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Accreting X-Ray Binaries display a wide range of behaviours. Some of them areobserved to spin up steadily, others to alternate between spin-up and spin-downstates, sometimes superimposed on a longer trend of either spin up or spindown. Here we interpret this rich phenomenology within a new model of thedisk-magnetosphere interaction. Our model, based on the simplest version of apurely material torque, accounts for the fact that, when a neutron star is inthe propeller regime, a fraction of the ejected material does not receiveenough energy to completely unbind, and hence falls back into the disk. We showthat the presence of this feedback mass component causes the occurrence ofmultiple states available to the system, for a given, constant value of themass accretion rate dot{M}_* from the companion star. If the angle chi of themagnetic dipole axis with respect to the perpendicular to the disk is largerthan a critical value chi_crit, the system eventually settles in a cycle ofspin-up/spin-down transitions for a constant value of dot{M}_* and independentof the initial conditions. No external perturbations are required to induce thetorque reversals. The transition from spin up to spin down is often accompaniedby a large drop in luminosity. The frequency range spanned in each cycle andthe timescale for torque reversals depend on dot{M}_*, the magnetic field ofthe star, the magnetic colatitude chi, and the degree of elasticity regulatingthe magnetosphere-disk interaction. The critical angle chi_crit ranges from\~25-30 deg for a completely elastic interaction to ~40-45 deg for a totallyanelastic one. For chi ~< chi_crit, cycles are no longer possible and thelong-term evolution of the system is a pure spin up. We specifically illustrateour model in the cases of the X-ray binaries GX 1+4 and 4U 1626-67.Comment: 37 pages, 11 figures, accepted to the Astrophysical Journa

On the Spin‐up/Spin‐down Transitions in Accreting X‐Ray Binaries