Spin-down of an oblique rotator with a current-starved outer magnetosphere

A variant of the vacuum-dipole model of rotation-powered pulsars is presented that accounts for the observed spin-down properties of all three pulsars with braking indices measured from absolute pulse numbering (the Crab, PSR B0540 — 69 and PSR B1509 — 58). In the model, the neutron star and inner magnetosphere are treated phenomenologically as a single unit, a magnetized, perfectly conducting sphere of radius rv rotating rigidly in vacuo. The ‘vacuum radius’ rv corresponds to the innermost point in the magnetosphere where field-aligned flow breaks down and the plasma becomes three-dimensional, that is, the point where cyclotron losses occur slowly enough to allow electrons (or positrons) to move an appreciable distance before decaying to the l = 0 Landau state. For young, Crab-like pulsars, one typically finds r* ≪rv <rL, where r* is the stellar radius, and rL is the light-cylinder distance. The model therefore differs from standard vacuum-dipole theories, in which the dipole has radius r* and is treated as point-like. Three observable pulsar parameters — the rotation frequency ω, its time derivative ώ), and the angle α between the rotation and magnetic axes — uniquely determine rv and hence the electromagnetic braking torque exerted on the star, calculated from the Deutsch radiation fields of the rotating dipole. With no free parameters, the theory yields braking index values for the Crab, PSR B0540 — 69 and PSR B1509 — 58 that agree with timing data to 4 per cent. The second deceleration parameter is also predicted for each object, but cannot be verified to useful precision using data available at present. The relationship between our idealized yet successful spin-down model and a genuine pulsar magnetosphere is discussed, and further observational tests of the theory are proposed.