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

We present a new study of the Rayleigh-Taylor unstable regime of accretion on to rotating magnetized stars in a set of high grid resolution three-dimensional magnetohydrodynamic simulations performed in low-viscosity discs. We find that the boundary between the stable and unstable regimes is determined almost entirely by the fastness parameter omega(s) = Omega(star)/Omega(K)(r(m)), where Omega(star) is the angular velocity of the star and Omega(K)(r(m)) is the angular velocity of the Keplerian disc at the disc-magnetosphere boundary r = r(m). We found that accretion is unstable if omega(s) less than or similar to 0.6. Accretion through instabilities is present in stars with different magnetospheric sizes. However, only in stars with relatively small magnetospheres, r(m)/R-star less than or similar to 7, do the unstable tongues produce chaotic hotspots on the stellar surface and irregular light curves. At even smaller values of the fastness parameter, omega(s) less than or similar to 0.45, multiple irregular tongues merge, forming one or two ordered unstable tongues that rotate with the angular frequency of the inner disc. This transition occurs in stars with even smaller magnetospheres, r(m)/R-star less than or similar to 4.2. Most of our simulations were performed at a small tilt of the dipole magnetosphere, Theta = 5 degrees, and a small viscosity parameter alpha = 0.02. Test simulations at higher alpha values show that many more cases become unstable, and the light curves become even more irregular. Test simulations at larger tilts of the dipole Theta show that instability is present, however, accretion in two funnel streams dominates if Theta greater than or similar to 15 degrees. The results of these simulations can be applied to accreting magnetized starswith relatively smallmagnetospheres: Classical T Tauri stars, accretingmillisecond X-ray pulsars, and cataclysmic variables.