Magnetars versus Radio Pulsars: MHD Stability in Newborn Highly Magnetized Neutron Stars

We study the stability/establishment of dipolar magnetostatic equilibrium configurations in new--born neutron stars (NSs) in dependence on the rotational velocity $\Omega$ and on the initial angle $\alpha$ between rotation and magnetic axis. The NS is modeled as a sphere of a highly magnetized ($B \sim 10^{15}$G) incompressible fluid of uniform density which rotates rigidly. For the initial dipolar background magnetic field, which defines the magnetic axis, two different configurations are assumed. We solve the 3D non--linear MHD equations by use of a spectral code. The problem in dimensionless form is completely defined by the initial field strength (for a fixed field geometry), the magnetic Prandtl number $\Pm$, and the normalized rotation rate. The evolution of the magnetic and velocity fields is considered for initial magnetic field strengths characterized by the ratio of ohmic diffusion and initial \Alf{} travel times $\ttOhm/\ttAO \approx 1000$, for $\Pm = 0.1, 1, 10$, and the ratio of rotation period and initial \Alf{} travel time, $P/\ttAO = 0.012, 0.12, 1.2, 12$. We find hints for the existence of a unique stable dipolar magnetostatic configuration for any specific $\alpha$, independent of the initial field geometry. Comparing NSs possessing the same field structure at the end of their proto--NS phase, it turns out that sufficiently fast rotating NSs ($P\la6 $ms) with $\alpha \la 45^0$ retain their magnetar field, while the others lose almost all of their initial magnetic energy by transferring it into magnetic and kinetic energy of relatively small--scaled fields and continue their life as radio pulsars with a dipolar surface field of $10^{12...13}$G.