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Bumpy Spin-Down of Anomalous X-Ray Pulsars: The Link with Magnetars

“…strong coupling) provided B in is moderately larger than B 0 as expected (Thompson & Duncan 1993). Under these circumstances, the star executes an anharmonic wobble, called radiative precession, with period τ pr (≈ τ nf ), and the near-field torque changes Ω on the precession time-scale (Melatos 1998).…”

“…Hydromagnetic stresses arising from non-radial gradients of the superstrong internal magnetic field, e.g. between the magnetic poles and equator if the field is dipolar, deform the star, inducing matter-density perturbations δρ ∼ B 2 in /µ 0 c 2 s and hence a fractional difference ǫ ∼ δρR 5 /I 1 ≈ 2 × 10 −9 (B in /10 10 T) 2 between the principal moments of inertia (Goldreich 1970;de Campli 1980;Melatos 1998). Here, B in is the characteristic magnitude of the internal magnetic field, c s is the isothermal sound speed (c s ≈ 3 −1/2 c), and R is the stellar radius; one has B in ≈ B 0 if the internal field is confined to the stellar crust and B in ∼ > B 0 if it is generated deep inside the star, e.g.…”

“…in a convective dynamo (Thompson & Duncan 1993). In a rotation-powered pulsar with B 0 ∼ < 10 9 T, the hydromagnetic deformation is comparable to the elastic deformation arising from shear stresses in the crystalline stellar crust, and the principal axes of inertia are oriented arbitrarily with respect to the axis m of the external magnetic dipole (Goldreich 1970;de Campli 1980;Melatos 1998). In a magnetar, however, the hydromagnetic deformation is much larger, and m is approximately parallel to one of the principal axes (e 3 , say); the alignment is not exact due to the complicated structure of the internal field near its generation site, cf.…”

“…The Eulerian precession is not free. It couples to a component of the vacuum magnetic-dipole torque associated with the asymmetric inertia of the near-zone radiation fields outside the magnetar; the electromagnetic energy density (and hence inertia) of the near-zone radiation fields is greater at the magnetic poles than at the equator by an amount ∼ B 2 0 /µ 0 , resulting in an oscillatory, precession-inducing torque (Goldreich 1970;Good & Ng 1985;Melatos 1998). The near-field torque exceeds the familiar braking torque (∝ Ω 3 ) by a factor c/ΩR ≫ 1 and acts on a commensurately shorter time-scale, τ nf ∼ τ 0 ΩR/c ≈ 6(B 0 /10 10 T) −2 (Ω/1 rad s −1 ) −1 yr, where τ 0 = µ 0 c 3 I 1 /2πB 2 0 R 6 Ω 2 ≈ 2 × 10 5 (B 0 /10 10 T) −2 (Ω/1 rad s −1 ) −2 yr is the characteristic electromagnetic braking time.…”

“…( 3) In (1)-(3), subscripts denote components along the principal axes of inertia, χ is the angle between m and e 3 , and we have a = 0.33, b = 0.094c/Ω 0 R, and Ω 0 = Ω(t = t 0 ), where t 0 is an arbitrary origin; for derivations of the equations, see Goldreich (1970) and Melatos (1998). Terms in (1)-(3) proportional to ǫ give rise to Eulerian precession, terms proportional to bτ −1 0 are associated with the near-field torque, and terms proportional to aτ −1 0 cause secular braking.…”