The inner disc radius in the propeller phase and accretion-propeller transition of neutron stars

We have investigated the critical conditions required for a steady propeller effect for magnetized neutron stars with optically thick, geometrically thin accretion discs. We have shown through simple analytical calculations that a steady-state propeller mechanism cannot be sustained at an inner disc radius where the viscous and magnetic stresses are balanced. The radius calculated by equating these stresses is usually found to be close to the conventional Alfven radius for spherical accretion, r(A). Our results show that: (1) a steady propeller phase can be established with a maximum inner disc radius that is at least similar to 15 times smaller than rA depending on the mass-flow rate of the disc, rotational period and strength of the magnetic dipole field of the star, (2) the critical accretion rate corresponding to the accretion-propeller transition is orders of magnitude lower than the rate estimated by equating rA to the co-rotation radius. Our results are consistent with the properties of the transitional millisecond pulsars that show transitions between the accretion powered X-ray pulsar and the rotational powered radio-pulsar states.