Magnetospheric radius of an inclined rotator in the magnetically threaded disk model

The estimate of the magnetospheric radius in a disk-fed neutron star X-ray binary is a long standing problem in high energy astrophysics. We have reviewed the magnetospheric radius calculations in the so-called magnetically threaded disk model, comparing the simplified approach originally proposed by Ghosh & Lamb (1979, ApJ, 232, 259) with the revised version proposed by Wang (1987, A&A, 183, 257), Wang (1995, ApJ, 449, L153), and Wang (1997, ApJ, 475, L135). We show that for a given set of fixed parameters (assuming also a comparable screening factor of the neutron star magnetic field by the currents induced on the disk surface) the revised magnetically threaded disk model predicts a magnetospheric radius that is significantly smaller than that derived from the Ghosh & Lamb (1979) treatment. For a fixed value of the neutron star magnetic field and a wide range of mass accretion rates, the inclusion of a large inclination angle between the neutron star rotation and magnetic field axes (χ ≳ 60 deg) leads to a further decrease of the magnetospheric radius. To illustrate the relevance of these calculations, we consider, as an example, the case of the transitional pulsars. During the so-called high mode of their sub-luminous accretion disk state, these sources have shown X-ray pulsations interpreted as due to accretion at an unprecedented low luminosity level compared to other neutron stars in X-ray binaries. In the context of the magnetic threaded disk model, we show that accretion at luminosities of ∼10^33 erg s^-1 (and thus accretion-driven X-ray pulsations) can be more easily explained when the prescription of the magnetospheric radius provided by Wang (1997) is used. This avoids the need to invoke very strong propeller outflows in the transitional pulsars, as proposed in other literature works