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Galactic cosmic rays

M.-B. Kallenrode

University of Luneburg, 21332 Luneburg, Germany

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First author: Kallenrode 1

Galactic cosmic rays

M.-B. Kallenrode

University of Luneburg, 21332 Luneburg, Germany

This presentation gives a brief review of galactic cosmic rays. It starts withobservations made on the Earth, in near-Earth space and at larger distancesand higher heliographic latitudes. In the subsequent sections, the physicalprocesses leading to modulation, that are scattering at magnetic field irregu-larities, convection with the solar wind, drifts in the large-scale heliosphericmagnetic field, and adiabatic deceleration, will be introduced. These effectscan be combined to yield a transport equation. Different attempts to solvethis equation will be presented. A comparison with observations shows thatthe basic understanding of the modulation process seems to be fairly welldeveloped, although details, in particular the relative importance of drifts,still are subject to debate.

1 What we see

Galactic cosmic rays (GCRs) originate outside the heliosphere. Before be-ing detected in near-Earth space, they travel through interplanetary space.Owing to this propagation process, we expect a spatial dependence of GCRintensities. In addition, because of the high variability of the solar wind andthe embedded magnetic field we also expect temporal variations. The observ-able quantities therefore are time sequences at fixed positions in space andspatial gradients in GCR intensities between observers at different positions.In addition, an easy access of GCRs over the poles of the Sun was expected:in the plane of the ecliptic, the magnetic field line is tightly wound to anArchimedian spiral while it is much less wound up above the poles, cf. Fig.1. Thus it was expected that in the inner heliosphere GCR intensities arehigher over the poles than in the plane of the ecliptic.

The main questions regarding GCRs are: ‘Where do they come from?’,‘How are they accelerated to such high energies?’, and ‘How do they prop-agate through the interstellar and interplanetary medium towards the inner

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Fig. 1. Travel paths for particles from the heliopause to the inner heliosphere are longerin the plane of the ecliptic than above the poles of the Sun. Thus an easy access of GCRswas expected at high heliographic latitudes [Lee, 1995].

heliosphere?’. This paper deals with the low energetic part of cosmic rays,namely the energy range from some tens of MeV to some tens or hundredsof GeV and focuses on the latter question. Thus we mainly are concernedwith the interaction between particles and background plasma. The particlesthan are used as probes for the structure of the interplanetary medium.

1.1 The early years

The first observations of the energetic particle component which later was tobecome the galactic cosmic radiation date back to 1912 when Victor Hess flewan ion chamber on a manned balloon up to an altitude of 5 km. The radiationincreased with height, in contrast to what was expected from its supposed ter-restrial origin. Since Hess found no difference between day and night side, heruled out the Sun as cause for the increased ionization. Instead, he suggesteda penetrating radiation from the outside which he called ‘Hohenstrahlung’. In1927 Clay was the first to report a latitudinal effect: close to the equator, theradiation was lower than at higher latitudes. When Størmer’s calculations ofparticle trajectories in the geomagnetic field became available in 1930, thelatitudinal effect could be understood as due to shielding by the geomagneticfield. The next corner stone in cosmic ray research was the discovery of amaximum in cosmic ray intensity in an altitude of about 15 km by Pfotzerin 1936. This Pfotzer maximum results from the interaction between GCRsand atmosphere. In 1937 Forbush observed a world-wide decrease in GCRsduring a strong magnetic storm (Forbush decrease), giving the first evidencefor a relation between solar activity and GCRs. Subsequently, GCR energyspectra, composition and temporal variations have been studied, the mostimportant results and their interpretation will be described below.

Today, there are basically two means of observation: ground-based obser-vations by a world-wide net of neutron monitors and satellite observations.Neutron monitors provide informations at rather high energies integral abovetheir cut-off rigidity of some GV, at one position in space with an angular res-olution (anisotropy) derived from the combination of neutron monitors from

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Fig. 2. Spectrum of galactic cosmic rays [Meyer et al. (1974)]

different places. Spacecraft, on the other hand, give measurements fromdifferent positions in space, including the outer heliosphere and higher heli-ographic latitudes. Spacecraft measurements can be made in rather small,well defined energy or rigidity bands well below neutron monitor energies, aswell as in integral channels above a certain threshold. For limited small timeintervals, measurements are also made from balloons. Integral GCR fluxesare given in counts/s, fluxes in limited energy bands as differential flux.

1.2 Observations from Earth

Galactic cosmic rays are ionized nuclei with energies above ∼100 MeV/nucl.Very few of them can have energies up to 1020 eV, corresponding to about 20J (that is the kinetic energy of an apple of 200 g moving at a speed of about50 km/h). Galactic cosmic rays hit the Earth at a rate of about 1000/(m2

s), their source is outside the solar system but within the galaxy, probably itis shock acceleration at super-nova remnants. Only the very highest energiesmight originate in extragalactic sources.

1.2.1 Energy spectra and Composition

The longest time records of GCRs are from neutron monitors. They provideintegral measurements above their respective cut-off rigidity ranging from afew GV to about 15 GV with cut-off rigidities being lower at high geomagnetic

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Fig. 3. Yearly running averages of Mount Washington neutron monitor GCR intensities(dashed) and monthly mean sunspot numbers (solid) from 1954 to 1996 [Lockwood andWebber (1997)]. Note the reversed scale in sunspot numbers: sunspot numbers are highduring GCR minimum and low during GCR maximum.

latitudes and at higher altitudes. The lower energies only can be measuredeither from balloons or from space.

Figure 2 shows a composite of such early measurements giving the energyspectra for four different particle species: hydrogen, helium, C+O nuclei, andiron. Note that the hydrogen spectrum is multiplied by a factor of 5. Thebasic constituents of galactic cosmic radiation are protons and α-particles,and, of course, the same amount of electrons. Heavier ions, such as C, O,and Fe can be observed in much smaller numbers.

Energetically, the galactic cosmic radiation starts at energies of some tenMeV/nucl. At lower energies, the spectrum is dominated by particles accel-erated on the Sun or locally at traveling interplanetary shocks or corotatinginteraction regions. At energies above some ten MeV/nucl, the spectrumhas a positive slope, i.e. the intensity increases with increasing energy. Thispositive slope can be observed up to some hundred MeV/nucl, it then turnsover to a power-law Eγ with a slope γ = −2.5.

1.2.2 Modulation with the solar cycle

At energies below a few GeV/nucl, GCRs show a strong dependence on solaractivity with maximum intensities during solar minimum, cf. Figure 3. Thismodulation is also indicated by the split of the spectrum in Fig. 2. Withincreasing solar activity (going from the upper to the middle and then to thelower curve in Fig. 2), the maximum of the energy spectrum shifts towardshigher energies. At proton energies of about 100 MeV/nucl the modulation ismaximal, while at energies of about 4 GeV/nucl the modulation only is 15%–20%. The energy/rigidity dependence can also be seen in the comparison ofneutron monitors with different cut-off rigidities, cf. Fig. 4. Up to about10 GeV galactic electrons show a spectrum similar to that of the protons,

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Fig. 4. Energy/rigidity dependence of the solar modulation of GCRs as observed inneutron monitor data [Popielawska (1995)]. Data are 27-day averages, rigidities are 2.3 GV(Kiel), 3 GV (Climax), 9.3 GV (Tsumeb) and ∼ 13 GV (Huancayo). Features marked ‘GT’are related to global transients and can also be observed in the outer heliosphere.

modulation of galactic electrons is observed between 0.1 and 1 GeV.From Fig. 3 it is obvious that GCRs are modulated by an 11-year cycle

with intensity maxima during solar minimum (e.g. in 1964, 1976, 1986, 1996)and vice versa (as in 1969, 1980, 1990). In addition, the shapes of succes-sive maxima alternate between peaked and flat (or mesa-like): the maximumof cycle 20 (1972–1978.0) is a broad plateau upon which several large de-creases are superimposed while in cycles 19 (1965) and 21 (1987) the peakedmaximum lasts only a little longer than a single solar rotation.

1.2.3 Short-term variations: Forbush decreases

The data in Fig. 3 are yearly averages, giving the general trend in GCRs.Superposed on these long-term development, short-term variations last forhours to days and are directly related to solar activity, in particular to coro-nal mass ejections (CMEs); for a statistical analysis of Forbush decreasesand the related solar activity see Cane et al. (1996). Occasionally, followinga very energetic flare, enhancements in neutron monitor counting rates canbe observed, starting within a few hours after the flare. These ground levelevents (GLEs) indicate the acceleration of protons up to energies of someGeV or even higher in the flare. More interesting, however, is a feature start-ing between one and three days after the flare: a depression in GCRs by a fewpercent. These Forbush decreases often occur in two steps, cf. Fig. 5 [Barn-den (1973); Bavassano et al. (1994); Cane et al. (1994); Fluckiger (1985)]:a rather slow decrease starts at the passage of the interplanetary shock, most

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Fig. 5. Two-step Forbush decreases observed in hourly averaged neutron monitor data(Deep River, Mt. Wellington, Kerguelen). The smooth line indicates the suggested shapeof the Forbush decrease after subtraction of the local ejecta effect (shaded). The bottompanel gives the standard deviation in counting rates [Wibberenz (1998)].

likely related to the enhanced turbulence down-stream of the shock. A fewhours later, a more abrupt depression in GCRs follows when the observerenters the ejecta driving the shock. This depression ends when the ejectahas passed by and is followed by a slow recovery period. If the observer en-counters the shock but not the ejecta, the second step (hatched area in Fig.5) is missing and a smooth profile with a long-lasting recovery phase can beobserved. Typical values at 500 MeV are ∼2% for the shock decrease and∼5% for the decrease related to the arrival of the ejecta [Cane et al. (1993)].A recent summary about two-step decreases and possible interpretations interms of modified scattering conditions can be found in Wibberenz (1998).

1.2.4 Recurrent decreases

Evidence for the existence of a 27-day GCR intensity variation was reportedas early as 1938 by Forbush, based on measurements with the world-wide net-work of ionization chambers (for a historical account see Simpson (1998b)).These depressions in GCRs can be related to corotating interaction regions(CIRs) where slow and fast solar wind streams interact. CIRs form outside1 AU and are bounded by a pair of shocks at which MeV protons are acceler-ated while GCRs are depressed [Kunow et al. (1995); Simpson et al. (1995a);Simpson (1998a)].

A surprising result stems from Ulysses observations at high latitudes: whenUlysses is well above the streamer belt completely embedded in the fast solarwind, the recurrent modulation of GCRs continues, although neither theshocks nor changes in the solar wind speed are observed. Note that despitethe increase of the differential rotation period of the Sun from ∼ 26 daysnear the equator to ∼ 32 days in polar regions, Ulysses still observes 26-dayrecurrent decreases. This indicates that the modulation region is mainly at

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low latitudes, cf. Simpson (1998a). Two explanations for this remote sensingof CIRs exist: Kota and Jokipii (1995) suggested that particle transportperpendicular to the magnetic field carries the particle signatures of CIRsfrom low to higher latitudes. Fisk (1996), on the other hand, points to theinterplay between the differential rotation of the footpoints of the magneticfield lines in the photosphere and the subsequent non-radial expansion ofthese same field lines with the solar wind from rigidly corotating coronalholes, which may result in extensive excursions of magnetic field over a widerange of heliographic latitudes. Thus field lines at high latitudes might bedirectly connected to CIRs at lower latitudes and larger radial distances. Themodel also predicts an tighter than expected spiral-angle (over-winding) ofthe field at high latitudes, as observed by Forsyth et al. (1996).

1.3 The outer heliosphere and high latitudes

GCRs in the outer heliosphere are modulated with the solar cycle, too. Fig-ure 6 shows 26 day averages of GCRs for two differential and one integralchannel observed by IMP 7/8 orbiting Earth and by Pioneer 10 and Voy-ager 2 propagating outward. The distances of the latter two spacecraft aregiven at the top. The data span two solar cycles, the modulation is visiblein all channels at all radial distances, although the amplitude of modulationis smaller in the integral channel, as is expected from the spectrum in Fig. 2.Note the general trend: GCRs increase towards larger radial distances. Thusa radial gradient exists. In addition, there is a systematic time delay betweenwell-defined modulation features at 1 AU, such as step decreases and the 1987peak intensity, and their counterparts at greater heliospheric distances. Thisdelay is produced by outward propagating interplanetary disturbances withapproximately the solar wind velocity [McDonald et al. (1981)].

1.3.1 Temporal variations: MIRs, GMIRs, CMIRs and LMIRs

As in the inner heliosphere, in the outer heliosphere variations on time scalesshorter than the solar cycle are observed, related to variations in the plasma.Due to solar wind evolution and interactions between different streams, thevariations are not related to individual shocks but to merged interactionregions (MIRs). Burlaga et al. (1993) identified three different types: globalMIRs (GMIRs), corotating MIRs (CMIRs) and local MIRs (LMIRs).

Global MIRs are shell-like regions with intense magnetic fields extendingaround the Sun and to high latitudes. They are associated with the inter-action of transient and corotating MIRs and produce step-like intensity de-creases in GCRs throughout the heliosphere which, in turn, produce most ofthe modulation [Perko and Burlaga (1992); Potgieter et al. (1993)]. Systemsof transient flows, which are likely to be the reason of GMIRs, may be re-lated to multiple flares and CMEs on rather short time scales [Cliver (1987)],which in turn can be related to super-events [Droge et al. (1992)]. Besideslarge-scale gradient and curvature drifts in the interplanetary magnetic field,

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Fig. 6. Time series (26 day averages) of H, He and 10-22 MeV/nucl anomalous heliumfrom IMP (1 AU) and V2/P10 (large heliocentric distances) [Fujii and McDonald (1997)].

the cumulative effects of long-lived GMIRs are the principle source of GCRmodulation over the 22-year heliomagnetic cycle [McDonald et al. (1993);van Allen and Randall (1997)]. This implies a long GMIR lifetime (1.5–1.8years) and a magnetic structure that effectively extends over the solar poles.The long GMIR life-time than would imply a modulation boundary of theorder of ∼ 175 AU.

Corotating MIRs are MIRs with spiral forms associated with the coales-cence of two or more CIRs. CMIRs and rarefaction regions generally produceseveral successive decreases and increases in GCRs over several month, whilethe background intensity stays roughly constant. Thus CMIRs do not leadto appreciable net modulation [Burlaga et al. (1985, 1991)].

Local MIRs are non-corotating MIRs with a limited longitudinal and lati-tudinal extend. Most likely, they are formed by interactions among transientand perhaps corotating flows. Their effect on GCRs is local, comparable tothe typical Forbush decrease observed at Earth.

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Fig. 7. Daily averages of the magnetic field strength normalized by the Parker magneticfield strength Bp and cosmic ray intensity > 70 MeV protons observed by Voyager 1 andVoyager 2 [Burlaga et al. (1993)].

Figure 7 shows daily averages of the intensities of > 70 MeV/nucl GCRsand the magnetic field from the beginning of 1986 to the end of 1989 forVoyagers 1 and 2. Note that the intensities are measured on outward prop-agating spacecraft, thus the long-term variations include effects due to theradial gradient evident in Fig. 6. The most important results are: (1) GCRsdecrease when a strong GMIR moves past the spacecraft (time periods D andD’). (2) GCRs tend to increase over periods of several month when MIRs areweak and the strength of the magnetic field is relatively low (R and R’). (3)GCRs fluctuate about a plateau when MIRs are of intermediate strength andare balanced by rarefaction regions (time period P when CMIRs passed Voy-ager 1). Some LMIRs (L1–L3) produce step-like intensity decreases, however,they are observed locally on one Voyager spacecraft only.

From the close correlation between increasing magnetic field strength anddecreasing GCR intensities, Burlaga et al. (1985) suggested the followingrelation, called CR-B relation, between the change in GCR intensity J andthe magnetic field strength B relative to the Parker value Bp:

dJ

dt= −D

(B

Bp− 1

)for B > Bp (1)

and

dJ

dt= R for B < Bp (2)

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Fig. 8. Values of the radial gradients and its radial dependence (12 month averages) for180–450 MeV/nucl He and 130–220 MeV H (left) and the estimated value of gr for GCR Heand H over the 1974–1996 time period for 1, 10, and 75 AU [Fujii and McDonald (1997)].

with D and R being constant. Thus GCRs show complex time profiles when adisturbance increasing the field strength is swept past the observer while therecovery during rarefaction regions is at a constant rate. The CR-B relationis a quantitative description, it does not make any assumptions about theunderlying processes. Examples for its application to GCR fluctuations ondifferent scales are given in Burlaga et al. (1993). The success of the CR-Brelation points to the importance of the magnetic field for modulation andthe CR-B relation can be used as a test for modulation models.

1.3.2 Radial Gradients

The cosmic ray intensity gradient can be represented as

1J

dJ = gr dr + gλ dλ (3)

with the local radial (latitudinal) intensity gradient gr (gλ) defined as

gr =1J

dJ

drand gλ =

1J

dJ

dλ. (4)

Actual measurements, however, are made between two often widely separatedspacecraft. Thus only an average gradient can be determined, the non-localradial gradient [Potgieter et al. (1989)]

Gr =1

r2 − r1ln

J2

J1. (5)

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Data sampled over successive solar minima by the Pioneer and Voyager space-craft indicate that the radial gradient might be a function of heliospheric dis-tance [Webber and Lockwood (1985); Cummings et al. (1990); McDonaldet al. (1992); Fujii and McDonald (1997)], conveniently expressed as

gr = Go · rα (6)

or for measurements between two spacecraft

Gorα+12 − rα+1

1

α + 1= ln

J2

J1. (7)

Note that both, Go and α, are unknown quantities, thus a minimum of threespacecraft is required to determine both unknowns. After 1993, when Voyageris beyond ∼ 55 AU, second-order corrections to this relation seem to berequired [Fujii and McDonald (1997)].

The annual values of Go and α for the period 1973–1995 (most of cycle 21and 22) are shown in the left panel of Fig. 8. For the 1980 and 1990 solarmaxima Go and α are roughly the same, but there is a significant change inα that is believed to be due to drift effects. For both cycles there are largechanges in Go as well as α between solar maxima and solar minima: at solarmaximum the radial gradients are larger than at solar minimum and there is avery significant change in the radial dependence of gr. Note, in particular, thereversal of sign in α in the course of the solar cycle. The right panel of Fig. 8gives the local radial gradient gr for GCR He and H determined from Go andα over the period 1974–1996 at 1, 10, and 75 AU. These gradients show acomplex temporal variation depending on position in space. For instance, forHe the 1 AU gradients decrease from 1977 to 1982 while at larger distancesthe gradients increase with the three data sets converging at a value of about3%/AU in 1982. In H a cross-over is observed in 1979. In addition, there isno significant change in gr for either species associated with the reversal ofthe solar magnetic field in 1980 and 1990. At solar minimum there is a strongdecrease in gr with increasing r and the magnitude of gr is appreciably largerin qA < 0 (1981) than in qA > 0 epochs (1977). The above discussion is truefor the integral rate of GCR ions above 70 MeV. Note that this channel is lesssensitive to modulation (cf. Fig. 6) than the differential channels, thus thedifferential gradients are likely to be more sensitive to spatial and temporalchanges in modulation conditions. From a recent re-examination of spacecraftdata, Webber and Lockwood (1999) suggested that (a) two distinct regimesexist with a strong decrease (in 1996 by an order of magnitude) of Gr withr inside 10–20 AU and a very weak dependence on r at larger distances, (b)Gr depends stronger on r during A > 0 cycles, and (c) Gr ∼ 1/J in A < 0cycles while Gr depends only weakly on J in A > 0 cycles.

1.3.3 Latitudinal gradients

Latitudinal gradients are less well studied than radial gradients: Ulysses,launched in 1990, is the first spacecraft to reach heliographic latitudes of 80◦,the only other spacecraft outside the plane of ecliptic is Voyager 1 at ∼35◦.

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Fig. 9. Daily averaged and 26-day running mean quiet time counting rates of > 106 MeVprotons observed by KET on board Ulysses and by the UoC instrument on board IMP from1993 to the end of 1997 [Heber et al. (1998)]. Ulysses radial distances and heliographiclatitude are given at the bottom of the figure, IMP is at 1 AU in the plane of ecliptic. Thefluctuations are caused by CIRs.

Figure 9 shows a comparison of IMP and Ulysses > 106 MeV countingrates between early 1993 and the end of 1996. From the IMP data, the re-covery of GCRs in the declining phase of solar cycle 22 is evident. GCRs onUlysses show a more complex time development, owing to the orbit of thespacecraft. At the beginning of the time period, Ulysses slowly moves inwardand to higher heliographic latitudes, passing the Sun’s south pole in fall 1994at a maximum southern latitude of 80◦ S at a radial distance of 2.3 AU.Within 11 month, Ulysses performs a fast latitude scan up to 80◦ N. After-wards, Ulysses slowly descends in heliographic latitude and moves outwards.During the fast latitude scan, Ulysses crosses the ecliptical plane at a radialdistance of 1.3 AU. At this time, GCR intensities on both spacecraft agree,their difference is largest when Ulysses is over the poles. The time of the fastlatitude scan is most suitable to study latitudinal gradients because Ulyssesis at rather small radial distances, thus the radial gradient does not influencethe counting rates significantly, and the time period is rather short and GCRintensities at the Earth’s orbit are nearly constant, indicating that temporaleffects are of minor importance, too. From these data, Heber et al. (1998) ob-tain a latitudinal gradient of ∼ 0.3%/degree, however, significant latitudinaleffects are only observed when Ulysses is totally embedded in the high speedsolar wind streams of the coronal holes. As long as slow and fast streams canbe observed, the latitudinal gradient vanishes; above latitudes of about 60◦

little variation of GCRs with latitude is observed [Simpson et al. (1995a)].There are also indications that the latitudinal gradient is slightly different atsouthern and northern high latitudes, cf. McDonald et al. (1997). Amaz-ingly, the GCR variation is not symmetric around the heliographic equatorbut has an offset to 7–10◦ S [Heber et al. (1996b,a); Simpson et al. (1996)],which even may be time dependent [Heber et al. (1997)]. Such an asymme-try might indicate an offset of the heliospheric current sheet (HCS) towardsthe south. Flux conservation than requires the average magnetic field to be

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Fig. 10. Proton energy spectra between 10 and 10 000 MeV during the 1977 and 1987periods of GCR maximum intensity [Lockwood and Webber (1996)].

larger in the southern than in the northern hemisphere by a factor of 1.3 fora shift of 7◦. Ulysses and WIND magnetic field data indeed show a roughly30% change in the radial magnetic field strength [Smith et al. (1997)], cf.discussion in Fisk et al. (1998b).

The fast latitude scan of Ulysses provides only a snapshot of the latitudinalgradient in the declining phase of cycle 22. Indications for the temporalvariability of gλ can be inferred for a limited latitudinal range of about 35◦

from the Voyager 1 observations. McDonald et al. (1997) found that over alarge range of rigidities (0.3–1.3 GV) the latitudinal gradient in the recoveryphase of cycle 22 is very small in both inner and outer heliosphere. In therecovery phase of cycle 21, the latitudinal gradients observed in the outerheliosphere (at about 30 AU compared to about 60 AU in cycle 22) havebeen larger and have been negative, indicating a decrease in intensity towardshigher latitudes compared to the intensity increase towards the poles observedduring this qA > 0 cycle [Cummings et al. (1987); McDonald et al. (1997)]while in the recovery phase of cycle 20 positive gradients have been observedup to latitudes of 16◦ N [McKibben (1989)]. This behavior is confirmed bythe gradients derived at much higher rigidities (16–134 GV) from neutronmonitor measurements [Ahluwalia (1994a, 1996); Hall et al. (1994)].

In electrons, on the other hand, no apparent variation of intensity withlatitude can be observed, in particular, the reversal of the latitude depen-dence predicted by drift-dominated modulation models can not be observed[Ferrando et al. (1996)].

1.3.4 Spectra/Rigidity dependence

Sofar, we have seen some examples for GCR temporal profiles and gradientsfrom neutron monitor and/or spacecraft observations. This gives a rough im-pression of phenomena. A detailed understanding, however, requires a careful

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Fig. 11. Comparison between electrons and nuclei over two solar cycles. Electron dataare from different balloon measurements, the energy is approximately 1.2 GeV. Nucleondata are from Climax Neutron Monitor [Evenson et al. (1995)].

analysis of the energy/rigidity dependence of these quantities and thereforethe temporal changes in the GCR spectrum. In Fig. 2 we had seen the varia-tion in the GCR spectrum during the solar cycle, indicating that modulationis stronger at lower rigidities than at higher ones. Nonetheless, the spectraat successive solar minima also show differences: Figure 10 shows the GCRproton spectrum between 10 MeV and 10 GeV for the solar minima/GCRmaxima 1977 and 1987. The spectra are different for energies below 3 GeV,a cross-over occurs at 400 MeV (P = 0.95 GV). The 1996 spectrum, albeitat a lower intensity, is more similar to the 1977 than to the 1987 spectrum[McDonald (1998)]. This, as gradients and the shape of GCR maxima, sug-gests a dependence on the 22-year heliomagnetic cycle. A formal descriptionof these spectra and its evidence for modulation models is discussed in Lock-wood and Webber (1996, 1997), a discussion of the rigidity dependence ofother modulation parameters can be found in McDonald (1998).

1.4 Electrons

Sofar, we have occasionally pointed to a 22-year modulation cycle, such as inthe sign of the latitudinal gradient or in the shape of the intensity profile, anddescribed the cycles as qA positive or negative. If the sign q of the particlesunder study is changed, the properties of their profiles and gradients shouldbe more similar to the nucleon properties in the preceding or subsequent cyclethan in the actual one. Therefore, the measurement of galactic electrons iscrucial for our understanding. Unfortunately, measurements are only sparse:balloon flights have provided measurements in the GeV range at isolatedtimes, cf. Fig. 11. Longer times series, albeit at lower energies, exist from theISEE-3 spacecraft (e.g. Tuska et al. (1991)) and the Ulysses spacecraft (e.g.Ferrando et al. (1995, 1996); Rastoin et al. (1995)). Summaries on electronmodulation can be found in Evenson (1998) and Clem et al. (1996).

In general, the modulation of cosmic ray electrons is similar to the one

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Fig. 12. Ratios between electrons and He in solar cycles with different polarity [Garcia-Munoz et al. (1991)]

of nuclei, although some clear differences exist. For instance, the electronspectrum below 100 MeV has a negative slope which might be caused bymodulation or Jovian electrons. Differences between electrons and nuclei canbe used to identify charge-sign dependent effects in modulation. Observa-tions indicate differences; for instance, the ratio of electrons to protons orhelium is different depending on the polarity of the solar magnetic field, cf.Fig. 12 and Garcia-Munoz et al. (1986); Ferrando et al. (1995). Thesedifferences can be understood either in terms of a drift model or due to dif-ferent scattering of positively and negatively charged particles due to helicity[Bieber et al. (1987)]. The drift model here faces difficulties insofar, as itwould predict flat electron maxima in cycles with peaked nuclei maxima andvice versa. In the 1980s, the electron profile is flatter than that of helium,however, in the 1970s both fluxes track each other quite well, cf. Fig. 11.Note that the interpretation of electron data is complicated by the fact thatthe relative number of positrons is not known. However, the net charge ofthe electron/positron mixture is assumed to be negative [Evenson (1998)].

2 How do we understand this? Basic physical processes

Galactic cosmic rays stem from outside the heliosphere. Sofar, we have noreason to assume an incidence other than isotropic and constant in time. Butthe outer boundary of the heliosphere is far, in the order of 100 AU. Thus thegalactic cosmic rays have to propagate through the heliosphere before reach-ing our detectors. Since the heliosphere is a highly structured medium, filledand shaped by the solar wind, particles will interact with waves and disconti-nuities embedded in this plasma. The resulting processes are (a) pitch-anglescattering of particles at magnetohydrodynamic waves, (b) convection of par-

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ticles with the solar wind and adiabatic deceleration in the expanding solarwind plasma, (c) drifts due to changes of field properties during a gyro-periodof the particles, and (d) blocking and reflection at inhomogeneities such asmagnetic clouds.

2.1 Diffusion

Diffusion is a stochastic process, resulting from pitch-angle scattering ofcharged particles at magnetic field irregularities. Individual interactions be-tween particles and fields lead to small changes in pitch-angle,pitch-anglescattering therefore can be described as a random walk process. A reversalof the direction of motion along the field line requires a large number of suchsmall-angle scatterings. If the particle is in resonance with the wave, thescattering is more efficient because all the small-angle changes work into onedirection instead of trying to cancel each other. Thus pitch-angle scatteringwill mainly occur from interactions with field-fluctuations in resonance withthe particle motion along the field (resonance scattering) as described by theresonance condition

λ‖ = v‖ · Tg or k‖ =ωc

v‖=

ω‖µv

(8)

with λ‖ (k‖) the wave-length (wave-number) of the fluctuations parallel to thefield, v (v‖) the particle speed (parallel to the average field), ωc the particle’scyclotron frequency, and µ = cos α the pitch-cosine. For a full treatment ofthe theory with application to the scattering of particles in interplanetaryspace see Jokipii (1966) or Hasselmann and Wibberenz (1968).

The amount of scattering a particle experiences basically depends on thepower density f(k‖) of the waves at the resonance frequency. Thus for fixedenergy/rigidity the scattering coefficient κ depends on pitch-cosine µ. If themagnetic field power density spectrum is described by a power-law

f(k‖) = C · k−q‖ (9)

with q being the spectral shape, k‖ the wave number parallel to the field, andC the power at a certain frequency, the pitch-angle diffusion coefficient canbe written as

κ(µ) = A(1 − µ2)|µ|q−1 (10)

with A being a constant related to the level C of the turbulence. The particlemean free path, that is the distance traveled before the direction of motionis reversed, then is given as

λ‖ =38v

+1∫−1

(1 − µ2)2

κ(µ)dµ . (11)

The mean free path depends on particle rigidity as λ‖ ∼ P 2−q as long asq < 2. The above discussion holds for the so-called ‘slab model’ where

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the fluctuations are assumed to be waves with wave vectors parallel to thefield and axially symmetric traverse fluctuations. Discussions of discrepancieswith observations and modifications to this model can be found in Kunowet al. (1991) and Bieber et al. (1994). Particle mean free paths λ‖ can bedetermined from fits on solar particle events, see e.g. Bieber et al. (1994);Kallenrode (1993); Palmer (1982) and references therein.

In the presence of large-scale regular magnetic fields, diffusion becomesanisotropic and a diffusion tensor κij is used [Jokipii and Parker (1969)]:

κij =

⎛⎝ κ‖ 0 0

0 κ⊥ κT

0 −κT κ⊥

⎞⎠ . (12)

Here κ‖ (κ⊥) is the diffusion coefficient parallel (perpendicular) to the fieldwith κ⊥ being a few percent of κ‖ [Palmer (1982); Bieber et al. (1994)],that is perpendicular diffusion is less efficient than field-parallel diffusion.Perpendicular diffusion most easily can be understood as due to hard spherescattering: during its gyro-orbit, the particle hits an obstacle (hard sphere),leading to a change in the particle’s direction of motion. The particle thenencircles a different field line: it’s gyro-center has performed a cross-field mo-tion. Alternative, and for interplanetary space more realistic interpretationsassume wave-particle interactions (for an overview see e.g. Bieber (1998);Giacalone (1998); Potgieter (1998)). The perpendicular diffusion coefficientcan be written as [Forman et al. (1974)]

κ⊥ =vRL

3λ⊥/RL

1 + (λ⊥/RL)2or κ⊥ =

vRL

3RL

λ⊥for λ⊥ � RL . (13)

Here v is the particle speed, RL the particle’s Larmor radius, and λ⊥ thescattering mean free path perpendicular to the magnetic field. Parallel andperpendicular diffusion can be combined to yield a radial diffusion coefficient

κrr = κ‖ cos2 Ψ + κ⊥ sin2 Ψ (14)

with Ψ being the angle between the radial direction and the Parker Spiral.The diffusion tensor (12) contains also an anti-symmetric term, κT , which

changes sign with the reversal of the magnetic field polarity. This term isrelated but not identical with particle drifts and can be expressed as

κT = sign(qB)κ⊥λ⊥RL

or κT =vpc

3qB(15)

with v being the particle speed, c the speed of light, q the particle charge,p the particle momentum, and B the magnetic field strength. As soon asthe scattering mean free paths and gyro-radii become comparable, particlescannot complete several cycles around the field before being scattered and κT

diminishes. The latter form of (15) is commonly used in drift models, whileempirical forms of κ‖ and κ⊥ are chosen to fit the data. Using the moregeneral form, drift effects can be reduced substantially [Burger (1990)].

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2.2 Convection and adiabatic deceleration

The magnetic field irregularities scattering the energetic particles are frozen-in into the solar wind and thus convected outwards with the solar wind speed.Therefore, the energetic particles, too, are convected outwards with the solarwind speed. During solar wind expansion the ‘cosmic ray gas’ also expands,resulting in an adiabatic cooling, which is equivalent to a deceleration of theenergetic particles. This process is called adiabatic deceleration. Adiabaticdeceleration formally is equivalent to a Betatron effect due to the reductionof the interplanetary magnetic field strength with increasing radial distance.

2.3 Drifts

Drifts are systematic processes acting in the gyro-center of the energeticparticle. Except for curvature drift, particle drifts in electromagnetic fieldsresult form changes in the Larmor radius during one gyration, either becauseof changes in the particle speed or because magnetic field changes, for atutorial see e.g. Alfven and Falthammar (1963) or Kallenrode (1998).

In the heliosphere, the most important drifts are curvature and gradientdrift. Curvature drift results from the centrifugal force a particle experienceswhen traveling along a curved magnetic field line. The drift speed is propor-tional to the square of the particle speed parallel to the field and inverselyproportional to both curvature radius and magnetic field strength. The gradi-ent drift results from changes in the particle gyro-radius during one gyrationbecause of changes in magnetic field strength. Thus the gyro-orbit is notclosed but after one gyration the particle is offset with respect to its startingposition. A very efficient form of gradient drift develops in the configurationof two opposing magnetic fields: when the particle crosses the neutral linebetween the fields, its sense of gyration is reversed. Thus during one gyrationthe particle experiences a displacement by 4 Larmor radii. In interplanetaryspace this efficient form of gradient drift takes place along the heliosphericcurrent sheet. Note that positive and negative charges drift into oppositedirection. In addition, drift depends on the magnetic field orientation, thusthe drift direction is reversed when the magnetic field polarity is reversed.

Drifts such as the gradient and curvature drift depend on the particle’spitch-angle. For a nearly isotropic particle distribution, the average drift canbe derived as the divergence of the anti-symmetric part of the diffusion tensor.Drift effects in the mean Archimedian spiral pattern can be characterized[Parker (1957); Fisk and Schwadron (1995)] by a drift velocity

�vD =cvp

3q

[∇×

�Bo

B2o

](16)

leading to an average streaming

〈�S〉 = �vD ×∇fo or ∇ · 〈�S〉 = �vD · ∇fo . (17)

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Drift than is described as convection of particles with the drift velocity �vD,cf. the transport equation (18). Jokipii et al. (1977) point out that in astandard Parker field the drift speed can be several times the solar windspeed and thus drift effects by far can exceed convection with the solar wind,making drift an important effect in the transport equation.

Note that the drift speed (16) is derived for an undisturbed magnetic field,that is neither the influences of scattering nor mitigating effects on the driftare considered.

3 Putting it all together: Modulation models

The first attempt to describe modulation dates back to Parker (1958, 1965).He suggested a transport equation of the form

∂U

∂t= ∇ · (κs · ∇U)) − (�vsowi + �vd) · ∇U +

13∇ · �vsowi

d(αTU)dT

. (18)

Here U is the cosmic ray density, �vsowi the solar wind velocity, T the particlekinetic energy, α = (T +2To)/(T +To) with To being the particle rest energy,κs the symmetric part of the diffusion tensor, and �vd the drift velocity. Theterms on the right hand side then give the diffusion of particles in the irregularmagnetic field, bulk motion due to the outward convection of particles withthe solar wind and particle drifts, and adiabatic deceleration resulting fromthe divergence of the solar wind flow. Recent summaries on modulationtheory and modeling can be found e.g. in Potgieter (1993, 1998).

Particle interactions with inhomogeneities are not included in this equation.Thus a modulation model based solely on Parker’s equation can be appliedonly at times around solar minimum when MIRs are rare and balanced byrarefaction regions.

Note that the above transport equation holds also at and behind the ter-mination shock. The energy gain of particles at the shock is accounted forby the energy-change term: the sign of ∇�vsowi determines whether particlesare accelerated (negative, implying compression) or decelerated (positive, ex-pansion).

3.1 The diffusion convection model

The simplest application of Parker’s transport equation neglects drifts andreduces the particle propagation to a diffusion-convection model with adia-batic deceleration. For quasi-stationary conditions ∂U/∂t ≈ 0 and a roughlyisotropic cosmic ray flux in a spherical-symmetric heliosphere, Gleeson andAxford (1968) derived a modulation parameter Φ

Φ =

R∫r

vsowidr

3κ(r, P )(19)

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with r the radius at which the observer is located, R the outer boundaryof the modulation region, and κ(P, r) the diffusion coefficient as function ofrigidity P and observer’s distance r. Thus the same value of Φ describes themodulation of all particle species with the same rigidity. Physically, Φ roughlycorresponds to the average energy loss of inward propagating particles dueto adiabatic deceleration, which might be several hundred MeV for particlestraveling from the outer boundary to 1 AU. Thus particles with energies belowsome hundred MeV/nucl in the interstellar medium are completely excludedfrom the vicinity of Earth [Urch and Gleeson (1972)]. Therefore, near-Earthobservations at energies below a few hundred MeV/nucl do not provide anyinformation regarding the spectrum of the local interstellar particles at theseenergies. The part of the interstellar spectrum blocked by modulation is notnegligible because it contains ∼ 1/3 of the GCR pressure or energy density[McKibben (1990)] and thus has a strong influence on the dynamics andenergetics of the interstellar medium.

3.2 Including the large scale magnetic field: drifts

Although drifts were included in the original transport equation (18), theywere generally neglected until in the late seventies Jokipii et al. (1977)pointed out that the inclusion of drifts may profoundly alter our pictureof modulation. In particular, since most drifts are sensitive to the polarityof the global magnetic field, drift is expected to induce a charge asymmetry.

In the heliosphere, the following drift pattern arises. In an A > 0 cycle(the Sun’s magnetic field in the northern hemisphere is directed outwards,the configuration in the seventies and nineties) positively charged particlesdrift inward in the polar regions, downward to the heliomagnetic equatorand outward along the neutral sheet. The sense of drift is reversed if themagnetic polarity is reversed (A < 0) or the particle charge is negative. Atthe termination shock there is a fast drift upward along the shock. Drift itselfwould not cause modulation, it only changes the path along which particlesenter the heliosphere [Jokipii et al. (1977); Kota (1990)]. Modulation canonly happen due to transient disturbances (MIRs) or due to changes in thetilt anglewhich alters the drift path of the particles.

The relative roles of drift and diffusion are crucial for our understandingof modulation. For typical conditions, diffusion dominates drift on smalltime-scales. On longer time-scales, however, drift effects can accumulate andtherefore become important compared to diffusion. Two conditions have tobe fulfilled: (a) noticeable effects of drift can only be expected if the particlespends enough time in the heliosphere to drift at least a significant portionof π/2 in latitude. (b) Perpendicular diffusion should not be too strong towash out the drift pattern. If perpendicular diffusion would be too strong,particles would not drift along the polar axis or neutral sheet but will bespread in latitude. This spread depends on the ratio κ⊥/κT : if κ⊥ � κT ,drift dominates while for κT � κ⊥ diffusion destroys the drift pattern. Inthe intermediate case, however, both effects have to be considered.

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The inclusion of drifts into the transport of cosmic rays leads to the fol-lowing consequences, cf. Jokipii and Thomas (1981); Kota (1990); Kota andJokipii (1983); Potgieter (1993, 1998); Potgieter and Moraal (1985):(A) A shift in the anisotropies of cosmic rays. This is observed; in fact, ashift in neutron monitor anisotropies was the first phenomenon explained bydrifts [Levy (1976)].(B) A polarity-dependent 11-year cycle with a pronounced maximum in aqA negative cycle and a flat plateau-like maximum in a qA positive cycle[Kota and Jokipii (1983)], which can also be found in the neutron monitordata, cf. Fig. 3. In addition, because in an A > 0 cycle positively chargedparticles drift inward through the polar regions, they are rather insensitive toconditions in the equatorial region. The good correlation between GCRs andturbulence in the ecliptic plane in qA < 0 cycles compared to the poor one inqA > 0 cycles [Shea and Smart (1981)] is in agreement with this prediction.(C) A correlation of the modulation with the tilt-angleis expected for qA < 0cycles when positively charged particles travel inward along the heliosphericcurrent sheet and a larger tilt-angle automatically implies a longer drift path.In numerical models [Kota and Jokipii (1983)], this dependence on tilt-anglealone can explain the alternating peaked and plateau-type profiles in qA < oand qA > 0 cycles.(D) A charge asymmetry, for instance in the Helium-to-electron ratio. Al-though evidence for a charge-asymmetry can be found in the change of theelectron/helium ratio in opposite cycles [Garcia-Munoz et al. (1986), cf.Fig. 12], the most important charge-dependent signature is missing: elec-trons fail to show the shifted peaked-plateau difference, cf. Fig. 11.(E) Changes in radial gradients are a little bit controversial in drift models.Early calculations predicted large differences between cycles with markedlysmaller gradients during qA > 0 when positively charged particles enter theheliosphere over the poles. However, assumptions about scattering condi-tions, in particular the size of κ⊥, or modifications in the field geometry, inparticular in the polar field as suggested by Jokipii and Kota (1989), influencethe model predictions [Kota (1990)]. In addition, observations are difficult tointerpret because measurements of radial gradients were performed by widelyseparated spacecraft with the separation varying with time.(F) The latitudinal gradients have opposite signs in the two cycles. Here theobservational support perhaps is strongest, although the observed gradientsare much smaller than the ones predicted from standard modulation theory.In addition, latitudinal gradients should be maximal when the inclination ofthe current sheet is smallest, which is evident from observations reported inCummings et al. (1990); McDonald et al. (1992).

In sum, the observations give strong evidence for the influence of driftsin the large-scale heliospheric magnetic field on the modulation of GCRs,however, the model predictions seem to overestimate the importance of driftswhich is most obvious in the smaller than expected latitudinal gradients.

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3.3 Modifications to modulation models

Before we will discuss modifications to modulation models, the relevant pa-rameter and assumptions of standard models are summarized here:(a) the heliosphere is assumed to be spherical-symmetric.(b) diffusion is described by the diffusion tensor under the following assump-tions: (i) κ⊥ � κ‖, (ii) perpendicular diffusion is isotropic, that is κ⊥ is thesame inside the plane of ecliptic and perpendicular to it, and (iii) κ ∼ 1/B.(c) the magnetic field is either assumed as a Parker field or as a Parker fieldwith a modified polar geometry as suggested by Jokipii and Kota (1989).The effect of these two geometries on modulation is discussed in Haasbroekand Potgieter (1995).

The modeling attempts described above are not able to describe all the ob-served aspects of modulation. Nonetheless, it is agreed [cf. introduction andarticles in Fisk et al. (1998a)] that the basic understanding of the modula-tion process – particles undergo diffusion, convection, adiabatic deceleration,and large-scale drifts – appears to be correct. There is no requirement to addany new physical processes nor to discard one of the previously consideredprocesses. However, sofar we have not been able to agree on a single set of pa-rameters to describe the basic processes. Thus some suggestions are made tomodify assumptions made in modeling, supported either by observations orby theoretical considerations. The main effort is to suppress drifts to reducelatitudinal gradients.

3.3.1 The diffusion tensor

A modification of the diffusion tensor allows variations of the relative impor-tance of drifts. Increasing perpendicular diffusion will smear out the signa-ture of drifts, in particular, latitudinal gradients will decrease. Significantperpendicular diffusion also can explain the fact that CIR related increasesand decreases are observed at heliographic latitudes well above the streamerbelt, cf. Jokipii et al. (1995); Kota and Jokipii (1998).

Reinecke et al. (1997) emphasized the importance of κrr and κλλ as basicparameters of Parker’s transport equation. Using a 2-D non-drift modelwith κrr independent of radius and κλλ = 0.1κrr combined with the Ulysses-observed variation of the solar wind with heliolongitude, fits on 1977 and 1987data were possible, the model was even superior to the ones incorporatingdrift. However, a further modification of the model including drift is requiredto account for the observed change in sign of the latitudinal gradient.

The influence of changes in the diffusion tensor on modulation in mod-els considering drift has been demonstrated in many works. For instance,Burger (1990) studied the effects of scattering and the random walk of fieldlines on drift and found that fluctuations on all scales equal or greater theresonant wavelength can reduce drift. Fisk and Schwadron (1995) considervariations in the heliospheric magnetic field with scales intermediate betweenthe gyro-radii of GCRs and the heliocentric radial distance. The authors

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demonstrate that a correlation between such fluctuations and GCR inten-sities exist: intermediate-scale fluctuations alter the transport patterns ofGCRs such that during a A > 0 cycle radial gradients will increase andlatitudinal gradients will decrease compared to the gradients derived with asimple Parker field. Thus gradients would come closer to the Ulysses obser-vations reported e.g. in Smith et al. (1995) and Heber et al. (1998).

Potgieter (1997) and Potgieter et al. (1997) assumed an anisotropic dif-fusion perpendicular to the magnetic field with the latitudinal diffusion co-efficient increased to 30% of the parallel diffusion coefficient, which leads toa marked reduction of drift effects. More elaborate methods for the theoret-ical derivation of diffusion coefficients are described in Bieber et al. (1995)and Hattingh et al. (1995), general discussions about the knowledge of thediffusion tensor can be found in Bieber (1998); Giacalone (1998); Potgi-eter (1998). There are also suggestions that the commonly used radial de-pendence of the diffusion coefficient might be too simple because it neglectsthe evolution of turbulence. This has far reaching consequences, in partic-ular, a correct consideration of the evolution of magnetic field turbulencemight require different treatment of the inner and outer heliosphere [Zank etal. (1996, 1998)].

3.3.2 Geometry and radial evolution

In standard modulation models, the heliosphere is assumed to be sphericalsymmetric. The latitudinal variation of the solar wind speed as observed byUlysses [Phillips et al. (1995); McComas et al. (1998)] has to be considered,however, this effect is not nearly as important as changes in the diffusioncoefficient and would tend to enhance rather than to diminish the modeledlatitudinal gradients [Potgieter (1998)]. The distribution of solar wind speedsalso might influence the shape of the heliosphere, leading to a poleward elon-gation. Although the latitudinal gradients are diminished by this assumption[Fichtner (1996); Haasbroek and Potgieter (1997)], reductions seem to bemuch smaller than required to explain the Ulysses observations.

If the over-winding of the south polar interplanetary magnetic field [Forsythet al. (1995)] is considered, the offset of the plane of symmetry in GCRs canbe reproduced in modulation models [Hattingh et al. (1997)].

But asides from the deviation of the actual shape of the heliosphere from aParker field, there is also the evolution of properties of the solar wind and theembedded magnetic field as it propagates outwards. Although the averageazimuthal angle is consistent with the Parker angle even at distances around30 AU at times of low solar activity [Burlaga and Ness (1993)], there is alarge variation around this average, indicating the presence of fluctuations.With increasing solar activity, the distribution becomes even broader and theaverage no longer coincides with the theoretical Parker angle. In addition tothe spiral angle, the sector pattern, too, evolves with increasing distance fromthe Sun, as does the heliospheric current sheet. There are large-scale fluctua-tions of the magnetic field strength about the average Parker value which do

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not appear to be periodic even though the observations were made during thedeclining phase of the solar cycle when corotating streams and coronal holesare expected (and indeed observed) near the Sun [Burlaga and Ness (1996)].In addition, magnetic field fluctuations can be described by a Pareto distri-bution [Burlaga (1995); Burlaga and Ness (1996)], a lognormal distributionwith an exponential tail at high magnetic field strengths, suggesting a multi-fractal structure of the interplanetary magnetic field. Transient disturbances(ejecta) tend to modify this pattern further by modifying the separation be-tween successive sector boundaries in an irregular way, but do not destroythe individual sectors. In addition, the Voyager 2 observations suggest that,at least for a few years around solar minimum, shocks are not important indetermining the structure, dynamics, and thermodynamics of the equatorialsolar wind beyond 20 AU, because only few shocks were observed and thesewere weak [Burlaga (1994)]. Because on the other hand the solar wind tem-peratures are surprisingly high in the outer heliosphere beyond about 20 AU,at least much higher than expected from simple adiabatic expansion, a sourceof heating in the outer heliosphere appears to exist [Gazis et al. (1994)]: mostlikely the interaction between high-speed and low-speed solar wind streams.Other speculated sources are the interaction of the solar wind with the localinterstellar medium LISM due to mass loading of the solar wind with pick-upions. Thus, as with turbulence, in solar wind energetics a distinction betweenthe inner and outer heliosphere might be necessary.

3.3.3 Transient effects

Transient effects, in which cosmic rays respond to localized disturbances inthe solar wind are interesting near solar minimum but provide little impacton solar modulation. In contrast, during the rise to and during solar max-imum transient effects are likely to be the governing mechanism for cosmicray modulation. The transport equation (18) directly does not include theeffects of inhomogeniteis such as ejecta or, more important, MIRs of anykind. However, from the study of Forbush decreases and the magnetic fieldproperties of MIRs, there is strong evidence that enhanced scattering is ofcrucial importance [Wibberenz et al. (1998) and references therein]. Thusthe simplest way to incorporate such regions in modulation models is to in-crease scattering as a function of distance and time to simulate the MIR andits outward propagation.

Some attempts to incorporate such disturbances into modulation mod-els exist, cf. Potgieter (1998). The first attempt goes back to Perko andFisk (1983), see also Perko (1993), who introduced propagating barriers(which correspond to MIRs) in a time-dependent numerical model. le Rouxand Potgieter (1995) [see also Haasbroek et al. (1995a); Haasbroek et al.(1995b); Potgieter et al. (1993)] used a drift-model with 4 major GMIRsto model the complete proton cycle from 1977-1987. This model also sug-gests that an extended plateau-like maximum is not only tied to drift butalso to a rather long time period void of major MIRs. Despite this success,

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one fundamental problem remains: standard drift models considering MIRsproduce GCR recovery times behind the MIR which are strongly dependenton the polarity of the interplanetary magnetic field – the observations, on theother hand, suggest that recovery mainly depends on whether solar activityis increasing or decreasing.

3.3.4 Is cycle 22 different from other cycles?

The Ulysses observations have been made during solar cycle 22, in particularduring the declining phase of solar activity and therefore during the recoveryphase of GCRs. Some observers have suggested that this cycle may exhibitfeatures different from other cycles. Cycle 22 has been a positive cycle (qA >0). From neutron monitor data Ahluwalia (1979), Ahluwalia (1994a), andAhluwalia (1994b) showed that during odd cycles (qA < 0), the recovery ofGCRs is completed within 5 to 8 years while less than half the time is requiredfor recovery during even cycles. Solar cycle 22, however, violates this rule,showing a rather slow recovery at neutron monitor energies [Ahluwalia andWilson (1996)]. Usoskin et al. (1998) also note differences between cycle 20and 22, in particular a different rigidity dependence in GCRs, however, fromthe analysis of four solar cycles these authors suggest that cycle 20 mighthave been the unusual one while cycle 22 fits well into the general picture.The authors note that this holds only for GCRs while solar activity shows asimilar behavior in all four cycles under study.

4 Summary

The modulation of GCRs is determined by diffusion, convection with thesolar wind, adiabatic deceleration, and drifts in the large-scale heliosphericmagnetic field. Although it is agreed that all these processes are importantand no additional process are required to understand modulation, so far it hasnot been possible to agree an a parameter set describing all observations. Inparticular, the relative importance of drifts is subject to debate. Our under-standing of modulation, however, is complicated by the fact that GCRs aretemporally and spatially variable while our observations, except for the con-tinuous observations from the orbit of Earth, are made at different positionsat different times. In these observations, it is difficult to distangle spatial andtemporal effects. Missions to the outer heliosphere greatly might improve ourunderstanding of modulation: radial distances already visited by the Voyagerand Pioneer spacecraft will be re-visited at a later time and in a differentphase of the solar cycle, which will help to distangle spatial and temporalvariations. A spacecraft propagating outwards combined with Earth-boundand Pioneer/Voyager observations will help to determine gradients more ac-curate. And finally, a spacecraft crossing the heliopause will measure thelocal interstellar spectrum of cosmic rays, which also is an important inputparameter in modulation models and sofar can be only guesstimated.

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