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Heliospheric Physics and Cosmic Rays

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Lecture notes

Fall term 2003

Prepared by Kalevi Mursula and Ilya UsoskinUniversity of Oulu

Lectured in 2003 by K. Mursula

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Preface

This is the second time that a course under the title “Heliospheric Physicsand Cosmic Rays” is lectured at the University of Oulu. The course isstill in an evolutionary phase and, inevitably, limited to basics of the fields.The topics selected out of the wide range of cosmic ray physics reflect thebiased view of the authors. Moreover, we would like to note that muchof the material presented here is still a subject of discussion and intensiveresearch. Therefore, the point of view presented here is not always theonly one possible. While we try to emphasize a physically consistent andgenerally accepted view, other interpretations may also be taken on sometopics.

Chapter 1

Introduction to Cosmic Rays

1.1 Milestones of cosmic ray research

The study of cosmic rays has a long story. The first experimental discovery

related to cosmic rays was made more than 100 years ago.

By measuring the accummulated static charge, C.T.R. Wilson dis-

covered in 1900 the continuous atmospheric ionisation. It was then (erro-

neously) believed to be only due to the natural radioactivity of the Earth.

In order to check that, Victor Hess (Nobel Prize 1936) from the Univer-

sity of Vienna launched in 1912 an electrometer (a charge collector) aboard

a balloon to the altitude of 5 km (Fig. 1.1).

He discovered that the ionization rate first decreased up to about 700 m

as expected, but then increased with altitude showing thus an outer space

origin for ionisation. During subsequent experiments, Hess showed that the

ionising radiation was not of solar origin since it was similar for day and

night time. The term ”cosmic radiation” became common. It was then

believed that the radiation consists of γ-rays . However, this assumption

was soon questioned, and in 1925 Robert Millikan from Caltech, USA,

introduced the term cosmic rays.

1

1.1. MILESTONES OF COSMIC RAY RESEARCH 3

The later developments showed that cosmic rays (CR) consist of charged

particles:

1928: J. Clay discovered that the ionisation rate increased with latitude,

suggesting that the sources of ionisation were charged particles de-

flected by the geomagnetic field.

1929: Using a newly invented cloud chamber, D. Skobelzyn observed the

first ghostly tracks left by cosmic rays.

1929: Bothe and Kolhorster verified that the cloud chamber tracks are

curved. This showed that CR are charged particles.

1937: Seth Neddermeyer and Carl Anderson discovered muons (first

erroneously called µ mesons) in cosmic rays. Particle physics devel-

oped and used cosmic rays as the main experimental method until

the advent of particle accelerators in the 1950’s.

1938: T.H. Johnson et al. discovered that the ionisation rate increased

from east to west viewing angle, indicating that the ionisation was

due to positively charged particles (correctly assumed to be protons).

(Charged particles drift in the Earth’s inhomogeneous magnetic field

due to the so called gradient drift).

1938: Pierre Auger, who had positioned particle detectors high in the

Alps, noticed that two detectors located many meters apart both

detected the arrival of particles exactly at the same time. Auger

had discovered “extensive air showers”, showers of secondary nuclei

produced by the collision of a primary high-energy particle with air

molecules (Fig. 1.2). In this way, changing the distance between

the detectors, he could observe CR with energies up to 1015 eV - ten

million times higher than reached by then in laboratory experiments,

and still several orders of magnitude above the highest laboratory

energies reached today.

4 CHAPTER 1. INTRODUCTION TO COSMIC RAYS

Figure 1.2: Artist’s view of an atmospheric shower.

1.1. MILESTONES OF COSMIC RAY RESEARCH 5

28 Feb. 1942: First detection of solar cosmic rays as a high increase in

ionisation chambers connected with a flare and radio disturbances.

1948: Phyllis Frier et al. discovered He nuclei and heavier elements in

CR.

May 11, 1950: U.S. Naval Research Lab fired the first research rocket to

collect cosmic ray, and air pressure and temperature data.

1959: Konstantin Gringauz flew “ion traps” on the Soviet Luna 2 and

3 missions. The NASA Explorer VII satellite was launched into a

low-Earth orbit with a particle detector.

1977: The Voyager 1 and 2 spacecraft were launched to an interstellar orbit

carrying, e.g., cosmic ray detectors.

1977-1982: Bogomolov et al. made a series of balloon experiments and

found antiprotons in CR.

1990: As the first spacecraft, Ulysses probe was launched into a high he-

liospheric latitude orbit to study the 3D picture of solar wind and

cosmic rays. So far, it has twice passed close to both the two solar

poles.

Cosmic rays have provided and still provide a unique opportunity to study

nuclear and particle physics in the energy range unreachable in present or

near-future laboratories. It is hard to overestimate the contribution of cos-

mic ray studies (including neutrino and γ-ray observations) to nuclear and

particle physics.

The astrophysical aspects of cosmic rays are another important connec-

tion. CR studies have nourished several theoretical investigations, such as

the theory of novas and supernovas, magnetohydrodynamics (MHD) and

other plasma theories in astrophysics. In these areas, the following mile-

stones can be noted:


1934: W. Baade and F. Zwicky suggested that supernova explosions (Fig.

1.3) are the sources of cosmic rays.

1949: Enrico Fermi suggested that the cosmic rays are accelerated in their

interactions with magnetic field irregularities (so called 2nd order

Fermi acceleration).

1977: Ian Axford et al. suggested that the cosmic rays are accelerated by

first-order Fermi acceleration in supernova shocks in a hot interstellar

medium.

In addition, works of other famous scientists, e.g., such asAlfven, Ginzburg,

Parker, and Zeldovich should be acknowledged in the development of CR

physics.

1.2 What is a Cosmic Ray?

Cosmic ray is not a ray, but a particle. (A small fraction of primary cosmic

rays consist of energetic γ-quanta and neutrinos but there are left beyond

the scope of this course.) Most cosmic rays are ionised atoms, ranging from

the proton up to the iron nucleus and even beyond to heavier nuclei. Cosmic

rays originate from space, being produced by a number of different sources,

such as the Sun, other stars, and more exotic objects, such as supernova

and their remnants, neutron stars and black holes, as well as active galactic

nuclei and radio galaxies.

Most cosmic ray particles are travelling very close to the speed of light.

The most energetic CR particle ever observed had an energy of about 150

Joules, equivalent to the kinetic energy of a fast baseball. The number

density of primary CR integrated over energy (> 100 MeV/nucleon) is about

NCR ≈ 10−10 cm−3 in the vicinity of the Earth. The total energy density ofprimary CR particles is WCR ≈ 1 eV cm−3.

The Earth’s atmosphere and the geomagnetic field protect us from be-

1.2. WHAT IS A COSMIC RAY? 7

Figure 1.3: Cygnus loop supernova remnant. Image by ROSAT.


ing excessively exposed to these particles. As a cosmic ray enters the at-

mosphere, it will collide with an atmospheric particle (usually a nitrogen or

oxygen molecule), generating a series of secondary particles.

It is common to separate three kinds of cosmic rays:

Galactic cosmic rays (GCR) originate far outside of our solar system.

They are the most energetic CR particles with the energy extending

up to 1021 eV. Composition is mostly protons with ≈ 7− 10% of He

and ≈ 1% of heavier elements. The source of the very energetic GCRis not exactly known. The GCR flux in the solar system is modulated

by solar activity: enhanced solar activity shields the Earth from these

particles.

Solar cosmic rays (SCR), also called solar energetic particles, originate

mostly from solar flares. Coronal mass ejections and shocks in the

interplanetary medium can also produce energetic particles. SCR

particles have energies typically up to several hundred MeV/nucleon,

sometimes up to a few GeV/nucleon. SCR composition is roughly

similar to GCR: mostly protons, ≈ 10% of He, < 1% of heavier

elements. During strong solar flares that are optimally located on

the Sun, the flux of CR at the Earth can increase by a few hundred

percent for hours/days because of the increase of SCR. This is called

a Solar Particle Event.

Anomalous cosmic rays (ACR) originate from the interstellar space be-

yond the heliopause. We will discuss later the mechanism of ACR

production. The composition of ACR is quite different from GCR

and SCR, including, e.g., more helium than protons, and much more

oxygen than carbon.

Chapter 2

Galactic Cosmic Rays

2.1 Composition

About 90% of the cosmic ray nuclei are hydrogen nuclei (protons), next

common are helium nuclei (α-particles), and all other elements make up

only about 1%. Within this one percent there are also very rare elements

and isotopes. These species require large detectors in order to collect enough

particles to observe their “fingerprint”.

For instance, the HEAO 3 (High Energy Astrophysical Observatory)

Heavy Nuclei Experiment, launched in 1979, collected only about 100 cos-

mic rays with charges between 75 and 87 during almost 1.5 years of mea-

surements. It was one of the biggest space borne astroparticle instruments.

Good measurements require a large instrument but, unfortunately, the cost

of space instruments increases greatly with the size (mass) of the instru-

ment. Ground-based experiments may have a much larger effective area

and a greatly higher sensitivity but they cannot measure the chemical com-

position of CR because of atmospheric shielding.

All galactic cosmic ray particles are fully ionised, i.e., consist of nuclei

only. The violent processes accelerating charged particles strip off the elec-

9

10 CHAPTER 2. GALACTIC COSMIC RAYS

Table 2.1: Relative and absolute CR abundance (E > 2.5GeV/nuc, [9])particlegroup

nucleuscharge

integral parti-cle intensity

number of particles per 105 protons

m−2 s−1 sr−1 in CR in the Universe

protons 1 1300 10000 10000helium 2 94 720 1600L 3-5 2 15 10−4

M 6-9 6.7 52 14H 10-19 2 15 6VH 20-30 0.5 4 0.06SH > 30 10−4 10−3 7 · 10−5electrons 1 13 100 10000antiprotons 1 > 0.1 5 ???

trons from atoms, leaving isolated nuclei and electrons.

The abundance of primary CR is essentially different from the standard

abundance of nuclei in the Universe (Table. 2.1). The difference is biggest

for light nuclei (L = Li, Be, B) which are mainly produced by CR collisions

with interstellar matter in the Galaxy. The relative abundance of different

elements in cosmic rays is shown also in Fig. 2.1.

Among normal matter nuclei, there are also some antimatter nuclei. Nu-

merous balloon experiments devoted to search for antimatter in space took

place since 1970s. They have collected, in total, several hundred antipro-

tons. A big astroparticle experiment AMS (Alpha Magnetic Spectrometer)

was lunched onboard the Space Shuttle Discovery and flew during 10 days in

June 1998 (Fig. 2.2). It collected about 200 antiprotons with energy above

1 GeV. According to the standard theory, the antiprotons do not originate

in the birth of the universe but were produced inside the Galaxy in nu-

clear collisions of the CR particles with the interstellar matter. However,

some ideas of possible extra-galactic origin of antiprotons have also been

presented. Unfortunately, data collected so far do not allow to distinguish

2.1. COMPOSITION 11

Figure 2.1: Relative abundance of elements in cosmic rays and in the solarsystem.


reliably between the alternative hypotheses.

So far, not a single antihelium nucleon (not to speak about heavier anti-

nuclei) has been detected in CR although the sensitivity of AMS was just

high enough to catch the theoretically expected 1 antihelium nucleon during

the 10-day flight. This gives an upper limit for the He/He ratio of 1.1·10−6.In a few years a new bigger detector, AMS-2, is to be installed onboard the

International Space Station for years. Hopefully, it will resolve the above

mentioned questions related to antinuclei.

2.2 Energy spectrum

The differential energy spectrum of GCR is based on measurements from

different instruments covering the energy range from ≤ 109 to ≥ 1020 eV.(The highest energy of a CR particle detected so far was 1021 eV.) Actually,

the spectrum of primary CR particles below some 10 GeV/nucleon can not

be directly measured because of solar modulation.

The differential energy spectrum (Fig. 2.3) shows the CR flux (number

of cosmic ray particles passing through a unit area surface in a unit time

from a unit space angle per energy unit) at different energies. The unit is

particles per cm2 s sr GeV . The graph is double logarithmic; a straight line

indicates that the number of cosmic rays with some energy is proportional

to the energy to some power.

Note that the energy spectrum in Fig. 2.3 is not exponential. Ac-

cordingly, the GCR spectrum is harder than the thermal energy spectrum

(Gaussian distribution). This means that particles have experienced consid-

erable (nonthermal) acceleration. (The different acceleration mechanisms

will be discussed later).

As a first approximation, the flux of energetic CR can be considered to

be isotropic near the Earth.

2.2. ENERGY SPECTRUM 13

Figure 2.2: Inflight view of the Discovery shuttle and the AMS detector.


Figure 2.3: Differential energy spectrum of galactic cosmic rays.


It is therefore reasonable and common to approximate the differential

energy spectrum of GCR with the power law:

I(E) ∝ E−γ cm−2s−1sr−1GeV −1, (2.1)

where γ is called the spectral index, which is the main characteristics of the

spectrum.

The ultra high energy particles are very rare and can be detected as

extensive air showers on the ground. In this method, the atmosphere is a

major part (moderator) of the detector. (This will be discussed later in

more detail.)

Figure 2.4 shows the differential energy spectra for different GCR species.

One can see that the shapes of the spectra are fairly similar to each other,

which indicates that the particles were generated/accelerated in similar

processes.

The spectrum of GCR as measured in the vicinity of the Earth can be

divided into the following parts (see Figure 2.3):

• Particles with energy below about 20-50 GeV are subject to solar mod-ulation. Here the spectrum deviates from the power law.

• Within the range 1010 − 1015 eV, the spectrum is a power law with

the spectral index γ ≈ 2.7.

• Around 1015 eV, the spectrum changes and becomes steeper, with

γ ≈ 3.1 This is known as the “knee” of the spectrum. Actually, thechange is very small, almost imperceptible when viewed in normal

units. However, the number of cosmic rays observed at these energies

is large enough to make the measurements reliable with great accuracy.

The knee is more visible in the enlarged and scaled view of Fig. 2.5.


Figure 2.4: Differential spectra of some spices of GCR near the Earth.


Figure 2.5: The differential GCR spectrum multiplied by E2.5.


The knee is believed to arise because the acceleration mechanism in su-

pernova shocks becomes less efficient at this energy, probably because

of particle’s gyroradius exceeds the thickness of the shock.

• Within the range of 1015 − 1020 eV, the spectrum is a power law with

γ ≈ 3.1.

• The spectrum becomes flatter (harder) again at around 1020 eV but

the data are rather poor in this range to estimate this change reliably.

This change is known as the “ankle” of the spectrum. The source of the

particles above this energy range (the cause of additional acceleration)

is not clear so far.

2.3 Origin of Cosmic Rays

Any theory of GCR acceleration must account for the above described energy

spectrum.

2.3.1 Ultra High Energy Cosmic Rays

The standard view is that some cosmic rays are mainly accelerated in our

Galaxy (Milky Way) and some are accelerated outside it. The origin of

the very (ultra) high energy cosmic rays above the knee is still a mystery.

Because of low statistics at such high energies it is hard, e.g., to tell exactly

where they come from. For example, cosmic ray particles with energies

greater than 1019 eV hit the Earth at a rate of one particle per square

kilometre per century (Fig. 2.3).

So far, only a few particles with energy exceeding 1020 eV have been de-

tected. For a number of reasons, it is suspected that the cosmic rays above

the ankle are of extragalactic origin, perhaps generated in the cores of Ac-

tive Galactic Nuclei, in powerful radio galaxies, or by the speculated cosmic

strings. These sources can offer tremendous amounts of energy needed to

2.3. ORIGIN OF COSMIC RAYS 19

accelerate particles to such high energies. However, a direct correlation has

not yet been established. As more sensitive detectors gather more evidence,

scientists will have a better picture of where these extraordinarily high en-

ergy particles are generated.

2.3.2 High Energy Cosmic Rays

The particles below the ankle are generally thought to be mainly produced

in our Galaxy. Furthermore, there are reasons to believe that at least up to

about 1014 eV, if not up to the knee or even to the ankle, most CR particles

are accelerated in the shocks of supernova remnants (SNR). In this model,

particles are scattered across the moving shock fronts of a SNR, gaining

energy at each crossing (Fermi shock acceleration; see later).

Until recently, evidence supporting this idea was only circumstantial,

based on theory rather than on observations. It seemed theoretically rea-

sonable that SNR shocks could accelerate particles to the desired energies.

The kinetic energy released in a supernova explosions is more than enough

to account for the galactic cosmic rays at least up to 1015 eV. Supernovae

are fairly common and occur throughout the Galaxy, so it is reasonable that

they could be responsible for these energetic cosmic rays. However, even

more direct evidence is found for shock acceleration of particles in SNR.

X-ray evidence for SNR acceleration of GCR

Synchrotron radiation is emitted when fast charged particles are moving

in the presence of magnetic fields. The magnetic field will force an energetic

particle to travel in a helical path thereby experiencing circular acceleration

and emitting radiation. It is known that strong magnetic fields exist near

and around SNR. So if there are fast charged particles, they should produce

synchrotron radiation which could be observed.


The energy of synchrotron radiation depends on the mass and energy of

the charged particles and the strength of the magnetic field. For the cosmic

ray energies observed on Earth and magnetic field strengths deduced from

radio measurements, cosmic ray synchrotron radiation should be in the X-

ray range. For instance, the X-ray source of the Crab nebula (Fig. 2.6) is

believed to be due to synchrotron radiation of electrons accelerated up to

1014 − 1015 eV. In addition, observations of high energy (10 MeV - 1000

MeV) γ-rays resulting from cosmic ray collisions with interstellar gas show

that most cosmic rays are confined to the disk of the Galaxy (Fig. 2.7),

presumably by its magnetic field.

2.3.3 Lifetime of GCR in the Galaxy

Because of the magnetic field of the Galaxy, cosmic rays are trapped in it for

a long time. During their travel inside the Galaxy (they spend most of the

time in the halo), they can gain more energy or lose energy, they can collide

with other particles, etc. Thus, their propagation in the Galaxy is diffusive.

Since they are not moving along straight lines, we cannot trace their origin

directly, and have to use indirect methods like synchrotron radiation in order

to study it.

How long are the cosmic rays trapped in the Galaxy? There are several

ways to estimate that.

• Collisions of cosmic ray nuclei with the interstellar matter (or with eachother) can produce lighter nuclear fragments, including radioactive

isotopes such as 10Be, which has a half-life time of 1.6 million years.

The measured amount of 10Be in cosmic rays implies that, on an

average, cosmic rays spend about 10 million years in the Galaxy.

• CR particles are lost in collisions. Assuming the mean free path of aCR particle before absorption to be several g/cm2, and keeping in mind


Figure 2.6: The ROSAT image of the Crab nebula.


Figure 2.7: A schematic side view of the Galaxy.


that the average density of the galactic disk is about 10−24 g/cm3, one

can estimate the distance traversed by the particle before absorption

to be of the order of 1024 cm, which corresponds to the time of about

1014 sec (or several millions of years) for relativistic particles. The

corresponding time for the Halo, where the density is 10−26 g/cm2 , is

of the order of 108 years.


Chapter 3

Acceleration of cosmic rays

The question of how, where and when CR are accelerated is most important

both for galactic, solar and anomalous cosmic rays.

How CR are accelerated to ultra high energies (UHE) is still a subject

of intensive study. At present, there is no standard theory to explain the

origin of these extremely relativistic particles.

However, in the lower part of CR energy spectrum, the theory is more

or less established.

Let us note that only charged particles can be accelerated. Thus, e.g.,

the energetic solar neutrons sometimes detected during solar flares are sec-

ondaries.

3.1 Fermi acceleration

In order to explain the origin of cosmic rays, Enrico Fermi (1949) suggested

an effective mechanism of particle acceleration. Fermi exploited the idea

of magnetic clouds moving in the interstellar medium (ISM). These clouds

can be rather large, several light years wide, with the density 10-100 times

higher than the average ISM density and an enhanced “frozen-in” magnetic

field. Such clouds are believed to occupy several per cent of ISM. When

25

26 CHAPTER 3. ACCELERATION OF COSMIC RAYS

Figure 3.1: Collisions of a charged particle with magnetic field.

a fast moving particle collides with a random irregularity of the field, the

particle can change its momentum, gaining or losing some energy.

Figure 3.1 shows two types of “collisions” or, rather, elastic scatterings,

leading to the reflection of a particle. The upper case is called the magnetic

mirror. In the lower case the particle is guided along sharply bent field lines.

Since the magnetic field is very stable and remains unchanged during the

time of scattering, the scattering of a particle with the field irregularity is

“kinetically equal” to the collision of a fast ball with a wall.

If the “wall” (i.e., the magnetic field irregularity) is moving (with the

velocity V ), the particle may gain or lose energy during such a reflection.

During a frontal (V towards the incoming particle) collision the particle

would gain energy, while it would lose it during an overtaking (V away

from the incoming particle) collision. (The motion of a particle scattered

on random magnetic irregularities inside the cloud can be considered as a

3.1. FERMI ACCELERATION 27

θ1

θ2

V

E1

E2

Figure 3.2: Sketch of a collision of a charged particle with moving magneticcloud.

random walk.) Note that since the probability of frontal collisions is higher

than that of overtaking, the whole particle population gains more energy

then loses energy.

Let us consider a fast moving particle with (laboratory frame) energy E1

entering a slowly moving magnetic cloud (Fig. 3.2). Assuming the particle

to be relativistic, i.e., E ≈ pc, one can obtain

E01 = γE1(1− βcosθ1) (3.1)

where β = Vc and γ = 1√

1−β2 are the relative speed β and Lorentz factor of

the cloud.

Primes denote quantities measured in the cloud rest frame and θ1 is the

angle between the speed vectors of the particle and the cloud.


When the particle comes out of the cloud, it has energy E02 and angle θ02with respect to the cloud velocity, both in the cloud rest frame. Going back

to the laboratory frame, one can obtain

E2 = γE 02(1 + βcosθ02) (3.2)

We assume that there are no collisions with the matter, only elastic scatter-

ing on the magnetic field irregularities. Therefore, the total energy of the

particle should be conserved in the rest frame of the moving cloud, E 01 = E 02.

Therefore,

E2 = γ2E1(1− βcosθ1)(1 + βcosθ02) (3.3)

and

E2 − E1E1

' ∆EE

=1− βcosθ1 + βcosθ02 − β2cosθ1cosθ

02

1− β2− 1. (3.4)

Since the particle motion inside the cloud is random, all θ02 have equal proba-

bility, resulting to < cosθ02 >= 0. Because of the movement of the cloud, the

probability of a particle to enter the cloud with cosθ1 is (for relativistic par-

ticle and slow cloud) proportional to c−V cosθ12c , leading to < cosθ1 >= −13β.

Averaging Eq. 3.4 over angles, one can obtain that

∆E

E=1 + 1

3β2

1− β2− 1 ≈ 4

3β2. (3.5)

Thus, the net energy gain (averaged per collision) is

dE ∝ β2 · E, (3.6)


where β = V/c is constant. Note, that the energy gain increases with the

particle’s energy. Thus, the energy attained by the particle after n collisions

is

E = Ei · exp(β2n) (3.7)

where Ei is the initial or “injection” energy of the particle. Let us assume

the average time between collisions to be τc, hence the number of collisions

during time interval t is n = t/τc, and the energy is

E(t) = Ei · exp(β2t

τc) = Eiexp(t/tc) (3.8)

where tc = τc/β2.

A particle may also be lost in inelastic collisions with the ISM or simply

leak out of the system. Let the mean time of that be tl. The probability of

a particle to survive until a time greater than t is

P (> t) = exp(−t/tl) (3.9)

Combining Eqs. (3.8) and (3.9) one can obtain an integral spectrum of CR

(the total number of particles with energy greater than E):

J(> E) = K ·E−α (3.10)

where α = tc/tl. This is the integral form of the differential energy spectrum

with a power law behaviour as discussed earlier. In this case the index of

the differential spectrum is

γ = α+ 1 (3.11)


Figure 3.3: Dependence of energy gain and loss upon proton’s energy.


-u 1

E1

E2

V=-u 1+u 2

downstreamupstream

Figure 3.4: Sketch of a collision of a charged particle with a moving shock.

However, particles also lose their energy by means of ionisation. The

comparison of gains vs. losses for protons is shown in Fig. 3.3. One can see

that, effectively, the Fermi acceleration mechanism has a threshold energy.

For protons, the threshold energy is about 200 MeV, for oxygen about 20

GeV and for iron as high as 300 GeV because the heavier ions have higher

ionisation losses. Thus, this mechanism cannot produce the similar shape

of differential spectra for different nuclei at these energies (see Section 2.2.

The above mechanism is called the 2nd order Fermi acceleration be-

cause the mean energy gain per collision is dependent on the mirror velocity

squared (Eq. 3.6). Bell (1978) and Blandford and Ostriker (1978) inde-

pendently showed that Fermi acceleration by supernova remnant shocks is

particularly efficient because the motions are not random. A charged particle

ahead of the shock front can pass through the shock and then be scattered

by magnetic inhomogeneities behind the shock (see Fig. 3.4). Assume a

large plane shock front moving with velocity −u1. The shocked gas flows


away from the shock with a velocity u2 relative to the shock front, and

|u2| < |u1|. Thus, in the laboratory frame the gas behind the shock movesto the left with velocity V = −u1 + u2. Eq. 3.4 applies also to this sit-uation with with β = V/c now interpreted as the velocity of the shocked

gas (“downstream”) relative to the unshocked gas (“upstream”). Since the

shock is planar, the probability of a particle to hit it with cosθ1 is propor-

tional to 2 cosθ1 (−1 ≤ cosθ1 ≤ 0) leading to < cosθ1 >= −2/3. Similarly,< cosθ02 >= 2/3. Therefore,

∆E

E=1 + 4

3β +49β

2

1− β2− 1 ≈ 4

3β (3.12)

One can see that this acceleration is more effective (β << 1) than the 2nd

order mechanism. The particle gains energy from this ”bounce” and flies

back across the shock, where it can be scattered by magnetic inhomogeneities

ahead of the shock. This enables the particle to bounce back and forth,

gaining energy each time. This repeated bouncing process is now called the

1st order Fermi acceleration because the mean energy gain is dependent on

the shock velocity only to the first power. The 1st order Fermi acceleration

(also called Fermi shock acceleration) is also used to explain the SCR and

ACR acceleration.

3.2 Magnetic pumping

This mechanism was first described by Alfven (1963).

Let us consider a particle with momentum p in a homogenous magnetic

field (to be called here H). If the field changes slowly, the perpendicular

and parallel components of the particle’s momentum with respect to the

magnetic field line obey the laws:

p2⊥H= const and p2k = const (3.13)

3.2. MAGNETIC PUMPING 33

These equations express the conditions of the adiabatic motion and the

conservation of the adiabatic invariant. They are valid if the typical scale

of magnetic inhomogeneities is greater than the gyroradius of the particle.

(Note that in the non-relativistic limit, the ratio of p2⊥/H becomes a constant

times the ratio W⊥/H = 12mv

2⊥/H, which is the non-relativistic magnetic

moment of a charged particle in a magnetic field.)

As long as Eq. (3.13) is valid, fluctuations of the field do not result in

energy gain. However, if the size of magnetic irregularities is small with

respect to the particle’s gyroradius, Eq. (3.13) does not have to be valid.

In such a case, the guiding center of the particle will be randomly (if the

irregularities are random) moved to another magnetic field line. Thus, the

particle’s guiding center will perform a random walk (scatter) at the small

irregularities.

Averaging over a big ensemble of particles moving through a field with

small size irregularities, one can assume an equipartition of the particle’s

momentum over the three degrees of freedom (one parallel, two perpendic-

ular to the magnetic field):

p2⊥ =2

3p2o and p

2k =

1

3p2o (3.14)

In order to obtain this average momentum distribution, particles must spend

a long time (with respect to the interval between scatterings) in the field.

Let us consider a low density flux of particles traversing a region with

field H to a neighboring region with field k · H and back. Both regions

contain small-size irregularities. The scheme of the changing H is shown in

Fig. 3.5 for k > 1.

The necessary conditions for the magnetic pumping mechanism are:

τg ¿ (t2 − t1)¿ τe ¿ (t3 − t2), (3.15)


Figure 3.5: One cycle of magnetic pumping in arbitrary time units.

where τg is the gyro period of the particle around its guiding centre, and τe

is the equipartitioning time. The process starts at time t1 with a random

distribution of momentum described by Eq. (3.14). Between t1 and t2, as

well as between t3 and t4, the distribution is described by the conservation

laws of Eq. (3.13). Then, between t2 and t3, as well as between t4 and

t5, the momenta are equipartitioned by the scatterings to the form of Eq.

(3.14). Since there is no change of the total momentum between t2 and t3,

and t4 and t5, we assume p2 = p3, and p4 = p5. Thus

p21 =1

3p2o +

2

3p2o

p22 =1

3p2o +

2k

3p2o

p23 =1

3

µ1

3+2k

3

¶p2o +

2

3

µ1

3+2k

3

¶p2o

p24 =1

3

µ1

3+2k

3

¶p2o +

2

3k

µ1

3+2k

3

¶p2o =

µ5

9+2k

9+2

9k

¶p2o

3.2. MAGNETIC PUMPING 35

or, after one complete cycle:

p5 =po3·r5 + 2k +

2

k= κpo (3.16)

One can see that κ ≥ 1, and a momentum gain will take place always when

k 6= 1. After n cycles, the momentum becomes

pn = po · κn (3.17)

It was shown by Alfven (1959) that the differential spectrum of CR

accelerated by this mechanism and diffusing thereafter, is proportional to

p−2+φ, where φ is some unknown factor within the range [−1, 1].This mechanism is very effective and can take place wherever there are

such magnetic structures with different intensities and small scale inho-

mogeneities. These structures are known to exist, e.g., in the interstellar

medium, in the interplanetary space formed by plasma clouds created by

the Sun, etc. However, this mechanism also requires some pre-acceleration

of particles to be injected because thermal particles do not satisfy the nec-

essary conditions.

The three above considered mechanisms are slow but very effective. They

are believed to be the most important sources of acceleration of GCR. Be-

sides, the Fermi shock acceleration is considered to be the main source of

anomalous cosmic rays. For solar cosmic rays the situation is different. Ac-

cording to observations, SCR must be accelerated very fast, within seconds

or minutes. For SCR, the mechanism of magnetic reconnection at the top

of a magnetic loop is thought to be the main source (see later), while Fermi

acceleration and magnetic pumping are only taken into account as a factor

changing the SCR spectrum during their propagation through the interplan-

etary space.


There are also some other mechanisms of particle acceleration but we

will not consider them within the scope of this course.

Chapter 4

Solar Cosmic Rays

Solar Cosmic Rays, also called Solar Energetic particles (SEP), were first

discovered on February 28, 1942. The sudden increase of Geiger counter’s

counting rate was associated with a large solar flare. Since then CR detectors

have occasionally seen sudden increases in CR intensity, sometimes as large

as several hundred per cent, associated with an outburst (mostly flares) on

the Sun.

The cosmic ray intensity returns to normal within tens of minutes to

days, as the acceleration process ends and accelerated ions disperse through-

out the interplanetary space. These short increases of cosmic ray count

rates associated with SEP are called GLEs (Ground Level Enhancements

or Ground Level Events). So far, more than 60 GLEs have been registered

since 1942 (see Table 4.1).

The time profile of the Oulu Cosmic Ray Station (Neutron monitor)

during the strong GLE of 24 Oct 1989 (# 45 in Table4.1) is shown in Fig. 4.1.

During this GLE the maximum count rate was nearly twice the normal count

rate level.

Compared to GCR’s, SCR’s have relatively low energies, generally below

1 GeV and only rarely around 10 GeV. That is why such events are often

missed by cosmic ray detectors near the equator where the lowest energies

37

38 CHAPTER 4. SOLAR COSMIC RAYS

Table 4.1: The list of GLEsGLE # date GLE # date GLE # date

1 28/02/1942 26 29/04/1973 51 11/06/19912 07/03/1942 27 30/04/1976 52 15/06/19913 25/07/1946 28 19/09/1977 53 25/06/19924 19/11/1949 29 24/09/1977 54 02/09/19925 23/02/1956 30 22/11/1977 55 06/11/19976 31/08/1956 31 07/05/1978 56 02/05/19987 17/07/1959 32 23/09/1978 57 06/05/19988 04/05/1960 33 21/08/1979 58 24/08/19989 03/09/1960 34 10/04/1981 59 14/07/200010 12/11/1960 35 10/05/1981 60 15/04/200111 15/11/1960 36 12/10/1981 61 18/04/200112 20/11/1960 37 26/11/1982 62 04/11/200113 18/07/1961 38 07/12/1982 63 26/12/200114 20/07/1961 39 16/02/1984 64 24/08/200215 07/07/1966 40 25/07/198916 28/01/1967 41 16/08/198917 28/01/1967 42 29/09/198918 29/09/1968 43 19/10/198919 18/11/1968 44 22/10/198920 25/02/1969 45 24/10/198921 30/03/1969 46 15/11/198922 24/01/1971 47 21/05/199023 01/09/1971 48 24/05/199024 04/08/1972 49 26/05/199025 07/08/1972 50 28/05/1990

39

Figure 4.1: Count rate (in per cent) of the Oulu NM during the GLE of 24Oct 1989.


Table 4.2: Average integral fluxes of SCR in the vicinity of the Earth duringsolar maximum and minimum years (units for particles cm−2s−1).

energy range solar maximum solar minimum

above 30 MeV 3 · 102 2 · 10−2above 100 MeV 20 2 · 10−3

are excluded by the Earth’s magnetic field. The best detectors for observing

solar particles are therefore those at high-latitude regions (like Oulu Cosmic

Ray Station) which are more sensitive to the lowest CR energies.

On an average, the average integral flux of solar cosmic ray particles in

the vicinity of the Earth is shown in Table 4.2. Note that the flux varies

greatly with solar activity. GLEs seldom occur during solar activity minima

and have their maximum occurrence most typically some 1-3 years after the

sunspot maximum.

Note also that during a GLE the flux can be several orders of magnitude

larger than the average. The flux during SEP events is high enough to be

dangerous for astronauts and also for the crews of high-altitude airplanes

over polar regions. Therefore, SEPs are an important factor in the new

concept of Space Weather which, e.g., tries to predict the short-term solar

activity, including SEP events, and its effects in the near-Earth space.

Note also that SCR particles are primary cosmic rays, i.e., their char-

acteristics (energy spectrum, time profile of intensity, direction of arrival,

pitch angle distribution, etc.) are not very much disturbed during their

propagation through the interplanetary space. (Actually they are somewhat

disturbed already at 1 AU but this can be taken into account by models of

propagation.) Thus, the in situ conditions in the acceleration site of the

solar atmosphere can be diagnosed by studying SCR.

4.1. ENERGY SPECTRUM AND COMPOSITION OF SCR 41

4.1 Energy Spectrum and Composition of SCR

4.1.1 Chemical Composition and Ionisation State

SCR are considered to consist of three main components:

- proton-nucleon component;

- electron-positron component;

- electromagnetic component.

The electromagnetic component, although is not a “cosmic ray” as we

define here, is closely connected with the the other two components. The

electron-positron component is also accelerated to relativistic energies but

it is almost absent in SCR observed at Earth because it has large energy

losses.

The energy losses can be separated into nuclear and radiative losses. The

nuclear losses (collisions with other nuclei) depend on the amount of matter

traversed and can be neglected in a typical acceleration process of a solar

flare. The radiative energy losses include, e.g., the synchrotron radiation

and the bremsstrahlung radiation. They are related to the acceleration of

the particle, and therefore are much higher for the electron-positron compo-

nent than for protons. (E.g., synchrotron radiation is³mp

me

´4 ' 1013 timesstronger for an electron than for a proton of the same energy).

The radiative energy losses dominate the high-energy part of the electron-

positron component so that almost all accelerated electrons lose most of

their energy in the solar corona or photosphere, producing the X− and

γ−ray emission of typical solar flares. Moreover, it is more difficult for

the electron-positron component to propagate through the interplanetary

medium. Therefore we will only consider here the proton-nucleon compo-

nent which is mainly responsible for GLEs and other terrestrial phenomena.

The difference in the chemical composition between SCR and GCR is

mainly due to the different amounts of matter passed by the two groups


Table 4.3: Relative abundances of SCR (10-47 MeV/nucl) for the event of03/06/1982

H/He He/O C/O N/O O/O Ne/O Mg/O Si/O Fe/O

03/06/198211:30-18:00

132 102 0.38 - 1 0.87 0.62 0.2 2.5

baseline 66 53 0.45 0.13 1 0.13 0.18 0.15 0.07

enrichmentfactor

2 1.9 0.84 - 1 6.7 3.4 1.3 38

of particles (≤ 0.1 g cm−2 for SCR and ≈ 7 g cm−2 for GCR). This resultsin the lack in SCR of light nuclei like Li, Be, B and other elements and

isotopes that are absent in the source and produced by weaker collisions.

Another peculiarity is that the SCR composition depends on particle’s

energy. At energy ≥ 100 MeV/nucleon the relative composition of SCR issimilar to that in the solar atmosphere. However, SCRs of lower energy

are enriched with heavy nuclei. For instance, Table 4.3 shows the relative

abundances and enrichment factors (ratios with respect to the baseline)

during the famous solar energetic particle event of 3 June 1982.

The composition varies significantly from one SEP event to another. For

instance, for 150 SCR events in the energy range 1.9-2.8 MeV/nucl detected

by instruments onboard the IMP-8 and ISEE-3 satellites in 1978-1983, the

relative abundances varied significantly: H/O - 200-30000; He/O - 30-200;

Si/O - 0.07-1.00; Fe/O - 0.03-30000 (see Fig. 4.2).

Heavier ions are not fully ionised in the SCR source. This was verified

by numerous measurements of the charge state of SCR near the Earth.

Table 4.4 shows the measured average charge states of some SCR elements.

One can see that they are not fully ionised and the relative ionisation level

decreases with the charge number and mass. (Note that the charge state

does not change from event to event as much as abundance.)


Figure 4.2: Distribution of the ratios H/O, He/O, C/O, Si/O, Fe/O in theenergy range of 1.9-2.8 MeV/nucl using data from IMP-8 and ISEE-3 forthe period 1978-1983.


Table 4.4: Mean charge state of SCR (0.4-2.6 MeV) as measured by ISEE-3in 1978-1979

element C N O Ne Mg Si S Fe

full ionisation 6 7 8 10 12 14 16 26measured 5.7 6.37 7.0 9.05 10.7 11.0 10.9 14.9

Since the charge state of plasma ions in equilibrium is determined by

plasma temperature, one can estimate the plasma temperature in that part

of solar atmosphere where SEPs are accelerated. (However, that estimate

must be modified because the charge state changes during a fast acceleration,

which is not an equilibrium process, and during coronal and interplanetary

propagation.) The estimated equilibrium temperature is 1 · 106 − 7 · 106K.These are typical temperatures in solar corona.

Let us also briefly mention the so called 3He-rich events when the

3He/4He ratio in SCR is 2-3 orders of magnitude higher than in the so-

lar atmosphere. These events are associated with impulsive flares.

4.1.2 SCR Energy spectrum

The energy spectrum of SCR decreases with particle’s energy. This is the

only similarity with the GCR energy spectrum.

The first remarkable difference is the maximum energy. Solar protons

can be accelerated up to some 20 GeV only. This is in dramatic difference

with the maximum observed GCR energy of about 1021 eV.

The other dramatic difference is that while the GCR flux is roughly

constant and exists permanently, SCRs appear rather rarely and very irreg-

ularly in time and the SCR flux levels vary accordingly. Moreover, often

there are two components in the SCR spectrum, the so called prompt and

delayed components with different spectral and temporal characteristics.


The SCR energy spectrum usually cannot be expressed by a single power

law. Often either a broken power law (a compound of several power law

pieces) in energy, or an exponential in rigidity (rigidity P = pc/q, scaled

ratio of particle’s momentum and charge), or a Bessel function in energy is

fitted to the observed energy spectrum.

Usually, the SCR spectrum is softer than that of GCR. The SCR spectral

index γ (see Eq. 2.1) varies between 2 and 5. A “typical” SCR spectrum

is shown in Fig. 4.3 for the SCR event of 15 June 1991. (Here, “typical”

does not mean that spectra of other events look similar, but rather that

spectra of most events are as complicated.) For this particular event the

spectrum was seen to consist of a soft component with a Bessel-function

type spectrum (curves 1a and 1b) and a hard component with a power law

spectrum (line 2). One can see that a single power law (line 3) would be a

very rough approximation since the discrepancy would be about 20.

The lower energy part of the SCR spectrum below some hundred MeV

can only be determined reliably using either direct space-borne observations

or observations of secondary emissions (microwave or X-ray/γ−radiation, orneutrons).

In order to estimate the higher energy (1-10 GeV) flux of SCR, it is usual

to make use of the world-wide network of neutron monitors (NM). Every NM

has a certain geomagnetic rigidity cut-off Pc (see later) and hence its count

rate can be written as

N(Pc, t) =

Z ∞Pc

dJ

dP(P, t)SNM (P )dP, (4.1)

where dJdP (P, t) is the differential rigidity spectrum of primary SCR, and

SNM (P ) is the known specific yield function of the NM.

Hence, knowing normalised count rates of different NMs with different

Pc, one can estimate the original spectrumdJdP (P, t) by fitting the data with


Figure 4.3: SCR spectrum during the flare of 15 June 1991. 1a, 1b and 2 aremodel spectra of in situ protons (the hatched area denotes their difference)for the interval 08:37-09:02 UT, 3 is the best fitting power law.

4.2. SOLAR NEUTRONS 47

the model spectrum. A sample of such a reconstruction is shown in Fig.

4.4 for the GLE of 15 June 1991. This reconstruction corresponds to line 2

in Fig. 4.3. This technique cannot be applied below ≈ 0.9 GV of rigidity

because of the atmospheric cut-off (particles with lower rigidity cannot pass

through the Earth’s atmosphere and reach the ground level). This is seen

as the saturation of the line in Fig. 4.4 at low Pc.

Fig. 4.5 shows the relative importance of GCR and SCR fluxes at dif-

ferent CR energies. At high energies (above some GeV/nucleon) GCR are

the dominant part of the CR, showing a general anticorrelation with so-

lar activity. At low energies (below some hundred MeV), SCR dominate

the overall CR flux. This part varies in concert with solar activity. In the

energy range between some hundred MeV/nucleon and some GeV/nucleon

either the GCR or SCR component may dominate, and the flux variations

in this energy range have a very complicated pattern.

4.2 Solar neutrons

Another important component in solar cosmic rays is solar neutrons. Since

neutrons are neutral, they cannot be accelerated by electric fields or on

magnetic structures and therefore are not “primary” cosmic rays. Why can

we still see solar energetic neutrons in the Earth’s vicinity?

Depending on the magnetic configuration in the flare site, some acceler-

ated protons/or alpha particles can be trapped in a magnetic loop (bottle)

and interact with the solar matter. Let us consider one leg of a magnetic

loop (see Fig. 4.6). Magnetic field lines are depicted in dash.

Within the more dense matter below the visible solar surface (in the

photosphere), the density of magnetic field lines becomes higher producing

a magnetic “mirror” for particles (p1) which enter the region from above

with large enough pitch angle. Before flying backward (upward) they spend

some time in a relatively dense matter. This makes it probable for the


Figure 4.4: Neutron monitor increases in per cent at the maximum of a GLEon 15 June 1991.


Figure 4.5: CR energy spectrum. Solid lines denote the GCR spectrumfor maximum and minimum solar activity. Dashed line gives the averagespectrum.


particle p1 to interact with a nucleus (most likely proton as well, or ≈ 10 %of α-particles) of the matter, shown as an explosion in the Figure, producing

the neutron n1.

It is important to note that the direction of the neutron is quite close to

that of the interacting proton. Therefore neutrons are almost excluded in

the upward direction. After being produced, neutrons move straight. Note

that neutrons (n2) produced by protons (p2) with a small pitch angle cannot

be seen in the Earth as they cannot escape the dense matter region. Thus,

only neutrons from flares which are located close to the solar limb can be

seen in the Earth. This has been verified by direct measurements of neutrons

and solar flare γ-rays. (Note also the the mechanism of γ-ray production in

solar flares is similar to the solar neutron production described above.)

Thus, solar neutrons carry unique information about the conditions at

the flare site. After being produced, they move on straight lines preserving

their kinetic energy without being disturbed by solar, interplanetary or ge-

omagnetic fields. The first solar neutrons were detected by the world wide

network of neutron monitors during the big flare of 3 June 1982. Later,

ground-based and space borne instruments have detected solar neutrons in

several other events.

Unfortunately we can not detect galactic or extragalactic neutrons which

could locate the remote sources of GCR acceleration because a free neutron

is unstable. A neutron decays with a β−decay to a proton, an electron andan antineutrino with a mean (e−fold) life time of about 920 sec. The time oflight to travel from the Sun to the Earth is 1AUc ≈ 500 sec. Hence, roughlyonly a half of energetic solar neutrons can reach the Earth. (This estimate is

not true for relativistic neutrons because of the relativistic time dilatation.

However, only few star neutrons are strongly relativistic.

It is also interesting to note that protons originating from the decay of

solar neutrons before 1 AU have been detected as a small increase of the


Figure 4.6: Sketch of secondary neutron production in a solar flare. Dashedlines denote magnetic field lines and solid lines particle trajectories.


proton flux shortly before the onset of a major GLE. (Try to figure out why

those neutron-decay protons come BEFORE SCR!)

4.3 SCR acceleration: flares and CMEs

During GLEs, SCR particles have been accelerated up to some 10 GeV

energy within a very short time. All details of SCR acceleration have not

yet been solved. Traditionally, it has been supposed that all solar particle

events originate from solar flares. Recently, the concept of a coronal mass

ejection (CME) has been introduced as another solar phenomenon causing

particle acceleration. It is now known that CMEs, not flares as earlier

thought, are mainly responsible for large magnetic storms in the Earth’s

magnetosphere. (The earlier view is now called “the solar flare myth”.)

CMEs and flares have very different properties, CMEs have a much larger

spatial scale, involving huge amounts of coronal mass. Flares are of much

smaller spatial scale. Both are related to the active regions in the Sun and

often appear closely connected. However, it is probable that flares, not

CMEs, are mainly responsible for large SCR events, in particular for those

leading to GLEs.

Traditionally, solar flares are divided into impulsive and gradual flares.

Similarly, SCR events are divided into impulsive and gradual events which

have significantly different characteristics, such as duration, composition

and energy spectrum (see also Table 4.5). This implies that mechanisms

responsible for the two SCR types are also different. Usually, impulsive

events can be reliably associated with the impulsive phase of a flare while

gradual events can mostly be associated with a shock driven by a CME in

the corona and in the interplanetary space. However, sometimes gradual

SCR events show evidence of flare origin as was the case, e.g., on Oct 16,

2000 (see Figs. 4.7, 4.8)

4.3. SCR ACCELERATION: FLARES AND CMES 53

Figure 4.7: Time profile of the X-ray flux in different wavelength bandsduring a gradual flare.


Figure 4.8: Time profile of the integral proton flux in different channelsduring a gradual flare.


Table 4.5: Properties of impulsive and gradual SCR events

Impulsive Gradual

Particles: e-rich p-rich3He/4He ≈ 1 ≈ 0.005Fe/O ≈ 1 ≈ 0.1H/He ≈ 10 ≈ 100Charge of Fe ≈ 20 ≈ 14Temperature ≈ 107K ≈ 2 · 106KDuration Hours DaysLongitude cone ≈ 30o ≈ 180oCoronograph - CMESolar wind - IP ShockEvents per year ≈ 1000 ≈ 10

4.3.1 Solar flares

Solar flares are sudden, huge explosions on the surface of the Sun. They

were first observed in the visible light (so-called white light flares) already

in 1860. Flares are very fast processes, with the smallest time scales of only

a few minutes. Usually they occur near sunspots, along the dividing line

(neutral line) between the areas of oppositely directed magnetic fields where

the magnetic field structures get twisted and sheared, releasing energy after

magnetic reconnection.

The energy released in solar flares can be distributed in many forms:

hard electromagnetic radiation (γ- and X-rays), energetic particles (protons

and electrons), and mass flow.

Flares are usually characterized and classified by their brightness in X-

ray radiation. The biggest flares are called X-class flares. The brightness of

M-class flares is some ten times smaller than in the X-class. Next weaker

classes are C, A and B-classes. Within these main classes the flares are

further divided into subclasses. For instance, a X3 flare is stronger than X2.


The strongest observed flares were of X12-X13 class. Fig. 4.9 shows how

the solar atmosphere looks at the flare site in the wavelength band of the

Hα line emission.

A schematic sketch of particle acceleration during a solar flare is shown

in Fig. 4.10. A stable pre-flare loop in the solar atmosphere may experience

a pressure force by the surrounding plasma (horizontal arrows in the left

panel). This may lead to an interaction between the oppositely directed

magnetic field lines and, finally, to a complete reconfiguration of the mag-

netic structure. This interaction is called the magnetic reconnection and

results in an explosion-like release of energy, seen as the impulsive phase of

a flare.

In the reconnection process, a huge amount of magnetic energy is released

very rapidly and transformed to thermal and kinetic energy of particles.

Reconnection can accelerate particles to a high energy within a short time

as required by the very impulsive SEP events.

Energetic particles accelerated in the reconnection region are guided

away from this region along the newly reconnected magnetic field lines.

Those particles that are ejected upward (escaping protons; see right panel

of Fig. 4.10) may either remain trapped or escape into the interplanetary

space if the upper magnetic configuration becomes open, and cause an im-

pulsive SEP event.

Note that theories based on magnetic reconnection can satisfy all the

observational facts of impulsive events. Note also that magnetic reconnec-

tion occurs, in addition to the Sun’s atmosphere, in many other plasma

environments, including the Earth’s magnetosphere.

Those accelerated protons that are ejected downward can be trapped

in the magnetic bottle of the smaller loop formed within the original loop,

populating it with very energetic particles. They are trapped in this bottle

bouncing between the two ends (“feet”) of the loop. When inside either


Figure 4.9: Hα image of the flare of 10 Oct 1971 by the Big Bear SolarObservatory.


Figure 4.10: A schematic view of particle acceleration during a solar flare.


foot, they produce secondary emissions, in particular neutrons and γ− andX-rays as described earlier (see Fig. 4.6).

The properties of the observed secondary emissions from solar flares are

fairly well described by this model. The trapped protons can also gradually

escape from the new bottle through coronal diffusion across magnetic field

lines, finally reaching open magnetic lines and escaping into interplanetary

space. This process may lead to flare-associated gradual events. A more

detailed analysis of possible solar flare scenarios is beyond the scope of the

present course.

4.3.2 Coronal Mass Ejection

Coronal mass ejections (CMEs) are spatially larger and temporarily slower

events than flares, in which huge amounts of plasma initially trapped in

closed coronal magnetic field lines are ejected into interplanetary space.

During active times, several CMEs may occur daily. CMEs involve typi-

cally 1012 to 1013 kg of mass, and kinetic energy on the order of 1024 to 1025

J.

The disruption of a large stable, magneticallyclosed structure still poses

fundamental questions for the magnetohydrodynamic theory (MHD). How-

ever, it is probable that large-scale magnetic reconnection is involved in the

formation of a CME. Fig. 4.11 shows a CME of 24 Oct 1989. Note that

although flares and CMEs are often connected, this is not always so. There

are flares that are not followed by CMEs and CMEs without flares.

A CME often leads to a huge hot plasmoid (a closed magnetic structure)

moving with a high speed in the interplanetary space, and to an interplan-

etary shock located at the front edge of the plasmoid. The shock can also

accelerate particles by the Fermi shock acceleration in analogy with super-

nova shocks (see Section 3.1). Of course, energies related to the acceleration

by the interplanetary shocks are much lower than those related to supernova


Figure 4.11: CME of 24 Oct 1989 as seen by the Solar Maximum Mission.

4.4. INTERPLANETARY PROPAGATION 61

shocks. Most CME-related SCR particles are sub-relativistic and the events

have a time scale of a day.

Note also the difference between CMEs and the quiet solar wind. Solar

wind consists of particles with a frozen-in magnetic field, i.e., the prop-

agation is mainly determined by plasma motion. In other words, kinetic

energy of solar wind particles dominates over magnetic energy of IMF. The

situation for CMEs is opposite since the CME evolution is defined by the

magnetic field or, in other words, magnetic energy dominates the particles’

kinetic energy.

Summarizing, impulsive SEP events are associated with solar flares.

Gradual SEP events may be either of flare or CME origin.

4.4 Interplanetary propagation

Interplanetary propagation is an important factor affecting the SCR. Even

for the biggest solar events, particles can miss the Earth if the relative

geometry is not favorable.

As will be discussed in Section 7.3 in more detail, the interplanetary

magnetic field has a shape of a spiral (see Fig. 4.12) in the ecliptic plane.

(However, strong solar events can disturb this picture significantly.) Let us

assume that there is a solar event containing both a flare (impulsive event)

and a CME (gradual event). If the event is located in the middle of the

solar disk (see Fig. 4.13, top panel), particles accelerated in the flare cannot

reach the Earth since the Earth and the flare site are not connected by IMF

lines. However, CME which moves radially reaches the Earth, resulting in

a gradual SEP or GLE.

On the other hand, if the solar event is located near the western limb of

the solar disk (see Fig. 4.13, bottom panel), flare accelerated particles can

reach the Earth along the IMF lines, resulting in an impulsive SEP/GLE


Figure 4.12: Spiral lines of the interplanetary magnetic field in the eclipticplane.


event while CME misses the Earth. If the event is located on the eastern

limb or on the back side of the Sun, it cannot cause a SEP/GLE event.

Thus, SEP/GLE events observed in the Earth depend very much on the

solar location of the event and the interplanetary propagation of SCR.

The interplanetary propagation of SCR causes temporal differences for

fluxes of particles different energies. Even if ejected simultaneously, particles

of different energies come to the Earth at different times due to different

speeds. This concerns mainly non-relativistic particles. Fig. 4.14 shows the

arrival times of protons of different energies (different shaded boxes) for the

solar flare of 15 June 1991. The slope of lines is c/L, where L is the length

of the corresponding IMF line (≈ 1.2 AU). The vertical axis is 1/β = c/v.One can see that although the arrival times of SCR with different energies

were spread over half an hour, they have been ejected simultaneously, and

the time differences was due to the differenr speeds.


Figure 4.13: Schematic view of a GLE associated with a solar event locatedin the central meridian (top panel) and the western limb (bottom panel) ofthe Sun.


Figure 4.14: The times of first arrival of particles vs β−1.


Chapter 5

Anomalous Cosmic Rays

Anomalous Cosmic Rays (ACRs) are the third primary component of cosmic

rays (along with GCRs and SCRs). ACRs were first discovered in 1973 as

a ”bump” in the spectra of certain elements (He, N, O, Ne) at energies of

about 10 MeV/nucleon. By now, ACRs have also been observed in H, Ar,

and C.

A schematic view of ACR origin is shown in Fig. 5.1. ACRs arise primar-

ily from neutral interstellar atoms which are swept into the solar magnetic

field dominated space (called the heliosphere) by the motion of the Sun

through the interstellar medium. At ∼1-3 AU, these neutral atoms becomesingly ionized either by photoionization by solar UV photons or by charge

exchange collisions with solar wind protons.

Once the particles are charged, IMF picks them up and carries them,

together with the outward flowing solar wind, up to the solar wind termi-

nation shock which is expected to be located at the radial distance of about

70—100 AU. The ions are called pickup ions during this part of their trip.

The ions repeatedly collide with the termination shock, gaining energy

in the process, and being accelerated from solar wind energies of about 1

keV/nucleon to higher energies of tens of MeV/nucleon. This continues

67

68 CHAPTER 5. ANOMALOUS COSMIC RAYS

until they escape from the shock. Some of them then diffuse into the inner

heliosphere.

Recent observations from the SAMPEX satellite indicate that singly ion-

ized ions are accelerated to the maximum kinetic energy of about 250—350

MeV. However, collisions in the termination shock region cause some ions to

become further stripped off electrons, thereby reaching higher ionic charge

states (+2, +3, +4, etc.). The electric fields in the termination shock acceler-

ate these higher charge state ions to even higher energies. In fact, SAMPEX

has observed ACR oxygen ions at Earth with energies up to at least 100

MeV/nucleon, albeit with a very steep energy spectrum (see Fig. 5.2). (Note

also that because anomalous cosmic rays are less than fully ionized, they are

not as effectively deflected by the Earth’s magnetic field as galactic cosmic

rays at the same energies.)

ACRs are thought to represent a sample of the very local interstellar

medium. The atoms with a high first-ionization potential (typically light

atoms, such as H, He) are ionized, on an average, closer to the Sun than

those atoms (typically heavier atoms) which have a low ionization potential.

Accordingly, the heliosphere acts as a kind of elemental filter for the inter-

stellar atoms allowing a larger amount of high-ionization potential atoms to

pass through the heliosphere untouched, and favouring the low-ionization

potential atoms to be ionized and to become ACRs. This filtering process

explains the above mentioned elemental distribution of ACRs. Therefore,

ACRs are a tool to study the motion of energetic particles within the so-

lar system, to learn about the general properties of the heliosphere, and to

study the nature of interstellar matter.

69

Figure 5.1: Schematic view of anomalous cosmic rays.

70 CHAPTER 5. ANOMALOUS COSMIC RAYS

Figure 5.2: ACR energy spectra at the positions of Voyager-1 (57 AU) andVoyager-2 (44 AU) spacecraft in 1994. a) ACR H, b) ACR He, c) ACR C,d) ACR N, e) ACR O, and f) ACR Ne.

Chapter 6

Solar Wind

6.1 General facts

S. Chapman ja V. Ferraro proposed in 1931 that bursts of particles emitted

from the Sun would cause brief compression of the Earth’s magnetic field

called the SSC (Sudden Storm Commencement), often preceding large ge-

omagnetic disturbances called magnetic storms. According to their model

(now known to be erroneous), solar wind would only occur temporarily in

connection with flares or other specific solar phenomena.

In 1951 L. Biermann studied cometary tails (see Fig. 6.1) and showed

that the pressure of solar radiation alone can not explain his observations.

Biermann suggested that solar wind exists always and essentially affects

the formation of cometary tails. His estimate of about 500 km/s for the

velocity of the continuously blowing solar wind, based on his observations of

cometary tails, proved later to be amazingly accurate. Biermann’s proposal

is now considered to form the start of the modern view of the solar wind (as

well as the cometary research). However, the name ”solar wind” was coined

by E. N. Parker only in 1958 when developing the theory of the (continuous)

solar wind.

The existence of solar wind was finally proven by the Soviet Lunnik-2

71

72 CHAPTER 6. SOLAR WIND

Figure 6.1: Comet Tail. This photograph shows the curved dust tail andstraight ion tail on Comet Myros; both tails point away from the Sun. Thepressure of the Sun’s light gives the dust particles an outward push, creatinga broad arc. In contrast, the solar wind accelerates the ions to high velocitiesand pushes them into the relatively straight ion tails. (Coutresy of LickObservatory)

6.1. GENERAL FACTS 73

and 3 probes in 1960 after reaching out from the Earth’s magnetosphere.

Moreover, the Mariner-2 probe confirmed the continuous flow of solar wind

during its 4-month trip to the planet Venus in 1962.

Solar wind has the following characteristics (at 1 AU in the ecliptic plane,

i.e. at the Earth’s orbit, unless otherwise mentioned; see Fig. 6.2):

• The average velocity is ca. 400 km/s, but varies between 200—800

km/s.

• Solar wind consists mainly of protons and electrons but there are about5—20% of α-particles (He++-ions), and several heavier ion species at

a much smaller percentage. The average charge of solar wind is zero.

• Particle density is ca. 5 · 106 m−3 (or 5 cm−3), varying between (1—20)·106 m−3.

• The mean particle flux from the Sun is therefore

φ = nv ≈ 2 · 1012 m−2s−1 (6.1)

from which one can calculate the amount of particles that the Sun

loses in a second

N = 4πr2φ = 5, 6 · 1035 s−1 (6.2)

• The average energy of protons and electrons is about 1 keV and 1 eV,respectively.

• The average temperature of protons is ca. 104 − 2 · 105 K, i.e., thecorresponding thermal energy is about 1—20 eV. (Note! 1 eVb=1, 16·104K). The temperature of electrons is roughly the same as that of protons

during disturbed times but about 3—4 times higher during quiet times.


Figure 6.2: Histograms of occurrence frequency for the values of the solarwind velocity, proton number density and proton temperature in interplan-etary space (from Hundhausen et al., 1970).

6.2. DELAVAL NOZZLE 75

• The temperature of charged particles in a magnetic field is generallyanisotropic so that the temperature Tk in the direction of the mag-

netic field (or in opposite direction) is higher than the temperature

T⊥ against it . This difference arises from the fact that the motion

of charged particles along the magnetic field is more free than in the

perpendicular direction. In the solar wind Tk ≈ 2 ·T⊥.

• The speed of sound in the solar wind is

cs =

sγkT

mp≈ 1, 2 · 104 m/s (6.3)

where γ = 5/3 is the adiabatic constant for a monoatomic gas. Thus,

the velocity of solar wind is about 40 times the speed of sound. Solar

wind is therefore extremely supersonic at 1 AU.

• At the base of the corona the solar wind speed is still below the soundspeed, i.e., it is subsonic. However, it is rapidly accelerated so that at

about 2—6 solar radii it reaches the speed of sound, and beyond that

remains supersonic until at the outer bondary of the heliosphere called

the termination shock it becomes subsonic again.

• The outward motion of the solar wind, i.e., of the coronal plasma, fol-lows from the fact that the pressure of the solar atmosphere is greater

than the counter-acting pressures due to the solar gravitation and the

interstellar matter.

6.2 DeLaval nozzle

Already at the distance of about 10 RS away from the Sun the solar wind is

about 300 km/s, i.e., very close to the average value observed at the Earth’s

orbit. Accoringly, a very effective acceleration mechanism must exist close

to the Sun which can speed up the solar wind from a subsonic motion to a


Figure 6.3: Mass flow through a nozzle used to explain the acceleration ofthe flow speed

strongly supersonic flow over a rather short distance. In order to understand

this mechanism we first study the idea of the so called deLaval nozzle where a

subsonic flow can be transformed to supersonic. This is the same mechanism

that is, e.g., behind the principle of a jet engine.

Let us now examine the flow of (neutral) gas through a tube whose cross

section is decreasing (see Fig. 6.3). If the flow speed of the gas is v and the

mass density is ρ at a point with cross section A, the mass flux

φm = ρvA (6.4)

announces the amount of mass that passes through the tube per time unit.

Since, in a steady flow, this flux is the same at each point, we find

dφmdr

=d(ρvA)

dr= vA

dρ

dr+ ρA

dv

dr+ ρv

dA

dr= 0 (6.5)

6.2. DELAVAL NOZZLE 77

Multiplying this by dr and dividing by φm leads to

dρ

ρ+dv

v+dA

A= 0. (6.6)

The flow is sustained by a pressure difference between the two ends of the

tube. A local pressure difference enhances the flow (so called Bernoulli law):

dp

dr= −ρ · a = −ρdv

dt= −ρdv

dr

dr

dt= −ρv dv

dr(6.7)

or

dp = −ρvdv (6.8)

Dividing this by ρ we obtain

dp

ρ=dp

dρ

dρ

ρ= −vdv (6.9)

Let us assume that we have an ideal gas and an adiabatic process (no ex-

change of heat with surroundings) for which we have the following equation

of state:

p · ρ−γ = const (6.10)

(γ is the adiabatic constant). Differentiating this we have

ρ−γdp = γpρ−γ−1dρ. (6.11)

Multiplying eq. (6.11) by ργ we obtain an equation for the speed of sound

cs:dp

dρ= γ

p

ρ= c2s. (6.12)

inserting this to eq. 6.9) and solving dρ/ρ we find

dρ

ρ=−vdvc2s

. (6.13)

From this and eq. (6.6) we finally obtain:

dA

A= −dρ

ρ− dvv=

Ãv2

c2s− 1

!dv

v. (6.14)


Figure 6.4: Mass flow through a nozzle with a minimum cross-section toexplain the presence of a critical region in the mass flow in order for the lowspeed to become supersonic.

When the cross section decreases (dA < 0), the velocity increases (dv >

0) until the velocity is below the sound speed, i.e., subsonic (v < cs). If

the velocity reaches the sound speed (v = cs), we must have dA = 0, i.e.,

the cross section must not decrease any longer. If we want the velocity still

to grow after exceeding the sound speed (v > cs), the cross section must

increase (dA > 0)! Thus, the tube must continued by a additional section

(see Fig. 6.4) which gives us the de Laval nozzle.

On the other hand, if the gas does not reach the sound speed at the

narrowest point of the (continued) tube, the velocity must, according to eq.

(6.14), turn to decrease during the enlarging part of the tube. This situation

is called the Venturi tube. Whether the gas is accelerated to be supersonic

according to the principle of the deLaval nozzle, or whether it remains a

6.3. ACCELERATION OF SOLAR WIND 79

subsonic Venturi tube is determined by the pressure ratio between the ends

of the tube. E.g., if the tube ends in a vacuum, it is always a deLaval nozzle.

This fact is exploited in the jet engines of space satellites and probes.

6.3 Acceleration of solar wind

E. Parker presented in 1958 a modern theory of solar wind according to

which a subsonic gas is accelerated supersonic with a mechanism whose

principle is quite analogous to the deLaval nozzle. In fact, there is only an

analogy between the two since, of course, there is no tube with a decreasing

cross section in the solar atmosphere but, rather, the particle density is

continuously decreasing as 1/r2 when moving away for the Sun. We will see

that the strong gravitation field of the Sun has a great influence and leads to

a situation which really is analogous to the principle of the deLaval nozzle.

Let us now assume that solar wind is a steadily flowing ideal gas and

forget, e.g., viscosity and the effect of magnetic field. As above, the con-

stancy of mass flux at different distances leads to the equations (6.4)—(6.6).

However, the Bernouli equation (6.7) must be added by a term taking into

account the effect of solar gravitation:

dp

dr= −ρvdv

dr− ρ

GM¯r2

(6.15)

where r is now the distance from the center of the Sun, M¯ is the solar mass

and G = 6, 67 · 10−11 Nm2kg−2 is the gravitational constant. Dividing eq.(6.15) by ρ and multiplying by dr we find

dp

ρ= −vdv − GM¯

r2dr. (6.16)

On the other hand, the left-hand side can be written in the form (cf. eqs.

(6.9) and (6.12)):dp

ρ=dp

dρ

dρ

ρ=dρ

ρc2s. (6.17)


Dividing eq. (6.17) by c2s we obtain, with the help of eqs. (6.16) and (6.6):

dρ

ρ= − v

c2sdv − GM¯

c2s

dr

r2= −dA

A− dvv. (6.18)

Here we can join the dv terms and solve for

dA

A=

Ãv2

c2s− 1

!dv

v+GM¯c2s

dr

r2. (6.19)

The solar wind is spreading spherically away from the Sun whence A ∼ r2and

dA

A= 2

dr

r. (6.20)

Substituting this in eq. (6.19) we finally obtain an equation between dr and

dv which resembles eq. (6.14):

µ2− GM¯

c2sr

¶dr

r=

Ãv2

c2s− 1

!dv

v(6.21)

Equation (6.21) has several different solutions (6.5) which are briefly

treated below.

— If solar wind speed grows (dv > 0) away from the Sun (dr > 0) but is

subsonic (v < cs), the two sides of the equation are both negative and the

equation has a solution. When r reaches the so called critical distance

rc =GM¯2c2s

(6.22)

where the left side of the equation vanishes, the right side must also vanish.

This can either occur so that the velocity reaches the sound speed (v = cs;

solution #1) at the critical distance, or so that the velocity reaches an

extremum dv = 0; solution #2). In the first case, the velocity may still

grow even as supersonic when moving outwards, since both sides of the

equation are positive. This solution corresponds to the situation which is

valid for the solar wind from our Sun.


Figure 6.5: Solution of the solar wind acceleration equation (6.21).


— In the second solution the solar wind attains at the critical distance

a maximum which is smaller than the sound speed, and decreases outside

the critical distance. Accordingly, in this case the solar wind remains as a

subsonic ”solar breeze”.

— The above two solutions are physically the most interesting and are

realized in nature as stellar winds of various types of stars. However, eq.

(6.21) has also other, more exotic mathematical solutions.

— The third solution is when an initially subsonic solar wind reaches the

sound speed before the critical distance rc. Then the right side of eq. (6.21)

is zero and the left side vanishes only if dr = 0. Thus, the solar wind attains

its maximum distance from which it turns back toward the Sun (dr < 0)

while the velocity grows as supersonic.

— If the solar wind speed on solar surface is supersonic, the left side of

eq. (6.21) is negative and the only way to make the right side negative is to

decrease velocity. At the critical distance the left side becomes positive. If

the wind is then still supersonic, the velocity starts growing again beyond

the critical distance (solution #4). If the velocity decreased to sound speed

by the critical distance, it must continue decreasing even outside it (solution

#5).

— There is also a solution for a subsonic flow approaching the Sun. Then

dr < 0 and the left side of eq. (6.21) is negative. The right side gets negative

if the velocity increases. If the velocity attains the sound speed at a distance

r > rc, the velocity continues growing as supersonic but the distance must

start growing (dr > 0; solution #6).

In order for the solution #1 to exist, the critical distance rc must exist

outside the solar surface (rc > R¯). This yields a condition for the sound

speed:

c2s <GM¯2R¯

(6.23)


and, using eq. (6.3), leads to the so called critical temperature of the Sun:

T < Tc ' 7 · 106 K (6.24)

Accordingly, stars with a very high surface temperature have a subsonic

stellar breeze. This can also be seen straight from eq. (6.21) which gives, in

the limit cs →∞:2dr

r= −dv

v(6.25)

Thus, the velocity decreases when going out from the stellar surface. The

physical explanation is the following. Since the mass flow ρvA is constant,

v grows only if ρ decreases with r faster than 1/r2. Such a large density

gradient can not exist in a very hot star where a high pressure p = nkT

pushes the matter outward and decreases the density gradient.

The importance of solar gravitation for the birth of a solar wind with

observed properties can be studied for example by leaving the gravitatinal

term away, whence eq. (6.21) attains the form:

2v

r=

Ãv2

c2s− 1

!dv

dr(6.26)

So, if the solar wind on the solar surface is subsonic, the derivative dv/dr < 0

and the wind always remains subsonic. Accordingly, a supersonic solar wind

can not exist without solar gravitation. The physical reason is that he strong

solar gravitation yields the required density gradient, i.e., ρ decrases faster

than 1/r2.

We can calculate the solar wind speed in the Parker’s model. Assuming

a typical coronal temperature of 106 K, the sound speed is

cs =√RT =

√8.3 · 103 · 106 ≈ 105 m/s = 100 km/s

(for comparison, the sound speed in the air is ≈ 300 m/s). The critical

radius is

rc =GM¯2c2s

≈ 10R¯,


while the Earth’s orbit is 1 AU ≈ 214R¯. The speed of solar wind at theEarth’s vicinity (in the framework of the Parker’s model) can be calculated

from Eq. (6.26) as

v = 3.45cs ≈ 310 km/s.

Observation at 1 AU give the speed of quiet solar wind as 300-400 km/s.

However, the average speed of solar wind is different at the ecliptic plane

and at polar regions. Due to the recent space missions, in particular Ulysses

which traveled outside the ecliptic plane, it is now possible to measure the

latitudinal distribution of solar wind speed (Fig. 6.6). One can see that solar

wind is slower in the near-ecliptic regions (about 400 km/s), and about twice

faster (about 800 km/s) in polar regions. (Sometimes, during strong solar

events (CMEs, flares), the solar wind can be as fast as 1000 km/s in the

ecliptic plane.) The fast solar wind from the polar regions can sometimes

extend to close to the solar equator and overtake the earlier emitted slow

stream, resulting in a ”corotating interaction region” to be discussed in next

sections.

The exact mechanism of coronal heating required for the existence of

solar wind is not precisely known. Open coronal magnetic structures called

”coronal holes” (see Fig. 6.7) emit the fast solar wind, while slow solar wind

comes from closed magnetic structures. Coronal holes are located mainly

at high heliographic latitutes and polar regions around the solar mimimum

times. During solar activity maxima, only small and short-lived coronal

holes are observed, mainly at low latitudes. Plasma outflowing from regions

of magnetic field can spread this field to wherever they arrive. This happens

by ”field line preservation,” a property derived from the equations of an ideal

plasma. By those equations, in an ideal plasma ions and electrons which

start out sharing the same magnetic field line continue to do so later on, as if

the line were a (deformable) wire and the particles beads threaded by it. If

the energy of the magnetic field is dominant, its field lines keep their shapes


Figure 6.6: Diagram of the solar wind speed.


Figure 6.7: Yohkoh soft X-ray image for 02 Nov. 2000.


and particle motion must conform to them. On the other hand, if the energy

of the particles is dominant - that is, if the field is weak and the particles

dense - the motion of the particles is only slightly affected, whereas the field

lines are bent and dragged to follow that motion. That is the case with the

solar wind, and the magnetic field is ”frozen in” (see the next Chapter).


Chapter 7

Heliosphere andInterplanetary MagneticField

The heliosphere is the region controlled by the solar magnetic field, similarly

to the Earth’s magnetosphere that is the region dominated by the Earth’s

internal magnetic field. The heliosphere is a big magnetic bubble in the

interstellar wind formed by solar wind and the solar magnetic field trans-

ported with it. The size of the heliosphere is believed to be about 100—150

AU. The heliosphere studied directly by mankind is expanding because the

space missions like Pioneers or Voyagers explore a larger and larger part of

the heliosphere. In late 1970’s, when the most distant spacecraft was only

about 10 AU away, the size of the heliosphere was assumed to be only 20—25

AU, later it increased to about 50 AU, and then to about 70 AU. Now,

when the most distant spacecraft is close to 90 AU, there is first evidence

for a termination shock. Within a few years we will hopefully verify this evi-

dence and also observe the heliopause. A scheme of the heliosphere is shown

in Fig. 7.1. The heliospheric structure reminds loosely that of the Earth’s

magnetosphere. Bow shock is the most outer boundary of the heliospheric

influence. The interstellar wind does not ”know” about the presence of the

89

90CHAPTER 7. HELIOSPHERE AND INTERPLANETARYMAGNETIC FIELD

heliosphere beyond the bow shock. Heliosheath is a transition region be-

tween the areas dominated by the solar wind (inside the termination shock)

and interstellar wind (outside the bow shock).

91

Figure 7.1: Artistic view of the Heliosphere.


7.1 General facts about IMF

The Sun is a magnetic star whose magnetic field is formed at the bottom

of the convection layer. Magnetic flux tubes rise from the source region

towards the surface, forming local regions of very strong magnetic field.

During sunpot maximum years such active regions are observed in a large

part of solar surface, forming a complex structure for the solar magnetic field.

On the other hand, during sunspot minimum years the weak background

magnetic field with a roughly dipolar form is dominating, and large areas of

open magnetic flux, so called coronal holes, are observed at polar regions.

The magnetic structure of the Sun (corona) also regulates the properties

of the solar wind. A slower (ca. 300—400 km/s) but denser solar wind is

emitted from regions close to the active areas of the Sun, while the polar

coronal holes emit a faster (ca. 700—800 km/s) wind. This leads, e.g., to

strong latitudinal gradients in solar wind speed during sunspot minimum

years (see Fig. 7.2). The solar wind carries the magnetic field of the so-

lar corona as the so called interplanetary magnetic field (IMF). As will be

discussed later in more detail, the IMF is said to be ”frozen in” the solar

wind.

Here we list some basic properties of the IMF:

• The intensity of the IMF at 1 AU is about 5 · 10−9 T = 5γ. It variesbetween 1γ−15γ but can temporarily attain much greater values, evenbeyond 100γ (see Fig. 7.3). The largest IMF values are observed inside

so called shock fronts which can be due, e.g., to coronal mass ejections

(CME) or solar flares, or due to the interaction between a low and

a fast solar wind region. The latter are called corotating interaction

regions (CIR).

• The direction of IMF also varies greatly. The mean IMF direction at1 AU is about 45 off the direction of the Sun. The IMF field lines

7.1. GENERAL FACTS ABOUT IMF 93

Figure 7.2: Average distribution in the ecliptic plane of the solar wind ve-locity (scale on left) in 1976 (O) and 1977 (∆) as a function of the angulardistance in degrees to the neutral sheet (λ) (Bruno et al., 1986). Scaleon right: the velocity associated aa index according to Svalgaard (1977).The hatched area shows the slower solar wind (V ≤ 450 km s−1) and thecorrelated thickness of its coronal source: the ”slow wind sheet”. The com-parison between the two series of data (1976 and 1977) shows that the sheetthickness depends upon the latitudinal gradient of the wind velocity.


Figure 7.3: Histogram of occurrence frequency for magnetic field strengthvalues in interplanetary space (from Ness, 1969; e.g., Falthammar, 1973).


(in a quiet situation) have a spiral structure, forming the so called

Archimedean spiral (see Fig. 7.4). We will derive the structure of this

spiral later in this Chapter.

• The IMF field lines directed toward the Sun and away from it occur

intermittently, dividing the IMF to two sectors where either direction

is dominating. The sector directed toward the Sun is called the T-

sector (Toward) and the one away from it the A-sector (Away).

• While the Sun is rotating, the two types of IMF sectors are found oneafter the other. The pattern of sectors observed (e.g. at 1 AU) during

one solar rotation form the momentary sector structure of the IMF.

• The so called heliospheric current sheet (HCS), also called the he-liospheric neutral sheet, is located between the T- and A-sectors. The

generally wavy form of the HCS resembles that of the skirt of a bal-

lerina dancer (see Fig. 7.5). The projection of the HCS on the solar

surface marks the solar magnetic equator.

• The field lines of the T- and A-sectors come, by definition, from the

southern and northern magnetic hemispheres of the Sun, respectively.

The two magnetic hemispheres may greatly deviate from the location

of the two heliographic hemispheres. Moreover, the magnetic hemi-

spheres change from one heliographic hemisphere to another every

solar cycle, forming the roughly 22-year magnetic cycle of the Sun.

• The nature of the sector structure varies within the solar cycle. Duringthe solar minimum, a 2-sector structure with one T-and one A-sector

dominates. Other frequent patterns are 3-sector and 4-sector struc-

tures which mainly occur during high solar activity times.


Figure 7.4: Interplanetary magnetic field forming the Archimedian spiral.


Figure 7.5: The ”ballerina skirt” of the heliospheric neutral sheet.


7.2 Magnetic field frozen in the plasma

The good electric conductivity of the plasma leads to the fact that an ini-

tial magnetic field is carried out together with the plasma, leading to the

concept of the so called “frozen in” magnetic field. We will study now how

this really comes about. In addition to the solar wind and IMF, a similar

situation occurs in a number of space (e.g., ionosphere, magnetosphere) and

laboratory plasmas, making the concept very useful.

Let us first examine the Ohm law in a coordinate system moving with

the plasma (primed variables denote this system):

j0 = σE0 (7.1)

where j0 is the electric current density, E 0 electric field and σ electric con-

ductivity. In another coordinate system with respect to which the plasma

is moving with a velocity v, these variables are transformed as follows when

| v |<< c:j ≈ j0 (7.2)

E ≈ E 0 − v × B0 = E0 − v × B (7.3)

B ≈ B0 (7.4)

(These are the non-relativistic versions of the complete relativistic Lorentz-

transformations for fields. We do not derive them here).

Using the transformations (7.2)—(7.4) eq. (7.1) yields

j = σ(E + v × B) (7.5)

which is the generalized Ohm’s law. Dividing this by σ we find

E + v × B = j/σ. (7.6)

7.2. MAGNETIC FIELD FROZEN IN THE PLASMA 99

Since plasmas have a very good conductivity (σ →∞) there is an approxi-mate relation:

E = −v × B. (7.7)

This is the formula of the so called convection electric field, and one form

for the equation of frozen-in fields. Substituting eq. (7.7) into the Faraday

law (one of the four Maxwell equations)

∂B

∂t= −∇× E (7.8)

we obtain the following condition for the magnetic field:

∂B

∂t= ∇× (v × B). (7.9)

In order to study the implications of this equation, we look at the tem-

poral change in the magnetic flux φ which flows through a surface S moving

at the velocity v

dφ

dt=d

dt

ZS

ZB · dS =

ZS

Z∂B

∂t· dS +

IL

B · (v × dl) (7.10)

where L is the boundary of the surface. The first term on the right-hand side

describes the flux change due to the temporal change in the magnetic field,

the second term describes the change due to the motion of the surface across

B. The second term can, using Stokes’ law, be modified to the following

form:IL

B · (v × dl) = −IL

(v × B) · dl = −ZS

Z∇× (v × B) · dS. (7.11)

With eq. (7.9) we finally obtain

dφ

dt=

ZS

Z Ã∂B

∂t− ∇× (v × B)

!· dS = 0 (7.12)


which proves that the magnetic flux through a surface moving with plasma

remains constant. Therefore the magnetic field is, in a way, frozen in the

motion of the plasma. Figure 7.6 illustrates two simple effects of this rule.

In the first example a plasma is formed inside a background magnetic field.

When the plasma is forced to move, the frozen-in condition leads to the

bending of magnetic field lines. In the second example a plasma moves to a

region of magnetic field whose field lines are subsequently bent in order to

conserve the, initially, zero magnetic flux through the leading surface of the

intruding plasma.

7.3 Archimedean spiral

Let us study a solar wind plasma flowing at speed v radially outward from a

point at the solar equator whose longitude (with respect to some arbitrary

direction) is φ0. Then, after time t, the polar coordinates of plasma in the

rotating coordinate system are

r = v · t+ r0 (7.13)

φ = Ω · t+ φ0 (7.14)

where Ω is the angular velocity of solar rotation. Eliminating time t, one

obtains

r = v · φ− φ0Ω

+ r0. (7.15)

This equation which relates the radial distance of the plasma with the lon-

gitude of its rotating source, is called the Archimedean spiral (see Fig. 7.7).

The situation is analogous to a spinning garden sprinkler whose emitted

water forms a similar spiral.

Next we will study what kind of structure the IMF moving with solar

wind will attain. Remaining still in the equatorial plane for simplicity, the

velocity vector and the magnetic field only have the r- and φ-components:

v = (vr, 0, vφ) (7.16)

7.3. ARCHIMEDEAN SPIRAL 101

Figure 7.6: An illustration of the ”frozen-in” field concept. (a) A magneticfield B is assumed to be penetrating a region of highly conducting plasma.(b) When the plasma starts to move, the magnetic field lines will be ”frozen-in” and follow the motion of the plasma. (c) A highly conducting plasma isapproaching an area of magnetic field. (d) Due to the high conductivity thefield cannot penetrate into the plasma and is pushed ahead of the plasmablob.


Figure 7.7: The solar wind plasma streams radially out from a rotating Sun,and its motion can be described as an Archimedian spiral (garden hose). Atthe position of the Earth the angle (δ) between the plasma velocity and theSun-Earth line is close to 45o. The Earth’s orbit and the eastward directionare indicated.


B = (Br, 0, Bφ) (7.17)

and their absolute values only depend on the distance r:

B = |B| = B(r) (7.18)

v = v(r). (7.19)

The magnetic field must always fulfill the following Maxwell equation

(sourcelessness of magnetic flux):

∇ · B = 0 (7.20)

The divergence of of a vector field F in spherical coordinates is

∇ · F = 1

r2∂

∂r(r2Fr) +

1

r

∂Fθ∂θ

+cot θ

rFθ +

1

r sin θ

∂Fφ∂φ

(7.21)

In the present case we have an axially symmetric case ( ∂∂φ = 0) and also

Bθ = 0, so only the first term in (7.21) applied for B remains:

1

r2∂

∂r(r2Br) = 0 (7.22)

or

r2Br = r20B0 = constant (7.23)

This equation only says that a constant amount of magnetic flux flows

through the surface of a sphere with any radius r. One can solve for Br:

Br = B0

µr0r

¶2(7.24)

In a steady (temporally constant) flow

∂B

∂t= 0 (7.25)

and so the frozen-in field equation (7.9) attains the form:

∇× (v × B) = 0 (7.26)


The vector v × B only has a component in the direction of θ:

v × B = (0, vφBr − vrBφ, 0) (7.27)

and its curl only has a component in the direction of φ which can be trans-

formed, using eq. (7.26), in the form:

1

r

∂

∂r

hr(vφBr − vrBφ)

i= 0. (7.28)

Integrating this we obtain

r(vφBr − vrBφ) = constant. (7.29)

If the magnetic field is initially radial

Bφ0 = 0, Br0 = B0 (7.30)

eq. (7.29) can be cast in a form

r0vφ0B0 = rvφBr − rvrBφ. (7.31)

The azimuthal velocity at distance r0 due to the Sun’s rotation is

vφ0 = r0Ω (7.32)

whence eq. (7.31) becomes

r20ΩB0 = rvφBr − rvrBφ (7.33)

from which Bφ can be solved:

Bφ =rvφBr − r20ΩB0

rvr=vφBr − rΩ(r0r )2B0

vr(7.34)

or, using eq. (7.24),

Bφ =vφ − rΩvr

·Br. (7.35)


Far from the Sun rΩ >> vφ, whence

Bφ = −rΩvrBr = −r

20Ω

rvrB0. (7.36)

Since vr is roughly constant far from the Sun, we see that the azimuthal com-

ponent of the IMF decreases with increasing r as 1/r, i.e., much slower than

the radial component Br which decreases as 1/r2 (eq. (7.24)). Accordingly,

far from the Sun the IMF turns more and more azimuthal.

Let us determine the angle δ, which the IMF makes with the radius

vector:

tan δ =|Bφ|Br

. (7.37)

Using eqs. (7.24) and (7.36) we find

tan δ =r20Ω

rvrB0 · 1

B0(r0r )2=rΩ

vr. (7.38)

At large distances from the Sun tan δ → ∞, i.e., δ → 90. As the Earth’s

orbit tan δ ≈ 1, i.e., δ ≈ 45. This prediction is in a good agreement withthe average angle of the observed IMF.

Note that, remembering the radial dependence of the two IMF compo-

nents and the average value of IMF intensity and angle δ at one distance

(e.g., at 1 AU), the corresponding IMF values can be easily obtained at any

distence from the Sun.


Chapter 8

Solar modulation of galacticcosmic rays

GCR are influenced by the solar wind and IMF when entering the he-

liosphere. This influence which is seen, e.g., in the change of GCR intensity

and spectrum is called the solar (or heliospheric) modulation of GCR. The

amplitude of modulation depends on the level of solar activity, resulting in

the 11-year cyclicity of GCR intensity.

The amplitude of modulation is very different for different GCR energies.

For instance, the modulation is only a few percent over the solar cycle for

particles with an energy of several tens of GeV/nucleon, while it can be a

factor of 100 for 300 MeV particles.

The theory of solar modulation is well developed by now although all

details are not yet completely understood. All modulation studies are based

on the (now standard) transport theory suggested by Parker in 1965. The

transport equation is written as follows:

∂f

∂t= −(V+ < vD >) ·∇f +∇ · (K(S) ·∇f) + 1

3(∇ ·V) ∂f

∂lnP, (8.1)

107

108CHAPTER 8. SOLARMODULATIONOFGALACTIC COSMIC RAYS

where f(r, P, t) is the distribution function of GCRs (the phase space density

of GCRs, or the number of particles per unit volume of phase space averaged

over particle directions), P is rigidity, r is distance from the Sun and t is

time.

The differential flux is then obtained as jE = P2f and gives the number

of particles in units of m−2 sr−1 s−1 MeV−1. V is the radially directed solar

wind velocity. (Usually V is taken to be 400 km/s in the ecliptic plane and

is assumed to increase with increasing latitude). K is the so called diffusion

tensor, which can be divided into a symmetric part K(S) and antisymmetric

part K(A). The symmetric part relates to diffusion and the antisymmetric

part describes the gradient and curvature drifts. The vector

< vD >= ∇×K(A) · BB

(8.2)

is the pitch-angle averaged guiding center drift velocity.

The first term on the right-hand side of Eq. 8.1 describes the outward

convection of GCR due to the solar wind and the GCR drift due to the

curvature and gradient of IMF. The second term describes the diffusion,

and the third term the adiabatic energy loss.

In the coordinate system determined by IMF, the diffusion tensor can

be written in the form

K =

¯¯κk 0 00 κ⊥ κT0 −κT κ⊥

¯¯ = K(S) +K(A)

The symmetric part of the diffusion tensor includes the diffusion coefficients

parallel (κk) and perpendicular (κ⊥) to the mean magnetic field. The anti-

symmetric term κT represents the drift coefficient.

Usually, the transport equation is solved numerically starting from a

model interplanetary spectrum of GCR particles at the outer boundary of

8.1. SPHERICALLY SYMMETRIC FORCE-FIELD APPROXIMATION109

the heliosphere. Modulation parameters are then determined by fitting the

model to observational data. Usually the heliosphere is considered to be a

sphere with a radius of about 100 AU. (However, it has been recently shown

that the shape of the heliosphere may significantly deviate from the sphere.)

The transport equation (8.1) is difficult to solve, and usually this is done

by means of finite difference techniques. This method is quite developed

and sophisticated (3D time-dependent model) but it cannot deal with in-

finite derivatives (δ-functions in spatial/time domains, e.g., shocks, or in

energy spectrum, e.g., monoenergetic fluxes). Recently, an alternative tech-

nique has been developed. This is the stochastic simulation technique which

allows for step- or δ-function like changes and can trace individual particles

inside the heliosphere. The latter technique is more difficult numerically and

requires more CPU time.

There are some simplified approaches to the theory of CR propagation

in the heliosphere that are discussed in this Chapter.

8.1 Spherically symmetric force-field approxima-tion

Let us consider a very simple but still interesting approximation of GCR

modulation which can be solved analytically. Assuming spherical symme-

try in GCR distribution, the terms in Eq. (8.1) are greatly simplified, in

particular the diffusion tensor is symmetric in this case, < vD >= 0 (Eq.

8.2). Then the transport equation (8.1) can be written in the spherically-

symmetric form as

∂f

∂t= −V ∂f

∂r+1

r2∂

∂r

µr2κ

∂f

∂r

¶+1

r2∂

∂r(r2V )

P

3

∂f

∂P(8.3)

Gleeson and Axford showed in late 1960s that under some reasonable

assumptions (e.g., constant solar wind speed, roughly power-law differential

energy spectrum of particles, slow spatial change of f , etc.) one can simplify


Eq. 8.3 even further. First, let us consider a steady-state case with ∂f∂t = 0.

Next, assuming V and κ to be constant with radius, Eq. 8.3 can be written

as follows:

κ∂2f

∂r2+2κ

r

∂f

∂r− V ∂f

∂r+2V P

3r

∂f

∂P= 0. (8.4)

Let us now estimate the relative importance of the first three terms of

Eq. 8.4. First, we assume a simple diffusion case, i.e., the inward diffusive

flux is equal to the outward convection flux:

κ∂f

∂r= V f (8.5)

Therefore, at zero approximation the shape of f(r) is

f = fo · expµV

κr

¶,

whence∂f

∂r∝ V

κf

∂2f

∂r2∝ V

2

κ2f (8.6)

Substituting (8.6) into (8.4), we can estimate the first three term to be of

the order of V2

κ f ,2Vr f and

V 2

κ f , respectively.

The relative importance of the first two terms is Vκ vs.2r . For diffusion,

κ = 13λv, where λ ≈ 1 AU for 1 GeV protons, and we should therefore

compare Vc and

6λr . One can see that

V

c= 1.3 · 10−3 <<

6λ

r= 0.1− 10

Therefore, first and third terms of Eq. 8.4 can be neglected when compared

with the second one.

Thus, we came to the so-called force-field approximation:

∂f

∂r+V P

3κ

∂f

∂P= 0 (8.7)

Then the validity of the assumptions can be verified since now

r

f

∂f

∂r<< 1.

8.1. SPHERICALLY SYMMETRIC FORCE-FIELD APPROXIMATION111

The solution of the partial differential equation (8.7) can be presented in

the form of characteristic curves, which are lines of constant f in the (r, P )

plane. Along these lines the following condition must be fulfilled

df

dr=

∂f

∂r+dP

dr

∂f

∂P= 0 (8.8)

From (8.7) and (8.8) one can see that the characteristic curves in this case

are given by the expressiondP

dr=V P

3κ. (8.9)

Following the so called quasilinear theory, we can take

κ = κ0 · βP, (8.10)

where β = v/c is the particle velocity in the units of the light speed. Sub-

stituting (8.10) into Eq. 8.9, one can obtain that

dP

dr=

V P

3κ0βP=

V

3κ0β

and then

βdP =V

3κ0dr.

From the relativistic conversions for a proton (and expressing energy in

MeV and rigidity in MV) E =pP 2 + T 2o one can obtain that β · dP = dE.

Therefore,

dE =V

3κ0dr (8.11)

which can be solved in the following form

ER − E(r) = TR − T (r) = Φ · R− rR− 1 (8.12)

where TR and T (r) is the kinetic energy of a particle at the distance R and

r in units of AU, respectively; R is the outer heliospheric boundary, and Φ

is the so called modulation strength:

Φ =V (R− 1)3κ0

(8.13)


For r = 1 AU, Eq. 8.12 yields that

T (1 AU) = TR − Φ

which means that the modulation strength describes the loss of kinetic en-

ergy from the outer limit to 1 AU.

Moreover, since f is constant on the characteristic curve, we have

f1AU (TR −Φ) = fR(TR). (8.14)

Note that the radial dependence is denoted here as a subscript so that, e.g.,

f1AU (TR − Φ) = f(1AU, TR − Φ). (8.15)

Using P 2 = T (T +2To), one can find that the corresponding differential

flux, j = P 2f , is given at 1 AU as:

j1AU(TR − Φ) = jR(TR)(TR − Φ)(TR − Φ+ 2To)TR(TR + 2To)

(8.16)

where To = 938 MeV is the rest energy of a proton.

Thus, for fixed Φ, one can easily calculate the spectrum of particles at the

Earth’s orbit once the LIS spectrum is known. This approximation works

quite well for high energy particles.

However, from the observations of the mean free path of solar cosmic

rays, the diffusion coefficient is known to be rigidity independent for low

energies:

κ = κ0 · βP, for P > Pc (8.17)

κ = κ0 · βPc, for P < Pc

where Pc ≈ 1 GV. In this case, the equation for the characteristic curve,

f = constant, attains the form (exercise)

dP

dr=

V

3κ0PcE (8.18)

8.2. DIFFUSION-DOMINATED APPROACH 113

where E is the particle’s total energy.

The solution of this equation for r = 1 AU is

ER −Ec = Pc · lnµP1AU + E1AUPc + Ec

¶+ Φ, (8.19)

where rigidity is expressed in MV, and energy in MeV, respectively. Thus,

the expression for P1AU can be obtained in the form of

P1AU =1

2

³φ− T 2o φ

´(8.20)

where φ(TR, Pc,Φ) = (Pc + Ec) · exp³TR−Tc−Φ

Pc

´Note that the force-field approximation is valid in the energy range above

a few hundred MeV.

8.2 Diffusion-dominated approach

This approach is based on the assumption that the diffusion coefficient de-

pends inversely on large-scale fluctuations of the magnetic field strength.

This agrees with the observations carried out by Voyager 1 (V1), Voyager

2 (V2) and Pioneer 10 (P10) spacecraft. It was shown that large decreases

in CR intensity can be associated with regions where the magnetic field is

enhanced with respect to the standard spiral magnetic field which, in the

ecliptic plane and for V=400 km/s is

BP = 4.75 ·√1 + r2

r2

where r is the radial distance in AU.

Regions of enhanced magnetic field, B > BP , are called interaction re-

gions (IR), while regions with relatively small magnetic field, B < BP , are

called rarefaction regions (RR). Although such a decomposition of the inter-

planetary magnetic field is an oversimplification which does not fully take


into account the actual multi-fractal structure, it is still a very useful ap-

proach. At the distance of 1 AU two types of IRs can be distinguished:

corotating IRs associated with corotating streams and transient IRs associ-

ated with transient ejecta like CMEs.

The IRs evolve dramatically with the distance from the Sun. The struc-

ture of the solar wind changes qualitatively at the distance between 5 AU

and ∼15 AU. Isolated interaction regions may grow in size (both radial andlatitudinal extent) with increasing distance. Neighboring IRs may coalesce

to form larger regions of enhanced magnetic field, called merged interaction

regions (MIR). So, MIRs result from the dynamical merging of interaction

regions formed between solar wind streams of different velocities, or shock

waves from solar flares, or CMEs.

In brief, three main types of MIRs can be identified:

(i) Global MIR is a shell-like quasi-spherical MIR extending 360 in

heliolongitude around the Sun in the ecliptic plane and at least ±30 inheliolatitude. Global MIRs are mostly responsible for the long term (e.g.,

solar cycle related) decrease of CR intensity. Global MIRs are produced

by systems of transient flows. However, not every system of transient flow

produces a global MIR.

(ii) Corotating MIR is a spiral MIR produced by the coalescence of coro-

tating IRs. They can produce several successive decreases of CR intensity

(often with a period close to the solar rotation period) but they are not

strong enough to significantly influence upon the long-term modulation.

(iii) Local MIRs are not corotating and have a more limited longitudinal

and latitudinal extent. They are also formed by the interaction of transient

flows.

MIRs are considered to propagate with a constant solar wind velocity,

leading to propagating ”barriers” against inward diffusion.

8.3. DRIFT-DOMINATED APPROACH 115

The effect of MIRs is taken into account as a decrease of the diffusion

coefficient due to stronger scattering of particles in MIRs:

κ = D(B

BP)−α, (8.21)

where D and α ∼ 1 are constants. The value of B is taken from spacecraft

observations (usually V 2). This approach shows generally a good agreement

with CR intensity variations but has some weak points: (i) the results are

valid mainly in the ecliptic plane, (ii) the models assume the solar wind

velocity to be constant in the whole heliosphere, (iii) disturbances of the

magnetic field are assumed to propagate unchanged, (iv) inside ∼10 AUthe results are less reliable because MIRs are not completely formed at this

distance, and (v) inhomogenities of solar wind and magnetic field cannot be

taken into account.

8.3 Drift-dominated approach

Another direction of modulation models is to consider the effect of large-

scale drifts on the transport of CRs. In the heliosphere the most important

are the curvature and gradient drifts.

Curvature drift results from the centrifugal force that the particle expe-

riences when traveling along a curved magnetic field line. The drift velocity

is therefore proportional to the factor (centrifugal force normalized by local

gyrofrequency):

vcurv ∝v2k

Rcurv ·B

The gradient drift results from the change in the particle’s gyroradius

during one rotation because of the changing magnetic field intensity. This

is particularly efficient around the heliospheric current sheet.


B -

B +

V drift

Figure 8.1: Sketch of the particle drift in the heliospheric neutral sheet.

Drifts depend on the particle’s pitch angle. For an isotropic particle

distribution fo, the average drift can be estimated for the mean Parker

spiral as

vd =cvp

3q

·∇× Bo

B2o

¸(8.22)

Then this effective drift velocity is treated as ”convection” of particles in the

transport equation. The importance of drifts for heliospheric modulation

was shown in late 1970s.

A major role in this approach is played by the heliospheric current sheet.

When IMF is directed towards the Sun in the northern hemisphere (so called

negative solar polarity; A < 0), positively charged particles can drift to-

wards the Sun along the sheet (Fig. 8.1). During positive IMF polarity,

the drift sweeps positive particles away from the Sun along the sheet. The

drift behaviour of negative particles is opposite to that of positive particles.

Therefore one often combines the particle charge and IMF polarity together

to form the polarity factor qA. The HCS drift effect depends on this factor.

The drift model can reproduce the observed GCR modulation only for

8.4. THE MODULATION EFFECT 117

periods when the waviness of HCS is small or moderate, i.e., when the tilt

angle α ≤ 35, and only during negative polarity conditions (qA < 0).

The drift models can also explain the plateau-like CR intensity behaviour

during positive polarity minima (qA > 0) when CRs are considered to come

mostly from polar to equatorial regions. Of course, drifts do not dominate

when the HCS is largely disrupted by a system of transient ejecta.

More recent and sophisticated models of CR modulation combine all the

mechanisms described briefly above, explaining pretty well the observed CR

intensity at different heliodistances and during different phases of the solar

magnetic cycle. In general, the modulation inside ∼10 AU (so called innerheliosphere) is driven mainly by transients and small scale IRs. At larger

distances, drift is responsible for the long-term modulation during periods of

low and moderate solar activity. When solar activity is moderate, both drift

and MIR/RR effects should be taken into account. Finally, during periods of

high solar activity the CR modulation is mostly driven by MIR/RR effects.

8.4 The modulation effect

In the following, some effects of modulation (only convection and diffusion,

no drifts included) are shown, as calculated using stochastic simulation tech-

niques in a 2D case. The results are shown for two different levels of modu-

lation:

Weak modulation corresponds approximately to the heliospheric condi-

tions in 1977, i.e., near minimum solar activity, when the heliosphere is

quiet.

Medium modulation corresponds to the heliospheric conditions in 1992,

i.e., in the middle of declining phase of solar cycle 22.


Figure 8.2: Average time spent by GCR protons in the heliosphere beforereaching the Earth as function of energy for medium (1992) and weak (1977)modulation.

Fig. 8.2 shows the average time needed for GCR particle to reach from the

termination shock to the Earth’s orbit as function of energy. One can see

that this time varies from a fortnight (10 GeV for weak modulation) to half

a year (300 MeV protons for medium modulation). This results in the well

known time lag between solar activity and corresponding long-term changes

in GCR intensity.

Note that only a small fraction of cosmic rays entering the heliosphere

finally do reach the Earth’s orbit, because of the geometric factor and

IMF/solar wind effects. The results discussed in this section only deal with

particles which succeed to reach the Earth’s orbit.


Figure 8.3: Energy spread of monoenergetic (δ-function in energy) fluxesfor fixed initial proton energy, T= 0.3, 0.7, 1, 3, 10 GeV, for medium (1992,panel a) and weak (1977, panel b) modulation.


Figure 8.4: Spectra of galactic cosmic rays. Thick line presents the localinterstellar spectrum (LIS), solid line the modulated spectrum at 1 AU forweak modulation, and dashed line the modulated spectrum at 1 AU formedium modulation.

Fig. 8.3 presents the energy losses and the energy spread of a monoen-

ergetic particle flux for medium and weak modulation. GCR protons with

the same initial energy (T = 0.3, 0.7, 1, 3, or 10 GeV) enter the heliosphere,

and their energy spectrum at 1 AU is shown in the Figure.

High energy (10 GeV) protons only lose a few percent of their initial

energy during propagation in the heliosphere. For lower energy protons

(below 1 GeV), the fraction of lost energy becomes significant even for weak

modulation. Thus, energy losses of GCR when propagating across the IMF

are very important in the lower energy part of the spectrum.


The modulated spectrum of GCR at 1 AU is shown in Fig. 8.4 for weak

and medium modulation together with the local interstellar spectrum (LIS).

While LIS is continuously decreasing with energy, the GCR spectrum has

a maximum at 0.1-1 GeV. Note however, that solar cosmic rays dominate

over GCR in the low energy part of the spectrum (see Fig. 4.5).

Fig. 8.5 shows the history of sample protons with 1 GeV and 9.2 GeV

initial energy during medium modulation. Time evolution of the heliocentric

distance of protons, their energy losses as well as 2D trajectories are shown

in the Figure. One can see that particles diffuse long at middle heliocentric

distances until they reach the polar region. After that, they move fairly

rapidly to the distance of 1 AU (or escape from the heliosphere).


0

10

20

30

40

50

60

70

80

90

100

0 50

R, au

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 50

T, Ge

V

0

10

20

30

40

50

60

70

80

90

100

0 50 100

0

10

20

30

40

50

60

70

80

90

100

0 10 20

time, days

R, au

8

8.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

10

0 10 20

time, days

T, Ge

V

0

10

20

30

40

50

60

70

80

90

100

0 50 100

Figure 8.5: Tracing of sample protons with the initial energy of 1 GeV (upperpanels) and 9.2 GeV (lower panels) for medium modulation. Panels fromthe left to the right present the heliodistance as function of time spent inthe heliosphere; energy losses as function of time; trajectory of the particlein the heliosphere.

Chapter 9

Variations of Cosmic RayIntensity

CR intensity is not constant but changes continuously at different time

scales. Two groups of CR intensity variations are shown in Tables 9.1-9.2.

The first group of variations in CR intensity detected at the Earth in-

cludes the variations of terrestrial origin (Table 9.1). Seasonal and diurnal

variations are due to the differences in atmospheric structure between winter

and summer seasons and daytime and night-time, respectively. This effect

is significant for the muon component but small for the neutron component.

The asymmetric shape of the Earth’s magnetosphere results to a small

diurnal change of the local geomagnetic cut-off and, correspondingly, to a

small diurnal variation of CR intensity on the Earth’s surface.

We are more interested here in the extra-terrestrial variations of CR

intensity, i.e., CR variations whose origin is outside the Earth’s magne-

tosphere. There are periodic and sporadic extra-terrestrial CR variations.

Periodic extra-terrestrial variations include diurnal, solar rotation re-

lated and solar cycle related variations. The solar rotation related varia-

tions last, on an average, 27 days. They are due to the effect of strong,

123

124 CHAPTER 9. VARIATIONS OF COSMIC RAY INTENSITY

Table 9.1: Cosmic ray intensity variations: Terrestrial effects

type amplitude nature

Periodic variation

seasonal < 1 %Variations of the absorption of secondaryparticles in the atmosphere due to sea-sonal changes of the atmospheric struc-ture.

diurnal < 1%Variations similar as above but due tothe day-night difference in the the at-mospheric structure.

diurnal small Asymmetry of the magnetosphere leadingto a daily variation of the local geomag-netic cutoff.

Sporadic variation

increase during amagnetic storm

up to 10% Decrease of local geomagnetic cutoff dueto the disturbed magnetosphere.

125

Table 9.2: Cosmic ray intensity variations: Extra-terrestrial effects

type amplitude nature

Periodic variation

11- and 22-year up to 30 % Solar modulation of GCR in the he-liosphere.

27-day < 2% Long-lived longitudinal asymmetry inIMF or solar wind structure.

diurnal few %Anisotropy of CR fluxes due to convec-tion by solar wind and diffusion along IMFlines.

Sporadic variations

GLE 1-300% Increase of CR intensity due to arrival ofsolar cosmic rays.

Forbush decreases up to 30% GCR decrease due to the shielding by aninterplanetary shock passing the Earth.

increase beforeForbush decrease

< 2% CR increase due to “collection” of CR par-ticles in front of the interplanetary shockcausing a Forbush decrease.

magnetic cloudeffect

few % GCR decrease due to the shielding by amagnetic cloud passing the Earth.


Figure 9.1: Diurnal variations of CR intensity recorded by Oulu NM.

long-lived sunspot groups or due to persistent streams of fast solar wind

from longitudinally asymmetric coronal holes.

Sporadic variations include GLEs, Forbush decreases and decreases due

to magnetic clouds.

9.1 Extra-terrestrial diurnal variations

Diurnal variations are often nearly sinusoid-like variations of GCR intensity

with an amplitude of 1-2% (see Fig. 9.1). The largest contributions to the

diurnal variation come from extra-terrestrial causes.

The diurnal variation is mainly due to a local anisotropy of CR fluxes.

Let us consider the idealised picture of Fig. 9.2. The anti-sunward convection

and the spiral-directed diffusion affect the GCR flux, causing a minimum

in the post-midnight and a maximum in the early afternoon LT sector (see

also Fig. 9.1).

9.2. SPORADIC VARIATIONS 127

The phase and amplitude of the diurnal variation (the direction of the

GCR anisotropy) may differ from this idealised picture because of drifts,

local latitudinal gradients of CR intensity, varying level of diffusion, changing

solar wind conditions, etc. (The Earth’s orbital motion also gives a minor

contribution).

9.2 Sporadic variations

Forbush decreases (Fig. 9.3) are sudden decreases (up to 30% during several

hours) of CR intensity followed by a gradual recovery during several days

to weeks.

A Forbush decrease is due to an interplanetary shock passing the Earth’s

orbit and producing an effective barrier of intense magnetic field to cosmic

ray particles. Such a shock “collects” CR particles in front of it because of

enhanced scattering of the particles. This is often seen as a small increase

of CR intensity immediately before a Forbush decrease.

Sporadic CR intensity variations are also caused by magnetic clouds

passing the Earth (Fig. 9.4). They result in a steady, small decrease of CR

intensity of a few percent during several days.

9.3 Solar cycle variations

CR intensity depicts an 11-year variation in anti-phase with solar activity

(Fig. 9.5). There is a time lag between the changes of solar activity and the

corresponding changes in CR intensity. This time lag is due to the large size

of the heliosphere and the finite propagation time of the solar wind (and

the IMF disturbances moving with it), as well as the finite diffusion time of

GCR particles (see earlier discussion).

The overall time lag is several months but the momentary time lag varies

in time (see Fig. 9.6) from zero (or even negative) values to about 2 years.


Figure 9.2: A scheme of the diurnal anisotropy of galactic cosmic rays.

9.3. SOLAR CYCLE VARIATIONS 129

Figure 9.3: A Forbush decrease, a GLE and the diurnal variation of CRintensity as recorded by Oulu NM in July 2000.

Figure 9.4: A magnetic cloud passing the Earth, as observed by Oulu NMin March 2000.


0

100

200

1950 1960 1970 1980 1990 2000

su

nsp

ot

nu

mb

ers

70

80

90

100

1950 1960 1970 1980 1990 2000

co

un

tra

te,p

er

cen

t

Climax

Huanc/Hal

Oulu

b)

a)

Figure 9.5: Solar modulation of cosmic rays at neutron monitor energies.(a) Monthly sunspot numbers as index of solar activity. (b) Monthly countrates of different neutron monitors.

9.3. SOLAR CYCLE VARIATIONS 131

-30

-20

-10

0

10

20

30

1950 1960 1970 1980 1990

tim

ela

g,

mo

nth

s

Climax

Huancayo

Oulu

19 20 21 22

Figure 9.6: Momentary time lag in months between solar activity cycle andthe corresponding cosmic ray cycle.


One can also see a 22-year variation of CR intensity, e.g. in the different

shape of CRmaxima during positive and negative polarity minima (Fig. 9.5),

as well as in the significantly different time lags at these times (Fig. 9.6).

The 22-year cycle in cosmic rays can be understood in terms of the drift

effects around solar minimum times.

Chapter 10

Cosmic Rays and the Earth

10.1 Atmospheric cascade

The matter weighted mean free path of energetic protons (neutrons) in the

air is about 100 g · cm−2 (140 g · cm−2). This mean free path is mainlydetermined by the nuclear collisions of the proton (neutron) with the nuclei

of atmospheric atoms and molecules.

On the other, the amount of matter in the Earth’s atmosphere is 1033

g ·cm−2, leading to the 1 atm pressure of the normal atmosphere. Therefore,it is very improbable that a primary CR could reach the Earth’s surface.

Instead, they suffer a series of successive collisions and interactions, form-

ing the so-called atmospheric cascade (Fig. 10.1). The cascade consists of

three main components.

One is called the “soft” or electromagnetic component and it consists of

electrons, positrons and photons (electromagnetic quanta).

The second component is called the “hard” or muon component, con-

sisting of muons. Note that sometimes this component is, unfortunately,

also called the meson component. This erroneous naming dates back to

early history of CR research when the properties of the muon and other

elementary particles were not yet well known. We know now that muons

133

134 CHAPTER 10. COSMIC RAYS AND THE EARTH

Table 10.1: Active particles in a cosmic ray cascadeInteraction Atmospher.

Particle electromagn. strong weak mass (MeV) lifetime absorb.length(g/cm2)

Pion x x ≈134 ≈26 ns ≈115Muon x x ≈106 ≈ 2 µs ≈260Neutron x 932 12 min ≈140Proton x x 938 stable ≈110Electron x 0.511 stable ≈100Photon x stable

do not interact strongly but belong, together with electrons, tau leptons

and neutrinos, to the group of weakly interacting particles that are also

called leptons. On the other hand, mesons are strongly interacting particles

(hadrons) with integer spin number. The most common meson produced in

a CR shower is the pion. Pions are very short-lived and decay before reach-

ing the ground. Charged pions mainly decay to muons, producing most of

those muons observed on the ground. Taking this into account the term

“meson” component is partially justified.

The third CR component is the nucleonic component which, on the

Earth, mostly consists of suprathermal neutrons. The characteristics of

the constituent particles are given in Table 10.1.

The three components of the cascade have different spatial (horizontal)

widths. The relative widths are shown in Fig. 10.2. The absolute widths

depend on the energy of incoming particle: the more energetic the particle

is, the wider cascade it generates. Recently, muon showers of many tens

of kilometers wide have been detected, implying that the energy of the CR

particle was of the order of 1020 eV.

The flux of different components at the Earth’s surface is shown in

Fig. 10.3. The neutron component containing of suprathermal neutrons is

very significant. This component is detected by most ground based cosmic

10.1. ATMOSPHERIC CASCADE 135

Figure 10.1: A scheme of the atmospheric cascade.


Figure 10.2: Lateral spread of the three components of the atmosphericcascade.

10.1. ATMOSPHERIC CASCADE 137

Figure 10.3: Energy spectra of different components of the atmosphericcascade at the sea level in NYC.


ray stations. Therefore they are also called neutron monitors.

Another important component is the muon component. It dominates the

flux at energies above 100 MeV. At lower energies the flux of muons decreases

fairly fast because of their short lifetime of only 2 µs. (Relativistic muons

can travel longer distances than nonrelativistic muons due to faster speed

and time dilatation.)

Muon flux is also sensitive to the atmospheric structure, in particular

to the altitude of the first collision. The higher, on an average, the first

collision of the primary CR particle occurs, the less of muons are seen on

the ground. This leads, e.g., to the above mentioned diurnal and seasonal

changes in the muon component due to the changes in the atmosphere.

The flux of protons is similar to that of neutrons at energies above 1

GeV but is much smaller at lower energies because thermal protons cause

ionisation (while neutorons do not).

10.2 Magnetospheric propagation and geomagneticcutoff

Since CR are charged particles, their propagation close to the Earth is af-

fected by the geomagnetic field. The direction of a particle entering the

atmosphere is often very much different from the original direction of the

particle outside the Earth’s magnetosphere in the interplanetary space. A

sample of a particle trajectory is shown in Fig. 10.4

The idea of finding the original direction of the CR particle by trajec-

tory tracing results in the concept of the asymptotic direction (also called

asymptotic cone). This is the direction that the incoming CR particle must

have in the interplanetary space in order to reach a certain location on the

Earth.

10.2. MAGNETOSPHERIC PROPAGATIONANDGEOMAGNETIC CUTOFF139

Figure 10.4: A sample trajectory of CR particle’s trajectory in the Earth’smagnetosphere.


Often one calculates the asymptotic directions in a dipolar magnetic field

for those protons that enter vertically the atmosphere above the station at

the altitude of 20 km. The asymptotic directions are calculated for protons

in the energy range between 1—10 GeV and plotted as a curved line in the

latitude-longitude plot. (Note that the asymptotic direction points in the

direction from where the particle is coming. The particle’s velocity must

therefore have an opposite direction.)

Examples of asymptotic directions are shown in Fig. 10.5. The asymp-

totic directions are typically around the equator, somewhat east of the sta-

tion because of the left-handed curvature of the positively changed CR pro-

ton. The 1 GeV end of the asymptotic curve is typically more eastward and

more southward than the 10 GeV end because the more energetic particles

fly more directly to the station.

Figure 10.6 depicts the asymptotic directions of some NM stations in the

GSE coordinate system at 14 UT on May, 2, 1998. The forward end (plus)

and the tail end (cross) of the direction of the momentary IMF vector are

depicted in the figure, together with the equi-pitch angle curves with respect

to the IMF. Such plots can be used to study the question what must the

pitch angle of those particles be that can reach any of the NM stations. The

plot reveals, e.g., that only particles whose velocity is roughly perpendicular

with respect to the IMF can be seen at Oulu.

The geomagnetic field prevents low-energy particles from reaching the

ground level. This leads to the concept of geomagnetic cutoff which means

a rigidity threshold for CR particles that can reach a certain geographical

location. As a first approximation, the geomagnetic cutoff rigidity can be

estimated by a simple empirical formula

Pcut(γ,λ) = 60

Ã1−p1− cosγcos3λ

cosγcosλ

!2(10.1)

where Pcut is expressed in GV, λ is the geomagnetic latitude of the station,

10.2. MAGNETOSPHERIC PROPAGATIONANDGEOMAGNETIC CUTOFF141

Figure 10.5: Asymptotic directions (curved lines) for several NMs stations,such as Apatity (Ap), Iniuvik (In), Thule (Th) and Tixie (Ti).


Figure 10.6: Asymptotic directions for NMs Apatity (Ap), Goose Bay (G-B), Iniuvik (In), McMurdo (M-M), Oulu (Ou), South Pole (S-P), Thule (Th)and Tixie (Ti) for 14 UT 02.05.1998. Plus and cross denote the IMF forwardand tail ends.

10.3. TERRESTRIAL EFFECTS OF COSMIC RAYS 143

and γ is the angle between the incoming particle velocity and the direction

of geomagnetic east.

Note that the rigidity is smaller for those particles coming from the east

(whose angle γ is larger than 90). This reflects the fact that the particles

which have a velocity close to the asymptotic direction find it easier to

approach the station.

For γ = 90, cos(γ) = 0, the expression for the geomagnetic cutoff

rigidity becomes very simple

Pcut = 15(cosλ)4

and is called the vertical geomagnetic cutoff. The distribution of the vertical

geomagnetic rigidity cutoff is shown in Fig. 10.7.

Note that the geomagnetic coordinates do not coincide with the geo-

graphical coordinates because of two effects. First, the axis of the magnetic

dipole is inclined with respect to the Earth’s rotation axis by about 11.2o.

Second, the center of the dipole is shifter with respect to the Earth’s center

by about 534 km in the direction of ≈ 22 North and ≈ 144 East (to-

wards India). On the opposite side of the globe, there is a region of reduced

geomagnetic field (enhanced CR flux) which is called the South-Atlantic

Anomaly.

The total CR intensity per neutron monitor counter as a function of the

geomagnetic cutoff rigidity is shown in Fig. 10.8.

10.3 Terrestrial effects of cosmic rays

A list of some terrestrial effects of cosmic rays is given in Fig. 10.9.


Figure 10.7: Isolines of vertical geomagnetic cutoff rigidity for the epoch of

10.3. TERRESTRIAL EFFECTS OF COSMIC RAYS 145

Figure 10.8: CR intensity per NM counter versus geomagnetic rigidity cutoffin 1992.


Figure 10.9: Terrestrial and human effects of cosmic rays.

Chapter 11

Detection of Cosmic Rays

Cosmic rays are measured at very different locations: from underground and

underwater detectors up to the far edge of the Solar system. CR particles

can be observed by the following types of interactions:

• Inelastic scattering caused by the Coulomb force between the CR par-ticle and the orbital electrons of the detector material.

• Elastic scattering of CR particle from nuclei of the detector by the

electromagnetic or strong force.

• Emission of Cherenkov radiation by the CR particle moving faster

than light in matter.

• Emission of transition radiation. Transition radiation is produced

when a charged particle passes through media of different dielectric

properties. A charged particle approaching such a boundary between

two media (e.g., from vacuum to a dielectric medium) represents to-

gether with its mirror charge an electric dipole, whose field strength

changes in time as the particle moves along and vanishes when the

particle enters the dielectric medium. This produces electromagnetic

radiation, called the transition radiation.

147

148 CHAPTER 11. DETECTION OF COSMIC RAYS

• Nuclear reactions (inelastic scattering by the strong force) between theCR particle and the detector nuclei.

• Bremsstrahlung caused by the CR particle in the detector material.

This is negligibly small for CR protons or heavier CR particles.

11.1 Space-borne detectors

Space-borne detectors can be divided into two groups by their location. One

group is located onboard satellites which have a fixed orbit near the Earth.

Such satellites are, e.g., IMP, GOES, SOHO, and AMS. The other group of

space-borne experiments is located on space probes which explore different

parts of the heliosphere. Such probes are, e.g., Pioneer, Voyager, and Ulysses

spacecraft.

As a sophisticated and state-of-the-art example, let us consider the AMS-

02 detector which is to be installed onboard the ISS in 2006. The scheme of

the detector is shown in Fig. 11.1. This detector is actually a combination

of many types of detectors. The main part is a tracker which consists of 6

orthogonal silicon strip planes (< 2 mm wide, more than 34,000 channels

for each plane) in a permanent magnetic field of about 2 T produced by a

superconductive magnet with He cooling. The tracker is able to precisely

reconstruct the trajectory of a particle in the magnetic field which allows to

determine the particle’s rigidity, mass and the incoming direction.

Additional devices are:

• Synchrotron radiation detector (SRD) measures the synchrotron radi-ation and is primarily devoted to detect electrons.

• Transition radiation detector (TRD) measures the transition radiationof particles.

11.1. SPACE-BORNE DETECTORS 149

Figure 11.1: The scheme of the AMS-02 detector.


• Time of flight (TOF) system measures the time that the particle needsto fly through the detector. This allows to estimate the velocity of the

particle and to reject fake events.

• Veto counter is a simple electronic counter used as a trigger to rejectparticles whose trajectories pass through the sides of the detector.

• Ring Cherenkov detector (RICH) measures the Cherenkov emission ofparticles in an aerogel, allowing their energy to be estimated.

• Electromagnetic calorimeter (Ecal) also measures the energy of parti-cles.

This combination allows to reconstruct the arrival direction, energy, charge

and mass of the CR particle, i.e., identify it completely. For instance, 3He+

and 4He+ can be reliably distinguished from each other at the confidence

level better than 99%.

11.2 Balloon detectors

Modern balloons allow to lift detectors to the altitude of 40-70 km. Earlier,

rather small and simple detectors (see, e.g., Fig. 11.2) were flown on balloons.

However, nowadays rather big and complicated telescopes such as the BESS

(Balloon Borne Experiment with Superconducting Solenoidal Spectrometer)

detector are flown on balloons (see Fig. 11.3). At these high altitudes,

the atmosphere above the balloon is negligible for CR, and therefore the

balloon-borne detectors can measure primary CR particles, unlike ground

based detectors. In this sense they are like low-orbit satellites, only much

cheaper and easier to operate. For instance, the first cosmic antiprotons

were discovered by the group of Prof. Bogomolov in late 1970s using a

balloon-borne spectrometer.

11.2. BALLOON DETECTORS 151

Figure 11.2: A standard radio-sonde for CR observations in the atmosphere,consisting of (a) Geiger counters, (b) radio-transmitter, (c) altitude sensor,(d) power supply.

Figure 11.3: A scheme of the BESS balloon detector.


The geomagnetic rigidity cutoff is still a significant effect for balloon

observations. Moreover, the atmospheric albedo particles (particles reflected

or scattered back into space from the atmosphere) play a role and have

to be taken into account in balloon observations. The main disadvantage

of balloon-borne experiments is that they are campaign-like experiments,

operating only for a short time interval.

11.3 Ground-based detectors

Ground based cosmic ray experiments can be divided into different sub-

groups according to the component of the atmospheric cascade (see Fig. 10.1)

that they measure.

11.3.1 Neutron monitor

The nucleonic component of the atmospheric cascade is measured by neutron

monitors (also called cosmic ray stations). The scheme of a neutron monitor

(NM) is shown in Fig. 11.4. The sensor tubes are filled with BF3 gas which

is enriched with the B10 isotope. The paraffin layer surrounding the tubes

is used as a pre-moderator decelerating atmospheric neutrons. The lead

layer decelerates neutrons further and produces still more neutrons from

the atmospheric neutrons and protons. There is also a plastic layer around

the tubes as the final moderator, making particles almost thermal (< 1 eV)

so that the cross section for neutron capture by boron is optimal inside the

counter tube:

10B + n → 7Li + α

The produced fast helium and lithium ions strip electrons from the neu-

tral atoms in the tube, producing charge inside the gas tube. The charge

avalange enhanced by a high negative voltage in the tube is detected by the

amplifier as one count in the central wire inside the tube.

11.3. GROUND-BASED DETECTORS 153

Figure 11.4: The scheme of a neutron monitor.

Recently, a new type of NM counter tube was developed. It is filled with

3He and uses the reaction

3He + n → 3H + p + γ(7.65 MeV )

These counters are presently in test and calibration phase.

There is a global network consisting of 50-70 NMs (depending on the year

since some stations have been closed down and others have been opened) lo-

cated at different geomagnetic latitudes (Fig. 11.5) and, correspondingly, at

different geomagnetic cutoff rigidities. This allows to use the network as one

unique, global CR spectrometer “spaceship Earth” (Fig. 10.8). Typically,

the bulk of CR observed at NMs are in the energy range 0.3-20 GeV. This

is also called the “neutron monitor energy range”, and it is closely similar

to the energy range of effective solar modulation of GCR.

11.3.2 Extensive Air Shower arrays

Extensive atmospheric (air) shower arrays detect the muon component of

the atmospheric cascade. Usually, it is an array of simple muon detectors

working in coincidence. The size of air shower array varies from hundreds of


Figure 11.5: The worldwide network of neutron monitors.

11.3. GROUND-BASED DETECTORS 155

Figure 11.6: A couple of units of the extensive atmospheric shower array(EASA) and the Cherenkov array in Canary Islands.

meters to tens of kilometers. A part of the EASA array in Canary Islands is

shown in Fig. 11.6. Such arrays allow to measure primary cosmic rays with

energy between 1012 − 1021 eV. The larger the array is the higher primaryenergies can be measured.

11.3.3 Cherenkov detectors

Fig. 11.6 shows also a couple of units of the Cherenkov array in Canary

Islands. Relativistic electrons and positrons, produced in the atmospheric

cascade, generate Cherenkov emission in the visible light range when propa-

gating through the air. The Cherenkov array collects such light pulses from

a large volume (thousand cubic kilometers). A similar technique is also used

to study neutrinos but then the Cherenkov light pulses are produced and

detected in deep water or ice.


11.3.4 Underground muon experiments

By underground experiments one can study the high-energy part of the

muon component. Such experiments use the good penetration capability of

muons in matter which allows to easily separate them from the other CR

components (except for neutrinos). The underground muon detector may

be either a single detector or a small array. (Note also that atmospheric,

solar and cosmic neutrinos can also be studied deep underground. However,

the size of the detector must be very large in order to compensate the small

cross section of neutrinos.)

11.4 Paleoastrophysics

Direct measurements of CR intensity have been carried out on a regular

basis only since 1930-1940s. However, it would be interesting to know the

CR intensity level on even longer time scales, in particular in order to study

the heliosphere and IMF in the past. There are two main methods to the

extend CR studies to earlier times.

11.4.1 Meteorites

The cosmic age of a meteorite, i.e., the time interval from the formation of

the meteorite as a cosmic body to the moment of their collision with the

Earth depends on the meteorite type. Stone meteorites have a typical age

of 5-40 million years. Chondrites (consisting of granules of silicate minerals)

can reach the age of up to 108 years, while iron meteorites are sometimes as

old as 109 years.

During their cosmic life time, meteorites are irradiated by cosmic rays,

resulting in the production of radioactive isotopes inside the meteorite.

These isotopes have very different decay times from a month (37Ar) to

millions (10Be) and billions (40K) of years. Comparing the abundance of

11.4. PALEOASTROPHYSICS 157

different isotopes with different decay times in a meteorite, one can estimate

the total dose and the radiation rate.

Orbits of some meteorites are rather well known from the measurement of

the last part of their trajectory or from their belonging to a large meteorite

stream (e.g., Perseides or Leonides) whereby they can be associated to a

certain region of the heliosphere.

Thus, the study of radioactive isotopes in meteorites can help in recon-

structing the average cosmic ray intensity on very long time scales.

11.4.2 Cosmogenic Isotopes

The method of cosmogenic isotopes is in principle fairly similar to the mete-

orite study but it deals with isotopes in some natural archives. In this case,

in order to reconstruct the history of CR intensity, one needs to date of

the archival samples. There are two commonly used archives of cosmogenic

isotopes.

One is the radioactive carbon 14C isotope which is mostly produced

in the atmosphere in the capture of a thermal neutron by an atmospheric

nitrogen includes:

n + 14N → 14C + p.

14C decays by β-decay with a half-life of 5370 years.

After being produced, radiocarbon is soon oxidized to a carbon dioxide

(CO2) molecule and experiences different processes of atmospheric circu-

lation and reservoir exchange (Fig. 11.7). During this cycle, it can, e.g.,

be consumed by plants or trees, and stored in the natural archive of tree

rings where it can only decay. Tree rings are a good sample of an archive

because of the possibility for an independent dating based on the counting

(and thickness) of tree rings.

The relative abundance of 14C/12C is a measure of radiocarbon varia-

tions. The radiocarbon abundance is proportional to the CR intensity at the


Figure 11.7: The scheme of the radiocarbon cycle. Numbers on reservoirsshow the normalized relative abundance (in promille) of radiocarbon for twodifferent models.

time of archive formation. The radiocarbon method allows to reconstruct

CR intensity of up to 104 years in the past. Once calibrated, the radiocar-

bon method can be used for independent dating of other samples, e.g., in

archaeology.

The other possible isotope is the 10Be isotope which can, e.g., be stored

in polar ice. This radioactive isotope of beryllium is produced in a chain of

reactions in the atmosphere between the CR particle and the atmospheric

nitrogen or oxygen nucleus.

Then, 10Be is attached to aerosols and precipitates within a couple of

months to the ground with rain or snow and may be stored in ice. The

11.4. PALEOASTROPHYSICS 159

half-life time of 10Be is ≈ 1.5 · 106 years. Since polar (e.g., Antarctic orGreenland) ice can be independently dated, this is a good tool to study

long-term variations of CR. On the other hand, berillium concentration in

ice is affected by local weather conditions which may distort short-time

variations.


Bibliography

Books

[1] V.S. Berezinskii, S.V. Bulanov, V.A. Dogiel, V.L.Ginzburg (editor), V.S.

Ptuskin, Astrophysics of Cosmic Rays, North-Holland, Amsterdam, 1990.

[2] R. Clay and B. Dawson, Cosmic Bullets: high energy particles in astro-

physics, Allen & Unwin, Sidney, 1997.

[3] L.I. Dorman, Cosmic Rays: variations and space explorations, North-

Holland Publishing Co., Amsterdam, 1974.

[4] T.K. Gaisser, Cosmic Rays and Particle Physics, Cambridge University

Press, Cambridge, 1990.

[5] P.K.F. Grieder, Cosmic Rays at the Earth, Elsevier, Amsterdam, 2001.

[6] M. Pomeranz, Cosmic Rays, Van Nostrand Reinhold Co., NY, 1971.

[7] K. Sakurai, Physics of Solar Cosmic Rays, University of Tokio Press,

Tokyo, 1974.

[8] A. E. Sandstrom, Cosmic Ray Physics, North-Holland Publishing Co.,

Amsterdam, 1965.

[9] I.N. Toptygin, Cosmic Rays in Interplanetary Magnetic Fields, D.Redel

Publ.Co., Dordrecht, 1985.

[10] A.W. Wolfendale, Cosmic Rays, George Newnes Ltd., London, 1963.

161

162 BIBLIOGRAPHY

Review papers

[11] Balogh, A., J.T. Gosling, J.R. Jokipii, R.Kallenbach, H. Kunow, (eds.)

Corotating Interaction Regions, Space Sci.Rev., 89(1/2), 1999.

[12] L.I. Dorman and D. Venkatesan, Solar Cosmic Rays, Space Sci. Rev.,

v.64, 183-363, 1993.

[13] Fisk, L.A., J. R. Jopikii, G. M. Simnett, R. von Steiger, and K.-P.

Wenzel, (eds.) Cosmic Rays in the Heliosphere, Space Sci.Rev., 83(1/2),

1998.

[14] J. A. Lockwood, H. Debrunner, Solar Flare Particle Measurements with

Neutron Monitors, Space Sci. Rev., v.88, 501-528, 1999.

[15] J. A. Miller, Particle Acceleration in Impulsive Solar Flares, Space

Sci.Rev., v.86, 79-105, 1998.

[16] K. Sakurai, High-energy Phenomena on the Sun: an Introductory Re-

view, Space Sci. Rev., v.51, 1-9, 1989.

[17] P.H. Stoker, Relativistic Solar Events, Space Sci. Rev., v.73, 327-385,

1994.

[18] I.G.Usoskin, G.A.Kovaltsov, H.Kananen, P.Tanskanen, The World

Neutron Monitor Network as a Tool for the Study of Solar Neutrons,

Annales Geophysicae, v.15, p.375-386, 1997.

Relevant web-sites

[19] Cambridge University Press Handbook of Space Astronomy and Astro-

physics (http : //adsbit.harvard.edu/books/hsaa/idx.html)

[20] Oulu Space Physics Textbook

(http : //www.oulu.fi/ spaceweb/textbook/)

BIBLIOGRAPHY 163

[21] High Energy Astrophysics at MSSL

(http : //www.mssl.ucl.ac.uk/wwwastro/lecturenotes/hea/hea.html)

[22] Solar Physics at MSSL

(http : //www.mssl.ucl.ac.uk/wwwsolar/homepage.html)

[23] Cosmic Rays at NGDC

(http : //web.ngdc.noaa.gov/stp/SOLAR/COSMICRAY S/cosmic.html)

[24] The Cosmic Web at Utah University

(http : //www.physics.utah.edu/research/cosmicweb/index.html)

[25] Martindale’s ”virtual” astronomy, astrophysics and space science center

(http : //www.martindalecenter.com/GradSpace.html)

[26] High Energy Astrophysics

(http : //dustbunny.physics.indiana.edu/ dzierba/HEPA/)

[27] Cosmic Ray Learning Center by NASA

(http : //helios.gsfc.nasa.gov/cosmic.html)

164 BIBLIOGRAPHY

Units and definitions

Units

eV (electron volt) 1 eV = 1.6 · 10−19 J = 1.6 · 10−12 erg: The energy gainedby an electron falling through a potential difference of 1 volt.

AU (astronomical unit) 1 AU = 1.495 · 1011 m: The mean distance to theSun from the Earth.

Definitions

CME Coronal mass ejection is a huge bubble of plasma ejected from the

solar corona during several hours. CMEs seem to be more related to

prominence eruptions than solar flares.

Energy By energy we mean here the kinetic energy (not the total energy)

of the particle unless specially mentioned.

GCR Galactic cosmic rays are cosmic rays of galactic or extra-galactic ori-

gin.

Gyroradius Radius of gyration, or cyclotron radius. The radius of the

circular orbit of a charged particle gyrating around its guiding center:

r =mv⊥|q|B = P⊥/B,

where m, q, P⊥, v⊥ are the mass, charge, rigidity and the velocity

perpendicular to the magnetic field line of the gyrating particle.

165

166 BIBLIOGRAPHY

IMF Interplanetary Magnetic Field lines have the average shape of an Archi-

median spiral due to radial solar wind and the solar rotation.

Pitch angle The angle α between magnetic field B and velocity vector of

a charged particle, v:

sinα =v⊥vtotal

,

where v⊥ is the velocity component perpendicular to B.

Rigidity Magnetic rigidity of a charged particle is defined as

P =p

|q| ,

where p and |q| are the momentum (either classical or relativistic)

and charge of the particle, respectively.

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