29
The theory of cosmic-ray and high-energy
solar-particle transport in the atmosphere
K. O’Brien
Northern Arizona University, Flagstaff, AR 86011-6010, USA
An essentially analytical theory of the transport of high-energy radiation through the Earth’s
atmosphere is presented here. The transport of primary and secondary particles is described by
a solution of the Boltzmann transport equation. The transport of secondary particles is based
on a solution of the Boltzmann equation separable into longitudinal and transverse components, applicable to high-energy hadron–nucleus collisions, and based on work by Passow [1]
and reported by Alsmiller [2], and on Elliott [3] and Williams [4]. All secondary particles
other than hadrons are mediated by meson production and decay. The breakup of primary
nuclei as a result of collisions with atoms of air is treated by means of a generalized Rudstam
[5] formula. The theory described here has also been applied to the atmospheres of other solar
system bodies.
1. Introduction
This is a necessarily abbreviated account (due to the limitations of space) of theoretical developments in atmospheric cosmic-ray transport that have spanned some decades since the first
publication in 1970 [6]. These developments have allowed the calculation of the propagation
and attenuation of primary cosmic rays and the generation and propagation of secondary cosmic rays in the Earth’s atmosphere and the atmospheres of other solar system bodies [7]. These
results have been also been generalized to treat the propagation of energetic solar particles in
the Earth’s atmosphere.
2. Cosmic-ray and energetic solar-particle radiation propagation through the
atmosphere
2.1. Atmospheric structure
The atmospheric structure, its pressure, temperature and density as a function of elevation
was taken from Beranek [8]. Its composition was taken from CIRA [9], yielding an average
RADIOACTIVITY IN THE ENVIRONMENT
VOLUME 7 ISSN 1569-4860/DOI 10.1016/S1569-4860(04)07004-4
© 2005 Elsevier Ltd.
All rights reserved.
30
K. O’Brien
atomic number of 7.22, an average atomic weight of 14.485 and an ionization potential of
93 eV.
2.2. The transport equation
The Boltzmann equation is an integro-differential equation describing the behavior of a dilute
assemblage of corpuscles. It was derived by Ludwig Boltzmann in 1872 to study the properties
of gases. It applies equally to the behavior of radiation.
Boltzmann’s equation is a continuity equation in phase space in terms of the angular flux
t), the number of particles of a given type (nucleons, leptons, mesons, etc.) at a
x , E, Ω,
φi (
at a time t. It is given by
location x , with an energy E, a direction Ω,
t =
Qij ,
B̂i φi r, E, Ω,
j
• ∇ + σi + di − (∂/∂Ei )Si ,
B̂i = Ω
∞
dΩ
dEB σij EB → E, Ω → Ω φj r, EB , Ω, t ,
Qij =
j
di =
4π
E
1 − β 2 (τi cβ),
(1)
where B̂i is the Boltzmann operator; σi is the absorption cross section for particles of type i;
di is the decay probability per unit flight path of radioactive particles (such as muons or
mesons) of type i; Si is the stopping power for charged particles of type i (assumed to be zero
t) is the particle flux of type-j particles at location x ,
x , EB , Ω,
for uncharged particles); φi (
energy E, direction Ω and time t; Qij is the “scattering-down” integral, the production rate
an energy E at a location x , by collisions with nuclei
of particles of type-i with a direction Ω,
at a higher energy EB ; σij is the doublyor decay of type-j particles having a direction Ω
differential inclusive cross section for the production of type-i particles with energy E and
from nuclear collisions or decay of type-j particles with an energy EB and a
a direction Ω
; βi is the speed of a particle of type i with respect to the speed of light (= ν/c);
direction Ω
τi is the mean life of a radioactive particle of type i in the rest frame; and c is the speed of
light in vacuo.
The most significant of the reactions that result from cosmic-ray and solar-particle collisions with the nuclei that comprise the Earth’s atmosphere are:
Ai + Air → N Z n n + N Z p p + N Z π π + N Z K K + N Z A Aj ,
p + Air → N p n n + N p p p + N p π π + N p K K,
n + Air → N n n n + N n p p + N n π π + N n K K,
π± → μ± + ν,
π0 → 2γ → electromagnetic showers,
μ± → e± + 2ν → electromagnetic showers.
(2)
The theory of cosmic-ray and high-energy solar-particle transport in the atmosphere
31
Equation (2) implies that atmospheric cosmic rays propagate by means of the nucleonic
cascade (the first three lines are recursive), and that all other secondaries ultimately result
from nucleon-nucleus collisions. High-energy nuclei and nucleons collide with nuclei, producing other high-energy nucleons, which in turn collide with other nuclei, losing energy to
secondary mesons and eventually to leptons and photons. Ai and Aj are primary and secondary cosmic-ray nuclei, and the N i j are the multiplicities of j -type particles resulting from
the collisions of nuclei, protons and neutrons with nuclei of air.
3. The hadronic component
The solution to equation (1) is assumed to be separable into a longitudinal component and a
transverse component:
= Li (x, E)Ti (y, z, E).
φi x, E, Ω
(3)
3.1. The longitudinal component of the hadronic transport equation
The theory of the longitudinal component depends most critically on 3 terms in equation (1):
(1) the total collision cross section;
(2) the partial inelasticities of each of the emitted hadrons;
(3) the multiplicity of secondary hadrons and nuclei emitted from a collision.
3.1.1. The total collision cross section
The total collision cross section, σ , is constant and geometric. As, at high energies, elastic
scattering is primarily forward, it is set to zero. An elastic scattering that results in the colliding
particle traveling undeviated from its original direction involves no exchange of energy and
hence does not affect its transport through the medium. The total collision cross section is
given by
σ = ξ πri2 + πrt2 L/At cm2 g−1
(4)
where
1/3
ri = 1.28 Ai − 1 fm,
1/3
rt = 1.28At
fm
and ξ =
1, At 4,
2, At > 4,
(5)
L is Avogadro’s number, A is the atomic weight, and the subscripts i and t correspond to the
incident and target nucleus, respectively.
3.1.2. The partial inelasticities
The partial inelasticities, Kj , are defined as
EB
→ Ω
dE
(E + mj )Fij EB → E, Ω
Ki =
(6)
η
where mj is the rest mass of the j -type particle and η is the lowest energy an i-type particle
can be emitted from a nucleus struck by a j -type particle.
32
K. O’Brien
Table 1
Partial inelasticities for nucleon–air collisions
Hadron
p-nucleus
n-nucleus
P
N
π+
π0
π−
K+
K0
K−
Ki
0.211
0.211
0.180
0.180
0.112
0.034
0.034
0.022
Ki
0.211
0.211
0.112
0.180
0.180
0.022
0.034
0.034
The analytical form for the production spectrum required by the theory is
−Ω
,
Fij = αi EBl /E (l+1) δ Ω
(7)
where δ is Dirac’s improper function, implying that the secondary particles emitted from a
collision proceed in the same direction as the incident primary.
Substituting equation (7) in equation (6), integrating, and taking the limit as EB goes to
infinity yields
Ki = αi /(1 − l)
(8)
and therefore
−Ω
.
Fij = (1 − l)Ki EBl /E (l+1) δ Ω
(9)
The values of the partial inelasticities to be used are taken from O’Brien [10]. These values
are based on both accelerator and cosmic-ray data and are exhibited in Table 1.
3.1.3. The secondary particle multiplicity
The secondary particle multiplicity is defined thus:
EB
→ Ω
dE.
Fij EB → E, Ω
nj (EB ) =
(10)
η
Representing Fij by equation (10), the index l can be determined by fitting equation (10)
to the shower-particle data of Meyer, Teucher and Lohrmann [11].
The stationary solution φ for secondary hadrons for a unit incident flux along the x-coordinate with an energy EB [1,2]:
Li (x, EB → E) = Ai σ (1 − l)Kj EBl /E l+1 U (EB − ηi ) x/B(EB , E)
(11)
× I1 2 rB(EB , E) ,
(1 − l)Ki ln EB − ln EU (E − ηj ) + ηi U (ηj − E) ,
B(EB , E) = σ
(12)
U (x) =
j =n,p
1,
0,
x > 0,
x < 0.
The theory of cosmic-ray and high-energy solar-particle transport in the atmosphere
33
The Heaviside function, U , shuts off secondary particle production below a bombarding
energy ηj to take into account the effect of charged-particle slowing down on secondary particle production by a charged incident particle, such as a proton. I1 is the hyperbolic Bessel
function of the first kind.
Ai is a correction factor that accounts for secondary meson decay and charged particle
slowing down.
The solution of the Boltzmann equation for a unit incident flux of energy EB and atomic
number Z at a depth x1 > x0 in a direction given by some zenith angle Ω is
0
EZ
1
0
1
σZ /SZ (E ) dE
φZ EZ , x1 = SZ EZ , x0 /SZ EZ , x1 exp −
(13)
EZ1
where
x=
E0
E1
dE /SZ (E )
(14)
is the distance in the medium where the incident nucleus at an energy E 0 has been reduced to
→ Ω)
in equation (1) is given
an energy E 1 . For secondary nuclei, the term σij (EB → E, Ω
by a modified Rudstam CDMD equation.
3.2. The transverse component of the hadronic transport equation
The cosmic-ray flux is incident on the upper boundary of the atmosphere, which is treated as
a free surface with a radius of curvature of 6.378 × 108 cm. The curvature is introduced by
means of the Chapman Function. The unit of r is g cm−2 , which simplifies the format of the
Boltzmann equation, though, of course, the detailed density structure of the atmosphere must
be used to properly account for pion and muon decay and allows the specification of a free
surface, which would otherwise be impossible, as the atmosphere has an exponential density
dependence.
Elliott [3] and Williams [4] have, by applying a Fourier transform to the transport equation,
shown how the linear, one-dimensional form might be applied to two- or three-dimensional
problems. It has been applied to account for the spreading of the cosmic-ray beam and thus to
provide an improved boundary condition for the cosmic-ray problem.
Equation (1) can be rewritten in Cartesian coordinates, where x has the same meaning as
above:
B̂i = ∂/∂x + (1 − μ2 ) cos ϕ(∂/∂y) + sin ϕ(∂/∂z) + σ + di − ∂Si (E)/∂E,
(15)
t =
Qij ,
B̂i φi r, E, Ω,
j
with the flux entering at some point x1 and traveling along x. The upper free surface is located
along z at some point z0 .
Taking the Fourier transform of equation (15) gives
∞
∞
exp(iBy y + Bz z),
dy
dz φi r, E, Ω
φi (B) =
(16)
−∞
−∞
34
K. O’Brien
which yields
φi B = ξ φi Li (r , E) exp(iBy + iBz ),
(17)
where ξ is chosen so that
∞
∞
dy
dz φi B = Li
(18)
−∞
−∞
and Li is the longitudinal component of the solution as given by equation (7).
The transverse momentum distribution from a high-energy hadron–nucleus collision is
given by [12]:
√
Ni (θ ) = Cpi /π exp −Cpi2 θ 2
(19)
where pi = E 2 + 2mi E is the momentum of the secondary, mi is its mass and C = 3.11 ×
10−6 .
The mean angle of emission from the cascade as a function of angle can be gotten from the
transverse momentum distribution by multiplying Ni by the Jacobian that converts fluxes per
unit momentum to fluxes per unit energy (J = p/W ) where W is the total energy.
Thus the mean angle of emission is
∞
θ̄i = 2π
(20)
θ J Ni dθ = π/C (Ei + m),
0
and since this is a nucleonic cascade, m is the nucleonic mass.
Instead of inverting the Fourier transform, −σtr,i is substituted for iB yielding
φi (E, r) = ξ Li exp −σtr,i y 2 + z2 = Li (E, x)Ti (E, y, z) + φp (x, EB )
(21)
where
σtr,i = σ 1 − cos(π/2 − θ̄i )
(22)
is the transport cross section.
For a component of the cosmic-ray flux entering the atmosphere at some zenith angle
greater than zero, the contribution to the flux at the origin, at a depth z0 in the atmosphere
is given by the integral
∞
z0
dy
dz φi (E, r)
φi (E, x) =
(23)
−∞
−∞
or
φi (E, x) = Li 2 − (1 + σtr,i z0 ) exp(−σtr,i z0 ) /2 + φp (EB , x).
(24)
Equation (19) takes into account the existence of the upper free surface on the radiation intensity in the atmosphere, a boundary above which there is no contribution to the atmospheric
cosmic-ray fluxes except from the primary fluxes; that is, no secondary particles above the
free surface contribute to the flux at x.
The theory of cosmic-ray and high-energy solar-particle transport in the atmosphere
35
3.3. The low-energy neutron component
Equation (24) is valid for high-energy hadrons. Charged and radioactive hadrons are largely
restricted to high energies because of the effects of charged-particle slowing down and radioactive decay at low energies. Low-energy hadron transport is therefore low-energy neutron
transport. Observing from the work of Armstrong [13] and Hess et al. [14] that the shape
of the low-energy cosmic-ray neutron spectrum changes but slowly with depth, the neutron
spectrum of Rösler [15] was applied to the neutron spectrum calculated using equation (19)
between 500 MeV and 0.5 eV. Below 0.5 eV, the Hess et al. [14] spectrum was made piecewise
continuous with that neutron spectrum.
4. The lepton and photon component
4.1. The muon component
The source for the muon flux, Qπμ is
Qπμ = (mπ± /2πmμ )φπ (mπ /mμ )Ej dπ .
(25)
The flux at a depth x1 > x0 in a direction given by some zenith angle Ω is, in the
continuous-slowing-down approximation,
1
0
1
φμ Eμ , x1 = Qπμ Sμ Eμ , x0 /Sμ Eμ , x1 exp −
0
Eμ
1
Eμ
1 dπ /Sμ E dE .
(26)
The relationship between E 0 , E 1 and x = x0 − x1 is given by
x =
E0
E1
dE /Sμ (E )
(27)
4.2. The electromagnetic component
Muons and pions are radioactive. Muons have a mean life of 2.197 μs, charged pions, a mean
life of 26 ns and neutral pions a mean life of 84 as. Muons decay into two neutrinos and an
electron, and neutral pions into two photons (cf. equation (2)). Charged pions give rise to the
muon component of cosmic rays. In this approach, the energy of the two photons from neutral
pion decay is deposited where the pion is produced and decays. The energy from a decaying
muon is equally shared among the three light particles, so that the electron gets one third, and
this energy is also deposited at the point where the electron is produced. Secondary negaton,
positon and photon scalar spectra were calculated using CASCADE [16] and coupled to the
locally deposited energy.
36
K. O’Brien
5. Cosmic rays
5.1. Composition
Peters [17] represents the integral cosmic-ray spectrum by
log Φ = a − 0.0495[11.9 + log(1.7 + E)]2
(28)
where Φ is the number of particles with energies greater than E GeV, per (m2 s sr). The
differential spectrum is therefore
log ϕ = a − 0.0495 11.9 + log 1.7 + E 2
+ log 0.0990 11.9 + log(1.7 + E) /(1.7 + E)
(29)
where ϕ is now the flux per (GeV m2 s sr) per nucleon.
The constant a governs the magnitude and intensity of ϕ.
Gaisser and Stanev [18] (1998) gave a table of relative particle intensities at 10.6 GeV
normalized to the oxygen flux (≡ 1). The oxygen flux per nucleon at that energy is 3.26 ×
10−6 per (cm2 s sr GeV). Their data appear in Table 2, along with absolute intensities obtained
by multiplying their data in column 3 by the oxygen flux. Equation (29) is solved for each of
these components.
5.2. The primary spectrum
The primary cosmic-ray spectrum used in the calculations described below is divided into
twelve groups:
• the protons in the hydrogen flux, the unbound or free protons, and
• eleven groups of primary nuclei.
Table 2
Composition of cosmic rays at 10.6 GeV per nucleon
Z
Element
Relative abundances
1
2
3–5
6–8
9–10
11–12
13–14
15–16
17–18
19–20
21–25
26–28
H
He
Li–Be
C–O
F–Ne
Na–Mg
Al–Si
P–S
Cl–Ar
K–Ca
Sc–Mn
Fe–Ni
730
34
0.4
2.20
0.3
0.22
0.19
0.03
0.01
0.02
0.05
0.12
The theory of cosmic-ray and high-energy solar-particle transport in the atmosphere
37
The elements above helium in the periodic table are represented here by the astronomer’s
term, “metals”, rather than the traditional cosmic-ray physicist’s term, HZEs. The metals are,
after all, generated from the circumstellar material in the vicinity of a Type II supernova and
are proper material for study by astronomers, and HZE is so clumsy, not to say ugly, a term.
Cosmic-ray proton spectra below 10 GeV are represented by the following equation [19]:
−2.65
,
ϕ = 9.9 × 104 E + 780 exp −2.5 × 10−4
(30)
where E is in MeV/nucleon. The eleven groups of nuclei are obtained by multiplying equation (30) by the values in column three of Table 2.
The cosmic-ray spectra above 10 GeV are represented by Peters’ model, equation (29).
5.3. The geomagnetic field
A particle entering the Earth’s magnetic field must have sufficient momentum per unit charge,
or rigidity to penetrate the Earth’s magnetic field and collide with air nuclei and produce
atmospheric cosmic rays. That rigidity, a function of the particle’s charge, and the zenith
and azimuthal angles it makes with the Earth’s surface is called the “cutoff” rigidity and is
expressed in GV (GeV c−1 per unit charge).
Shea and Smart [20–23] have calculated vertical cutoff rigidities for a number of magnetic
epochs using numerical methods. These are effective cutoffs, taking into account the shielding
of the solid Earth and the penumbra.
Rösler [15] has normalized Störmer’s equation to the vertical cutoff given by the Shea–
Smart calculations. Thus the details of the moments of the geomagnetic field are carried by
the vertical cutoff distribution, but otherwise the field is locally dipole. In this theory, the
cutoff rigidity is treated like a high-pass filter, though for certain angles and locations this
may not be absolutely correct.
5.4. Solar modulation
Gleeson and Axford [24] have shown theoretically that the effect on the galactic cosmic-ray
spectrum of passage through the interplanetary medium is approximately the same as would
be produced by a heliocentric potential with a magnitude at the Earth’s orbit equal to the
energy lost per unit charge to that point by interacting with the solar wind.
The energy spectrum at the Earth’s orbit is then obtained from the unmodulated energy
spectrum outside the heliopause:
2
ϕi (E) dE = ϕi (T ) dE r(E)/p(T ) , T = E + ZV
(31)
where r is the rigidity, and V is the voltage.
As ϕi is isotropic outside the heliosphere, it is, by Liouville’s theorem, isotropic at the
Earth’s orbit.
5.5. Determining the heliocentric potential
Neher [25] measured the cosmic-ray proton spectrum during a series of 31 balloon flights
between July 31 and August 4, 1965. Half were made from shipboard going north from Peru
38
K. O’Brien
to Greenland. The other half were simultaneous flights made from Bismarck, North Dakota.
The former were used to analyze the cosmic-ray spectrum by virtue of its interaction with the
Earth’s magnetic field and the latter to correct for changes in its intensity. The importance of
this experiment to the determination of the heliocentric potential is that the timing of each
of these balloon flights is precisely known and could be related to the Deep River neutron
monitor counting rate at the time of the flight.
Equation (31) fits the measured spectrum with V = 500 MV.
The further assumption was made that the NM-64 monitors respond in the same way to
cosmic-ray fluxes as the Deep River and Goose Bay neutron monitors, that purely local effects are negligible, and that small changes in the cosmic-ray spectra with modulation and
with changes in ground elevation are also negligible, enabling the use of other high-latitude
monitors. This has become important, as the Deep River monitor has been taken down.
6. Comparison with experiment
6.1. The cosmic-ray ionization profile
A comparison with ionization measurements over Durham, New Hampshire in 1969(26) is
shown in Fig. 1. The ionization, or ion-pair production rate, is given as I , ion-pairs per (cm3 s)
of air at NTP. Agreement is seen to be quite good.
6.2. The cosmic-ray neutron profile
Atmospheric neutron fluxes were calculated for Aire sur l’Adour, France, for the geomagnetic
and solar conditions of 1966 and compared with a variety of measurements [15,27,33]. For
convenience, the experimental results were characterized in terms of the geomagnetic latitude.
Fig. 1. Calculated and measured [26] cosmic-ray ionization over Durham, New Hampshire in 1969.
The theory of cosmic-ray and high-energy solar-particle transport in the atmosphere
39
Fig. 2. Calculated and measured [27–33] cosmic-ray neutron profile at 42◦ geomagnetic latitude.
Despite the scatter in the measurement results, agreement is good (Fig. 2). The conditions of
the calculation were chosen to correspond to the measurements of Boella et al. [27], which
were made at the highest altitude and, therefore, most sensitive to geomagnetic and solar
effects.
7. Energetic solar particles
High-energy solar particles, produced in association with solar flares and coronal mass ejections, occasionally bombard the Earth’s atmosphere, resulting in radiation intensities additional to the already-present cosmic radiation. Access of these particles to the Earth’s vicinity
during times of geomagnetic disturbance are not adequately described by using static geomagnetic field models. These solar fluxes are also often distributed non-uniformly in space,
so that fluxes measured by satellites at great distances from the Earth and which sample large
volumes of space around the Earth do not accurately predict fluxes locally at the Earth’s surface.
A method is described here, which uses the ground-level neutron monitor counting rates as
adjoint sources of the flux in the atmosphere immediately above them to obtain solar-particle
ionization rates as a function of position over the Earth’s surface. This approach has been
applied to the large September 29–30, 1989 event (GLE 42). It was applied to determine the
magnitude and distribution of the solar-particle ionization from this atypically large event.
7.1. The acceleration mechanism
Solar flares and coronal mass ejections (CMEs) can accelerate hydrogen (and some helium
and heavier nuclei) ions to high energies by means of the shock-acceleration mechanism.
The shock-acceleration mechanism yields a spectrum, which is a power-law per unit of
rigidity or (since protons alone were measured and therefore alone will be considered) momentum, and can be converted to the flux per unit energy by multiplying, again, by the appro-
40
K. O’Brien
priate Jacobian:
ϕ(r) = a(t)r −γ (t) ,
φ(E) = ϕ(r)r/(E + m).
(32)
When the Earth is intercepted by a shock or other interplanetary disturbance, the induced
current in the magnetosphere affects the geomagnetic field in a complex manner, changing the
distribution of cutoff rigidities and usually reducing them.
If the Earth is in the right position, it may be intercepted by the plasma accelerated by a
prior shock. This plasma will affect the geomagnetic field in a complex manner, changing the
distribution of cutoff rigidities and usually reducing them.
Further, the flux may be distributed non-uniformly over the Earth’s surface. These factors
make a straightforward calculation of the resulting radiation distributions from satellite data
impossible.
However, since there are a number of cosmic-ray neutron monitors distributed over the
land surface of the Earth, they may be used to obtain a(t) whereas γ (t) can be obtained from
satellite data.
7.2. The computational method
In principle, one could use the ground-level neutron monitor data as adjoint sources and solve
the adjoint form of the transport equation, applying the satellite spectra as boundary conditions. However, since many radiation components contribute to the response of a neutron
monitor, an equal number of adjoint calculations would have to be calculated for each value
of γ . A simpler and more straightforward approach is to execute a forward calculation of the
neutron monitor response for a range of values of γ and interpolate among them. The value
of a(t) in the equation for the flux is determined by setting the integral flux above 100 MeV
to unity for each value of γ (t).
7.3. Neutron monitor data
The neutron monitor stations that were in existence and had useful data during ground-level
event 42 (GLE 42, September 29–30, 1989) and which were used to obtain the necessary
adjoint source data are listed in Table 3.
These data and the associated counting rates were obtained from WDC-C2 for Cosmic Rays
at Ibaraki University by ftp [34].
7.4. Satellite particle spectra
The satellite particle energy spectra used here were derived from data obtained by particle
detectors aboard the GOES-7 satellite maintained by the NOAA Space Environment Center. These detectors measure the flux of energetic protons at geostationary orbit from energies of 600 keV to greater than 700 MeV (or momenta of 330 MeV c−1 to greater than
1300 MeV c−1 ) in 11 discrete channels. The observations of protons of greater than about
9 MeV (130 MeV c−1 ) are representative of those that would be obtained outside the magnetosphere in that protons of higher energy have full access to geostationary orbit (or alternatively, in that the geomagnetic cutoff at geostationary orbit has never been observed to exceed
9 MeV (130 MeV c−1 )) [35].
The theory of cosmic-ray and high-energy solar-particle transport in the atmosphere
41
Table 3
Neutron monitor stations used in the analysis of GLE 42
Latitude
−78.3
−75.5
−63.5
−57.4
−33.2
−27.3
−18.0
−0.4
25.3
28.4
33.3
36.2
40.9
42.6
44.2
47.9
48.0
48.2
50.5
50.8
51.3
52.0
54.8
54.9
57.6
58.3
60.3
61.8
62.4
62.7
64.5
70.4
88.2
Longitude
(degrees geomagnetic)
0.0
230.9
43.8
127.7
80.3
90.1
82.6
354.0
205.5
184.5
150.7
122.1
174.4
92.1
157.8
89.5
102.4
315.4
210.2
193.7
351.7
87.9
357.7
95.6
348.9
301.8
191.3
117.1
225.7
125.6
12.6
264.1
2.0
Station
South Pole
Terre
Sanae8
Kerguelen
Hermanus
Potchefstroom
Tsumeb
Huancayo
Tokyo
Beijing
Alma Ata
Tbilisi
Irkutsk
Rome
Novosibirsk
Jungfraujoch
Lomnicky Stit
Climax
Magadan
Yakutsk
Newark
Dourben
Durham
Kiel
Deep River
Calgary
Tixie
Oulu
Cape Schmidt
Apatity
Goose Bay
Inuvik
Thule
The principal correction that had been applied to those data were a correction for the HEPAD response to backward fluxes through it, and subtraction of the background counting rates
in each channel due primarily to galactic cosmic rays, their progeny, and to a lesser extent,
instrument noise.
7.5. Results
Ionization calculations were carried out for 35 hours following the event, for elevations of sea
level: 30 000; 40 000; 50 000; 60 000; 70 000 and 80 000 feet for each of the sites using the
theory described above but designed for solar-particle events. Figure 3 exhibits the hourly-
42
K. O’Brien
Fig. 3. Cosmic-ray ionization at a depth of 100 g cm−2 in the atmosphere at the peak of GLE 42, 0130 hrs, September 30, 1989.
averaged distribution of ionization due to the solar-particle event at the period of maximum
intensity of the event at an atmospheric depth of 100 g cm−2 [35].
Any irregularities in the apparent solar particle flux distributions are presumably due to
the approximations described above and the existence of irregularities and/or gradients in the
interplanetary medium as well as to variations in the access of these particles to the magnetosphere. It is demonstrable that significant perturbations of magnetospheric access can occur
(at least at energies of the order of 500 MeV or less) as a result of the interference of interplanetary shocks and/or magnetic structures with the uniform propagation of solar energetic
particles into the magnetosphere.The ionization distribution from this large event has a maximum at the highest latitudes at an atmospheric depth of 100 g cm−2 of almost 2000 I .
8. Conclusions
A two-component analytical solution to the Boltzmann equation has been successfully applied to the propagation of cosmic rays in the Earth’s atmosphere. Its adequacy has been
demonstrated by comparison with experimental data.
This equation has also been applied to the propagation of high-energy solar particles. Unfortunately there are no data, to the best of the author’s knowledge, with which these calculations
can be compared. The solar-particle calculations are more difficult and more approximate than
the cosmic-ray calculations due to magnetospheric effects, time dependence and the sparseness of the data base on which these calculations are based. Nonetheless the approach and the
results are felt to be reasonable.
The theory of cosmic-ray and high-energy solar-particle transport in the atmosphere
43
Acknowledgements
I wish to acknowledge many useful discussions with Ernst Felsberger, Wolfgang Heinrich,
Brent Lewis and Herbert H. Sauer. Stefan Rösler generously supplied me with his code for
calculating non-vertical cutoffs, and with his low-energy neutron spectrum, both of which
play important parts in these calculations. I wish especially to acknowledge M.A. Shea and
D.F. Smart for generously giving me their calculations of the 1995 Epoch of the vertical cutoff
distribution in advance of its publication.
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