Leroy C., Rancoita P.-G. Principles Of Radiation Interaction In Matter And Detection

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310 Principles of Radiation Interaction in Matter and Detection

equator [see Eq. (4.17)]. Finally, since ∇·

B = 0, the magnetic ﬁeld at the point (r,

θ, φ) is [Parker (1958)]







(r, θ, φ) = B(r

, θ, φ

) (r

/r)

(r, θ, φ) = 0 ,

(r, θ, φ) = −B(r

, θ, φ

) (r

/r)

(r − r

) (ω

) sin θ.

(4.22)

B(r, θ , φ)| is given by

B(r, θ , φ)| = B(r

, θ, φ

)

1 +

(r − r

)

sin θ

, (4.23)

consequently, the magnetic-ﬁeld strength decreases as 1/r

for θ ≈ 0

◦

or ≈ 180

◦

for radial distances not too large with respect to r

; whereas for r À r

it varies as

(sin θ)/r. B(r

, θ, φ

) represents the ﬁeld at r = r

; under the assumption that only

the solar dipole ﬁeld threads the escaping gas, then

B(r

, θ, φ

) = B

cos θ, (4.24)

where B

is the ﬁeld strength at the poles. For more complicated ﬁeld structures,

B(r

, θ, φ

) can be approximated in a diﬀerent way, but in either case, the lines of

force follow Eq. (4.22). For instance, if the magnitude of ﬁeld strength at r

≈ 10 R

is ≈ 10

−6

T (e.g., see Table 5.2 at page 180 of [Cravens (1997)]), then from the

computation of Eq. (4.23) at Earth orbit (i.e., at ≈ 215 R

, see Appendix A.2)

we get a ﬁeld strength of a few nT. This latter value is in agreement with typical

observed ones. In general, the Parker model agrees with suitable averages of the IMF

over a wide range of heliospheric distances and latitudes. However, for instance, it

has to be noted that the ﬁeld instantaneous-orientation deviates from that predicted.

Experimental observations allowed one to determine that i) while during solar-

minimum conditions the solar magnetic-ﬁeld is approximately dipole-like with a

magnetic dipole closely aligned to the solar rotation axis (e.g., see [Smith, Tsurutani

and Rosenberg (1978); Smith (1979); Jokipii and Thomas (1981)]), ii) during the

declining phase of solar cycle the dipole is more tilted (e.g., see [Hundhausen (1977);

Pizzo (1978)]). In addition, as the solar activity approaches its maximum, the large-

scale ﬁeld seems to be not well described by a dipole ﬁeld alone. Furthermore, in

1976 observations made using the Pioneer 11 spacecraft up to about 16

◦

above

the solar equatorial plane have shown that a) the magnetic-ﬁeld sector structure

††

(e.g., see Chapter V of [Hundhausen (1972)]) was dependent on latitude, b) this

sector structure tends to vanish at higher latitudes, where c) the IMF has the sign of

the solar magnetic-ﬁeld of the appropriate pole. The HMF ﬁeld-lines consist of those

coming i) from the northern solar magnetic hemisphere and directed away from the

Sun, and ii) from the southern hemisphere and directed towards the Sun. Field

††

At 1 AU, close to the ecliptic plane, the magnetic ﬁeld typically maintains one direction (either

towards or away from the Sun) for many days. The region of space in which an orientation is

maintained is called sector. Usually, the number of sectors varies from about two to four and it is

related to the solar activity (e.g., see Section 6.4.3 of [Cravens (1997)]).

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Radiation Environments and Damage in Silicon Semiconductors 311

lines

∗∗

from the two hemispheres are separated by a near-equatorial current sheet

([Smith, Tsurutani and Rosenberg (1978)] and references therein), also termed the

heliospheric current sheet (HCS), which is eﬀectively the extension of the solar

magnetic equator into the SW. The average position of the HCS is tilted relative to

the solar equator and warped

. Thus, as the Sun rotates, the Earth passes through

the current sheet and, consequently, experiences periods of alternating magnetic-

ﬁeld polarities. In practice, the maximum solar latitude of the current sheet is almost

equal to the tilt angle of the magnetic dipole axis relative to the rotation axis.

Furthermore, the solar corona is frequently dominated by long-lived structures

which can be reﬂected by large-ﬂow patterns of the SW. An example is provided

by the so-called corotating interaction regions (CIRs) [Pizzo (1978)]. This dynam-

ical process is related to inhomogeneities (in the coronal expansion), which cou-

ple with the solar rotation, resulting in signiﬁcant rearrangement of the material

in the interplanetary space: fast streaming material may catch up with slower

streams. Thus, Parker-spiral interaction structures may occur at large heliocen-

tric distances (e.g., see [Pizzo (1978)], Section 12.5 of [Gombosi (1998)], Sec-

tion 6.3 of [Aschwanden (2006)]). This mechanism is responsible for introducing

azimuthal gradients and, in addition, meridional gradients transverse to the eclip-

tic plane. These latter arise because latitude variations stem partly from intrinsic

variations in the corona and partly from the latitudinal dependence of the solar

rotation [e.g., see Eq. (4.17)].

High-speed SW is known to originate from regions close to centers of the so-

called coronal holes

†

[Wang and Sheeley (1993)]. These latter, in turn, are known

to exhibit the unusual property of rotating almost rigidly with an angular rotation

speed similar to that of the solar equator [Timothy, Krieger and Vaiana (1975);

Nash, Sheeley and Wang (1988)], in spite of the diﬀerential rotation observed in

the photosphere. Large variations of the SW-structure

∗

(Fig. 4.7) during the solar

cycle [McComas, Elliott, Schwadron, Gosling, Skoug and Goldstein (2003)] and

recurrent energetic particle events (see references in [Fisk (1996)]) were observed

for solar latitudes of ±80

◦

by Ulysses spacecraft, which allowed one to observe SW

properties a) through the declining phase and solar minimum (ﬁrst orbit) and b)

through the rise to solar maximum, solar maximum and immediately post-maximum

(second orbit).

Fisk (1996) pointed out that there were experimental evidences on how some

modulation and particle acceleration eﬀects can be induced by mechanisms related

∗∗

The conﬁgurations of ﬁeld-lines may exhibit the so-called helmet streamers [Pneuman and Kopp

(1971)] (e.g., see also Section V.6 of [Hundhausen (1972)]), which develop over active regions. The

legs of the helmet streamer connect regions of opposite magnetic polarity.

The HCS resembles a ballerina skirt.

†

Coronal holes are regions where the corona is dark and exhibit low X-ray intensity. Coronal holes

are associated with “open” magnetic-ﬁeld lines [Pneuman and Kopp (1971); Wang and Sheeley

(1993)] and are often found at Sun’s poles (in this case, they are also termed polar holes).

∗

Additional results from Ulysses mission can be found in [McComas et al. (2000); Schwadron,

McComas, Elliott, Gloeckler, Geiss and von Steige (2005)].

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312 Principles of Radiation Interaction in Matter and Detection

Fig. 4.7 Polar plots of SW speed (top panel) as a function of latitude for the ﬁrst two orbits of the Ulysses spacecraft; sunspot number (bottom

panel) shows that the ﬁrst orbit occurred through the solar cycle declining phase and minimum while the second orbit spanned solar maximum

(reprinted with permission from McComas, D.J., Elliott, H.A., Schwadron, N.A., Gosling, J.T., Skoug, R.M. and Goldstein, B.E. (2003), The three-

dimensional solar wind around solar maximum, Geophys. Res. Lett. 30 (no. 10), 1517, doi: 10.1029/2003GL017136,

°2003 American Geophysical

Union). Both are plotted over solar images characteristic of solar minimum and maximum. The SW speed is color coded by IMF orientation: red

indicates outward pointing IMF, while blue indicates inward pointing.

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Radiation Environments and Damage in Silicon Semiconductors 313

to CIRs at solar latitudes larger than ±30

◦

. It has to be noted that, within this range

of latitudes, CIRs patterns were observed in the SW by Ulysses. In the Fisk model,

the open coronal magnetic-ﬁeld follows the treatment of Wang and Sheeley (1993)

and results from the superposition of an axisymmetric ﬁeld, which is related to pho-

tospheric material undergoing diﬀerential rotation, and a non-axisymmetric ﬁeld,

which i) originates from decaying active regions near the solar equator, thus, ii)

rotates at the equatorial speed and iii) determines the mechanism by which the

boundary of the coronal holes rotates almost rigidly at the equatorial speed. The

magnetic ﬁeld in the inner corona of the polar hole can be represented by a tilted

dipole, which has a magnetic axis oﬀset from the solar rotation axis, and which

rotates at the equatorial speed. The interplay among a) the diﬀerential rotation

of photospheric material

‡

, b) the SW expansion through the more rigidly rotating

polar holes (which are oﬀset from the solar rotation axis) and c) the tilted mag-

netic dipole results in a SW expansion and, thus, an IMF conﬁguration which can

vary from those predicted by Parker [Smith et al. (1995); Fisk (1996); Zurbuchen,

Schwadron and Fisk (1997); Fisk (1999); Burger and Hattingh (2001); Fisk (2001);

Reinard and Fisk (2004); Schwadron, McComas, Elliott, Gloeckler, Geiss and von

Steige (2005); Zurbuchen (2007)]. In addition, ﬁeld lines may be moved in latitude

by the CIR mechanism. As a consequence, ﬁeld lines are allowed to execute large

latitudinal excursions which, for instance, naturally explain 26-day recurrences in

cosmic ray events observed at high latitudes. Experimental observations have given

indication for the validity of this revised heliospheric magnetic-ﬁeld ([Zurbuchen,

Schwadron and Fisk (1997)] and references therein).

Smith (2004) (see also references therein) has reviewed the characteristics of the

radial, azimuthal and North–South components of the magnetic ﬁeld in the outer

heliosphere. He pointed out that recent Ulysses measurements, at both solar mini-

mum and maximum, indicate that i) the radial component is almost independent of

solar latitude, ii) the azimuthal component deviates from the Parker values at high

latitudes (e.g., see [Banaszkiewicz, Axford and McKenzie (1998)]), iii) a turning of

the spiral angle toward the radial direction by tens of degrees is often observed inside

the so called corotating rarefaction regions (CCR) [Murphy, Smith and Schwadron

(2002)] and iv) the North–South component can depart from zero for many days as

a result of the tilting of the interface between fast and slow SW streams.

4.1.2.2 Extension of the Heliosphere and the Earth Magnetosphere

Parker (1961b) was the ﬁrst to investigate the extension of the heliosphere. Al-

though since then more work has been done on this topic, details of the SW inte-

raction with the interstellar medium are mostly speculative because we lack ex-

The reader can ﬁnd additional information in [Sheeley, Nash and Wang (1987); Nash, Sheeley

and Wang (1988); Wang, Sheeley, Nash and Shampine (1988)] and references therein.

‡

The heliospheric magnetic-ﬁeld lines in high sp eed streams are assumed to be anchored in the

diﬀerentially rotating photosphere.

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314 Principles of Radiation Interaction in Matter and Detection

tensively direct observations of this space region (e.g., see Chapter IX of [Parker

(1963)], Section 1.3 of [Toptygin (1985)], [Suess (1990)], Section 6.61 of [Cravens

(1997)], Section 12.7 of [Gombosi (1998)], Section 9 of [Gosling (2006)] and refe-

rences therein). Parker (1961b,1963) predicted the existence of i) the heliospheric

termination shock which results because the SW, that is ﬂowing supersonically away

from the Sun, must make a transition to subsonic

∗∗

and ii) the heliopause, which

is the boundary surface separating the interstellar and SW plasmas and it is lo-

cated outside the termination shock (Fig. 4.8). A rough estimate of the heliopause

location may be obtained from balancing the SW pressure, P

, and the interstellar

medium pressure, P

, i.e.,

= P

. (4.25)

Since the thermal and magnetic pressure

††

components of the SW can be neglected

{e.g., see Equations (7.17–7.21) at page 349 of [Meyer-Vernet (2007)]}, P

is prac-

tically given by the SW dynamic pressure P

d,w

(also termed the ram pressure of the

SW ), i.e.

∼ P

d,w

(4.26)

with (e.g., see Section 7.2.1 of [Meyer-Vernet (2007)])

d,w

= ρv

, (4.27)

where ρ and v

are the SW-plasma mass density and speed, respectively. Under

steady-state spherically symmetric condition, the continuity equation [Eq. (4.5)]

becomes

= constant ∼ n

1AU

w,1AU

× (1 AU)

∗∗

The behavior of gas depends on the sp eed of sound v

[Eq. (4.6)], which controls the propagation

of disturbances. In the SW, particles are also linked by their embedded magnetic-ﬁeld, thus, by

the “magnetic pressure” and, in turn, by the “Alfv´en speed”, v

√

where ρ is the plasma mass-density and µ

is the permeability of the free space. The local fast

mode speed is the characteristic speed with which small amplitude pressure signals propagate in a

plasma and it is given by

+ v

At 1 AU, the mean values of v

and v

in units of km/s are 63 and 50, respectively (Table 1 at

page 101 of [Gosling (2006)]).

††

The magnetic pressure is given by:

2µ

where µ

is the permeability of the free space (e.g., see Section 4.5.1 of [Cravens (1997)]). In cgs

units it takes the form

8π

dyn/cm

with

B in units of Gauss. It can be described as the tendency of neighboring ﬁeld lines to repulse

each other. Note that, in contrast to the gas-dynamic pressure, the magnetic pressure is not

isotropic but is always perpendicular to the ﬁeld (e.g., see Section 3.3.1 of [Kallenrode (2004)].

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Radiation Environments and Damage in Silicon Semiconductors 315

where n

≈ ρ/m

is the number of protons

‡‡

per cm

in the SW plasma [e.g., see

Eq. (4.4)]; n

1AU

and v

w,1AU

are the SW-plasma density and speed at 1 AU, respec-

tively. Thus, as discussed at page 308, under the assumption that the SW speed is

already almost constant beyond the Earth, i.e.,

≈ v

w,1AU

for r & 1 AU,

we have

∼ n

1AU

1 AU

. (4.28)

Using Eqs. (4.4, 4.28), the ram pressure can be expressed in terms of the SW

characteristics at 1 AU by re-writing Eq. (4.27) as

d,w

≈ n

1AU

w,1AU

1 AU

= P

1AU

1 AU

, (4.29)

where {see Equation (7.18) at page 349 of [Meyer-Vernet (2007)]}

1AU

= (2.1–2.6) × 10

−8

dyn/cm

(4.30)

is the dynamic pressure

at 1 AU. The interstellar medium pressure P

is given by

the sum of i) the magnetic-ﬁeld pressure, ii) the thermal pressure of the interstellar

plasma, iii) the dynamic pressure of the interstellar plasma and iv) the cosmic rays

pressure. Suess (1990) estimated that

≈ (1.3 ± 0.2) × 10

−12

dyn/cm

thus, from Eqs. (4.25, 4.26, 4.29), we obtain that the distance R

of the heliopause

from the Sun is related by

= P

1AU

1 AU

(1.3 ± 0.2) × 10

−12

≈ (2.1–2.6) × 10

−8

1 AU

and, consequently, we get

∼

(2.1–2.6) × 10

−8

(1.3 ± 0.2) × 10

−12

≈ (1.18–1.54) × 10

AU. (4.31)

Since the distance of the heliopause from the Sun is large [Eq. (4.31)], the IMF

embedded in the SW cannot be considered to instantaneously propagate through

‡‡

For this approximate calculation the small portion of helium in the SW was neglected.

Equation (4.30) can be equivalently expressed as: P

1AU

= (2.1–2.6) × 10

−9

Pa.

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316 Principles of Radiation Interaction in Matter and Detection

the solar cavity. In fact using Eq. (4.31) (see also Appendix A.2), a rough estimate

of the traveling time (τ

) through the heliosphere up to the heliopause is given by:

≈

w,1AU

(1.18–1.54) × 10

× 1.496 × 10

470 km/s

= (3.76–4.90) × 10

s ∼ (1.19–1.55) y, (4.32)

where v

w,1AU

≈ 470 km/s is the average value of SW speed already almost constant

beyond the Earth (e.g., see pages 300, 308). As a consequence, the structure of

the IMF depends on the past solar-activity occurred during more than a year. Fur-

thermore, an additional complexity is added to the IMF structure by the delayed

propagation of phenomena aﬀecting the SW, for instance those observed at large

solar latitudes and discussed at pages 311-313.

Voyager 1 spacecraft has already veriﬁed the existence of the termination shock

at a heliocentric distance of ≈ 94 AU roughly in the direction of the relative motion

of the heliosphere relative

∗

to the interstellar medium (e.g., see [Cummings, Stone,

McDonald, Heikkila, Lal and Webber (2005)]) and entered the heliosheath, which is

located between the termination shock and the heliopause (Fig. 4.8).

Outside the heliopause the interstellar ﬂow is deﬂected around the helio-

sphere. Furthermore, if the external ﬂow speed is larger than the fast mode speed

(see footnote at page 314), it is expected the formation of a bow shock to deﬂect the

interstellar ﬂow around the heliosphere. The shape of the heliosphere is asymmet-

ric (Fig. 4.8), because of its motion relative

∗

to the interstellar medium, and it is

largely elongated in the opposite direction, thus, generating the so-called heliotail.

Furthermore inside the heliosphere, the Earth’s magnetosphere is the region of

space to which the Earth’s magnetic-ﬁeld

is conﬁned by the SW plasma and, thus,

shaped (Fig. 4.9) by its interaction with the SW itself (e.g., see [Knecht and Shuman

(1985); Kivelson and Bagenal (2006); Luhmann and Solomon (2006); Schulz (2007)]

and references therein). This cavity was named magnetosphere by T. Gold in 1959 to

provide a description of the space region above the ionosphere. The ﬁrst suggestion

of such a cavity (sometimes referred to as the Chapman–Ferrero cavity) was made

by S. Chapman and V. Ferraro in 1930.

The space environment of the Earth is determined by its almost dipolar inter-

nal magnetic-ﬁeld that forms an obstacle to the SW with its embedded magnetic-

ﬁeld. Thus, to ﬁrst approximation, the magnetic ﬁeld close to the Earth surface has a

dipole shape, but moving outwards other contributions become important. Among

these we have to mention, for instance, those resulting from the several currents

(e.g., the ring current, Birkeland’s currents and tail currents) due to charged par-

ticles trapped inside the magnetosphere (Sect. 4.1.2.5). Moreover the latitudinal

∗

The Sun and heliosphere move at a speed of ≈ 23 km/s relative to the interstellar medium.

Note that a number of coordinate systems are employed in the description of geomagnetic

phenomena; among the commonly used are geographic, geomagnetic, geocentric solar-ecliptic,

geocentric solar-magnetospheric and solar-magnetic coordinate systems. The reader can ﬁnd the

deﬁnitions, for instance, in Section 4.1.2 of [Knecht and Shuman (1985)].

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Radiation Environments and Damage in Silicon Semiconductors 317

Fig. 4.8 Schematic representation of the heliosphere with Voyager 1 spacecraft crossing into the

heliosheath, the region where interstellar gas and solar wind start to mix. (Courtesy NASA/JPL-

Caltech).

Fig. 4.9 Schematic representation of the magnetosphere to which the Earth’s magnetic-ﬁeld is

conﬁned and, thus, shaped by the SW plasma (Courtesy of NASA’s Marshall Space Flight Center

and Science@NASA).

dependence is not geographically symmetric, because the Earth magnetic dipole is

tilted with respect to the Earth rotation axis and shifted from the Earth center. The

magnetic ﬁeld of the Earth magnetosphere can be described (e.g., see Appendix A

of [Bobik et al. (2006)] and Section 7.2 of [Schulz (2007)]) using i) the Interna-

tional Geomagnetic Reference Field (IGRF) [Barton (1997)] for representing the

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318 Principles of Radiation Interaction in Matter and Detection

main contribution due to the inner Earth and ii) the external magnetic ﬁeld model

implemented by Tsyganenko and Stern (e.g., see [Tsyganenko (1995); Tsyganenko

and Stern (1996)]) for representing the other contributions.

Similarly to the formation of the heliopause discussed above, the magnetopause

is the location where the outward magnetic pressure (see footnote at page 314) of the

Earth’s magnetic-ﬁeld is counterbalanced by the SW pressure, which is practically

given by the SW dynamic pressure at 1 AU (P

1AU

) [see Eq. (4.30)], since the thermal

and magnetic combined pressure is less than 1% of the SW ram-pressure [e.g., see

Equations (7.17–7.21) at page 349 of [Meyer-Vernet (2007)]]. The subsolar distance

‡

) of the magnetopause can b e estimated in SI units from

1AU

≈

, (4.33)

where F ≈ 2 is a magnetic-ﬁeld compression factor (e.g., see Section 14.2.2 of [Gom-

bosi (1998)] and, also, Section 8.2.2 of [Kallenrode (2004)]), µ

is the permeability

of the free space, B

≈ 3.11 ×10

−5

T is the equatorial magnetic-ﬁeld

at the surface

of the Earth and Re is the radius of the Earth (Appendix A.2). From Eq. (4.33),

we get:

≈

1AU

Re (4.34)

≈ (9.2–9.5) Re.

Thus, on the dayside of the planet, for normal SW conditions the subsolar distance of

the magnetopause is ≈ 10 Re. However, when the SW pressure is particularly strong,

this distance can be reduced to ≈ (6–7) Re. On the nightside, the magnetosphere is

stretched into a long magnetotail to a distance, which might extend up to ≈ 1000 Re

(Fig. 4.9). The magnetotail has an approximate cylindrical shape of ≈ 40 Re in

diameter.

Since the SW is a supersonic ﬂow, a standing shock wave, the bow shock

, de-

velops in front of the magnetopause. Its subsolar distance (R

) is typically located

(2–3) Re ahead of this latter and is approximately given by (e.g., see Equation 14.18

at Page 284 of [Gombosi (1998)])

≈ 1.275 × R

At the bow shock the SW plasma is slowed down to subsonic speed and a substantial

fraction of its energy is converted into thermal energy. The thermalised SW ﬂow

passes through the bow shock and enters the region called magnetosheath, but can-

not easily penetrate through the magnetopause. However, Dungey (1961) suggested

‡

The subsolar distance is the radial distance towards the Sun to the nose of magnetopause. The

subsolar point on Earth is where the Sun is perceived to be directly overhead.

The dipolar magnetic-ﬁeld ﬂux-density falls oﬀ with the distance r as 1/r

A further discussion about the bow-shock formation and location can be found in [Schwartz

(1985)].

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Radiation Environments and Damage in Silicon Semiconductors 319

that the magnetopause i) is not a complete barrier to the SW and ii) the magneto-

sphere structure itself is also controlled by its reconnection with the IMF. Further-

more, the magnetic ﬁeld exhibits the polar cusps, which separate closed ﬁeld lines

on the dayside of the magnetosphere from ﬁeld lines swept to its nightside and to

which the ﬁeld lines of the magnetopause converge (Fig. 4.9). The cusp regions are

ﬁlled with plasma particles from the magnetosheath and, thus, at the cusps these

particles can penetrate deep into the Earth’s atmosphere. Close to the Earth, in the

inner magnetosphere the dipolar shape of the magnetic ﬁeld is almost preserved.

4.1.2.3 Propagation of Galactic Cosmic Rays through Interplanetary Space

The Galactic cosmic radiation [e.g., the Galactic Cosmic Rays (GCRs), see

Sect. 4.1.2.4] is incident on the solar cavity isotropically and constantly and is af-

fected by the outwards ﬂowing SW with its embedded magnetic-ﬁeld and magnetic-

ﬁeld irregularities. For instance, at energies

∗

below a few GeV/nucleon, the inten-

sity shows a strong dependence on solar activity (e.g., see Sect. 4.1.2.1) with a

maximum at the solar minimum and decreases with increasing the solar sunspot

numb er [Smart and Shea (1985)]. This eﬀect is called modulation of GCRs. Fur-

thermore, diﬀerential energy spectra of all high-energy isotopes exhibit a maximum

[Fig. 4.12(a) and Fig. 9.37], which shifts towards higher energies and, at the same

time, decreases in intensity with increasing solar activity: this eﬀect was observed at

1 AU for protons and other isotopes up to iron [Meyer, Parker and Simpson (1974);

Webber and Lezniak (1974); Simpson (1983); Smart and Shea (1985, 1989)].

Parker (1963,1964,1965) proposed a theory for the general properties of the prop-

agation of GCRs through the interplanetary space, considering transit time, energy-

loss and the outward convection of solar particles. Furthermore, he reached the con-

clusion that the SW, and in general the IMF conﬁguration which results, lead to

important cosmic rays modulation-eﬀects dependant on the solar activity. He based

his theory on IMF observations made by the Explorer XVIII spacecraft: the IMF

recorded-variations were magnetic-ﬁeld spatial irregularities transported rigidly in

the SW (Figure 1 of [Parker (1965)]). These irregularities appear with dimensions

of 10

–10

km, which, for instance, are comparable with the radius of gyration of

protons with kinetic energies larger than ≈ 100 MeV and lower than ≈ 10 GeV in a

typical IMF of a few nT. He succeeded in showing that a charged particle, moving

in a large-scale ﬁeld containing small-scale irregularities, is most eﬀectively scat-

tered by irregularities which have a scale comparable to the radius of gyration of

the particle [Parker (1964)]. When this occurs, the eﬀect of such irregularities is to

cause (in the reference frame of the magnetic irregularities) the GCRs to a random

walk, which can be considered as a Markhoﬀ process

∗

The reader can ﬁnd the deﬁnition of kinetic energies per nucleon in Sect. 1.4.1.

A Markhoﬀ process is when the value of a physical parameter at the next measuring time

depends only on its value at the present time and not on its value at any previous measuring time,

i.e., the system has one time step (or less) “memory”.