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High-frequency behavior of w-mode pulsations of compact stars

We study the asymptotic behavior of the quasi-normal modes (QNMs) of w-mode pulsations of compact stars in the high-frequency regime. We observe that both the axial and polar w-mode QNMs attain similar asymptotic behaviors in spite of the fact that they are described by two totally different differential equation systems. We obtain robust asymptotic formulae relating w-mode QNMs of different polarities and different angular momenta. To explore the physical reason underlying such similarity, we first derive a high-frequency approximation for the polar w-mode oscillations to unify the descriptions for both cases. Then, we develop WKB-type analyses for them and quantitatively explain the observed asymptotic behaviors for polytropic stars and quark stars. We also point out that such asymptotic behaviors for realistic stars are strongly dependent on the equation of state near the stellar surface.