  As a follow-up to our previous work on particle acceleration simulation near
neutron stars, in this paper, we discuss the impact of radiation reaction on
test particles injected into their magnetosphere. We therefore neglect the
interaction between particles through the electromagnetic field as well as
gravitation. We integrate numerically the reduced Landau-Lifshitz equation for
electrons and protons in the vacuum field of a rotating magnetic dipole based
on analytical solutions in a constant electromagnetic field. These expressions
are simple in a frame where the electric and magnetic field are parallel.
Lorentz transforms are used to switch back and forth between this frame and the
observer frame. We found that, though due solely to the Lorentz force,
electrons reach Lorentz factors up to $\gamma=10^{14}$ and protons reach them
up to $\gamma=10^{10.7}$. When radiation reaction is enabled, electrons reach
energies up to $\gamma=10^{10.5}$ and protons reach energies up to
$\gamma=10^{8.3}$. The second set of values are more realistic since the
radiation reaction feedback is predominant within the magnetosphere. Moreover,
as expected, symmetrical behaviours between the north and south hemispheres are
highlighted, either with respect to the location around the neutron star or
with respect to particles of opposite charge to mass ratio~$(q/m)$. The study
of the influence of the magnetic dipolar moment inclination shows similar
behaviours regardless of whether radiation reaction is enabled. Protons
(respectively electrons) impact the surface of the neutron star less as the
inclination angle increases (decreases for electrons), while if the rotation
and magnetic axes are aligned, all the protons impact the neutron star, and all
the electrons impact the surface if the rotation and magnetic axes are
anti-aligned.
