Using data on the proton fluxes of the Earth's radiation belts
(ERBs) with energy ranging from 0.2 to 100 MeV on the drift L shells ranging
from 1 to 8, the quasi-stationary distributions over the drift frequency
fd of protons around the Earth are constructed. For this purpose, direct
measurements of proton fluxes of the ERBs during the period from 1961 to 2017 near the
geomagnetic equator were employed. The main physical processes in the ERB
manifested more clearly in these distributions, and for protons with
fd>0.5 mHz at L>3, their distributions in the {fd,L} space have a more regular shape than
in the {E,L} space. It has also been found that
the quantity of the ERB protons with fd∼ 1–10 mHz at L∼2
does not decrease, as it does for protons with E> 10–20 MeV (with
fd>10 mHz), but increases with an increase in solar
activity. This means that the balance of radial transport and loss of
ERB low-energy protons at L∼2 is disrupted in favor of transport
of these protons: the effect of an increase in the radial diffusion rates
with increasing solar activity overpowers the effect of an increase in the
density of the dissipative medium.
Introduction
The Earth's radiation belts (ERBs) mainly consist of charged particles with
energy from E∼100 keV to several hundreds of
megaelectronvolts (MeV). In the field of the geomagnetic trap, each particle
of the ERBs with energy E and equatorial pitch angle α0
(α is the angle between the local vector of the magnetic field and
the vector of a particle velocity) makes three periodic movements: Larmor rotation, oscillations along the magnetic field line, and drift
around the Earth (Alfvén and Fälthammar, 1963; Northrop, 1963).
Three adiabatic invariants (μ, K, Φ) correspond to these
periodic motions of trapped particles as well as three periods of time or
three frequencies: a cyclotron frequency (fc), a frequency of particle
oscillations along the magnetic field line (fb), and a drift frequency of
particles around the Earth (fd). For the near-equatorial ERB protons, we
have fc∼ 1–500 Hz, fb∼ 0.02–2 Hz, and fd∼ 0.1–20 mHz. The frequency fc increases by tens to hundreds of times
with the distance of the particle from the plane of the geomagnetic equator
(in proportion to the local induction of the magnetic field), and the
frequency fb decreases by almost 2 times with increasing amplitude of
particle oscillation.
The number of particles with a given frequency fc decreases rapidly with
an increase in L and refers to higher and higher geomagnetic latitudes. For
each given frequency fb, particles become more and more energetic with
an increase in L (E∞L2), and their number becomes smaller.
Compared with the frequencies fc and fb, the drift frequency fd
for one particle species has a much narrower range of values; it does not
depend on the mass of the particles, and it very weakly depends on the
amplitude of their oscillations (∼20 % variation). In this case, there are a significant number of particles corresponding to
a certain value of fd on each L shell.
Therefore, it can be expected that the distributions of the ERB particles in
the {fd,L} space will have a more regular
shape than in the {E,L} space, and the main
physical processes in these belts will manifest themselves more clearly in
these distributions. Furthermore, it can also be expected that more fine features, which would not appear
in the {E,L} space, can be revealed on this more
ordered background.
Despite the importance of the drift frequency fd for the mechanisms of
the ERB formation, reliable and sufficiently complete distributions of
particles in the ERBs (over the frequency fd) have not been presented
nor analyzed; indeed, this is the first time.
The analysis presented in this paper is limited to the protons of the ERB
during magnetically quiet periods of observations (Kp < 2), when the
proton fluxes and their spatial-energy distributions were quasi-stationary.
In the following sections, the distributions of the ERB protons over their
drift frequency fd are constructed from experimental data (Sect. 2) and
analyzed (Sect. 3). The main conclusions of this work are given in Sect. 4.
Constructing the distributions of the ERB protons over their drift
frequency
The problem of methodical differences in measurements of the fluxes of
protons of the radiation belts on different satellites was one of the main
issues in this work. From the available published experimental data, those
that are in good agreement with one another were selected, and
all unreliable measurement results (with admixture of electrons and various
ionic components of the ERB to the protons) were excluded from consideration. The reliable
experimental results for proton fluxes and their anisotropy near the
equatorial plane were then represent in the {E,L} space;
this space is very efficient with respect to organizing experimental data
obtained in different ranges of E and L.
In such a representation of experimental data, there is no need for
interpolation and extrapolation of fluxes on the energy (in other
representations, such necessity arises due to differences in channel widths
and their positions on the energy scale for instruments installed on
different satellites). In addition, with such a representation – the data from various experiments, in one
figure – it is possible to construct the
isolines of fluxes (and anisotropy of fluxes); these isolines cannot
intersect with each other and, thus, allow for the exclusion of data that
sharply fall out of the general picture (for more details, see
Kovtyukh, 2020).
Spatial-energy distributions of the ERB protons near the
equatorial plane
To construct the distributions of the ERB particles over the drift
frequency, it is necessary to have reliable distributions of the
differential fluxes of the ERB protons in the {E,L} space, where E is the kinetic energy of protons, and L is the drift
shell parameter.
From the data of averaged satellite measurements of the differential fluxes
of protons with an equatorial pitch angle α0≈90∘, the aforementioned distributions are constructed in Kovtyukh (2020)
during quiet periods (Kp < 2) near the solar activity maximum in the 20th
(1968–1971), 22th (1990–1991), 23th (2000), and 24th (2012–2017) solar
cycles. Such distributions, separately for minima and maxima of the 11-year
solar activity cycles, are also constructed from satellite data for other
ionic components of the ERB (near the equatorial plane), but the most
reliable and detailed picture was obtained for protons (see Kovtyukh,
2020).
In Fig. 1, one of these distributions is reproduced for periods near solar
maxima (on the data from 1968 to 2017); here, data from different satellites
are associated with different symbols. The numbers on the curves (isolines)
refer to the values of the decimal logarithms of the differential fluxes
J (cm2 s sr MeV)-1 of protons (with equatorial pitch angle
α0≈90∘). The red lines correspond to the
dependences fd (mHz) =0.379⋅L⋅E (MeV) for the drift
frequency of the near-equatorial protons in the dipole approximation of the
geomagnetic field.
Distribution of the differential fluxes J in the
{E,L} space for protons with α0≈90∘ near maxima of the solar activity (from Kovtyukh, 2020). Data from
satellites are associated with different symbols. The numbers on the curves
refer to the values of the decimal logarithms of J. Fluxes are given in units
of (cm2 s sr MeV)-1. The red lines correspond to the drift
frequency fd (mHz), and the green line corresponds to the maximum energy of
the trapped protons.
During the quiet periods considered in this work, the geomagnetic field at L<5 is close to the dipole configuration and L≈L∗
(see Roederer and Lejosne, 2018). At large L, the magnetic field differs from
the dipole configuration, even in quiet periods; this leads to the flattening of the
isolines of the proton fluxes at L>5 in Fig. 1.
Only protons with energies less than some maximum values determined by the
Alfvén criterion can be trapped on the drift shells. The Alfvén criterion is calculated as follows: ρc(L,E)≪ρB(L), where ρc is the gyroradius of protons, and ρB is the radius of the curvature of the magnetic field (near the
equatorial plane). According to this
criterion and to the theory of stochastic motion of particles, the
geomagnetic trap in the dipolar region can only capture and durably hold
protons with E (MeV) <2000⋅L-4 (Ilyin et al., 1984). The
green line in Fig. 1 represents this boundary.
The distribution of the ERB proton fluxes, shown in Fig. 1, refers to the
years of the solar maximum, but the solar-cyclic variations in the ERB
proton fluxes are small and localized at L<2.5 (see Kovtyukh,
2020).
Spatial-energy distributions of the ERB protons outside the equatorial plane
The quasi-stationary fluxes J of the ERB particles with given energy and
local pitch angle α usually decrease when the point of observation
is shifted from the equatorial plane to higher latitudes along a certain
magnetic field line. In the inner regions of the ERB, on L<5,
angular distributions of protons usually have a maximum at the local
pitch angle α=90∘. In the wide interval near this maximum, these
distributions are well-described by the function J(α,B/B0)∝B/B0-A/2sinAα
(Parker, 1957), where A is the index of an anisotropy of a fluxes, B is the
induction of a magnetic field at the point of measurements of these fluxes,
and B0 is the induction of a magnetic field at the equatorial plane on the
same magnetic line.
The empirical model of an anisotropy A(E,L) for the proton fluxes with E∼ 0.1–2 MeV on L∼ 2–5 near the equatorial plane for the
quasi-stationary ERB (for quiet periods with Kp < 2) is presented in
Fig. 2. The anisotropy index A of these fluxes is shown in Fig. 2, in the {E,L} space, in the form of isolines with the
same values of A from 1.5 to 8.0 and with a step ΔA=0.5. The integer
values of this index are plotted on the corresponding isolines as
red numbers.
Empirical model of the anisotropy index A(E,L) of the ERB proton
fluxes averaged on the data from the satellites obtained near the plane of the
geomagnetic equator. Values of A are given on isolines of the anisotropy: A= 1.5–8.5 with the step ΔA=0.5.
When constructing this model, we consider and analyze the data from the
following satellites: Explorer 12 (Hoffman and Bracken, 1965), Explorer 14
(Davis, 1965), Explorer 26 (Søraas and Davis, 1968), OV1-14 and OV1-19
(Fennell et al., 1974), Explorer 45 (Williams and Lyons, 1974; Fritz and
Spjeldvik, 1981; Garcia and Spjeldvik, 1985), ISEE-1 (Garcia and Spjeldvik,
1985; Williams and Frank, 1984), Van Allen
Probes (Shi et al., 2016), and other satellites. These data were obtained from
1961 to 2015.
Figure 2 shows that the anisotropy
of a proton flux monotonically increases with decreasing L (from A∼3.5 to A∼8.0) for rather high energy (>1 MeV). For E>0.3 MeV on L<3, anisotropy monotonically increases with increasing energy, but for E>0.5 MeV on L>3, it is almost energy independent.
Some small irregularities of the isolines in Fig. 2 are due to the fact that
the experimental data used to construct this figure were obtained in
different years, with different instruments, and during different
solar activity intensities. At the same time, Fig. 2 demonstrates the important
regularities of the pitch angle distributions of the quasi-stationary ERB
proton fluxes.
In the {E>0.5 MeV, L>3} region, the isolines of the anisotropy index are almost parallel to
each other and to the energy axis. This adiabatic regularity refers to
protons belonging to the power-law tail of their energy spectra, the
exponent of which practically does not change when L changes (at L>3). In Fig. 2, the red lines correspond to the lower boundary
of the power-law tail of the ERB protons' energy spectra: Eb=(36±11)L-3 MeV (see Kovtyukh, 2001, 2020).
The pattern of A(E,L) in the region on L>3 at E∼ 0.2–0.5 MeV and the local minimum at L∼3 (E∼0.2 MeV) are connected
with the local maximum in the quasi-stationary proton energy spectra of the ERB,
which corresponds to E=(17±3)L-3 MeV (see Kovtyukh, 2001,
2020).
These regularities in the pattern of A(E,L) are explained within the framework
of the theory of radial transport (diffusion) of the ERB protons with
conservation of the adiabatic invariants μ and K of their
periodic motions (these issues were most fully considered in Kovtyukh,
1993).
Both the local maximum at L∼2.5 (E<0.1 MeV) and the region
of low anisotropy at L∼2 (E∼0.1 MeV) in Fig. 2 are related to
the ionization losses of protons.
With respect to the data from the satellites, the pitch angle distributions of the ERB
proton fluxes strongly depend on magnetic local time (MLT) at L>5: the average index
A values on the dayside are larger than on the nightside, and this
dependence becomes more distinct with increasing energy (see, e.g., Shi et
al., 2016). These results indicate that drift shells splitting (Roederer,
1970) play an important role in the formation of these distributions at L>5.
In the calculations performed here, it was assumed that the pitch angle distributions of the ERB proton fluxes
at L>6, averaged over MLT, at α0∼90∘
are nearly isotropic near
the equatorial plane.
High anisotropy for the fluxes of protons at E= 5–50 MeV and a strong
dependence A(L) at the inner boundary of the inner belt (L= 1.15–1.40,
B/B0= 1.0–1.7) were obtained on the DIAL satellite (Fischer et al.,
1977). According to these data, there is an anisotropy index increase from A∼12 at L=1.25 to A∼60 at L=1.15, but this value is not dependent on L at L= 1.25–1.40. These results are supported by the data from the
Resurs-01 N4 satellite for protons with E= 12–15 MeV, which were obtained at h∼800 km (Leonov et al., 2005). These data will be taken into account in our
calculations.
The experimental results on the pitch angle distributions of the ERB proton
fluxes and their anisotropy indexes were discussed in detail in Kovtyukh (2018).
Drift frequency distributions of the ERB protons
Based on the results shown in Figs. 1 and 2, one can calculate the
distributions of the ERB protons over the drift frequency fd. In these
calculations, the dipole model of the geomagnetic field was used, according
to which (see, e.g., Roederer, 1970) the point of the magnetic field line at
a given L and a geomagnetic latitude λ is located from the center of
the dipole at a distance
R(L,λ)=RELcos2λ.
Here, RE is the Earth's radius, and the field induction at a given L
changes with changing λ as follows:
B(L,λ)=4-3cos2λcos6λB0(L),
where B0(L)=0.311G×L-3.
The fact that the drift frequency fd of the
nonrelativistic particles essentially only depends on their kinetic energy
E and on L was also taken into account. This value depends very slightly on the particle pitch angle: with
an increase in the geomagnetic latitude of the mirror point of the particle
trajectory from 0 to 10∘, it increases by only 1.5 %, and in the range
from 0 to 20–30∘, it increases by 5.8 %–12.5 %.
The number of protons with energies from E to E+dE per unit volume n is equal
to the differential flux of these particles J (falling per unit time per unit
area of the detector per unit solid angle) divided by the velocity v of these
particles: n=J/v. For nonrelativistic protons with mass m, this velocity is
(2E/m)1/2.
Thus, in the near-equatorial region, between L and L+dL and within geomagnetic
latitudes from 0 to ±λ0, the total number of
nonrelativistic protons with mirror points within this region and with
energy from E to E+dE, drifting on a given L with frequency fd(L,E) around
the Earth, is
ΔN(L,fd)=2∫0λ02πRE2LdLB0(L)B(L,λ)RELcosλ4-3cos2λdλ×4π∫α01α02JL,E(L,fd)dE2E(L,fd)/msinAα0cosα0dα0,
where m is the rest mass of a proton, J(L,E(L,fd)) is the differential
fluxes, and E(L,fd) is the protons' energy. The first integral takes into account that the magnetic flux in the layer between shells L and L+dL is
conserved when latitude λ changes, i.e., 2πRELcosλREdL=2πRELB0(L)B(L,λ)REdL.
As a result of integrating the last expression over α0 and
replacing cosλ≡t, we obtain
ΔN(L,fd)=4πRE3L2dLJL,E(L,fd)dE2E(L,fd)/m×4πA+1×∫cosλ01t7t64-3t2A+12-(0.565)A+1dt.
When integrating over equatorial pitch angles α0, Liouville's
theorem and the conservation of the first adiabatic invariant (μ) are
taken into account: sin2α01=B0(L)/B(L,λ0) and sin2α02=B0(L)/B(L,λ), where B(L,0)=B0(L).
With an increase in λ from 0 to λ0=30∘, the
value of the function 4-3t2 increases from 1 to 1.32, i.e.,
deviates from the average value (1.16) by only 16 %. Most parts of the ERB
protons are concentrated at these latitudes. Therefore, when calculating the
last integral, we will assume that 4-3t2≈1.16.
Thus the following expression is obtained:
ΔN(L,fd)=kJL,E(L,fd)E(L,fd)F(A)L2dLdE,
where
F(A)=1A+1[(1.16)-(A+1)/23A+111-0.21⋅0.65A-0.085(0.565)A+1]
and
k=(4π)2RE3m/2=2.945×1019cm2ssrMeV1/2.
When calculating the values of ΔN, we assume that dL/L=dE/E=0.1.
Finally, for the indicated ERB region near the equatorial plane, we obtain
ΔN(L,fd)=2.945×1017JL,E(L,fd)×E(L,fd)F(A)L3,
where J, the differential fluxes of protons with equatorial pitch angle
α0≈90∘, is given in units of (cm2 s sr MeV)-1, and the energy of protons E is given in megaelectronvolts. The dependence
F(A) is shown in Fig. 3.
Dependence of the factor F(A) in Eq. (1) on the anisotropy index
A of the proton fluxes.
For protons of the ERB, the radial profiles ΔN(L,fd) for
fd=0.2, 0.3, 0.5, 1, 2, 3, 5, 10, 20, and 30 mHz, calculated using Eq. (1) and Figs. 1–3, are shown in Fig. 4, and the
frequency spectra ΔN(fd,L) at L=2, 2.5, 3, 4, 5, and 6 are
shown in Fig. 5. Near each curve in Fig. 4, the corresponding value of
fd (mHz) is indicated, and each spectrum in Fig. 5 shows the corresponding
L value (these values are highlighted in red). In Figs. 4 and 5, for the sake of clarity,
thin curves alternate with thick curves, and spectra at L=2 and
2.5 are highlighted in red in Fig. 5.
Radial profiles ΔN(L,fd) for protons of the ERB with
drift frequencies fd=0.2, 0.3, 0.5, 1, 2, 3, 5, 10, 20, and 30 mHz,
plotted for periods of maximum solar activity. The fd values
corresponding to each curve are highlighted in red. For clarity, thin curves
are interspersed with thick curves.
Frequency spectra ΔN(fd,L) for protons of the ERB at
L=2, 2.5, 3, 4, 5, and 6, plotted for periods of maximum solar activity.
The values of L corresponding to each spectrum and spectra at L=2 and 2.5 are
highlighted in red. The red dotted line shows the spectrum ΔN(fd,L) of the ERB protons at L=2, constructed from data during
minimum periods of solar activity (see Kovtyukh, 2020). For clarity, thin
curves are interspersed with thick curves.
The errors in these calculations mainly consist of the errors in the
averaged experimental data shown in Figs. 1 and 2 (these errors are most
significant at L<2) and the deviations of the
geomagnetic field from the dipole model at L>5.
As λ0 decreases, the errors in our calculations will decrease.
These errors can also be reduced by using numerical computer calculations.
However, it should be taken into account that, even in very quiet periods of
observations, the fluxes of the ERB protons, as well as the energy spectra
and pitch angle distributions of these fluxes, may experience changes that
exceed the errors in our calculations.
Discussion
In agreement with the results of experimental and theoretical studies, at
L>2, the main mechanism for the formation of the ERB protons is
the radial diffusion of particles from the outer boundary of the geomagnetic
trap to the Earth under conservation the adiabatic invariants μ and
K (see, e.g., Lejosne and Kollmann, 2020; Kovtyukh, 2016, 2018).
Figures 1 and 2 make it possible to determine the regions of
the {E,L} space near the equatorial plane in
which the ionization losses of ions during their radial diffusion can be
neglected and where they cannot.
The isolines of proton fluxes in Fig. 1 at sufficiently large E and L ascend
with decreasing L, in the direction of increasing energy, in strict agreement
with the adiabatic laws of the radial transport of particles. At lower L these
isolines do change the direction of their course under the influence of
ionization losses, which increase rapidly with decreasing L (see Kovtyukh,
2020, for details).
At sufficiently large values of E and L, isolines of the anisotropy index in
Fig. 2 pass practically parallel to each other and parallel to the energy
axis, in agreement with the laws of adiabatic transport of particles with
power-law energy spectra (see Kovtyukh, 1993). At lower E and L, a more
complex picture is formed under the influence of ionization losses (for more
details, see Kovtyukh, 2001, 2018).
With decreasing L, the radial diffusion is decreased very rapidly, and the
belt of protons with E> 10–20 MeV on L<2 is generated
mainly as result of the decay of neutrons of albedo which are knocked from the
atmospheric atoms' nuclei by the galactic cosmic ray's (GCR) protons. This
mechanism (CRAND) is simulated in many contemporary studies based on
experimental data (see, e.g., Selesnick et al., 2007, 2013, 2014, 2018).
The mechanisms of formation of the ERB under the action of radial diffusion
and CRAND are manifested and clearly differ both in the radial profiles
ΔN(L,fd) and in the frequency spectra ΔN(fd,L) of
protons.
Let us consider the manifestations of these mechanisms in Figs. 4 and 5 as well as the
related effects.
In contrast to the radial profiles of fluxes J(L,E), the radial profiles
ΔN(L,fd) for protons with fd<5 mHz (see Fig. 4)
have much less steep outer edges, and their steepness decreases with
decreasing frequency fd. This effect is mainly connected with an
increase in the volume of magnetic tubes (factor L3 in Eq. 1 from
Sect. 2.3) and with a decrease in the anisotropy index of proton fluxes
with increasing L.
At the same time, in comparison with the radial profiles J(L,E), the radial
profiles ΔN(L,fd) have much steeper inner edges. This effect is
mainly connected to the large anisotropy of proton fluxes in the
corresponding region of {E,L} space and to the
rapid growth of the anisotropy index with decreasing L in this region. It is
especially expressed in the radial profiles ΔN(L,fd) at fd∼ 0.3–1 mHz (see Fig. 4); this is due to the fact that the
anisotropy index of proton fluxes strongly depends on E and L in the
corresponding region of {E,L} space (see Fig. 2).
Radial profiles ΔN(L, fd) at fd>10 mHz are formed
by the CRAND mechanism. They have a maximum at L∼ 1.5–2.0, and the
steepness of their inner and outer edges does not differ as much as for
lower frequencies fd (see Fig. 4). When constructing these profiles, it
was taken into account that an anisotropy index A of proton
fluxes at E= 5–50 MeV does not depend on L at L= 1.25–1.40: A=12±2 (Fischer et
al., 1977; Leonov et al., 2005).
The shape of the spectra ΔN(fd,L) at L>3 is
determined, first of all, by the shape of the energy spectra of proton
fluxes J(E,L) at the outer boundary of the geomagnetic trap. Gradually, as the
particles diffuse to the Earth, their energy spectra are transformed under
the action of betatron acceleration and ionization losses of particles.
In contrast to the energy spectra of proton fluxes J(E,L), distributions
ΔN(fd,L) of the ERB protons over their drift frequency fd
(Fig. 5) differ much less from each other at L>3. Such
convergence of the spectra ΔN(fd,L) is driven by an increase in the
volume of magnetic tubes and a decrease in the anisotropy index of the ERB
proton fluxes with increasing L. Figure 5 demonstrates the closeness to the
adiabatic transformations of the spectra ΔN(fd,L) when L changes
at L>3.
The energy spectra of near-equatorial proton fluxes J(E,L) with E>10×L-3 MeV at L>3 in quiet periods have a local
maximum at E=(17±3)⋅L-3 MeV and a power-law tail (J∝E-γ, where γ=4.25±0.75) at E>(36±11)⋅L-3 MeV (Kovtyukh, 2001, 2018, 2020).
The frequency spectra of the ERB protons at L>3 weakly depend on
L and over a wide range of fd have a close to power-law
shape with an exponent γ=4.71±0.43 (at fd>fd∗, where fd∗∼0.5 mHz at L∼ 3–6, ∼2 mHz at L=2.5 and ∼5 mHz at L=2). Note that the
spread of the parameter γ for the frequency spectra of protons is
almost 2 times less than for their energy spectra. These spectra become more
rigid (flattened) at fd<fd∗.
Thus, the average exponents of the power-law tail of the energy and
frequency spectra of protons differ by Δγ=0.46, and
there is no local maximum in the frequency spectra at fd>2 mHz at L>2.5. The main role in such differences in the shape of
the energy and frequency spectra of protons was played by the factor
F(A) in Eq. (1), in which the anisotropy index A is a function of E and L
(see Figs. 2 and 3). Note that in the {E>0.5 MeV, L>3} region the anisotropy index A, as well as the
protons energy, is transformed according to adiabatic laws when L changes
(see Fig. 2 and the related text).
These results confirm our hypothesis about the ordering of the distributions
of protons over their drift frequency fd in the outer regions of the
ERB, at L>3, where most of the ERB protons are located and where
the radial diffusion of protons overpowers their ionization losses.
At all L, the frequency spectra ΔN(fd,L) become more flat at small
fd (at small E) under the influence of ionization losses. However, in the range
of high fd (from 3–5 to 30 mHz), for protons with high energies and
low ionization losses, the protons frequency spectra even have a power-law tail at L=2 (see Fig. 5).
For protons with fd<0.5 mHz, which correspond to the ERB
protons of the lowest energies, ionization losses lead to the same
consequences at higher L shells: the radial profiles ΔN(L,fd)
approach each other, and the spectra ΔN(fd,L) flatten out (see
Figs. 4 and 5).
In the region of the steep inner edge of the radial distributions ΔN(L,fd), spectra ΔN(fd,L) of the ERB protons gradually become
increasingly rigid with decreasing L and rapidly diverge from each other
(see Figs. 4 and 5). In the range of small fd at L<2.5, the
connection between these distributions and the shape of the boundary energy
spectra of protons is gradually lost.
These results indicate a violation of the order in the distributions of
protons under the influence of ionization losses.
In Fig. 5, the dotted line also shows the spectrum ΔN(fd,L) of
the ERB protons at L=2, constructed from experimental data for periods of
low solar activity between the 19th–20th, 20th–21th, 21th–22th, and
22th–23th solar cycles (see Fig. 1 in Kovtyukh, 2020). Figure 5 shows that there were more protons at the minimum
of solar activity at
L=2 for fd>10 mHz, whereas there were more protons
at the maximum of solar activity for fd∼ 1–10 mHz.
The effect of a decrease in the ΔN(fd, L) values for protons with
fd>10 mHz at L<2 with an increase in solar
activity is mainly connected with a decrease in the fluxes of protons with
E> 10–20 MeV here. This effect is well-known; it is described by
the CRAND mechanism (see, e.g., Selesnick et al., 2007) and was considered
in detail in Kovtyukh (2020). With an increase in solar activity, the
densities of atmospheric atoms and ionospheric plasma on small L shells
significantly increase, which leads to an increase in ionization losses of
the ERB protons, whereas the power of their main source (CRAND) practically does
not change. As a result, the equilibrium fluxes and ΔN(fd,L) for protons with fd>10 mHz are established at
lower levels.
However, the effect of an increase in ΔN(fd,L) for fd∼ 1–10 mHz at low L with increasing solar activity, corresponding to the
protons of lower energies, was discovered here for the first time.
With decreasing E (and fd) of protons, their ionization losses increase,
and if the fluxes of low-energy protons in the inner belt were also formed
by the CRAND mechanism, one would have observed an even stronger increase in
their fluxes with decreasing solar activity compared with protons with E> 10–20 MeV (fd>10 mHz). However, for protons with
fd∼ 1–10 mHz, we see the opposite effect in the
spectra ΔN(fd,L) at L=2 (Fig. 5), which is not described by the CRAND
mechanism.
On the other hand, it was proved that quasi-stationary fluxes of protons
with E<15 MeV at L∼2 are mainly formed by the mechanism of
protons' radial diffusion from the external region of the ERB (Selesnick et
al., 2007, 2013, 2014, 2018). These fluxes and ΔN(fd,L) values
for fd∼ 1–10 mHz at L=2 are formed as a result of a balance
of the competing processes of radial diffusion of protons and their ionization
losses.
The rates of transport of the ERB protons to the Earth (radial diffusion)
rapidly increase with decreasing particles energy (see Kovtyukh, 2016). In
addition, with an increase in solar activity, the average level of
geomagnetic fluctuations in the ERB increases. Under the influence of these
factors, one can expect a significant increase in the intensity of radial
diffusion of the low-energy protons at low L with an increase in solar
activity. As a result, the effect of the increasing density of a
dissipative medium with an increase in solar activity is overpowered by the
more significant effect of the increasing rates of radial diffusion of
protons.
According to numerous experimental data, a wide
variety of complex spectra of powerful pulsations of magnetic and electric
fields in the considered ultra-low frequency (ULF) range, which are non-regularly distributed over L, can be generated in the
geomagnetic trap during magnetic storms; these
pulsations can lead to local acceleration and losses of the ERB particles
(see, e.g., Sauvaud et al., 2013). Such effects will violate the regular
characteristics of the proton distributions shown in Figs. 4 and 5. However, during quiet periods (Kp < 2), the amplitudes of such pulsations are
small and lead only to the radial diffusion of particles.
Conclusions
From the data on near-equatorial ERB proton fluxes (with energy from 0.2 to
100 MeV and drift L shells ranging from 1 to 8), their quasi-stationary
distributions over the drift frequency of particles around the Earth
(fd) were constructed. The results of calculations of the number ΔN of the ERB protons within 30∘ of geomagnetic latitude at different
L and fd for periods of maximum solar activity are presented. They differ
from the corresponding distributions of the ERB protons for periods of low
solar activity only at L<2.5 (for comparison, the spectra of these
distributions are given at L=2).
The radial profiles of these distributions ΔN(L,fd) have only one
maximum that shifts toward the Earth with increasing fd. In comparison
to the proton fluxes' profiles J(L,E), the radial profiles ΔN(L,fd) at fd<5 mHz have steeper inner edges and flatter outer
edges. However, the radial profiles ΔN(L,fd) at fd>10 mHz, which are formed by the CRAND mechanism, have inner and
outer edges with only a slightly difference from each other with respect to
the steepness of their profiles.
In contrast to the energy spectra of proton fluxes J(E,L), the frequency
spectra ΔN(fd, L) of the ERB protons at L>3 are weakly
dependent on L, and for sufficiently large fd they have a nearly
power-law shape with an exponent γ=4.71±0.43. There is no
local maximum in these spectra in the {fd>2mHz,L>2.5} region, as in the
corresponding J(E,L) spectra.
The main physical processes in the ERB (radial diffusion, ionization losses
of particles, and the CRAND mechanism) manifested clearly in these distributions.
Distributions ΔN(L,fd) and ΔN(fd,L) of the ERB protons
in the {fd>0.5mHz,L>3} region have a more regular shape than in the corresponding region
of the {E,L} space. The majority of the ERB protons exist in this region, and their radial diffusion overpowers their
ionization losses during the transport of particles to Earth.
In the region of the steep inner edges of the radial distributions ΔN(L,fd), the spectra ΔN(fd,L) of protons rapidly diverge from
each other with decreasing L, and these spectra become
flattened at low frequencies. These results indicate a violation of the order in these
distributions of protons under the influence of ionization losses.
With increasing solar activity, the number of protons ΔN(fd,L) at L∼2 decreases for fd>10 mHz and
increases for fd∼ 1–10 mHz. The effect at high fd,
corresponding to protons with E>15 MeV, is well-known and is
described in the framework of the CRAND mechanism.
However, the opposite effect at low fd, corresponding to the
lower-energy protons, is discovered here for the first time. This effect can
be associated with the fact that the low-frequency part of the spectrum
ΔN(fd,L) of protons, even at L∼2, is mainly formed by the
mechanism of proton transport from the outer regions of the ERB. This
effect may indicate that the average rates
of radial diffusion of protons also increase with increasing solar activity. For low-energy protons at
L∼2, the effect of increasing density of a dissipative medium with
increasing solar activity is overpowered by the increase in the rates of
radial diffusion of particles.
Comparing this result with the results for ions with Z≥2 at L>2.5 (see Kovtyukh, 2020), one can conclude that the amplitude
of solar-cyclic variations of the radial diffusion coefficient DLL
increases with decreasing E and L (Z is the charge of the atomic nucleus with
respect to the charge of the proton).
Data availability
All data from this investigation are presented in Figs. 1–5.
Competing interests
The author declares that there is no conflict of interest.
Acknowledgements
The author is very grateful to the reviewers for their important and
fruitful comments and proposals regarding the paper and to topical editor,
Elias Roussos, for editing this paper.
Review statement
This paper was edited by Elias Roussos and reviewed by two anonymous referees.
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