Stability of the Magnetopause of Disk-accreting Rotating Stars

We discuss three modes of oscillation of accretion disks around rotating magnetized neutron stars which may explain the separations of the kilohertz quasi-periodic oscillations (QPOs) seen in low-mass X-ray binaries. The existence of these compressible, nonbarotropic magnetohydrodynamic (MHD) modes requires that there be a maximum in the angular velocity Ω_phi(r) of the accreting material larger than the angular velocity of the star Ω_*, and that the fluid be in approximately circular motion near this maximum rather than moving rapidly toward the star or out of the disk plane into funnel flows. Such a maximum in Ω_phi occurs naturally in disk accretion to a slowly rotating magnetized star due the magnetic braking of the disk rotation by the stellar field. Our MHD simulations show this type of flow and Ω_phi(r) profile. The first mode is a Rossby wave instability (RWI) mode which is radially trapped in the vicinity of the maximum of a key function g(r)F(r) at r_R . The real part of the angular frequency of the mode is ω _r = mΩ_phi(r_R ), where m = 1, 2, ... is the azimuthal mode number. We argue that the nonlinear saturation of the RWI occurs when the trapping frequency of fluid particles in the Rossby vortex equals the RWI growth rate. The second mode is a mode driven by the rotating, nonaxisymmetric component of the star's magnetic field. It has an angular frequency equal to the star's angular rotation rate Ω_*. This mode is strongly excited near the radius of the Lindblad resonance which is slightly outside of r_R . The third mode arises naturally from the interaction of a flow perturbation with the rotating nonaxisymmetric component of the star's magnetic field. It has an angular frequency Ω_*/2. We suggest that the first mode with m = 1 is associated with the upper QPO frequency, ν _u ; that the nonlinear interaction of the first and second modes gives the lower QPO frequency, ν_ell = ν _u - ν_* and that the nonlinear interaction of the first and third modes gives the lower QPO frequency ν_ell = ν _u - ν_*/2, where ν_* = Ω_*/2π.