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 kilo-Hertz quasi periodic oscillations (QPO) seen in low mass X-ray binaries. The existence of these compressible, non-barotropic magnetohydrodynamic (MHD) modes requires that there be a maximum in the angular velocity $Ω_ϕ(r)$ of the accreting material larger than the angular velocity of the star $Ω_*$, and that the fluid is in approximately circular motion near this maximum rather than moving rapidly towards the star or out of the disk plane into funnel flows. Our MHD simulations show this type of flow and $Ω_ϕ(r)$ 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 $g(r){\cal F}(r)$ at $r_{R}$. The real part of the angular frequency of the mode is $ω_r=mΩ_ϕ(r_{R})$, where $m=1,2...$ is the azimuthal mode number. The second mode, is a mode driven by the rotating, non-axisymmetric 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 flow perturbation with the rotating non-axisymmetric 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 $ν_*=Ω_*/2π$.