Implications of the narrow period distribution of anomalous X-ray pulsars and soft gamma-ray repeaters

The spin periods of 10 observed anomalous X-ray pulsars and soft gamma-ray repeaters lie in the very narrow range of 6-12 s. We use a point-likelihood technique to show that the observed period clustering is not simply a statistical fluctuation, and we quantify the constraints it imposes on the birth period and on the final period of such systems. We consider a general law for their spin evolution described by a constant braking index. We find that, for positive values of the braking index, the observed clustering requires an upper cutoff period that is very close to the maximum observed period of ≃ 12 s. We also show that the constraint on the birth period depends very strongly on the assumed value of the braking index n, ranging from a few milliseconds for n ≳ 2 to a few seconds for n ≲ 2. We discuss possible ways of tightening these constraints based on similarities with the population of radio pulsars and by future observations of such sources with current X-ray observatories.