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

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300 Principles of Radiation Interaction in Matter and Detection

in space, one of the primary eﬀects of degraded operations is the reduction of the

current gain.

In the following sections, we present a review about i) the origin of the solar wind

and heliospheric (or interplanetary) magnetic-ﬁeld (Sect. 4.1.2.1), ii) the extension

of the heliosphere and the Earth magnetosphere (Sect. 4.1.2.2), iii) how the solar

wind aﬀects (modulates) the propagation of galactic cosmic rays in the heliosphere

(Sect. 4.1.2.3), iv) ﬂuxes of solar, heliospheric and galactic cosmic rays (Sect. 4.1.2.4)

and v) a description of trapped particles in the Earth magnetosphere (Sect. 4.1.2.5).

4.1.2.1 Solar Wind and Heliospheric Magnetic Field

Nowadays, we know that the solar wind (SW) is a plasma

††

that permeates the

interplanetary space and constitutes the interplanetary medium. It is, as discussed

later, the outer part of the Sun’s corona

streaming through the solar system and

creating the so-called solar cavity (also termed heliosphere). It is highly variable in

both time and space; it consists

†

of protons (about 95% of the positively charged

particles), double charged helium ions, a very small number of other positively

charged particles and electrons, so that the plasma is electrically neutral. At the

orbit of Earth, i.e., at 1 AU from the Sun (see Appendix A.2), experimental obser-

vations of SW allowed one to determine that i) the proton density and temperature

are ≈ (8–9) proton/cm

and ≈ 1.2 × 10

K, respectively, ii) the mean wind speed

is ≈ 470 km/s, iii) the mean speed of sound is (50–63) km/s (e.g., see Table 3-1

of [Feynman (1985)], Table 1 of [Gosling (2006)] and references therein) and, thus,

iv) the SW speed is supersonic

‡‡

. It carries an embedded weak magnetic-ﬁeld, which

results in modulating the ﬂuxes of GCRs entering the heliosphere. It has to b e noted

that until the early 1990s our knowledge of the heliosphere was limited to the eclip-

tic plane. In fact, the investigation of plasma, particles and ﬁeld in the polar regions

of the Sun were among the main goals of the Ulysses mission, launched in 1990.

In 1859, the ﬁrst SW (indirect) observation was made by Carrington, who wit-

nessed what is now called a solar ﬂare

∗

and noted that a major geomagnetic storm

began about 17 hours after the ﬂare. In 1896, Birkeland suggested that the Earth

environment is bombarded by “rays of electric corpuscles emitted by the Sun”. In

the early 1900s, Lindemann suggested that geomagnetic storms may result from an

interaction of the geomagnetic-ﬁeld and plasma clouds ejected by the solar acti-

††

The plasma is an ionized gas, electrically quasi-neutral and exhibiting collective behavior

(e.g., see Chapter 2 of [Meyer-Vernet (2007)]).

The corona with temperatures over 10

K is a highly rareﬁed region above a small region called

chromosphere, which extends for ≈ 2000 km above the photosphere. The photosphere, in turn, is

a thin layer at the surface of the Sun. The reader may ﬁnd an introduction to the Sun structure,

for instance, in [Aschwanden (2006)].

†

The reader can ﬁnd details of solar wind composition, for instance, in [Feynman (1985); Gosling

(2006)]).

‡‡

The sonic point, i.e., the position at which the solar wind speed becomes equal to the speed of

sound, is located at several solar radii (R

∗

The solar ﬂare is a giant explosive energy release on the Sun.

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Radiation Environments and Damage in Silicon Semiconductors 301

vity. In the 1930s, Forbush noted a rapid decrease (the so-called Forbush decrease)

in the observed GCR intensity following a coronal mass ejection. It o ccurs, as we

know now, due to the magnetic ﬁeld of the SW plasma sweeping some of the galac-

tic cosmic rays away from Earth. In the early 1950s, Biermann concluded that

there must be a steady stream of charged particles from the Sun to account for the

fact that ionic tails of comets always point away from the Sun. In the mid 1950s,

Chapman constructed a static solar corona model and calculated the properties of

a low-density gas at ≈ 10

K: he determined that this latter is an excellent con-

ductor of heat and must extend way out into space, beyond the orbit of Earth. In

other words, the Earth is expected to be inside the solar corona. These results and

observations inspired Parker (1958)

to formulate a radically new model of the so-

lar corona in which this later is continuously expanding outward, thus solving the

discrepancy between the estimated interstellar pressure of (10

−12

–10

−13

) dyn/cm

(e.g., see [Parker (1958)], pages 8–11 of [Toptygin (1985)] and Table 1 of [Suess

(1990)]) and that of ≈ 0.6 ×10

−5

dyn/cm

[Parker (1958)], computed under the as-

sumption of a static equilibrium in the solar corona (see, also, [Parker (2007)]). Since

then, Parker’s model has demonstrated to provide a general and elegant framework

for heliospheric features. However (e.g., see [Fisk (2001); Cranmer (2002)] and re-

ferences therein), subsequently this model was modiﬁed to account additional ob-

servations, like for instance those obtained with i) the Ulysses spacecraft (launched

by the Space Shuttle Discovery in 1990, as already mentioned) designed to reach

high solar latitudes, thus out of the ecliptic plane and ii) the Solar and Heliospheric

Observatory (SOHO, launched in 1995) to study the Sun from its deep core to outer

corona and the SW.

As mentioned above, Parker (1958) assumed that i) the solar corona was not in

hydrostatic equilibrium, ii) there is a stationary expansion with the coronal plasma

ﬂowing out, iii) the stationary expansion is spherically symmetric, iv) the kinetic

temperature of the gas is maintained (by heating mechanisms) at a uniform value T

from the base of the corona at radial distance r

≈ 10

km to some radius beyond

which the heating vanishes and T is negligible, v) electrons and protons are the

dominant (and equal in number) particles of the solar coronal plasma, and vi) the

stationary expansion satisﬁes the equation of motion

ρv

= −

− Gm

, (4.3)

where r is the radial distance; G and m

are the Newtonian constant and solar

mass, respectively (see Appendix A.2); P is the pressure; v is the radial velocity of

the corona expansion; the mass density is given by

ρ = n

+ m

) = n

m ≈ n

, (4.4)

This article on Parker’s theory about the SW was submitted to Astrophysical Journal, but was

rejected by the two reviewers. It was saved by then editor Chandrasekhar (1983 Nobel Prize in

physics). Parker received the Kyoto and James Clerk Maxwell Prizes in 2003.

This equation is the Bernoulli equation for the solar corona expansion and reduces to the so-

called barometric equation, when the ﬁrst term is 0 (i.e., for dv/dr = 0). The barometric equation

accounts for hydrostatic equilibrium.

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302 Principles of Radiation Interaction in Matter and Detection

in which m

) is the proton (electron) rest-mass and, thus,

m = m

+ m

≈ m

;

is the number of protons (or electrons, since the gas is assumed to b e neutral

and fully ionized) per cm

and satisﬁes the equation of continuity

= 0 (4.5)

for conservation of matter. For an ideal (mono-atomic) gas with equal electron and

proton temperatures, the pressure is given by

P = 2 n

where k

is the Boltzmann constant (Appendix A.2), and the (local) sound speed

dρ

. (4.6)

Equation (4.3) yields a p ositive expansion velocity, v, as a function of the radial

distance, r, for r ≥ r

when [Parker (1958)]

2 k

− ln

2 k

= −3 − 4 ln

Gmm

4 r

+4 ln

Gmm

. (4.7)

In Fig. 4.4, it is shown the expansion velocity, v(r), obtained using Eq. (4.7) for

an isothermal solar corona with temperatures T

= 10

, 1.5 × 10

and 2 × 10

and r

≈ 10

km at the base. Furthermore, Parker extended his treatment to i) the

case in which the pressure and temperature are related by the polytropic law [Parker

(1960, 1963)]

P = P

, (4.8)

where α is the polytropic index (for instance, α = 1 and α = 5/3 for isothermal

and adiabatic processes, respectively) and ii) the case of sudden corona-expansion

following a large solar ﬂare [Parker (1961a)]. The hydrodynamic expansion of the

coronal plasma was termed the solar wind for the Sun and the stellar wind for the

other stars by Parker (1960).

For approaching the description of a more complex three-dimensional SW struc-

ture as presently observed ([McComas, Elliott, Schwadron, Gosling, Skoug and

Goldstein (2003)] and references therein), further discussions, criticisms and, also,

improvements to Parker’s model can be found, e.g., in [Parker (1961a, 1963);

Using Eqs. (4.6, 4.8), the local sound speed can be rewritten as v

α(P/ρ). Thus, the

wavespeed depends on the conditions under which the compression takes place. For example,

in case of an adiabatic compression of a monoatomic gas we have α = 5/3 and, consequently,

(5P /3ρ) (e.g, see Sections 17.3 and 21.5 of [Ingard and Kraushaar (1960)]).

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Radiation Environments and Damage in Silicon Semiconductors 303

Fig. 4.4 Spherically symmetric hydrodynamic expansion velocity (i.e., the velocity of the solar

wind) in km/s as a function of the radial distance (in AU) from the Sun, for an isothermal corona

with temperatures T

of 10

, 1.5 × 10

and 2 × 10

K and a radius of ≈ 10

km at the base.

Hundhausen (1972); Feynman (1985)], Chapter 6 of [Cravens (1997)], Chapter 12

of [Gombosi (1998)], Chapter 6 of [Kallenrode (2004)] and Chapter 6 of [Parks

(2004)] (see, in addition, references therein). Limitations of this model are related,

for instance, to i) observations that the pressure is less isotropic as the plasma

moves away from the Sun, ii) the assumption that the ﬂow is considered uniform

and radial and iii) proton and electron temperatures, which were assumed equal and

isotropic. For example, measurements have determined the presence of a fast solar

wind component, where electrons have a core component with a nearly Maxwellian

(kinetic) energy distribution and a high-energy tail (the so-called halo component),

which has an approximate Maxwellian distribution with a temperature larger by

about an order of magnitude with respect to that of the core component. Although

not all issues can be treated from the particle point of view, some of these features

motivated some authors (e.g., [Lemaire and Scherer (1973); Maksimovic, Pierrard

and Lemaire (2001)] and Section 6.7.4 of [Parks (2004)]) to investigate the so-called

kinetic models of the SW. However, detailed treatments of alternative or extended

solar wind models go beyond the purpose and ﬁelds of applications of the present

bo ok.

As already mentioned, in Parker’s model the magnetic-ﬁeld lines are supposed

to be embedded in the streaming particles of the SW. The notion of frozen-in

magnetic-ﬁeld originates from the work on electromagnetic hydrodynamic waves

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304 Principles of Radiation Interaction in Matter and Detection

by Alfv´en

∗

(1942) and plays a relevant role in the description of interplanetary

magnetic-ﬁeld [Parker (1957, 1958)] and planet magnetospheres. Alfv´en noted that

the motion of matter can be coupled to the modiﬁcation of the magnetic-ﬁeld lines,

so that these latter follow the matter motion. In magnetohydrodynamics (MHD), we

can express the time derivative of the magnetic ﬁeld

B as a function of the electric

ﬁeld,

E, and the center-of-mass plasma-velocity, ~v

, in the laboratory

††

. In fact

using the so-called generalized Ohm’s law and since the Hall, ambipolar polarization

and inertial terms can be neglected (e.g., see Sections 4.3.1 and 4.3.2 of [Cravens

(1997)]), we have

∂

∂t

= −∇ ×

℘

−~v

= −∇ ×

℘

+ ∇ ×

, (4.9)

where ℘ is the conductivity and

J is the current density. For slow time-variations

and non-relativistic plasma bulk-speeds

‡

, the term (1/c

) ∂

E/∂t can be neglected

in Maxwell’s equation regarding ∇ ×

B, i.e., we have

∇ ×

B = µ

J +

∂

∂t

≈ µ

⇒

J ≈

∇ ×

where µ

is the permeability of the free space (see Appendix A.2). Thus, using

a vector calculus identity we can rewrite the ﬁrst term of the right-hand side of

Eq. (4.9) as

−∇ ×

℘

≈ −

℘

∇ × ∇ ×

= −

℘

∇

∇ ·

− ∇

∇

℘

since

∇ ·

B = 0.

∗

Alfv´en, who got the Nobel prize in 1970, criticized how the concepts of frozen-in magnetic-ﬁeld

lines and ﬁeld-line reconnection were used in the theory of magnetosphere (e.g., see [Alfv´en (1976)]

and references therein).

††

For the interplanetary magnetic-ﬁeld, the laboratory reference-frame is that of the solar cavity.

‡

For example, these conditions are satisﬁed for a velocity (v

), a time-scale (τ

) and a length-

scale (L

) such that

;

for a discussion, the reader can see, for instance, Section 2.3.3 of [Meyer-Vernet (2007)].

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Radiation Environments and Damage in Silicon Semiconductors 305

Finally, Eq. (4.9) can be rewritten as

∂

∂t

∇

℘

+ ∇ ×

. (4.10)

For a plasma composed by i) electrons with rest-mass m

and concentration n

and

ii) (an equal amount of) single ionized ions with rest-mass À m

and concentration

ion

= n

the electrical conductivity

is given by

℘ ≈

coll

, (4.11)

where ν

coll

is the collision frequency and e is the electron charge. In addition

{e.g., see Equation (2.102) of [Meyer-Vernet (2007)]}, ℘ is approximated by

℘ ≈ 6 × 10

−4

× T

3/2

[Ω m]

−1

, (4.12)

whenever the magnetic-ﬁeld eﬀect can be neglected, i.e., in the direction i) parallel

B and ii) perpendicular to

B, if the particle mean free-path is much smaller than

the radius of gyration. In general along this latter direction, the opposite condition

occurs and, consequently, the electric conductivity results to be strongly reduced

perp endicularly to

B. The quantity

℘µ

coll

(4.13)

is the so-called magnetic diﬀusion coeﬃcient. Equation (4.10) is known as the

magnetic convection-diﬀusion equation and accounts for two contributions to the

magnetic-ﬁeld variation: these are due to the a) diﬀusion, produced by conductive

losses (i.e., the ﬁrst term), and b) convection, produced by the plasma bulk mo-

tion (i.e., the second term). Furthermore, the ratio of the magnetic convection and

diﬀusion term

magnetic convection term

magnetic diﬀusion term

(4.14)

is termed magnetic Reynolds number and can be estimated from the expression

(e.g., see Section 4.4.2 of [Cravens (1997)] or page 88 of [Meyer-Vernet (2007)])

≈

= v

℘µ

where the length-scale L

is deﬁned in the footnote at page 304. For R

À 1 the

magnetic diﬀusion term can be neglected in Eq. (4.10), which reduces to

∂

∂t

≈ ∇ ×

. (4.15)

The reader can see, e.g., Section 4.3.2 of [Cravens (1997)], Section 4.3.2 of [Gombosi (1998)], Sec-

tions 2.1.3 and 2.3.3 of [Meyer-Vernet (2007)].

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306 Principles of Radiation Interaction in Matter and Detection

When the magnetic convection-diﬀusion equation is reduced to Eq. (4.15), we are in

the condition of a perfectly conducting-ﬂuid moving in a magnetic ﬁeld, for which it

can be proven

∗

that the magnetic ﬂux through any closed contour moving with the

plasma is constant. Thus, the ﬂuid can move freely along the magnetic-ﬁeld lines

and these latter are frozen in the ﬂuid (e.g., see [Parker (1957)] and, also, Chapter 4

of [Cravens (1997)], Section 14.4.2 of [Gombosi (1998)], Section 3.4 of [Kallenrode

(2004)], Section 5.5.1 of [Parks (2004)], [Lundin, Yamauchi, Sauvaud and Balogh

(2005)] and references therein, Section 2.3.2 of [Meyer-Vernet (2007)]).

For the SW non-relativistic plasma, it can be shown that ℘ is large and R

À 1

(e.g., see page 291 of [Meyer-Vernet (2007)]) and, thus, the SW can ﬂow with a

frozen-in magnetic-ﬁeld. Furthermore, in the SW at 1 AU from the Sun the mean

free path among streaming particles is ≈ 1 AU, as a consequence, particle collisions

can hardly occur (e.g., see page 53 of [Meyer-Vernet (2007)]).

The ﬁrst indication of a solar magnetic-ﬁeld was obtained in 1905 by Hale. He

could demonstrate the presence of strong magnetic-ﬁelds in sunspots by means of

the Zeeman eﬀect. These sunspots are supposed to be created deeper in the Sun

†

and, afterwards, moved to the surface by magnetic-buoyancy eﬀects. A large spot

is about 2 × 10

km across and may have a central umbra-region with a tempera-

ture of ≈ 4.1 × 10

K (e.g., see Section 1.3 in [Priest (1985)]). Daily observations

were started at the Zurich Observatory in 1749. Their number varies and increases

with the solar activity following a cycle of ≈ 11 years duration termed the sunspot

cycle: the cyclic variation of the number of sunspots [also called the (Schwabe)

solar cycle or Schwabe–Wolf cycle] was ﬁrst observed by Heinrich Schwabe between

1826 and 1843 and led Rudolf Wolf to make systematic observations starting in

1848 (Fig. 4.5)). The relative sunspot number

‡

) remains an important index for

determining the solar activity level; it is computed using the formula (collected as

a daily index of sunspot activity):

= k

(s + 10 × g), (4.16)

where s is the number of individual spots, g is the number of sunspot groups and,

ﬁnally, k

is a factor that varies with location and instrumentation (also known as

the observatory factor). Furthermore, sunspots are used to trace the Sun surface-

rotation since Galileo in 1611. Nowadays, it is determined that the Sun rotation-

period is a function of the solar latitude and is called (sidereal) diﬀerential rota-

∗

This is known as Alfv´en’s theorem.

†

The reader can ﬁnd a general description of the Sun characteristics, for instance, in [Altrock et

al. (1985)] and in [Aschwanden (2006)].

‡

is also known as the International sunspot number or Z¨urich number or Wolf num-

ber. Note that there are actually at least two ”oﬃcial” sunspot numbers reported. The In-

ternational Sunspot Number is compiled by the Sunspot Index Data Center in Belgium

(http://sidc.oma.be/index.php). The NOAA sunspot number is compiled by the US Na-

tional Oceanic and Atmospheric Administration. The sunspot numbers and smoothed monthly

mean sunspot numbers can be found at the web site: ftp://ftp.ngdc.noaa.gov/ftp.html or

http://www.ngdc.noaa.gov/stp/SOLAR/ftpsunspotnumber.html#international (the reader can

also see: http://solarscience.msfc.nasa.gov/SunspotCycle.shtml).

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Radiation Environments and Damage in Silicon Semiconductors 307

1750 1760 1770 1780 1790 1800 1810 1820 1830 1840 1850

DATE

100

200

300

SUNSPOT NUMBER

1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950

DATE

100

200

300

SUNSPOT NUMBER

1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050

DATE

100

200

300

SUNSPOT NUMBER

NASA/MSFC/NSSTC/HATHAWAY 2008/06

Fig. 4.5 Monthly averages (updated monthly) of the sunspot numbers show that the number

of sunspots visible on the sun waxes and wanes with an approximate 11-year cycle (Courtesy

NASA; see the web site http://solarscience.msfc.nasa.gov/SunspotCycle.shtml).

tion. The sidereal rotation can be approximated by

ω = 14.522 − 2.84 × sin

b [deg/day], (4.17)

where b is the heliographic latitude (see page 77 of [Aschwanden (2006)] and

also [Brajˇsa et al. (2001)]). An eﬀective rotation period of ≈ 27 days is usually

satisfactory to indicate most of the Sun rotation-recurrences. In addition, the Sun’s

rotation axis is tilted by ≈ 7

◦

from the axis of the Earth’s orbit.

H.D. and H.W. Babcock (1954) observed the Sun magnetic-ﬁeld over a two

years period (1952–1954) using an instrument (called the solar magnetograph) uti-

lizing the Zeeman eﬀect and established that the Sun has a general magnetic dipole

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308 Principles of Radiation Interaction in Matter and Detection

Fig. 4.6 Conﬁguration of the interplanetary magnetic-ﬁeld (following the the so-called

Archimedean spiral shape) for the Parker model as a function of the radial distance (in AU)

from the Sun in the solar equatorial plane for i) a steady solar wind with velocity of 470 km/s from

≈ 10 R

and ii) an angular velocity of 2.9 × 10

−6

rad/s for the solar rotation at equator.

ﬁeld of ≈ 10

−4

T distributed over the visible photosphere

∗

. The magnetic-dipole

tilt with respect to the rotation axis varies and depends on the solar cycle: appar-

ently the magnetic polarity is reversed (e.g., see [Jones, Balogh and Smith (2003)]

and references therein) about every 11 years with the evolution of the solar cycle,

i.e., a similar magnetic conﬁguration is found about every 22 years (the so-called

Hale cycle or solar magnetic cycle). Parker (1958) (see also [Parker (1957, 1960,

1961a, 1963, 2007)]) suggested that the solar magnetic-ﬁeld is frozen-in to the ﬂow

of the SW non-relativistic plasma, which carries the ﬁeld with it into interplane-

tary space and generates the so-called interplanetary magnetic-ﬁeld (IMF) or helio-

spheric magnetic-ﬁeld (HMF). He assumed i) a constant solar rotation with angular

velocity ω

, ii) a simple spherically symmetric emission of the SW (see page 301)

and iii) a constant (or approaching an almost constant) wind speed, v

, at larger

radial distances, e.g., for r > r

≈ 10 R

, since beyond r

the wind speed varies

slowly with the distance (see Fig. 4.4). This latter assumption is justiﬁed because

both the solar gravitation and the outward acceleration by high coronal tempera-

ture may be neglected or aﬀect marginally the wind speed for r & r

. Thus, in a

∗

The photosphere is the apparent solar surface with a diameter usually considered to be the

diameter of the Sun. The photosphere thickness is ≈ 0.014% of R

≈ 4.65 × 10

−3

AU is the solar radius, see Appendix A.2.

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Radiation Environments and Damage in Silicon Semiconductors 309

spherical system of coordinates (r, θ , φ) rotating with the Sun, the outward velocity

of a ﬂuid element carrying the magnetic ﬁeld is approximately

given by [Parker

(1958)]







= v

= 0,

= −(r − r

) ω

sin θ.

(4.18)

Since from Eq. (4.18) and for r À r

we can write

= r

dφ

−rω

sin θ

, (4.19)

the streamline with azimuth φ

at r

is obtained after integrating the previous

expression, i.e.,

= −

sin θ

dφ

⇒ r = r

−

sin θ

(φ − φ

) . (4.20)

The “spiral angle” (ψ

) is the angle that the heliospheric magnetic-ﬁeld line makes

to the radial direction and, from Eq. (4.19), is given by

= tan

−1

; (4.21)

for instance, in the ecliptic plane it becomes

sp,e

' tan

−1

rω

Furthermore, since the plasma velocity is parallel to the magnetic ﬁeld, the ratios

of the corresponding φ and r components are same [e.g., see Eqs. (4.18), (4.22)]:

(r, θ, φ)

−(r − r

) ω

sin θ

Thus, for large r the curve describing a magnetic-ﬁeld line a) can be found using

Eq. (4.20) and b) has the shape of an Archimedean spiral also termed the Parker

spiral. In this model, the IMF ﬁeld-line wraps around a cone, whose surface has

an angle θ with respect to the solar rotation axis; i.e., equivalently the expected

spiral pattern consists of ﬁeld lines on cones of constant heliographic latitude. In

Fig. 4.6, it is shown the Parker spiral for interplanetary magnetic-ﬁeld lines in the

solar equatorial plane (θ = 90

◦

) as a function of the radial distance (in AU) from

the Sun for i) a steady solar wind with velo city of 470 km/s from r

≈ 10 R

and

ii) angular velocity (ω

) of 2.9 ×10

−6

rad/s, that is the one of the solar rotation at

A complete derivation of the curve describing a magnetic-ﬁeld line, i.e., the Archimedean spiral,

can be found, for instance, in Section 6.3.3 of [Cravens (1997)], Section 12.3.1 of [Gombosi (1998)]

and Section 6.3 of [Kallenrode (2004)].