Magnetospheric Structure and Non-Thermal Emission of AXPs and SGRs

In the framework of the magnetar model for the Soft Gamma Repeaters and Anomalous X-ray Pulsars, we consider the structure of neutron star magnetospheres threaded by large-scale electrical currents. We construct self-similar, force-free equilibria under the assumption of axisymmetry and a power law dependence of magnetic field on radius, B ∝ r−(2+p). A large-scale twist of the field lines softens the radial dependence to p < 1, thereby accelerating the spindown torque with respect to a vacuum dipole. A magnetosphere with a strong twist (Bφ/Bθ = O(1) at the equator) has an optical depth ∼ 1 to resonant cyclotron scattering, independent of frequency (radius), surface magnetic field strength, or the charge/mass ratio of the scattering charge. We investigate the effects of the resonant Compton scattering by the charge carriers (both electrons and ions) on the emergent X-ray spectra and pulse profiles. 1. Nature of the AXPs and SGRs The Anomalous X-ray Pulsars and Soft Gamma Repeaters are neutron stars which share similar spin periods, P = 5 − −12 s, characteristic ages, P/Ṗ = 3×103−4×105 yr, and X-ray luminosities, LX = 3×10 34−1036 erg s−1, well in excess of the spin-down luminosities (Thompson et al. 2001). They are almost certainly young and isolated: no evidence for a binary stellar companion has yet been detected in any of these sources, and a few are convincingly associated with young supernova remnants. The overlap between the SGR and AXP sources in a three-dimensional parameter space (P , Ṗ and LX), and the observed variabilDepartment of Physics, McGill University, Montréal, QC Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 CITA National Fellow Canadian Institute for Theoretical Astrophysics, 60 St. George St., Toronto, ON M5S 3H8 California Institute of Technology, 105-24, Pasadena, CA 91125 1 2 Lyutikov, Thompson & Kulkarni ity in their X-ray output, provides circumstantial evidence that they share a common energy source: the decay of a very strong (∼> 10 15 G) magnetic field. The persistent emission of the SGRs and AXPs has both a thermal and a non-thermal component. In the case of the SGRs, this emission becomes brighter after periods of bursting activity and involves a comparable release of energy to the outbursts (averaged over time). 2. Twisted Neutron Star Magnetospheres The magnetic fields of magnetars are most likely generated by a hydromagnetic dynamo as the star is born, and may be associated with rapid initial rotation (Duncan & Thompson 1992). A strong twist in a ∼ 1015 G magnetic field will relax at intervals as the field is transported through the deep interior of the neutron star. Even if the electrical current were initially confined to interior of the star, the Lorentz force would become strong enough to deform its crust, thereby twisting up the external magnetic field. In such a situation, a persistent current will flow through the magnetosphere, supported by emission of light ions (e.g. H, He, C) and electrons from the neutron star surface. The decay of this current outside of the star is an efficient mechanism for converting magnetic energy to X-rays, and for inducing rapid variations in the X-ray flux. To see how the persistent current will modify the structure of the magnetosphere, we find axisymmetric force-free equilibria outside a (non-rotating) spherical surface, ∇ × B = α(P)B. These equilibria form a one-dimensional sequence labeled by the flux parameter P = P(R, θ), with poloidal magnetic field BP = ∇P × φ̂/R sin θ. As a major simplification we consider self-similar configurations P = P0(R/RNS) F (θ), B(R, θ) ∼ F (θ)× (R/RNS) , (1) following Lynden-Bell & Boily (1994). The radial index p is uniquely determined by a single parameter C, which is related to the strength of the current: p(p+ 1)F + sin θ ∂2F ∂(cos θ)2 = −CF . (2) Choosing a dipole field at a zero current, p(C) is determined by the three boundary conditions, Br(R, θ = π/2) = 0, Br(RNS, θ = 0) = const, Bφ(R, θ = 0) = 0. The index p is most conviently expressed as a function of the net twist ∆φN−S between the north and south magnetic poles (Fig. 1a). 3. Resonant Scattering The current-currying charges also provide a significant optical depth to resonant cyclotron scattering. For a particle of charge Ze and massM , the resonant crosssection is σres(ω) = (π 2Ze2/Mc) (1 + cos2 θkB)δ(ω − ωc). The optical depth is determined by relating the particle density nZ to the twist in the magnetic field through (Ze)nZvZ = ǫZ(c/4π)|∇ ×B|. In our self-similar model,