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Modelling of `barrier' modulation for cosmic ray protons in the outer heliosphere
Advances in Space Research 34 (2004) 138–143 www.elsevier.com/locate/asr
Modelling of ‘barrier’ modulation for cosmic ray protons in the outer heliosphere U.W. Langner a, M.S. Potgieter
a,*
, W.R. Webber
b
a
b
School of Physics, North-West University, 2520 Potchefstroom, South Africa Department of Astronomy, New Mexico State University, Las Cruces, NM 88005, USA
Received 5 November 2002; received in revised form 7 February 2003; accepted 7 February 2003
Abstract Voyager and Pioneer spacecraft observations over 22 years and out to 82 AU have found two particular features of cosmic ray proton modulation in the heliosphere: (1) Markedly different behavior for minimum modulation conditions between the radial intensity profiles for periods of opposite magnetic polarities. (2) Most of the residual modulation for these periods was found to take place in the outer heliosphere, near and beyond where the termination shock (TS) is expected to be, and seems to differ for the two polarity cycles. Since contemporary models do not provide satisfactory quantitative explanations for these features, they are studied with a numerical model including a solar wind TS, a heliosheath and drifts. Results show interesting features caused by the TS and the heliosheath which differ markedly for the two polarity cycles, with energy and with increasing solar activity. Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Outer heliosphere; Cosmic ray protons; Heliosheath modulation; Termination shock; Heliospheric cosmic ray modulation
1. Introduction Observations near the predicted location of the solar wind termination shock (85–90 AU) enabled us to study the modulation of galactic cosmic rays (GCRs) in the outer heliosphere in detail, especially how modulation may occur in the heliosheath, the region between the termination shock (TS) and the heliopause. The structure of the heliospheric magnetic field (HMF) and the width of the heliosheath in this region is not well known. It is estimated to be at least 30–50 AU. The heliopause is a contact discontinuity and because it separates the solar and interstellar plasmas, it can be considered the outer boundary of the heliosphere, at least from a cosmic ray point of view. The possibility of significantly large modulation in the very distant heliosphere was first addressed by Webber and Lockwood (1987). Recently, several observational publications addressed the issue of helio*
Corresponding author. Tel.: +27-18-299-2406; fax: +27-18-2992421. E-mail address: [email protected] (M.S. Potgieter).
sheath (‘barrier’) modulation (McDonald et al., 2000, 2002; Webber and Lockwood, 2001a,b; Webber et al., 2001). Webber and Lockwood (2001a,b) used IMP, Voyager and Pioneer spacecraft data to derive radial intensity profiles of GCRs out to 70 AU at several energies during the minimum modulation periods in 1987 and 1997, followed by a similar study for solar maximum conditions in 1990–1991. They concluded that modulation in the outer heliosphere dominates the overall residual modulation at low energies, even at solar minimum, and that there is a clear difference between the modulation for the A > 0 and A < 0 solar magnetic field polarity cycles, with the heliosheath ‘barrier’ being more effective during A > 0 periods. Some of these effects were addressed over the years in modelling papers by Potgieter and le Roux (1989a,b), Quenby et al. (1990) and more comprehensively by Jokipii et al. (1993). However, with more reliable local interstellar spectra (LIS), a new approach to diffusion coefficients (e.g., Burger et al., 2000) and good observations closer to the TS around 90 AU, these aspects can be done more quantitatively. For this study a timedependent, two-dimensional numerical model, including
0273-1177/$30 Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2003.02.058
U.W. Langner et al. / Advances in Space Research 34 (2004) 138–143
the solar wind TS, a heliosheath and drifts is used to establish: (1) Compatibility between the model and CR observations, at Earth and as a function of radial distance, and (2) To illustrate the effects of the TS and the heliosheath on proton modulation at different energies, for the two field polarity cycles, and as modulation changes from minimum to maximum conditions.
2. Modulation model and parameters The model is based on the numerical solution of the Parker’s (1965) time-dependent transport equation: of 1 ¼ ðV þ hmD iÞ $f þ $ ðK s $f Þ þ ð$ VÞ ot 3 of þ Jsource ; ð1Þ o ln R where f ðr; R; t) is the CR distribution function, R is rigidity, r is position, and t is time, with V the solar wind velocity. Terms on the right-hand side represent convection, gradient and curvature drifts, diffusion, adiabatic energy changes and a source, respectively. The symmetric part of the tensor K s consists of a parallel diffusion coefficient ðKk Þ, and perpendicular diffusion coefficients (K? ).The anti-symmetric element ðKT Þ of the tensor describes gradient and curvature drifts in the large scale HMF. The function Jsource represents any local source, e.g., the Jovian electrons, the pick-up ions, etc. For this work we concentrate on GCR protons, neglecting all local sources, using as input to the model only the LIS for galactic protons. Eq. (1) is solved time-dependently as a combined diffusive shock acceleration and drift modulation model with two spatial dimensions, neglecting any azimuthal dependence. Similar models were described by e.g., Jokipii et al. (1993), Steenkamp (1995), le Roux et al. (1996), Haasbroek (1997), and Haasbroek et al. (1997). The outer modulation boundary was assumed at 120 AU, where the proton LIS of Moskalenko et al. (2001) is specified. Eq. (1) was solved in a spherical coordinate system with the current sheet tilt angles, a ¼ 10° and a ¼ 75°, representing solar minimum and maximum modulation during so-called A > 0 (e.g., 1990 to present) and A < 0 (e.g., 1980 to 1990) HMF polarity cycles. The HMF, assumed to have a basic Parkerian geometry in the equatorial plane was modified in the polar regions similar to Jokipii and K ota (1989) which is qualitatively supported by Ulysses measurements. A TS is assumed at rs ¼ 90AU with a compression ratio s ¼ 3:2, and a shock precursor scale length of L ¼ 1:2 AU. The magnetic field and diffusion coefficients ‘jump’ by a factor s at the TS. The solar wind speed changes from 400 km s1 in the equatorial plane ðh ¼ 90°Þ to 800 km s1 in the polar regions. This increase of a factor of two happens when 120° 6 h 6 60°
139
for solar minimum conditions, but it is reduced to 1.10 when 170° 6 h 6 10° for solar maximum conditions. For the precursor scale length in front of the shock, V decreases, for example, in the equatorial plane from the upstream value of V1 according to the relationship (le Roux et al., 1996) r r V1 ðs þ 1Þ V1 ðs 1Þ s tan h VðrÞ ¼ ð2Þ 2s 2s L This means that up to the shock, V decreases by 0.5s, then abruptly to the downstream value, in total to V1 =s. Beyond the TS, V decreases further as 1=r2 up to the outer boundary. The diffusion coefficients Kk ; K? , and ðKT Þ are the same as given by Burger et al. (2000) except for minor changes; see Burger et al. (2000) for a motivation of these diffusion coefficients based on diffusion and turbulence theory. In order to fit the relevant proton data as shown below, the following changes to the constants in the Burger et al. expressions for Kk ; K? ,and KT , respectively are made: j0k ¼ 0:9 and d ¼ 20:0R; ¼ 0:185;
j0e ¼ 0:026 and j0p
j0T ¼ 0:55;
ð3Þ
for a ¼ 10°, where d is valid for both polarity cycles. For increasing a up to a ¼ 75°: j0k ! 0:7; j0e ! 0:05 and j0p ! 0:356, respectively. Diffusion perpendicular to the HMF was enhanced in the polar direction by assuming j0p > j0e (K ota and Jokipii, 1995; Potgieter, 1996; Burger et al., 2000; Ferreira et al., 2000). For the A < 0 polarity cycle an additional adjustment had to be made to the radial dependence of Kk given by j0k ð45:0 r=r0 Þ0:218aþ0:289
when 0:1 < r < 45:0 AU ð4Þ
in order to fit the radial intensity in the inner heliosphere and at Earth (r0 ¼ 1 AU). This modification decreases to a value of 1.0 j0k for a ¼ 75°, at all radial distances. These diffusion coefficients are optimal for a numerical model without an azimuthal dependence and solar maximum effects like global merged interaction regions. This set can also be used without additional changes for both polarity cycles for electrons, anti-protons, positrons and protons (see e.g., Langner and Potgieter, 2004) to give reasonable fits to a variety of data sets. However, it must be noted that a quantitative fitting of the data was not the purpose of this study, but rather to find a set of diffusion coefficients that is generally compatible to the observations made for a variety of CR species.
3. Results The first objective was to find a set of diffusion coefficients/modulation parameters that would establish
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good compatibility between the model solutions and the major observed features of CR modulation. In Fig. 1 the computed modulation for GCR protons at Earth is shown as spectra with respect to the LIS at 120 AU, for both polarity cycles with a ¼ 10°, in comparison with observations for 1987 and 1997. Although it is easily done in principle, note that the A > 0 spectrum crosses the A < 0 spectrum (only slightly visible) as observations require, for this model at 700:0 MeV and again at 4:0 GeV (see Reinecke et al., 1997). For the fit of a more comprehensive data set at Earth, see also Moskalenko et al. (2001). Not shown is that the computed modulation for a ¼ 55° and a ¼ 75° at Earth in the A > 0 polarity cycle also gives reasonable fits to observations for these periods of increasing modulation. The model predicts that increasing a from 10° to 75° causes a drop in CR intensity (< 1:0 GeV) by a factor of 3:8 for the A > 0 period but a factor of 10:0 for the A < 0 polarity cycles. This behaviour is also a characteristic feature of steady-state drift models and will be illustrated from a different point of view in Fig. 4. In Fig. 2 the computed latitudinal gradients (in %/ degree) between co-latitudes 10° and 90° are shown as a function of rigidity for a ¼ 10° and a ¼ 75° for the A > 0 polarity cycle at a fixed distance of 3.0 AU. This is in comparison with observations by the Ulysses spacecraft based on the maximum intensities in 1995 (Heber et al., 1996; Heber, 1997; for modeling also see Burger et al., 2000 and Potgieter et al., 2001). The computed latitudinal gradients become smaller with increasing solar activity as suggested by Ulysses observations. This observation, concerning both the value of the latitudinal gradient as well as its rigidity dependence, put severe constraints on the model and requires that K? must be enhanced in the polar direction as described above. Fig. 3 depicts the computed radial intensities for GCR protons, for both polarity cycles, in the equatorial plane with a ¼ 10°, at 0.2, 0.5, 1.0 and 5.0 GeV, re-
Fig. 1. Computed differential intensities for protons as a function of kinetic energy for both polarity cycles at Earth for a ¼ 10° in comparison with solar minimum data for 1987 and 1997. The LIS is at 120 AU, with a TS at 90 AU.
Fig. 2. Computed latitudinal gradients, in %/degree, for the A > 0 polarity cycle at 3.0 AU for a ¼ 10° and a ¼ 75° between colatitudes 10° and 90°. Data are from Ulysses in 1995 (see Heber et al., 1996; Heber, 1997; Burger et al., 2000).
spectively. The filled circles represents A > 0 data, the squares represents A < 0 data from IMP, Pioneer 10, Voyager 1 and 2 (Webber and Lockwood, 2001a) for solar minimum conditions. The modulation effects of the TS, of the heliosheath and the differences between the two polarity cycles are illustrated. These effects clearly diminish with increasing energy to the extent that ‘barrier’ effects remain effective only for A > 0 cycles. The model and observations are most reasonably compatible, illustrating that the model can represent the observed radial gradients for both polarity cycles, with the radial gradients for the A > 0 always less than for the A < 0 cycles for r < 60 AU at all energies. In the A > 0 cycles significant modulation happens beyond the TS, evidently decreasing with increasing energy, with the intensity at the TS always less than for the LIS at 120 AU. For the A < 0 polarity cycle, there is a more gradual increase-in the radial intensities up to the TS, with the intensity higher at the TS than at 120 AU but this changes significantly at 0.2 GeV where the modulation is conspicuously different beyond the TS compared to the other energies. To establish to what extent drifts play a role in the heliosheath, solutions are repeated for 0.2 and 0.5 GeV with no drifts beyond the TS, instead of reducing drifts only by a factor of s as in the previous cases. The sudden changes at the TS then make way for more gradual increases in the non drift case. Fig. 4 shows the radial intensities for protons for both polarity cycles in the equatorial plane at 0.2, 0.5, 1.0 and 5.0 GeV, respectively, but now for a ¼ 75°, representing solar maximum conditions with accompanying changes in the diffusion coefficients as explained in the model section. The data points are from IMP, Pioneer 10 Voyager 2 for solar maximum conditions in 1990–1991 (Webber and Lockwood, 2001b). As for solar minimum conditions, the modulation inside the TS is different for A > 0 and A < 0 cycles but these differences diminish
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Fig. 3. Computed radial intensities for GCR protons for both polarity cycles in the equatorial plane at 0.2, 0.5, 1.0 and 5.0 GeV, respectively, and for a ¼ 10°. The LIS is specified at 120 AU. The filled circles represent A > 0 data, the squares represent A < 0 data of IMP, Pioneer 10, Voyager 1 and 2 (Webber and Lockwood, 2001a) for solar minimum conditions. For 0.2 and 0.5 GeV modelling was also done for solutions with no drift in the heliosheath, instead of a reduction of a factor s in drifts as done for the other solutions.
Fig. 4. Radial intensities for protons for both polarity cycles in the equatorial plane at 0.2, 0.5, 1.0 and 5.0 GeV, respectively, for a ¼ 75° as function of radial distance. The squares represent A < 0 data of IMP, Pioneer 10, Voyager 2 (see Webber and Lockwood, 2001b) for solar maximum conditions, 1990–91.
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with increasing energy. The reason is that the A < 0 intensities respond more significantly to changes in the tilt angle. In contrast to solar minimum, the modulation difference beyond the TS between A > 0 and A < 0 is insignificant at lower energies but becomes more noticeable at higher energies. The ‘barrier’ effect has also changed significantly, in this case without the large increases for the A < 0 cycle. The modulation beyond the TS is still substantial for 0.2 GeV but becomes less significant with increasing energy compared to the amount of modulation inside the TS. For these conditions, the effect of the TS is the largest for the A > 0 cycle.
4. Discussion and conclusions The inclusion of a TS in the modulation model for solar minimum conditions causes abrupt changes in the CR radial gradients, gr , at the TS, at almost all energies of interest to modulation studies. For A > 0 cycles, gr becomes typically very large (positive) inside of the TS but almost zero at energies > 1 GeV beyond the TS, to increase again significantly close to the outer boundary; at lower energies, gr progressively becomes larger positive behind the TS. The CR proton intensity is lower at the TS than the value of the assumed LIS at 120 AU for the energies considered here. For the A < 0 cycle, the abrupt changes at the TS in gr are qualitatively similar, but they differ quantitatively, especially the lower the energy. For these cycles the intensity at the TS at energies >0.5 GeV can be higher than the corresponding value of the LIS, so that gr may even be negative beyond the TS. For increased solar activity, the modulation in the heliosheath is quit different from minimum activity. The heliosheath does not any longer play the role of a distinguished ’barrier’, although abrupt changes in gr at the TS may still occur, especially for A > 0 cycles, surprising more so for higher energies. The role of drifts manifests itself in the clear differences between the polarity cycles. The spectral form of the LIS is not primarily important but the value of the LIS at a given energy may change the total ‘barrier’ effect. Increasing the outer boundary from 120 to 140 AU enhances the ‘barrier’ effect on modulation but only at the lower energies during minimum modulation periods. Qualitatively our results are consistent to those of Jokipii et al. (1993) but there are quantitatively marked differences. Although the aim of this study was not to study the diffusion coefficients, it is quite evident that the chosen set gives reasonable fits to the observations. They can also be used without significant alterations for both polarity cycles, and for different species of cosmic rays (e.g., electrons, positrons and anti-protons). Can the heliosheath be considered as a distinguishable modulation ‘barrier’? First, the overall effect is clearly energy dependent. At solar minimum during
A > 0 cycles the ‘barrier’ effect at lower energies is quite clear, e.g., it contributes 40% to the overall modulation at 0.5 GeV. However, for A < 0 cycles, the contribution at this energy is negligible. At maximum modulation the ‘barrier’ effect is also different and surprisingly less distinguishable below 1 GeV. Incorporating huge transient ‘barriers’ in the model may alter this conclusion.
Acknowledgements We thank R.A. Burger and participants of the Cosmic Ray Workshop in South Africa in March 2002, for many useful discussions, and the SA National Science Foundation for partial financial support.
References Burger, R.A., Potgieter, M.S., Heber, B. Rigidity dependence of cosmic-ray proton latitudinal gradients measured by the Ulysses spacecraft: implication for the diffusion tensor. J. Geophys. Res. 105, 27447–27455, 2000. Ferreira, S.E.S., Potgieter, M.S., Burger, R.A., Heber, B. Modulation effects of anisotropic perpendicular diffusion on cosmic ray electron intensities in the heliosphere. J. Geophys. Res. 18, 18305–18314, 2000. Haasbroek, L.J. The transport and acceleration of charged particles in the heliosphere, Ph.D. Thesis, Potchefstroom University for C.H.E., South Africa, 1997. Haasbroek, L.J., Potgieter, M.S., le Roux, J.A. Some modulation effects of termination shock accelerated galactic and Jovian electrons, in: Proceedings of the 25th International Cosmic Ray Conference, vol. 2, pp. 29–32, 1997. Heber, B. Modulation galaktischer kosmicher Protonen und a Teilchen in der inneren drei-dimensionelen Heliosph€are, Messungen des Kiel Electron Telescopes an Bord der Raumsonde Ulysses, Ph.D. Thesis, Kiel University, Germany, 1997. Heber, B., Dr€ oge, W., Ferrando, P., Haasbroek, L.J., Kunow, H., M€ uller-Mellin, R., Paizis, C., Potgieter, M.S., Raviart, A., Wibberenz, G. Spatial variation of >40 MeV/n nuclei fluxes observed during the Ulysses rapid latitude scan. Astron. Astrophys. 316, 538–546, 1996. Jokipii, J.R., K ota, J. The polar heliospheric magnetic field. Geophys. Res. Lett. 16, 1–4, 1989. Jokipii, J.R., K ota, J., Merenyi, E. The gradient of galactic cosmic rays at the solar wind termination shock. Astrophys. J. 405, 782–786, 1993. K ota, J., Jokippi, J.R. 3-D distribution of cosmic rays in the outer heliosphere, in: Proceedings of the 24th International Cosmic Ray Conference, vol. 4, pp. 680–683, 1995. Langner, U.W., Potgieter, M.S. Effects of the solar wind termination shock on charge-sign dependent cosmic ray modulation. Adv. Space Res., this issue, 2004, doi:10.1016/j.asr.2003.01.031. le Roux, J.A., Potgieter, M.S., Ptuskin, V.S. A transport model for the diffusive acceleration and modulation of anomalous cosmic rays in the heliosphere. J. Geophys. Res. 101, 4791–4804, 1996. McDonald, F.B., Heikkila, B., Lal, N., Stone, E.C. 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, 2000. McDonald, F.B., Klecker, B., McGuire, R.E., Reames, D.V. Relative recovery of galactic and anomalous cosmic rays at 1 AU: further
U.W. Langner et al. / Advances in Space Research 34 (2004) 138–143 evidence for modulation in the heliosheath. J. Geophys. Res. 107 (SHH 2), 1–9, 2002. Moskalenko, I.V., Strong, A.W., Ormes, J.F., Potgieter, M.S., Langner, U.W. Secondary antiprotons in cosmic rays, in: Proceedings of the 27th International Cosmic Ray Conference, pp. 1868– 1871, 2001. Parker, E.N. The passage of energetic charged particles through interplanetary space. Planet. Space Sci. 13, 9–49, 1965. Potgieter, M.S. The heliospheric modulation of galactic electrons: consequences of new calculations for the mean free path of electrons between 1 and 10 GeV. J. Geophys. Res. 101, 24,411– 24,422, 1996. Potgieter, M.S., le Roux, J.A. A numerical model for a cosmic-ray modulation barrier in the outer heliosphere. Astron. Astrophys. 209, 406–410, 1989a. Potgieter, M.S., le Roux, J.A. More on a possible modulation barrier in the outer heliosphere. Adv. Space Res. 9 (4), 121–124, 1989b. Potgieter, M.S., Burger, R.A., Ferreira, S.E.S. Modulation of cosmic rays in the heliosphere from solar minimum to maximum: a theoretical perspective. Space Sci. Rev. 97, 295–307, 2001. Quenby, J.J., Lockwood, J.A., Webber, W.R. Cosmic ray gradient measurements and modulation beyond the inner solar wind termination shock. Astrophys. J. 365, 365–371, 1990.
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Reinecke, J.P.L., Moraal, H., Potgieter, M.S., McDonald, F.B., Webber, W.R. Different Crossovers? in: Proceedings of the 25th International Cosmic Ray Conference, vol. 2, pp. 49–52, 1997. Steenkamp, R. Shock acceleration as source of the anomalous component of cosmic rays in the heliosphere, Ph.D. Thesis, Potchefstroom University for C.H.E., Potchefstroom, 1995. Webber, W.R., Lockwood, J.A. Interplanetary radial cosmic ray gradients and their implication for a possible large modulation effect at the heliospheric boundary. Astrophys. J. 317, 534–542, 1987. Webber, W.R., Lockwood, J.A. Voyager and Pioneer spacecraft measurements of cosmic ray intensities in the outer heliosphere: toward a new paradigm for understanding the global modulation process. 1. minimum solar modulation (1987 and 1997). J. Geophys. Res. 106, 29,323–29,331, 2001a. Webber, W.R., Lockwood, J.A. Voyager and Pioneer spacecraft measurements of cosmic ray intensities in the outer heliosphere: toward a new paradigm for understanding the global modulation process. 2. maximum solar modulation (1990–1991). J. Geophys. Res. 106, 29333–29340, 2001b. Webber, W.R., Lockwood, J.A., McDonald, F.B., Heikkila, B. 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, 2001.
Modelling of ‘barrier’ modulation for cosmic ray protons in the outer heliosphere U.W. Langner a, M.S. Potgieter
a,*
, W.R. Webber
b
a
b
School of Physics, North-West University, 2520 Potchefstroom, South Africa Department of Astronomy, New Mexico State University, Las Cruces, NM 88005, USA
Received 5 November 2002; received in revised form 7 February 2003; accepted 7 February 2003
Abstract Voyager and Pioneer spacecraft observations over 22 years and out to 82 AU have found two particular features of cosmic ray proton modulation in the heliosphere: (1) Markedly different behavior for minimum modulation conditions between the radial intensity profiles for periods of opposite magnetic polarities. (2) Most of the residual modulation for these periods was found to take place in the outer heliosphere, near and beyond where the termination shock (TS) is expected to be, and seems to differ for the two polarity cycles. Since contemporary models do not provide satisfactory quantitative explanations for these features, they are studied with a numerical model including a solar wind TS, a heliosheath and drifts. Results show interesting features caused by the TS and the heliosheath which differ markedly for the two polarity cycles, with energy and with increasing solar activity. Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Outer heliosphere; Cosmic ray protons; Heliosheath modulation; Termination shock; Heliospheric cosmic ray modulation
1. Introduction Observations near the predicted location of the solar wind termination shock (85–90 AU) enabled us to study the modulation of galactic cosmic rays (GCRs) in the outer heliosphere in detail, especially how modulation may occur in the heliosheath, the region between the termination shock (TS) and the heliopause. The structure of the heliospheric magnetic field (HMF) and the width of the heliosheath in this region is not well known. It is estimated to be at least 30–50 AU. The heliopause is a contact discontinuity and because it separates the solar and interstellar plasmas, it can be considered the outer boundary of the heliosphere, at least from a cosmic ray point of view. The possibility of significantly large modulation in the very distant heliosphere was first addressed by Webber and Lockwood (1987). Recently, several observational publications addressed the issue of helio*
Corresponding author. Tel.: +27-18-299-2406; fax: +27-18-2992421. E-mail address: [email protected] (M.S. Potgieter).
sheath (‘barrier’) modulation (McDonald et al., 2000, 2002; Webber and Lockwood, 2001a,b; Webber et al., 2001). Webber and Lockwood (2001a,b) used IMP, Voyager and Pioneer spacecraft data to derive radial intensity profiles of GCRs out to 70 AU at several energies during the minimum modulation periods in 1987 and 1997, followed by a similar study for solar maximum conditions in 1990–1991. They concluded that modulation in the outer heliosphere dominates the overall residual modulation at low energies, even at solar minimum, and that there is a clear difference between the modulation for the A > 0 and A < 0 solar magnetic field polarity cycles, with the heliosheath ‘barrier’ being more effective during A > 0 periods. Some of these effects were addressed over the years in modelling papers by Potgieter and le Roux (1989a,b), Quenby et al. (1990) and more comprehensively by Jokipii et al. (1993). However, with more reliable local interstellar spectra (LIS), a new approach to diffusion coefficients (e.g., Burger et al., 2000) and good observations closer to the TS around 90 AU, these aspects can be done more quantitatively. For this study a timedependent, two-dimensional numerical model, including
0273-1177/$30 Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2003.02.058
U.W. Langner et al. / Advances in Space Research 34 (2004) 138–143
the solar wind TS, a heliosheath and drifts is used to establish: (1) Compatibility between the model and CR observations, at Earth and as a function of radial distance, and (2) To illustrate the effects of the TS and the heliosheath on proton modulation at different energies, for the two field polarity cycles, and as modulation changes from minimum to maximum conditions.
2. Modulation model and parameters The model is based on the numerical solution of the Parker’s (1965) time-dependent transport equation: of 1 ¼ ðV þ hmD iÞ $f þ $ ðK s $f Þ þ ð$ VÞ ot 3 of þ Jsource ; ð1Þ o ln R where f ðr; R; t) is the CR distribution function, R is rigidity, r is position, and t is time, with V the solar wind velocity. Terms on the right-hand side represent convection, gradient and curvature drifts, diffusion, adiabatic energy changes and a source, respectively. The symmetric part of the tensor K s consists of a parallel diffusion coefficient ðKk Þ, and perpendicular diffusion coefficients (K? ).The anti-symmetric element ðKT Þ of the tensor describes gradient and curvature drifts in the large scale HMF. The function Jsource represents any local source, e.g., the Jovian electrons, the pick-up ions, etc. For this work we concentrate on GCR protons, neglecting all local sources, using as input to the model only the LIS for galactic protons. Eq. (1) is solved time-dependently as a combined diffusive shock acceleration and drift modulation model with two spatial dimensions, neglecting any azimuthal dependence. Similar models were described by e.g., Jokipii et al. (1993), Steenkamp (1995), le Roux et al. (1996), Haasbroek (1997), and Haasbroek et al. (1997). The outer modulation boundary was assumed at 120 AU, where the proton LIS of Moskalenko et al. (2001) is specified. Eq. (1) was solved in a spherical coordinate system with the current sheet tilt angles, a ¼ 10° and a ¼ 75°, representing solar minimum and maximum modulation during so-called A > 0 (e.g., 1990 to present) and A < 0 (e.g., 1980 to 1990) HMF polarity cycles. The HMF, assumed to have a basic Parkerian geometry in the equatorial plane was modified in the polar regions similar to Jokipii and K ota (1989) which is qualitatively supported by Ulysses measurements. A TS is assumed at rs ¼ 90AU with a compression ratio s ¼ 3:2, and a shock precursor scale length of L ¼ 1:2 AU. The magnetic field and diffusion coefficients ‘jump’ by a factor s at the TS. The solar wind speed changes from 400 km s1 in the equatorial plane ðh ¼ 90°Þ to 800 km s1 in the polar regions. This increase of a factor of two happens when 120° 6 h 6 60°
139
for solar minimum conditions, but it is reduced to 1.10 when 170° 6 h 6 10° for solar maximum conditions. For the precursor scale length in front of the shock, V decreases, for example, in the equatorial plane from the upstream value of V1 according to the relationship (le Roux et al., 1996) r r V1 ðs þ 1Þ V1 ðs 1Þ s tan h VðrÞ ¼ ð2Þ 2s 2s L This means that up to the shock, V decreases by 0.5s, then abruptly to the downstream value, in total to V1 =s. Beyond the TS, V decreases further as 1=r2 up to the outer boundary. The diffusion coefficients Kk ; K? , and ðKT Þ are the same as given by Burger et al. (2000) except for minor changes; see Burger et al. (2000) for a motivation of these diffusion coefficients based on diffusion and turbulence theory. In order to fit the relevant proton data as shown below, the following changes to the constants in the Burger et al. expressions for Kk ; K? ,and KT , respectively are made: j0k ¼ 0:9 and d ¼ 20:0R; ¼ 0:185;
j0e ¼ 0:026 and j0p
j0T ¼ 0:55;
ð3Þ
for a ¼ 10°, where d is valid for both polarity cycles. For increasing a up to a ¼ 75°: j0k ! 0:7; j0e ! 0:05 and j0p ! 0:356, respectively. Diffusion perpendicular to the HMF was enhanced in the polar direction by assuming j0p > j0e (K ota and Jokipii, 1995; Potgieter, 1996; Burger et al., 2000; Ferreira et al., 2000). For the A < 0 polarity cycle an additional adjustment had to be made to the radial dependence of Kk given by j0k ð45:0 r=r0 Þ0:218aþ0:289
when 0:1 < r < 45:0 AU ð4Þ
in order to fit the radial intensity in the inner heliosphere and at Earth (r0 ¼ 1 AU). This modification decreases to a value of 1.0 j0k for a ¼ 75°, at all radial distances. These diffusion coefficients are optimal for a numerical model without an azimuthal dependence and solar maximum effects like global merged interaction regions. This set can also be used without additional changes for both polarity cycles for electrons, anti-protons, positrons and protons (see e.g., Langner and Potgieter, 2004) to give reasonable fits to a variety of data sets. However, it must be noted that a quantitative fitting of the data was not the purpose of this study, but rather to find a set of diffusion coefficients that is generally compatible to the observations made for a variety of CR species.
3. Results The first objective was to find a set of diffusion coefficients/modulation parameters that would establish
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good compatibility between the model solutions and the major observed features of CR modulation. In Fig. 1 the computed modulation for GCR protons at Earth is shown as spectra with respect to the LIS at 120 AU, for both polarity cycles with a ¼ 10°, in comparison with observations for 1987 and 1997. Although it is easily done in principle, note that the A > 0 spectrum crosses the A < 0 spectrum (only slightly visible) as observations require, for this model at 700:0 MeV and again at 4:0 GeV (see Reinecke et al., 1997). For the fit of a more comprehensive data set at Earth, see also Moskalenko et al. (2001). Not shown is that the computed modulation for a ¼ 55° and a ¼ 75° at Earth in the A > 0 polarity cycle also gives reasonable fits to observations for these periods of increasing modulation. The model predicts that increasing a from 10° to 75° causes a drop in CR intensity (< 1:0 GeV) by a factor of 3:8 for the A > 0 period but a factor of 10:0 for the A < 0 polarity cycles. This behaviour is also a characteristic feature of steady-state drift models and will be illustrated from a different point of view in Fig. 4. In Fig. 2 the computed latitudinal gradients (in %/ degree) between co-latitudes 10° and 90° are shown as a function of rigidity for a ¼ 10° and a ¼ 75° for the A > 0 polarity cycle at a fixed distance of 3.0 AU. This is in comparison with observations by the Ulysses spacecraft based on the maximum intensities in 1995 (Heber et al., 1996; Heber, 1997; for modeling also see Burger et al., 2000 and Potgieter et al., 2001). The computed latitudinal gradients become smaller with increasing solar activity as suggested by Ulysses observations. This observation, concerning both the value of the latitudinal gradient as well as its rigidity dependence, put severe constraints on the model and requires that K? must be enhanced in the polar direction as described above. Fig. 3 depicts the computed radial intensities for GCR protons, for both polarity cycles, in the equatorial plane with a ¼ 10°, at 0.2, 0.5, 1.0 and 5.0 GeV, re-
Fig. 1. Computed differential intensities for protons as a function of kinetic energy for both polarity cycles at Earth for a ¼ 10° in comparison with solar minimum data for 1987 and 1997. The LIS is at 120 AU, with a TS at 90 AU.
Fig. 2. Computed latitudinal gradients, in %/degree, for the A > 0 polarity cycle at 3.0 AU for a ¼ 10° and a ¼ 75° between colatitudes 10° and 90°. Data are from Ulysses in 1995 (see Heber et al., 1996; Heber, 1997; Burger et al., 2000).
spectively. The filled circles represents A > 0 data, the squares represents A < 0 data from IMP, Pioneer 10, Voyager 1 and 2 (Webber and Lockwood, 2001a) for solar minimum conditions. The modulation effects of the TS, of the heliosheath and the differences between the two polarity cycles are illustrated. These effects clearly diminish with increasing energy to the extent that ‘barrier’ effects remain effective only for A > 0 cycles. The model and observations are most reasonably compatible, illustrating that the model can represent the observed radial gradients for both polarity cycles, with the radial gradients for the A > 0 always less than for the A < 0 cycles for r < 60 AU at all energies. In the A > 0 cycles significant modulation happens beyond the TS, evidently decreasing with increasing energy, with the intensity at the TS always less than for the LIS at 120 AU. For the A < 0 polarity cycle, there is a more gradual increase-in the radial intensities up to the TS, with the intensity higher at the TS than at 120 AU but this changes significantly at 0.2 GeV where the modulation is conspicuously different beyond the TS compared to the other energies. To establish to what extent drifts play a role in the heliosheath, solutions are repeated for 0.2 and 0.5 GeV with no drifts beyond the TS, instead of reducing drifts only by a factor of s as in the previous cases. The sudden changes at the TS then make way for more gradual increases in the non drift case. Fig. 4 shows the radial intensities for protons for both polarity cycles in the equatorial plane at 0.2, 0.5, 1.0 and 5.0 GeV, respectively, but now for a ¼ 75°, representing solar maximum conditions with accompanying changes in the diffusion coefficients as explained in the model section. The data points are from IMP, Pioneer 10 Voyager 2 for solar maximum conditions in 1990–1991 (Webber and Lockwood, 2001b). As for solar minimum conditions, the modulation inside the TS is different for A > 0 and A < 0 cycles but these differences diminish
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Fig. 3. Computed radial intensities for GCR protons for both polarity cycles in the equatorial plane at 0.2, 0.5, 1.0 and 5.0 GeV, respectively, and for a ¼ 10°. The LIS is specified at 120 AU. The filled circles represent A > 0 data, the squares represent A < 0 data of IMP, Pioneer 10, Voyager 1 and 2 (Webber and Lockwood, 2001a) for solar minimum conditions. For 0.2 and 0.5 GeV modelling was also done for solutions with no drift in the heliosheath, instead of a reduction of a factor s in drifts as done for the other solutions.
Fig. 4. Radial intensities for protons for both polarity cycles in the equatorial plane at 0.2, 0.5, 1.0 and 5.0 GeV, respectively, for a ¼ 75° as function of radial distance. The squares represent A < 0 data of IMP, Pioneer 10, Voyager 2 (see Webber and Lockwood, 2001b) for solar maximum conditions, 1990–91.
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with increasing energy. The reason is that the A < 0 intensities respond more significantly to changes in the tilt angle. In contrast to solar minimum, the modulation difference beyond the TS between A > 0 and A < 0 is insignificant at lower energies but becomes more noticeable at higher energies. The ‘barrier’ effect has also changed significantly, in this case without the large increases for the A < 0 cycle. The modulation beyond the TS is still substantial for 0.2 GeV but becomes less significant with increasing energy compared to the amount of modulation inside the TS. For these conditions, the effect of the TS is the largest for the A > 0 cycle.
4. Discussion and conclusions The inclusion of a TS in the modulation model for solar minimum conditions causes abrupt changes in the CR radial gradients, gr , at the TS, at almost all energies of interest to modulation studies. For A > 0 cycles, gr becomes typically very large (positive) inside of the TS but almost zero at energies > 1 GeV beyond the TS, to increase again significantly close to the outer boundary; at lower energies, gr progressively becomes larger positive behind the TS. The CR proton intensity is lower at the TS than the value of the assumed LIS at 120 AU for the energies considered here. For the A < 0 cycle, the abrupt changes at the TS in gr are qualitatively similar, but they differ quantitatively, especially the lower the energy. For these cycles the intensity at the TS at energies >0.5 GeV can be higher than the corresponding value of the LIS, so that gr may even be negative beyond the TS. For increased solar activity, the modulation in the heliosheath is quit different from minimum activity. The heliosheath does not any longer play the role of a distinguished ’barrier’, although abrupt changes in gr at the TS may still occur, especially for A > 0 cycles, surprising more so for higher energies. The role of drifts manifests itself in the clear differences between the polarity cycles. The spectral form of the LIS is not primarily important but the value of the LIS at a given energy may change the total ‘barrier’ effect. Increasing the outer boundary from 120 to 140 AU enhances the ‘barrier’ effect on modulation but only at the lower energies during minimum modulation periods. Qualitatively our results are consistent to those of Jokipii et al. (1993) but there are quantitatively marked differences. Although the aim of this study was not to study the diffusion coefficients, it is quite evident that the chosen set gives reasonable fits to the observations. They can also be used without significant alterations for both polarity cycles, and for different species of cosmic rays (e.g., electrons, positrons and anti-protons). Can the heliosheath be considered as a distinguishable modulation ‘barrier’? First, the overall effect is clearly energy dependent. At solar minimum during
A > 0 cycles the ‘barrier’ effect at lower energies is quite clear, e.g., it contributes 40% to the overall modulation at 0.5 GeV. However, for A < 0 cycles, the contribution at this energy is negligible. At maximum modulation the ‘barrier’ effect is also different and surprisingly less distinguishable below 1 GeV. Incorporating huge transient ‘barriers’ in the model may alter this conclusion.
Acknowledgements We thank R.A. Burger and participants of the Cosmic Ray Workshop in South Africa in March 2002, for many useful discussions, and the SA National Science Foundation for partial financial support.
References Burger, R.A., Potgieter, M.S., Heber, B. Rigidity dependence of cosmic-ray proton latitudinal gradients measured by the Ulysses spacecraft: implication for the diffusion tensor. J. Geophys. Res. 105, 27447–27455, 2000. Ferreira, S.E.S., Potgieter, M.S., Burger, R.A., Heber, B. Modulation effects of anisotropic perpendicular diffusion on cosmic ray electron intensities in the heliosphere. J. Geophys. Res. 18, 18305–18314, 2000. Haasbroek, L.J. The transport and acceleration of charged particles in the heliosphere, Ph.D. Thesis, Potchefstroom University for C.H.E., South Africa, 1997. Haasbroek, L.J., Potgieter, M.S., le Roux, J.A. Some modulation effects of termination shock accelerated galactic and Jovian electrons, in: Proceedings of the 25th International Cosmic Ray Conference, vol. 2, pp. 29–32, 1997. Heber, B. Modulation galaktischer kosmicher Protonen und a Teilchen in der inneren drei-dimensionelen Heliosph€are, Messungen des Kiel Electron Telescopes an Bord der Raumsonde Ulysses, Ph.D. Thesis, Kiel University, Germany, 1997. Heber, B., Dr€ oge, W., Ferrando, P., Haasbroek, L.J., Kunow, H., M€ uller-Mellin, R., Paizis, C., Potgieter, M.S., Raviart, A., Wibberenz, G. Spatial variation of >40 MeV/n nuclei fluxes observed during the Ulysses rapid latitude scan. Astron. Astrophys. 316, 538–546, 1996. Jokipii, J.R., K ota, J. The polar heliospheric magnetic field. Geophys. Res. Lett. 16, 1–4, 1989. Jokipii, J.R., K ota, J., Merenyi, E. The gradient of galactic cosmic rays at the solar wind termination shock. Astrophys. J. 405, 782–786, 1993. K ota, J., Jokippi, J.R. 3-D distribution of cosmic rays in the outer heliosphere, in: Proceedings of the 24th International Cosmic Ray Conference, vol. 4, pp. 680–683, 1995. Langner, U.W., Potgieter, M.S. Effects of the solar wind termination shock on charge-sign dependent cosmic ray modulation. Adv. Space Res., this issue, 2004, doi:10.1016/j.asr.2003.01.031. le Roux, J.A., Potgieter, M.S., Ptuskin, V.S. A transport model for the diffusive acceleration and modulation of anomalous cosmic rays in the heliosphere. J. Geophys. Res. 101, 4791–4804, 1996. McDonald, F.B., Heikkila, B., Lal, N., Stone, E.C. 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, 2000. McDonald, F.B., Klecker, B., McGuire, R.E., Reames, D.V. Relative recovery of galactic and anomalous cosmic rays at 1 AU: further
U.W. Langner et al. / Advances in Space Research 34 (2004) 138–143 evidence for modulation in the heliosheath. J. Geophys. Res. 107 (SHH 2), 1–9, 2002. Moskalenko, I.V., Strong, A.W., Ormes, J.F., Potgieter, M.S., Langner, U.W. Secondary antiprotons in cosmic rays, in: Proceedings of the 27th International Cosmic Ray Conference, pp. 1868– 1871, 2001. Parker, E.N. The passage of energetic charged particles through interplanetary space. Planet. Space Sci. 13, 9–49, 1965. Potgieter, M.S. The heliospheric modulation of galactic electrons: consequences of new calculations for the mean free path of electrons between 1 and 10 GeV. J. Geophys. Res. 101, 24,411– 24,422, 1996. Potgieter, M.S., le Roux, J.A. A numerical model for a cosmic-ray modulation barrier in the outer heliosphere. Astron. Astrophys. 209, 406–410, 1989a. Potgieter, M.S., le Roux, J.A. More on a possible modulation barrier in the outer heliosphere. Adv. Space Res. 9 (4), 121–124, 1989b. Potgieter, M.S., Burger, R.A., Ferreira, S.E.S. Modulation of cosmic rays in the heliosphere from solar minimum to maximum: a theoretical perspective. Space Sci. Rev. 97, 295–307, 2001. Quenby, J.J., Lockwood, J.A., Webber, W.R. Cosmic ray gradient measurements and modulation beyond the inner solar wind termination shock. Astrophys. J. 365, 365–371, 1990.
143
Reinecke, J.P.L., Moraal, H., Potgieter, M.S., McDonald, F.B., Webber, W.R. Different Crossovers? in: Proceedings of the 25th International Cosmic Ray Conference, vol. 2, pp. 49–52, 1997. Steenkamp, R. Shock acceleration as source of the anomalous component of cosmic rays in the heliosphere, Ph.D. Thesis, Potchefstroom University for C.H.E., Potchefstroom, 1995. Webber, W.R., Lockwood, J.A. Interplanetary radial cosmic ray gradients and their implication for a possible large modulation effect at the heliospheric boundary. Astrophys. J. 317, 534–542, 1987. Webber, W.R., Lockwood, J.A. Voyager and Pioneer spacecraft measurements of cosmic ray intensities in the outer heliosphere: toward a new paradigm for understanding the global modulation process. 1. minimum solar modulation (1987 and 1997). J. Geophys. Res. 106, 29,323–29,331, 2001a. Webber, W.R., Lockwood, J.A. Voyager and Pioneer spacecraft measurements of cosmic ray intensities in the outer heliosphere: toward a new paradigm for understanding the global modulation process. 2. maximum solar modulation (1990–1991). J. Geophys. Res. 106, 29333–29340, 2001b. Webber, W.R., Lockwood, J.A., McDonald, F.B., Heikkila, B. 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, 2001.