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761656S
.
Heliospheric Physics and Cosmic Rays
.
Lecture notes
Fall term 2003
Prepared by Kalevi Mursula and Ilya UsoskinUniversity of Oulu
Lectured in 2003 by K. Mursula
ii
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:
6 CHAPTER 1. INTRODUCTION TO COSMIC RAYS
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.
8 CHAPTER 1. INTRODUCTION TO COSMIC RAYS
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.
12 CHAPTER 2. GALACTIC COSMIC RAYS
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.
14 CHAPTER 2. GALACTIC COSMIC RAYS
Figure 2.3: Differential energy spectrum of galactic cosmic rays.
2.2. ENERGY SPECTRUM 15
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.
16 CHAPTER 2. GALACTIC COSMIC RAYS
Figure 2.4: Differential spectra of some spices of GCR near the Earth.
2.2. ENERGY SPECTRUM 17
Figure 2.5: The differential GCR spectrum multiplied by E2.5.
18 CHAPTER 2. GALACTIC COSMIC RAYS
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.
20 CHAPTER 2. GALACTIC COSMIC RAYS
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
2.3. ORIGIN OF COSMIC RAYS 21
Figure 2.6: The ROSAT image of the Crab nebula.
22 CHAPTER 2. GALACTIC COSMIC RAYS
Figure 2.7: A schematic side view of the Galaxy.
2.3. ORIGIN OF COSMIC RAYS 23
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.
24 CHAPTER 2. GALACTIC COSMIC RAYS
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.
28 CHAPTER 3. ACCELERATION OF COSMIC RAYS
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)
3.1. FERMI ACCELERATION 29
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)
30 CHAPTER 3. ACCELERATION OF COSMIC RAYS
Figure 3.3: Dependence of energy gain and loss upon proton’s energy.
3.1. FERMI ACCELERATION 31
-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
32 CHAPTER 3. ACCELERATION OF COSMIC RAYS
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)
34 CHAPTER 3. ACCELERATION OF COSMIC RAYS
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.
36 CHAPTER 3. ACCELERATION OF COSMIC RAYS
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.
40 CHAPTER 4. SOLAR COSMIC RAYS
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
42 CHAPTER 4. SOLAR COSMIC RAYS
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.)
4.1. ENERGY SPECTRUM AND COMPOSITION OF SCR 43
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.
44 CHAPTER 4. SOLAR COSMIC RAYS
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.
4.1. ENERGY SPECTRUM AND COMPOSITION OF SCR 45
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
46 CHAPTER 4. SOLAR COSMIC RAYS
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
48 CHAPTER 4. SOLAR COSMIC RAYS
Figure 4.4: Neutron monitor increases in per cent at the maximum of a GLEon 15 June 1991.
4.2. SOLAR NEUTRONS 49
Figure 4.5: CR energy spectrum. Solid lines denote the GCR spectrumfor maximum and minimum solar activity. Dashed line gives the averagespectrum.
50 CHAPTER 4. SOLAR COSMIC RAYS
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
4.2. SOLAR NEUTRONS 51
Figure 4.6: Sketch of secondary neutron production in a solar flare. Dashedlines denote magnetic field lines and solid lines particle trajectories.
52 CHAPTER 4. SOLAR COSMIC RAYS
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.
54 CHAPTER 4. SOLAR COSMIC RAYS
Figure 4.8: Time profile of the integral proton flux in different channelsduring a gradual flare.
4.3. SCR ACCELERATION: FLARES AND CMES 55
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.
56 CHAPTER 4. SOLAR COSMIC RAYS
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
4.3. SCR ACCELERATION: FLARES AND CMES 57
Figure 4.9: Hα image of the flare of 10 Oct 1971 by the Big Bear SolarObservatory.
58 CHAPTER 4. SOLAR COSMIC RAYS
Figure 4.10: A schematic view of particle acceleration during a solar flare.
4.3. SCR ACCELERATION: FLARES AND CMES 59
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
60 CHAPTER 4. SOLAR COSMIC RAYS
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
62 CHAPTER 4. SOLAR COSMIC RAYS
Figure 4.12: Spiral lines of the interplanetary magnetic field in the eclipticplane.
4.4. INTERPLANETARY PROPAGATION 63
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.
64 CHAPTER 4. SOLAR COSMIC RAYS
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.
4.4. INTERPLANETARY PROPAGATION 65
Figure 4.14: The times of first arrival of particles vs β−1.
66 CHAPTER 4. SOLAR COSMIC RAYS
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.
74 CHAPTER 6. SOLAR WIND
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
76 CHAPTER 6. SOLAR WIND
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)
78 CHAPTER 6. SOLAR WIND
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)
80 CHAPTER 6. SOLAR WIND
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.
6.3. ACCELERATION OF SOLAR WIND 81
Figure 6.5: Solution of the solar wind acceleration equation (6.21).
82 CHAPTER 6. SOLAR WIND
— 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)
6.3. ACCELERATION OF SOLAR WIND 83
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¯,
84 CHAPTER 6. SOLAR WIND
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
6.3. ACCELERATION OF SOLAR WIND 85
Figure 6.6: Diagram of the solar wind speed.
86 CHAPTER 6. SOLAR WIND
Figure 6.7: Yohkoh soft X-ray image for 02 Nov. 2000.
6.3. ACCELERATION OF SOLAR WIND 87
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).
88 CHAPTER 6. SOLAR WIND
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.
92CHAPTER 7. HELIOSPHERE AND INTERPLANETARYMAGNETIC FIELD
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.
94CHAPTER 7. HELIOSPHERE AND INTERPLANETARYMAGNETIC FIELD
Figure 7.3: Histogram of occurrence frequency for magnetic field strengthvalues in interplanetary space (from Ness, 1969; e.g., Falthammar, 1973).
7.1. GENERAL FACTS ABOUT IMF 95
(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.
96CHAPTER 7. HELIOSPHERE AND INTERPLANETARYMAGNETIC FIELD
Figure 7.4: Interplanetary magnetic field forming the Archimedian spiral.
7.1. GENERAL FACTS ABOUT IMF 97
Figure 7.5: The ”ballerina skirt” of the heliospheric neutral sheet.
98CHAPTER 7. HELIOSPHERE AND INTERPLANETARYMAGNETIC FIELD
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 ̶B
∂t− ∇× (v × B)
!· dS = 0 (7.12)
100CHAPTER 7. HELIOSPHERE AND INTERPLANETARYMAGNETIC FIELD
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.
102CHAPTER 7. HELIOSPHERE AND INTERPLANETARYMAGNETIC FIELD
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.
7.3. ARCHIMEDEAN SPIRAL 103
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)
104CHAPTER 7. HELIOSPHERE AND INTERPLANETARYMAGNETIC FIELD
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)
7.3. ARCHIMEDEAN SPIRAL 105
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.
106CHAPTER 7. HELIOSPHERE AND INTERPLANETARYMAGNETIC FIELD
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
110CHAPTER 8. SOLARMODULATIONOFGALACTIC COSMIC RAYS
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)
112CHAPTER 8. SOLARMODULATIONOFGALACTIC COSMIC RAYS
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
114CHAPTER 8. SOLARMODULATIONOFGALACTIC COSMIC RAYS
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.
116CHAPTER 8. SOLARMODULATIONOFGALACTIC COSMIC RAYS
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.
118CHAPTER 8. SOLARMODULATIONOFGALACTIC COSMIC RAYS
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.
8.4. THE MODULATION EFFECT 119
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.
120CHAPTER 8. SOLARMODULATIONOFGALACTIC COSMIC RAYS
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.
8.4. THE MODULATION EFFECT 121
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).
122CHAPTER 8. SOLARMODULATIONOFGALACTIC COSMIC RAYS
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.
126 CHAPTER 9. VARIATIONS OF COSMIC RAY INTENSITY
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.
128 CHAPTER 9. VARIATIONS OF COSMIC RAY INTENSITY
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.
130 CHAPTER 9. VARIATIONS OF COSMIC RAY INTENSITY
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.
132 CHAPTER 9. VARIATIONS OF COSMIC RAY INTENSITY
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.
136 CHAPTER 10. COSMIC RAYS AND THE EARTH
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.
138 CHAPTER 10. COSMIC RAYS AND THE EARTH
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.
140 CHAPTER 10. COSMIC RAYS AND THE EARTH
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).
142 CHAPTER 10. COSMIC RAYS AND THE EARTH
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.
144 CHAPTER 10. COSMIC RAYS AND THE EARTH
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.
146 CHAPTER 10. COSMIC RAYS AND THE EARTH
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.
150 CHAPTER 11. DETECTION OF COSMIC RAYS
• 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.
152 CHAPTER 11. DETECTION OF COSMIC RAYS
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
154 CHAPTER 11. DETECTION OF COSMIC RAYS
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.
156 CHAPTER 11. DETECTION OF COSMIC RAYS
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
158 CHAPTER 11. DETECTION OF COSMIC RAYS
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.
160 CHAPTER 11. DETECTION OF COSMIC RAYS
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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.
Recommended