Solar wind termination shock and heliosheath effects on the modulation of protons and antiprotons

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi:10.1029/2003ja010158, 2004 Solar wind termination shock and heliosheath effects on the modulation of protons and antiprotons U. W. Langner and M. S. Potgieter Unit for Space Physics, School of Physics, Potchefstroom University for CHE, Potchefstroom, South Africa Received 23 July 2003; revised 15 September 2003; accepted 23 October 2003; published 6 January 2004. [1] The interest in the role of the solar wind termination shock (TS) and heliosheath in cosmic ray modulation studies has increased significantly as the Voyager 1 and 2 spacecraft approach the estimated position of the TS. For this work the modulation of cosmic ray protons (p) and antiprotons (p), and the consequent charge-sign dependence, is studied with a numerical model including a TS with diffusive shock acceleration, a heliosheath, and drifts. The model allows a comparison of modulation with and without a TS. A more fundamental and comprehensive set of diffusion coefficients is used, applicable to a number of cosmic ray species during both magnetic polarity cycles of the Sun. Newly computed and improved local interstellar spectra for p and p are used. The modulation of p with an anomalous component is also done to establish charge-sign dependence at low energies. The modulation effects of the heliosheath and TS are illustrated for the different species and how they affect the computed p/p. We found that the computed modulation for p is surprisingly different from p and that the heliosheath is important for cosmic ray modulation. The local proton interstellar spectrum may not be known at energies <1 GeV until a spacecraft actually approaches the heliopause because of the strong modulation that occurs in the heliosheath, the effect of the TS, and the presence of anomalous protons. For antiprotons, in contrast, these effects are less pronounced. INDEX TERMS: 2104 Interplanetary Physics: Cosmic rays; 2124 Interplanetary Physics: Heliopause and solar wind termination; 2152 Interplanetary Physics: Pickup ions; 2162 Interplanetary Physics: Solar cycle variations (7536); 2114 Interplanetary Physics: Energetic particles, heliospheric (7514); KEYWORDS: cosmic rays, heliosphere, termination shock, protons, antiprotons Citation: Langner, U. W., and M. S. Potgieter (2004), Solar wind termination shock and heliosheath effects on the modulation of protons and antiprotons, J. Geophys. Res., 109,, doi:10.1029/2003ja010158. 1. Introduction [2] The solar wind termination shock (TS) and the heliosheath are prominent and interesting features of the heliosphere. The interest in the role of these features in cosmic ray modulation studies has increased significantly as the Voyager 1 and 2 spacecraft approach the estimated position of the TS [e.g., Webber et al., 2001]. Observations near the predicted location of the TS [e.g., McDonald et al., 2000] enable us to study cosmic ray modulation in the outer heliosphere in more detail, especially the effects of the TS and what level of modulation may occur in the heliosheath. There is consensus that the TS should be near 90 AU [e.g., Stone and Cummings, 2001], but the position of the heliopause is uncertain, probably at least 30 50 AU beyond the TS. For our purposes, this region between the heliopause (outer boundary) and the TS is called the heliosheath. Recently, Langner et al. [2003] described a numerical model that incorporated a TS with a heliosheath and with drifts. They illustrated some characteristics of the model, e.g., that the TS could cause significant changes in the Copyright 2004 by the American Geophysical Union. 0148-0227/04/2003JA010158 cosmic ray proton radial gradients at almost all energies of interest to modulation studies, and that the heliosheath might contribute, e.g., up to 60% to the overall modulation for protons at 0.5 GeV, but it depends on the polarity of the heliospheric magnetic field (HMF). The TS combined with charge-sign dependent effects caused by drifts should have clear observable effects on cosmic ray modulation. These aspects are studied in more detail in this paper using the TSdrift model of Langner et al. [2003] applied to cosmic ray protons (p) and antiprotons (p). Because of large-scale gradient, curvature and current sheet drifts, p for example, will drift inward primarily through the polar regions of the heliosphere during so-called A < 0 polarity cycles, when the HMF is directed toward the Sun in the northern hemisphere. Protons, on the other hand, will then drift inward primarily through the equatorial regions of the heliosphere, encountering the wavy heliospheric current sheet in the process. During the A > 0 polarity cycles the drift directions for the differently charged species reverse, so that a clear 22-year cycle is caused [e.g., Burger and Potgieter, 1999]. [3] Modulation of the different cosmic ray species inside the heliosphere hides the value of the different local interstellar spectra (LIS) for p and p especially below a few GeV. However, improved calculations for the p and in 1of12

particular the p LIS based on sophisticated models for the propagation of cosmic rays in the Galaxy were published by Moskalenko et al. [2002, 2003]. With these more reliable LIS, a fresh and more fundamental approach to diffusion coefficients [e.g., Burger et al., 2000; Giacalone and Jokipii, 1999, 2001; Langner et al., 2003], and good observations closer to the TS, the numerical study of the modulation of cosmic rays in the outer heliosphere can be done more quantitatively. [4] The following topics are addressed in more detail: (1) The effects of the TS on the modulation of p, with and without an anomalous component, and for the first time also for p, for both HMF polarity cycles, and as solar activity changes from minimum to moderate maximum conditions. (2) The differences in modulation with and without a TS. (3) The level of modulation in the simulated heliosheath and the importance of this barrier modulation for the different species, and (4) to establish the consequent charge-sign dependent effects by means of the modulated p/p. The difference between minimum and moderate maximum modulation conditions is contained in this model through changing the current sheet tilt angle a from 10 to 75, but it is also accompanied by changes in the solar wind with increasing modulation and changes in the value of perpendicular diffusion, where the latter implies decreasing drift with increasing solar activity as described by Langner et al. [2003]. 2. Modulation Model [5] The model is based on the numerical solution of the time-dependent cosmic ray transport equation [Parker, 1965]: @f @t ¼ V ð þ h v DiÞrf þrðk s rfþþ 1 3 ðrvþ @f @ ln p þ J source ; where f (r,p,t) is the omnidirectional cosmic ray distribution function, p is the particle momentum, r is position, and t is time, with V the solar wind velocity. The averaged guiding center drift velocity for a near isotropic cosmic ray distribution is given by hv D i = r(k T e B ), with e B = B/B m and B m the magnitude of the modified background HMF as given below. The unit for distance is given in Astronomical Units (AU). Terms on the right-hand side represent convection, gradient and curvature drifts, diffusion, adiabatic energy changes and a source function, respectively. The symmetric part of the tensor K S consists of a parallel diffusion coefficient (k k ), and perpendicular diffusion coefficients (k? ). The antisymmetric elements (k T or k A ) of the tensor describes gradient and curvature drifts in the large scale HMF. The function J source represents any local source, e.g., the Jovian electrons, the pick-up ions, etc. For this work, this function specifies a source of anomalous protons, similar to le Roux et al. [1996]. ð1þ Figure 1. Computed differential intensities for protons, antiprotons, and for protons with an anomalous component as function of kinetic energy for both polarity cycles with different tilt angles as indicated in the graphs. Spectra are shown at Earth (top two panels), compared to data from various experiments for protons and antiprotons as compiled by Moskalenko et al. [2002, 2003], and at r = 60 AU in the equatorial plane (bottom panel) in comparison with IMP and Pioneer 10 data as compiled by Steenberg [1998]. The TS is at 90 AU and the LIS at 120 AU. Figure 2. Left panels: Computed differential intensities for galactic protons as a function of kinetic energy for both polarity cycles and solar minimum and moderate maximum conditions, at radial distances of 1, 60, 90 and 115 AU (bottom to top) in the equatorial plane. Right panels: Corresponding differential intensities as function of radial distance for 0.016, 0.2 and 1.0 GeV, respectively. Solutions without a TS are given here as dotted lines. In all panels the TS is at 90 AU, as indicated, and the LIS specified at 120 AU. 2of12

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Figure 3. Similar to Figure 2 for antiprotons. 4of12

Figure 4. Similar to Figure 2 but for protons with an anomalous component. Here the dotted lines represent solutions with s = 2.0 instead of s = 3.2. 5of12

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[6] The details of the model were described by Langner et al. [2003] with the essentials repeated here for the convenience of the reader. Equation (1) is solved timedependently in a spherical coordinate system as a combined diffusive shock acceleration and drift modulation model with two spatial dimensions, neglecting any azimuthal dependence and is symmetric around the equatorial plane. Similar numerical models were described by, e.g., Jokipii et al. [1993], Steenberg and Moraal [1996], and Potgieter and Ferreira [2002]. [7] The outer modulation boundary was assumed at r b = 120 AU, where the LIS for the different galactic species are specified. The proton LIS of Moskalenko et al. [2002, 2003] is given by 8 exp 4:64 0:08ðln EÞ 2 p 2:91 ffiffiffi >< E if E < 1:0GeV j LIS ¼ exp 3:22 2:86 ln E 1:50 ; >: if E 1:0 GeV E ð2þ with E kinetic energy in GeV, j LIS = R 2 f the differential intensity in particles.m 2.s 1 sr 1.MeV 1 and R = pc/q rigidity in GV, p the particle momentum, c the speed of light in space, q the particle charge and f the distribution function as in equation (1). The antiproton LIS of Moskalenko et al. [2002, 2003] is given by j LIS ¼ exp 9:60 0:10ðln EÞ 2 1:91 expð EÞ if E 0:94 GeV and j LIS ¼ 2:4 10 3 E ð 2:81Þ ð0:81 þ 7:74E 1:81 Þ 2 : ð3þ [8] For protons with an anomalous component a source was injected at the TS position at a energy of 86.0 kev as a delta function with a magnitude set to give reasonable fits to anomalous p data at 60 AU and to p data at Earth [see also Steenberg, 1998]. The solutions are independent of this injection energy as long as it is lower than the cutoff energy for anomalous protons [see also Steenkamp, 1995]. [9] The tilt angles were assumed to represent solar minimum and moderate maximum modulation conditions with a =10 and a =75, respectively, during A > 0 (e.g., 1990 2001) and A < 0 (e.g., 1980 1990) magnetic polarity cycles. The HMF was assumed to have a basic Parkerian geometry in the equatorial plane but was modified in the polar regions similar to Jokipii and Kóta [1989]. [10] The TS was assumed at r S = 90 AU with a compression ratio s = 3.2 for p and p, and a shock precursor scale length of L = 1.2 AU. For the precursor scale length in front of the shock, V decreases in the equatorial plane, for example, from the upstream value of V 1 according to the relationship given by Langner et al. [2003] [see also le Roux et al., 1996]. This means that up to the shock, V(r) decreases by 0.5s starting at L, then abruptly as a step function to the downstream value, in total to V 1 /s. Beyond the TS, V decreases further as 1/r 2 to the outer boundary. For p with an anomalous component solutions with s = 2.0 are also shown. The HMF increases by a factor s at the TS and V changes from 400 km s 1 in the equatorial plane (q =90 )to 800 km s 1 in the polar regions. This increase of a factor of 2 happens in the whole heliosphere for 120 q 60 for solar minimum conditions, but it is reduced to a factor of 1.10 with 170 q 10 For moderate maximum conditions. [11] The diffusion coefficients k k, k?, and k T are similar to those given by Burger et al. [2000] for a steady-sate model, except for minor changes to their values caused by the introduction of the TS in this model. Perpendicular diffusion is assumed to enhance toward the poles in order to fit the observed the latitudinal gradients [e.g., Burger et al., 2000; Ferreira et al., 2000]. For a complete description of these diffusion coefficients, see Langner et al. [2003]. They are optimal for a numerical TS model without an azimuthal dependence and without solar maximum transient effects, e.g., global merged interaction regions. This set can also be used by changing only the rigidity dependences of k k accordingly at low rigidities for electrons and positrons to give reasonable fits to a variety of data sets and is the same for both polarity cycles [Langner et al., 2003; Langner and Potgieter, 2003; Potgieter and Langner, 2003c]. However, it must be noted that a quantitative fitting of the data was not the purpose of this study, but rather to find one set of diffusion coefficients that is generally compatible to the basic observations made for a variety of cosmic ray species, applicable for both polarity cycles and for a model including a TS. 3. Modeling Results [12] The results shown in this section concentrate on five aspects of heliospheric modulation: (1) Can the TS model produce realistic modulation at Earth. (2) The difference in the modulation of p and p, given the vastly different LIS. (3) How the inclusion of a TS in the model alters the modulation of p and p and the subsequent effects on chargesign dependence. (4) The nature of modulation effects to be expected near the TS and in the heliosheath. (5) The effects of increased solar activity. [13] In Figure 1 it is illustrated that the set of diffusion coefficients based on fundamental arguments gives solutions reasonably consistent with proton and antiproton data at Earth and for anomalous protons at 60 AU. The data are from various experiments (e.g., BESS, IMAX, CAPRICE, etc.) for p and p and were compiled by Moskalenko et al. [2002, 2003], while the data for p with an anomalous Figure 5. Intensity ratios of solutions with a TS compared to those without a TS as a function of kinetic energy at radial distances of 1, 60, 90 and 115 AU (left panels) and as function of radial distance at energies of 0.016, 0.2 and 1.0 GeV (right panels) for both polarity cycles in the equatorial plane. Top four panels are for protons and bottom four for antiprotons, all with a = 10. 7of12

component (e.g., from Pioneer 10) were compiled by Steenberg [1998]. The model also gives results compatible to the observed radial dependence and the latitudinal dependence of protons as shown by Langner et al. [2003]. To obtain the same reasonable compatibility with the anomalous p at 60 AU, the compression ratio had to be decreased from s = 3.2 to s = 2.0. The solutions in the inner heliosphere (r < 40 AU) are largely insensitive to this change (see Figure 4). Decreasing s causes the peak in the modulated anomalous p spectrum to shift to lower energies as the data seem to require and could be caused by a decreasing shock strength with increasing solar activity. These quantitative aspects of anomalous p modulation were discussed in detail by Steenberg and Moraal [1996] and Potgieter and Langner [2003a, 2003b] and is not pursued further in this work. The main purpose of Figure 1 is to simply illustrate that the modeling parameters are not arbitrary but can reproduce the main features of observed p and p modulation. For our purpose any further fine-tuning would not contribute significantly to the better understanding of modulation in the heliosphere. [14] The left panels of Figures 2 to 4 show the modulation obtained with the TS model with respect to the LIS for galactic p and p, and for p with an anomalous component, as a function of kinetic energy, respectively. This is done at 1, 60, 90 and 115 AU in the equatorial plane for the A >0 and A < 0 polarity cycles with a = 10 and a = 75, respectively. The right panels of Figures 2 to 4 show the corresponding differential intensities at 0.016, 0.20 and 1.00 GeV as function of radial distance in the equatorial plane, respectively for solutions with a TS and without a TS. In Figure 4 the proton solutions are repeated with an anomalous component with s = 3.2, as the rest but also with s = 2.0 for reasons given above. Comparing the energy spectra and radial dependence of the intensities for the chosen energies in these three figures, one can see how: (1) The modulation differs from solar minimum to moderate solar maximum for the given species. (2) The effect of switching the HMF polarity from A >0toA < 0. (3) The difference between p and p modulation and how it is affected by incorporating a TS; (4) The effect on proton modulation when an anomalous component is added, and (5) the barrier type modulation caused by the heliosheath. [15] In Figure 5 the effects of the TS on p and p modulation are illustrated by depicting the ratio of intensities obtained with and without a TS as a function of kinetic energy at radial distances of 1, 60, 90 and 115 AU, and as a function of radial distance at energies of 0.016, 0.2 and 1.0 GeV, respectively, in the equatorial plane for both polarity cycles when a =10. The ratios as a function of energy converge naturally at E > 10 GeV because the TS has progressively less modulation effects the higher the energy. The ratios as function of radial distance approach unity at 120 AU where the LIS were specified. [16] The modulation differences between p and p are emphasized by plotting p/p, and p/p with an anomalous p component, as a function of kinetic energy. This charge-sign dependence is shown in Figure 6 for both polarity cycles in the equatorial plane at 1 AU and 90 AU, with a =10 and a = 75, respectively. As a reference, all the modulated ratios are compared to the corresponding (unmodulated) LIS ratios. [17] Next, the modulation computed to take place in the heliosheath, between r b and r s, is compared to what happens between r b and 1 AU (LIS to Earth) and between r s and 1 AU (TS to Earth). This comparison is emphasized by showing in Figure 7 the intensity ratios j LIS /j 1, j LIS /j 90 and j 90 /j 1 for the three species under consideration as a function of kinetic energy in the equatorial plane for both polarity cycles with a =10. Note that for a few cases the ratios become less than unity. 4. Discussion [18] In this section the lesser known modulation features shown above will be discussed individually. From Figure 1 it follows that the p modulation is significantly different from galactic p modulation primarily caused by the p LIS below 2 GeV. Shown in Figure 5, the effect of the TS on the modulation of p and p with respect to the relevant LIS is profound, it decreases the intensities at lower energies (e.g., at 100 MeV) but increases it at higher energies (e.g., at 1 GeV), because the lower energy particles are being accelerated to higher energies. A remarkable feature with the TS is that the modulated p spectra at large radial distances for the A < 0 cycle can actually exceed the corresponding LIS between 200 MeV and a few GeV, which cannot happen without a TS. This effect is not pronounced for p, seem absent for larger a s and clearly depends on the drift direction. [19] The energy spectra in Figures 2 to 4 also depict how the slopes of the modulated p spectra obtain the characteristic spectral index (energy slope) caused by adiabatic cooling at higher energies than for protons. The reason is that the p LIS already has an almost (E) 1 dependence, with E the kinetic energy. This causes much larger modulation with respect to the relevant LIS for p than for p at energies below 1 GeV but the level clearly depends on the polarity cycle. Beyond the TS (r > r s ), the spectra obtain a much steeper energy slope and can cause rather strong negative radial gradients at very low energies as is evident from the top left panel of Figure 2. This is caused by the assumed divergence free solar wind speed in the heliosheath (V / 1/r 2 ). These low energy particles obviously experience increased modulation primarily caused by k? / R 1/3 at these radial distances. This implies that the p LIS may not be known at these low energies until a spacecraft actually approaches the heliopause. This feature is not present in the solutions without a TS, as expected. For p with an anomalous component, shown in Figure 4, the intensities at the Figure 6. Ratios, p/p, and p/p with an anomalous proton component (bottom panels), as a function of kinetic energy in the equatorial plane at 1 AU (top panels) and at the TS (r s = 90 AU) for both polarity cycles with a =10 (left panels) and a = 75 (right panels), respectively. All ratios are compared to the LIS p/p ratio (at 120 AU) as a reference. The ratios without a TS are shown in the third row panels only at 90 AU. 8of12

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Figure 7. Intensity ratios j LIS /j 1, j LIS /j 90 and j 90 /j 1 (120 to 1 AU, 120 to 90 AU and 90 to 1 AU) for protons, antiprotons, and for protons with an anomalous component as a function of kinetic energy in the equatorial plane with a =10 ; left panels: for A > 0, right panels for A < 0. The LIS is at 120 AU and the TS at 90 AU. TS where the anomalous source is injected follow the characteristic (E ) 1.2 spectrum with s = 3.2 and (E ) 2.0 with s = 2.0, dictated by the acceleration of the anomalous protons at the TS with E < 100 MeV. [20] The modulation in the heliosheath is clearly an important part of the total modulation for p and p as shown in the right panels of Figures 2 to 4. The TS plays in this regard a prominent role. For both species its effect becomes more pronounced the lower the energy. At higher energies, the barrier effect progressively diminishes; the radial dependence beyond the shock may vanish or even become negative to create a conspicuous shock effect on the radial intensity profiles. This effect is strongly dependent on the HMF polarity cycles, and also on the level of drifts allowed 10 of 12

beyond the TS, e.g., the peculiar radial dependence for 200 MeV protons in the A < 0 cycle and pinthea > 0 cycle for a =10. For an elaborate discussion on these effects, see also Langner et al. [2003]. [21] The radial dependence for p is noteworthy because it is very weak except for E < 300 MeV where it becomes somewhat stronger. The shock effects (the abrupt changes in the radial intensities) above this energy are almost negligible. Despite the large error bars of the p data, our modeling indicates that the computed LIS may be too low, causing peculiar little modulation but with interesting consequences, e.g., that measurements in the inner heliosphere below 1 GeV may already indicate the spectral shape of the LIS in sharp contrast to protons. [22] From Figure 4 follows that the inclusion of an anomalous p component has a profound effect on the proton intensities at larger radial distances (r > 60 AU) at E < 100 MeV, but an almost negligible effect on the intensities at Earth. Near the TS the spectrum is of course substantially different because of the injected anomalous source. The radial dependence of the 16 MeV intensity is consequently significantly different, but at the higher energies the effect diminishes as the acceleration cut-off is approached. Our results indicate that a s between 3.2 and 2.0 is preferred when anomalous p are also considered, in fact, this ratio cannot be determined effectively using only galactic p and p spectra. Clearly, a strong shock with s = 4 is most unlikely [see also Potgieter and Langner, 2003a, 2003b]. [23] The computed ratios in Figures 5 and 6 indicate quantitatively how much the computed modulation for p and p changes when a TS is present. The differences between the two models can be significant, especially with E < 100 300 MeV and r > 60 AU. The computed charge-sign dependence of p and p becomes more significant in the outer heliosphere, manifesting at higher energies, than at 1 AU and more pronounced with a =75. As the modulation process becomes dominated by adiabatic energy losses in the inner heliosphere, p/p reaches steady values with E < 100 MeV, but somewhat depending on the polarity cycle. At r s = 90 AU this also happens but at lower energies. These steady values are significantly different when a = 75. For p with an anomalous part, the p/p exhibits similar behavior at 1 AU, but differs significantly at r s, as expected. The p/p curves as a function of energy also crosses for the A > 0 and A < 0 cycles, at 1 AU at a much higher energy than at r s, and the effect seems to move to higher energies with increasing solar activity. For the no shock ratios at r s this cross-over still occurs but at much lower energies, indicating that it is strongly imbedded in the drift modulation and cannot easily be overwhelmed by TS effects. [24] According to Figure 7 a significant level of modulation occurs in the heliosheath for p when A > 0 with E < 200 MeV, to a somewhat lower energy with A < 0 when j LIS /j 90 drops below unity because the intensity at the TS becomes larger than the LIS value. For p the equivalent happens in the A > 0 cycle although clearly not as pronounced as for galactic protons (note scale differences), e.g., at 10 MeV the heliosheath (barrier) modulation for galactic p is a factor of 100 but only a factor of 7forp. Obviously, these ratios all converge at a high enough energy where no modulation is present (not shown). The addition of the anomalous protons changes the ratios significantly for energies up to 1 2 GeV; j LIS /j 1 is only slightly changed as expected. 5. Conclusions [25] Five aspects of heliospheric modulation were highlighted: (1) The differences in the modulation of galactic protons and antiprotons. (2) How the inclusion of a TS in the model alter this modulation and the consequent chargesign dependence. (3) How the inclusion of anomalous protons changes the modulation for protons. (4) The kind of modulation effects to be expected near the TS and in the heliosheath and (5) The effects of increased solar activity. Qualitatively, our results for protons are consistent to those of Jokipii et al. [1993] but there are quantitatively marked differences as noted before [Langner et al., 2003]. The antiproton TS modulation results and the effects of the TS on charge-sign dependent modulation are new. The results confirm that this numerical model with a TS can reasonably reproduce the modulation between the outer boundary and Earth for p, p and anomalous p. This will also be illustrated for electrons, positrons and helium in follow-on papers. Although our results are most reasonable it seems unavoidable that the diffusion coefficients should change time-dependently, together with the tilt angle and parameters like the compression ratio. Our results indicate that a TS compression ration between 3.2 and 2.0 is preferred when anomalous p are also considered, in fact, this ratio cannot be determined effectively using only galactic p and p spectra. A strong shock with s = 4 is most unlikely. [26] The modulation produced by a model with and without a TS can differ significantly, depending on the species and HMF polarity. These differences increase toward lower energies and larger radial distances. The p/p approaches a steady value at all radial distances for lower energies which is a manifestation of the adiabatic cooling the species experience in the heliosphere and is independent of the shape of the LIS. Strong charge-sign dependent effects occur for protons and antiprotons enhanced by the vastly different LIS and the different effect the TS has on these cosmic ray particles. [27] The heliosheath can be considered a distinguishable modulation barrier for both p and p with the overall effect clearly energy, polarity cycle and solar activity dependent, e.g., for galactic p most of the modulation may occur in the heliosheath for E < 200 MeV at solar minimum during A < 0 cycles. For p the equivalent happens in the A > 0 cycle but not as pronounced as for protons, e.g., at 10 MeV the heliosheath (barrier) modulation for galactic p is a factor of 100 but only a factor of 7forp. This aspect could make it easier to detect in the inner heliosphere any additional galactic antiproton component [Moskalenko et al., 2003] given that the p LIS indeed has the illustrated spectral form at lower energies and that the chosen energy is not too low. [28] This study indicates that the proton LIS may not be known at E < 1 GeV until a spacecraft actually approaches the heliopause because of the strong modulation that occurs in the heliosheath, the effect of the TS and the presence of anomalous protons. For antiprotons, in contrast, these effects are less pronounced. 11 of 12

[29] Acknowledgments. We thank Stefan Ferreira, Harm Moraal and Adri Burger for useful discussions, and the SA National Research Foundation and the SA Department of Labor (DoL) for partial financial support. [30] Shadia Rifai Habbal thanks Jonathan F. Ormes and Joe Giacalone for their assistance in evaluating this paper. References Burger, R. A., and M. S. Potgieter (1999), The effect of large current sheet tilt angles in numerical modulation models: A theoretical assessment, Conf. Pap. Int. Cosmic Ray Conf. 26th, 7, 13 16. 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, 27,447 27,455. Ferreira, S. E. S., M. S. Potgieter, R. A. Burger, and B. Heber (2000), Modulation effects of anisotropic perpendicular diffusion on cosmic ray electron intensities in the heliosphere, J. Geophys. Res., 105, 18,305 18,314. Giacalone, J., and J. R. Jokipii (1999), The transport of cosmic rays across a turbulent magnetic field, Astrophys. J., 520, 204 214. Giacalone, J., and J. R. Jokipii (2001), The transport of energetic particles and cosmic rays in the heliosphere, Adv. Space Res., 27, 461 469. Jokipii, J. R., and J. Kóta (1989), The polar heliospheric magnetic field, Geophys. Res. Lett., 16, 1 4. Jokipii, J. R., J. Kóta, and E. Merényi (1993), The gradient of galactic cosmic rays at the solar wind termination shock, Astrophys. J., 405, 782 786. Langner, U. W., and M. S. Potgieter (2001), Differences in proton and antiproton modulation in the heliosphere, Conf. Pap. Int. Cosmic Ray Conf. 27th, 4, 3657. Langner, U. W., and M. S. Potgieter (2003), Effects of the solar wind termination shock on charge-sign dependent cosmic ray modulation, Adv. Space, in press. 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. le Roux, J. A., M. S. Potgieter, and V. S. Ptuskin (1996), A transport model for the diffusive acceleration and modulation of anomalous cosmic rays in the heliosphere, J. Geophys. Res., 101, 4791 4804. McDonald, F. B., B. Heikkila, N. Lal, and E. C. Stone (2000), The relative recovery of galactic and anomalous cosmic rays in the distant heliosphere: Evidence for modulation in the heliosheath, J. Geophys. Res., 105, 1 8. Moskalenko, I. V., A. W. Strong, J. F. Ormes, and M. S. Potgieter (2002), Secondary antiprotons and propagation of cosmic rays in the galaxy and heliosphere, Astrophys. J., 565, 280 296. Moskalenko, I. V., A. W. Strong, S. G. Mashnik, and J. F. Ormes (2003), Challenging cosmic-ray propagation with antiprotons: Evidence for a fresh nuclei component?, Astrophys. J., 586, 1050 1066. Parker, E. N. (1965), The passage of energetic charged particles through interplanetary space, Planet. Space Sci., 13, 9 49. 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 the heliosphere, J. Geophys. Res., 107(A7), 1089, doi:10.1029/ 2001JA009040. Potgieter, M. S., and U. W. Langner (2003a), Modulation and acceleration of anomalous protons in the outer heliosphere, Adv. Space Res, in press. Potgieter, M. S., and U. W. Langner (2003b), Modulation of anomalous protons with increasing solar activity, Adv. Space Res., 32(4), 687 692. Potgieter, M. S., and U. W. Langner (2003c), Heliospheric modulation of cosmic ray electrons and positrons: Effects of the heliosheath and the solar wind termination shock, Astrophys. J., in press. Steenberg, C. D. (1998), Modelling of anomalous and galactic cosmic ray modulation in the outer heliosphere, Ph.D. thesis, Potchefstroom Univ., Potchefstroom, South Africa. Steenberg, C. D., and H. Moraal (1996), An acceleration/modulation model for anomalous cosmic ray hydrogen in the heliosphere, Astrophys. J., 463, 776 783. Steenkamp, R. (1995), Shock acceleration as source of the anomalous component of cosmic rays in the heliosphere, Ph.D. thesis, Potchefstroom Univ., Potchefstroom, South Africa. Stone, E. C., and A. C. Cummings (2001), Estimate of the location of the solar wind termination shock, Conf. Pap. Int. Cosmic Ray Conf. 27th, 10, 4263. 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, 253 260. U. W. Langner and M. S. Potgieter, Unit for Space Physics, School of Physics, Potchefstroom University for CHE, 2520, Potchefstroom, South Africa. (fskuwl@puk.ac.za; fskmsp@puk.ac.za) 12 of 12