Introduction
In the first stage (in the 1960s), exploration of Earth's radiation belts was
very active and culminated with the construction of a general dynamic
picture of these belts and the creation of a classical theory of this natural
particle accelerator.
In the 1970s and 1980s measurements of fluxes and energy spectra of the
trapped particles were continued. Detailed measurements of
pitch-angle distributions of electrons and protons were carried out. The following were studied in detail: the
dynamics of the belts during storms, cyclotron instability and precipitation
of particles from the belts, dynamics of the ion composition of the belts, the ring current during storms and substorms, and stochastic effects of
drift motion of trapped particles. However, in these decades it seemed that all
the basic problems of the physics of the Earth's radiation belts were solved, at
least for proton belts, and it remained only to clarify some of the details,
accurately carrying out mathematical modeling of the belts and constructing a
dynamic mathematical and empirical models.
In the early 1990s surprising dynamical effects of electrons and protons with energies of tens of megaelectronvolt were suddenly discovered in the depths of the
Earth's radiation belts (Blake et al., 1992), and further studies showed very complex and in many respects uncertain dynamics of the outer belt of
relativistic electrons. These discoveries led to a revision of the classical
theory, including problems related to the transport and acceleration of
particles. Since the basic properties of the mechanisms of this transport and
acceleration are universal for all particles of the Earth's radiation belts,
such a revision also concerns the ion belts.
According to the classical theory, the Earth's radiation belts are formed by
mechanisms of the radial diffusion of particles under the action of fluctuations
of electric and magnetic fields in the range of the drift periods of trapped
particles, i.e., in the range from several minutes to some hours (Tverskoy,
1969; Roederer, 1970; Schulz and Lanzerotti, 1974; Walt, 1994). Only protons with E > 10 MeV and
electrons with E < 0.8 MeV at L < 2
related to the mechanism of cosmic rays albedo neutron decay (CRAND) are an exception.
At the same time, the first (μ) and the second (I
or K= I / p, where
p is the momentum of a particle) invariants of the drift motion of
particles are conserved, and the third invariant (Φ) is
violated. The first invariant is associated with the gyration of charged
particles in a magnetic field, the second invariant is associated with the
oscillations of the particles between the mirror points, and the third
invariant is associated with the drift of the particles around the Earth in a
magnetic trap. The drift shell parameter L is related to the
invariant Φ by a well-known linear expression (Roederer,
1970).
Radial diffusion of trapped particles is determined by their resonant
interaction with the fluctuations of electric and magnetic fields on the
drift frequencies of these particles. The main parameter of a radial
diffusion (DLL) determines the rates of radial transport of the
trapped particles and in the general case, such as the drift frequency, DLL
depends on L, μ, K and the electric charge of
the particles. If small relativistic corrections are neglected, the drift frequency
of the particles and DLL do not depend on the rest mass of the
particles and are applicable to both protons and electrons.
The parameter DLL is determined by the specific mechanisms of
diffusion and is changed with the level and pattern of magnetic activity, as
well as changes in solar wind parameter and the interplanetary magnetic field (IMF). The value of DLL can
be increased by several orders of magnitude during strong magnetic activity
(e.g., Tverskoy, 1969; Lanzerotti et al., 1978; Walt, 1994). It also depends
on the phase of the solar cycle, the state of the ionosphere, and the
spectral density of electromagnetic fluctuations (pulsations) in the range
of ultralow frequency (ULF).
The first evaluations of DLL was obtained by ground-based data of
low-frequency fluctuations of the magnetic field (Nakada and Mead, 1965;
Tverskoy, 1965). These estimates differ from each
other by 1 order of magnitude.
The spectra of the fluctuations of magnetic and
electric fields in the range of ULF were also obtained from satellites (e.g.,
Lanzerotti et al., 1978; Holzworth and Mozer, 1979; Lanzerotti and Wolfe,
1980; Ali et al., 2015). The results of these estimates of DLL differ
from each other by several orders of magnitude.
In recent years, in connection with the problem of the dynamics of the outer
belt of relativistic electrons, this work intensified. On the basis of spectra of
pulsations of the magnetic and electric fields in the range of ULF (Pc4–Pc5), values of DLL have been calculated in many recent works (e.g., Tu et al.,
2012; Ozeke et al., 2012, 2014; Ali et al., 2015; Liu et al., 2016). For
this purpose, data from the Geostationary Operational Environmental Satellite (GOES),
Active Magnetospheric Particle Tracer Explorers (AMPTE), the Combined Release and Radiation Effects Satellite (CRRES),
the Time History of Events and Macroscale Interactions during Substorms (THEMIS) mission, Van Allen Probes, etc., for
spectra of pulsations are used. The results of these calculations of DLL also differ significantly from each other.
The parameter DLL was also evaluated as a result of a numerical solution
of the radial diffusion equation and fitting it to the experimental data on
the fluxes and energy spectra of the Earth's radiation belts. This work was
done in the same way as for electrons (e.g., Newkirk and Walt, 1968; Lanzerotti et al., 1970; Tomassian et al., 1972;
West et al., 1981; Chiu et al., 1990; Brautigam and Albert, 2000; Brautigam
et al., 2005; Ma et al., 2016), protons and other ions/nuclei (e.g.,
Spjeldvik, 1977; Fritz and Spjeldvik, 1981; Jentsch, 1981; Westphalen and
Spjeldvik, 1982; Panasyuk, 2004; Alinejad and Armstrong, 2006; Selesnick et
al., 2016). The values of DLL obtained by this method differ from each
other by 2 and more orders of magnitude.
In overall mathematical modeling of the Earth's radiation belts, as in the project SALAMMBO, DLL is a result of the selection and variation of certain classes of functions in the framework of a set of calculations that
takes into account all known factors affecting the belts (e.g., Beutier
et al., 1995). The results of such computations depend on many free
parameters which vary during the calculations.
I solved the inverse problem: the values of DLL are derived directly
from experimental data on the fluxes and spectra of the trapped protons.
Methods of mathematical modeling are not used here. There are no free parameters here. In the discussion of the obtained results, I consider only the
most reliable conclusions found from experimental data and the most general
physical reasons. After deriving DLL from the proton data (Sect. 2), I discuss the obtained results and compare them with data on the
fluctuations (pulsations) of electric and magnetic fields in the range of
ULF (Sect. 3).
The calculation of DLL(μ,L) from the structure of the proton belt
To extract DLL from the data on trapped particles, using the diffusion
equation, it is necessary to have complete and reliable values of the rate
of loss of these particles depending on L at various fixed values
of μ. For the proton belt near the equatorial plane such
dependences are presented in Kovtyukh (2016) for quiet periods. These
dependences were calculated on the basis of modern models of distributions of
cold plasma and atoms in the geomagnetic trap.
These calculations take into account that for the quiet belt the main loss
mechanism of protons is the ionization losses. During the quiet periods,
proton precipitation and the influence of ion cyclotron and other waves on
the lifetimes of protons can be neglected (e.g., Schulz and Lanzerotti,
1974; Lyons and Williams, 1984).
The values of DLL(μ,L) for the proton belt are calculated here on the basis of results in Kovtyukh (2016), satellite data and a diffusion equation.
For these calculations, I used data of the International Sun–Earth Explorer 1 (ISEE-1) for protons with an energy
of 24 to 2081 keV at L = 2–10 (Williams, 1981; Williams and
Frank, 1984) and data of Explorer-45 for protons with an energy of 78.6 to
872 keV at L = 2–5 (Fritz and Spjeldvik, 1981). These data are
verified in different studies and are in good agreement with each other.
Radial diffusion of the particles is described by the Fokker–Planck equation
(e.g., Tverskoy, 1964; Roederer 1970; Schulz and Lanzerotti, 1974). Under
certain conditions, which are fully implemented for these protons, the equation is
reduced to the ordinary diffusion equation (e.g., Tverskoy, 1965;
Fälthammar, 1968; Roederer, 1970; Schulz and Lanzerotti, 1974).
The values of DLL are most simply derived from the data obtained near
the equatorial plane. Here, I will consider only protons with equatorial pitch angles of α0 close to 90∘ (particles with the
second adiabatic invariant K≈ 0).
On the basis of numerous experimental results, I believe that in quiet (Kp < 2) periods the belt of protons with α0∼ 90∘ and E ∼ 0.1–1 MeV is almost stationary. I also
believe that local sources of these protons are absent at 2 < L < 10.
In this case, radial diffusion and losses of the protons are described by
the following equation:
L2∂∂LDLLL2∂f∂L=-∂f∂tcc-∂f∂tce,
where f(μ, L) is the distribution function of
protons in the phase space. The functions f and DLL in this
equation refer to the particles with given values of μ.
Equation (1) shows that for each L shell of the stationary
radiation belt, diffusion and losses of protons with given values of
μ are completely balanced.
The first term on the right-hand side of Eq. (1) describes Coulomb losses of
protons, and the second term describes the charge exchange of protons with
atoms. Coulomb scattering of protons by pitch angles is neglected in Eq. (1)
according to Schulz and Lanzerotti (1974).
The proton loss rate depends on the distributions of cold plasma and atoms
in the geomagnetic trap. Modern models of these distributions are the most
reliable for magnetically quiet periods. During geomagnetic disturbances the
distributions change (the distribution of cold plasma changes very much, and
the density of atoms varies within 20 %).
Losses related to ion-cyclotron waves are also added during geomagnetic disturbances.
With the increase in geomagnetic activity, the values of DLL increase, and the magnitude of the effect may depend on L.
Thus, in order to finding DLL(μ,L) for the trapped protons, functions
f(μ, L) of protons and the rates of
ionization losses of protons with different μ on different
L must first be calculated with the satellite data obtained near the plane of the geomagnetic
equator in the magnetically quiet periods.
I will consider the protons with μ from 0.2 to 7 keV nT-1 (from 20 to 700 MeV G-1) and L ≈ 2.5–10. These
particles are adjacent and in part overlap with less energetic particles,
which are usually attributed to the storm ring current (e.g., Williams,
1987). During the quiet periods, the protons considered here are the major
contributors to the pressure of the trapped particles in the geomagnetic
trap. Therefore, they can be regarded as a quiet ring current (see Kovtyukh,
2001). This belt of protons is called the ring current also in Williams (1981),
the results of which are used in my work. However, at μ > 0.5 ± 0.2 keV nT-1 at L > 3, a
belt of protons (and other trapped ions) in quiet and slightly disturbed
periods is maintained in the stationary state mainly due to the radial
diffusion of particles from the outer boundary of the trap to the Earth with
conservation μ and K (Kovtyukh, 2001).
The calculation of f(μ, L) for the belt
of protons
For calculations of f(μ, L), I took the data
of Explorer-45 and ISEE-1 for a protons with α0 ≈ 90∘ (K≈ 0). The data of
Explorer-45 are obtained for June 1972, near the maximum (early fall) of the
20th cycle of solar activity; the data of ISEE-1 are obtained for November 1977, at a minimum (at the beginning of the growth) of the
21st cycle.
For nonrelativistic protons with α0= 90∘
(K= 0)
f(μ,L)=kj[L,E(μ,L)]E(μ,L)=kf∗(μ,L),
where j[L, E(μ, L)] is the
measured fluxes of protons, E is the kinetic energy of protons, and
μkeV⋅nT-1=EB0(L)=3.215× 10-5L3EkeV,
where B0(L) is the magnetic induction near the
equatorial plane. The values of μ were calculated here for the
dipole magnetic field.
A value of the coefficient k depends on dimensions of variables in
Eq. (2). For j given as (cm2 s × ster × keV)-1,
E given in kiloelectronvolt and f given as s3 cm-6, the
value of k =5.447× 10-31. The coefficient k plays no
role in our calculations, so I will use f∗(μ, L) instead of f(μ, L). Equation (1) is invariant under this replacement.
In Kovtyukh (2016) functions f∗(μ, L) of the trapped protons were calculated on the basis of the
ISEE-1 data for the quiet period (Kp ≤ 1), from 20:27 UT 24 November
1977 to 01:30 UT 25 November 1977, given in Williams (1981). For
completeness, here I used also ISEE-1 data for a weakly disturbed period from
17:52 to 21:05 UT 17 November 1977 (Williams and Frank, 1984). In this
period the index Dst has changed from -17 to -18 nT, and the index Kp = 1- (Kp ≤ 2+ for 12 h and Kp ≤ 3- for 24 h
prior to this period UT). The measurements were carried out near the noon
sector, in the following eight energy channels:
24–45.5–65.3–95.5–142–210–333–849–2081 keV.
Functions f∗(L) for protons with
different μ (from 0.3 to 7 keV nT-1) calculated with ISEE-1
data from Williams and Frank (1984) for a slightly disturbed (Kp ≤ 2)
period 17 November 1977. Here μ is given in kiloelectronvolt per nanotesla. The
value of f∗ (s3 cm-6) is 1.836 × 1030 f.
Functions f∗(L) for protons with
different μ (from 0.2 to 3 keV nT-1) calculated with Explorer-45 data from Fritz and Spjeldvik (1981) for the quiet period of 1–15
June 1972. Here μ is given in kiloelectronvolt per nanotesla. The value of f∗ (s3 cm-6) is 1.836 × 1030 f. The curve for μ= 0.2 keV nT-1 is raised
above the other curves by 1 order of magnitude.
For calculating the functions f∗(μ, L), it is necessary to have the differential fluxes of particles
(see Eq. 2). As a rule, to find these fluxes the count rate of particles in
each channel is divided by the geometric factor of the instrument and by the
width of the corresponding channel. The values thus obtained refer to
the midpoints of the channels, i.e., to the arithmetic mean values of the
energy channels, E¯=E1+E2/2, where
E1 and E2 are the lower and upper bounds of this
channel. But this is true only for a flat spectrum or for the linear
spectrum, i.e., j(E)∝E. Sometimes the energy of the particles in the
channel is defined as the geometric mean, E¯=E1E2,
but this is true only for the power spectrum j(E)∝E-2. For a
more accurate binding of the experimental data to a specific energy of the particles
(within each channel of the spectrometer) and calculations of the functions
f∗(μ, L), I have developed a special
method based on successive approximations to the integrating fluxes within
the energy channels of the device. This method is described in detail in
Kovtyukh (2016).
Using the ISEE-1 data of Williams and Frank (1984) for the period
17:52–21:05 UT 17 November 1977, I have calculated the
f∗(μ, L) for protons with
μ from 0.3 to 7 keV nT-1 by this method and constructed a radial
dependences of f∗(μ, L), which are
shown in Fig. 1. The crosses in Fig. 1 show our calculated points between
which the interpolation was performed by the method of least squares.
I have also made the same calculations of the functions f∗(μ, L) for protons with α0= 90∘ for Explorer-45 data, averaged in Fritz and Spjeldvik (1981) over
60 orbits for the quiet period 1–15 June 1972. The measurements of proton
fluxes on this satellite were carried out at L < 5.25 in
the nine energy channels: 78.6–138.5–195.5–300 keV and
363.5–375–390–430–533–674–872 keV. The results of our calculations of
the functions f∗(μ, L) for these
data are shown in Fig. 2. To avoid overlapping, the curve for μ= 0.2 keV nT-1 is raised above the other curves by 1 order of
magnitude. As in Fig. 1, crosses in Fig. 2 show our calculated points
between which the interpolation was performed by the method of least
squares.
In overlapping ranges of L and at the same μ the
results of calculations of the functions f∗(μ, L) for the proton belt, shown in Figs. 1 and 2, are in good
agreement with each other, both in shape and in absolute values. A significant
difference is obtained only in a narrow interval, at 0.7 ≤ μ≤1 keV nT-1
on 3 ≤ L ≤ 4, where according to
Explorer-45, we have flatter spectra and a radial dependence of
f∗. The discrepancy can apparently be related to the
solar-cyclic variations of the belts, with some geomagnetic activity in
November 1977, and to the difference in the averaging of the data of Explorer-45
and ISEE-1.
Positive radial gradients of the functions f∗(μ, L) in Figs. 1 and 2 show that the trapped
protons diffuse mainly to the Earth.
The calculation of the rates of ionization losses of the trapped protons
Eq. (1) can be represented as follows:
∂∂LDLLL2∂f∗∂L=f∗L2τ,
where
τ-1(μ,L)=-1f∗∂f∗∂tcc+∂f∗∂tce=-∂lnf∗∂t.
Without updating the belts by the radial diffusion, f∗(μ, L) decays exponentially with time constant τ(μ,L). I calculate the ionization losses of the protons (Coulomb losses and the
losses to charge exchange) on the basis of experimental cross
section of charge exchange presented in Claflin (1970) and Lindsay and
Stebbings (2005) and on the real experimental spectra of protons (for the
same data of the ISEE-1 and Explorer-45).
The radial dependence of the rates of the ionization losses of
protons with various μ calculated with regard to the shape of
the energy spectra of protons based on the ISEE-1 data from Williams and Frank (1984)
for a period of 17 November 1977. Here μ is given in kiloelectronvolt per nanotesla. The jump on these curves at L = 5.0–5.5 corresponds
to a sharp drop in electron density near the plasmapause. The vertical cuts
on these curves mark L, at which rate the Coulomb loss is equal to
the charge exchange rate of the protons.
The same as in Fig. 3 for the Explorer-45 data from Fritz and
Spjeldvik (1981) for the period of 1–15 June 1972. Here μ is given in kiloelectronvolt per nanotesla.
Coulomb losses and the losses to charge exchange were calculated for the
protons with specific values of μ, and then these losses were
summed. As a result, I found L dependences of the rates of
ionization losses for the protons with different values of μ
(from 0.2 to 7 keV nT-1). For these calculations, I used the modern
empirical models of the plasmasphere (Østgaard et al., 2003; Zoennchen et
al., 2013) and exosphere (Moldwin et al., 2002; Ozhogin et al., 2012). The
methodology of these calculations is described in detail in Kovtyukh (2016).
The radial dependences of the rates of the ionization losses of the trapped
protons were calculated for 17 November 1977 are shown in Fig. 3. Coulomb losses
of protons calculated with regard to the functions f∗(μ, L) for this period (see Fig. 1). They correspond
to ISEE-1 data. The dotted plots of these curves result from the
extrapolation of the ISEE-1 data on low L. For protons with
μ≥ 1.5 keV nT-1, the jump on these curves at
L = 5.0–5.5 reflects a sharp drop in electron density and, as a
result, the drop in the rate of the Coulomb losses of protons near the
plasmapause.
The vertical cuts on these curves mark L, at which rate the
Coulomb loss is equal to the charge exchange rate of the protons; i.e., it is
the boundary between the small L area dominated by Coulomb losses and the larger L area dominated by the charge exchange loss of protons. The
position of this boundary (Lb) depends on μ of
the protons: Lb≈ 4.71×μ0.32, where
μ is given in kiloelectronvolt per nanotesla (Kovtyukh, 2016). This means that
Eb≈ 300 keV and that protons with E < 300 keV dominate the charge exchange with atoms and protons
with E > 300 keV dominate the Coulomb losses. At L ∼ 3–10 this boundary is almost independent of the proton
energy. This is mainly due to the fact that the ratio of the density of the electrons of cold plasma to the density of hydrogen atoms does not change very much with changing L (with the exception of the region of the
plasmapause).
Radial dependences of the rates of the ionization losses of the trapped
protons, calculated for 17 November 1977 (Fig. 3) and 24–25
November 1977 (Fig. 8 in Kovtyukh, 2016), are in good agreement with each
other. This is due to the similarity of the shape of the proton spectra in a
quiet and a weakly disturbed periods. Some of the differences are associated
with slight differences in the spectra of protons measured in these periods,
which leads to differences in the rate of the Coulomb losses of protons.
Figure 4 shows the radial dependence of the rates of ionization losses of the
trapped protons calculated for the quiet period 1–15 June 1972. Coulomb
losses of protons are calculated with regard to the functions f∗(μ, L) for this period (see Fig. 2). They correspond to Explorer-45
data. From these data, in the region dominated by the Coulomb losses of
protons, at L < 4, the spectra of protons were flatter than the
spectra measured on ISEE-1 for the period 17 November 1977, and therefore the
losses of protons were less. On another hand, Explorer-45 data, compared to
ISEE-1 data, were obtained in a period of higher solar activity; in this
period the density of the plasmasphere and exosphere was apparently somewhat
higher and, therefore the losses of protons, especially the Coulomb losses at
L < 5, was significantly more (this effect was not considered in our calculations).
In the region of the plasmapause, methodical errors of our calculations of
the rates of the Coulomb losses of the trapped protons can be more than in
the other regions of the belts. However, from further consideration it will
be seen that this circumstance can have an effect only on the calculations
of DLL on 5 ≤ L ≤ 6 for protons with μ∼ 1.5 keV nT-1.
For protons with μ < 1.5 keV nT-1, the charge exchange is dominated in the region of the
plasmapause, and for protons with μ > 1.5 keV nT-1, reliable calculations of DLL can be done only for L > 6 (for the ISEE-1 data).
The calculation of rates of the radial transport of the trapped protons
We divide the radial dependences of f∗(μ, L) shown in Figs. 1
and 2 into separate segments and integrate Eq. (4) within each such segment
taking into account Figs. 3 and 4. As a result, for each value of μ in
Figs. 1 and 2, we obtain the following chain of integrodifferential
equations:
DLL(μ,Li+1)Li+12∂f∗∂LLi+1-DLL(μ,Li)Li2∂f∗∂LLi=∫LiLi+1f∗(μ,L)L2τ(μ,L)dL,
where Li and Li+1 (i = 1,
2, …, n) are the lower and upper boundaries of the corresponding
segment of the radial profile f∗(μ, L). After calculating all the derivatives and integrals in Eq. (5),
we obtain a system of the linear algebraic equations for a given μ.
The values of the two terms on the left part of Eq. (5) are very close to
each other and their difference strongly depends on the radial dependence
of DLL. In addition, the system of Eq. (5) is incomplete: the number
of unknowns DLL(μ,Li) is one more than the number of equations.
It can be solved only if we exclude one of these unknowns in each system of
equations (for each given value of μ).
By summing these equations, we exclude from the system of Eq. (5) all
intermediate terms, and we get the complete equation. The difference between the normalized diffusion flows on the biggest
L and on the smallest L is on the left side of this
is, and the normalized integral of the rates of losses of the protons between these
extreme L (for a given μ) is on the right-hand side.
For general physical reasons, it follows that DLL rapidly decreases with decreasing L. This fact is reflected in all the proposed
mechanisms of the radial diffusion of particles in the Earth's radiation belts
and in the belts of other planets (see, e.g., Kollmann et al., 2011).
Primarily, this is due to the fact that the magnetic field increases rapidly
with decreasing L.
Therefore, the diffusion flow for the smallest L is much less than
for the largest L, and we can leave the flow for the largest L on the left side of the
complete equation. As a result we obtain a
linear equation with one unknown variable and we find from it the value
DLL at the external boundary of the L range (for a given
μ). Substituting this value DLL in system of Eq. (5),
we obtain the complete system of equations and gradually find all the other
values of DLL at different L (for a given μ).
Similarly, one can create and resolve the system Eq. (5) for other values of
μ. However, I do not want to make any preliminary assumptions about the radial
dependence of DLL.
For all values of μ and L, shown in Figs. 1 and 2, I calculated L-2(∂f∗/∂L) from the left part of
Eq. (5). According to our calculations, for protons with μ ≥ 1.5 keV nT-1 from ISEE-1 data (Fig. 1), the value
L-2(∂f∗/∂L) monotonically decreases with
decreasing L: 10.4 times when reducing L from 10 to 4 for
μ= 1.5 keV nT-1, 27.3 times when reducing L
from 10 to 5 for μ= 3 keV nT-1, 40.2 times when
reducing L from 10 to 5 for μ= 5 keV nT-1,
and 45.4 times when reducing L from 10 to 6 for μ= 7 keV nT-1.
So even if we assume that DLL does not depend on L, the
smaller of the two terms in the left part of the total equations of systems
(5), for smallest L, can be neglected for protons with
μ≥ 1.5 keV nT-1. The error of the calculations of
DLL at L = 10, related to this, ranges from ∼ 10 % for
μ= 1.5 keV nT-1 to ∼ 2 % for μ = 7 keV nT-1 (if we posit that DLL decreases with decreasing
L, this error will be much less). Of course, when approaching the
lower boundary on L (for a given value of μ), the error
of our calculations of DLL increased. To this error we must add the
errors of calculations f∗ (Figs. 1 and 2) and the ionization
losses of protons (Figs. 3 and 4).
However, for protons with μ= 1 keV nT-1, the value
L-2(∂f∗/∂L) is reduced only 2.9 times when
L is reduced from 9 to 3. In this case, my method is valid only
under the assumption that DLL quite strongly decreases with
decreasing L. Even more important is the assumption for protons
with μ < 1 keV nT-1 for which the dependence
L-2(∂f∗/∂L) on L is non-monotonic.
For protons with μ= 0.7 keV nT-1, the value of L-2(∂f∗/∂L) decreases by 10.4 times with
L decreasing from 5 to 2.5, and for protons with μ = 0.3 keV nT-1, it decreases by 23.4 times with L decreasing
from 4.5 to 2. However, at L > 5 for μ= 0.7 keV nT-1 and at L > 4.5 for μ= 0.3 keV nT-1, the value of L-2(∂f∗/∂L)
increases with decreasing L.
Thus, for protons with μ < 1 keV nT-1, our
calculations of DLL according to ISEE-1 data are less reliable than
calculations for protons with μ≥ 1.5 keV nT-1.
For protons with μ < 1.5 keV nT-1, the ISEE-1 data
are well complemented by Explorer-45 data obtained at smaller L.
According to the Explorer-45 data (Fig. 2), the value of L-2(∂f∗/∂L)
monotonically decreases with decreasing L for protons with μ= 0.2–3 keV nT-1: 54.4 times when reducing
L from 4 to 2 for μ= 0.2 keV nT-1, 8.6 times
when reducing L from 4.5 to 2.5 for μ= 0.3 keV nT-1, 1.6 times when reducing L from 4 to 3 for μ= 0.7 keV nT-1, 4.5 times when reducing L from 5 to 4 for
μ= 1.5 keV nT-1, and 1.6 times when reducing
L from 5 to 4.5 for μ= 3 keV nT-1. Therefore, this method appears to be applicable to Explorer-45 data for protons
with μ= 0.2–0.7 keV nT-1.
In calculating the derivatives on the left-hand side and integrals on the
right-hand side of Eq. (5), I divided the scale of L into fairly short
intervals where the functions f∗(μ, L) and τ(μ,L) are well approximated by a power law with
different exponents. All the approximation functions were joined together at the boundaries of these intervals.
The results of our calculations of DLL(μ,L) based on Figs. 1 and 3
(ISEE-1) are shown in Fig. 5, and those based on Figs. 2 and 4 (Explorer-45) are
shown in Fig. 6. The numbers on the right-hand side of Figs. 5 and 6 refer to the
values of μ (in kiloelectronvolt per nanotesla).
From Figs. 5 and 6, we see that the results of our calculations of DLL(μ,L) based on the data from ISEE-1 and Explorer-45 are in good agreement with each
other for μ= 0.3 keV nT-1 (they differ by no more than
∼ 2.5 times their value). For protons with μ= 0.7 keV nT-1, the functions of DLL(L) are sewn together well at L = 4.
For protons with 0.7 < μ < 1 keV nT-1
in the region 3 ≤ L ≤ 4, where according to ISEE-1 and
Explorer-45 (see Figs. 1 and 2) the radial gradients of the functions
f∗(μ, L) are significantly different, good agreement was also obtained between the calculated values
of DLL(μ,L). However, for μ= 1.5 keV nT-1 the
values of DLL calculated at L = 4.5–5.0 on the basis of the Explorer-45
data were ∼ 7–8 times smaller than the values calculated with the
ISEE-1 data.
This discrepancy is reduced if we consider that the data from Explorer-45 are
obtained in a period of higher solar activity. In this period the density of
the plasmasphere and exosphere was apparently higher than during the period
of measurements on ISEE-1. Therefore, the losses of protons were greater than
our calculated values, especially at L < 5. In this regard,
the values of DLL shown in Fig. 6 should be increased (see Eqs. 4 and 5).
The errors of my method of calculating DLL depend on the width of the
range of L in which we have conducted the calculations: the narrower
this range, the more errors there are in our calculations. For different μ the width
of the range L is different. For sufficiently long
series of calculations (for large numbers of Eq.
5), when the maximum (at the upper limit of the range L) and the minimum (at
the lower end of this range) values of DLL differ by more than 1 order
of magnitude, i.e., for μ from 0.3 to 5 keV nT-1 based on the ISEE-1 data and for μ∼ 0.2 keV nT-1 based on the Explorer-45 data, errors of our calculations of DLL do not exceed 10 %
at large L. With decreasing L and with increasing
μ of protons, these errors increase to some tens of percent.
The results of calculations of the values of DLL(μ,L) near
the equatorial plane based on the ISEE-1 data in Williams and Frank (1984) for
the weakly disturbed period from 17:52 to 21:05 UT, 17 November 1977. In this
period Kp = 1- (Kp ≤ 2+ for 12 h prior to this period
UT). These results take into account Figs. 1 and 3. Red signs show values
of DLL for protons with μ= 1 keV nT-1,
calculated according to ISEE-1 data in Williams (1981) for the quiet period of 24–25
November 1977 (Kp ≤ 1). Here, DLL is given in values per second and μ is given in kiloelectronvolt per nanotesla.
The results of calculations of the values of DLL(μ,L) near
the equatorial plane based on the averaged Explorer-45 data in Fritz and Spjeldvik (1981)
for the quiet period of 1–15 June 1972. These results take into account
Figs. 2 and 4. Here DLL is given in values per second and μ is given in kiloelectronvolt per nanotesla.
Since the Explorer-45 data are limited to a maximum available L
∼ 5, correct calculations of DLL from these data are
possible only for μ < 1 keV nT-1. For large
values of μ the ranks of our calculations of DLL on the scale of L are short, which leads to large methodical errors and to
a significant underestimation of DLL in the calculations for protons
with μ > 1 keV nT-1 based on the Explorer-45 data.
For L > 5 the magnetosphere is asymmetric in magnetic local time (MLT), and
with the growth of L this asymmetry increases. Because in the quiet
periods the asymmetry of the magnetosphere for 5 < L < 10 is
not very large and the function f(μ, L) in Eq. (1) as the function
f∗(μ, L) in Eq. (5) is averaged over the drift of particles around the
Earth, the average values of L are close to the values given in
Fig. 5. According to our estimates, the associated error does not exceed
other methodical errors in our calculations.
The transition from a dipole model to a more realistic mathematical model of the
geomagnetic field leads to some changes in the calculated values of DLL(μ,L). However, the experimental data used here were obtained in quiet and
slightly disturbed (Kp < 2) periods. The ISEE-1 data relate to
near-noon sector of MLT, and the Explorer-45 data were obtained at L < 5.
So, we can hope that the deviations of the geomagnetic field from
the dipole configuration in the outer regions of the trap will not lead to
significant changes in the calculated values of DLL(μ,L), and our main conclusions will not change.
Taking into account all possible errors, the calculated values of DLL(μ,L) are shown in Figs. 5 and 6. These deviate from the real values by no more
than ∼ 2.5 times their own value and, as a rule, do not exceed the size of the
symbols in Figs. 5 and 6 (for all values of L and μ of protons considered here, except for μ= 1.5 keV nT-1
in Fig. 6).
This is confirmed by a comparison between our calculations according to data
from ISEE-1 for the period 17 November 1977 and according to those from 24–25 November 1977. For
these periods, we obtained values of DLL(μ,L) that were close to each other at the same values of μ and L. For protons with
0.5 ≤ μ≤ 3 keV nT-1, they differ by no more
than 1.5–2.0 times their own value. For comparison, red signs in Fig. 5 show several
values of DLL for protons with μ= 1 keV nT-1,
calculated according to ISEE-1 data (Williams, 1981) for the quiet period of 24–25
November 1977 (Kp ≤ 1).
In the total errors of our calculations of DLL, the errors associated with
models of the plasmasphere (see Kovtyukh, 2016) play a major role. To this
rather large value (up to ∼ 2.5 times)
much smaller errors of the measurements of proton fluxes based on ISEE-1 and Explorer-45 and
methodical errors (see above) are added. Some errors can also be added by physical processes unaccounted for here, such as plasma instability and
the interaction of protons with electromagnetic waves and micro-injections of
hot plasma from the tail of the magnetosphere (for low-energy protons at
large L). However, our calculations are carried out for quiet
and weakly disturbed periods in Earth's magnetosphere when the plasma distribution is
stable, and fast dynamic processes can be neglected in comparison with radial
diffusion.
Discussion
It has been shown that at L > 3, all stationary distributions
(spatial, energy and pitch angle) of protons (and other ions) of the Earth's
radiation belts are interrelated and should be formed by mechanisms which
provide the radial transport of these particles while conserving μ and
K. Kovtyukh (1984, 1985a, b, 1989, 1994, 1999a, b, 2001) has given the fullest
and most comprehensive justification of this interrelation in the
distributions of protons (and other ions) as a result of the data analysis
of 22 missions (Explorer-12, Explorer-14, Mariner-4, Explorer-33, European Space
Research Organisation satellite 2 (ESRO-2), Injun-4, Injun-5, 1968-26B,
Orbiting Vehicle 1-19 (OV1-19) , Explorer-45, 1972-076B, Molnija-1, Applications
Technology Satellite 6 (ATS-6), Molnija-2, ISEE-1, Spacecraft Charging at
High Altitudes (SCATHA), AMPTE/Charge Composition Explorer (CCE), Gorizont-21,
Akebono, CRRES, Gorizont-35 and the Engineering Test Satellite VI (ETS-VI))
for 34 years of space research (1961–1994). For protons with μ > 0.5 ± 0.2 keV nT-1, such a situation can only be
provided by mechanisms of the radial diffusion of particles to the Earth from the outer
boundary of belts while conserving μ and K (Kovtyukh,
2001).
The main result of our calculations is the strong dependence of DLL not
only on L but also on μ. Figures 5 and 6 show that for all considered L, the values of DLL decrease rapidly
with increasing values of μ.
This result can be seen from Figs. 1–4 and Eq. (4) before DLL(μ,L) is calculated: for any given L, the rates of the ionization losses
of protons decrease with increasing μ (see Figs. 3 and 4),
but the value (∂lnf∗/∂L) increases or remains
almost unchanged (see Figs. 1 and 2). Therefore, to keep the balance of
radial diffusion and loss of particles, the coefficient of DLL should
decrease with μ increasing. If other possible losses (primarily, the interaction of protons with the
waves) are take into account, the dependence of DLL on μ only increases.
The effect of reducing DLL with increasing μ is clearly expressed in the ISEE-1 data presented in Fig. 2a in
Williams (1981). From this figure, it is seen that the radial gradient of f(μ, L) increases sharply in the transition from high to low L. The greater μ is, the more
L there is, where this happens. This effect indicates a decrease in DLL
with μ increasing. However, this effect was not discussed in
Williams (1981). It clearly contradicts the theory of the radial transport
of trapped particles under the influence of sudden impulses (SIs) of the magnetic
field, which was dominant at the time when Williams was
writing.
The mechanism of particle transport under the influence of SI was proposed
by Kellogg (1959), and in many works it has been used as the main mechanism. It is
implemented when fluctuations in the dynamic pressure of
the solar wind influence the magnetosphere and is usually called magnetic
diffusion. I denote the diffusion coefficient for this mechanism by
DLLM. Models of the Earth's radiation belts based on the mechanism
of magnetic diffusion were created by Nakada and Mead (1965) and Tverskoy (1965,
1969). In the model of Nakada and Mead (1965), DLLM= 2.3 × 10-15×L10 s-1.
In the model of Tverskoy (1965, 1969), DLLM= 5 × 10-14×L10 s-1. In these models it is supposed
that the spectrum of magnetic fluctuations has a power-law form with an
exponent of -2; in this case DLLM does not depend on the drift
frequency of the particles or their energy and μ but only on
L.
The mechanism of magnetic diffusion is efficient only for traps with a
strong azimuthal asymmetry of the geomagnetic field. But in the depths of
the geomagnetic trap the magnetic field is almost symmetric and, therefore,
the efficiency of the magnetic diffusion should be very small.
Another popular mechanism of radial transport of trapped particles is their
diffusion under the action of the fluctuations of an electric field in the
magnetosphere during substorms (Fälthammar, 1965, 1966, 1968; Cornwall,
1968, 1972). In contrast to the vortex electric fields generated in the
magnetosphere during SI, the electric field of a substorm can be described
with an electric potential, and such a mechanism of particle transport is
usually called electric diffusion. It does not depend on the azimuthal
asymmetry of the magnetic field. In this mechanism DLL depends on
μ and on the charge of the particles. I denote the diffusion
coefficient for this mechanism by DLLE. According to
Cornwall (1968, 1972), for protons
DLLE=(1.5× 10-10-1.5× 10-9)L10L4+μ2,
where DLLE is measured as values per second and μ is measured in megaelectronvolt per gauss.
For μ > 5 keV nT-1 (> 500 MeV G-1)
at L > 5, μ2≫L4 and, according
to Eq. (6), DLLE∝μ-2L10, but for lower
values of μ, the dependence of DLLE on L is
weakened at large L.
Equation (6) for DLLE and the expression for DLLM were
parameterized for Kp by Brautigam and Albert (2000). With Kp = 1 the
expression for DLLE in Brautigam and Albert (2000) corresponds to
Eq. (6) with the coefficient on the right-hand side of the expression ∼ 9× 10-10,
and DLLM∼ 1.8 × 10-14 ×L10 s-1; i.e., they correspond to the average
of DLLM given by Nakada and Mead (1965) and Tverskoy (1969). According
to Brautigam and Albert (2000), with Kp increasing from 1 to 6, the values of DLLE increase ∼ 200 times and the values of DLLM
increase ∼ 340 times.
The functions of DLLM and DLLE depend on L and
μ in different ways, and therefore in different regions of
{L, μ}, space their ratio is different. As DLLE depends on μ and decreases with
decreasing L to less than DLLM, if DLLM dominates
at large L, DLLE can dominate at small L. If
DLLE dominates for small μ, DLLM can dominate for large μ. In addition, this ratio can change depending on
magnetic activity.
These circumstances lead to different conclusions for the ratio of DLLM to DLLE. The conclusions were drawn in different articles and are sometimes incompatible with each other. So, for electrons with μ= 5–50 keV nT-1 and related fluctuations of the electric field at L = 3–7, it has been argued based on CRRES data that DLLE≪DLLM (Brautigam et al., 2005), but based on the
fluctuations of the magnetic field at ground stations and based on AMPTE and
GOES data, it has been argued, for the same L and for the same
μ of electrons, that DLLM≪DLLE (Ozeke et al., 2012) for both Kp = 1 and Kp = 6 (see Fig. 11 in Ozeke et al., 2012). In Ozeke et al. (2014), the conclusions of Ozeke
et al. (2012) were supported by the analysis of data from CRRES and GOES, and another parameterization, different from that of Brautigam and Albert (2000), was proposed for DLLE and DLLM for Kp (and L) where these
parameters do not depend on μ.
In many works, DLLM and DLLE have been considered to be modes of radial diffusion that are independent
of each other. In our calculations of DLL, I only used data for the particles and did not carry out a separation of DLL for different modes. In the course of further discussion it will be
shown that the function of DLL(μ,L) calculated here corresponds to the uniform diffusion mode which operates in a broad band on L
and the energies of protons.
The values of DLL are determined by the spectral density of the fluctuations
(pulsations) of the electric and magnetic fields (Schulz and Lanzerotti, 1974).
In theoretical works a power dependence of DLL on L is usually postulated: DLL∝Ln. This corresponds to
the power spectra of the fluctuations of the fields. For different mechanisms of
the radial diffusion of the particles, the parameter n takes different
values. Therefore, in sufficiently wide ranges of L, the dependence of DLL on L for particles with a fixed μ is not
described by a simple power law. This is evident in the evaluation of the
parameter n, obtained from the experimental data: the parameter
n takes significantly different values in different intervals of
L and μ (e.g., Schulz and Lanzerotti, 1974).
According to Fig. 5, in the range of μ∼ 1–7 keV nT-1, values of the parameter n ∼ 7.5, 7.4, 8.2, 10 and 7.7 for
μ∼ 1, 1.5, 3, 5 and 7 keV nT-1 (average n ∼ 8.2).
Figure 7 presents the dependence of DLL(μ) on L = 7 based
on the results shown in Fig. 5. Here DLL is given in values per second and μ is given in kiloelectronvolt per nanotesla. This dependence is shown by a thick
curve, and the signs on it correspond to the signs in Fig. 5. In
addition, in Fig. 7 thin lines with dots show the dependences of
DLL(μ) on L = 7 for electric diffusion, which are
calculated according to Eq. (6). Thin horizontal lines show DLL at L = 7 for magnetic
diffusion, as given in Nakada and Mead (1965) and Tverskoy (1965, 1969).
Figure 7 shows that the results of our calculations are not consistent, not
only with magnetic diffusion but also with electric diffusion, as described by
Eq. (6). Different linear combinations of DLLM and DLLE
do not lead to reasonable agreement between these results and our
calculations.
Figure 8 presents the dependences of DLL(μ) on different L,
in the range from 4 to 10, based on the results shown in Fig. 5. Here, as
in Fig. 7, DLL is given in values per second and μ is given in kiloelectronvolt per nanotesla.
The thick lines in this figure are calculated using the method of least squares. The numbers near these lines correspond to the values of L.
The dependence of the DLL on μ for L = 7. The thick curve corresponds
to the results given in Fig. 5. The thin
curves correspond to Eq. (6) for the electric diffusion. The
lower horizontal line is the value of DLL in Nakada and Mead (1965), and
the upper line is the value of DLL in Tverskoy (1965, 1969). Here DLL is given in values per second.
The dependences of DLL(μ) on different L (in the
range from 4 to 10), which corresponds to the results shown in Fig. 5. Here DLL is given in values per second.
According to Fig. 8, DLL(μ)∝μ-m,
where the parameter m depends on L. The maximum value of m (∼4.4–4.5) is achieved at L ∼ 6–7; the
minimum m (∼ 2.4) is achieved at L = 4. The value
of this parameter averaged over L equals 3.9 (or 4.1 if excluding the value of m for L = 4).
The results of our calculations of DLL, shown in Figs. 5 and 8, for
protons with μ from ∼ 0.7 to ∼ 7 keV nT-1 at L≈ 4.5–10, are most adequately described by the
following expressions:
DLL(μ,L)≈ 4.9× 10-14μ-4.1L8.2,
where DLL is measured in values per second and μ is measured in kiloelectronvolt per nanotesla. Replacing μ by
E in Eq. (7), we get
DLL(E,L)≈ 1.3× 105EL-4.1,
where DLL is measured in values per second and E is measured in kiloelectronvolt. Replacing EL by drift
frequency in Eq. (8), we get
DLL(fd)≈ 1.2× 10-9fd-4.1,
where fd is the azimuthal drift frequency of nonrelativistic
protons (inverse value of the drift period of a particle around the Earth),
fd is measured in millihertz and DLL is measured in values per second.
In Eqs. (7)–(9) the results
of the calculations of DLL for protons with μ= 0.3 keV nT-1 are not taken into account. These differ greatly (see Fig. 5) from the results
of the calculations for protons with μ≥ 0.7 keV nT-1.
They correspond to the low-energy part of the spectra of protons, and even in quiet periods, this
part of the spectra of trapped particles is very sensitive to the cyclotron and other plasma instabilities that were not
taken into account in our calculations.
The Eqs. (7)–(9) contradict the theory of diffusion under the
action of SI (magnetic diffusion). According to this theory, DLL does
not depend on the μ, E and fd of particles
(Nakada and Mead, 1965; Tverskoy, 1965, 1969; Schulz and Lanzerotti, 1974).
In the dipole approximation, the azimuthal drift frequency of
nonrelativistic trapped particles is fd=11.8×μL-2,
where fd is given in millihertz and μ is given in kiloelectronvolt per nanotesla. The
results for protons with μ= 0.3, 0.7, 1, 1.5, 3, 5 and 7 keV nT-1 shown in Fig. 5 correspond to fd∼ 0.14–0.57, 0.13–0.52, 0.15–0.74, 0.18–0.87, 0.35–0.98, 0.59–1.20 and
0.83–1.68 mHz. These frequencies belong to the range of Pc6.
For magnetic field fluctuations DLL∝fd2PM(fd)L10 (Fälthammar, 1965; Nakada and Mead, 1965; Tverskoy, 1965), and
for electric field fluctuations DLL∝PE(fd)L6
(Fälthammar, 1966, 1968), where PM and PE represent the spectral
density of these fluctuations (pulsations). Therefore, to satisfy the
Eqs. (7)–(9), the spectrum of magnetic field fluctuations should
have an exponent of about -6 and should be attenuated with increasing
L as L-10. The spectrum of electric field fluctuations should
have an exponent of about -4 and should be attenuated with increasing
L as L-6.
Unfortunately, in the range of f < 1 mHz the spectra of
fluctuations (pulsations) of the electric and magnetic fields in the
geomagnetosphere and in the surrounding area are insufficiently studied. In
the range of 0.1–1.7 mHz (Pc6) spectra of these fluctuations are irregular and vary strongly. Compared to higher-frequency ranges, there were few reliable
measurements of the pulsations of the electric and magnetic fields in this interval until
the recent publication of the results from experiments on THEMIS.
According to the measurements of the Geotail and Wind satellites in the near-Earth foreshock (Berdichevsky et al., 1999), typical spectra of the
magnetic field fluctuations in the range of ∼ 0.5–100 mHz can be
approximated by a power law with an exponent of -4 to -2. In the upper
part of this range (> 10 mHz), some of these spectra can be
approximated by a power law with an exponent of -6, but at the bottom of the
range (< 10 mHz) the spectrum is flatter and the exponent is close to
the value -2 (see Fig. 8 in this article). This is confirmed by CRRES data
(Ali et al., 2015): in the range of ∼ 1–8 mHz averaged spectra of the
magnetic field fluctuations are very hard and almost flat.
In the range of 1–100 mHz the spectra of the fluctuations of the magnetic field
were also measured at ground stations associated with cusp/cleft (Posch et
al., 1999). These spectra are irregular and vary strongly depending on the
speed of the solar wind. For a power-law approximation, the average exponent
of these spectra is close to the value -4 in the range of 10–100 mHz, but in
the lower part of the range (1–10 mHz) an exponent of the spectral density
of these fluctuations is close to -5/3 (see Fig. 6 in this article).
According to GOES (on geosynchronous orbit) and Wind (in the solar wind) in
the range of 0.2–1.7 mHz (Kepko and Spence, 2003), the amplitude spectra of
magnetic fluctuations are irregular (fine structure with narrow peaks), but
the average amplitude of these fluctuations decreases with increasing
frequency by a power law with an exponent of -1.8 to -1.5 (see Figs. 4,
9, 11, 13 and 15 in this article); i.e., for the average spectral density we
have an exponent of -4.6 to -4.0.
Thus, the experimental spectra of the magnetic field fluctuations
(pulsations) are not consistent with Eqs. (7)–(9) obtained here.
On the basis of over 7 years of averaged data from THEMIS for all MLT, for
different Kp (from 0 to 5) and for different L (from 3.5 to 7.5),
Liu et al. (2016) constructed the spectra of electric field
fluctuations (pulsations) in the range of ∼ 0.5–10 mHz (unfortunately,
the data at f < 0.5 mHz are not given). In the range of ∼ 0.5–2 mHz these spectra are very soft, especially during quiet and
slightly disturbed (Kp = 0–2) periods. For Kp = 0–2, according to Fig. 2 from Liu et al. (2016), in the range of ∼ 0.5–2 mHz the dependence of
the spectral density of the fluctuations on the frequency can be
approximated by a power law with an exponent that varies from
-(3.3–3.9)
at L > 5.5 to -5.4 at L = 4.5 and -10.5 at L = 3.5; i.e., at L ≥ 4.5 the average value of this
exponent is close to -4. Our calculations also show that the slope of the
radial dependence of DLL(μ,L) usually increases with decreasing
L (see Figs. 5 and 6).
According to Fig. 2 in Liu et al. (2016), at a frequency of ∼ 0.5 mHz
the spectral density of the electric field fluctuations decreases also with
increasing L (approximately as Lk, where mean square value of
the exponent k is changed from -6.5 when Kp = 0 to -7.3 when Kp = 2). But already at a frequency of ∼ 1.0 mHz this spectral density
does not depend on L, and at frequencies of ∼ 1 to 10 mHz this value increases with L.
Thus, according to THEMIS data averaged over 7 years for all MLT, for
periods with Kp = 0–2 the spectral density of the electric field
fluctuations (pulsations) at L ≥ 4.5 for f∼ 0.5–1 mHz can be described by the following expression: PE∝fd-4L-6. Substituting this expression into the formula
DLL∝PE(fd)L6 (Fälthammar, 1966, 1968) for
the radial diffusion of particles influenced by the electric field
fluctuations, we obtain DLL∝fd-4. This result
corresponds to our Eq. (9), and, hence, it is consistent with Eq. (8) and
Eq. (7), which refer to the main cluster of calculated points in Fig. 5
and satisfy the ranges fd∼ 0.5–1.2 mHz (Pc6) and Kp = 0–2.
On the basis of the spectra of electric field fluctuations, Liu et al. (2016)
calculated DLL for relativistic electrons with μ= 5-40 keV nT-1 (500–4000 MeV × G-1) at L = 3.5–7.5.
The drift frequency of these electrons corresponds to the range
fd∼ 1–4 mHz (Pc5), where the spectrum of fluctuations
is flatter and the spectral density increases with L. For these
reasons, in comparison to our calculations for protons for a range
fd∼ 0.15–1.2 mHz (Pc6), in Liu et al. (2016)
the dependence of DLL on μ is much weaker and the dependence of DLL on L is slightly stronger.
Note that for electrons with μ= 1–50 keV nT-1 in the
range of L = 3–7 , the functions of DLL are calculated also for the
spectra of electric field fluctuations in the range of 0.2–15.9 mHz measured
by CRRES (Brautigam et al., 2005). For Kp = 1, about the
same dependence of DLL on L is obtained as in Eq. (7) (DLL∝L8), but the dependence of DLL on μ was much
weaker (see Fig. 9 in Brautigam et al., 2005).
Conclusion
I calculate the value DLL, which is a measure of the rate of the radial transport
(diffusion) of the particles of the Earth's radiation belts,
directly from data on the fluxes and energy spectra of protons with an equatorial pitch angle of α0≈ 90∘ during quiet and
slightly disturbed (Kp ≤ 2) periods.
This is done by successively solving the equations of the balance of the radial
transport/acceleration and ionization losses of protons for the stationary
belt. Calculations of the ionization losses of protons (Coulomb losses and
charge exchange) were carried out on the basis of modern models of the
plasmasphere and the exosphere.
To find DLL I calculated the radial dependences of the distribution
function f∗(L, μ) for different
μ. For each of the given values of μ, these
dependences were divided into short segments and the systems (chains)
of integrodifferential equations describing the balance of radial transport
and losses of protons were solved. This work is carried out here for the first time.
For these calculations I used the data of ISEE-1 for protons with an
energy of 24 to 2081 keV at L≈2–10 and the data of
Explorer-45 for protons with an energy of 78.6 to 872 keV at L≈2–5. The values of DLL calculated from the data of ISEE-1 and
Explorer-45, in the overlapping intervals of L and μ,
are in good agreement with each other.
As a result of the calculations, I found that in the range of L≈2.5–10 the dependences of DLL(L) are significantly different for
protons with different μ (from 0.2 to 7 keV nT-1). The
values of DLL decrease rapidly with decreasing L as well as with
increasing μ.
It is shown that for protons with μ from ∼ 0.7 to ∼ 7 keV nT-1
at L≈ 4.5–10, the functions of DLL can be approximated by the following equivalent expressions:
DLL≈ 4.9× 10-14μ-4.1L8.2, or DLL≈ 1.3× 105EL-4.1, or DLL≈ 1.2× 10-9fd-4.1, where fd
is the drift frequency of the protons (in mHz), DLL is given in values per second, E is given in kiloelectronvolt and μ is given in kiloelectronvolt per nanotesla. These
expressions are obtained for quiet and weakly disturbed conditions in the
magnetosphere (Kp ≤ 2). During magnetic storms DLL increases, and
the expressions obtained here for DLL can change completely.
These results contradict the mechanism of the radial diffusion of
particles under the influence of sudden impulses (SI) of the magnetic field and
also under the influence of substorm impulses of the electric field, as was suggested by Conwall (1968, 1972).
It is shown that the bulk of the calculations of DLL, in the range of ∼ 0.5–1.2 mHz (Pc6), is consistent with spectra
of the fluctuations (pulsations) of the electric field at L ∼ 4.5–7.5 during quiet and weakly disturbed periods (Kp ≤ 2) averaged
over 7 years for all MLT according to THEMIS data in Liu et al. (2016. These ranges of fd and L correspond to the trapped protons with energies from ∼ 0.18 to ∼ 0.7 MeV and electrons with energies from ∼ 0.21 to ∼ 1.19 MeV.
The comparison DLL for protons in a certain region {μ, L} with electric field pulsations in
the appropriate frequency range shows a close relationship between the radial
diffusion of particles and the pulsations of the electric field. For higher
frequencies (in the range of Pc5), the experimental spectra of these
pulsations are flatter and the spectral density increases by about 1 order of magnitude with the increase in L from 3.5 to 7.5. Therefore,
for more energetic particles of the radiation belts corresponding to higher
drift frequencies, other dependences of DLL on μ
and L may exist. This applies in particular to the relativistic and
ultra-relativistic electrons of the radiation belts.
Because the values of DLL were calculated here only for two short periods
(quiet and weakly disturbed) according to only two missions (ISEE-1 and
Explorer-45) near the equatorial plane and only for protons in the limited
ranges of μ and L, they cannot of course be regarded as
complete and final. This work should be continued.