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Trapped, Two-Armed, Nearly Vertical Oscillations in Polytropic Disks

We have examined trapping of two-armed nearly vertical oscillations in polytropic disks. Two-armed nearly vertical oscillations are interesting in the sense that they are trapped in an inner region of disks with proper frequencies, if the inner edge of disks is a boundary that reflects oscillations. The frequencies of the trapped oscillations cover the frequency range of kHz QPOs to low frequency QPOs in LMXBs, depending on the modes of oscillations. Low frequency trapped oscillations are particularly interesting since their trapped region is wide. These low frequency oscillations are, however, present only when $Γ(\equiv 1+1/N)$ is close to but smaller than 4/3 (when spin parameter $a_*$ is zero), where $N$ is the polytropic index. The above critical value 4/3 slightly increases as $a_*$ increases.