Abstract. The integral invariant coordinate I and Roederer's L or L* are proxies for the second and third adiabatic invariants, respectively, that characterize charged particle motion in a magnetic field. Their usefulness lies in the fact that they are expressed in more instructive ways than their counterparts: I is equivalent to the path length of the particle motion between two mirror points, whereas L*, although dimensionless, is equivalent to the distance from the center of the Earth to the equatorial point of a given field line, in units of Earth radii, in the simplified case of a dipole magnetic field. However, care should be taken when calculating the above invariants, as the assumption of their conservation is not valid everywhere in the Earth's magnetosphere. This is not clearly stated in state-of-the-art models that are widely used for the calculation of these invariants. The purpose of this work is thus to investigate where in the near-Earth magnetosphere we can safely calculate I and L* with tools with widespread use in the field of space physics, for various magnetospheric conditions and particle initial conditions.

More particularly, in this paper we compare the values of I and L* as calculated using LANL*, an artificial neural network developed at the Los Alamos National Laboratory, SPENVIS, a space environment online tool, IRBEM, a software library dedicated to radiation belt modeling, and ptr3D, a 3-D particle tracing code that was developed for this study. We then attempt to quantify the variations between the calculations of I and L* of those models. The deviation between the results given by the models depends on particle initial position, pitch angle and magnetospheric conditions. Using the ptr3D v2.0 particle tracer we map the areas in the Earth's magnetosphere where I and L* can be assumed to be conserved by monitoring the constancy of I for energetic protons propagating forwards and backwards in time. These areas are found to be centered on the noon area, and their size also depends on particle initial position, pitch angle and magnetospheric conditions.