Boundary between stable and unstable regimes of accretion. Ordered and chaotic unstable regimes

We search for the boundary between stable and Rayleigh-Taylor unstable regimes of accretion to magnetized stars in a new set of high grid resolution simulations. We found that the boundary between stable and unstable regimes is mainly determined by the ratio of the corotation radius r_cor (where the Keplerian angular velocity in the disc matches the angular velocity of the star) to the magnetospheric radius r_m (where the magnetic stress in the magnetosphere matches the matter stress in the disc). Instability is stronger when r_cor is larger with respect to r_m, that is, when the gravitational force is larger than the centrifugal force at the inner disc. In the cases of a small tilt of the magnetosphere, Theta=5 deg, and a small alpha-parameter of viscosity, alpha=0.02, the boundary is located at r_cor approx. 1.4 r_m. Instability becomes stronger at higher values of viscosity, and occurs at lower values of r_cor/r_m. At higher values of Theta, the variability associated with instability decreases. Simulations show two types of unstable accretion: chaotic and ordered. In the chaotic regime, several tongues form randomly and the light-curve has a few peaks per dynamical time-scale. In the ordered unstable regime, one or two tongues form and rotate with the frequency of the inner disc. The ordered unstable regime is found in the cases of relatively small magnetospheres, r_m 1.7 r_m. Results of our simulations are applicable to accreting magnetized stars with relatively small magnetospheres: Classical T Tauri stars (CTTSs), some accreting magnetized white dwarfs and neutron stars. The variability associated with the unstable regimes may explain the quasi-periodic oscillations (QPOs) in different types of stars, such as accreting millisecond pulsars. In observations of young stars, this QPO frequency may be mistaken for the period of the star.