From the data on the fluxes and energy spectra of protons with an equatorial pitch angle of

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

At the same time, the first (

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 (

The parameter

The first evaluations of

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

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

The parameter

In overall mathematical modeling of the Earth's radiation belts, as in the project SALAMMBO,

I solved the inverse problem: the values of

To extract

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

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

On the basis of numerous experimental results, I believe that in quiet (Kp < 2) periods the belt of protons with

In this case, radial diffusion and losses of the protons are described by
the following equation:

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

Thus, in order to finding

I will consider the protons with

For calculations of

For nonrelativistic protons with

A value of the coefficient

In Kovtyukh (2016) functions

Functions

Functions

For calculating the functions

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

I have also made the same calculations of the functions

In overlapping ranges of

Positive radial gradients of the functions

Eq. (1) can be represented as follows:

The radial dependence of the rates of the ionization losses of
protons with various

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

Coulomb losses and the losses to charge exchange were calculated for the
protons with specific values of

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

The vertical cuts on these curves mark

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

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

We divide the radial dependences of

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

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

For general physical reasons, it follows that

Therefore, the diffusion flow for the smallest

For all values of

So even if we assume that

However, for protons with

Thus, for protons with

For protons with

In calculating the derivatives on the left-hand side and integrals on the
right-hand side of Eq. (5), I divided the scale of

The results of our calculations of

From Figs. 5 and 6, we see that the results of our calculations of

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

The errors of my method of calculating

The results of calculations of the values of

The results of calculations of the values of

Since the Explorer-45 data are limited to a maximum available

For

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

Taking into account all possible errors, the calculated values of

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

In the total errors of our calculations of

It has been shown that at

The main result of our calculations is the strong dependence of

This result can be seen from Figs. 1–4 and Eq. (4) before

The effect of reducing

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

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

Equation (6) for

The functions of

These circumstances lead to different conclusions for the ratio of

In many works,

The values of

Figure 7 presents the dependence of

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

Figure 8 presents the dependences of

The dependence of the

The dependences of

According to Fig. 8,

The results of our calculations of

In Eqs. (7)–(9) the results
of the calculations of

The Eqs. (7)–(9) contradict the theory of diffusion under the
action of SI (magnetic diffusion). According to this theory,

In the dipole approximation, the azimuthal drift frequency of
nonrelativistic trapped particles is

For magnetic field fluctuations

Unfortunately, in the range of

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

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

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

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

According to Fig. 2 in Liu et al. (2016), at a frequency of

Thus, according to THEMIS data averaged over 7 years for all MLT, for
periods with Kp

On the basis of the spectra of electric field fluctuations, Liu et al. (2016)
calculated

Note that for electrons with

I calculate the value

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

For these calculations I used the data of ISEE-1 for protons with an
energy of 24 to 2081 keV at

As a result of the calculations, I found that in the range of

It is shown that for protons with

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

The comparison

Because the values of

The author would like to thank P. Kollmann (Applied Physics Laboratory, Johns Hopkins University) for very important and fruitful comments on and proposals regarding the paper and E. Roussos (Max Planck Institute for Solar System Research) for editing the paper. The author thanks the Kyoto World Data Center for Geomagnetism for providing the Dst indices. The topical editor, E. Roussos, thanks P. Kollmann, S. Bourdarie, and one anonymous referee for help in evaluating this paper.