In this paper, we discuss the results of a new particle pusher in realistic ultra-strong electromagnetic fields such as those encountered around rotating neutron stars. After presenting the results of this algorithm in simple fields and comparing them to expected exact analytical solutions, we present new simulations for a rotating magnetic dipole in vacuum for a millisecond pulsar by using the Deutsch solution. Particles are injected within the magnetosphere, neglecting radiation reaction, interaction among them and their feedback on the fields. Our simulations are therefore not yet fully self-consistent because the Maxwell equations are not solved according to the current produced by these particles. The code highlights the symmetrical behaviour of particles of opposite charge to mass ratio, $q/m$ , with respect to the north and south hemispheres. The relativistic Lorentz factor $\gamma$ of the accelerated particles is proportional to this ratio $q/m$ : protons reach up to $\gamma _p \simeq 10^{10.7}$ , whereas electrons reach up to $\gamma _e \simeq 10^{14}$ . Our simulations show that particles could either be captured by the neutron star, trapped around it or ejected far from it, well outside the light cylinder. Actually, for a given charge to mass ratio, particles follow similar trajectories. These particle orbits show some depleted directions, especially at high magnetic inclination with respect to the rotation axis for positive charges and at low inclination for negative charges because of symmetry. Other directions are preferred and loaded with a high density of particles, some directions concentrating the highest or lowest acceleration efficiencies.

Context. Neutron stars are known to be efficient accelerators that produce particles with ultra-relativistic energies. As a by-product, they also emit copious amounts of photons from radio wavelengths up to gamma rays. Aims. 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. Methods. 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. Results. We found that, though due solely to the Lorentz force, electrons reach Lorentz factors up to γ = 1014 and protons reach them up to γ = 1010.7. When radiation reaction is enabled, electrons reach energies up to γ = 1010.5 and protons reach energies up to γ = 108.3. 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). Consequently, it is useless to simulate the full set of geometrical parameters in an effort to obtain an overview of all possibilities. Conclusions. 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. Similarly, we still find that particles are ejected away from the neutron star, in some preferred directions and Lorentz factors.