Theory of quasi-spherical accretion in X-ray pulsars

A theoretical model for quasi-spherical subsonic accretion onto slowly rotating magnetized neutron stars is constructed. In this model the accreting matter subsonically settles down onto the rotating magnetosphere forming an extended quasi-static shell. This shell mediates the angular momentum removal from the rotating neutron star magnetosphere during spin-down episodes by large-scale convective motions. The accretion rate through the shell is determined by the ability of the plasma to enter the magnetosphere. The settling regime of accretion can be realized for moderate accretion rates $\dot M< \dot M_*\simeq 4\times 10^{16}$ g/s. At higher accretion rates a free-fall gap above the neutron star magnetosphere appears due to rapid Compton cooling, and accretion becomes highly non-stationary. From observations of the spin-up/spin-down rates (the angular rotation frequency derivative $\dot \omega^*$, and $\partial\dot\omega^*/\partial\dot M$ near the torque reversal) of X-ray pulsars with known orbital periods, it is possible to determine the main dimensionless parameters of the model, as well as to estimate the magnetic field of the neutron star. We illustrate the model by determining these parameters for three wind-fed X-ray pulsars GX 301-2, Vela X-1, and GX 1+4. The model explains both the spin-up/spin-down of the pulsar frequency on large time-scales and the irregular short-term frequency fluctuations, which can correlate or anti-correlate with the X-ray flux fluctuations in different systems. It is shown that in real pulsars an almost iso-angular-momentum rotation law with $\omega \sim 1/R^2$, due to strongly anisotropic radial turbulent motions sustained by large-scale convection, is preferred.