Asymptotic approximations of low-frequency gravity modes in stars with a convective core and a radiative envelope

We develop lowest-order asymptotic approximations for the eigenfrequencies and eigenfunctions of linear, isentropic, low-frequency g-modes in stars consisting of a convective core and radiative envelope. The approximations are derived from a fourth-order system of differential equations in the divergence and radial component of the Lagrangian displacement without neglecting the Eulerian perturbation of the gravitational potential. Hitherto, asymptotic theories of low-frequency g-modes in stars with a convective core and radiative envelope have invariably treated the boundary between the convective core and radiative envelope as a fixed turning point in the differential equation for the divergence of the Lagrangian displacement. Here, we take into account the frequency dependence of the turning point and derive a generalized eigenvalue equation for the frequencies of low-frequency g-modes. From the application of the asymptotic theory to a 5M⊙ zero-age main-sequence stellar model, it follows that the generalized eigenvalue equation generally yields more accurate asymptotic eigenfrequencies than eigenvalue equations neglecting the frequency dependence of the turning point. We also find the accuracy of the asymptotic eigenfrequencies to depend strongly on the choice of an appropriate polytropic index describing the surface layers of the star.