  (Abridged) Following particle trajectories in the intense electromagnetic
field of a neutron star is prohibited by the large ratio between the cyclotron
frequency $\omega_{\rm B}$ and the stellar rotation frequency $\Omega$. No
fully kinetic simulations on a macroscopic scale and with realistic field
strengths have been performed so far due to the huge computational cost implied
by this enormous scale of separation. In this paper, we derive new expressions
for the particle velocity subject to strong radiation reaction that are
intended to be more accurate than the current state-of-the-art expression in
the radiation reaction limit regime, the so-called Aristotelian regime. We
shortened the timescale hierarchy by solving the particle equation of motion in
the radiation reaction regime, where the Lorentz force is always and
immediately balanced by the radiative drag, and including a friction not
necessarily opposite to the velocity vector, as derived in the Landau-Lifshitz
approximation. Starting from the reduced Landau-Lifshitz equation (i.e.
neglecting the field time derivatives), we found expressions for the velocity
depending only on the local electromagnetic field configuration and on a new
parameter related to the field strength that controls the strength of the
radiative damping. The new velocity expressions produce results similar in
accuracy to the radiation reaction limit approximation. We therefore do not
expect this new method to improve the accuracy of neutron star magnetosphere
simulations. The radiation reaction limit is a simple but accurate, robust, and
efficient way to follow ultra-relativistic particles in a strong
electromagnetic field.
