Non-Axisymmetric Precession of Magnetars and Fast Radio Bursts

The repeating FRBs 180916.J0158 and 121102 are visible during periodically-occuring windows in time. We consider the constraints on internal magnetic fields and geometry if the cyclical behavior observed for FRB~180916.J0158 and FRB 121102 is due to precession of magnetars. In order to frustrate vortex line pinning we argue that internal magnetic fields must be stronger than about $10^{16}$ Gauss, which is large enough to prevent superconductivity in the core and destroy the crustal lattice structure. We conjecture that the magnetic field inside precessing magnetars has three components, (1) a dipole component with characteristic strength $\sim 10^{14}$ Gauss; (2) a toroidal component with characteristic strength $\sim 10^{15}-10^{16}$ Gauss which only occupies a modest fraction of the stellar volume; and (3) a disordered field with characteristic strength $\sim 10^{16}$ Gauss. The disordered field is primarily responsible for permitting precession, which stops once this field component decays away, which we conjecture happens after $\sim 1000$ years. Conceivably, as the disordered component damps bursting activity diminishes and eventually ceases. We model the quadrupolar magnetic distortion of the star, which is due to its ordered components primarily, as triaxial and very likely prolate. We address the question of whether or not the spin frequency ought to be detectable for precessing, bursting magnetars by constructing a specific model in which bursts happen randomly in time with random directions distributed in or between cones relative to a single symmetry axis. Within the context of these specific models, we find that there are precession geometries for which detecting the spin frequency is very unlikely.