Evolutions of Magnetized and Rotating Neutron Stars Steven L. Liebling, 1 Luis Lehner, 2, 3, 4 David Neilsen, 5 and Carlos Palenzuela 6, 7 $^{1}$Department of Physics, Long Island University – C.W. Post Campus, Brookville, NY 11548 2 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada 3 Department of Physics, University of Guelph, Guelph, Ontario, Canada 4 Canadian Institute for Advanced Research, Cosmology & Gravity Program $^{5}$Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602 6 Canadian Institute for Theoretical Astrophysics (CITA), Toronto, Ontario, Canada $^{7}$Max-Planck-Institut f¨ ur Gravitationsphysik, Albert-Einstein-Institut, Golm, Germany (Dated: August 7, 2018) We study the evolution of magnetized and rigidly rotating neutron stars within a fully general relativistic implementation of ideal magnetohydrodynamics with no assumed symmetries in three spatial dimensions. The stars are modeled as rotating, magnetized polytropic stars and we examine diverse scenarios to study their dynamics and stability properties. In particular we concentrate on the stability of the stars and possible critical behavior. In addition to their intrinsic physical significance, we use these evolutions as further tests of our implementation which incorporates new developments to handle magnetized systems. I. INTRODUCTION Neutron stars play a key role in some of the most in- teresting astrophysical events observed, from supernova remnants and pulsars to a less certain role in long gamma ray bursts (GRBs). As such they have attracted signif- icant research into their dynamics (for a recent review see [1]). In this paper, we revisit the dynamics of rotat- ing, and possibly magnetized, neutron stars modeled as polytropic stars within a fully nonlinear, general relativis- tic model of ideal magnetohydrodynamics (GR+MHD). We study the stability properties of these stars and high- light possible critical behavior exhibited by the system. Furthermore, our studies serve to demonstrate the effec- tiveness of our code and certain new developments dis- cussed here. This code has been applied to other astro- physical problems [2–7]. In recent years, a number of fully relativistic evolutions (as opposed to those using Newtonian gravity or other ap- proximations to general relativity) of rotating polytropes have appeared studying gravitational wave production and black hole formation in astrophysically relevant sys- tems. To this end, studies beyond spherical symmetry are required which are computationally more demand- ing. For instance, rotating stars require moving beyond spherical symmetry and many interesting axisymmetric scenarios can be addressed using 2D implementations al- lowing for excellent resolution without requiring major resources (e.g. [8–10]). In contrast, studies of the most general flows and instabilities require 3D simulations and are the most expensive realistically accessible scenarios being currently considered. Future efforts, including ra- diation transport mechanisms, will move beyond 3D sce- narios and require efficient use of petaflop (and beyond) resources (e.g. [11]). Another important distinction in these studies is whether the modeled stars are rigidly rotating or allow for differential rotation. Rigidly rotating stars support more mass than non-rotating ones (so-called TOV stars), but generally do not support the largest masses achieved with differential rotation nor do they demonstrate some of the more interesting instabilities such as the bar mode instability [12]. Instead, uniformly rotating stars tend to demonstrate one of two behaviors; stability or instability to collapse to a black hole. Previous studies suggest that significant disks do not form from such collapse [13–15]. Recall that stars rotating differentially are expected to settle into rigidly rotating configurations on short time scales, and hence a normal neutron star observed today is generally expected to be rigidly rotating. Achieving ever more realism, models of neutron stars have also begun to consider stars with magnetic field [9, 16–18]. Magnetic fields provide, in particular, an effec- tive pressure which generally supports greater mass [19] as well as an efficient way to transport angular momen- tum. In differentially rotating stars, even small mag- netic fields can be amplified by the magnetorotational instability. Generally these studies begin with a non- magnetized neutron star to which a small, seed magnetic field is added. However, in [20], fully consistent, magne- tized stars are evolved, although only nonrotating results are presented. Another approach is to study the modes through a perturbation approach [21, 22] and analyze the growth of these modes with respect to a given stationary solution. The dynamical behavior of magnetized stars is important also for their role in explaining strong electro- magnetic emissions. Indeed, isolated stars with strong magnetic fields, so called magnetars , are suspected to be the engines powering anomalous X-ray pulsars (AXPs) and soft gamma ray repeaters (SGRs) [23]. At least 10% of all neutron stars [24] are born as magnetars. In the context of single stars, it is thus interesting to exam- ine if strong magnetic fields may deform stars away from axisymmetry making them strong producers of gravita- tional waves [25]. Here we present results with uniformly rotating neu- tron stars which possess a fully consistent magnetic dipole moment. That is to say, the initial data used here arXiv:1001.0575v2 [gr-qc] 24 May 2010