Modulation effects of anisotropic perpendicular diffusion on cosmic ray electron intensities in the heliosphere

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 05, NO. A8, PAGES 8,305-8,34, AUGUST, 2000 Modulation effects of anisotropic perpendicular diffusion on cosmic ray electron intensities in the heliosphere S. E. S. Ferreira, M. S. Potgieter, and R. A. Burger Research Unit for Space Physics, School of Physics, Potchefstroom University for Christian Higher Education, Potchefstroom, South Africa B. Heber Max-Planck-Institut far Aeronomy, Katlenburg-Lindau, Germany Abstract. The modulation of cosmic ray electrons provides a useful tool to study the diffusion tensor applicable to heliospheric modulation. Electron modulation responds directly to the assumed energy dependence of the diffusion coefficients below ~500 MeV in contrasto protons which experience large adiabatic energy losses below this energy. As a result of this and because drifts become unimportant for electrons at these low energies, conclusions can be made about the appropriate diffusion coefficients. Using a modulation model, we illustrate the role of anisotropic perpendicular diffusion on electron modulation. In general, we find that perpendicular diffusion dominates electron modulation below ~00 MeV. Enhancing it in the polar direction typically produced an increase in modulation for both the A > 0 (e.g., ~990 to ~2000) and A < 0 (e.g., ~980 to ~990) solar magnetic polarity cycles. It also causes the radial dependence of the intensity to become more uniform throughouthe heliosphere, and causes a significant reduction in the latitude dependence of the intensities at all radial distances, with the largest effects in the inner heliosphere and at low energies. This agrees with studies of cosmic ray protons, which suggest that perpendicular diffusion enhanced in the polar direction of the heliosphere is required in conventional drift models to explain the small latitudinal gradients observed for protons on board the Ulysses spacecraft. The role of enhanced perpendicular diffusion was further investigated by examining electron modulation as a function of the "tilt angle" c of the wavy current sheet. In general, a reduction occurred between the modulation differences caused by drifts as a function of c for both polarity cycles. This work illustrates that anisotropic perpendicular diffusion has profound effects on the modulation of galactic cosmic ray electrons during both polarity cycles.. Introduction distinguish between perpendicular diffusion in the radial/azimuthal direction and in the polar direction by using Diffusion perpendicular to the heliospheric magnetic field here the symbols K r and K 0, respectively. The effects of (HMF) plays an important role in the heliospheric modulation enhancing K 0 on cosmic ray modulation have been of galactic cosmic rays. This role became directly evident occasionally studied using numerical modulation models from latitude-dependent modulation first studied more than [e.g., K6ta and Jokipii, 995; Potgieter, 996, 997], and it 24 years ago with a two-dimensional model by Fisk [976]. was illustrated that the enhancement is necessary to make Even with the introduction of global and current sheet drifts these model solutions compatible with the small-latitude in models of increasing complexity [K6ta and Jokipii, 983; effects observed for protons on board the Ulysses spacecraft Potgieter and Moraal, 985; le Roux and Potgieter, 990; [Heber et al., 996]. As a matter of fact, in a comprehensive Burger and Hattingh, 998], the importance of perpendicular drift model an enhanced K 0 produces latitude effects for both diffusion remained. K6ta and Jokipii [995] revived the protons and electrons as a function of rigidity that are concept that perpendicular diffusion may not be isotropic, compatible to observations made by Ulysses during the fastgiven the manner in which field fluctuations expand as they latitude scan period [Potgieter et al., 997; Burger et al., are convected with the solar wind in the regions over the solar 999]. The reason why this enhancement is required in poles. The original concept was described by Jokipii and models based on a modified Parker HMF geometry is that for Parker [969]. The expectation is that in these polar regions so-called A > 0 epochs (e.g., 990 to ~2000) the original drift the perpendicular diffusion should be larger, even models predict, in general, a very large positive latitudinal significantly larger, in the polar direction than in the gradient for cosmic ray protons. This is because they allow for radial/azimuthal direction. It therefore becomes necessary to very little modulation in the polar zones of the heliosphere during this epoch, especially if large parallel diffusion is used Copyright 2000 by the American Geophysical Union. Paper number 2000JA in the polar regions of the simulated heliosphere [Potgieter and Haasbroek, 993; Haasbroek and Potgieter, 995]. No comprehensive theory exists as yet for the perpendicular /00/2000JA $09.00 diffusion coefficient, although it is arguably the most 8,305

2 8,306 FERREIRA ET AL.: HELlOSPHERIC MODULATION OF COSMIC RAY ELECTRONS important element of the diffusion tensor. The best that can be antisymmetric element KA describes gradient and curvature done at this stage is to make reasonable assumptions about its drifts in the large-scale magnetic HMF where the pitch angle value, spatial, and rigidity (energy) dependence and study its averaged guiding center drift velocity for a near isotropic effects on heliospheric modulation using numerical models. cosmic ray distribution given by vo)= v x (KA eb), with Of particular relevance for our effort of continually improving eb = B/B, and B is the magnitude of the background HMF. our working knowledge of cosmic ray modulation is that the Equation () was solved in a spherical coordinate system heliospheric modulation of cosmic ray electrons provides a assuming azimuthal symmetry, and for a steady state, that is, useful tool in understanding and in determining the tyf/tyt = O. This two-dimensional (2-D) model which appropriate diffusion coefficients. Computed electron emulates the effects of the wavy HCS was developed by modulation responds directly to what is assumed for the Hattingh and Burger [995a] as an improvement to the wellenergy dependence of the diffusion coefficients below-500 known models of Potgieter and Moraal [985] and Burger MeV. This is in contrast to protons which experience and Potgieter [989]. Using this 2-D model is still well significant adiabatic energy changes below this energy and which consequently obscure the effects of changing the justified based on a comparison with a three-dimensional (3- D) wavy HCS model developed by Hattingh and Burger energy dependence of any of the diffusion coefficients. [995b]; for details on the numerical approaches of the Another aspect is that drifts become progressively less important with decreasing electron energy to have almost no comparative study and in particular how the wavy HCS is handled and what the relation is between (x and the waviness effect on electron modulation below-00 MeV. of the HCS, see Burger and Hattingh [ 995], Hattingh et al. For the present work, cosmic ray electrons are used to illustrate how important perpendicular diffusion is to their heliospheric modulation. This is the completion of the topic [997], Hattingh [ 998], Ferreira [ 998], and Ferreira et al. [999a]. For the parallel and perpendicular diffusion coefficients, first presented by Potgieter and Ferreira [999a] during the and the "drift" coefficient K^, the following general forms Cospar Meeting in Nagoya, Japan. For a detailed description were assumed, respectively: of the importance of the various parameters in electron modulation, the reader is also referred to Potgieter [996]. Kii = Koflf (R)f2(O,r ), glr '- a gll, Preliminary reports were given by Potgieter et al. [999a],?R (2) Ferreira and Potgieter [999], and Ferreira et al. [999b]. K o = b Kii, K,4 = (K,4)o 3B' For reviews on observed electron modulation, see Ferrando [997] and Evenson [998]. Here,8 is the ratio of the speed of the cosmic ray particles to In the next sections the role of enhancing Kx0 compared to the speed of light; Ji(R) gives the rigidity dependence in GV; K,are illustrated on () the computed electron intensities and Ko is a constant in units of 6 x 020 cm 2 s 'l, with a a constant spectr as a function of radial distance and polar angle, (2) which determines the value of Kxr, and b is a constant modulation as a function of the latitudinal extent of the determining the value of Kx0. Diffusion perpendicular to the heliospheric current sheet which can be approximated by the HMF was enhanced in the polar direction by assuming K00 = "tilt angle" a. It is well known that the wavy heliospheric Kño- bkii with b > a in (2). The details of the various current sheet (HCS) is a very important modulation parameter constants will be given below with the corresponding results. as were illustrated with drift models [dokipii and Thomas, (KA)o specifies the amount of drifts allowed, with (K )o =.0 98; K6ta and dokipii, 983; le Roux and Potgieter, 990; a maximum, which is assumed for this work. The effective Burger and Hattingh, 998]. The computed modulation radial sin2w, diffusion with W the coefficient angle between is given the by radial Krr '- direction Kii cos2w and + K, the effects of a are, however, dependent on other modulation parameters, in particular the parallel and perpendicular averaged HMF direction. Note that W -> 90 ø when r > 0 AU diffusion coefficients, as will be illustrated below. Finally, the with the polar angle 0 -> 90% and W -> 0 ø when 0 -> 0 ø, role of K 0 and K, in electron modulation is further which means that Kii dominates Krr in the inner and polar investigated by examining the modulation effects of their regions and K r dominates in the outer equatorial regions of assumed rigidity dependence. the heliosphere. The differential intensities, j oc R2f, are calculated as particles m '2 sr 'l s 4 MeV 4. For the spatial 2. Modulation Model dependence, J (O,r) = + r/ro was assumed, with r0 = AU. This assumption is based on the work by Haasbrock [997] The modulation model that was used for this study is based and Haasbrock et al. [995], who illustrated that simulating on the numerical solution of the cosmic ray transport equation the long-term modulation of 220 MeV protons observed by [Parker, 965]: Ulysses from 99 to 998 is the best represented by assuming Kii independent of latitude, and Kll ocr n, with 0 < n 8t 8nR ' < 2. The value of n has been refined by Potgieter [997] to be ctf=(v+(vd)).vf +V.(Ks.Vf)+I (v.v) ctf () where f (r,r,t) is the cosmic ray distribution function, r is position, and t is time; R - pc/q is rigidity, where p is momentum, c is the speed of light, and q is particle charge. Terms on the right-hand side represent convection, gradient and curvature drifts, diffusion and adiabatic energy changes respectively, with V the solar wind velocity. The symmetric part of the tensor Ks consists of diffusion coefficients parallel (Kii) and perpendicular (Kñ0 and K ) to the average HMF. The between 0.5 and.0. The first set of solutions were computed with a simple rigidity dependence for Kii, Kñr and K 0 given by Ji(R) = It R/R0 with R > 0.4 GV, and Ji(R) = It (0.4 GV)/R0, with R < 0.4 GV and R0 = GV. Different assumptions for Ji(R) may change the slopes of the spectra at low energies as was illustrated in detail by Potgieter [996], but it is not principally important for the results and conclusions of the present study of anisotropic perpendicular diffusion. Note that

3 FERREIRA ET AL.' HELIOSPHERIC MODULATION OF COSMIC RAY ELECTRONS 8,307 Ki/fl and Kfffl have the same rigidity dependence that becomes flat and constant below 0.4 GV. This feature causes the electron modulation a given position in the heliosphere to become almost constant at energies <-50 MeV. At energies <-0 MeV, Jovian electrons may contribute to the computed spectra but are neglected for this study. The effect of a solar wind termination shock is also neglected. Based on the termination shock modeling by Haasbroek et al. [997a, b], there are no compelling reasons to believe that the introduction of a termination shock will change the qualitative conclusions of the present work. It also requires an inherently time-dependent code which demands on computer time make a study as presented here impractical. For the basic modeling details of termination shock effects on modulation, see are shown Haasbroek [997]. For new attempts on modeling Jovian from electron propagation, see Fichtner et al. [2000] and Ferreira et al. [2000]. The HMF magnitude as described by Parker [965] was modified significantly, but only in the polar regions of the heliosphere according to Jokipii and Kdta [989]; see also because Haasbroek and Potgieter [995]. This modification is?otgieter supported qualitatively by measurements of the HMF in the polar regions of the heliosphere by Ulysses [Forsyth et al., 995]. The so-called Fisk field [Fisk, 996] was not considered here because of its complex geometry and basic dynamic nature applicable only to the polar regions with fast solar wind speeds which when combined with the slow solar wind regions requires in the modeling a different approach than used so far for numerical modulation models [see, e.g., Kdta and Jokipii, 997, 999]. The outer boundary of the simulated heliosphere was assumed at 00 AU, which seems plane a reasonable consensus value. The solar wind speed V was assumed to change from 400 km s - in the equatoorial plane (0 = 90 ø) to a maximum of 800 km s - when 0 < 60, which is in quantitative agreement with measurements of the solar wind speed on board Ulysses [Phillips et al., 995]. The galactic electron spectrum published from the COMPTEL results [Strong et al., 994] was used as the local interstellar o loo rigidities (energies) between 0.4 GV (0.4 GeV) and-.0 MV (-0.6 MeV). Clearly, increasing K 0 leads to more modulation at all relevant energies; a factor of 5 increase causes more than a factor of l0 increase in modulation energies below - GeV at AU. The shapes of the modulated spectra in Figures l a and lb are characteristic of the respectivepochs; the relatively sharper changes in the slopes of the spectra in 0 0 Figure l a compared to Figure lb is a characteristic of drift modulation because the electrons propagate through different regions of the hellosphere to AU during the different HMF ' a) ø'2 'K.... ' : polarity epochs. The differences are less conspicuous at 60 AU. As Kño was increased, a shift in the tum-ups from MeV for b = 0.05 to -40 MeV for b followed for the A lo 2 lo lo 0 > 0 epoch as shown in Figure l a, but this shift is almost negligible for the A < 0 epoch as shown in Figure lb. For the A > 0 cycle at 60 AU in Figure l c, the increase in Kñ0 only 0-2 affects energies between-30 MeV and- GeV, while for the 0 '3 A < 0 cycle at 60 AU in Figure l d, the increase has little 0-4,,C,!,, d),...,...,...,, effect on the modulation spectra at all energies. The enhancement of Kñ0 has less significant effects in the middle Kinetic Energy ( GeV ) Kinetic Energy ( GeV ) to outer hellospheric regions. The consequences of increasing Kñ0 on the radial Figure. Computed electron spectra for the A > 0 and A < 0 polarity cycles shown at (a, b) AU and at (c, d) 60 AU, redependence of electron differential intensities are shown for spectively. Here Kñ0 was increased from 5% to 25% of the the A < 0 cycle in Figure 2a with b = 0.05 and in Figure 2b value of Kii. Solutions are shown in the equatorial plane (0 = for b = 0.5. Intensities are shown for 0.30 GeV electrons 90 ø) with b = 0.05, 0.0, 0.5, and 0.25 in (2), respectively, from 0 = 90 ø to 0 = 5 ø. Figure 2a shows a weak radial from top to bottom. dependence for intensity at 0 = 90 ø at r < -80 AU, but for... ß > :;..;: j:/'.,./...:,:.5.'.-...: ' / I I I I I Radial distance ( AU ) Radial distance ( AU ) Figure 2. Electron differential intensities at 0.30 GeV as a function of radial distance for the A < 0 polarity cycle. Here Kñ0 was increased from (a) 5% to (b) 5% of Kll. Intensities for 0 = 90 ø, 80 ø,..., 20 ø, 0 ø, and 5 ø, respectively, bottom to top. Units are particles m -2 sr - s 'l MeV 'l. spectrum (LIS). Newer calculations by Strong and Moskalenko [999] have not been considered for this study they still seem controversial; in this regard, see also [ 996] and Peterson et al. [ 999]. 3. Results and Discussions For the first results the following constants were used in (2): Ko ; a- 0.05; b , 0.0, 0.5, and 0.25 respectively; (KA)o-.0; and ct - l0 ø, which representsolar minimum conditions. The consequences electron spectra of increasing K 0 from 5% to 25% of KiI are shown in Figures la and lb, from top to bottom for AU in the equatorial (0-90ø), for the A > 0 and A < 0 HMF polarity cycles, respectively. Figures lc and ld show the corresponding spectr at 60 AU. The significant change in the slopes of the spectra, that is, sharp turn-ups in the shape of the spectra that occur especially in Figure l a for all cases are due to the assumed constant rigidity dependence of K r, K 0, and Kll at

4 8,308 FERREIRA ET AL.' HELlOSPHERIC MODULATION OF COSMIC RAY ELECTRONS t I I I I I -] I I I I I shown in Figure 4a, the ratio converges to the same value in the innermost heliosphere irrespective of latitude. This does ß not occur for the A < 0 cycle, indicating that the near-earth latitude dependence is more sensitive to enhanced latitudinal diffusion during A < 0 solar polarity cycles than A > 0 cycles >.-.:..--' Figures 4c and 4d show the ratio as a function of kinetic energy at and 60 AU. For the A > 0 cycle, shown in Figure c, the dominant feature at AU is the increase in the ratio as the particle energy increases. For 60 AU the effect of Polar Angle (degrees) Polar Angle (degrees) increasing K o becomes negligible at the low and high energies but not at intermediatenergies. For the A < 0 cycle, Figure 3. Electron differential intensities at 0.30 GeV as a shown in Figure 4d, the dominant feature at AU is again the function of polar angle for the A < 0 polarity cycle. Here K o increase in the ratio as the particle energy increases. However, was increased from (a) 5% to (b) 5% of Kii, from r = AU (bottom line) to r = 90 AU (top line), for -->30 AU and for in this cycle an increase in K o has an almost negligibl effect 60-->90 AU in steps of 0 AU. Units are particles m -2 sr ' s ' at 60 AU. Increasing K o clearly causes subtle differences MeV -. between the two polarity epochs. These differences are parameter dependent and illustrate the intrinsic play-off between drifts and perpendicular diffusion as it changes with position in space and with energy. This makes it difficult to larger distances the radial dependence increased significantly. predict what the effects of increasing K 0 are without doing This is a characteristic feature of drift models for this cycle the actual computations for each case. and can be of interest in establishing when a spacecraft In what follows we illustrate the effect that an enhanced K 0 approaches the heliospheric boundary. At 0 = 5 ø a strong has on profiles of the differential intensity as a function of radial dependence in the inner heliosphere (r <-0 AU) is In Figure 5 the differential intensities are shown for.94 GeV evident with a relatively weak dependence at larger radial electrons at 0 = 90 ø (equatorial plane) for both polarity cycles. distances. By increasing Kxo, as shown in Figure 2b, the The solutions are shown at three radial distances and for two latitude dependence was decreased, most significantly in the different values of K o. Figures 5a and 5b show solutions at inner heliosphere. The radial dependence increased AU; Figures 5c and 5d show solutions for 5 AU, and Figures significantly for r <-80 AU in the equatorial plane but 5e and 5f show solutions for 80 AU, for b = 0.05 and b = decreased for r > -80 AU. In the polar regions it decreased 0.5, respectively. This figure shows that at AU the for r <-0 AU but increased beyond this value. Globally, the intensity for the A < 0 polarity cycle is higher than for the A > radial dependence of electrons tends to be more uniform 0 cycle. As K o was enhanced by increasing b from 0.05 to throughout the heliosphere when using this approach. 0.5, a reduction occurs in the differences between the two The effects of enhancing K o on the electron differential epochs. The intensities for both epochs are now lower and do intensities as a function of polar angle are shown in Figure 3a not have such a strong dependence on ot as for a smaller Kño. for b = 0.05 and in Figure 3b for b = 0.5. Intensities are At 5 AU the intensities for the A > 0 and A < 0 cycles cross at shown for the A < 0 cycle for 0.30 GeV electrons from r = ot, 5 ø with the A < 0 intensities lower than those for the A > AU to r = 90 AU in steps of 0 AU for -->30 AU and again 0 for ot < 5 ø. As for AU the increase in Kl0 led to a for 60-->90 AU. From Figure 3a it follows that there is a decrease in the dependence on or, especially with ot < 40 ø. relative strong global positive latitude dependence which increases with radial distance. (The latitude dependence must be zero at 0 = 90 ø due to the symmetry conditions in the model.) This will change when a north-south asymmetry is assumed for the HMF [see, e.g., Hattingh et al., 997]. By increasing Kx0 an overall reduction in the latitude dependence of the intensities occurs in the inner heliosphere as shown in [o ao,..v,...," / ^>0 Figure 3b. The peculiar effect at 90 AU is due to the close proximity of the outer boundary. The latitude dependence, however, stays positive throughouthe heliosphere, although Radial distance ( AU ) Radial distance (AU) quite small in the inner heliosphere. The effects and importance of the enhancement of Kxo are '":..:.::.'.'.'i..'... ;:i......:.:'::::" summarized in Figure 4 by depicting the ratio of intensities O. - / AU'-' - obtained with b = 0.25 to b = This is done in Figure 4a for the A > 0 cycle as a function of radial distance for.94 --AU and 0.30 GeV electrons, respectively. In Figure 4b the AU f.... 6o ^u ' '4 0 ø 0 corresponding situation is shown for the A < 0 cycle. The solid lines represent values in the equatorial plane (0 = 90ø), Kinetic Energy ( GeV ) Kinetic Energy ( GeV ) and the dotted lines represent values in the polar regions (0 = Figure 4. The ratio of electron differential intensities for.94 5ø). The ratio converges to unity at the outer boundary for and 0.30 GeV, obtained for b = 0.05 and b = 0.25 as in (), both 0 = 90 ø and 0 = 5 ø, as it should. The two figures depicted as a function of radial distance and kinetic energy: emphasize that in general the effect of increasing Kxo is more (a, b) for the equatorial plane (0 = 90 ø, solid lines) and for the important in the inner and middle heliosphere, where the ratio is significantly less than. Moreover, in the A > 0 cycle, pola regions (0 = 5 ø, dotted lines); (c, d) Solid lines represent AU, and the dotted lines represent 60 AU.

5 , ß ß FERREIRA ET AL.: HELlOSPHERIC MODULATION OF COSMIC RAY ELECTRONS 8, , , l: 0.08 >, (a) 0.'05... A U i i i i i i (c), (e) i i i i i [ Tilt Angle (degrees) (b) ' ' A- (d) 0'. ' ' ' sau I I I i I i I (f) ' Ab , Tilt Angle (degrees) Figure 5. Electron differential intensity at.94 GeV as a function of the HCS "tilt angle" ct, shown in the equatorial plane (0 = 90 ø) for A > 0 and A < 0 polarity epochs. Solutions are shown at, 5, and 80 AU for two different values of K 0 in (2): (a, c, e) with b = 0.05 and panels (b, d, f) with b = ]. The more important aspect is that enhancing KLe reduces the differences between the two polarity cycles as a function of or, except for the outer heliospheric regions. The next results mainly concern the rigidity dependence of the diffusion coefficients, in particular, perpendicular diffusion, and its effect on modulated electron spectra.?otgieter and Ferreira [999a] and?otgieter et al. [999b] reported on this issue and concluded that perpendicular diffusion dominates electron modulation energies below a few hundred MeV. To emphasize this aspect, Figure 7 depicts the computed cosmic ray electron spectrat, 5, and 60 AU in the equatorial plane of the simulated heliosphere. The spectr are shown with respecto the assumed LIS at 00 AU and for the A > 0 HMF polarity cycle. The inset shows the energy (rigidity) dependence of Kii and KL (that is, the same for both K r and K o) used for the present study. Again, we assumed J (O,r) = + r/ro in (2), with ro = AU. The other parameters were ct = 5 ø, Ko = 25, a = 0.05, and b = 0.5, which means that K o = 3K r. The values shown in the inset are for Kttl fl and K fl, respectively. The computed spectrat and 5 AU are compatible with the 996 spectrum observed with the Kiel Electron Telescope (KET) on board the Ulysses spacecraft when it was at ~4.5 AU and ~22 ø heliolatitude as shown in the figure [Raviart et al., 997]. (Note that the data at a few MeV are consistent to previous observations Earth [see, e.g., Evenson 998]. Huber [ 998] pointed out that there is a discrepancy between the KET results and the ICE data between 300 MeV and GeV. However, both instruments are in agreement in the 2-5 GeV range.) Figures 8 and 9 depicthe results of Potgieter and Ferreira [999a] where they used for Ktl the sophisticated rigidity dependence of Bieber et al. [994] that was also discussed in While the profiles shown no longer cross at 5 AU, they do cross at a slightly larger radial distance. For 80 AU, the intensities for the A > 0 are consistently higher than for the A < 0 epoch and the increase in K 0 has little or no effect on the differential intensities as a function of or. This again indicates that the increase in K 0 is more important in the inner and middle heliosphere. An interesting feature is that the intensities increase with decreasing ot as shown in Figures 5e and 5f; see Hattingh [998] for a detailed account of this effect. This shows that the combination of all the modulation mechanisms in the model may produce results that are difficult to anticipate and sometimes even counterintuitive. The procedure was repeated for 0.30 GeV electrons and the results are shown in Figure 6. Qualitatively, the response to changes in K 0 is similar to the higher energy case in Figure 5; the ct dependence of the A > 0 intensities is less linear and the differences between the two epoch solutions are evidently larger than for the higher electron energie shown in Figure 5. Increasing b from 0.05 to 0.5 again reduces the driftoriginate differences between the two epochs. The crossover shown in Figure 5c now occurs at r > 5 AU. The situation for 0 = 5 ø was also studied but not shown here. The effects of increasing ot at this polar angle were found negligible compared to the equatorial regions, as one may anticipate. The only significant ot dependence occurs when ct > 60 ø during the A < 0 cycle. This is understandable because when ot becomes very large the modulation of intensities must respond to the presence of the wavy HCS at these highlatitude regions of the simulated heliosphere [Hattingh, (a) o.2o '""":'""'"'A' 'l/ U o.. e :... o o 8 A0.0 (c) 0.6 '..: ,08 o.o4 0. A (e) 4,,, 3 A'"'"'"""-- u 2..., ' o 0;05,... Tilt Angle (degrees) (b) 0'.5'... A'U 0.2 A , 0.08 A +... (d) o.o,,, 0.6 o.2 A (f) ' A+ U 0;5, Tilt Angle (degrees) Figure 6. As in Figure 5, but for 0.30 GeV electrons.

6 . 8,30 FERREIRA ET AL.: HELIOSPHERIC MODULATION OF COSMIC RAY ELECTRONS - 04 > 03 ',. 02 v E o 0- (, / 60 ALaN 04 ' 4 \ 2 04,,, [- I \\\X ø 0 L \\ K.E.(GeV) 5A >0 %- 0 4 > ; 03 ',. 02 I: 0 u) c: 00 0-,- 0-2,(t) 0.[ -I - % 0-,,, I- 0 ' K.E. (GeV) ii x - / >0 0-3,,,,,,,,I,,,,,,,,I,,,,,,,,I,,,,,,I, Kinetic Energy ( GeV ) Figure 7. Computed electron modulation at, 5, and 60 AU in the equatorial plane compared to the LIS at 00 AU for an A > 0 epoch. Insert shows the values of Kii/fl and K fl (depicted as KD in units of 6 x 020 cm 2 s '. For this case, Kñ0-3K r but with an energy dependence identical to that of K r. Electron data (solid circles) are from Ulysses/KET for 996 [Raviart et al., 997]. detail by Potgieter [996], but for K r and Kxo a straightforwardependence oc fir, as shown in the insets to the two figures denoted by Ki/fl and K /fl. In these insets the value of K /fl was multiplied by 0 for illustrative purposes. The subsequent spectra in Figure 8, with Kñ0 = 3K r illustrate generally that when the rigidity dependence of K r and Kx0 is taken independently from that for Kii at energies below ~ o 0 ' LIS 0ø (KII ' ' :: - - t 0' - 4 "ø" 04 ',, - 0 ' A K.E. (OeV) /// -. /,>o/ Kinetic Energy ( GeV ) Figure 8. Computed electron modulation at, 5, and 60 AU in the equatorial plane with the LIS at 00 AU for an,4 > 0 epoch. Insert shows the values of Ki/fl and K r/fl (depicted as Kñ) in units of 6 x 020 cm 2 s 'l, with K. L0 = 3K r, but with an energy dependence identical to that of K r. 0-3 lo lo Kinetic Energy ( GeV ) Figure 9. Similar to Figure 8, but with K 0 = 8K r. Note how the situation changed at the low energies, especially in the inner heliosphere. MeV, perpendicular diffusion evidently dominates modulation at these energies. The computed spectra in the inner heliosphere are still compatible to the data at higher energies, but below ~00 MeV the modulation becomes unreasonably large (see also the discussion below). For Figure 9 it was assumed that Kx0 = 8K r which means that Kñ0 was further enhanced by a factor The consequence is that a significant increase occurs for intensities below 50 MeV without a preceding change in the rigidity dependence of K r and Kñ0. The main conclusion is that although K r and Kx0 are only 5% and 5% of the value of Kii, respectively, perpendicular diffusion dominates electron modulation below ~00 MeV and that it is as such a very important parameter. If the increase in the low-energy part of observed electron spectra with decreasing energy was taken as a characteristic of modulated electron spectra, as in Figure 7, then K r and Kñ0 oc fir seems not a viable option for the rigidity dependence of perpendicular diffusion. To extend the study on the modulation aspect shown in Figures 7 to 9, Ferreira [998] constructed an analytical expression for Kii, applicable to electrons, using the theoretical work of Hattingh [998] and Burger and Hattingh [998] where they on their part used the approach and formalism of Bieber et at. [994], especially the random sweeping model for dynamical turbulence with pure slab geometry [see also e.g., Zank et at., 997]. This expression, which is considerably more complicated than the simple rigidity dependence used in the previous part of this study, is depicted as a function of kinetic energy for, 0, 50 and 00 AU in the equatorial plane in Figure 0. No explicit latitude dependence was built into it. It is evident that the radial dependence of Kll is much more sophisticated with the main feature the changing slopes of the function and that Kii is much larger in the outer heliosphere at high energies than at low energies, with the opposite at AU. The corresponding computed spectra are shown in Figure. In this case, a = 0.05 and b = 0.5 in (2). The electron data from the

7 FERREIRA ET AL.: HELlOSPHERIC MODULATION OF COSMIC RAY ELECTRONS 8,3 Ulysses/KET experiment for 996 are again shown to provide a general reference for inner heliospheric electron intensities during minimum modulation. This approach, which is more theoretically based than the simple function used above, clearly gives very large modulation at low energies. The compatibility between the Ulysses data and the model is reasonable at energies > 400 MeV, but not at energies < 00 MeV. Although we used a very sophisticated function for and then assumed K[r, and K 0 oc Kll, it apparently does not give electron modulation reasonably compatible to Ulysses 0 2 % 'E, 0ø. - ' data at low energies. In this case the only way to improve compatibility is to make Kñr/ and especially K o/fl nearly independent of kinetic energy below- 00 MeV, because as ' 0-2 shown before it dominates electron modulation at these low energies. However, it is importanto note that the measured == spectra at low energies, up to - 30 MeV, contains a Jovian contribution so that the galactic intensities are less than shown by the data in Figure. As we will discuss below, r' o 0 this Jovian contribution will not change the negative slope in the data to a positive slope as in the computations. Kinetic Energy (GeV) In addition to the results addressed in this paper, two modulation issues warrant some additional discussion. One Figure. Computed electron spectra in the equatorial plane at, 5, 0, 50, 70, 80, and 90 AU, corresponding to the funcissue concerns the observation of low-energy cosmic ray tion shown for Ktl in Figure 0, and based on the assumption electrons, and the lack of good observations throughouthe that K r and K 0 are both oc Kll. Data are at- 5 AU for 997 heliosphere. In contrasto nucleons, only a few low-energy from the Ulysses/KET experiment. electron measurements are available. These measurements have typically large uncertainties as discussed by Heber et at. [999]. On top of that, the issue remains of how many Voyager 2 [Lukasiak et al., 999] together with the positrons are measured together with the electrons [e.g., measurements from Pioneer 0 [Lopate, 99], and from Potgieter and Burger, 990]. At low energies (< ~30 MeV) Ulysses, indicate almost no radial dependence for-0 MeV the uncertainty about how dominant the Jovian source is cosmic ray electrons between AU and AU. This inhibits firm conclusions based on the kind of approach used means that if the LIS used in this paper was correct, enormous in this paper, which is to find compatibility between models modulation might occur for galactic electrons at these lowand data. A proper understanding of what the true level is of energies in the outermost regions of the heliosphere. Lowcosmic ray (galactic) electron modulation at low energies in energy galactic electrons might essentially be eliminated over the inner heliosphere is needed. The first results of a new 3-D the first 0-20 AU inwards from the outer boundary of the model concentrating on this issue is already available heliosphere, and consequently this might cause a "barrier" [Fichtner et at., 2000]. A second aspect is that recent upper effect that has been addressed by several authors before limits for low-energy galactic electrons at-70 AU from [Webber and Lockwood, 987, 997; Potgieter and te Roux, 989; Quenby et at., 990]. However, the large uncertainty 07 - the LIS, especially at low energies, has been reemphasized recently by Strong and Moskatenko [999] and will certainly 0 5Equatorial plane (90 ø) /00ALF influence conclusions about how large this "barrier" type E modulation is and will be studied further. The maximum of v.,, 0 s 5..///50AU // about 2 + galactic electrons m -2 sr 4 s ' MeV - at 0 MeV near -70 AU puts a severe constraint on models, and a first O 04 5 ==- '/"//.. 0AU attempt to understand this by Potgieter and Ferreira [999b] -./// ß led to the conclusion that Kii, K r, and K 0 should have a o O /.//,..., : complicated rigidity/radial dependence similar to what was ß.o 02 / --= shown in Figure 0 but modified to improve compatibility with the measurements. This means that the rigidity it: 0 dependence may change with radial distance and even polar angle as one moves outward in the heliosphere. Fundamental theoretically based efforts to improve our understanding have o3 I I I I I I I-- increased but are not yet at a comprehensive level for all three o O of the diffusion coefficients under consideration [e.g., Bieber Kinetic Energy (GeV) et at., 999; Giacatone and Jokipii, 999; te Roux et at., 999]. Another issue concerns specifically the rigidity Figure 0. The parallel diffusion coefficient KIi for cosmic dependence of Kii, Kxr, and Kx0. In this study we find that ray electrons in units of 6 x 020 at, 0, 50, and 00 AU, in these diffusion coefficients may have different rigidity the equatorial plane, as a function of kinetic energy. Note the dependencies. Work supporting this conclusion was done by changes from to 00 AU. Burger et at. [ 999], who studied the Ulysses observation of a

8 8,32 FERREIRA ET AL.: HELlOSPHERIC MODULATION OF COSMIC RAY ELECTRONS small-latitude effect for cosmic ray protons in the inner causes an overall reduction in the latitude dependence of the heliosphere using one of the most advance diffusion tensors electron intensities in the inner heliosphere as is required by in modulation studies. Apart from being small, the latitudinal Ulysses measurements [Ferrando, 997], which is arguably gradient exhibited a particular rigidity dependence, increasing the most important effect for modulation. The latitude up to ~.5 GV to then decreasing significantly with increasing dependence, however, stays positive for the A > 0 epoch rigidity [Heber et al., 996]. In principle, a model with an (Figure 3). We conclude that the effect of the enhancement of enhanced K,0 can easily reproduce the small gradient K,0 is the largest on model solutions at low energies in the [Potgieter et al., 997], but a straightforward approach could inner heliosphere (Figure 4). not give the correct detail of the rigidity dependence. Burger 2. Enhancing K,0 in the polar direction reduces the et al. [999] illustrated that this could be done if the three differences between the modulated intensities as a function of mentioned diffusion coefficients had different rigidity ct for the two magnetic polarity cycles. The increase in K,0 dependencies, but more importantly, that K 0 should have a causes a decrease in the tilt-angle dependence of the flat rigidity dependence at low rigidity. That particular study differential intensities for ct < 40 ø for the inner heliosphere in would therefore rule out the strong rigidity dependence of K,0 the equatorial regions (Figures 5 to 7). For the polar regions shown in Figures 8 and 9. Unfortunately, their diffusion the increase in K 0 caused little or no change in the ct tensor for protons, which does not include the effects of the dependence of the intensities for ct < 60 ø, but it caused a dissipation range of the magnetic field power spectrum vital significant reduction in the global latitude dependence of for electrons, does not give the correct almost-zero radial electron modulation as mentioned before. gradient for electrons in the ecliptic plane in the mid to outer 3. It was argued that if the increase in intensity with heliosphere, reported by Lopate [99] and Lukasiak et al. decreasing energy below ~00 MeV in the observed electron [999]. This interesting aspect is being studied further and spectra in the inner heliosphere is taken as a characteristic of will be reported elsewhere [Ferreira et al., 2000]. modulated electron spectra, as shown in Figures 7 and, then K*r/fl and especially K,o/fl have to be nearly independent of kinetic energy below-00 MeV. A main conclusion is that 4. Summary and Conclusions the three diffusion coefficients, K,, K 0, and Kii probably Cosmic ray (galactic) electron modulation in a simulated have different rigidity dependencies. Work supporting this conclusion was done by Burger et al. [999] who studied the heliosphere was used to illustrate the role and importance of anisotropic perpendicular diffusion, the latter a feature Ulysses observation of a small-latitude effect for cosmic ray protons in the inner heliosphere. (Note that Potgieter et al. emphasized by K6ta and Jokipii [995]. The expectation is [999b] suggested that the rigidity dependence may also be that the perpendicular diffusion coefficient in the polar time dependent, meaning that it may change over the 22-year direction of the heliosphere (K 0) should be larger, even modulation cycle.) significantly larger, than in the radial/azimuthal direction This work illustrates that the enhancement of K,0 in the (K r). We found that with K r and K 0 only 5% and 5% of polar direction of the heliosphere, which makes perpendicular the value of Ktl, respectively, perpendicular diffusion diffusion anisotropic, has profound effects on the modulation dominates electron modulation below-00 MeV. In of galactic cosmic ray electrons during both magnetic polarity particular, we illustrate () the effect of enhancing K 0 on the cycles of the Sun. Anisotropic perpendicular diffusion is computed electron intensities and spectra as a function of clearly of great importance to cosmic ray modulation and radial distance and polar angle, (2) the effect of changing its requires further study, especially at the fundamental rigidity dependence on the solutions, and (3) how enhancing theoretical level. K,0 affects the response of electron intensities when the HCS "tilt angle" ct is changed. Acknowledgments. The authors thank the SA National Research Electron modulation is a handy tool for such a study Foundation and the Potchefstroom University for C.H.E. for their fibecaus electrons respond directly to the assumed energy nancial support. M.S.P. also thanks the Intemational Space Science dependence of the diffusion coefficients below-500 MeV. Institute (Bern, Switzerland) for their hospitality and support during April to July 999. Protons, in contrast, experience significant adiabatic energy Janet G. Luhmann thanks Paul A. Evenson and another referee for changes below this energy which consequently obscure the their assistance in evaluating this paper. effects of changing the energy dependence of any of the diffusion coefficients. Another contrasting aspect is that drifts References become progressively less important with decreasing electron energy to have almost no effect on electron modulation below Bieber, J. W., et al., Proton and electron mean free paths: The Palmer consensus revisited, Astrophys. J., 420, 294, 994. ~00 MeV. Because no comprehensive theory exists for the Bieber, J. W., R. L. Mace, and W. H. Matthaeus, Perpendicular perpendicular diffusion, several assumptions had to be made diffusion of charged test particles in magnetostatic slab about its radial and rigidity dependence, the simplest being turbulence, Conf. Pap. Int. Cosmic Ray Conf. 26th, 7, 6, 999. that K,r and Kñ0 are proportional to Kii. Our main conclusions Burger, R. A., and M. Hattingh, Steady-state drift-dominated are the following. modulation models for galactic cosmic rays, Astrophys. Space Sci., 230, 375, Increasing K,0 in the polar direction with respect to Burger, R. A., and M. Hattingh, Toward a realistic diffusion tensor typically resulted in an increase in modulation for both HMF for galactic cosmic rays, Astrophys. J., 505, 244, 998. polarity cycles at almost all energies (Figure ). An important Burger, R. A., and M. S. Potgieter, The calculation of neutral sheet consequence of this approach is that the radial dependence drift in two-dimensional cosmic ray modulation models, Astrophys. d., 339, 50, 989. also increased general, except in the innermost heliosphere Burger, R. A., M. S. Potgieter, and B. Heber, Rigidity dependence of (Figure 2) so that drift models can in principle produce large near-earth latitudinal proton density gradients, Conf. Pap. Int. radial gradients even in A > 0 polarity epochs. This approach Cosmic Ray Conf. 26th, 7, 242, 999.

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10 8,34 FERREIRA ET AL.: HELlOSPHERIC MODULATION OF COSMIC RAY ELECTRONS Raviart, Implications of the heliospheric modulation of cosmic ray Webber, W. R., and J. A. Lockwood, Intensities of anomalous and electrons observed by Ulysses, Adv. Space Res., 23(3), 467, galactic cosmic rays in the outer heliosphere near the heliospheric 999b. equator in 987 and during the period from 994 and 996, J. Quenby, J. J., J. A Lockwood, and W. R. Webber, Cosmic-ray Geophys. Res., 02, 9773, 997. gradient measurements and modulation beyond the inner solar Zank, G. P., W. H. Matthaeus, J. W. Bieber, and H. Moraal, The wind termination shock, Astrophys. J., 365, 365, 990. cosmic ray diffusion tensor in the heliosphere, Conf Pap. Int. Raviart, A., et al., Evolution of cosmic ray electron spectra above Cosmic Ray Conf 25th, 2, 9, MeV along the Ulysses trajectory, Conf. Pap. Int. Cosmic Ray Conf. 25th, 2, 37, 997. R. A. Burger, S. E. S. Ferreira, and M. S. Potgieter, Research Unit Strong, A. W., et al., Diffuse continuum gamma rays from the galaxy for Space Physics, School of Physics, Potchefstroom University for observed by COMPTEL, Astron. Astrophys. 292, 82, 994. CHE, 2520 Potchefstroom. South Africa. (fskrab puknet.puk.ac.za; Strong, A. W., and I. V. Moskalenko, Evidence for a discrete source fsksesf puknet. ac.za; fskmsp puknet.ac.za) contribution to low-energy continuum galactic y-rays, Conf. Pap. B. Heber, Max Planck Institut flir Aeronomy, Max Planck Strasse Compton Syrup. 5th, in press, , D-379 Katlenburg - Lindau, Germany. (heber linmpi.mpg.de) Webber, W. R., and J. A. Lockwood, Interplanetary radial cosmic ray gradients and their implication for a possible large modulation (Received February 0, 2000; revised March 22, 2000; effect at the heliosphere boundary, Astrophys. J., 37, 534, 987. accepted March 22, 2000.)