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Interstellar-Terrestrial Relations: Variable Cosmic Environments, The Dynamic Heliosphere, and Their Imprints on Terrestrial Archives and Climate


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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.
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E. FL ¨
1Institut f¨
ur theoretische Physik, Weltraum- und Astrophysik, Ruhr-Universit¨
at Bochum,
D-44780 Bochum, Germany
ossische Anstalt f¨
ur Wasserversorgung, Abwasserreinigung und Gew¨
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,
8Hebrew University Jerusalem, 91904 Israel
9Ottawa-Carleton Geoscience Center, University of Ottawa, 140 Louis Pasteur, Ottawa, Canada
K1N 6N5
(Author for correspondence: E-mail:
(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
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
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,
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.
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.
Possible astronomical or geological explanations of the different periods observed in Figure 1
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.
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
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
The transport of cosmic rays is calculated by solving the transport equation Parker
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:
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
(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.
The model of the dynamical heliosphere is in most cases based on the following
(normalized) magneto-hydrodynamical equations
ρvv+pth +1
e+pth +1
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
Bis the magnetic field and ˆ
Ithe unity tensor. The terms Qρ,
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:
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,
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
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.
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
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
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
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).
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:
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 vlmaps 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 vlmaps 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.
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
1kpc1, while the other half are spread between 4 and 5 km s1
kpc1. In fact, one analysis revealed that both p=5 and 11.5km s1kpc1
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
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 s1
kpc1and 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 s1kpc1. 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 s1kpc1.
For comparison, a combined average of the 7 previous measurements of the
1kpc1range, which appears to be an established fact for both the
Perseus and Sagittarius-Carina arms, gives p=11.1±1kms
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.
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
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).
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 a5×107yr), and with the same iron group
classification, were replaced by a single effective meteor with the average exposure
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
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
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:
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
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:
xDcos θf
ln p(7)
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
C: Continuity of differential energy flow at the shock, i.e.:
ln p3κdf
ln p3κdf
Far from the shock one may assume unmodulated spectra with asymptotic so-
lutions given by f±(p,x→±)=f±∞(p). Downstream of the shock (x0) it
is expected that fis independent on x, i.e.: f=f+∞(p).
Upstream of the shock (x0), however, fmust be expected to be modulated,
i.e. given by:
f=f−∞(p)+(f+∞(p)f−∞(p))exp ux
The full solution for f+∞(p) matching all the above requirements then is given by
the following formal solution:
where the power index qis given by the expression: q=3s/(s1) with the shock
compression ratio sgiven by: s=u/u+.
Given the spectral distribution far upstream of the shock in the form f−∞(p)
pwith qthen Equation (9) yields:
Assuming, on the other hand, that f−∞(p)p, with =0qfor pp0
only, and with qfor pp0,then Equation (9) in contrast gives:
with the solutions for
which finally evaluates to:
and simply is of the twin-power law form:
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:
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.
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 =EB0has the dimension [cm5sec2],
one can thus introduce the following self-similar normalization:
The special point Rsof the shock front location with the normalized value ξsas
function of time hence behaves like:
As consequence from the above relation one easily derives the expansion velocity
of the SN shock front by:
dt =2
The upstream Mach number of the SN shock is permanently decreasing with
time after the explosion event according to:
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
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:
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:
exp u
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 uand 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 fhas 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):
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: uf1
s11p2dp =ε(p0)un0(22)
The above expression can finally be evaluated with the distribution function given
by Equation (20):
3(s2s)f0p2dp =s2+6s1
s1dx =ε(p0)n0(23)
which delivers for the quantity C:
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
and Fahr, 1995, 2000). For this to happen the following relation simply needs
to be fulfilled:
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;
exp(x2)dx =11
where x2
s. Here the following nota-
tions have been used: g(s)=(1 (1 s1)1/2) with the Mach number of the
upstream plasma defined by M2
This finally delivers for Cthe expression:
This result expresses the fact that the absolute value of fgiven 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
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):
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:
where M1(t) denotes the upstream Mach number depending on SN shock evolution
time tand is given by:
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
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
and thus the average GCR spectrum should be obtainable by the following
Xmaxtmax Xmax
×exp U1(XRx(t))
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
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
turbulences which act as scattering centers for GCRs bouncing to and fro through
the shock. From Equation (30) one thus derives:
tmax 2ESN
which for values given by Hartquist and Morfill (1983) (i.e. ESN =1051erg; ρ1/m=
10 cm3;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
tmin 2ESN
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
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.
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
r=const =rκ
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:
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
rand |fi|Ra=|fa|Ra(37)
where κaand κidenote 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:
ln Ra
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:
where q=3s/(s1) 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−χn1p2, for details see Lerche and Schlickeiser (1982a,b,c). The second
identity follows with q4 and n110 cm3and 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:
from which with the help of Equation (36) one furthermore derives
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
ln Ra
ln r
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)
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=
0.1κicorresponding 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
;ξ=aP +bP
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.
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
aa=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
ln Ra
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
8. Propagation of Cosmic Rays Inside the Heliosphere
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
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
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
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)
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.
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.
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 10at solar minimum to
75at solar maximum (theoretically 90is 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
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
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.
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.
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.
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)
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.
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).
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.
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
Figure 11. Left panels: Computed differential intensities for galactic protons with α=10as 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).
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
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=10and 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).
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 E100 MeV/nuc and r60 AU. Similar results were
found for CR protons and helium (He) (Langner et al., 2003; Langner and Potgieter,
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 E200 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
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,
role for CR intensities at Earth, because at low energies most of the modulation of
CRs happens in this region.
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
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
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).
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
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).
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.
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,
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