Mapping the transitions of Proton Adiabaticity in Jupiter's Inner Radiation Belt: A comprehensive approach

The dynamics of energetic charged particles in Jupiter's inner magnetosphere are governed by the conservation of adiabatic invariants, yet the transition to non-adiabatic motion has not been systematically quantified across particle energy, $M$-shell, and pitch angle. Prior work relies on an empirical gyro-radius threshold ($r_L<1000$~km) tested at only two $M$-shells, leaving the intermediate, partially adiabatic regime uncharacterized. We develop a fully relativistic, three-dimensional test-particle model (Boris pusher) coupled to the JRM33 and Con2020 magnetic field models to investigate ultra-relativistic proton dynamics in Jupiter's inner radiation belt.A simulation design experiment encompassing 135 simulations (five energies,1--100~GeV; nine $M$-shells, 3--7~$R_J$; three pitch angles, $alpha_0=30^\circ,60^\circ,89^\circ$) was performed, yielding three formally defined regimes: fully adiabatic, partially adiabatic, and fully non-adiabatic. The Boris pusher is found to be more reliable for long-term invariant studies compared to the Runge-Kutta scheme. A field-geometric curvature parameter, $\kappa=(R_c/r_L)^{1/2}$, to be effective in adiabaticity break down criteria which is validated via simulations. It is found that the adiabatic--partial boundary shifts to lower energy with increasing $M$-shell and pitch angle; at $M=6.0\,R_J$, even 1~GeV protons exceed the gyro-radius threshold, and the partially adiabatic regime is most prominent at $M=3.0$--$4.5\,R_J$ for 1--10~GeV. The transitional zone for 10--20~GeV protons overlaps substantially with the interchange-event range reported from independent electron pitch-angle observations, suggesting a shared association with reduced magnetic confinement in this region. These results provide a validated, framework, with direct relevance to the radiation hazard environment for future Jovian spacecraft missions.