Størmer's particles' trapping regions for a planet with an intrinsic dipolar magnetic field are considered, taking into account the ring current which arises due to the trapped particles' drift for the case of the Earth. The influence of the ring current on the particle trapping regions' topology is investigated. It is shown that a critical strength of the ring current exists under which further expansion of the trapping region is no longer possible. Before reaching this limit, the dipole field, although deformed, retains two separated Størmer regions. After transition of critical magnitude, the trapping region opens up, and charged particles, which form the ring current, get the opportunity to leave it, thus decreasing the ring current strength. Numerical calculations have been performed for protons with typical energies of the Earth's radiation belt and ring current. For the Earth's case, the Dst index for the critical ring current strength is calculated.

One of the first scientists who studied the problem of finding trajectories of charged particles moving from the Sun to the Earth was Norwegian physicist Carl Størmer. Since it is rather difficult to solve this problem directly without numerical methods, he used several simplifying assumptions: only a magnetic field of terrestrial origin (magnetic dipole) acts on a charged particle; magnetic and electric fields from other sources, including charged particle fluxes, are not taken into account

For a complete analytical solution of the problem, one has to find three integrals of motion. In the given problem the law of conservation of the generalized angular momentum of a particle follows from the isotropy of space (the Lagrangian does not change relative to the coordinate system rotations); the law of conservation of energy is the result of the fact that the Lagrangian does not change with respect to the time shift. The third integral of motion cannot be found analytically. Størmer solved this problem by obtaining an expression of a special parameter

The effect of the equatorial ring current on the intensity of cosmic rays was considered by

Also, Størmer's analyses have been expanded for the external homogeneous field case.

The main sources which form the magnetospheric magnetic field are the internal field of the planet, interplanetary magnetic field (IMF) and the field of the magnetospheric current systems (Chapman–Ferraro currents at the magnetopause, magnetospheric tail currents and ring current). In the Størmer problem in connection with its specificity (axial symmetry), we will consider only the contributions of the dipole field (the contributions of the quadrupole and other multipoles have been considered in

So far, Størmer's trapping regions have been considered separately from real trapping regions in the Earth's magnetosphere. We consider allowed regions of charged particles' motion in a wide energy range for different ring current strength magnitudes. We show that, using the Størmer analysis, one can find trapping regions for particles of the ring current and radiation belts energies, and also the fact that the forbidden region of particles' motion can break as the current increases, and then particles get an opportunity to leave the trapping region.

The present paper consists of the following sections: Sect.

The dipole configuration of the Earth's magnetic field creates a region that is a kind of a “magnetic vessel” inside which charged particles are trapped. For the first time the possibility of the existence of such a region was shown by Carl Størmer

For the first time the hypothesis of a ring current was made by Carl Størmer to explain the location of the aurora and its shift to the equatorial latitudes under magnetic storm conditions

Magnetic storms on the Earth are directly related to the ring current. The increase in its strength leads to compression of the inner magnetic field of the planet. During quiet times, the ring current is localized near the geomagnetic equator, directed to the west, and depends little on local time. A geomagnetic storm usually begins with the impact of a strong and permanent southward-oriented IMF along with the increased dynamic pressure of the solar wind on the dayside magnetosphere. The increased pressure moves the magnetopause inward by several Earth radii, increasing currents on the magnetopause and temporarily causing an increase in the surface magnetic field. This phenomenon is known as the sudden commencement of a storm. During the main phase of the magnetic storm, which is usually associated with the southward-oriented IMF magnitude growth, the electric field applied to the magnetosphere increases, which leads to an increase in the number of particles injected into the inner magnetosphere due to the

The strength of the storm-time current is estimated by the Dst index and can reach 10 MA

During the main phase of the storm, the hot plasma is injected into the inner magnetosphere. The stronger the storm, the closer the ring current comes to the Earth and the slower it relaxes. Storm-time ring current essentially differs from radiation belts, consisting of more energetic particles, in composition, dynamics and generation mechanisms. Unlike radiation belt ions, whose fluxes are weakened during storms

The ring current is not the only source which contributes to the Dst index during the storm.

There are several loss processes which act on decay of the ring current ions: the charge exchange of ions with the exospheric atoms, the Coulomb drag, the Coulomb scattering, pitch angle scattering due to interaction with electromagnetic waves, the violation of the first adiabatic invariant and the convection outflow. All these sources are considered in detail in

We propose a new loss mechanism associated with the disappearance of Størmer's forbidden region of particle motion between two allowed regions (outer, which is connected with the outer space, and inner, which is not) due to an increase in the ring current strength. This consideration allows us to find the upper limit on the maximum current, which can be created by trapped particles. In this threshold case, the ring current particles are allowed to leave the trapping region, which leads to a weakening of the ring current and Dst index decrease. The existing mechanisms of scattering and precipitation of particles can lead to the fact that the equilibrium value of the ring current will be smaller (in the case of the Earth by more than an order of magnitude) than this threshold value. But can this mechanism limit the maximum field depression during large magnetic storms?

In the zeroth approximation, the ring current can be represented as a ring with current in the equatorial plane at a certain distance from the Earth. The magnetic field in the center of the ring with a current of radius

From this expression, we can obtain an approximation of the magnetic field perturbation in the Earth's center:

Equations of the charged particle motion in an electromagnetic field in the case of axial symmetry are more convenient to consider in cylindrical coordinates. We consider a cylindrical coordinate system in which the positive

We will use the Lagrangian formulation; the equation of motion can then be written as

The Lagrangian function

The scalar potential of an electromagnetic field will not be taken into account due to the absence of an electric field in Størmer's problem. Since the field we are considering has the axial symmetry, in a cylindrical coordinate system the Lagrangian does not depend on the azimuth angle

The total energy of a particle in the Lagrangian formulation is

Since we neglect the particle radiation, the total energy

In other words, the square of the velocity of the particle

It is convenient to integrate over the arc of the trajectory

We express

From the relativistic relations, we can obtain the relation

After substituting this relation into Eq. (

For clarity, we rewrite this equation using the expression from Eq. (

As can be seen from Eq. (

Under these conditions, trapping regions, separated from infinity by forbidden regions (where particles cannot get due to the preservation of exact integrals of motion for any initial conditions), are formed in the phase space. Depending on the initial conditions, the trajectories of particles of a given energy may have a scattering pattern – particles come from infinity, change their direction of motion near the dipole, and again go to infinity, deviating by the scattering angle, or if their position and the generalized angular momentum satisfy Størmer's conditions, the particles make a finite motion, filling the allowed Størmer regions and forming the trapped particles' fluxes. This moment is given little attention in Størmer's original work

The surface

The vector potential of the magnetic field produced by a magnetic dipole with moment

The vector potential of a homogeneous magnetic field

The vector potential of the ring current has a complex form and depends on the assumed radius of its cross section. In our problem we assume that the ring current has the form of a ring with radius

One can separate the considered region into two parts: when

Substituting all expressions for

We define the Størmer radius

In the case of the Earth,

Dependence of the Størmer radius of the particle (in Earth radii) on its energy.

Størmer's parameter

Having two integrals of motion and the Størmer parameter

The size of the allowed regions of motion, as can be seen from Eq. (

Using Eq. (

After solving Eq. (

As mentioned above, the allowed regions of a particle's motion are determined by the

Using Eq. (

In our problem

It is easy to show that Eq. (

If we consider the case where

If we consider the motion in the equatorial plane of a dipole, then

If we put Eq. (

The boundaries of the allowed region of motion are determined by the equality to

Allowed regions of motion of a 100 keV proton for two cases:

To describe the particle trajectory in the whole space, we can change from cylindrical coordinates (

According to Eq. (

This situation is shown in Fig.

The allowed region shown in Fig.

To find

Dependence of Størmer's parameter

In the original Størmer analysis the magnetic field is stationary and a particle does not receive energy from outside, so the parameter

In modeling, we assumed the following.

The Earth's magnetic field is constant and has only a dipole component due to the axial symmetry of the problem. This is quite reasonable, because (1) the dipole magnetic field is responsible for the basic motion of the ring current particles, that is, the bounce and drift motion, and (2) the higher degree of the coefficient becomes dominant at a radial distance of less than 1.5

The ring current has the form of an infinitely thin ring in the equatorial plane at a distance of 4

The

Charged particles in both the radiation belt and ring current are protons (

Radiation belt particles with energies of 1 and 100 MeV.

Ring current particles with energies of 10 and 100 keV.

For particles with different values of

All the following figures show the meridional cross sections of the near-Earth region; the allowed regions of motion are highlighted in white, forbidden in black. Magnetic field lines are highlighted in cyan. Coordinates are measured in the Earth radii. The spatial configuration of forbidden regions is obtained by rotating the meridional sections around the

At first we consider the energies of the ring current protons of 10 and 100 keV (Figs.

The ring current strength

The ring current strength

The ring current strength

As can be seen from Figs.

The obtained current strengths are consistent with the observations (see Sect.

Now let us consider what happens to the protons of the radiation belts (with energies of 1 and 100 MeV) after increase in the ring current magnitude (Figs.

The ring current strength

The ring current strength

As can be seen from Figs.

However, the geometry of the allowed–forbidden regions and the current strength at which the inner and outer Størmer trapping regions get connected strongly depend on two parameters – the external magnetic field

The strength of the ring current is 9.17 MA; the energy of the proton is 10 keV.

We consider an initial state for a trapped particle under external field

Now let us consider how the configuration of the particles' allowed regions of motion with the same parameters varies for different radii

The strength of the ring current is 9.17 MA; the particle energy is 10 keV.

One can see that Figs.

Størmer analysis is not a self-consistent approximation; the magnetic field in the problem is axially symmetric and specified – independent of the trapped particles' number. When searching for the critical value of the ring current, we set the distance based on the available observations. Nevertheless, for a complete solution of the problem it would be useful to have a generalization of the Størmer analysis to the case of a self-consistent field. We will try to show how to come to this solution by the method of successive approximations. We will calculate the threshold current for particles with fixed energy at a given distance, changing the external field. Afterwards, we will calculate the threshold current for particles with fixed energy with a given external field, changing the radius

Let us consider a proton with energies of 10 and 100 keV at

Critical strength of a ring current for different magnitudes of the external field

It can be seen from Table

Let us also consider the influence of the radius

As one can see from Table

Critical strength of a ring current for different ring radii

The change in the radius

As a result of particles' injection into the inner magnetosphere, their population and consequently the ring current strength increase. Størmer's analysis shows that at a certain point, when the ring current strength reaches a critical magnitude, the forbidden region, which separates the internal and external allowed regions, disappears, and thus particles get the opportunity to leave the internal trapping region. As a result, the number of particles, which comprise the ring current, and consequently the current strength, begin to decrease. In the limit in this case, the particle density near the dipole and at infinity must be the same. The critical magnitude of the ring current's strength depends on the energy of the particles composing the ring current, but at a certain moment the maximum strength is reached, when particles even with the lowest energy are able to leave the trapping region. The maximum possible number of particles that can be trapped in a dipole field is determined by a number of particles, forming a ring current, which changes the “connectivity” of Størmer's regions. For the doubly connected geometry of Størmer's regions (when internal and external allowed regions of motion are not connected), the particles' flux at infinity and the flux of trapped particles close to the Earth can differ arbitrarily. The mixing of passing (coming and going to infinity) and trapped particles does not occur. Therefore, it becomes possible to form regions of trapped radiation, the fluxes of which are determined by the injection and loss mechanisms and are not associated with particles' fluxes at infinity. The deformation of Størmer's regions by the self-consistent field of the trapped radiation (ring current) leads to the formation of connected allowed regions of motion, and in this case the particles' fluxes in the allowed region are uniquely determined by the particles' flux and distribution over the pitch angles at infinity. We come to the conclusion that the ring current strength has a finite upper limit. At a certain current strength magnitude, the inner trapping region becomes connected to the outer region associated with infinity and further accumulation of particles in the internal trapping region is impossible. At present, there are several recognized main processes, leading to the ring current decay. We showed that a new mechanism, described in the paper, can also be added to these mechanisms. This mechanism leads to the limiting of the ring current strength during a magnetic storm irrespective of the particle loss mechanism from the internal allowed trapping region.

Størmer's analysis shows that the existence of radiation belts follows from the existence of a geomagnetic field. Particles with particle energies of radiation belts and with different parameters

No datasets were used in the preparation of this paper. All the physical quantities' values are public knowledge.

ASL and IIA were responsible for the planning, coordination, revisions, and responses to the referees. ASL was responsible for the writing of the entire paper and IIA proposed the original idea; IVT was responsible for calculations and plotting the corresponding results.

The authors declare that they have no conflict of interest.

The work was supported by the Ministry of Science and Higher Education of the Russian Federation (grant RFMEFI61617X0084).

The work was supported by the Ministry of Science and Higher Education of the Russian Federation (grant RFMEFI61617X0084).

This paper was edited by Elias Roussos and reviewed by two anonymous referees.