Period clustering of the anomalous X-ray pulsars

In this paper we address the question of why the observed periods of the Anomalous X-ray Pulsars (AXPs) and Soft Gamma-ray Repeaters (SGRs) are clustered in the range 2-12 s. We explore a possibility to answer this question assuming that AXPs and SGRs are the descendants of High Mass X-ray Binaries (HMXBs) which have been disintegrated in the core-collapse supernova explosion. The spin period of neutron stars in HMXBs evolves towards the equilibrium period, averaging around a few seconds. After the explosion of its massive companion, the neutron star turns out to be embedded into a dense gaseous envelope, the accretion from which leads to the formation of a residual magnetically-levitating (ML) disk. We show that the expected mass of a disk in this case is 10^-7 - 10^-8 M_sun which is sufficient to maintain the process of accretion at the rate 10^14 - 10^15 g/s over a time span of a few thousand years. During this period the star manifests itself as an isolated X-ray pulsar with a number of parameters resembling those of AXPs and SGRs. Period clustering of such pulsars can be provided if the lifetime of the residual disk does not exceed the spin-down timescale of the neutron star.