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INTERSTELLAR-TERRESTRIAL RELATIONS: VARIABLE COSMIC

ENVIRONMENTS, THE DYNAMIC HELIOSPHERE, AND THEIR

IMPRINTS ON TERRESTRIAL ARCHIVES AND CLIMATE

K. SCHERER1,∗, H. FICHTNER1, T. BORRMANN1, J. BEER2, L. DESORGHER3,

E. FL ¨

UKIGER3, H.-J. FAHR4, S. E. S. FERREIRA5, U. W. LANGNER5,

M. S. POTGIETER5, B. HEBER6, J. MASARIK7, N. J. SHAVIV8and J. VEIZER9

1Institut f¨

ur theoretische Physik, Weltraum- und Astrophysik, Ruhr-Universit¨

at Bochum,

D-44780 Bochum, Germany

2Eidgen¨

ossische Anstalt f¨

ur Wasserversorgung, Abwasserreinigung und Gew¨

asserschutz,

133 D¨

ubendorf, CH-8600, Switzerland

3Physikalisches Institut, Abteilung f¨

ur Weltraumforschung und Planetologie, Sidlerstr. 5,

CH-3012 Bern, Switzerland

4Institut f¨

ur Astrophysik und extraterrestrische Forschung der Universit¨

at Bonn, Auf dem H¨

ugel 71,

53121 Bonn, Germany

5Unit for Space Physics, School of Physics, North-West University, 2520 Potchefstroom, South Africa

6Institut f¨

ur Experimentelle und Angewandte Physik, Leibnizstraße 19, 24098 Kiel, Germany

7Commenius University, Department of Nuclear Physics & Bio Physics, 84248 Bratislava 4,

Slovakia

8Hebrew University Jerusalem, 91904 Israel

9Ottawa-Carleton Geoscience Center, University of Ottawa, 140 Louis Pasteur, Ottawa, Canada

K1N 6N5

(∗Author for correspondence: E-mail: kls@tp4.rub.de)

(Received 1 September 2006; Accepted in ﬁnal form 20 November 2006)

Abstract. In recent years the variability of the cosmic ray ﬂux has become one of the main issues

interpreting cosmogenic elements and especially their connection with climate. In this review, an

interdisciplinary team of scientists brings together our knowledge of the evolution and modulation of

the cosmic ray ﬂux from its origin in the Milky Way, during its propagation through the heliosphere, up

to its interaction with the Earth’s magnetosphere, resulting, ﬁnally, in the production of cosmogenic

isotopes in the Earth’ atmosphere. The interpretation of the cosmogenic isotopes and the cosmic

ray – cloud connection are also intensively discussed. Finally, we discuss some open questions.

Keywords: interstellar-terrestrial relations, variable cosmic ray ﬂuxes, dynamical heliosphere,

cosmogenic isotopes, climate

Space Science Reviews (2006) 127: 327–465

DOI: 10.1007/s11214-006-9126-6 C

Springer 2007

.

Part I

Introduction to the Problem

.

INTERSTELLAR-TERRESTRIAL RELATIONS 331

1. Interstellar-Terrestrial Relations: Deﬁnition and Evidence

There is evidence that the galactic environment of the Solar System leaves traces on

Earth. Well-known are supernova explosions, which are responsible for an increased

3He abundance in marine sediments (O’Brien et al., 1991), or catastrophic cometary

impacts, which are considered as causes for biological mass extinctions (Rampino

et al., 1997; Rampino, 1998). These and other events, to which also gamma ray

bursts (Thorsett, 1995) or close stellar encounters (Scherer, 2000) can be counted,

can be considered as ‘quasi-singular’ and belong to so-called stellar-terrestrial

relations. From those one should distinguish ‘quasi-periodic’ events, which are

connected to encounters of different interstellar gas phases or molecular clouds

(Frisch, 2000), to the crossing of the galactic plane (Schwartz and James, 1984),

and to the passage through galactic spiral arms (Leitch and Vasisht, 1998). As will

be explained in the following, these quasi-periodic changes inﬂuence the Earth

and its environment and are, therefore, called interstellar-terrestrial relations. The

mediators of such environmental changes are the interstellar plasma and neutral

gas as well as the cosmic rays, all of which affect the structure and dynamics of

the heliosphere. The heliosphere, however, acts as a shield protecting the Earth

from the direct contact with the hostile interstellar environment. From all particle

populations that can penetrate this shield, only the ﬂux variations of cosmic rays

can be read off terrestrial archives, namely the depositories of cosmogenic isotopes,

i.e. ice-cores, sediments, or meteorites.

The typical periods of interstellar-terrestrial relations seen in these archives are

determined by external (interstellar) triggers on time-scales longer than about ten-

thousand years, while those for shorter time-scales are governed by an internal

(solar) trigger. The latter results from solar activity, which leads to variations of the

cosmic ray ﬂux with periods of the various solar cycles, like the Hale-, Schwabe-

and Gleissberg-cycle amongst others.

The interpretation of the cosmogenic archives is of importance for our under-

standing of variations of the galactic cosmic ray spectra and of the solar dynamo

and, therefore, of high interest to astrophysics. Moreover, the correlation of cos-

mogenic with climate archives gives valuable information regarding the question

to what extent the Earth climate is driven by extraterrestrial and extraheliospheric

forces. Candidates for such climate drivers are the variable Sun (solar forcing), the

planetary perturbations (Milankovitch forcing), the variable cosmic ray ﬂux (cos-

mic ray forcing), and the varying atomic hydrogen inﬂow into the atmosphere of

Earth (hydrogen forcing).

The current debate concentrates on solar and cosmic ray forcing, because the

Milankovitch forcing is well understood and the hydrogen forcing is highly specu-

lative. While there exists a vast amount of literature, especially reviews and mono-

graphs, concerning the solar forcing, the work on cosmic ray forcing is still largely

scattered and no comprehensive overview has been compiled so far. This review

intends to make the ﬁrst step to change that situation by bringing together our

332 K. SCHERER ET AL.

knowledge about cosmogenic archives, climate archives, cosmic ray transport and

heliospheric dynamics.

2. Cosmic Ray Forcing

The idea that cosmic rays can inﬂuence the climate on Earth dates back to

Ney (1959) who pointed out that if climate is sensitive to the amount of tro-

pospheric ionization, it would also be sensitive to solar activity since the so-

lar wind modulates the cosmic ray ﬂux (CRF), and with it, the amount of

tropospheric ionization. These principal considerations have been revived by

Svensmark and Friis-Christensen (1997) and Svensmark (1998), who found from

a study of satellite and neutron monitor data a correlation between cosmic ray

intensity and the global cloud coverage on the 11-year time-scale of the solar

activity cycle. While Marsh and Svensmark (2000a,b), Palle Bago and Butler

(2000) have signiﬁcantly reﬁned this correlation analysis, Usoskin et al. (2004b)

have found that the CRF/low altitude cloud cover is as predicted. Namely, the

amount of cloud cover change over the solar cycle at different latitudes is pro-

portional to the change in tropospheric ionization averaged over the particular

latitudes. Others have started to identify the physical processes for cloud forma-

tion due to high-energy charged particles in the atmosphere (Tinsley and Deen,

1991; Tinsley and Heelis, 1993; Eichkorn et al., 2002; Yu, 2002; Harrison and

Stephenson, 2006). There is, however, also severe doubt regarding the signiﬁcance

of the correlation, see, e.g. Gierens and Ponater (1999), Kernthaler et al. (1999),

Carslaw et al. (2002), Sun and Bradley (2002), Kristj´ansson et al. (2004), and

Sun and Bradley (2004) for the latest development see Svensmark et al. (2006) and

Kanipe (2006).

The critics rather favour the most evident external climate driver, namely the

solar irradiance. While on the 11-year time-scale (Schwabe cycle) both the cosmic

ray forcing and the solar forcing act in an indistinguishable manner, on the 22-year

time-scale (Hale cycle), there should be a difference because, in contrast to the

solar irradiation, the cosmic ray ﬂux is sensitive to the heliospheric magnetic ﬁeld

polarity as a consequence of drift-related propagation (Fichtner et al., 2006).

Other clues result from the study of the climate and cosmogenic archives for

intermediate and very long time-scales. Regarding the former, the so-called grand

minima of solar activity have been investigated (van Geel et al., 1999a; Caballero-

Lopez et al., 2004; Scherer and Fichtner, 2004) because temperature was generally

lower during these periods (Grove, 1988). There is evidence from historical sunspot

observations and cosmogenic archives that both forcing processes could have been

responsible for this climate variation so that, unfortunately, no decision can be

expected unless the 22-year Hale cycle is detected in the data, a claim that has been

made already (Miyahara et al., 2005).

The situation is different on very long time-scales. Opposite to the shorter time-

scales, on which the cosmic ray ﬂux variations are dominated by solar activity,

INTERSTELLAR-TERRESTRIAL RELATIONS 333

on longer time scales they are inﬂuenced by processes external to the heliosphere,

like interstellar environment changes (Yabushita and Allen, 1998) or spiral arm

crossings (Shaviv, 2003a). So, one should expect corresponding climate variations

on time-scales of millions of years. Indeed, Shaviv and Veizer (2003) have found

a correlation between the cosmic ray ﬂux and Earth temperature for the last 500

million years that can be related to the spiral arm crossings of the heliosphere

occuring with a quasi-period of about 135 million years. Because there is no reason

to expect that solar activity and, in turn, solar irradiance is triggered by spiral arm

crossings or interstellar environment changes, any cosmic ray climate correlation

on such time-scales is a strong argument in favour of cosmic ray forcing.

3. Known Astronomical Effects

Quite early the inﬂuence of interstellar clouds on the climate on Earth has been

discussed (Shapley, 1921; Hoyle and Lyttleton, 1939; McCrea, 1975; Eddy, 1976;

Dennison and Mansﬁeld, 1976; Begelman and Rees, 1976; McKay and Thomas,

1978) and revisited by Yeghikyan and Fahr (2004a, b). A possible inﬂuence of

interstellar dust particles on the climate was discussed in Hoyle (1984). A review

of the possible long-term ﬂuctuations of the Earth environment and their possible

astronomical causes was given by McCrea (1981). The inﬂuence of neutral inter-

stellar particle ﬂuxes on the terrestrial environment was studied by Bzowski et al.

(1996)

In the middle of the last century (Milankovitch, 1941) discussed the planetary

inﬂuence on terrestrial climate, especially on the ice ages. The secular variations of

the Earth’s orbital elements caused by the other planets, lead to periodical changes

in the inclination and eccentricity (with the most signiﬁcant periods of: 19, 23, 41,

100, 400 kyr ), which in turn affect the absorption of solar irradiation (the latitudinal

dependence), insolation, the length of the seasons, etc., causing climatic changes,

see e.g. Berger (1991) and Ruddiman (2006). These and other periods can be found

in Figure 1 taken from Mitchell (1976). Concentrating on variations longer than one

year in Figure 1 the different periods can be identiﬁed in the following ways: While

the Milankovitch cycles are more or less conﬁrmed, all periods for the external

forcing of the climate listed above are still under debate. Recently, Lassen and

Friis-Christensen (1995) pointed to the connection of the solar cycle length and

the temperature variation in the northern hemisphere. These external effects have

the major drawback, that up to now no detailed process is known which drives

the related climate changes. The 2400-year period is probably connected with the

relative motion of the Sun around the center of mass (barycentre) of the solar system

(Charvatova, 1990). The 30-Myr peak coincides with the galactic plane crossing of

the heliosphere, and the (220–500)-Myr peak corresponds to the revolution period of

the Sun around the galaxy (see Section 6). In Table I, some alternative explanations

are also listed.

334 K. SCHERER ET AL.

TABLE I

Possible astronomical or geological explanations of the different periods observed in Figure 1

Explanation

Years Astronomical Geological

10–20 Solar cycle variations

100–400 Long term solar variations

2400 Motion of Sun around solar system Deep-sea thermohaline

barycentre circulations

19000, 23000 Precession parameter (Milankovitch cycle)

41000 Obliquity (Milankovitch cycle)

100000 Eccentricity (Milankovitch cycle)

(30–60) ×106Galactic plane crossing Tectonism

(200–500) ×106Orbital revolution of the Sun around Tectonism

galactic center

Figure 1. Compilation of the climatic changes on Earth on all times scales (after Mitchell, 1976).

Other astronomical effects of sporadic nature are, for example, supernovae ex-

plosions (Ruderman, 1974), gamma-ray bursts (Thorsett, 1995), and stellar encoun-

ters (Scherer, 2000) and will not be discussed further.

INTERSTELLAR-TERRESTRIAL RELATIONS 335

4. Structure of the Review

The general physical ideas for cosmic ray acceleration and modulation together

with magneto-hydrodynamic (MHD) concepts are brieﬂy presented in part II.

In part III the problem of determining the local interstellar cosmic ray spectra

is considered. This is done in two sections: First, in Section 6 the distribution of

matter and stars in the galaxy along the orbit of the Sun and their inﬂuences on

the cosmic ray ﬂux is discussed (N. J. Shaviv). Second, in Section 7 the galactic

cosmic ray spectra inside and outside of galactic spiral arms are computed (H.-J.

Fahr, H. Fichtner, K. Scherer).

The heliospheric modulation of present-day interstellar spectra due to solar ac-

tivity cycle is subject of part IV. While in Section 8 the time dependence of the modu-

lation processes are described for the 11- and 22-year solar cycles (M.S. Potgieter),

Section 9 concentrates on the spatial aspect of the modulation, in particular its

dependence on the outer heliospheric structure (U.W. Langner, M.S. Potgieter).

For the considerations in part III and IV a stationary heliosphere was assumed.

This approximation is dropped in part V. A general description of hydrodynamic

modeling of heliospheric plasma structures given in Section 10 (H. Fichtner, T.

Borrmann) is followed by Section 11 with a presentation of results of hybrid

modeling, including the kinetic transport equation of cosmic rays (S.E.S. Ferreira,

K. Scherer).

The interaction of cosmic rays with the environment of the Earth is studied

in part VI. After discussing the magnetospheric and atmospheric propagation of

cosmic rays as well as the corresponding ionization and energy deposition in the

atmosphere in Section 12 (B. Heber, L. Desorgher, E. Flckiger), the production of

cosmogenic nuclei is described in Section 13 (J. Masarik, J. Beer).

The imprints of cosmic rays on Earth and their implications for climate pro-

cesses are subject of part VII. The emphasis in Section 14 is put on the storage

of cosmogenic isotopes in various archives (K. Scherer), while in Section 15 the

evidence of cosmic ray driven climate effects on different time scales is presented

(J. Veizer).

In the ﬁnal part VIII an attempt is made to identify and formulate the crucial

questions in this new interdisciplinary ﬁeld.

.

Part II

General Theoretical Concepts

.

INTERSTELLAR-TERRESTRIAL RELATIONS 339

5. The Fundaments for the Quantitative Modelling

The fundamental equations for quantitative studies are presented in the following

two sections. The transport equation of cosmic rays discussed in the Section 5.1 is

used to describe the acceleration and propagation of cosmic rays through the galaxy

as well as through the heliosphere. For the latter plasma structure the magneto-

hydrodynamic (MHD) equations are presented in Section 5.2 with their general

assumptions.

5.1. COSMIC RAY TRANSPORT

The transport of cosmic rays is calculated by solving the transport equation Parker

(1965)

∂f

∂t=∇·(↔

κ∇f)−(v+vdr)·∇f+p

3(∇·v)∂f

∂p+S(r,p,t) (1)

The description is based on the isotropic phase space distribution function f(r,p,t)

depending on location r, magnitude of momentum pand time t. Often instead of

the momentum pthe rigidity R=pc/qis used, with cand qdenoting the speed

of light and the particle charge, respectively. The equation contains, in addition to

the effects of convection velocity vand drift vdr in the magnetic ﬁeld

Ba fully

anisotropic diffusion tensor:

↔

κ=⎛

⎜

⎝

κ⊥r00

0κ⊥θ0

00κ

⎞

⎟

⎠(2)

This tensor, denoted here in spherical polar coordinates (r,θ,ϕ), is formulated

with respect to the local magnetic ﬁeld, see Figure 2. Various suggestions for the

explicit form of its elements have been made, see, e.g., Burgerand Hattingh (1998),

Fichtner et al. (2000), Ferreira et al. (2001), Matthaeus et al. (2003), Bieber et al.

Figure 2. Illustration of the elements of the diffusion tensor. The coefﬁcient κdescribes the diffusion

along the local magnetic ﬁeld

B.

340 K. SCHERER ET AL.

(2004), or Shalchi and Schlickeiser (2004). The transport equation is generally

solved numerically using mixed boundary conditions.

For quantitative studies of interstellar-terrestrial relations it is necessary to have

a model of a three-dimensional heliosphere, which is immersed in a dynamic local

interstellar medium. There are at least two reasons why such model should be three-

dimensional. First, a comprehensive and self-consistent treatment of the cosmic ray

transport must take into account the three-dimensional structure of the turbulent

heliospheric plasma and, second, the heliosphere can be in a disturbed state for

which no axisymmetric description can be justiﬁed. The present state-of-the-art

of the modelling of a dynamic heliosphere with a self-consistent treatment of the

transport of cosmic rays is reviewed in Fichtner (2005). As is pointed out in that

paper, the major challenge is the development of a three-dimensional hybrid model.

This task requires, on the one hand, the generalisation of the modelling discussed in

the following section and, on the other hand, the formulation of three-dimensional

models of the heliospheric plasma dynamics.

5.2. THE DYNAMICAL HELIOSPHERE

The model of the dynamical heliosphere is in most cases based on the following

(normalized) magneto-hydrodynamical equations

∂

∂t⎛

⎜

⎜

⎜

⎝

ρ

ρv

e

B

⎞

⎟

⎟

⎟

⎠+∇·⎛

⎜

⎜

⎜

⎜

⎜

⎝

ρv

ρvv+pth +1

2B2ˆ

I−

B

B

e+pth +1

2B2v−

B(v·

B)

v

B−

Bv

⎞

⎟

⎟

⎟

⎟

⎟

⎠=⎛

⎜

⎜

⎜

⎝

Qρ

Qρv

Qe

0

⎞

⎟

⎟

⎟

⎠(3)

for each thermal component taken into account. Here, ρis the mass density, v

the velocity, ethe total energy density and pth the thermal pressure of a given

component.

Bis the magnetic ﬁeld and ˆ

Ithe unity tensor. The terms Qρ,

Qρvand

Qedescribe the exchange of mass, momentum and energy between the thermal

components and with the cosmic rays if present. For the closure of Equation (3)

an equation of state for each component is needed, for which usually the ideal gas

equation is taken.

Alternatively, the treatment of hydrogen atoms can be based on their kinetic

transport equation:

∂fH

∂t+w·∇fH+

F

mp·∇

wfH=P−L(4)

Here fHis the distribution function of hydrogen atoms with velocity w. The force

Fis the effect of gravity and radiation pressure, while Pand Ldescribe the sources

and sinks, respectively. This equation takes into account, that the atoms may not

collide sufﬁciently frequent, to allow a single-ﬂuid approach (Baranov and Malama,

INTERSTELLAR-TERRESTRIAL RELATIONS 341

1993; Lipatov et al., 1998; M¨uller et al., 2000; Izmodenov, 2001). Heerikhuisen

et al. (2006) have demonstrated, however, that a multiﬂuid approach for hydrogen

leads to a reasonable accurate description of the global heliosphere, comparable to

the kinetic models.

To keep computing time for the solution of Equations (3) affordable, in most

cases the number of species in 3-D models is restricted to protons and neu-

tral hydrogen atoms (Zank, 1999; Fahr, 2004; Izmodenov, 2004; Borrmann and

Fichtner, 2005). In sophisticated MHD models, which nowadays have been devel-

oped (Ratkiewicz et al., 1998; Opher et al., 2004; Pogorelov, 2004; Pogorelov et al.,

2004; Washimi et al., 2005), computing time is even more critical and therefore

only protons are treated, except in Pogorelov and Zank (2005) who include also

hydrogen atoms.

In order to include more species the space dimension has to be reduced. In the

2-D hydrodynamic codes so far up to ﬁve species could simultaneously and self-

consistently be included, namely in addition to protons and hydrogen also pickup

ions (PUIs) as seed for the anomalous cosmic ray (ACR) component and the galactic

cosmic rays (GCRs) (Fahr et al., 2000).

Recent developments allow to combine the kinetic modeling of the cosmic ray

transport equation (1) with the ﬁve species approach, resulting in a hybrid model

(Scherer and Ferreira, 2005a,b; Ferreira and Scherer, 2005).

The dynamics of the heliosphere includes time varying boundary conditions

for both the solar activity cycle and the changing interstellar medium. The inner

boundary condition determines the structure of the global heliosphere as well as

the cosmic ray ﬂux at the Earth on time scales of tens to thousands of years. For the

longer periods, i.e. millions of years, the changes of the outer boundary conditions

is more important. Details of modelling and its support by data are discussed in the

following sections.

.

Part III

Galactic Cosmic Rays

.

INTERSTELLAR-TERRESTRIAL RELATIONS 345

6. Long-Term Variation

The galactic cosmic ray ﬂux reaching the outskirts of the Milky Way (MW) often

regarded as a constant. However, on long enough time scales, the galactic environ-

ment varies, and with it so does the density of cosmic rays in the vicinity of the

solar system. In this section, we will concentrate on these variations, which are

larger than the short term modulations by the solar wind. In particular, we expect

variations from spiral arm passages over the 108yr time scale, while Star Formation

Rate (SFR) variations in the Milky Way are expected to be a dominant cause of

Cosmic Ray Flux (CRF) variability on even longer time scales. We discuss here the

expected variability over these scales, together with the empirical evidence used to

reconstruct the actual variations. On shorter time scales, local inhomogeneities in

the galactic environment or the occurrence of a nearby supernova can give rise to

large variations. These variations will not be discussed since no deﬁnitive predic-

tions yet exist nor do reliable reconstructions of the CRF on these shorter scales,

which are still long relative to the cosmogenic records on Earth.

6.1. STAR FORMATION RATE

The local and overall SFR in the MW is not constant. Variations in the SFR will

in turn control the rate of supernovae. Moreover, supernova remnants accelerate

cosmic rays (at least with energies 1015 eV), and inject fresh high-Z material into

the galaxy. Thus, cosmic rays and galactic nuclear enrichment, is proportional to

the SFR.

Although there is a lag of several million years between the birth and death

of massive stars, this lag is small when compared to the relevant time scales at

question. Over the “galactic short term”, i.e., on time scales of 108yr or less, the

record of nearby star formation is “Lagrangian”, i.e., the star formation in the

vicinity of the moving solar system. This should record passages through galactic

spiral arms. On longer time scales, of order 109yr or longer, mixing is efﬁcient

enough to homogenize the azimuthal distribution in the Galaxy (Wielen, 1977). In

other words, the long-term star formation rate, as portrayed by nearby stars, should

record the long term changes in the Milky Way SFR activity. These variations may

arise, for example, from a merger with a satellite or a nearby passage of one.

Scalo (1987), using the mass distribution of nearby stars, concluded that the SFR

had peaks at 0.3 Gyr and 2Gyr before present (BP). Barry (1988), and a more elab-

orate and recent analysis by Rocha-Pinto et al. (2000), measured the star formation

activity of the Milky Way using chromospheric ages of late type dwarfs. They found

a dip between 1 and 2 Gyr and a maximum at 2–2.5 Gyr b.p. (see also Figure 3).

The data in Figure 3 are not corrected for selection effects (namely, the up-

ward trend with time is a selection effect, favorably selecting younger clusters

more of which did not yet dissolve). Since the clusters in the catalog used are

346 K. SCHERER ET AL.

Figure 3. The history of the SFR. The squares with error bars are the SFR calculated using chromo-

spheric ages of nearby stars (Rocha-Pinto et al., 2000), which is one of several SFR reconstructions

available. These data are corrected for different selection biases and are binned into 0.4 Gyr bins.

The line and hatched region describe a 1-2-1 average of the histogram of the ages of nearby open

clusters (using the Loktin et al., 1994, catalog), and the expected 1-σerror bars.

spread to cover two nearby spiral arms, the signal arising from the passage of

spiral arms is smeared, such that the graph depicts a more global SFR activity

(i.e., in our galactic ‘quadrant’). On longer time scales (1.5 Gyr and more), the

galactic azimuthal stirring is efﬁcient enough for the data to reﬂect the SFR in

the whole disk. There is a clear minimum in the SFR between 1 and 2 Gyr BP,

and there are two prominent peaks around 0.3 and 2.2 Gyr BP. Interestingly, the

Large Magellanic Cloud (LMC) perigalacticon should have occurred sometime

between 0.2 and 0.5 Gyr BP in the last passage, and between 1.6 and 2.6 Gyr

BP in the previous passage. This might explain the peaks in activity seen. This

is corroborated with evidence of a very high SFR in the LMC about 2 Gyr BP

and a dip at 0.7–2 Gyr BP (Gardiner et al., 1994; Lin et al., 1995). Also depicted

are the periods during which glaciations were seen on Earth: The late Archean

(3 Gyr ) and early Proterozoic (2.2–2.4 Gyr BP ) which correlate with the previ-

ous LMC perigalacticon passage (Gardiner et al., 1994; Lin et al., 1995) and the

consequent SFR peak in the MW and LMC. The lack of glaciations in the in-

terval 1–2 Gyr BP correlates with a clear minimum in activity in the MW (and

LMC). Also, the particularly long Carboniferous-Permian glaciation, correlates

INTERSTELLAR-TERRESTRIAL RELATIONS 347

with the SFR peak at 300 Myr BP and the last LMC perigalacticon. The late Neo-

Proterozoic ice ages correlate with a less clear SFR peak around 500–900 Myr

BP. Since both the astronomical and the geological data over these long time

scales have much to be desired, the correlation should be considered as an as-

suring consistency. By themselves, they are not enough to serve as the basis of ﬁrm

conclusions.

Another approach for the reconstruction of the SFR, is to use the cluster age dis-

tribution. A rudimentary analysis reveals peaks of activity around 0.3 and 0.7 Gyr

BP, and possibly a dip between 1 and 2 Gyr (as seen in Figure 3). A more re-

cent analysis considered better cluster data and only nearby clusters, closer than

1.5 kpc (de La Fuente Marcos and de La Fuente Marcos, 2004). Besides the above

peaks which were conﬁrmed with better statistical signiﬁcance, two more peaks

were found at 0.15 and 0.45 Gyr. At this temporal and spatial resolution, we are

seeing the spiral arm passages. On longer time scales, cluster data reveals a notable

dip between 1 and 2 Gyr (Shaviv, 2003a; de La Fuente Marcos and de La Fuente

Marcos, 2004).

6.2. SPIRAL ARM PASSAGES

On time scales shorter than those affecting global star formation in the Milky Way,

the largest perturber of the local environment is our passages through the galactic

spiral arms.

The period with which spiral arms are traversed depends on the relative angular

speed around the center of the galaxy, between the solar system with and the

spiral arms with p:

T=2π

m|−p|,(5)

where mis the number of spiral arms.

Our edge-on vantage point is unfortunate in this respect, since it complicates

the determination of both the geometry and the dynamics of the spiral arms. This

is of course required for the prediction of the spiral arm passages. In fact, the

understanding of neither has reached a consensus.

Claims in the literature for a 2-armed and a 4-armed structure are abundant.

There is even a claim for a combined 2 +4 armed structure (Amaral and Lepine,

1997). Nevertheless, if one examines the v−lmaps of molecular gas, then it is hard

to avoid the conclusion that outside the solar circle, there are 4 arms1(Blitz et al.,

1983; Dame et al., 2001). Within the solar circle, however, things are far from clear.

This is because v−lmaps become ambiguous for radii smaller than R, such that

each arm is folded and appears twice (Ris the present distance of the Sun from

1Actually, 3 are seen, but if a roughly symmetric set is assumed, then a forth arm should simply be

located behind the galactic center.

348 K. SCHERER ET AL.

the galactic center). Shaviv (2003a) has shown that if the outer 4 arms obey the

simple density wave dispersion relation, such that they cannot exist beyond the 4:1

Lindblad resonances then two sets of arms should necessarily exist. In particular,

the fact that these arms are apparent out to rout ≈2Rnecessarily implies that

their inner extent, the inner Lindblad radius, should roughly be at R. Thus, the

set of arms internal to our radius should belong to a set other than the outer 4 arms.

The dynamics, i.e., the pattern speed of the arms, is even less understood than the

geometry. A survey of the literature (Shaviv, 2003a) reveals that about half of the

observational determinations of the relative pattern speed −pcluster around

9to13kms

−1kpc−1, while the other half are spread between −4 and 5 km s−1

kpc−1. In fact, one analysis revealed that both −p=5 and 11.5km s−1kpc−1

ﬁt the data equally well (Palous et al., 1977).

Interestingly, if spiral arms are a density wave (Lin and Shu, 1964), as is com-

monly believed (e.g., Binney amd Tremaine, 1987), then the observations of the

4-armed spiral structure in HI outside the Galactic solar orbit (Blitz et al., 1983)

severely constrain the pattern speed to satisfy −p9.1±2.4kms

−1kpc−1,

since otherwise the four armed density wave would extend beyond the outer 4:1

Lindblad resonance (Shaviv, 2003a).

This conclusion provides theoretical justiﬁcation for the smaller pattern speed.

However, it does not explain why numerous different estimates for pexist. A

resolution of this “mess” arises if we consider the possibility that at least two spiral

sets exist, each one having a different pattern speed, Indeed, in a stellar cluster birth

place analysis, which allows for this possibility, it was found that the Sagittarius-

Carina arm appears to be a superposition of two arms (Naoz and Shaviv, 2006).

One has a relative pattern speed of −P,Carina,1=10.6+0.7

−0.5sys ±1.6stat km s−1

kpc−1and appears also in the Perseus arm external to the solar orbit. The second set

is nearly co-rotating with the solar system, with −P,Carina,2=−2.7+0.4

−0.5sys ±

1.3stat km s−1kpc−1. The Perseus arm may too be harboring a second set. The Orion

“armlet” where the solar system now resides (and which is located in between the

Perseus and Sagittarius-Carina arms), appears too to be nearly co-rotating with us,

with −p,Orion =−1.8+0.2

−0.3sys ±0.7stat km s−1kpc−1.

For comparison, a combined average of the 7 previous measurements of the

9to13kms

−1kpc−1range, which appears to be an established fact for both the

Perseus and Sagittarius-Carina arms, gives −p=11.1±1kms

−1kpc−1.At

reasonable certainty, however, a second set nearly co-rotating with the solar system

exists as well.

The relative velocity between the solar system and the ﬁrst set of spiral arms

implies that every ∼150 Myr , the environment near the solar system will be that

of a spiral arm. Namely, we will witness more frequent nearby supernovae, more

cosmic rays, more molecular gas as well as other activity related to massive stars.

We will show below that there is a clear independent record of the passages through

the arms of the ﬁrst set. On the other hand, passages through arms of the second

set happen infrequently enough for them to have been reliably recorded.

INTERSTELLAR-TERRESTRIAL RELATIONS 349

Figure 4. The components of the diffusion model constructed to estimate the Cosmic Ray ﬂux varia-

tion. We assume for simplicity that the CR sources reside in Gaussian cross-sectioned spiral arms and

that these are cylinders to ﬁrst approximation. This is permissible since the pitch angle iof the spirals

is small. The diffusion takes place in a slab of half width lH, beyond which the diffusion coefﬁcient

is effectively inﬁnite.

To estimate the variable CRF expected while the solar system orbits the galaxy,

one should construct a simple diffusion model which considers that the sources

reside in the Galactic spiral arms. A straight forward possibility is to amend the basic

CR diffusion models (e.g., Berezinski˘ı et al., 1990) to include a source distribution

located in the Galactic spiral arms. Namely, one can replace a homogeneous disk

with an arm geometry as given for example by Taylor and Cordes (1993), and solve

the time dependent diffusion problem as was done by Shaviv (2003a). Heuristically,

such a model is sketched in Figure 4.

The main model parameters include a CR diffusion coefﬁcient, a halo half width

(beyond which the CRs diffuse much more rapidly) and of course the angular

velocity −pof the solar system relative to the spiral arm pattern speed.

The latter number is obtained from the above observations, while typical diffusion

parameters include a CR diffusion coefﬁcient of D=1028cm2/s, which is a typical

value obtained in diffusion models for the CRs (Berezinski˘ı et al., 1990; Lisenfeld

et al., 1996; Webber and Soutoul, 1998), or a halo half-width of 2 kpc, which again

is a typical value obtained in diffusion models (Berezinski˘ı et al., 1990). Note that

given a diffusion coefﬁcient, there is a relatively narrow range of effective halo

widths which yields a Be age consistent with observations (Lukasiak et al., 1994).

For the nominal values chosen in the diffusion model and the pattern speed

found above, the expected CRF changes from about 25% of the current day CRF

350 K. SCHERER ET AL.

Figure 5. The cosmic-ray ﬂux variability and age as a function of time for D=1028 cm2/s and

lH=2 kpc. The solid line is the cosmic-ray ﬂux, the dashed line is the age of the cosmic rays as

measured using the Be isotope ratio. The shaded regions at the bottom depict the location, relative

amplitude (i.e., it is not normalized) and width of the spiral arms as deﬁned through the free electron

density in the Taylor and Cordes model. The peaks in the ﬂux are lagging behind the spiral arm crosses

due to the SN-HII lag. Moreover, the ﬂux distribution is skewed towards later times.

to about 135%. Moreover, the average CRF obtained in units of today’s CRF is

76%. This is consistent with measurements showing that the average CRF over the

period 150–700 Myr BP, was about 28% lower than the current day CRF (Lavielle

et al., 1999).

Interestingly, the temporal behavior is both skewed and lagging after the spiral

arm passages (Figure 5). The lag arises because the spiral arms are deﬁned through

the free electron distribution. However, the CRs are emitted from which on average

occur roughly 15 Myr after the average ionizing photons are emitted. The skew-

ness arises because it takes time for the CRs to diffuse after they are emitted. As

a result, before the region of a given star reaches an arm, the CR density is low

since no CRs were recently injected in that region and the sole ﬂux is of CRs that

succeed to diffuse to the region from large distances. After the region crosses the

spiral arm, the CR density is larger since locally there was a recent injection of

new CRs which only slowly disperse. This typically introduces a 10 Myr lag in the

ﬂux, totaling about 25 Myr with the delay. This lag is actually observed in the syn-

chrotron emission from M51, which shows a peaked emission trailing the spiral arms

(Longair, 1994).

INTERSTELLAR-TERRESTRIAL RELATIONS 351

6.3. COSMIC RAY RECORD IN IRON METEORITES

Various small objects in the solar system, such as asteroids or cometary nuclei,

break apart over time. Once the newly formed surfaces of the debris are exposed to

cosmic rays, they begin to accumulate spallation products. Some of the products are

stable and simply accumulate with time, while other products are radioactive and

reach an equilibrium between the formation rate and their radioactive decay. Some

of this debris reaches Earth as meteorites. Since chondrites (i.e., stony meteorites)

generally “crumble” over 108yr, we have to resort to the rarer iron meteorites,

which crumble over 109yr, if we wish to study the CRF exposure over longer

time scales.

The cosmic ray exposure age is obtained using the ratio between the amount of

the accumulating and the unstable nuclei. Basically, the exposure age is a measure

of the integrated CRF, as obtained by the accumulating isotope, in units of the CRF

“measured” using the unstable nucleus. Thus, the “normalization” ﬂux depends on

the average ﬂux over the last decay time of the unstable isotope and not on the

average ﬂux over the whole exposure time. If the CRF is assumed constant, then

the ﬂux obtained using the radioactive isotope can be assumed to be the average

ﬂux over the life of the exposed surface. Only in such a case, can the integrated

CRF be translated into a real age.

Already quite some time ago, various groups obtained that the exposure ages

of iron meteorites based on “short” lived isotopes (e.g., 10Be) are inconsistent with

ages obtained using the long lived unstable isotope 40K, with a half life of ∼1 Gyr.

In essence, the ﬁrst set of methods normalize the exposure age to the ﬂux over a

few million years or less, while in the last method, the exposure age is normalized

to the average ﬂux over the life time of the meteorites. The inconsistency could be

resolved only if one concludes that over the past few Myr, the CRF has been higher

by about 28% than the long term average (Hampel and Schaeffer, 1979; Schaeffer

et al., 1981; Aylmer et al., 1988; Lavielle et al., 1999).

More information on the CRF can be obtained if one makes further assump-

tions. Particularly, if one assumes that the parent bodies of iron meteorites tend

to break apart at a constant rate (or at least at a rate which only has slow varia-

tions), then one can statistically derive the CRF history. This was done by Shaviv

(2003a), using the entire set of 40K dated iron meteorites. To reduce the probabil-

ity that the breaking apart is real, i.e., that a single collision event resulted with a

parent body braking apart into many meteorites, each two meteorites with a small

exposure age difference (with a≤5×107yr), and with the same iron group

classiﬁcation, were replaced by a single effective meteor with the average exposure

age.

If the CRF is variable, then the exposure age of meteorites will be distorted.

Long periods during which the CRF was low, such that the exposure clock “ticked”

slowly, will appear to contract into a short period in the exposure age time scale.

This implies that the exposure ages of meteorites is expected to cluster around

352 K. SCHERER ET AL.

Figure 6. The exposure age of iron meteorites plotted as a function of their phase in a 147 Myr

period. The dots are the 40K exposure ages (larger dots have lower uncertainties), while the stars are

36Cl based measurements. The K measurements do not suffer from the long term “distortion” arising

from the difference between the short term (10 Myr ) CRF average and the long term (1 Gyr) half

life of K (Lavielle et al. 1999). However, they are intrinsically less accurate. To use the Cl data, we

need to “correct” the exposure ages to take into account this difference. We do so using the result

of Lavielle et al. (1999). Since the Cl data is more accurate, we use the Cl measurement when both

K and Cl are available for a given meteorite. When less than 50 Myr separates several meteorites

of the same iron group classiﬁcation, we replace them with their average in order to discount for

the possibility that one single parent body split into many meteorites. We plot two periods such that

the overall periodicity will be even more pronounced. We see that meteorites avoid having exposure

ages with given phases (corresponding to epochs with a high CRF). Using the Rayleigh Analysis,

the probability of obtaining a signal with such a large statistical signiﬁcance as a ﬂuke from random

Poisson events, with any period between 50 and 500 Myr , is less than 0.5%. The actual periodicity

found is 147 ±6 Myr, consistent with both the astronomical and geological data.

(exposure age) epochs during which the CRF was low, while there will be very few

meteors in periods during which the CRF was high.

Over the past 1 Gyr recorded in iron meteorites, the largest variations are ex-

pected to arise from our passages through the galactic spiral arms. Thus, we expect

to see cluster of ages every ∼150 Myr . The actual exposure ages of meteorites are

plotted in Figure 6, where periodic clustering in the ages can be seen. This clus-

tering is in agreement with the expected variations in the cosmic ray ﬂux. Namely,

iron meteorites recorded our passages through the galactic spiral arms.

Interestingly, this record of past cosmic ray ﬂux variations and the determination

of the galactic spiral arm pattern speed is different in its nature from the astronomical

determinations of the pattern speed. This is because the astronomical determinations

assume that the Sun remained in the same galactic orbit it currently occupies.

The meteoritic measurement is “Lagrangian”. It is the measurement relative to a

INTERSTELLAR-TERRESTRIAL RELATIONS 353

moving particle, the heliosphere, which could have had small variations in its orbital

parameters. In fact, because of the larger solar metalicity than the solar environment,

the solar system is more likely to have migrated outwards than inwards. This radial

diffusion gives an error and a bias when comparing the effective, i.e., “Lagrangian”

measured ˜

p, to the “Eulerian” measurements of the pattern speed:

˜

p−p=0.5±1.5kms

−1kpc−1(6)

Taking this into consideration, the observed meteoritic periodicity, with P=147±

6 Myr , implies that −p=10.2±1.5sys ±0.5stat, where the systematic error

arises from possible diffusion of the solar orbital parameters. This result is consistent

with the astronomically measured pattern speed of the ﬁrst set of spiral arms.

7. Cosmic Ray Spectra Inside and Outside of Galactic Arms

In this section we want to follow the line of argumentations of the previous one, but

shall approach the problem based on more fundamental physical considerations.

The passage of the heliosphere through dense interstellar clouds has many interest-

ing direct effects (see e.g. Yeghikyan and Fahr, 2003, 2004a, b) and also inﬂuences

via decreased modulation the near-Earth ﬂux intensities of GCRs and of anomalous

cosmic rays (ACRs) (see Scherer, 2000; Scherer et al., 2001a, b). Here we study

the problem of GCR spectra which are to be expected inside and outside of galactic

arms.

7.1. ACCELERATIONS AT SHOCKS

Shocks, for a long time already, have been recognized as effective astrophysical

sites for particle acceleration. This is because particles, which strongly interact with

scattering centers embedded in astrophysical magnetohydrodynamic plasma ﬂows

can easily and effectively proﬁt from strong velocity gradients occuring in these

ﬂows. Most effective in this respect are velocity gradients which are established

at astrophysical MHD shocks. One may characterize the transition from upstream

to downstream velocities at such a shock by a typical transition scale δand by the

extent Hof the whole region over which the acceleration procedure is considered.

Then the particle transport equation (1) given in Section 5 needs to be solved for the

case δrgλHwith rgand λbeing the gyroradius and the mean scattering

length parallel to the background magnetic ﬁeld, respectively. For a quasi one-

dimensional shock, and for stationary conditions, at positions not too far from the

shock it transforms into the following one-dimensional equation:

u∂f

∂x−∂

∂xDcos θ∂f

∂x=1

3(u+−u−)δ(x)∂f

∂ln p(7)

354 K. SCHERER ET AL.

where ±denote the plasma parameters upstream (+) and downstream (−) of the

shock structure, respectively, uis the corresponding plasma bulk speed, and Dthe

coefﬁcient of spatial diffusion along the magnetic ﬁeld.

Criteria, that in any case should be fulﬁlled by a formal solution of the above

equation, are:

A: steadiness of differential particle density at the shock, i.e.:

f+(p,x=0) =f−(p,x=0)

B: Continuity of differential streaming at the shock, i.e.:

uf −κdf

dx+,0=uf −κdf

dx−,0

C: Continuity of differential energy ﬂow at the shock, i.e.:

−u∂f

∂ln p3−κdf

dx+,0=−u∂f

∂ln p3−κdf

dx−,0

Far from the shock one may assume unmodulated spectra with asymptotic so-

lutions given by f±(p,x→±∞)=f±∞(p). Downstream of the shock (x≥0) it

is expected that fis independent on x, i.e.: f=f+∞(p).

Upstream of the shock (x≤0), however, fmust be expected to be modulated,

i.e. given by:

f=f−∞(p)+(f+∞(p)−f−∞(p))exp u−x

0

dx

κ(8)

The full solution for f+∞(p) matching all the above requirements then is given by

the following formal solution:

f+∞(p)=qp−qp

0

f−∞(p)p(q−1)dp(9)

where the power index qis given by the expression: q=3s/(s−1) with the shock

compression ratio sgiven by: s=u−/u+.

Given the spectral distribution far upstream of the shock in the form f−∞(p)∼

p−with ≤qthen Equation (9) yields:

f+∞(p)∼qp−qp

0

p−pq−1dp=qp−qpq−

q−

p

0=q

q−p−(10)

Assuming, on the other hand, that f−∞(p)∼p−, with =0≤qfor p≤p0

only, and with ≥qfor p≥p0,then Equation (9) in contrast gives:

f+∞(p)∼qp−qp0

0

pq−0−1dp+p

p0

pq−−1dp(11)

with the solutions for

INTERSTELLAR-TERRESTRIAL RELATIONS 355

f+∞(p)∼⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

q

q−0

p−0;p≤p0

qp−q1

q−0

pq−0

0+1

q−pq−−pq−

0;p≥p0

(12)

which ﬁnally evaluates to:

f+∞(p)∼q

q−p

p0−q

+q(0−)

(q−0)(

q−)p

p0−q

and simply is of the twin-power law form:

f+∞(p)=Ap

p0−q

+Bp

p0−

(13)

One should keep in mind that here ≥qwas assumed, which makes it evident

that the ﬁrst term clearly is the leading term for pp0meaning that here one

obtains a simple mono-power law:

f+∞(p≥p0)∼q

−qp

p0−q−q

q−0

p−0

0+p−

0∼p

p0−q

(14)

In the following this solution for the shock-related GCR distribution is to be

applied to giant astrophysical shock waves like supernova blast waves sporadically

running out from collapsing stars.

7.2. SELF-SIMILAR BLAST WAVES

Supernova shock waves are considered in terms of spherical blast waves under

the assumption of self-similarity (see Sedov, 1946). For the purpose of justify-

ing this concept the outside pressure must be expected to be equal to P00.

The consideration starts with the adiabatic Sedov phase which implies the initial

explosion-induced SN energy release EBis converted into kinetic energy of the

dynamics of the mass-accumulating SN shell. The problem in this adiabatic phase

is fully determined by two quantities, namely EBand the mass density ρ0of the

unperturbed, pristine interstellar medium.

In a spherically symmetric problem all hydrodynamic functions only are func-

tions of the distance rfrom the SN explosion center and of the time telapsed

since the explosion event, and all solutions should allow a self-similar scaling by

r(t)=α(t)r(t0). Since the quantity =EB/ρ0has the dimension [cm5sec−2],

one can thus introduce the following self-similar normalization:

ξ=r/x(t)=rρ0

EBt21/5

(15)

356 K. SCHERER ET AL.

The special point Rsof the shock front location with the normalized value ξsas

function of time hence behaves like:

Rs(t)=ξsEB

ρ01/5

t2/5(16)

As consequence from the above relation one easily derives the expansion velocity

of the SN shock front by:

u−=dR

s

dt =2

5

Rs

t=ξs

2

5EB

ρ01/5

t−3/5(17)

The upstream Mach number of the SN shock is permanently decreasing with

time after the explosion event according to:

M(t)=M0t

t0η−1

=ηR0

t0C0t

t0η−1

(18)

where η=2/5 in a homogeneous low-pressure medium and M0and C0are the

initial SN shock Mach number and the sound velocity of the unperturbed interstellar

medium. Roughly it can be estimated that the adiabatic Sedov expansion starts,

when the initial SN explosion energy is converted into kinetic energy of the shell

matter, i.e. when (4π/3)ρ0R3

s0C2M2

0=ESN holds. This yields the time t0after the

explosive event t=0 when the adiabatic phase of the shock expansion starts as

related to the initial shock distance by:

Rs0=13.5mESN

ρ01/5

t2/5

0[pc](19)

7.3. GALACTIC COSMIC RAY SPECTRA

Based on a stochastic occurrence of SN events within the spiral arm regions it may be

necessary, before an inner-arm particle spectrum can be estimated, to inspect various

important time periods characterizing the course of relevant physical processes, like

the SN-occurrence period, the SN shock passage time to the borders of the arm, the

mean capture time of energetic particles within the arm region or the diffusion time,

and the average particle acceleration time near the expanding SN shock surface.

Starting from theoretical solutions of the cosmic ray transport equation as pre-

sented by Axford (1981), O’C Drury (1983) or Malkov and O’C Drury (2001),

where, as described above, a one-dimensional shock geometry is assumed, one ﬁnds

the following upstream solution f−(x,p) for the spectrum of shock-accelerated en-

ergetic particles:

f−(x,p)=C

Ap

p0−q

exp u−

κ(p)x(20)

INTERSTELLAR-TERRESTRIAL RELATIONS 357

Here Cis a constant and the coordinate xdenoting the linear distance from the

planar shock surface is counted negative in the direction upstream of the shock. The

speed by which the shock passes over the galactic material amounts to u−and may

be of the order of 1000 to 2500 km/s. Downstream of the shock it is assumed that

the spatial derivative of f+vanishes, i.e. ∂f+/∂ x0, meaning that f+const.

The absolute value of the distribution function f−has not yet been speciﬁed.

Thus the value Cneeds to be ﬁxed such as to fulﬁll ﬂux continuity relations

at the shock expressing the fact that the total outﬂow of the GCR ﬂuxes to

the left and to the right side of the SN shock (i.e. the sum of the upstream and

downstream streamings, respectively, e.g. see Jokipii, 1971; Gleeson and Axford,

1968) has to be identical with the ﬂux of particles above the injection thresh-

old p=p0which are convected from the upstream side into the shock and can

serve as the seed of SN-accelerated GCRs. This requirement expresses in the form

(see Fahr, 1990):

1

3f−u−−1

3u−p∂f−

∂p−κ−

∂f−

∂xp2dp

+1

3f+u+−1

3u+p∂f+

∂p−κ+

∂f+

∂xp2dp =ε(p0)u+n+(21)

where ε(p0)n+is the number of particles with momenta pμ≤−p0upstream of

the shock which can serve as seed of the GCRs. Evaluating the above equation with

the expression for f±given in Equation (20) then, when reminding that at x=0 the

upstream and downstream distribution functions are identical, i.e. f−=f+leads

to: u−f−1

31+1

s4s−1

s−1−1p2dp =ε(p0)u−n0(22)

The above expression can ﬁnally be evaluated with the distribution function given

by Equation (20):

s2+6s−1

3(s2−s)f0p2dp =s2+6s−1

3(s2−s)Cp3

0∞

1

x−2+s

s−1dx =ε(p0)n0(23)

which delivers for the quantity C:

C=3sε(p0)n0

(3s2+2s−3)p3

0

(24)

As a surprise the above result does not anymore show the explicit dependence

of Con the upstream plasma velocity u−. This dependence, however, implicitly is

hidden in the value p0for the critical momentum of the particle injection into

the shock acceleration. In order to inject particles into the diffusive accelera-

tion process, it is necessary that these particles have the dynamic virtue due to

which they are not simply convected over the electric potential wall of the SN

shock but become reﬂected at this wall at least for the ﬁrst time (see e.g. Chalov

358 K. SCHERER ET AL.

and Fahr, 1995, 2000). For this to happen the following relation simply needs

to be fulﬁlled:

1

2mu−−p0

m2

≤1

2m(u2

−−u2

+)⇒p0≥mu−1−1−1

s2(25)

The percentage of particles with momenta pμ≤−p0in the shifted Maxwellian

distribution function, describing particles comoving with the upstream plasma ﬂow,

is then given by;

(p0)=1

π1/2∞

x0

exp(−x2)dx =1−1

√πerf(x0)=1−1

√πerf(κ(s)Ms)

(26)

where x2

0=p2

0/2KT

0m=mu2

−g2(s)/2KT

0=κ2(s)M2

s. Here the following nota-

tions have been used: g(s)=(1 −(1 −s−1)−1/2) with the Mach number of the

upstream plasma deﬁned by M2

s=mu2

−/γ KT

0.

This ﬁnally delivers for Cthe expression:

C=3sn0

1−π−1/2erf(κ(s)Ms)

(3s2+2s−3)p3

0

(27)

This result expresses the fact that the absolute value of f−given by Cis de-

termined by the upstream ﬂow velocity u−, the upstream Mach number Ms, the

compression ratio sas function of Msand the upstream plasma density n0which is

known to be greater by a factor of about 10 in the spiral arms compared to inter-arm

regions.

To describe the evolution in time and space of spectra for GCRs originating at SN

shock waves one furthermore needs to know something about the evolution of the

SN shock at its propagation in circumstellar space. Relying on the Sedov solution

for the SN blast wave evolution at its propagation into the ambient interstellar

medium one can describe the propagation velocity U1=U1(t) as a function of

time by the following relation (see Krymskii, 1977a,b):

U1(t)=2

52ESN

ρ11/5

t−3/5(28)

where ESN denotes the total energy released by the SN explosion, and ρ1is the

ambient interstellar gas mass density ahead of the propagating shock.

Keeping in mind that the compression ratio sas given by the Rankine-Hugoniot

relations writes:

s(t)=(γ+1)M2

1(t)

(γ−1)M2

1(t)+2(29)

INTERSTELLAR-TERRESTRIAL RELATIONS 359

where M1(t) denotes the upstream Mach number depending on SN shock evolution

time tand is given by:

M2

1(t)=ρ1U2

1(t)

γP1=4

25

ρ3/5

1

P1

(2ESN)2/5t−6/5(30)

one can predict the temporal change ds/dt of the SN shock compression ratio. It

then clearly turns out that the typical period τsby which the strength of the SN

shock changes in time is large with respect to τa(p), i.e. that:

τs=− s

ds/dt ≥τa(p)=6s

s−1

κ1

U2

1

(31)

7.4. THE AVERAGE GCR SPECTRUM INSIDE GALACTIC ARMS

To calculate the average GCR spectrum for a casually placed space point within

the galactic arm regions we shall assume that such a point is at a random distance

with respect to casually occuring SN shock fronts, the latter being true as conse-

quence of stochastic occurrences of SN explosions at random places in the arms.

We shall denote the casual x-axis position of an arbitrary space point with respect

to the center of a stochastic SN explosion by X. At time t, after the explosion took

place, the SN shock front has an actual x-axis position of Rx(t)=t

0U1(t)dt

and thus the average GCR spectrum should be obtainable by the following

expression:

f(p)=1

Xmaxtmax Xmax

Xmin

dXtmax

tmin

dtC(t)p

p0−q(t)

+Bp

p0−

×exp −U1(X−Rx(t))

κ(p)H(Rx(t)−X)(32)

Here the function H(λ) is the well known step function with H(λ)=0 for positive

values of λ.

The quantity Xmax Rais to determine the maximum distance which a stochasti-

cally placed detector point may have to the SN explosion center. This maximum dis-

tance, for physical reasons and in order to make the expression (32) statistically rele-

vant, should be selected such that within the counted arm volume Vmax =πR2

aXmax

during a time tmax one obtains the probability “1” for a next SN explosion to occur.

With an SN-explosion rate ςper unit of time and volume within the arm region

one then ﬁnds Xmax =[πR2

aςtmax]−1. The quantity tmax is taken as the time after

SN explosion till which the evolving SN shock front has upstream Mach numbers

larger than or equal to 1 and thus accelerates GCRs. One can conclude that diffu-

sive acceleration of GCRs can continue till the propagation speed U1(t) of the SN

shock front falls below the local Alfv´en speed vA1impeding the pile-up of MHD

360 K. SCHERER ET AL.

turbulences which act as scattering centers for GCRs bouncing to and fro through

the shock. From Equation (30) one thus derives:

tmax 2ESN

ρ11/35

2vA1−5/3

(33)

which for values given by Hartquist and Morﬁll (1983) (i.e. ESN =1051erg; ρ1/m=

10 cm−3;vA1=106cm/s ) evaluates to tmax 6 Myr. The distance Xmin denotes the

SN shock distance from the SN explosion center at time tmin after explosion given

by:

tmin 2ESN

ρ11/35

2U1,max−5/3

(34)

where U1,max is the maximum SN shock speed just after shock formation. For esti-

mate purposes we may assume here that the following connection can be assumed

4π

3X3

minρ1U2

1,max =ESN and that a maximum shock speed of U1,max =3500 km/s

can be adopted at the beginning of the Sedov phase.

7.5. ESCAPE INTO THE INTERARM REGION

Assuming that the expression for f(p) given by the Equation (32) is valid for all

space points located within a cylindrical tube along the central axis of the spiral

arm, i.e. f(p) represents an axially and temporally averaged GCR spectrum for

all near axis points within a galactic arm, and adopting an arm-parallel magnetic

ﬁeld, then in addition to the very efﬁcient spatial diffusion parallel to the magnetic

ﬁeld a much less efﬁcient diffusion perpendicular to the ﬁeld operates everywhere

which eventually lets GCR particles escape into the interarm region. We describe

this diffusion with respect to the cylindric coordinate ras a source-free, time-

independent diffusion (∇·(↔

κf)=0) which gives in cylindrical coordinates

rκ⊥

∂f

∂r=const =rκ⊥

∂f

∂r0=−πr2

0

f0

τe

(35)

where r0is the radius of an inner tube within which the distribution function f0

prevails, and where τeis the period of GCR escape into the interarm region. Then

the solution for f=f(r) is obtained from the expression:

f(r,p)=f(r0,p)+r

r0

const

rκ⊥

dr=f(r0,p)−πr2

0

τeκ⊥

ln r

r0 (36)

At the border r=Raof the arm to the interarm region the identity at both sides

of both GCR ﬂux and the spectral intensity is required yielding the following two

relations:

Raκi⊥

∂fi

∂r=Raκa⊥

∂fa

∂rand |fi|Ra=|fa|Ra(37)

INTERSTELLAR-TERRESTRIAL RELATIONS 361

where κa⊥and κi⊥denote spatial diffusion coefﬁcients in the arm and the interarm

region, respectively. With these requirements one obtains the distribution function

fi(r,p) in the interarm region as given in the form:

fi(r,p)=f(r0,p)1−πr2

0

τe1

κa⊥

ln Ra

r0+1

κi⊥

ln r

Ra (38)

To achieve consistency with the assumptions made in the derivations above one

should be able to justify a time-independence of the GCR distribution function, i.e.

the fact that ∂f/∂t=0 is assumed. From a simpliﬁed phase-space transport equa-

tion one can then derive the requirement that time-independence of fis achieved,

if the average galactic arm SN occurrence period τSN and the escape period τeare

related by:

τe=τSN

1−1

p3qχn1τSN=τSN

1+4pi0

p3τSN

τi0(39)

where q=3s/(s−1) is the power index of the GCR spectrum and where the

momentum loss of GCR particles due to gas ionisations has been assumed as

˙

pi−χn1p−2, for details see Lerche and Schlickeiser (1982a,b,c). The second

identity follows with q4 and n110 cm−3and τi0=τi(pi0)=108s and pi0=

p(100 MeV). The standard period τSN might be quantiﬁed by: τSN 1010s.

Now we try to obtain a reasonably well supported value for the dimension r0

within the above derived calculation. Going back to Equation (35) one ﬁrst ﬁnds:

rκ⊥

∂f

∂rRaRaκ⊥

f0−fRa

Ra=πr2

0

f0

τe

(40)

from which with the help of Equation (36) one furthermore derives

πr2

0=κ⊥τe

f0−fRa

f0=κ⊥τe1−1+πr2

0

τeκ⊥

ln Ra

r0 (41)

simply requiring r0=Ra/exp(1).

With help of Equation (39) one now can use Equation (38) to display the spectral

ﬂux intensity of GCRs as function of the off axis-distance rfrom the axis of the

galactic arms.

Based on formula (38) one can estimate the variation of the galactic cosmic

ray spectra along the trajectory of the Sun, in particular inside and outside galactic

spiral arms. In a ﬁrst step, we compute an arm spectrum from the expression

ja(r0,p)=j(r,p)1−πr2

0

τe1

κa⊥

ln Ra

r0+1

κi⊥

ln r

Ra−1

(42)

assuming that the present-day local interstellar spectrum derived from observations

can be represented as (Reinecke et al., 1993)

j(r,p)=p2f(r,p)=12,41 v/c

(Ek+0.5E0)2.6part./m2/s/srad/MeV (43)

362 K. SCHERER ET AL.

where vis the speed of a proton with kinetic energy Ekin GeV and E0is the proton

rest energy in GeV.

For the present location of the Sun relative the next main spiral arm with radius

Ra=0.35 kpc we use r=1 kpc. Interpreting the interarm diffusion coefﬁcient

as that one considered in galactic propagation models we select a typical value of

κi⊥=3·1028 cm2/s. For the diffusion coefﬁcient inside an arm we adopt κa⊥=

0.1κi⊥corresponding to about three times higher turbulence level inside an arm

than outside.

As we are computing spectral rather than just total ﬂux variations, we have to

take into account the dependence of the diffusion on rigidity P.Weuse

κi⊥=1.5P

P0ξ

;ξ=aP +bP

0

P+P0

(44)

which avoids the spectral break of the expression given by B¨usching et al. (2005)

and approximate the latter with the values a=0.51 and b=−0.39.

Figure 7. Galactic cosmic ray spectra inside and outside galactic spiral arms: the solid line gives the

present-day spectrum according to Reinecke et al. (1993), the upper dashed line is the arm spectrum

computed from formula 45 assuming that the Sun is located 1 kpc outside the next main spiral arm,

the lower dashed line shows the spectrum in the middle between two arms, and the dash-dotted line

is the ratio of the arm to the interarm spectrum for a spiral arm radius of Ra=0.35 kpc, r0=0.1 kpc,

κi⊥=3·1028 cm2/s, and κa⊥=0.1κi,⊥, and τe=7.1·106a. The other lines give the corresponding

spectra for a 20% wider spiral arm.

INTERSTELLAR-TERRESTRIAL RELATIONS 363

Because the time scale τeresulting from Equation (41) is even shorter than

τSN, its use in Equation (38) would not be consistent with the diffusion time scale

R2

a/κa⊥=7.1·106yr, which we therefore use instead of τe.

The resulting arm spectrum is shown as the upper dashed line in Figure 7. From

the latter we subsequently computed the spectrum approximately in the middle

between to spiral arm from

ji(rm,p)=j(ra,p)1−πr2

0

τe1

κa⊥

ln Ra

r0+1

κi⊥

ln rm

Ra (45)

with rm=3 kpc resulting in the lower dashed curve in the ﬁgure. The dotted lines

are at the same locations inside and outside an spiral arm but for a 20% greater Ra.

That there is not much variation of the spectra in the interarm region is consistent

with the rather high diffusion coefﬁcient which cannot result in strong modulation

over a few kpc.

Obviously, we obtain the expected variation of factors two to seven depending on

parameters, compare with the chapter 6. In our approach, however, this variation is

computed as a function of kinetic energy, see the dash-dotted lines in the Figure 7.

Interestingly, the maximum variation occurs at around 3 GeV, which means that

also the modulated spectra at Earth should exhibit a variation. This modulation of

the interstellar spectra within the heliosphere is the subject of the following part,

while the interactions of CRs in the atmosphere are described in part VI.

.

Part IV

Heliospheric Modulation

.

INTERSTELLAR-TERRESTRIAL RELATIONS 367

8. Propagation of Cosmic Rays Inside the Heliosphere

8.1. SOLAR ACTIVITY: 11-YEAR AND 22-YEAR CYCLES IN COSMIC RAYS

In the heliosphere three main populations of cosmic rays, deﬁned as charged par-

ticles with energies larger than 1MeV, are found. They are: (1) Galactic cosmic

rays, mainly protons and some fully ionized atoms, with a spectral peak for protons

at about 2 GeV at Earth. (2) The anomalous component, which is accelerated at

the solar wind termination shock after entering the heliosphere as neutral atoms

that got singly ionized. For a review of these aspects, see Fichtner (2001). (3) The

third population is particles of mainly solar origin, which may get additionally

accelerated by interplanetary shocks. A prominent strong electron source of up to

50 MeV is the Jovian magnetosphere, with the Saturnian magnetosphere much less

pronounced.

We are protected against CRs by three well-known space “frontiers”, the ﬁrst

one arguably the less appreciated of the three: (1) The solar wind and the accom-

panying relatively turbulent heliospheric magnetic ﬁeld extending to distances of

more than 500 AU in the equatorial plane and to more than 250AU in the polar

plane. The heliospheric volume may oscillate signiﬁcantly with time depending on

solar activity, and where the solar system is located in the galaxy, see part V. (2)

The Earth’s magnetic ﬁeld, which is not at all uniform, e.g. large changes in the

Earth’s magnetic ﬁeld are presently occurring over southern Africa. This means that

signiﬁcant changes in the cut-off rigidity at a given position occur. These changes

seem sufﬁciently large over the past 400 years that the change in CRF impacting

the Earth may approximate the relative change in ﬂux over a solar cycle (Shea and

Smart, 2004). The magnetosphere also withstands all the space weather changes that

the Sun produces, and can reverse its magnetic polarity on the long-term. (3) The

atmosphere with all its complex physics and chemistry. The cosmic ray intensity

decreases exponentially with increasing atmospheric pressure. The Sun contributes

signiﬁcantly to atmospheric changes through, e.g. variations in solar irradiance,

and variations in the Earth’s orbit (Milankovitch cycles).

The dominant and the most important variability time scale related to solar

activity is the 11-year cycle. This quasi-periodicity is convincingly reﬂected in the

records of sunspots since the early 1600’s and in the GCR intensity observed at

ground and sea level since the 1950’s when neutron monitors (NMs) were widely

deployed, especially as part of the International Geophysical Year (IGY). These

monitors have been remarkably reliable, with good statistics, over ﬁve full 11-year

cycles. An example of this 11-year cosmic ray cycle is shown in Figure 8, which is

the ﬂux measured by the Hermanus NM in South Africa. The intensity is corrected

for atmospheric pressure to get rid of seasonal and daily variations. This means that

atmospheric pressure must also be measured very accurately at every NM station.

In Figure 8 another important cycle, the 22-year cycle, is shown. This cycle

is directly related to the reversal of the solar magnetic ﬁeld during each period

368 K. SCHERER ET AL.

Figure 8. Cosmic ray ﬂux measured by the Hermanus NM (at sea-level with a cut-off rigidity of

4.6 GV) in South Africa. Note the 11-year and 22-year cycles.

of extreme solar activity and is revealed in CR modulation as the alternating ﬂat

and sharp proﬁles of consecutive solar minimum modulation epochs when the CR

intensity becomes a maximum (minimum modulation). The causes and the physics

of the 11-year and 22-year cycles will be discussed below, but ﬁrst a short discussion

in the context of this paper will be given about other variabilities related to CRs in

the heliosphere.

Short periodicities are evident in NM and other cosmic ray data, e.g. the 25–

27-day variation owing to the rotational Sun, and the daily variation owing to the

Earth’s rotation. These variations seldom have magnitudes of more than 1% with

respect to the previous quite times. The well-studied corotating effect is caused

mainly by interaction regions (CIRs) created when a faster solar wind overtakes a

previously released slow solar wind. They usually merge as they propagate outwards

to form various types of interaction regions, the largest ones are known as global

merged interaction regions – GMIRs (Burlaga et al., 1993). Such a GMIR caused

the very large cosmic ray decrease in 1991, shown in Figure 8. They are related

to what happened to the solar magnetic ﬁeld at some earlier stage and are linked

to coronal mass ejections (CMEs), which are always prominent with increased

solar activity but dissipate completely during solar minimum. They propagate far

outward in the heliosphere with the solar wind speed, even beyond the solar wind

termination shock around 90–95 AU. Although CIRs may be spread over a large

region in azimuthal angle, they cannot cause long-term periodicities on the scale

(amplitude) of the 11 year cycle. An isolated GMIR may cause a decrease similar in

INTERSTELLAR-TERRESTRIAL RELATIONS 369

magnitude than the 11-year cycle but it usually lasts only several months to about a

year. A series or train of GMIRs, on the other hand, may contribute signiﬁcantly to

modulation during periods of increased solar activity, in the form of large discrete

steps, increasing the overall amplitude of the 11-year cycle (le Roux and Potgieter,

1995). The Sun also occasionally accelerates ions to high energies but with a highly

temporal and anisotropic nature, which are known as solar energetic particle (SEP)

events.

The 11-year and 22-year cycles are modulated by longer term variability on

time scales from decades to centuries, perhaps even longer. There are indications

of periods of 50–65 years and 90–130 years, also for a periodicity of about 220 and

600 years. It is not yet clear whether these variabilities should be considered “pertur-

bations”, stochastic in nature or truly time-structured to be ﬁgured as superpositions

of several periodic processes. Cases of strong “perturbations” of the consecutive

11-year cycles are the “grand minima” in solar activity, with the prime example the

Maunder Minimum (1645–1715) when sunspots almost completely disappeared.

Assuming the solar magnetic ﬁeld to have vanished or without any reversals during

the Maunder minimum would be an oversimpliﬁcation as some studies already

seem to illustrate (Caballero-Lopez et al., 2004; Scherer and Fichtner, 2004). The

heliospheric modulation of CRs could have continued during this period but much

less pronounced (with a small amplitude). It is reasonable to infer that less CMEs,

for example, occurred so that the total ﬂux of CRs at Earth then should have been

higher than afterwards. However, to consider the high levels of sunspot activity for

the last few 11-year cycles as unprecedented is still inconclusive. From Figure 8

follows that the maximum levels of CRs seem to gradually decrease.

The CRF is also not expected to be constant along the trajectory of the solar

system in the galaxy. Interstellar conditions, even locally, should therefore differ

signiﬁcantly over long time-scales, for example, when the Sun moves in and out

of a spiral arm (Shaviv, 2003a, see also part III). The CRF at Earth is therefore

expected to be variable over time scales of 105to 109years (e.g. Scherer, 2000;

Scherer et al., 2004, and the references therein).

It is accepted that the concentration of 10Be nuclei in polar ice exhibits temporal

variations in response to changes in the ﬂux of the primary CRs (Beer et al., 1990;

Masarik and Beer, 1999, and references therein). McCracken et al. (2002, 2004)

showed that the 10Be response function has peaked near 1.8 GeV/nucleon since

1950. They also claim that the NM era represents the most extreme cosmic ray

modulation events over the past millennium and that this period is not the typical

condition of the heliosphere. There is the hypothesis that short-term (one month or

less) increases in the nitrate component of polar ice are the consequence of SEPs

(Shea et al., 1999). The observed concentration of 10Be is also determined by both

production and transport processes in the atmospheric, and a terrestrial origin for

many of the noticeable enhancements in 10Be is possible, a major uncertainty that

inhibits the use of cosmogenic isotopes for the quantitative determination of the

time variations of galactic CRs on the same scales for which 10Be is available.

370 K. SCHERER ET AL.

Exploring cosmic ray modulation over time scales of hundreds of years and

during times when the heliosphere was signiﬁcantly different from the present

epoch is a very interesting development. Much work is still needed to make the

apparent association (correlations) more convincing, being very complex is well

recognized, than what e.g. McCracken et al. (2004) and Usoskin and Mursula (2003)

discussed. However, the association between the 10 Be maxima and low values of

the sunspot number is persuasive for the Maunder and Dalton minima.

8.2. CAUSES OF THE 11- AND 22-YEAR MODULATION CYCLES

Although there is a large number of solar activity indices, the sunspot number is

the most widely used index. From a CR modulation point of view, sunspots are not

very useful, because the large modulation observed at Earth is primarily caused by

what occurs, in three-dimensions, between the outer boundary (heliopause) and the

Earth (or any other observation point). In this sense the widely used “force-ﬁeld”

modulation model (e.g. Caballero-Lopez and Moraal, 2004) is very restricted, ig-

noring all the important latitudinal modulation effects e.g., perpendicular diffusion,

gradient and curvature drifts.

Our present understanding of cosmic ray modulation is based on the cosmic

ray transport equation (1). For this equation, with a full description of the main

modulation mechanisms and the main physics behind them, the reader is referred

to Potgieter (1995, 1998) and Ferreira and Potgieter (2004), and the references

therein, for more details see Section 5. The individual mechanisms are well-known

but how they combine to produce cosmic ray modulation, especially with increasing

solar activity, is still actively studied. Basically it works as follows. GCRs scatter

from the irregularities in the heliospheric magnetic ﬁeld as they attempt to diffuse

from the heliospheric boundary toward the Earth. With these irregularities frozen

into the solar wind, the particles are convected outward at the solar wind speed. In

the process, they experience adiabatically energy losses, which for nuclei can be

quite signiﬁcant. Gradient and curvature drift is the fourth major mechanism, and

gets prominent during solar minimum conditions when the magnetic ﬁeld becomes

globally well structured. In the A>0 drift cycle (see Figure 8) the northern ﬁeld

points away from the Sun, consequently positively charged particles drift mainly

from high heliolatitudes toward the equatorial plane and outward primarily along

the current sheet, giving the typical ﬂat intensity-time proﬁles. The current (neutral)

sheet separates the ﬁeld in two hemispheres and becomes progressively inclined

and wavy, due to solar rotation, with increasing solar activity (Smith, 2001). The

extent of inclination or “tilt angle” changes from about 10◦at solar minimum to

75◦at solar maximum (theoretically 90◦is possible but the current sheet on the Sun

becomes unrecognizable long before then; Hoeksema, 1992). In the A<0 cycle

the drift directions are reversed, so that when positive particles drifting inward along

the wavy current sheet, the intensity at Earth becomes strongly dependent on the tilt

INTERSTELLAR-TERRESTRIAL RELATIONS 371

angle and consequently exhibits a sharp intensity-time proﬁle for about half of the

11-year cycle. For negatively charged particles the drift directions reverse so that a

clear charge-sign dependent effect occurs, a phenomenon that has been conﬁrmed

by observations from the Ulysses mission for more than a solar cycle (Heber et al.,

2003). The CRF thus varies in anti-correlation with the 11-year solar activity cycle

indicating that they are indeed modulated as they traverse the heliosphere. The

extent of this modulation depends on the position and time of the observation, and

strongly on the energy of the cosmic rays. The 22-year cycle, originating from the

reversal of the solar magnetic ﬁeld roughly every 11 years, is superimposed on

the 11-year cycle with an amplitude less than 50% of the 11-year cycle. As shown

in Figure 8, the NM intensity-time proﬁles exhibit the expected peak-like shapes

around the solar minima of 1965 and 1987 (A<0), while around 1954, 1976

and 1998 (A>0) they were conspicuously ﬂatter. Shortly after the extraordinary

ﬂat proﬁle around 1976 was observed, two research groups, in Arizona (Jokipii

et al., 1977) and in South Africa, quickly recognized that gradient and curvature

drifts, together with current sheet drifts, could explain these features (Potgieter

and Moraal, 1985, and references therein). After the revealing of drifts as a major

modulation mechanism, the “tilt angle” of the current sheet, being a very good

proxy of its waviness which on its turn is directly related to solar activity, has

became the most useful solar activity “index” for cosmic ray studies.

While the cosmic ray intensity at NM energies are higher in A<0 cycles at

solar minimum than in the A>0 cycles – see Figure 8 – the situation is reversed

for lower energies e.g., for 200MeV protons, conﬁrmed by spacecraft observa-

tions. This requires the differential spectra of consecutive solar minima to cross

at energies between 1 and 5 GeV (Reinecke and Potgieter, 1994). The maxima in

these spectra also shift somewhat up or down in energy depending on the drift

cycle because the energy losses are somewhat less during A>0 cycles than during

A<0 cycles. Convincing experimental evidence of drift effects followed since the

1970’s, e.g. when it was discovered that NM differential spectra based on latitude

surveys showed the 22-year cycle, and when the intensity-time proﬁles of cosmic

ray electrons depicted the predicted “opposite” proﬁles. It further turned out that

the A>0 minimum in the 1990’s was not as ﬂat as in the 1970’s, by allowing the

solar minima modulation periods to be less drift dominated, as predicted (Potgi-

eter, 1995). This fortuitous ﬂat shape during of the 1970’s is therefore not entirely

owing to drifts but also to the unique unperturbed way in which solar activity

subsided after the 1969–70 solar maximum. The period from 1972–1975 became

known as a “mini-cycle”, interestingly close to the 5-year cycle that McCracken

et al. (2002) reported. It is also known that the sharp proﬁles are consistently

asymmetrical with respect to the times of minimum modulation, with a faster in-

crease in cosmic ray ﬂux before than after the minima (about 4 years to 7 years,

respectively). The 11-year solar cycle thus has an asymmetric shape, also evident

from “tilt angle” calculations, and should therefore be evident in the cosmogenic

archives.

372 K. SCHERER ET AL.

In the mid-1990’s, le Roux and Potgieter (1995) illustrated that the waviness of

the current sheet cannot be considered the only time-dependent modulation parame-

ter because large step decreases occurred in the observed CR intensities (McDonald

et al., 1981). These steps are prominent during increased solar activity when the

changes in the current sheet are no longer primarily responsible for the modulation.

In order to successfully model CR intensities during moderate to higher solar activ-

ity requires some form of propagating diffusion barriers (PDBs). The extreme forms

of these diffusion barriers are the GMIRs, mentioned above. They also illustrated

that a complete 11-year modulation cycle could be reproduced by including a com-

bination of drifts and GMIRs in a time-dependent model. The addition of GMIRs

convincingly explains the step-like appearance in the observed cosmic ray intensi-

ties. The periods during which the GMIRs affect long-term modulation depend on

the radius of the heliosphere, their rate of occurrence, the speed with which they

propagate, their amplitude, their spatial extent, especially in latitude, and ﬁnally

also on the background turbulence (diffusion coefﬁcients) they encounter. Drifts,

on the other hand, dominate the solar minimum modulation periods so that during

an 11-year cycle there always is a transition from a period dominated by drifts

to a period dominated by diffusive propagating structures. During some 11-year

cycles these periods of transition happen very gradually, during others it can be

very quickly, depending on how the solar magnetic ﬁeld transforms from a domi-

nating dipole structure to a complex higher order ﬁeld. For reviews on long-term

modulation, see e.g. Heber and Potgieter (2000) and Potgieter et al. (2001).

If there is a direct relation between 10Be concentrations and CRs impacting

Earth, large decreases like the one in 1991 which reduced the ﬂux of relatively high

energy signiﬁcantly, should show up in the time-proﬁles of 10Be.

A third improvement in our understanding of 11-year and 22-year cycles came

when Potgieter and Ferreira (2001) generalized the PDBs concept by varying also all

the relevant diffusion coefﬁcients with an 11-year cycle, in a fully time-dependent

model directly reﬂecting the time-dependent changes in the measured magnetic

ﬁeld magnitude at Earth. These changes were propagated outwards at the solar

wind speed to form effective PDBs throughout the heliosphere, changing with the

solar cycle. This approach simulated an 11-year modulation cycle successfully for

cosmic ray at energies >10 GeV, but it resulted in far less modulation than what

was observed at lower energies. They therefore introduced the compound approach,

which combines the effects of the global changes in the heliospheric magnetic ﬁeld

magnitude, related to all diffusion coefﬁcients, with global and current sheet drifts

in a complex manner, not merely approximately proportional to 1/B, with Bthe

magnetic ﬁeld magnitude, to produce realistic time-dependent relations between

the major modulation parameters (Ferreira and Potgieter, 2004). This approach has

so far provided the most successful modeling of the 11-year and 22-year cycles. An

example is given in Figure 9, where the 11-year simulation done with the compound

numerical model is shown compared to the Hermanus NM count rates expressed

as percentage values for the period of 1980–1992.

INTERSTELLAR-TERRESTRIAL RELATIONS 373

Figure 9. Model computations, based on the compound approach (Ferreira and Potgieter, 2004),

shown with the Hermanus NM count rates expressed as percentage values for 1980–1992. Shaded

areas indicate when the solar magnetic ﬁeld polarity was not well deﬁned.

This inversion CR-B method is used to derive values of the solar magnetic

ﬁeld back in time, after the modulation model is calibrated to CR observations,

typically for minimum modulation like in May 1965, and further by assuming a

direct relation between CRs and the long-term cosmogenic isotope time-proﬁles.

This produces interesting results but further investigation is required because these

computations are highly model dependent. It is apparent that for the reconstruction

of sunspot numbers from the rate of cosmogenic isotopes, one needs to take into

account drift effects described above. Using sunspot numbers as a proxy for the

long-term changes in the interplanetary magnetic ﬁeld over long periods of time

and hence the cosmic ray intensity is not reasonable.

The structural features and geometry of the heliosphere, including the solar

wind termination shock, the heliosheath and heliopause, especially their locations,

also inﬂuence the cosmic ray ﬂuxes at Earth. This is the topic of the next section.

Together with these features, one has to take into account the possible variability of

the local interstellar spectrum for the various cosmic ray species as the heliosphere

moves around the galactic center as discussed in part III. The impact of these global

heliospheric features on very long-term cosmic ray modulation will be intensively

studied in future, with the interest already being enhanced by the recent encounter

(Stone et al., 2005) of the solar wind termination shock of the Voyager 1 spacecraft.

374 K. SCHERER ET AL.

9. Effects of the Heliospheric Structure and the Heliopause on the

Intensities of Cosmic Rays at Earth

As the heliosphere moves through interstellar space, various changes in its envi-

ronment could inﬂuence and change its structure. In this section the purpose is to

show how changes in the geometrical structure of the heliosphere can affect the

modulation of cosmic rays at Earth from a test particle model point of view. The

next two subsections will discuss the hydrodynamic point of view. The main focus

will be on the modulation effects of the outer heliospheric structures: (1) The solar

wind termination shock (TS) where charged particles are getting re-accelerated to

higher energies. (2) The outer boundary (heliopause) where the local interstellar

spectra (LIS) of different particle species are encountered; and (3) the heliosheath,

the region between the TS and the heliopause. The TS is described as a collisionless

shock, i.e. a discontinuous transition from supersonic to subsonic ﬂow speeds of the

solar wind, in order for the solar wind ram pressure to match the interstellar thermal

pressure, accompanied by discontinuous increases in number density, temperature

and pressure inside the heliosheath. The heliopause is a contact discontinuity; a

surface in the plasma through which no mass ﬂow occurs, and which separates the

solar and interstellar plasmas. For a review of these features, see Zank (1999) and

also part V.

With the recent crossing of the TS by the Voyager 1 spacecraft at ≈94 AU

a compression ratio, between the upstream and downstream solar wind plasmas,

was measured between ≈2.6 (Stone et al., 2005) and ≈3 (Burlaga et al., 2005).

This implies that the TS is rather weak, as assumed in our modeling. The TS may

move signiﬁcantly outwards and inwards over a solar cycle (Whang et al., 2004).

Many factors inﬂuence the position of the heliopause, making it less certain, but

it is probably at least 30–50 AU beyond the TS in the nose direction, the region

in which the heliosphere is moving, but signiﬁcantly larger in the tail direction

of the heliosphere, because the dimensions of the heliosphere should be affected

by its relative motion through the local interstellar medium (Scherer and Fahr,

2003; Zank and M¨uller, 2003). The conﬁguration and position of the TS and the

heliopause will also change if the heliosphere would move in and out of a denser

region in the interstellar medium, like a crossing of the galactic spiral arm.

The effects on the intensities of CRs at Earth of some assumptions and unknowns

in heliospheric modeling are shown in this part; these effects may just as well be

interpreted as caused by changes in the local interstellar space.

9.1. MODULATION MODELS

Modulation models are based on the numerical solution of the time-dependent CR

transport equation (Parker, 1965), see also Section 5. The details of the model

used to obtain the results shown below, were discussed by Langner et al. (2003)

INTERSTELLAR-TERRESTRIAL RELATIONS 375

and Langner and Potgieter (2005c). Equation (1) was solved time-dependently as

a combined diffusive shock acceleration and drift modulation model, neglecting

any azimuthal dependence. The heliospheric magnetic ﬁeld (HMF) was assumed

to have a basic Archimedian geometry in the equatorial plane, but was modiﬁed in

the polar regions similar to the approach of Jokipii and Kota (1989). The solar wind

was assumed to be radially outward, but with a latitudinal dependence. The current

sheet tilt angle αwas assumed to represent solar minimum modulation conditions

when α=10◦, and solar maximum when α=75◦, for both the magnetic polarity

cycles, respectively called A>0 (e.g. ≈1990–2001) and A<0 (e.g. 1980–1990).

The position of the outer modulation boundary (heliopause) was assumed at rHP =

120 AU, except where explicitly indicated, where the proton LIS of Strong et al.

(2000) was speciﬁed, or the interstellar spectra of Moskalenko et al. (2002, 2003)

for boron (B) and carbon (C). The position of the TS was assumed at rs=90 AU,

with a compression ratio s=3.2 and a shock precursor scale length of L=1.2AU

(Langner et al., 2003), except where explicitly indicated.

9.2. CHANGES IN THE SHAPE OF THE HELIOSPHERE

An example of the effects on galactic CR protons at Earth due to a change in the

shape of the heliosphere is illustrated in Figure 10 for both HMF polarity cycles for

α=10◦. The shape of the heliosphere is changed from symmetrical, with rHP =

120 AU and rs=90 AU, to asymmetrical with rHP =120 AU and rs=90 AU in

the nose direction and rHP =180 AU and rs=100 AU in the tail direction. In the

left panels the energy spectra are shown at radial distances of 1AU, 60 AU, and at

rsand rHP. In the right hand panels the differential intensities are shown at energies

of 16 MeV, 200 MeV, and 1 GeV, respectively. The 16 MeV proﬁles are shown for

illustrative purposes only.

The comparison of these spectra illustrates that no signiﬁcant difference occurs

for the A>0 cycle for solar minimum between a symmetrical and asymmetrical

heliosphere, despite a difference of a factor of 1.5 in the position of the heliopause

in the equatorial tail direction; even when the heliopause is moved from 120 AU to

200 AU and the TS from 90 AU to 105AU. For the A<0 polarity cycle differences

remain insigniﬁcant in the nose direction, but they increase towards the Sun with

decreasing radial distances, for all latitudes. Changes in the shape of the heliosphere

therefore have an inﬂuence on the CR intensities at Earth, although relatively small

(Langner and Potgieter, 2005c).

9.3. CHANGES IN THE SIZE OF THE HELIOSHEATH

In Figure 11 the computed spectra for galactic protons are shown for both magnetic

polarity cycles and for solar minimum conditions with α=10◦. The spectra and

differential intensities are shown at the same distances and energies as in Figure 10.

376 K. SCHERER ET AL.

Figure 10. Solutions for a symmetric (red curves) and an asymmetric heliosphere (black curves)

shown for the nose region (θ=90◦), for solar minimum conditions (α=10◦), and for the A>0

polarity cycle (top panels) and the A<0 polarity cycle (bottom panels), respectively. Left panels:

Energy spectra at radial distances of 1 AU, 60 AU, at the TS position and at the LIS position. Right

panels: Differential intensities as a function of radial distance at energies of 16MeV, 200 MeV, and

1 GeV, respectively. Here rs=90 AU and rHP =120 AU for both heliospheric shapes, but only in the

nose direction, for the asymmetrical shape rs=100 AU and rHP =180 AU in the tail direction. The

LIS is speciﬁed at rHP (from Langner and Potgieter, 2005b).

The LIS is speciﬁed ﬁrst at rHP =120 AU and then with rHP =160 AU. All the

modulation parameters including the diffusion coefﬁcients were kept the same for

both situations. Qualitatively the results for the different heliopause positions look

similar, but quantitatively they differ, especially as a function of radial distance.

The spectra for rHP =120 AU in all four panels are higher than for the 160 AU

position. The differences between the differential intensities are most prominent

for energies ≤1 GeV and increase with decreasing energy indicative of the wider

heliosheath. In the equatorial plane the TS effects are most prominent in the A<0

cycle judged by the amount and at what energies the spectra at 90AU and even at

60 AU exceed the LIS value. This “excess” effect is reduced when the heliopause

is moved further out. As a function of radial distance these effects are quite evi-

dent for the chosen energies, e.g. the 0.20 GeV intensities are lower at all radial

distances.

INTERSTELLAR-TERRESTRIAL RELATIONS 377

Figure 11. Left panels: Computed differential intensities for galactic protons with α=10◦as a

function of kinetic energy for both polarity cycles, at 1AU, 60 AU, and the TS location (bottom to

top) in the equatorial plane (θ=90◦). Right panels: The corresponding differential intensities as

function of radial distance for 0.016, 0.2 and 1.0 GeV, respectively at the same latitude as in the left

panels. The TS is at 90 AU, as indicated, with the LIS speciﬁed at 120 AU (red lines) and 160AU

(black lines), respectively (from Langner and Potgieter, 2005a).

The “barrier” effect, the sharp drop in intensities over relatively small radial

distances in the outer heliosphere, becomes more prominent (covers a larger dis-

tance) when the heliopause is moved outward, especially during the A>0 cycles

when it happens over an extended energy range. The width of this modulation “bar-

rier” is dependent on the modulation conditions (diffusion coefﬁcients) close to the

outer boundary. For energies ≤200 MeV most of the modulation happens in the

heliosheath for both cycles, but especially because of the barrier covering relatively

small distances near the heliopause during the A>0 cycle. For CR intensities at

Earth the position of the TS proved to be not as signiﬁcant as the position of the

heliopause (Langner and Potgieter, 2004, 2005a,b).

9.4. CHANGES IN THE TERMINATION SHOCK COMPRESSION RATIO

The modulation obtained with the TS model with respect to the carbon LIS, as a typ-

ical example of the modulation of CR nuclei, is shown in the left panels of Figure 12

(Potgieter and Langner, 2004) for boron spectra, with a detailed discussion. The

378 K. SCHERER ET AL.

Figure 12. Left panels: Computed spectra for galactic carbon for both polarity cycles, at 1 AU, 60 AU

and 90 AU (bottom to top) in the equatorial plane. Right panels: Corresponding differential intensities

as a function of radial distance for 0.016, 0.2 and 1.0 GeV, respectively. The TS is at 90AU, as

indicated, with the LIS (blue lines) at 120 AU, with α=10◦and 75◦, respectively. Solutions without

a TS are indicated by black lines for the same radial distances and energies. Note the scale differences

(from Potgieter and Langner, 2004).

INTERSTELLAR-TERRESTRIAL RELATIONS 379

spectra and differential intensities are now also shown for α=75◦, for a model with

a TS and then without a TS, respectively. The modulation of C is clearly affected by

incorporating a TS. Note the manner in which the modulation changes from solar

minimum to moderate solar maximum activity and how the effects increase with

solar activity.

The effect of the TS on the modulation of C is for the larger part of the helio-

sphere signiﬁcant; it drastically decreases the intensities at lower energies (e.g. at

100 MeV/nuc) but increases it at higher energies (e.g. at 1GeV/nuc), as the lower

energy particles are being accelerated to higher energies. The adiabatic spectral

slopes are also altered in the process. The intensities at low energies are, therefore,

lower at Earth with the TS than without it in the A>0 polarity cycle, but not

for the A<0 cycle, because in this cycle the low energy particle population are

supplemented by the modulation of the larger population of high energy particles

at the TS, emphasizing the role of particle drifts. These differences can be seen

at Earth, and it is clear that a change in the compression ratio will have conse-

quences on the intensities at Earth. The differences between the two approaches

are most signiﬁcant with E≤100 MeV/nuc and r≥60 AU. Similar results were

found for CR protons and helium (He) (Langner et al., 2003; Langner and Potgieter,

2004).

9.5. MODULATION IN THE HELIOSHEATH

Also shown in the right panels of Figure 12 is that the modulation in the heliosheath

is an important part of the total modulation for C. Barrier type modulation is caused

by the heliosheath as was previously mentioned for galactic protons. It differs

signiﬁcantly for different energies, from almost no effect at high energies to the

largest effect at low energies, and with changes in HMF polarity cycle. The TS

plays in this regard a prominent role and can be regarded as a main contributor to

the barrier modulation effect at low energies. For a discussion of these effects for

protons, see Langner et al. (2003).

In Figure 13 the computed modulation to take place in the heliosheath, between

rband rs, is compared to what happens between rband 1 AU (LIS to Earth) and

between rsand 1 AU (TS to Earth). This comparison is emphasized by showing in

this ﬁgure the intensity ratios jLIS/j1,jLIS/j90 and j90/j1for B and C in the equatorial

plane for both polarity cycles with α=10◦. Note that for a few cases the ratios

become less than unity. Obviously, all these ratios must converge at a high enough

energy where no modulation takes place. According to this ﬁgure a signiﬁcant level

of modulation occurs in the heliosheath when A>0 with E≤200 MeV/nuc for

solar minimum (α=10◦). This is also true for A<0 but at a somewhat lower

energy. The level of modulation in the heliosheath decreases signiﬁcantly for E>

200 MeV/nuc in contrast with that of j90/j1for the A<0 cycle but to a lesser extent

for the A>0 cycle. From this it is clear that the heliosheath can play an important

380 K. SCHERER ET AL.

Figure 13. Intensity ratios jLIS/j1,jLIS /j90 and j90/j1(120 to 1 AU, 120 to 90 AU and 90 to 1 AU)

for boron and carbon as a function of kinetic energy in the equatorial plane with α=10◦; left panels:

for A>0, right panels for A<0. Interstellar spectra are considered local interstellar spectra (LIS)

at 120 AU and the TS is positioned at 90 AU. Note the scale differences (from Potgieter and Langner,

2004).

role for CR intensities at Earth, because at low energies most of the modulation of

CRs happens in this region.

9.6. CHANGES IN THE LOCAL INTERSTELLAR SPECTRUM

By comparing the energy spectra and radial dependence of the intensities for the

chosen energies in Figure 14 it can be seen that the modulation for B and C differs as

a function of radial distance. This is primarily because of the much steeper spectral

slope for the local interstellar spectrum (LIS) below 100 MeV/nuc for B compared

to C. This implies that the C modulation should have a much larger radial gradient

below ≈200–500 MeV/nuc in the outer heliosphere than for B. The spectral slopes

at low energies change with increasing radial distance as the adiabatic energy loss

effect gets less. Despite the rather ﬂat LIS for C below 100 MeV/nuc, the modulated

spectra at 1 AU look very similar for B and C, a characteristic of large adiabatic

“cooling”. The computed differential intensities for B and C are also shown at Earth

INTERSTELLAR-TERRESTRIAL RELATIONS 381

Figure 14. Top and middle panels: Computed differential intensities for boron (top) and carbon

(middle) at Earth for both polarity cycles compared to observations. Computations are done with the

IS for boron and carbon by Moskalenko et al. (2002) (left panels) and by Moskalenko et al. (2003)

(right panels). Bottom panel: B/ C as a function of kinetic energy for both polarity cycles with α=10◦

compared to corresponding observations. The computations are compared to the interstellar B/ C at

120 AU as a reference (blue lines). The data compilation is taken from Moskalenko et al. (2003) (from

Potgieter and Langner, 2004).

for both polarity cycles compared to B and C observations. These comparisons are

shown for two sets of LIS as mentioned in the ﬁgure caption. This second approach

contains a new, local component to spectra of primary nuclei and is probably closer

to what can be considered a LIS. The B to C ratios as functions of kinetic energy

are also shown compared to the observations, with the interstellar B/C at 120AU

as a reference (Potgieter and Langner, 2004).

As noted before the spectral shapes at 1 AU are very similar for B and C owing to

adiabatic energy loses between 120 AU and 1 AU. This causes a steady B/C below

382 K. SCHERER ET AL.

200–300 MeV/nuc. This ratio will systematically decrease with increasing radial

distances to eventually coincide with the LIS ratios. However, the spectral slopes

at 1 AU are slightly different for the two polarity epochs owing to the different

particle drift directions during the two magnetic polarity cycles. This causes the

well-known crossing of the spectra for successive solar minima, seen here between

100–200 MeV/nuc (Reinecke and Potgieter, 1994). The LIS of Moskalenko et al.

(2002) is most reasonable above 500MeV/nuc, although a more reasonable ﬁt

is obtained below 300 MeV/nuc by using the second LIS of Moskalenko et al.

(2003), which from 200 MeV/nuc to ≈4 GeV/nuc is higher than the previous one.

Unfortunately these modiﬁed LIS produce modulated spectra that do not represent

the observations well between ≈200 MeV/nuc and ≈1 GeV/nuc for both B and

Figure 15. A comparison of the two sets of interstellar spectra for boron (black lines) and carbon

(blue lines); lower values (LIS1; solid lines) by Moskalenko et al. (2002), higher values (LIS2-dashed

lines) by Moskalenko et al. (2003). The latter contains a local interstellar contribution to spectra

of primary nuclei as proposed by Moskalenko et al. (2003) and is probably closer to what can be

considered a LIS for carbon. In the lower panel the corresponding ratios (LIS2/LIS1) are shown as a

function of energy/nuc (from Potgieter and Langner, 2004).

INTERSTELLAR-TERRESTRIAL RELATIONS 383

C, with the ﬁt to the low-energy B/C still in place. This aspect is emphasized

in Figure 15 by showing the two sets of LIS, with the changes introduced by

Moskalenko et al. (2003), and the corresponding ratios as a function of energy.

These differences in the intensities at Earth, caused by different local interstellar

spectra, are therefore a clear indication that even small changes in the spectral shape

of the LIS can play an important role in the measured intensities of CRs at Earth, if

it would occur at high enough energy not to be hidden by adiabatic energy losses.

Changes in the heliospheric structure and in the heliosheath can play a mea-

surable part on the CR intensities at Earth. Qualitatively the modulation for B, C,

protons, and He are similar, with certainly quantitative differences. Although these

studies were done with a different compression ratio and position for the TS than

what was recently observed, the results will qualitatively stay the same. Even though

each of the discussed changes cause only small effects at Earth, which alone may

seem insigniﬁcant, it is clear that a superposition of changes, strongly dependent

on energy and on the HMF polarity cycle, may cause a signiﬁcant effect on the

intensities of CRs at Earth.

.

Part V

Effects of the Dynamical Heliosphere

.

INTERSTELLAR-TERRESTRIAL RELATIONS 387

10. 3D (Magneto-)Hydrodynamic Modelling

For quantitative studies of interstellar-terrestrial relations it is necessary to have a

model of a three-dimensional heliosphere, which is immersed in a dynamic local

interstellar medium. There are at least two reasons why such model should be three-

dimensional. First, a comprehensive and self-consistent treatment of the cosmic ray

transport must take into account the three-dimensional structure of the turbulent

heliospheric plasma and, second, the heliosphere can be in a disturbed state for

which no axisymmetric description can be justiﬁed. The present state-of-the-art

of the modeling of a dynamic heliosphere with a self-consistent treatment of the

transport of cosmic rays is reviewed in Fichtner (2005). As is pointed out in that

paper, the major challenge is the development of a three-dimensional hybrid model.

This task requires, on the one hand, the generalisation of the modeling discussed in

the following section and, on the other hand, the formulation of three-dimensional

models of the heliospheric plasma dynamics. The fundamental equations are dis-

cussed in Section 5 for both the cosmic ray transport as well as the MHD-ﬂuid

equations. In the following we discuss different approaches based on these funda-

mental Equations (1) to (3).

10.1. 3D MODELS WITHOUT COSMIC RAYS

Several three-dimensional models without cosmic rays have been presented. Fol-

lowing early work, which is reviewed in Zank (1999), Fichtner (2001), Fahr (2004),

and Izmodenov (2004), nowadays sophisticated MHD models have been developed,

see Washimi et al. (2005), Opher et al. (2004), Pogorelov (2004), Pogorelov et al.

(2004), and Pogorelov and Zank (2005). Their results are not discussed further,

because this review is focused on models containing cosmic rays.

10.2. 3D MODELS WITH COSMIC RAYS

So far, a truly dynamical, three-dimensional model for the large-scale helio-

sphere that also includes self-consistently a sophisticated cosmic ray trans-

port comprising fully anisotropic diffusion and drifts is still missing. For the

existing three-dimensional models including the cosmic ray transport rather over-

simplifying approximations had to be made. Common to all these models is their

pure hydrodynamical character, i.e. the fact that the heliospheric magnetic ﬁeld is

included only kinematically. Further simpliﬁcations depend on the type of approach

being used.

10.2.1. Models Based on a Kinetic Description of Cosmic Rays

Those models that include the kinetic cosmic ray transport equation, are not self-

consistent by prescribing the heliospheric plasma structure. This has been done,

388 K. SCHERER ET AL.

Figure 16. The (normalized) spatial distribution of anomalous protons with 31 MeV for the no-

drift case (corresponding to solar activity maximum) in a non-spherical heliosphere. Both cuts are

containing the upwind-downwind axis (horizontal solid line): the left panel is a cut perpendicular to

the symmetry axis of the heliospheric magnetic ﬁeld and the right panel is a cut containing it. The

outermost dashed line indicates the heliospheric shock in these planes. The contours have, from the

shock inwards, the values 1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, 0.05, 0.01 (taken from Sreenivasan

and Fichtner, 2001).

in extension of earlier work, by Sreenivasan and Fichtner (2001), who treated the

kinetic, drift-free transport of anomalous cosmic rays within a three-dimensionally

structured stationary heliosphere with a Parker ﬁeld and excluded the region beyond

the asymmetric termination shock. Despite these simpliﬁcations the resulting spatial

cosmic ray distribution (see Figure 16) gives a ﬁrst impression of what one should

expect quantitatively for the outer heliosphere.

The ﬁgure shows the spatial distribution of anomalous protons with a kinetic

energy of 31 MeV for a non-spherical heliospheric shock (outermost dashed line)

in the ‘equatorial’ plane (left), which is perpendicular to the symmetry axis of the

heliospheric magnetic ﬁeld and contains the upwind-downwind axis (horizontal

solid line), and in a meridional plane (right) containing both the symmetry axis of the

heliospheric magnetic ﬁeld and the upwind-downwind axis. The shock is elongated

in the polar and the downwind direction by factors of 1.3 and 1.5, respectively, as is

found with the above-mentioned (M)HD studies. The resulting spectra are compared

with those for a spherical heliosphere in Figure 17.

From the ﬁgures it is obvious that the three-dimensional structure of the he-

liosphere is manifest in the spatial and spectral distributions of anomalous cosmic

rays only in the outer heliosphere beyond