Anisotropic Magnetized Asteroseismic Waves

We solve for waves in a polytropic, stratified plasmas with a spatially varying background magnetic field that points along a horizontal $x$-direction, and with gravity that is directed along the vertical $z$-direction. Force balance determines the magnitude of the background magnetic field, $B_0^2\sim z^{n+1}$, where $n$ is the polytropic index. Using numerical and asymptotic methods, we deduce an explicit dispersion relation for fast pressure-driven waves: ${\mathrm{\Omega }}^2\sim K(2m+n)[1+(1\mathrm{/}{M}_{\mathrm{A}}{)}^2(4-2\gamma +{\cos }^2\theta -3{\cos }^4\theta )\mathrm{/}4]$. Here, $\mathrm{\Omega }$ is the frequency, $K$ the wavenumber, $\theta$ the angle the wave-vector makes with the background magnetic field, $M_{\mathrm{A}}$ the Alfvénic Mach number, and $m$ an integer representing the eigenstate. We discuss roles of such an explicit formula in asteroseismology.