The integral invariant coordinate

More particularly, in this paper we compare the values of

The motion of charged particles in the geomagnetic field is complicated, even
if one approximates that field with only its dipole component. It is helpful
to break down the total motion of the particle into three individual
components: gyration around a guiding magnetic field line, bounce along the
magnetic field line between magnetic mirror points, and gradient and
curvature drift across the magnetic field line in an azimuthal direction
around the Earth. Because these components evolve over very different time
scales, they are nearly independent of each other and can thus be summed
linearly to obtain the total motion

The first invariant,

In calculations involving the adiabatic invariants, it is often instructive
to use proxy invariant parameters. In the case of calculations concerning the
second adiabatic invariant, the integral invariant coordinate

A practical way to calculate

In general, calculation of

The artificial neural network consists of two layers. The first layer
provides 19 nodes, one for each input parameter for the TS05 model plus
additional nodes to help specify the drift shell especially for low Earth
orbit. The hidden layer in the neural network contains 20 neurons that are
connected to each input node and one output node to produce

LANL* V2.0 can be downloaded at

IRBEM-lib (formerly known as ONERA-DESP-LIB) is a freely distributed library
of source codes dedicated to radiation belt modeling put together by the
Office National d'Etudes Aérospatiales (ONERA-DESP). The library allows the
computation of magnetic coordinates and fields for any location in the
Earth's environment for various magnetic field models. It is primarily
written in Fortran with access to a shared library from IDL or Matlab

As seen in the IRBEM
source code, e.g.,

As found in the IRBEM source code, also in

The latest version of IRBEM-lib can be found at

Calculations of

The European Space Agency (ESA) Space Environment Information System
(SPENVIS) provides standardized access to models of the hazardous space
environment through a World Wide Web interface

Reference: UNILIB source code.

.The SPENVIS web interface can be accessed at

The calculations of

ptr3D v2.0 calculates

The integral invariant

Calculations of

Calculations of

In Figs.

In Fig.

In Fig.

In Fig.

There is generally good agreement between the results from IRBEM and SPENVIS
and those from ptr3D for 500

Calculations of

Calculations of

Calculations of

Calculations of

Calculations of

Calculations of

In the following,

The Lorenz trace of the forwards (blue) and backwards (red)
propagating particle is plotted. The region where

Plots of the regions of constant

Generally, for the quiet conditions case, the results from all the models
tend to agree more at smaller distances (4–6

For the disturbed conditions case, similar trends are observed, albeit more
accentuated. The results from LANL* and IRBEM agree relatively well, as do
those from SPENVIS, where available. The results from ptr3D deviate
significantly from those of the other models for distances greater than 4

Similarly to the simulations above, some of the 12 particles of various
initial gyrophases precipitated or otherwise failed to complete a full
revolution around the Earth; these particles were not taken into
consideration when averaging the results for each initial gyrophase. If more
than half of the particles failed to complete a rotation around the Earth, no

Next, we demonstrate at which magnetic longitude the conservation of

Using ptr3D,

In the case of the

In the case of disturbed solar wind conditions, the regions of constant

Using the ptr3D v2.0 particle tracer, LANL*, IRBEM-lib and SPENVIS, we
quantified the variations in the calculations of

The results for the calculations of

Generally, the same trends are observable for the calculations of

Using ptr3D we mapped the areas in the Earth's magnetosphere where

In the discussions of particle transport, energization and loss in the Earth's radiation belts, a major question concerns the relative contribution between wave–particle interactions vs. radial diffusion, which is generally best discussed in terms of phase-space density, calculated at constant adiabatic invariants. From the discussions herein, it is evident that caution should be exercised when considering the second and third adiabatic invariants to remain constant across all L-shells and local times within the radiation belts as well as for all particle energies and all geomagnetic conditions. In particular, in regions where the results from the various models diverge from the results from the particle tracer, which most closely follows the calculations of the invariants, we can conclude that the models should be used with caution, the lack of confidence in them being analogous to the magnitude of this divergence. It has been demonstrated that under extreme curvature of the magnetospheric magnetic field, particles of high energy and low pitch angles cannot be considered to remain adiabatic in terms of their second and third invariants.

The physical mechanism that leads to breaking of the invariants in the regions illustrated does not involve temporal variations in the magnetic field of timescales shorter than the associated timescales of the second and third invariants, i.e., the bounce period and drift period, as the fields used in the simulations above are all static. Instead, the breaking of the invariants in the above is associated with deviations of the magnetic field from a dipole configuration: in the definition of the invariants, in order for the second adiabatic invariant to remain constant it is required that the magnetic field between two mirroring points does not change much in one bounce period as the particle's guiding center drifts across field lines. Similarly, in order for the third adiabatic invariant to remain constant, it is required that the magnetic flux through the guiding center orbit of a particle around the Earth should remain constant. However during active geomagnetic conditions the curvature of the field lines in the nightside of the Earth in combination with the large gyro-radii of large-energy particles leads to deviations from these conditions that need to be taken into account.

The present paper by no means aims to serve as a guideline of the adiabaticity of particles at all energies, pitch angles and geomagnetic conditions; instead, it aims to raise awareness and caution in using general-purpose models and tools, such as IRBEM, LANL* and SPENVIS, to calculate the values of the adiabatic invariants in regions and cases where they are not well defined.

Instructions on downloading or accessing third-party software used in this work are given in their respective sections. ptr3D V2.0 is a particle tracing code developed by the authors based on the equations of charged particle motion under the Lorenz force, as described in detail in the respective chapter of this paper, and its results can be verified by any other particle tracers. In its current version (V2.0) it has been tuned to work accurately and efficiently within the region, times and energies of the particles under investigation, and hence at this stage it is not a generic code that can be provided for use as a general particle tracer; it is envisioned that its next version (ptr3D V3.0) will be released as a general particle tracer code that can be used for any range of particle energies, times or regions.

This study was supported by NASA grants (THEMIS, NNX10AQ48G and NNX12AG37G) and NSF grant ATM 0842388. This research has also been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) Research Funding Program: Thales. Investing in knowledge society through the European Social Fund. Edited by: J. Koller