J. Geophys. Res., 99(A7), 13,307. Stone, E. C., and A. C. Cummings (2001), Estimate of the location of the solar wind termination shock, Proc. 27th Int. Conf. Cosmic Rays, 10, 4263. Stone, E. C., and A. C. Cummings (2003), The approach of Voyager 1 to the termination shock, Proc. 28th Int. Conf. Cosmic, 3889. Webber, W. R., and J. A. Lockwood (2004), Onset and amplitude of the 11-year solar modulation of cosmic ray intensities at the Earth and at Voyagers 1 and 2 during the period from 1997 to 2003, J. Geophys. Res., 109, A09103, doi:10.1029/2004JA010492. Webber, W. R., J. A. Lockwood, F. B. McDonald, and B. Heikkila (2001), Using transient decreases of cosmic rays observed at Voyagers 1 and 2 to estimate the location of the heliospheric termination shock, J. Geophys. Res., 106(A1), 253. Whang, Y. C., L. F. Burlaga, Y. M. Wang, and N. R. Sheeley (2004), The termination shock near 35 latitude, Geophys. Res. Lett., 31, L03805, doi:10.1029/2003GL018679. Zank, G. P. (1999), Interaction of the solar wind with the local interstellar medium: A theoretical perspective, Space Sci. Rev., 89(3/4), 413. Zank, G. P., and H. R. Mu¨ller (2003), The dynamical heliosphere, J. Geophys. Res., 108(A6), 1240, doi:10.1029/2002JA009689. Zank, G. P., H. L. Pauls, L. L. Williams, and D. T. Hall (1996), Interaction of the solar wind with the local interstellar medium: A multifluid approach, J. Geophys. Res., 101, 21,639.

T. Borrmann, H. Fichtner, and U. W. Langner, Institut fu¨r Theoretische Physik IV, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany. ([email protected]) M. S. Potgieter, Unit for Space Physics, School of Physics, North-West University, Potchefstroom 2520, South Africa.

14 of 14