Galactic cosmic rays

Transcription

1 Galactic cosmic rays M.-B. Kallenrode University of Lüneburg, Lüneburg, Germany Camera-ready Copy for Summer school Outer Heliosphere Manuscript-No.??? Offset requests to: Kallenrode

2 First author: Kallenrode 1 Galactic cosmic rays M.-B. Kallenrode University of Lüneburg, Lüneburg, Germany This presentation gives a brief review of galactic cosmic rays. It starts with observations made on the Earth, in near-earth space and at larger distances and higher heliographic latitudes. In the subsequent sections, the physical processes leading to modulation, that are scattering at magnetic field irregularities, convection with the solar wind, drifts in the large-scale heliospheric magnetic field, and adiabatic deceleration, will be introduced. These effects can be combined to yield a transport equation. Different attempts to solve this equation will be presented. A comparison with observations shows that the basic understanding of the modulation process seems to be fairly well developed, 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 being detected in near-earth space, they travel through interplanetary space. Owing to this propagation process, we expect a spatial dependence of GCR intensities. In addition, because of the high variability of the solar wind and the embedded magnetic field we also expect temporal variations. The observable quantities therefore are time sequences at fixed positions in space and spatial 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 an Archimedian 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 are higher 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 propagate through the interstellar and interplanetary medium towards the inner

3 First author: Kallenrode 2 Fig. 1. Travel paths for particles from the heliopause to the inner heliosphere are longer in the plane of the ecliptic than above the poles of the Sun. Thus an easy access of GCRs was 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 hundreds of GeV and focuses on the latter question. Thus we mainly are concerned with the interaction between particles and background plasma. The particles than 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 to become the galactic cosmic radiation date back to 1912 when Victor Hess flew an ion chamber on a manned balloon up to an altitude of 5 km. The radiation increased with height, in contrast to what was expected from its supposed terrestrial origin. Since Hess found no difference between day and night side, he ruled out the Sun as cause for the increased ionization. Instead, he suggested a penetrating radiation from the outside which he called Höhenstrahlung. In 1927 Clay was the first to report a latitudinal effect: close to the equator, the radiation was lower than at higher latitudes. When Størmer s calculations of particle trajectories in the geomagnetic field became available in 1930, the latitudinal effect could be understood as due to shielding by the geomagnetic field. The next corner stone in cosmic ray research was the discovery of a maximum in cosmic ray intensity in an altitude of about 15 km by Pfotzer in This Pfotzer maximum results from the interaction between GCRs and atmosphere. In 1937 Forbush observed a world-wide decrease in GCRs during a strong magnetic storm (Forbush decrease), giving the first evidence for a relation between solar activity and GCRs. Subsequently, GCR energy spectra, composition and temporal variations have been studied, the most important results and their interpretation will be described below. Today, there are basically two means of observation: ground-based observations by a world-wide net of neutron monitors and satellite observations. Neutron monitors provide informations at rather high energies integral above their cut-off rigidity of some GV, at one position in space with an angular resolution (anisotropy) derived from the combination of neutron monitors from

4 First author: Kallenrode 3 Fig. 2. Spectrum of galactic cosmic rays [Meyer et al. (1974)] different places. Spacecraft, on the other hand, give measurements from different positions in space, including the outer heliosphere and higher heliographic latitudes. Spacecraft measurements can be made in rather small, well defined energy or rigidity bands well below neutron monitor energies, as well as in integral channels above a certain threshold. For limited small time intervals, measurements are also made from balloons. Integral GCR fluxes are 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 ev, corresponding to about 20 J (that is the kinetic energy of an apple of 200 g moving at a speed of about 50 km/h). Galactic cosmic rays hit the Earth at a rate of about 1000/(m 2 s), their source is outside the solar system but within the galaxy, probably it is shock acceleration at super-nova remnants. Only the very highest energies might originate in extragalactic sources Energy spectra and Composition The longest time records of GCRs are from neutron monitors. They provide integral measurements above their respective cut-off rigidity ranging from a few GV to about 15 GV with cut-off rigidities being lower at high geomagnetic

5 First author: Kallenrode 4 Fig. 3. Yearly running averages of Mount Washington neutron monitor GCR intensities (dashed) and monthly mean sunspot numbers (solid) from 1954 to 1996 [Lockwood and Webber (1997)]. Note the reversed scale in sunspot numbers: sunspot numbers are high during GCR minimum and low during GCR maximum. latitudes and at higher altitudes. The lower energies only can be measured either from balloons or from space. Figure 2 shows a composite of such early measurements giving the energy spectra for four different particle species: hydrogen, helium, C+O nuclei, and iron. Note that the hydrogen spectrum is multiplied by a factor of 5. The basic 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 ten MeV/nucl. At lower energies, the spectrum is dominated by particles accelerated on the Sun or locally at traveling interplanetary shocks or corotating interaction regions. At energies above some ten MeV/nucl, the spectrum has a positive slope, i.e. the intensity increases with increasing energy. This positive slope can be observed up to some hundred MeV/nucl, it then turns over to a power-law E γ with a slope γ = Modulation with the solar cycle At energies below a few GeV/nucl, GCRs show a strong dependence on solar activity with maximum intensities during solar minimum, cf. Figure 3. This modulation is also indicated by the split of the spectrum in Fig. 2. With increasing solar activity (going from the upper to the middle and then to the lower curve in Fig. 2), the maximum of the energy spectrum shifts towards higher energies. At proton energies of about 100 MeV/nucl the modulation is maximal, 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 of neutron monitors with different cut-off rigidities, cf. Fig. 4. Up to about 10 GeV galactic electrons show a spectrum similar to that of the protons,

6 First author: Kallenrode 5 Fig. 4. Energy/rigidity dependence of the solar modulation of GCRs as observed in neutron 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 successive maxima alternate between peaked and flat (or mesa-like): the maximum of cycle 20 ( ) is a broad plateau upon which several large decreases are superimposed while in cycles 19 (1965) and 21 (1987) the peaked maximum lasts only a little longer than a single solar rotation 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 for hours to days and are directly related to solar activity, in particular to coronal mass ejections (CMEs); for a statistical analysis of Forbush decreases and the related solar activity see Cane et al. (1996). Occasionally, following a very energetic flare, enhancements in neutron monitor counting rates can be observed, starting within a few hours after the flare. These ground level events (GLEs) indicate the acceleration of protons up to energies of some GeV or even higher in the flare. More interesting, however, is a feature starting between one and three days after the flare: a depression in GCRs by a few percent. These Forbush decreases often occur in two steps, cf. Fig. 5 [Barnden (1973); Bavassano et al. (1994); Cane et al. (1994); Flückiger (1985)]: a rather slow decrease starts at the passage of the interplanetary shock, most

7 First author: Kallenrode 6 Fig. 5. Two-step Forbush decreases observed in hourly averaged neutron monitor data (Deep River, Mt. Wellington, Kerguelen). The smooth line indicates the suggested shape of the Forbush decrease after subtraction of the local ejecta effect (shaded). The bottom panel gives the standard deviation in counting rates [Wibberenz (1998)]. likely related to the enhanced turbulence down-stream of the shock. A few hours later, a more abrupt depression in GCRs follows when the observer enters the ejecta driving the shock. This depression ends when the ejecta has passed by and is followed by a slow recovery period. If the observer encounters 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 be observed. 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 in terms of modified scattering conditions can be found in Wibberenz (1998) Recurrent decreases Evidence for the existence of a 27-day GCR intensity variation was reported as early as 1938 by Forbush, based on measurements with the world-wide network 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 outside 1 AU and are bounded by a pair of shocks at which MeV protons are accelerated while GCRs are depressed [Kunow et al. (1995); Simpson et al. (1995a); Simpson (1998a)]. A surprising result stems from Ulysses observations at high latitudes: when Ulysses is well above the streamer belt completely embedded in the fast solar wind, the recurrent modulation of GCRs continues, although neither the shocks nor changes in the solar wind speed are observed. Note that despite the increase of the differential rotation period of the Sun from 26 days near the equator to 32 days in polar regions, Ulysses still observes 26-day recurrent decreases. This indicates that the modulation region is mainly at

8 First author: Kallenrode 7 low latitudes, cf. Simpson (1998a). Two explanations for this remote sensing of CIRs exist: Kota and Jokipii (1995) suggested that particle transport perpendicular to the magnetic field carries the particle signatures of CIRs from low to higher latitudes. Fisk (1996), on the other hand, points to the interplay between the differential rotation of the footpoints of the magnetic field lines in the photosphere and the subsequent non-radial expansion of these same field lines with the solar wind from rigidly corotating coronal holes, which may result in extensive excursions of magnetic field over a wide range of heliographic latitudes. Thus field lines at high latitudes might be directly connected to CIRs at lower latitudes and larger radial distances. The model also predicts an tighter than expected spiral-angle (over-winding) of the 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. Figure 6 shows 26 day averages of GCRs for two differential and one integral channel observed by IMP 7/8 orbiting Earth and by Pioneer 10 and Voyager 2 propagating outward. The distances of the latter two spacecraft are given at the top. The data span two solar cycles, the modulation is visible in all channels at all radial distances, although the amplitude of modulation is smaller in the integral channel, as is expected from the spectrum in Fig. 2. Note the general trend: GCRs increase towards larger radial distances. Thus a radial gradient exists. In addition, there is a systematic time delay between well-defined modulation features at 1 AU, such as step decreases and the 1987 peak intensity, and their counterparts at greater heliospheric distances. This delay is produced by outward propagating interplanetary disturbances with approximately the solar wind velocity [McDonald et al. (1981)] Temporal variations: MIRs, GMIRs, CMIRs and LMIRs As in the inner heliosphere, in the outer heliosphere variations on time scales shorter than the solar cycle are observed, related to variations in the plasma. Due to solar wind evolution and interactions between different streams, the variations are not related to individual shocks but to merged interaction regions (MIRs). Burlaga et al. (1993) identified three different types: global MIRs (GMIRs), corotating MIRs (CMIRs) and local MIRs (LMIRs). Global MIRs are shell-like regions with intense magnetic fields extending around the Sun and to high latitudes. They are associated with the interaction of transient and corotating MIRs and produce step-like intensity decreases in GCRs throughout the heliosphere which, in turn, produce most of the modulation [Perko and Burlaga (1992); Potgieter et al. (1993)]. Systems of transient flows, which are likely to be the reason of GMIRs, may be related to multiple flares and CMEs on rather short time scales [Cliver (1987)], which in turn can be related to super-events [Dröge et al. (1992)]. Besides large-scale gradient and curvature drifts in the interplanetary magnetic field,

9 First author: Kallenrode 8 Fig. 6. Time series (26 day averages) of H, He and MeV/nucl anomalous helium from 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 GCR modulation over the 22-year heliomagnetic cycle [McDonald et al. (1993); van Allen and Randall (1997)]. This implies a long GMIR lifetime ( years) and a magnetic structure that effectively extends over the solar poles. The long GMIR life-time than would imply a modulation boundary of the order of 175 AU. Corotating MIRs are MIRs with spiral forms associated with the coalescence of two or more CIRs. CMIRs and rarefaction regions generally produce several successive decreases and increases in GCRs over several month, while the background intensity stays roughly constant. Thus CMIRs do not lead to appreciable net modulation [Burlaga et al. (1985, 1991)]. Local MIRs are non-corotating MIRs with a limited longitudinal and latitudinal extend. Most likely, they are formed by interactions among transient and perhaps corotating flows. Their effect on GCRs is local, comparable to the typical Forbush decrease observed at Earth.

10 First author: Kallenrode 9 Fig. 7. Daily averages of the magnetic field strength normalized by the Parker magnetic field strength B p and cosmic ray intensity > 70 MeV protons observed by Voyager 1 and Voyager 2 [Burlaga et al. (1993)]. Figure 7 shows daily averages of the intensities of > 70 MeV/nucl GCRs and the magnetic field from the beginning of 1986 to the end of 1989 for Voyagers 1 and 2. Note that the intensities are measured on outward propagating spacecraft, thus the long-term variations include effects due to the radial gradient evident in Fig. 6. The most important results are: (1) GCRs decrease when a strong GMIR moves past the spacecraft (time periods D and D ). (2) GCRs tend to increase over periods of several month when MIRs are weak 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 and are balanced by rarefaction regions (time period P when CMIRs passed Voyager 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 and decreasing GCR intensities, Burlaga et al. (1985) suggested the following relation, called CR-B relation, between the change in GCR intensity J and the magnetic field strength B relative to the Parker value B p : ( ) dj B dt = D 1 for B>B p (1) B p and dj dt = R for B<B p (2)

11 First author: Kallenrode 10 Fig. 8. Values of the radial gradients and its radial dependence (12 month averages) for MeV/nucl He and MeV H (left) and the estimated value of g r for GCR He and H over the 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 a disturbance increasing the field strength is swept past the observer while the recovery during rarefaction regions is at a constant rate. The CR-B relation is a quantitative description, it does not make any assumptions about the underlying processes. Examples for its application to GCR fluctuations on different scales are given in Burlaga et al. (1993). The success of the CR-B relation points to the importance of the magnetic field for modulation and the CR-B relation can be used as a test for modulation models Radial Gradients The cosmic ray intensity gradient can be represented as 1 J dj = g r dr + g λ dλ (3) with the local radial (latitudinal) intensity gradient g r (g λ ) defined as g r = 1 J dj dr and g λ = 1 J dj dλ. (4) Actual measurements, however, are made between two often widely separated spacecraft. Thus only an average gradient can be determined, the non-local radial gradient [Potgieter et al. (1989)] G r = 1 r 2 r 1 ln J 2 J 1. (5)

12 First author: Kallenrode 11 Data sampled over successive solar minima by the Pioneer and Voyager spacecraft indicate that the radial gradient might be a function of heliospheric distance [Webber and Lockwood (1985); Cummings et al. (1990); McDonald et al. (1992); Fujii and McDonald (1997)], conveniently expressed as g r = G o r α (6) or for measurements between two spacecraft r2 α+1 r1 α+1 G o α +1 =ln J 2 J 1. (7) Note that both, G o and α, are unknown quantities, thus a minimum of three spacecraft is required to determine both unknowns. After 1993, when Voyager is beyond 55 AU, second-order corrections to this relation seem to be required [Fujii and McDonald (1997)]. The annual values of G o and α for the period (most of cycle 21 and 22) are shown in the left panel of Fig. 8. For the 1980 and 1990 solar maxima G o and α are roughly the same, but there is a significant change in α that is believed to be due to drift effects. For both cycles there are large changes in G o as well as α between solar maxima and solar minima: at solar maximum the radial gradients are larger than at solar minimum and there is a very significant change in the radial dependence of g r. Note, in particular, the reversal of sign in α in the course of the solar cycle. The right panel of Fig. 8 gives the local radial gradient g r for GCR He and H determined from G o and α over the period at 1, 10, and 75 AU. These gradients show a complex temporal variation depending on position in space. For instance, for He the 1 AU gradients decrease from 1977 to 1982 while at larger distances the gradients increase with the three data sets converging at a value of about 3%/AU in In H a cross-over is observed in In addition, there is no significant change in g r for either species associated with the reversal of the solar magnetic field in 1980 and At solar minimum there is a strong decrease in g r with increasing r and the magnitude of g r is appreciably larger in qa < 0 (1981) than in qa > 0 epochs (1977). The above discussion is true for the integral rate of GCR ions above 70 MeV. Note that this channel is less sensitive to modulation (cf. Fig. 6) than the differential channels, thus the differential gradients are likely to be more sensitive to spatial and temporal changes in modulation conditions. From a recent re-examination of spacecraft data, Webber and Lockwood (1999) suggested that (a) two distinct regimes exist with a strong decrease (in 1996 by an order of magnitude) of G r with r inside AU and a very weak dependence on r at larger distances, (b) G r depends stronger on r during A>0 cycles, and (c) G r 1/J in A<0 cycles while G r depends only weakly on J in A>0cycles 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.

13 First author: Kallenrode 12 Fig. 9. Daily averaged and 26-day running mean quiet time counting rates of > 106 MeV protons observed by KET on board Ulysses and by the UoC instrument on board IMP from 1993 to the end of 1997 [Heber et al. (1998)]. Ulysses radial distances and heliographic latitude are given at the bottom of the figure, IMP is at 1 AU in the plane of ecliptic. The fluctuations are caused by CIRs. Figure 9 shows a comparison of IMP and Ulysses > 106 MeV counting rates between early 1993 and the end of From the IMP data, the recovery of GCRs in the declining phase of solar cycle 22 is evident. GCRs on Ulysses show a more complex time development, owing to the orbit of the spacecraft. At the beginning of the time period, Ulysses slowly moves inward and to higher heliographic latitudes, passing the Sun s south pole in fall 1994 at 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. Afterwards, Ulysses slowly descends in heliographic latitude and moves outwards. During the fast latitude scan, Ulysses crosses the ecliptical plane at a radial distance 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 fast latitude scan is most suitable to study latitudinal gradients because Ulysses is at rather small radial distances, thus the radial gradient does not influence the counting rates significantly, and the time period is rather short and GCR intensities at the Earth s orbit are nearly constant, indicating that temporal effects are of minor importance, too. From these data, Heber et al. (1998) obtain a latitudinal gradient of 0.3%/degree, however, significant latitudinal effects are only observed when Ulysses is totally embedded in the high speed solar wind streams of the coronal holes. As long as slow and fast streams can be 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 at southern and northern high latitudes, cf. McDonald et al. (1997). Amazingly, the GCR variation is not symmetric around the heliographic equator but 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 asymmetry might indicate an offset of the heliospheric current sheet (HCS) towards the south. Flux conservation than requires the average magnetic field to be

14 First author: Kallenrode 13 Fig. 10. Proton energy spectra between 10 and MeV during the 1977 and 1987 periods of GCR maximum intensity [Lockwood and Webber (1996)]. larger in the southern than in the northern hemisphere by a factor of 1.3 for a shift of 7. Ulysses and WIND magnetic field data indeed show a roughly 30% 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 latitudinal gradient in the declining phase of cycle 22. Indications for the temporal variability of g λ can be inferred for a limited latitudinal range of about 35 from the Voyager 1 observations. McDonald et al. (1997) found that over a large range of rigidities ( GV) the latitudinal gradient in the recovery phase of cycle 22 is very small in both inner and outer heliosphere. In the recovery phase of cycle 21, the latitudinal gradients observed in the outer heliosphere (at about 30 AU compared to about 60 AU in cycle 22) have been larger and have been negative, indicating a decrease in intensity towards higher latitudes compared to the intensity increase towards the poles observed during this qa > 0 cycle [Cummings et al. (1987); McDonald et al. (1997)] while in the recovery phase of cycle 20 positive gradients have been observed up to latitudes of 16 N [McKibben (1989)]. This behavior is confirmed by the gradients derived at much higher rigidities ( GV) from neutron monitor measurements [Ahluwalia (1994a, 1996); Hall et al. (1994)]. In electrons, on the other hand, no apparent variation of intensity with latitude can be observed, in particular, the reversal of the latitude dependence predicted by drift-dominated modulation models can not be observed [Ferrando et al. (1996)] Spectra/Rigidity dependence Sofar, we have seen some examples for GCR temporal profiles and gradients from neutron monitor and/or spacecraft observations. This gives a rough impression of phenomena. A detailed understanding, however, requires a careful

15 First author: Kallenrode 14 Fig. 11. Comparison between electrons and nuclei over two solar cycles. Electron data are from different balloon measurements, the energy is approximately 1.2 GeV. Nucleon data are from Climax Neutron Monitor [Evenson et al. (1995)]. analysis of the energy/rigidity dependence of these quantities and therefore the temporal changes in the GCR spectrum. In Fig. 2 we had seen the variation in the GCR spectrum during the solar cycle, indicating that modulation is stronger at lower rigidities than at higher ones. Nonetheless, the spectra at successive solar minima also show differences: Figure 10 shows the GCR proton spectrum between 10 MeV and 10 GeV for the solar minima/gcr maxima 1977 and The spectra are different for energies below 3 GeV, a cross-over occurs at 400 MeV (P =0.95 GV). The 1996 spectrum, albeit at 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, suggests a dependence on the 22-year heliomagnetic cycle. A formal description of these spectra and its evidence for modulation models is discussed in Lockwood and Webber (1996, 1997), a discussion of the rigidity dependence of other modulation parameters can be found in McDonald (1998). 1.4 Electrons Sofar, we have occasionally pointed to a 22-year modulation cycle, such as in the sign of the latitudinal gradient or in the shape of the intensity profile, and described the cycles as qa positive or negative. If the sign q of the particles under study is changed, the properties of their profiles and gradients should be more similar to the nucleon properties in the preceding or subsequent cycle than in the actual one. Therefore, the measurement of galactic electrons is crucial for our understanding. Unfortunately, measurements are only sparse: balloon flights have provided measurements in the GeV range at isolated times, cf. Fig. 11. Longer times series, albeit at lower energies, exist from the ISEE-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 electron modulation can be found in Evenson (1998) and Clem et al. (1996). In general, the modulation of cosmic ray electrons is similar to the one

16 First author: Kallenrode 15 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 electron spectrum below 100 MeV has a negative slope which might be caused by modulation or Jovian electrons. Differences between electrons and nuclei can be used to identify charge-sign dependent effects in modulation. Observations indicate differences; for instance, the ratio of electrons to protons or helium is different depending on the polarity of the solar magnetic field, cf. Fig. 12 and Garcia-Munoz et al. (1986); Ferrando et al. (1995). These differences can be understood either in terms of a drift model or due to different scattering of positively and negatively charged particles due to helicity [Bieber et al. (1987)]. The drift model here faces difficulties insofar, as it would predict flat electron maxima in cycles with peaked nuclei maxima and vice 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 that the relative number of positrons is not known. However, the net charge of the 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 no reason to assume an incidence other than isotropic and constant in time. But the outer boundary of the heliosphere is far, in the order of 100 AU. Thus the galactic cosmic rays have to propagate through the heliosphere before reaching our detectors. Since the heliosphere is a highly structured medium, filled and shaped by the solar wind, particles will interact with waves and discontinuities embedded in this plasma. The resulting processes are (a) pitch-angle scattering of particles at magnetohydrodynamic waves, (b) convection of par-

17 First author: Kallenrode 16 ticles with the solar wind and adiabatic deceleration in the expanding solar wind plasma, (c) drifts due to changes of field properties during a gyro-period of the particles, and (d) blocking and reflection at inhomogeneities such as magnetic clouds. 2.1 Diffusion Diffusion is a stochastic process, resulting from pitch-angle scattering of charged particles at magnetic field irregularities. Individual interactions between particles and fields lead to small changes in pitch-angle,pitch-angle scattering therefore can be described as a random walk process. A reversal of the direction of motion along the field line requires a large number of such small-angle scatterings. If the particle is in resonance with the wave, the scattering is more efficient because all the small-angle changes work into one direction instead of trying to cancel each other. Thus pitch-angle scattering will mainly occur from interactions with field-fluctuations in resonance with the particle motion along the field (resonance scattering) as described by the resonance condition λ = v T g or k = ω c = ω (8) v µv with λ (k ) the wave-length (wave-number) of the fluctuations parallel to the field, v (v ) the particle speed (parallel to the average field), ω c the particle s cyclotron frequency, and µ = cosα the pitch-cosine. For a full treatment of the theory with application to the scattering of particles in interplanetary space see Jokipii (1966) or Hasselmann and Wibberenz (1968). The amount of scattering a particle experiences basically depends on the power density f(k ) of the waves at the resonance frequency. Thus for fixed energy/rigidity the scattering coefficient κ depends on pitch-cosine µ. Ifthe magnetic 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, and C the power at a certain frequency, the pitch-angle diffusion coefficient can be written as κ(µ) =A(1 µ 2 ) µ q 1 (10) with A being a constant related to the level C of the turbulence. The particle mean free path, that is the distance traveled before the direction of motion is reversed, then is given as λ = v 1 (1 µ 2 ) 2 dµ. (11) κ(µ) The mean free path depends on particle rigidity as λ P 2 q as long as q < 2. The above discussion holds for the so-called slab model where

18 First author: Kallenrode 17 the fluctuations are assumed to be waves with wave vectors parallel to the field and axially symmetric traverse fluctuations. Discussions of discrepancies with observations and modifications to this model can be found in Kunow et al. (1991) and Bieber et al. (1994). Particle mean free paths λ can be determined 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 becomes anisotropic and a diffusion tensor κ ij is used [Jokipii and Parker (1969)]: κ ij = κ κ κ T. (12) 0 κ T κ Here κ (κ ) is the diffusion coefficient parallel (perpendicular) to the field with 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 sphere scattering: during its gyro-orbit, the particle hits an obstacle (hard sphere), leading to a change in the particle s direction of motion. The particle then encircles a different field line: it s gyro-center has performed a cross-field motion. Alternative, and for interplanetary space more realistic interpretations assume wave-particle interactions (for an overview see e.g. Bieber (1998); Giacalone (1998); Potgieter (1998)). The perpendicular diffusion coefficient can be written as [Forman et al. (1974)] κ = vr L 3 λ /R L 1+(λ /R L ) 2 or κ = vr L R L for λ R L. (13) 3 λ Here v is the particle speed, R L the particle s Larmor radius, and λ the scattering mean free path perpendicular to the magnetic field. Parallel and perpendicular diffusion can be combined to yield a radial diffusion coefficient κ rr = κ cos 2 Ψ+κ sin 2 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, κ T,which changes sign with the reversal of the magnetic field polarity. This term is related but not identical with particle drifts and can be expressed as κ T λ = sign(qb)κ or κ T = vpc (15) R L 3qB 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 as the scattering mean free paths and gyro-radii become comparable, particles cannot complete several cycles around the field before being scattered and κ T diminishes. The latter form of (15) is commonly used in drift models, while empirical forms of κ and κ are chosen to fit the data. Using the more general form, drift effects can be reduced substantially [Burger (1990)].

19 First author: Kallenrode Convection and adiabatic deceleration The magnetic field irregularities scattering the energetic particles are frozenin into the solar wind and thus convected outwards with the solar wind speed. Therefore, the energetic particles, too, are convected outwards with the solar wind speed. During solar wind expansion the cosmic ray gas also expands, resulting in an adiabatic cooling, which is equivalent to a deceleration of the energetic particles. This process is called adiabatic deceleration. Adiabatic deceleration formally is equivalent to a Betatron effect due to the reduction of the interplanetary magnetic field strength with increasing radial distance. 2.3 Drifts Drifts are systematic processes acting in the gyro-center of the energetic particle. Except for curvature drift, particle drifts in electromagnetic fields result form changes in the Larmor radius during one gyration, either because of changes in the particle speed or because magnetic field changes, for a tutorial see e.g. Alfvén and Fälthammar (1963) or Kallenrode (1998). In the heliosphere, the most important drifts are curvature and gradient drift. Curvature drift results from the centrifugal force a particle experiences when traveling along a curved magnetic field line. The drift speed is proportional to the square of the particle speed parallel to the field and inversely proportional to both curvature radius and magnetic field strength. The gradient drift results from changes in the particle gyro-radius during one gyration because of changes in magnetic field strength. Thus the gyro-orbit is not closed but after one gyration the particle is offset with respect to its starting position. A very efficient form of gradient drift develops in the configuration of two opposing magnetic fields: when the particle crosses the neutral line between the fields, its sense of gyration is reversed. Thus during one gyration the particle experiences a displacement by 4 Larmor radii. In interplanetary space this efficient form of gradient drift takes place along the heliospheric current sheet. Note that positive and negative charges drift into opposite direction. In addition, drift depends on the magnetic field orientation, thus the drift direction is reversed when the magnetic field polarity is reversed. Drifts such as the gradient and curvature drift depend on the particle s pitch-angle. For a nearly isotropic particle distribution, the average drift can be 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 v D = cvp 3q [ B o B 2 o leading to an average streaming ] (16) S = v D f o or S = v D f o. (17)

20 First author: Kallenrode 19 Drift than is described as convection of particles with the drift velocity v D, cf. the transport equation (18). Jokipii et al. (1977) point out that in a standard Parker field the drift speed can be several times the solar wind speed 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 drift are 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)) ( v sowi + v d ) U v d(αt U) sowi. (18) dt Here U is the cosmic ray density, v sowi the solar wind velocity, T the particle kinetic energy, α =(T +2T o )/(T +T o ) with T o being the particle rest energy, κ s the symmetric part of the diffusion tensor, and v d the drift velocity. The terms on the right hand side then give the diffusion of particles in the irregular magnetic field, bulk motion due to the outward convection of particles with the solar wind and particle drifts, and adiabatic deceleration resulting from the divergence of the solar wind flow. Recent summaries on modulation theory 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 applied only at times around solar minimum when MIRs are rare and balanced by rarefaction regions. Note that the above transport equation holds also at and behind the termination shock. The energy gain of particles at the shock is accounted for by the energy-change term: the sign of v sowi determines whether particles are accelerated (negative, implying compression) or decelerated (positive, expansion). 3.1 The diffusion convection model The simplest application of Parker s transport equation neglects drifts and reduces the particle propagation to a diffusion-convection model with adiabatic deceleration. For quasi-stationary conditions U/ t 0 and a roughly isotropic cosmic ray flux in a spherical-symmetric heliosphere, Gleeson and Axford (1968) derived a modulation parameter Φ Φ= R r v sowi dr 3κ(r, P ) (19)

21 First author: Kallenrode 20 with r the radius at which the observer is located, R the outer boundary of the modulation region, and κ(p, r) the diffusion coefficient as function of rigidity P and observer s distance r. Thus the same value of Φ describes the modulation of all particle species with the same rigidity. Physically, Φ roughly corresponds to the average energy loss of inward propagating particles due to adiabatic deceleration, which might be several hundred MeV for particles traveling from the outer boundary to 1 AU. Thus particles with energies below some hundred MeV/nucl in the interstellar medium are completely excluded from the vicinity of Earth [Urch and Gleeson (1972)]. Therefore, near-earth observations at energies below a few hundred MeV/nucl do not provide any information regarding the spectrum of the local interstellar particles at these energies. The part of the interstellar spectrum blocked by modulation is not negligible because it contains 1/3 of the GCR pressure or energy density [McKibben (1990)] and thus has a strong influence on the dynamics and energetics of the interstellar medium. 3.2 Including the large scale magnetic field: drifts Although drifts were included in the original transport equation (18), they were generally neglected until in the late seventies Jokipii et al. (1977) pointed out that the inclusion of drifts may profoundly alter our picture of modulation. In particular, since most drifts are sensitive to the polarity of the global magnetic field, drift is expected to induce a charge asymmetry. In the heliosphere, the following drift pattern arises. In an A>0cycle (the Sun s magnetic field in the northern hemisphere is directed outwards, the configuration in the seventies and nineties) positively charged particles drift inward in the polar regions, downward to the heliomagnetic equator and outward along the neutral sheet. The sense of drift is reversed if the magnetic polarity is reversed (A <0) or the particle charge is negative. At the termination shock there is a fast drift upward along the shock. Drift itself would not cause modulation, it only changes the path along which particles enter the heliosphere [Jokipii et al. (1977); Kota (1990)]. Modulation can only happen due to transient disturbances (MIRs) or due to changes in the tilt anglewhich alters the drift path of the particles. The relative roles of drift and diffusion are crucial for our understanding of modulation. For typical conditions, diffusion dominates drift on small time-scales. On longer time-scales, however, drift effects can accumulate and therefore become important compared to diffusion. Two conditions have to be fulfilled: (a) noticeable effects of drift can only be expected if the particle spends enough time in the heliosphere to drift at least a significant portion of π/2 in latitude. (b) Perpendicular diffusion should not be too strong to wash out the drift pattern. If perpendicular diffusion would be too strong, particles would not drift along the polar axis or neutral sheet but will be spread in latitude. This spread depends on the ratio κ /κ T : if κ κ T, drift dominates while for κ T κ diffusion destroys the drift pattern. In the intermediate case, however, both effects have to be considered.

22 First author: Kallenrode 21 The inclusion of drifts into the transport of cosmic rays leads to the following consequences, cf. Jokipii and Thomas (1981); Kota (1990); Kota and Jokipii (1983); Potgieter (1993, 1998); Potgieter and Moraal (1985): (A) A shift in the anisotropies of cosmic rays. This is observed; in fact, a shift in neutron monitor anisotropies was the first phenomenon explained by drifts [Levy (1976)]. (B) A polarity-dependent 11-year cycle with a pronounced maximum in a qa 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 monitor data, cf. Fig. 3. In addition, because in an A>0 cycle positively charged particles drift inward through the polar regions, they are rather insensitive to conditions in the equatorial region. The good correlation between GCRs and turbulence in the ecliptic plane in qa < 0 cycles compared to the poor one in qa > 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 < 0 cycles when positively charged particles travel inward along the heliospheric current sheet and a larger tilt-angle automatically implies a longer drift path. In numerical models [Kota and Jokipii (1983)], this dependence on tilt-angle alone can explain the alternating peaked and plateau-type profiles in qa < o and qa > 0cycles. (D) A charge asymmetry, for instance in the Helium-to-electron ratio. Although evidence for a charge-asymmetry can be found in the change of the electron/helium ratio in opposite cycles [Garcia-Munoz et al. (1986), cf. Fig. 12], the most important charge-dependent signature is missing: electrons 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 markedly smaller gradients during qa > 0 when positively charged particles enter the heliosphere over the poles. However, assumptions about scattering conditions, in particular the size of κ, or modifications in the field geometry, in particular in the polar field as suggested by Jokipii and Kota (1989), influence the model predictions [Kota (1990)]. In addition, observations are difficult to interpret because measurements of radial gradients were performed by widely separated spacecraft with the separation varying with time. (F) The latitudinal gradients have opposite signs in the two cycles. Here the observational support perhaps is strongest, although the observed gradients are much smaller than the ones predicted from standard modulation theory. In addition, latitudinal gradients should be maximal when the inclination of the current sheet is smallest, which is evident from observations reported in Cummings et al. (1990); McDonald et al. (1992). In sum, the observations give strong evidence for the influence of drifts in the large-scale heliospheric magnetic field on the modulation of GCRs, however, the model predictions seem to overestimate the importance of drifts which is most obvious in the smaller than expected latitudinal gradients.

23 First author: Kallenrode Modifications to modulation models Before we will discuss modifications to modulation models, the relevant parameter 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 assumptions: (i) κ κ, (ii) perpendicular diffusion is isotropic, that is κ is the same 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 field with a modified polar geometry as suggested by Jokipii and Kota (1989). The effect of these two geometries on modulation is discussed in Haasbroek and Potgieter (1995). The modeling attempts described above are not able to describe all the observed aspects of modulation. Nonetheless, it is agreed [cf. introduction and articles in Fisk et al. (1998a)] that the basic understanding of the modulation process particles undergo diffusion, convection, adiabatic deceleration, and large-scale drifts appears to be correct. There is no requirement to add any new physical processes nor to discard one of the previously considered processes. However, sofar we have not been able to agree on a single set of parameters to describe the basic processes. Thus some suggestions are made to modify assumptions made in modeling, supported either by observations or by theoretical considerations. The main effort is to suppress drifts to reduce latitudinal gradients The diffusion tensor A modification of the diffusion tensor allows variations of the relative importance of drifts. Increasing perpendicular diffusion will smear out the signature of drifts, in particular, latitudinal gradients will decrease. Significant perpendicular diffusion also can explain the fact that CIR related increases and decreases are observed at heliographic latitudes well above the streamer belt, cf. Jokipii et al. (1995); Kota and Jokipii (1998). Reinecke et al. (1997) emphasized the importance of κ rr and κ λλ as basic parameters of Parker s transport equation. Using a 2-D non-drift model with κ rr independent of radius and κ λλ =0.1κ rr combined with the Ulyssesobserved variation of the solar wind with heliolongitude, fits on 1977 and 1987 data were possible, the model was even superior to the ones incorporating drift. However, a further modification of the model including drift is required to account for the observed change in sign of the latitudinal gradient. The influence of changes in the diffusion tensor on modulation in models considering drift has been demonstrated in many works. For instance, Burger (1990) studied the effects of scattering and the random walk of field lines on drift and found that fluctuations on all scales equal or greater the resonant wavelength can reduce drift. Fisk and Schwadron (1995) consider variations in the heliospheric magnetic field with scales intermediate between the gyro-radii of GCRs and the heliocentric radial distance. The authors