Euler ’ s equations of motion are derived exactly for a rigid , triaxial , internally frictionless neutron star spinning down electromagnetically in vacuo .
It is shown that the star precesses , but not freely : its regular precession relative to the principal axes of inertia couples to the component of the radiation torque associated with the near-zone radiation fields and is modified into an anharmonic wobble .
The wobble period \tau _ { 1 } typically satisfies \tau _ { 1 } \lesssim 10 ^ { -2 } \tau _ { 0 } , where \tau _ { 0 } is the braking time-scale ; the wobble amplitude evolves towards a constant non-zero value , oscillates , or decreases to zero , depending on the degree of oblateness or prolateness of the star and its initial spin state ; and the ( negative ) angular frequency derivative \dot { \omega } oscillates as well , exhibiting quasi-periodic spikes for triaxial stars of a particular figure .
In light of these properties , a young , Crab-like pulsar ought to display fractional changes of order unity in the space of a few years in its pulse profile , magnetic inclination angle , and \dot { \omega } .
Such changes are not observed , implying that the wobble is damped rapidly by internal friction , if its amplitude is initially large upon crystallization of the stellar crust .
If the friction is localized in the inner and outer crusts , the thermal luminosity of the neutron star increases by a minimum amount \Delta L \approx 3 \times 10 ^ { 31 } ( \epsilon / 10 ^ { -12 } ) ( \omega / 10 ^ { 3 } { rad s ^ % { -1 } } ) ^ { 2 } ( \tau _ { d } / 1 { yr } ) ^ { -1 } { erg s ^ { -1 } } , where \epsilon is the ellipticity and \tau _ { d } is the damping time-scale , with the actual value of \Delta L determined in part by the thermal conduction time \tau _ { cond } .
The increased luminosity is potentially detectable as thermal X-rays lasting for a time \approx { max } ( \tau _ { d } , \tau _ { cond } ) following crystallization of the crust .